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PhysRevB.84.245441.pdf

PHYSICAL REVIEW B 84, 245441 (2011) Coulomb drag in monolayer and bilayer graphene E. H. Hwang, Rajdeep Sensarma, and S. Das Sarma Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA (Received 13 June 2011; revised manuscript received 5 November 2011; published 22 December 2011) We theoretically calculate the interaction-induced frictional Coulomb drag resistivity between two graphene monolayers as well as between two graphene bilayers, which are spatially separated by a distance d.W e show that the drag resistivity between graphene monolayers can be significantly affected by the intralayermomentum-relaxation mechanism. For energy-independent intralayer scattering, the frictional drag induced byinterlayer electron-electron interaction goes asymptotically as ρ D∼T2/n4d6andρD∼T2/n2d2in the high- density ( kFd/greatermuch1) and low-density ( kFd/lessmuch1) limits, respectively. When long-range charge impurity scattering dominates within the layer, the monolayer drag resistivity behaves as ρD∼T2/n3d4andT2ln(√nd)/nfor kFd/greatermuch1a n dkFd/lessmuch1, respectively. The density dependence of the bilayer drag is calculated to be ρD∝T2/n3 both in the large and small layer separation limit. In the large layer separation limit, the bilayer drag has a strong 1 /d4dependence on layer separation, whereas this goes to a weak logarithmic dependence in the strong interlayer correlation limit of small layer separation. In addition to obtaining the asymptotic analytical formulafor Coulomb drag in graphene, we provide numerical results for arbitrary values of density and layer separationinterpolating smoothly between our asymptotic theoretical results. DOI: 10.1103/PhysRevB.84.245441 PACS number(s): 72 .80.Vp, 81 .05.ue, 72.10.−d, 73.40.−c I. INTRODUCTION Much attention has recently focused on multilayer sys- tems in graphene, where carrier transport properties maybe strongly affected by interlayer interaction effects. 1–6 In particular, temperature and density dependent Coulomb drag properties have recently been studied in the spatially separated double-layer graphene systems.2Frictional drag measurements of transresistivity in double-layer systems haveled to significant advances in our understanding of density and temperature dependence of electron-electron interac- tions in semiconductor-heterostructure-based parabolic 2Dsystems. 7–12While electron-electron interactions have indirect consequences (for example, through carrier screening) fortransport properties of a single isolated sheet of monolayer or bilayer graphene, the Coulomb drag effect provides an oppor- tunity to directly measure the effects of electron-electron inter-actions through a transport measurement, where momentum istransferred from one layer to the other layer due to interlayerCoulomb scattering. In Coulomb drag measurements the role of electron interaction effects can be controlled by varying, in a systematic manner, the electron density ( n), the layer separation ( d), and the temperature ( T) since the interlayer electron-electron interaction obviously depends on n,d, and T. (There is also a rather straightforward dependence of the drag on the background dielectric constant κ, arising trivially through the interaction coupling constant or equivalently thegraphene fine-structure constant, which we do not discussexplicitly.) In view of the considerable fundamental significance of the issues raised by the experimental observations, 2we present in this paper a careful theoretical calculation of frictionaldrag resistivity, ρ D(T), both between monolayer graphene (MLG) sheets and bilayer graphene (BLG) sheets, usingthe Boltzmann transport equation. The current work is ageneralization of the earlier theoretical work on graphenedrag by Tse et al. , 13where the drag is calculated within the canonical many-body Fermi liquid theory assuming anenergy-independent intralayer momentum relaxation time. We would like to note that, although an energy-independentrelaxation time captures the effects of both short-range andscreened Coulomb impurities in 2D electron gas with parabolicdispersion, it does not correspond to any disorder model inMLG with its linear dispersion. In this paper, we generalizethe formalism for calculating drag resistivity by including anarbitrary energy-dependent intralayer scattering mechanismwithin the linear response Boltzmann equation description.One new feature of our work is that we consider both MLGand BLG drag on equal theoretical footing, providing detailedtheoretical drag results for both systems. We find that, for MLG, the energy-dependent intralayer transport scattering time is the key to understanding the density dependence of drag resistivity induced by interlayer electron- electron interaction. In the presence of an energy-independent scattering time, the low-temperature drag goes asymptotically asρ D∼T2/n4d6andρD∼T2/n2d2in the high-density (kFd/greatermuch1, where kFis the Fermi wave vector) and low- density ( kFd/lessmuch1) limits, respectively. These interlayer drag results for energy-independent intralayer momentum relax- ation, however, change qualitatively in the presence of an energy-dependent intralayer relaxation. For example, if the intralayer scattering is dominated by the charged impurities, which is believed to be the dominant intralayer momentum relaxation mechanism in most currently available graphene samples on a substrate,14–16the MLG drag resistivity be- haves asymptotically as ρD∼T2/n3d4andT2ln(√nd)/nfor kFd/greatermuch1 and kFd/lessmuch1, respectively. In the low-temperature regime ( T/T F/lessmuch1), the enhanced phase space for q=2kF Coulomb backscattering leads to ∼T2ln(T) corrections to the usual T2dependence of the drag resistivity in ordinary 2D electron systems; i.e., ρD∝T2lnT. We find that, due to the chirality-induced suppression of the q=2kFbackscattering in MLG, these ∼T2ln(T) corrections to the drag resistivity are absent in this system. 245441-1 1098-0121/2011/84(24)/245441(9) ©2011 American Physical SocietyE. H. HW ANG, RAJDEEP SENSARMA, AND S. DAS SARMA PHYSICAL REVIEW B 84, 245441 (2011) We also investigate the Coulomb drag resistivity for two bilayer graphene sheets separated by a distance d.T h e low-temperature behavior of BLG drag follows the usualT 2dependence found in MLG systems. Both in the weakly correlated large layer separation limit, qTFd/greatermuch1, and the strongly correlated small separation limit, qTFd/lessmuch1(qTF being the Thomas-Fermi screening wave vector in BLG), the bilayer drag shows an inverse cubic dependence on the carrierdensity; i.e., ρ D∝T2/n3. In the large layer separation limit, the BLG drag has a strong 1 /d4dependence on layer separa- tion, whereas this goes to a weak logarithmic dependence inthe strong interlayer correlation (small layer separation) limit.The BLG drag, in contrast to MLG drag, does not manifest anyqualitative dependence on the intralayer momentum relaxationmechanism with both short-range disorder scattering andlong-range charged impurity scattering producing the sameinterlayer BLG drag. The paper is organized as follows. In Sec. IIwe present the general formula for drag resistivity in 2D materialsfrom a Boltzmann transport approach, considering arbitrarymomentum-dependent scattering times. The analysis here isfor arbitrary 2D chiral electron systems with two gaplessbands, and hence is applicable to both MLG and BLG dragresistivity. In Sec. III, we study in detail the temperature, density, and layer separation dependence of the drag resistivityin MLG. We provide both analytic low-temperature asymptoticforms and numerical results for the MLG drag resistivity, usingan energy-independent as well as a linearly-energy-dependentscattering time. In Sec. IV, we study the drag resistivity of BLG, focusing both on analytic asymptotic forms andnumerical results, within an energy-independent scatteringtime approximation. Finally, we conclude our study in Sec. V with a summary of our work and a comparison of our resultswith other works in this field. II. DRAG RESISTIVITY IN 2D CHIRAL ELECTRON SYSTEMS In this section, we will derive the general formula for drag resistivity of chiral 2D electron-hole systems. We considertwo layers ( aandp) of the material, which are kept separated by an insulating barrier of thickness d, such that the carriers in the different layers are only coupled through the Coulombinteraction and there is no tunneling between the two layers.In drag experiments, an electric field E ais applied to layer a(the driven or active layer) and causes a current density ja, which induces the electric field Epin layer p(the dragged or passive layer), where the current jpis set to zero. Then the drag resistivity is defined by ρD=Eα p/jα a, where αis the direction along which the current jaflows.8,9 We will consider each layer to be composed of a chiral two-band (electron and hole) material where the density ofthe carriers can be changed independently. Let the dispersionof these bands be given by ε sk, and the corresponding two- component electron wave functions be given by ψsk, where s=± 1 denotes the band indices. The group velocity of the bands are given by vsk=∇ kεsk. Let ˜fi skdenote the nonequilibrium distribution function of the band sin the layer i, where ( i=a,p). The Boltzmannequation for the distribution function is eEi·vsk∂˜fi sk ∂εsk=Ii sk, (1) where Iskis the collision integral. Within a linearized relaxation-time approximation, ˜fi sk=fi sk−eτi kEi·vsk∂fi sk ∂εsk=fi sk+φi skfi sk/parenleftbig 1−fi sk/parenrightbig T,(2) where φi sk=eτi kEi·vsk,τi k=τi kis the transport scattering time in the layer i,fi sk=1/[e(εsk−μi)/T+1] is the equilibrium Fermi distribution function in layer i,μiis the chemical potential in the layer i, and Tis the temperature of the system. We have set ¯ h=1 and the Boltzmann constant kB=1 throughout this paper. The current in layer ican then be written as ji=−ge/summationdisplay skvsk˜fi sk=ge/summationdisplay skτi kvskIi sk, (3) where g=4 is the spin and valley degeneracy of the excitations. We will now focus on the form of the collisionintegral in the presence of the two layers. The collisionintegral in each layer has two contributions: (i) from impurityscattering in the same layer and (ii) from Coulomb scattering with electrons of the other layer; i.e., I i sk=Ii(imp) sk+Ii(C) skand hence ji=j(imp) i+j(C) i. The impurity scattering contribution to the current j(imp) i=σiEi,σibeing the usual conductivity of the layer iin absence of the second layer. From now on, we will focus on the Coulomb scattering contribution and dropthe superscript ( C) in the collision integral and the current. The electron-electron scattering between the layers is mediated through a dynamically screened interlayer Coulombinteraction V(q,ω). We will discuss the precise form of the screened Coulomb interaction at the end of this section. Thecollision integral is given by I i sk=2πg/summationdisplay s/primer,r/prime/summationdisplay k/primeq/integraldisplay dω|V(q,ω)|2Fss/prime k,qFrr/prime k/prime,−q ×/bracketleftbig/parenleftbig 1−˜fi sk/parenrightbig˜fi s/primek+q/parenleftbig 1−˜fl rk/prime/parenrightbig˜fl r/primek/prime−q −˜fi sk/parenleftbig 1−˜fi s/primek+q/parenrightbig˜fl rk/prime/parenleftbig 1−˜fl r/primek/prime−q/parenrightbig/bracketrightbig ×δ(ω+εsk−εs/primek+q)δ(ω+εr/primek/prime−q−εrk/prime),(4) where r,r/prime=± 1,lis the layer other than i, and Fss/prime k,q= |ψ† s/primek+qψsk|2is the wave function overlap between the bands. It is easy to verify that the collision integral vanishes if we replace ˜fiby the equilibrium approximation, ˜fi=fi.U s i n g Eq. ( 2), to linear order in the deviations from the equilibrium distribution, the collision integral is given by Ii sk=2πg T/summationdisplay s/primer,r/prime/summationdisplay k/primeq/integraldisplay dω|V(q,ω)|2Fss/prime k,qFrr/prime k/prime,−q ×fi sk/parenleftbig 1−fi s/primek+q/parenrightbig fl rk/prime/parenleftbig 1−fl r/primek/prime−q/parenrightbig ×/parenleftbig φi sk−φi s/primek+q+φl rk/prime−φl r/primek/prime−q/parenrightbig ×δ(ω+εsk−εs/primek+q)δ(ω+εr/primek/prime−q−εrk/prime).(5) 245441-2COULOMB DRAG IN MONOLAYER AND BILAYER GRAPHENE PHYSICAL REVIEW B 84, 245441 (2011) Multiplying Eq. ( 5)b yegτi kvskand summing over sandk,t h e terms with φivanish. Further, using the identity fsk(1−fs/primek/prime)δ(ω+εsk−εs/primek/prime)=fsk−fs/primek/prime 1−e−ω/T, (6) we get the current in layer i, ji=πg2e2 2T/summationdisplay q/integraldisplay dω|V(q,ω)|2 sinh2ω 2T/summationdisplay ss/primekFss/prime k,qτi kvsk ×/parenleftbig fi sk−fi s/primek/prime/parenrightbig δ(ω+εsk−εs/primek+q) ×/summationdisplay rr/primek/primeFrr/prime k/prime,qEl·/parenleftbig τl k/primevrk/prime−τl k/prime+qvr/primek/prime+q/parenrightbig ×/parenleftbig fl rk/prime−fl r/primek/prime+q/parenrightbig δ(ω+εrk/prime−εr/primek/prime+q). (7) Rearranging the terms with fi, we finally obtain jα i=σDEα l, (8) where σD=1 4πT/summationdisplay q/integraldisplay dω|V(q,ω)|2Im/bracketleftbig χα a(q,ω)/bracketrightbig Im/bracketleftbig χα p(q,ω)/bracketrightbig sinh2ω 2T (9) and the nonlinear drag susceptibility is given by χi(q,ω)=e/summationdisplay ss/primekFss/prime k,q/parenleftbig fi sk−fi s/primek+q/parenrightbig/parenleftbig τi kvsk−τi k+qvs/primek+q/parenrightbig ω+εsk−εs/primek+q+i0+. (10) Then the drag resistivity is given by ρD=−σD σaσp−σ2 D∼−σD σaσp. (11) We will now focus on the last piece of information required to calculate the drag resistivity of chiral electron-holesystems, the dynamically screened interaction, V(q,ω), as shown in Fig. 1(a). The intralayer bare Coulomb interaction is given by V aa(q)=Vpp(q)=2πe2/κq, where κis the dielectric constant. The interlayer bare Coulomb interactionis given by V ap(q)=Vpa(q)=Vaa(q)e−qd, where dis the layer separation. Within random phase approximation (RPA),the dynamically screened interlayer interaction is given byV(q,ω)=V ap(q)//epsilon1(q,ω), with the dielectric function of cou- pled layer systems given by5 |/epsilon1(q,ω)|=[1−Vaa(q)/Pi1a(q,ω)][1−Vpp(q)/Pi1p(q,ω)] −Vap(q)Vpa(q)/Pi1a(q,ω)/Pi1p(q,ω), (12) where /Pi1iis the polarizability of the layer i. /Pi1i(q,ω)=−g/summationdisplay ss/primekFss/prime k,q/parenleftbig fi sk−fi s/primek+q/parenrightbig ω+εsk−εs/primek+q. (13) Equations ( 8)–(13) thus completely define the drag resistivity of chiral electron-hole systems in terms of the band dispersionand wave functions. In the next two sections, we willadapt these general equations to study the drag resistivity inmonolayer and bilayer graphene. So far we have used the linearized Boltzmann equation to derive the drag conductivity which relates the induced electric FIG. 1. (a) Screened interlayer Coulomb interaction in the RPA. The thin and thick lines are the bare and the screened interactions,respectively. The bare bubble represents the polarizability /Pi1(q,ω). (b) The leading-order diagrams contributing to the drag conductivity. /Gamma1indicates the nonlinear susceptibility. field in one layer to the driving current in the other layer in a double-layer system. Before moving on to the specific caseof SLG and BLG, we would like to show that the linearizedBoltzmann result captures the leading-order result for dragconductivity within a diagrammatic expansion of the linearresponse Kubo formula. When an external field is applied tolayer 1 and the induced current is measured in layer 2, the dragconductivity is given by the Kubo formula 17 σD(ω)=1 ωA/integraldisplay∞ 0dteiωt/angbracketleft[J† 1(t),J2(0)]/angbracketright, (14) where Ais the area of the sample and Jiis the current operator in the ith layer. The nonvanishing leading-order diagrams corresponding to Eq. ( 14)a r eg i v e ni nF i g . 1(b). The two leading-order diagrams can be written in a symmetric form σD(i/Omega1n)=1 2i/Omega1n1 T/summationdisplay q,iω m/Gamma11(q,ωm+/Omega1n)/Gamma12(q,ωm) ×V(q,ωm+/Omega1n)V(q,ωm), (15) where /Omega1n=2πinT andωm=2πimT are boson frequen- cies,V(q,ω) is the interlayer screened Coulomb interaction [Fig. 1(a)], and /Gamma1i(q,ω) is the three-point vertex diagrams (or the nonlinear susceptibility) and given by17–19 /Gamma1(q,ω)=T/summationdisplay /epsilon1nTr/braceleftbig G/epsilon1nG/epsilon1n+ωJ(q)G/epsilon1n+ω/bracerightbig +Tr/braceleftbig G/epsilon1nG/epsilon1n−ωJ(q)G/epsilon1n−ω/bracerightbig , (16) where G/epsilon1nis the Green’s function, J(q) stands for the current, and “Tr” the trace. To get the dc drag conductivity we need toperform an analytical continuation of external frequencies toa real value, i/Omega1 n→/Omega1, and the limit /Omega1→0 should be taken. After summing over the boson frequencies, ωm, and performing an analytical continuation to a real value of /Omega1, we have σD=1 16πT/summationdisplay q/integraldisplaydw sinh2ω 2T/Gamma11(q,ω)/Gamma12(q,ω)|V(q,ω)|2. (17) 245441-3E. H. HW ANG, RAJDEEP SENSARMA, AND S. DAS SARMA PHYSICAL REVIEW B 84, 245441 (2011) The nonlinear susceptibility with real frequencies is given by /Gamma1(q,ω)=1 4πi/integraldisplay d/epsilon1/parenleftbigg tanh/epsilon1 2T−tanh/epsilon1 2T/parenrightbigg ×/summationdisplay pTr[(G− /epsilon1−G+ /epsilon1)G− /epsilon1+ωJ(p)G+ /epsilon1+ω] +{(q,ω)→(−q,−ω)}, (18) where G± /epsilon1=(/epsilon1−H±iγ)−1denotes the retarded ( −) and advanced ( +) Green function for a given system with the Hamiltonian H, respectively. Here we use a damping constant γ=1/2τto include the disorder scattering in the Green’s function. In the Boltzmann regime ( ωτ/greatermuch1o rkFl/greatermuch1, where kFis the Fermi wave vector and l=vFτis the mean-free path), which corresponds to weak impurity scattering andthe actual experimental regime of the high-mobility graphenesamples, we can treat the vertex correction in the currentJ(q) within the impurity ladder approximation. 18,19Then, the impurity-dressed current vertex becomes J=(τtr/τ)∂H/∂k , where τtris the transport time. By using the following equation, G− /epsilon1+ω(p)G+ /epsilon1+ω(p)=2τImG+ /epsilon1+ω(p), (19) and expressing the matrix form of the Green’s function in the chiral basis, finally, we have the nonlinear susceptibility /Gamma1(q,ω)=τ/summationdisplay ss/prime,k[Js(k)−Js/prime(k+q)]Fss/prime(k,k+q) ×Imfsk−fs/primek+q ω+/epsilon1sk−/epsilon1s/primek+q+i0+. (20) Comparing the above equation with Eq. ( 10)w eh a v e /Gamma1(q,ω)=2Imχ(q,ω). (21) Thus we recover the Boltzmann equation drag conductivity result by employing the leading-order diagrammatic expansionwithin the Kubo formalism. III. DRAG RESISTIVITY IN MONOLAYER GRAPHENE Monolayer graphene (MLG) is characterized by the pres- ence of gapless linearly dispersing electron and hole bandswith dispersion /epsilon1 sk=vFk, (22) where s=± 1 denotes the electron and hole bands and vFis the Fermi velocity of graphene. The linear dispersion of MLGimplies that the group velocity of the bands are given by v sk= vFk/|k|. Thus, contrary to usual semiconductor systems, or the case of bilayer graphene to be treated later, the groupvelocity is constant in magnitude and does not scale with themomentum of the band states. As we will show, this leads toprofound qualitative differences in the variation of the MLGdrag resistivity with the carrier densities in the layers andthe distance between the layers. The chirality of the electrons(holes) are encoded in the band wave functions ψ sk=1√ 2/parenleftbigg e−iθk s/parenrightbigg , (23)where θkis the azimuthal angle in the 2D kspace. This leads to the wave function overlap factor Fss/prime k,q=(1/2)[1+ ss/primecos(θk+q−θk)]. Before we go on to a detailed discussion of the non- linear drag susceptibility and the drag resistivity in MLG,let us first discuss the finite-temperature polarizability inMLG which will control the finite-temperature screeningof the Coulomb potential. Although analytic expressionsfor graphene polarizability at T=0 have been worked out before, 20the finite-temperature versions of the polarizability have not been calculated analytically. The full expression offinite-temperature polarizability is necessary to understandmore precisely the temperature-dependent drag including theplasmon enhancement effects. 9Here we provide a finite- temperature generalization of our earlier work on zero-temperature graphene polarizability. 20The MLG polarizability of layer iis given by /Pi1i(q,ω, ) =2EFi πv2 F/parenleftBigg π 8y2 /radicalbig y2−z2/integraldisplay∞ 0dx[f(x)+g(x)] +⎡ ⎣/parenleftbigz2−y2 4+zx+x2/parenrightbig sgn(a+)/radicalBig (z2−y2)2 4+(z2−y2)(zx+x2) +/parenleftbigz2−y2 4−zx+x2/parenrightbig sgn(a−)/radicalBig (z2−y2)2 4+(z2−y2)(−zx+x2)⎤ ⎦⎞ ⎠,(24) where x=k/kFi,y=q/kFi, and z=ω/E Fi,f(x)= [e−(x−μi)/ti−1]−1,g(x)=[e(x+μi)/ti+1]−1,a±=z2−y2± 2zx, with EFiandkFibeing the Fermi energy and the Fermi wave vector in layer i. Here, μiis the chemical potential in layeriin units of EFi(to be calculated self-consistently) and ti=T/E Fi. We now turn our attention to the nonlinear drag susceptibil- ity and drag resistivity in MLG systems. The drag resistivityin MLG crucially depends on the variation of the transportscattering time with energy (or equivalently momentum). Inthis paper, we will consider the drag resistivity of MLG usingtwo different models of scattering time: (a) a momentum-independent scattering time τ k=τand (b) a scattering time scaling linearly with momentum (energy), τk=τ0|k|, which results from the unusual screening of charge impurity potentialin this linearly dispersive material. 1,14–16 We first consider the energy-independent scattering time approximation. In this case, the nonlinear drag susceptibilityin MLG can be written as χ i(q,ω)=4τivF/summationdisplay ss/primek/parenleftbigg sk+q |k+q|−s/primek |k|/parenrightbigg ×Fss/prime k,q/parenleftbig fi sk+q−fi s/primek/parenrightbig ω−εsk+q+εs/primek+i0+. (25) Within the energy-independent scattering time approxi- mations, the intralayer conductivities are given by σi= e2EFiτi/4. We focus on the analytic asymptotic behaviors of the drag resistivity in MLG at low temperatures, both in the large layer 245441-4COULOMB DRAG IN MONOLAYER AND BILAYER GRAPHENE PHYSICAL REVIEW B 84, 245441 (2011) separation weak-coupling limit ( kFd/greatermuch1) and in the small layer separation strong-coupling limit( kFd/lessmuch1). To calculate the asymptotic behavior of drag resistivity first we investigatethe nonlinear susceptibility for ω<v Fq<E F. Due to the phase-space restriction the most dominant contribution to thedrag resistivity arises from ω<v Fqat low temperatures. When we neglect the energy dependence in the transport times,i.e.,τ k=τ, then we obtain, for ω<v Fq, χ(q,ω)∼τq2 πEFω vFq/radicalBig 1−q2/4k2 F. (26) With assumptions of a large interlayer separation [ kFd/greatermuch1, or qTFd/greatermuch1, with qTFbeing the Thomas-Fermi (TF) screening wave vector] and the random-phase approximation (RPA) inwhich /Pi1 iiis replaced by its value for the noninteracting electrons, we have the drag resistivity at high density andlow temperature, ρ D=h e25!ζ(5) 3×28(kBT)2 EF1EF21 /parenleftbig qTF 1d/parenrightbig/parenleftbig qTF 2d/parenrightbig/parenleftbig kF1d/parenrightbig2/parenleftbig kF2d/parenrightbig2, (27) where qTF=4rskFis the TF wave vector with the graphene fine-structure constant rs=e2/κvFandζ(x)i st h eR i e - mann zeta function. This result shows that ρD(n)∝n−4and ρD(T)∝T2. For large layer separation (i.e., kFd/greatermuch1) the backscattering q≈2kFis suppressed due to the exponential dependence of the interlayer Coulomb interaction v12(q)∝ exp(−qd)/qas well as the graphene chiral property. In this case the drag is dominated by small-angle scattering and oneexpects ρ D∝T2/(n4d6). For the strong interlayer correlation ( kFd/lessmuch1) in the low- density or small-separation limit, the asymptotic behavior ofthe drag resistivity becomes ρ D=h e21 6(kBT)2 EF1EF2r2 s/parenleftbig kF1d/parenrightbig/parenleftbig kF2d/parenrightbig. (28) We have the same temperature dependence, ρD(T)∼T2, but the density dependence becomes much weaker, ρD(n)∼ 1/(nd)2. At low densities (or strong interlayer correlation, kFd/lessmuch1) the exponent in the density-dependent drag differs from−4. In an ordinary 2D system, at low carrier densities and for closely spaced layers the backward scattering canbe important since k Fd∼1. In the low-temperature range T/T F/lessmuch1 the enhanced phase space for q=2kFbackward Coulomb scattering leads to ln( T) corrections to the usual T2dependence of the drag; i.e., ρD∝T2lnT. However, due to the suppression of the q=2kFbackscattering due to the chirality of graphene there is no ln( T) correction in the drag resistivity of monolayer graphene. The drag resistivity within this approximation is plotted as a function of temperature and density in Fig. 2for different layer separations. The parameters corresponding to the experimentalsetup of Ref. 2are used. In Fig. 2(a) we show the calculated Coulomb drag as a function of temperature for an equalcarrier density, n 1=n2=1012cm−2, for two different layer separations d=50˚A and d=200 ˚A. The overall temperature dependence of drag is close to the quadratic behavior, ρD∝ T2. But we find a small corrections at low temperatures, especially at low values of kFd. In regular 2D systems there is10 100 T (K)10-310-210-1100ρ (Ω)Dd=50A d=100A 0.1 1 n (10 cm )10-410-310-210-1100101ρ (Ω)D d=200A10050 12 -2(a) (b) FIG. 2. (Color online) The calculated drag resistivity by con- sidering the energy-independent scattering approximation. (a) The temperature dependence of Coulomb drag for two different layerseparations d=50˚A,d=100 ˚A and the equal electron densities n 1=n2=1012cm−2. (b) The density-dependent Coulomb drag for different layer separations d=5, 10, 20 nm and at T=200 K. al n (T) correction to the T2dependence of the drag. However, due to the suppression of the backscattering in graphene suchlogarithmic correction does not show up in our numericalresults except perhaps at extremely low temperatures. InFig. 2(b) the density-dependent Coulomb drag is shown for different layer separations. Our calculated Coulomb dragresistivity follows a n αdependence with α∼− 2a tl o w carrier densities (or, kFd< 1), but as the density increases the exponent decreases to α∼− 4. So far, we have considered the energy-independent scat- tering time. However, it is known that the scattering bythe charged impurity disorder which inevitably exists in thegraphene environment dominates and the scattering time dueto the charged impurity is linearly proportional to the energy,τ i∼ε=τ0 ik.1,14–16In this approximation, the nonlinear drag susceptibility is given by χi(q,ω)=4τ0 ivF/summationdisplay ss/primek[s(k+q)−s/primek]Fss/prime k,q/parenleftbig fi sk+q−fi s/primek/parenrightbig ω−εsk+q+εs/primek+i0+. (29) In this case, we will mainly focus on the low-temperature asymptotic for the drag resistivity. For linearly-energy-dependent scattering time due to the charged impurities, thenonlinear susceptibility for w<v Fqis χi(q,ω)=2τ0 i πkF vFω EF1/radicalBig 1−q2/4k2 F. (30) Then the drag resistivity for a large interlayer separation (kFd/greatermuch1) is given by ρD=h e2ζ(3) 23(kBT)2 EF1EF21/parenleftbig kF1d/parenrightbig/parenleftbig kF2d/parenrightbig/parenleftbig qTF 1d/parenrightbig/parenleftbig qTF 2d/parenrightbig,(31) and for the strong interlayer correlation limit ( kFd/lessmuch1), we have ρD=h e224r2 s 3(kBT)2 EF1EF2ln/bracketleftBigg 2/parenleftbig qTF 1+qTF 2/parenrightbig d+1 2/parenleftbig qTF 1+qTF 2/parenrightbig d/bracketrightBigg . (32) Thus we see that when we consider the energy-dependent intralayer scattering time we have very different asymptoticbehaviors compared to the results from energy-independent 245441-5E. H. HW ANG, RAJDEEP SENSARMA, AND S. DAS SARMA PHYSICAL REVIEW B 84, 245441 (2011) 10 100 T (K)10-210-1100101ρ (Ω)Dd=50A d=100A 0.1 1 n (10 cm )10-210-1100101102103ρ (Ω)D d=200A10050 12 -2(a) (b) FIG. 3. (Color online) The calculated drag resistivity by con- sidering the linearly-energy-dependent scattering time. (a) The temperature dependence of Coulomb drag for two different layerseparations d=50˚A,d=200 ˚A and the equal electron densities n 1=n2=1012cm−2. (b) The density-dependent Coulomb drag for different layer separations d=5, 10, 20 nm and at T=200 K. scattering time; i.e., we find for kFd/greatermuch1,ρD∼T2/(n3d4) and for kFd/lessmuch1,ρD∼T2ln(√nd). Thus in the presence of the strong charged impurity scattering the Coulomb dragresistivity follows a n αdependence with α/greaterorsimilar−3a th i g h densities but as the density decreases the exponent ( α) increase. Based on our calculation we believe that the experimentaldeparture from the n −3behavior reported in Ref. 2is essentially a manifestation of the fact that the asymptotic n−3 regime is hard to reach in low-density electron systems where thekFd/greatermuch1 limit simply cannot be accessed. We predict a weak ln( n)/ndensity dependence in the low-density or small separation limit. In Fig. 3(a) we show the calculated Coulomb drag as a function of temperature for two different layer separationsd=50 ˚A and d=100 ˚A by considering the linearly- energy-dependent scattering time. The overall temperaturedependence of drag increases quadratically and there isno logarithmic correction due to the suppression of thebackscattering. In Fig. 3(b) the density-dependent Coulomb drag is shown for different layer separations. The density-dependent Coulomb drag follows a n αdependence with α∼− 2. Based on our calculation the consideration of the energy-dependent scattering time is crucial to understand theexperiment measurements of Ref. 2. IV. DRAG RESISTIVITY IN BILAYER GRAPHENE In this section we will study the drag resistance in a heterostructure made of two bilayer graphene (BLG) layersseparated by an insulating barrier and study its dependenceon temperature, density, and separation of the layers. BothBLG and MLG have chiral gapless electron and hole bands.However, BLG differs from MLG in two crucial aspects:(i) The bands have a parabolic dispersion as opposed to linear dispersion in MLG and (ii) the chiral angle is doublethat in MLG leading to enhanced rather than suppressedbackscattering between the quasiparticles. We will show in thissection how these two differences lead to dramatic changes inthe density and layer separation dependence of the BLG dragresistivity compared to MLG drag resistivity. BLG consists of an electron and a hole band with quadratic dispersion /epsilon1 sk=sk2/2m (33) [s=± 1 for electron (hole) band] and the BLG mass is m= 0.033me, where meis the electron mass. The group velocity in the two bands is vsk=sk/m, which scales linearly with momentum. The wave functions in these bands are given by ψsk=1√ 2/parenleftbigg e−2iθk s/parenrightbigg , (34) which gives the overlap factor Fss/prime k,q=(1/2)(1+ss/prime)− ss/primeq2sin2φ/|k+q|2, where φis the angle between k andq. Without loss of generality, we will assume that the applied electric field, the induced electric field, and the resultantcurrents are all in the ˆxdirection. We will also assume that the chemical potential in both layers is in the conduction (electron)band; i.e., they are electron doped. The chemical potential asa function of temperature is obtained by solving the integralequation g/summationdisplay k1 e(k2/2m−μi)/T+1+1 e−(k2/2m+μi)/T+1−1=ρi, (35) where ρiis the density of the layer i. It is an interesting feature of the BLG dispersion that in the noninteractingapproximation, the chemical potential is independent of thetemperature and is given by μ i=Efi, where Efiis the Fermi energy of the electrons in layer i. We will first focus on the polarizability and hence the screened Coulomb interaction in BLG systems. The analyticform for the BLG polarizability at zero temperature hasbeen derived before, 21but we need to take into account the temperature dependence of the screening to study the detailedbehavior of the drag resistivity with temperature. Here, wegeneralize our earlier results on BLG polarizability to finitetemperature. The polarizability can be broken up into theintraband [ s=s /primeterms in Eq. ( 13)] and interband [ s/negationslash=s/prime terms in Eq. ( 13)] contributions. Working out the azimuthal integrals analytically we obtain /Pi1intra i(q,ω)=gm 4π/integraldisplay∞ 0dx(fi +x−fi −x) x(x2+z)⎡ ⎣|x2−y2|−(x2+z)+sgn(ζ1+2xy)(2x2+ζ1)2 /radicalBig ζ2 1−4x2y2⎤ ⎦+(y,z→−y,−z) (36) and /Pi1inter i(q,ω)=−gm 4π/integraldisplay∞ 0dx(fi +x−fi −x) x(x2+z)/bracketleftbig |x2−y2|+(x2+z)−sgn(ζ2+2xy)/radicalBig ζ2 2−4x2y2/bracketrightbig +(y,z→−y,−z), (37) 245441-6COULOMB DRAG IN MONOLAYER AND BILAYER GRAPHENE PHYSICAL REVIEW B 84, 245441 (2011) where y=q/kFi,x=k/kFi,z=ω/E Fi,ζ1=z−y2, ζ2=z+2x2+y2, and fsx=1/(e(sx2−1)/ti+1) with ti= T/E Fi. We will now shift our attention to the nonlinear drag susceptibility. For BLG systems, both charge impurity scat-tering and short-range impurity scattering lead to a transportscattering time independent of momenta, and hence, we willonly consider an energy-independent scattering time, τ i k=τi, in this case. With this approximation the nonlinear dragsusceptibility is given by χi(q,ω)=4τi m/summationdisplay ss/primek[s(k+q)−s/primek]Fss/prime k,q/parenleftbig fi sk+q−fi s/primek/parenrightbig ω−εsk+q+εs/primek+i0+, (38) where mis the BLG mass. m=0.033me, where meis the free electron mass. The nonlinear susceptibility χx ican be separated into an intraband contribution and an interband contribution. The azimuthal integrals can be done analytically and we get Im[χx(intra) i (q,ω)]=τikFiycosφq π/bracketleftBigg/integraldisplay∞ |z−y2| 2ydxfi +x+fi −x x(x2+z)(2x2+z−y2)2 /radicalbig 4x2y2−(z−y2)2−(z→−z)/bracketrightBigg (39) and Im[χx(inter) i (q,ω)]=τikFicosφq πy/bracketleftBigg /Theta1(b)/integraldisplay|y+√ b| 2 |y−√ b| 2dxfi +x+fi −x x(x2−z)(2x2−z)/radicalbig 4x2y2−(2x2+y2−z)2−(z→−z)/bracketrightBigg , (40) where b=2z−y2andφqis the azimuthal angle related to the vector q. We note that the intraband drag susceptibility is propor- tional to the intraband polarizability of BLG [ χintra(q,ω)= (4τ/m )q/Pi1intra(q,ω)] only in the T=0 limit. This relation breaks down at finite temperatures. Finally, with a momentum-independent scattering time and a quadratic dispersion, theintralayer conductivity can be written in the simple formσ i=nie2τi/m. Note that the scattering times cancel in the expression for drag resistivity which is purely dominatedby electron-electron interactions. In Fig. 4(a),w ep l o tt h e temperature dependence of the BLG drag resistivity for twodifferent layer separations with a common layer carrier densityofn 1=n2=1012cm−2. The drag shows a quadratic behavior with logarithmic corrections at low temperatures. In Fig. 4(b), we plot the density dependence of the bilayer drag at T= 100 K for two different layer separations. 1 n ( 1012 cm-2)10100 ρD ( Ω) d = 50 A d = 70 A(b) 10 100 T (K)110100 ρD ( Ω)d = 50 A d = 70 A (a) FIG. 4. (Color online) (a) The Coulomb drag resistivity in BLG as a function of temperature for two layer separations, d=50˚A (thick black line) and d=70˚A(dashed red line). The density of the two layers n1=n2=1012cm−2. (b) The density dependence of bilayer Drag at a temperature T=100 K. The thick black line is for d=50˚Aand the dashed red line is for d=70˚A.W eh a v eu s e da dielectric constant κ=4 corresponding to boron nitride substrate.We now focus on the low-temperature asymptotic behavior of the BLG drag resistivity. To obtain the leading temperaturedependence of ρ Dat low temperatures, both χand/Pi1can be replaced by their intraband contributions at T=0i nt h e limit of small q,ω, i.e., Im[ χi(q,ω)]∼(gm/ 2π)τiω/kF, and the screened Coulomb interaction is replaced by its staticvalue with /Pi1 1=/Pi12∼gm/ (2π). The screening in BLG is controlled by the Thomas-Fermi wave vector qTF=e2gm/κ , where κis the background dielectric constant. For large layer separation, i.e., qTFd/greatermuch1, the screened interaction has the formV(q)∼q/[D0qTFsinh(qd)] in which D0=gm/ 2πis the density of states of bilayer graphene at Fermi level, and the leading-order drag resistivity is given by ρD∼1 e2π2ζ(3) 16T2 EF1EF21 (kF1d)(kF2d)(qTFd)2.(41) We note that contrary to MLG, qTFin BLG is a constant and is independent of the density. Thus the large layer separationlimit is not the same as the high-density limit ( k Fd/greatermuch1). In the opposite limit of small layer separation ( qTFd/lessmuch1) and strong interlayer correlations, the screened interactiontakes the form V(q)∼(q TF/D 0)e x p (−qd)/(q+2qTF) and the leading-order drag resistivity is given by ρD∼1 e2π2 24T2 EF1EF2q2 TF kF1kF2[−lnqTFd+γ−1−ln 4 +··· ], (42) where γis the Euler constant. Thus the drag resistance ∼T2 both in the large and small layer separation limit. The density dependence of the coefficient of the T2term is 1 /(n1n2)3/2 in both limits. In the large separation limit, the leading order term∼1/d4, whereas in the small separation limit, the drag 245441-7E. H. HW ANG, RAJDEEP SENSARMA, AND S. DAS SARMA PHYSICAL REVIEW B 84, 245441 (2011) resistivity shows a weak logarithmic dependence on the layer separation. We also note that, since backscattering is enhancedin BLG, there will be an additional T 2lnTcorrection to the formulas derived here. V. CONCLUSION In this paper, we have theoretically studied the frictional drag between two spatially separated MLG and BLG layersto lowest nonvanishing order in the screened interlayerelectron-electron interaction using Boltzmann transport theory(which is equivalent to a leading-order diagrammatic pertur-bation theory). We find that the low-temperature drag mostlyshows a quadratic temperature dependence, both in MLGand BLG, regardless of the layer separation and density ofcarriers. However the density and layer separation dependenceof the coefficient of the T 2term is very different for MLG and BLG. The density and layer separation dependence of low- temperature MLG drag resistivity crucially depends on the variation of the intralayer momentum scattering time with energy. For energy-independent intralayer scattering time(which does not correspond to any model of disorder in MLG, but captures the effects of both short-range and screened Coulomb impurities in standard 2DEG), the drag variesfromρ D∝T2/(nd)2forkFd/lessmuch1t oρD∝T2/(n4d6)f o r kFd/greatermuch1. However, for energy-dependent intralayer scattering times (corresponding to intralayer Coulomb scattering byrandom charged impurities in the environment) the power law of density dependence is significantly changed. Thus, an accurate measurement of interlayer MLG drag is in principle capable of distinguishing the main disorder scattering. In most currently available graphene samples the scattering due tocharged impurity disorder dominates. In this case the intralayer scattering time depends linearly on the energy and the drag resistivity becomes ρ D∝ln(√nd)/nforkFd/lessmuch1 and ρD∝ n−3d−4forkFd/greatermuch1. We note that the density dependence of drag resistivity is very sensitive to the experimental setup, so the density dependence does not have any universal powerlaw behavior because one is never in any asymptotic regime with clear-cut analytical power law behavior. Experimental measurements 2therefore may not find any clear-cut power law behavior in the density dependence of MLG drag since one is always in the crossover regime. We also find that due to the suppression of the q=2kFbackscattering in graphene there are no ∼T2ln(T) corrections to the low- temperature MLG drag resistivity. We would like to take this opportunity to compare this work, which is a more complete and updated version of Ref. 22, with recent works on low-temperature MLG drag resistivity13,23–25 which predict widely varying density and layer separation dependence of the low-temperature MLG drag resistivity.Within the energy-independent scattering time approximation,Tse et al. 13studied the drag in MLG in the high-density limit and obtained ρD∼T2/n3d4. Katsnelson25also obtained the same result in the high-density limit ( kFd/greatermuch1) and additionally considered the kFd/lessmuch1 limit of Ref. 13finding aρD∼T2ln(nd2)/ndependence in the low-density limit. Narozhny,23on the other hand, found that the drag vanishesfor linearly dispersive Dirac particles. In the current paper, within the energy-independent scattering time approximation,we find that ρ D∼T2/n4d6in the high-density limit and ρD∼T2/n2d2in the low-density limit. Narozhny23computes the drag to linear order in the interlayer potential where itis known to vanish even for ordinary 2D electron systems.The drag resistivity is at least quadratic in the interlayerpotential and its vanishing to linear order is a rather trivialresult. The main difference between our present work andRefs. 13and 25arises because both these papers use a form of the nonlinear drag susceptibility where the band groupvelocity scales linearly with the momentum kand results in a leading-order qdependence of the vertex in the susceptibility. While this is true for quadratic band dispersions as inregular 2D and BLG systems, the group velocity in linearlydispersive MLG is constant in magnitude ( v F), and combined with the overlap functions, gives rise to a leading-order q2 momentum in the nonlinear susceptibility in our case. This accounts for the discrepancy between our results and earlierresults. Finally Peres et al. 24consider the drag resistivity of MLG within a linearly-energy-dependent scattering time approxima-tion and obtain ρ D∼T2/n4d6in the high-density limit and ρD∼T2/n2d2in the low-density limit. These results match with our energy-independent scattering time approximationresults, whereas within linearly-energy-dependent scattering time, we get ρ D∼T2/n3d4in the high-density limit and ρD∼T2ln(nd2)/nin the low-density limit. This discrepancy can also be understood in terms of the scaling of thecurrent vertex factors. The vertex factor going into the dragsusceptibility is v skτk. Peres et al.24use a drag susceptibility where the vertex factor scales as q2and they recover our energy-independent scattering time scaling. However, with a constant group velocity, as is the case for MLG, the energy-dependent scattering time approximation should leadto a leading-order qdependence (coming from the momentum dependence of the scattering time) of the vertex and thisaccounts for the discrepancy between our results and Pereset al. The conceptual element of our work is the new role of group velocity in the nonlinear susceptibility defining the drag, which turns out to be crucial to the calculation of Coulomb drag in monolayer graphene (with its linear energy-momentumdispersion). We have also studied the drag resistivity in twospatially separated bilayer-graphene structures. We find thatthe drag resistivity shows a quadratic temperature dependenceat low temperatures, as for monolayer graphene. The densitydependence of the BLG drag is independent of the layer separation with ρ D∝T2/n3both in the large and small layer separation limit. In the large layer separation limit(weak interlayer correlation), the drag has a strong 1 /d 4 dependence on layer separation, whereas this goes to a weak logarithmic dependence in the strong interlayer correlationlimit. ACKNOWLEDGMENTS The authors gratefully thank Andre Geim for useful discussions and for asking several penetrating questions. Thiswork is supported by the US-ONR and NRI-SW AN. 245441-8COULOMB DRAG IN MONOLAYER AND BILAYER GRAPHENE PHYSICAL REVIEW B 84, 245441 (2011) 1S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. 83, 407 (2011). 2S. Kim, I. Jo, J. Nah, Z. Yao, S. K. Banerjee, and E. Tutuc, Phys. Rev. B 83, 161401 (2011). 3B. E. Feldman, J. Martin, and A. Yacoby, Nature Phys. 5, 889 (2009). 4H. Min, G. Borghi, M. Polini, and A. H. MacDonald, P h y s .R e v .B 77, 041407 (2008). 5E. H. Hwang and S. Das Sarma, P h y s .R e v .B 80, 205405 (2009). 6R. Nandkishore and L. Levitov, P h y s .R e v .L e t t . 104, 156803 (2010). 7T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, P h y s .R e v .L e t t . 66, 1216 (1991). 8L. Zheng and A. H. MacDonald, Phys. Rev. B 48, 8203 (1993). 9K. Flensberg and B. Y .-K. Hu, P h y s .R e v .B 52, 14796 (1995). 10A. G. Rojo, J. Phys. Condens. Matter 11, R31 (1999). 11E. H. Hwang, S. Das Sarma, V . Braude, and A. Stern, Phys. Rev. Lett.90, 086801 (2003). 12S. Das Sarma and E. H. Hwang, P h y s .R e v .B 71, 195322 (2005).13W.-K. Tse, B. Y .-K. Hu, and S. Das Sarma, Phys. Rev. B 76, 081401 (2007). 14K. S. Novoselov et al. ,Nature (London) 438, 197 (2005). 15Y .-W. Tan, Y . Zhang, K. Bolotin, Y . Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, P h y s .R e v .L e t t . 99, 246803 (2007). 16J. H. Chen, C. Jang, M. S. Fuhrer, E. D. Williams, and M. Ishigami,Nature Phys. 4, 377 (2008). 17K. Flensberg, B. Y .-K. Hu, A.-P. Jauho, and J. M. Kinaret, Phys. Rev. B 52, 14761 (1995). 18A. Kamenev and Y . Oreg, Phys. Rev. B 52, 7516 (1995). 19W.-K. Tse and S. Das Sarma, P h y s .R e v .B 75, 045333 (2007). 20E. H. Hwang and S. Das Sarma, P h y s .R e v .B 75, 205418 (2007). 21R. Sensarma, E. H. Hwang, and S. Das Sarma, Phys. Rev. B 82, 195428 (2010). 22E. H. Hwang and S. D. Sarma, e-print arXiv:1105.3203v1 . 23B. N. Narozhny, P h y s .R e v .B 76, 153409 (2007). 24N. M. R. Peres, J. M. B. Lopes dos Santos, and A. H. Castro Neto, Europhys. Lett. 95, 18001 (2011). 25M. I. Katsnelson, P h y s .R e v .B 84, 041407 (2011). 245441-9

PhysRevB.81.045319.pdf

Interlayer and interfacial exchange coupling in ferromagnetic metal/semiconductor heterostructures M. J. Wilson,1M. Zhu,1R. C. Myers,2D. D. Awschalom,2P. Schiffer,1and N. Samarth1,* 1Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 2Department of Physics, University of California, Santa Barbara, California 93106, USA /H20849Received 21 May 2009; revised manuscript received 16 December 2009; published 25 January 2010 /H20850 We describe a systematic study of the exchange coupling between a magnetically hard metallic ferromagnet /H20849MnAs /H20850and a magnetically soft ferromagnetic semiconductor /H20849Ga1−xMn xAs/H20850in bilayer and trilayer hetero- structures. An exchange spring model of MnAs /Ga1−xMn xAs bilayers accounts for the variation in the exchange-bias field with layer thickness and composition. We also present evidence for hole-mediated inter-layer exchange coupling in MnAs /p-GaAs /Ga 1−xMn xAs trilayers and study the dependence of the exchange- bias field on the thickness of the spacer layer. DOI: 10.1103/PhysRevB.81.045319 PACS number /H20849s/H20850: 75.50.Pp, 75.75. /H11002c, 81.16. /H11002c I. INTRODUCTION The systematic study of interlayer and interfacial ex- change coupling in ferromagnetic /H20849FM/H20850metal multilayers has led to important advances in condensed-matter physics,in addition to guiding rapid progress in magnetic storagetechnologies. 1,2In a similar manner, fundamental inquiry into the exchange coupling in FM semiconductor hetero-structures could have an important influence on the develop-ment of semiconductor spintronics, where one can envisionan additional level of optoelectronic control over the under-lying exchange interaction. 3Interlayer and/or interfacial ex- change coupling in FM semiconductor heterostructures hasindeed been unequivocally observed in a variety of experi-ments, including neutron-scattering measurements ofGa 1−xMn xAs /GaAs multilayers,4,5magnetometry, and mag- netoresistance measurements of MnO /Ga1−xMn xAs bilayers6,7and MnAs /Ga1−xMn xAs bilayers,8and x-ray mag- netic circular dichroism studies of Fe /Ga1−xMn xAs heterostructures.9However, despite long-standing theoretical interest in this topic,10there is little systematic experimental data that examines the dependence of this coupling on rel-evant parameters; the availability of such data is critical fordeveloping a deeper theoretical understanding of interfacial/interlayer exchange coupling in FM semiconductor hetero-structures. This paper presents systematic studies of the interlayer and interfacial exchange coupling between a magneticallysoft FM semiconductor /H20849Ga 1−xMn xAs/H20850and a magnetically hard FM metal /H20849MnAs /H20850. The juxtaposition of these materials is particularly convenient because the interlayer and interfa-cial exchange coupling are readily measured using standardmagnetometry techniques, 8rather than requiring more elabo- rate methods such as neutron scattering4,5or x-ray magnetic circular dichroism /H20849XMCD /H20850.9We systematically map out the variation in the exchange coupling as a function of manysample parameters in both bilayer and trilayer geometries.Our group has previously examined exchange coupling inbilayers as a function of Ga 1−xMn xAs thickness /H20849tGMA/H20850, showing that the exchange field /H20849HE/H20850varies inversely with tGMA.8The present work first confirms this result in an addi- tional set of samples and then goes on to study the depen-dence of the exchange coupling on a wide range of previ-ously unexplored parameters such as MnAs thickness, magnetization, and spacer thickness in trilayer systems, all ofwhich give important insights into the physics of this system.The results are consistent with the formation of an exchangespring in bilayers due to FM interfacial coupling withMnAs. 11Additionally, in trilayers, we find evidence for FM hole-mediated exchange coupling that decays exponentiallywith spacer-layer thickness, persisting over a length scale of/H110115 nm. We do not find evidence for antiferromagnetic /H20849AFM /H20850exchange coupling over the entire space of param- eters examined. This is in contrast to the reported observa-tion of AFM interlayer coupling in neutron-scattering studiesof a Ga 1−xMn xAs /p-GaAs /Ga1−xMn xAs multilayer sample.5 Finally, we find an enhancement of the Curie temperature /H20849TC/H20850of Ga 1−xMn xAs layers by the overgrowth of MnAs. Al- though superficially resembling a “proximity effect” whereintheT Cof a weak ferromagnet might be enhanced by interfa- cial exchange coupling with a strong ferromagnet,9control measurements show that this is an extrinsic effect stemmingfrom the unintentional annealing during growth. II. EXPERIMENTAL DETAILS This paper focuses on four different series of samples. Series A consists of three bilayer samples with 12 nm“type-A” MnAs on top of a Ga 1−xMn xAs layer with x/H110116% and with Ga 1−xMn xAs layer thicknesses of tGMA=30, 50, and 80 nm. Series B consists of three MnAs /Ga1−xMn xAs bilay- ers where the composition of the Ga 1−xMn xAs is varied /H208490.05/H11349x/H113490.16/H20850while keeping the Ga 1−xMn xAs and MnAs thicknesses constant at 30 and 8 nm, respectively. SeriesC consists of several MnAs /Ga 1−xMn xAs bilayer samples /H20849x/H110116%, tGMA=30 nm /H20850in which the MnAs layer is pur- posely varied in thickness /H208491/H11351tMA/H113514n m /H20850across the wafer by exposing the static substrate to a spatially inhomogeneousMn flux. Series D consists of MnAs /p-GaAs /Ga 1−xMn xAs trilayers /H20849with x/H110116%/H20850in which the MnAs and Ga 1−xMn xAs layer thicknesses are kept fixed /H20849tMA=10 nm and tGMA =30 nm /H20850while the spacer thickness is varied /H20849tspacer =1,2,3,4,5 nm /H20850. The heterostructures used in our studies are fabricated by low-temperature molecular-beam epitaxy on semi-insulatingPHYSICAL REVIEW B 81, 045319 /H208492010 /H20850 1098-0121/2010/81 /H208494/H20850/045319 /H208496/H20850 ©2010 The American Physical Society 045319-1/H20849001/H20850GaAs substrates, after first depositing a 170-nm-thick high-temperature grown GaAs buffer layer at 580 °C. This isfollowed by the growth of a Ga 1−xMn xAs layer at a substrate temperature in the range 235–250 °C; the optimal substratetemperature depends on the Mn concentration. In particular,we note the use of distinct conditions during the growth ofthe highest Mn composition bilayer sample used in seriesB. 12For trilayer samples, the epitaxial growth proceeds with deposition of a p-doped GaAs:Be layer at the same substrate temperature. The carrier concentration is approximately 3/H1100310 19cm−3as determined by Hall-effect measurements of control samples. After the growth of the Ga 1−xMn xAs layer /H20849or Ga 1−xMn xAs /p-GaAs heterostructure /H20850, the substrate tem- perature is lowered to /H11011200 °C with the As shutter open to initiate the growth of a few monolayers of MnAs under As-rich conditions. The substrate temperature is then raised to/H11011230 °C and MnAs growth is resumed; this procedure con- sistently yields MnAs in the type-A orientation with the c axis aligned with the /H208511 ¯10/H20852axis of the Ga 1−xMn xAs layer. Cross-sectional transmission electron microscopy /H20849TEM /H20850 shows an atomically abrupt and smooth interface betweenMnAs and Ga 1−xMn xAs, despite a large lattice mismatch /H20851Fig. 1/H20849a/H20850/H20852. Atomic force microscopy measurements show that the freshly grown MnAs surface exhibits a relativelysmooth surface with an rms roughness of about 1 nm /H20851Fig. 1/H20849b/H20850/H20852. Some trenches with depth of /H110112 nm are observed as a result of the transition from three-dimensional island growthto two-dimensional layer growth. Note that the MnAs layeroxidizes quite readily, thus necessitating sample storage invacuum for observing consistent physical properties with ag-ing. The magnetic properties of the samples are characterizedusing a dc superconducting quantum interference device /H20849SQUID /H20850magnetometer. For temperature-dependent mea- surements of the magnetization M/H20849T/H20850, samples are first cooled down from room temperature in a 20 kOe field ap-plied along the easy axis of MnAs; unless otherwise stated,measurements are then taken while warming up in a field of30 Oe. For magnetization hysteresis measurements, we focushere only on minor loops of the Ga 1−xMn xAs layers in order to determine the exchange field. The minor loops are taken toa positive field large enough to switch the Ga 1−xMn xAs layer /H20849typically 500–1200 Oe /H20850but smaller than the coercivity of MnAs, which is around 2 kOe. The hysteresis loops are mea-sured after first saturating the MnAs layer in a negative 20kOe field. All hysteresis loops unless noted are measured at4.2 K and the applied field is along the /H20851110/H20852GaAs crystal- line axis. This corresponds to the /H20851112 ¯0/H20852direction in MnAs, which is the easy axis of the type-A MnAs layer. III. EXCHANGE SPRING MODEL OF BILAYERS We begin by discussing the interfacial exchange coupling in MnAs /Ga1−xMn xAs bilayers. To calculate the exchange field experienced by the Ga 1−xMn xAs layer, we use a partial domain wall /H20849PDW /H20850model analogous to the one used in AFM/FM systems13and hard/soft metallic FM bilayers.14 The magnetization of MnAs is considered to be fixed alongits easy axis in the positive field direction along the /H20851110/H20852 axis of Ga 1−xMn xAs; this corresponds to the /H20851112¯0/H20852direction of the hexagonal MnAs crystal. The magnetization of theGa 1−xMn xAs layer is free to switch in an external magnetic field and its direction is designated with reference to thefixed MnAs magnetization, as illustrated in Fig. 1/H20849c/H20850.W e assume that a PDW of thickness t 1is formed in the Ga1−xMn xAs layer near the interface. The angle between the MnAs magnetization and the Ga 1−xMn xAs magnetization at the interface is defined as /H92721while/H92722is the angle between the MnAs magnetization and the bulk Ga 1−xMn xAs magneti- zation. Due to the strong coupling at the interface and therelatively strong anisotropy constant of MnAs, the interfacialspin alignment of the Ga 1−xMn xAs layer should be very close to that in the MnAs layer. For this strong interfacial cou-pling, where /H92721/H110150 and t1/H11270t2/H11015tGMA, the energy density per unit area can be written as E=2/H20881AK/H208491 − cos /H92722/H20850−Aex+KutGMA sin2/H92722 +1 /4KctGMA cos22/H92722−HMt GMA cos/H92722. /H208491/H20850 The first term is the energy of the PDW, where Ais the spin stiffness of Ga 1−xMn xAs and Kis the effective aniso- tropy constant; the second term /H20849Aex/H20850is the exchange cou- pling at the interface; the third and fourth terms are theuniaxial and biaxial anisotropy energy in Ga 1−xMn xAs. The terms KuandKcare the uniaxial and biaxial anisotropy con- stants, respectively. The last term is the Zeeman energywhere His the externally applied magnetic field and Mis the saturated magnetization of the Ga 1−xMn xAs layer. When considering a strong cubic anisotropy, the energy minimumoccurs at 45° and 135°. Thus, the two switching fieldsare determined by using the following two conditions:MnAs[110] /.notdef.g00011 /.notdef.g00012+H -H (Ga,Mn)As thickness: t GMAPartial DW in (Ga,Mn)As: t1 Complete domain in(Ga,Mn) As: t2=tGMA-t1(c) (b) 1µ mx1 µ m (a) 5n m(Ga,Mn)AsMnAs FIG. 1. /H20849Color online /H20850/H20849a/H20850High-resolution TEM images of a bilayer sample. /H20849b/H20850Atomic force microscope image of the top MnAs layer. /H20849c/H20850Depiction of a partial domain-wall configuration in Ga1−xMn xAs, with spins continuously rotating as a function of the distance from the interface. Beyond a certain depth t1, a complete domain /H20849t2/H20850forms.WILSON et al. PHYSICAL REVIEW B 81, 045319 /H208492010 /H20850 045319-2/H115092E /H11509/H927222/H20849/H92722=/H9266/4/H20850/H110220,/H115092E /H11509/H927222/H20849/H92722=3/H9266/4/H20850/H110220. This yields the fol- lowing switching fields: H/H11022HC1=−2/H208812AK−4Kct /H208812Mt, /H208492/H20850 H/H11022HC2=−2/H208812AK+4Kct /H208812Mt. /H208493/H20850 The exchange field, given by HE=/H20849HC1+HC2/H20850/2 =−2/H20881AK /Mt, shows an inverse dependence on both the thickness and magnetization of the Ga 1−xMn xAs layer. In ad- dition, the model also predicts that the coercive field, given byHc=/H20849HC2−HC1/H20850=8Kc//H208812M, shows an inverse depen- dence on the magnetization of the Ga 1−xMn xAs layer. The validity of this model is tested by studying the exchangecoupling in MnAs /Ga 1−xMn xAs bilayers as a function of sample geometry and composition. IV . V ARIATION IN EXCHANGE FIELD AND COERCIVE FIELD WITH Ga 1−xMnxAs THICKNESS IN BILAYERS We first address the effect of varying the Ga 1−xMn xAs layer thickness /H20849series A /H20850. Figure 2/H20849a/H20850shows the temperature dependence of the remanent magnetization M/H20849T/H20850in three bilayer samples with tGMA=30, 50, and 80 nm, measured in a field of 30 Oe after cooling down from room temperature ina −20 kOe field. Note that the plot depicts the magnetizationof the films normalized to area /H20849not volume /H20850since the samples are bilayer stacks of two different ferromagnets. Weclearly observe two distinct FM phase transitions at T C /H1101175 K for Ga 1−xMn xAs and TC/H11011318 K for MnAs. The major magnetization hysteresis loops /H20849data not shown /H20850are similar to the data shown on other samples in a previousreport, 8revealing two different coercivities for Ga1−xMn xAs/H20849/H11011100 Oe /H20850and MnAs /H20849/H110112 kOe /H20850. Based upon our SQUID measurements of these major hysteresis loops,we do not find any obvious indications of a biquadratic cou-pling. Figure 2/H20849b/H20850shows the minor loops for two bilayers with different Ga 1−xMn xAs thicknesses; the displacement of the center of the minor loop is always opposite to the mag-netization of the MnAs layer, indicating a “negative ex-change bias” due to FM coupling between the two layers,where a parallel alignment of the two layers is favored. Notethat this FM coupling contrasts with the AFM coupling re-ported in XMCD studies of Fe /Ga 1−xMn xAs bilayers.9The opposite sign of the interfacial exchange coupling inMnAs /Ga 1−xMn xAs and Fe /Ga1−xMn xAs is very intriguing but we do not presently have an explanation for this differ-ence in behavior. We find that the exchange field H E /H11011/H20849tGMA/H20850−1, in agreement with our model /H20851Fig.2/H20849c/H20850/H20852and con- sistent with earlier measurements on a different sampleseries. 8With typical parameters for Ga 1−xMn xAs samples of comparable composition /H20849A/H110110.4 pJ /m,K/H110110.3 kJ /m3, and M/H1101116 000 A /m3/H20850,15we calculate HE/H11011440, 264, and 165 Oe for the bilayers with tGMA=30, 50, and 80 nm, respec- tively. This agrees reasonably well with the respective ex-perimental values of 455, 235, and 120 Oe. The PDW modelalso predicts that the coercive field H cshould be independent oftGMA; our data are in agreement with this, with no obvious experimental dependence found in all three samples in Fig.2/H20849c/H20850. Studies of ferromagnets exchange biased by an antiferro- magnet typically show a correlation between H Eand the co- ercive field Hc. This relationship can be studied by examin- ing the temperature dependence of HEand Hc. For this purpose, we chose a MnAs /Ga1−xMn xAs bilayer with high Mn concentration /H20849x=0.16 /H20850since this leads to a higher TC /H20849=160/H110065K/H20850,12thus allowing us to cover a wider tempera- ture range. In contrast to conventional exchange biasing us-ing an antiferromagnet, we find that the H candHEare not correlated while Hcdecreases with increasing temperature, HEstays relatively constant /H20851Fig. 2/H20849d/H20850/H20852. The decrease in co- ercivity is expected and is also seen in nonexchange-biasedGa 1−xMn xAs samples where it is attributed to a weakening of the magnetization and anisotropies of the Ga 1−xMn xAs layer. However, the constant exchange field is surprising; the PDWmodel predicts that as the magnetization of the Ga 1−xMn xAs layer decreases with increasing temperature, HEshould in- crease. However, we note that the anisotropy constants ofGa 1−xMn xAs are also a function of temperature, decreasing with increasing temperature.15,16Thus, a possible reason for our observation is that the temperature dependence of boththe magnetization and anisotropy cancel out any temperaturevariation in H E. V . V ARIATION IN EXCHANGE FIELD AND COERCIVE FIELD WITH Ga 1−xMnxAs MAGNETIZATION IN BILAYERS A second prediction of the model is that HEshould de- crease inversely with the saturated magnetization /H20849Msat/H20850of500 400 300 200 100 0HE(Oe) 80 60 40 20 tGMA(nm)(c)1/tGMAfit-70-60-50-40 MArea(10-5emu/cm2) 500 0 H(Oe)(b)60 40 20 0MArea(10-5emu/cm2) 300 200 1000 T (K)30nm 50nm 80nm(a) 100 80 60 40 20 0Hc(Oe) 120 80 40 0 T(K)250 200 150 100 50 0 HE(Oe) (d) FIG. 2. /H20849Color online /H20850/H20849a/H20850Temperature-dependent remanent magnetization M/H20849T/H20850for bilayer samples with three different Ga1−xMn xAs thicknesses as indicated on the plot /H20849series A /H20850. The Ga1−xMn xAs composition is x=0.06 and the MnAs thickness is tMA=12 nm. /H20849b/H20850Minor hysteresis loop for the bilayer samples with tGMA=50 nm /H20849green squares /H20850and tGMA=80 nm /H20849blue triangles /H20850. Data are taken at T=4.2 K. /H20849c/H20850Exchange field versus Ga 1−xMn xAs thickness, showing that HE/H11008/H20849tGMA/H20850−1. Data are taken at T=4.2 K. /H20849d/H20850Coercivity and exchange field as a function of temperature for a high Mn composition MnAs /Ga1−xMn xAs bilayer with tGMA =50 nm and x=0.16.INTERLAYER AND INTERFACIAL EXCHANGE COUPLING … PHYSICAL REVIEW B 81, 045319 /H208492010 /H20850 045319-3the Ga 1−xMn xAs layer. This prediction is tested with a series of samples of varied Mn content /H20849Series B /H20850. Figure 3/H20849a/H20850 shows the minor loops of three samples with varying satura-tion of Ga 1−xMn xAs layer with nominal values of x /H110150.05,0.07,0.16. Figure 3/H20849b/H20850shows that the data are quali- tatively consistent with the model /H20849i.e., HEdecreases with increasing Msat/H20850but deviate from the predicted inverse de- pendence. A plausible explanation for this deviation is thatthe changing Mn composition will also change the aniso-tropy constants and not just saturated magnetization. Alsoshown in Fig. 3/H20849b/H20850is the coercivity as a function of magne- tization, which is qualitatively consistent with the /H20849M sat/H20850−1 dependence predicted by the model. VI. V ARIATION IN EXCHANGE FIELD WITH MnAs THICKNESS IN BILAYERS In the PDW model, the MnAs layer is treated as being essentially infinitely thick. In order to investigate the limita-tions of this assumption, we now address the behavior ofbilayer samples in which the thickness of the MnAs layer/H20849t MA/H20850is varied, keeping tGMA fixed /H20849series C /H20850. These samples were grown by stopping the rotation of the wafer during thegrowth of the MnAs layer, allowing for a spatial variation inMnAs thickness across a single wafer. We grew two wafersand cut each into five separate samples. Figure 3/H20849c/H20850shows M/H20849T/H20850for the first set of samples with 1.4 /H11351t MA/H113512 nm,where we estimate tMAusing the saturated magnetization. The second set has estimated values 3 /H11351tMA/H113514 nm. Note that all these thicknesses are significantly thinner than ourother sets of samples, which had t MA/H113508 nm. Figure 3/H20849d/H20850 shows HEvstMAfor both sets of samples /H20849differentiated by the color of the data points /H20850; the plot indicates that HEshows little dependence on tMAfor bilayer samples with at least 3 nm of MnAs, which is consistent with the PDW model.However, Fig. 3/H20849d/H20850also shows that H Erapidly decreases for very thin layers of MnAs /H20849tMA/H113512n m /H20850. The observed variation in the exchange field on the thick- ness of the biasing MnAs layer is reminiscent of the behaviorin the conventional exchange-biasing effect provided by anantiferromagnet. 17Using the simple Meiklejohn-Bean model, exchange biasing is obtained under the conditionK AFMtAFM/H11271Aex, where KAFMandtAFMare the anisotropy and the thickness of the AFM layer and Aexis the interfacial exchange coupling. This model /H20849and its more sophisticated extensions /H20850thus predict that a critical thickness of the AFM layer is needed for exchange biasing with a value propor- tional to the ratioAex KAFM. Studies of AFM/FM bilayers have confirmed in some detail the expectations of this picture, showing both the quenching of exchange bias below a criti-cal value as well as a saturation of the exchange bias at largeAFM layer thickness. 18It is tempting to state that a similar picture could explain the variation in the exchange field withthe MnAs layer thickness; the only difference between ourhard/soft FM bilayers and conventional AFM/FM bilayers isthat the term describing the energy of the biasing layer de-pends upon the anisotropy of a ferromagnet rather than anantiferromagnet. Thus, our observation of an exchange fieldthat saturates for rather small values of the biasing layerthickness /H20849t MA/H114072n m /H20850could be viewed as being qualita- tively consistent with the relatively large anisotropy of theMnAs layer compared with the interfacial exchange-coupling energy. VII. INTERLAYER EXCHANGE COUPLING IN Ga 1−xMnxAs ÕGaAs ÕMnAs TRILAYERS: V ARIATION WITH THICKNESS AND DOPING OF SPACER LAYER Next, we address the propagation of the exchange cou- pling through a nonmagnetic spacer by studying the behaviorof MnAs /p-GaAs /Ga 1−xMn xAs trilayers /H20849series D /H20850. Our data provide evidence that the exchange coupling between thetwo FM layers is mediated by holes in the spacer. Figure 4/H20849a/H20850 shows a comparison between minor hysteresis loops for twosamples wit ha3n m spacer, one of which is undoped and the other doped with a nominal hole concentration of 3/H1100310 19cm−3. There is no evidence for exchange biasing in the sample with the undoped spacer while the doped sampleshows a clear shift, thus strongly suggesting that the ex-change between the two magnetic layers is hole mediated. Asystematic study of this coupling as a function of the dopingdensity is beyond the scope of this paper. Instead, we focuson the spacer thickness /H20849t spacer/H20850for a fixed p-doping level in the spacer. Figure 4/H20849b/H20850shows both HEandHcas a function oftspacer, indicating that the exchange coupling becomes280 260 240 220HE(Oe) 4 3 2 1 tMA(nm)(d)16 12 8 4 0MArea(10-5emu/cm2) 300 200 1000 T(K)1 2 3 4 5(c)-50-2502550MGMA(emu/cc) 1000 0 H(Oe)(a) 600 400 200 0H(Oe) 60 40 20 0 MSat(emu/cc)(b)HE HC16% 5%7% 5% 7% 16% FIG. 3. /H20849Color online /H20850/H20849a/H20850Minor hysteresis loops of three bilayer samples with varying composition x=0.05,0.07,0.16 /H20849series B /H20850.I n these three samples, tGMA=30 nm and tMA=8 nm. The magnetiza- tion is shown per unit volume with the MnAs signal subtracted out.Data are taken at T=4.2 K. /H20849b/H20850Exchange field /H20849H E/H20850and coercivity /H20849Hc/H20850as a function of the saturated magnetization. Data are taken at T=4.2 K. /H20849c/H20850Temperature-dependent remanent magnetization M/H20849T/H20850for bilayer samples with different MnAs layer thicknesses in the range 1 /H11351tMA/H113514n m /H20849series C /H20850. The Ga 1−xMn xAs layer has a thickness tGMA=30 nm and composition x=0.06. The magnetiza- tion is shown per unit area. /H20849d/H20850Exchange field versus MnAs layer thickness showing critical thickness between 2 and 3 nm. The dif-ferent symbols refer to two different nonrotated growths. Data aretaken at T=4.2 K.WILSON et al. PHYSICAL REVIEW B 81, 045319 /H208492010 /H20850 045319-4negligible when the spacer is larger than 5 nm /H20849around the anticipated spin-diffusion length in Be-doped GaAs /H20850. The plot also shows that HEdecays exponentially with tspacer /H20851as indicated by the fit in Fig. 4/H20849b/H20850/H20852while Hcis rela- tively constant. The robustness of this behavior has beenconfirmed in a second set of trilayer samples /H20849not shown /H20850. Note there is no evidence for AFM exchange couplingover the entire space of parameters examined; this differsfrom the findings of a recent neutron-scattering study of aGa 1−xMn xAs /p-GaAs /Ga1−xMn xAs superlattice sample that revealed hole-mediated AFM coupling between theGa 1−xMn xAs layers.5We speculate that this difference prob- ably arises from the very different electronic structure anddensities of states in these two systems; our studies probe theexchange coupling between a metallic FM and a FM semi-conductor while the neutron-scattering measurements centeraround the exchange coupling mediated by band holes in apurely FM semiconductor superlattice. We now discuss the monotonic ferromagnetic decay of the interlayer exchange within the context of the correspond-ing phenomenon in metallic multilayers where oscillatoryinterlayer exchange coupling is commonly observed in well-prepared samples. In that case, the oscillatory dependence ofthe coupling on spacer thickness is understood using a modelthat relates the interlayer exchange to spin-dependent reflec-tion at interfaces and resultant quantum-confined stateswithin the spacer. 1As the spacer-layer thickness is changed, the energy of these quantum-well states changes. The oscil-lation period is then determined by the filling and emptyingof these states as they pass through the Fermi energy of thespacer. The oscillation period is thus directly related to criti-cal spanning vectors of the spacer-layer Fermi surface and in a simple free-electron gas picture is given by /H9266 kF. Such a model also predicts that the amplitude of the oscillatory cou- pling will be damped with an inverse dependence on tspacer.I f these concepts are applied to an ideal, disorder-freeGa 1−xMn xAs /GaAs /MnAs trilayer, it is apparent that the os- cillation period will be much longer than in a metallic sys-tem, simply because of the smaller carrier density in thesemiconductor spacer. For instance, for a hole density p /H1101110 19cm−3, the Fermi wave vector kF/H110110.67 nm−1so that the oscillation period is /H110115 nm. Thus, even in an ideal sample, we would not expect to observe an oscillatory cou-pling over the spacer thicknesses studied in our experiments.It is however difficult to ignore the presence of disorder inour samples; the low-temperature growth of the p-GaAs spacer results in a low carrier mobility and a short Drudemean-free path /H20849/H110114n m /H20850. Under these circumstances, the smearing of the Fermi surface can rapidly quench the oscil-latory Ruderman-Kittel-Kasuya-Yoshida interaction. Thiscould also account for the exponential decay in the amplitudeof the coupling rather than the weaker inverse dependence ont spacer expected in the ideal case. Nonoscillatory FM coupling in multilayers can also arise from extrinsic effects; the most trivial example is that ofdirect FM coupling through pin holes. This is ruled out bydetailed TEM studies of our samples that show that thespacer layer is continuous with no obvious pin holes. An-other possible extrinsic effect arises from the interdiffusionof magnetic ions into the nominally nonmagnetic spacer.Experimental studies of Fe/Si/Fe trilayers with a thin/H20849/H110211.6 nm /H20850undoped Si spacer showed exponentially decay- ingantiferromagnetic coupling with both bilinear and biqua- dratic terms. 19This AF coupling was interpreted using a model that attributes the coupling to the polarization of para-magnetic loose spins in the spacer layer. 20The FM nature of the coupling observed in our samples is however contrary tothe predictions of this model. In addition, previous studies 21 of Mn interdiffusion in GaAs/MnAs superlattices suggestthat the interdiffusion is limited to a several monolayers andis thus not extensive enough to produce the observed effect.Finally, an exponentially decaying exchange coupling wasobserved in exchange-biased trilayer systems wherein anAFM biasing layer is separated from a FM layer by a noblemetallic spacer layer. 22Again, due to the vast differences in electronic structure and the density of states, it seems un-likely that there would be a common underlying physicalmechanism that can describe the exponentially decaying cou-pling in both our samples and these metallic AFM/noblemetal/FM trilayers. VIII. MODIFICATION OF TCIN Ga 1−xMnxAs ÕMnAs HETEROSTRUCTURES The final section of this paper addresses an intriguing pos- sibility: is it possible that the exchange coupling between400 300 200 100 0H (Oe) 6 4 2 0 tSpacer(nm)HE Hc (b) -90-80-70-60 MArea(10-5emu/cm2) -500 0 500 H (Oe)Undoped Doped (a) 15 10 5 0MArea(10-5emu/cm2) 200 150100500 T (K)Etched Annealed(d)15 10 5 0MArea(10-5emu/cm2) 200150100500 T (K)Bilayer Epilayer(c) FIG. 4. /H20849Color online /H20850/H20849a/H20850Minor loops for two trilayer samples, one with an undoped spacer and one with a Be-doped spacer. Dataare taken at T=4.2 K. The Ga 1−xMn xAs, MnAs, and spacer-layer thicknesses are tGMA=30 nm, tMA=12 nm and tspacer =3 nm, re- spectively. /H20849b/H20850Exchange field /H20849HE/H20850and coercivity /H20849Hc/H20850in Ga1−xMn xAs /p-GaAs /MnAs trilayers versus p-GaAs spacer-layer thickness /H20849series D /H20850. Here, tGMA=30 nm and tMA=10 nm. Data are taken at T=4.2 K. The solid line shows a fit of the variation in HE vstspacer to an exponential decay: HE/H11008exp/H20849−/H9251tspacer /H20850./H20849c/H20850 Temperature-dependent magnetization M/H20849T/H20850for bilayer and single- layer control sample. These measurements are taken while warmingup in a field of 200 Oe. /H20849d/H20850S a m ea si n /H20849c/H20850with the added etched sample showing that T Cdid not change and with added annealed control sample showing very similar increase in TC. The unusual shape of the M/H20849T/H20850at lower temperatures results from temperature- dependent changes in the easy axis.INTERLAYER AND INTERFACIAL EXCHANGE COUPLING … PHYSICAL REVIEW B 81, 045319 /H208492010 /H20850 045319-5MnAs and Ga 1−xMn xAs could “bootstrap” the onset of ferro- magnetism in the latter via a “proximity” effect? Recentx-ray magnetic circular dichroism studies have suggestedthat such a proximity effect results in room-temperature fer-romagnetism in a very thin region of Ga 1−xMn xAs within Fe /Ga1−xMn xAs bilayers, although no direct evidence for such an effect is observed in magnetometry.9The growth of MnAs on top of Ga 1−xMn xAs consistently enhances TCof the latter typically by /H1101125 K compared to single epilayers of Ga1−xMn xAs grown under similar conditions /H20851an example is shown in Fig. 4/H20849c/H20850/H20852. The as-grown bilayer samples can show TCof up to 150 K, much higher than can normally be achieved before annealing. To better understand the nature ofthis effect and to see if it is intrinsic to the exchange cou-pling, such bilayer samples were measured after removingthe top MnAs with a chemical etch. Figure 4/H20849d/H20850shows the results of this control experiment; the T Cof the Ga 1−xMn xAs layer remains elevated after etching the sample, instead ofdropping as would be expected if the enhancement origi-nated in a proximity effect. Our results thus suggest that theenhancement of T Cis likely an extrinsic effect, stemming from very effective annealing of Mn interstitial defects dur-ing the overgrowth of MnAs. To further confirm this hypoth-esis, the single epilayer control sample was annealed undersimilar conditions and found to have a very similar increaseinT C/H20851Fig. 4/H20849d/H20850/H20852. We note that our magnetometry measure- ments cannot of course rule out the existence of exchange-enhanced ferromagnetism in a thin interfacial region ofMnAs /Ga 1−xMn xAs bilayers like what was seen in iron- capped Ga 1−xMn xAs bilayers.9IX. SUMMARY In summary, we have reported a comprehensive study of exchange coupling in hybrid FM metal/semiconductor het- erostructures. Our study maps out the variation in the inter-facial exchange coupling between MnAs and Ga 1−xMn xAs in bilayers as a function of a variety of system parameters. Theresulting data are consistent with the formation of a partialexchange spring configuration in the soft Ga 1−xMn xAs layer. Studies of trilayer samples show that this exchange couplingcan propagate through a p-doped nonmagnetic spacer layer, resulting in an interlayer exchange coupling. Using a metal-lic FM layer to exchange-bias Ga 1−xMn xAs offers a new test bed for studying exchange coupling between FM metals andsemiconductors, and it also possibly provides a model sys-tem to study spin-dependent transport in nonuniform magne-tization configurations. 23,24As an engineering tool, it opens up opportunities for tailoring the coercivity of FM semicon-ductors for proof-of-concept device applications. ACKNOWLEDGMENTS This research is supported by the ONR MURI program under Contract No. N0014-06-1-0428. 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Tonomura, P. D. Brown, R. P. Cam- pion, K. W. Edmonds, B. L. Gallagher, J. Zemen, and T. Jung-wirth, Phys. Rev. Lett. 100, 047202 /H208492008 /H20850. 16M. Cubukcu, H. J. von Bardeleben, K. Khazen, J. L. Cantin, M. Zhu, M. J. Wilson, P. Schiffer, and N. Samarth, J. Appl.Phys. 105, 07C506 /H208492009 /H20850. 17J. Nogues and I. Schuller, J. Magn. Magn. Mater. 192, 203 /H208491999 /H20850. 18M. S. Lund, W. A. A. Macedo, K. Liu, J. Nogués, I. K. Schuller, and C. Leighton, Phys. Rev. B 66, 054422 /H208492002 /H20850. 19G. J. Strijkers, J. T. Kohlhepp, H. J. M. Swagten, and W. J. M. de Jonge, Phys. Rev. Lett. 84, 1812 /H208492000 /H20850. 20J. Slonczewski, J. Appl. Phys. 73, 5957 /H208491993 /H20850. 21M. J. Wilson, G. Xiang, B. L. Sheu, P. Schiffer, N. Samarth, S. J. May, and A. Bhattacharya, Appl. Phys. Lett. 93, 262502 /H208492008 /H20850. 22N. J. Gökemeijer, T. Ambrose, and C. L. Chien, Phys. Rev. Lett. 79, 4270 /H208491997 /H20850. 23J. Foros, A. Brataas, Y . Tserkovnyak, and G. E. W. 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PhysRevB.93.014404.pdf

PHYSICAL REVIEW B 93, 014404 (2016) Nonlocal topological magnetoelectric effect by Coulomb interaction at a topological insulator-ferromagnet interface Stefan Rex,1Flavio S. Nogueira,2,3and Asle Sudbø1 1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 2Institute for Theoretical Solid State Physics, IFW Dresden, PF 270116, 01171 Dresden, Germany 3Institut f ¨ur Theoretische Physik III, Ruhr-Universit ¨at Bochum, Universit ¨atsstraße 150, DE-44801 Bochum, Germany (Received 14 October 2015; published 6 January 2016) The interface between a topological insulator and a ferromagnetic insulator exhibits an interesting interplay of topological Dirac electrons and magnetism. As has been shown recently, the breaking of time-reversal invarianceby magnetic order generates a Chern-Simons term in the action, that in turn leads to a Berry phase and amagnetoelectric effect of topological origin. Here, we consider the system in the presence of a long-rangeCoulomb interaction between the Dirac electrons, and find that the magnetoelectric effect of the fluctuatingelectric field becomes nonlocal. We derive a Landau-Lifshitz equation for the fluctuation-induced magnetizationdynamics and the Euler-Lagrange equation of the Coulomb field by explicit one-loop calculations. Via theCoulomb interaction, divergences in the in-plane magnetization affect the magnetization dynamics over largedistances in a topologically protected way. DOI: 10.1103/PhysRevB.93.014404 I. INTRODUCTION In a topological insulator (TI), the bulk band structure gives rise to gapless surface states that are protected by symmetryvia a bulk-boundary correspondence [ 1,2]. These conducting states have a linear dispersion (Dirac electrons) arising mainlydue to strong spin-orbit coupling. In addition, spin-momentumlocking makes surface currents on a TI a promising tool forspintronics applications [ 3,4]. However, not all materials that feature a Dirac dispersion and a strong spin-orbit couplingare TIs. For instance, pure bismuth is a Dirac-like materialfeaturing a strong spin-orbit coupling, which is not a TI,since its surface states are not protected by symmetry. Theprotecting symmetry in most TIs is time-reversal invariance(TRI). In three-dimensional (3D) TIs, the electromagnetic re- sponse is characterized by a magnetoelectric term in theLagrangian [ 5,6]. Unlike the magnetoelectric term arising in other materials, for example, multiferroics, the magnetoelec-tric term in TI electrodynamics is intrinsically topological,both due to the topological properties in reciprocal latticespace and in real space. This can be seen by applying anexternal magnetic field perpendicular to the surface of a 3D TIof thickness L. A computation of the vacuum polarization of two-dimensional Dirac fermions in the presence of an externalfield for each TI surface yields the action [ 7,8] S vpol=e2 8π/integraldisplay dt/integraldisplay dxdyε μνλ ×(Aμ∂νAλ|z=L−Aμ∂νAλ|z=0), (1) where Aμis the gauge potential corresponding to the external field and we have adopted a covariant notation. In the aboveequation z=0 and z=Lcorrespond to the lower and upper surfaces, respectively. We work in units where c=1 and /planckover2pi1=1. The above action yields the difference between Chern- Simons (CS) terms generated by the vacuum polarizationon both surfaces. It can rewritten as the integral of a totalderivative, S vpol=e2 16π/integraldisplay dt/integraldisplay dxdy/integraldisplayL 0dz ∂ z(εμνλAμFλν) =e2 32π/integraldisplay d4xεμνλρFμνFλρ, (2) where Fμν=∂μAν−∂νAμ, and in passing from the first to the second line, the expression has been made fullycovariant by introducing an additional spacetime index ( ρ) to accommodate the third spatial coordinate in the covariantnotation. The above equations follow from the assumptionthat the Fermi level of each surface state lies precisely atzero, i.e., at the Dirac point of the Dirac spectrum. Moreover,spin-momentum locking implies that the Dirac fermions atthe upper surface have a helicity opposite to the lower ones.Thus, we have obtained a magnetoelectric term that is overalltime-reversal invariant. A more general form is given by S vpol=e2θ 32π2/integraldisplay d4xεμνστFμνFστ, (3) where θis given by [ 5] θ=1 8π/integraldisplay d3ktr/bracketleftbigg a(k)∧f(k)−2 3a(k)∧a(k)∧a(k)/bracketrightbigg ,(4) where the 2-form f(k) yields the Berry curvature, f(k)=da(k)+ia(k)∧a(k), (5) with aαβ(k)=−i/angbracketleftα,k|∇k|β,k/angbracketright, (6) being the non-Abelian Berry vector potential associated with the Bloch state |α,k/angbracketright. Thus, the electromagnetic response of 3D TIs yields an interesting interplay between the differentialgeometry of the Bloch states and the topology of electromag-netic gauge fields in the form of a so-called axionic [ 9]t e r m , Eq. ( 3), with θrepresenting a uniform axion field. The axion is periodic and we find that for θ=πTRI holds, since under a time-reversal transformation θ→−θ[5]. 2469-9950/2016/93(1)/014404(6) 014404-1 ©2016 American Physical SocietySTEFAN REX, FLA VIO S. NOGUEIRA, AND ASLE SUDBØ PHYSICAL REVIEW B 93, 014404 (2016) In terms of electric and magnetic field components, the axion term ( 3) becomes Svpol=e2θ 4π2/integraldisplay d4xE·B. (7) This magnetoelectric contribution is a topological term in real space, as it is more easily seen from the covariant writing,Eq. ( 3), which clearly exhibits its independence of the metric. Furthermore, in view of Eq. ( 4) it is also topological in Bloch momentum space due to the induced gauge structure in theHilbert space of Bloch states. If TRI at the TI surface is broken in the presence of a magnetically ordered phase, then B=H+4πM, and a topological magnetoelectric effect (TME) has been predicted[5,6]. This has inspired many proposals of magnetic TI devices [10–17]. In the TME, an electric field causes a magnetic polarization in the same direction as the field. The TMEis the consequence of a CS term generated via the vacuumpolarization due to proximity with a ferromagnetic insulator(FMI). If the FMI is epitaxially grown on only one of the TI surfaces, there is only one CS term, in contrast to Eq. ( 1). The CS term yields an additional Berry phase that modifies thedynamics of the magnetization [ 13,14]. While previous studies have focused on the magnetic polarization generated by an externally applied electric field,in this paper we address a different important consequence ofthe TME, namely, its interplay with long-range Coulomb in- teraction among the Dirac electrons. The Coulomb interaction will generate a fluctuating electric field that interacts with themagnetization. Consequently, a nonlocal TME emerges thatsignificantly impacts on the magnetization dynamics by aneffective coupling over large distances. Taking a similar approach as in Ref. [ 14], we will carry out explicit calculations of the vacuum polarization contributions to the effective action at zero temperature to leading order in the quantum fluctuations (one-loop diagrams) to derive thedynamics of both the magnetization and the Coulomb electricfield at the TI/FMI interface. II. MODEL SYSTEM We consider the interface between a FMI layer on top of a TI, as shown in Fig. 1, which we assume to lie in the xyplane. As a starting point, we use on the one hand the Lagrangiandensity of a bulk FMI, L FMI=b·∂tn−κ 2[(∇n)2+(∂zn)2]−m2 2n2−u 24(n2)2, (8) where nis the magnetization, bis the Berry connection, m2<0 for temperatures below the critical temperature of the FIG. 1. Magnetization nat the interface of a TI and a FMI.magnetically ordered phase in the bulk, and κ,uare positive constants. Note that throughout this paper, ∇= (∂x,∂y,0). On the other hand, the topological Dirac electrons on the surface of a TI are described by LTI=/Psi1†(r)[i∂t−ivF(σy∂x−σx∂y)+Jσ·n(r)]/Psi1(r),(9) where /Psi1†(r) creates an electron at position rin the xyplane, vFis the Fermi velocity, σ=(σx,σy,σz) is the vector of Pauli matrices, and J> 0 is the strength of the coupling of the electron spin to the magnetization natz=0. In addition, we account for long-range Coulomb interaction between the Dirac electrons at the interface, V=1 2/summationdisplay qρ(q)vCou(q)ρ(−q), (10) where the summation is over the two-dimensional momentum q, the density operator is ρ(q)=/summationtext k,s/Psi1† k+q,s/Psi1k,s, with spin denoted s, and vCou(q) is the Fourier transform of the Coulomb potential vCou(r−r/prime)=e2/|r−r/prime|for two electrons at positions randr/prime, where eis the elementary charge and the dielectric constant is 1 /(4π) in Gaussian units. In two dimensions, the potential in reciprocal space takes the form vCou(q)=2πe2 |q|. (11) The interaction can be made linear in electron density by a Hubbard-Stratonovich decoupling. With an auxiliary scalarfieldϕ, that we define to have the unit of an electric potential, one finds the decoupled Lagrangian L Cou=/summationdisplay q/bracketleftbigg eϕ(q)ρ(q)−1 4πϕ(−q)|q|ϕ(q)/bracketrightbigg , (12) which combined with Eq. ( 9) gives the complete real-space Lagrangian density of conduction electrons at the interface, Lc=/Psi1†[i∂t+ivFˆez·(σ×∇)+Jσ·n+eϕ]/Psi1 −1 8π2[∇rϕ(r)]·/integraldisplay d2r/prime∇r/primeϕ(r/prime) |r−r/prime|. (13) where vFis the Fermi velocity. In total, the bilayer system is described by L=Lc+LFMI. III. FLUCTUATION EFFECTS In this section, the quantum fluctuations will be evaluated to leading order by integrating out the electrons. First, we rewritethe fermionic part L f cofLcin a form reminiscent of quantum electrodynamics [ 14]. With the definitions γ=(γ0,γ1,γ2)= (σ0,−iσx,−iσy),a=(e Jϕ,ny,−nx),∂=(∂t,vF∇), and the common notations /Psi1=/Psi1†γ0and/A=γμAμ, we get Lf c=/Psi1[i/∂+J(nz−/a)]/Psi1. (14) The mean-field value nMF=σ0ˆezof the magnetization leads to an effective mass m/Psi1=Jσ0of the fermion field, while ˜ σ= nz−σ0describes the out-of-plane fluctuations. Integrating out the fermions in the standard way [ 18] then leads to the action Sc=SMF−J2 2Tr[G(˜σ−/a)]2(15) 014404-2NONLOCAL TOPOLOGICAL MAGNETOELECTRIC EFFECT . . . PHYSICAL REVIEW B 93, 014404 (2016) with the propagator G=(i/∂+m/Psi1)−1. We relinquish an anal- ysis of the mean-field action SMF, which has been discussed in detail in Ref. [ 14], and focus instead on the fluctuation effects. These are contained in the second term δSof Eq. ( 15), where we have already restricted ourselves to leading order. Theoperation Tr implies integration over space-.time and tracingout all quantum numbers. Diagrammatically, δScontains four contributions to the vacuum polarization: δS=/integraldisplayd3λ (2π)3/bracketleftBig + + +/bracketrightBig . (16) The fields /Psi1,a, and ˜σare represented by solid, wiggly, and dashed lines, respectively, and λcomprises both frequency and momentum. Some details of the calculation of the diagramscan be found in the Appendix. Each of the mixed diagrams inthe second line vanishes, and the remaining processes yield inthe long-wavelength limit δS=J 2 8π/integraldisplay dt/integraldisplay d2r ×/bracketleftbigg (a×∂)·a−(∂×a)2 3m/Psi1−4m/Psi1˜σ2+(∂˜σ)2 3m/Psi1/bracketrightbigg .(17) Note that scalar products are to be taken in Minkowski space, with signature ( +,−,−). As has been discussed earlier [13,14], the term ( a×∂)·ais a fluctuation-induced CS term. In total, we arrive at the following effective Lagrangian for thecoupled FMI-TI bilayer system: L eff=−σxy 2v2 F(n×∂tn)·ˆez+σxye vFJn·∇ϕ −NJ2 24πm/Psi1[(∇·n)2+(∇nz)2]+NJ2 24πv2 Fm/Psi1(∂tn)2 +Ne2 24πm/Psi1(∇ϕ)2−NJe 12πvFm/Psi1[(∇ϕ)×(∂tn)]·ˆez −NJ2m/Psi1 2πv2 Fn2 z+NJm2 /Psi1 πv2 Fnz +LFMI−1 8π2[∇rϕ(r)]·/integraldisplay d2r/prime∇r/primeϕ(r/prime) |r−r/prime|, (18) where ˆezis the unit vector in the zdirection. Furthermore, we assumed Norbital degrees of freedom of the Dirac electrons and defined the Hall conductance σxy=NJ2/(4π)i nt h et w o contributions from the CS term. The first one describes a Berryphase that adds up with the FMI Berry phase, while the secondone leads to the TME. Derivatives of ˜ σhave been replaced by derivatives of n z, since σ0is constant. Applying the Euler-Lagrange formalism on Leffyields the Landau-Lifshitz equation (LLE) for the magnetization atthe interface and the equation of motion for the fluctuatingCoulomb potential ϕ. We arrange the LLE such that all first-order time derivatives of the magnetization are on theleft side, such that it takes the form A·∂ tn=dwith a matrix Aand a vector dthat depends on ϕand any other instanceofn. Since Athen collects precisely the Berry phase terms, it is antisymmetric and we can rewrite A·∂tn=v×∂tn, where we find v=n n2+σxy v2 Fˆez. (19) The first term stems from the FMI Berry connection b, which satisfies the condition ∂n×b=−n/n2. The second term originates with the CS term and enhances the overall Berryphase. If the magnetization is strong, the Berry phase mayeven be dominated by this topologically protected term. Bytaking the cross product with nin both sides of the equation v×∂ tn=d, we obtain v 2∂tn2−(n·v)∂tn=n×d. (20) Assuming that n2is time independent, Eq. ( 20) becomes ∂tn=d×n 1+σxy v2 F(n·ez). (21) We split d=dn+dϕinto the magnetization-dependent part dn=ρs·∇2n+NJ2 12πm/Psi1/bracketleftbigg∂2 tn v2 F+∇(∇·n)/bracketrightbigg +NJm /Psi1 πv2 F(Jnz−m/Psi1)ˆez+/parenleftBig m2+u 6n2/parenrightBig n,(22) where the stiffness matrix is ρs=κ1+(NJ2/12π m/Psi1)diag(0 ,0,1), and the contribution from the Coulomb interaction dϕ=−σxye vFJ∇ϕ−NJe 12πvFm/Psi1ˆez×∂t∇ϕ. (23) In addition, we obtain the Euler-Lagrange equation for the fieldϕ. To make the physics more transparent, we write it in terms of the fluctuating electric field E=− ∇ ϕ, 0=2πσxye vFJn/parallelshort+Ne 6m/Psi1/parenleftbigg eE−J vF∂tn׈ez/parenrightbigg −1 4π/integraldisplay d2r/primeE(r/prime) |r−r/prime|, (24) where n/parallelshortdenotes the in-plane part of the magnetization. This is an explicit form of the fluctuation-induced TME, where theelectric field will be aligned with the magnetization, up toa dynamical correction depending on ∂ tn. For the net field and magnetization this correction is irrelevant, since the timeaverage of ∂ tnvanishes. The first terms in Eqs. ( 23) and (24) stem from the contribution proportional to n·Ein the Lagrangian, Eq. ( 18), representing the usual TME, which is a local effect. In contrast, the last term in Eq. ( 24)i sa direct consequence of the long-range Coulomb interaction,and clearly makes the TME nonlocal by integration over thefield at each point in the plane. The motion of the magnetization becomes more clear when the bosonic field ϕin Eq. ( 18) is integrated out as well. The part of the Lagrangian density that depends on the Coulombinteraction then becomes L ϕ(r,t)=1 2ρn(r,t)/integraldisplay d2r/primeρn(r/prime,t) |r−r/prime|, (25) 014404-3STEFAN REX, FLA VIO S. NOGUEIRA, AND ASLE SUDBØ PHYSICAL REVIEW B 93, 014404 (2016) with the induced magnetic charge density, ρn(r,t)=σxye vFJ∇·n(r,t)−NJe 12πvFm/Psi1[∇×∂tn(r,t)]·ˆez. (26) Note that to leading order in momentum, the term involving(∇ϕ) 2is negligible compared to the last term in Eq. ( 18). We observe that the fluctuation-induced magnetic charge containsan additional contribution besides the usual one. Typically,the magnetic charge density is proportional to ∇·nand usually arises in studies of magnetic skyrmions [ 19]. We also obtain a contribution ∼(∇×∂ tn)·ˆez, which does not have a topological origin. From the continuity equation we derivealso the magnetic current density, j n=−σxy vF∂tn +NJe 24π2vFm/Psi1/integraldisplay d2r/primer−r/prime |r−r/prime|2[∇r/prime×∂tn(r/prime,t)]·ˆez. (27) The magnetization dynamics is now determined by the integro-differential equation ∂tn=Dn×n 1+σxy v2 F(n·ez), (28) where Dn=dn+σxye 2vFJE+NJe 24πvFm/Psi1ˆez×∂tE, (29) and the electric field is now given explicitly by E(r)=−/integraldisplay d2r/primeρn(r/prime,t)(r−r/prime) |r−r/prime|3. (30) The equation of motion can be simplified by an approximation ofρn. Namely, in the low-frequency regime we can expect the second term in Eq. ( 26) to be small compared to the first term. Consequently, we find that the Coulomb interactionmainly acts via the CS term. The induced electric field is thenindependent of ∂ tn, and the equation can be brought into an explicit form similar to Eq. ( 21). An important consequence of Eqs. ( 26) and ( 28)i s that the Coulomb interaction does not directly couple themagnetizations at different points in the plane. Rather, itis the divergence of the magnetization that enters into the magnetization dynamics over long distances. This can beunderstood by the duality of magnetic and electric chargeson the surface of a TI [ 20], where ∇·nis equivalent to an electric charge of the magnetic texture. This charge generatesa Coulomb field. In the case of a uniform magnetization,where both ρ nandEare absent, the Coulomb interaction will thus not affect the magnetization dynamics. We are then leftwith the LLE ( 21) with d=d n,w h e r ea l s oi n dn, all spatial derivatives vanish. From the remaining terms, we simplyobtain a precession of the magnetization around the zaxis by Eq. ( 21). To illustrate how the long-range Coulomb interaction affects the dynamics, we turn to a simple example of a nonuni-form magnetization. Assume that the system is prepared witha magnetic texture, where the phase of the precession changeswithin a narrow region about x=0. The divergence of nwill then be nonzero within that region. The corresponding terms ind nwill locally alter the magnetization dynamics, trying to align the magnetization at neighboring sites. This will smoothen thetransition at x=0 and evoke spin waves spreading in both half planes. However, via the Coulomb interaction, there isan instantaneous impact on the magnetization even far fromthe texture. For large x, we can assume ρ n=ρn,0δ(x), where ρn,0oscillates with the precession frequency at x=0, and one readily verifies that E=2ˆexρn,0/x. The second and third terms in Eq. ( 29) lead to in-plane components of the effective field of precession in the xandydirections, respectively, where the latter can be neglected in the low-frequency limit. Thus, theeffective field at arbitrary xis already tilted away from the z direction before the spin waves due to the local stiffness termsarrive. As a final remark, we note that only the in-plane inho- mogeneities of the magnetization participate in the Coulombdriven dynamics, while the out-of-plane magnetization doesnot enter. Therefore, we can expect a similar nonlocal effect ifwe replace the texture discussed above by a domain wall, aslong as the rotation of the magnetization within the transitionregion happens in a way that involves an in-plane divergence.Apart from evoking Coulomb terms in the magnetizationdynamics, the presence of a domain wall in a magnetic layeron a TI also leads to other effects, e.g., chiral currents, thatwe have not discussed in this paper, but have been subject to anumber of previous studies [ 11,16,21–23]. IV . CONCLUSION We have analytically studied a TI-FMI interface in the presence of long-range Coulomb interaction and derived thefluctuation-induced dynamics of both the magnetization andthe electric field mediating Coulomb interactions, to secondorder in gradients and fields. We have found that, as a result oflong-range interactions, the TME becomes nonlocal, such thatthe magnetization is coupled to the electric field anywherein the plane. The CS term in the effective action enhancesthe overall Berry phase and thus modifies the magnitude ofthe effective field of the magnetization precession. Magnetictextures involving a divergence of the in-plane magnetizationtilt the effective field of the precession in a nonlocal way. ACKNOWLEDGMENTS A.S. and S.R. acknowledge support from the Norwe- gian Research Council, Grants No. 205591/V20 and No.216700/F20. F.S.N. acknowledges support from the Collabora-tive Research Center SFB 1143 “Correlated Magnetism: FromFrustration to Topology” of the German Research Foundation(DFG). APPENDIX: CALCULATION OF THE DIAGRAMS In this Appendix, we present the zero-temperature calcula- tion that leads to Eq. ( 17). From Eq. ( 15), we have δS=−J2 2/integraldisplay dt/integraldisplay d2x/summationdisplay /kappa1/angbracketleft/kappa1|tr[G(˜σ−/a)]2|/kappa1/angbracketright, (A1) 014404-4NONLOCAL TOPOLOGICAL MAGNETOELECTRIC EFFECT . . . PHYSICAL REVIEW B 93, 014404 (2016) where the trace tr is taken in spin space, /kappa1denotes all other quantum numbers, and the propagator is G=−i/∂+m/Psi1 ∂2+m2 /Psi1. (A2) We go to imaginary time by the Wick rotation, τ=it, which makes space-time Euclidean. The Dirac γmatrices are thenidentical to the Pauli matrices. We find i/∂→−/∂and/a→ /α, where we defined α=(a0,ia1,ia2). Furthermore, δSis transformed to reciprocal space and the sum over electronquantum numbers is carried out in a basis of plane-wave states,/kappa1=(ω,k), with frequency ωand two-dimensional momentum k. The frequency and momentum of the bosonic fields αand ˜σin reciprocal space are denoted as λ=(/Omega1,q). We get δS=iJ2 2/integraldisplayd3λ (2π)3/integraldisplayd3/kappa1 (2π)3tr[(m/Psi1+i//kappa1)(−/α(λ)+˜σ(λ))(m/Psi1+i(//kappa1−/λ))(−/α(−λ)+˜σ(−λ))]/parenleftbig /kappa12+m2 /Psi1/parenrightbig/parenleftbig (/kappa1−λ)2+m2 /Psi1/parenrightbig , (A3) and the matrix structure inside the remaining trace is now determined by products of Dirac matrices. As one caneasily verify by using the commutation and anticommutationrelations of the Euclidean Dirac matrices [ 24,25], tr(γ μγν)= 2δμν,t r (γμγνγλ)=2iεμνλ, and tr( γμγνγλγρ)=2(δμνδλρ− δμλδνρ+δμρδνλ). Inserting these formulas into the numerator of the integrand in Eq. ( A3) yields tr[...]= 2αμ(λ)αν(−λ)/bracketleftbig m/Psi1εμρνλρ+δμν/parenleftbig m2 /Psi1+/kappa1·(/kappa1−λ)/parenrightbig −2/kappa1μ/kappa1ν+/kappa1νλμ+/kappa1μλν/bracketrightbig +2˜σ(λ)˜σ(−λ)/bracketleftbig m2 /Psi1−/kappa1·(/kappa1−λ)/bracketrightbig +2i˜σ(λ)αμ(−λ)[−m/Psi1(/kappa1μ−λμ)−m/Psi1/kappa1μ +ερνμ/kappa1ρ(/kappa1ν−λν)] +2iαμ(λ)˜σ(−λ)[−m/Psi1(/kappa1μ−λμ)−m/Psi1/kappa1μ +ερμν/kappa1ρ(/kappa1ν−λν)], (A4) corresponding to the four diagrams in Eq. ( 16). Let these diagrams be called D1,..., D 4, in the same order as in Eq. ( 16). Next, the integral over /kappa1will be carried out. As has been discussed in Appendix of Ref. [ 25], one can rewrite the first diagram to take the form D1(λ)=iJ2aμ(λ)aν(−λ)/bracketleftbigg εμρνm/Psi1λρI1(λ) +Pμν(λ)/parenleftbigg m/Psi1I1(λ)−λ2 4I1(λ)+1 2I2/parenrightbigg/bracketrightbigg ,(A5)with the projector Pμν(λ)=δμν−λμλν/λ2and the integrals I1(λ)=/integraldisplayd3/kappa1 (2π)31/parenleftbig /kappa12+m2 /Psi1/parenrightbig/bracketleftbig (/kappa1−λ)2+m2 /Psi1/bracketrightbig =1 4π|λ|arctan/parenleftbigg|λ| 2m/Psi1/parenrightbigg , (A6) I2=/integraldisplayd3/kappa1 (2π)31 /kappa12+m2 /Psi1=−m/Psi1 4π, (A7) with the result D1(λ)=iNJ2αμ(λ)αν(−λ)/bracketleftbiggεμρνλρ 8π−λ2Pμν(λ) 24πm/Psi1/bracketrightbigg (A8) to second order in λ. Note that I2requires dimensional regularization [ 24], since it is formally divergent. By simple manipulations, one can reduce the second diagram to the sameintegrals: D 2(λ)=iNJ2˜σ(λ)˜σ(−λ)/bracketleftbigg/parenleftbigg 2m2 /Psi1+1 2λ2/parenrightbigg I1(λ)−I2/bracketrightbigg =iNJ2˜σ(λ)˜σ(−λ)/bracketleftbiggm/Psi1 2π+λ2 24πm/Psi1/bracketrightbigg +O(λ3).(A9) In the two diagrams mixing αand ˜σ, performing the /kappa1 integration leads to D3(λ)=D4(λ)=0. Summing up the contributions from D1andD2and transforming back to real space and real time finally yields Eq. ( 17). [1] M. Z. Hasan and C. Kane, Rev. Mod. Phys 82,3045 (2010 ). [2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys 83,1057 (2011 ). [3] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Nat. Phys. 4,273 (2008 ). [4] T. Yokoyama, Y . Tanaka, and N. Nagaosa, Phys. Rev. B 81, 121401(R) (2010 ). [5] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008 ). [6] A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett. 102,146805 (2009 ). [7] A. J. Niemi and G. W. Semenoff, Phys. Rev. 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Tokura, Nat. Nanotech. 8,899(2013 ). [20] K. Nomura and N. Nagaosa, P h y s .R e v .B 82,161401(R) (2010 ). [21] J. Linder, P h y s .R e v .B 90,041412(R) (2014 ). [22] Y . Ferreiros and A. Cortijo, Phys. Rev. B 89,024413 (2014 ). [23] C. Wickles and W. Belzig, P h y s .R e v .B 86,035151 (2012 ). [24] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw- Hill College, Singapore, 1980). [25] F. S. Nogueira and I. Eremin, Phys. Rev. B 88,085126 (2013 ). 014404-6

PhysRevB.100.195119.pdf

PHYSICAL REVIEW B 100, 195119 (2019) 4fconduction in the magnetic semiconductor NdN W. F. Holmes-Hewett ,1R. G. Buckley ,2B. J. Ruck,1F. Natali,1and H. J. Trodahl1 1The MacDiarmid Institute for Advanced Materials and Nanotechnology and The School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand 2The MacDiarmid Institute for Advanced Materials and Nanotechnology and Robinson Research Institute, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand (Received 11 September 2019; revised manuscript received 30 October 2019; published 13 November 2019) We report the growth of films of the intrinsic ferromagnetic semiconductor NdN and investigate their optical and transport properties. There is clear evidence of a strong anomalous Hall effect as expected from a 4 f conduction channel, supported by an optical absorption into a 4 for 4f/5dhybridized tail at the base of the conduction band. The results reveal a heavy-fermion 4 f/5dband lying where it can be occupied at controllable levels with nitrogen-vacancy donors. DOI: 10.1103/PhysRevB.100.195119 I. INTRODUCTION Strongly correlated electron systems have been of fun- damental interest since their identification, with many oftheir unusual properties still evading understanding [ 1,2]. The most exotic behavior is encountered when there exists a flat, heavy-mass band in the vicinity of an extended band. The anticrossing where these bands meet now creates two heavyfermion bands separated by a narrow hybridization gap [ 3,4]. For the vast majority of materials these anticrossings occur farfrom the Fermi energy thus for the hybridization gap to haveinfluence over the transport properties is uncommon. In spiteof this and other material science challenges the engineering of strongly correlated materials in general remains an area of active research [ 5]. The rare-earth nitrides (LN, L a lanthanide element) provide a class of simply structured materials ideallysuited to this study. The LN series of magnetic semiconductors have empty 4 f states which lie variously throughout the conduction band andfilled 4 fstates throughout the valence band. This allows the selection of (i) the location of the 4 fin the conduction and valence bands via the choice of rare-earth element, and (ii)the location of the Fermi energy via the doping of the materialwith electrons though nitrogen vacancy sites. Members of the LN series form in the simple NaCl struc- ture comprising L 3+and N3−ions. They are dopable with electrons via the introduction of nitrogen vacancies, eachvacancy freeing three electrons of which two are predictedto remain localized on the vacancy site while the third findsan extended state in the conduction band at modest tem-peratures [ 6]. The control of the Fermi energy via nitrogen vacancy concentration in the LN has been shown in numer-ous experimental studies [ 7–9]. In contrast to the electronic properties the magnetic properties of the LN are dominatedby the occupation in the 4 fshell and are largely unaffected by nitrogen vacancies. Various calculations exist regarding the band structure of members of the LN series [ 10–17], most using the LSDA +Umethod. Calculations find the LN as insulating orsemimetallic at zero temperature with optical band gaps of ∼1 eV. The Hubbard Uparameter is generally adjusted to fit the experimental optical band gap of GdN [ 18]. Calculations largely agree on the form and location of the 5 dbands which are predicted as forming the conduction-band minimum inmost members. The location and hybridization of the 4 f bands show much less agreement, with different calculationson the same LN member spanning several eV in regard to theirenergy. GdN is the most studied of the series largely due to the simplicity of the half filled 4 fshell and its electronic configuration 4 f 7. It serves as a valuable comparison to the more complex members of the series. Calculations [ 10–12] place the empty minority spin 4 fbands ∼5 eV above the conduction-band minimum. The filled majority spin bands areplaced ∼7 eV below, keeping them well away from, and with little influence over, the conduction-band minimum and thusthe transport properties of the material. The transport behavior of GdN is largely understood in terms of its single spin-split Gd 5 dconduction band and N 2pvalence band. Calculations [ 10–12] find a ∼1.3-eV gap between the conduction-band minimum and valence bandat the Xpoint, which is consistent with many experiential studies [ 18–21]. The 70-K Curie temperature is the highest among the LN materials, raised from 50 K in stoichiomet-ric samples via the enhanced exchange offered by magneticpolarons formed around nitrogen vacancy sites [ 22,23]. The half filled 4 fshell of GdN has no orbital contribution to the magnetization and results in a saturation magnetizationof 7μ Bper Gd3+ion in the ferromagnetic phase [ 9,24]. The large magnetization causes a strong Zeeman interaction andcontributes to the small coercive field on the order of 100 Oeat low temperatures. As one moves to lighter LN materials a change can be seen in the band structure, driven largely by the location ofthe unoccupied majority spin 4 fbands and their hybridization with the 5 dbands above the Fermi energy. EuN has a single unoccupied majority spin 4 fband which calculations place from 1 to 4 eV above the conduction-band 2469-9950/2019/100(19)/195119(6) 195119-1 ©2019 American Physical SocietyW. F. HOLMES-HEWETT et al. PHYSICAL REVIEW B 100, 195119 (2019) minimum [ 13,14]. Experiments, however, show that electrons are doped into this 4 fband implying it forms the conduction- band minimum [ 14,25]. SmN has two unoccupied majority spin 4 fbands, which calculations again place variously above the 5 d conduction-band minimum [ 11,15–17]. In contrast transport results [ 26,27]i m p l ya4 fband forms the conduction-band minimum. A spectroscopy study [ 28] has recently located the lowest unoccupied 4 fband as forming the conduction- band minimum, in close vicinity to the 5 d.T h e4 f5config- uration of SmN results in a nonzero orbital contribution tothe magnetization which opposes the spin contribution [ 29]. Experiment has shown that the magnetization of SmN is closeto extinguished by this opposition. A value of 0 .03μ Bper Sm3+ion is found that is furthermore dominated by the orbital contribution [ 30,31]. Continuing down the series NdN has four unoccupied majority spin 4 fbands above the Fermi energy and electronic configuration 4 f3. NdN has been identified as ferromagnetic with a Curie temperature of ∼50 K and the less than half filling of the 4 fshell suggests an orbital dominated mag- netization [ 11,32]. A semiconducting ground state has been identified via transport measurements and an optical band gapof∼1 eV has been found via optical transmission measure- ments on moderately doped films ( n=5×10 20cm−3)[32]. The only calculations of the band structure propose threesolutions largely differing on the location of the 4 flevels and their hybridization with the 5 d. These range from a 4 fband some few eV above the valence-band maximum at /Gamma1with a 5dconduction-band minimum at X,t ot h e4 fforming the minimum optical band gap at /Gamma1before hybridizing with the 5 d near X, both comprising the conduction-band minimum [ 11]. In the present paper we report optical and transport mea- surements of NdN. We identify a ∼1 eV minimum optical band gap between the valence band and lowest unoccupied4fband at /Gamma1. The conduction-band minimum is found at X where there is a ∼1.5 eV band gap between the valence band and a hybridized 4 f/5dband. II. EXPERIMENTAL METHODS Thin NdN films, on the order of 100 nm, were grown by thermal evaporation of metallic Nd sources inside an ultrahighvacuum chamber. Films were grown in an atmosphere of1×10 −4mb of molecular N 2at a rate of ∼1Å/s−1. Films intended for optical measurements were grown at ambienttemperature to limit the number of nitrogen vacancies. Filmsproduced for electrical measurement were grown under thesame conditions but at an elevated substrate temperature of∼400 ◦C to intentionally dope the films with nitrogen vacan- cies. Films were grown simultaneously on various substratessuited to each measurement required. All films were cappedwith∼100 nm of AlN to protect the NdN from the damaging effects of atmospheric water vapor and oxygen. The structural properties and quality of the films was first investigated via x-ray diffraction. All films showed the ex-pected NaCl structure with lattice parameters in line with theliterature and recent experimental results [ 9]. Film thicknesses were measured by both electron microscopy and profilometryafter growth.Samples for electrical transport measurements were grown on 10 ×10×0.5m m 3c-plane sapphire substrates with Cr/Au contacts predeposited in a van der Pauw configura- tion. Electrical transport measurements were conducted in aQuantum Design physical property measurement system attemperatures from 300 to 2 K and in magnetic fields of ±9T . Magnetic measurements were conducted in a Quantum De-sign magnetic property measurement system at temperaturesfrom 300 to 5 K and a field of up to ±7T . Hall-effect measurements above the Curie temperature were conducted in positive and negative field then treatedusing the usual technique to separate the even parasitic longi-tudinal signal from the odd transverse Hall signal. Below themagnetic transition the subtraction technique is more involvedas there is now the additional contribution from the anomalousHall effect. This term is odd in the magnetization of thesample rather than the applied field and thus must be treatedaccordingly [ 27]. Optical measurements were conducted using a Bruker Vertex 80v Fourier transform interferometer at ambient tem-perature on films grown on both Si and sapphire substrates.Measurements from 0.01 to 1.2 eV were conducted on filmsgrown on Si substrates as Si is largely transparent in thisregion, higher energy measurements from 1 to 4 eV wereconducted on sapphire where sapphire is largely transparent.These measurements combined gave a picture of the LNmaterial over the entire energy range of 0.01–4 eV . Reflectionmeasurements were performed using a 250-nm Al film as areference and data from Ehrenreich et al. [33] to adjust for the reflectivity of Al. Measurements of the substrates andcapping layers were analyzed separately which enabled theindependent modeling of LN layers. The software package RefFIT [ 34], which uses an in- herently Kramers-Kronig consistent sum of Lorentzians torepresent the dielectric function of a material, was used torecreate reflection and transmission spectra for each filmsimultaneously on Si and sapphire substrates. To account forabsorption above the measurement range a constant value /epsilon1 ∞ was added. When reproducing LN layers, Lorentzians were placed every 100 cm−1from 300 to 40 000 cm−1each with a width of 100 cm−1. The amplitudes of each of these were then adjusted to reproduce the measured spectra. A singleLorentzian was used to reproduce the phonon absorption near250 cm −1. III. RESULTS AND DISCUSSION A. Magnetic and electrical transport We begin by discussing the magnetic and electrical trans- port results on an intentionally doped NdN film. Magneticmeasurements showed a Curie temperature of ∼45 K. Similar to previous reports the inverse susceptibility showed a Curie-Weiss–like relationship from 120 to 60 K, while below this adeviation is caused by the crystal field [ 32]. A fit to the data from 120 to 60 K results in an effective paramagnetic momentof∼3.6μ B, very close to the moment calculated for the Hund’s rules ground state of gjμB√J(J+1)=3.62μBwith the Landé gfactor gj=8/11. When saturated at the lowest temperatures the moment in the ferromagnetic phase reduces 195119-24fCONDUCTION IN THE MAGNETIC SEMICONDUCTOR … PHYSICAL REVIEW B 100, 195119 (2019) FIG. 1. Plot of the Hall resistivity in a NdN film as a function of applied field at various temperatures above and below the Curie tem- perature. The 300-K data show a simple negative linear dependenceindicating negative charge carriers. Below the magnetic transition the anomalous component of the Hall effect shows a positive sign with hysteresis developing at the lowest temperatures. The inset shows themagnetization as a function of filed applied parallel to the film plane at 10 K which saturates at ∼0.9μ Bper Nd3+ion. to∼0.9μBper Nd3+ion, as has been previously reported [ 32]. A measurement of the magnetization as a function of fieldapplied parallel to the film plane at 10 K is shown in the insetof Fig. 1. Hall-effect measurements above and below the Curie tem- perature can be seen for this sample in Fig. 1. To begin we can consider the measurement at 300 K which shows the ordinaryHall effect. A linear trend with a negative slope is seen as is thecase for all semiconducting rare-earth nitride members. This isindicative of negative charge carriers present in the conductionband caused by doping the material with nitrogen vacancies.The slope at 300 K gives a carrier concentration of 4 .9± 1×10 21cm−3. This large carrier concentration is consistent with the 300-K resistivity of 0 .35±0.07 m/Omega1cm. The carrier concentration and resistivity then result in a relaxation time of2.1±0.9×10 −15s. The influence of the anomalous Hall effect can be seen below the magnetic transition temperature. The 30-K datain Fig. 1show that the anomalous Hall effect has a clear positive sign, opposite to that of the ordinary Hall effect.The magnitude of the anomalous Hall effect continues toincrease as temperature decreases, saturating at the lowesttemperatures as is expected from the increased spin imbalancewhich drives the anomalous Hall effect. We now move to the magnitude of the anomalous Hall effect in NdN and compare this to similar measurements inSmN and GdN films of comparable carrier concentration. Theanomalous Hall effect in Fig. 1saturates at ∼6μ/Omega1cm in the 2-K measurement. Figure 2shows the NdN measurement along with measurements in a SmN film and a GdN film,all taken well below the Curie temperature of each materialwhere the anomalous Hall effect is well saturated. The sign ofthe anomalous Hall effect in NdN and SmN is positive while FIG. 2. Measurements of the Hall resistivity in SmN (blue), NdN (red), and GdN (orange). The sign of the anomalous component of the Hall effect is positive in SmN and NdN while negative inGdN. The enhanced magnitude of the anomalous Hall effect over the GdN measurement indicates a 4 fconduction channel in both SmN and NdN. negative in GdN. Comparing the NdN data to measurements in SmN and GdN we see that NdN reaches ∼60% of the SmN magnitude, and is at least an order of magnitude larger thancomparable GdN measurements. The strong spin-orbit interaction of the lanthanide materi- als causes the intrinsic contribution to dominate the anoma-lous Hall effect in the LN. The intrinsic anomalous Hall effecthas been described with quantitative success in both GdN [ 35] and SmN [ 27]; it is this semiclassical description that we now extend to NdN. The anomalous Hall effect can, under thecubic symmetry of the LN, be written in terms of an integralover the unit cell of the product of the charge density ρ(r) and the square of the conduction electron’s wave function |u k|2, ρxy∝/integraldisplay d3rρ(r)|uk|2. (1) The 5 dand 4 fwave functions, the potential candidates for conduction in the LN, both have no weight at the nucleuscausing the integral to be negative (positive) if the conductionelectron’s magnetic moment is aligned (antialigned) with thenet sample magnetization. The 4 fwave function has more weight at smaller radius than the 5 d, closer to the majority of the core electron charge density. Conduction in a 4 fband is then expected to result in a larger value for the integralin Eq. ( 1) and an enhanced anomalous Hall effect when compared to 5 dconduction. A simple calculation [ 27]o f this integral shows that for the case of conduction in a 4 f band the anomalous Hall effect in NdN should be ∼90% of the magnitude of the anomalous Hall effect in SmN. Incomparison for the case of 5 dconduction the anomalous Hall effect in NdN is expected to be smaller than in GdN. The enhanced experimental value for the anomalous Hall effect in NdN over GdN, when viewed in the context ofEq. ( 1), points to a 4 fcontribution to the transport channel. The ratio of the experimental values of the magnitude of 195119-3W. F. HOLMES-HEWETT et al. PHYSICAL REVIEW B 100, 195119 (2019) FIG. 3. Measurement (black) along with a model (red) of the absorption (1- R-T) as a function of energy constructed from mea- surements of a NdN sample capped with AlN on both Si and sapphire substrates. Interband absorption begins in NdN above ∼1e Va n da phonon absorption can be seen centered around 0.03 eV (250 cm−1). Features between these are caused by the substrate and capping layer. the anomalous Hall effect for NdN and SmN in Fig. 2is ∼6μ/Omega1cm/9μ/Omega1cm, close to the 0.9 calculated. The exper- imental value is in any case much larger than the measuredvalue for GdN, where the nearest experimental values areclose to an order of magnitude below the present NdN value.The enhancement over the GdN value shows that the 4 fband contributes to the transport channel in NdN, and thus must lienear, if not form, the conduction-band minimum. The sign of the anomalous Hall effect in NdN is positive, as is the case for SmN, which indicates that the orientation of thenet magnetization opposes that of the conduction electrons’magnetic moment [ 27,35]. This is consistent with the spin- orbit opposition in the 4 fshell observed via X-ray magnetic circular dichroism [ 32] and expected from the 4 f 3electronic configuration of NdN. B. Optical spectroscopy We now turn to optical measurements on an undoped NdN film. Measurements of reflection and transmission were usedto determine the absorption (1- R-T) which can be seen in Fig. 3. Beginning in the interband region above 1 eV , Fig. 3 shows strong absorption indicating direct optical transitionsacross an energy gap of ∼1 eV. Moving to lower energy we see the transparent region below the minimum optical gap.The small peak near 0.35 eV and the structure near 0.08 eV iscaused by absorption in the substrate and capping layer. Atthe lowest energy there is a broad phonon absorption near0.03 eV (250 cm −1) which is at a similar location to the infrared phonon modes in SmN, GdN, and DyN [ 28,36]. The reflection and transmission measurements were used to determine the real part of the optical conductivity σ1(ω), which is shown in Fig. 4along with the optical conductivity of GdN [ 28] for comparison. The optical conductivity of NdN is qualitatively similar to the absorption shown in Fig. 3but is FIG. 4. Plot of the real part of the optical conductivity σ1(ω) as a function of energy for NdN and GdN. Both show a phonon absorption at low energy and interband absorption above ∼1e V .T h e NdN data show an additional contribution below ∼2.5 eV indicating absorption into a band not present in GdN. The inset shows the area directly above the band gap with the additional contribution to the absorption in NdN identified with red hatching. now free from absorption in the substrates and capping layer. It is notable that the interband absorption above 2 eV is afactor of ∼2 weaker in NdN than in GdN. Such a contrast can be expected to follow from the increased lattice constantof NdN, roughly 3% larger than GdN. The already smalloverlap between the N 2 pand L 4 f/5dwave functions in GdN is further reduced by the increased separation, leadingto a decrease in the transition rate and in turn the opticalconductivity in the interband region. The GdN data in Fig. 4show a nearly featureless rise from the band edge to ∼4 eV. In contrast the NdN data appear to have two contributions with the initial absorption turningover above 1.5 eV before an additional contribution to theabsorption appears near 2.5 eV . The calculated band structureof GdN features the parabolic 5 dband alone [ 10–12] which is matched well by experiments [ 18–21,28]. Calculations of the band structure of NdN show a similar 5 dband to GdN, but in addition four unfilled majority spin 4 fbands. It is clear that any additional structure in the NdN data must be due tothese 4 fbands not present in GdN and predicted to lie near the conduction-band minimum in NdN [ 11]. The comparison between NdN and GdN can be seen more clearly in the insetof Fig. 4which shows σ 1(ω) for each in the region of the band gap. The slightly convex form of σ1(ω) between 1 and 2.5 eV is qualitatively expected from transitions into a flat band, ashas been seen in SmN [ 28]. These results need to be considered in the context of (i) the electrical transport measurements which showed anenhanced anomalous Hall effect indicating a 4 fcontribution to the conduction channel in an intentionally doped film, (ii)calculations [ 11] which place the lowest unoccupied majority spin 4 fband in NdN variously from the bottom of the con- duction band to some few eV higher, and (iii) recent opticalmeasurements on SmN [ 28] showing a similar low-energy 195119-44fCONDUCTION IN THE MAGNETIC SEMICONDUCTOR … PHYSICAL REVIEW B 100, 195119 (2019) FIG. 5. Schematic of the band structure of NdN based on the data shown in Fig. 4. The minimum optical gap is ∼1 eV between the va- lence band and Nd 4 fband at /Gamma1while the conduction-band minimum is at X. Various hybridization scenarios are possible between the 4 f and 5 dbands near X, none of which are shown. feature in the optical conductivity, identified as transitions into a4fband. The most natural description is that the additional absorption in NdN is caused by a 4 fband lying near the conduction-band minimum. A schematic band structure which is consistent with the data in Fig. 4is presented in Fig. 5which shows the N 2 p valence band and the lowest unoccupied Nd 5 dand Nd 4 f bands. Guided by calculations [ 10–12] and experiments on both GdN [ 18–21] and SmN [ 28,37]t h e5 dband is parabolic with a minimum at X.T h e4 fband is shown in the disper- sionless limit and the N 2 pband has ∼0.5 eV of dispersion between the maximum at /Gamma1and the minimum at X. The data in Fig. 4indicate that the initial absorption involves transitions from the valence band to the 4 fband with a direct gap of ∼1 eV. In the dispersionless limit this takes place at /Gamma1.T h e direct band gap at Xis difficult to determine from the data available, but extrapolating from the higher energy absorptiongives a value of ∼1.5e V .T h e4 fand 5 dbands then meet at theXpoint with a number of hybridization scenarios possible, none of which are shown. The only existing calculations of the NdN band structure offer three possible scenarios [ 11]. Two of these result in the 4fbands at /Gamma1being raised several eV above the conduction-band minimum. This would rule out any optical transitions between these bands in the energy range shown in Fig. 4. With only the 5 dband remaining one would expect the form of σ 1(ω) for NdN to then be very similar to that of GdN. The third scenario offered, where the 4 felectrons are housed in t2u↑states, is the most consistent with the present data. This calculation shows a minimum optical gap of ∼0.6 eV between the valence band and lowest unoccupied majority spin 4 f band at /Gamma1.T h i s4 fband then falls in energy monotonically, hybridizing with the 5 dband near X, where the direct gap is now some 30% larger than the gap at /Gamma1. Although the placement of the bands here seems to reflect the experimentaldata this calculation still reaches a semimetallic conclusion,in contrast to the present results which clearly indicate asemiconducting ground state in undoped films. IV. SUMMARY In summary, the electrical transport, magnetic, and optical properties of NdN have been measured and compared withsimilar results on SmN and GdN films. Electrical measure-ments on an intentionally doped film show that NdN has ananomalous Hall effect with a magnitude similar to that ofSmN and roughly an order of magnitude larger than GdN.A simple calculation shows that this is expected for the caseof the 4 fband contributing to the conduction channel in NdN. These transport measurements highlight that NdN canbe doped appropriately such that the Fermi energy can lieinside the lowest majority spin 4 fband. Optical measurements of both reflectivity and transmission were conducted on an undoped NdN film and modelled toproduce the optical conductivity. A comparison to similarresults on GdN films has found additional structure at lowenergy, near a predicted location of the lowest unoccupiedmajority spin 4 fband in NdN. The optical measurements show the presence of a low- lying majority spin 4 fband near the conduction-band minimum. The minimum optical gap of ∼1 eV occurs at /Gamma1between the valence band and a 4 fband while the conduction-band minimum is formed from a hybridized4f/5dband at X. ACKNOWLEDGMENTS This research was supported by the New Zealand Endeav- our fund (Grant No. RTVU1810). The MacDiarmid Instituteis supported under the New Zealand Centres of ResearchExcellence Programme. [1] P. Coleman, in Handbook of Magnetism and Advanced Magnetic Materials , edited by H. Kronmuller and S. Parkin, Fundamen- tals and Theory V ol. 1 (John Wiley and Sons, 2007), pp. 95-148. [2] L. Degiorgi, Rev. Mod. Phys. 71,687(1999 ). [3] P. S. Riseborough, Adv. Phys. 49,257(2000 ). [4] M. Dzero, J. Xia, V . Galitski, and P. Coleman, Annu. Rev. Condens. Matter Phys. 7,249(2016 ). [5] R. Adler, C.-J. Kang, C.-H. Yee, and G. Kotliar, Rep. Prog. Phys. 82,012504 (2018 ).[6] A. Punya, T. Cheiwchanchamnangij, A. Thiess, and W. R. L. Lambrecht, MRS Proc. 1290 ,4(2011 ). [7] F. Ullstad, G. Bioletti, J. R. Chan, A. Proust, C. 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PhysRevB.85.140509.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 85, 140509(R) (2012) Metal-insulator transition and superconductivity induced by Rh doping in the binary pnictides RuPn(Pn=P, As, Sb) Daigorou Hirai,1,*Tomohiro Takayama,1Daisuke Hashizume,2and Hidenori Takagi1,2,3 1Department of Advanced Materials, University of Tokyo and JST-TRIP, Kashiwa 277-8561, Japan 2RIKEN Advanced Science Institute, Wako 351-0198, Japan 3Department of Physics, University of Tokyo, Hongo 113-0033, Japan (Received 3 October 2011; revised manuscript received 17 March 2012; published 16 April 2012) Binary ruthenium pnictides, RuP and RuAs, with an orthorhombic MnP structure, were found to show a metal to a nonmagnetic insulator transition at TMI=270 and 200 K, respectively. In the metallic region above TMI,a structural phase transition, accompanied with a weak anomaly in the resistivity and the magnetic susceptibility,indicative of a pseudogap formation, was identified at T s=330 and 280 K, respectively. These two transitions were suppressed by substituting Ru with Rh. We found superconductivity with a maximum Tc=3.7 and 1.8 K in a narrow composition range around the critical point for the pseudogap phase, Rh content xc=0.45 and 0.25 for Ru1−xRhxPa n dR u 1−xRhxAs, respectively, which may provide us with a nonmagnetic route to superconductivity at a quantum critical point. DOI: 10.1103/PhysRevB.85.140509 PACS number(s): 74 .70.Xa, 71 .30.+h, 74.25.Dw, 74 .40.Kb The relationship between superconductivity and other collective electronic states has been a long-standing enigma incondensed-matter physics. In a variety of systems with distinctchemical characters, including cuprates, 1heavy fermions,2 organics,3and, more recently, iron pnictide,4superconduc- tivity was found in a narrow region near the critical boarder tomagnetism as a function of pressure and doping. Superconduc-tivity has also been observed at a critical border to classes ofelectronic orderings other than magnetic ordering, includingcharge ordering 5–7and charge density wave,8,9although Tc remains relatively low. Stimulated by the discovery of iron pnictide superconduc- tors, we have been exploring possible superconductivity in Rupnictides. 4 dRu has the same d-electron number as 3 dFe and is in general less magnetic. A series of binary compoundsRuPn(Pn=P, As, and Sb) has been reported to crystallize in a MnP-type orthorhombic structure (space group Pnma ). 10–12In this crystal structure (see the inset to Fig. 1), RuPn 6octahedra form a face-sharing chain along the aaxis. The chains are connected by the edges and Ru forms a distorted triangularlattice within the bcplane. We discovered two sequential phase transitions in RuP and RuAs: a weak transition from a metal to a pseudogapphase accompanied with the superstructure formation at hightemperature ( T s=330 K for RuP and 280 K for RuAs) and a first-order transition to a nonmagnetic insulator at lowtemperature ( T MI=270 K for RuP and 200 K for RuAs). We suppressed those two transitions by Rh doping for Ru andfound superconductivity at the critical point for the “pseudogapphase.” Although the microscopic origin of the transitionsremains yet to be clarified, the discovery should provide aplayground for superconductivity at a nonmagnetic criticalpoint. Herein, we present the transport, magnetic, thermal,and structural properties of Ru 1−xRhxP and Ru 1−xRhxAs, with emphasis on the discovery of two phase transitions andsuperconductivity at a critical point, and discuss the possibleorigin of the phase transitions. Polycrystalline samples of Ru Pn(Pn=P, As, and Sb) and Rh-doped samples Ru 1−xRhxP and Ru 1−xRhxAs wereprepared by a conventional solid-state reaction. A mixture of Ru metal, Rh metal, and pnictogen elements was sinteredin an evacuated quartz tube initially at 550 ◦C for 10 h and then at 1050◦Cf o rR u 1−xRhxP, 950◦Cf o rR u 1−xRhxAs, and 900◦C for RuSb for 48 h. An excess of pnictogen elements was added to compensate for the loss due to volatilization.The sintered pellet was reground, repelletized, and sinteredagain for 72 h. The x-ray diffraction (XRD) pattern of theobtained sample [Fig. 2(a)] indicated the formation of a single phase within the given resolution, except for heavily Rh-dopedRuP containing a trace amount of elemental Rh of the orderof 1%. Magnetic, transport, and thermal measurements wereconducted by a superconducting quantum interference device(SQUID) magnetometer and a Physical Property MeasurementSystem (PPMS: Quantum Design). Electrical resistivity ρ(T) above 350 K was measured separately by a four-probe methodin a furnace with flowing N 2gas. Very small single crystals of RuP, less than 100 μm in size, were grown out of Sn flux and used for the structural analysis. All of the three pnictides, RuP, RuAs, and RuSb, were found to be metallic at room temperature with a magnitude ofresistivity ∼1m/Omega1cm. On cooling, a metal-insulator transition was clearly observed for RuP and RuAs at T MI=270 and 200 K, shown in Fig. 2(b) and, below the TMI,ρ(T) shows an insulating behavior. The presence of tiny but clear hysteresisaround the T MIindicates that the metal-insulator transitions are of first order. In the case of RuAs, we observe a muchbroader transition than in RuP, which we believe representsthe presence of some inhomogeneity in the RuAs sample.RuSb was found to be metallic down to the lowest temperaturemeasured. The magnetic susceptibility χ(T) [Fig. 2(b)] in the metallic phase above T MIis less than 10−4emu/mol, which may be ascribed to the Pauli paramagnetic susceptibility of ametal with a moderate density of states (DOS). At T MIfor RuP and RuAs, χ(T) shows an almost discontinuous drop to a negative value with hysteresis, which is comparable tothe expected core diamagnetism. 13This suggests that the low-temperature insulating state is nonmagnetic. A recent 140509-1 1098-0121/2012/85(14)/140509(5) ©2012 American Physical SocietyRAPID COMMUNICATIONS HIRAI, TAKAY AMA, HASHIZUME, AND TAKAGI PHYSICAL REVIEW B 85, 140509(R) (2012) RuPn a bc R u P n a b b c c 0.1110100 0.6 0.4 0.2 0.0x10 Ru1-xRhxP xc = 0.45SCTc = 3.7 KMetal Pseudo-gap Non-mag Insulator 0.1110100 T (K) 0.6 0.4 0.2 0.0 Rh content x600 MetalRuSb SCTc = 1.2 K0.1110100 0.6 0.4 0.2 0.0x102 Ru1-xRhxAs Metal SCPseudo -gap Non-mag InsulatorTc = 1.8 K xc = 0.25 FIG. 1. (Color online) Electronic phase diagrams of Ru 1−xRhxP, Ru1−xRhxAs, and RuSb as functions of Rh doping. Solid squares and triangles in Ru 1−xRhxPa n dR u 1−xRhxAs correspond to the transition temperatures to the pseudogap phase Tsdetermined from the minima inρ(T) curves. Solid and open circles in Ru 1−xRhxPr e p r e s e n t the superconducting transition temperatures Tcdetermined from the magnetization and the specific-heat measurements, respectively. Open triangles in Ru 1−xRhxAs indicate Tcdetermined from the heat capacity data. The inset shows the crystal structure of Ru Pn(Pn= P, As, and Sb). muon spin relaxation ( μSR) experiment on RuAs (Ref. 14) also supports the presence of a nonmagnetic ground state.The systematic suppression of a metal-insulator transition ongoing from P, As, to Sb would reflect the increased bandwidthdue to the enhanced p-dhybridization. The increase of Pauli paramagnetic susceptibility from P, As, to Sb, however,suggests the increased density of states, which contradictsthe increased bandwidth and therefore requires invokingadditional ingredients. Closely inspecting the poorly metallic state of RuP and RuAs above T MI, we notice an additional anomaly at Ts=330 and 280 K, respectively. As shown in the inset to Fig. 2(b), atTs, there is a minimum in ρ(T) and a maximum in χ(T). It appears that the anomaly at Tsrepresents a precursor to the metal to nonmagnetic insulator transition in that ρ(T) increases andχ(T) decreases below Ts. From ρ(T) and χ(T) data alone, it is not clear whether or not Tsrepresents a well-defined phase transition. However, the single-crystal x-ray structural analysis on a RuP shown inFig. 2(c) indicates clearly that it is a phase transition. The 101001000 ρ (mΩ cm )RuP RuAs Intensity (arb. unit) 32 30 28 2θ (degrees, Cu K α)002 011 200 102x = 0.4 x = 0.3 x = 0.2 x = 0.1 x = 0.0Intensity (arb. unit) 34 32 302θ (degrees, Cu K α)002 011 200x = 0.6 x = 0.45 x = 0.3 x = 0.1 x = 0.0 1.05 1.04 1.03ρ (mΩcm ) 360 320 280 T (K)Intensity (arb. unit) 120 100 80 60 40 20 2θ (degrees, Cu K α)RuP RuAs 8.0 4.0 0.0 -4.0χ (10-5emu/mol ) 400 300 200 100 0 T (K)1ρ RuSb RuAs RuPRuSb 2.8 2.6 2.4 χ (10-5emu/mol ) 360 320 280 T (K) 400K 300K 250K 0 1/4 1/4011002111102101 FIG. 2. (Color online) (a) XRD patterns of (upper panel) RuP and (lower panel) RuAs. The open circles, solid line, and lowersolid line represent observed, calculated, and difference XRD patterns, respectively. Tick marks indicate the position of allowed reflections. The insets show an enlarged area of the XRD pattern,showing a systematic change with Rh concentration for Ru 1−xRhxP and Ru 1−xRhxAs. (b) Temperature dependence of (upper panel) resistivity ρ(T) in zero applied field and (lower panel) dc magnetic susceptibility χ(T) under an applied field of 1 T for RuP, RuAs, and RuSb. Open and solid arrows indicate the metal to nonmagnetic insulator transitions and high-temperature structural transitions inRuP and RuAs, respectively. The insets show the ρ(T)a n d χ(T) anomaly associated with the phase transition at T s=330 K in RuP. (c) Single-crystal x-ray diffraction patterns for RuP measured at 400,300, and 250 K. The reflections at 400 K are indexed based on the orthorhombic cell. 140509-2RAPID COMMUNICATIONS METAL-INSULATOR TRANSITION AND ... PHYSICAL REVIEW B 85, 140509(R) (2012) crystal structure of RuP at 400 K (above Ts) was refined well with an orthorhombic Pnma space group, as reported in Ref. 10. On cooling, superlattice spots hk/4l/4 appear just below Ts=330 K, indicating the fourfold structural modulation within the bcplane along the [011] direction. By lowering the temperature further, additional spots indicativeof tripling of the aaxis, the chain direction, emerge at T MI= 270 K. The crystal structures below Tsand below TMIremain yet to be refined. In the RuAs polycrystalline powder, weobserved the superlattice peaks in the powder pattern at T sand TMI, analogous to those observed for the RuP single crystal. Considering the three-dimensional crystal structure of RuP, a nesting-driven charge density wave (CDW) in its simplest form is highly unlikely to describe the insulating groundstate with the whole Fermi surface gapped. A band-structurecalculation indeed indicated the presence of complicatedand multiple Fermi surfaces and a nesting-driven CDWis highly improbable to occur. A more elaborate picture,such as local spin dimer formation associated with orbitalordering, should be invoked to account for the nonmagneticinsulating state. A metal to nonmagnetic insulator transitionin three-dimensional complex transition-metal oxides hasbeen observed, for example, in Magn ´eli phase vanadium and titanium oxides, 15Tl2Ru2O7,16CuIr 2S4,17MgTi 2O4,18 and LiRh 2O4.19In all these compounds, orbital ordering is believed to play a key role in realizing the nonmagnetic, spinsinglet ground state. Interestingly, in LiRh 2O4, the orbital ordering, with weak ρ(T) and χ(T) anomalies similar to those observed in Ru pnictides, occurs at a higher tempera-ture than the first-order metal-insulator transition and givesrise to a reduced dimensionality of the itinerant electrons,which acts as a precursor to the metal-nonmagnetic insulatortransition. 19 Inspired by the close link between electronic order and superconductivity recognized in a variety of systems, we haveattempted to suppress the two transitions in RuP and RuAsby doping. We found that Rh doping for Ru systematicallysuppresses the two transitions. As seen from ρ(T) andχ(T) shown in Fig. 3, upon Rh doping, the first-order transition at T MIis rapidly suppressed and is absent already at 10% doping level for both RuP and RuAs. The transition at Tsappears to be much more robust against doping than the metal-insulatortransition. Even with more than 10% doping, we see a broadpeak in χ(T) and a minimum in ρ(T) representing T s.B e l o w Ts, an anomalous and poorly metallic state is realized. First of all,ρ(T) shows a very weak increase on cooling but appears to approach a finite value. χ(T) shows a pronounced decrease on cooling sometimes even to a diamagnetic regime, suggestingreducing DOS. The reduction of magnetic susceptibility isanalogous to those observed, for example, in the underdopedcuprates, 20and indicative of the presence of a pseudogap. In this Rapid Communication, we call this poorly metallic phasebelow T sas a “pseudogap” phase. This is again suggestive of the transition at Tsbeing a precursor to the nonmagnetic insulator phase observed in the undoped compounds. Eventually the pseudogap transition disappears at xc=0.45 for RuP and at xc=0.25 for RuAs, as clearly seen in Fig. 3.I n support of the presence of a well-defined critical point, the veryclear anomaly in the doping dependence of Debye temperature/Theta1 Dand the electronic specific-heat coefficient γwas observed6 4 2 0 -2 -4χ (10-5emu/mol ) 300 200 100 0 400 300 200 100x = 0.0 x = 0.1 x = 0.2 x = 0.3 x = 0.4 x = 0.5x = 0.0 x = 0.05 x = 0.15 x = 0.25 x = 0.3 x = 0.4Ru1-xRhxPR u1-xRhxAsρ (arb. unit) 600 400 200 0 T (K)400 300 200 100 T (K)x = 0.0 x = 0.1 x = 0.2 x = 0.3 x = 0.4 x = 0.45x = 0.0 x = 0.1 x = 0.2 x = 0.25 x = 0.4 Ru1-xRhxP Ru1-xRhxAs 300 200 100 0 T (K)400 300 200 100 T (K) FIG. 3. (Color online) Temperature-dependent (a) resistivity (ρT) and (b) magnetic susceptibility ( χT)f o rR u 1−xRhxPn(Pn= P and As). dc magnetic susceptibility was measured under an appliedmagnetic field of 1 T. The arrows in (a) indicate the minima of the ρ(T) curve, defined as the pseudogap transition temperature T s. atxc, indicative of the presence of a phase transition involving both electrons and lattices. We discovered superconductivity at the critical point for the pseudogap phase. As shown in Fig. 4(a), zero resistance and full diamagnetic shielding, indicative of a superconductingtransition, are observed below T c=3.7 and 1.8 K for the samples with the critical Rh content xc,R u 0.55Rh0.45P and Ru 0.75Rh0.25As, respectively. The electronic specific heat Ce(T) of those two samples were estimated by subtracting the normal-state CN(T) under 9-T magnetic field, which is well above the upper critical field μ0Hc2(0), and adding the γTterm with γobtained from the extrapolation of CN(T)/T toT=0. The electronic specific-heat coefficient γwas estimated as 1.3 mJ /mol K2for Ru 0.55Rh0.45P and 3.0 mJ /mol K2for Ru 0.75Rh0.25As, which is quite moderate for a 4 d intermetallic compound. Ce(T) shows a large jump at Tc both for Ru 0.55Rh0.45P and Ru 0.75Rh0.25As, evidencing the bulk superconductivity. The rapid decrease of Ce(T)/Tbelow Tcin Ru 0.55Rh0.45P suggests fully gapped superconductivity, which is very likely an swave. The slow decrease of Ce(T)/Tin Ru 0.75Rh0.25As at a glance appears to imply a gapless superconductivity, but considering the pronouncedinhomogeneity in the RuAs system, we suspect that it reflects adistribution of an inhomogeneous gap rather than gap node(s). As seen from the specific-heat data C(T) and magne- tization data χ(T) for the samples with different doping 140509-3RAPID COMMUNICATIONS HIRAI, TAKAY AMA, HASHIZUME, AND TAKAGI PHYSICAL REVIEW B 85, 140509(R) (2012) 0.5 0.4 0.3 0.2 0.1 0.0ρ (mΩ cm) 6 4 2Ce / T (mJ /mol K2) -2.0-1.5-1.0-0.50.04π M/H (emu /cm3) 4 3 2 1 0 T (K)x = 0.15 x = 0.25 x = 0.35 x = 0.45 x = 0.35 x = 0.40x = 0.50x = 0.45x = 0.10 Ru1-xRhxPRu1-xRhxAs Ru0.55Rh0.45PRuSbRu0.55Rh0.45P Ru0.75Rh0.25As FIG. 4. (Color online) Superconducting transitions observed in Ru pnictides. (a) Temperature-dependent resistivity ρ(T)o f Ru0.55Rh0.45P, Ru 0.75Rh0.25As, and RuSb. (b) Electronic specific heat divided by temperature Ce/Tfor Ru 0.55Rh0.45Pa n dR u 1−xRhxAs (x=0.1, 0.15, 0.25, 0.35, and 0.45). (c) dc magnetization data at low temperatures under applied magnetic field of 20 Oe for Ru 1−xRhxP (x=0.35, 0.40, 0.45, and 0.50). levels shown in Figs. 4(b) and 4(c), superconductivity was observed in a limited region around the critical point xcand transition temperature Tcpeaked at xcboth for Ru 1−xRhxP and Ru 1−xRhxAs. The interplay between the criticality and superconductivity in doped RuP and RuAs can be illustratedvisually as a phase diagram shown in Fig. 1. The rapid collapse of the nonmagnetic insulating phase upon dopingmay suggest that the accommodation of an integer numberof electrons is an important ingredient for the emergence of anonmagnetic insulator phase below T MI. On the other hand, the insensitivity of Tsand systematic suppression of the pseudogap behavior upon doping might mean a local character of thephase transition. The presence of a superconducting domecentered at the critical point clearly indicates the link betweenthe criticality to the ordering below T sand superconductivity. Comparing Ru 1−xRhxP and Ru 1−xRhxAs, it is clear that theTsordering is suppressed more readily for Ru 1−xRhxAs in which Tsfor the undoped compound and xcare much lower for RuAs than RuP. Possibly reflecting this, the “optimum”T cis higher for Ru 1−xRhxP(Tc=3.7 K) than Ru 1−xRhxAs (Tc=1.8 K). It might be interesting to infer here that Ru0.55Rh0.45P has a smaller electronic specific-heat coefficient (γ∼1.3 mJ /mol K2) than that of Ru 0.75Rh0.25As (γ∼ 3.0 mJ /mol K2). We argue that such an anticorrelation between DOS and Tc, opposite to what is predicted from BCS theory, might imply the vital role of the energy scale of criticality. TheT sordering appears to be suppressed completely for RuSb, but superconductivity with a lower Tcthan Ru 1−xRhxP and Ru1−xRhxAs,Tc=1.2 K, was still observed, as seen in Fig. 4(a). This might suggest that RuSb is located not far away from the hidden critical point. In conclusion, we found two sequential transitions in binary pnictides RuP and RuAs: a high-temperature transition toa pseudogap phase at T sand a low-temperature metal to nonmagnetic insulator at TMI. To clarify the physics behind those two transitions, the refinement of the lattice distortionpattern below T sandTMIshould have a high priority. Rh doping was found to suppress those two transitions. In anarrow doping region around a critical point for the pseudogapphase, superconductivity was discovered with maximum T cof 3.7 K for Ru 0.55Rh0.45P and 1.8 K for Ru 0.75Rh0.25As, giving rise to a playground for the superconductivity at a criticalpoint. We emphasize here that the critical point here is neitherantiferromagnetic nor ferromagnetic, as is usually the case inwidely discussed superconductivity at a critical point. We thank A. Mackenzie, P. Radaelli, T. Mizokawa, Y . Katsura, D. Nishio-Hamane, and A. Yamamoto for stim-ulating discussions. This work was partly supported by aGrant-in-Aid for Scientific Research (S) (Grant No. 19104008)and a Grant-in-Aid for Scientific Research on Priority Areas(Grant No. 19052008). *Present address: Department of Chemistry, Princeton University, Princeton, New Jersey 08540, USA. 1S. Huffner, M. A. Hossain, A. Damaceli, and G. A. Sawatzky, Rep. Prog. Phys. 71, 062501 (2008). 2N .D .M a t h u r ,F .M .G r o s h e ,S .R .J u l i a n ,I .R .W a l k e r ,D .M .F r e y e , R. K. W. Haselwimmer, and G. G. Lonzarichand, Nature (London) 394, 39 (1998). 3T. Lorenz, M. Hofmann, M. Gr ¨uninger, A. Freimuth, G. S. Uhrig, M. Dumm, and M. Dressel, Nature (London) 418, 614 (2002). 4Y . Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008). 5T. Yamauchi, Y . Ueda, and N. Mori, P h y s .R e v .L e t t . 89, 057002 (2002).6M. Ito, H. Kurisaki, F. Nakamura, T. Fujita, T. Suzuki,J. Hori, H. Okada, and H. Fujii, J. Appl. Phys. 97, 10B112 (2005). 7H. Suzuki, T. Furubayashi, G. Cao, H. Kitazawa, A. Kamimura,K. Hirata, and T. Matsumoto, J. Phys. Soc. Jpn. 68, 2495 (1999). 8E. 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 (2006). 9A. F. Kusmartseva, B. Sipos, H. Berger, L. Forro, and E. Tutis, Phys. Rev. Lett. 103, 236401 (2009). 10S. Rundqvist, Acta Chem. Scand. 16, 287 (1962). 11R. D. Heyding and L. D. Calvert, Can. J. Chem. 39, 955 (1961). 140509-4RAPID COMMUNICATIONS METAL-INSULATOR TRANSITION AND ... PHYSICAL REVIEW B 85, 140509(R) (2012) 12K. Endresen, S. Furuseth, K. Selte, A. Kjekshus, T. Rakke, and A. F. Andresen, Acta Chem. Scand. A 31, 249 (1977). 13P. W. Selwood, Magnetochemistry (Interscience, New York, 1956), 2nd ed., p. 78. 14Y . Ishii (private communication). 15M. Marezio, D. B. McWhan, P. D. Dernier, and J. P. Remeika, Phys. Rev. Lett. 28, 1390 (1972). 16S. Lee, J.-G. Park, D. T. Adroja, D. Khomskii, S. Streltsov, K. A. Mcewen, H. Sakai, K. Yoshimura, V . I. Anisimov, D. Mori,R. Kanno, and R. Ibberson, Nat. Mater. 5, 471 (2006).17P. G. Radaelli, Y . Horibe, M. J. Gutmann, H. Ishibashi, C. H. Chen, R. M. Ibberson, Y . Koyama, Y .-S. Hor, V . Kiryukhin, and S.-W.Cheong, Nature (London) 416, 155 (2002). 18M. Schmidt, W. Ratcliff, P. G. Radaelli, K. Refson, N. M. Harrison, and S. W. Cheong, P h y s .R e v .L e t t . 92, 056402 (2004). 19Y . Okamoto, S. Niitaka, M. Uchida, T. Waki, M. Takigawa, Y . Nakatsu, A. Sekiyama, S. Suga, R. Arita, and H. Takagi, Phys. Rev. Lett. 101, 086404 (2008). 20T. Watanabe, T. Fujii, and A. Matsuda, Phys. Rev. Lett. 84, 5848 (2000). 140509-5

PhysRevB.71.012413.pdf

Orbital magnetism in the half-metallic Heusler alloys I. Galanakis * Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany sReceived 21 July 2004; published 24 January 2005 d Using the fully relativistic screened Korringa-Kohn-Rostoker method I study the orbital magnetism in the half-metallic Heusler alloys. Orbital moments are almost completely quenched and they are negligible withrespect to the spin moments. The change in the atomic-resolved orbital moments can be easily explained interms of the spin-orbit strength and hybridization effects. Finally, I discuss the orbital and spin momentsderived from x-ray magnetic circular dichroism experiments. DOI: 10.1103/PhysRevB.71.012413 PACS number ssd: 75.50.Cc, 71.20.Be, 71.20.Lp Introduction . Half-metallic ferromagnets are a new class of materials which attracted a lot of attention due to theirpossible applications in spintronics. 1In these materials the two spin bands have a completely different behavior. Whilethe majority spin band sreferred to also as spin-up band d shows the typical metallic behavior, the minority spin bandsspin-down band dis semiconducting. The spin-polarization at the Fermi level is 100% and these compounds could maxi-mize the efficiency of the magnetoelectronic devices. 2 de Groot and collaborators were the first to predict the existence of half-metallicity in the case of NiMnSb.3Since then a lot of materials have been predicted to be half-metals:other half-Heusler alloys se.g., PtMnSb d, 4,5a large number of the full-Heusler alloys se.g., Co 2MnGe d,6,7the quaternary Heusler alloys,8,9some oxides se.g., CrO 2and Fe 3O4d,10the manganites se.g., La 0.7Sr0.3MnO3d,10the double perovskites se.g., Sr 2FeReO 6d,11the pyrites se.g., CoS 2d,12the transition metal chalcogenides se.g., CrSe dand pnictides se.g., CrAs din the zinc-blende or wurtzite structures,13–15and the diluted magnetic semiconductors se.g., Mn impurities in Si or GaAs d.16,17Heusler alloys are particularly interesting due to their very high Curie temperature and the similarity between their crystal structure and the zinc-blende structure adoptedby the III-V and II-VI binary semiconductors like GaAs orZnS. Several papers have been devoted to the calculation of the electronic structure of the half-metallic Heusler alloys. Allthese studies produced a similar description of their magneticproperties. 7,18,19In 2002 Galanakis et al.have shown that the appearance of the gaps in these alloys is directly connectedto the magnetic spin moments and moreover that the totalspin magnetic moment M tscales linearly with the total num- ber of valence electrons Ztfollowing the relation: Mt=Zt −18 for the half-Heusler alloys like NiMnSb and Mt=Zt −24 for the full Heusler alloys like Co 2MnGe.4,6The orbital magnetic moments of these alloys on the other hand haveattracted much less attention and results are scarce. Also ex-perimentally only in few cases the orbital magnetic momentshave been determined via the x-ray magnetic circular di-chroic sXMCD dspectra of thin films. 20,21 In this contribution I will present a study of the atomic- resolved orbital magnetic moments of several Heusler alloysusing first-principles calculations. This investigation wasmotivated by the experiments in Ref. 21 where the orbitalmoments derived from the XMCD spectra were one order ofmagnitude larger than the ones normally found in materials with cubic symmetry like the Heusler materials. In this casethe spin-orbit coupling appears only in the fourth order of theperturbation theory contrary to materials crystallizing inhighly anisotropic structures like the L1 0, e.g., CoPt, where the spin-orbit coupling appears in the second order and or-bital moments are much higher. 22My results show that or- bital moments are much smaller than the experimental val-ues, as expected by symmetry reasons. In the last section Idiscuss the discrepancy between the experimental and thetheoretical results. Calculations details . To calculate the orbital and spin magnetic moments I used the fully relativistic sFRdversion of the Korringa-Kohn-Rostoker sKKR dmultiple-scattering Green function method where the Dirac equation for the cell-centered potentials in the atomic spheres sASA dis solved. 23 The Vosko, Wilk, and Nusair parametrization24of the local density approximation sLDA dis used for the exchange and correlation potential. This method has been already em-ployed to calculate the effect of the spin-orbit coupling onthe minority band gap in the case of half-metallicferromagnets. 25In the case of NiMnSb and similar half- Heusler alloys it was shown that the spin-orbit induces stateswithin the gap but the effect is very weak and the alloysshow a region of very high spin-polarization s,99% din- stead of a gap; defects have a much more pronounced effect on the destruction of the gap. 26 If I compare the results obtained in this contribution by using the FR-KKR-ASA with the results obtained in Refs. 4and 6 using the full-potential sFPdKKR method where the scalar-relativistic approximation is employed sthe spin-orbit coupling is not taken into account d, both versions of the KKR method reproduce a similar description of the spinmagnetic moments; the differences are restricted to smalldeviations in the absolute values of the spin magnetic mo-ments. Both C1 bandL21structures of the half- and full- Heusler alloys, respectively, are close-packed structures andASA is expected to give a good description of their elec-tronic structure with respect to FP. Moreover, spin-orbit is aweak effect and only marginally changes the spin moments. Ishould also note that LDA is known to underestimate theorbital moments by as much as 50% but reproduces the cor-rect trends. 22,27 Half-Heusler alloys containing Mn -Sb. The first family IPHYSICAL REVIEW B 71, 012413 s2005 d 1098-0121/2005/71 s1d/012413 s4d/$23.00 ©2005 The American Physical Society 012413-1will study is the MnSb-based half-Heusler alloys and in Table I I have gathered their magnetic moments.To this fam-ily belong the Fe-, Co-, Ni-, and PtMnSb which are half-metallic sHMd. RhMnSb and IrMnSb are isoelectronic to CoMnSb but the Fermi level falls within the minority va-lence band and the HM is lost sthe total spin moments are slightly above the ideal value of 3 mBd. The Cu-, Ag-, and AuMnSb have 23 valence electrons and if they were HMthey should have a total spin moment of 5 mB, but as it was shown in Ref. 4 this value is practically impossible to obtain;it is energetically more favorable to lose the HM.As a resultalso the spin moments of the Sb atoms are now parallel tothe spin moments of the Mn atoms contrary to the othercompounds. The orbital moments are small with respect to the spin moments and only in the case of IrMnSb the m orbitIrap- proaches the 0.1 mB. In the case of the Sb atoms, the sp-bands lay low in energy and are almost completely filled for bothspin directions. 4There is only a very small majority spin p-weight around the Fermi level due to the antibonding p-d hybrids. As a result the antimonium orbital moment is prac-tically zero for all compounds. Mn atoms possess a large spin-magnetic moment in all Heusler alloys. The Mn spin-up states are practically com-pletely occupied while Mn admixture in the occupied minor-itydstates is limited; it is mainly the X atom which domi- nates the minority bonding dstates. 4Mn orbital moment is less than 0.1 mBin all cases and remains parallel to the spin moment following the 3rd Hund rule. The latter rule, al-though derived for atoms, stands also for solids with fewexceptions. 28It states that if the dband is more than half- filled sMn has 7 d-electrons dthen the spin and orbital mo- ments should be parallel. Increasing the valence of the Xatom by one electron either following the 3 dseries sFe !Co!Ni!Cudor the 4 dseries sRh!Pd!Agdonly scarcely changes the Mn orbital moment while there are sig- nificant variations in the value of the Mn spin moment. Ifnow the X-atom changes along the 5 d-elements series sIr !Pt!Aud, the increase of the Mn spin moment by ,0.5 mB at every step is accompanied by a large decrease of the Mn orbital moment which is practically halved. The increase ofthe spin moment is expected since the hybridization betweenMn and a datom decreases as the valence of the datom increases leading to a more atomiclike electronic structurearound the Mn site. The large effect on the Mn orbital mo-ment in the case of the 5 datoms has been already discussed in Ref. 28, where using perturbation theory it was shown thatthe large spin-orbit coupling of the 5 delements has a large effect on the orbital moment of the 3 dneighboring atoms in the case of alloys. Finally for the X atom the orbital moment follows the Hund’s rules and is always parallel to the spin magnetic mo-ment. Note that the Fe, Co, Rh, and Ir spin moments areantiparallel with respect to the Mn atom. The orbital momentfollows the changes of the spin moment and it increases asthe number of valence electrons increase. As I substitute Cofor Fe the orbital moment increases from −0.06 mBto −0.04 mBand then to 0.015 mBfor Ni in NiMnSb. The abso- lute value of the orbital moment depends strongly also on thespin-orbit coupling. This is clearly seen if I compare Ir withCo. Both atoms have similar spin moments; −0.16 mBfor Co and −0.20 mBfor Ir. On the other hand, cobalt’s orbital mo- ment is −0.04 mBwhile the Ir orbital moment is double as much s−0.09 mBd. Also hybridization plays an important role on the value of the orbital moment, e.g., in FePt Fe has a spin moment of 2.9 mBinstead of −1.0 mBin FeMnSb but the Fe orbital moment is similar in both cases; its absolute value is0.07 mBfor FePt and 0.06 mBfor FeMnSb.29 Orbital moments from first-principle calculations exist for the Ni-, Pd-, and PtMnSb compounds obtained using thefull-potential linear muffin-tin orbitals method sFPLMTO d. 30 While results for NiMnSb are similar to the present calcula- tions this is not the case for the Pd and Pt atoms in PdMnSband PtMnSb compounds. FPLMTO predicts that their orbitalmoment is antiparallel to the spin moment contrary to thepresent calculations. This difference can arise from the treat-ment of the spin-orbit coupling. Whilst in the present calcu-lations the Dirac equations are solved, in the case of theFPLMTO study the spin-orbit coupling is treated as a pertur-bation and since orbital moments are very small this can leadto such small deviations. Safe conclusions could me madeonly if the KKR calculations were performed in the sameway as the FPLMTO ones. Finally it was shown in Ref. 31 that the orbital moment is proportional to the difference between the number of statesTABLE I. Spin smspindand orbital smorbitdmagnetic moments in mBfor the XMnSb half-Heusler com- pounds. The last three columns are the total spin and orbital magnetic moment and their sum, respectively. mspinXmorbitXmspinMnmorbitMnmspinSbmorbitSbmspintotalmorbittotalmtotal FeMnSb −0.973 −0.060 2.943 0.034 −0.040 −0.002 1.958 −0.028 1.930 CoMnSb −0.159 −0.041 3.201 0.032 −0.101 −0.001 2.959 −0.010 2.949 NiMnSb 0.245 0.015 3.720 0.027 −0.071 −0.001 3.951 0.040 3.991 CuMnSb 0.132 0.006 4.106 0.032 0.028 −0.006 4.335 0.032 4.367RhMnSb −0.136 −0.033 3.627 0.035 −0.141 −0 3.360 0.001 3.361 PdMnSb 0.067 0.007 4.036 0.028 −0.117 −0 4.027 0.035 4.062 AgMnSb 0.106 0.006 4.334 0.031 0.040 −0.007 4.556 0.029 4.585 IrMnSb −0.201 −0.094 3.431 0.092 −0.109 −0.001 3.130 −0.004 3.126 PtMnSb 0.066 0.006 3.911 0.057 −0.086 0 3.934 0.063 3.997 AuMnSb 0.134 0.021 4.335 0.027 0.056 −0.006 4.606 0.044 4.650BRIEF REPORTS PHYSICAL REVIEW B 71, 012413 s2005 d 012413-2of majority and minority spin at the Fermi level: morbit ~n"sEFd−n#sEFd. In the case of the half-metallic systems n#sEFd=0 and thus the total orbital moment should be paral- lel to the total spin moment. This is not the case always as can be seen in Table I. In Ref. 31 it was assumed that the t2g andegstates are degenerate and the local DOS of all atoms is a Lorentzian; thus the applicability of this relation is re-stricted. Half-metallic full-Heusler alloys . In the second part of my study I will concentrate on the half-metallic full-Heusler al-loys and in Table II I have gathered my results. The orbitalmoments are quite small like the half-Heuslers. In all caseswith the exception of Rh atom in Rh 2MnAl the Hunds rules are obeyed; note that for V in Mn 2VAl the spin and orbital moments are antiparallel since V dvalence shell is less than half-filled. The orbital moments of the spatoms sZ sites dare almost zero for all cases as in the half-Heuslers. The Co 2Mn-Z type compounds are the most interesting since they present the highest Curie temperature among theknown half-metals. ?The comparison between the Al and Si compounds, which have one valence electrons difference, re-veals large changes in their magnetic properties. The Co spinmoment increases by nearly 0.25 mBand the Co orbital mo- ment follows this change since it is more than double for theSi compound. The increase in the Mn spin moments is pro-portionally smaller and so do the orbital moments. Substitut-ing now Ge or Sn for Si, which are isovalent systems, hasonly a weak effect on the spin moments. Co spin momentslightly decreases while the Mn spin moment slightly in-creases. For both atoms the orbital moments show a smallincrease with the atomic number. The next step is to substitute Cr for Mn in Co 2MnAl. Co spin moment is not affected by this substitution and so doesits orbital moment. Thus the Co orbital moment is mostlyinduced by the spin-orbit coupling at the Co moment and isinsensitive to hybridization with the neighboring sites. Crmoments on the other hand have to account for the missingelectron and are considerably smaller than the Mn ones. Sub-stituting Fe for Mn in Co 2MnAl has a more pronounced ef- fect. Co spin moment increases by 0.35 mBwhile its orbital moment is more than tripled. Its also interesting to compareCo 2FeAl to the isoelectronic Co 2MnSi. Co spin moment in the case of Co 2FeAl is slightly larger while the Co orbital moment is increased by ,50%.Comparing Co 2MnAl with Fe 2MnAl reveals only small changes at the Mn site and the decrease in the total numberfor valence electrons is taken care by Fe atoms. Substitutingnow Rh for Co in the same compound leads to an increase ofboth the spin and orbital moments of Mn since the hybrid-ization between Mn and Rh dstates is considerably smaller than between the Mn and Co dstates. Finally I also calcu- lated the properties of Mn 2VAl. The increased hybridization between the Mn and its neighboring Mn and V atoms leadsto a large orbital moment at the Mn site although its spinmoment is halved with respect to the cases above where Mnoccupied the Y site. To my knowledge, calculations of the orbital moment ex- ist only by Picozzi et al. 19for the Co 2Mn-Si, -Ge, and -Sn compounds. The orbital moment at the Co site was found tobe around 0.02 mBand at the Mn site around 0.008 mB. These moments are slightly smaller than my values.The differencescan arise from the treatment of the spin-orbit coupling asperturbation in their calculations. Experiments . Few experiments dedicated to the orbital magnetism exist on these compounds. These experiments in-volve the obtaining of the XMCD spectra of thin films. XMCD is the difference between the absorption spectra forleft- and right-circular polarized light involving 2 pcore states excitations towards unoccupied dstates. Elmers and collaborators 21derived orbital moments of 0.12 mBfor Co, 0.04mBfor Cr and 0.33 mBfor Fe in the case of a Co2Cr0.6Fe0.4Al thin film. If I compare these values with my calculations for the Co 2CrAl and Co 2FeAl compounds they are one order of magnitude larger. LDAusually gives orbitalmoments only halved with respect to experiments. 22Also the XMCD derived spin moments are half of the theoretical pre-dicted values. On the other hand, Kimura et al. 20studied the NiMnSb and PtMnSb films and found that morbittotal/mspintotal ,0.05 while in my calculations this ratio is around 0.01.The spin moments derived by Kimura et al.experiments are also comparable to the theoretical results. Thus the deviation be-tween the present theoretical results and the experiments inRef. 20 is considerably smaller than when compared to theones in Ref. 21. In both sets of experiments the orbital and spin moments are derived by applying the sum rules to the XMCD spectra.The sum rules have been derived using an ionic model 32and their application to itinerant systems, in particular to lowsymmetry systems, is strongly debated 33since XMCDTABLE II. Same as Table I for the X2YZfull-Heusler compounds. mspinXmorbitXmspinYmorbitYmspinZmorbitZmspintotalmorbittotalmtotal Co2MnAl 0.745 0.012 2.599 0.013 −0.091 −0 3.998 0.038 4.036 Co2MnSi 0.994 0.029 3.022 0.017 −0.078 0.001 4.932 0.076 5.008 Co2MnGe 0.950 0.030 3.095 0.020 −0.065 0.001 4.931 0.081 5.012 Co2MnSn 0.905 0.038 3.257 0.025 −0.079 0 4.988 0.101 5.089 Co2CrAl 0.702 0.012 1.644 0.008 −0.082 0 2.966 0.033 2.999 Co2FeAl 1.094 0.045 2.753 0.060 −0.095 −0 4.847 0.149 4.996 Fe2MnAl −0.311 −0.015 2.633 0.014 −0.016 0.001 1.994 −0.014 1.980 Mn2VAl −1.398 −0.034 0.785 −0.009 0.013 0.005 −1.998 −0.073 −2.071 Rh2MnAl 0.304 −0.011 3.431 0.034 −0.037 −0.001 4.002 0.011 4.013BRIEF REPORTS PHYSICAL REVIEW B 71, 012413 s2005 d 012413-3probes mainly the region near the surface of a film. Thus their application to experimental spectra is not straightfor-ward. Elmer’s and collaborators sum-rule derived total spinmoment is halved not only with respect to the theoreticalresults but most importantly also with respect to the valuederived from the SQUID measurements. This inconsistencyeven between XMCD and SQUID measurements on thesame sample shows that the application of sum rules to de-rive the moments in the case of XMCD experiments on filmsis not really adequate. Summary . I have studied the orbital magnetism in the half-metallic half- and full-Heusler alloys using the Diracformalism within the framework of the Korringa-Kohn- Rostoker Green’s function method. The quenching of the or-bital moments is pretty complete and their values are verysmall with respect to the spin moments. The change in theatomic-resolved orbital moments can be easily explained interms of the spin-orbit strength and hybridization effects.Moments derived by applying the sum rules to the experi-mental x-ray dichroic spectra of thin films should be treated with caution. 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PhysRevB.103.075143.pdf

PHYSICAL REVIEW B 103, 075143 (2021) Two-dimensional Dirac dispersion in the layered compound BaCdSb 2 Dong-Yun Chen,1,2,3,*Jin Cao ,2,*Botao Fu,2,4Yongkai Li,2,3Xiaoxiong Wang,1JunXi Duan,2,3Junfeng Han,2,3 Yun-Ze Long,1Bing Teng,1Dong Chen ,1,†Zhonghao Liu,5,6,‡and Yugui Yao2,3,§ 1College of Physics, Qingdao University, Qingdao 266071, China 2Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, 100081, Beijing, China 3Micronano Centre, Beijing Key Lab of Nanophotonics and Ultrafine Optoelectronic Systems, Beijing Institute of Technology, Beijing, 100081, China 4College of Physics and Electronic Engineering, Center for Computational Sciences, Sichuan Normal University, Chengdu, 610068, China 5State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China 6College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China (Received 5 December 2020; accepted 2 February 2021; published 24 February 2021) We report a comprehensive study of the electronic structure of layered compound BaCdSb 2by using electrical transport measurements, first-principles calculations, and angle-resolved photoemission spectroscopy (ARPES).The samples show semiconductorlike temperature dependence of resistivity and low carrier density. TheShubnikov-de Haas (SdH) oscillations with a single frequency reveal a small two-dimensional (2D) cylinderlikeFermi surface (FS), a light cyclotron effective mass, and trivial Berry’s phase. Quantum limit can be achievedunder a moderate magnetic field. The calculated bands without any renormalization are well consistent with theARPES results, showing the Dirac band crossing with 2D character near the Fermi energy and a gap of about34 meV at the Dirac point induced by spin-orbit interaction. The SdH oscillations can be identified to the Diracband by the similar cross sectional areas of FSs. Our findings indicate BaCdSb 2is a promising material for researching the quantum phenomena of the 2D Dirac fermions. DOI: 10.1103/PhysRevB.103.075143 I. INTRODUCTION The Dirac fermions in condensed matter systems, featured by a cone-shaped electronic dispersion, have been found inthe so-called Dirac materials, such as graphene, topologicalinsulators, and Dirac semimetals [ 1–5]. This kind of energy spectrum can be described by the relativistic Dirac equation.When the Dirac cones are close to the Fermi level ( E F), the low-energy quasiparticle excitations are drastically differentfrom the usual Schrödinger-type fermions. Many peculiartransport phenomena have been found in the Dirac materials,such as quantum Hall effect with nontrivial Berry’s phase andlarge linear magnetoresistance [ 6–9]. Recently, the layered compounds AMnX 2(A=Ca, Sr, Ba, Eu, and Yb; X=Bi, and Sb) have been found to possess Dirac fermions near EF[10–33]. They have tetrag- onal or orthorhombic structures with three different spacegroups: I4/mmm ,P4/nmm , and Pnma . All the structures are stacked by Mn X 4tetrahedral layers and Xsheets alterna- tively, with Aions between them. The Xsheets are square nets for tetragonal structures and are distorted into zigzag *These authors contributed equally to this work. †dchen@qdu.edu.cn ‡lzh17@mail.sim.ac.cn §ygyao@bit.edu.cnchains for the orthorhombic structure. Magnetoresistance andquantum oscillation measurements on the single crystals ofCaMnBi 2and SrMnBi 2have revealed large linear magne- toresistance and nontrivial Berry’s phase, which stronglysuggest the existence of the Dirac fermions in the vicinityofE F[11–13]. Unlike graphene, the Dirac fermions in these layered compounds are usually highly anisotropic, show-ing the momentum-dependent velocity of Dirac cones. TheDirac dispersions have been found in CaMnBi 2,S r M n B i 2, and BaMnBi 2by angle-resolved photoemission spectroscopy (ARPES) [ 16,17]. When taking the spin-orbit coupling (SOC) into account, a massive Dirac fermion can be formed witha small band gap. Moreover, the magnetism of Mn atomsinduces much more rich physics and makes the systemmore sophisticated at the same time. It has been reportedthat the canted antiferromagnetic order in Sr 1−yMn 1−zSb2 and YbMnBi 2breaks the time-reversal symmetry and allows the Dirac point to separate into a pair of the Weyl points[23,24]. The interlayer magnetic exchange in CaMnBi 2drives a charge-gap opening on the Dirac state besides the SOC[34]. The electronic structures of some other nonmagnetic isoelectronic compounds have also been studied in such 112-family materials recently, revealing the Dirac dispersions nearE F, such as BaZnBi 2,S r Z n S b 2, and LaAgSb 2[35–40]. The nonmagnetic compounds are believed to be better to investi-gate the Dirac band structure without the interference frommagnetic order. It’s meaningful to find more nonmagnetic 2469-9950/2021/103(7)/075143(7) 075143-1 ©2021 American Physical SocietyDONG-YUN CHEN et al. PHYSICAL REVIEW B 103, 075143 (2021) counterparts of the Mn-based compounds hosting Dirac dis- persions around EF. In this paper, we report the electrical transport, first- principles calculations, and ARPES studies on BaCdSb 2,a nonmagnetic layered compound isostructural to SrMnBi 2. The resistivity of the sample shows a semiconductorlike tem-perature dependence, which is rarely found in the 112-familymaterials. Shubnikov-de Haas (SdH) oscillations observed inmagnetoresistance have only one frequency, indicating a smalltwo-dimensional (2D) cylinderlike Fermi surface (FS). Thequantum limit can be achieved under a relatively low field.Light cyclotron effective mass and trivial Berry’s phase can beextracted from the oscillations. The band structure calculatedfrom first-principles calculations shows a linear band cross-ing near E Falong the /Gamma1-Mline, which is gapped with SOC turned on. The cross-sectional area of the FS formed by theDirac band is close to the one obtained by SdH oscillations.The ARPES observations are in good agreement with theband calculations. We find that BaCdSb 2has lighter cyclotron effective mass, smaller FS cross-sectional area, and closerlocation of the Dirac point to E Fthan most of the 112-family materials. II. EXPERIMENT AND METHODS The single crystals of BaCdSb 2were grown by slow- cooling method. Ba, Cd, and Sb with a ratio of Ba : Cd :Sb=1 : 1 : 2 were loaded into an alumina crucible, which was then sealed into an evacuated quartz tube. The tube washeated at 720 ◦C for 10 hours and then cooled down to 400◦C with the rate of 2◦C/hour. After that, the tube was cooled down naturally to the room temperature. Plate-shaped singlecrystals with typical dimensions of 2 ×2×0.1m m 3were obtained by cleaving. The crystal structure of the obtainedcrystals was characterized by x-ray diffraction (XRD) on aRigaku Smartlab x-ray diffractometer with Cu Kαradiation at room temperature. The atomic ratio was determined byOxford energy-dispersive x-ray (EDX) spectroscopy analy-sis. The electrical transport properties of the samples weremeasured using a Quantum Design physical property mea-surement system (PPMS). The first-principles calculations were performed in Vienna ab initio simulation package [ 41] based on the projec- tor augmented-wave pseudopotential method [ 42], using the generalized gradient approximation in the scheme of Perdew-Burke-Ernzerhof functional [ 43,44]. The primitive cell was used for all calculations. The energy cutoff was set to be400 eV for the plane-wave basis, and the Brillouin zone (BZ)is sampled by a /Gamma1-centered Monkhorst-Pack grid (6 ×6×6) [45]. All atomic positions and lattice parameters were fully relaxed before further calculations. The SOC has been takeninto account. The Wannier orbital based tight-binding Hamil-tonian was constructed by the WANNIER90 package [ 46]. The FS calculations were implemented in the WannierToolspackage [ 47]. ARPES measurements were performed at the Dreamline of Shanghai Synchrotron Radiation Facility (SSRF). Samplessmaller than 1 ×1m m 2were cleaved in situ , yielding flat mir- rorlike (001) surfaces. During measurements, the temperature20 40 60 80 0 100 200 3000.60.91.2 0369-4-20(b) SbCd ba(0 0 16)(0 0 12) ( 004 )Intensity (arb. units) 2(deg)( 008 ) cBa(a) xx(10-2Ωcm) T(K)(c) yx(mΩcm) B(T)T=2K FIG. 1. (a) Crystal structure of BaCdSb 2. (b) XRD pattern of BaCdSb 2single crystal with (00 l) reflections. (c) Temperature de- pendence of resistivity with current flowing in the abplane. The inset shows the Hall resistivity as the function of magnetic field at T=2K . was kept at T=20 K, and the pressure was maintained at less than 4 ×10−11Torr. III. RESULTS AND DISCUSSION The crystal structure of BaCdSb 2is formed by the stacked CdSb 4tetrahedrons, Sb square nets, and Ba ions, as shown in Fig. 1(a). It belongs to the space group of I4/mmm with lattice parameters a=b=4.558 Å and c=24.160 Å [ 48]. Figure 1(b)shows the XRD pattern of the maximum surface of the plate-shaped single crystal. All the peaks can be indexed asthe (00 l) reflections of BaCdSb 2, suggesting that the crystals grow along the abplane. The phase of BaCdSb 2can be further confirmed by the powder XRD and EDX measurements. Fig-ure1(c)shows the temperature dependence of the longitudinal resistivity with current flowing in the abplane of the sample. The resistivity increases with the decreasing temperature. Thissemiconducting behavior is rarely found in other 112-familymaterials except for EuMnSb 2[30]. The dominant charge car- riers can be indicated as electrons by the negative slope of Hallresistivity curve [the inset of Fig. 1(c)]. With the single-band model, the electron density is estimated as 1 .96×10 18cm−3. Figure 2(a) shows the magnetic field dependence of resis- tivity of BaCdSb 2with various rotating angles ( ϕs) at T= 2 K. The experimental geometry is sketched in the inset ofFig. 2(a), where the magnetic field rotates in the plane per- pendicular to the current and ϕis defined as the angle between magnetic field and the caxis. The magnetoresistance curve for ϕ=0 ◦has a positive field dependence at low field and shows obvious SdH oscillations above 5 T. The oscillations have anobvious peak and valley at B=6.4 and 12.1 T, respectively. The few oscillating extremes suggest the small FS as dis-cussed below. With the rotating angle increases, the positions 075143-2TWO-DIMENSIONAL DIRAC DISPERSION IN THE … PHYSICAL REVIEW B 103, 075143 (2021) 05 1 01.11.21.31.4 02 0 4 0 6 05101520 0 . 00 . 10 . 20 . 30 . 40246 05 1 01.11.21.31.4 0 5 10 15 20230.0 0.1 0.201275°60° 45° 15°30°xx(10-2Ωcm) B(T)0°(a) peak valleyextreme B(T) (deg)(b) 0°15°30°45°60°dxx/dB(arb. units) 1/Bv(T-1)=7 5°(c) 20 Kxx(10-2Ωcm) B(T)2K(e) Δxx fitting curveΔxx(mΩcm) T(K)(f)xxminima xxmaxima linear fitn 1/B(T-1)(d) FIG. 2. (a) Field dependence of resistivity with different field orientations. The inset depicts the measurement configuration. (b) Angle dependence of the extreme positions of oscillations. The red and blue dashed lines are the fitting curves using the 2D re-lation B(ϕ)=B 0/cosϕ. (c) Resistivity derivative dρxx/dBversus 1/Bv=1/(Bcosϕ). (d) Landau fan diagram of Landau level index nversus 1 /B. (e) Field dependent resistivity at temperatures varying from 2 K to 20 K. (f) The difference between the peak and valley values of resistivity as the function of temperature. The dashed line is the fitting curve. of the peak and valley move to higher fields and eventually can’t be detected within 14 T for ϕ=75◦. Considering the 2D nature of the crystal structure of this compound, it isreasonable to check whether the SdH oscillations have the 2Dnature. Figure 2(b) shows the angle dependence of the field positions of the peak and valley. Both of the extremes can bewell fitted by the 2D reference curves of B 0/cosϕ. The 2D feature of oscillations can be further confirmed by the resis-tivity derivative dρ xx/dBas the function of 1 /Bv[Fig. 2(c)], where Bvis the field’s component perpendicular to the ab plane. For all the ϕs, the oscillations have the same extreme 1/Bvpositions with a single frequency, which suggests that the SdH oscillations originate from a cylinderlike FS. To extract more information of the band structure based on the SdH oscillations, we resort to the Lifshitz-Kosevitchformula [ 49]: /Delta1ρ xx∼λ sinhλcos[2π(F/B+1/2−γ+δ)], (1) where λ=2π2kBTm∗/eB¯h,m∗is the cyclotron effective mass, and Fis the oscillating frequency. γis related to the Berry’s phase φBbyγ=φB/2π, andδis 0 or ±1 8for 2D orthree-dimensional (3D) systems. The Berry’s phase φBis 0 for a conventional Schrödinger dispersion or πfor the massless Dirac case. To extract γ−δ, the Landau fan diagram can be used, as shown in Fig. 2(d), where the maximum of ρxx is assigned to the integer indices and the minimum of ρxxis assigned to the half integer indices. γ−δcan be identified as the intercept of the line on the naxis, which is 0.01. With the 2D nature of the oscillations, leading to δ=0, a trivial Berry’s phase is obtained, which suggests the absence of masslessband crossing. It should be noted that, from the Landau fandiagram, the quantum limit is achieved above 6.4 T, whereonly n=0 Landau level is occupied. The cross-sectional area of the FS can be calculated by the Onsager relation [ 49]:S F=2πe ¯hF. The oscillation frequency can be obtained from the linear slope of the Landau level fandiagram, which gives F=6.6 T. Thus, the cross-sectional area of the FS is 6 .3×10 −4Å−2, about only 0.034% of the area of the first BZ. This small FS is consistent with the relatively weak field for achieving the quantum limit. The cyclotron effective mass can be obtained by fitting the tem-perature dependence of oscillation amplitude with /Delta1ρ xx∼ 2π2kBT/¯hωc sinh(2 π2kBT/¯hωc). Figure 2(e) shows the magnetic field depen- dent resistivity at different temperatures. Due to the few periods of the oscillations in the curves, it is hard to accu-rately determine the backgrounds and extract the oscillationparts. By setting the difference of the resistance at the peakatB 1=6.4 T and the valley at B2=12.1Ta s /Delta1ρ xx,1/B= (1/B1+1/B2)/2, the fitting gives m∗/similarequal0.06m0, where m0is the bare mass of electron, as shown in Fig. 2(f). The FS cross- sectional area and effective mass of BaCdSb 2are smaller than most of the 112-family materials, as shown in Table I. To further investigate the electronic structure of BaCdSb 2, we performed the first-principles calculations. Figures 3(a) and3(b) display orbital-projected density of states (DOS) and momentum-resolved dispersions along the high-symmetry di- rections, respectively. The calculations indicate that the DOS near EFmainly comes from the contribution of Sb p(blue), Cdd(green), and Ba s(red) orbitals. The low DOS at EF is consistent with semiconducting behavior indicated by the temperature-dependent resistivity. From the band dispersions along the high-symmetry directions, one can clearly see an electronlike band at X(R) point crossing EFand the Dirac band on the /Gamma1-M(Z-A) line in the vicinity of EF. The latter is similar to that observed in other 2D layered Dirac materials mentioned above. From the calculation results, we obtain that the cone is located at ∼80 meV above EFwith a momentum of∼0.39 Å−1along the /Gamma1-Mdirection. The velocity of the cone is about v/Gamma1=7.9±0.02 eV Å ( ∼1.2×106m/s) for the branch near /Gamma1andvM=9.6±0.03 eV Å ( ∼1.4×106m/s) for the branch near M, showing a band asymmetry of the cone [15]. The energy gap of about 34 meV is opened when SOC is taken into account. Figure 3(c) shows the orbital-resolved band structure along the /Gamma1-Mpath, revealing that the Dirac band is mainly contributed by Sb porbital hybridized with partial Cd dorbital. Figure 3(d) shows the calculated 3D FS map. The electronlike FS at the X(R) point is indicated by red, and the holelike FS indicated by green on the /Gamma1-M(Z-A) line corresponds to the Dirac bands. The two types of FSs are both tubelike, and the cross sections cut by the /Gamma1-X-Mplane 075143-3DONG-YUN CHEN et al. PHYSICAL REVIEW B 103, 075143 (2021) TABLE I. Parameters of 112-family materials extracted from quantum oscillations, ARPES, and band calculations. m∗is the cyclotron effective mass, Fis the oscillation frequency, φBis the Berry’s phase, vDis the velocity of the Dirac cone, ED-EFis the energy of the Dirac point ( ED) relative to EF.F o r vD,vM(v/Gamma1) denotes the velocity for the branch near M(/Gamma1) point along the /Gamma1-Mline. m∗(me) F(T) φB vD(eV Å) ED-EF(meV) Reference CaMnBi 2 0.35 101, 181 0.9 π 10.6( vM), 2.1( v/Gamma1)2 5 [ 12,13,16] CaMnSb 2 0.05, 0.06 8.3, 15.4 1.16 π,−0.26π [25] SrMnBi 2 0.29 152 1.2 π 10.9( vM), 2.4( v/Gamma1)7 5 [ 10,14,16] SrMnSb 2 0.14 67 1.1 π 5.5(vM) 200 [ 24,27] SrZnSb 2 0.1, 0.1, 0.09 103, 127, 160 0.06 π,1.2π,0.74π 200 [ 38] BaMnBi 2 0.105 33.3 0.4–0.6 π 6(vM), 1.9( v/Gamma1)8 0 [ 17,19] BaMnSb 2 0.05 25 π [28] BaZnBi 2 0.17 168 0.5 π 8.1(vM), 4.4( v/Gamma1)8 0 [ 17,37] EuMnBi 2 23–26 1.2 π 9(vM)[ 21,23] YbMnBi 2 0.27 130 0.42 π 9(vM)[ 22,23] YbMnSb 2 0.134 73 0.97 π 80 [ 32,33] LaAgBi 2 0.056 67 [40] LaAgSb 2 3.75( vM), 5.24( v/Gamma1) −100 [ 39] BaCdSb 2 0.06 6.6 0.02 π 9.6(vM), 7.9( v/Gamma1) 80 This work are shown in Fig. 3(e). The crescent cross section of the Dirac bands indicates a highly anisotropic Dirac dispersion along and perpendicular to the /Gamma1-Mline and a much lower velocity perpendicular to the /Gamma1-Mdirection. This highly anisotropicDirac dispersion is commonly found in most of the 112 com- pounds. The cross-sectional area is about 4 .6×10−2Å−2and 1.5×10−3Å−2for the electronlike and holelike FSs, respec- tively. By comparing the cross-sectional area obtained from -2-1012 -2-1012 10 5 0 -2 -1 0 1 2Total p d s -101 -1 0 1ΓM X DOS (states/unit cell/eV)(a) (e) (d)(b)E-EF (eV) X ΓM ZA R~34 meV (g) (f)Γ X M ΓZ A R Z kx (Å-1)ky (Å-1)E-EF (eV) (c) MΓ M LowHigh High LowΓX M DOS FIG. 3. (a) Total and orbital-projected DOS around EF. The different colors indicate the different orbitals. (b) Calculated bands with SOC along the high-symmetry directions. The Dirac band is marked by the dashed circle. SOC opens the energy gap of ∼34 meV at the cone point. (c) Orbital-projected dispersions along the /Gamma1-Mpath. The Dirac band is mainly contributed by Sb porbital (blue) hybridized with partial Cd d orbital (green). (d) Calculated bulk FSs in the 3D BZ with marked high-symmetry points. (e) Calculated 2D FSs. The shadow at the BZ centeris induced by the holelike band at /Gamma1(Z) point below E F. (f) Measured FSs ( EF±20 meV) taken with 100-eV photons in σgeometry. (g) Sketch of the experimental polarization setup. The πandσpolarizations are for electric fields parallel and perpendicular to the mirror plane (red), respectively. 075143-4TWO-DIMENSIONAL DIRAC DISPERSION IN THE … PHYSICAL REVIEW B 103, 075143 (2021) -2-10 -1-0.5 00.5 1 -0.5 0 0.5EF EF-0.07 eV -0.2-0.10 -0.5 0 0.5 -2-10 -1-0.5 00.5 1 -2-10 -1-0.5 00.5 1 -2-10 -1-0.5 00.5 1 MΓ MM MM Γ Γπ σ MM M Γ(a) (f) (d)(e) (c) (b) π σE-EF (eV) E-EF (eV) E-EF (eV) Intensity (a.u.) k// (Å-1) k// (Å-1)High Low π DPLB k// (Å-1)k// (Å-1) k// (Å-1) k// (Å-1) FIG. 4. (a),(b) Intensity plot and corresponding second derivative plot along the /Gamma1-Mdirection in πgeometry, taken with 100-eV photons. The red solid lines are calculated bands without renormalization and without shift. The linear band (LB) and the Dirac point (DP) are indicatedby the white arrows. (c) and (d) correspond to (a) and (b), respectively, but taken in σgeometry. (e) Enlarged view of (b) shows the Dirac bands near E F, as indicated by the dashed white lines. (f) Momentum distribution curves correspond to (a) showing the Dirac bands near EF. The black sticks are guides to the bands. SdH oscillations, it is preferred to assign the oscillations to the Dirac bands, if we assume the EFis slightly lifted. The lifted EFcauses smaller hole FSs and larger electron FSs, which is consistent with the electron-dominant Hall resistivity. This scenario can also explain the weak SdH oscillations in themagnetoresistance curves. We applied the projected 2D BZ ( /Gamma1-M-X) in our ex- perimental studies due to the weak k zdispersions and the limitation of kzresolution in ARPES experiments. Comparing between Fig. 3(e) and Fig. 3(f), the measured FSs are consis- tent with the calculated ones. The shadow at the BZ centerin Fig. 3(e) is induced by the holelike band at /Gamma1(Z) point below E F. Here we note that the ARPES intensity near EFis very low, due to semiconducting behavior with a small amount of electron excitations at EF. The experimental setup shows the high-symmetry directions and the normal of the sample surface defines a mirror plane [Fig. 3(g)]. The π(σ) geometry refers to the electric fields of the incident photons within (normal to) the mirror plane. The even (odd) orbitals with respect to the mirror plane are detected in π(σ) geometry. With selected experimental symmetries, certain orbitals com- binations could be enhanced or suppressed individually. Thus the Dirac bands could be more prominent under certain con- ditions, because the bands come mainly from Sb px,yorbitals. Figures 4(a) and4(c) show ARPES intensity plots taken with 100-eV photons along the /Gamma1-Mdirection in πandσgeometries, respectively. The corresponding second derivative plots are shown in Figs. 4(b) and4(d), respectively. The red solid lines are calculated band without renormalization andwithout shift. On the whole, the calculation results well matchthe experimental data, except that the bands with bindingenergy of about 1 eV at the Mpoint need to be shifted down as indicated by the white circle in Fig. 4(b). As mentioned above, the intensity near E Fis very low due to the low DOS and semiconducting behavior. The weak linear band indicatedby the white arrow in Fig. 4(a)corresponds to the Dirac band. In order to clearly see the Dirac bands near E F, we present the enlarged view and momentum distribution curves aroundthe Dirac bands, as shown in Figs. 4(e) and4(f). By linear fitting the observed band, we estimate that the Fermi velocityis about 7 .5±0.05 eV Å ( ∼1.1×10 6m/s), which nearly agree with the calculated value. We thus demonstrate that the layered BaCdSb 2hosts 2D Dirac dispersion near EF. Here, we have a closer look at Bi/Sb-based 112 materials with crystal structures similar to that of BaCdSb 2.D u et o A(A=Ca, Sr, Ba, Eu, and Yb) atoms being below and above the Bi /Sb square nets, there are two identical Bi /Sb atoms in the unit cell. The Dirac bands along the /Gamma1-Mline are hybridized by the px,yand dorbitals, and gapped by SOC [ 10,16]. Therefore, the entire electronic structure of BaCdSb 2is similar to that of the other ones, ex- cept that the valence band at the BZ center is below EFleading 075143-5DONG-YUN CHEN et al. PHYSICAL REVIEW B 103, 075143 (2021) to a low DOS and a rare semiconducting behavior. Comparing with the other materials listed in Table I, the velocities and energy positions of the Dirac cones are comparable amongmost of them. The SOC gap of ∼34 meV from the calcula- tion is less than that of SrMnBi 2(∼40 meV) and CaMnBi 2 (∼50 meV) [ 10,15] and more than that of SrMnSb 2and BaMnSb 2(∼20 meV) [ 18]. In addition, it is noticeable that the asymmetry of the cones along the /Gamma1-Mline in BaCdSb 2 and LaAgSb 2is weaker than that of the Bi-based compounds, as listed in Table I. The asymmetry of the Dirac cone in this material system is attributed to the SOC of Bi /Sb square nets [15,16]. Due to the weaker SOC of Sb than that of Bi, our results also support this viewpoint. IV. CONCLUSION In conclusion, we have presented a comprehensive study of the low-energy electronic structure of BaCdSb 2by using electrical transport, first-principle calculations, and ARPES.The resistivity shows a semiconducting temperature depen-dence and obvious SdH oscillations, which is rarely found inother 112 Dirac materials. The SdH oscillations have a low frequency and 2D angle dependence, indicating a small 2DFS. Band structure calculations show a Dirac band crossingnear E F, and a gap is opened by SOC. The Dirac band has a 2D FS, which has a cross-sectional area close to the one obtainedfrom SdH oscillations. The gap opened at the Dirac pointdue to SOC can be directly reflected by the semiconductingbehavior and trivial Berry’s phase. Our ARPES results canwell prove the reliability of the calculated band structure. Thiswork enriches the 112 Dirac material family and provides apromising platform for researching the 2D Dirac fermions. ACKNOWLEDGMENTS This work was supported by the National Natural Sci- ence Foundation of China (Grants No. 11804176 and No.11734003), the National Key Research and DevelopmentProgram of China (Grants No. 2020YFA0308800 and No.2016YFA0300600), Shandong Provincial Natural ScienceFoundation of China (Grant No. ZR2018BA030), and ChinaPostdoctoral Science Foundation (Grant No. 2018M632609). [1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009) . [2] Y . Chen, J. G. Analytis, J.-H. Chu, Z. Liu, S.-K. Mo, X.-L. Qi, H. Zhang, D. Lu, X. Dai, Z. Fang, S. C. Zhang,I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science 325, 178 (2009) . [ 3 ]Y .X i a ,D .Q i a n ,D .H s i e h ,L .W r a y ,A .P a l ,H .L i n ,A .B a n s i l , D. Grauer, Y . S. Hor, R. J. Cava, and M. Z. Hasan, Nat. Phys. 5, 398 (2009) . [4] Z. Liu, B. Zhou, Y . Zhang, Z. Wang, H. Weng, D. Prabhakaran, S.-K. Mo, Z. Shen, Z. Fang, X. Dai, Z. Hussain, and Y . L. 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PhysRevB.58.1424.pdf

Many-particle effects in Be- d-doped GaAs/AlxGa12xAs quantum wells M. Kemerink, P. M. M. Thomassen, P. M. Koenraad, P. A. Bobbert, J. C. M. Henning, and J. H. Wolter COBRA Interuniversity Research Institute, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands ~Received 17 October 1997 ! We have performed photoluminescence ~PL!and photoluminescence excitation measurements on two series of center- d-dopedp-type GaAs/AlxGa12xAs quantum wells, with variable well width and doping concentra- tion. The experimental data are compared with self-consistent field calculations. The effects of exchange andcorrelation were found to be extremely important and various models for the hole exchange and correlation arecompared with the experimental data. It is found that the model recently proposed by Bobbert et al. @P. A. Bobbertet al., Phys. Rev. B 56, 3664 ~1997!#consistently describes our experimental observations. Further- more, for well widths w>600 Å clear excitonic effects were observed, for hole densities as high as 12 310 12cm22, which is explained in terms of small spatial overlap between the screening particles and the exciton along the growth direction. In contrast to earlier work on similar samples, we found no indication fora Fermi-edge singularity in the PL spectra of our samples. Peaked structures at the high-energy side of the PLspectra are shown to arise from bulk transitions. @S0163-1829 ~98!04324-0 # I. INTRODUCTION The collective behavior of mobile carriers in a semicon- ductor lattice has attracted a lot of attention ever since the early days of semiconductor physics. With the advent of~quasi- !two-dimensional ~2D!systems, most of the many- body effects that had previously been studied in bulk semi-conductors or metals became subjects of intense research inthese structures. Among these are such well-known effects asscreening, the Fermi-edge singularity ~FES!or Mahan exci- ton, exchange and correlation, etc. Since exchange and cor-relation effects lead to a reduction of the effective band gapin degenerate systems, their effect is often denoted as band-gap renormalization ~BGR!. Although the basic concepts are not very new, all of these effects are still of great currentinterest. 1–4Both experimental and theoretical studies on many-particle effects have predominantly focused on n-type systems, mainly to avoid the complications arising from thevalence-band coupling. However, due to the very differentcharacteristics of the valence bands as compared to those ofthe conduction band, i.e., the coexistence of heavy- andlight-hole ground states, the high effective masses, and the strong nonparabolicity, the study of p-type systems can greatly enhance the general understanding of many-bodyphysics. 5–10 In this paper, we will report on photoluminescence ~PL! and photoluminescence excitation ~PLE!measurements on GaAs/Al 0.20Ga0.80As quantum wells ~QW’s !with a Be- d- doping spike placed in the center of the well. These struc-tures are ideal to study band-gap renormalization because ofthe high carrier densities that can be achieved, and the occu-pation of multiple subbands. We will compare our resultswith the results of self-consistent calculations, in which theeffects of exchange and correlation have been incorporatedby means of the local-density approximation ~LDA!. In par- ticular, we will compare various models for the holeexchange-correlation potential with our experiments and cal-culations without these many-body corrections. We find thatthe inclusion of exchange and correlation effects in self-consistent calculations is essential for a meaningful compari- son with experiments, and that the model that has recentlybeen developed by Bobbert et al. 1consistently describes our experimental findings. In a recent series of papers, Wagner and co-workers11–13 report the observation of a FES in the luminescence spectraof structures that are very similar to the ones discussed here.From a theoretical point of view this observation is veryremarkable. 4,14,15Both the small effective mass of the minor- ity carriers and the large energetic separation between thehighest occupied state and the lowest unoccupied state in thevalence band, reported in Refs. 12 and 13, make the occur-rence of a FES surprising. In our experiments, we do notobserve any indication for a Fermi-edge singularity. In Sec.V we will briefly discuss this negative result in the light ofrecent theoretical work on this subject, and we will compareour results to those of Wagner and co-workers. This paper is organized as follows. In Sec. II we will describe the samples used in this work. Experimental resultswill be reported in Sec. III and discussed in Sec. V. Thenumerical model used in the interpretation of our data ispresented in Sec. IV, along with a comparison of measuredand calculated transition energies. Section VI will summa-rize our conclusions. II. SAMPLES The structures investigated were grown on semi- insulating GaAs substrates by conventional molecular-beam-epitaxy techniques. On top of the substrate a 100-periodGaAs/AlAs superlattice was grown, followed by a 200 Å Al 0.20Ga0.80As barrier layer. Both the superlattice and the Al0.20Ga0.80As barrier were grown at 690 °C. Subsequently, the growth temperature was lowered to 480 °C to avoid Bediffusion during the growth of the active layers. The activelayers consist of ten periods of the following structure: an undoped GaAs layer of width w/2, a Be- d-doping spike with a Be surface concentration p, deposited during a growth in- terrupt, another undoped GaAs layer of width w/2, and a 75-PHYSICAL REVIEW B 15 JULY 1998-I VOLUME 58, NUMBER 3 PRB 58 0163-1829/98/58 ~3!/1424 ~12!/$15.00 1424 © 1998 The American Physical SocietyÅA l0.20Ga0.80As barrier. The total structure was terminated with another 125 Å of Al 0.20Ga0.80As and a 100 Å GaAs cap layer. In addition, a bulk Be-doped reference sample wasgrown. This structure consists of a single, 2 mm-thick bulk GaAs layer that had an aimed doping concentration of 2310 18cm23. It was grown at 630 °C directly on top of a semi-insulating GaAs substrate. The motivation for growing a multiple quantum-well structure, instead of a single QW, which is expected to showless broadened optical spectra, is twofold. Apart from theobvious increase in signal strength, two points are worthsome further discussion. First, the two outer d-doped wells will screen possible depletion fields, arising from mid-gappinning of the Fermi level at the surface and in thesubstrate. 16Therefore, the larger central part of the total structure will be unaffected by these uncontrollable fields,and thus will have a symmetric potential profile. This sym-metry is essential in calculating the self-consistent solutionof the coupled Poisson and Schro ¨dinger equations. Further-more, it has been shown by Rodriguez and Tejedor 4that the symmetry of the confining potential can strongly affect theappearance of Fermi-edge singularities. Also for this reason,uncontrollable and possibly illumination-dependent electricfields are undesirable. Second, since the surroundings of theactive layers are screened by the two outer wells, the Fermilevel in the central eight wells is solely determined by the doping in the dlayers. The small ~ptype!background dop- ing concentration in the order of a few times 1014cm23, resulting from contaminations in the molecular-beam-epitaxysystem, is fully negligible with respect to the amount of dop-ing in the dlayers. This also facilitates numerical simula- tions, as it allows for a restriction of the calculation intervalto the active layers. Furthermore, as d-doped samples are not expected to show very sharp optical lines, some broadeningdue to fluctuations over the ten periods of active layers isacceptable. Two series of d-doped samples were grown. One with a variable doping concentration p, ranging from 2 to 12 31012cm22, at a fixed well width w5600 Å, and one with a variable well width, in between 150 and 1200 Å, at a dop- ing level of p5831012cm22. In Table I the relevant growth parameters are listed. In Figs. 1 ~a!–1~c!~self-consistent po- tentials and wave functions of three representative structuresare displayed. The model used for the calculations is de-scribed in Sec. IV. III. EXPERIMENTAL RESULTS The experiments described in this paper are performed with the sample mounted in a continuous-flow He cryostat,in which temperatures from 4 to 300 K can be reached. Un-less stated otherwise, all reported data are taken at 5 K. Thesamples were excited using either a tunable Ti:sapphire laser, FIG. 1. Valence-band self-consistent confining potential and wave functions of samples 1 ~a!,3~b!, and 5 ~c!, calculated using the model outlined in the text. The thick solid line denotes the confining potential, the thin solid and dashed lines denote the heavy- and light-holeenvelope functions, respectively. The envelope functions are offset by their energy at k50. The dash-dotted line indicates the position of the Fermi level E F.TABLE I. Growth parameters of the investigated samples. The Be-d-doping spike is placed in the center of the GaAs well region. The confining Al0.2Ga0.8As barriers are 75 Å thick. Sample no. Well width ~Å!Doping (1012cm22) 1 150 8 2 300 83 600 84 1200 85 600 26 600 47 600 12 Reference 2 mm2 31018cm23PRB 58 1425 MANY-PARTICLE EFFECTS IN Be- d-DOPED...or the yellow 594 nm ~2.087 eV !line from a He–Ne laser. The former source excites below the band gap of the 20% AlxGa12xAs barriers ~1800 meV !, the latter above. Both the excitation and detection beams were aligned perpendicular tothe sample surface ~backscattering configuration !. Using po- larization selective excitation and detection, i.e., using left ( s2) or right ( s1) circularly polarized light, heavy- and light-hole contributions to the optical spectra of the 2D struc-tures could be separated. The excitation densities were ap- proximately 1 and 0.15 W cm 22for PL and PLE, respec- tively. The luminescence signal was dispersed by a double0.75 m Spex monochromator and detected using a cooledGaAs photomultiplier, connected to a dc electrometer. At the high-energy side of the PL spectra of all d-doped samples, the structure appears to be independent of the wellwidth and doping concentration. Also, the PLE spectra ofthese samples show a 2D structure-independent backgroundsignal, which is sensitive to the detection wavelength. Wewill first identify these structure independent features in thePL~E!spectra of the d-doped samples as being due to bulk GaAs. Then, the 2D-related PL ~E!spectra will be discussed. A. Bulk-related PL E Figure 2 displays the polarized luminescence spectra of sample 1 ~w5150 Å,p5831012cm22!, taken with the ex- citing laser at 1530 meV ~a!and 1560 meV ~b!. Both spectra exhibit the features labeled A,B, andC~at 1491, 1494, and 1513 meV, respectively !, that, from their energetic position, can be identified as donor-acceptor ( D,A), band-acceptor (e,A), and acceptor-bound exciton ( A,X) recombinations in bulk GaAs, respectively. The identification of these featuresas being related to bulk transitions is confirmed by the fol-lowing observations. First, as stated above, the lack of de-pendence on ~2D!structural parameters. If one of the fea- tures were due to an enhancement of emission intensity atthe Fermi level, or any other 2D-related transition, its posi-tion should definitely depend on the well width and dopingconcentration. Second, when exciting in the Moss-Bursteingap, i.e., at 1530 meV, see also Fig. 3, absorption in the 2D structure is forbidden due to the phase-space filling in thevalence bands. No luminescence from the structure is there-fore to be expected at this excitation energy. Third, the ab-sence of polarization when the excitation is in the Moss-Burstein gap of the 2D structure is characteristic for bulk PL.When the PL signal of the wells is superimposed on the bulklines, a polarized signal is to be expected, as is shown in Fig.2~b!. Fourth, Ferreira et al.demonstrated that for 150 Å wells with doping concentrations above p56310 11cm22, the bulk and 2D PL signals start to overlap.6 The excitation spectra of sample 1, detected either near the maximum of the bulk luminescence ~1495 meV !or be- low the bulk PL ~1480 meV !, are shown in Figs. 3 ~a!and 3~b!, respectively. Clearly, when the detection coincides with the bulk PL lines, three dominant lines are added to the ex-citation spectrum. The total disappearance of these lineswhen the detection is below the bulk PL indicates that, in- deed, the emission features A,B, andCoriginate from a different region in the sample than the remainder of the lu-minescence. If this were not the case, carriers excited at any of the features D,E,o rF, should be able to recombine at energies below feature A, i.e., in the quantum-well layers. PeaksE~1512 meV !andF~1516 meV !can now be identi- fied as the absorption peaks of the acceptor bound and freeexcitons in bulk GaAs, respectively. The origin of structure Dis not fully clear at present. From measurements with other detection energies we found that it is extended to, atleast, 1485 meV, and that it is always of the same intensity as theEpeak. It should be stressed that also the features D, E, andFare equally present in all samples. B. 2D-related PL E Polarization resolved PL and PLE spectra for samples 1, 3, and 5 are shown in Figs. 4 ~a!–4~c!. The spectra of the other samples are fully consistent with the ones shown. ThePL spectra are all taken using the Ti:sapphire laser as theexcitation source. It is worthwhile to point out that none of FIG. 2. Polarization-resolved photoluminescence spectra of sample 1, excited at 1530 meV @in the Moss-Burstein gap, panel ~a!#and at 1560 meV @above the Moss-Burstein gap, panel ~b!#. The solid and dashed lines are taken in parallel ( s11) and cross ( s12) polarization configuration, respectively. The identification of peaks A,B, andCis discussed in the text.1426 PRB 58 M. KEMERINK et al.the PL spectra showed significant changes when the wave- length of the exciting light was changed, nor when theHe-Ne laser was used for excitation, apart from an obviousscaling of the PL spectrum. This is in marked contrast withthe observations in Refs. 11–13 where the appearance of aFES in the PL spectrum of similar structures is reported,once the photon energy of the exciting laser is above theband gap of the confining barriers. In Sec. V we will comeback to this point. From a comparison of the PL spectra of Figs. 4 ~a!and 4~b!it can be seen that the main effect of the increasing well width is a redshift of the 2D-related spectrum, due to thedecreased confinement energy of the ground-state electron levelE0. As the lowest light- and heavy-hole levels ~L0 and H0!are solely confined by the notch potential of the d- doping layer, see Fig. 1, their confinement energy is not ex-pected to change significantly with increasing well width.This is reflected by the almost constant separation between theH0-E0 andL0-E0 luminescence lines, assumed to equal the separation between the positive and negative extrema inthe polarization curve ~dotted line !. For the same reason, also the width of the PL line, reflecting the separation between H0 andE Fremains constant with increasing well width. Obviously, this is not the case when, at constant well width,the carrier density is decreased, as shown in Figs. 4 ~b!and 4~c!. Here, the experiments indicate that the L0 subband is not significantly occupied, since the negative extremum in the polarization curve ( s11-s12) has disappeared, in agreement with our calculations ~we calculate EF-L0 50.6 meV !. The apparent blueshift with decreasing carrier density is due to the decreasing depth of the dpotential and the reduction of the strength of the exchange-correlation po-tentials with decreasing doping concentration, again in ac-cordance with the calculations depicted in Fig. 1. FIG. 3. Photoluminescence excitation spectra of sample 1, detected at 1480 meV ~a!and at 1495 meV ~b!. The solid and dashed lines are taken in parallel ( s11) and cross ( s12) polarization configuration, respectively. The identification of features D,E, andFis discussed in the text. The units of the vertical axis of the ~a!and~b!panels are identical. FIG. 4. PL and PLE spectra of samples 1 ~a!,3~b!, and 5 ~c!. The PLE spectra are detected below the bulk PL signal. Solid and dashed lines denote s11ands12polarizations, respectively. The dash-dotted line is the polarization curve, i.e., the difference between the polarized PL spectra.PRB 58 1427 MANY-PARTICLE EFFECTS IN Be- d-DOPED...The broad tail on the low-energy side of the PL spectra is assigned to transitions from background acceptor states tothe lowest confined conduction-band level. 17Since the en- ergy gap between the acceptor level and the E0 level is dependent on the position of the acceptor in the well, thelength of the low-energy tail is expected to correlate with thedepth of the dpotential, i.e., with the doping concentration. Our measurements indeed show a monotonic increase of thetail length with increasing doping concentration, cf. Figs. 4~b!and 4 ~c!. It is, however, extremely hard to quantify this effect, although the order of magnitude of the tail lengthcompares favorably with the calculated potential profiles inFig. 1. The PLE spectra of our structures are hardly dependent on the doping concentration, as can be seen from a comparisonof Figs. 4 ~b!and 4 ~c!. This can easily be understood by realizing that the PLE spectrum is determined by the higher,unoccupied, hole levels and the empty electron levels. Sincethese states ‘‘feel’’ relatively little of the dpotential, they are hardly affected by an increase in the doping concentration,which only changes the central region of the notch potential. Although the L0 level seems unoccupied at a doping level of 2310 12cm22,L0-E0 is not observable in the PLE spec- trum, due to either the small matrix element for L0-E0 tran- sitions ~from calculations, the step at 1517 meV is expected to be a factor 3 higher than the L0-E0 absorption !or a small Moss-Burstein shift, resulting from a slight occupation of the L0 subband. Increasing, at constant-doping concentration, the well width from 150 to 600 Å, the steps at 1545 and 1565meV @Fig. 4 ~a!#shift to lower energy, lose intensity, and seem to merge at 1518 meV @Fig. 4 ~b!#. Since these steps are due toH2-E0 andH1-E1 absorption, respectively, this be- havior can be understood from two points. First, the obviousshifts of the hole and electrons with well width, which causesa strong reduction of the separation between these transi-tions. Second, the increasing well width causes a reduction of theH1-E1 matrix element by more than a factor 5, effec- tively removing the H1-E1 absorption step from the PLE spectrum. The step at 1527 meV in Fig. 4 ~b!is assigned to H2-E2 absorption.Apart from the 150 and 300 Å samples ~nos. 1 and 2 !, all samples show a sharp peak at the absorption onset. Based onits position and its small width ~2.5 meV for sample 3 !,w e assign this to a H2-E0 exciton. In order to validate this assignment, temperature-dependent PLE measurements wereperformed, see Fig. 5. The upper inset shows the activation plot of the height 18of the excitonic PLE signal, normalized on the band-to-band PLE signal, as illustrated in the lowerinset. The latter correction is needed to account for the tem-perature dependence of nonradiative losses that affect thePLE signal strength. Clearly visible is the activated behavior, with an activation energy of 1.5 60.2 meV. Taking the large density of mobile carriers that screen the Coulomb interac-tion into account, this value seems reasonable. 19,8In Sec. V this feature will be discussed in more detail. Further confirmation of the assignment of the peak at the absorption onset to an excitonic transition of an unoccupiedsubband comes from the temperature dependence of the PLsignal that is shown in Fig. 6. Clearly visible, apart from theredshift due to the shrinkage of the band gap with tempera-ture, is the luminescence arising from one or more thermallyoccupied subbands. From our band-structure calculations we find for the H1-E F,H2-EF, andE0-E1 separations, re- spectively, 9.7, 14.1, and 3.9 meV. Comparing this to kBT 57 meV at 80 K, the assignment of this line to either H1-E1o rH2-E0 recombination is possible, since they have calculated transition energies of 1517 and 1518 meV, respec- tively. Since the matrix element for the H2-E0 transition in sample 5 is a factor 2 larger than for the H1-E1 transition, the former transition is expected to dominate. The coinci-dence of the PL maximum with the half-height point of thepeak at the PLE onset shows that both peaks share the sameorigin. IV. NUMERICAL CALCULATIONS A. Model The model used to obtain the self-consistent wave func- tions and confinement potentials of electrons and holes has FIG. 5. ~a!Main panel: The PLE onset of sample 5 in s11 polarization at 20, 40, 60, 80, and 100 K. Upper inset: Activation plot of the normalized peak height. The activation energy of theexciton unbinding is 1.5 60.2 meV. Lower inset: Schematic repre- sentation of the spectra plotted in the main panel and definition ofthe normalized peak height, used in the upper inset. FIG. 6. PL spectra and PLE onsets at 5 and 80 K for sample 5. Note the strong line in the 80 K PL spectrum associated withH2-E0 recombination, which is visible due to the thermal popula- tion of the H2 subband. The thick vertical lines are calculated PLenergies; the meaning of all other lines is the same as in Fig. 4.1428 PRB 58 M. KEMERINK et al.been described in detail in an earlier publication,20so only a brief outline will be given here. Additions that were made tothe model described in Ref. 20, and that are essential for aproper calculation of the d-potential solutions, will be dis- cussed in more detail. Hole wave functions and energies were obtained as nu- merically exact solutions of the 4 34 Luttinger Hamiltonian, including the valence-band anisotropy. The used Luttinger parameters are g156.85, g252.1, and g352.9 for GaAs and g153.45, g250.68, and g351.29 for AlAs,21the used elec- tron masses in GaAs and AlAs are 0.067 m0and 0.15m0. The parameters for Al xGa12xAs were calculated using the virtual-crystal approximation. The band gap of Al xGa12xAs was calculated as Eg(x)51519.2 11360x1220x2meV, where the common 40:60 rule was used for the distributionof the band-gap discontinuity over the valence and conduc- tion bands at the GaAs/Al xGa12xAs interfaces. The Hartree potential was calculated by solving the Poisson equation nu-merically. Only the ionized beryllium atoms and the occu-pied valence-band states are contributing to this potential, forreasons explained in Sec. II. The degree of ionization of theBe acceptors is assumed to be 100%. 22The broadening of the Be dlayer is assumed to be rectangular, with a width of 25 Å.12,23In Ref. 23 it is shown by cleaved-edge scanning tunnel microscopy that Be- d-doping layers in GaAs, grown under exactly the same circumstances as the present ones,have doping profiles with a full width at half maximum of about 10 Å, for doping concentrations up to 1 310 13cm22. We found that, within the experimental uncertainty, the cal-culated energy of all optical transitions is independent of theassumed broadening for widths up to 25 Å, which we used asa safe upper limit. In contrast to our experiences with barrier- dopedp-type single and double QW’s, it turned out that assuming parabolic valence bands in calculating the chargedistribution gives erroneous results in these center- d-doped wells. We therefore used the actual hole dispersions from theLuttinger Hamiltonian to calculate the Fermi level, and cal- culated the charge distribution by summing all ~kdependent ! hole wave functions up to the Fermi level. The density ofstates was used as weight function in the latter procedure. Inorder to expedite this part of the calculations, the axial ap-proximation was applied in determining the Fermi level. Inall other calculations the full warping was taken into ac-count. Absorption spectra are calculated as indicated in Ref.19. For the structural parameters needed in the self-consistent calculations, i.e., the well width wand acceptor concentra- tionp, the nominal values of the growth menu are taken. For the dopant concentration, the error made by this procedurecan be estimated from the characterization data of the refer-ence sample. This sample had a nominal Be concentration of 2310 14cm22permm GaAs; characterization with van de Pauw measurements showed an actual doping level of 1.97310 14cm22permm. The error in the subband calculations, caused by an error of this size in the dopant concentration, isfully negligible with respect to the experimental resolution.Deviations from the nominal value of the well width areusually a few percent. For 150 Å-wide wells this may causemeasurable deviations in the calculations; for wider wellsthis will not pose a problem. It is important to note that noadjustable fitting parameters have been used in our model.B. Exchange and correlation Inclusion of many-particle corrections in the subband cal- culations beyond the direct Coulomb interaction or Hartreeterm turned out to be essential. In recent literature, variousattempts have been made to capture the complications aris-ing from the coexistence of light and heavy holes in calcu- lations of the BGR in p-type systems. 24,25Because of the high hole density and the occupation of multiple subbands,the present samples are extremely suited as a test system forvarious hole-BGR models. In this work, calculations basedon the models proposed by Reboredo and Proetto 24and Bob- bertet al.1will be compared with calculations without ex- change and correlation corrections, and with calculationsbased on the one-component plasma model of Hedin andLundqvist. 26The model proposed by Sipahi et al.25is only applicable to homogeneous systems, and can therefore not be applied to the present samples. However, as far as k50 en- ergies are concerned, this model is similar to the one pro-posed by Reboredo and Proetto, in the sense that holes with umJu53 2and umJu51 2experience different exchange- correlation corrections. For details concerning the variousBGR models, the reader is referred to the original publica-tions. However, for the sake of self-containedness, the basicassumptions of the models by Hedin and Lundqvist, Re-boredo and Proetto, and Bobbert et al.will be briefly out- lined below. All three models apply the LDA for extending results ob- tained for a homogeneous bulk system to a quasi-2D system,by calculating an effective exchange-correlation potential V xcthat only depends on the local carrier density, i.e., Vxcp(z). The Hedin and Lundqvist model was originally derived for n-type systems. It therefore assumes that the car- rier plasma consists of one type of ~parabolic !carriers only. By applying this model directly to a hole gas, characterized by the effective heavy-hole mass mh*, one indirectly as- sumes that all holes are heavy holes. The validity of this assumption is further discussed in Sec. V. Although this as-sumption totally ignores the actual valence-band structure,favorable comparisons with experiments have been reportedfor BGR calculations that treat the valence bands as a single,parabolic band. 27–29Since the Hedin-Lundquist model is only used for comparison with more sophisticated models,there is no particular reason for choosing this model insteadof any other parametrized model available in the literature for calculating the BGR in n-type systems, 30,31apart from the fact that the Hedin-Lundquist model appears to be themost popular. The model proposed by Reboredo andProetto 24is based on an analogy with the spin-density func- tional formalism. The exchange-correlation potential is made dependent on umJu,VxcumJup3/2(z),p1/2(z). By ignoring ex- change and correlation between light holes and heavy holes, the final exchange-correlation potential becomes only depen- dent on the local density of particles with umJu53 2or1 2, VxcumJupumJu(z). The quantity pumJu(z) is defined as the den- sity in the heavy- ( umJu53 2) or light- ( umJu51 2) hole sub- bands. For the functions VxcumJua parametrized expression, de- rived for a one-component plasma, is used, where the carrier mass is chosen equal to an effective heavy- or light-hole mass, depending on umJu. To summarize, the most important feature of this model is that heavy and light holes experiencePRB 58 1429 MANY-PARTICLE EFFECTS IN Be- d-DOPED...different exchange-correlation potentials. In Sec. V A, we will discuss the inconsistency and omissions of this model.The model derived by Bobbert et al.does not assign a par- ticular character ~heavy or light !or mass to individual sub- bands, as Ref. 24, or to particular spinor components, as Ref.25, of the 2D structure. Rather, it is based on the notice that,within the LDA formalism, the quasi-2D structure locallyis treated as bulkand that therefore the bulkdispersion relations must be used to determine the localamount of heavy and light holes. Based on this idea, Bobbert et al.calculate the exchange and correlation energy exc(r) of a hole in a homo- geneous hole gas of density r, where all heavy-hole–light- hole interactions, i.e., heavy-hole–heavy-hole, light-hole–light-hole, and heavy-hole–light-hole exchange andcorrelation, are taken into account. Since the Coulomb inter-action is nondiagonal with respect to hole character, it isfundamentally impossible to identify a single hole as ‘‘light’’or ‘‘heavy.’’ The exchange-correlation energy is therefore identical for all holes in the system. Subsequently, exc(r)i s converted to the exchange-correlation potential, using Vxc(z)52(d/dr)@rexc(r)#(z). Note that also Vxc(z)i s equal for ‘‘heavy’’- and ‘‘light’’-hole subbands. Unlessstated otherwise, all calculations shown are obtained using the model of Bobbert et al.forV xc(z). To arrive at the total band-gap renormalization, also the correlation of the photogenerated electron with the sea ofholes must be taken into account. Independent of the modelused for the hole exchange-correlation potential, we use theparametrized expression of Ref. 1 for the electron-correlation potential V c(z), again using the LDA. C. Numerical results Because of the strong overlap of the electron and hole wave functions with the ionized acceptors, relaxation of the k-selection rules could easily result from localization or strong scattering. Before a meaningful comparison of experi-ments and calculations can be performed, it is therefore es- sential to determine whether or not the measured optical transitions are direct in kspace. To do so, an experimental PL and PLE spectrum will be compared with numericalsimulations, see Fig. 7. The numerical spectra ~PLE!and energies ~PL!are calculated with full kconservation. The good agreement between measured and calculated PLE spec-tra, both in position and steepness of onsets, shows that the relaxation of k-conservation selection rules indeed is negli- gible, as far as absorption is concerned. In emission this isdefinitely not the case, which is illustrated in Fig. 8, where the experimental and calculated H0-E Fseparations are plot- ted versus carrier density. Experimentally, the H0-EFsepa- ration is determined by taking the energy at the maximum inthe polarization curve as the transition energy associated withH0(k50) toE0(k50) recombination, 32and the half- maximum point at the high-energy side of the emission spec- trum as due to H0(k5kF)o rL0(k5kF)t oE0(k50) re- combination. The very favorable comparison with calculated FIG. 7. Comparison between experimental and simulated PLE spectra of sample 2. The experimental PL spectrum can be com-pared with calculated transition energies for recombination at k 50~thick vertical lines !. Solid and dashed lines denote s11and s12polarizations, respectively. The dash-dotted line is the differ- ence between the polarized PL spectra. The thick, smooth lines aresimulations, the thinner lines are experimental curves. The arrowsidentify the origin of features in the simulated PLE spectra. FIG. 8. Experimentally and numerically determined separation between the ground-state heavy-hole energy at k50 and the Fermi energy versus doping concentration for w5600 Å. The large ~small !squares denote experimental ~numerical !points; the thin line connects the numerical points. FIG. 9. Dispersion relations of the lowest subbands of sample 2. The solid and dashed lines denote heavy- and light-hole dispersions,respectively. The dash-dotted line indicates the position of theFermi level E F.1430 PRB 58 M. KEMERINK et al.values shows that this assignment is correct, and that transi- tions in emission can either be direct or indirect in k. How- ever, the maximum emission intensity still seems to arisefrom direct transitions, and positive and negative extrema inthe polarization curve will in the following be assumed to indicateH0(k50) toE0(k50) andL0(k50) toE0(k 50) transitions, respectively. It is worthwhile to point out that the onset of absorption does not correspond to H0(k 5k F)o rL0(k5kF)t oE0(k5kF) transitions, as in most modulation-doped heterostructures, but to H1-E1o rH2-E0 transitions at the zone center. This is due to the large holedensities in the present samples, which cause extremely largeMoss-Burstein shifts. As an example, the Moss-Burstein shifts for sample 3 are 56 meV for the L0-E0 transition, and more than 250 meV for the H0-E0 transition. As a typical example, Fig. 9 shows the dispersion relation for the same sample as shown in Fig. 7 ~no. 2!, together with the calculated Fermi energy E F. The well-known nonpara- bolicity of the valence-band states is apparently visible.However, extreme nonparabolicities as negative masses, which are generally found for the L0 band in heterostruc- tures based on the Al xGa12xAs system,19are absent. Never- theless, the remaining nonparabolicities still strongly affectthe optical spectra, which is manifested mainly in the nonstep-like behavior of the absorption spectra, see Fig. 7. The summary of our experimental and numerical data is shown in Figs. 10 and 11, where measured and calculatedtransition energies are plotted versus the well width ~Fig. 10 ! and the dopant concentration ~Fig. 11 !. Experimental ener- gies were determined at half height for steplike onsets, and atpeak maximum for sharply peaked onsets. No correction forexciton binding energy has been applied, since the involvedbinding energies are relatively uncertain and small comparedto the overall experimental error, see also the discussion inSec. III B. Comparing the measured energies with the onescalculated without exchange and correlation corrections di-rectly shows the need for these corrections, in contrast withthe claim in Ref. 12. Furthermore, it shows that in these d-doping layers the BGR cannot be accounted for by a rigid shift of all valence bands, like in, for instance, single quan-tum wells. Clear differences can also be found between theresults obtained with the BGR models by Bobbert et al.and by Reboredo and Proetto, which both aim to account for the coexistence of light and heavy holes. These differences aremost pronounced in the light-hole subbands, particularly atlow doping concentrations and narrow well widths, cf. the FIG. 10. Comparison of measured transition energies with calculated values, using various models for the hole exchange and correlation interactions discussed in the text, versus the width of the confining well. The large symbols denote experimental points, the small symbolsdenote calculations. The lines connect the calculated energies. The experimental error is usually less than 2 meV. The meaning of thesymbols is as follows: solid squares, H0-E0; solid circles, L0-E0; open squares, H2-E0; open circles, H1-E1; open up triangle, L1-E1; open down triangle, H2-E2.PRB 58 1431 MANY-PARTICLE EFFECTS IN Be- d-DOPED...upper-right and lower-left panels of Figs. 10 and 11. Both the energies of the L0-E0 andL1-E1 transitions calculated with the Reboredo-Proetto model are far more than the ex-perimental error of about 2 meV above the experimentallyfound values. Moreover, the Reboredo-Proetto model incor-rectly predicts one occupied subband for a hole density of p54310 12cm22. A surprisingly good correspondence with experimental data is found when the one-particle model ofHedin and Lundqvist is used in the calculations. Comparingwith the model by Bobbert et al.with ‘‘fitting to the experi- mental data’’ as criterion for success, the Bobbert modelseems to prevail for all samples, except those with the lowestdensity and most narrow well width, for which the Hedin-Lundqvist model seems to prevail. The differences are, how-ever, quite small. On physical grounds, the success of the model by Hedin and Lundqvist for p-type systems is acci- dental and based on a cancellation of errors, 1to which we will come back in the next section. V. DISCUSSION A. Exchange and correlation In the previous section, we have shown that the BGR model by Reboredo and Proetto systematically underesti-mates the renormalization of the light-hole-related transi-tions. These deviations are directly due to the fact that thismodel uses different exchange-correlation potentials forheavy and light holes, and that these potentials are only a function of the local density in the heavy- orlight-hole sub- bands. The analogy with the spin-density functional formal-ism, on which this model is based, is tempting but invalid.The local ‘‘heavy’’- and ‘‘light’’-hole densities, obtained byReboredo and Proetto from the envelope functions of the umJu53 2andumJu51 2spinor components, are easily shown to be dependent on the direction of the quantization axis,which, of course, should not be the case. Furthermore, the exchange and correlation interactions between ‘‘light’’ and‘‘heavy’’ holes are, by definition, ignored in this model.Even in the hypothetical case of an infinitesimally small light-hole mass, when no states with umJu51 2are occupied, these interactions cannot be ignored due to the nondiagonalcharacter of the Coulomb interaction with respect to the holecharacter. 1 The surprisingly good correspondence between experi- mentally determined energies and those calculated with theHedin-Lundquist model for exchange and correlation is, asstated above, due to a cancellation of errors. To be moreprecise, the implicit assumption that all holes are ‘‘heavy,’’made by applying an ‘‘electron gas’’ model to a hole gas,leads to an overestimation of the exchange energy and anunderestimation of the correlation energy. The qualitativereason why the exchange energy for the hole gas is smallerin magnitude than that for the electron gas is the fact that,besides the spatial degrees of freedom, there are four instead FIG. 11. As Fig. 10, but as a function of the total dopant concentration.1432 PRB 58 M. KEMERINK et al.of two internal degrees of freedom ~mJ563 2,61 2for a hole gas, and only Jm561 2for an electron gas !. Consequently, it is easier to fulfill the Pauli exclusion principle, which re-duces the exchange interaction. The underestimation of thehole-correlation energy, when applying the Hedin-Lundquistmodel to a hole gas is also related to the number of internaldegrees of freedom. Due to the coexistence of light andheavy holes, the number of possible excitations at the Fermilevel is increased, leading to a higher dielectric constant for ahole gas than for an electron gas of the same density. A highdielectric constant means that the system reacts efficiently ona perturbation ~strong screening !, which implies a strong cor- relation with the perturbation. In this specific case the per-turbation is just the Coulomb potential of any hole in thesystem. Therefore, the correlation energy of a system withfour internal degrees of freedom is higher than that of asystem with a lower degree of freedom. The only method of defining local heavy- and light-hole fractions that is consistent with the LDA formalism is em-ployed in the model by Bobbert et al.It states that, since LDA treats the quasi-2D charge distribution locally as a bulkdensity, also the bulk dispersion relations have to be used indetermining the local heavy- and light-hole fractions. Thesuccess of the model of Bobbert et al.for the present samples therefore, mainly shows the validity of the local-density approximation in the calculation of the effects ofhole exchange and correlation in quasi-2D systems of highdegeneracy. B. Exciton screening The observation of strong excitonic features in the present, highly degenerate, samples may appear surprising atfirst sight. Although it is well known that the efficiency ofCoulomb screening in 2D systems is strongly reduced ascompared to that in 3D systems, straightforward extrapola-tion of available theoretical results to our samples suggeststhat only an infinitesimally small binding energy shouldremain. 33,34However, as was pointed out in a few earlier publications,35,8,9standard 2D screening theory ignores the differences in probability distributions of various subbandsalong the growth direction. To be more specific, the rela- tively strong confinement of the occupied ground states ~H0 andL0!, leads to a poor screening of excitons formed by more extended states like H2 andE0. 35,8,9Furthermore, it has been suggested6that the presence of ionized impurities inside the quantum well makes the screening of excitons lessefficient, which will facilitate the survival of excitons up tothe current doping concentrations. The overlap argument isin full agreement with the observation that no significantexcitonic absorption enhancement is visible for the 150- and 300 Å wells. Due to the confinement by the Al 0.20Ga0.80As barriers, the extension of the H2 andE0 wave functions along the growth direction is not much larger than that of the H0 andL0 wave functions, causing a relatively efficient screening of the H2-E0 exciton. There are two points that we would like to stress concern- ing the screening of excitons in these samples. The first isthat it has been shown that peaked structures in absorptionspectra are generally spoken an unreliable indicator for thepresence or absence of excitons. 19,8See, e.g., the peaks at1540 and 1565 meV in the PLE spectra of the 300- and 150 Å wells, respectively @Figs. 7 and 4 ~a!#, which can fully be accounted for by the valence-band structure only.36The ab- sence of a peak in the absorption spectrum, on the otherhand, does not imply the total bleaching of the exciton. 8The second point, which is strongly related to this, is that we do not claim that, apart from the H2-E0 exciton, all excitons are unbound. The apparent dominance of the H2-E0 exciton in the spectra of wide wells is due to the subtle interplaybetween the exciton binding strength and the optical matrix element. The latter is extremely large for the H2-E0 transi- tion in wide wells, as can directly be concluded from a com- parison of the strengths of the H0-E0 andH2-E0 PL lines at 80 K, see Fig 6. As an example, the H1-E1 subbands are also expected to form an exciton of significant binding en- ergy for wide wells ( w>600 Å), but the optical matrix ele- ment ofH1-E1 transitions is at least a factor two smaller than that of the H2-E0 exciton for these well widths, which prevents the exciton from being identifiable in PLE. C. The absence of a Fermi-edge singularity In contrast to what is reported by Wagner and co-workers11–13for samples very similar to the ones dis- cussed here, we do not find any indication for a Fermi-edgesingularity in the PL spectra of our samples. It has beenshown that such a FES can arise from either strong localiza-tion of the minority carriers 7,15,37–40or a near-resonance con- dition between states at the Fermi level and those of a nearbyexcitonic level. 3,4,41–43Due to the small electron mass in GaAs, the former condition is not very likely to be fulfilled,as was also noticed in Ref. 12. Furthermore, the small effec-tive mass of the unlocalized electron will inhibit the obser-vation of any FES-like features in emission spectra, due tothe large recoil of the scattered electron. 4,14,15The latter con- dition requires an excitonic level that is almost at resonance with the Fermi level. For all samples but the one with p 5231012cm22the separation between EFand the lowest occupied hole level is at least 13 meV, which is too far for causing any significant coupling in n-type systems.41–44For p-type systems, this is even more unlikely, which can be understood from a consideration of the lifetime broadeningof the minority carriers g. Apart from the separation between the Fermi level and the lowest unoccupied state gis an es- sential parameter determining the strength of the FES.43,4In first order gis proportional to the effective mass, leading to a smaller value for minority electrons than for minorityholes, which, in turn, leads to a decrease of the FES intensity inp-type systems. 43Moreover, it was shown by Rodriguez and Tejedor4that no such coupling at all can occur in sym- metric potentials. Summarizing the above, the absence of aFES in the PL spectra of our samples appears to be in goodagreement with most earlier work on the FES in emissionspectra. There are two significant differences between the present samples and the ones used by Wagner and co-workers. The first is the density range, which runs from 3 310 12to 4 31013cm22in Refs. 12 and 13. However, the separations betweenEFand the lowest unoccupied subband reported for these samples are of the same size as in our samples. Thesecond, and probably most significant, difference concernsPRB 58 1433 MANY-PARTICLE EFFECTS IN Be- d-DOPED...the fact that Wagner and co-workers employ samples with a single d-doped well. As was pointed out by Wagner et al.,13 this might easily lead to a breaking of the symmetry of the confining potential, which can strongly enhance the forma-tion of a FES. 4Finally, we should stress that exciting above the band gap of the confining barriers did not, in any sense,change our spectra, in contrast to what is reported in Ref. 11. It is interesting to mention the observation of a FES in a highly disordered system of Be- d-doped bulk GaAs by Fritze et al.7In these samples, the extremely high Be coverage in thedlayer ~up to 0.35 ML or p52.131014cm22!caused the formation of Be clusters. The resulting high disorder leads tothe localization of the minority carriers ~electrons !, despite their low effective mass. The Be coverages used in oursamples, maximally 0.02 ML, are far below the density atwhich the so-called surface phase transition occurs, and Beclusters start to form. VI. CONCLUSIONS Summarizing, we have studied many-body interactions in Be-d-doped quantum wells, by means of a careful compari- son between PL ~E!experiments and self-consistent calcula- tions. Different LDA models for the exchange and correla-tion potentials of an interacting hole gas have been comparedwith experiments. It is found that the model that was recentlyderived by Bobbert et al. 1consistently describes the experi-mental band-gap renormalization, both for empty and filled subbands. Furthermore, our results indicate that ‘‘heavy’’-and ‘‘light’’-hole subbands experience the same exchange-correlation potential, which is consistent with the assump-tions of the LDA formalism. For well widths above 600 Å asharp peak dominates the absorption spectra of our samples, which is attributed to a H2-E0 exciton. The dominance of this exciton is explained in terms of a strong optical matrixelement and a reduced screening efficiency of higher sub-bands, due to a small overlap with the screening particles. Incontrast to earlier work on similar samples, no indication fora Fermi-edge singularity in PL was found, which is dis-cussed and understood in the framework of other theoreticaland experimental work on this singularity. During the preparation of this manuscript, a paper was published by Enderlein et al., 45in which they theoretically discuss exchange and correlation interactions in degenerate p-type heterostructures. In this paper, a matrix expression on the Luttinger-Kohn basis is constructed for describing theexchange and correlation interactions. In spirit, this methoddiffers significantly from the model proposed by Bobbertet al. 1Moreover, one can show that the partial local heavy- and light-hole densities, as defined in this paper, do not addup to the total local-hole density. Therefore we stick to ourconclusion that the model by Bobbert et al.is the only model that, within the LDA, consistently describes the effects ofexchange and correlation. 1P. A. Bobbert, H. Wieldraaijer, R. van der Weide, M. Kemerink, P. M. Koenraad, and J. H. Wolter, Phys. Rev. B 56, 3664 ~1997!. 2S. A. Brown, J. F. Young, Z. Wasilewski, and P. T. Coleridge, Phys. Rev. B 56, 3937 ~1997!. 3S. J. Xu, S. J. Chua, X. H. Tang, and X. H. Zhang, Phys. Rev. B 54, 17 701 ~1996!. 4F. J. Rodriguez and C. Tejedor, J. Phys.: Condens. Matter 8, 1713 ~1996!. 5I. A. Buyanova, W. M. Chen, A. Henry, W.-X. Ni, G. V. Hans- son, and B. Monemar, Phys. Rev. B 53, 1701 ~1996!. 6A. C. Ferreira, P. O. Holz, B. E. Sernelius, I. Buyanova, B. Mon- emar, O. Mauritz, U. Ekenberg, M. Sundaram, K. Campman, J. L. Merz, and A. C. Gossard, Phys. Rev. B 54, 16 989 ~1996!. 7M. Fritze, A. Kastalsky, J. E. Cunningham, W. H. Know, R. N. Pathak, and I. E. Perakis, Solid State Commun. 100, 497 ~1996!. 8M. Kemerink, P. M. Koenraad, P. C. M. Christianen, R. van Schaijk, J. C. Maan, and J. H. Wolter, Phys. Rev. B 56, 4853 ~1997!. 9M. Kemerink, P. M. Koenraad, A. Parlangeli, P. C. M. Chris- tianen, R. van Schaijk, J. C. Maan, and J. H. Wolter, Phys.Status Solidi A 164,7 3~1997!. 10M. Kemerink, P. M. Koenraad, and J. H. Wolter, Phys. Rev. B 57, 6629 ~1998!. 11J. Wagner, A. Ruiz, and K. Ploog, Phys. Rev. B 43, 12 134 ~1991!. 12D. Richards, J. Wagner, H. Schneider, G. Hendorfer, M. Maier, A. Fischer, and K. Ploog, Phys. Rev. B 47, 9629 ~1993!. 13J. Wagner, D. Richards, H. Schneider, A. Fischer, and K. Ploog, Solid-State Electron. 37, 871 ~1994!.14P. Hawrylak, Phys. Rev. B 44, 3821 ~1991!. 15A. E. Ruckenstein and S. Schmitt-Rink, Phys. Rev. B 35, 7551 ~1987!. 16Using the formula for the electrostatic potential of a charged plate V5ped/2e0e1, withdthe separation between the topmost d layer and the surface ~600 Å !, we find that 1.8 31012cm22holes have to be transferred to surface states to overcome the mid-gappinning of the Fermi level at the surface. So, even at the lowestdoping concentration, the doping in the first dlayer is enough to accomplish this. 17M.-L. Ke, L. S. Rimmer, B. Hamilton, J. H. Evans, M. Missous, K. E. Singer, and P. Zalm, Phys. Rev. B 45, 14 114 ~1992!. 18In fact, one should measure the integrated excitonic intensity. However, a large degree of arbitrariness exists in separating theexcitonic and free-particle contributions to the total PLE spec-trum, which makes a quantitative analysis along this line impos-sible. The followed method does not suffer from this problem. 19M. Kemerink, P. M. Koenraad, and J. H. Wolter, Phys. Rev. B 54, 10 644 ~1996!. 20M. Kemerink, P. M. Koenraad, P. C. M. Christianen, A. K. Geim, J. C. Maan, J. H. Wolter, and M. Henini, Phys. Rev. B 53, 10 000 ~1996!. 21Physics of Group IV Elements and III-V Compounds , edited by O. Madelung, M. Schultz, and H. Weiss, Landolt-Bo ¨rnstein, New Series, Group III, Vol. 17, Pt. a ~Springer-Verlag, Berlin, 1982 !; ibid., Vol. 22, Pt. a ~Springer-Verlag, Berlin, 1987 !. 22E. F. Schubert, J. M. Kuo, R. F. Kopf, H. S. Luftman, L. C. Hopkins, and N. J. Sauer, J. Appl. Phys. 67, 1969 ~1990!. 23P. M. Koenraad, M. B. Johnson, H. W. M. Salemink, W. C. van der Vleuten, and J. H. Wolter, Mater. Sci. Eng., B 35, 485 ~1995!.1434 PRB 58 M. KEMERINK et al.24F. A. Reboredo and C. R. Proetto, Phys. Rev. B 47, 4655 ~1993!. 25G. M. Sipahi, R. Enderlein, L. M. R. Scolfaro, and J. R. Leite, Phys. Rev. B 53, 9930 ~1996!. 26L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 ~1971!. 27G. Tra¨nkle, H. Leier, A. Forchel, H. Haug, C. Ell, and G. We- imann, Phys. Rev. Lett. 58, 419 ~1987!. 28S. Haacke, R. Zimmermann, D. Bimberg, H. Kal, D. E. Mars, and J. N. Miller, Phys. Rev. B 45, 1736 ~1992!. 29S. Das Sarma, R. Jalabert, and S.-R. E. Yang, Phys. Rev. B 41, 8288 ~1990!. 30O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 ~1976!. 31J. P. Pedrew and A. Zunger, Phys. Rev. B 23, 5048 ~1982!. 32This is always a point of discussion. However, to get rid of the unpolarized bulk signal one has to look at the polarization of thePL signal. Under the reasonable assumptions that the electrondistribution is ~almost !thermalized and that k-conserving tran- sitions are more efficient than non- k-conserving ones, the ex- trema in the polarization will correspond to transitions at k50, independent of a slight ~in!homogeneous broadening. 33G. D. Sanders and Y.-C. Chang, Phys. Rev. B 35, 1300 ~1987!. 34D. A. Kleinman, Phys. Rev. B 32, 3766 ~1985!. 35A. B. Henriques, Phys. Rev. B 44, 3340 ~1991!. 36Peaked structures can arise in absorption spectra due to an anti-crossing in the corresponding valence-band state. E.g., the L1-E1 transition in Fig. 7 is allowed for small k.A tk510 3106m~Fig. 9 !L1 experiences an anticrossing with a higher subband, which changes the character of the L1 subband, and makes the L1-E1 transition parity forbidden, resulting in a sud- den drop of the PLE signal. 37Y. H. Zhang, N. N. Ledentsov, and K. Ploog, Phys. Rev. B 44, 1399 ~1991!. 38M. S. Skolnick, J. M. Rorison, K. J. Nash, D. J. Mowbray, P. R. Tapster, S. J. Bass, and A. D. Pitt, Phys. Rev. Lett. 58, 2130 ~1987!. 39G. Livescu, D. A. B. Miller, D. S. Chemla, M. Ramaswamy, T. Y. Chang, N. Sauer, A. C. Gossard, and J. H. English, IEEE J.Quantum Electron. QE-24, 1677 ~1988!. 40S. Schmitt-Rink, C. Ell, and H. Haug, Phys. Rev. B 33, 1183 ~1986!. 41W. Chen, M. Fritze, W. Walecki, A. V. Nurmikko, D. Ackley, J. M. Hong, and L. L. Chang, Phys. Rev. B 45, 8464 ~1992!. 42M. Fritze, W. Chen, A. V. Nurmikko, J. Jo, M. Santos, and Shayegan, Phys. Rev. B 45, 8408 ~1992!. 43J. F. Mueller, Phys. Rev. B 42, 11 189 ~1990!. 44P. Hawrylak, Phys. Rev. B 44, 6262 ~1991!. 45R. Enderlein, G. M. Sipahi, L. M. R. Scolfaro, and J. R. Leite, Phys. Rev. Lett. 79, 3712 ~1997!.PRB 58 1435 MANY-PARTICLE EFFECTS IN Be- d-DOPED...

PhysRevB.56.13503.pdf

Ab initio study of the anomalies in the He-atom-scattering spectra of H/Mo 110and H/W 110 Bernd Kohler, Paolo Ruggerone, and Matthias Scheffler Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14 195 Berlin-Dahlem, Germany ~Received 29 July 1996 ! Helium-atom-scattering ~HAS!studies of the H-covered Mo ~110!and W ~110!surfaces reveal a twofold anomaly in the respective dispersion curves. In order to explain this unusual behavior we performed density-functional theory calculations of the atomic and electronic structure, the vibrational properties, and the spec-trum of electron-hole excitations of those surfaces. Our work provides evidence for hydrogen-adsorptioninduced Fermi-surface nesting. The respective nesting vectors are in excellent agreement with the HAS dataand recent angle resolved photoemission experiments of the H-covered alloy system Mo 0.95Re0.05~110!. Also, we investigated the electron-phonon coupling and discovered that the Rayleigh phonon frequency is loweredfor those critical wave vectors. Moreover, the smaller indentation in the HAS spectra can be clearly identifiedas a Kohn anomaly. Based on our results for the susceptibility and the recently improved understanding of theHe-scattering mechanism we argue that the larger anomalous dip is due to a direct interaction of the He atomswith electron-hole excitations at the Fermi level. @S0163-1829 ~97!00343-3 # I. INTRODUCTION The interest in the ~110!surfaces of Mo and W has been fostered in recent years by the discovery of deep and ex-tremely sharp indentations in the energy loss spectra of He-atom scattering at H/Mo ~110!and H/W ~110!. 2,3As depicted in Fig. 1 those anomalies are seen at an incommensurate wave vector, Q¯ expc,1, along the @001#direction ( GH) and ad- ditionally at the commensurate wave vector Q¯ expc,25S¯at the boundary of the surface Brillouin zone ~SBZ!. At those points two simultaneous anomalies develop out of the ordi- nary Rayleigh mode. One, v1, is extremely deep, and is only seen by helium-atom scattering ~HAS!.2,4–6,3The other, v2, is instead modest, and is observed by both HAS and highresolution electron energy loss spectroscopy ~HREELS !. 1,7,8 Since the @001#components of both critical wave vectors Q¯c,1andQ¯c,2are approximately the same, it was suggested that the anomaly runs parallel to GNthrough the SBZ as illustrated in Fig. 2. Various models have been brought forward in order to account for the unusual vibrational properties of H-coveredMo~110!and W ~110!: For instance, a pronounced but less sharp softening of surface phonons is also seen on the ~001! surfaces of W ~Ref. 10 !and Mo. 11There, the effect is caused by marked nesting properties of the Fermi surface.12–15How- ever, angular resolved photoemission ~ARP!experiments of H/W~110!and H/Mo ~110!gave no evidence for nesting vec- tors comparable to the HAS determined critical wave vectors~Refs. 9, 16, and 17 !~see Fig. 3 !. Thus, for the phonon anomalies of those two systems there seems to be no con-nection between the electronic structure and the vibrationalproperties. Moreover, even if such a relation existed it wouldbe unclear whether the phenomena has to be interpreted as aKohn anomaly or as the phason and amplitudon modes as-sociated with the occurance of a charge-density wave. In fact, the experimental finding of two modes at S ¯as well as some theoretical arguments rule out the phason-amplitudonidea. 18Furthermore, any model which links the phononanomalies to the motion of the hydrogen atoms7has to be ruled out because the HAS spectra remain practically un-changed when deuterium is adsorbed instead of hydrogen. 2,3 Still unexplained, but probably not directly correlated to the anomalies, is the peculiar behavior of hydrogen vibra- tions observed in HREELS experiments by Balden et al.7,8 At a H coverage of one monolayer the adsorbate modes par- allel to the surface form a continuum interpreted as a liquid-like phase of the adsorbate. An additional puzzle to the already confusing picture is added by the observation of a symmetry loss in the low-energy electron diffraction ~LEED !pattern of W ~110!upon H adsorption. The phenomena was believed to be caused bya H-induced displacement of the top layer W atoms along the@1 ¯10#direction19@see Figs. 4 ~b!and 4 ~c!#. For H/Mo ~110! similar studies do not provide any evidence for a top-layer-shift reconstruction. 20 The goal of our work is to give a comprehensive expla- nation for the observed anomalous behavior and clarify theconfusing picture drawn by the different experimental find-ings. We show that the anomalous behavior observed in theHe and electron loss experiments is indeed governed by H-induced nesting features in the Fermi surface. The remainder of the paper is organized as follows. First, Sec. II gives a brief introduction into the ab initio method used. The results of our work are presented in Sec. III: InSec. III A we study the atomic geometry of the clean andH-covered Mo ~110!and W ~110!surfaces. In particular, we check whether the hydrogen adsorption induces a top-layer-shift reconstruction. Following in Sec. III B, we focus on theelectronic structure of those surface systems. In this context,we are particularly interested in how the Fermi surfacechanges upon H adsorption: Is there evidence for Fermi-surface nesting? If yes, how does it connect to the ARPmeasurements of the Fermi surfaces and the critical wavevectors detected by HAS and HREELS? Along the same linewithin the model of a Kohn anomaly, Sec. III C digs into thecoupling between electronic surface states and surfacephonons. There, we discuss the mechanism which causes thePHYSICAL REVIEW B 15 NOVEMBER 1997-II VOLUME 56, NUMBER 20 56 0163-1829/97/56 ~20!/13503 ~16!/$10.00 13 503 © 1997 The American Physical Societysmall anomaly v1. In Sec. III D we examine the He- scattering mechanism, study the spectrum of electron-holeexcitations, and explain the physical nature of the huge anomalous branch v2. Finally, the major aspects of our work are summarized in Sec. IV. The Appendix presentsseveral detailed test calculations giving an insight into theproperties and the accuracy of our method.II. METHOD All our calculations employ the density-functional theory ~DFT!. They are performed within the framework of the local-density approximation ~LDA!for the exchange- correlation energy functional.21,22For the self-consistent so- lution of the Kohn-Sham ~KS!equations we employ a full- FIG. 2. Position of the HAS anomalies for the H/W ~110!adsor- bate system within the SBZ. The dotted line indicates the form of the anomaly apart from the symmetry directions GHandGSas suggested by Hulpke and Lu ¨decke ~Ref. 4 !. FIG. 3. ARP-measured Fermi surface of the H-covered W ~110! surface ~Ref. 9 !. The shaded areas are the ~110!projection of the bulk Fermi surface. Surface states at the Fermi level are marked bydots and repeated in the other equivalent sections by crosses. The arrows represent the HAS critical wave vectors Q¯ expc,1andQ¯ expc,2. FIG. 1. HREELS spectra of the clean W ~110! surface ~upper panel !and its H-covered (1 31) phase ~lower panel !from Ref. 1: Shown are the dispersion of the Rayleigh wave ~triangles !and the longitudinally polarized surface phonons ofthe first ~circles !and second ~squares !layer. The dots represent the results of Hulpke and Lu ¨- decke’s HAS measurements ~Refs. 2 and 3 !.13 504 56 KOHLER, RUGGERONE, AND SCHEFFLERpotential linearized augmented plane-wave ~FP-LAPW ! code23–25which we enhanced by the direct calculation of atomic forces.26,27Combined with damped Newton dynamics28this enables an efficient determination of fully relaxed atomic geometries. Also, it allows the fast evaluationof phonon frequencies within a frozen-phonon approach. A. LAPW formalism There exists a variety of different LAPW pro- grams.24,29–38We therefore briefly specify the main features of our code. This is also in order to clarify the meaning ofthe LAPW parameters which determine the numerical accu-racy of the calculations.In the augmented plane-wave method space is divided into the interstitial region ~IR!and nonoverlapping muffin-tin ~MT!spheres centered at the atomic sites. 39–41This division accounts for the atomiclike character of the wave functions,the potential, and the electron density close to the nuclei andthat the behavior of these quantities is smoother in between the atoms. The basis functions fK(r) which we employ in our code for the expansion of the electron wave functions ofthe KS equation ck,i~r!5( uKu<KwfCi~K!fK~r! ~1! are defined as fK~r!5HV21/2exp~iKr!,rPIR ( lm l<lwf@almI~K!ulI~elI,rI!1blmI~K!u˙lI~elI,rI!#Ylm~rˆI!,rI<sI.~2! Here,K5k1Gdenotes the sum of a reciprocal lattice vector Gand a vector kwithin the first Brillouin zone. The wave function cutoff Kwflimits the number of those Kvectors and thus the size of the basis set. The other symbols in Eq. ~2! have the following meaning: Vis the unit cell volume, sI specifies the MT radius, and rI5r2RIrepresents a vector within the MT sphere of the Ith atom. Note that Ylm(rˆ)i sa complex spherical harmonic with Yl2m(rˆ)5(21)mYlm*(rˆ). The radial functions ul(el,r) andu˙l(el,r) are solutions of the equations Hsphul~el,r!Ylm~rˆ!5elul~el,r!Ylm~rˆ!, ~3! Hsphu˙l~el,r!Ylm~rˆ!5@elu˙l~el,r!1ul~el,r!#Ylm~rˆ!,~4! which are regular at the origin. Here, the KS operator Hsph contains only the spherical average of the effective potential within the respective MT. The expansion energies elare cho- sen somewhere within the respective energy bands with l character, whereas the coefficients almI(K) andblmI(K) are fixed by requiring that value and slope of the basis functions are continuous at the surface of the Ith MT sphere. Obvi- ously, this can only be fulfilled for l<lwf, but aslwfis typi- cally high ~e.g.,lwf>8!this does not represent a problem. The representation of the potentials and densities re- sembles the one employed for the wave functions, i.e., V~r!55( uGu<GpotVGexp~iGr!,rPIR ( lm l<lpotVlm,I~rI!Ylm~rˆI!,rI<sI.~5! Thus, no shape approximation is introduced. The quality of this full-potential description is controlled by the cutoff pa-rameterGpotfor the reciprocal lattice vectors Gand the maximum angular momentum lpotof the (l,m) representa- tion inside the MT’s. B. Slab systems and parametrization The substrate surfaces are modeled by five-, seven-, and nine-layer slabs repeated periodically and separated bya vacuum region whose thickness is equivalent to foursubstrate layers. The MT radii for the W and Mo atoms are chosen to be s Mo5sW51.27 Å. Note that the experimen- tal interatomic distances of bulk Mo and W are 2.73 Å and2.74 Å, respectively. For hydrogen the MT radius is set to s H50.48 Å. In the case of the W surfaces the valence and semicore electrons are treated scalar-relativistically while the coreelectrons are handled fully relativistically. For Mo all calcu-lations are done nonrelativistically. In both metals we intro-duce a second energy window for the treatment of the semi- core electrons ~4sand 4pfor Mo, 5 sand 5pfor W !. The in-plane lattice constants ~compare Fig. 4 !a Wtheo53.14 Å and aMotheo53.13 Å are calculated to be without including zero- point vibrations. They are in good agreement with the re- spective measured bulk lattice parameters at room tempera-ture and other theoretical results ~see Appendix A1 !. For the potential the ( l,m) representation within each MT sphere is taken up to l pot54 while the kinetic-energy cutoff for the interstitial region is set to ( \2/2m)(Gwf)25100 Ry. Generally, we choose the plane-wave cutoff for the wave functions to be ( \2/2m)(Kwf)2512 Ry; only for the frozen- phonon calculations we use a slightly smaller cutoff of (\2/2m)(Kwf)2510 Ry which is indeed sufficient. The (l,m) representation of the wave functions within the MT’s is taken up to lwf58. Also, we employ Fermi smearing with a broadening of kBTel568 meV in order to stabilize the self- consistency and the kisummation. All energies given below are however extrapolations to zero temperature Tel.42,43We56 13 505 AB INITIO STUDY OF THE ANOMALIES IN THE He -...performed systematic tests comparing the LDA and the gen- eralized gradient approximation44,45exchange-correlation XC functionals. For the quantities reported in this paper ~to- tal energy differences of different adsorption sites, bondlengths, etc. !both treatments give practically the same re- sults. All ksets used were determined by using the special k-point scheme of Monkhorst and Pack. 46Also, we checkedthekipoint convergence: In the case of atomic and elec- tronic structure calculations a two-dimensional uniform mesh of 64 kipoints within the (1 31) SBZ gives stable results. For the evaluation of frozen-phonon energies a set of 56kipoints is employed within the SBZ of the enlarged (132) and (2 31) surface cells ~see the Appendix, Sec. 3b!. In Table I we summarize the main features of our LAPW parameter setting. For more details the reader is re-ferred to the discussion in the Appendix. III. RESULTS A. Atomic structure Using the directly calculated forces in combination with damped Newton dynamics of the nuclei the atomic structureof stable and metastable geometries is obtained quite auto-matically. It is important that the starting configuration for anoptimization run is of low ~or no !symmetry. Otherwise the system may not relax into a low-symmetry structure of pos-sibly lower energy. Also, independent of the optimizationmethod used, there is always a chance that the minimizationof the total energy leads into a local minimum instead of the global one. In order to reduce and hopefully abolish this riskone has to conduct several optimizations with strongly vary-ing starting configurations. We consider a system to be in astable ~or metastable !geometry if all force components are smaller than 20 meV/Å. The error in the structure parameters of the relaxed system is then 60.02 Å '61.0%d 0for the W systems and 60.01 Å '60.5%d0for the Mo surfaces ~see the Appendix, Sec. 2 !. The relaxation parameters calculated for the clean and H-covered ~110!surfaces of Mo and W are presented in Table II @Fig. 4 ~d!serves to clarify the meaning of these parameters #. The converged nine-layer-slab results are re- markably similar for both transition metals. Moreover, theyshow excellent quantitative agreement with the results of arecent LEED analysis also presented in Table II. For a de-tailed comparision between experiment and theory we referto Ref. 47. As the energetically most favorable hydrogen-adsorption site on both W ~110!and Mo ~110!we identify a quasithreefold position @indicated as ‘‘H’’ in Fig. 4 ~a!#. The FIG. 4. ~a!Adsorbate positions within the (1 31) unit cell of the W ~110!and Mo ~110!surfaces: Shown are the long-bridge ( L), short-bridge ( S), hollow ( H), and on-top ( T) site. ~b!and~c! Structure of the H-covered W ~110!surface as suggested by Chung et al. ~Ref. 19 !:~b!low H-coverage Q,0.5 ML; ~c!top-layer-shift reconstruction for a H-coverage Q.0.5 ML. The white ~shaded ! circles represent the W atoms in the surface ~subsurface !layer. The hydrogen positions are indicated by full dots. ~d!Illustration of the structure parameters used in Table II. The in-plane lattice constantis denoted by a 0.TABLE I. LAPW parameter setting used in Sec. III for the bulk studies, the calculation of the atomic and electronic surface struc- tures, and the evaluation of frozen phonons. LAPW Parameter BulkSurface Structure Phonons lpot44 3 \2 2m\2 2m~Gpot!2~Ry! 64 100 100 lwf88 8 \2 2m~Kwf!2~Ry! 12 12 10 Fermi smearing ~meV! 68 68 68 Number of kpoints (231)13 506 56 KOHLER, RUGGERONE, AND SCHEFFLERcalculated adsorption energies for other possible positions shown in Fig. 4 ~a!are several 100 meV less favorable.49 Our investigations also throw light on the suggested model of a H-induced structural change: For both materials the calculated shift y1is only of the order of 0.01 Å and thus there is no evidence for a pronounced top-layer-shift recon-struction. Moreover, this subtle change in the surface geom-etry is unlikely to be resolved experimentally due to zero-point vibrations. This aspect becomes evident by evaluatingthe total energy with respect to a rigid top-layer shift. In Fig.5 we present the results of such a calculation for H/W ~110! and depict the first three vibrational eigenstates obtained from a harmonic expansion of the total energy E tot(y1); for H/Mo ~110!the energetics is similar. Since we have kBT'25 meV at room temperature thermal fluctuations of y1are of the order of 0.1 Å. This is considerably larger than the theoretically predicted ground state value of y1. In Ref. 47 Arnold et al.offer an explanation of how the previous LEED work was probably misled by assuming that scatteringfrom the H layer is negligible. B. Electronic structure In Sec. I we already pointed out that one needs to find pronounced nesting features in the Fermi surface of the H-covered surfaces in order to make the model of a Kohnanomaly work. After the determination of the relaxed surfaceconfigurations we therefore turn our attention towards theelectronic structure of those systems. In particular, we focuson surface states close to the Fermi level. One should keep in mind that DFT is not expected to give the exact Fermi surface, i.e., the self-energy operator mayhave ~a strong or weak !kdependence, different from that of the exchange-correlation potential. Moreover, we encounteradditional problems because it is rather difficult in slab sys-tems to distinguish between surface resonances and pure sur-TABLE II. Calculated relaxation parameters for the clean and H-covered ~110!surfaces of Mo and W. The height of the hydrogen above the surface and its @1¯10#offset from the @001#bridge position are denoted bydHandyH, respectively. The shift of the surface layer with respect to the substrate is y1. The parameters Ddijdescribe the percentage change of the interlayer distance between the ith and the jth substrate layers with respect to the bulk interlayer spacing d0. For each system the results of a five-, seven-, and nine-layer calculation as well as of a recent LEED analysis ~Refs. 47 and 48 !are presented ~labeled as ‘‘5,’’ ‘‘7,’’ ‘‘9,’’ and ‘‘L’’!. SystemyH ~Å!dH ~Å!y1 ~Å!Dd12 (%d0)Dd23 (%d0)Dd34 (%d0) Mo~110! 5 25.9 20.8 7 24.5 10.5 0.0 9 25.0 10.7 20.3 L 24.060.6 10.260.8 0.0 61.1 H/Mo ~110!5 0.63 1.08 0.05 22.7 20.4 7 0.60 1.07 0.03 22.1 10.1 20.1 9 0.62 1.09 0.04 22.5 10.3 20.2 L 0.55 60.4 1.3 60.3 0.0 60.1 22.060.4 0.0 60.5 0.0 60.8 W~110! 5 24.1 20.2 7 23.3 20.1 20.4 9 23.6 10.2 20.3 L 23.160.6 0.0 60.9 0.0 61.0 H/W~110!5 0.68 1.12 0.01 21.4 0.0 7 0.67 1.11 0.05 21.3 10.3 10.3 9 0.67 1.09 0.02 21.4 20.3 20.1 L 0.56 60.4 1.2 60.25 0.0 60.1 21.760.5 0.0 60.6 0.0 60.9 FIG. 5. Change of total energy Etotversus top-layer-shift y1 calculated for a H/W ~110!seven-layer slab system. For each data point the whole surface is relaxed keeping only the substrate @11¯0# coordinates y1andy25y350 Å fixed. Also shown are the first three oscillator eigenstates calculated from a harmonic expansion ofthe total energy E tot(y1).56 13 507 AB INITIO STUDY OF THE ANOMALIES IN THE He -...face states. Due to interactions between the two faces on either side of the slab resonances can be shifted into a bulkband gap. There, they are easily mixed up with real surfacestates. By contrast, in a semi-infinite substrate only true sur-face states are localized in the bulk band gap and decayexponentially into the bulk. Therefore, our slab method lacksa clear indicator for the characterization of electronic states. However, we found that the localization w MTwithin the MT spheres of the top two surface layers of a seven-layer slab is a useful and suitable measure ~60% ,wMT: surface-state like; 30% ,wMT,60%: surface-resonance like; wMT,30%: bulklike !. Nevertheless, this approach is anything but exact. Also, the calculated kispace location is only accurate if the respective band is strongly localized at the surface. Experimentalists face similarly serious problems: In ARP the Fermi surface is determined by extrapolation peaks foundbelow the Fermi level. However, the value of the Fermi en-ergy itself is only known within an uncertainty of about 6100 meV. This uncertainty can amount to a significant er- ror in the extrapolated k ispace position of states, especially if the respective bands are flat.15 In Figs. 6 and 7 we report the calculated Fermi surfaces and compare them to experimental ones obtained by theKevan 9,17and Plummer50groups. Let us first focus on the theoretical results which are—as in the case of the surfacegeometries—again very similar for Mo and W. For both systems the H adsorption induces the shift of a band with ( d 3z22r2,dxz) character to lower binding energies.49,51This effect which is illustrated in Fig. 8 moves the Fermi line associated with this band into the band gap ofthe surface projected band structure. Subsequently, the re-spective states become true surface states. It is important tonote that the band shift is due not to a hybridization betweenhydrogen orbitals and ( d 3z22r2,dxz) bands but to a hydrogen- induced modification of the surface potential. The bondingstates of the hydrogen-substrate interaction are about 5 eVbelow the Fermi energy and the antibonding states are 4 eVabove. Therefore, they are not involved in this process whichtakes place at the Fermi level. The shifted ( d 3z22r2,dxz) band is characterized by a high density of states at the Fermi level. For the clean Mo ~110! surface one finds that the ( d3z22r2,dxz) band has a MT lo- calization of wMT'30%. Due to the H adsorption this value increases to more than 60%. This modification is also visiblein the charge-density plots in Fig. 9. More important, thenew Fermi contour reveals pronounced nesting features. InFigs. 6 ~c!and 7 ~c!one sees the dramatic changes due to the H-induced shift. Segments of the two-dimensional Fermi contour of the ( d 3z22r2,dxz) band are now running parallel to GNand normal to GS. There are two nesting vectors Q¯c,1 andQ¯c,2which connect those segments in different parts of the SBZ. As can be seen in Table III they agree very wellwith the HAS measured critical wave vectors. With respect to those nesting vectors our results contra- dict the photoemission studies by Kevan andco-workers 9,16,17,52which were depicted in Figs. 6 ~b!and 6~d!~for W !and Fig. 2 in Ref. 49 ~for Mo !. Apart from that Kevan’s and our data compare rather well. Therefore, it isdifficult to give a plausible reason for the discrepancies.However, in view of the fact that for the clean surface thetheoretical and experimental Fermi surfaces are in very goodagreement and that our calculated Fermi nesting vectorsagree very well with the HAS and HREELS anomalies wedared to suggest that those differences may be due to thealready discussed problems within the experimental FIG. 6. Theoretical and ARP Fermi surfaces of the clean ~upper part !and the H-covered ~lower part !W~110!surface. The solid ~dotted ! lines denote surface resonances or surface stateswhich are localized by more than 60% ~30%!in the MT’s of the two top W layers. Shaded areasrepresent the ~110!projected theoretical W bulk Fermi surface. The arrows Q ¯ expc,1andQ¯c,2are the critical wave vectors found in HAS. The ARPdata stem from Refs. 9 and 17.13 508 56 KOHLER, RUGGERONE, AND SCHEFFLERanalysis.49We also note that recent ARP studies50of a re- lated system which we present in Fig. 7 seem to support ourconclusion. This experimental work deals with the ~110!sur- face of the alloy Mo 0.95Re0.05but the surface physics of Mo0.95Re0.05~110!and Mo ~110!should be practically the same because in both cases the top layer consists only of Moatoms. This assumption is also backed by test calculations where we simulated a Mo 0.95Tc0.05alloy be the virtual crystal approximation and found only minor quantitative changes. From Figs. 7 ~c!and 7 ~d!it becomes clear that in particu- lar for the important ( d3z22r2,dxz) surface band, which was not seen by Kevan’s group, experiment and DFT now agreevery well. There are, however, still differences: Theory pre- dicts bands centered at S¯which are not seen by ARP whereas the band circle at G¯in Fig. 7 ~d!is only observed experimen- tally. Also, in the calculations we find an elliptical band cen- tered at the G¯point whereas in ARP the same band has the shape of a rectangle. Those discrepancies are probably due tomatrix elements of the photoemission process and to the in-herent inaccuracies in theory and experiment which we men-tioned above. Nevertheless, the theoretical results are en-couraging because they provide strong evidence for a linkbetween the HAS anomalies and the Fermi surfaces. C. Vibrational properties The study of the electronic structure revealed pronounced nesting features for the H-covered surfaces. At this point, thefollowing question arises: How does the surface react to thisapparent electronic instability? At the kvectors of the Fermi- surface nesting the coupling between electrons and phononsis expected to become significant which implies a possiblebreakdown of the Born-Oppenheimer approximation. This leads to a softening of the related phonons. If the electron-phonon coupling is strong and the energetic cost of a surfacedistortion is small the nesting could even trigger a recon-struction combined with the build-up of a charge-densitywave as in the case of the ~001!surfaces of W 10and Mo.11 Then, at the reconstructed surface, the Born-Oppenheimer approximation is valid again. It is clear that one needs toperform frozen-phonon calculations in order to determine theactual strength of the electron-phonon coupling and the re-sulting phonon softening. 18 The experimental vibrational spectra of the ~110!surfaces of Mo and W show two distinct reactions to the adsorption of hydrogen. Along GHandGSthe H adsorption induces a softening of the Rayleigh and ~to a smaller amount !of the longitudinal wave53while a stiffening of these modes is ob- served along GN. Our goal is to investigate both effects theoretically. We use an enlarged surface unit cell together with a five- layer slab. The plane-wave cutoff for the wave functions is reduced to ( \2/2m)(Kwf)2510 Ry ~see test caculations in the Appendix, Sec. 3 a!. The geometries are defined by the five-layer relaxation parameters presented in Table II. Some relaxation parameters, i.e., Ddij, change considerably when we perform a calculation with a slab of five instead of sevenor nine metal layers. We note, however, that the values of thecritical wave vectors and the position of the hydrogen withrespect to the substrate surface are practically insensitive tothe thickness of the slab. This indicates that the physicalproperties we are interested in, e.g., the nesting features, arewell localized surface phenomena and that the results of ourfive-layer-slab studies can be trusted. As shown in the Ap- FIG. 7. Theoretical Fermi surfaces of the ~a! clean and ~c!H-covered Mo ~110!surface. Also presented are data points ~1!which stem from an ARP study of the ~b!clean and ~d!H-covered Mo0.95Re0.05~110!surface ~Ref. 50 !. The presen- tation is equivalent to Fig. 6.56 13 509 AB INITIO STUDY OF THE ANOMALIES IN THE He -...pendix, Sec. 3 afor theS¯-point phonon the calculated fre- quencies are relatively insensitive to the SBZ kipoint sam- pling, and we conclude that a uniform mesh of 56 points issufficient for the present study. Being zone boundary phonons in the Rayleigh waves at S¯ andN¯the displacements of the surface atoms are along the direction normal to the surface. Moreover, the vibrations arestrongly localized in the surface region. We assume that evenwith hydrogen adsorbed the coupling to modes parallel to thesurface is small. Therefore, we can confine our study to thevibrational components normal to the surface. Besides thefirst substrate layer we also include the vibrations of the sec-ond layer in our calculations. Due to the small mass thehydrogen atom follows the substrate vibrations adiabatically.In the calculations this adsorbate relaxation lowers the dis-tortion energy by several meV and thus reduces the phononenergy significantly. Therefore, all phonon frequencies arecalculated for a fully relaxed hydrogen adsorbate.For the study of the vibrational properties of the systems we displace the substrate atoms in the surface ~according to Fig. 10 !, relax the hydrogen atom, and calculate the resulting atomic forces. The same is done for the atoms in the subsur-face layer. From a third-order fit to the calculated forces forfive different displacement steps between 0% and 5% of thelattice constant we obtain the matrix elements of the dynami-cal matrix and then via diagonalization the respective pho-non energies. The calculated frequencies are collected in Table IV. At the N ¯point our results reproduce the experi- mentally observed increase of the Rayleigh-wave frequencyas hydrogen is adsorbed. At theS¯point we find that the strong coupling to elec- tronic states at the Fermi level leads to a lowering of thephonon energy in good agreement with the experimental re-sults. In Figs. 11 ~a!–~c!we schematically illustrate the mechanism which is responsible for this effect. It is, in fact,a textbook example of a Kohn anomaly due to Fermi-surfacenesting: 54,55~a!Within the unperturbated system the (d3z22r2,dxz) band cuts the Fermi level exactly midway be- tween G¯andS¯.~b and c !The nuclear distortion associated with the S¯-point phonon modifies the surface potential and hence removes the degeneracy at the backfolded zone bound- ary point S8. The occupied states are shifted to lower ener- gies~see Fig. 12 !. This amounts to a negative contribution of the electronic band structure energy to the total energy andthus a lowering of the phonon energy. A frozen-phonon study for the second nesting vector along GHis not performed because the respective Q¯c,1 is highly noncommensurate. Thus, such a calculation would be very expensive. However, since the character of the (d3z22r2,dxz) band does not change when shifting from the GSto the GHnesting we expect similar results; this was recently confirmed by Bungaro56within the framework of a perturbation theory. The calculation of the electron-phonon interaction at the Mo~110!and W ~110!surfaces and its change due to hydro- gen adsorption pinpoints the phonon character of the small anomaly v2and identifies the interplay between the elec- tronic structure and the vibrational spectra of the transitionmetal surfaces. Thus, our results clearly support the interpre-tation that the small dip observed by both HAS and HREELSis due to a Kohn anomaly. Furthermore, we find that theelectron-phonon coupling is not strong enough to induce astable reconstruction. Thus, the system remains in a some-what limbolike state. D. Electron-hole excitations For the deep and narrow anomaly v1the interpretation appears to be less straightforward. It is particularly puzzlingthat it is only seen in the HAS spectra and not in HREELS.This difference, together with the above noted ‘‘limbo state’’of the surface yield the clue to our present interpretation. Atfirst, it is necessary to understand the nature of rare-gas atomscattering. In a recent study we found that those scattering processes are significantly more complicated ~and more interesting ! than hitherto assumed: 58The reflection of the He atom hap- pens in front of the surface at a distance of 2–3 Å. This is FIG. 8. Hydrogen-induced shift of the ( d3z22r2,dxz) band at the Mo~110!surface. The upper part of the figure represents a one- dimensional cut through the two-dimensional surface band struc-ture. Shaded areas illustrate the ~110!projected bulk band structure. The dotted ~solid!lines depict the position of the ( d 3z22r2,dxz) band for the clean ~H-covered !surface. The arrows indicate how the band structure translates into the two-dimensional Fermi surfaceshown in the lower part. In Fig. 9 the charge density of the twostates marked by the arrows is presented.13 510 56 KOHLER, RUGGERONE, AND SCHEFFLERillustrated in Fig. 13. More important, it is not the totalelec- tron density of the substrate surface which determines theinteraction but the electronic wave functions close to theFermi level. In the case of the H/W ~110!and H/Mo ~110! systems it is thus plausible to assume that the He atom couples directly to the ( d 3z22r2,dxz) surface states mentioned above and excites electron-hole pairs. By contrast, the elec-trons in HREELS scatter at the atomic cores and interactonly weakly with the electron density at the surface. In order to analyze the spectrum of those excitations as seen by HAS in some more detail we evaluate the local den-sity of electron-hole excitations, P ~qi,\v!5E SBZdkiwki1qiwki~fki1qi2fki! 3d~eki1qi2eki2\v!, ~6! which is a measure of the probability that a He atom loses the energy \vand the momentum qiwhen interacting with the surface electron density. In our approach we use allei- genvalues ekiand occupation numbers fkiobtained via anine-layer-slab calculation. In order to refer to HAS we take into account the localization wkiof the respective state at a distance of 2.5 Å in front of the substrate surface. The results obtained for Mo ~110!and H/Mo ~110!with \v'6.8 meV are presented in Fig. 14. For the clean surface we find that the intensity of the electron-hole excitations de- creases continuously as we move away from the G¯point towards the zone boundaries. This relatively smooth behav-ior of the susceptibility is modified considerably when hy-drogen is adsorbed on the clean surface: Pronounced peaks appear, in particular one which stretches from the GHline through the SBZ to the symmetry point S¯. This is in excel- lent agreement with the HAS results ~see also Fig. 2 !.W e have carried out calculations with increasing \v. The result- ing function P(q,v) loses progressively its structures, and the peaks are smeared out. This is not surprising, since anincreased \ vmeans a larger number of accounted electronic transitions. Also, there is a peak located at the upper half of FIG. 10. Distortion pattern of the atoms of the top metal layer for the Rayleigh phonons at the symmetry points S¯andN¯. FIG. 9. Charge-density plot along the ~010! plane for Mo ~110!and H/Mo ~110!: Shown are the two states marked in Fig. 8 by arrows. Onlythe plane-wave part of the wave functions isshown. The positions of the Mo atoms aremarked by dots. The square represents the hydro-gen atom in the hollow adsorption site. TABLE III. Theoretical Fermi-surface nesting vectors compared to critical wave vectors obtained by HAS and HREELS experi-ments ~Refs. 2, 3, 7, and 8 !. Direction SystemuQ¯cu(Å21) Theory Experiment GH H/Mo ~110! 0.86 0.90 H/W~110! 0.96 0.95 GS H/Mo ~110! 1.23 1.22 H/W~110! 1.22 1.2256 13 511 AB INITIO STUDY OF THE ANOMALIES IN THE He -...theGNsymmetry line and a large structure at and close to G¯. They cannot be associated with anomalies in the HAS spec- trum. These results may be due to the fact that selection rulesor matrix elements for the interaction between the scatteringHe atoms and the electron-hole excitations are not consid- ered inP(k,\ v). For the large structure around G¯the ab- sence in the HAS spectra of an anomaly may be caused also by the fact that close to and at G¯for\v56.8 meV the fea- ture is buried in a phonon band, whereas at the critical wave vectors along the @001#direction ( GH) and atS¯the excita- tions lie in a region without additional phonon branches. Inthe HAS time-of-flight spectra the the deepest anomaly ischaracterized by identifiable but small peaks that can bewashed out by the background of the several real phonon branches close to G ¯in the range of energy around \v56.8 meV. In Fig. 15 the respective data for the clean and FIG. 11. Schematic representation of the mechanism which causes the Kohn anomaly of the H/Mo ~110!and H/W ~110!. Shown is the band structure along SGSparallel to the nesting vector Q¯c,2in Figs. 6 ~c!and 7 ~c!. The form of the ( d3z22r2,dxz) band is indicated by dashed lines. See text for a detailed description. FIG. 12. H/W ~110!: Band gap Dinduced by a S¯-point frozen- phonon distortion. Shown are the KS eigenvalues of the (d3z22r2,dxz) band at the S8point versus the amplitude of the atomic distortion t1zof the first substrate layer. FIG. 13. Schematic illustration of the scattering of He atoms ~HAS, upper part !and electrons ~HREELS, lower part !at a metal surface ~from Ref. 57 !.TABLE IV. Comparison of calculated frozen-phonon energies and experimental values obtained by HAS Refs. 2 and 3 andHREELS ~Ref. 8 !. The theoretical phonon energies are obtained using a five-layer slab. Their numerical accuracy is about 65%, i.e.,61 meV. Phonon SystemE ph~meV! Theory Experiment N¯ W~110! 15.4 14.5 H/W~110! 17.6 17.0 S¯ Mo~110! 22.7 ;21 H/Mo ~110! 17.2 ,16 W~110! 18.3 16.1 H/W~110! 12.0 11.013 512 56 KOHLER, RUGGERONE, AND SCHEFFLERH-covered W ~110!surfaces are displayed. In general, our findings are in agreement with the experimental results ofLu¨decke and Hulpke that the anomaly stretches through the SBZ. However, we do not support the idea of a fully one-dimensional nesting mechanism with a constant @001#com- ponent of the critical wave vector as depicted in Fig. 2. In view of the simplicity of our approach, the calculated spectrum of electron-hole excitations as predicted by P(q,\ v) seems to be well-connected to the HAS measure- ments. Moreover, it is surprising how much of the Fermi-surface nesting @compare to Figs. 6 ~c!and 7 ~c!#is still present at a ~rather large !distance from the surface. In fact, if FIG. 15. Contour plot of the local probability function P(qi,\v)o fW ~110!and H/W ~110!calculated for \v 50.5 mRy '6.8 meV. The notation is equivalent to Fig. 14. FIG. 14. Contour plot of the local probability function P(qi,\v)o fM o ~110!and H/Mo ~110!calculated for \v50.5 mRy '6.8 meV. The dashed line represents a value of three arbitrary units while each full line denotes an additional in-crease by 1 arbitrary units ~with increasing linewidth !. The posi- tions of the HAS measured anomalies within the SBZ are indicatedby dots.56 13 513 AB INITIO STUDY OF THE ANOMALIES IN THE He -...the localization wkiwere not included into the calculations or if it were taken closer to the surface the H-induced peaks would be hidden in the background noise of other electron-hole excitations. The clear image in Figs. 14 and 15 onlyappears if we take the position of the classical turning pointinto account. To our knowledge these are the first theoretical studies of the rare-gas atom scattering spectrum using a first-principleselectronic band structure. Our findings support the earliersuggestion 49that the giant indentations, seen only in HAS, are due to the excitation of electron-hole pairs. In HREELSsuch excitations are less likely, and thus the strong anomalyremains invisible. IV. SUMMARY AND OUTLOOK In conclusion, the major results of our work are as fol- lows: We demonstrate that for both W ~110!and Mo ~110!the (131) surface geometry is stable upon hydrogen adsorption. There is no evidence for a pronounced H-induced top-layer-shift reconstruction, a result confirmed by a recent LEEDanalysis. 47,48Also, we give a consistent explanation for the H-induced anomalies in the HAS spectra of W ~110!and Mo~110!. The H-adsorption induces surface states of (d3z22r2,dxz) character that show pronounced Fermi-surface nesting. The modestly deep anomaly is identified as a Kohnanomaly due to those nesting features. 49,51,18For an under- standing of the huge dip we stress that the scattering of rare-gas atoms is crucially influenced by interactions with sub-strate surface wave functions at the classical turning point. 58 Assuming that the He atom couples efficiently to the H-induced surface states on H/W ~110!and H/Mo ~110!we therefore conclude that the deep HAS anomaly is predomi-nantly caused by a direct excitation of electron-hole pairsduring the scattering process. We hope that our calculations and interpretations stimu- late additional theoretical and experimental work: For in-stance, it is highly important to better understand the detailsof rare-gas atom scattering processes at ‘‘real’’ surfaces.Also, the liquidlike behavior of the H adatoms observed inthe HREELS measurements 8remains an unresolved puzzle. One would also like to know whether the HAS and HREELS spectra of the Mo 0.95Re0.05~110!alloy system reveal H-induced anomalies as expected. Finally, we call for a newARP study of the clean and H-covered Mo ~110!and W~110!surfaces in order to experimentally identify the Fermi surface. We also note that scattering experiments with atoms or molecules like Ne or H 2might provide additional insight into the interesting behavior of these surfaces. ACKNOWLEDGMENTS We have profitted considerably from discussions with Erio Tosatti. In particular, they have enlightened our under-standing of phason and amplitudon modes as a possible ex-planation of the two indentations. APPENDIX: TEST CALCULATIONS 1. Bulk In Table V we list the calculated equilibrium lattice con- stantsa0and the bulk moduli B0[2V]2Etot/]2V, and com-pare them to published results from experiment and theory. Our values were obtained using the LAPW parameter settingpresented in Table I. 2. Surfaces a. Test system The focus of this paper lies on the Mo ~110!and W ~110! surfaces. In order to test the numerical accuracy of our cal-culations for such systems it is important to go beyond typi-cal tests for the crystal bulk. For instance, we need to knowthe error bars for atomic geometries obtained via forces ascompared to a pure total energy calculation. Five- and more-layer systems are not suitable and in fact for most cases alsonot necessary for such a detailed study. Therefore, questionsreferring to the size of the vacuum region, to the accuracy ofthe directly calculated LAPW forces, and to the appropriatewave function cutoff will be answered by studying a simpleMo~110!double-layer-slab system. Starting from the LAPW parameter set listed in Table I we vary one parameter andstudy its influence on the evaluated total energy and theatomic force. These quantities are calculated with respect to the interlayer distance z. Again, we note that we are inter- ested not in the physics of this double-layer system but in theaccuracy and the convergence of our method. Therefore, the calculations are done using just four k ipoints within the SBZ. Employing a least-square fit which has the form of a Morse potential,65TABLE V. Theoretically and experimentally obtained crystal parameters for Mo and W. The percentage deviations from experi-ment are given in parentheses. The results of this work are calcu-lated nonrelativistically ( N) and scalar relativistically ( R). Also, the different XC potentials used in the theoretical studies are given:Wigner ~Ref. 59 !~W37!, Barth and Hedin ~Ref. 60 !~BH72 !, and Ceperley and Alder ~Ref. 21 !~CA80 !parametrized by Perdew and Zunger ~Ref. 22 !. The abbreviation ‘‘NL-PP‘‘ stands for ‘‘nonlocal pseudopotential‘‘ method. All theoretical numbers ignore the influ-ence of zero-point vibrations. Metal Method XC a 0~Å!B0~MBar ! Mo experimenta3.148 2.608 NL-PPbBH72 3.152 ( 10.1) 3.05 ~15.5! LAPWcBH72 3.131 ( 20.5) 2.91 ~11.6! this work ( N) CA80 3.126 ( 20.7) 2.88 ~10.4! this work ( R) CA80 3.115 ( 21.1) 2.89 ~10.8! W experimentd3.163 3.23 NL-PPbBH72 3.173 ( 10.3) 3.45 ~6.8! LAPWeW34 3.149 ( 20.4) 3.46 ~7.1! LAPWcBH72 3.162 ( 60.0) 3.40 ~5.3! this work ( N) CA80 3.194 ( 11.0) 2.92 ( 29.6) this work ( R) CA80 3.137 ( 20.8) 3.37 ~4.3! aFrom Ref. 61. bFrom Ref. 62. cFrom Ref. 35. dFrom Refs. 63 and 64. eFrom Ref. 34.13 514 56 KOHLER, RUGGERONE, AND SCHEFFLEREtot~z!5D@12e2b~z2d2d0!#2, ~A1! we calculate the equilibrium relaxation parameter d. Here, d0represents ~110!layer distance in the bcc crystal. Addi- tionally, the vibrational properties are of considerable inter- est. From the Morse parameters Dandbone extracts the oscillator frequency by using the expression v254Db MI, ~A2! where the atomic masses MIare those of the Mo and W nuclei. A corresponding fit is also employed for the directlycalculated forces normal to the ~110!surface: F ~z!52dEtot~z! dz522D˜b˜@e22b˜~z2d˜2d0!2e2b˜~z2d˜2d0!#. ~A3! This enables a quantitative study of the accuracy of the evaluated LAPW forces with respect to the total energy Etot(z). Within the Tables VI–IX the two alternatives are marked by ‘‘ F’’~force!and ‘‘E’’~energy !. We consider a calculation to be converged if the relax- ation parameter dand the frequency vdiffer by less the Dd50.25%dconv'0.005 Å and \Dv5\5%vconv'2 meV from the fully converged results dconvandvconv. In the fol- lowing discussion we focus on these two quantities dandv. They provide detailed information about the accuracy ofstructure-optimization and frozen-phonon calculations per-formed in this paper.b. Vacuum size The first set of test calculations serves to determine the appropriate size of the vacuum region. The results presentedin Table VI show that an equivalent of four substrate layers (5ˆ8.8 Å) is sufficient to decouple the two substrate surfaces across the vacuum. Both relaxation parameter dand the fre- quency vvary only within the given error range upon further increasing the vacuum region. c. LAPW parameters Another important parameter is the wave function energy cutoff ( \2/2m)(Kwf)2. It has a decisive influence on theTABLE VI. Mo ~110!double layer: Morse parameter dand the energy \vversus the number of vacuum layers. The arrow indi- cates the parameter choice which yields converged results. Number of layersd(%d0) \v~meV! EFE F 2 21.7 22.1 36.5 35.3 3 21.7 22.1 36.7 35.2 )4 22.0 22.4 37.3 35.6 5 22.1 22.4 37.4 35.7 6 22.0 22.3 36.9 35.4 TABLE VII. Mo ~110!double layer: Variation of the wave func- tion cutoff parameter ( \2/2m)(Kwf)2. The notation is equivalent to Table VI. \2/2m(Kwf)2 ~Ry!d(%d0) \v~meV! EFE F 9 20.5 21.0 35.0 33.4 10 21.5 22.1 36.0 34.6 11 21.6 22.2 36.1 35.2 )12 22.0 22.4 37.3 35.6 13 21.9 22.2 37.1 35.5 14 21.9 22.1 36.5 35.4TABLE VIII. Mo ~110!double layer: Variation the potential pa- rameters ( \2/2m)(Gpot)2andlpot. The notation is equivalent to Table VI. \2/2m(Gpot)2 ~Ry!d(%d0) \v~meV! EFE F 64 22.0 22.3 37.3 35.6 81 21.9 22.3 36.6 35.5 )100 22.0 22.4 37.3 35.6 121 21.9 22.3 37.0 35.6 144 21.9 22.3 37.0 35.6 lpotd(%d0) \v~meV! EFE F 2 21.9 21.7 37.5 34.6 )3 22.0 22.4 37.3 35.6 4 22.0 22.5 37.1 35.2 5 22.1 22.6 37.1 35.3 TABLE IX. W ~110!double layer: Accuracy of the calculation in dependence of the number of kipoints for the ~a!valence elec- trons and ~b!semicore electrons. The respective fixed ki-point set is made up by one special kipoint in the irreducible part of the SBZ ~Ref. 46 !. The arrows indicate the converged ki-point set used for the~110!surface calculations. The notation is equivalent to Table VI. ~a!Valence electrons No. kipointsd(%d0) \v~meV! EFE F 36 28.5 29.5 20.5 21.1 49 29.0 210.1 23.8 23.1 )64 28.4 29.3 19.2 20.3 81 28.4 29.4 18.9 20.4 ~b!Semicore electrons No. kipointsd(%d0) \v~meV! EFE F 4 21.0 21.8 27.4 26.2 )9 21.0 21.9 27.4 26.2 16 21.0 21.9 27.3 26.2 25 21.0 21.9 27.4 26.2 36 21.0 21.9 27.4 26.256 13 515 AB INITIO STUDY OF THE ANOMALIES IN THE He -...computer time. With respect to the structure parameter dour results in Table VII indicate that the smallest possible value is (\2/2m)(Kwf)2512 Ry. However, for \valready a cal- culation with ( \2/2m)(Kwf)2510 Ry provides converged results. The two potential parameters ( \2/2m)(Gpot)2andlpotare of little influence on the calculated quantities. This can be seen in Table VIII. The values of the relaxation parameter d and the frequency vremain relatively stable within the pa- rameter range 64 Ry ,(\2/2m)(Gpot)2,144 Ry. Both quantities are more sensitive to changes in the ( l,m) expan- sion of the MT potential. However, a value of lpot53 seems to be sufficient. In general we note that energy ( E) and force ( F) calcu- lations demonstrate a similar convergence behavior. How-ever, the magnitude of the relaxation in the force calculations is overestimated by about Dd50.5%d conv'0.01 Å; for the frequencies one finds a deviation of about \Dv55%\vconv'1.8 meV from the converged total en- ergy values. Those differences are probably due to numericalinaccuracies in the force calculation. d.ki-point set Similar calculations for a W ~110!double-layer system lead to a converged LAPW parameter set which is compa-rable to the one obtained for Mo. Only the MT potential parameter l pothas to be chosen higher: lpot54 is slightly different. The agreement between force and total energy cal- culations ~Dd51%dconv'0.02 Å and Dv55%vconv!is nearly as good as in the case of Mo. The following calcula- tion is aimed to determine a two-dimensional ki-point set which can be used for the calculation of the atomic and elec-tronic structure of Mo ~110!and W ~110!surfaces ~see Table IX!. Because of the size of ~110!surface slab systems it is important to keep the number of k ipoints as small as pos- sible. For the calculations we use the LAPW parameter set given in Table I and vary the ki-point set for either the va- lence or the semicore electrons. These sets always consist ofuniform two-dimensional point meshes. Converged results are obtained by using 64 k ipoints for the valence and 9 ki points for the semicore electrons. 3. Phonons a. LAPW parameter Our final series of tests is meant to determine the accuracy of the frozen-phonon calculation. In order to keep the com-puter time for those elaborate calculations within pleasant limits one has to keep the number of kipoints and basis functions as small as possible. As a sample case we study the S¯-point Rayleigh phonon of the H/Mo ~110!system. The calculations are performed as follows: We distort the atoms of the first substrate layer by t1z(S¯)50.08 Å accord- ing to the pattern in Fig. 10 and determine the total energy and force changes DEtotandDF1z. These quantities have the same convergence behavior as the phonon frequencies: Therefore, if DEtotandDF1zare stable with respect to a particular LAPW parameter so are the phonon frequencies.Converged results for Mo as well as for W are obtained byusing the following setting: wave function plane-wave cutoff (\ 2/2m)(Kwf)2510 Ry;lpot53 for the ( l,m) expansion of the MT potential; uniform ki-point mesh consisting of 56 points; five substrate layers; a vacuum region equivalent tofour substrate layers; XC potential from Refs. 21,22. b.ki-point mesh In order to accurately describe the electronic structure and hence the electron-phonon coupling for the S¯-point phonon one has to provide a detailed ki-point set. For instance, for the study of the (2 31) reconstructions of the diamond ~111! surface Vanderbilt and Louie employ a ki-point mesh which becomes logarithmically denser close to the zoneboundary. 66One cannot rely on the results for the two Mo~110!and W ~110!double-layer slabs in the Appendix Sec. 2dbecause in these systems nesting effects are totally unimportant. We perform tests for the H/Mo ~110!system using up to 232kipoints in the SBZ and find that a uniform ki-point mesh of 56 points is sufficient in order to obtain convergedresults ~see Table X !. For all those frozen-phonon calcula- tions the Monkhorst-Pack k i-point meshes46are shifted in order to include the S¯8point of the (1 31) unit cell ~see Fig. 11!. For the ki-point summation we employ a Fermi smear- ing ofkBTel'68 meV. Only for the 232 ki-point calculation listed in Tabl e X a smaller value of kBTel'14 meV is used. However, this parameter is apparently uncritical because the energies are obtained for the kBTel 0 K limit as discussed in Ref. 42.TABLE X. Frozen-phonon distortion for H/Mo ~110!: Change of the total energy DEtotand the force DF1zversus the number of ki points. All ki-point sets are uniform meshes which include the sym- metry point S8of the backfolded SBZ ~see Fig. 11 !. Given are also the calculated values for the clean Mo ~110!surface ~with 56ki points !. No.kipoints DEtot~meV!D F1z~meV/Å ! 16 110 672 56 136 75680 144 797 120 135 790232 136 777 Clean 161 938TABLE XI. Frozen-phonon distortion for H/Mo ~110!: Change of the total energy DEtotand the force DF1zwith respect to the wave function plane-wave cutoff ( \2/2m)(Kwf)2. \2/2m(Kwf)2~Ry!D Etot~meV!D F1z~meV/Å ! 10 136 756 12 137 767 TABLE XII. Frozen-phonon distortion for H/Mo ~110!: Varia- tion of the XC potential: LDA Refs. 21,22, GGA ~Ref. 44 !. XC potential DEtot~meV!D F1z~meV/Å ! LDA 136 756 GGA 141 76413 516 56 KOHLER, RUGGERONE, AND SCHEFFLERc. Basis functions Next, we check whether the results are sensitive to an increase in the wave function cutoff, i.e., whether the chosen value ( \2/2m)(Kwf)2510 Ry is large enough. The data listed in Table XI demonstrates that this is indeed the case.The results remain stable for the higher cutoff (\ 2/2m)(Kwf)2512 Ry. d. LDA versus GGA For some systems involving hydrogen the results of a to- tal energy calculation are not reliable if the LDA is used. Inthose cases the bonding—especially between hydrogen atoms—is only described accurately if gradient correctionsare considered. 67In Table XII we compare results obtained with the generalized gradient approximation44,45~GGA !and those found within the LDA. The latter is identical with the 56ki-point calculation in Table X. There, the XC potential of Ceperley and Alder21parametrized by Perdew and Zunger22is used. 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PhysRevB.101.094426.pdf

PHYSICAL REVIEW B 101, 094426 (2020) Magneto-dielectric effect in relaxor superparaelectric Tb 2CoMnO 6film R. Mandal ,1,2,*M. Chandra ,3V . Roddatis,4P. Ksoll,2M. Tripathi,3R. Rawat,3R. J. Choudhary,3and V . Moshnyaga2,† 1Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India 2Erstes Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 3UGC-DAE Consortium for Scientific Research, Indore Centre, University Campus, Khandwa Road, Indore 452017, India 4Institut für Materialphysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany (Received 17 October 2019; revised manuscript received 23 January 2020; accepted 2 March 2020; published 19 March 2020) We report magneto-dielectric properties of partially B-site ordered monoclinic Tb 2CoMnO 6double perovskite thin film epitaxially grown on Nb : SrTiO 3(100) substrates by metalorganic aerosol deposition technique. Transmission electron microscopy and electron energy loss spectroscopy mapping shows the presence anddistribution of both Co 2+and Co3+ions in the film, evidencing a partial B-site disorder, which was further confirmed by the observation of reduced saturation magnetization at low temperatures. The ferromagnetic Curietemperature, T C=110 K, is slightly higher as compared to the bulk value (100 K) probably due to an in plane epitaxy tensile strain. Temperature dependent dielectric study reveals an unexpected high temperature dipolarrelaxor-glass-like transition at a temperature T ∗∼190 K >TC, which depends on the applied frequency and indicates a superparaelectric behavior. Two different dielectric relaxation peaks have been observed; they mergeatT ∗where likely a coupling to the disorder-induced short range charge-spin correlations results in a 4% magneto-dielectric coupling. DOI: 10.1103/PhysRevB.101.094426 I. INTRODUCTION Magneto-dielectrics and magneto-electric materials with the coupled magnetic and electric dipolar order parameters areof fundamental as well as of technological importance. Rare-earth-based perovskite oxides have been proved to be poten-tial candidates for the next generation memory and spintronicdevice applications [ 1–7].A 2BB/primeO6(Ais a rare earth cation, B andB/primeare transition metal ions) double perovskites with the layered ABO3/AB/primeO3cation-ordered structure along the [111] axis represent themselves an emerging and promising plat-form to study strong electronic correlations, complex mag-netic structure, spin-lattice interaction, and magneto-dielectriccoupling [ 8,9]. R 2(Co/Ni)MnO 6(where R=La to Lu) system having a monoclinic structure with P21/nspace group is espe- cially attractive as they possess an insulating ferromagnetic(FMI) behavior with relatively high Curie temperatures, T C∼ 200–300 K, allowing a high temperature magneto-dielectriccoupling. FMI originates from a 180°- superexchange in-teraction between high spin Co 2+/Ni2+and Mn4+ions, described by the second Goodenough-Kanamori-Anderson rule [ 10–12]. The FM ordering as well as dielectric be- havior depend strongly on the B-site ordering which con-trols the superexchange interaction along with hoping ofcharge. The fully and partially B-site ordered La 2CoMnO 6 has been well explored due to a reasonably high ferro-magnetic Curie temperature, T C=230 K [ 13], spin-phonon coupling [ 11]; they reveal a weak magneto-dielectric ef- *rajesh.mandal@students.iiserpune.ac.in †vmosnea@gwdg.defect (3%) [ 14]. Partially disordered La 2NiMnO 6has been viewed as a promising multiglass material where two differentglassy states (spin and dipolar) were observed simultane-ously, resulting in a stronger magneto-dielectric coupling con- stant ( є MD=є/prime(8T)−є/prime(0T)є/prime(0T)×100)∼16% at room tempera- ture [ 15]. Considering the smaller radii rare-earth ions in the A-site ( R=Pr to Lu) the ferromagnetic TCdecreases consid- erably (down to TC=48 K for Lu) [ 8] along with spin-phonon interaction [ 16]. The trend in the dielectric behavior also remains unchanged from bigger to smaller A-site cations as found in bulk (La /Tb/Y)2CoMnO 6where dielectric constant decreases monotonically with lowering temperature [ 17]. The La2CoMnO 6(LCMO) from the concerned rare earth double perovskite family has been reported to possess a large dielec-tric constant at room temperature, which gradually decreaseswith lowering the temperature [ 17]. Taking ions with smaller ionic radii, like Y 3+and Tb3+in the A-site, the trend of decreasing dielectric constant monotonically towards lower- ing temperature remains similar with no significant deviation [18]. The overall behavior is the same in the case of epitaxial LCMO thin films though the dielectric constant becomes verylow [ 14]. A controllable disorder in a perovskite or double per- ovskite system can create a dipolar glass that can couple withits magnetic subsystem, inducing a novel magneto-dielectricor rare multiglass behavior [ 19]. Tb 3+with a small cation radius in the A-site has a special significance as it can tune and stabilize a different hexagonal structure in a strained thin film, which cannot sustain in the bulk form as observed in TbMnO 3 [20]. With all these experimental observations and intuitions, the Tb-based double perovskite thin films with controllableB-site (partial) ordering could be suggested as an importantand exclusive playground for studying magnetic and dielectrictransitions along with possible coupling between them. 2469-9950/2020/101(9)/094426(7) 094426-1 ©2020 American Physical SocietyR. MANDAL et al. PHYSICAL REVIEW B 101, 094426 (2020) Here we report the epitaxial growth of monoclinic Tb2CoMnO 6/Nb : SrTiO 3(100) thin film by using a metalor- ganic aerosol deposition (MAD) technique [ 21]. The es- tablished partial B-site disorder in the film is accompaniedby an unexpected high temperature relaxor glassy transitionalong with a superparaelectric behavior. The asymmetricalnature of the temperature dependent dielectric constant ismanifested with the three different polarized nanoregions(PNR) present in the system. Moreover, two different dielec-tric relaxation peaks in frequency domain along with a 4%magneto-dielectric coupling was observed probably due to theinteraction of the B-site-disorder-induced local spin momentswith the PNRs. II. EXPERIMENTAL SECTION Tb2CoMnO 6(TCMO) films have been grown by a met- alorganic aerosol deposition technique on commercial elec-trically conducting 0.5% Nb-doped SrTiO 3(100) substrates (Crystal GmbH). Acetylacetonates of Tb, Mn, and Co wereused as precursors. Precursor solutions in dimethylformamidewith concentration 0.02 M (for both Co and Mn precursor)and empirically found molar ratio Tb /(Co+Mn)=1.1w e r e prepared. The films with thickness, d=80 nm, were grown by spraying the precursor solution by using dry compressedair onto a substrate heated to T sub∼900◦C. The films were grown with an average growth rate of v=15 nm/min and were cooled down to room temperature in 20 min after de-position. X-ray diffraction (XRD) characterization was per-formed by using the “Bruker D8” spectrometer with Cu K α1,2 radiation in a /Theta1-2/Theta1Bragg-Brentano geometry. Magnetization as a function of temperature and magnetic field, appliedparallel to the film surface, was measured using commercial7T-superconducting quantum interference device-vibrating-sample magnetometer (Quantum Design Inc., USA) system.Magnetization vs temperature was measured following theconventional protocols of zerofield-cooled warming (ZFC)and field-cooled warming (FCW) cycles in an applied mag-netic field H=100 Oe. The local structure of TCMO films was studied by scanning transmission electron microscopyand electron energy loss spectroscopy (EELS) using a FEITitan 80–300 G2 environmental transmission electron mi-croscope (TEM), operated at an acceleration voltage of 300kV . The TEM is equipped with a Gatan Imaging FilterQuantum 965 ER. EELS spectra were taken with a dis-persion of 0.05 eV /channel. The convergence and collec- tion semiangles were about 10 and 22 mrad, respectively.TEM lamellas were prepared by a focused ion beam lift-outtechnique using a Thermo-Fischer (former FEI) Helios 4UCinstrument. The temperature- and magnetic-field-dependentcomplex dielectric measurements were performed using ahomemade insert coupled with 9-T superconducting magnetand a Keysight E4980A LCR-meter operating at frequencyrange f=20 Hz–2 MHz. III. RESULTS AND DISCUSSION Temperature dependence of the ZFC and FC mag- netic susceptibility, χ(T), of the TCMO film, measured for μ0H=100 Oe, is shown in Fig. 1(a). One can see a phase FIG. 1. (a) Zero- (blue, ZFC) and field-cooled (red, FC) mag- netic susceptibility [ χ(T)] as a function of temperature ( T)i nt h e TCMO film; (b) Isothermal magnetization with applied externalmagnetic field at 5 K. transition at TC=110 K, below which the long range ferro- magnetic ordering develops due the superexchange interactionbetween Co 2+and Mn4+ions [ 12]. The transition seems to be of a second order as it is not apparently sharp andno warming/cooling hysteresis was observed. The transitiontemperature in our TCMO film is a bit higher than thatobserved in a Tb 2CoMnO 6single crystal (100 K) [ 22], likely due to an in-plane epitaxial tensile strain in the TCMO thinfilm,ε=− 0.6%, evaluated from the XRD pattern [ 23] (see Fig. SM-1), which shows an out-of-plane epitaxy. The bifur-cation between ZFC and FC curves denotes the magnetic irre-versibility in the system due to the presence of antiferromag-netic/FM competing interactions among magnetic domains,which are characteristic for ferro- and ferrimagnetic systemswith large coercivity in the ordered phase, irrespective of whatorigin. It means that along with the dominating Co 2+/Mn4+ FM superexchange the AFM interactions of Mn4+/Mn4+ and/or Co2+/Co2+type could be present, although no addi- tional features were observed at low temperatures due to thedomain-wall depinning process similar to that observed in asingle crystal [ 22]. The anomalous increase in FC curve at 18 K is due to the ordering of spins of Tb 3+ions, which are likely FM coupled to the Co2+/Mn4+sites. In Fig. 1(b) the field dependence of the isothermal magnetization, M(H), 094426-2MAGNETO-DIELECTRIC EFFECT IN RELAXOR … PHYSICAL REVIEW B 101, 094426 (2020) FIG. 2. (a) Dielectric constant [ /epsilon1/prime(T)] vs temperature with different applied frequencies (1–100 KHz); (b) Curie-Weiss fit for the /epsilon1/prime(T) above the relaxor glass transition for different frequencies; (c) Temperature dependence of the dielectric loss part /epsilon1/prime/prime(T) with different applied frequencies showing relaxation peaks; (d) Arrhenius fit for the relaxation time vs peak temperature for the dielectric constant /epsilon1/prime(T)a n d dielectric loss /epsilon1/prime/prime(T) (inset). measured at 5 K is shown. The evaluated magnetization close to saturation, MS∼5.5μB/f.u., is significantly smaller than the theoretical value (6 μB/f.u.) for a fully B-site ordered Co2+/Mn4+system. Moreover, the Tb+3ions possess an even higher moment of 9 .72μB/Tb3+due to the spin-orbit coupling and, considering their additive contribution, the totalmagnetic moment of the TCMO system should be even muchlarger. Recently, the TCMO single crystal has been shownto have a strong magnetic anisotropy of Tb 3+ions, which prefers to order along the caxis yielding a large value of MS∼9.73μB/f.u.[22]. As for our TCMO/Nb:STO(001) film the caxis according to XRD [ 23] (see Fig. SM-1) stays perpendicular to the film plane, and the observed value ofM S∼5.5μB/f.u.seems to be quite small though we do not know the exact orientation of the magnetic easy axisfor our film. From the trend and comparison to the singlecrystal data it could be concluded that the direction of appliedmagnetic field is somewhere in between the magnetic easyand hard axis of the film. A very small value of remnantM r=1.1μB/f.u.and a large coercive field of H C=0.35 T as compared to that of the TCMO single crystal is in linewith significant amount of Mn 3+disorder along with Co2+- O-Co2+and Mn4+-O-Mn4+interactions, contributing to the AFM phase boundaries and developing FM/AFM competitiveinteractions [ 12]. The presence of Mn 3+/Co3+ions, making a finely distributed partial disorder at nm scale within thefilm, as well as a dominance of Co 2+/Mn4+oxidation states responsible for the main FM phase has been further confirmedby EELS mapping in high-resolution TEM [ 24], shown in Fig. SM-2. In Fig. 2(a) we present the temperature dependence of the real part of dielectric constant, /epsilon1/prime(T), measured for different frequencies, showing a broad maxima of /epsilon1/prime(T), which is of the same order as that observed in the LCMO film [ 14]. The temperature of the maximum, Tm, depends on the frequency and shifts to higher temperatures with increasing frequency.This kind of glassy behavior is very new for the A 2CoMnO 6 double perovskites and probably indicates a ferroelectric re- laxor behavior, which is not associated with any structuraltransition in the system. With further lowering the temperature(T/lessorequalslant100–120 K) the /epsilon1 /prime(T) starts to increase again possibly due to the electronic contribution from the conducting Nb:STOsubstrate and the substrate/film interface. The observed ferroelectric relaxor behavior can be fitted with the Curie-Weiss law, /epsilon1 /prime=C/(T−θ) in the paraelectric region [ 25] above the frequency dependent relaxor transition temperature, Tm≈150–200 K, as shown in Fig. 2(b).T h e fitting parameters are presented in Table SM-1 [ 26]. At low frequencies the data were fitted well with the Curie constant, C=8×103, and the Curie temperature, θ∼150 K, respec- tively. With increasing frequency, the data start to deviate fromthe Curie-Weiss law. As for high frequencies the transitiontemperatures, T m, increase, the fitting range becomes to be not far away from the transition temperature. The short rangecorrelations among electric dipoles emerge, being frequencydependent and causing deviation from an ideal ferroelectric 094426-3R. MANDAL et al. PHYSICAL REVIEW B 101, 094426 (2020) FIG. 3. (a) Gaussian fit for the temperature dependent dielectric constant /epsilon1/prime(T) and deconvolution of three peaks signifying the presence of three different types of PNRs; (b), (c) and (d) are Arrhenius fits for three relaxation times related to these three different peak temperatures: T1 m,T2 mandT3 m, respectively. behavior. The plausible interpretation is the formation of small Polar Nano Regions (PNR), having different responses forsufficiently high frequency. For an ideal ferroelectric case theCurie-Weiss fit should not depend on the applied frequency. In Fig. 2(c) one can see a broad relaxation peak ( T m) in the temperature dependence of the dielectric loss, /epsilon1/prime/prime(T), which also shows a frequency dispersion. The observed re-laxation in both real, /epsilon1 /prime(T), and imaginary part, /epsilon1/prime/prime(T), of dielectric constant could be interpreted in the frameworkof a dipolar glass model [ 27]. It considers small dipolar regions, induced in the system by the B-site disorder, the dipole moment of which fluctuates/vibrates thermally at high temperatures T>T m. Analogously to the spin glass, the dipoles are expected to be frozen at low temperatures. Thefreezing temperature ( T f) is finite if the interaction between dipoles is strong enough. The dynamics of a PNR can bedescribed by the so called V ogel-Fulcher (VF) formalism f −1=τ0exp[Ua/KB(Tm−Tf)], where fis the frequency of an applied electric field, and Tmis the relaxation peak maxima [28–31]. If the electrostatic interaction among the dipoles is not strong enough to freeze them cooperatively, then dipolescan vibrate with external ac electric field at any finite temper-ature and can show a thermally activated Arrhenius behaviordown to T f→0 K. Both VF and the Arrhenius law f−1= τ0exp[Ua/KB(Tm)] as well as a power law were tried to fit the data. The best fit was obtained with Arrhenius behavior,shown in Fig. 2(d) for both /epsilon1and/epsilon1. The activation energies, calculated from the fits are Ua=0.25 eV and 0.16 eV for /epsilon1and/epsilon1, respectively; they look physically reasonable and are of the same order of magnitude as activation energies∼0.1 eV for a typical relaxor ferroelectric [ 26]. The evaluated relaxation time, τ 0=1.59×10−12s, corresponds roughly to characteristic phonon frequencies of few THz, obtained fromRaman spectra of similar double perovskites films [ 11]. The observed asymmetry in the relaxor peak in /epsilon1 /prime(T) has been analyzed in terms of the diffuse phase transitionmodel, which describes the temperature dependent dielectricpermittivity as well as the size distribution of PNRs by means of a Gaussian function: 1√2πσ2exp[−(T−Tm)2 2σ2][26]. Our exper- imental data can be fitted well by this distribution function and the /epsilon1/prime(T) relaxor behavior was found to be a superposi- tion of three different maxima, denoted as T1 m,T2 m, and T3 m, shown in Fig. 3(a) forf=10 kHz. The experimental data were fitted for all frequencies and three frequency dependenttemperatures are presented in Table SM-2 Ref. [ 32]. Again VF, activation law and power law were tried to fit these threetemperatures and we found the Arrhenius behavior providesthe best fits as shown in Figs. 3(b),3(c), and 3(d). The cal- culated activation energies, 0.22, 0.30, and 0.32 eV , look alsophysically reasonable. From the deconvolution of the /epsilon1peaks we obtained three types of PNRs, which may differ in theirmicroscopic origin and have distinguishable size distributions. 094426-4MAGNETO-DIELECTRIC EFFECT IN RELAXOR … PHYSICAL REVIEW B 101, 094426 (2020) These three distinct classes of dipoles can be related to their microscopic origin, taking into account that electric dipolesin TCMO may originate from the oxygen bonds with Co 2+ and Mn4+ions in the CoO 6and MnO 6octahedrons, which are getting distorted/polarized in an applied electric field. Thethird contribution could be caused by the disorder-inducedpresence of the Mn 3+as well as of Co3+ions. All these PNRs started to interact with lowering temperature. Due tothe difference in their distributions as well as in the activationenergies they respond differently with external frequency andwith temperature. The overall macroscopic response hence,shows a ferroelectric relaxation. Instead of the ferroelectricrelaxor-glass-like behavior, described by the VF formalism asa mostly suitable for a dipolar glass model, so our systemcould be more accurately interpreted by the Arrhenius law.This scenario is an indication that partially disordered TMCOfilm could be considered as a superparaelectric, i.e., a blockedrelaxor at low temperatures and nonzero frequencies, ratherthan as a dipolar glass. In order to inspect the mechanisms of interactions among these dipoles, the dielectric loss ( ε´´) has been analyzed in the frequency domain for different temperatures. For the used frequency range, f=20 Hz–2 MHz, two main mechanisms are known to be responsible for the dipolar relaxation: (1)the Maxwell-Wagner (MW) mechanism originated from thelocal charge accumulation at grain boundaries [ 33] and (2) the Debye relaxation, which is the dipolar contribution from thehopping of charge carriers among asymmetric sites (Mn 4+, Co2+and Mn3+)[34]. Figure 4(a) shows the dielectric re- laxation over the frequency range for the temperatures inthe relaxor transition regime. At very low frequencies, f< 200 Hz, one can see a MW behavior, which is manifestedby a linear decrease of /epsilon1with increasing frequency in the logarithmic ωscale. With further increasing frequency the data start to deviate from the MW behavior and displaytwo distinct Debye relaxation peaks at frequencies, denotedasf 1=1/τ1and f2=1/τ2. These two relaxation behaviors could be recognized as β(τ1)−andα(τ2)-like processes in a glassy system [ 35]. Indeed, the frequency f2increases with in- creasing temperature and merges with the f1at a temperature TC, which is comparable with the glass transition temperature in an amorphous material [ 36]. The slower α(τ2) relaxation is, analogously to glasses, a primary relaxation process in thisdipolar glassy system and can be assigned to the major chargetransfer between the Co 2+-Mn4+sites. A secondary, or faster βprocess, develops from the localized disorder or minor sites, occupied likely by Mn3+and/or Co3+ions. Interestingly, one can see that these two processes merge at a temperaturearound 190 K giving a single relaxation peak at around 100kHz as shown in Fig. 4(b). MD analysis has been done by dielectric measurements in an applied magnetic field, B=8T ,a t f=100 kHz. Here we can observe a characteristic change in the dielectric constantin the vicinity of the relaxor transition at T m∼195 K at this frequency as shown in Fig. 5(a). The temperature dependence of the MD coupling constant єMD=є/prime(8T)−є/prime(0T) є/prime(0T)×100 along with the derivative of dielectric constant (δє/prime δT)i nF i g . 5(b) shows that |єMD|increases by cooling down the system and takes the highest value of 4% close to the relaxor transition. FIG. 4. (a) Frequency dependence of the dielectric loss part /epsilon1/prime/prime(T) measured for different temperatures in the frequency range of f=20 Hz–2 MHz; (b) Temperature dependence of the two different Debye relaxation times τ1=1/f1andτ2=1/f2, evaluated from the positions of peaks in 4(a). Moreover, the MD effect is negative, i.e., ε´ decreases in applied magnetic field. By further lowering the temperatureMD coupling decreases again as dipoles are started freezing.The absence of the magnetoresistance [ 37]( s e eF i g .S M - 3 ) confirms that the MD coupling is intrinsic to the dipolespresent in the material. Moreover, the temperature dependenceof electrical resistivity, ρ(T), can be fitted by a variable-range- hopping Mott’s behavior [ 38],r(T)=r 0×exp(T0/T)1/4, with T0=211 K and ρ0=7×10−4/Omega1cm, which illustrates a disorder-dominated charge transport. The so-called Motttemperature T 0=211 K is given [ 38] by the formula, kBT0= b/(g(EF)×R3lo c), where β=21 and g(EF) and Rlocare the density of states at the Fermi level and localization radius ofcharge carriers, respectively. Considering characteristic val-ues of g(E F)∼1027–1028(eV∗m3)−1we get Rloc∼10–5 nm, which is in a good agreement with the nm-scale Co3+/Mn3+ disorder in the EELS spectra [ 24,39] (see Fig. SM-2). As indicated previously, mostly the Debye processes [ 35], originating from a charge transfer between dipoles, contribute 094426-5R. MANDAL et al. PHYSICAL REVIEW B 101, 094426 (2020) FIG. 5. (a) Temperature dependence of the dielectric constant /epsilon1/prime(T) at 100 KHz without and with applied external magnetic field of 8 T; (b) Temperature dependence of derivative of dielectric constant (δє/prime δT) along with the magneto-dielectric coupling constant єMD(%) showing a maximum 4% magneto-dielectric coupling. to the dielectric constant at high frequencies. Dielectric con- stant increases up to the relaxor transition mainly from thecontribution of activated dipoles due to the charge transferfrom Mn 3+to other sites. Below the relaxor transition Mn4+ and Co2+start interacting magnetically with the other dis- order sites and dielectric relaxation peak splits at the sametemperature indicating a charge transfer between them [seeFig. 4(b)]. As the charge transfer is coupled to and depends on the spin arrangement, the short-range spin-spin interactiontries to restrict the charge hopping and we can see the secondrelaxation peak in Fig. 4is not that much pronounced as the first one. With applied strong magnetic field this processis further interrupted and dielectric constant further reduce,causing a 4% negative MD coupling at the same temperature. We have to say that the exact physical mechanism of the observed MD coupling is not known. However, by comparingit with a quite similar high temperature MD effect in the La 2NiMnO 6[15], a hint into a probable coupling between the disorder-induced local spins in the paramagnetic regionand electric dipoles can be given. One can speculate on thecoexistence of very similar temperature scales, i.e., for α-β merging around 190 K, for relaxor transition, T m∼195 K, and for the resistivity disorder scale T0∼211 K [ 37] (see Fig. SI-3). Surprisingly, the latter is very close to the chargeordering (CO) temperature in manganites [ 40,41], i.e., T 0∼ TCO. Note that a peak in a permittivity in THz region was also observed at T∼TCOin PrCaSrMnO [ 41]. Likely, the close- ness of these temperature scales, where a disorder-inducedshort range Mn 3+/Co3+charge correlations, stabilized at T< T0∼TCO, coexist and interplay with superparaelectric dipole regions could result in an unusual MD coupling. It is alsohighly unexpected to have a spin glass transition at such hightemperature that can explain this MD coupling in terms ofmultiglass behavior. Due to the huge background signal fromthe substrate and the interface we were unable to resolveany short range spin correlations for T/greatermuchT Cfrom the Curie- Weis fit of the 1 /χ(T) curve. Further detailed studies of the relationship between the B-site disorder and MD coupling are necessary to elucidate its mechanism. In summary, we have grown monoclinic phase of Tb2CoMnO 6double perovskite thin film on Nb : SrTiO 3(100) by using the MAD technique. TEM/EELS mapping shows thepresence and distribution of both Co 2+as well as Co3+ions in the film, evidencing a partial B-site disorder, further confirmed by the observed reduction of the saturation magnetization atlow temperatures. The ferromagnetic T C=110 K was slightly higher as compared to the bulk value due to an in planetensile strain. Two different dielectric relaxation peaks ( βand α) have been observed that merge at a temperature close to the relaxor glass transition. Moreover, we observed anunexpected high temperature relaxor-glass-like transition anda superparaelectric behavior, at which a probable couplingto short range correlated local spin moment results in a 4%magneto-dielectric coupling. ACKNOWLEDGMENTS R.M. acknowledges financial support from the Erasmus Plus programme, European Union, Georg-August-UniversitätGöttingen, and IISER Pune. R.M. and M.C. are grateful to S.Yadav for helping in Dielectric measurement. V .R. and V .M.acknowledge financial support from Deutsche Forschungsge-meinschaft (DFG) via SFB 1073 (TP A02, TP Z02) as well asvia DFG Projects No. MO-2254-4 and No. RO-5387/2-1. [1] Y . Tokura, S. Seki, and N. Nagaosa, Rep. Prog. Phys. 77, 076501 (2014 ). [2] N. A. Spaldin and R. Ramesh, Nat. Mater. 18,203 (2019 ). [3] M. Bibes and A. Barthélémy, Nat. Mater. 7,425(2008 ). [4] M. 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PhysRevB.97.115123.pdf

PHYSICAL REVIEW B 97, 115123 (2018) Abelian Chern-Simons theory for the fractional quantum Hall effect in graphene Christian Fräßdorf Dahlem Center for Complex Quantum Systems and, Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany (Received 10 December 2017; published 12 March 2018) We develop a theory for the pseudorelativistic fractional quantum Hall effect in graphene, which is based on a multicomponent Abelian Chern-Simons theory in the fermionic functional integral approach. Calculations areperformed in the Keldysh formalism, directly giving access to real-time correlation functions at finite temperature.We obtain an exact effective action for the Chern-Simons gauge fields, which is expanded to second order in thegauge field fluctuations around the mean-field solution. The one-loop fermionic polarization tensor as well asthe electromagnetic response tensor in random phase approximation are derived, from which we obtain the Hallconductivities for various FQH states, lying symmetrically around charge neutrality. DOI: 10.1103/PhysRevB.97.115123 I. INTRODUCTION The integer quantum Hall effect (IQHE) is a remarkable experimental discovery of the early 1980s, since it provesquantum mechanics at work on macroscopic scales [ 1]. In a nonrelativistic two-dimensional electron gas (2DEG) atlow temperatures and in high external magnetic fields, theHall conductivity shows a plateau structure as a function ofthe magnetic field or chemical potential occurring at integermultiples of the “conductance quantum” e 2/h. Remarkably, the existence of these plateaus can already be understood insimple noninteracting models by the formation of discrete,equidistant energy levels, the Landau levels (LLs) [ 2]. In sharp contrast to an ordinary 2DEG with its parabolic band structure, in the vicinity of the charge neutrality point,the band structure of graphene mimics the energy-momentumdispersion of massless, relativistic Dirac particles [ 3–7]. When subjected to strong magnetic fields such a pseudorelativisticdispersion relation has profound consequences on the LLs,which, in turn, influences the measurable Hall conductivity[6,8]. In theoretical studies one finds an anomalous quantiza- tion, where each of the four fermionic flavors in graphene con-tributes a half-integer, n+1/2, to the total Hall conductivity [8–10] σ 0,xy=±4/parenleftbigg n+1 2/parenrightbigge2 h,n=0,1,2,.... (1) The additional fraction of 1 /2 can be traced back to the existence of a half-filled Landau level located directly at thecharge neutrality point, which has only half the degeneracyof the other levels (the spectral anomaly), while the factor offour is a direct consequence of the four independent SU(4)symmetric flavors of charge carriers in the low-energy Diracmodel. With the recent success of graphene’s experimentalisolation, these theoretical predictions became experimentallyaccessible and could indeed be verified [ 11,12]. Shortly after the IQHE was discovered in nonrelativistic semiconducting devices, measurements on high quality sam-ples revealed the occurrence of additional plateaus at certainfractional fillings [ 13,14], and more recently this effect has also been observed in graphene [ 15–17]. For this fractional quantum Hall effect (FQHE), electron-electron interactionsare an essential ingredient in the theoretical treatment to gainfurther understanding of the underlying physics. The maindifficulty here is that the noninteracting Landau levels, formingthe basis of the analysis, are macroscopically degenerate. As aconsequence, conventional perturbative approaches inevitablyfail, making the FQH system a prime example for stronglycorrelated matter, which has to be analyzed by truly nonper-turbative methods. Based on the seminal work of Laughlin [ 18], Jain introduced the idea that physical electrons/holes and magnetic flux quanta,or vortices, form bound states, so-called “composite fermions”[19]. Due to the process of flux nucleation, the magnetic field is reduced, leading to a new set of effective Landau levels thatare occupied by the composite fermions. The integer fillingsof those effective LLs map to the fractional fillings observedin the experiments. Thus, the fractional QHE of ordinaryfermions can be understood as an integer QHE of compositefermions [ 2,20]. This intuitive, albeit rather unconventional picture led to a vast body of theoretical predictions, which could be verified experimentally to a large extent [ 21–36]. Applying these ideas to the Dirac electrons in graphene leadsto the notion of “composite Dirac fermions.” Accordingly,one might expect that their pseudorelativistic spectrum, whichleads to the anomalous quantization of the Hall conductivityin the noninteracting case, leaves its marks in the FQHE. In the theoretical treatment of the FQHE, there are several slightly different approaches to realize Jain’s idea of flux-binding. Within the trial wave function approach, vortices areattached in the form of Jastrow factors multiplying the many-body wave function of noninteracting fermions in an IQH state[2,20]. To make use of this strategy in graphene, one considers a completely empty or completely filled lowest LL—usuallythe one at the charge neutrality point—as the vacuum stateand attaches flux quanta to the physical electrons/holes thatpartially fill/deplete this energy level. Thereby it is assumedthat the effective LLs and their associated single-particle wave 2469-9950/2018/97(11)/115123(20) 115123-1 ©2018 American Physical SocietyCHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) functions, which make up the IQH state, are notof the Dirac type, but coincide with the nonrelativistic Schrödinger typeones [ 8,37,38]. This assumption is justified by the fact that the quenched Hamiltonian of graphene projected to the lowestLL is identical to the Hamiltonian encountered in systemswith a nonrelativistic parabolic dispersion [ 8,37–39]. Loosely speaking, graphene electrons confined to the lowest LL losetheir identity as Dirac fermions upon projection, such that theonly impact of graphene’s unconventional band structure is theSU(4) symmetry of the ansatz wave function, which derivesfrom the SU(4) symmetry of the individual fermionic flavors.This construction leads to the conventional Jain sequenceand wave functions. Straightforward generalizations of thisapproach are given by Halperin wave functions [ 8,40–44], which potentially break the SU(4) symmetry down to SU(2) ⊗2 or even U(1)⊗4. Despite its indisputable successes, the trial wave function approach has several drawbacks, two of which we want tocomment on further. First, it crucially depends on projectedHamiltonians, which typically neglect LL mixing. Whilefor nonrelativistic systems for the most part this is only aminor issue, since at large magnetic fields LL mixing issuppressed as 1 /√ B[45], in graphene, it is a substantially more severe problem. Here, LL mixing is controlled by the fine structure constant α, which is independent of the magnetic field and—more importantly—genuinely large ( α≈2.2i n suspended graphene), making LL mixing a nonperturbativeproblem already on the level of the Hamiltonian [ 46,47]. Hence, although the kinetic energy may be quenched within apartially filled LL, the electrons in graphene still feel their Dirac heritage. Yet, if LL mixing is taken into account, at least per- turbatively, Refs. [ 46,47] reported—quite surprisingly—that it has practically no effect on the wave functions in the zerothLL. Not entirely decoupled from the above, the second mainproblem is concerned with particle-hole symmetry, or ratherits strong breaking inherent in the construction of trial wavefunctions. The origin of paticle-hole symmetry is different for nonrelativistic and relativistic systems. For the former, it is only an emergent symmetry of the lowest LL projectedHamiltonian, but for the latter, it is an exact symmetry ofthe unprojected Hamiltonian (and, hence, is a good symmetryeven if LL mixing is taken into account). The constructionof particle-hole conjugated wave functions is still possible,but the explicit symmetry breaking is not only unsatisfying but also comes with its own complications, see, for example, Refs. [ 48,49] for a more elaborate discussion. A complementary approach to the construction of explicit wave functions is the Chern-Simons field theory, which doesnot rely on a projection to the lowest LL. Here, magneticflux tubes—which should be distinguished from the vorticesof the wave-function approach—are attached to the fermionic degrees of freedom either via a singular gauge transformation [20,50], or equivalently via a minimal coupling of a Chern- Simons gauge field to the kinetic action in addition to akinetic Chern-Simons term [ 51–55]. (See also Ref. [ 56]f o r a similar treatment involving bosons.) In the process, ordinaryfermions are transformed into composite fermions, whosenature—Schrödinger or Dirac—is determined by the structure of the kinetic action. Hence, as opposed to the picture drawn inRef. [ 37], the Chern-Simons composite fermions in graphene are actual Dirac-type particles. Accordingly, one might expect that the spectral anomaly of the composite Dirac fermions (thehalf-integer quantization of the filling fractions) enters the an-alytical formulas for the total filling fraction/Hall conductivityof the electronic system. However, the graphene Chern-Simonstheories proposed in Refs. [ 57,58] attach flux to the physical electrons/holes with respect to the bottom/top of the lowest LL, which eliminates the spectral anomaly and yields predictions for the total filling fraction that are in accordance with thewave-function approach. Concerning LL mixing, the Chern-Simons approaches reside on the other side of the spectrum,meaning there is a large amount of LL mixing [ 59], which is a result of the Chern-Simons transformation and the absence ofprojection. Regarding the nonperturbative nature of LL mixing in graphene, this feature should not necessarily be considered a flaw, but the question, if the Chern-Simons induced LL mixingdescribes the physical reality accurately, remains. Although the non-Dirac nature of the composite fermions in graphene’s lowest LL appears to be fully establishedby the results of Ref. [ 47], the conclusion that theoretical frameworks that employ Dirac-type composite fermions,such as the aforementioned pseudorelativistic Chern-Simonstheories of Refs. [ 57,58], lose their viability would be too hasty as Son’s work, Ref. [ 49], impressively shows. Focussing on the conventional nonrelativistic FQH system, Son proposeda manifestly particle-hole symmetric, pseudorelativisticeffective model, which declares Jain’s composite fermion tobe a Dirac particle by nature. Specifically, the ν=1/2 state is described by a charge neutral Dirac particle interacting withan emergent gauge field ( notof the Chern-Simons type) that forms a Fermi liquid, while Jain’s principal sequence aroundhalf-filling can be explained as the IQHE of those Diracquasiparticles, fully incorporating the particle-hole symmetryof the lowest Landau level. In contrast to Son’s effective model, in the present paper, we employ a rather standard microscopic Chern-Simons theory,similar to Refs. [ 57,58]. The crucial difference to those works is the reference point at which we implement Chern-Simonsflux-attachment, namely the particle-hole symmetric Diracpoint at charge neutrality. This shift in the reference pointshould not be underestimated as a mere shift in the total fillingfraction, since it allows for a flux-attachment scheme that isdistinctively different from the aforementioned approaches.Instead of attaching flux to the physical electrons/holes, itis possible to attach flux to the charge carrier density, thatis, electron- or hole-like quasiparticles measured from thecharge neutrality point. In particular, we obtain a mean-fieldequation which involves the charge carrier density, instead ofthe electron/hole density, and within the calculation of Gaus-sian fluctuations, we naturally encounter pseudorelativisticpropagators experiencing an effective magnetic field, whichincorporate the spectral anomaly. Our central result is theelectromagnetic polarization tensor in linear response to anexternal perturbation, which—among others—gives accessto the Hall conductivity of the multicomponent fractionalquantum Hall system: σ xy=/summationdisplay ασα 0,xy−/summationdisplay α,βσα 0,xy(ˆσ0,xy+ˆK−1)−1 αβσβ 0,xy.(2) 115123-2ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) Here,σα 0,xyis the Hall conductivity of a noninteracting, single flavour α, which is half-integer quantized at low temperatures, due to the Dirac nature of the composite fermions, ˆ σ0,xyis a diagonal matrix containing these single flavour conductivities,and ˆKis an integer-valued symmetric matrix accounting for the flux-attachment [ 60]. We show that Eq. ( 2) leads to particle-hole symmetric Hall plateaus around charge neutrality, if positive flux-attachmentto electronlike and negative flux-attachment to hole-like quasi-particles is considered. This observation enables us to constructmanifestly particle-hole symmetric filling fractions as func-tions of the chemical potential. This result seems surprising,since the Chern-Simons term explicitly breaks particle-holesymmetry—which is why Son discarded such a term inhis effective theory [ 49]—irrespective of the reference point where the Chern-Simons flux is attached. Since this symmetrycannot be generated dynamically, particle-hole symmetric Hallplateaus are not expected to occur in such a symmetry brokentheory. The puzzle is resolved as follows. The standard def-inition of the particle-hole transformation involves fermionicand bosonic fields only, but leaves the Chern-Simons couplinguntouched. By allowing the coupling to depend on the sign ofthe carrier density, we have altered the flux-attachment pre-scription in such a way to make it consistent with the standardparticle-hole symmetry transformation. One may also interpretit the other way around: we use the standard flux-attachmentbut change the symmetry transformation to involve a sign flip of the Chern-Simons coupling. Thus, one may argue that the Chern-Simons term only breaks particle-hole symmetryin a weak sense, since it can be circumvented altogether bysufficiently modifying the symmetry transformation or theflux-attachment prescription, alleviating the seeming incom-patibility of particle-hole symmetry and Chern-Simons theory.Furthermore, we show that the above formula reproduces theHall conductivities proposed in Refs. [ 61,62] as special cases, as well as several other filling fractions that have been obtainedin the wave-function approach. In this paper, we employ the real-time Keldysh formal- ism, which offers several technical advantages in comparisonto the conventional real-time ground-state formalism. Thisformulation will allow for a natural regularization of theotherwise ill-defined mean-field equations, upon which theflux-attachment interpretation is based, and it additionallyyields results that are valid at finite temperature, which comewithout further calculational costs. Our exposition is inspiredby the original work of Refs. [ 51,53], where the fermion Chern-Simons theory for the FQHE of nonrelativistic matterhas been introduced. Since there are several subtle differencesdue to the Dirac nature of the quasiparticles and the Keldyshformulation, we will present the theory in a self-containedmanner. The outline of the article is as follows. In Sec. II, we describe the field theory of interacting Dirac fermionscoupled to statistical Chern-Simons fields with the Abeliangauge group U(1) ⊗4. In the subsequent section, we derive an exact effective action for the statistical gauge fields and discussits Gaussian approximation around the mean-field solution ofthe quantum Hall liquid. Section IVcontains our main results. We address the topic of gauge fixing and calculate the full elec-tromagnetic response tensor together with Hall conductivitiesfor a selected set of states. We conclude in the final section.Further technical details of the computation are given in two appendices. II. ABELIAN CHERN-SIMONS THEORY The starting point of our considerations is the second quan- tized low-energy Hamiltonian for interacting Dirac electronsin monolayer graphene (¯ h=1), H=/integraldisplay /vectorx/Psi1†(/vectorx)ˆHD/Psi1(/vectorx)+1 2/integraldisplay /vectorx,/vectoryδn(/vectorx)V(/vectorx−/vectory)δn(/vectory),(3) withδn(/vectorx)=/Psi1†(/vectorx)/Psi1(/vectorx)−¯n(x). The fermionic field opera- tors/Psi1and/Psi1†are, in fact, eight-component spinors /Psi1≡ (/Psi1↑/Psi1↓)/intercal, with /Psi1σ≡(ψAK+ψBK+ψBK−ψAK−)/intercal σ. The indices A/B,K ±, and↑,↓represent sublattice, valley, and spin degrees of freedom, respectively. The first term—the Dirac part of the Hamiltonian— describes the dynamics of the four flavors of Dirac electronsα=(K +↑,K−↑,K+↓,K−↓). Within the basis chosen above, the single-particle Hamiltonian ˆHDassumes a diagonal form in flavor space, ˆHD=diag( HD,K +↑,HD,K −↑,HD,K +↓,HD,K −↓), (4) where the Hamiltonian for each individual flavor reads HD,α=−καivF/vectorσ·/vector∇. (5) Here,κα=±1 distinguishes between the two valleys K±, and vFis the Fermi velocity with the numerical value vF≈c/300. Note that we indicated the 4 ×4 matrix structure of the flavor space in Eq. ( 4) with a hat symbol explicitly, while the 2 ×2 matrix structure of the sublattice space is implicit. The second term of Eq. ( 3) describes two-particle interac- tions between the Dirac fermions. The interaction amplitudeis given by the instantaneous, U(4) symmetric Coulomb inter-action V(/vectorx−/vectory)=e 2 /epsilon1|/vectorx−/vectory|. (6) The term ¯n(/vectorx)=/summationtext α¯nα(/vectorx) in the definition of the bosonic operator δn(/vectorx) is a background density. In general, it is space- and possibly even time-dependent, but for our purposes, how-ever, will be constant. It acts as a counterterm, that cancels thezero momentum singularity of the bare Coulomb interaction.Furthermore, /epsilon1is the dielectric constant of the medium (being unity in vacuum), which describes the influence of a substrateon the bare Coulomb interaction. In this paper, we employ the Keldysh formalism to formu- late a real-time theory at finite temperature and density for thefour interacting flavors of Dirac particles in graphene, that aresubject to an external magnetic field and coupled to four sta-tistical U(1) gauge fields. Within the Keldysh formulation, thedynamical degrees of freedom of the theory are defined on theSchwinger-Keldysh contour, which is a closed contour in thecomplex time plane [ 63,64]. The time arguments of the field op- erators are elevated to contour-time and correlation functionsare derived as the expectation value of their “path ordered”products. As shown in Fig. 1, the time contour starts at a refer- ence time t 0—at which the initial density matrix is specified— extends into the infinite future along the real axis and returns tothe reference time eventually. Here, we are mainly interested in 115123-3CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) FIG. 1. Schwinger-Keldysh closed time contour in the complex time plane with forward ( C+) and backward time branch ( C−). Here, t0is a reference time, where an initial density matrix enters the theory. Since we are only interested in the system’s linear response properties close to thermal equilibrium, we send the reference time to the remote past (t0=− ∞ ) outright. the thermal equilibrium state in linear response to an external electromagnetic perturbation. Therefore we send the referencetimet 0to the infinite past, which erases all the information about possible nontrivial initial correlations and transientregimes [ 65,66]. As a consequence, the quantum kinetic equations are of no further concern, since they can be triviallysolved by the well-known thermal distribution functions [ 64]. Before we discuss the field theoretic model in its action formulation, a few remarks concerning notational conventionsare in order. First, to a large extent, we will work within the abstract contour-time representation, and only switch to a physical real-time representation at the end of Sec. IIIwhen we discuss Gaussian fluctuations of the bosonic effective actionaround the mean-field solution of the fractional quantum Hall liquid. The major advantage of the contour-time representation is that it allows for a compact and concise notation, resemblingthe zero temperature vacuum (or ground state) theory, yetencoding the full information of thermal fluctuations [ 63]. Furthermore, we employ a covariant notation, where upper and lower case greek letters μ, ν, andλdenote contra- and covariant components of a Minkowski three-vector,respectively. As usual, a repeated index implies summationaccording to the Einstein summation convention. Thissummation rule is lifted if a repeated index is bracketed. [This statement will only apply for repeated flavor space indices αandβ,s e eE q .( 13) for instance.] The convention for the flat Minkowski metric is chosen to be η μν=diag(1 ,−1,−1), and natural units (¯ h=c=1) are used throughout the article. Lastly, space-time integrations will be denoted by /integraldisplay C,x≡/integraldisplay Cdt/integraldisplay d2r, (7) where Cindicates that the time integration is performed along the Schwinger-Keldysh contour, and x=xμ=(t,/vectorr) labels (contour-)time and spatial variables. After introducing these general notational conventions, we proceed to describe the details of the model. The entire physical content of the theory is summarized by the coherent state functional integral [ 63,64,67] Z/bracketleftbig eAμ+Aα μ/bracketrightbig =/integraldisplay DψDψ†DaeiS[ψ,eA μ+Aα μ,aα μ],(8) which is a generating functional of correlation functions. The action Sin the exponential can be written as a sum of three terms: S/bracketleftbig ψ,eA μ+Aα μ,aα μ/bracketrightbig =SD/bracketleftbig ψ,eA μ+Aα μ+aα μ/bracketrightbig +SCoul[ψ]+SCS/bracketleftbig aα μ/bracketrightbig , (9)where the first two terms, involving the fermionic fields, are readily obtained from the Heisenberg picture HamiltonianH(t) by the definition S D[ψ]+SCoul[ψ]=/integraldisplay C,t/parenleftbigg/integraldisplay /vectorr/Psi1†(x)i∂t/Psi1(x)−H(t)/parenrightbigg .(10) The bosonic fields within the Dirac part of the action, Aμ,Aα μ, andaα μ, are introduced via the minimal coupling pre- scription. They represent an external electromagnetic potential,local two-particle source fields, and the statistical gauge fields,respectively. The source fields will later be used to generatethe desired correlation functions. The Dirac action can bewritten compactly as a quadratic form of an eight-componentGrassmann spinor /Psi1[5,9] S D/bracketleftbig ψ,eA μ+Aα μ+aα μ/bracketrightbig =/integraldisplay C,xy/Psi1†(x)ˆG−1 0(x,y)/Psi1(y).(11) The matrix ˆG−1 0is the inverse contour-time propagator, which inherits the flavor diagonal structure from the single-particleHamiltonian ( 4): ˆG −1 0=diag/parenleftbig G−1 0,K+↑,G−1 0,K−↑,G−1 0,K+↓,G−1 0,K−↓/parenrightbig . (12) According to Eq. ( 5), the dynamics of each flavor is governed by the pseudorelativistic, massless Weyl operator G−1 0,α(x,y)=δC(x−y)/parenleftbig iσμ (α)D(α) μ+μα/parenrightbig . (13) Here,δC(x−y)=δC(x0−y0)δ(/vectorx−/vectory) involves the contour- time delta function [ 63] and σμ α≡(σ0,καvFσ1,καvFσ2)i sa three-vector of Pauli matrices, acting in sublattice space. Thegauge covariant derivative D α μ=∂μ+ieAμ(xμ)+iAα μ(xμ)+iaα μ(xμ) (14) contains the aforementioned covariant vector potentials Aμ,Aα μ, andaα μ. For the external potential Aμ, we choose the Landau gauge, Aμ(xμ)=(0,Bx2,0)=(0,By, 0), to describe a uniform and static magnetic field Bperpendicular to the graphene plane. Note that it does not depend on the flavorindex α, so that all flavors universally couple to the same field. The source fields A α μand the statistical gauge fields aα μ, on the other hand, do carry a flavor index and, thus, couple to each fermionic flavor individually. Such a couplingbreaks the global U(4) symmetry of the theory without Chern-Simons fields down to a local U(1) ⊗4symmetry. Finally, we introduced a flavor dependent chemical potential μα, allowing for independent doping of the individual flavors. Physically,this flavor dependence may be thought of as originating froma generalized Zeeman term [ 8]. The Coulomb interaction part requires no further discussion as it is directly obtained from the interaction part of theHamiltonian ( 3), S Coul[ψ]=−1 2/integraldisplay C,xyδn(x)V(x−y)δn(y), (15) withV(x−y)=V(/vectorx−/vectory)δC(x0−y0). The third term in the action ( 9) is the kinetic term for the four statistical gauge fields, which is given by a generalized 115123-4ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) Chern-Simons action [ 51,53,54,59] SCS/bracketleftbig aα μ/bracketrightbig =1 2(ˆK−1)αβ/integraldisplay C,xεμνλaα μ(x)∂νaβ λ(x). (16) Herein εμνλis the total antisymmetric Levi-Civita tensor (we use the convention ε012=1), and ˆKis a regular, i.e., invertible, symmetric 4 ×4 matrix, ˆK=2π⎛ ⎜⎜⎝2k1m1n1n2 m12k2n3n4 n1n32k3m2 n2n4m22k4⎞ ⎟⎟⎠, (17) with integers ki,mi,ni. For those configurations of integers where ˆKhappens to be singular, Eq. ( 16) needs to be regu- larized. This may be achieved by adding a diagonal matrix ˆR=2πdiag(+iη,−iη,+iη,−iη)t oE q .( 17), where ηis an infinitesimal (the signs therein are purely conventional).The physical meaning of the Kmatrix is to attach statistical magnetic flux to the fermions. This feature will become moreclear in the next section when we discuss the stationary phaseapproximation. The theory we described above possesses a local U(1) ⊗4 symmetry, in comparison to the symmetry of the original model of interacting electrons in graphene, being a global U(4) flavor symmetry (U(2)×U(2), respectively, if one takes into account a Zeeman term [ 5]). It has to be emphasized that the symmetry is broken explicitly by considering the flavordependent chemical potential in addition to the U(1) ⊗4sym- metric gauge field coupling. As pointed out by the authors ofRef. [ 54], who studied the FQHE for nonrelativistic fermions in bilayers, as well as SU(2) symmetric monolayers, the originalU(4) symmetry may only be generated dynamically (once theflavor dependence of the chemical potential is neglected [ 54]). Therefore it is expected that some of the fractional quantumHall states we obtain in this work—after certain necessaryapproximations have been made—may not be realized in theexact theory, as they could be destabilized by higher-orderfluctuations. A manifestly U(4) [respectively, U(2) ×U(2)] symmetric theory, on the other hand, could be constructedin analogy to Refs. [ 52,54], by considering an appropriate non-Abelian generalization of Eq. ( 16), with a corresponding set of non-Abelian statistical gauge fields, coupling gaugecovariantly to holon and spinon fields; see Sec. Vfor a brief discussion. Clearly, such a non-Abelian gauge theory is inmany aspects significantly more complex than the Abeliantheory of the present article and we leave its construction andanalysis for future work. As a final remark we want to stress that the partition function (8) as it stands is not well-defined. Since the Chern-Simons fieldsa α μare gauge fields, the functional integral contains an infinite summation over all, physically equivalent orbits ofpure gauge, leading to a strong divergence. In order to extractphysically meaningful information from the partition function,the gauge equivalent orbits have to be removed, such that eachgauge field configuration in the functional integral uniquelycorresponds to a physical field configuration. To this end,we employ the well-known Fadeev-Popov procedure [ 68], but we postpone the details of the discussion to Sec. IV. For nowwe work with Eqs. ( 8) and ( 9) as they are, but keep in mind that they need to be modified. III. EFFECTIVE BOSONIC ACTION, MEAN-FIELD THEORY, AND GAUSSIAN FLUCTUATIONS In this section, we derive an exact expression for the effective action of the gauge fields aα μ, following Ref. [ 51]. Subsequently, the nonpolynomial action we obtain will beexpanded to second order in the fluctuations around its mean-field solution, resulting in an exactly solvable Gaussian model.The quadratic action will be stated in its real-time form inKeldysh basis. Due to the Coulomb interaction being quartic in the fermionic fields, an integration of these microscopic degreesof freedom is not readily possible. For this reason, we rewritethe problematic interaction term by means of a Hubbard-Stratonovich transformation in the density-density channel[64], which introduces an auxiliary boson φ: e iSint[ψ]=/integraldisplay DφeiSHS[φ]+iSint[ψ,φ]. (18) The quadratic action of the Hubbard-Stratonovich boson is given by SHS[φ]=1 2/integraldisplay C,xyφ(x)V−1(x−y)φ(y), (19) with the inverse Coulomb interaction V−1, which, of course, has to be understood in the distributional sense. The secondterm contains a trilinear Yukawa-type interaction and a linearterm, describing the interaction of the auxiliary boson with thebackground density ¯n: S int[ψ,φ]=−/integraldisplay C,xφ(x)(/Psi1†(x)/Psi1(x)−¯n(x)). (20) Note that the fluctuating Bose field φin the Yukawa interaction appears on the same footing as the zero component of theexternal gauge potential A μ, coupling to all flavors identically, see Eqs. ( 11)–(14). As a consequence of the above manip- ulation, the Grassmann fields ψappear only quadratically, such that the fermionic integral can be performed exactly. Ourintermediate result for the effective action now only containsbosonic degrees of freedom: S /prime eff/bracketleftbig eAμ+Aα μ,aα μ,φ/bracketrightbig =−itr ln ˆG−1 0/bracketleftbig eAμ+Aα μ+aα μ+φδ0μ/bracketrightbig +SHS[φ]+φ¯n+SCS/bracketleftbig aα μ/bracketrightbig . (21) Remarkably, the Hubbard-Stratonovich boson φcan be integrated exactly after shifting the statistical gauge fields asfollows: a α μ→aα μ−φδ0μ[51,54]. The result is the desired effective action of the Chern-Simons gauge fields in thepresence of the two-particle source fields A α μ: Seff/bracketleftbig eAμ+Aα μ,aα μ/bracketrightbig =−itr ln ˆG−1 0/bracketleftbig eAμ+Aα μ+aα μ/bracketrightbig +SV/bracketleftbig aα μ/bracketrightbig +SCS/bracketleftbig aα μ/bracketrightbig . (22) 115123-5CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) The term SV[aα μ] is a quadratic functional of the gauge fields that is generated by the φintegration: SV/bracketleftbig aα μ/bracketrightbig =−1 2/integraldisplay C,xy/parenleftbig (ˆK−1)α1β1ε0μ1ν1∂μ1aβ1 ν1−¯nα1/parenrightbig (x) ×Vα1α2(x−y)/parenleftbig (ˆK−1)α2β2ε0μ2ν2∂μ2aβ2 ν2−¯nα2/parenrightbig (y). (23) Here we have defined Vαβ(x−y)≡V(x−y), where the ad- ditional flavor-space indices keep track of the correct summa-tion. Note that Eq. ( 23) is nothing but the Coulomb interaction term ( 15), in which the density of flavor α,/Psi1† (α)(x)/Psi1(α)(x), is substituted by ( ˆK−1)αβε0μν∂μaβ ν(x). In the above derivation, no approximations were involved. Yet, due to the nonpolynomialtracelog term, the residual functional integral over the gaugefields cannot be performed exactly. A common strategy to dealwith this problem, which we adopt here as well, is to findthe field configuration in which the effective action becomesstationary and, subsequently, expand in powers of fluctuationsaround the mean. The variation of Eq. ( 22) in the absence of two-particle sources A α μyields δSeff δaαμ(z)=−jμ α(z)+(ˆK−1)αβεμνλ∂νaβ λ(z) −/integraldisplay C,xy/parenleftbig (ˆK−1)α1β1ε0μ1ν1∂μ1δμ ν1δβ1 αδC(x−z)/parenrightbig ×Vα1α2(x−y)/parenleftbig (ˆK−1)α2β2ε0μ2ν2∂μ2aβ2 ν2−¯nα2/parenrightbig (y). (24) Here, jμ αis the particle 3-current density per flavor αin the presence of an external gauge potential Aμand the Chern- Simons fields aα μ: jμ α(x)=−iδ δaαμ(x)tr ln ˆG−1 0/bracketleftbig eAμ+aα μ/bracketrightbig . (25) We have to stress at this point that Eqs. ( 24) and ( 25)h a v e to be treated with special care as they demand a properregularization. First, in the infinite system, the particle currentis not well defined, since its μ=0 component—being the particle density—diverges. This fact is a direct consequenceof the Dirac approximation of the tight-binding graphenespectrum. Another issue is related to the fact that the definitionof the particle current involves the average of a (contour-)timeordered product of two fermionic fields evaluated at the sametime. However, these problems are immediately resolved oncethe theory is mapped to the physical real-time representationin Keldysh basis. Hereto, one splits the Schwinger-Keldyshcontour into a forward and backward branch and defines adoubled set of fields, /Psi1 ±and (a±)α μ, which are associated to the respective branch [ 63,64]. In a next step, one performs a rotation from ±basis to Keldysh basis by defining “classical” and “quantum” fields, indexed by candq, respectively, as symmetric and antisymmetric linear combinations of the ± fields [ 63,64]. The net result is that the derivative in Eq. ( 24) is performed with respect to the quantum components of thegauge fields, the particle 3-current densities are replaced bythe well-defined charge carrier 3-current densities ¯j α μ(x), seeEq. ( A12), and the gauge fields on the right-hand side are replaced by their classical components [ 69]. The requirement of a vanishing first variation defines the mean-field equations for the Chern-Simons fields. As pointedout by the authors of Refs. [ 51,54], these mean-field equa- tions allow for several physically different scenarios such asWigner crystals and solitonic field configurations. FollowingRefs. [ 51,54], we here concentrate on those solutions, which lead to a vanishing charge carrier current and a uniformand time independent charge carrier density ¯n α, describing a quantum Hall liquid. In that case, Eq. ( 24) reduces to the relation ¯nα=(ˆK−1)αβε0μν∂μ¯aβ ν=−e(ˆK−1)αβbβ. (26) Here, the second equality defines the (uniform) Chern-Simons magnetic field bα, experienced by the flavor αcharge carriers, in terms of the expectation value of the Chern-Simons fields ¯aα μ≡/angbracketleft(ac)α μ/angbracketright. Inverting the above relation yields the statistical magnetic fields bαas functions of the densities ¯nα: bα=−1 eKαβ¯nβ. (27) Writing the mean-field equation in this form reveals the physical meaning of the K-matrix, as it defines the precise flux-attachment procedure of the multicomponent quantumHall system. Each flavor βof charge carriers, contributes to the statistical magnetic field for the flavor αwith a magnetic flux K α(β)¯n(β). Hence the component Kαβrepresents the contribu- tion to the statistical flux per flavor βas seen by flavor α. Thus Eq. ( 27) may be interpreted as a “flux-binding” relation, which transforms ordinary Dirac fermions into “composite Diracfermions.” Furthermore, it is important to notice that Eq. ( 27) is well-defined even for singular Kmatrices, in contrast to Eq. (26). Such singular configurations should not be discarded, however, as the following discussion shows. Consider, forexample, the special case, where all components of ˆKequal the same constant 2 k. In that case, the four equations ( 27) reduce to a single one, yielding a unique statistical field b associated to the density of charge carriers ¯n=/summationtext α¯nα.T h i s scenario corresponds to a Chern-Simons theory, where onlya single dynamical gauge field, a μ=/summationtext αaα μ, is present that couples to the different flavors identically [ 57]. The other three eigenvectors one obtains by diagonalizing Eq. ( 17) span a triply degenerate subspace with eigenvalue zero, and thus decouple.Likewise, for other singular K-matrix configurations, one would obtain a theory with only two or three dynamical gaugefields and a correspondingly reduced parameter space. (In theextreme case where ˆKis identically zero, all gauge fields would decouple and no flux binding could occur, which leads back tothe integer quantum Hall regime.) With this physical picturein mind we now continue our discussion. By virtue of the gauge covariant derivative ( 14), each one of the statistical magnetic fields ( 27) adds to the external magnetic fieldBindividually, resulting in a flavor-dependent effective magnetic field [ 70] B α eff=B+bα=B−1 eKαβ¯nβ. (28) It is this effective magnetic field, rather than the external field Balone, which enters the fermionic propagators, such that 115123-6ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) Eqs. ( 27) and ( 28), in fact, represent self-consistency equations. A straightforward calculation of the free propagator for Diracfermions moving in the effective magnetic field B α effyields the charge carrier density for the flavor αas a function of the chemical potential μα, the effective magnetic field Bα eff, and temperature T(see Appendix Afor details): ¯nα/parenleftbig μα,Bα eff,T/parenrightbig =1 2π/lscript2 (α)ν(α)/parenleftbig μα,Bα eff,T/parenrightbig . (29) Here, we have introduced the magnetic length /lscriptα=1//radicalbig |eBα eff| and the filling fraction per flavor [ 9,10], να=1 2/parenleftBigg tanhμα 2T+∞/summationdisplay n=1/summationdisplay λ=±1tanhλ√nωα c+μα 2T/parenrightBigg ,(30) where ωα c=√ 2vF//lscriptαdenotes the pseudorelativistic cyclotron frequency. The charge carrier density as a function of aneffective magnetic field B α effat constant chemical potential μα and the filling fraction as a function of the chemical potential at constant field are shown in Fig. 2. At large magnetic fields and low temperatures, the filling fraction shows the typicalplateau structure that is characteristic for the (anomalous)integer quantum Hall effect. This issue will be discussed inmore detail at the end of this section, once we have obtainedthe Gaussian approximation to the exact action ( 22).From the above mean-field equation, one may calculate the possible fractional fillings at which the spectrum is gapped,leading to a plateau structure for the “interacting Hall con-ductivity,” which is the hallmark of the fractional quantumHall effect. However, we prefer to extract the filling fractionsdirectly from the interacting Hall conductivity, which will bederived in the next section. To continue, we only need toknow that the mean-field equation has a nontrivial solution,which depends on the Kmatrix, the external magnetic field, temperature, and chemical potential. Furthermore, observe thatthe effective magnetic field is invariant upon changing the signofK αβand ¯nαsimultaneously. This is a first hint, how to construct manifestly particle-hole symmetric filling fractionsin the presence of a Chern-Simons term. The stationary field configuration we found above serves as a reference point around which one should expand the effectiveaction ( 22) in powers of field fluctuations. To this end, one writes a α μ=¯aα μ+/Delta1aα μand expands the effective action to the desired order in the fluctuation /Delta1aα μand the source Aα μ.A s mentioned before, here we are only interested in an expansionup to second order. Terms linear in the fluctuation vanish sincethe effective action is evaluated at the saddle point, whereaslinear source terms do not vanish. However, since the latter onlycouple to the above mean-field 3-currents, their contributionis not interesting for the further analysis and will be omitted.We state the result in the physical real-time representation inKeldysh basis: Seff/bracketleftbig Aα μ,/Delta1aα μ/bracketrightbig =/integraldisplay xy/parenleftbig(/Delta1ac)α μ+(Ac)α μ(/Delta1aq)α μ+(Aq)α μ/parenrightbig (x)/parenleftBigg0( /Pi1A)μν αβ (/Pi1R)μν αβ (/Pi1K)μν αβ/parenrightBigg (x,y)/parenleftbigg(/Delta1ac)β ν+(Ac)β ν (/Delta1aq)β ν+(Aq)β ν/parenrightbigg (y) +/integraldisplay xy/parenleftbig(/Delta1ac)α μ(/Delta1aq)α μ/parenrightbig (x)/parenleftBigg0( CA)μν αβ (CR)μν αβ (CK)μν αβ/parenrightBigg (x,y)/parenleftbigg(/Delta1ac)β ν (/Delta1aq)β ν/parenrightbigg (y) ≡/integraldisplay xy/bracketleftbig/parenleftbig /Delta1aα μ+Aα μ/parenrightbig/intercal(x)/Pi1μν αβ(x,y)/parenleftbig /Delta1aβ ν+Aβ ν/parenrightbig (y)+/Delta1aα μ(x)Cμν αβ(x,y)/Delta1aβ ν(y)/bracketrightbig . (31) As discussed in the paragraph following Eq. ( 24), the additional degrees of freedom are a consequence of the mapping fromabstract contour to physical real time. The second line definesa compact notation, where the Keldysh degrees of freedomare indicated by bold symbols. While /Delta1a α μandAα μare two- dimensional vectors in Keldysh space with “classical” and“quantum” components, /Pi1μν αβandCμν αβare triangular 2 ×2 matrices with retarded, advanced, and Keldysh components.The latter contain the statistical information of the theory.Since in this article we are only interested in the linear responseregime at finite temperatures, the Keldysh components ( /Pi1 K)μν αβ and (CK)μν αβboth obey the bosonic fluctuation-dissipation theorem [ 64,71]. In Eq. ( 31),/Pi1μν αβ(x,y) is the one-loop fermionic polarization tensor /Pi1μν αβ(x,y)=−i 2δ2 δaβ ν(y)δaαμ(x)tr ln ˆG−1 0/bracketleftbig eAμ+aα μ/bracketrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle a=¯a, (32)in which ˆG−1 0[eAμ+aα μ] is the inverse fermionic propagator (12) mapped to Keldysh space. Referring to Appendix Bfor details, we calculated this tensor at nonvanishing temperatures.Since the free propagators are diagonal in the flavor index, wefind that the polarization tensor is diagonal in flavor space aswell,/Pi1 μν αβ=/Pi1μν (α)δ(α)β. However, this may need not be the case at higher orders, so for now we keep both indices. The tensor Cμν αβ(x,y), which we refer to as Chern-Simons–Coulomb tensor, is the integral kernel of SCS[/Delta1aα μ]+SV[¯aα μ+/Delta1aα μ]. Its real space representation reads (CR/A)μν αβ(x,y) =(ˆK−1)αβεμλνδ(x−y)→ ∂λ −← ∂μ1ε0μ1μ(ˆK−1)αα1Vα1β1(x−y)(ˆK−1)β1βε0ν1ν→ ∂ν1, (33) where the arrows above the partial derivatives indicate the direction in which they operate. 115123-7CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) FIG. 2. (a) Charge carrier density at constant chemical potential as a function of the effective magnetic field Bα effatT=10 K (blue). The straight dotted lines (orange) indicate the first few Landau levels. At vanishing magnetic field, the charge carrier density scales quadratically with the chemical potential [ 5,10]. (Without loss of generality the sign of the chemical is assumed to be positive). Upon increasing (the absolute value of) the effective magnetic field Bα eff—while keeping the chemical potential fixed—the carrier density shows oscillations in the regime ωα c<μ α, whereas for ωα c/greatermuchμαit grows linearly as a function of the magnetic field. This behavior is readily explained by the formation of Landau levels, and the dependence of their degeneracy and relative energetic separation on the magnetic field. (b) Filling fraction per flavor ναat constant effective magnetic field Bα eff=15 T and temperature T=10 K as a function of the chemical potential. The plateaus occur at half-integer filling fractions να=±(nα+1/2). The transitions between the plateaus are smeared out due to temperature. (c) Derivative of ναwith respect to the chemical potential as a measure for temperature-induced Landau level broadening at Bα eff=15 T for the temperatures T=5,25,50,and 75 K. Increasing the temperature clearly leads to a broadening of the discrete energy levels. Since the relativistic Landau levels are not equidistant in energy space, the level broadening causes the Landau levels to overlap significantly away from the charge neutrality point. The Landau level located directly at the charge neutrality point, however, remains well-defined up to rather large temperatures. Both the fermionic polarization tensor /Pi1μν αβ,a sw e l la st h e Chern-Simons–Coulomb tensor Cμν αβ, are transverse, which may be expressed by the identities → ∂μ/Pi1μν αβ=0,/Pi1μν αβ← ∂ν=0, (34a) → ∂μCμν αβ=0,Cμν αβ← ∂ν=0. (34b) As is well-known, this property is a consequence of gauge invariance [ 68]. Furthermore, for the polarization tensor, it is possible to factorize its tensorial structure and expand it intothree distinct scalar kernels /Pi1 0 αβ,/Pi11 αβ, and/Pi12 αβ[51,54]. Trans- forming to Fourier space, the 2 +1-dimensional representation of this expansion, where timelike ( μ,ν=0) and spacelike (μ,ν=i,j=1,2) indices are separated, reads /Pi100 αβ(ω,/vectorq)=− /vectorq2/Pi10 αβ(ω,/vectorq), (35a) /Pi10i αβ(ω,/vectorq)=−ωqi/Pi10 αβ(ω,/vectorq)+iε0ijqj/Pi11 αβ(ω,/vectorq),(35b) /Pi1i0 αβ(ω,/vectorq)=−ωqi/Pi10 αβ(ω,/vectorq)−iε0ijqj/Pi11 αβ(ω,/vectorq),(35c) /Pi1ij αβ(ω,/vectorq)=−ω2δij/Pi10 αβ(ω,/vectorq)+iε0ijω/Pi11 αβ(ω,/vectorq) +(δij/vectorq2−qiqj)/Pi12 αβ(ω,/vectorq). (35d) One can readily check that the above expansion fulfills the transversality condition ( 34a). We close this section by a short discussion about the (anomalous) integer quantum Hall effect in graphene. Al-though the electromagnetic response of the interacting systemto an external perturbation is encoded in the electromagneticresponse tensor to be derived in the next section, the responseproperties of the noninteracting system are already contained inthe fermionic polarization tensor /Pi1 μν αβ. In fact, it is established that the essential physics of the integer quantum Hall effectcan largely be understood within a noninteracting model andinteractions only play a minor role [ 72]. We therefore only needto consider /Pi1 μν αβ, in particular its retarded component. The dc conductivity tensor per fermionic flavor αcan be obtained as the limit [ 9] (σ0)ij α=lim ω→0lim /vectorq→0e2 iω(/Pi1R)ij α(ω,/vectorq), (36) where i,jare the aforementioned spacelike indices ( i,j= 1,2). Recall that to one-loop order the polarization tensor is diagonal in flavor space, hence we dropped the secondflavor index. Furthermore, we here concentrate on the off-diagonal Hall conductivity, which reduces Eq. ( 36)t ot h e kernel ( /Pi1 R)1 α(0,0), /parenleftbig σxy 0/parenrightbig α=e2(/Pi1R)1 α(0,0)=sign/parenleftbig eBα eff/parenrightbige2 2πνα. (37) The second equality follows after a lengthy, but straightforward calculation. As we mentioned earlier in this section, at largemagnetic fields and zero temperatures, the filling fraction ν α is quantized into plateaus located at ±(nα+1 2), with nα= 0,1,2,...,s e eE q .( 30) and Fig. 2. Consequently, Eq. ( 37) describes the anomalous integer quantum Hall effect of thesingle fermionic flavor α. A summation of the remaining flavor index then yields the Hall conductivity of the entire system ofDirac particles. For simplicity, we may assume the absenceof Zeeman terms and flux-binding for the moment by settingμ α=μandBα eff=B. In that case, the contributions from the individual flavors are identical, giving rise to the well-knownfactor of four after summing over all flavors. Restoring ¯ hwe, thus, obtain the anomalous integer quantization of the Hallconductivity in graphene [ 9,10]: σ xy 0=±sign(eB)e2 2π¯h4/parenleftbigg n+1 2/parenrightbigg ,n=0,1,2,.... (38) A finite temperature leads to a smearing of these plateaus, due to the thermal broadening of the Landau levels. However, sincethe Landau levels are not equidistant in energy because of the 115123-8ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) linear Dirac spectrum, even a small temperature eventually washes out the plateau structure at large fillings. Only thelowest levels are relatively robust against the thermal smearing.Taking into account the rather large value of the relativisticcyclotron frequency ω c, it is possible to observe the quantum Hall effect experimentally at room temperature [ 12]. By now, this is a well-known fact, but still it is insofar astonishing, asthe quantum Hall effect for ordinary, nonrelativistic fermionscan only be observed at low temperatures, close to absolutezero. IV . ELECTROMAGNETIC RESPONSE TENSOR AND HALL CONDUCTANCE In order to obtain the electromagnetic polarization tensor, we need to perform the residual functional integration overthe statistical gauge fields, which—according to the rulesof Gaussian integration—involves the inverse of ( /Pi1+C) μν αβ. However, since both /Pi1μν αβandCμν αβare transverse, neither their individual inverse nor the inverse of their sum does exist. Thisproblem is rooted in the gauge invariance of the partitionfunction ( 8). As advertised at the end of Sec. II, we here discuss the issues of the gauge fixing procedure—resorting tothe contour-time representation for the moment—and derivethe electromagnetic response tensor, from which we obtainthe dc Hall conductivity. We emphasize that the techniquedescribed below is not limited to the Gaussian approximationof the effective action ( 22). The problematic gauge equivalent orbits, causing Eq. ( 8) to diverge, can be factorized from the nonequivalent physicalfield configurations by the well-known Fadeev-Popov gaugefixing procedure. Referring to Ref. [ 68] for details, we obtain the intermediate result Z/bracketleftbig A α μ/bracketrightbig =N/integraldisplay D/Delta1a δ/bracketleftbig G/parenleftbig /Delta1aα μ/parenrightbig/bracketrightbig eiSeff[Aα μ,/Delta1aα μ]. (39) Here, the divergent integral over pure gauge fields as well as the so-called Fadeev-Popov determinant have been absorbed intothe formally infinite normalization constant N. Since it does not enter any correlation function, this constant may safelybe omitted [ 77]. The functional delta distribution enforces the gauge constraint G(/Delta1a α μ)=0 within the functional integral, such that only physically inequivalent field configurationscontribute to the amplitude. The gauge fixing function can bechosen at will, but for definiteness, we consider the generalizedLorentz gauge condition, G/parenleftbig /Delta1a α μ/parenrightbig =∂μ/Delta1aμ α(x)−ω(x), (40) where ω(x) is an arbitrary function, in the remainder of this paper. In its present form, Eq. ( 39) can in principle be employed to calculate the desired correlation functions, yet it is beneficial tomake use of Feynman’s trick of “averaging over gauges” [ 68]. Hereto one averages the partition function ( 39) over different field configurations ω(x) with a Gaussian “probability mea- sure.” This procedure closely resembles a Gaussian disorderaverage of the partition function, albeit a disorder potentialwould couple in a different manner [ 63,64,78,79]. The netresult is the gauge fixed partition function Z GF/bracketleftbig Aα μ/bracketrightbig =/integraldisplay D/Delta1a eiSeff[Aα μ,/Delta1aα μ]+iSGF[/Delta1aα μ], (41) where the additional contribution in the exponent is the gauge fixing action SGF/bracketleftbig /Delta1aα μ/bracketrightbig =1 2ξ/integraldisplay C,x/parenleftbig ∂μ/Delta1aμ α(x)/parenrightbig2 ≡1 2/integraldisplay C,xy/Delta1aα μ(x)Gμν αβ(x,y)/Delta1aβ ν(x). (42) Here, ξis a real-valued parameter, which may be chosen at will to simplify calculations. In the end, for any physical—that is gauge invariant—observable the dependence on ξhas to drop out. After mapping this contour-time action to thephysical real-time representation and performing the Keldyshrotation, the additional gauge fixing term effectively leads tothe substitution C μν αβ→(C+G)μν αβin the effective action ( 31). Since Gμν αβis invertible so is the sum ( /Pi1+C+G)μν αβ, resulting in a well-defined functional integral over the statistical gaugefields. For nonvanishing source fields, the residual Gaussian inte- gration yields the generating functional of connected correla-tion functions [ 64,67] W/bracketleftbig A α μ/bracketrightbig =−ilnZGF/bracketleftbig Aα μ/bracketrightbig =/integraldisplay xyAα μ(x)Kμν αβ(x,y)Aβ ν(y). (43) In this expression, Kμν αβ(x,y) defines the electromagnetic po- larization tensor. Accordingly, it represents the linear electro-magnetic response of the system to an external perturbation.We state its explicit form in terms of the fermionic polarizationtensor and the Chern-Simons–Coulomb tensor of the precedingsection by employing a condensed matrix notation. For themoment the hat symbol not only indicates the flavor-spacematrix structure, but also covers the discrete Minkowskiindices μ,ν and the continuous space-time variables x,y,i f not stated otherwise, ˆK=ˆ/Pi1−ˆ/Pi1(ˆ/Pi1+ˆC+ˆG) −1ˆ/Pi1. (44) In this expression, matrix multiplication is defined naturally by implying summation over discrete and integration overcontinuous degrees of freedom. This tensor has the usualtriangular Keldysh space structure, with retarded, advanced,and Keldysh components [ 64] K μν αβ(x,y)=/parenleftBigg 0( KA)μν αβ(x,y) (KR)μν αβ(x,y)(KK)μν αβ(x,y)/parenrightBigg . (45) Transforming to frequency-momentum space, these compo- nents read ˆKR/A ω,/vectorq=ˆ/Pi1R/A ω,/vectorq−ˆ/Pi1R/A ω,/vectorq/parenleftbig (ˆ/Pi1+ˆC+ˆG)R/A ω,/vectorq/parenrightbig−1ˆ/Pi1R/A ω,/vectorq,(46a) ˆKK ω,/vectorq=coth/parenleftBigω 2T/parenrightBig/parenleftbigˆKR ω,/vectorq−ˆKA ω,/vectorq/parenrightbig . (46b) Here, the frequency and momentum dependence has been written as an index, flavor and Minkowski indices are still 115123-9CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) covered by the hat symbol. The second equation is just a manifestation of the bosonic fluctuation-dissipation theorem. Although the electromagnetic response tensor as given by Eq. ( 44) contains the gauge fixing kernel ˆGexplicitly, any reference of it drops out in the final expression for ˆK. In fact, the electromagnetic response tensor is a physical observable and,thus, has to be gauge-invariant. We have checked explicitlyfor a single flavor that other common choices, such as theCoulomb and axial gauge, indeed, lead to the same result.As a consequence of gauge invariance, the electromagneticresponse tensor ˆKis transverse and, hence, admits the verysame decomposition as the fermionic polarization tensor ˆ/Pi1, see Eqs. ( 35a)–(35d). The only difference are the kernels ˆK 0/1/2, which are now complicated functions of the kernels ˆ/Pi10/1/2,t h e K-matrix and the (Fourier transformation of the) Coulomb interaction matrix ˆV(/vectorq). Recall that the latter is a 4×4 matrix in flavor space, with all its components being equal to the same Coulomb interaction amplitude ( 6); see also Eq. ( 23), the comments thereafter and Eq. ( 33). Suppressing frequency and momentum labels (the hat symbol only indicatesflavor space here), we obtain for the retarded kernels ˆKR 0=ˆK−1(ˆDR)−1ˆK−1, (47a) ˆKR 1=ˆK−1+1 2ˆK−1/bracketleftbig/parenleftbig /vectorq2ˆVˆK−1−/parenleftbigˆ/Pi1R 0/parenrightbig−1/parenleftbigˆ/Pi1R 1+ˆK−1/parenrightbig/parenrightbig (ˆDR)−1+(ˆDR)−1/parenleftbigˆK−1/vectorq2ˆV−/parenleftbigˆ/Pi1R 1+ˆK−1/parenrightbig/parenleftbigˆ/Pi1R 0/parenrightbig−1/parenrightbig/bracketrightbigˆK−1,(47b) ˆKR 2=−1 /vectorq2ˆK−1/parenleftbig /vectorq2ˆVˆK−1−/parenleftbigˆ/Pi1R 0/parenrightbig−1/parenleftbigˆ/Pi1R 1+ˆK−1/parenrightbig/parenrightbig (ˆDR)−1/parenleftbigˆK−1/vectorq2ˆV−/parenleftbigˆ/Pi1R 1+ˆK−1/parenrightbig/parenleftbigˆ/Pi1R 0/parenrightbig−1/parenrightbigˆK−1 +1 /vectorq2ˆK−1/parenleftbig/parenleftbigˆ/Pi1R 0/parenrightbig−1−/vectorq2ˆV+ω2(ˆDR)−1/parenrightbigˆK−1, (47c) with ˆDR/A=−(ω±i0)2ˆ/Pi1R/A 0+/vectorq2/parenleftbigˆ/Pi1R/A 2−ˆK−1ˆVˆK−1/parenrightbig +/parenleftbigˆ/Pi1R/A 1+ˆK−1/parenrightbig/parenleftbigˆ/Pi1R/A 0/parenrightbig−1/parenleftbigˆ/Pi1R/A 1+ˆK−1/parenrightbig .(48) The advanced kernels are obtained by Hermitian conjugation just as usual. We have to emphasize at this point, that—incontrast to the one-loop fermionic polarization tensor ˆ/Pi1— the electromagnetic polarization tensor ˆKis in general not diagonal in flavor space, but a symmetric matrix. This factderives from the Kmatrix, which is also not necessarily diagonal, but symmetric. The above equations, together with the results for the fermionic polarization tensor ˆ/Pi1given in Appendix B, represent the main result of this work. Given a particular K-matrix con- figuration, the electromagnetic polarization tensor ˆKcontains the full information about the system’s response to a weak,external electromagnetic perturbation. The kernel ˆK R 0, when multiplied with −/vectorq2, equals the density response function, cf. Eq. ( 35a), K00 αβ(ω,/vectorq)=− /vectorq2K0 αβ(ω,/vectorq), (49) and as such determines the dynamical screening properties, as well as the collective modes. The latter can be obtained by theroots of the denominator matrix ˆD R,E q .( 48). Furthermore, in the zero temperature and long wavelength limit, it is possibleto calculate the absolute value square of the ground-statewave function and corrections thereof (as an expansion inq/B) ,w h i c hw a ss h o w ni nR e f .[ 80]. The current response tensor is given by the spatial components, μ, ν=1,2, of the polarization tensor, encoding the information about the(dynamical) conductivity tensor. In the remainder of this paper,we focus on the dc Hall conductivity. A further investigationof the above mentioned quantities will be left for future work. In close analogy to the noninteracting case, we need to investigate the zero frequency and momentum limit of thekernel ˆK R 1to obtain the Hall conductivity. Using Eqs. ( 47b) and ( 48), as well as lim /vectorq→0/vectorq2ˆV(/vectorq)=0, we obtain ˆKR 1(0,0)=lim ω→0lim /vectorq→01 iω(ˆKR)12(ω,/vectorq) =/bracketleftbigˆK+/parenleftbigˆ/Pi1R 1(0,0)/parenrightbig−1/bracketrightbig−1. (50) Clearly, if ˆKis identically zero, the kernel ˆKR 1reduces to the noninteracting kernel ˆ/Pi1R 1, leading back to the integer quantum Hall regime, Eq. ( 37). For the most general Kmatrix, Eq. ( 50) reads ˆKR 1=1 2π⎛ ⎜⎜⎜⎝2k 1+1 ν1m1 n1 n2 m1 2k2+1 ν2n3 n4 n1 n3 2k3+1 ν3m2 n2 n4 m2 2k4+1 ν4⎞ ⎟⎟⎟⎠−1 . (51) Observe that the temperature dependence only enters via the kernel ˆ/Pi1R 1, i.e., via the filling fractions να. A finite temperature does not modify the Kmatrix in any way, as it should be. Only the composite Dirac fermions, filling the effective Landaulevels, are subject to thermal fluctuations, the flux-bindingitself, as described by Eq. ( 27), is not influenced. Furthermore, note that we absorbed the sign of the effective magnetic fieldinto the filling fractions ν α. As discussed above, the kernel ˆKR 1 is a nondiagonal but symmetric matrix. In order to obtain the Hall conductivity, one has to sum over all of its components: σxy=e2/summationdisplay α,β/parenleftbigˆKR 1/parenrightbig αβ(0,0). (52) This fact becomes clear by taking into account that a physical electromagnetic fluctuation should couple identically to allflavors. Therefore one has to neglect the flavor index of thesource fields A α μ(x)i nE q .( 43), which, in turn, leads to 115123-10ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) TABLE I. Filling fraction νGfor three distinct K-matrix configurations, leading to a Hall conductivity σxy=e2 2π¯hνG(¯hrestored). The examples (2a) and (2b), respectively, (3a) and (3b), correspond to states with the same analytical properties but interchanged spin and valley degrees of freedom. The temperature dependence, contained within the composite-fermion filling fractions να, is suppressed. For the singular Kmatrices (1), (2a), and 2(b), there exists an equivalent Abelian gauge theory with a reduced set of Chern-Simons fields. The associated gauge groups are shown in the last column. Kmatrix Total filling fraction νG Gauge symmetry 1⎛ ⎜⎜⎝2k2k2k2k 2k2k2k2k 2k2k2k2k 2k2k2k2k⎞ ⎟⎟⎠ν1+ν2+ν3+ν4 2k(ν1+ν2+ν3+ν4)+1U(1) 2a⎛ ⎜⎜⎝2k12k1nn 2k12k1nn nn 2k22k2 nn 2k22k2⎞ ⎟⎟⎠/parenleftbig 2k1+1 ν1+ν2/parenrightbig −n/parenleftbig 2k1+1 ν1+ν2/parenrightbig/parenleftbig 2k2+1 ν3+ν4/parenrightbig −n2+/parenleftbig 2k2+1 ν3+ν4/parenrightbig −n/parenleftbig 2k1+1 ν1+ν2/parenrightbig/parenleftbig 2k2+1 ν3+ν4/parenrightbig −n2U(1) ↑⊗U(1) ↓ 2b⎛ ⎜⎜⎝2k1n 2k1n n 2k2n 2k2 2k1n 2k1n n 2k2n 2k2⎞ ⎟⎟⎠/parenleftbig 2k1+1 ν1+ν3/parenrightbig −n/parenleftbig 2k1+1 ν1+ν3/parenrightbig/parenleftbig 2k2+1 ν2+ν4/parenrightbig −n2+/parenleftbig 2k2+1 ν2+ν4/parenrightbig −n/parenleftbig 2k1+1 ν1+ν3/parenrightbig/parenleftbig 2k2+1 ν2+ν4/parenrightbig −n2U(1) K+⊗U(1) K− 3a⎛ ⎜⎜⎝2k1m1 00 m12k2 00 00 2 k3m2 00 m22k4⎞ ⎟⎟⎠/summationtext i=1,2/parenleftbig 2ki+1 νi/parenrightbig −m1/parenleftbig 2k1+1 ν1/parenrightbig/parenleftbig 2k2+1 ν2/parenrightbig −m2 1+/summationtext i=3,4/parenleftbig 2ki+1 νi/parenrightbig −m2/parenleftbig 2k3+1 ν3/parenrightbig/parenleftbig 2k4+1 ν4/parenrightbig −m2 2U(1)⊗4 3b⎛ ⎜⎜⎝2k1 0m1 0 02 k2 0m2 m1 02 k3 0 0m2 02 k4⎞ ⎟⎟⎠/summationtext i=1,3/parenleftbig 2ki+1 νi/parenrightbig −m1/parenleftbig 2k1+1 ν1/parenrightbig/parenleftbig 2k3+1 ν3/parenrightbig −m2 1+/summationtext i=2,4/parenleftbig 2ki+1 νi/parenrightbig −m2/parenleftbig 2k2+1 ν2/parenrightbig/parenleftbig 2k4+1 ν4/parenrightbig −m2 2U(1)⊗4 a summation over all matrix components rather than, e.g., taking a trace. Equation ( 52) is the simplest form of the Hall conductivity. Alternatively, our result could be written interms of the (anomalous) integer quantum Hall conductivitiesof the noninteracting system, which may be slightly morecomplicated but possibly more appealing in physical terms.As advertised in the introduction, we get Eq. ( 2), σ xy=/summationdisplay ασα 0,xy−/summationdisplay α,βσα 0,xy(ˆσ0,xy+ˆK−1)−1 αβσβ 0,xy. Continuing the parallels with the noninteracting case, the Hall conductivity should be proportional to some filling factorν G(adopting here the notation of Ref. [ 8]). This filling factor can easily be extracted from Eq. ( 52), using the equality σxy=e2 2πνG. It is a complicated rational function of all the components of the Kmatrix and the filling factors of the individual composite fermions να. Clearly, for such a large pa- rameter space some of its input will be mapped to the exactsame filling fraction ν G. In other words, several different FQH states produce the same filling fraction, respectively, the sameHall conductivity. Hence the measurement of a Hall plateauat a particular filling fraction alone does not identify a singleFQH state. In order to distinguish from the theoretical sidewhich state realizes a certain filling fraction in an actualexperiment, one should estimate the energy associated to allof the states in question. In principle, this should lead toa unique lowest energy state, which realizes that particularFQH plateau. In addition, one could investigate—theoreticallyand experimentally—the screening properties and/or collectivemodes of the respective states to gain a deeper understanding and potentially exclude a certain subset of states. Considering the complexity of the matrix inverse in Eq. ( 51) for the most general K-matrix configuration, it becomes clear that a complete analysis of the full parameter space is highlyinvolved. For its systematic study, it is advisable to partiallyrestrict the parameter space and collect the correspondingK-matrix configurations into several distinct classes, which should have some overlap in their restricted parameter space.In this context, recall our discussion of singular Kmatrices in the preceding section. Employing this strategy it is not onlypossible to explore the full parameter space eventually, but italso simplifies the identification of the underlying physics thatis described by a particular class of K-matrix configurations considerably. In the remainder of this paper we outline thisstrategy, concentrating on a few special cases. Those K-matrix configurations we decided to investigate further, together withtheir resulting Hall conductivities are listed in Table I. We encountered the first of these examples already in our discussion of singular Kmatrices. The states described by this particular Kmatrix belong to the simplest possible class of FQH states, which can be described by a simplerChern-Simons gauge theory, where only a single local U(1)gauge field is present. The structure of the Kmatrix indicates a residual global SU(4) flavor symmetry, which is weakly brokenby the Zeeman terms. Once the symmetry breaking termsare neglected—that is, equating all composite-fermion fillingfractions ν α=ν—we obtain a hierarchy of states described by the filling fractions νG=4ν 2k·4ν+1, which have also been 115123-11CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) FIG. 3. Total filling fraction νG=4ν 2k·4ν+1as a function of the chemical potential μatBeff=15 T and T=10 K for k=+1 (blue) andk=−1 (orange). The finite temperature smears out the transitions from one Hall plateau to another, similar to the noninteracting case, cf.Fig.2. Note that the plateaus occur in pairs, lying symmetrically around the charge neutrality point νG=0. In this simple case, one can construct a manifestly particle-hole symmetric filling fraction by considering the regimes μ< 0a n d μ> 0 and flip the sign of kat μ=0, see Fig. 4, which yields the two branches |νG|<|1/2k|and |νG|>|1/2k|. obtained in Ref. [ 62]. This total filling fraction as a function of the chemical potential μis shown in Fig. 3at the effective magnetic field Beff=15 T and temperature T=10 K for k=±1. We remind the reader that, if one wishes to change the charge carrier density via the chemical potential, but keepthe effective magnetic field B effto be constant, then, according to the mean-field equation ( 28), one has to change the external magnetic field Bas well. For a fixed flux attachment prescribed by the integer k,i ti s obvious that the filling fraction νGis not manifestly particle- hole symmetric. Yet, the Hall plateaus occur in particle-holesymmetric pairs, when considering kand−ksimultaneously. This observation suggests that one can construct a manifestlyparticle-hole symmetric filling fraction by distinguishing thetwo regimes μ< 0 andμ> 0, and flip the sign of katμ=0, which yields the two branches ν ph G=4ν −2|k|4ν+1/Theta1(−μ)+4ν 2|k|4ν+1/Theta1(μ),(53a) νph∗ G=4ν 2|k|4ν+1/Theta1(−μ)+4ν −2|k|4ν+1/Theta1(μ),(53b) where |νph G|<|1/2k|and|νph∗ G|>|1/2k|. Note that the latter branch, νph∗ G, appears to have the wrong overall sign. (Naively, one would expect the sign of the total filling fraction νGto coincide with the sign of μ.) But recall that we absorbed the sign of the effective magnetic field into the compositeDirac fermion filling fractions ν α, meaning that this “wrong sign” should be interpreted as an effective magnetic fieldbeing antiparallel to the external one. In Fig. 4, we show the branch ( 53a)f o r|k|=1,..., 4, as well as a generalization of Eq. ( 53a) when a finite spin Zeeman coupling is present, with Zeeman energies E Z=0.1,..., 0.4ׯhωeff cfor|k|=1. We have chosen such large Zeeman energy scales, which vastlyexceed the ones found in a realistic graphene sample [ 6,8],FIG. 4. Particle-hole symmetric total filling fractions for νG= (ν↑+ν↓) 2k·(ν↑+ν↓)+1, with ν↑=ν1+ν2,ν↓=ν3+ν4, as a function of the chemical potential μatBeff=15 T and T=10 K. (a) Zero Zeeman splitting, implying ν↑=ν↓=2ν,f o r|k|=1,..., 4. (b) Finite spin Zeeman term with Zeeman energies EZ=0.0,0.1,..., 0.4ׯhωeff c for|k|=1. The finite Zeeman term leads to the formation of new plateaus, with the νG=0 plateau being the most dominant one. for demonstrational purposes to make the additional plateau structure at νG=0 visible. The examples (2) and (3) of Table Iare best discussed comparatively. Each of these examples comes in two varia-tions, where the flux attachment to spin and valley degreesof freedom are interchanged. Without loss of generality wemay limit our comparative discussion to the (2a) and (3b)configuration, simply referring to them as (2) and (3) if notstated otherwise. While the (2) configuration is another important example of a singular Kmatrix, and as such can be represented in terms of a reduced Chern-Simons theory [in this case, a U(1) ↑⊗ U(1) ↓], the (3) configuration is regular. The two K-matrix configurations represent very different physical scenarios. Thestates associated to (2) are the analog of the nonrelativisticbilayer FQH states found in Ref. [ 53], where an additional internal degree of freedom—the valley polarization—in each “spin-layer” is present. Neglecting the Zeeman couplings in thevalley subspace, that is, equating ν 1=ν2andν3=ν4,r e s t o r e s the global valley SU(2) symmetry. The states associated tothe (3) configuration on the other hand, can be interpreted astwo independent, decoupled “bilayers,” one for each valleydegree of freedom. Once again, the bilayer structure is formedby the spin degree of freedom, but the valley now appears 115123-12ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) as an external degree of freedom. [For comparison, the (3a) configuration would yield a bilayer structure formed by thevalley and the spin would appear as an external degree offreedom.] The difference of internal and external valley polarization is also reflected in the filling fraction itself, as can be seenfrom Table I. For simplicity we set all composite fermion filling fractions equal, ν α=ν, and, furthermore, we may also setk1=k2=kfor the (2) configuration, and kα=k,m 1= m2=mfor the (3) configuration. In those cases the K-matrix configurations (2a) and (2b), respectively, (3a) and (3b) yieldthe same filling fraction. If the valley appears as an internaldegree of freedom, we obtain ν int G=4ν (2k+n)2ν+1, (54) whereas if the valley is an external degree of freedom we get νext G=22ν (2k+m)ν+1. (55) The two filling fractions coincide by setting n=0i nE q .( 54) andm=2kin Eq. ( 55). It is this special case, which has been proposed in Ref. [ 61]. Manifestly particle-hole symmetric total filling fractions can be constructed for Eqs. ( 54) and ( 55)i nt h es a m ew a y as was done before, but this time at νG=0 one has to flip the sign of kandn, respectively, m, simultaneously. Similarly, the other filling fractions in Table Ialso lead to particle-hole symmetric Hall plateaus. [In general, sending ˆK→− ˆKand να→−ναresults in νG→−νG,c f .E q s .( 50) and ( 52).] Hence it is expected that for those filling fractions a similar, albeitmore involved construction can be performed to present themin a manifestly particle-hole symmetric form. As a final example we show how the prominent ν G= ±1/3 filling fraction, which has recently been observed in an experiment [ 7,15,16], arises in our theory. To produce such a filling fraction, there are several possible candidatesfor the Kmatrix, the simplest of which is given by the configuration (1) of Table Iupon choosing/summationtext ανα=±1 and k=±1. Note that this K-matrix configuration also gives rise to the prominent filling fractions νG=±2/3 andνG=±2/5, which are obtained by setting/summationtext ανα=±2 and k=±1, or k=±2, respectively. Another possible choice for the Kmatrix is configuration (2), where nandk2are set to zero (one may also set k1=0 instead). In that case, the total filling fraction simplifies to νG=ν1+ν2 2k1(ν1+ν2)+1+ν3+ν4. Choosing the composite fermion filling fractions and the remainingflux-attachment parameter k 1appropriately, that is, ν1+ν2= ±1,k1=±1 andν3+ν4=0, yields νG=±1/3 likewise. Lastly, configuration (3) can also be employed to yield a total filling fraction of one third and it seems to us that this isthe analogous configuration of the one discussed in Ref. [ 43], which employs the conventional wave-function approach. Inthis work, it was argued, that, from the four spin-valley Landaulevels, two are completely filled, one is completely empty, and alast one is filled to one third. Taking this statement literally, onecan interpret it as follows: while the completely filled (empty)levels each contribute to the filling fraction with + 1 2(−1 2), the last level should be empty to a sixth ( −1 2+1 3=−1 6). Setting m1=m2=0, as well as k2=k3=k4=0i nt h econfiguration (3) of Table I, meaning that flux is attached to one flavor only, we obtain νG=ν1 2k1ν1+1+ν2+ν3+ν4. Clearly, for ν1=ν2=−1 2,ν3=ν4=1 2, and k1=−2, we reproduce the above situation νG=−1 6−1 2+1 2+1 2=1 3. However, the actual wave function proposed in Ref. [ 43] has an (mmm )-like structure, meaning the Jastrow factor contains an “off-diagonal” vortex-attachment accounting for interflavorcorrelations between two of the four flavors, which we believeis not realized by the above simple flux attachment. Althoughthe precise correspondence between the Kmatrix and the electron/hole wave function is not yet clear, since our flux-attachment scheme refers to the charge carriers rather thanelectrons/holes, such off-diagonal correlations between twoflavors are achieved by relaxing the constraint that, say, m 1 vanishes. Referring to the (3a) configuration for definiteness, one may set k1=k2=k,k 3=k4=0, and ν1=ν2=ν.I n this special case, the total filling fraction becomes νG= 2ν (2k+m)ν+1+ν3+ν4.I fn o w k=−1 and m=2k−1=−3 is considered (which resembles a Kmatrix that is used in the nonrelativistic Chern-Simons theory to describe a (333) state[54]), and ν=ν 3=ν4=1/2, we obtain the desired filling fraction νG=−2/3+1=1/3. V . CONCLUSIONS AND OUTLOOK In the present work, we developed a finite temperature theory for the pseudorelativistic fractional quantum Hall effectof monolayer graphene, employing the real-time Keldyshformalism in the functional integral approach. We consideredaU ( 1 ) ⊗4Chern-Simons gauge theory, which is minimally coupled to the system of interacting Dirac fermions. In thistheory, each fermionic flavor interacts with any other flavorthrough Coulomb interactions, in addition to an individualU(1) gauge field. The latter transforms ordinary into compositeDirac fermions. After integrating the fermionic degrees of free-dom, we obtained an exact effective action for the gauge fieldsthat has been analyzed in the random phase approximation. Wederived the electromagnetic response tensor from which the dcHall conductivities have been extracted. Our research could be extended into several different directions. One obvious extension concerns a more detailedanalysis of the electromagnetic response tensor for the variousFQH states as presented here. The density-density response,given by K 00 αβ, allows for an investigation of the dynamical screening properties of the system together with the spectrumof collective modes. The current-current response, given by Kij αβ, may be studied beyond the static case, which gives infor- mation about the optical conductivity σij(ω). In this context, we also want to mention the straightforward generalizations of theresponse tensor, which result from modifications of the linearand isotropic Dirac spectrum. Here we only considered a non-vanishing (generalized) Zeeman term implicitly through theflavor dependent chemical potentials μ α. Other modifications, such as trigonal warping, anisotropies in strained graphene,or finite mass terms (gaps), could lead to interesting effectsand can be obtained by adding the respective term to thenoninteracting Dirac action ( 11). Note that such alterations do not invalidate our general result given by Eqs. ( 47) and (48), if the analysis is restricted to the Gaussian approximation, 115123-13CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) but would enter via a modification of the kernels /Pi10/1/2 αβ. [The kernel expansion of Kμν αβis based on gauge invariance and therefore exact, but a higher-order expansion in gaugefluctuations prior to integration may not only be manifestedin a modification of the ˆ/Pi1kernels, but also in the form of the ˆKkernels, Eqs. ( 47) and ( 48).] Only minor modifications are involved to describe spin- or valley-polarized bilayer graphene,in the limit of weak interlayer coupling. Once the Gaussian theory and, especially, the associated collective excitations are fully understood, the next logicalstep would be to consider a higher-order expansion in thefluctuating gauge fields. Such a calculation requires a greatamount of effort and, therefore, needs to be properly mo- tivated. While the renormalization of the collective modes in a nonrelativistic system is constrained to some degreeby virtue of Kohn’s theorem [ 81], this is not the case for (pseudo)relativistic systems like graphene. Kohn’s theoremonly applies to Galilei invariant systems, stating that in thelong wavelength limit the entire electron system performsa cyclotron motion with the bare cyclotron frequency ω c= eB/m , where mis the band mass. In other words, in the limit /vectorq→0, the spectra of the collective modes in the IQH and FQH system converge to the gap ωc, which is not renormalized by Coulomb interactions [ 51,53,54,80]. The breakdown of Kohn’s theorem in graphene and the associated renormalization ofcyclotron resonances may be understood intuitively by con-sidering the renormalization of the spectrum due to Coulomb interactions in the absence of external magnetic fields. In this case the Coulomb interaction leads to logarithmic momentumcorrections to the Fermi velocity, which renormalize thelinear spectrum and diverge in the infrared regime at zerotemperature [ 6,7,82–85]. Such deviations from linearity should influence the Landau level spectrum when finite magneticfields are considered, even for large fields in the IQH regime. Indeed, perturbative calculations of the self-energy at finite Blead to similar logarithmic corrections, which, in turn, renormalize the noninteracting cyclotron resonances [ 7,73– 76], showing the breakdown of Kohn’s theorem. Since here the FQHE is viewed as an IQHE of composite Dirac fermionsfor which Kohn’s theorem does not hold, it is natural toexpect that self-energy corrections to the composite Dirac fermions also lead to observational consequences in the FQH regime. The analysis of such corrections and their impact on the electromagnetic response in particular, would require more elaborate calculations that are going beyond the RPA of the present paper. Recall that within the RPA the polarizationtensor ˆ/Pi1is computed with noninteracting Green functions, which take into account the finite gauge field expectation valuesbut neglect exchange self-energy contributions. One way toinclude self-energy effects would be to expand the tracelog inEq. ( 22) to higher orders in the fluctuations as stated above. The resulting effective action ( 31) would then contain non- Gaussian contributions, which renormalize the propagator ofgauge fluctuations as well as the electromagnetic response ten-sorˆK,E q .( 44). Alternatively, one may reintroduce fermionic degrees of freedom by writing the exponentiated tracelog inEq. ( 22) as a fermionic functional integral. Such a Fermi-Bose theory allows for a more systematic approach to study themutual effects of gauge fluctuations on the fermionic self- energy, and, vice versa, self-energy effects on the bosonicpolarization tensor, even beyond perturbation theory [ 83–85]. An important aspect in the study of the integer and fractional quantum Hall effect constitutes the role of disorder [ 2]. As is well-known, scalar potential disorder leads to a broadening ofthe noninteracting Landau levels, which enter the calculationof the fermionic polarization tensor and, in turn, lead to observ-able consequences in the electromagnetic response spectrum,such as new kinds of collective modes (typically, diffusionmodes). Apart from the simple scalar potential disorder, thereare other types of disorder potentials, which allow scatteringprocesses between different flavors, causing the fermionicpropagators to be nondiagonal in flavor space and may evenlead to another set of collective diffusion modes [ 86–89]. Given the large variety of possible microscopic scatteringchannels among the different flavors of Dirac particles andthe mutual interactions between the possible collective modes,the study of disorder in graphene is a highly nontrivial task.The Keldysh formulation we employed here has proven to bean efficient computational tool for these kinds of problems,as one can perform a disorder average directly on the level ofthe partition function, assuming that the disorder potentials aredelta correlated, which results in a fermionic pseudointeraction[64,78,79]. In contrast to the Matsubara formulation, there is no need of the replica trick and a subsequent analyticalcontinuation. The pseudointeraction term may then be ana-lyzed by standard techniques, such as Hubbard-Stratonovichbosonization and/or the Wilsonian/functional renormalizationgroup [ 64,78,79]. Another particularly interesting research direction concerns the gauge group of the Chern-Simons field itself. Here weformulated an Abelian U(1) ⊗4CS theory, where SU(2)⊗2and SU(4) invariant states only arise as a subset of all possibleFQH states obtained from the U(1) ⊗4theory. The symmetry of the exact theory may only be generated as a dynamicalsymmetry in a more elaborate calculation, going well beyondthe Gaussian fluctuations around a mean-field solution. Analternative route, where the non-Abelian SU(2) ⊗2, respec- tively SU(4) symmetry is manifest, would be to formulate acorresponding non-Abelian gauge theory in analogy to theone proposed in Refs. [ 52,54]. In these works the electron is regarded as a compound object, consisting of a chargecarrying holon and a charge neutral spinon that carries thespin degree of freedom, which are bound together by an RVB (resonating valence bond) gauge field. The holon interactswith a U(1) Chern-Simons gauge field (in addition to thecharge density–charge density Coulomb interaction) which isresponsible for the actual FQHE and yields the allowed fillingfractions, whereas the spinon interacts with a non-AbelianSU(2) Chern-Simons gauge field assigning the correct spinstructure to the respective states at each filling fraction. Asa consequence, the states for each filling fraction naturallyform irreducible representations of the non-Abelian gaugegroup [ 90]. It should be possible to apply these ideas also for graphene and it is expected that analogous features willarise in this pseudorelativistic framework. The spin sectorof such theories, however, would be much more difficult toanalyze than the corresponding Abelian theory, in particularbeyond a mean-field approximation, due to the additional 115123-14ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) cubic gauge field term required by gauge invariance, and propagating Fadeev-Popov ghosts, arising from gauge fixing[68]. Nevertheless, such a model is worth studying as it may lead to interesting insights in the fractional quantum Hall effectin graphene. As a last remark, we want to point out that our Chern-Simons theory may be of use in the conventional nonrelativistic FQHE.In this context we remind the reader of Son’s proposal of apseudorelativistic theory to explain the physics of a half-filledLandau level, Ref. [ 49]. Naively applying our framework for a single Dirac flavor under the assumption that charge neutralityof this relativistic model maps to half-filling of the nonrela-tivistic one, ν NR=1 2+νCDF 2kνCDF+1, we made an interesting ob- servation: not only this formula reproduces all the particle-holesymmetric filling fractions of Jain’s primary sequence aroundhalf-filling, but also those filling fractions that are found in theHaldane-Halperin hierarchy and/or Jains secondary sequence(such as 5 /13, 4/11, and 7 /11, for example) [ 36,92,93], as long as kis restricted to be an even integer. Of course, it could very well be the case that this feature is a mere accident, but themore appealing possibility is that there is a deeper connectionbetween our Chern-Simons framework and Son’s idea thanexpected. In any case, it is worthwhile investigating this issue. ACKNOWLEDGMENTS The author wants to thank Piet Brouwer and Zhao Liu for helpful discussions and Piet Brouwer for support in thepreparation of the manuscript. This work is supported bythe German Research Foundation (DFG) in the framework ofthe Priority Program 1459 “Graphene.” APPENDIX A: FERMION PROPAGATOR IN EXTERNAL MAGNETIC FIELD In this first Appendix, we derive the noninteracting propa- gator of two-dimensional Dirac particles in graphene, movingin a homogeneous magnetic field at finite temperature inKeldysh basis. This propagator has already been calculated by several authors using different methods, see, for example, theRefs. [ 94–96], but in order to make the article self-contained, we present one of those calculations, adapted to our notationalconventions, here again. The problem of inverting the operator ˆG −1 0in the quadratic form ( 11) is simplified by the fact that it is diagonal in flavor space [see Eq. ( 12)]. Therefore the propagator itself has to be flavor diagonal, ˆG=diag/parenleftbig G+↑,G−↑,G+↓,G−↓/parenrightbig , (A1) withGα=(G−1 α)−1. Thus the problem is reduced to finding the inverse of G−1 α, which describes the propagation of a single flavor. Slightly abusing language, we refer to the propagator foreach individual flavor G αas “the propagator” in what follows. Based on the results of the mean-field approximation, Eqs. ( 27) and ( 28), we assume that each of the flavors is subject to an individual magnetic field Bα eff=B+bα, and we allow each flavor to be doped individually. The propagator we obtain hereoccurs in the derivation of the one-loop polarization tensor(32). The latter will be derived in detail in Appendix B. In order to lighten the notation, a repeated flavor space index does notimply summation. Furthermore, calculations are performed inthe mixed frequency-position space. After mapping from contour to physical time and rotating to Keldysh basis, the propagator obeys the triangular Keldyshstructure G α(/vectorr,/vectorr/prime,ε)=/parenleftbiggGK α(/vectorr,/vectorr/prime,ε)GR α(/vectorr,/vectorr/prime,ε) GA α(/vectorr,/vectorr/prime,ε)0/parenrightbigg . (A2) As mentioned in the main text, we are only interested in the linear response regime at finite temperature. Hence thefluctuation-dissipation theorem can be employed to expressthe Keldysh propagator as G K α(/vectorr,/vectorr/prime,ε)=tanh/parenleftBigε 2T/parenrightBig/parenleftbig GR α(/vectorr,/vectorr/prime,ε)−GA α(/vectorr,/vectorr/prime,ε)/parenrightbig .(A3) The retarded and advanced propagators will be constructed from the exact solutions of the stationary Dirac equation.Working in Landau gauge with the effective vector potential /vectorA α eff(/vectorr)=(−Bα effy,0)/intercal, these solutions read /Psi10 α,kx(x,ξ)=eikxx/parenleftbigg0 ψ0(ξ)/parenrightbigg ,/Psi1λ α,kxn(x,ξ)=1√ 2eikxx/parenleftbigg−λκαψn(ξ) ψn+1(ξ)/parenrightbigg ifeBα eff<0, (A4a) /Psi10 α,kx(x,ξ)=eikxx/parenleftbigg ψ0(ξ) 0/parenrightbigg ,/Psi1λ α,kxn(x,ξ)=1√ 2eikxx/parenleftbigg ψn+1(ξ) +λκαψn(ξ)/parenrightbigg ifeBα eff>0, (A4b) where kxis a momentum quantum number, nis a positive integer including zero, ξ=y /lscriptα+sign(eBα eff)kx/lscriptαis a dimensionless real-space coordinate, and /lscriptα=1√ |eBα eff|is the magnetic length associated to the effective magnetic field Bα eff. The spinor /Psi10 α,kx(x,ξ) is the zero-energy Landau level located at the Dirac point, and /Psi1λ α,kxn(x,ξ) are Landau levels in the conduction ( λ=+1) and valence band ( λ=−1), respectively, whose spectrum is symmetric around the Dirac point. Recall that κα=±1 in the definition of the above spinors refers to the valleys K±. Furthermore, ψn(ξ) are the normalized harmonic oscillator wave functions ψn(ξ)=1√ 2nn!1 π1/4e−1 2ξHn(ξ), (A5) withHn(ξ) being the Hermite polynomial of degree n. 115123-15CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) In terms of the above exact solution the retarded and advanced propagators admit the following spectral decomposition: GR/A α(/vectorr,/vectorr/prime,ε)=1 /lscriptα/integraldisplaydkx 2π/bracketleftBigg /Psi10 α,kx(x,ξ)/Psi10† α,kx(x/prime,ξ/prime) ε+μα±i0+/summationdisplay λ=±1∞/summationdisplay n=0/Psi1λ α,kxn(x,ξ)/Psi1λ† α,kxn(x/prime,ξ/prime) (ε+μα±i0)−λ√n+1ωαc/bracketrightBigg , (A6) with the cyclotron frequency ωα c=√ 2vF /lscriptα. The momentum integration therein can be performed analytically with the help of the integral identity (Ref. [ 97], Eq. 7.377), /integraldisplay xe−x2Hm(y+x)Hn(z+x)=2n√πm!zn−mLn−m m(−2yz),m/lessorequalslantn. (A7) Here, Lk n(x) are the associated Laguerre polynomials of degree n. As a result of the momentum integration, we find that the propagators can be written as a product of a translation- and gauge noninvariant phase χα(/vectorr,/vectorr/prime)=−e/integraltext/vectorr/prime /vectorr/vectorAα(/vectorr/prime/prime)·d/vectorr/prime/prime—which is nothing but a Wilson line—and a translation- and gauge-invariant part SR/A α(/vectorr−/vectorr/prime,ε), GR/A α(/vectorr,/vectorr/prime,ε)=eiχα(/vectorr,/vectorr/prime)SR/A α(/vectorr−/vectorr/prime,ε). (A8) Introducing the relative coordinate /Delta1/vectorr=/vectorr−/vectorr/prime, and the projection operators P±=1 2/parenleftbig σ0±sign/parenleftbig eBeff α/parenrightbig σ3/parenrightbig , (A9) the translation- and gauge-invariant part of the propagators can be written compactly as SR/A α(/Delta1/vectorr,ε)=exp/parenleftbig −/Delta1/vectorr2 4/lscript2α/parenrightbig 4π/lscript2α∞/summationdisplay n=0/summationdisplay λ=±1/bracketleftbigg P+L0 n/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbigg +P−L0 n−1/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbigg +iλκ√ 2/lscriptα/vectorσ·/Delta1/vectorr√nL1 n−1/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbigg/bracketrightbigg SR/A α,λn(ε), (A10) with SR/A α,λn(ε)=1 (ε+μα±i0)−λ√nωαc. (A11) Here we have defined L0 −1,L1 −1≡0. The charge carrier 3-current per flavor, ¯jμ α, is given by ¯jμ α(/vectorr,t)=−i 2trσμ αGKα(/vectorr,t,/vectorr,t)=−i 2/integraldisplay εtanh/parenleftBigε 2T/parenrightBig trσμ α/parenleftbig GR α(/vectorr,/vectorr,ε)−GA α(/vectorr,/vectorr,ε)/parenrightbig . (A12) In thermal equilibrium, only its zero component, being the charge carrier density, ¯j0 α≡¯nα, acquires a finite value ¯nα(/vectorr,t)=1 2π/lscript2ανα. (A13) Here,ναdefines the filling fraction per flavor as a function of the chemical potential μα, the effective magnetic field Bα eff, and temperature T: να=1 2/bracketleftBigg tanh/parenleftBigμα 2T/parenrightBig +∞/summationdisplay n=1/parenleftbigg tanh/parenleftbigg√nωα c+μα 2T/parenrightbigg +tanh/parenleftbigg−√nωα c+μα 2T/parenrightbigg/parenrightbigg/bracketrightBigg . (A14) Near absolute zero temperature the filling fraction is quantized into plateaus of half-integers να=±(nα+1 2),nα=0,1,2,..., see Fig. 2. The anomalous additional fraction occurs due to the presence of a Landau level at charge neutrality ( μα=0). APPENDIX B: FERMIONIC ONE-LOOP POLARIZATION TENSOR In this second Appendix, we derive the one-loop polarization tensor for Dirac fermions experiencing a homogeneous, flavor- dependent effective magnetic field Bα eff=B+bα. See also Ref. [ 98] for a calculation of the polarization function [the 00 component of Eq. ( B1)], with which our result coincides. Displaying the Keldysh structure explicitly, Eq. ( 32) reads /Pi1μν αβ(x,y)=−i 2δ2 δaβ ν(y)δaαμ(x)tr ln/parenleftBigg 0/parenleftbigˆGA 0/parenrightbig−1 /parenleftbigˆGR 0/parenrightbig−1−/parenleftbigˆGR 0/parenrightbig−1/parenleftbigˆGK 0/parenrightbig/parenleftbigˆGA 0/parenrightbig−1/parenrightBigg /bracketleftbig eAμ+aα μ/bracketrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle a=¯a. (B1) Recall that ¯ais the field expectation value of the statistical gauge field, which possesses a classical component only. Performing the functional derivatives and evaluating the result at the mean-field values of the statistical gauge fields, we obtain the following 115123-16ABELIAN CHERN-SIMONS THEORY FOR THE … PHYSICAL REVIEW B 97, 115123 (2018) retarded, advanced, and Keldysh components: (/Pi1R/A)μν αβ(x−y)=i 2tr/parenleftbig σμ αSR/A α(x−y)σν αSK α(y−x)+σμ αSK α(x−y)σν αSA/R α(y−x)/parenrightbig δαβ, (B2a) (/Pi1K)μν αβ(x−y)=i 2tr/parenleftbig σμ αSR α(x−y)σν αSA α(y−x)+σμ αSA α(x−y)σν αSR α(y−x)+σμ αSK α(x−y)σν αSK α(y−x)/parenrightbig δαβ.(B2b) The repeated flavor space index αdoes notimply summation, as was the case in Appendix A. Recall that the Pauli 3-vector therein is given by σμ α≡(σ0,καvFσ1,καvFσ2). First, observe that the polarization tensor is diagonal in flavor space, /Pi1μν αβ=/Pi1μν αδαβ, which is a consequence of the free propagator being diagonal, see Eq. ( A1). Second, note that the gauge- and translation-noninvariant phaseχα(/vectorr,/vectorr/prime) drops out, such that the polarization tensor can be expressed solely in terms of the propagators SR/A/K α , proving its manifest gauge and translation invariance. In Fourier space, the above equations for the flavor diagonal components /Pi1αbecome (/Pi1R/A)μν α(ω,/vectorq)=i 2/integraldisplay /Delta1/vectorre−i/vectorq·/Delta1/vectorr/integraldisplay εtr/parenleftbig σμ αSR/A α(/Delta1/vectorr,ε+ω)σν αSK α(−/Delta1/vectorr,ε)+σμ αSK α(/Delta1/vectorr,ε)σν αSA/R α(−/Delta1/vectorr,ε−ω)/parenrightbig , (B3a) (/Pi1K)μν α(ω,/vectorq)=coth/parenleftBigω 2T/parenrightBig/parenleftbig (/Pi1R)μν α(ω,/vectorq)−(/Pi1A)μν α(ω,/vectorq)/parenrightbig . (B3b) Equation ( B3b) is a manifestation of the (bosonic) fluctuation-dissipation theorem. In order to arrive at this form, one has to rewrite the first line of Eq. ( B2b) according to σμSR xyσνSA yx+σμSA xyσνSR yx=−σμ(SR xy−SA xy)σν(SR yx−SA yx), which holds true because of the causality properties of the retarded and advanced propagators, cf. Ref. [ 64]. Next, one has to employ Eq. ( A3) and finally make use of the identity tanh( x)tanh(y)−1=coth(x−y)(tanh( y)−tanh(x)). By substituting the propagators ( A3) and ( A10) into Eq. ( B3a), the polarization tensor acquires the form (/Pi1R/A)μν α(ω,/vectorq)=1 32π2/lscript4α/summationdisplay n,n/prime/summationdisplay λ,λ/primeFλλ/prime nn/prime(T,μα) (ω±i0)−λ√nωαc+λ/prime√ n/primeωαc/integraldisplay /Delta1/vectorre−i/vectorq·/Delta1/vectorre−/Delta1/vectorr2 2/lscript2αtr/parenleftbig σμ αMα n(λ/Delta1/vectorr)σν αMα n/prime(−λ/prime/Delta1/vectorr/prime)/parenrightbig ,(B4) with Fλλ/prime nn/prime(T,μα)=tanh/parenleftbiggλ/prime√ n/primeωc α−μα 2T/parenrightbigg −tanh/parenleftbiggλ√nωc α−μα 2T/parenrightbigg (B5) and Mα n(λ/Delta1/vectorr)=P+L0 n/parenleftbigg/Delta1/vectorr2 2/lscriptα/parenrightbigg +P−L0 n−1/parenleftbigg/Delta1/vectorr2 2/lscriptα/parenrightbigg +iλκ√ 2/lscriptα/vectorσ·/Delta1/vectorr√nL1 n−1/parenleftbigg/Delta1/vectorr2 2/lscriptα/parenrightbigg . (B6) Performing the trace for each tensor component and comparing the resulting expressions with the kernel expansion ( 35), we can extract the following scalar quantities: (/Pi1R/A)0 α(ω,/vectorq)=−1 32π2/lscript4α1 /vectorq2/summationdisplay n,n/prime/summationdisplay λ,λ/primeFλλ/prime nn/prime(T,μα) (ω±i0)−λ√nωαc+λ/prime√ n/primeωαc/parenleftbigg I0 n−1,n/prime(Qα)+I0 n,n/prime−1(Qα)+2λλ/prime √ nn/primeI1 n−1,n/prime−1(Qα)/parenrightbigg , (B7a) (/Pi1R/A)1 α(ω,/vectorq)=−sign(Beff α) 32π2/lscript4αv2 F ω/summationdisplay n,n/prime/summationdisplay λ,λ/primeFλλ/prime nn/prime(T,μα) (ω±i0)−λ√nωαc+λ/prime√ n/primeωαc/parenleftbig I0 n−1,n/prime(Qα)−I0 n/prime,n−1(Qα)/parenrightbig , (B7b) (/Pi1R/A)2 α(ω,/vectorq)=+1 32π2/summationdisplay n,n/prime/summationdisplay λ,λ/primeFλλ/prime nn/prime(T,μα) (ω±i0)−λ√nωαc+λ/prime√ n/primeωαc/parenleftbigg2λλ/prime √ nn/primev2 F /lscript2α∂2 Qα˜I1 n−1,n/prime−1(Qα)/parenrightbigg . (B7c) Here we have defined the integral expressions Ik n,n/prime(Qα)=/integraldisplay /Delta1/vectorre−i/vectorq·/Delta1/vectorre−/Delta1/vectorr2 2/lscript2α/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbiggk Lk n/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbigg Lk n/prime/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbigg =2π/lscript2 αQn>−n< αe−Qα(n<+k)! n>!Ln>−n< n<(Qα)Ln>−n< n<+k(Qα),k=0,1, (B8a) ˜I1 n,n/prime(Qα)=/integraldisplay /Delta1/vectorre−i/vectorq·/Delta1/vectorre−/Delta1/vectorr2 2/lscript2αL1 n/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbigg L1 n/prime/parenleftbigg/Delta1/vectorr2 2/lscript2α/parenrightbigg =n/summationdisplay m=0n/prime/summationdisplay m/prime=0I0 m,m/prime(Qα), (B8b) 115123-17CHRISTIAN FRÄßDORF PHYSICAL REVIEW B 97, 115123 (2018) where Qα=/vectorq2/lscript2 α 2is a dimensionless momentum variable, and n>=max{n,n/prime},n<=min{n,n/prime}. Note that both Ik n,n/primeand˜I1 n,n/primeare symmetric in their Landau indices n,n/prime. Hence, without loss of generality, we can assume n/lessorequalslantn/primein the following proof. First, let us show how ˜I1 n,n/prime(Qα) can be reduced to a sum of I0 n,n/prime(Qα). In order to prove this equality, we only have to make use of the property Lk+1 n(x)=/summationtextn m=0Lk m(x), see Ref. [ 97], Eq. 8.974.3, and interchange integration and summation. We immediately arrive at the second line of Eq. ( B8b). The proof of Eq. ( B8a) is more involved. First of all, one has to work in polar coordinates, substituting t=/Delta1/vectorr2 2/lscript2α, and perform the angle integration, which yields the Bessel function of the first kind J0: Ik n,n/prime(Qα)=2π/lscript2 α/integraldisplay∞ 0dt e−ttkJ0(2/radicalbig Qαt)Lk n>(t)Lk n<(t). (B9) Next, we rewrite Lk n<(t)=(−t)−k(n<+k)! n<!L−k n<+k(t), see Ref. 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PhysRevB.93.165410.pdf

PHYSICAL REVIEW B 93, 165410 (2016) Energy levels of hybrid monolayer-bilayer graphene quantum dots M. Mirzakhani,1,2,*M. Zarenia,1,†S. A. Ketabi,2,‡D. R. da Costa,3and F. M. Peeters1,3,§ 1Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium 2School of Physics, Damghan University, P . O. Box: 36716-41167, Damghan, Iran 3Departamento de F ´ısica, Universidade Federal do Cear ´a, 60455-900 Fortaleza, Cear ´a, Brazil (Received 28 January 2016; published 8 April 2016) Often real samples of graphene consist of islands of both monolayer and bilayer graphene. Bound states in such hybrid quantum dots are investigated for (i) a circular single-layer graphene quantum dot surrounded by an infinitebilayer graphene sheet and (ii) a circular bilayer graphene quantum dot surrounded by an infinite single-layergraphene. Using the continuum model and applying zigzag boundary conditions at the single-layer–bilayergraphene interface, we obtain analytical results for the energy levels and the corresponding wave spinors. Theirdependence on perpendicular magnetic and electric fields are studied for both types of quantum dots. The energylevels exhibit characteristics of interface states, and we find anticrossings and closing of the energy gap in thepresence of a bias potential. DOI: 10.1103/PhysRevB.93.165410 I. INTRODUCTION Quantum dots (QDs) in both single-layer and bilayer graphene have been the subject of intensive research during thelast few years, owning to their unique electronic and opticalproperties [ 1–14]. Single-layer graphene (SLG) QDs are small flakes cut out from graphene in which carrier confinement isdue to the quantum size effect. The electronic and optical properties of such QDs depend on the shape and edges of the dot. For example, in the presence of zigzag edges, the energyspectrum of SLG QDs exhibits zero-energy levels, while witharmchair edges the spectrum displays an energy gap [ 2,8,16]. Electrostatic confinement of electrons in integrable grapheneQDs was also proposed in which the effect of edges is nolonger important [ 15]. Bilayer graphene (BLG) consists of two van der Waals (vdW) coupled layers of SLG and has a gapless and parabolic-like spectrum at low energies [ 17]. Unlike SLG, an external electric field, realized by external gate potentials, can induce atunable band gap in the energy spectrum of BLG. Engenderingthis gap using nanostructured gates led to the realization ofelectrostatic defined BLG QDs [ 5,18]. In such BLG QDs, the confinement is due to the electrostatic potentials and therefore the effect of edges are no longer important. Such QDs were recently realized by two different experimental groups [ 19,20]. Most graphene samples exfoliated from graphite consist of islands of one or few layers of graphene [ 21–23]. In these samples, the influence of junctions between different grapheneregions play a significant role in their transport and electronicproperties. It was shown that SLG-BLG hybrid systems exhibit unusual transport properties due to the different quantum Hall (QH) states in the SLG and BLG regions [ 24]. An unconventional Landau quantization was recently observed atthe interface of such hybrid systems [ 25]. The Landau levels of an infinite SLG-BLG system were theoretically studiedfor both zigzag and armchair boundary conditions at the 1Dinterface [ 26]. *mohammad.mirzakhani@uantwerpen.be †mohammad.zarenia@uantwerpen.be ‡saketabi@du.ac.ir §francois.peeters@uantwerpen.beHere, we investigate a very different geometry and demon- strate that carriers can be confined in SLG and BLG islandsin a hybrid QD-like structure made of SLG-BLG junctions.This novel type of QDs can be realized by the (accidental)nanostructuring of one of the graphene layers in bilayergraphene. For convenience, we will restrict ourselves tocircular QDs. This will allow us to present analytical results,which we will compare with a pure numerical approach. Wepropose the following two types of hybrid QDs: SLG-infinite BLG —a circular SLG QD surrounded by an infinite BLG sheet [see Fig. 1(a)] and BLG-infinite SLG —a circular BLG QD surrounded by an infinite SLG sheet [see Fig. 1(b)]. Taking the circular geometry with radius Rfor the QDs, we employ the continuum model, i.e., solving the Dirac-Weylequation, and obtain analytical results for the energy levelsand corresponding wave functions. We study the effect of bothperpendicular electric and magnetic fields on the energy levels.For zero-magnetic field, we demonstrate that SLG-infiniteBLG, in contrast to the BLG-infinite SLG QD, exhibit confinedstates in the presence of an external electric field. If such acircular QD is cut out of BLG, one will have both armchairand zigzag edges. However, in order to obtain analyticalresults for the energy levels and to observe features broughtby the zigzag edges in the spectrum, we will implement thezigzag boundary condition at the SLG-BLG junction in ourcontinuum approach. Breaking the inversion symmetry due tothe interface and breaking the time reversal symmetry witha magnetic field, the two Dirac valleys KandK /prime, should be studied separately. The paper is organized as follows. In Sec. II, we consider SLG-infinite BLG QDs in the (A) absence and (B) presenceof a perpendicular magnetic field. In both cases, the effectof an external electric field is studied. Section IIIconcerns numerical results for BLG-infinite SLG QDs. We concludethe manuscript in Sec. IV. II. SLG-INFINITE BLG QUANTUM DOTS A. Zero magnetic field First, we investigate the energy levels of a circular SLG QD embedded in infinite BLG [see Fig. 1(a)]. This system can be 2469-9950/2016/93(16)/165410(11) 165410-1 ©2016 American Physical SocietyMIRZAKHANI, ZARENIA, KETABI, DA COSTA, AND PEETERS PHYSICAL REVIEW B 93, 165410 (2016) A1 B1 A2 B2 R ABR AB(a) (b)2R 2R FIG. 1. Schematic pictures of the proposed circular SLG-BLG hybrid QDs with radius R. (a) SLG-infinite BLG QD: circular SLG dot surrounded by an infinite BLG. (b) BLG-infinite SLG QD: circular BLG dot surrounded by an infinite SLG. The upper pictures show a side view of the systems. considered as an infinite BLG sheet where a circle of radius R is cut out from its upper graphene layer. So we assume that onelayer of BLG, containing A1 and B1 sublattices, seamlessly continues to the SLG with AandBsublattices, while the other graphene layer composed of A2 andB2 sublattices is sharply cut at the boundary r=R. We obtain the corresponding Hamiltonian in both SLG and BLG regions and by implementing zigzag boundary conditionsto one of the graphene layers at the SLG-BLG interface, wecalculate the energy levels. The dynamics of carriers in thehoneycomb lattice of covalent-bond carbon atoms of singlelayer graphene can be described by the following Hamiltonian,which in zero magnetic field is given by [ 27] H=v Fp·σ+U1I, (1) where vF≈106m/s is the Fermi velocity, p=(px,py)i s the two-dimensional momentum operator, σdenotes the Pauli matrices, and U1is the potential applied to SLG. We assume that the carriers are confined in a circular area of radius R, with zigzag boundary. In polar coordinates and dimensionlessunits, the Hamiltonian ( 1) reduces to the form H=/parenleftbigg u 1π+ π−u1/parenrightbigg , (2) with the momentum operator π±=−ie±iτϕ/bracketleftbigg∂ ∂ρ±iτ ρ∂ ∂ϕ/bracketrightbigg , (3) where the dimensionless variables are ρ=r/R andu1= U1R//planckover2pi1vF.randϕare the radial and azimuthal coordinates of the cylindrical coordinate system, respectively. The twovalleys are labeled by the quantum number τ, which is τ=+1 for the Kvalley and τ=−1f o rt h e K /primevalley. The Schr ¨odinger equation becomes H/Psi1(ρ,ϕ)=ε/Psi1(ρ,ϕ), (4) where the carrier energy E, is written in dimensionless units asε=ER//planckover2pi1vF. The two-component wave function has theform [ 28] /Psi1τ(ρ,ϕ)=eimϕ/parenleftbiggφτ A(ρ) ie−iτϕφτ B(ρ)/parenrightbigg , (5) where m=0,±1,±2,... denotes the angular momentum label. The components φAandφBcorrespond to different sublattices AandB, respectively. Solving Eq. ( 4), the radial dependence of the spinor components is described by /bracketleftbiggd dρ−mτ−1 ρ/bracketrightbigg φτ B(ρ)=(ε−u1)φτ A(ρ), /bracketleftbiggd dρ+τm ρ/bracketrightbigg φτ A(ρ)=−(ε−u1)φτ B(ρ). (6) Decoupling the above equations, we arrive at the Bessel differential equation for φτ A: ρ2d2φτ A(ρ) dρ2+ρdφτ A(ρ) ρ+[(ε−u1)2ρ2−m2]φτ A(ρ)=0, (7) with the solution φτ A(ρ)=CτJm(aρ), (8) where a=ε−u1andCτis the normalization constant. The second component of the wave function can be obtained fromEq. ( 6)a s φ τ B(ρ)=−τCτJm−τ(aρ). (9) Thus the wave function becomes /Psi1τ(ρ,ϕ)=eimϕ/parenleftbiggCτJm(aρ) ieiτϕτCτJm+τ(aρ)/parenrightbigg . (10) The BLG region can be described in terms of four sublattices, labeled A1,B1,for the lower layer and A2,B2,for the upper layer [see Fig. 1(a)]. TheA1 andB2 sites are coupled via a nearest-neighbor interlayer hopping term t≈0.4e V .T h e BLG Hamiltonian in the vicinity of the Kpoint, is given by (in dimensionless units) [ 18,29] HK=HK 0+(/Delta1u/2)σz, (11) with HK 0=⎛ ⎜⎝u0π+t/prime0 π−u0 00 t/prime0u0π− 00 π+u0⎞ ⎟⎠, (12) where t/prime=tR//planckover2pi1vF,u0=(u1+u2)/2,/Delta1 u=u1−u2, and u1,2=U1,2R//planckover2pi1vF, with U1andU2the potentials at the two layers. The operator σzis defined as σz=/parenleftbigg I 0 0−I/parenrightbigg , (13) where Iis the 2 ×2 identity matrix. The Hamiltonian at the K/primepoint is obtained by interchanging π+andπ−in Eq. ( 12). The eigenstates of Hamiltonian ( 11) are four-component spinors [ 30] /Phi1K(ρ,ϕ)=⎛ ⎜⎜⎜⎝φK A1(ρ)eimϕ iφK B1(ρ)ei(m−1)ϕ φK B2(ρ)eimϕ iφK A2(ρ)ei(m+1)ϕ⎞ ⎟⎟⎟⎠, (14) 165410-2ENERGY LEVELS OF HYBRID MONOLAYER-BILAYER . . . PHYSICAL REVIEW B 93, 165410 (2016) where mis the angular momentum label. Solving the Schr ¨odinger equation, the radial dependence of the spinor components are described by /bracketleftbiggd dρ−m−1 ρ/bracketrightbigg φK B1(ρ)=(α−δ)φK A1(ρ)−t/primeφK B2, /bracketleftbiggd dρ+m ρ/bracketrightbigg φK A1(ρ)=−(α−δ)φK B1(ρ), /bracketleftbiggd dρ+m+1 ρ/bracketrightbigg φK A2(ρ)=(α+δ)φK B2(ρ)−t/primeφK A1(ρ), /bracketleftbiggd dρ−m ρ/bracketrightbigg φK B2(ρ)=−(α+δ)φK A2(ρ), (15) where α=ε−u0andδ=(u1−u2)/2. These equations can be decoupled and we obtain for φK A1: /bracketleftbiggd2 dρ2+1 ρd dρ−m2 ρ2/bracketrightbigg φK A1(ρ)=γ2 ±φK A1(ρ), (16) where the potential-dependent eigenvalues are γ±={ − (α2+δ2)±[(α2−δ2)t/prime2+4α2δ2]1/2}1/2.(17) The differential equation ( 16) is the known modified Bessel equation. Here we choose the modified Bessel function of thesecond kind K m(γ±), as the appropriate solutions vanishing at r→∞ . Thus we have φK A1(ρ)=CK 1Km(γ+ρ)+CK 2Km(γ−ρ). (18) Using Eqs. ( 15), we obtain the other spinor components: φK B1(ρ)=1 α−δ/bracketleftbig CK 1γ+Km−1(γ+ρ)+CK 2γ−Km−1(γ−ρ)/bracketrightbig , φK B2(ρ)=1 (α−δ)t/prime/bracketleftbig CK 1((α−δ)2+γ2 +)Km(γ+ρ) +CK 2((α−δ)2+γ2 −/parenrightbig Km(γ−ρ)/bracketrightbig , φK A2(ρ)=1 (α2−δ2)t/prime/bracketleftbig CK 1γ+((α−δ)2+γ2 +)Km+1(γ+ρ) +CK 2γ−((α−δ)2+γ2 −)Km+1(γ−ρ)/bracketrightbig , (19) where CK j(j=1,2) are the normalization constants. The wave function for the K/primevalley can be written as /Phi1K/prime(ρ,ϕ)=⎛ ⎜⎜⎜⎝φK/prime A1(ρ)eimϕ iφK/prime B1(ρ)ei(m+1)ϕ φK/prime B2(ρ)eimϕ iφK/prime A2(ρ)ei(m−1)ϕ⎞ ⎟⎟⎟⎠. (20) Solving the Schr ¨odinger equation ( 4)f o rt h e K/primevalley and comparing with the differential equations ( 15), we find (φK/prime A1,φK/prime B1,φK/prime B2,φK/prime A2)=(φK B2,φK A2,φK A1,φK B1). Now, we apply zigzag boundary conditions [ 26] at the SLG- BLG interface. These conditions yield /Psi1τ A(ρ,ϕ)=/Phi1τ A1(ρ,ϕ)|ρ=1, /Psi1τ B(ρ,ϕ)=/Phi1τ B1(ρ,ϕ)|ρ=1, 0=/Phi1τ B2(ρ,ϕ)|ρ=1, (21)where τ=±1, distinguishes the boundary conditions for the two valleys. The above conditions lead to a system of equationsfrom which we obtain the eigenvalues. For the Kpoint, with the help of the wave functions ( 10), (14), (18), and ( 18), we arrive at M K⎛ ⎝CK CK 1 CK 2⎞ ⎠=⎛ ⎝−Jm(a) Km(γ+) Km(γ−) Jm−1(a)b+Km−1(γ+)b−Km−1(γ−) 0 c+Km(γ+)c−Km(γ−)⎞ ⎠ ×⎛ ⎝CK CK 1 CK 2⎞ ⎠=0, (22) where b±=γ±/(α−δ) andc±=((α−δ)2+γ2 ±)/(α−δ)t/prime. The corresponding calculations for the K/primepoint leads to MK/prime⎛ ⎜⎝CK/prime CK/prime 1 CK/prime 2⎞ ⎟⎠=⎛ ⎝−Jm(a)c+Km(γ+)c−Km(γ−) −Jm+1(a)d+Km+1(γ+)d−Km+1(γ−) 0 Km(γ+) Km(γ−)⎞ ⎠ ×⎛ ⎜⎝CK/prime CK/prime 1 CK/prime 2⎞ ⎟⎠=0, (23) where d±=c±γ±/(α+δ). The nonzero eigenenergies are the solutions of det |MK|=0 and det |MK/prime|=0. The zero-energy states can be investigated separately bysolving Eqs. ( 6) and ( 15) in the case of /epsilon1=0 andU 1=U2=0. Applying the boundary conditions ( 21), one finds zero-energy states at the K(K/prime) valley only for m/greaterorequalslant0(m/lessorequalslant0). Returning to the nonzero eigenenergies for the unbiased caseU1=U2=0, we find γ±=(−ε2±√ ε2t/prime2)1/2which is pure imaginary when |ε|>t/prime. In the interval |ε|<t/prime,γ+ is real while γ−is pure imaginary. Requiring det |MK|= det|MK/prime|=0, we find a continuum energy band with no discrete levels and thus no confined states in the QD.In the presence of bias, γ +=γ∗ −when |ε|<u 1. For this interval we can select the real (imaginary) part of the modifiedBessel functions K m(γ+)(Km(γ−)) as our solutions. Figure 2 shows the energy levels with the angular momenta m= 0,±1,±2,±3 as a function of the dot radius R, for the biased potential U1=−U2=0.1 eV . An energy gap appears between the conduction and valence bands. Notice that the band gap is given by [ 17]/Delta1g=| /triangleU|t//radicalbig /triangleU2+t2(see the solid black horizontal lines in Fig. 2), coming from the Mexican-hat shaped low-energy dispersion in pristine BLG. For /Delta1U/lessmucht, the band gap is /Delta1g≈/Delta1U. For both valleys, the number of energy levels increases, as the dot radius increases and the band gap decreases to zero. In both cases, each set of energy levelsare approximately equally spaced for fixed m. The energy spectrum corresponding to the Kvalley [Fig. 2(a)] exhibits the symmetry E K(m)=EK(−m), which does not hold for the K/prime valley levels. This is different from SLG [ 28] and BLG QD [ 5] flakes in which the KandK/primeenergy levels are degenerate at zero magnetic field. This difference is due to the zigzagboundary condition applied to the SLG-BLG interface whichremoves the layer symmetry and thus the valley symmetry inBLG. The energy spectrum in Fig. 2(b)shows groups of energy 165410-3MIRZAKHANI, ZARENIA, KETABI, DA COSTA, AND PEETERS PHYSICAL REVIEW B 93, 165410 (2016) 0 10 20 30 40 50 60 700.100.050.000.050.10 (b) )Ve( E R (nm)3 4(a) m=0 m=±1 m=±2 m=±3)Ve( E1 2 FIG. 2. Energy levels of SLG-infinite BLG QD for m= 0,±1,±2,±3 as a function of the dot radius R, in the presence of bias U1=−U2=0.1e Va tt h e( a ) Kand (b) K/primevalleys. The energy levels of the Kvalley (a) satisfy EK(m)=EK(−m). The solid (dashed) curves are for m> 0(m< 0). levels with E∼+/Delta1g/2 and m> 0. These levels correspond to states that are mainly confined at the interface [see Fig. 3(c)]. Figures 3(a)–3(d) show the electron densities in each layer corresponding to the points labeled by (1)–(4) in Figs. 2(a) and2(b). The energy levels at the Kvalley [see Figs. 3(a) and3(b) for the levels (1) and (2)] show that the confinement is mostly in the SLG QD, while for the energy levelscorresponding to the K /primevalley [see Figs. 3(c) and3(d)]t h e 0.054 0.026 0.082 0.054(a) (b) (d) (c) FIG. 3. Probability density corresponding to the states indicated by (1), (2), (3), and (4) in the energy spectrum of Fig. 2forR=30 nm. Blue solid curves refer to layer 1, and red dashed curves denotelayer 2.carriers are confined in both SLG QD and at the SLG-BLG interface. The set of energy levels with E∼+/Delta1g/2a r e interface states that are predominantly confined along thezigzag edge of the second layer with low density in the dotarea of the first layer [see Fig. 3(c)]. B. Nonzero magnetic field In the presence of a perpendicular magnetic field B=Bˆez, one needs to replace the canonical momentum pby the gauge- invariant kinetic momentum p+eA(r) in the Hamiltonians ( 2) and ( 12).A(r)=(0,Br/ 2,0) is the vector potential in the symmetric gauge. In the case of B/negationslash=0, it is more convenient to use the magnetic length lB=√/planckover2pi1/eB as the unit of length, facilitating the interpretation of our numerical results. This results in the following dimensionless quantities, ξ=r/√ 2lB, ε=ElB//planckover2pi1vF, and u1=U1lB//planckover2pi1vF. In these units, π±in Eqs. ( 2) and ( 12) takes the form π±=−ie±iτϕ1√ 2/bracketleftbigg∂ ∂ξ±iτ ξ∂ ∂ϕ∓τξ/bracketrightbigg , (24) and the radial Dirac-Weyl equation for the spinor components of the wave function ( 5) becomes 1√ 2/bracketleftbigg∂ ∂ξ−(τm−1) ξ−τξ/bracketrightbigg φτ B(ξ)=(ε−u1)φτ A(ξ), 1√ 2/bracketleftbigg∂ ∂ξ+τm ξ+τξ/bracketrightbigg φτ A(ξ)=−(ε−u1)φτ B(ξ).(25) After decoupling the above equations, we obtain /bracketleftbigg −1 2/parenleftbiggd2 dξ2+1 ξd dξ−m2 ξ2/parenrightbigg +(m−τ)+ξ2 2/bracketrightbigg φτ A(ξ) =(ε−u1)2φτ A(ξ). (26) Using the ansatz φτ A(ξ)=ξ|m|e−ξ2/2f(ξ2),Eq. ( 26) yields the confluent hypergeometric ordinary differential equation ˜ξd2f(˜ξ) d˜ξ2+(|m|+1−˜ξ)df(˜ξ) d˜ξ −1 2/parenleftbig |m|+m−τ+1−(ε−u1)2/parenrightbig f(˜ξ)=0,(27) where ˜ξ→ξ2. The solutions are the confluent hypergeometric functions of the first kind 1˜F1(a,b,ξ2) with a=1 2/parenleftbig |m|+m−τ+1−(ε−u1)2/parenrightbig ,b=|m|+1.(28) Then/Psi1τ Abecomes /Psi1τ A(ξ,ϕ)=Cτeimϕξ|m|e−ξ2/2 1˜F1(a,b,ξ2), (29) where Cτis the normalization constant. The second compo- nent of the wave function is extracted from the second equationof Eq. ( 25): /Psi1 τ B(ξ,ϕ)=−iCτei(m−τ)ϕ √ 2(ε−u1)ξ|m|e−ξ2/2 ×/bracketleftbigg 2ξa 1˜F1(a+1,b+1,ξ2) +/parenleftbigg|m|+τm ξ+(τ−1)ξ/parenrightbigg 1˜F1(a,b,ξ2)/bracketrightbigg .(30) 165410-4ENERGY LEVELS OF HYBRID MONOLAYER-BILAYER . . . PHYSICAL REVIEW B 93, 165410 (2016) Using Eqs. ( 4), (11), (12), (14), and ( 24) in the new dimensionless units the radial dependence of the spinorcomponents in BLG are described by 1 √ 2/bracketleftbiggd dξ−(m−1) ξ−ξ/bracketrightbigg φK B1(ξ) =(α−δ)φK A1(ξ)−t/primeφK B2(ξ), 1√ 2/bracketleftbiggd dξ+m ξ+ξ/bracketrightbigg φK A1(ξ)=−(α−δ)φK B1(ξ), 1√ 2/bracketleftbiggd dξ+(m+1) ξ+ξ/bracketrightbigg φK A2(ξ) =(α+δ)φK B2(ξ)−t/primeφK A1(ξ), 1√ 2/bracketleftbiggd dξ−m ξ−ξ/bracketrightbigg φK B2(ξ)=−(α+δ)φK A2(ξ), (31) where α=ε−u0,δ=(u1−u2)/2,u0=(u1+u2)/2, u1,2=U1,2lB//planckover2pi1vF,ε=ElB//planckover2pi1vF, andt/prime=tlB//planckover2pi1vF.U1and U2are the potentials on the two different graphene layers. These equations can be decoupled to obtain for φK A1 /bracketleftbigg −1 2/parenleftbiggd2 dξ2+1 ξd dξ−m2 ξ2/parenrightbigg +m+ξ2 2/bracketrightbigg φK A1(ξ) =γ±(ε)φK A1(ξ), (32) where γ±=α2+δ2±[(α2−δ2)t/prime2+(1−2αδ)2]1/2. (33) Using the ansatz φK A1(ξ)=ξ|m|e−ξ2/2f(ξ2),similar for the SLG region, we arrive at the confluent hypergeometric or-dinary differential equation with solutions 1˜F1(a±,b,ξ2) and U(a±,b,ξ2), where a±=1 2(|m|+m+1−γ±),b=|m|+1. (34) Now for the BLG region, we need to take the confluent hypergeometric functions of the second kind U(a,b,x ) which decays exponentially for r→∞ . Then φK A1becomes φK A1(ξ)=ξ|m|e−ξ2/2/bracketleftbig CK 1U(a+,b,ξ2)+CK 2U(a−,b,ξ2)/bracketrightbig , (35) where CK j(j=1,2) are the normalization constants. The other spinor components of the wave function can be obtained usingEq. ( 31) by inserting φ K A1and employing the properties of the confluent hypergeometric function. The wave functionat the K /primepoint can be obtained from ( φK/prime A1,φK/prime B1,φK/prime B2,φK/prime A2)= (φK B2,φK A2,φK A1,φK B1). Applying the boundary conditions ( 21)f o rt h e Kpoint at the interface ξ=ξR=R/(√ 2lB), we arrive at MK ZZ⎛ ⎝CK CK 1 CK 2⎞ ⎠=0, where MK ZZ=⎛ ⎝m11m12m13 m21m22m23 m31m32m33⎞ ⎠, (36)with the matrix elements m11=− 1˜F1/parenleftbig a;b;ξ2 R/parenrightbig , m12=U/parenleftbig a+,b,ξ2 R/parenrightbig , m13=U/parenleftbig a−,b,ξ2 R/parenrightbig , m21=1 ε−u1/bracketleftbigg 2ξRa1˜F1/parenleftbig a+1;b+1;ξ2 R/parenrightbig +(|m|+m) ξR 1˜F1/parenleftbig a;b;ξ2 R/parenrightbig/bracketrightbigg , m22=1 α−δ/bracketleftbigg 2ξRa+U/parenleftbig a++1;b+1;ξ2 R/parenrightbig −(|m|+m) ξRU/parenleftbig a+;b;ξ2 R/parenrightbig/bracketrightbigg , m23=1 α−δ/bracketleftbigg 2ξRa−U/parenleftbig a−+1;b+1;ξ2 R/parenrightbig −(|m|+m) ξRU/parenleftbig a−;b;ξ2 R/parenrightbig/bracketrightbigg , m31=0, m32=4ξ2 Ra+(a++1)U/parenleftbig a++2,b+2,ξ2 R/parenrightbig +4/bracketleftbig ξ2 R−|m|−1/bracketrightbig a+U/parenleftbig a++1,b+1,ξ2 R/parenrightbig +2[(α−δ)2−|m|−m]U/parenleftbig a+,b,ξ2 R/parenrightbig , m33=4ξ2 Ra−(a−+1)U/parenleftbig a−+2,b+2,ξ2 R/parenrightbig +4/bracketleftbig ξ2 R−|m|−1/bracketrightbig a−U/parenleftbig a−+1,b+1,ξ2 R/parenrightbig +2[(α−δ)2−|m|−m]U/parenleftbig a−,b,ξ2 R/parenrightbig . (37) One can similarly obtain the corresponding matrix for the K/prime valley MK/prime ZZ. The nonzero-energy levels are obtained from the condition det |MK ZZ|=0 In order to find the solutions of the zero-energy states, one can solve Eqs. ( 25) and( 31) in the case of /epsilon1=0(U1=U2= 0). Applying the boundary conditions ( 21) at the interface shows that zero-energy states exist for all angular momenta m, at theKvalley, and only for m< 0a tt h e K/primevalley. 1. Unbiased system Figure 4shows the energy levels as a function of magnetic field at the K(solid curves) and K/prime(dashed curves) valleys when U1=U2=0. The energy levels are shown for the angular momenta, m=−1 (blue), m=0 (green), and m=1 (red) with QD radius R=30 nm. As mentioned before, the SLG-infinite BLG QD in the absence of unbiased magnetic field displays a continuum energy band which is also apparentin Fig. 4. In this case, there are many degenerate zero-energy states at zero magnetic field corresponding to all angularmomenta for both KandK /primevalleys. Increasing the magnetic field, zero-energy degenerate levels for each mis lifted due to the breaking of the time reversal symmetry. Breaking ofthe inversion symmetry due to the interface removes thedegeneracy of the KandK /primevalleys. The spectrum shows anticrossings, which is due to the influence of the SLG-BLGinterface. At high magnetic fields, the energy levels merge into 165410-5MIRZAKHANI, ZARENIA, KETABI, DA COSTA, AND PEETERS PHYSICAL REVIEW B 93, 165410 (2016) 0 5 10 15 20 25 30 350.00.51.01.52.02.53.0 B (T)m=-1 m=0m=1 U=0K K’ 51 011.31.41.5 a bc)l/ v (ћB F FIG. 4. Energy spectrum of unbiased SLG-infinite BLG QD (i.e., U1=U2=0) as a function of magnetic field for m=−1 (blue), m=0, (green) and m=1 (red) with the dot radius R=30 nm at theK(solid curves) and K/prime(dashed curves) valleys. The inset is an enlargement of the black square box for a particular anticrossing of energy levels in the K/primevalley. the Landau levels (LLs) of SLG, ( ε=±√ 2n,nis the LLs’ number [ 26]). In a strong magnetic field, the carriers become localized at the center of the SLG dot and will not be influencedby effects due to the dot interface. It should be mentioned thatelectron-hole symmetry exists for both valleys, because allmatrix elements in M K ZZ[Eq. ( 36)] and MK/prime ZZare even function of the dimensionless energy ε, when U1=U2=0. An enlargement around a particular anticrossing point is given in the inset of Fig. 4. The wave functions as well as the probability densities for the points labeled by (a), (b), and(c) in Fig. 4are shown in Fig. 5. The upper panels show the wave functions of layer 1 which are continuous in both SLGand BLG. The wave functions of layer 2 are plotted in themiddle panels. Point (a) corresponds to confinement in bothBLG regions and the SLG-BLG interface, while for the energystate (c) the electrons are confined inside the SLG QD. Right r (nm) r (nm) r (nm)(a) (b) (c)).u .a( noitcnuF eva WK' AK' B1 K' A1K' B K' B2K' A2 K' 2 Layer1||K' 2 Layer 2|| FIG. 5. The wave functions corresponding to the points (a), (b), and (c) of the inset in Fig. 4. The upper panel shows the wave functions for layer 1, the middle panel displays the wave functions for layer 2, and the lowest panel shows the density of the bound states in the twolayers for the K /primevalley.B (T))l/ v (ћB FA1B1 A2B2R ABR 0 15 30 B (T)01 5 3 00.00.51.01.52.02.53.0 ’K K(a) (b) (c) FIG. 6. (a) Schematic pictures of the terminated systems, SLG QD (left) and BLG antidot (right). Lower panel displays the energyspectrum of SLG-infinite BLG QD (black solid curves) and the terminated systems, SLG QD (blue dashed curves) and BLG antidot (red dashed curves) as a function of magnetic field for the two valleys(b)Kand (c) K /prime. The dot radius is R=30 nm and m=1. at the anticrossing [i.e., point (b)] one finds confinement in both SLG and BLG. Plateaulike features appear in the energy spectrum, that can be understood when comparing the energy levels of terminatedSLG QD and BLG antidot. Consider two terminated systems,SLG QD and BLG antidot with zigzag edges as shown inFig.6(a). The boundary condition is /Psi1 τ A=0 for SLG QD and /Phi1τ B1=/Phi1τ A2=0 for BLG antidot. Lower panels of Figs. 6(b) and 6(c) show the energy spectrum of SLG-infinite BLG QD (black solid curves) and those of the terminated systems(dashed curves) at the two valleys KandK /primeform=1. The spectrum of SLG-infinite BLG QD resembles that ofthe terminated systems with an energy gap opened at everycrossing point. Such energy gaps at the crossing points can beinterpreted as due to the hybridization between SLG QD andBLG states. The energy levels of the lowest bound states for the unbiased system are shown in Fig. 7as a function of the dot radius forB=10T, and m=−1,0,1a tt h e K(solid curves) andK /prime(dashed curves) valleys. For R→0, the energy levels correspond to the LLs of unbiased bilayer graphene givenby [31] ε=±1 E0/radicalBigg t2 2+(2n/prime+1)E2 0±/radicalbigg t4 4+(2n/prime+1)t2E2 0+E4 0, (38) where E0=/planckover2pi1vF/lB,n/prime=n+(|m|+m)/2, and n= 0,1,2,..., with m> 0 and m< 0 states being degenerate. 165410-6ENERGY LEVELS OF HYBRID MONOLAYER-BILAYER . . . PHYSICAL REVIEW B 93, 165410 (2016) R (nm)m=-1m=0m=1 U=0K K’)l/ v (ћB F FIG. 7. Energy states of unbiased SLG-infinite BLG QD ( U1= U2=0) as a function of the dot radius with B=10 T, for m=−1,0,1 at the valleys K(solid curves) and K/prime(dashed curves). Increasing R, the LLs of BLG split for different valleys (breaking of inversion symmetry) as well as for differentangular momenta (breaking of time reversal symmetry) andapproach the LLs of SLG. The energy levels correspondingto the Kvalley in Fig. 7, demonstrate the merging of the nth BLG LLs at R→0t ot h e nth SLG LLs at larger R.F o rt h e K /prime energy levels, the nth LLs of BLG approach the ( n+1)th LLs of SLG. Similar analysis is also available for the terminatedsystems, illustrated in Fig. 6, exhibiting the appearance of plateau feature in the spectrum. 2. Biased system The dependence of the spectrum on the magnetic field, at the two valleys KandK/primefor a biased system with U1=−U2= 0.1 eV , are shown respectively in Figs. 8(a) and8(b).T h e results are presented for m=−1 (blue), m=0 (green), and m=1 (red). The energy levels show a band gap between the conduction and valence bands at the KandK/primepoints. For small magnetic field ( B→0), this band gap exhibits a divergence when expressed in the units of /planckover2pi1vF/lB. Applying a gate potential breaks the degeneracy of the lowest-energy states,and the spectrum becomes strongly dependent on m. However, as the magnetic field increases, the magnetic confinementbecomes important, as seen by the lifting of the degeneracyof the states, and the energy levels approach the LLs of SLG(see black dashed curves, i.e., ε=±√ 2n+u1). Furthermore, bothKandK/primespectra show electron-hole asymmetry because of the breaking of inversion symmetry due to the presence of the external gate potentials. The behavior of the electron states in both valleys is qualitatively similar, but the hole statesdisplay different behavior which for the Kvalley are nearly degenerate. Results for the spectrum of localized states as a function of Rare shown in Fig. 9for the biased case, U 1=−U2=0.1e V . The energy levels are plotted for angular momentum labels,m=−1 (blue), m=0 (green), and m=1 (red) with B= 10 T at the valleys (a) Kand (b) K /prime. When R→0, for both valleys, energy levels coincide with the LLs of biased BLGas it should be. The LLs of biased BLG is determined by the0 10 20 30 403210123 m=-1 m=0 m=1SLG LLs + u B (T))l/ v (ћB F0 10 20 30 403210123 m=-1m=0 m=1 SLG LLs + u(a) (b))l/ v (ћB F FIG. 8. Energy spectrum of biased SLG-infinite BLG QD as a function of magnetic field for m=−1,0,1a tt h e( a ) Kand (b) K/prime valleys with the dot radius R=30 nm. The black dashed curves depict the SLG LLs +u1. The applied bias is U1=−U2=0.1e V . equation [ 31] [(α+δ)2−2(n/prime+1)][(α−δ)2−2n/prime]−(α2−δ2)t/prime2=0. (39) For small R, the lowest energy levels become nearly degen- erate, forming two energy bands around ε=±u1=±1.23. AsRincreases, degeneracy of the energy levels is lifted and finally they connect to the LLs of SLG subject to the externalpotential u 1. Figure 10shows the spectrum for terminated systems (dashed curves) and biased SLG-infinite BLG QD(black solid curves) as a function of Rat the valleys KandK /prime form=1. The resemblance of the two different spectra are also evident for the biased case. III. BLG-INFINITE SLG QUANTUM DOTS A. Zero magnetic field In this section we choose the inverse of the previous QD system in which a BLG QD is surrounded by an infinite SLG[see Fig. 1(b)]. This system can be considered as an infinite BLG sheet in which a circle of radius Rfrom its upper layer is left and the other part is removed. Similar to the previoussection the Hamiltonian is solved for both parts of the systemand then we choose the appropriate wave functions to satisfythe extreme conditions when r→0 for bilayer dot and r→∞ for SLG. Here, the modified Bessel function of the first kind I m, 165410-7MIRZAKHANI, ZARENIA, KETABI, DA COSTA, AND PEETERS PHYSICAL REVIEW B 93, 165410 (2016) 0 10 20 30 40 50 60 7032101234 m=-1 m=0 m=1SLG LLs + u R (nm))l/ v (ћB F(a) (b)m=-1 m=0 m=1SLG LLs + u)l/ v (ћB F FIG. 9. Energy spectrum of biased SLG-infinite BLG QD ( U1= −U2=0.1 eV) as a function of dot radius with B=10 T and for m=−1,0,1 at the valleys K(a) and K/prime(b). Black dashed lines are the LLs of SLG +u1. and the Bessel function of the second kind Ym, are the solutions for regions of the BLG QD and infinite SLG, respectively.As discussed before (see Sec. II A), there is no unique linear combination of real or imaginary parts of I m(γ±) andYm(γ±) from which unique discrete energies can be obtained even 32101234 )l/ v (ћB F R (nm) R (nm)07 0 03 570 35’K K(a) (b) FIG. 10. Energy spectrum of SLG-infinite BLG QD (black solid curves) and the terminated systems, SLG QD (blue dashed curves) and BLG antidot (red dashed curves) as a function of dot radius R, for the two valleys (a) Kand (b) K/prime. The magnetic field is B=10 T andm=1.B (T)m=-1 m=0m=1 U=0K K’ BLG LLs 0.00.51.01.52.0 0 0.1 0.22.5)l/ v (ћB F FIG. 11. Energy spectrum of unbiased BLG-infinite SLG QD (i.e.,U1=U2=0) as a function of magnetic field for R=30 nm andm=−1,0,1 at the two valleys, K(solid curves) and K/prime(dashed curves). Black dot-dashed curves are the LLs of unbiased infinite BLG. The inset shows an enlargement of the energy levels for lowmagnetic fields. in the presence of bias. Thus there are no bound states in BLG-infinite SLG QDs for B=0. B. Nonzero magnetic field In the presence of a magnetic field, the calculations are similar to those presented in Sec. II B. To avoid repetition, we limit ourselves here to the numerical results. Similar analysisfor the case of zero-energy levels as in previous section, onefinds that zero-energy states exist only for m/lessorequalslant0a tt h e K valley, and for all momenta at the K /primevalley. 1. Unbiased system The dependence of the spectrum on magnetic field, for R= 30 nm and m=−1 (blue), m=0 (green), and m=1 (red) at the two valleys K(solid curves) and K/prime(dashed curves) is shown in Fig. 11. The inset shows an enlargement of the states for small magnetic fields. As we see, for B=0, the energy levels coincide with the LLs of SLG. The energy levels arenearly degenerate for m> 0 andm< 0 at very low magnetic fields (0–0 .1 T). As the magnetic field increases, the states merge to form the LLs of an unbiased infinite BLG sheet[black dashed curves, see Eq. ( 38)], which indicates that the carriers become strongly localized at the center of the dot. Theenergy levels in Fig. 11show that the LLs of SLG connect to LLs of BLG with the same Landau level indices for theKvalley. However, for the K /primevalley, the SLG and BLG LLs connect by nSL→nBL−1. The spectrum of terminated systems [in this case, BLG QD (red dashed) and SLG antidot (blue dashed)], and theBLG-infinite SGL QD (black solid curves) are shown in Fig. 12 at the two valleys KandK /primeform=−1. The oscillatory feature in the energy levels can be understood in relation tothe terminated systems as explained in Fig. 6. In Fig. 13, we show the dependence on Rof the low energy spectrum, at the two valleys K(solid curves) and K /prime(dashed curves) for m=−1 (blue), m=0 (green), and m=1 (red) withB=10 T. The R→0 limit corresponds to the case of 165410-8ENERGY LEVELS OF HYBRID MONOLAYER-BILAYER . . . PHYSICAL REVIEW B 93, 165410 (2016) B (T) B (T)K K’(v / l )ћB F(a) (b) FIG. 12. Energy levels of BLG-infinite SLG QD (black solid curves) and the terminated systems, BLG QD (blue dashed curves) and SLG antidot (red dashed curves) as a function of magnetic field at the two valleys K(a) and K/prime(b) for m=−1. The dot radius is R=30 nm. SLG sheet, and the spectrum agrees with the LLs of SLG with the states being degenerate for all angular momenta and forboth valleys. When Rincreases, angular momentum as well as valley-degenerate levels split, showing flat plateau featuresfor certain ranges of dot radius, and eventually merge into theLLs of unbiased BLG. The way of the LLs of SLG and BLGare connected is similar to those discussed for Fig. 11. 2. Biased system Figure 14displays the energy levels for the biased case withU1=−U2=0.1 eV and R=30 nm as a function of magnetic field for (a) Kand (b) K/primevalleys. As in the unbiased case, the spectrum is plotted for m=−1 (blue), m=0 (green), and m=1 (red). The energy levels show a band gap between the conduction and valence bands. For very smallvalues of B, the spectrum becomes degenerate and forms a continuum band. However, as the magnetic field increases, thedegeneracy of the levels is lifted for each angular momentum m=-1 m=0m=1 ’K K’ R (nm))l/ v (ћB F FIG. 13. Energy levels of unbiased BLG-infinite SLG QD ( U1= U2=0) as a function of dot radius R,f o rm=−1,0,1 at the two valleys, K(solid curves) and K/prime(dashed curves). The magnetic field isB=10 T.m=-1 m=0 m=1Biased BLG LLs B (T)(b) )l/ v (ћB F0 10 20 30 403210123 m=0 m=1Biased BLG LLsm=-1(a) )l/ v (ћB F FIG. 14. Energy spectrum of biased BLG-infinite SLG QD (i.e., U1=−U2=0.1 eV) as a function of the magnetic field for R= 30 nm and m=−1,0,1a tt h et w ov a l l e y s , K(a) and K/prime(b). Black dashed curves are the LLs of a biased infinite BLG. (with the large shift for low lying states) and eventually approaching the LLs of biased BLG (blacked dashed curves).For each value of m, the hole energy levels show anticrossings when the energy levels approach the LLs of biased BLG.There are some other features, e.g., electron-hole asymmetry,degeneracy lifting between the states of the different valleys,and different behavior of the hole states in the Kvalley, similar to SLG-infinite QDs. The spectrum as a function of Rfor the biased case with U 1=−U2=0.1 eV , is plotted in Fig. 15for the valleys (a) Kand (b) K/prime. The other parameters are the same as for the unbiased case shown in Fig. 13.F o r R=0, the spectrum corresponds to the LLs of the SLG sheet, being degenerate forallm. With increasing dot radius, the degeneracy of the levels for different mis lifted, and the levels connect to different LLs of BLG. The results show that the low-energy LLs of SLG converge to the low-energy LLs of BLG, as the dot radius increases, and they form two bands around these energies. Itis also seen that the hole states, in the case of the Kvalley are approximately degenerate. In graphene QDs, it is possible to define a scaling factor for the maximum of the magnetic field by setting the dotsizeRequal to the cyclotron radius at the Fermi energy, i.e.,R=l 2 BkF[32]. Having E=/planckover2pi1vFkFandE=(/planckover2pi1vFkF)2/t for the low energy dispersion of monolayer and bilayergraphene, respectively, one can obtain k Fand consequently the scaling factor with respect to the magnetic field and size 165410-9MIRZAKHANI, ZARENIA, KETABI, DA COSTA, AND PEETERS PHYSICAL REVIEW B 93, 165410 (2016) m=-1m=0 m=1 R (nm)(b) )l/ v (ћB Fm=-1m=0m=1(a) )l/ v (ћB F FIG. 15. Energy spectrum of biased BLG-infinite SLG QD as a function of dot radius, for m=−1,0,1 at the two valleys, K(a) and K/prime(b). The magnetic field is B=10 T and U1=−U2=0.1e V .T h e black dashed lines show the SLG LLs +u1. of the confinement area, i.e., RSLG=E/(evFB) for SLG- infinite BLG QD and RBLG=√ Et/(evFB) for BLG-infinite SLG QD. IV . CONCLUSION Using the continuum model, i.e., solving the Dirac-Weyl equation, we obtained analytical results for the energy levelsand corresponding wave functions for two new types of QDsin hybrid SLG-BLG systems: SLG-infinite BLG QDs andBLG-infinite SLG QDs. We implemented the zigzag boundarycondition at the SLG-BLG junction in order to observe featuresbrought by the zigzag edges in the spectrum. Both Diracvalleys were investigated separately because of the breakingof inversion symmetry due to the presence of the SLG-BLGinterface.No bound states are observed in both types of QDs when B=0 and when no bias potential is applied to the graphene layers. In biased SLG-infinite BLG QDs with B=0, bound states and the opening of an energy gap between the electronand hole energy levels are found, and this gap closes whenincreasing the dot radius. The energy spectrum correspondingto the K valley exhibits the symmetry E K(m)=EK(−m), which no longer holds for the K/primevalley levels. In the presence of a perpendicular magnetic field without bias, degeneracy of energy levels is lifted. Increasing themagnetic field, the spectrum exhibits anticrossings and thelevels merge into the LLs of SLG which indicates that thecarriers become strongly localized at the center of the SLG QD.The anticrossings are due to the interplay of confined states inboth SLG and BLG region and at the SLG-BLG interface. Inthe presence of an external electric field, our results showedthat the degeneracy of the lowest-energy states are lifted atlow magnetic fields and the opening of an energy gap is found.Furthermore, both the KandK /primespectrum show electron-hole asymmetry because of the breaking of the inversion symmetryby the external potential. For BLG-infinite SLG QDs, we found that there are no bound states with or without external electric field whenB=0. Energy spectrum of unbiased BLG-infinite SLG QDs forB/negationslash=0 showed that the degeneracy of the states is lifted, and for high magnetic fields, the energy levels merge into theLLs of unbiased BLG. The biased case showed an opening ofenergy gap. Dependence of spectrum on dot radius in the SLG-infinite BLG QDs (BLG-infinite SLG QDs) shows that in the R→0 limit, the energy levels correspond to the unbiased LLs of BLG(SLG). Increasing R, the LLs of BLG (SLG) split for different angular momenta and approach the LLs of SLG (BLG). Forthe case of biased SLG-infinite BLG QDs with B/negationslash=0, the energy gap for R→0 closes with increasing R. The energy levels of the proposed new QDs can be investigated experimentally by STM, which measures the localdensity of states (LDOS). From the LDOS, one can obtaininformation about the position of the bound states and thelocalized electron/hole distribution. ACKNOWLEDGMENT This work was supported by the Fonds Wetenschappelijk Onderzoek (FWO)-CNPq project between Flanders and Braziland the Brazilian Science Without Borders program. [1] A. V . Rozhkov, G. Giavaras, Y . P. Bliokh, V . Freilikher, and F. Nori, Phys. Rep. 503,77(2011 ). [2] Z. Z. 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PhysRevB.102.235302.pdf

PHYSICAL REVIEW B 102, 235302 (2020) Editors’ Suggestion Gate-controlled unitary operation on flying spin qubits in quantum Hall edge states Takase Shimizu ,*Taketomo Nakamura , Yoshiaki Hashimoto, Akira Endo , and Shingo Katsumoto Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan (Received 7 October 2020; revised 30 November 2020; accepted 1 December 2020; published 16 December 2020) The spin and orbital freedoms of electrons traveling on spin-resolved quantum Hall edge states (quantum Hall ferromagnets) are maximally entangled. The unitary operations on these two freedoms are hence equivalent,which means that one can manipulate the spins with nonmagnetic methods through the orbitals. Taking thequantization axis of the spins along the magnetization axis, the zenith angle is determined by the partitionrate of spin-separated edges, while the azimuth angle is defined as the phase difference between the edges.Utilizing these properties, we have realized an electrically controlled unitary operation on the electron spinson quantum Hall ferromagnets. The zenith angle of the spin was controlled through the radius of gyration ata corner by applying voltage to a thin gate placed at one edge. The subsequent rotation in the azimuth anglewas controlled via the distance between the edge channels also by a gate voltage. The combination of the twooperations constitutes a systematic electric operation on spins in quantum Hall edge channels. DOI: 10.1103/PhysRevB.102.235302 I. INTRODUCTION Electron spins in semiconductor nanostructures such as quantum dots are expected to serve as qubits for practicalquantum computation [ 1]. The flying qubit (FQ) scheme, which is the usage of traveling electron spins as qubits, isnot only indispensable for long-distance entanglement [ 2,3] between localized arrays of qubits, but is also usable for uni-tary operations on qubits [ 4]. The flying spin qubits (FSQs) for electrons considered and tested so far are mostly basedon quantum wires with Rashba and Dresselhaus spin-orbitinteractions (SOIs) [ 5–7], which work as a magnetic field effective only on spins [ 8]. A convenient way to describe unitary transformations on an FQ is obtained by viewing thetraveling electron from the coordinate fixed at the center ofthe electron wave packet (center coordinate). Then, a spa-tially local Hamiltonian H loccan be introduced to describe the dynamics other than translational motion. That is, thetravel of a wave packet through a quantum wire with spatialpotential modulation can be viewed as a process in which thelocal Hamiltonian evolves with time [ 9,10], i.e., the effective Schrödinger equation for the wave packet |/Phi1/angbracketrightat the center coordinate is written as i¯h∂|/Phi1/angbracketright ∂t=Hloc(t)|/Phi1/angbracketright. (1) In this picture, the SOI term in the Hamiltonian also varies with time, which can cause a unitary transformation of thespin freedom in |/Phi1/angbracketrightthrough a nonadiabatic transition. Here, we would like to show that there is another type of SOI without a term that explicitly contains ˆ p(momentum op- erator) and ˆ s(spin operator) in H loc. Generally, an interaction term in a Hamiltonian introduces quantum entanglement [ 11] *takase@issp.u-tokyo.ac.jpbetween initially unrelated subsystems. Conversely, when we prepare an initial state with finite quantum entanglement, theinteraction appears between subsystems without any interac-tion term in the Hamiltonian [ 12], which is well known as the Einstein-Podolsky-Rosen (EPR) paradox [ 13]. For example, let us consider a wave packet |/Phi1/angbracketrightwith Stern- Gerlach-type entanglement [ 14] between the orbital {|ξ/angbracketright,|η/angbracketright} and spin {|↑/angbracketright,|↓/angbracketright} as |/Phi1/angbracketright=|ξ/angbracketright/parenleftbigg cosθ 2|↑/angbracketright/parenrightbigg +|η/angbracketright/parenleftbigg eiφsinθ 2|↓/angbracketright/parenrightbigg (0/lessorequalslantφ< 2π,0/lessorequalslantθ/lessorequalslantπ), (2) where /angbracketleftξ|ξ/angbracketright=/angbracketleftη|η/angbracketright=1,/angbracketleftξ|η/angbracketright=0, and φandθare the azimuth and zenith angles of the spin, respectively. We as-sume that H loc(t) has no operator on {|↑/angbracketright,|↓/angbracketright}; thus, there is no explicit interaction term for a certain period, in whichthe orbital part evolves to e iχ(|ξ/angbracketright,eiϕ|η/angbracketright). Here, χis the phase developed as the wave packet travels, while ϕ,t h e phase difference resulting from the path difference, etc., canbe absorbed into the azimuth angle of the spin. The processhence can be viewed as translational motion with spin pre-cession, which is simply a phenomenon associated with aneffective magnetic field by an SOI [ 15]. The equivalence of the phenomena with a quite different appearance reflects theinseparability of systems with maximal entanglement. Thisidea also suggests the possibility of manipulating the electronspins through orbital motion, the architecture for which istheoretically proposed in Ref. [ 16]. The quantum coherence length of the one-dimensional channel for FQ propagation should be long enough to preservequantum information. In solids, the longest coherence lengthshave been reported for quantum Hall edge channels (QHECs)[17,18], which are thus strong candidates for FQ channels. QHECs are known to show spin separation at comparativelylow magnetic fields with the aid of an exchange interaction 2469-9950/2020/102(23)/235302(8) 235302-1 ©2020 American Physical SocietyTAKASE SHIMIZU et al. PHYSICAL REVIEW B 102, 235302 (2020) [19], transforming into ferromagnetic phases [ 20,21]. In such a ferromagnetic regime, the entanglement of spin and edgechannels occurs, naturally preparing for the realization of thescheme in ( 2). A preliminary experiment on such precession control was reported by Nakajima et al. [22,23]. For zenith angle tuning, a controlled splitting of the wave packet into twochannels is required. Such experiments have been reported byDeviatov et al. [24–26] with the use of current imbalance, and by Karmakar et al. [27,28] with the use of periodic magnetic gates. In this paper, we present experimental results regarding the unitary operations of FSQ in spin-polarized QHECs inthe above scheme with electrostatic gates. With respect toEq. ( 2), a rotation in the zenith angle corresponds to tunneling between spin-polarized QHECs. It is shown that this can beachieved by the sharp bending of the edge line. At such acorner, an angular momentum in the edge orbital emerges,and Landau-Zener-type tunneling brings about a rotation inthe zenith angle. The rate of Landau-Zener tunneling dependson the sharpness of the corner. With the addition of a thin gateto vary the sharpness, we show that the zenith angle can becontrolled more simply. We should note that the same physicscan be described in the terms for the quantum Hall effects. Webelieve the present description will bring a fresh perspectiveto the traditional quantum Hall effects. II. EXPERIMENTAL METHOD Figures 1(a)–1(c) describe the experimental setup in three different ways for a two-dimensional electron system (2DES)in the spin-split quantum Hall regime. For simplicity, thefilling factor νis chosen to be 2 in the figure, although the region of ν=4 was mostly used in the present experiment. Figure 1(a) is a schematic of wave-propagation paths, Fig. 1(b) shows the gate-electrode configuration for realizing them, and Fig. 1(c) illustrates a blowup of the down edges of the side gates (SL, SR) and center gate (C) along withthe propagation paths in Fig. 1(a). The sample edges have two QHECs for ν=2, which we here denote channel 1 and channel 2, in which spins are locked at ↑and↓, respectively. Then, we can write their wave packet as |1/angbracketright| ↑ /angbracketright and|2/angbracketright|↓/angbracketright , respectively, where |1/angbracketrightand|2/angbracketrightare the normalized wave func- tions of the orbital part. Let us trace a wave packet that is emitted from the right electrode. Beneath the gates L and R, the filling factors ν L andνRare tuned to 1, and only channel 1 goes through them [29,30]. Hence, the incident wave packet in Fig. 1(a) can be written as |ψ1/angbracketright|↑/angbracketright . Channels 1 and 2 meet at the lower right corner edge of the gate SR, where a partial transfer occursthrough a local SOI as a result of the orbits wrapping aroundthe sharp corner. This scattering process is written as |1/angbracketright| ↑ /angbracketright→| /Phi1/angbracketright SR=t11R|1/angbracketright| ↑ /angbracketright+ t12R|2/angbracketright|↓/angbracketright, (3) where tijRare the complex transmission coefficients of the processes |i/angbracketright→| j/angbracketrightat the right corner satisfying the unitary condition |t11R|2+|t12R|2=1 because of the perfect chirality of the QHEC. Hence, they can be written as t11R=cosθ/2 andt12R=eiφ0sinθ/2, where θreflects the amplitude ratio of the partial waves and φ0is the phase difference between t11R andt12R. Thus, |/Phi1/angbracketrightSRis prepared in the form of |/Phi1/angbracketrightin (2). AAB 10 μmgate SLgate SLgate Cgate Cgate SRgate SR gate Lgate L gate Rgate Rdrain L drain Bx y IL IB1 2 1 2VSL VC VSR IL IB(a) (b) (c) FIG. 1. (a) Schematic diagram of the “quantum circuit” for elec- tron wave packets (red and blue circles with arrows indicating spin),with an illustration of the Bloch sphere description of an FSQ. (b) Optical micrograph of the sample with an external circuit illus- tration. The orange regions are metallic gates, three of which areannotated. The 2DES substrate is trimmed at the white dashed lines. (c) illustrates the hybridization of (a) and (b) around the lower ends o ft h eg a t eS L ,C ,a n dS R . In QHEC, the orbital part of the wave function in the single-electron picture is written as a quasi-one-dimensionalplane wave in a real-space representation [ 31], ψ i(r)∝exp(ikix)e x p/bracketleftbigg −(y−yi)2 2l2/bracketrightbigg , (4a) yi=− l2ki, (4b) where l(=√¯h/eB,Bis the magnetic field) is the magnetic length, iis the channel index counted from outer (lower en- ergy in bulk) to inner, the x-axis direction is taken as along the one-dimensional channel, and yiis the guiding center position. In the edge states, ψiaccommodates the propagating wave packet |i/angbracketright.|i/angbracketrighttravels on ψialong the down edges of the gate SR, C, and SL, thereby gaining a kinetic phase. At the end ofthe travel over total length L, the difference in the acquired kinetic phase or the azimuth angle rotation of the spin is φ=(k 1−k2)L (5a) =(y2−y1)L l2=2πΔyLB h/e. (5b) 235302-2GATE-CONTROLLED UNITARY OPERATION ON FLYING … PHYSICAL REVIEW B 102, 235302 (2020) Because ΔyLB/(h/e) is the magnetic flux piercing the area between the two paths measured in units of flux quantum(h/e), the difference in kinetic phase acquired in the travel over the two channels is equal to the Aharonov-Bohm (AB)phase caused by the magnetic field. Thus, we can tune φusing both the magnetic field and the voltage supplied to gate C,which varies the interval Δybetween the edge states. This single-electron picture requires a correction with considera-tion of the screening effect, as will be discussed in the analysissections, although the above results can still be applied to realexperiments with some modifications, e.g., Bdependence of /Delta1yas given in Eq. ( 7). At the left corner of the gate SL, channel 1 with ↑goes up to go beneath the region of ν L=1, while channel 2 with ↓ goes down to turn around the region. Because both channelschange their directions abruptly, a crossing transition betweenthem by a local SOI occurs at the corner, as illustrated inFigs. 1(a) and1(c). In this two-in–two-out vertex, the parti- tion ratio is affected by the phase φ−φ 0, and the traverse across the sample ends up at drain L or drain B with theprobabilities determined by the ratio. Hence, the partition ratiocan be measured as the ratio of the current through L ( I L) to the total current ( IL+IR), i.e., current distribution ratio D≡IL/(IL+IR). In a simple model of the two-in–two-out vertex described in Sec. IV,Dis written as D=C0+C1sinθcos(φ+Δϕ), (6) where Δϕrepresents the phase shift associated with the in- teredge scattering at the two corners, including −φ0. Equation (6) is similar to the simplest Young’s double-slit approxima- tion of an AB interferometer because of the chirality or brokentime-reversal symmetry of the channels and multiterminalconfiguration [ 32]. The partition ratio of the input affects the visibility, giving the θdependence. A two-dimensional electron system (2DES) with an elec- tron density of 4 .4×10 11cm−2and a mobility of 86 m2/Vs in an Al xGa1−xAs/GaAs ( x=0.265) single heterostructure was used as the base system for the sample. The structureof the wafer was (from the front surface) 5-nm Si-dopedGaAs cap layer, 40-nm Si-doped ( N Si=2×1018cm−3) AlxGa1−xAs layer, 15-nm undoped Al xGa1−xAs spacer layer, and an 800-nm GaAs layer with a 2DES residing near theinterface with the upper layer. The terminal and Au /Ti gate configurations are shown in Fig. 1(b). We cooled the sample down to 20 mK and applied a perpendicular magnetic field B up to 9 T, at which the 2DES is in the quantum Hall state witha filling factor of ν=2. An AC voltage of typically 33 μV rms(except for the mea- surements in Fig. 5) at 170 Hz was applied to the right-side contact, and the current was measured at drain L and drain Bwith an I-Vamplifier by standard lock-in measurements. A representative difference of the contact and cable resistancebetween drains L and B was less than approximately 2%.Therefore, Dis nearly equal to the transmission probability from source to drain L. The voltage on gate C ( V C) modifies the potential gradient in the ydirection along gate C and thus the distance between neighboring edge states Δy, which leads to the modulation of φ[22,23]. 0.45 0.5 0.55 0.6 -0.9 -0.8 -0.7DB=4.84 T Minimum point 0.45 0.5 0.55 0.6 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 VC (V)(a) (b)Dcal124 3 Minimum point FIG. 2. (a) VCdependence of the current distribution ratio Dat B=4.84 T. Side gate voltages were set to VSR=VSL=− 0.6V , where the 2DES under the gates was completely depleted. (b) Calcu- lation example of Das a function of VCbased on the model of Eq. ( 7). d=66 nm from the one-dimensional Poisson-Schrödinger calcula- tion on the layered structure of the sample (see the Supplemental Material [ 33]). Other parameters are as follows: g=− 0.6[19];/epsilon1= 12.35 [ 34];C1sin(θ)=0.05,C0=0.53,/Delta1φ=0,LC=7.5μm; LSL=LSR=1.25μm; the offset in the gate voltage VC,VSL,a n d VSR resulting from the contact built-in potential is Voffset=− 0.05 V. See the text for n(y). III. ROTATION IN AZIMUTH ANGLE Figure 2(a) shows the VCdependence of Dmeasured at B=4.75 T, which corresponds to ν=4 in the nongated re- gion. The filling factors underneath gate L and gate R werekept at 1 (pinch-off conductance traces for gate L are in theSupplemental Material Sec. III[33]). In this situation, there are two extra edge states inside the 2DEG compared withthat illustrated in Fig. 1. In the following experiments, we believe the tunneling rate from the edge states 1 and 2 to theseextra states is small and they are negligible at the precisionlevel of the present research; then, the channel indices i=1 and 2 are under our consideration. The detailed discussion isgiven in Sec. VII of the Supplemental Material [ 33]. Side gate voltages were set to V SR=VSL=− 0.6 V, where the 2DES under the gates was completely depleted. The measured D shows an oscillation against VCin the range from −0.7t o −0.98 V, where four peaks are observable, as indicated by arrows. The oscillation period increased with negative VC.T h e region between peaks 3 and 4 is especially wide, and the lineshape shows the rewinding of oscillation. We tested severalother samples with essentially the same gate configuration,and such behavior was commonly observed. To verify the above phase modulation scenario (or equiv- alent rotation in φ), we need to know how Δyin Eq. ( 5b) depends on V Ctaking the electric screening effect into account. In the single-electron picture of Eq. ( 4), the one- dimensional channels are formed on the lines where theLandau levels cross the Fermi level. In more practicaltreatments in Refs. [ 35,36], the QHECs are described as 235302-3TAKASE SHIMIZU et al. PHYSICAL REVIEW B 102, 235302 (2020) “compressible” stripes separated by “incompressible” insu- lating regions. In the compressible stripes, the electrostaticpotential is kept constant by the screening effect, while thegroup velocity ∂E i/¯h∂kiis finite. Therefore, Eq. ( 4b) does not hold inside the stripes and the wave number kishould also be kept constant. In other words, Eq. ( 4b) only holds inside the incompressible regions. As the value of Δy, we should thus take the width of the incompressible stripes, which isgenerally much narrower than that of the compressible ones.In Refs. [ 35,36], such Δyis explicitly given for a simple classical electrostatic model of the QHEC as Δy≈/radicalBigg 8|ΔE|/epsilon1/epsilon10 πe2(dn/dy)|y=y/prime i, (7) where y/prime iis the position of the ith incompressible liquid strip, /epsilon1/epsilon10is the dielectric permittivity of the matrix semiconductor, n(y) is the electron sheet density profile, and ΔEis the energy difference between the levels of channels iand i+1. The model has been used in analyzing many experimental works[37–39].ΔEin the present case ( i=1) of exchange-aided Zeeman splitting can be written as gμ BB, where gis the effective Landé g-factor, and μBis the Bohr magneton. Figure 2(b) shows an example of the VCdependence of D, calculated from Eq. ( 7) with the parameters noted in the caption. These parameters are chosen to preserve semi-quantitative consistency with the analysis of the magneticresponse described later. To calculate n(y)a saf u n c t i o no f V C, we employ the “frozen surface” model and the self- consistent Thomas-Fermi approximation given in Ref. [ 36]. Then, ( dn/dy)|y=y/prime 1can be obtained numerically from n(y). The characteristic behavior of the oscillation in Fig. 2(a) is qualitatively reproduced, in that the oscillation phase ad-vances more rapidly with negative V Cat lower |VC|.T h e progress in the phase slows down, and the rewinding of theoscillation with increasing |V C|begins at the point indicated in the figure as the “minimum point.” This behavior is quali-tatively explained as follows (a schematic of this descriptionis in the Supplemental Material [ 33]). At low V C, the edge of the 2DES lies near the end of the center gate, and theelectrostatic confinement potential at the edge is soft, leadingto small ( dn/dy)| y=y/prime 1and large Δy. With increasing negative VC, the potential becomes steeper, lowering Δy. A further increase in negative VCcauses softening of the potential and an increase in Δyagain. Because Δymust be smooth as af u n c t i o no f VC,|d(Δy)/dVC|decreases with negative VC, i.e., the oscillation period becomes slower, until reaching theminimum point, roughly corresponding to the steepest edgeconfinement potential [maximum in ( dn/dy)| y=y/prime 1], and again increases with negative VC, resulting in the rewinding of the oscillation in D. In spite of the obvious resemblance between Figs. 2(a) and (b), quantitative fitting that is consistent with a responseto the magnetic field is difficult. We also investigated the“Fermi-level pinning” model for the surface states, although itdid not improve the quantitative agreement. This discrepancyindicates the necessity to take into account the effects notconsidered, e.g., the geometrical effect of the gate electrode.However, the close resemblance between Figs. 2(a) and2(b) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 4.4 4.5 4.6 4.7 4.8 4.9 50.78 V0.8 V0.82 V0.84 V0.86 V0.88 VD(a) (b)VC=0.9 V 1 2 3 4 4.4 4.5 4.6 4.7 4.8 4.9 5 B (T)Δy (nm) FIG. 3. (a) Current partition rate Das a function of B within the plateau regime encompassing the filling factor ν= 4 for several values of gate C voltage VC(traces are offset for clarity). Vac=32.8μVrms.( b )Δycalculated from the equa- tionΔy=(2/3)(h/e)/L/Delta1BforVC=− 0.86 V. /Delta1Bis given by twice the distance between the adjacent oscillation peak anddip. The blue line indicates Δy average=[Δy(VSL)LSL+Δy(VC)LC+ Δy(VSR)LSR]/(LSL+LC+LSR) from Eq. ( 7) with the parameters used in Fig. 2(b). still manifests the essential correctness of the scenario de- scribed so far. As in Eq. ( 5b), the azimuth angle rotation is locked to the AB phase acquired from the magnetic flux piercing theincompressible regions. This can be readily confirmed bythe oscillatory behavior of DvsB, as shown in Fig. 3(a) with V Cas a parameter. Because a single period ΔBof the oscillation corresponds to 2 πrotation in φ,Δyis given as Δy=(2/3)(h/e)/LΔBfrom Eq. ( 5b) in the local linear ap- proximation of the Bdependence of Δyin (7), where ΔE= gμBB(details of this calculation are described in the Supple- mental Material [ 33]). This gives Δyas a function of B,a s shown in Fig. 3(b) forVC=− 0.86 V. The obtained values ofΔy(1–3 nm) are much shorter than the magnetic length l=11 nm at B=5 T, which is consistent with the view of compressible /incompressible stripes, while the predicted B dependence of Δyfor constant gagainst Bdeviates from the experiment, as indicated by Fig. 3(b). To visualize the overall trend, the measured and calcu- lated values of Dare color plotted on B-VCin Figs. 4(a) and 4(b), respectively. The oscillation patterns appear as curved stripes in these plots. Such curving behavior is consistentwith the interpretation of Fig. 2as follows. With an in- crease in negative V Cfrom−0.7 V to approximately −0.8V for a fixed B,(dn/dy)|y=y/prime 1increases and φthus decreases, 235302-4GATE-CONTROLLED UNITARY OPERATION ON FLYING … PHYSICAL REVIEW B 102, 235302 (2020) 4.4 4.6 4.8 5 0.2 0.4 0.6 0.8 4.4 4.6 4.8 5 -0.95 -0.9 -0.8 -0.7 Distribution ratio DMagnetic Field B (T) -0.95 -0.9 -0.8 -0.7 Calculated DMagnetic Field B (T) 0.2 0.4 0.6 0.8 V(V)(a) (b) FIG. 4. (a) Color plot of the measured Das a function of Band VC.VSR=VSL=− 0.6 V. (b) Color plot of the theoretically given D as a function of BandVC.C0=0.42B−1.45 and C1sin(θ)=0.1 are used. The other parameters are the same as those in Fig. 2(b). corresponding to the up-going ridges. After reaching the max- imum at VC≈− 0.86 V, ( dn/dy)|y=y/prime 1declines with a further increase in negative VC. The observation of arclike curving strongly supports the legitimacy of the analysis so far. Similar arclike curves werealso observed at filling factors ν=2 and 3, but with smaller visibility. At smaller filling factors, Band hence Δyare larger, making θ, the zenith angle, smaller, as discussed later. Re- garding visibility, in Fig. 2, the oscillation under the present consideration is visible in the range of V Cfrom−0.7t o −1 V, and the visibility is highest around the minimum point. This tendency is common in the region Bin Fig. 4, and the visibility does not change very much with B. Because an increase in Bdenotes an enhancement in the rotation of φ, the possibility of dephasing by the number of φrotations is eliminated. Instead, we speculate that the simple model inFig. 1(c) is approximately realized only around the steepest edge potential condition. When the edge potential is soft, theQHECs have more chances to experience the effect of localpotential disorder. As a result, the effective edge line fluctu-ates spatially, creating local orbital angular momentum, whichcauses interedge state scattering [ 40]. The above discussion is a possible explanation of the dephasing in φrotation. 0.3 0.4 0.5 0.6 0.7 0.8 VSR=−0.594 V 0.4 0.5 0.6 0.7 VSR=−0.626 V 0.5 0.6 −0.8 −0.75 −0.7 −0.68 VC (V)VSR=−0.646 VDistribution ratio D FIG. 5. Three oscillation patterns corresponding to three dif- ferent values of VSR.B=4.5T a n d VSL=− 0.605 V. Vac= 23.8μVrms. The line for D=0.57 is indicated by dashed lines to show that there is almost no movement in the oscillation baseline. From the above results and analysis, we can safely say that the current partition ratio Dreflects the azimuth an- gle rotation of FSQ traveling along the down edges of thegates. The rotation angle can be tuned via center gate voltageelectrostatically. IV . ROTATION IN ZENITH ANGLE In Fig. 5, we compare the oscillation patterns for three representative values of VSR, which strongly affects the os- cillation amplitude. From VSR=− 0.594 V the amplitude gradually decreases with a further increase in negative VSR. The characteristic features of the oscillation versus VCob- served so far do not change with VSR, other than a phase shift, which is probably caused by a change in φ0. In the region VSR>−0.594 V, the oscillation pattern changed drastically with a slight difference in VSR. This is probably because chan- nel 2 penetrates the spatial gap between gate C and gate SR.Hence, this region is excluded from the present discussion. We should thus look for the origin of the amplitude mod- ulation in the zenith angle θ, which is determined when the wave packet turns the down-right corner of the gate SR. Atthe turning point, the time-dependent local Hamiltonian in ( 1) for the wave packet should contain SOI terms: one from thein-plane potential gradient [ 41], the other from the Rashba and Dresselhaus effects commonly observed in 2DES [ 8]. In the present case of spin-polarized QHEC, the former affectsthe effective Zeeman energy by the spin-orbit effective field,while the latter kinematically rotates the spin. Figure 6(a) illustrates the time evolution of quasieigenenergies for spin-down and spin-up, i.e., E ↑=/angbracketleft ↑ |/angbracketleft 1|Hloc(t)|1/angbracketright| ↑ /angbracketright andE↓= 235302-5TAKASE SHIMIZU et al. PHYSICAL REVIEW B 102, 235302 (2020) 12 (a)Corner regionCorner levels V(ξ) ξEF E↑E↓ E tEmingμBB (b) FIG. 6. (a) Schematic time evolution of quasieigenenergies E↑= /angbracketleft↑|/angbracketleft 1|Hloc(t)|1/angbracketright| ↑ /angbracketright andE↓=/angbracketleft ↓ |/angbracketleft 2|Hloc(t)|2/angbracketright|↓/angbracketright . (b) Illustration of spin-polarized edge states for a straight edge (solid and dashed lines) and a corner (dotted lines). ξrepresents the distance from an edge of infinite potential [ V(0)=∞ ]. For simplicity, the Landau levels are drawn in the single-electron picture. /angbracketleft↓|/angbracketleft 2|Hloc(t)|2/angbracketright| ↓ /angbracketright . Around the center of the corner region, E=E↓−E↑takes the minimum value Emin. The transition in Eq. ( 3) can then be taken as par- tially nonadiabatic tunneling. By summarizing these effectivetime-localized SOIs as H SOI(t), the probability Pof the in- teredge channel transition is given by slightly modifying aLandau-Zener-type formula [ 42]a s P∝|/angbracketleft ↑ |/angbracketleft 1|H SOI|2/angbracketright|↓ /angbracketright|2exp/bracketleftbigg −2π(Emin/2)2 ¯h(dE/dt)/bracketrightbigg , (8) where dE/dtis the slew rate of E. As indicated by the arrows in Fig. 6(b), the total process from |1/angbracketright| ↑ /angbracketright to|2/angbracketright|↓/angbracketright consists of a nonadiabatic transition from |1/angbracketright| ↑ /angbracketright to|1/angbracketright| ↓ /angbracketright and an adiabatic transition from |1/angbracketright| ↓ /angbracketright to|2/angbracketright|↓/angbracketright . The expression in (8) indicates that the slew rate and the minimum energy difference strongly affect the transition probability. A simple explanation of the tendency in Fig. 5follows from Eq. ( 8) and the electrostatic model in Refs. [ 35,36]. With in- creasing negative VSR, QHECs move away from the “steepest potential” point, where the distance between the outer andthe inner edges is the minimum and the radius of gyrationr talso is the shortest. As rtdecreases, dE/dtincreases be- cause of the shorter interaction time, and the potential gradientis larger; thus, E minis smaller. From Eq. ( 8), the transition probability Pis the maximum for the steepest edge potential condition. As in Fig. 5, this scenario tells us that the steepest potential condition should correspond to VSR>−0.594 V, which is considerably smaller than −0.8Vf o r VC.T h i sd i f - ference may come from the geometrical complexity in the realgate configuration. As in the Supplemental Material [ 33], the equipotential lines around the corner change intricately, andthe maximum of Pmay appear at smaller V SRthan the value at the steepest potential. From the oscillation data in Fig. 5, we can estimate the zenith angle θ, assuming that the dephasing is ignorable at the largest amplitude region in VCas follows. Even if such ignorable dephasing is not the case, the lower limit of θcan be obtained from the analysis. Let tijLbe the complex trans-mission coefficients of the processes |i/angbracketright→| j/angbracketrightat the bottom left corner of the gate SL; then, from Eq. ( 3), the wave-packet state that turns the corner and enters channel 1 to go to drainL is written as |/Phi1/angbracketright L=/parenleftbig t11Reik1Lt11L+t12Reik2Lt21L/parenrightbig |1/angbracketright| ↑ /angbracketright. For simplicity of expression, we write the complex transmis-sion coefficients in the modulus-argument form as t 11Rt11L= t1cos(θ/2)eiϕ1andt12Rt21L=t2sin(θ/2)eiϕ2. This leads to the simple Young’s double-slit result of the transmission coeffi-cient T L=/angbracketleft/Phi1|/Phi1/angbracketrightLas TL=t2 1cos2(θ/2)+t2 2sin2(θ/2) +t1t2sinθcos(φ+Δϕ). (9) From the comparison with Eq. ( 6), C0=t2 1+/parenleftbig t2 2−t2 1/parenrightbig sin2(θ/2),C1=t1t2. (10) In Fig. 5, the baseline of oscillation C0shows almost no change, while C1sinθvaries widely. This fact is based on Eq. ( 10), where t1andt2happen to be close to each other: t1≈t2, in the present condition (the best visibility condition). Then, C0≈t2 1≈C1≈0.57. In Fig. 5, the largest amplitude gives C1sinθas 0.17, which corresponds to θ≈17.4◦.T h i si s the lower bound of the estimated θ, which inevitably contains an underestimation because of dephasing. To obtain a preciseestimation of θ, the oscillation of C 1sinθshould be observed. Unfortunately, in the present case, the maximum obtainedvalue of θis less than 90 ◦, and further analysis is difficult. For more precise control of FSQ in the present scheme, the cornergates should be designed to create a sharper corner potential.Furthermore, the dephasing should be reduced, e.g., by softseparation of the edges with an extra gate. V . CONCLUDING REMARK We have studied the unitary operation of FSQs in QHECs with electric voltages on metallic gates. This operation uti-lized the maximal entanglement between spin and edgechannel orbitals. 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PhysRevB.79.235323.pdf

Memory effect on the multiphoton coherent destruction of tunneling in the electron transport of nanoscale systems driven by a periodic field: A generalized Floquet approach Tak-San Ho,1,*Shih-Han Hung,2Hsing-Ta Chen,2and Shih-I Chu2,3,† 1Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA 2Department of Physics, Center for Quantum Science and Engineering, National Taiwan University, Taipei 106, Taiwan 3Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, USA /H20849Received 28 April 2009; revised manuscript received 2 June 2009; published 22 June 2009 /H20850 Time-dependent electron-transport processes are often studied in the wide-band limit. In this paper, a gen- eralized Floquet approach beyond the wide-band limit is developed for the general treatment of memory effecton the virtually unexplored multiphoton /H20849MP /H20850coherent destruction of tunneling /H20849CDT /H20850phenomenon of peri- odically driven electrode-wire-electrode nanoscale systems. As a case study, we apply the approach for adetailed analysis of the electron-transport dc current in the electrode-quantum double dot-electrode system,showing the significance of memory effect as well as illustrating the origin of the MP-CDT phenomenon. DOI: 10.1103/PhysRevB.79.235323 PACS number /H20849s/H20850: 73.23. /H11002b, 73.63. /H11002b, 72.10.Bg I. INTRODUCTION An accurate account of electron transport of nanoscale quantum systems driven by external fields is essential to theunderstanding of many fundamental time-dependentprocesses, 1including the photon-assisted tunneling,2,3coher- ent destruction of tunneling /H20849CDT /H20850,4,5and nonadiabatic elec- tron pumping.6Time-dependent electron-transport processes are often studied in the wide-band limit /H20849WBL /H20850.7In particu- lar, for temporal periodic driving fields, the Floquet theory8,9 can be invoked to enable efficient computations and analysis of the transport current and the current noise.1,7The CDT phenomenon has been studied extensively for electron-transport problems within the WBL Floquet formulation. 10–13 However, the WBL approximation neglects an important fact that the electrode-wire coupling is in general energydependent. 14–16As a result, the memory effect /H20849the effect of energy-dependent spectral density of electrodes /H20850on electron transport cannot be accounted for in the Floquet theory-based calculations. In this paper, the Floquet theory is generalized beyond the WBL to include memory effect. Specifically, we make use ofthis generalized Floquet method to study the memory effecton the virtually unexplored multiphoton resonance CDT phe-nomenon, supplementing the well-studied zero-biased CDT,with the aim to facilitate the understanding and possible de-velopment of new mesoscopic optoelectronic devices. Thegeneralized Floquet formulation is made possible by model-ing the energy-dependent electrode-wire coupling /H20849i.e., the spectral density of the electrodes /H20850in terms of sums of Lorentzian functions. 17,18Previously, the memory effect on the CDT of zero-biased double quantum dots driven by aperiodic field has been studied in the context of the sameLorentzian ansatz 18but without taking advantage of the Flo- quet theory. Following the seminal work of Jauho et al. on the time-dependent transport theory,19within the framework of the nonequilibrium Green’s function,20,21and with the adoption of the Lorentzian ansatz for the electrode spectraldensity, we show that the underlying time-dependent single-particle propagator for the electrode-wire-electrode systemcan be efficiently solved within the general framework of theFloquet theory without resorting to the WBL. Moreover, we find that the generalized Floquet method provides a compre-hensive physical picture of the multiphoton CDT phenom-enon. The paper is organized as follows. In Sec. IIwe describe the general formulation of memory effect for an electrode-wire-electrode quantum system /H20849using a spinless tight- binding model /H20850in the presence of a time-dependent external field based on the Lorentzian model for the electrode spectraldensity. To this end, a set of coupled ordinary differentialequations for the underlying single-particle propagator is de-rived. Section IIIpresents the general Floquet approach be- yond the wide-band limit for the electrode-double quantum-dot /H20849DQD /H20850-electrode system driven by a periodic external field. Section IVanalyzes the memory effect on the multi- photon coherent destruction of tunneling phenomena for theelectron transport involving double quantum dots in thehigh-frequency limit. Finally, a brief summary is given inSec. V. II. GENERAL FORMULATION OF MEMORY EFFECT Consider a nanoscale electrode-wire-electrode quantum system driven by a time-dependent external field. The corre-sponding single-particle Hamiltonian H/H20849t/H20850can be written as 11,12 H/H20849t/H20850=HC/H20849t/H20850+HL+HR+H/H11032, /H208491/H20850 where HC/H20849t/H20850is the externally driven central wire Hamil- tonian, HLandHRare, respectively, the left and right elec- trode Hamiltonians, and H/H11032is the interaction between the wire and the electrodes. In the spinless tight-bindingapproximation 12with Norbitals /H20849Fig. 1/H20850driven by a time- dependent external field, the central quantum wire Hamil-tonian H C/H20849t/H20850may be expressed in terms of single-electron states /H20841/H9251/H20856’s, HC/H20849t/H20850=/H20858 /H9251=1N /H20858 /H9252=1N H/H9251/H9252/H20849t/H20850/H20841/H9251/H20856/H20855/H9252/H20841, /H208492/H20850 where the wire contains Nsites /H20849orbitals /H20850designated as 1,2,..., N−1,N; the site 1 is in contact with the electrode LPHYSICAL REVIEW B 79, 235323 /H208492009 /H20850 1098-0121/2009/79 /H2084923/H20850/235323 /H208496/H20850 ©2009 The American Physical Society 235323-1on the left-hand side and the site Nis with the electrode Ron the right-hand side. The left and right electrode HamiltoniansH LandHRcan be written as HL=/H20858 q/H9280q/H20841Lq/H20856/H20855Lq/H20841, /H208493/H20850 and HR=/H20858 q/H9280q/H20841Rq/H20856/H20855Rq/H20841, /H208494/H20850 respectively, in terms of the electron states /H20841Lq/H20856’s and /H20841Rq/H20856’s. Finally, the spectral densities of the left and right electrodes/H20849taking into account the two spin states of electrons /H20850can be written as /H9003¯L/R/H20849/H9280/H20850=4/H9266/H20858 q/H20841VL/Rq/H208412/H9254/H20849/H9280−/H9280q/H20850, /H208495/H20850 which in turn lead to the electrode-wire coupling Hamil- tonian H/H11032=/H20858 q/H20849VLq/H20841Lq/H20856/H208551/H20841+VRq/H20841Rq/H20856/H20855N/H20841/H20850+ H.c., /H208496/H20850 with VLqand VRqbeing, respectively, coupling amplitudes between the sites /H208411/H20856,/H20841N/H20856, and their contacting electrodes. The underlying time-dependent equation for the single- particle propagator U/H20849t,t0/H20850of the N-site quantum wire can be written in matrix form,12 ı/H6036/H11509 /H11509tU/H20849t,t0/H20850=HC/H20849t/H20850U/H20849t,t0/H20850−ı 2/H20885 t0t /H9003¯/H20849t−t/H11032/H20850U/H20849t/H11032,t0/H20850dt/H11032, /H208497/H20850 where U/H20849t,t0/H20850is an N/H11003Nmatrix function and /H9003¯/H20849t,t/H11032/H20850is the corresponding response function /H20849the memory kernel /H20850, /H9003¯/H20849t−t/H11032/H20850=/H9003¯L/H20849t−t/H11032/H20850/H208411/H20856/H208551/H20841+/H9003¯R/H20849t−t/H11032/H20850/H20841N/H20856/H20855N/H20841, /H208498/H20850 in which /H9003¯L/R/H20849t−t/H11032/H20850are the Fourier transforms of the spectral densities /H9003¯L/R/H20849/H9280/H20850via the relation/H9003¯L/R/H20849t−t/H11032/H20850=1 2/H9266/H6036/H20885/H9003¯L/R/H20849/H9280/H20850e−ı/H9280/H20849t−t/H11032/H20850//H6036d/H9280. /H208499/H20850 In the WBL,11,12/H9003¯L/R/H20849/H9280/H20850becomes independent of the energy /H9280, i.e.,/H9003¯L/R/H20849/H9280/H20850→/H9003L/R, and as a result the response function /H9003¯/H20849t−t/H11032/H20850can be reduced to the simple form, /H9003¯/H20849t−t/H11032/H20850=/H20849/H9003L/H208411/H20856/H208551/H20841+/H9003R/H20841N/H20856/H20855N/H20841/H20850/H9254/H20849t−t/H11032/H20850. /H2084910/H20850 The memory effect on the electron transport can be expedi- ently studied by modeling the electrode spectral densities /H9003¯L/R/H20849/H9280/H20850as sums of MLorentzian functions18 /H9003¯L/R/H20849/H9280/H20850=/H20858 k=1MqkL/RbkL/R /H20849/H9280−/H9268kL/R/H208502+/H20849bkL/R/H208502/H11021/H11009, /H2084911/H20850 where qkL/R,/H9268kL/R, and bkL/Rare real positive fitting parameters chosen to mimic the smooth property of the spectral density. In the WBL /H20849i.e., in the limit of the spectral widths bkL/R →/H11009/H20850, the spectral densities become /H9280independent, lim bkL/R→/H11009/H9003¯L/R/H20849/H9280/H20850→/H9003L/R=/H20858 k=1MqkL/R bkL/R/H11021/H11009. /H2084912/H20850 Here we assume that the ratios qkL/R/bkl/Rare of finite magni- tude in the Lorentzian ansatz, cf. Eqs. /H2084911/H20850and /H2084912/H20850, so that the values of the electrode spectral densities /H9003¯L/R/H20849/H9280/H20850also maintain finite in the calculations. Using the ansatz of MLorentzian spectral-density func- tions, cf. Eq. /H2084911/H20850, and carrying out the integral in Eq. /H208499/H20850,w e immediately obtain the following closed form for the re-sponse functions associated with the left and right electrodes, /H9003¯L/R/H20849t−t/H11032/H20850=1 2/H6036/H20858 k=1M qkL/Re−ı/H20849/H9268kL/R−ıbkL/R/H20850/H20849t−t/H11032/H20850//H6036. /H2084913/H20850 Furthermore, by making use of Eq. /H2084913/H20850, we can recast Eq. /H208497/H20850into a set of N+2Mcoupled ordinary differential equa- tions /H11509U/H20849t,t0/H20850 /H11509t=1 ı/H6036/H20875HC/H20849t/H20850U/H20849t,t0/H20850+/H20858 k=1Mı 2/H20881QkYk/H20849t,t0/H20850/H20876,/H2084914/H20850 /H11509Yk/H20849t,t0/H20850 /H11509t=1 ı/H6036/H20875−ı 2/H20881QkTU/H20849t,t0/H20850+/H20849/H9268k−ıBk/H20850Yk/H20849t,t0/H20850/H20876, /H2084915/H20850 subject to the initial conditions U/H20849t0,t0/H20850=1/H20849theN/H11003Niden- tity matrix /H20850andYk/H20849t0,t0/H20850=0/H20849the 2/H11003Nzero matrix /H20850, where Yk/H20849t,t0/H20850,k=1,..., Mare 2/H11003Nrectangular matrix functions of the form Yk/H20849t,t0/H20850=−/H20881QkT 2/H6036/H20885 t0t e−ı/H20849/H9268k−ıBk/H20849t−t/H11032/H20850/H20850U/H20849t/H11032,t0/H20850dt/H11032, /H2084916/H20850 with Qk,/H9268k, and Bkbeing, respectively, N/H110032, 2/H110032, and 2 /H110032 matrices, namely,ω LΓ RΓ 12N1N−E E RµLµ ∆∆ FIG. 1. Schematic diagram of an electrode- N-site quantum wire- electrode nanoscale system driven by a time-dependent externalfield of frequency /H9275. The electrode-wire couplings are denoted by /H9003Land/H9003Rwhile the site-to-site tunneling is denoted by /H9004. The electrochemical potentials of the left and right electrodes are de-noted as /H9262Land/H9262R, respectively.HOet al. PHYSICAL REVIEW B 79, 235323 /H208492009 /H20850 235323-2Qk=/H20898qkL0 00 ]] 00 0qkR/H20899,/H9268k=/H20873/H9268kL0 0/H9268kR/H20874,Bk=/H20873bkL0 0bkR/H20874, /H2084917/H20850 andQkTbeing the matrix transpose of Qk. Coupled equations /H2084914/H20850and /H2084915/H20850may be effectively seen as an /H20849N+2M/H20850-level quantum system composed of an externally driven N-site quantum wire simultaneously coupled to Mnoninteracting two levels, as depicted in Fig. 2for the M=1 case. Each two level is composed of a left /H20849L/H20850level and a right /H20849R/H20850level, respectively, endowed with the energies /H9268kL/Rand widths bkL/R with k=1,..., M. The coupling between the quantum wire and the kth two levels is proportional to the square root of the fitting parameters qkLandqkR, which scale linearly in terms of the spectral widths bkLandbkR, respectively /H20849by keeping the ratios qkL/R/bkl/Rfinite /H20850. It should be pointed out that Eqs. /H2084914/H20850 and /H2084915/H20850not only address the memory effect but also are applicable to both periodically and nonperiodically drivenelectron-transport problems. The physical quantities of inter-est in the electron-transport calculations can be derived interms of the single-particle propagator U/H20849t,t 0/H20850in Eqs. /H2084914/H20850 and /H2084915/H20850. Specifically, the single-electron retarded Green’s function G/H20849t,t0/H20850can be computed from the relation11,12,19 G/H20849t,/H9280/H20850=/H20885 0/H11009 eı/H9280/H9270//H6036U/H20849t,t−/H9270/H20850d/H9270, /H2084918/H20850 and for periodically driven quantum wires, HC/H20849t+T/H20850=HC/H20849t/H20850, the Fourier components G/H20849k/H20850/H20849/H9280/H20850=1 T/H20885 0T G/H20849t,/H9280/H20850eık/H9275tdt /H2084919/H20850 of the periodical function G/H20849t,/H9280/H20850=G/H20849t+T,/H9280/H20850. Accordingly, the time-ensemble averaged dc electron-transport current,I¯/H110131 T/H20885 0T /H20855IL/H20849t/H20850/H20856dt, /H2084920/H20850 can in turn be evaluated via the equation,12 I¯=e 2/H9266/H6036/H20858 k=−/H11009+/H11009/H20885d/H9280/H20853TLR/H20849k/H20850/H20849/H9280/H20850fR/H20849/H9280/H20850−TRL/H20849k/H20850/H20849/H9280/H20850fL/H20849/H9280/H20850/H20854, /H2084921/H20850 where TLR/H20849k/H20850/H20849/H9280/H20850=/H208491/4/H20850/H9003¯L/H20849/H9280+k/H6036/H9275/H20850/H9003¯R/H20849/H9280/H20850/H20841G1N/H20849k/H20850/H20849/H9280/H20850/H208412and TRL/H20849k/H20850/H20849/H9280/H20850 =/H208491/4/H20850/H9003¯R/H20849/H9280+k/H6036/H9275/H20850/H9003¯L/H20849/H9280/H20850/H20841GN1/H20849k/H20850/H20849/H9280/H20850/H208412are, respectively, the right- to-left and left-to-right transmission coefficients. The Fermi-energy functions are f L/R/H20849/H9280/H20850=1 //H208531+e/H9252/H20849/H9280−/H9262L/R/H20850/H20854and the initial density matrix describing the state of the electrodes is /H92670 /H11011e−/H9252/H20849HLL+HRR−/H9262LNL−/H9262RNR/H20850, where /H9262Land/H9262Rare, respectively, the electrochemical potentials of the left and right electrodes,N L/Rare the numbers of electrons in the electrodes, and /H9252 =1 /kBTis the product of the Boltzmann constant kBand the temperature T. III. GENERALIZED FLOQUET APPROACH BEYOND THE WIDE-BAND LIMIT For illustration, consider a periodically driven electrode- DQD-electrode quantum system /H20849see Fig. 3/H20850, for which the electrode spectral densities /H9003¯L/R/H20849/H9280/H20850may be effectively mod- eled by single Lorentzian functions,17,18 /H9003¯L/R/H20849/H9280/H20850=qL/RbL/R//H20853/H20849/H9280−/H9268L/R/H208502+/H20849bL/R/H208502/H20854. /H2084922/H20850 The formulations in this section can be readily extended for the cases involving any finite number of Lorentzian func-tions. From Eqs. /H2084914/H20850and /H2084915/H20850, it is seen that the current case reduces to a four-level quantum system governed by thetime-dependent equation d dt/H20873U/H20849t,t0/H20850 Y/H20849t,t0/H20850/H20874=1 ı/H6036H/H20849t/H20850/H20873U/H20849t,t0/H20850 Y/H20849t,t0/H20850/H20874, /H2084923/H20850 whereωE E RµLµ Lq/2 σLRq/2 Rσ 12N1N−∆∆ FIG. 2. Schematic diagram of an effective /H20849N+2/H20850-level electrode- N-site quantum wire-electrode nanoscale system driven by a time-dependent external field of frequency /H9275and with the electrode spectral densities /H9003¯L/R/H20849/H9280/H20850modeled by single Lorentzian functions. The electrode-wire couplings are denoted by /H20881qL/2 and /H20881qR/2 while the site-to-site tunneling is denoted by /H9004. The electro- chemical potentials of the left and right electrodes are denoted as /H9262L and/H9262R, respectively.ω 12E Lµ ∆ σLE RµRσLq/2Rq/2 FIG. 3. Schematic diagram of an effective four-level electrode- DQD-electrode nanoscale system driven by a time-dependent exter-nal field of frequency /H9275and with the electrode spectral densities /H9003¯L/R/H20849/H9280/H20850modeled by single Lorentzian functions. The electrode-wire couplings are denoted by /H20881qL/2 and /H20881qR/2 while the site-to-site tunneling is denoted by /H9004. The electrochemical potentials of the left and right electrodes are denoted as /H9262Land/H9262R, respectively.MEMORY EFFECT ON THE MULTIPHOTON COHERENT … PHYSICAL REVIEW B 79, 235323 /H208492009 /H20850 235323-3Y/H20849t,t0/H20850=−/H20881Q 2/H6036/H20885 t0t e−ı/H20849/H9268−ıB/H20849t−t/H11032/H20850/H20850U/H20849t/H11032,t0/H20850dt/H11032, /H2084924/H20850 and the effective 4 /H110034 Hamiltonian of the electrode-DQD- electrode quantum system takes on the expression, H/H20849t/H20850=/H20873HC/H20849t/H20850ı/H20881Q/2 −ı/H20881Q/2/H9268−ıB/H20874. /H2084925/H20850 Here, the Hamiltonian of the periodically driven DQD is given as HC/H20849t/H20850=/H20898−E0+Acos/H9275t 2−/H9004 −/H9004 +E0+Acos/H9275t 2/H20899,/H2084926/H20850 in which the energy difference E/H20849t/H20850=E0+Acos/H9275tshifts pe- riodically with a frequency /H9275and amplitude A. Moreover, the electrode-DQD coupling can be written as /H20881Q=/H20881qL/H208411/H20856/H208551/H20841+/H20881qR/H208412/H20856/H208552/H20841, /H2084927/H20850 while the positions and widths of the left and right Lorentz- ian spectral-density functions are, respectively, given as /H9268=/H9268L/H208411/H20856/H208551/H20841+/H9268R/H208412/H20856/H208552/H20841, /H2084928/H20850 and B=bL/H208411/H20856/H208551/H20841+bR/H208412/H20856/H208552/H20841. /H2084929/H20850 By invoking the Floquet theory,8,9a solution F/H20849t/H20850of Eq. /H2084923/H20850 may be expressed as F/H20849t/H20850=/H9021/H20849t/H20850e−ı/H9011t//H6036, /H2084930/H20850 where /H9021/H20849t/H20850i sa4/H110034 matrix of periodic functions of time t, i.e.,/H9021/H20849t+T/H20850=/H9021/H20849t/H20850and/H9011i sa4/H110034 diagonal matrix of com- plex numbers satisfying the quasienergy eigenvalue equation /H20877H/H20849t/H20850−ı/H6036d dt/H20878/H9021/H20849t/H20850=/H9021/H20849t/H20850/H9011. /H2084931/H20850 From Eqs. /H2084930/H20850and /H2084931/H20850, it can be shown that the fundamen- tal solution U/H20849t,t0/H20850of Eq. /H2084923/H20850may be written as8,9 U/H20849t,t0/H20850=F/H20849t/H20850F−1/H20849t0/H20850=/H20873U11/H20849t,t0/H20850U12/H20849t,t0/H20850 U21/H20849t,t0/H20850U22/H20849t,t0/H20850/H20874, /H2084932/H20850 where F−1/H20849t0/H20850is the inverse of F/H20849t0/H20850, andU11,U21,U12, and U22are 2/H110032 subblocks. The single-particle propagator U/H20849t,t0/H20850can be readily identified as U/H20849t,t0/H20850=U11/H20849t,t0/H20850U/H20849t0,t0/H20850+U12Y/H20849t0,t0/H20850=U11/H20849t,t0/H20850, /H2084933/H20850 which is in turn used to compute the current I¯, cf. Eqs. /H2084918/H20850–/H2084921/H20850. On the other hand, the single-particle propagator U/H20849t,t0/H20850 may be expediently written as the productU/H20849t,t0/H20850=/H9021C/H20849t/H20850X/H20849t,t0/H20850, /H2084934/H20850 where /H9021C/H20849t+T/H20850=/H9021C/H20849t/H20850, such that X/H20849t,t0/H20850and Y/H20849t,t0/H20850to- gether satisfy the equation, d dt/H20873X/H20849t,t0/H20850 Y/H20849t,t0/H20850/H20874=1 ı/H6036/H20898/H9011Cı 2/H9021C†/H20849t/H20850/H20881Q −ı 2/H20881Q/H9021C/H20849t/H20850/H9268−ıB/H20899/H20873X/H20849t,t0/H20850 Y/H20849t,t0/H20850/H20874, /H2084935/H20850 in the quasienergy /H20849Floquet /H20850state representation. Here the 2/H110032 diagonal quasienergy matrix, /H9011C=/H92611,0C/H208411/H20856/H208551/H20841+/H92612,0C/H208412/H20856/H208552/H20841, /H2084936/H20850 contains the DQD quasienergies /H92611,0Cand/H92612,0Cbelonging to the quasienergy states /H20841/H92611,0C/H20856and /H20841/H92612,0C/H20856, respectively. The quasienergy state function /H9021C/H20849t/H20850associated with /H9011Csatisfies the quasienergy eigenvalue equation /H20877HC/H20849t/H20850−ı/H6036d dt/H20878/H9021C/H20849t/H20850=/H9021C/H20849t/H20850/H9011C, /H2084937/H20850 governing the time evolution of the DQD driven by the pe- riodic field. The DQD quasienergy state function /H9021C/H20849t/H20850con- tains the elements /H20849/H9021C/H20850/H9251/H11032/H9251/H20849t/H20850=/H20858 k=−/H11009/H11009 /H20855/H9251/H11032k/H20841/H9261/H9251,0C/H20856e+ık/H9275t, /H2084938/H20850 where /H9251,/H9251/H11032=1,2 and the summation is performed over the Fourier components of the field-dressed Floquet states /H20841/H92611,0C/H20856 and /H20841/H92612,0C/H20856. Figure 4shows that the peaks of the averaged current I¯are closely correlated with the underlying quasien- ergy avoided crossing patterns at the multiphoton resonanceconditions E 0=n/H6036/H9275,n=1,2,3,4. Here the parameters E0 =10/H9004=1 eV, A=6/H9004,/H9268L=/H9268R=0,qL=qR=q,bL=bR=b, and00.51I[ 1 0-1e/CID1//CID2h]WBL b=50/CID1 b=20/CID1 b=10/CID1 -10010 2 4 6 8 10 12 14 16Quasienergy [ /CID1] /CID2h/CID3[/CID1] FIG. 4. /H20849Color online /H20850Current I¯and quasienergies /H92611/2,0Cas func- tions of /H6036/H9275/H20849in/H9004/H20850.E0=10/H9004=1 eV, A=6/H9004,/H9268L=/H9268R=0,qL=qR=q, bL=bR=b, and/H9003L=/H9003R=q/b=0.5/H9004. Upper panel: the peaks of the current I¯from the right to left, respectively, correspond to one-, two-, three-, and four-photon resonances. Lower panel: solid /H20849—/H20850 curves /H92611,0Cand dashed /H20849--- /H20850curves /H92612,0C.HOet al. PHYSICAL REVIEW B 79, 235323 /H208492009 /H20850 235323-4/H9003L=/H9003R=q/b=0.5/H9004have been used in the calculations. Moreover, it is found that the dc current resonance peaks areblueshifted and their magnitudes slightly reduced at the de-creasing spectral width bof the electrodes, in accordance to the fact that a weaker electrode-wire coupling corresponds toa smaller b/H20849/H11011q/H20850value. IV . MEMORY EFFECT ON MULTIPHOTON CDT FOR ELECTRODE-DQD-ELECTRODE SYSTEM To understand the memory effect on the MP-CDT phe- nomena, we consider the cases in the high-frequency limitand at the n-photon resonance condition E 0=/H20841n/H20841/H6036/H9275. Under these situations, the corresponding generalized rotating waveapproximation /H20849GRWA /H20850solution can be derived with the help of the transformation 22 R/H20849t/H20850=e+ı/H20851E0t/2/H6036+/H9278/H20849t/H20850/H20852/H208411/H20856/H208551/H20841+e−ı/H20851E0/2/H6036+/H9278/H20849t/H20850/H20852/H208412/H20856/H208552/H20841, /H2084939/H20850 where /H9278/H20849t/H20850=/H20849A/2/H6036/H9275/H20850sin/H9275t. In the rotated frame, the corre- sponding Hamiltonian of the central DQD, HC/H11032/H20849t/H20850=R†/H20849t/H20850HC/H20849t/H20850R/H20849t/H20850−ı/H6036R†/H20849t/H20850dR/H20849t/H20850/dt, /H2084940/H20850 can be explicitly written as HC/H11032/H20849t/H20850=−/H9004/H20853e−ı/H20851E0t//H6036+/H9278/H20849t/H20850/H20852/H208411/H20856/H208552/H20841+e+ı/H20851E0t//H6036+/H9278/H20849t/H20850/H20852/H208412/H20856/H208551/H20841/H20854, /H2084941/H20850 which in turn results in the n-photon GRWA Floquet Hamil- tonian HC/H20851n/H20852=−/H9004/H11003 /H20841Jn/H20849A//H6036/H9275/H20850/H20841/H20849/H208411/H20856/H208552/H20841+/H208412/H20856/H208551/H20841/H20850, /H2084942/H20850 where Jn/H20849x/H20850is the nth order Bessel function of the first kind. The n-photon GRWA quasienergy state function /H9021C/H20851n/H20852/H20849t/H20850=1 /H208812R/H20849t/H20850/H2087311 −1 1/H20874 /H2084943/H20850 associated with the n-photon quasienergy matrix /H9011C/H20851n/H20852can be further expressed as 1 /H208812/H20898/H20858 k=−/H11009/H11009 Jk/H20873A 2/H6036/H9275/H20874eık/H9275t/H20858 k=−/H11009/H11009 Jk/H20873A 2/H6036/H9275/H20874eık/H9275t −/H20858 k=−/H11009/H11009 Jk+n/H20873A 2/H6036/H9275/H20874e−ık/H9275t/H20858 k=−/H11009/H11009 Jk+n/H20873A 2/H6036/H9275/H20874e−ık/H9275t/H20899, /H2084944/H20850 which has been derived using the well-known expansion22 eıxsin/H9275t=/H20858 k=−/H11009/H11009 Jk/H20849x/H20850eık/H9275t. /H2084945/H20850 It can be readily shown from Eq. /H2084942/H20850that the n-photon GRWA quasienergies are /H92611,0C=−E0/2+/H9004/H20841Jn/H20849A//H6036/H9275/H20850/H20841, /H2084946/H20850 and /H92612,0C=−E0/2−/H9004/H20841Jn/H20849A//H6036/H9275/H20850/H20841, /H2084947/H20850 which are separated by the Rabi oscillation frequency,/H9024n=2/H9004/H20841Jn/H20849A//H6036/H9275/H20850/H20841, /H2084948/H20850 of the n-photon driven resonant DQD.22Numerical results in Fig. 5, based on Eq. /H2084923/H20850, reveals that the averaged current I¯ is suppressed at the roots of Jn/H20849A//H6036/H9275/H20850=0,n/H113501/H20849due to the vanishing /H9024n/H20850, a clear manifestation of the MP-CDT similar to the well-known zero-biased CDT at the roots of J0/H20849A//H6036/H9275/H20850=0.10–13,18,22It is found that the averaged current I¯ in general decreases as the field amplitude increases. The memory effect /H20849due to a decreasing spectral-density width b, therefore, corresponding to a deceasing electrode-wire cou- pling qvalue /H20850is to reduce the average current I¯but without qualitatively altering the feature of the MP-CDT. Calcula- tions of the average current I¯/H20849not shown /H20850based on Eqs. /H2084935/H20850 and /H2084944/H20850showed good agreements with the numerically ex- act results /H20849Fig. 5/H20850, albeit of smaller I¯values /H20849especially at the large Alimit /H20850. In the following, we provide a qualitative picture of the MP-CDT phenomena. At the MP-CDT condition /H9024n=0, cf. Eq. /H2084948/H20850, the DQD ceases to oscillate and the corresponding quasienergy state function /H9021C/H20851n/H20852/H20849t/H20850, Eq. /H2084944/H20850, may be further approximated by its time-averaged counterpart /H9021¯ C/H20851n/H20852=1 T/H20885 0T /H9021C/H20851n/H20852/H20849t/H20850dt=1 /H208812/H20898J0/H20873A 2/H6036/H9275/H20874J0/H20873A 2/H6036/H9275/H20874 −Jn/H20873A 2/H6036/H9275/H20874Jn/H20873A 2/H6036/H9275/H20874/H20899. /H2084949/H20850 It is then seen that at the weak-field amplitude, J0/H20849A/2/H6036/H9275/H20850 /H110151 and Jn/H20849A/2/H6036/H9275/H20850/H110150,n/H113501 since Jn/H20849x/H20850/H11011xn/2nn! for x /H112701, where x=A/2/H6036/H9275. Therefore, the off-diagonal electrode- DQD coupling in Eq. /H2084935/H20850can be approximated as024(a)b=10 /CID1 b=20 /CID1 b=50 /CID1 WBL 024I [10-2e/CID1//CID2h] (b) 024 0 20 40 60 80 100 A[/CID1](c) FIG. 5. /H20849Color online /H20850The MP-CDT at the /H20849a/H20850one-, /H20849b/H20850two-, and /H20849c/H20850three-photon resonances at different spectral widths band at the WBL. The abscissa is the field amplitude A/H20849in/H9004/H20850. The minima correspond to the roots of J1/H20849A//H6036/H9275/H20850=0 /H20849for one photon /H20850, J2/H20849A//H6036/H9275/H20850=0 /H20849for two photon /H20850, and J3/H20849A//H6036/H9275/H20850=0 /H20849for three photon /H20850, coinciding with the respective degenerate quasienergies /H92611,0C=/H92612,0C =/H11006E0/2,E0=10/H9004=1 eV, where /H6036/H9275=E0,E0/2,E0/3, respectively, for one-, two-, and three-photon CDTs. See Fig. 4for all other parameters.MEMORY EFFECT ON THE MULTIPHOTON COHERENT … PHYSICAL REVIEW B 79, 235323 /H208492009 /H20850 235323-5/H20881Q/H11003/H9021¯ C/H20851n/H20852/H110151 /H208812/H20873/H20881qL/H20881qL 0 0/H20874, /H2084950/H20850 implying that the left-hand side electrode Land the degener- ate DQD Floquet states /H20841/H92611,0C/H20856and /H20841/H92612,0C/H20856form a closed current loop, which is separated from the right-hand side electrode R, or vice versa. In this case, corresponding to A/H112702E0 =20/H9004in Fig. 5, little current is allowed to flow through the electrode-DQD-electrode quantum system, as indicated inthe first minima of all panels in Fig. 5. On the other hand, at the strong-field amplitude, /H9021¯ C/H20851n/H20852=0, implying that the degen- erate DQD Floquet states are decoupled from both elec- trodes, thus I¯=0, as shown in the minima toward the far right end in Fig. 5in the limit A/H112712E0=20/H9004, since Jn/H20849x/H20850 /H11011/H208812//H9266xcos/H20851x−/H208492n+1/H20850/H9266/4/H20852forx/H11271n. However, the gener- ally time-dependent GRWA quasienergy state matrix /H9021C/H20851n/H20852/H20849t/H20850 may not be well approximated by its time-averaged counter- part/H9021¯ C/H20851n/H20852away from the weak-field/strong-field amplitude re- gimes. Consequently, the CDT phenomenon involving morethan one photon /H20849i.e.,n/H110221/H20850may be less pronounced at the intermediate field amplitude, as clearly shown /H20849here 20 /H9004 /H11021A/H1102160/H9004/H20850in the two- and three-photon CDT /H20851the middle /H20849b/H20850and lower /H20849c/H20850panels in Fig. 5/H20852. V . SUMMARY In summary, we have presented a generalized Floquet ap- proach beyond the wide-band limit for studying memory ef-fect on multiphoton coherent destruction of tunneling phe- nomena of a periodically driven electron transport ofnanoscale quantum systems. The general formulation of thememory effect is equally applicable to periodically and non-periodically driven electron-transport problems. In particular,the generalized Floquet approach can be extended to involveany number of quantum dots, including a single-quantumdot, and can be readily adopted to study a host of time-dependent electron-transport processes that may be of inter-est in nanoscale devices. In the high-frequency limit, simplephysical pictures have been given for the occurrence of theMP-CDT phenomena in the electrode-DQD-electrode quan-tum system. Numerical simulations at different spectralwidths, as well as in the wide-band limit, showed that thememory effect reduces the electron-transport dc currentwithout altering the feature of the MP-CDT phenomena, con-sistent with the Lorentzian spectral-density ansatz. ACKNOWLEDGMENTS This work was partially supported by the Chemical Sci- ences, Geosciences, and Biosciences Division of the Officeof Basic Energy Sciences, Office of Sciences, Department ofEnergy, by the National Science Foundation. We also wouldlike to acknowledge the partial support of the National Sci-ence Council of Taiwan /H20849Grant No. 97-2112-M-002-003- MY3 /H20850and National Taiwan University /H20849Grant No. 97R0066 /H20850. *tsho@princeton.edu †sichu@ku.edu 1G. Platero and R. Aguado, Phys. Rep. 395,1 /H208492004 /H20850. 2C. A. Stafford and N. S. Wingreen, Phys. Rev. Lett. 76, 1916 /H208491996 /H20850. 3C. Meyer, J. M. Elzerman, and L. P. Kouwenhoven, Nano Lett. 7, 295 /H208492007 /H20850. 4F. Grossmann, T. 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PhysRevB.79.245125.pdf

Many-body theory of electronic transport in single-molecule heterojunctions J. P. Bergfield College of Optical Sciences, University of Arizona, 1630 East University Boulevard, Arizona 85721, USA C. A. Stafford Department of Physics, University of Arizona, 1118 East Fourth Street, Tucson, Arizona 85721, USA /H20849Received 4 December 2008; revised manuscript received 1 May 2009; published 23 June 2009 /H20850 A many-body theory of molecular junction transport based on nonequilibrium Green’s functions is devel- oped, which treats coherent quantum effects and Coulomb interactions on an equal footing. The centralquantity of the many-body theory is the Coulomb self-energy matrix /H9018 Cof the junction. /H9018Cis evaluated exactly in the sequential-tunneling limit, and the correction due to finite tunneling width is evaluated self-consistently using a conserving approximation based on diagrammatic perturbation theory on the Keldyshcontour. Our approach reproduces the key features of both the Coulomb blockade and coherent transportregimes simultaneously in a single unified transport theory. As a first application of our theory, we havecalculated the thermoelectric power and differential conductance spectrum of a benzenedithiol-gold junctionusing a semiempirical /H9266-electron Hamiltonian that accurately describes the full spectrum of electronic excita- tions of the molecule up to 8–10 eV. DOI: 10.1103/PhysRevB.79.245125 PACS number /H20849s/H20850: 73.63. /H11002b, 85.65. /H11001h, 72.10.Bg, 31.15.xr I. INTRODUCTION Electron transport in single-molecule junctions1–3is of fundamental interest as a paradigm for nanosystems far from equilibrium and as a means to probe important chemical4and biological5processes, with myriad potential device applications.3,6,7A general theoretical framework to treat the many-body problem of a single molecule coupled to metallic electrodes does not currently exist. Mean-field approachesbased on density-functional theory 8–14—the dominant para- digm in quantum chemistry—have serious shortcomings15–17 because they do not account for important interaction effects such as Coulomb blockade. An alternative approach is to solve the few-body molecu- lar Hamiltonian exactly and treat electron hopping betweenmolecule and electrodes as a perturbation. This approach hasbeen used to describe molecular-junction transport in thesequential-tunneling regime, 15,18,19but describing coherent quantum transport in this framework remains an open theo-retical problem. Higher-order tunneling processes may betreated rigorously in the density-matrix formalism 20,21and the closely related superoperator Green’s functionapproach, 22,23but the expansion is typically truncated at second23or fourth order,20,24with the calculation of higher- order terms being prohibitively difficult. Furthermore, thisapproach has so far been limited to very small molecules,and it is unclear what the prospects are for the application tocomplex molecules of interest for potential device applica-tions. In this paper, we develop a many-body theory of molecu- lar junction transport based on nonequilibrium Green’sfunctions 25–27/H20849NEGF /H20850in order to utilize physically moti- vated approximations that sum terms of all orders. The junc-tion Green’s functions are calculated exactly in thesequential-tunneling limit, and the corrections to the electronself-energy due to finite tunneling width are included viaDyson-Keldysh equations. The tunneling self-energy is cal- culated exactly using the equations-of-motions method, 26,28while the correction to the Coulomb self-energy is calculated using diagrammatic perturbation theory. In this way, tunnel-ing processes are included to infinite order, meaning that any approximation utilized is a truncation in the physical pro-cesses considered rather than in the order of those processes. Our approach reproduces the key features of both the Coulomb blockade and coherent transport regimes simulta-neously in a single unified transport theory. Nonperturbativeeffects of intramolecular correlations are included, which arenecessary to accurately describe the highest occupied mo-lecular orbital /H20849HOMO /H20850-lowest unoccupied molecular orbital /H20849LUMO /H20850gap, essential for a quantitative theory of transport. As a first application of our many-body transport theory, we investigate the benchmark system of benzene /H208491,4/H20850dithiol /H20849BDT /H20850with gold electrodes. Two key parameters determin- ing the lead-molecule coupling—the tunneling width /H9003and the chemical potential offset /H9004 /H9262—are fixed by comparison to linear-response measurements of the thermoelectricpower 29,30and electrical conductance.31The nonlinear junc- tion response is then calculated. The differential conductancespectrum of the junction exhibits an irregular “moleculardiamond” structure analogous to the regular Coulombdiamonds 32observed in quantum dot transport experiments, as well as clear signatures of coherent quantum transport—such as transmission nodes due to destructive interference—and of resonant tunneling through molecular excited states. This paper is organized as follows. A detailed derivation of the many-body theory of molecular junction transport ispresented in Sec. II. Two useful approximate solutions for the nonequilibrium Coulomb self-energy are given in Secs.I ID1 andI ID2 . The details of the model used to describe a molecular heterojunction consisting of a /H9266-conjugated mol- ecule covalently bonded to metallic electrodes are presentedin Sec. III. The electric and thermoelectric response of a BDT-Au junction are calculated in Sec. IVand compared to experimental results. A discussion and conclusions are pre-sented in Sec. V.PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 1098-0121/2009/79 /H2084924/H20850/245125 /H2084910/H20850 ©2009 The American Physical Society 245125-1II. NONEQUILIBRIUM MANY-BODY FORMALISM A. Molecular junction Hamiltonian The Hamiltonian of a junction consisting of a molecule coupled to several metallic electrodes may be written as Hjunction =Hmol+Hleads+HT. /H208491/H20850 The molecular Hamiltonian can be formally divided into one-body and two-body terms Hmol=Hmol/H208491/H20850+Hmol/H208492/H20850. In general, neglecting spin-orbit coupling, the one-body term can bewritten as H mol/H208491/H20850=/H20858 n,m,/H9268/H20851Hmol/H208491/H20850/H20852n/H9268,m/H9268dn/H9268†dm/H9268, /H208492/H20850 where dn/H9268†creates an electron of spin /H9268on the atomic orbital nof the molecule and /H20851Hmol/H208491/H20850/H20852is a Hermitian matrix. For simplicity, the atomic basis orbitals are taken to be orthonor- mal so that the anticommutator /H20853dn/H9268†,dm/H9268/H11032/H20854=/H9254nm/H9254/H9268/H9268/H11032. Exten- sion to nonorthogonal bases is straightforward in principle. In a localized orthonormal basis, a general spin-rotation invariant two-body /H20849e.g., Coulomb /H20850interaction has the form33 Hmol/H208492/H20850=1 2/H20858 n,mUnm/H9267n/H9267m, /H208493/H20850 where /H9267n=/H20858/H9268dn/H9268†dn/H9268. Values of Unmfor/H9266-conjugated systems34,35are discussed in Sec. III. Because of their continuous and almost featureless den- sity of states around the Fermi energy, the macroscopic me-tallic electrodes, labeled /H9251/H33528/H208511,..., M/H20852, may be modeled as noninteracting Fermi gases, Hleads=/H20858 /H9251=1M /H20858 k/H33528/H9251 /H9268/H9280k/H9268ck/H9268†ck/H9268, /H208494/H20850 where ck/H9268†creates an electron of energy /H9280k/H9268in lead /H9251. The electrostatic interaction of molecule and electrodes due to theelectric dipoles formed at each molecule-electrode interface may be included in H mol/H208491/H20850, as discussed in Sec. III. Tunneling of electrons between the molecule and the electrodes is de-scribed by the Hamiltonian H T=/H20858 /H9251=1M /H20858 k/H33528/H9251/H20858 n,/H9268/H20849Vnkdn/H9268†ck/H9268+ H.c. /H20850. /H208495/H20850 B. Nonequilibrium Green’s functions The electronic system /H208491/H20850of molecule plus electrodes has an infinite Hilbert space. A formal simplification of the prob-lem is obtained through the use of the Green’s functions 25,26 Gn/H9268,m/H9268/H11032/H20849t/H20850=−i/H9258/H20849t/H20850/H20855/H20853dn/H9268/H20849t/H20850,dm/H9268/H11032†/H208490/H20850/H20854/H20856, /H208496/H20850 Gn/H9268,m/H9268/H11032/H11021/H20849t/H20850=i/H20855dm/H9268/H11032†/H208490/H20850dn/H9268/H20849t/H20850/H20856, /H208497/H20850 known as the retarded and Keldysh “lesser” Green’s func- tions, respectively. Steady-state physical observables in themolecular transport junction can be expressed in terms of G andG/H11021. Transient effects36–39are not considered in this pa- per. In this section, an exact Dyson equation for Gis derived, along with a corresponding Keldysh equation for G/H11021. Setting /H6036=1, the retarded Green’s function obeys the equation of motion26 i/H11509 /H11509tGn/H9268,m/H9268/H11032/H20849t/H20850=/H9254/H20849t/H20850/H9254nm/H9254/H9268/H9268/H11032−i/H9258/H20849t/H20850 /H11003/H20855/H20853/H20851dn/H9268/H20849t/H20850,Hjunction /H20852,dm/H9268/H11032†/H208490/H20850/H20854/H20856./H208498/H20850 For the purposes of this paper, only the spin-diagonal term Gn/H9268,m/H9268is needed /H20851cf. Eqs. /H2084919/H20850and /H2084920/H20850/H20852. Evaluating the commutators /H20851dn/H9268,Hmol/H20852and /H20851dn/H9268,HT/H20852and noting that /H20851dn/H9268,Hleads/H20852=0, Eq. /H208498/H20850becomes i/H11509 /H11509tGn/H9268,m/H9268/H20849t/H20850=/H9254/H20849t/H20850/H9254nm+/H20858 n/H11032/H20851Hmol/H208491/H20850/H20852n/H9268,n/H11032/H9268Gn/H11032/H9268,m/H9268/H20849t/H20850 +/H20858 /H9251/H20858 k/H33528/H9251Vnkgk/H9268,m/H9268/H20849t/H20850+/H20858 n/H11032Unn/H11032Gn/H11032,n/H9268m/H9268/H208492/H20850/H20849t/H20850, /H208499/H20850 where gk/H9268,m/H9268/H11032/H20849t/H20850=−i/H9258/H20849t/H20850/H20855/H20853ck/H9268/H20849t/H20850,dm/H9268/H11032†/H208490/H20850/H20854/H20856 /H20849 10/H20850 and Gn/H11032,n/H9268m/H9268/H11032/H208492/H20850/H20849t/H20850=−i/H9258/H20849t/H20850/H20855/H20853/H9267n/H11032/H20849t/H20850dn/H9268/H20849t/H20850,dm/H9268/H11032†/H208490/H20850/H20854/H20856. /H2084911/H20850 The equation of motion for gk/H9268,m/H9268/H20849t/H20850is i/H11509 /H11509tgk/H9268,m/H9268/H20849t/H20850=−i/H9258/H20849t/H20850/H20855/H20853/H20851ck/H9268/H20849t/H20850,Hjunction /H20852,dm/H9268†/H208490/H20850/H20854/H20856./H2084912/H20850 Evaluating the commutators /H20851ck/H9268,HT/H20852and /H20851ck/H9268,Hleads/H20852and noting that /H20851ck/H9268,Hmol/H20852=0, Eq. /H2084912/H20850becomes i/H11509 /H11509tgk/H9268,m/H9268/H20849t/H20850=/H9280k/H9268gk/H9268,m/H9268/H20849t/H20850+/H20858 n/H11032Vn/H11032k/H11569Gn/H11032/H9268,m/H9268/H20849t/H20850./H2084913/H20850 Fourier transforming Eqs. /H208499/H20850and /H2084913/H20850into the energy domain and eliminating gk/H9268,m/H9268/H20849E/H20850, one arrives at the follow- ing matrix equation for G/H20849E/H20850: /H208511E−Hmol/H208491/H20850−/H9018T/H20849E/H20850/H20852G/H20849E/H20850=1+ UG/H208492/H20850/H20849E/H20850, /H2084914/H20850 where the retarded tunneling self-energy matrix is /H20851/H9018T/H20849E/H20850/H20852n/H9268,m/H9268/H11032=/H9254/H9268/H9268/H11032/H20858 /H9251/H20858 k/H33528/H9251VnkVmk/H11569 E−/H9280k/H9268+i0+. /H2084915/H20850 Equation /H2084914/H20850may be recast in the form of Dyson’s equation G/H20849E/H20850=/H208511E−Hmol/H208491/H20850−/H9018T/H20849E/H20850−/H9018C/H20849E/H20850/H20852−1/H2084916/H20850 via the ansatz UG/H208492/H20850/H20849E/H20850/H11013/H9018C/H20849E/H20850G/H20849E/H20850, which defines40the retarded Coulomb self-energy matrix /H9018C/H20849E/H20850, the central quantity of the many-body theory, which must be determinedvia an appropriate approximation. A prescription to compute/H9018 Cis given below in Secs. II CandII D. Equation /H2084916/H20850is a general formal result, independent of the choice of basis, andJ. P. BERGFIELD AND C. A. STAFFORD PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 245125-2is the starting point for our theoretical analysis. For nonequilibrium problems, the Keldysh lesser self- energy and Green’s function are also needed. G/H11021is deter- mined by the Keldysh equation26 G/H11021/H20849E/H20850=G/H9018/H11021G†+/H208491+G/H9018/H20850Gmol/H11021/H208491+/H9018†G†/H20850, /H2084917/H20850 where G†/H20849E/H20850and/H9018†/H20849E/H20850are the advanced Green’s function and self-energy, respectively, /H9018/H11021=/H9018T/H11021+/H9018C/H11021, and Gmol/H11021 =lim /H9018T→0G/H11021. The second term on the right-hand side of Eq. /H2084917/H20850is infinitessimal and can be neglected, provided that all of the molecular resonances have a finite broadening due tothe lead-molecule coupling. The lesser tunneling self-energy is /H20851/H9018 T/H11021/H20849E/H20850/H20852n/H9268,m/H9268/H11032=i/H20858 /H9251f/H9251/H20849E/H20850/H20851/H9003/H9251/H20849E/H20850/H20852n/H9268,m/H9268/H11032, /H2084918/H20850 where f/H9251/H20849E/H20850=/H208531+exp /H20851/H20849E−/H9262/H9251/H20850/kBT/H9251/H20852/H20854−1is the Fermi-Dirac distribution for lead /H9251, and /H20851/H9003/H9251/H20849E/H20850/H20852n/H9268,m/H9268/H11032=2/H9266/H9254/H9268/H9268/H11032/H20858 k/H33528/H9251VnkVmk/H11569/H9254/H20849E−/H9280k/H9268/H20850/H20849 19/H20850 is the tunneling-width matrix for lead /H9251. Once the Green’s functions are known, the relevant physi- cal observables can be calculated. For example, the currentflowing into lead /H9251is given by25 I/H9251=ie h/H20885dETr/H20853/H9003/H9251/H20849E/H20850/H20853G/H11021/H20849E/H20850+f/H9251/H20849E/H20850/H20851G/H20849E/H20850−G†/H20849E/H20850/H20852/H20854/H20854, /H2084920/H20850 where I/H9251denotes the expectation value of the current opera- tor. C. Sequential-tunneling limit In the limit of infinitessimal lead-molecule coupling /H9018T/kBT→0, coherent superpositions of different energy eigenstates of the molecule can be neglected and the junctionGreen’s function becomes lim /H9018T→0/H20851G/H20849E/H20850/H20852n/H9268,m/H9268/H11013/H20851Gmol/H20849E/H20850/H20852n/H9268,m/H9268 =/H20858 /H9263,/H9263/H11032/H20851P/H20849/H9263/H20850+P/H20849/H9263/H11032/H20850/H20852/H20855/H9263/H20841dn/H9268/H20841/H9263/H11032/H20856/H20855/H9263/H11032/H20841dm/H9268†/H20841/H9263/H20856 E−E/H9263/H11032+E/H9263+i0+, /H2084921/H20850 where /H20841/H9263/H20856and /H20841/H9263/H11032/H20856are many-body eigenstates of the isolated molecule satisfying Hmol/H20841/H9263/H20856=E/H9263/H20841/H9263/H20856, etc. The nonequilibrium probabilities P/H20849/H9263/H20850can be determined by solving a system of semiclassical rate equations for sequential tunneling.15,18,19,41 For steady-state transport, they satisfy the principle of de- tailed balance /H20849see Fig. 1/H20850, P/H20849/H9263/H20850/H20858 /H9251/H9003˜ /H9251/H9263/H9263/H11032f/H9251/H20849E/H9263/H11032−E/H9263/H20850=P/H20849/H9263/H11032/H20850/H20858 /H9251/H9003˜ /H9251/H9263/H9263/H11032/H208511−f/H9251/H20849E/H9263/H11032−E/H9263/H20850/H20852. /H2084922/H20850 Here the rate constants are given by Fermi’s golden rule as/H9003˜ /H9251/H9263/H9263/H11032=T r /H20853/H9003/H9251/H20849E/H9263/H11032−E/H9263/H20850C/H9263/H9263/H11032/H20854, /H2084923/H20850 where /H9003/H9251/H20849E/H20850is given by Eq. /H2084919/H20850and /H20851C/H9263/H9263/H11032/H20852n/H9268,m/H9268/H11032=/H20855/H9263/H20841dn/H9268/H20841/H9263/H11032/H20856/H20855/H9263/H11032/H20841dm/H9268/H11032†/H20841/H9263/H20856/H20849 24/H20850 are many-body matrix elements.42–44From the normalization of the many-body wave functions, the total resonance width /H9003˜/H9263/H9263/H11032=/H20858/H9251/H9003˜ /H9251/H9263/H9263/H11032scales as /H11011/H20858/H9251Tr/H20853/H9003/H9251/H20854/N, where Nis the num- ber of atomic orbitals in the molecule. For strongly corre-lated systems, there is an additional exponential suppressionof Eq. /H2084924/H20850as N→/H11009due to the orthogonality catastrophe. 44,45 Linear-response transport is determined by the equilib- rium Green’s functions. In equilibrium, the solution of the setof Eqs. /H2084922/H20850reduces to P/H20849 /H9263/H20850=e−/H9252/H20849E/H9263−/H9262N/H9263/H20850/Z, /H2084925/H20850 where Zis the grand partition function of the molecule at inverse temperature /H9252and chemical potential /H9262. Equation /H2084921/H20850implicitly defines the Coulomb self-energy matrix /H9018C/H208490/H20850in the sequential-tunneling limit via Gmol/H20849E/H20850=/H208511E−Hmol/H208491/H20850−/H9018C/H208490/H20850/H20849E/H20850/H20852−1. /H2084926/H20850 /H9018C/H208490/H20850is the high-temperature limit of the Coulomb self-energy /H20849i.e., the limit kBT/Tr/H20853/H9003/H9251/H20854/H112711/H20850and describes intramolecular correlations and charge quantization effects /H20849Coulomb block- ade away from resonance /H20850. The nonperturbative treatment of intramolecular correlations provided by the exact diagonal-ization of H molin Eq. /H2084921/H20850allows for an accurate description of the HOMO-LUMO gap, which is essential for a quantita-tive theory of transport. D. Correction to the Coulomb self-energy In general, the Coulomb self-energy matrix /H9018C=/H9018C/H208490/H20850 +/H9004/H9018 C, where /H9004/H9018 Cdescribes the change in the Coulomb self- energy due to lead-molecule coherence emerging at tempera-tures k BT/H11351Tr/H20853/H9003/H9251/H20854. Using this decomposition of the Cou- lomb self-energy, Dyson’s Eq. /H2084916/H20850can be rewritten in the following useful form: G−1/H20849E/H20850=Gmol−1/H20849E/H20850−/H9018T−/H9004/H9018 C, /H2084927/H20850 where the self-energy terms /H9018T+/H9004/H9018 Cdescribe the effects of finite tunneling width. /H9018Tis given by Eq. /H2084915/H20850. Here we point out that /H9004/H9018 C—unlike /H9018C/H208490/H20850—can be evaluated perturbatively using diagrammatic techniques on the Keldysh time contour FIG. 1. /H20849Color online /H20850A schematic of detailed balance. The left-hand and right-hand sides of Eq. /H2084922/H20850are represented in /H20849a/H20850and /H20849b/H20850, respectively.MANY-BODY THEORY OF ELECTRONIC TRANSPORT IN … PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 245125-3/H20849cf. Fig. 2/H20850. Such a perturbative approach is valid, in principle, at temperature/bias voltages satisfyingmax /H20853T,eV /k B/H20854/H11022TK, where TK is the Kondo temperature46—or when there is no unpaired electron on the molecule /H20849such as within the HOMO-LUMO gap of conju- gated organic molecules /H20850. /H9018T+/H9004/H9018 Cmay be thought of as the response of the junc- tion to turning on the tunneling coupling /H9003/H9251. A subtlety in the perturbative evaluation of /H9004/H9018 Cis that the diagrams deter- mining the Coulomb self-energy are typically formulated40in terms of the Green’s functions of the noninteracting system G/H208490/H20850/H20849E/H20850=/H208491E−Hmol/H208491/H20850−/H9018T/H20850−1and Gmol/H208490/H20850/H20849E/H20850=/H208491E−Hmol/H208491/H20850+i0+/H20850−1, while the response of the junction is determined by the fullGreen’s function G/H20849E/H20850/H20851cf. Eq. /H2084920/H20850/H20852. Evaluating the dia- grams in Fig. 2using G /H208490/H20850/H20849E/H20850and Gmol/H208490/H20850/H20849E/H20850would yield a correction to the Coulomb self-energy with a pole structure unrelated to that of /H9018C/H208490/H20850so that adding the two together would not yield a physically meaningful result. Our strategyis thus to calculate /H9004/H9018 Cby reformulating the terms in the perturbative expansion in terms of the full Green’s functionG/H20849E/H20850via appropriate resummations. This procedure is in general nontrivial, but the result for the Hartree-Fock /H20849HF/H20850 correction is given in Sec. I ID2 below based on physical arguments. Higher-order self-energy diagrams can be in-cluded in a similar fashion. Electron-phonon coupling 27,47–51 can also be included in the many-electron theory via the self-energy terms /H9018e-ph+/H9004/H9018Ce-ph, where /H9018e-phis given by the usual self-energy diagrams,27,48–51and/H9004/H9018Ce-phis the corre- sponding correction to the Coulomb self-energy. 1. Elastic cotunneling approximation: /H9018=/H9018T+/H9018C(0) Far from transmission resonances and for T/H11271TK,/H9004/H9018 C can be neglected. This is the limit of elastic cotunneling .52,53 The Green’s functions are given by Eqs. /H2084916/H20850and /H2084917/H20850with /H9018C=/H9018C/H208490/H20850. Note that /H20851/H9018C/H208490/H20850/H20852/H11021does not make a finite contribu- tion to Eq. /H2084917/H20850when /H9018Tis finite. The full NEGF current expression /H2084920/H20850then reduces to the multiterminal Büttiker formula54 I/H9251=e h/H20858 /H9252=1M/H20885 −/H11009/H11009 dET/H9251/H9252/H20849E/H20850/H20851f/H9252/H20849E/H20850−f/H9251/H20849E/H20850/H20852, /H2084928/H20850 where the transmission function is given by55 T/H9251/H9252/H20849E/H20850=T r /H20853/H9003/H9251/H20849E/H20850G/H20849E/H20850/H9003/H9252/H20849E/H20850G†/H20849E/H20850/H20854. /H2084929/H20850The elastic cotunneling approximation is a conserving approximation —current is conserved /H20851cf. Eq. /H2084928/H20850/H20852and the spectral function obeys the usual sum rule. 2. Self-consistent Hartree-Fock correction: /H9018C=/H9018C(0)+/H9004/H9018CHF In the HF approximation, the retarded Coulomb self- energy matrix is real and is given by /H20851/H9018CHF/H20852n/H9268,m/H9268/H11032=/H9254/H9268/H9268/H11032/H20875/H9254nm/H20858 n/H11032Unn/H11032/H20855/H9267n/H11032/H20856−Unm/H20855dm/H9268†dn/H9268/H20856/H20876, /H2084930/H20850 and /H20851/H9018CHF/H20852/H11021=0. In general, /H20855dn/H9268†dm/H9268/H11032/H20856=−i 2/H9266/H20885 −/H11009/H11009 dE/H20851G/H11021/H20849E/H20850/H20852m/H9268/H11032,n/H9268 /H2084931/H20850 and lim /H9018T→0/H20855dn/H9268†dm/H9268/H11032/H20856=/H20858 /H9263P/H20849/H9263/H20850/H20855/H9263/H20841dn/H9268†dm/H9268/H11032/H20841/H9263/H20856. /H2084932/H20850 The Hartree-Fock correction is then /H9004/H9018CHF/H11013/H9018CHF−/H9018CHF/H20841/H9018T→0. The Feynman diagrams representing this correction are shown in Fig. 2. A self-consistent solution of Eqs. /H2084917/H20850,/H2084927/H20850, and /H2084930/H20850–/H2084932/H20850yields a conserving approximation in which the junction current is again given by Eq. /H2084928/H20850. The use of the interacting Green’s functions GandGmolin the evaluation of the Hartree self-energy is clearly justifiedon physical grounds, since this yields the classical electro-static potential due to the actual nonequilibrium charge dis-tribution on the molecule. The direct /H20849Hartree /H20850and exchange /H20849Fock /H20850contributions to /H9004/H9018 Cmust be treated on an equal footing in order to cancel the unphysical self-interaction, jus-tifying the use of the same interacting Green’s functions inthe evaluation of the exchange self-energy. The diagrams ofFig. 2would be quite complex if expressed in terms of the noninteracting Green’s functions because both GandG mol involve /H9018C/H208490/H20850, which includes all possible combinations of Coulomb lines and intramolecular propagators. The self-consistent Hartree-Fock correction for a diatomic molecule is shown in Fig. 3. The parameters were chosen so that the resonance width /H11011/H20858/H9251Tr/H20853/H9003/H9251/H20854/N=0.2 eV, where Nis the number of atomic orbitals in the molecule. The elements of the matrix /H9004/H9018CHFare largest near a transmission resonance but vanish on resonance. This behavior can be understood byconsidering the molecular correlation functions shown inFig.4. The inclusion of the tunneling self-energy without the corresponding correction to the Coulomb self-energy leads toa charge imbalance on the molecule near resonance. This inturn leads to a Hartree correction which tends to counteractthe charge imbalance. A corresponding behavior is found forthe exchange correction and off-diagonal correlation func-tion. A non-self-consistent calculation would yield a much larger correction /H9004/H9018 CHF, indicating the important role of screening. It should be pointed out that a treatment of screen-ing in linear response is not adequate near resonance. As shown in Fig. 5, the transmission peaks and nodes are not shifted by the self-consistent HF correction and the trans-mission phase is not changed significantly. Transport proper- FIG. 2. The correction to the Coulomb self-energy /H9004/H9018 Cis given in the self-consistent Hartree-Fock approximation by the sum of /H20849a/H20850 the Hartree /H20849direct /H20850term and /H20849b/H20850the Fock /H20849exchange /H20850term. The wavy line represents the Coulomb interaction, which has the formU nm/H9254/H20849/H9270−/H9270/H11032/H20850on the Keldysh time contour.J. P. BERGFIELD AND C. A. STAFFORD PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 245125-4ties /H20851cf. Eqs. /H2084928/H20850and /H2084929/H20850/H20852are therefore qualitatively better described in the elastic cotunneling approximation than arethe correlation functions, and the approximation is quantita-tively accurate in the cotunneling regime /H20841 /H9262−/H9262res/H20841/H11022/H9003, max /H20853T,eV /kB/H20854/H11022TK. The tendency toward charge quantization near resonance is significantly increased by the self-consistent HF correctionas shown in Fig. 4. The steepness of the self-consistent charging step is limited only by thermal broadening. Thisresult is consistent with previous theoretical studies 56–58of Coulomb blockade in metal islands and quantum dots butinconsistent with the behavior of the Anderson model, 46 where singular spin fluctuations modify this generic behav-ior. III. MOLECULAR HETEROJUNCTION MODEL Heterojunctions formed from /H9266-conjugated molecules are of particular interest both because of their relevance fornanotechnology 3,6and their extensive experimental characterization.1–3A semiempirical /H9266-electron Hamiltonian6,34,35can be used to model the electronic de- grees of freedom most relevant for transport, Hmol=/H20858 n,/H9268/H9255ndn/H9268†dn/H9268−/H20858 n,m,/H9268/H20849tnmdn/H9268†dm/H9268+ H.c. /H20850 +/H20858 n,mUnm 2QnQm, /H2084933/H20850 where dn/H9268†creates an electron of spin /H9268in the /H9266orbital of the nth carbon atom, /H9255nis the atomic orbital energy, and tnmis the hopping matrix element between orbitals nandm.O f f - diagonal interaction terms33arising from nonzero differential overlap of neighboring atomic orbitals are negligible in /H9266-conjugated systems.59In the /H9266-electron theory, the effect of different side groups is included through shifts of the or-bital energies /H9255 n. The effect of substituents /H20849e.g., thiol groups /H20850used to bond the leads to the molecule can be included60,61in the tunneling matrix elements Vnk/H20851cf. Eq. /H208495/H20850/H20852. The effective charge operator for orbital nis6,44-0.5-0.2500.25 -5.5 -5 -4.5 -4Self-energy correction (eV) µ-µ0(eV)∆Σ11∆Σ12 FIG. 3. /H20849Color online /H20850Self-consistent Hartree-Fock correction to the Coulomb self-energy matrix of a hypothetical C-C diatomicmolecule versus lead chemical potential, shown in the vicinity ofthe HOMO resonance. Here /H92620=/H20849/H9255HOMO +/H9255LUMO /H20850/2 is the chemi- cal potential of the isolated molecule and T=300 K. The molecular junction parameters U11=U22=8.9 eV, U12=4.4 eV, t12=2.0 eV, and/H90031=/H90032=0.2 eV were used /H20849see Sec. III/H20850. 0.50.60.70.80.9 -5.5 -5 -4.5 -4〈d† 1d2〉 µ-µ0(eV)11.21.41.61.82〈N〉Σ=ΣC(0)+ΣT+∆ΣCHF Σ=ΣC(0)+ΣT Σ=ΣC(0) FIG. 4. /H20849Color online /H20850Equilibrium correlation functions near the HOMO resonance for a diatomic molecule /H20849same parameters as in Fig.3/H20850. Top: total molecular charge /H20855N/H20856in three different approxi- mations: /H9018=/H9018C/H208490/H20850/H20849isolated molecule in the grand canonical en- semble /H20850;/H9018=/H9018C/H208490/H20850+/H9018T/H20849elastic cotunneling approximation /H20850; and /H9018 =/H9018C/H208490/H20850+/H9018T+/H9004/H9018CHF/H20849self-consistent HF correction /H20850. The charging step is broadened in the elastic cotunneling approximation, compared tothat of the isolated molecule, and acquires a “knee” near resonance.In the self-consistent HF approximation, the charge imbalance /H9004N =/H20855N/H20856−/H20855N/H20856/H20841 /H9018T→0is strongly screened near resonance, leading to a charging step as steep as that of the isolated molecule, but asymp- totically approaches the elastic cotunneling result away from reso-nance. Bottom: the correlation function /H20855d 1/H9268†d2/H9268/H20856in the same three approximations.-π-π/20 -7 -6 -5 -4 -3φ(E)E=µ µ-µ0(eV)10-810-610-410-2100T(E)E=µ Σ=ΣC(0)+ΣT+∆ΣCHF Σ=ΣC(0)+ΣT FIG. 5. /H20849Color online /H20850Transmission probability and phase cal- culated with and without the self-consistent HF correction. Notethat the transmission peaks and nodes are not shifted by /H9004/H9018 CHFbut the width of the transmission resonance is reduced asymmetrically./H20849Note: linestyles reversed in lower panel. /H20850MANY-BODY THEORY OF ELECTRONIC TRANSPORT IN … PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 245125-5Qn=/H20858 /H9268dn/H9268†dn/H9268−/H20858 /H9251Cn/H9251V/H9251/e−1 , /H2084934/H20850 where Cn/H9251is the capacitive coupling between orbital nand lead/H9251,eis the electron charge, and V/H9251is the voltage on lead /H9251. The lead-molecule capacitances are elements of a full ca- pacitance matrix C, which also includes the intramolecular capacitances determined by the relation Cnm=e2/H20851U−1/H20852nm. The values Cn/H9251are determined by the zero-sum rules required for gauge invariance62 /H20858 mCnm+/H20858 /H9251=1M Cn/H9251=0 , /H2084935/H20850 and by the geometry of the junction /H20849e.g., Cn/H9251inversely pro- portional to lead-orbital distance /H20850. The effective interaction energies for /H9266-conjugated sys- tems can be written as34,35 Unm=/H9254nmU0+/H208491−/H9254nm/H20850U0 /H9280/H208811+/H9251/H20849Rnm/Å/H208502, /H2084936/H20850 where U0is the on-site Coulomb repulsion, /H9251 =/H20849U0/14.397 eV /H208502, and Rnmis the distance between orbitals nand m. The phenomenological dielectric constant /H9280ac- counts for screening due to both the /H9268electrons and any environmental considerations such as nonevaporatedsolvent. 35With an appropriate choice of the parameters tnm, U0, and/H9280, the complete spectrum of electronic excitations up to 8–10 eV of the molecules benzene, biphenyl, and trans- stilbene in the gas phase can be reproduced with highaccuracy 35by exact diagonalization of Eq. /H2084933/H20850. An accurate description of excited states is essential to model transportfar from equilibrium. Larger conjugated organic moleculescan also be modeled 34,63via Eqs. /H2084933/H20850and /H2084936/H20850. /H9268orbitals can also be included in Eq. /H2084933/H20850as additional energy bands, and the resulting multiband extended Hubbardmodel can be treated using the same NEGF formalismsketched above. Tunneling through the /H9268orbitals may be important in small molecules,64,65especially in cases where quantum interference leads to a transmission node in the /H9266-electron system. The biggest uncertainty in modeling single-molecule het- erojunctions is the lead-molecule coupling.66For this reason, we take the two most uncertain quantities characterizinglead-molecule coupling—the tunneling width /H9003and the chemical potential offset /H9004 /H9262of isolated molecule and metal electrodes—as phenomenological parameters to be deter-mined by fitting to experiment. In the broadband limit 67for the metallic electrodes, and assuming that each electrode iscovalently bonded to a single carbon atom of the molecule,the tunneling-width matrix reduces to a single constant, /H9003 nm/H9251/H20849E/H20850=/H9003/H9251/H9254na/H9254ma, where ais the orbital connected to elec- trode /H9251. Typical estimates60,61,68indicate /H9003/H113511 eV for or- ganic molecules coupled to gold contacts via thiol groups. IV. BENZENE(1,4)DITHIOL JUNCTION As a first application of our many-body theory of molecu- lar junction transport, we consider the benchmark system ofbenzene /H208491,4/H20850dithiol /H20849BDT /H20850with two gold leads.29–31,69–73The Hamiltonian parameters for benzene are35U0=8.9 eV, /H9280 =1.28, and tnm=2.68 eV for nandmnearest neighbors and tnm=0 otherwise. We consider a symmetric junction /H20849sym- metric capacitive couplings and /H90031=/H90032/H11013/H9003/H20850at room tem- perature /H20849T=300 K /H20850. A. Linear electric and thermoelectric junction response Thermoelectric effects29,30,74provide important insight into the transport mechanism in single-molecule junctionsbut are particularly sensitive to correlations, 75,76calling into question the applicability of single-particle or mean-fieldtheory. We can now investigate thermoelectric effects in mo-lecular heterojunctions using many-body theory. The ther-mopower Sof a molecular junction is obtained by measuring the voltage /H9004V/H11013−S/H9004Tcreated across an open junction in response to a temperature differential /H9004T. In general, Scan be calculated by taking the appropriate linear-response limitof Eq. /H2084920/H20850, which includes both elastic and inelastic pro- cesses. However, for purely elastic transport /H20849cf. Secs. I ID1 andI ID2 /H20850, Eq. /H2084928/H20850can be used to derive the well-known result 77,78 S/H20849/H9262,T/H20850=−1 eT/H20885 −/H11009/H11009 T12/H20849E/H20850/H20849−/H11509f /H11509E/H20850/H20849E−/H9262/H20850dE /H20885 −/H11009/H11009 T12/H20849E/H20850/H20849−/H11509f /H11509E/H20850dE. /H2084937/H20850 Thermopower measurements29,30provide a means to determine74the lead-molecule chemical potential mismatch /H9004/H9262=/H9262Au−/H92620, where /H92620=/H20849/H9255HOMO +/H9255LUMO /H20850/2 and /H9255HOMO /H20849/H9255LUMO /H20850is the HOMO /H20849LUMO /H20850energy level. Figure 6shows the thermopower of a BDT junction as a function of /H9004/H9262 calculated from Eqs. /H2084929/H20850and /H2084937/H20850in the elastic cotunneling approximation. As pointed out by Paulsson and Datta,74Sis nearly independent of /H9003away from the transmission reso--100-80-60-40-20020406080100 -5 -4 -3 -2 -1 0 1 2 3 4 5Sjunction(V/K) Au-0(eV)=0.2eV =0.5eV =1.0eV 6.877.2 -3.3 -3.25 -3.2 -3.15 So u r c e Drain FIG. 6. /H20849Color online /H20850Thermoelectric power of a BDT-Au junc- tion at T=300 K as a function of the lead-molecule chemical po- tential mismatch for three different tunneling widths. Comparison tothe experimental value /H20849Ref. 30/H208507.0/H110060.2 /H9262V/K fixes /H9262Au=/H92620 −/H208493.22/H110060.04 /H20850eV/H20849see the inset showing close-up of experimen- tally relevant parameter range /H20850.J. P. BERGFIELD AND C. A. STAFFORD PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 245125-6nances, allowing /H9004/H9262to be determined directly by compari- son to experiment. The thermopower of a BDT-Au junctionwas recently measured by Baheti et al. , 30who obtained the result S=/H208497.0/H110060.2/H20850/H9262V/K. Equating this experimental value with the calculated thermopower shown in Fig. 6,w e find that −3.25 /H11349/H9262Au−/H92620/H11349−3.15 eV over a broad range of /H9003values. The Fermi level of gold thus lies about 1.8 eV above the HOMO resonance, validating the notion that trans-port in these junctions is hole dominated. Nonetheless, /H9262Au −/H9255HOMO is sufficiently large that the elastic cotunneling ap- proximation is well justified. The only other free parameter in the molecular junction model is the tunneling width /H9003, which can be found by matching the linear-response conductance to experiment. Al-though there is a large range of experimental values, 14the most reproducible and lowest resistance contacts were ob-tained by Xiao et al. , 31who reported a single-molecule con- ductance value of 0.011 G0. This fixes /H9003=/H208490.63/H110060.02 /H20850eV with/H9262Au−/H92620=−3.22 /H110060.04 eV. This value of /H9003is within the range predicted by other groups for similarmolecules. 60,61,66–68With the final parameter in the model fixed, we can now use our many-body transport theory topredict the linear and nonlinear responses of this molecularjunction. The transmission probability T/H20849E/H20850/H20841 E=/H9262of the BDT junc- tion is shown as a function of /H9262−/H9262Auin the upper panel of Fig.7. This is the linear-response conductance in units of the conductance quantum G0=2e2/h. The chemical potential /H9262 is related to the gate voltage Vgin a three-terminal junction via the gate capacitance Cg. The transmission spectrum shown in Fig. 7exhibits several striking features: a large but irregular peak spacing with an increased HOMO-LUMOgap, nonsymmetric, Fano-like resonance line shapes, andtransmission nodes due to the destructive interference in thecoherent quantum transport.B. Nonlinear junction transport We next calculate the differential conductance /H11509I//H11509Vbiasof the BDT-Au junction as a function of /H9262andVbias/H20849see Fig. 7, lower panel /H20850. The current was calculated in the elastic cotun- neling approximation using Eqs. /H2084928/H20850and /H2084929/H20850. This approxi- mation accurately describes nonresonant transport, includingthe transmission nodes, as well as the positions and heightsof the transmission resonances, as discussed in Sec. I ID2 . The differential conductance spectrum of the junction exhib-its clear signatures of excited-state transport 32,79and an ir- regular molecular diamond structure analogous to the regularCoulomb diamonds observed in quantum dot transportexperiments. 32The charge on the molecule is quantized within the central diamonds of Fig. 7, an important interac- tion effect inaccessible to the so-called ab initio mean-field calculations. In Fig. 7, the full spectrum is shown for com- pleteness, although the junction may not be stable over theentire range of bias and gate voltages shown. Apart from the central HOMO-LUMO gap, the widths of the diamonds in Fig. 7can be roughly explained via a ca- pacitive model in which the molecule is characterized by asingle capacitance C mol=e2//H20855Unm/H20856, where /H20855Unm/H20856=5.11 eV is the average over all molecular sites. The HOMO-LUMO gapof/H1101110 eV is significantly larger than this estimate of the charging energy, an indication of the significant deviations ofour theory from a simple constant interaction model. Charge quantization, also known as Coulomb blockade , has been observed in several different types of molecularheterojunctions 80–84but has not yet been observed in BDT junctions due to the difficulty of gating such smallmolecules. 85The unambiguous observation of Coulomb blockade in junctions involving larger molecules /H20849with smaller charging energies /H20850indicates that such interaction ef- fects, lying outside the scope of mean-field approaches, are -5-4-3-2-10 log10[|dI/dV|/G0] 40Vbias(Volts ) -30 -20 -10 0 10 20 30-4-2024-5-4-3-2-10 -6 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 -5 FIG. 7. /H20849Color online /H20850Linear-response conductance versus chemical potential /H20849top panel /H20850and differential conductance versus chemical potential and bias voltage /H20849bottom panel /H20850for a benzene /H208491,4/H20850dithiol-Au junction. The calculation was carried out within the elastic cotun- neling approximation using Eq. /H2084928/H20850with/H90031=/H90032=0.63 eV and T=300 K. Here G0=2e2/his the conductance quantum. Note the asym- metric Fano-like line shapes and the transmission nodes /H20849arising from destructive interference /H20850in the top panel. The numbers within the central diamonds of the bottom panel indicate the quantized charge on the molecule /H20849relative to neutral benzene /H20850. The full spectrum is shown for completeness.MANY-BODY THEORY OF ELECTRONIC TRANSPORT IN … PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 245125-7undoubtedly even more pronounced in small molecules such as BDT. Important aspects of this phenomenon remain to beunderstood in larger molecules such as the anomalously lowreported values of the charging energy. 80–84 A zero-bias cross section of the bottom panel of Fig. 7 reproduces the transmission spectrum shown in the top panelof the same figure. Increasing the bias voltage, we find thatthe resonances split into negatively sloped particlelike/H20849/H20841e/H20841dV /d /H9262=−/H20849C1+C2+Cg/H20850/C1/H20850lines and positively sloped holelike /H20849/H20841e/H20841dV /d/H9262=+/H20849C1+C2+Cg/H20850/C2/H20850lines, where C1and C2are the mean lead-molecule capacitances defined by C/H9251 =/H20855Cn/H9251/H20856and/H9251=1,2. In the symmetric coupling case /H20849C1 =C2with Cg/C1/H112701/H20850the lines therefore have slopes of −2 and +2 for particlelike and holelike lines, respectively.Within the V-shaped outline traced by the particlelike andholelike lines, we find signatures of resonant tunnelingthrough electronic excited states in the many narrow nearlyparallel resonance lines. While transport through electronicexcited states has not yet been unambiguously identified insingle-molecule heterojunctions, it has been observed inquantum dots 32,79and carbon nanotubes.86 An accurate description of the HOMO-LUMO gap is es- sential for a quantitative theory of transport in molecularheterojunctions. The central HOMO-LUMO gap shown inFig. 7is significantly larger than that predicted 1by density functional theory—which neglects charge-quantizationeffects—but is consistent with previous many-body calcula-tions in the sequential-tunneling regime. 15,18,19It should be emphasized that the transport gap /H9004/H9262trin a molecular junc- tion exceeds the optical gap /H6036/H9275minof an isolated molecule. Roughly speaking, /H9004/H9262tr/H11229/H6036/H9275min+e2/Cmol+Ex, where e2/Cmolis the charging energy /H20849see discussion above /H20850and −Exis the exciton binding energy. Excitonic states of the BDT-Au junction can be identified in the differential-conductance spectrum of Fig. 8as the lowest-energy /H20849i.e., smallest bias /H20850excitations outside the central diamond of the HOMO-LUMO gap, from which it is apparent that /H9004 /H9262tr /H110152/H6036/H9275minfor a BDT-Au junction. V. CONCLUSIONS In conclusion, we have developed a many-body theory of electron transport in single-molecule heterojunctions thattreats coherent quantum effects and Coulomb interactions onan equal footing. As a first application of our theory, we haveinvestigated the thermoelectric power and differential con-ductance of a prototypical single-molecule junction, ben-zenedithiol with gold electrodes. Our results reproduce the key features of both the coher- ent and Coulomb blockade transport regimes; quantum inter-ference effects, such as the transmission nodes predictedwithin mean-field theory, 6,65,87are confirmed while the dif- ferential conductance spectrum exhibits characteristiccharge-quantization “diamonds” 80–84—an effect outside thescope of mean-field approaches based on density-functional theory. The HOMO-LUMO transport gap obtained is consis-tent with previous many-body treatments in the sequential-tunneling limit. 15,18,19 The central object of the many-body theory is the Cou- lomb self-energy /H9018Cof the junction, which may be expressed as/H9018C=/H9018C/H208490/H20850+/H9004/H9018 C, where /H9018C/H208490/H20850is the result in the sequential- tunneling limit and /H9004/H9018 Cis the correction due to a finite tunneling width /H9003. In this paper, we have evaluated /H9018C/H208490/H20850ex- actly, thereby including intramolecular correlations at a non-perturbative level, while the direct and exchange contribu-tions to /H9004/H9018 Cwere evaluated self-consistently using a conserving approximation based on diagrammatic perturba-tion theory on the Keldysh contour. An important feature ofour theory is that this approximation for /H9004/H9018 Ccan be system- atically improved by including additional processes diagram-matically. In this way, important effects such as dynamicalscreening, spin-flip scattering, 46and electron-phonon coupling27,47–51can be included as natural extensions of the theory. Our general theory of molecular junction transport should be contrasted to results for transport in the Andersonmodel. 46,88The Anderson model provides important insight into nonperturbative interaction effects in quantum transportthrough nanostructures; however, the internal structure of themolecule is neglected. Moreover, since it is limited to asingle spin-degenerate level, it can only describe a singleCoulomb diamond with an odd number of electrons on themolecule and is therefore not applicable to transport withinthe HOMO-LUMO gap of conjugated organic molecules,where the number of electrons is even /H20849more precisely, the HOMO-LUMO gap is taken to be infinite in the Andersonmodel /H20850. ACKNOWLEDGMENT The authors thank S. Mazumdar and D. M. Cardamone for useful discussions. -3 -2 -1 0 1 2 3-6-4-20246Vbias(Volts ) -5-4-3-2-10 log10[|dI/dV|/G0] FIG. 8. /H20849Color online /H20850Differential conductance spectrum of a benzene /H208491,4/H20850dithiol-Au junction /H20849same parameters as in Fig. 7/H20850fo- cusing on the vicinity of the HOMO resonance. The arrows indicatefeatures corresponding to resonant tunneling through excitonicstates of the junction.J. P. BERGFIELD AND C. A. STAFFORD PHYSICAL REVIEW B 79, 245125 /H208492009 /H20850 245125-81A. Nitzan and M. A. 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PhysRevB.80.081105.pdf

Spin and orbital states in La 1.5Sr0.5CoO 4studied by electronic structure calculations Hua Wu1and T. Burnus2 1II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany 2Institut für Festkörperforschung, Forschungszentrum Jülich, 52425 Jülich, Germany /H20849Received 24 July 2009; published 26 August 2009 /H20850 Electronic structure of the layered perovskite La 1.5Sr0.5CoO 4with a checkerboard Co2+/Co3+charge order is studied, using the local-spin-density approximation plus Hubbard Ucalculations including also the spin-orbit coupling and multiplet effect. Our results show that the Co2+ion is in a high spin state /H20849HS, t2g5eg2/H20850and Co3+ low spin state /H20849LS, t2g6/H20850. Due to a small Co2+t2gcrystal field splitting, the spin-orbit interaction produces an orbital moment of 0.26 /H9262Band accounts for the observed easy in-plane magnetism. Moreover, we find that the Co3+intermediate spin state /H20849IS,t2g5eg1/H20850has a multiplet splitting of several tenths of eV and the lowest-lying one is still higher than the LS ground state by 120 meV , and that the Co3+HS state /H20849t2g4eg2/H20850is more unstable by 310 meV . Either the IS or HS Co3+ions would give rise to a wrong magnetic order and anisotropy. DOI: 10.1103/PhysRevB.80.081105 PACS number /H20849s/H20850: 71.20. /H11002b, 71.27. /H11001a, 71.70. /H11002d, 75.30. /H11002m Cobaltates comprise a group of interesting materials which display spectacular properties such assuperconductivity, 1giant magnetoresistance,2and large ther- moelectric power.3One aspect in the physics of cobaltats, which distinguishes them from other transition metal oxides,is the spin state issue, particularly for the Co 3+ions. It can be a low spin /H20849LS/H20850, a high spin /H20849HS/H20850, or even an intermediate spin /H20849IS/H20850state, depending on a subtle interplay among crystal field, Hund exchange, multiplet effects, and spin-orbit cou-pling /H20849SOC /H20850. 4–17One prototype material is the perovskite LaCoO 3, and its temperature-dependent spin state transition has been extensively studied but a consensus has not yetbeen reached so far: the magnetic excitations at about 100and 500 K in LaCoO 3have been ascribed to either an LS→HS transition,4–9an LS →IS transition,10–13or LS →/H20849LS+HS /H20850→IS transitions.14–17 Recently, the single layered perovskite La 2−xSrxCoO 4re- ceived considerable attention for its extremely insulating be-havior, peculiar magnetic correlations, and doping-dependent charge/spin superstructures, and its spin state issue becomesa vital topic. 18–26The parent compound La 2CoO 4has a nor- mal HS Co2+and is an antiferromagnetic insulator with a quite high TN=275 K.27Upon Sr or Ca doping, Co3+ions are introduced. Measurements of magnetic and transportproperties of La 2−xSrxCoO 4/H208490.4/H11349x/H113490.8/H20850led Moritomo et al.to a conclusion that the Co3+ions undergo a spin state transition from HS /H20849x/H113490.6/H20850to IS /H20849x/H113500.8/H20850.28Neutron scat- tering measurements of La 1.5Sr0.5CoO 4by Zaliznyak et al. showed a checkerboard Co2+-Co3+charge order and a strongly decreasing spin ordering temperature /H20849TSO/H1101530 K /H20850, and they suggested the Co3+ions to be in an IS but nonmag- netic state quenched by strong planar anisotropy.29,30The Co2+-Co3+charge order was also observed in La 1.5Ca0.5CoO 4 and the Co3+ions were suggested to be in a mixed IS+HS state.31,32In contrast, a magnetic susceptibility study of La2−xSrxCoO 4/H208490.3/H11349x/H113490.8/H20850and crystal-field model calcula- tions indicated that the Co3+must be in the LS state for x /H113500.4.21In particular, a very recent x-ray absorption spectro- scopic study by Chang et al. established a picture of HS Co2+ and LS Co3+for La 1.5Sr0.5CoO 4, and their study well ac- counts for the extremely insulating nature of theLa 2−xSrxCoO 4series, the high charge ordering temperature and low TSOof La 1.5Sr0.5CoO 4.26It is, however, a bit surprising that this interesting material and the spin state issue received much less theoretical inves-tigation. Previous unrestricted Hartree-Fock calculationsshowed that there are two spin-state transitions inLa 2−xSrxCoO 4in the doping range of 0 /H11021x/H110211.1, and that La1.5Sr0.5CoO 4is in the ferromagnetic HS state.33In view of the existing controversy, the spin state of the charge orderedLa 1.5Sr0.5CoO 4remains to be an open question, and therefore we will shed light on this issue by carrying out a set ofdensity functional electronic structure calculations. Our re-sults show that La 1.5Sr0.5CoO 4is indeed in the HS-Co2+/LS-Co3+ground state, which can explain the ex- periments as seen below. However, the energetically unfavor-able IS or HS Co 3+state would have a trouble in such an explanation. We used the structural data of La 1.5Sr0.5CoO 4measured by single-crystal neutron diffraction.34Our electronic struc- ture calculations were performed by using the full-potentialaugmented plane waves plus local orbital method. 35The muffin-tin sphere radii are chosen to be 2.5, 2.0, and 1.5Bohr for La/Sr, Co, and O atoms, respectively. A virtual atomwith an atomic number Z=56.75 /H208490.75Z La+0.25 ZBa/H20850is used for the /H20849La1.5Sr0.5/H20850sites since La and Sr /H20849Ba/H20850ions are in most cases simply electron donors. The cutoff energy of 16 Ryd isused for plane-wave expansion, and 600 kpoints for integra- tions over the Brillouin zone. To account for the strong elec-tron correlations, the local-spin-density approximation plusHubbard U/H20849LSDA+ U/H20850/H20849Ref. 36/H20850calculations were carried out, with U=5 eV and Hund exchange of 0.9 eV for the Co 3 delectrons. 37,38The SOC turns out to be quite important and it is included by the second-variational method with sca-lar relativistic wave functions. 35 Usually, LSDA+ Ucalculations may yield different orbital-polarized solutions, depending on the initialized oc-cupation number matrix. Talking about the spin state issue,we have done a set of LSDA+ U+SOC calculations which are initialized by assuming the LS, IS, and HS states, respec-tively. Our results show that while the Co 2+ion in La1.5Sr0.5CoO 4is always stabilized at the normal HS state, the Co3+can be stabilized at the LS, IS, or HS state, as detailed below. It is important to note that the total-energyresults reveal the Co 3+LS ground state. Figure 1shows the density of states in the HS-Co2+/LS-Co3+ground state. The HS Co2+has the t2g5eg2PHYSICAL REVIEW B 80, 081105 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 1098-0121/2009/80 /H208498/H20850/081105 /H208494/H20850 ©2009 The American Physical Society 081105-1configuration with one t2ghole on the xyorbital. This corre- sponds to the local crystal field: since the out-of-planeCo 2+-O2bbondlength of 2.192 Å is much bigger than the in-plane Co2+-O1 bondlength of 1.955 Å,34the xylevel should be higher than the xz/yzand it is the t2ghole orbital. The t2gcrystal field splitting /H20849CFS /H20850and especially electron correlations open a gap between the xz/yzand xyin the down-spin channel. For the LS Co3+ion, it has a closed t2g6 shell which allows one to calculate the t2ginterior small CFS by simply determining the center of gravity of each orbital-resolved density of states. The calculated ionic CFS betweenthe higher xylevel and lower xz/yzis 70 meV . However, for the HS Co 2+with an open t2gshell, the LSDA self-interaction error of the lower-lying occupied orbital obscures an esti- mate of a small t2gCFS /H20849For the closed t2g6shell, however, the self-interaction errors are common for each t2gorbital and thus counteracted /H20850. Instead, we used a /H208812/H11003/H208812/H110031 su- percell and replaced one Co2+by an immediate neighbor Ni2+/H20849S=1, t2g6eg2/H20850which has a closed t2gshell. Thus the Co2+t2gCFS is estimated to be about 30 meV between the xyand xz/yz. Note that the smaller CFS of the Co2+t2glev- els than that of the Co3+t2gis consistent with the larger Co-O bondlengths of the former.34Owing to the small CFS of the open t2gshell at the Co2+site, the SOC is operative and mixes the lower-lying xz/yzwith higher xylevel and thus produces an in-plane orbital magnetic moment. Our LSDA+U+SOC calculations show that La 1.5Sr0.5CoO 4has an easy in-plane magnetism and an orbital moment of 0.26 /H9262Bper Co2+/H20849reduced by Co-O covalent effects /H20850, as well as an al- most quenched orbital moment of 0.02 /H9262Bat the LS Co3+site due to the closed t2gshell, as seen in Table I. The solution with an easy out-of-plane magnetism was calculated to havea higher energy by 12 meV per Co 2+and the corresponding Co2+orbital moment is only 0.01 /H9262B. Thus, our results, based on the HS-Co2+/LS-Co3+ground state solution, account for the experimentally observed easy in-plane magnetism of La1.5Sr0.5CoO 4.21,28Moreover, this ground state displays a narrow band insulating behavior with a band gap of 1.2 eV/H20849Fig. 1/H20850which is in agreement with an optical conductivity measurement, 39and the observed spectral peak around 3 eV can be explained as a charge transfer excitation from thein-plane O1 2 pat about −1 eV to the Co 2+xyand Co3+ x2-y2both at 2 eV . TABLE I. Total energies /H20849in unit of meV per 2 f.u. /H20850of La 1.5Sr0.5CoO 4having the LS, IS, or HS Co3+state and a robust HS Co2+calculated by LSDA+ U+SOC. The HS Co2+configuration is shown only once in the LS-Co3+/HS-Co2+ground state. Except for the ground state having an easy in-plane magnetism /H20849the spin and orbital moments in unit of /H9262Bmarked by a subscript “ ab”/H20850, all other solutions have a wrong easy out-of-plane magnetism. State and configuration Energy Cospin3+Coorb3+Cospin2+Coorb2+Figure LS Co3+/H20849t2g6/H20850/HS Co2+/H20849t2g↑3eg↑2xz↓1yz↓1/H20850 0 0.29 ab 0.02 ab 2.52 ab 0.26 ab Figure 1 LS Co3+/H20849t2g6/H20850/HS Co2+12 0.29 0.05 2.52 0.01 IS Co3+/H20851t2g↓3/H208493z2−r2/H20850↓1xz↑1yz↑1/H20852/HS Co2+725 −1.50 0 2.48 0.02 Figure 2/H20849a/H20850 IS Co3+/H20851t2g↓3/H208493z2−r2/H20850↓1xy↑1/H20849xz−iyz/H20850↑1/H20852/HS Co2+298 −1.52 −1.08 2.51 0 Figure 2/H20849b/H20850 IS Co3+/H20851t2g↑3/H208493z2−r2/H20850↑1xy↓1/H20849xz+iyz/H20850↓1/H20852/HS Co2+122 2.02 1.30 2.48 0.03 Figure 2/H20849c/H20850 HS Co3+/H20851t2g↓3eg↓2/H20849xz−iyz/H20850↑1/H20852/HS Co2+311 −2.95 −0.96 2.45 0.0221012 -8 -6 -4 -2 0 2 4 621012 -8 -6 -4 -2 0 2 4 6 21012 -8 -6 -4 -2 0 2 4 6xy xz,yz3z −r 22 3z −r x− y222 2 22 xy xz,yzLS Co3+2+HS Co x− y xz,yz xy Ener gy (eV)Density of States (states/eV) 2xO1 2pO2b 2p O2a 2p FIG. 1. /H20849Color online /H20850Density of states /H20849DOS /H20850of the HS-Co2+/LS-Co3+ground state of La 1.5Sr0.5CoO 4calculated by LSDA+ U+SOC. The HS Co2+hast2g5eg2with a t2ghole on the xy orbital, and the LS Co3+closed t2g6shell. The 2 pDOS of the planar O1, Co3+-apical O2a, and Co2+-apical O2b are also shown. Fermi level is set at zero energy.HUA WU AND T. BURNUS PHYSICAL REVIEW B 80, 081105 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 081105-2Since the Co2+ion is always stabilized at the HS state as described above, in the following the spin state and elec-tronic structure of the Co 3+ion only are further discussed. From the middle panel of Fig. 1, it can be seen that the 3 z2 −r2level is lower than the x2−y2by 0.7 eV , in good agree- ment with an x-ray absorption spectroscopic study.26When a Co3+IS state is realized, most probably one t2gelectron will be promoted to the 3 z2−r2level rather than the much higher x2−y2. Simply according to the crystal field level diagram, it is the xyelectron which will be promoted to the 3 z2−r2,a s seen in Fig. 2/H20849a/H20850. It is, however, important to note that the IS state has a significant multiplet effect. Because the 3 z2−r2 electron density has a bigger overlap and thus a stronger Coulomb repulsion with xzand yzthan with xy,21Coulomb interaction will strongly favor a promotion of one xz/yzelec- tron, instead of the naively expected xyelectron, to the 3 z2 −r2level. Indeed, this solution /H20851see Fig. 2/H20849b/H20850and Table I/H20852 turns out to have a much lower total energy than the former/H20851Fig. 2/H20849a/H20850/H20852by about 430 meV per Co 3+, through our LSDA +U+SOC calculations for the antiferromagnetically coupled IS-Co3+/HS-Co2+states. Such a significant multiplet effect, which has been often omitted in ab initio electronic structure calculations, should be taken good care of when studying theintriguing spin state issue of cobaltates. 8 As seen in Table I, the Co3+IS state with one xz/yzhole lies higher in energy than the LS ground state by 298 meVand has a large out-of-plane orbital moment of −1.08 /H9262B/H20849par- allel to the spin moment of −1.52 /H9262B/H20850which consists of −0.8/H9262Bfrom the complex orbital xz−iyz /H20849namely, Y2−1or d−1/H20850and −0.28 /H9262Bfrom the xy+i/H20849x2−y2/H20850/H20849Y2−2ord−2/H20850due to the Coulomb interaction adjusted electron occupation.40 Moreover, the IS Co3+ion having an empty x2−y2orbital and HS Co2+having a singly occupied x2−y2are expected to be ferromagnetically coupled in the abbasal plane, accord- ing to Goodenough-Kanamori-Anderson superexchangerules. This is supported by our result that the ferromagneticIS-Co 3+/HS-Co2+state is lower than the antiferromagnetic state by 176 meV /H20849Table I/H20850, giving a strong in-plane ferro- magnetism which is however in disagreement with the ob-served low T SO/H1101530 K. Furthermore, this lowest-lying Co3+ IS state /H20851Fig.2/H20849c/H20850/H20852out of its multiplet is still higher than the LS ground state by 122 meV , and it has again a huge out-of-plane orbital moment of 1.3 /H9262B/H208510.8/H9262Bfrom xz+iyz/H20849Y21or d1/H20850and 0.5 /H9262Bfrom xy−i/H20849x2−y2/H20850/H20849Y22ord2/H20850/H20852, in contradic- tion with the observed easy in-plane magnetism. This sup-ports an analysis of magnetic anisotropy by Hollmann et al. 21 It might be a bit surprising that the Co3+ion in La1.5Sr0.5CoO 4with a large distortion of the CoO 6octahe- dron does not have the IS as its ground state. We note thatthe large difference of the Co 3+-O bondlengths, 1.888 Å /H110034 vs 2.077 Å /H110032,34does not signal a strong Jahn-Teller /H20849JT/H20850distortion which may stabilize the IS state with a half- filled egorbital, since in the isostructural La 2NiO 4the non-JT ion Ni2+has also very different bondlengths, 1.95 Å /H110034v s 2.22 Å /H110032.41The out-of-plane elongation of the CoO 6and NiO 6octahedra may well be a consequence of the reduction of internal strains in the single-layered perovskites. Althoughthe large distortion of the CoO 6octahedron yields a pro- nounced egsplitting of 0.7 eV as discussed above, it is far less than the required huge egsplitting of about 2 eV via a JTdistortion to stabilize the IS state as ground state.40In this sense, a stabilization of the IS state via the JT effect mayneed an astonishingly large distortion which seems howeverhardly to reach in real materials. Now we turn to a possible Co 3+HS state. We first note that the HS Co3+would have a strong antiferromagnetic cou- pling with the HS Co2+as in the parent compound La 2CoO 4 with a quite high TNof 275 K, in contrast to the low TSO /H1101530 K of La 1.5Sr0.5CoO 4. This already infers that a HS Co3+state is quite unlikely. Indeed, our calculations show that the HS Co3+state is higher in energy than the LS ground state by 311 meV and that it has a big out-of-plane orbital21012 -8 -6 -4 -2 0 2 4 621012 -8 -6 -4 -2 0 2 4 6 21012 -8 -6 -4 -2 0 2 4 6x− y22 3z −r22xyxz,yz x− y22 3z −r22xz+iyzIS Co3+ IS Co3+xz,yzxy Ener gy (eV)Density of States (states/eV) IS Co3+ xz,yzxz,yz xy x− y22 xyxz−iyzxz+iyzxy xz−iyz 3z −r22xy(a) (b) (c) FIG. 2. /H20849Color online /H20850Density of states of the IS Co3+ion in different configurations: /H20849a/H20850t2g↓3/H208493z2−r2/H20850↓1xz↑1yz↑1,/H20849b/H20850t2g↓3/H208493z2 −r2/H20850↓1xy↑1/H20849xz−iyz/H20850↑1, and /H20849c/H20850t2g↑3/H208493z2−r2/H20850↑1xy↓1/H20849xz+iyz/H20850↓1. The corre- sponding multiplet splitting is calculated to be several tenths of eV/H20849see Table I/H20850. Note that the most favorable IS Co 3+state /H20851configu- ration /H20849c/H20850/H20852, being ferromagnetically coupled with the robust HS Co2+/H20849not shown here but refer to Fig. 1/H20850, is still higher in energy than the LS Co3+ground state by 122 meV and has a wrong easy out-of-plane magnetism /H20849see Table Iand main text /H20850.SPIN AND ORBITAL STATES IN La 1.5Sr0.5CoO 4… PHYSICAL REVIEW B 80, 081105 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 081105-3moment of −0.96 /H9262B/H20849parallel to its spin moment of −2.95 /H9262B, see Table I/H20850being again in disagreement with the observed easy in-plane magnetism. To conclude, we confirm that the checkerboard charge order in La 1.5Sr0.5CoO 4consists of the HS Co2+and LS Co3+, through a set of LSDA+ Uband structure calculations in- cluding the spin-orbit coupling and multiplet effect. This so-lution accounts for the optical spectra. The small Co 2+t2g crystal field splitting makes the spin-orbit coupling opera- tive, which produces the observed easy in-plane magnetism.In contrast, either the higher-lying IS or highest HS Co3+ states would yield a wrong easy out-of-plane magnetism. Moreover, the IS /H20849HS/H20850Co3+would have strong ferromag- netic /H20849antiferromagnetic /H20850coupling with the robust HS Co2+, both in disagreement with the low spin-ordering temperatureof La 1.5Sr0.5CoO 4. Finally we note that the multiplet effect of the IS state is significant and should be taken good care of. This work was supported by the Deutsche Forschungsge- meinschaft through SFB 608. 1K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. A. Dilanian, and T. Sasaki, Nature /H20849London /H20850422,5 3 /H208492003 /H20850. 2I. O. Troyanchuk, N. V . Kasper, D. D. Khalyavin, H. Szymczak, R. Szymczak, and M. Baran, Phys. Rev. Lett. 80, 3380 /H208491998 /H20850. 3I. Terasaki, Y . Sasago, and K. Uchinokura, Phys. Rev. B 56, R12685 /H208491997 /H20850. 4J. B. Goodenough, J. Phys. Chem. Solids 6, 287 /H208491958 /H20850. 5M. Zhuang, W. Zhang, and N. Ming, Phys. Rev. B 57, 10705 /H208491998 /H20850. 6S. Noguchi, S. Kawamata, K. Okuda, H. Nojiri, and M. Mo- tokawa, Phys. Rev. B 66, 094404 /H208492002 /H20850. 7Z. Ropka and R. J. Radwanski, Phys. Rev. B 67, 172401 /H208492003 /H20850. 8M. W. Haverkort, Z. Hu, J. C. Cezar, T. Burnus, H. Hartmann, M. Reuther, C. Zobel, T. Lorenz, A. Tanaka, N. B. Brookes,H. H. Hsieh, H.-J. Lin, C. T. Chen, and L. H. Tjeng, Phys. Rev.Lett. 97, 176405 /H208492006 /H20850. 9A. Podlesnyak, S. Streule, J. Mesot, M. Medarde, E. Pom- jakushina, K. Conder, A. Tanaka, M. W. Haverkort, and D. I.Khomskii, Phys. Rev. Lett. 97, 247208 /H208492006 /H20850. 10M. A. Korotin, S. Yu. Ezhov, I. V . Solovyev, V . I. Anisimov, D. I. Khomskii, and G. A. Sawatzky, Phys. Rev. B 54, 5309 /H208491996 /H20850. 11R. F. Klie, J. C. Zheng, Y . Zhu, M. Varela, J. Wu, and C. Leigh- ton, Phys. Rev. Lett. 99, 047203 /H208492007 /H20850. 12S. K. Pandey, A. Kumar, S. Patil, V . R. R. Medicherla, R. S. Singh, K. Maiti, D. Prabhakaran, A. T. Boothroyd, and A. V .Pimpale, Phys. Rev. B 77, 045123 /H208492008 /H20850. 13J. M. Rondinelli and N. A. Spaldin, Phys. Rev. B 79, 054409 /H208492009 /H20850. 14T. Kyômen, Y . Asaka, and M. Itoh, Phys. Rev. B 71, 024418 /H208492005 /H20850. 15K. Knížek, Z. Jirák, J. Hejtmánek, P. Herry, and G. André, J. Appl. Phys. 103, 07B703 /H208492008 /H20850. 16M. Tachibana, T. Yoshida, H. Kawaji, T. Atake, and E. Takayama-Muromachi, Phys. Rev. B 77, 094402 /H208492008 /H20850. 17K. Knížek, Z. Jirák, J. Hejtmánek, P. Novák, and Wei Ku, Phys. Rev. B 79, 014430 /H208492009 /H20850. 18Y . Shimada, S. Miyasaka, R. Kumai, and Y . Tokura, Phys. Rev. B73, 134424 /H208492006 /H20850. 19A. V . Chichev, M. Dlouhá, S. Vratislav, K. Knížek, J. Hejt- mánek, M. Maryško, M. Veverka, Z. Jirák, N. O. Golosova, D. P.Kozlenko, and B. N. Savenko, Phys. Rev. B 74, 134414 /H208492006 /H20850. 20A. T. Savici, I. A. Zaliznyak, G. D. Gu, and R. Erwin, Phys. Rev. B75, 184443 /H208492007 /H20850. 21N. Hollmann, M. W. Haverkort, M. Cwik, M. Benomar, M. Re- uther, A. Tanaka, and T. Lorenz, New J. Phys. 10, 023018 /H208492008 /H20850. 22R. Ang, Y . P. Sun, X. Luo, C. Y . Hao, and W. H. Song, J. Phys.D: Appl. Phys. 41, 045404 /H208492008 /H20850. 23K. Horigane, T. Kobayashi, M. Suzuki, K. Abe, K. Asai, and J. Akimitsu, Phys. Rev. B 78, 144108 /H208492008 /H20850. 24N. Sakiyama, I. A. Zaliznyak, S.-H. Lee, Y . Mitsui, and H. Yoshizawa, Phys. Rev. B 78, 180406 /H20849R/H20850/H208492008 /H20850. 25M. Cwik, M. Benomar, T. Finger, Y . Sidis, D. Senff, M. Reuther, T. Lorenz, and M. Braden, Phys. Rev. Lett. 102, 057201 /H208492009 /H20850. 26C. F. Chang, Z. Hu, H. Wu, T. Burnus, N. Hollmann, M. Benomar, T. Lorenz, A. Tanaka, H.-J. Lin, H. H. Hsieh, C. T.Chen, and L. H. Tjeng, Phys. Rev. Lett. 102, 116401 /H208492009 /H20850. 27K. Yamada, M. Matsuda, Y . Endoh, B. Keimer, R. J. Birgeneau, S. Onodera, J. Mizusaki, T. Matsuura, and G. Shirane, Phys.Rev. B 39, 2336 /H208491989 /H20850. 28Y . Moritomo, K. Higashi, K. Matsuda, and A. Nakamura, Phys. Rev. B 55, R14725 /H208491997 /H20850. 29I. A. Zaliznyak, J. P. Hill, J. M. Tranquada, R. Erwin, and Y . Moritomo, Phys. Rev. Lett. 85, 4353 /H208492000 /H20850. 30I. A. Zaliznyak, J. M. Tranquada, R. Erwin, and Y . Moritomo, Phys. Rev. B 64, 195117 /H208492001 /H20850. 31K. Horigane, H. Hiraka, T. Uchida, K. Yamada, and J. Akimitsu, J. Phys. Soc. Jpn. 76, 114715 /H208492007 /H20850. 32K. Horigane, H. Nakao, Y . Kousaka, T. Murata, Y . Noda, Y . Murakami, and J. Akimitsu, J. Phys. Soc. Jpn. 77, 044601 /H208492008 /H20850. 33J. Wang, Y . C. Tao, Weiyi Zhang, and D. Y . Xing, J. Phys.: Condens. Matter 12, 7425 /H208492000 /H20850. 34M. Cwik, Ph.D. thesis, Universität zu Köln, 2007. The charge ordered structure data: space group Bmmm /H20849No. 65 /H20850; lattice con- stant a=b=5.435 Å and c=12.560 Å; atomic positions La1 /Sr1 4 i/H208490, 0, 0.3611 /H20850, La2 /Sr2 4 j/H208490.5, 0, 0.1379 /H20850,C o3+2a /H208490, 0, 0 /H20850,C o2+2c/H208490.5, 0.5, 0 /H20850,O 1 8 o/H208490.2463, 0.2450, 0 /H20850, O2a4i/H208490, 0, 0.1654 /H20850, and O2 b4j/H208490.5, 0.5, 0.1745 /H20850; bondlengths Co3+-O1=1.888 Å /H110034, Co3+-O2a=2.077 Å /H110032, Co2+-O1=1.955 Å /H110034, Co2+-O2b=2.192 Å /H110032. 35P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, WIEN2K package, http://www.wien2k.at 36V . I. Anisimov, I. V . Solovyev, M. A. Korotin, M. T. Czyzyk, and G. A. Sawatzky, Phys. Rev. B 48, 16929 /H208491993 /H20850. 37W. E. Pickett, S. C. Erwin, and E. C. Ethridge, Phys. Rev. B 58, 1201 /H208491998 /H20850. 38H. Wu, M. W. Haverkort, Z. Hu, D. I. Khomskii, and L. H. Tjeng, Phys. Rev. Lett. 95, 186401 /H208492005 /H20850. 39S. Uchida, H. Eisaki, and S. Tajima, Physica B 186-188 , 975 /H208491993 /H20850. 40M. W. Haverkort, Ph.D. thesis, Universität zu Köln, 2005. 41G. H. Lander, P. J. Brown, J. Spalek, and J. M. Honig, Phys. Rev. B 40, 4463 /H208491989 /H20850.HUA WU AND T. BURNUS PHYSICAL REVIEW B 80, 081105 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 081105-4

PhysRevB.83.060503.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 83, 060503(R) (2011) Macroscopic quantum tunneling in multigap superconducting Josephson junctions: Enhancement of escape rate via quantum fluctuations of the Josephson-Leggett mode Yukihiro Ota,1,3Masahiko Machida,1,3,4and Tomio Koyama2,3 1Center for Computational Science and e-Systems (CCSE), Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno Taito-ku, Tokyo 110-0015, Japan 2Institute for Materials Research, Tohoku University, 2-1-1 Katahira Aoba-ku, Sendai 980-8577, Japan 3Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan 4Japan Science and Technology Agency (JST), Transformative Research-Project on Iron Pnictides (TRIP), 5 Sambancho Chiyoda-ku, Tokyo 102-0075, Japan (Received 18 November 2010; published 9 February 2011) We theoretically study the macroscopic quantum tunneling (MQT) in a hetero Josephson junction formed by a conventional single-gap superconductor and a multigap superconductor such as iron-based superconductorsand MgB 2. In such a Josephson junction more than one phase difference is defined. We clarify their phase dynamics and construct a theory for the MQT in the multigap Josephson junctions. The dynamics ofthe phase differences are strongly affected by the Josephson-Leggett mode, which is the out-of-phase oscillationmode of the phase differences. The escape rate is calculated in terms of the effective action renormalized by theJosephson-Leggett mode at zero-temperature limit. We successfully predict drastic enhancement of the escaperate when the frequency of the Josephson-Leggett mode is less than the Josephson-plasma frequency. DOI: 10.1103/PhysRevB.83.060503 PACS number(s): 74 .50.+r Macroscopic quantum tunneling (MQT)1–4is a counter- intuitive phenomenon in quantum mechanics appearing at amacroscopic level, and has been observed in various fields ofphysics such as condensed matters, nuclei, cosmology, etc.This phenomenon has still attracted great interest in physicscommunities. In particular, the MQT in Josephson junctions,which is observed in a switching event at low temperature, 3,4 has been intensively studied because it is promising for applications to a Josephson phase qubit.5–7 In this Rapid Communication we investigate the physics of MQT in an unexplored type of Josephson junction, which has multiple tunneling channels. Such a Josephson junction can be fabricated by using recently discovered iron-based superconductors8–10or MgB2,11–13because these superconductors are multiband ones having more than one disconnected Fermi surfaces and the superconducting gap can be individually well defined on each Fermi surface. In a Joseph-son junction made of multigap superconductors one may expect that the superconducting tunneling current has multiple channels between the two superconducting electrodes. 14–17We construct a theory for the quantum switching (i.e., MQT) in Josephson junctions with multiple tunneling channels. We are unaware of any theory that has been formulated for MQTs in multigap systems. The theory predicts that the escape rate, i.e., the rate of quantum tunneling, is drastically enhanced compared with that in conventional single-channel systems. In multigap superconductors a collective mode called the Leggett’s mode18–20appears in the low-energy region, which is an out-of-phase oscillation mode of the superconductingphases. In Ref. 14a theory for the Josephson effect in supercon- ducting hetero junctions made of a single-gap superconductorand a two-gap superconductor is formulated. In such Joseph-son junctions, because two kinds of gauge-invariant phasedifferences can be defined, there are two phase oscillationmodes, i.e., the in-phase mode and the out-of-phase one, which correspond, respectively, to the Josephson-plasma and theJosephson-Leggett (JL) mode. In this Rapid Communicationwe construct a theory for the MQT in superconducting hetero-junctions, incorporating the degree of freedom of the JL modeinto the quantum switching event from nonvoltage to voltagestates. It is shown that the zero-point motion of the JL modesignificantly enhances the MQT escape rate when its frequencyis less than the Josephson-plasma frequency. We also pointout that the ratio E J/Einin addition to EJ/ECgoverns the boundary between the classical and quantum regimes, whereE C,EJ, and Einare, respectively, the charging energy, the Josephson coupling energy between the two superconductors,and the interband Josephson coupling energy in the two-gapsuperconductor. Consider a hetero Josephson junction made of a single- gap superconductor and a two-gap superconductor, 14,16,17as shown schematically in Fig. 1. Such a junction has been already fabricated by using multigap superconductors suchas MgB 2(Ref. 21) or iron-based superconductors.22,23In this system one can define two gauge-invariant phase differences,θ (1)andθ(2). Then, the Josephson current density between the two superconducting electrodes is given by the sum ofthe superconducting currents in the two tunneling channelsasj 1sinθ(1)+j2sinθ(2), where jiis the Josephson critical current density in the ith tunneling channel. When a voltage vappears between the two superconducting electrodes, the gauge-invariant phase differences show a temporal evolution,satisfying the generalized Josephson relation 14 α2 α1+α2˙θ(1)+α1 α1+α2˙θ(2)=2e/Lambda1 ¯hv, (1) withαi=/epsilon1μi/dand/Lambda1=1+α1α2/(α1+α2),where /epsilon1is the dielectric constant of the insulator with a thickness dandμi is the charge screening length owing to the electrons in the ith 060503-1 1098-0121/2011/83(6)/060503(4) ©2011 American Physical SocietyRAPID COMMUNICATIONS YUKIHIRO OTA, MASAHIKO MACHIDA, AND TOMIO KOY AMA PHYSICAL REVIEW B 83, 060503(R) (2011) FIG. 1. (Color online) Schematic view of a superconductor- insulator-superconductor (SIS) hetero Josephson junction. We havetwo tunneling channels between the two superconductors with critical current densities j 1andj2as indicated in the right-hand panel. In the upper two-gap electrode the interband Josephson coupling with thecoupling constant J inexists. band. The constant αiis related to the charge compressibility in the two-gap superconducting electrode.16 As shown in Ref. 14, the Lagrangian in the hetero Josephson junction with an in-plane area Wand capacitance C=/epsilon1W/ 4πdis expressed as L=1 2¯h2C (2e)2/parenleftbigg˙θ2 /Lambda1+˙ψ2 α1+α2/parenrightbigg −V+EJIex Icθ, (2a) V=−EJ1cosθ(1)−EJ2cosθ(2)−κE incosψ, (2b) under a bias current Iexin the absence of an external magnetic field, where θandψare the center-of-mass phase difference and the relative phase difference defined as θ=α2 α1+α2θ(1)+α1 α1+α2θ(2),ψ=θ(1)−θ(2). The first two terms in Eq. ( 2b) are the Josephson coupling energies with the coefficients EJi=¯hWj i/2eand the third term represents the interband coupling energy, where the coef-ficient E inis expressed as Ein=¯hW|Jin|/2e.14Because the “interband current” Jincan take both signs, depending on the gap symmetry, we introduce the sign factor κ=Jin/|Jin|. The total critical Josephson current Icand the coefficient EJ in the last term in Eq. ( 2a) are defined as Ic=W|j1+κj2| andEJ=¯hIc/2e. We note that the voltage vappearing in the junction is related to only θ, as seen in Eq. ( 1). From Eq. ( 2a) one can derive the Euler-Lagrange equation for the center-of-mass phase difference θas /Lambda1−1¨θ+ω2 P1sinθ(1)+ω2 P2sinθ(2)=ω2 PIex Ic, (3) with ¯hωPi=√2ECEJiand ¯hωP=√2ECEJ. We note that ωPiis the Josephson-plasma frequency in the ith tunneling channel. From Eq. ( 2a) we also have the Euler-Lagrange equation for ψas ¨ψ+κω2 JLsinψ=−α1ω2 J1sinθ(1)+α2ω2 J2sinθ(2),(4) where ωJLis the frequency of the JL mode14given as ¯ hωJL=√2(α1+α2)ECEin. The above two equations are coupled because θ(1)andθ(2)are functions of θandψ. We note that the bias current is the source for the time evolution of θbut not for ψ, which is consistent with the generalized Josephson relation(1). It should be noted also that we have two characteristic energy scales, the Josephson-plasma frequency ωJiand the JL oneωJL, in this system. Let us now study the macroscopic quantum effects in the Josephson junction with multiple tunneling channels on thebasis of the Lagrangian ( 2a) and evaluate the MQT escape rate. In the following we assume κ> 0, because the case of κ< 0 shows qualitatively no difference. Suppose that the switching to the voltage state is induced by the quantum tunneling of the phase differences θ (1)and θ(2)that are confined inside a potential well. When both θ(1) andθ(2)show the tunneling at the switching, its transition probability is given by the expectation value of the timeevolution operator with respect to the state |θ (1)=0,θ(2)= 0/angbracketright(=|θ=0,ψ=0/angbracketright),3which yields the formula for the MQT escape rate as /Gamma1=2 ¯hβImK({0},{0};β). (5) Here, the symbol {0}means ( θ,ψ)=(0,0), and βis the inverse temperature, β=1/kBT. The propagator K(X,X/prime;β) in Eq. ( 5) is expressed in terms of the imaginary time path integral K(X,X/prime;β)=/integraldisplayX(¯hβ)=X X(0)=X/primeDθDψe−/integraltext¯hβ 0dτ LE/¯h, where X=(θ,ψ) and LEis the Euclidean version of the Lagrangian ( 2a). Let us assume that ψis confined in a small region around ψ=0 at the tunneling, which is justified when the interband coupling is not so strong. In this case one canutilize the expansion with respect to ψ. Then, up to the order of ψ 2, the Euclidean Lagrangian LEis approximated asLE=LE cm+LE rlt+LE int, where LE cm=¯h2 4EC/parenleftbiggdθ dτ/parenrightbigg2 −EJ/parenleftbigg cosθ+Iex Icθ/parenrightbigg , (6a) LE rlt=¯h2 4(α1+α2)EC/parenleftbiggdψ dτ/parenrightbigg2 +1 2Einψ2, (6b) LE int=g+EJψ2cosθ−g−EJψsinθ. (6c) Here,EJ=EJ1+EJ2and/Lambda1≈1 is assumed. The coupling constants g+andg−in Eq. ( 6c) are defined as g+= (EJ1/2EJ)[α1/(α1+α2)]2+(EJ2/2EJ)[α2/(α1+α2)]2and g−=(EJ1/EJ)[α1/(α1+α2)]−(EJ2/EJ)[α2/(α1+α2)]. We note that in the fully quantum case we have the discreteenergy levels as schematically illustrated in Fig. 2. To calculate the escape rate /Gamma1in Eq. ( 5) we employ the mean field approximation for ψ, that is, ψ 2andψin Eq. ( 6c)a r e approximated with their expectation values. Then, at zerotemperature we find /angbracketleftψ/angbracketright th=0 and /angbracketleftψ2/angbracketrightth(T=0)=¯h 2mrltωJL,m rlt=¯h2 2(α1+α2)EC. The finite value of /angbracketleftψ2/angbracketrightoriginates from the zero-point motion of the “quantized” JL mode. Under this approximation we 060503-2RAPID COMMUNICATIONS MACROSCOPIC QUANTUM TUNNELING IN MULTIGAP ... PHYSICAL REVIEW B 83, 060503(R) (2011) FIG. 2. (Color online) Schematic energy diagram for the fully “quantized” system with two quantum variables θandψ. In the case where ψis weakly oscillating within a potential well, the energy levels ofψcoincide with those of a harmonic oscillator with frequency ωJL. The energy levels of θare corrected by the quantum oscillations ofψ. find the effective Lagrangian of single degree of freedom as follows: LE cm,eff=¯h2 4EC/parenleftbiggdθ dτ/parenrightbigg2 +Vcm,eff, (7a) where Vcm,effis the renormalized potential Vcm,eff=−EJ/bracketleftbigg (1−ε) cosθ+Iex Icθ/bracketrightbigg , (7b) ε=g+/angbracketleftψ2/angbracketrightth≈g+√ 2(α1+α2)ωP ωJL/radicalBigg EC EJ. (7c) Then, in this approximation the expectation value K({0},{0};β)i nE q .( 5) is reduced to K({0},{0};β=∞ )≈/integraltextθ(¯hβ=∞ )=0 θ(0)=0Dθexp(−¯h−1/integraltext¯hβ=∞ 0LE cm,effdτ), which can be evaluated in the standard instanton approximation.3Hence, the MQT escape rate corrected by the zero-point motion of ψ is /Gamma1=12ωP(I)/radicalBigg 3V0 2π¯hωP(I)exp/parenleftbigg −36V0 5¯hωP(I)/parenrightbigg , (8) where ωP(I)=ωP[(1−ε)2−I2]1/4,V0=¯h2[ωP(I)]2 cot2θ0/3EC,( 1−ε)s i nθ0=I, andI=Iex/Ic. Figure 3shows a contour map of the ratio /Gamma1//Gamma1 0in the (Iex/IcvsωP/ωJL) plane with /Gamma10being the escape rate without correction, i.e., ε=0. It is seen that the escape rate is FIG. 3. (Color online) Ratio /Gamma1//Gamma1 0in the case of EJ/E C=102. We assume that α1=α2=0.1a n dj1=j2for simplicity. FIG. 4. (Color online) Renormalized potential Vcm,eff. The poten- tial for θis modified by the zero-point fluctuation of the JL mode. drastically enhanced in a wide parameter region. In particular, the enhancement is pronounced in the region of large ωP/ωJL. As seen in Eqs. ( 7a) and ( 7b), the Josephson coupling energy is renormalized by the zero-point motion of ψand the renor- malized one is decreased from the bare one because ε> 0. As a result, the tunneling barrier for θis lowered as schematically shown in Fig. 4, which causes the strong enhancement of the escape rate. In fact, R(ε)≡V0/¯hωP(I) is smaller than R(ε=0) for fixed Iwhen 0 <ε< 1, that is, the exponent in Eq. ( 8) is decreased. Thus, the renormalization increases /Gamma1. Also, it should be noted that the zero-point fluctuation becomeslarger as the frequency of the JL mode decreases. Thus, theconsiderable enhancement of /Gamma1occurs for the system with a lower value of ω JL. The MQT in the conventional systems is subject to the ratio EJ/EC, which is an important parameter for designing a superconducting Josephson qubit.5In the system with multiple tunneling channels the ratio ωP/ωJL(∝EJ/Ein) also affects the characteristics of the MQT. In this Rapid Communication we have focused on the tunneling process, |θ=0,ψ=0/angbracketright→| 0,0/angbracketright, and clarified the effect of the JL mode on the MQT. We mention that sucha process is not the unique one that contributes to the MQTrate in this system, because a system with two degrees offreedom generally has many tunneling routes. For example,the tunneling process in which the quantum switching in theθ (1)channel takes place successively after the switching in the θ(2)channel will be also possible in the present system.24 In this case the escape rate can be calculated from thetransition process |θ (1)=0/angbracketright→| 0/angbracketrightwithθ(2)=f(t), where f(t) is a time-dependent c-number function. This tunneling process is analogous to the MQT under a periodically time-dependent perturbation. 25It is also noted that the relative phase difference ψmight play the role of an environmental variable for θthrough a term that is linear in ψin Eq. ( 6c). The MQT rate in this process can be evaluated by usingthe influential functional integral method. 1,26The competition between the zero-point fluctuation and the “dissipation” occursin this case. The enhancement via the JL mode may besuperior to the reduction from such a dissipation wheng +>|g−|. We also mention that our theory for the MQT in the hetero Josephson junctions can be extended to the case ofintrinsic Josephson junctions (IJJs) with multiple tunnelingchannels. 15The MQT in such systems will be observed in 060503-3RAPID COMMUNICATIONS YUKIHIRO OTA, MASAHIKO MACHIDA, AND TOMIO KOY AMA PHYSICAL REVIEW B 83, 060503(R) (2011) several recently discovered highly anisotropic layered iron- based superconductors.27–29In the IJJs, correction owing to the JL mode for the corporative MQT among the junctions30,31 will be expected. Finally, we remark that the present theoretical prediction relies on the coexistence of the Josephson-plasma and JLmodes. Because the observation of Leggett’s mode in a bulkMgB 2sample has been reported32and in junctions with MgB2,33,34we expect that such a collective mode can be detected in a junction system and the theory will be verifiedexperimentally.In summary, we have constructed a theory of the MQT in hetero Josephson junctions with multiple tunneling channels.We have clarified that the zero-point fluctuation of the relativephase differences brings about a drastic enhancement of theMQT escape rate. The enhancement is large when the JL modehas a lighter mass than that of the Josephson plasma. Y .O. thanks M. Nakahara, S. Kawabata, and Y . Chizaki for illuminating discussions. T.K. was partially supported by aGrant-in-Aid for Scientific Research (C) (No. 22540358) fromthe Japan Society for the Promotion of Science. 1A. O. Caldeira and A. J. Leggett, P h y s .R e v .L e t t . 46, 211 (1981). 2R. 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Kurth, S. Haindl, I. M ¨onch, and B. Holzapfel, Appl. Phys. Lett. 97, 172504 (2010). 24Y . Chizaki and S. Kawabata (private communication). 25M. P. A. Fisher, P h y s .R e v .B 37, 75 (1988). 26S. Kawabata, T. Bauch, and T. Kato, Phys. Rev. B 80, 174513 (2009). 27H. Ogino, Y . Matsumura, Y . Katsura, K. Ushiyama, S. Horii,K. Kishio, and J. Shimoyama, Supercond. Sci. Technol. 22, 075008 (2009). 28H. Nakamura, M. Machida, T. Koyama, and N. Hamada, J. Phys. Soc. Jpn. 78, 123712 (2008). 29H. Kashiwaya, K. Shirai, T. Matsumoto, H. Shibata, H. Kambara, M. Ishikado, H. Eisaki, A. Iyo, S. Shamoto, I. Kurosawa, andS. Kashiwaya, Appl. Phys. Lett. 96, 202504 (2010). 30M. Machida and T. Koyama, Supercond. Sci. Technol. 20,S 2 3 (2007). 31S. Savel’ev, A. O. Sboychakov, A. L. Rakhmanov, and F. Nori,Phys. Rev. B 77, 014509 (2008). 32G. Blumberg, A. Mialitsin, B. S. Dennis, M. V . Klein, N. D. Zhigadlo, and J. Karpinski, P h y s .R e v .L e t t . 99, 227002 (2007). 33Ya. G. Ponomarev, S. A. Kuzmicheva, M. G. Mikheeva, M. V . Sudakovaa, S. N. Tchesnokova, N. Z. Timergaleeva, A. V . Yarigina,E. G. Maksimovb, S. I. Krasnosvobodtsevb, A. V . Varlashkinb,M. A. Heinc, G. M ¨ullerc, H. Pielc, L. G. Sevastyanovad, O. V . Kravchenkod, K. P. Burdinad, and B. M. Bulychevd, Solid State Commun. 129, 85 (2004). 34A. Brinkman, S. H. W. van der Ploeg, A. A. Golubov, H. Rogalla, T. H. Kim, and J. S. Moodera, J. Phys. Chem. Soilds 67, 407 (2006). 060503-4

PhysRevB.98.081104.pdf

PHYSICAL REVIEW B 98, 081104(R) (2018) Rapid Communications Large entropy change derived from orbitally assisted three-centered two-electron σbond formation in metallic Li 0.33VS 2 N. Katayama,1,*S. Tamura,1T. Yamaguchi,2K. Sugimoto,3K. Iida,4T. Matsukawa,5A. Hoshikawa,5 T. Ishigaki,5S. Kobayashi,1Y. O h t a ,2and H. Sawa1 1Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan 2Department of Physics, Chiba University, Chiba 263-8522, Japan 3Center for Frontier Science, Chiba University, Chiba 263-8522, Japan 4Neutron Science and Technology Center, Comprehensive Research Organization for Science and Society (CROSS), Tokai, Ibaraki 319-1106, Japan 5Frontier Research Center for Applied Atomic Sciences, Ibaraki University, Tokai, Ibaraki 319-1106, Japan (Received 2 May 2018; published 7 August 2018) We discuss herein the emergence of a large entropy change in metallic Li 0.33VS 2derived from orbitally assisted looseσbond formation. Comprehensive structural studies based on synchrotron x-ray and neutron diffraction analyses clarify the fabrication of ribbon chains at 375 K, consisting of multiple three-centered two-electron σ bonds based on the viewpoint of local chemical bonding. Although the metallic conductivity persists down to thelowest temperature measured, an exceptionally large entropy change as a metal, as much as /Delta1S=6.6 J mol −1K−1, appears at the transition. The emergence of a large entropy change in a metallic state may possibly lead to differentfunctional materials, such as a heat-storage material with a rapid thermal response. DOI: 10.1103/PhysRevB.98.081104 The search for various self-organizing phenomena in which multiple degrees of freedom of electrons intertwine in solidshas been intensively studied. Examples include stripelikecharge ordering in copper and nickel oxides [ 1], and the spontaneous formation of various molecular clusters in low-dimensional and/or geometrically frustrated systems [ 2–9]. The driving force of the latter example is the formation ofmolecular orbitals accompanied by a periodic modulation ofthe lattice structure, which gives rise to strong metal-metalbonds between the adjacent transition-metal ions. The variety of molecular cluster patterns is strongly related to the number of delectrons. In d 1-electron systems, dimers usually form between adjacent transition-metal ions; MgTi2O4 with a spinel lattice is an example [ 3]. Ford2-electron systems, triangular-shaped trimers appear, as demonstrated by LiVO 2 and LiVS 2, which have two-dimensional triangular lattices [4,10,11]. When a noninteger number of delectrons is involved in the bonds, complicated cluster patterns appear, such asfor the octamer in CuIr 2S4[6]. Although the cluster patterns vary, the mechanism of the cluster formation can be generallyunderstood based on the concept of an orbitally inducedPeierls state, proposed by Khomskii and Mizokawa [ 12]. From the viewpoint of local bonding, the resulting clusters consistof multiple dimers with two-centered two-electron σbonds (2c-2e)[2–9]. For a two-dimensional triangular lattice system with a d-electron number of 4/3, ribbon chains of one-dimensional clusters consisting of multiple linear trimers appear [ 13]. The examples include MTe 2(M=V ,N b ,T a ) ,a sw ew i l l describe in detail later [ 14,15]. Whangbo et al. proposed *katayama@mcr.nuap.nagoya-u.ac.jpthat the ribbon chain formation can be understood based on the concepts of both hidden one-dimensional Fermi-surfacenesting in dorbitals connected to form a one-dimensional chain and local chemical bonding [ 13]. Although the former is basically consistent with the orbitally induced Peierls stateintroduced above [ 12], the resulting local bonds constituting linear trimers can be interpreted as a loose three-centeredtwo-electron σbond (3 c-2e), which sharply differs from the tight 2 c-2ebond [ 2–9]. Considering that various physical properties and functions derived from bonding are centralthemes in condensed matter physics [ 16,17], such as the control of magnetism [ 18], the anomalous metallic state [ 4,9,19], and bond breaking superconductivity [ 7,20], experimental investigations of loose 3 c-2ebonds should extend the possible methodologies, leading to the chances of obtaining differentfunctional materials. In this Rapid Communication, we present the crystal struc- ture and properties of the layered transition-metal dichalco-genide Li 0.33VS2, which, upon cooling, undergoes a trigonal- to-monoclinic phase transition accompanied by ribbon chainformation consisting of multiple 3 c-2ebonds. We show that the entropy of Li 0.33VS2changes by as much as /Delta1S= 6.6Jm o l−1K−1at the transition, although it has high electrical conductivity down to the lowest temperature measured. Wediscuss that the exceptionally large entropy change in a metallicstate is derived from orbitally assisted loose 3 c-2ebond formation, possibly leading to different functional materials,such as heat-storage materials with a rapid thermal response. Powder samples of Li 0.33VS2were prepared using a soft- chemical method followed by a solid-state reaction. Initially,Li-deficient Li ∼0.75VS2was obtained by reacting an appro- priate amount of Li 2S, V , and S in an evacuated quartz tube at 700◦C for 3 days. The products were immersed 2469-9950/2018/98(8)/081104(6) 081104-1 ©2018 American Physical SocietyN. KATAYAMA et al. PHYSICAL REVIEW B 98, 081104(R) (2018) in a 0.2 Mn-BuLi hexane solution for 2 days to obtain LiVS 2. Next, the Li content was quantitatively tuned to obtain Li0.33VS2by using an I 2acetonitrile solution, by using the equation LiVS 2+0.33I 2Li0.33VS2+0.67LiI. The samples were characterized by both neutron and synchrotron powderx-ray diffraction experiments. Neutron diffraction experimentwas done by using ∼10 g of powdered samples in iMATERIA beamline equipped at J-PARC, Japan [ 21]. For analysis, the Z-RIETVELD software [ 22,23] was used. Synchrotron powder x-ray diffraction experiments were done at BL5S2 beamlineat Aichi Synchrotron, Japan. RIETAN -FPsoftware [ 24]w a s employed for the Rietveld analysis. Differential scanningcalorimetry (DSC) was conducted by using DSC 204 F1Phoenix (Netzsch). Magnetic susceptibility was measuredby a superconducting quantum interference device (SQUID)magnetometer (Quantum Design). The electrical resistivitywas measured using the four-probe method. A first-principlescalculation was performed using the WIEN 2Kcode [ 25]. The computational details are available in the SupplementalMaterial [ 26]. Figures 1(a)–1(c) show the temperature dependence of the magnetic susceptibility, electrical resistivity, and DSCsignals, respectively, for Li 0.33VS2. This material is reported to undergo a first-order transition with a decrease in magneticsusceptibility at 375 K [ 27]. The present data show a similar temperature dependence, as shown in Fig. 1(a). The results of the neutron diffraction experiment show no sign of magneticordering at 200 K, as shown in the inset of Fig. 1(a).T h e magnetic susceptibility increases slightly at the lowest temper-ature, which may be attributed to paramagnetic impurities atconcentrations below 1%. To measure electrical resistivity, weused a low-temperature sintered body, because the interlayerLi ions are easily defected when the sample is sintered athigh temperature. Increasing the temperature through the phasetransition at 375 K leads to a jump in the electrical resistivity;the low-temperature phase maintains a metallic conductivityof several m /Omega1cm. Considering that a low-temperature sintered sample is used, the intrinsic electrical resistivity is empiricallyone order of magnitude lower, indicating that Li 0.33VS2is metallic over the entire temperature range. Therefore, bothphases above and below 375 K are Pauli paramagnetic phases,so the jump in magnetic susceptibility at the transition shouldbe attributed to the difference in the electronic density of states.The electrical resistivity measurements also show anomalies at294 K, which appear in all measured samples. However, theorigin of the anomaly is unclear because no anomalies appearat the corresponding temperature in other experiments, such asmagnetic susceptibility, DSC, and structural analysis, maybeindicating the anomaly is derived from the surface effect. As shown in Fig. 1(c), the DSC experiment clarifies that a huge change in entropy of /Delta1S=6.6Jm o l −1K−1is ac- companied by a first-order phase transition at 375 K. Whilethe large entropy change often appears in localized electronsystems, such as in LiVS 2with/Delta1S=6.4Jm o l−1K−1and in LiVO 2with/Delta1S=14.5Jm o l−1K−1[4], the entropy change of /Delta1S=6.6Jm o l−1K−1is extremely large for a metal-to-metal electronic phase transition, indicating the unusual feature ofthe transition in Li 0.33VS2. Assuming that Li 0.33VS2is a metal weakly correlated with a Wilson ratio RW∼1 in the whole temperature range, which gives an estimate of the maximum-202dDSC/d T (10-3) 360 320 280 T (K)3 2 1 0 (mΩcm) 400 300 200 100 0 T (K)300 250 200 150 100 50 0M/H (10-6emu/mol) -0.50.00.5DSC (mW/mg)(a) (b) (c) On cooling 6.64 J/mol K On heating 6.44 J/mol KIntensity (a.u.)3.5 3.0 2.5 2.0 1.5 1.0 d / Angstrom200 K Rwp = 2.99% Rp = 2.12% FIG. 1. (a) Magnetic susceptibility of Li 0.33VS 2. Magnetic sus- ceptibility data reported by Murphy et al. are taken from Ref. [ 27]. The inset shows the Rietveld refinement of neutron diffractiondata at 200 K. Although a small impurity peak appears (see the asterisk), magnetic Bragg peaks are absent. (b) Electrical resistivity of Li 0.33VS 2. Because a large change in volume occurs at the transition at 375 K, cracks are generated in the sintered body, and the electrical resistivity as a function of temperature upon cooling does not match that upon heating. Only the cooling process is shown in (b). (c) DSC ofLi 0.33VS 2. The inset shows the differentiated DSC data. No anomaly appears around 294 K. entropy change as opposed to assuming other values for RW, we can roughly estimate the entropy change to be /Delta1S= /Delta1γTc∼1.5J m o l−1K−1. It is extremely small compared with the experimental value of /Delta1S=6.6Jm o l−1K−1, which seems to indicate that the transition involves ordering of thedegrees of freedom, such as orbital. To clarify the ground state of the low-temperature phase based on the structural analysis, we made comprehensiveneutron and synchrotron x-ray diffraction studies. The neutrondiffraction experiment indicates a Li fraction of 0.3388(13),which is very close to 1/3. The neutron diffraction experimentalso clarifies that magnetic Bragg peaks do not emerge atthe 375-K transition, as discussed before. An analysis ofthe synchrotron x-ray diffraction data gives the structuralparameters at each temperature. As shown in Fig. 2(a),t h e high-temperature phase is refined as P¯3m1w i t har e g u l a rV triangular lattice. Although ∼4.5% of the Li 0.5VS2impurity is identified, it is successfully corefined with Li 0.33VS2.A ss h o w n in the inset of Fig. 2(a), the spectrum drastically changes below the transition temperature, and the low-temperature phase is 081104-2LARGE ENTROPY CHANGE DERIVED FROM ORBITALLY … PHYSICAL REVIEW B 98, 081104(R) (2018) 500 400 300 200 100 0Intensity (a.u.) 60 50 40 30 20 10 0 2θ / degrees370K λ = 0.65029 ÅRwp = 5.037% Rp= 4.316% S = 1.5957Li0.33VS2 C2/m a = 13.21413(14) b = 3.26583(3) c = 8.43693(9) β = 112.2998(8) Intensity (a.u.)20 18 16 14 2θ / degrees500 400 300 200 100 0Intensity (a.u.) 60 50 40 30 20 10 0 2θ / degrees400 K λ = 0.65029 Å Li0.33VS2 P3m1 a,b = 3.27068(3) c = 6.15062(7) Rwp = 4.976% Rp = 4.183% S = 1.5462 Intensity (a.u.) 1918171615 2θ / degrees(a) (b) FIG. 2. Synchrotron x-ray diffraction data at (a) 400 K and (b) 370 K. The inset of (a) shows the spectrum around 375 K. The insetof (b) shows the spectrum around 294 K. No significant change appears corresponding to the anomaly observed in the resistivity measurements. refined by using the monoclinic space group C2/m,a ss h o w n in Fig. 2(b). Table SI in the Supplemental Material summarizes the structural parameters [ 26]. Figures 3(a) and 3(b) show the low-temperature crystal structure obtained from the Rietveld analysis. At 375 K, Liordering appears to form a one-dimensional chain in the FIG. 3. (a), (b) Refined crystal structure of Li 0.33VS 2.AL,BL,a n d CLindicate the V1-V1, V1-V2, and interchain V2-V2 bond lengths, respectively. Ribbon chains are clearly apparent in (b). (c) V-V bond length as a function of temperature.direction of the baxis. In the low-temperature phase, the V site splits into two sites, V1 and V2 in a ratio of 1 : 2. Asshown in Fig. 3(b), the V1 sites form a one-dimensional chain in the direction of the baxis, whereas V2 approaches the one-dimensional V1 chain. The result is that the ribbon chainin the low-temperature phase is clearly apparent. Figure 3(c) shows the temperature dependence of V-V distances. Thedistance A Lbetween adjacent V1 sites remains essentially constant across the transition, whereas the V1-V2 distanceB Lwithin the ribbon chain decreases to 3.075 ˚A at 360 K, which is about 8% shorter than the V-V distance AHin the high-temperature phase. Relatively, the V2-V2 distance CLis significantly increased. The mechanism at work in the phase transition may be understood by invoking Whangbo’s theory [ 13]. In the high- temperature phase, the xy,yz, andzxbands constructed by the corresponding dorbitals are 4/9 filled. The yzandzxone- dimensional chains undergo a Peierls transition accompaniedby a trigonal-to-monoclinic transition, which also broadenstheyzandzxbands. As a result, the filling of yzandzx bands increases up to 2/3, forming linear trimers consistingof 3c-2ebonds. In contrast, the xyband becomes almost empty. The change in band filling, aided by Li-ion ordering,may be responsible for a large change in entropy of /Delta1S= 6.6Jm o l −1K−1. Although the contribution of the Li-ion order to the change in entropy is not clear, we consider it to benegligibly small based on previous studies of the surroundingmaterials [ 28,29]. A characteristic Fermi surface to bolster this argument is realized by a first-principles calculation. As shown inFig. 4(a), the Fermi surface at high temperature facilitates a good nesting feature with Q=a ∗/3, which corresponds to the lattice periodicity in the low-temperature phase. AlthoughLi ions one-dimensionally aligned parallel to the ribbon chainlead us to suspect that Li-ion ordering is a possible drivingforce behind the phase transition at 375 K, this is not thecase. To support this claim, we assume a 3 ×1×1 supercell, and structurally optimized the system using the virtual crystalapproximation (VCA) scheme, whereby Li ions are uniformlyaligned. The result indicates that the ribbon chain structureactually reduces the ground-state energy and yields bondlengths near the experimental lengths A L=3.28,BL=3.06, andCL=3.73. This means that the instability towards the ribbon chain structure is inherent in the electronic state of thetwo-dimensional VS 2plane and the observed Li-ion ordering is merely a consequence of ribbon chain formation. We should note that there are some tellurides, MTe2(M= V, Nb, Ta), which exhibit structural phase transitions from aregular triangular lattice with trigonal symmetry to monoclinicwith the formation of ribbon chains upon cooling [ 13–15]. It is discussed that the Te p-block bands increase in energy because of the interlayer Te-Te interactions, thereby leading to a partialelectron transfer from the top portion of the Te p-block bands to the d-block bands of the metal, resulting in the actual d 4/3 electron count and following 3 c-2ebond formation. However, the present Li 0.33VS2should be more appropriate than these tellurides for investigating the intrinsic nature derived fromloose 3 c-2eformations for some reasons. Initially, the formal delectron count for vanadium can be finely tuned to 4/3 in Li 0.33VS2. The diffraction study clearly shows that the 081104-3N. KATAYAMA et al. PHYSICAL REVIEW B 98, 081104(R) (2018) FIG. 4. (a) Fermi surface in the high-temperature phase of Li0.33VS 2. Partial density of states in (b) high- and (c) low-temperature regions. (d) Three molecular orbitals derived accompanied by the 3c-2eformation. Schematic view of band structures originating from dxy,dyz,a n ddzxorbitals at (e) high and (f) low temperatures. interlayer S-S distance of 3.788 ˚A at 400 K is significantly longer than the sum of the van der Waals radii of the S2− ion [ 30], which means that the interlayer p-phybridization is negligibly small compared with these tellurides. In turn,this indicates that the delectronic state can be finely tuned by controlling the Li fraction. The neutron diffraction resultindicates that the Li fraction is quite close to 1/3, whichensures the presence of a d 4/3electronic state in Li 0.33VS2. Second, we can expect weak p-dhybridization in Li 0.33VS2, which is compared with the strong p-dhybridization inherent to 1Ttellurides [ 31]. The previous band calculations clearly indicate the considerable contribution of a Te 5 pcharacter at the Fermi level [ 32], which prevents us from studying the intrinsic nature of the 3 c-2estate derived from a strong d character. In Li 0.33VS2, the first-principles calculation clearly indicates a negligibly small p-dhybridization. As shown in Figs. 4(b)and4(c),t h eS3 pband is completely separated from the V 3 dband in both the high- and low-temperature phases, and the strong V 3 dcharacter dominates near the Fermi energy. The calculation further clarifies the modification of the band structure derived from the 3 c-2eformation. As shown schematically in Fig. 4(e),t h exy,yz, andzxbands are triply degenerated at high temperatures. Thus, we expect the yz andzxbands to transform into bands composed of three components at a low temperature: bonding, nonbonding, andantibonding, as shown in Fig. 4(d). We thus divide the yzand zxbands into three parts, as shown in Fig. 4(f). Whereas the bonding orbitals are almost filled, the Fermi surface survivesdue to the small stabilization energy of the bonding orbitalfollowing the incomplete gap opening between the bonding and nonbonding orbitals, and the overlap between the yzand zxbands, with the xyband further generating a small charge transfer between them. The existence of the Fermi surfaceis consistent with the metallic conductivity observed in theresistivity measurement. The metallic conductivity at low temperatures differs sharply from the insulating behavior observed in conventionalmolecular cluster compounds with 2 c-2ebonds [ 2–9]. The reason is strongly related to the bonding nature. For example, LiVS 2, which is the d2analog of Li 0.33VS2, undergoes a metal-to-insulator transition at 314 K, accompanied by theformation of V triangular-shaped trimers consisting of multiple2c-2ebonds [ 4]. The change in V-V distance reaches ∼12% at the transition in LiVS 2, whereas the V-V distance is 3.39 ˚Aa t high temperatures and the intratrimer V-V distance decreases to 2.997 ˚A at low temperatures [ 4]. The ∼12% change in the V-V distance is much greater than the ∼8% change in Li0.33VS2, which indicates that the 2 c-2ebond is much stronger than the 3 c-2ebond, as can be understood intuitively. Because the bond strength is a function of the band gap, we consider that the weakness of the 3 c-2ebond may be explained by the incomplete gap, leading to the metallic conductivity inthe low-temperature phase of Li 0.33VS2. Conversely, however, a large change in entropy of /Delta1S=6.6J m o l−1K−1at the transition in Li 0.33VS2seldom appears in conventional itinerant electron systems, such as 2 H-TaSe 2and 4Hb-TaSe 2[33], which frequently appear in compounds in which molecular clusters form, accompanied by orbital ordering [ 4,34–36]. The coexistence of high electrical conductivity and a large changein entropy is a remarkable feature of Li 0.33VS2with 3 c-2e bonds, which sharply differs from that for 2 c-2ebonds, leading to different functional materials. One possible application is as a heat-storage material with a rapid thermal response. Many conventional heat-storage mate- rials exploit a large change in enthalpy in the solid-liquid phase transition, such as paraffin with 140 J cc−1at a melting point m.p.=64◦C[37] and polyethylene glycol with 165 J cc−1 at m.p. =20◦C[38]. Heat-storage materials that exploit a solid-liquid phase transition have a large enthalpy, whereas a low thermal conductivity leads to a difference betweenthe internal temperature and the surface temperature, makingit difficult to keep the surface temperature constant. Thus, fabricating heat-storage materials that exploit a solid-solid phase transition is strongly desired, and attention has focusedon heat-storage materials that exploit the enthalpy change that is accompanied by an electronic phase transition in strongly correlated electron systems [ 39–42]. Our estimate gives an enthalpy change /Delta1H=73.1J c c −1for Li 0.33VS2, which is derived from /Delta1S=6.6Jm o l−1K−1. This is a relatively large value for a heat-storage material that exploits a solid-solid transition, as summarized in Table SII in the SupplementalMaterial [ 26]. Note that the high electrical conductivity, which is a feature of Li 0.33VS2, improves the thermal conductivity and leads to a rapid thermal response. This characteristic is unique to Li 0.33VS2and is not found in conventional heat-storage materials that use the solid-solid phase transition. Thus, the3c-2ebond is an interesting topic for both basic and applied sciences. To summarize, we present physical and structural studies of the layered two-dimensional triangular lattice 081104-4LARGE ENTROPY CHANGE DERIVED FROM ORBITALLY … PHYSICAL REVIEW B 98, 081104(R) (2018) system Li 0.33VS2, and discuss how Li 0.33VS2embodies the 3c-2ebond. Compared with a conventional 2 c-2ebond, the 3c-2ebond is loose, which leads to an incomplete band gap and concomitant metallic conductivity at lowtemperatures. In contrast, the entropy change at the tran-sition is exceptionally large as a metal. The coexistenceof these features may make Li 0.33VS2a useful functional material. The authors are grateful to Y . Okamoto and N. Mitsuishi for valuable discussions. This work was partly supported by aGrant in Aid for Scientific Research (No. JP17K05530 andNo. JP17K17793), Fuji Science and Technology Foundation,The Thermal and Electric Energy Technology Inc. Foundation, and Daiko Foundation. This work was carried out under theVisiting Researcher’s Program of the Institute for Solid StatePhysics, the University of Tokyo. The synchrotron powderx-ray diffraction experiments were conducted at the BL5S2 ofAichi Synchrotron Radiation Center, Aichi Science and Tech-nology Foundation, Aichi, Japan (Proposals No. 201702049,No. 201702101, No. 201703027, No. 201704027, and No.201704099). 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PhysRevB.100.104440.pdf

PHYSICAL REVIEW B 100, 104440 (2019) Suppression of the orbital magnetic moment driven by electronic correlations in Sr 4Ru3O10 Filomena Forte ,1,2,*Lucia Capogna,3Veronica Granata,1,2Rosalba Fittipaldi,1,2Antonio Vecchione,1,2and Mario Cuoco1,2 1Consiglio Nazionale delle Ricerche, SPIN, Via G. Paolo II 132, I-84084 Fisciano, Italy 2Dipartimento di Fisica “E.R. Caianiello,” Università degli Studi di Salerno, Via G. Paolo II 132, I-84084 Fisciano, Italy 3Consiglio Nazionale delle Ricerche, IOM OGG, and Institut Laue Langevin, 71 avenue des Martyrs, F-38042 Grenoble, France (Received 30 January 2019; revised manuscript received 31 July 2019; published 30 September 2019) The coupling of spin and orbital degrees of freedom in the trilayer Sr 4Ru3O10sets a long-standing puzzle due to the peculiar anisotropic coexistence of out-of-plane ferromagnetism and in-plane metamagnetism. Recently,the induced magnetic structure by in-plane applied fields was investigated by means of spin-polarized neutrondiffraction, which allowed the extraction of a substantial orbital component of the magnetic densities at Rusites. It has been argued that the latter is at the origin of the evident layer-dependent magnetic anisotropy,where the inner layers carry larger magnetic momenta than the outer ones. We present a spin-polarized neutrondiffraction study in order to characterize the nature of the ferromagnetic state of Sr 4Ru3O10in the presence of a magnetic field applied along the caxis. The components of the magnetic densities at the Ru sites reveal a vanishing contribution of the orbital magnetic moment which is unexpected for a material system where orbitaland spin degeneracies are lifted by spin-orbit coupling and ferromagnetism. We employ a model that includes theCoulomb interaction and spin-orbit coupling at the Ru site to address the origin of the suppression of the orbitalmagnetic moment. The emerging scenario is that of nonlocal orbital degrees of freedom playing a significant rolein the ferromagnetic phase, with a Coulomb interaction that is crucial to making an antialigned orbital momentat short distance, resulting in a ground state with vanishing local orbital moments. DOI: 10.1103/PhysRevB.100.104440 I. INTRODUCTION Ruthenium oxide perovskites of the Ruddlesden-Popper (RP) family An+1RunO3n+1(n=1,2,3) are quite unique materials in the realm of transition-metal oxides, changingdrastically their electronic and magnetic properties as a func-tion of the number nof RuO 2layers in the unit cell [ 1]. A wide variety of collective phenomena has been observed, in-cluding unconventional superconductivity ( n=1) [2], heavy d-electron masses ( n=2,3) [3], colossal magnetoresistance effects ( n=2) [4], and itinerant ferromagnetism and metam- agnetism ( n=3) [5,6], as well as anomalous ferromagnetism (n=∞ )[7]. In those compounds, the extended nature of 4 d orbitals of the ruthenium ions leads to comparable energiesfor competing interactions, i.e., crystal field, Hund, spin-orbit, and electron-lattice couplings and p-dhybridization. Moreover, it renders the physical properties highly dependenton the dimensionality nand susceptible to perturbations such as applied magnetic fields and pressure, without the need forchemical doping [ 8–11]. Recently, the key role played by the orbital physics as it concerns the electronic and magnetic properties of layeredruthenates has been invoked for several Ca- and Sr-based RPcompounds. In such systems, the orbital degree of freedomis typically active and has a complex interplay with charge,spin, and lattice degrees of freedom, which turns out to bequite relevant in setting the unconventional superconductiv-ity in Sr 2RuO 4[2,12], band-dependent Mott metal-insulator *Corresponding author: filomena.forte@spin.cnr.ittransition [ 13], orbital ordering in Ca 2RuO 4[14,15], and metamagnetism and correlated effects in Sr 3Ru2O7[16,17]. Situated between n=2 and n=∞ ,S r 4Ru3O10is the n=3 member of the Sr-based RP series with triple layers of corner-sharing RuO 6octahedra separated by SrO rocksalt double layers [ 18]. It displays complex phenomena rang- ing from tunneling magnetoresistance and low-frequencyquantum oscillations to nonstandard switching behavior. Themost intriguing feature, however, is a borderline magnetism:while along the caxis (perpendicular to the Ru-O layers) Sr 4Ru3O10shows ferromagnetism with a saturation moment of 1.13μB/Ru and a Curie temperature Tcat 105 K, for the field in the abplane it exhibits a sharp peak in the magnetization at T∗=50 K and a first-order metamagnetic transition. The coexistence of the interlayer ferromagnetismand the intralayer metamagnetism, i.e., the anisotropy inthe field response, is not typically encountered in magneticmaterials, and it may arise from a peculiar electronic state with two-dimensional Van Hove singularity close to the Fermi level in conjunction with a distinct coupling of the spins to theorbital states and lattice [ 19–21]. Another important physical aspect emerging in the Sr 4Ru3O10metamagnetism is provided by the magnetoelastic coupling [ 22–24]. In particular, direct evidence of a strong spin-lattice interaction was obtained bymeans of neutron scattering, demonstrating that significant structural changes occur concomitantly with the metamag- netic transition [ 22]. Recently, it was proposed that a layer-dependent magnetic state may be allowed due to the interplay between octahedraldistortions and spin-orbit and Coulomb interactions [ 25]. In that experiment, a polarized neutron scattering study was 2469-9950/2019/100(10)/104440(8) 104440-1 ©2019 American Physical SocietyFILOMENA FORTE et al. PHYSICAL REVIEW B 100, 104440 (2019) performed in order to analyze the spin and orbital spatial components of the induced magnetization density M(r) with a magnetic field applied in the abplane and in the metam- agnetic regime ( B>2 T). It was found that there exists a distinct relation between spin and orbital moments and theiramplitudes in the unit cell since they are strongly linked to thelayers where the electrons are located. Specifically, the innerruthenium ions in the triple layer have larger spin and orbitalmagnetic moments than the outer ones. Remarkably, the inner-outer correspondence is robust with respect to temperaturevariations since it persists even above T c, thus indicating that these features are intrinsic to the high-field magnetic state. We present here the outcome of a polarized neutron scatter- ing study on a high-quality single crystal of Sr 4Ru3O10in the ferromagnetic (FM) regime, i.e., with a strong magnetic field applied along the easy axis ( caxis). This study is motivated by the need to clarify the nature of the FM phase, in particular therole of the orbital magnetic moment in setting the magneticproperties of this compound within the inequivalent layersof the unit cell. The refinement of our neutron scatteringdata reveals a vanishing contribution of the orbital componentto the magnetic density at Ru sites. Supported by modelingbased on an electron-correlated description that includes thecoupling between all the relevant spin-orbital degrees of free-dom at inequivalent Ru sites, we interpret this evidence asthe effect driven by the spin-orbital exchange energy gain inthe FM state. In particular, we show that robust short-rangeantiferro-orbital correlations, developing within the ( d zx,dyz) doublet, are responsible for the vanishing of the local orbital moment in the direction of the applied field. The proposed physical mechanism arises from the balance between elec-tronic correlations and kinetic energy in the presence of spin-orbit coupling and the crystal field (CF) splitting set by thetetragonal distortions, which is a common thread in Ru 4 d- electron-based oxides [ 14]. Remarkably, our analysis demon- strates a tight connection between the character of the orbitalangular momentum and the anisotropic magnetic responseof Sr 4Ru3O10: while the orbital component of the magnetic moment is suppressed in the case where a longitudinal field isapplied, it is substantial and at the origin of the inequivalentintralayer and interlayer magnetic response in the case of anin-plane field [ 25]. This paper is organized as follows. In Sec. II, we introduce the experimental setup and the polarized beam approach. In Sec. III, we present the experimental results, while Sec. IV is devoted to their interpretation and the description of atheoretical model which is able to explain the quenching of theorbital angular momentum in the high-field magnetic state. InSec. Vwe provide the concluding remarks. II. EXPERIMENTAL METHODOLOGY Single crystals of Sr 4Ru3O10were grown in an image furnace as described elsewhere [ 26]. The samples were cut into small rectangular slices with an average size of 4 × 4×0.2m m3. Similar samples were used in our previous studies [ 22,25]. X-ray diffraction, energy and wavelength dis- persive spectroscopy, and neutron Laue diffraction were usedto fully characterize the structure, quality, and purity of thecrystals. Magnetizations measurements on crystals from thesame batch identified the FM transition at T c∼=105 K and a metamagnetic transition at the temperature T∗∼=50 K, when a magnetic field is applied in the abplane. The metamagnetic transition appears for magnetic field up to 2 T [ 27]. The experiments were carried out at the D9 and D3 diffractometers at the Institute Laue-Langevin in Grenoble.Preliminary neutron diffraction measurements performed atD9 allowed us to determine the structural parameters and theextinction coefficients at T=115 K to use in the following re- finement of the magnetic structure. Polarized beam measure-ments were then performed with the D3 diffractometer. Sucha technique is a well-established probe of the magnetizationdensity via the measurement of the flipping ratio R, which is defined as the ratio of the cross sections with neutrons paralleland antiparallel to the applied magnetic field. In this exper-iment, a magnetic field of 9 T was applied on cooling alongthe [00 l] direction (i.e., the caxis), which is the magnetic easy direction for the crystal. Under these conditions, one could beconfident that the magnetic moments were completely alignedin the vertical axis, so that the following simplified expressionfor the flipping ratio is valid [ 28]: R=|F N(K)−(γr0/2μB)M(K)|2 |FN(K)+(γr0/2μB)M(K)|2, (1) where γr0=5.36×10−15m,Kis the scattering vector, FN(K) is the nuclear structure factor, and M(K) is the recipro- cal space magnetization density. Since the number structurefactors F N(K) are known from the crystal structure, from Eq. ( 1) one can directly get the amplitude of M(K). The real-space magnetization density M(r) is then extracted by doing the Fourier transform of M(K). A wavelength of about 0.825 Å was used to measure the scattering in the [ h00/0k0] plane with a tilting option for the detector in order to acquire the intensity of many ( hkl) reflections with a nonvanishing lindex. Since the crystal cell is quite extended along z,c∼28 Å, values of lup to 13 could be reached. A radio-frequency coil inserted between the monochro- mator and the sample was then used to flip the spin stateof the incident neutrons so that the intensity of about 65independent reflections could be measured at three differenttemperatures: 2, 50, and 120 K in both the up- and down-spinpolarizations, with a degree of polarization of 0.94 for eachspin channel. A 0.5-mm erbium filter allowed us to reducehigher-order contamination in the incident beam. A Heusslermonochromator was also used. III. SPIN-POLARIZED NEUTRON DIFFRACTION RESULTS In this section we present the outcome of the polarized neu- tron diffraction measurements and the resulting magnetizationprofile resolved within the layers of the unit cell. We followthe standard procedure which was already presented in aprevious study [ 25]. The magnetic moments were refined from the flipping ratios Rusing the program FULLPROF [29], assum- ing a spherical approximation for the electron density [ 30]. The magnetic form factor was expressed in terms of sphericalBessel functions [ 29], where only the first two terms were 104440-2SUPPRESSION OF THE ORBITAL MAGNETIC MOMENT … PHYSICAL REVIEW B 100, 104440 (2019) TABLE I. Total and orbital contributions to the magnetic densities at the ruthenium and oxygen sites at the three relevant temperatures of the system, as refined with the program FULLPROF . B/bardblc 2 K 50 K 115 K M(units of μB) Mtot Morb Mtot Morb Mtot Morb Ruin 1.75(7) 0.1(2) 1.5(7) 0.1(2) 0.95(7) 0.1(2) Ruout 1.10(6) −0.1(1) 1.0(1) −0.1(1) 0.61(6) −0.1(1) Obas 0.15(1) 0.13(1) 0.07(1) Oap 0.05(2) 0.04(2) 0.03(1) χ21.74 2.60 1.81 retained, the first corresponding to the total magnetic moment and the second being related to the orbital component, whichis expected in light of the moderate spin-orbit coupling inruthenates. Due to the large unit cell, the refinement procedurewas performed by reducing the number of inequivalent Ruions, which are sorted in two distinct classes: the rutheniumatoms located in the central layer, which are defined as Ru in (inner), and those in the outer layers, which are indicated asRu out(outer). It is worth pointing out that the neutron diffraction data acquired on D9 are compatible with two different nuclearstructures, the primitive Pbam space group which was al- ready adopted in the literature [ 31] and the face-centered structure, Acam . An extended analysis of the data was per- formed in order to further investigate the crystal symmetryof the sample and also to understand whether the resultingmagnetic outcome might be sensitive to the selected spacegroup. However, no significant differences could be obtainedin the least-squares refinement of the two models. Moreover,both crystal structures lead to the same results in terms ofmagnetization densities. Then the Pbam space group was adopted, in agreement with our previous paper and otheravailable data in the literature [ 25]. In Table I, the values of magnetic densities at the ruthenium sites located in the inner and outer octahedra of the triple-layerunit cell are reported, together with those of the oxygens, forthe three temperature values of 2, 50, and 120 K. Severalimportant observations can be extracted from the results inTable I. First of all, we observe that the orbital component of the magnetic moment appears to be vanishing in the presentfield configuration. Indeed, its value is undetectable withinthe experimental error for all temperatures which have beenconsidered. This result points out a major difference withrespect to what was previously found in the case of an in-planeapplied field, where a substantial orbital contribution to themagnetic densities was detected at Ru sites [ 25]. A discussion of the microscopic mechanism leading to this anisotropicorbital quenching in the high-field magnetic states will bepresented in the next section. Instead, the contribution of theoxygens in this kind of system is not negligible, as previouslyreported [ 25]. Another important observation deals with the layer depen- dence of the magnetic density, which is graphically depictedin the maps shown in Fig. 1. Those magnetization density maps were calculated in a direct way with the maximumentropy method [ 28] implemented in the program DYSNOMIA[32], which is now available in the FULLPROF suite. The most likely spin distribution is the one that maximizes theentropy among those which are compatible with the observedmagnetic structure factors. The latter were refined with theprogram FULLPROF . Our results show that the inner Ru ions carry a larger spin moment with respect to the outer Ru ions and that this orderrelation is robust against temperature variation. In Fig. 1,t h e present results ( B/bardblc) are compared to what was previously reported for B/bardblab[25]. The most noticeable effect of the field polarization along the caxis is to invert the trend of the magnetization density on the outer ruthenium ions whenthe temperature is swept across the metamagnetic transitionat around 50 K. Indeed, with the field in the abplane, the outer ruthenium ions are more intensely magnetized at 2 Kthan above T ∗. FIG. 1. Sections of the magnetization density in the acplane of Sr4Ru3O10calculated directly with the maximum entropy method. The two maps in the top panel are for a sample polarization in the ab plane, while the bottom ones correspond to a field applied along thecaxis. In the color scale, magnetic density ranges from white (zero magnitude) to red (maximal value at Ru inforB/bardblc). For quantitative estimates, refer to the values in Table I. 104440-3FILOMENA FORTE et al. PHYSICAL REVIEW B 100, 104440 (2019) The inequivalence of the inner and outer ruthenium sites also has crystallographic origins since the inner ruthenium sitsin a higher-symmetry site having a Wyckoff position with amultiplicity of 2, while the outer ruthenium has a multiplicityof 4 [ 31]. In zero field, the outer octahedra are slightly elon- gated compared to the inner ones, which are regular octahedra[31]. In addition, the inner octahedra are more rotated than the outer ones; in fact, they have an average rotation of 10 .6 ◦, while the outer ones 5 .25◦, with the former being larger than the critical angles, range from 6 .5◦to 9◦, which has been theoretically evaluated to be sufficient to stabilize theferromagnetism [ 33]. As a consequence, one can argue that the Ru inare supposed to be more prone to ferromagnetism than the Ru out. This is reflected in the value of FM moment along the caxis on the two sites, which has been measured with neutron scattering at low temperature (2 K), yielding1.59μ Bon the inner ruthenium ions and 0 .92μBon the outer ones [ 22]. In the next section, we focus on the role of the orbital component of the magnetic moment and discuss amechanism which leads to anisotropic suppression, whichdepends on the orientation of the applied field. IV . MODELING THE FM STATE WITH A VANISHING ORBITAL MAGNETIC MOMENT In this section, we propose a physical scenario which is able to account for the occurrence of a vanishing orbitalmagnetic moment in the high-field FM phase of Sr 4Ru3O10. The analysis is performed by focusing on the orbital characterof the spin-polarized ground state described by an effectivemicroscopic model that we solve on a cluster in order toinclude on equal footing all the interacting electronic degreesof freedom for the dstates at the Ru site. The physical context is set by our polarized neutron diffraction study, which revealsa layer-independent quenching of the orbital component ofthe magnetic density, thus manifesting both at the Ru ionsbelonging to the central plane and the outer RuO plane of theunit cell, assuming that an external field is applied along the c axis. This result has a completely different outcome and trendcompared to the experimental evidence of a substantial orbitalcomponent in the magnetic density at the Ru sites, when afield of equal strength is applied along the abplane [ 25]. Our aim is to provide a microscopic scenario to account for therealization of a spin-polarized phase with almost zero orbitalmoment. On general grounds, the magnetic moment carried by electrons in solid-state materials has two components: the onearising from its spin and the one originating from its orbitalcharacter. The local spin magnetic moment typically emergesas a consequence of the Coulomb interaction, and in particu-lar, Hund’s coupling is a key player at work in the majority ofmagnetic solids. On the other hand, concerning the formationof the orbital magnetic moment, it is ascribed to the spin-orbitcoupling, which lifts the quenching of the orbital moment in amagnetic solid. Hence, in a FM configuration, where the spindegeneracy is lifted, the spin-orbit coupling (SOC), H soc= λL·S, is expected to yield an orbital moment Lantiparallel to the spin moment S, with λbeing the strength of the SOC. We would then have a local orbital moment at each Ru site alongthecaxis in the spin-orbit coupled FM state of Sr 4Ru3O10.Hence, another mechanism has to be invoked to account for its suppression. Starting from this picture, we demonstratethat the formation of distinct nonlocal orbital correlationscould be the driving mechanism leading to the suppressionof the orbital moment. More specifically, a correlated FMstate with approximately isotropic short-range antiferrotypeorbital configurations, i.e., orbital moments aligning in sucha way that they are antiparallel on neighboring sites, canyield a quantum configuration where the average on-siteorbital moment is suppressed. In order to proceed further,we introduce a minimal model which is able to capturethe competition between the local and nonlocal microscopicmechanisms governing the formation of the magnetic orbitalmoment. Due to the correlated nature of Ru-based oxides weadopt a standard model Hamiltonian for the itinerant electronsof the t 2gRu bands close to the Fermi level, which includes all the relevant interaction terms at the Ru sites, and thekinetic part for the Ru-Ru connectivity, in the presence ofan applied magnetic field. The local Hamiltonian consistsof the complete Coulomb interaction for the t 2gelectrons (i.e., density-density coupling Uand Hund interaction JH), the spin-orbit coupling λ, and the tetragonal CF potential /Delta1 mimicking the static electron-lattice coupling [ 34]. We face the problem by performing an exact diagonalization study ofan effective cluster of two Ru sites. This approach allows usto solve the full quantum Hamiltonian in an unbiased way,so that we can capture the effects of the short-range spin-charge-orbital couplings, which are set by the competitionof local interactions and the kinetic energy set by the d-d hopping amplitude t[35]. The details of the microscopic model Hamiltonian are reported in the Appendix. Concerning the crystal field potential, we observe that pos- itive (negative) values of /Delta1correspond to an elongated (flat) RuO 6octahedron along the caxis and favor the occupation in thedxz,yz(dxy) sector. We start by considering as a reference the CF configuration at zero field and assume /Delta11is negative to simulate Ru inand/Delta12is positive to simulate Ru out. In a second instance, we will also vary the CF parameters in order toaddress possible magnetoelastic effects driven by the appliedfield and to assess the robustness of the obtained effects withrespect to the octahedral distortions. Since we deal with 4 doxides, it is also important to include the atomic spin-orbit coupling, which in several RP ruthenates has strength comparable to the CF splitting among the t 2g levels [ 14,36]. In particular, we adopt the Russel-Saunders scheme suitable for correlated ruthenates [ 34] and assume that total spin Siand orbital angular momentum Liare formed at the ionic level, which are coupled by the SOC Hamiltonian.Moreover, since the total angular momentum is not a goodquantum number, we consider the S iandLicomponents and evaluate the corresponding average expectation values to fully characterize the spin and orbital character of the ground state. Then, taking into account local-density approximation pre- dictions [ 1], we assign as a reference in the calculations the values t=0.4 eV and /Delta11/t=− 0.3,/Delta12/t=0.225 for the CF parameters of the central flat and outer elongatedoctahedra, respectively. Subsequently, we will also vary /Delta1 2/t from positive to negative values in order to explore differentregimes of octahedral distortions. Concerning the Coulombinteractions, we consider the ratio U/J H=5.0 and analyze 104440-4SUPPRESSION OF THE ORBITAL MAGNETIC MOMENT … PHYSICAL REVIEW B 100, 104440 (2019) FIG. 2. Evolution of (a) the spin and orbital components of the Ru magnetic density and (b) the Ru-Ru nonlocal orbital moment correlations projected along the caxis. Bis the amplitude of the applied Zeeman field oriented along the caxis. We assume that t=0.4e V , /Delta11/t=− 0.3,/Delta12/t=0.225, U/t=5.0,JH/U=0.2. AF-OM (CAF-OM) stands for a ferromagnetic ground state with antialigned orbital moments and inequivalent resulting orbital polar-ization with an averaged amplitude μ L av=(1/2) (/angbracketleftμL 1/angbracketright+/angbracketleftμL 2/angbracketright)t h a t is smaller (larger) than 0.1. B∗indicates the amplitude of the Zeeman field above (below) which the ground state is spin polarized with asmaller (larger) averaged orbital moment than a given reference that is set at 0.1. the regime of intermediate electronic correlations set by U/t=5.0[37]. Concerning the SOC term, we assume an amplitude λ/t=0.14 in the range of values that is expected to hold for Sr-based ruthenates [ 38]. We study the evolution towards the FM state as a function of an applied Zeeman field B(coupled to the spin and orbital angular momenta at the Ru site) along the caxis to assess the relation between the amplitude of the local spin and orbitalmagnetic moment and the nearest-neighbor orbital pattern. InFig.2(a) we present the field dependence of the c-axis projec- tion of the local spin, /angbracketleftμ S/angbracketright, and orbital, /angbracketleftμL/angbracketright, components of the magnetic density evaluated at the Ru 1and Ru 2sites. We also track the evolution of the nonlocal orbital correlationsbetween the orbital moments at the two Ru sites /angbracketleftμ L 1μL2/angbracketright [Fig. 2(b)]. The analysis is performed for a representative distortive state of the octahedra at the Ru 1and Ru 2sites being in flat and elongated configurations, respectively. There arevarious remarkable aspects of the correlated spin-polarizedstate that the presented investigation unveils. First, as onewould expect, we observe that inequivalent octahedral dis-tortions generally lead to different spin and orbital magneticmoments at the corresponding Ru sites. We have to recall thatthe orbital degree of freedom in the d 4Ru configuration is set by the position of the double occupancy (doublon) within thethree t 2gorbital states. Then, since flat (elongated) octahedra tend to favor an orbital occupancy with the doublon placedin the xy({xz,yz}) orbital sector, one has that the spin-orbit lifting of the orbital degeneracy leads to an orbital magneticmoment which is preferably oriented in the abplane ( caxis). As we can see in Fig. 2, the increase of the applied magnetic field drives the transition between two distinct spin and orbitalconfigurations. At low fields the ground state, labeled cantedantiferro-orbital moment (CAF-OM) phase, exhibits a nonsat-urated local spin moment /angbracketleftμ S/angbracketright∼1 on both Ru 1and Ru 2and an orbital component whose amplitude /angbracketleftμL/angbracketrightis comparable to the spin density at the Ru 2site (elongated octahedron). On the other hand, the Ru-Ru orbital correlations are negative,thus indicating that the orbital moments are antialigned. Theamplitude of /angbracketleftμ L 1μL2/angbracketrightdoes not reach the maximal value of −1 in the CAF-OM, and thus, it shows a sort of canting in theorbital configuration. A further growth of the applied field brings the transition to the fully polarized FM spin state, where the averagedorbital moment over the two Ru sites is suppressed. Indeed, asreported in Fig. 2(b), one can single out an effective amplitude of the Zeeman field B ∗that separates the CAF-OM state from the AF-OM one, characterized by a reduced averaged orbitalmoment μ L av=(1/2)(/angbracketleftμL 1/angbracketright+/angbracketleftμL 2/angbracketright), lower than the reference amplitude of μL av=0.1. The microscopic mechanism which governs the quenching of the orbital moment can be under-stood as follows: when an electron hops between neighboringsites in the spin-polarized background, the effective transferintegral between the degenerate ( d zx,dyz) orbitals is optimized if an antiferro-orbital (AFO) configuration is realized, wherethe double occupancy sits on alternating orbitals on neighbor-ing sites. Such an orbital configuration results in a vanishingvalue of the average angular momentum density. To assess the relation between the critical field amplitude B ∗and the character of the octahedral distortions, we explored the impact of the variation of the CF potential on the groundstate. In particular, we keep in mind that recent studies re-vealed that distinct structural mechanisms are associated withthec-axis magnetization, specifically an induced extension of thecaxis, which may result in an elongation of the apical RuO bonds and a contraction of the in-plane RuO bonds [ 23]. In our calculations, we mimic this trend by lowering the CFterm at the Ru 1site by keeping fixed the configuration at the Ru2site. In Fig. 3, we report the evolution of the CAF-OM and AF-OM phases as a function of /Delta11by moving from the zero- field compressed octahedral state ( /Delta11/t∼− 1.2) towards the symmetric configuration ( /Delta11/t=0). We do observe that, when the flattening at the Ru 1site is released by simulating a weakly distorted octahedron ( /Delta11/t∼0), the FM state with a vanishing orbital moment is settled at lower fields and holdsat any amplitude of the applied magnetic field. Finally, it is worth pointing out that the main objective of the performed computation is to unveil the mechanismsthat can account for the observed orbital quenching. Giventhe cluster size of the quantum simulation and the manycompeting energy scales in the problem, a tight quantita-tive correspondence between the theoretical outcome andthe experimental data is naturally beyond the scope of theeffective model. Still, our analysis is more suitable to singleout the range of the microscopic parameters where reasonablequalitative and quantitative agreement on the trend and evo-lution of the physical observables can be achieved. From thispoint of view, the specific choice of the Coulomb interaction 104440-5FILOMENA FORTE et al. PHYSICAL REVIEW B 100, 104440 (2019) FIG. 3. Evolution of the AF-OM and CAF-OM ferromagnetic phases as a function of the caxis Zeeman field and of the octahedral distortions at the Ru 1site/Delta11for a given amplitude of the CF potential (elongated configuration) at Ru 2.Bis the amplitude of the applied magnetic field. We assume that t=0.4e V , /Delta11/t=− 0.3, U/t=5.0,JH/U=0.2. AF-OM (CAF-OM) stands for a ground state with antialigned orbital moments and inequivalent resulting orbital polarization corresponding to an averaged amplitude μL av= (1/2) (/angbracketleftμL 1/angbracketright+/angbracketleftμL 2/angbracketright) that is smaller (larger) than 0.1. B∗indicates the amplitude of the Zeeman field above (below) which the ground state is spin polarized with a smaller (larger) averaged orbital moment than a given reference that is set at 0.1. parameters, Uand JH, is not crucial. However, the disen- tangling of spin and orbital degrees of freedom by the ap-plied magnetic field along c, which drives magnetic ordering coexisting with short-range antiferro-orbital correlations, canbe obtained only in a correlated picture. We expect that theoutcomes will not be qualitatively altered when consideringother correlated approaches with embedded (e.g., dynamicalfield theory) or coupled clusters (e.g., cluster perturbationmethods) to deal with the full lattice problem. V . CONCLUSIONS We used polarized neutron scattering diffraction to deter- mine the spin and orbital character of the magnetic state ofSr 4Ru3O10in the high-field FM phase with spin moments aligned along the caxis. Remarkably, our study revealed a vanishing contribution of the orbital component to themagnetic density at Ru sites. We discussed the microscopicmechanisms which are able to account for the suppression ofthe orbital moment as being due to the formation of robustantiferro-orbital correlations which are driven by the fullypolarized magnetic FM phase. Moreover, we showed that sucha microscopic mechanism may be assisted by the structuraldeformations (namely, expansion of the apical Ru-O distancesof inner and outer octahedra) which effectively separate theplanar d xybands from the longitudinal ( dxz,dyz) doublet and are allowed in a certain window for the CF parameters whichsimulate the tetragonal deformations. In particular, we showedthat the effective magnetic field to access the FM phase witha vanishing orbital moment is lower in the case where theoctahedra are more uniform and less compressed along thecaxis. The comprehensive view offered by the comparative stud- ies of the field-induced magnetic phases of Sr 4Ru3O10,f o r longitudinal and in-plane applied fields, highlights the roleof the orbital component in setting the anisotropic magneticresponse. In particular, we provided a self-contained physical scenario where the orbital moment is suppressed when thefield is applied along the caxis, while it is substantial and at the origin of the layer-dependent magnetic response in thecase of an in-plane applied field. ACKNOWLEDGMENTS We acknowledge insightful discussions with and valuable support during the experiments by A. Stunault, O. Fabelo, A.J. Rodriguez Velamazan, and A. Goukassov. APPENDIX: MODEL HAMILTONIAN In this Appendix, we report the details of the model Hamiltonian employed to simulate the evolution of the spinand orbital angular momentum in a cluster consisting of twoinequivalent Ru sites. The examined microscopic model Hamiltonian with two inequivalent atoms, Ru 1and Ru 2, is expressed as H=Hkin+Hel−el+Hcf+Hsoc+Hz. (A1) The first term in Eq. ( A1) is the kinetic operator between the t2gorbitals on different Ru sites, Hkin=−t/summationdisplay ij,σα(d† iασdjασ+H.c.), (A2) with d† iασbeing the creation operator for an electron with spinσat the isite in the αorbital. A similar modeling approach is commonly adopted to describe the kinetic termin transition metal compounds with partially filled d-bands having different orbital character [ 39,40]. The second term is the local Coulomb interaction between t 2gelectrons [ 34,41]: Hel−el=U/summationdisplay iαniα↑niα↓−2JH/summationdisplay iαβSiα·Siβ +/parenleftbigg U/prime−JH 2/parenrightbigg/summationdisplay iα/negationslash=βniαniβ+J/prime/summationdisplay iαβd† iα↑d† iα↓diβ↑diβ↓, (A3) where niασandSiαare the on-site charge for spin σand the spin operators for the αorbital, respectively. U(U/prime)i st h e intraorbital (interorbital) Coulomb repulsion, JHis Hund’s coupling, and J/primeis the pair-hopping term. Due to the invari- ance for rotations in the orbital space, the following relationshold: U=U /prime+2JH,J/prime=JH. The Hcfpart of the Hamiltonian His the crystalline field potential, controlling the symmetry lowering from the cubicto tetragonal one: H cf=/summationdisplay i/Delta1i/bracketleftbigg nixy−1 2(nixz+niyz)/bracketrightbigg . (A4) The SOC Hamiltonian reads Hsoc=λ/summationdisplay iLi·Si. (A5) Due to the cubic CF terms in RuO 6octahedra separating the lower t2gfrom the unoccupied eglevels, Listands for an effective L=1 angular momentum, projected onto the t2g 104440-6SUPPRESSION OF THE ORBITAL MAGNETIC MOMENT … PHYSICAL REVIEW B 100, 104440 (2019) subspace. Its components have the following expression in terms of orbital fermionic operators: Liz=i/summationdisplay σ[d† ixzσdiyzσ−d† iyzσdixzσ], Lix=i/summationdisplay σ[d† ixyσdixzσ−d† ixzσdixyσ], Liy=i/summationdisplay σ[d† iyzσdixyσ−d† ixyσdiyzσ]. (A6) The SOC term binds Liand spin Simomenta into the total an- gular momentum Ji=Li+Si. Although the spin-orbit term commutes with both total angular momenta J2 iandJz i,t h ef u l lHamiltonian of Eq. ( A1) has a reduced symmetry, for Jz iis not a conserved quantity due to the orbital anisotropy of thekinetic term [ 42]. 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PhysRevB.80.165123.pdf

Site-resolved oxygen K-edge ELNES of the layered double perovskite La 2CuSnO 6 M. Haruta, H. Kurata, H. Komatsu, Y. Shimakawa, and S. Isoda Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan /H20849Received 14 July 2009; revised manuscript received 17 September 2009; published 21 October 2009 /H20850 Electron energy-loss spectroscopy /H20849EELS /H20850combined with scanning transmission electron microscopy /H20849STEM /H20850allows for the investigation of the local electronic structure with atomic column resolution, owing to the development of spherical aberration correctors. In the present research, we report B-site-resolved oxygen K-edge energy-loss near-edge structure /H20849ELNES /H20850measured from a double perovskite La 2CuSnO 6/H20849LCSO /H20850 using STEM-EELS. There are two kinds of BO6/H20849B=Sn and Cu /H20850octahedrons, arranged in layers in the LCSO crystal. The observed site-resolved oxygen K-edge ELNES showed different features reflecting local chemical bonding around the Sn and Cu ions. In particular, it is demonstrated that the local electronic structure in thedistorted CuO 6octahedron caused by the Jahn-Teller effect can be detected by site-resolved ELNES. DOI: 10.1103/PhysRevB.80.165123 PACS number /H20849s/H20850: 82.80.Pv, 31.15.ae, 68.37.Ma, 79.20.Uv I. INTRODUCTION With recent developments in correcting electron optical aberrations, it is now possible to analyze structural imageswith subangstrom resolution. 1–4Electron energy-loss spec- troscopy /H20849EELS /H20850combined with scanning transmission elec- tron microscopy /H20849STEM /H20850has realized single atomic column analysis5–10and element-selective imaging with atomic resolution.11–13As a next step, detailed chemical bond map- ping at atomic resolutions is anticipated in the near future.Site-resolved electron energy-loss near-edge structure /H20849EL- NES /H20850is thus very important to realize chemical bond mapping. 14,15 Perovskite-structured oxides have the general formula ABO3, which can be described as a framework of corner- shared BO6octahedrons with 12-coordinated Acations. When two different cations are introduced at the Bsite, double perovskite structures with the formula A2BB/H11032O6are formed. Depending on differences in size and charge of theB-site cations, three types of structural arrangements of cat- ions are known: random, rocksalt, and layered. 16In the ma- jority of ordered double perovskite oxides /H20849A2BB/H11032O6/H20850,Band B/H11032sites have the rocksalt structure, whereas only Cu2+and Sn4+ions in La 2CuSnO 6/H20849LCSO /H20850are arranged in layers at ambient pressures.17,18The layered crystal structure deter- mined by powder x-ray diffraction analysis is shown sche-matically in Fig. 1. The La column–La column distance around the Cu ion is shorter than around the Sn ion becausethe CuO 6octahedrons are slightly distorted by the Jahn- Teller effect. The alternation of CuO 6and SnO 6octahedral layers and buckling of CuO 2and SnO 2sheets induce a monoclinic superstructure with the following lattice param-eters: a=0.8510, b=0.7815, c=0.7817 nm, and /H9252=91.151°, corresponding almost to 2 ap/H110032ap/H110032ap/H20849apis the lattice pa- rameter for a cubic perovskite /H20850. LCSO had been expected to show superconductivity by appropriate carrier doping of Sr2+ or Ca2+substitution for La3+. However, all attempts had tried unsuccessfully to find superconductivity because of heavilybuckled Cu-O-Cu bonds in the CuO 2sheets.19,20There are eight nonequivalent oxygen atoms /H20849two O1, four O2, two O3, four O4, four O5, two O6, four O7, and two O8 atoms /H20850 with different atomic coordinates in one unit cell. By takingadvantage of the electron channeling effect, it should be pos- sible to excite specific oxygen atoms coordinating to a B cation in the unit cell when the electron probe is placed at theB-site atom. The layered structure such as LCSO is suitable for examining the site-resolved EELS. In the present re-search, we demonstrate the B-site-resolved oxygen K-ELNES measured from LCSO using the STEM-EELS method, which can be used to extract the local electronicstructure arising from the Jahn-Teller distortion of the CuO 6 octahedron. Recently, Varela et al. also reported the notice- able change in the prepeak of O K-edge ELNES calculated for the nonequivalent O sites in a Jahn-Teller distorted MnO 6 of manganite perovskites.21Since the Jahn-Teller effect is important for the high-temperature cuprate superconductor, itshould be valuable to examine this effect at high spatial res-olution using the STEM-EELS method. II. METHODOLOGY The sample with a layered double perovskite LCSO film was fabricated using a pulsed laser deposition technique.22 FIG. 1. /H20849Color online /H20850/H20849a/H20850Structure of layered La 2CuSnO 6ana- lyzed by x-ray diffraction. There are eight nonequivalent oxygenatoms in the unit cell. The circles /H20849radius of 0.18 nm /H20850indicate delocalization of inelastic scattering by the excitation of O 1 selec- trons. Blue, red, and yellow circles show the O sites binding only toCu, only to Sn, and linking to both Cu and Sn, respectively. /H20849b/H20850 Atomic resolution HAADF image of LCSO. The scanning areas forsite-resolved EELS spectra are illustrated as rectangular areas en-closed by the white dashed lines.PHYSICAL REVIEW B 80, 165123 /H208492009 /H20850 1098-0121/2009/80 /H2084916/H20850/165123 /H208496/H20850 ©2009 The American Physical Society 165123-1LCSO thin films were grown heteroepitaxially on SrTiO 3 substrates. The growth temperature was 670 °C in an oxy- gen partial pressure of 0.1 Torr, which are optimized condi-tions for fabricating the layered structure. Cross-sectionalsamples were thinned down to electron transparency by ionmilling. The thickness of the observed area is about 30 nm,as estimated by EELS measurements. Atomic resolutionSTEM imaging and EELS measurements were performed atroom temperature using a 200 kV TEM/STEM /H20849JEM- 9980TKP1; C s=−0.025 mm, C5=15 mm /H20850equipped with a spherical aberration corrector for the illuminating lens sys-tem. This provided an incident electron probe of less than 0.1nm in diameter with a convergent semiangle of approxi-mately 23 mrad. Atomic resolution high-angle annular darkfield /H20849HAADF /H20850images were obtained with a detection semi- angle of 70–170 mrad. EELS spectra were obtained using anomega filter with a collection semiangle of 10 mrad. Theenergy resolution measured by the full width at half maxi-mum of a zero-loss peak was about 0.5 eV using a cold fieldemission gun. First-principles band structure calculations were per- formed by a full-potential linear augmented plane wave pluslocal orbital method using the WIEN2K code.23ELNES spec- tra were calculated using the TELNES.2 package incorporated in the WIEN2K code.24The effect of a core hole was taken into account in the calculations by introducing a hole in theoxygen 1 sstate at each nonequivalent oxygen site and add- ing an electron in the valence band. Since the unit cell of theLCSO is relatively large, as mentioned previously, calcula-tion including a core hole was carried out by using the primi-tive unit cell for execution within a practical time period. In order to evaluate the spatial resolution of site-resolved EELS, it is worth estimating the delocalization of inelasticscattering due to excitation of the oxygen 1 selectron. The delocalization, d E, of the inelastic electrons losing an energy of/H9004Ecan be simply represented by the following equation:25,26 dE=/H9261 2/H208732E /H9004E/H208743/4 =/H9261 2/H9258E3/4. /H208491/H20850 Here, /H9261andEare the wavelength and beam energy of the incident electrons, respectively, and /H9258E=/H9004E//H208492E/H20850is a char- acteristic inelastic scattering angle. The delocalization ofoxygen K-edge spectra /H20849/H9004E=532 eV /H20850excited by a beam energy of 200 keV is estimated to be 0.18 nm, which indi-cates that the delocalization is sufficiently narrow to dis-criminate contributions from oxygen atoms in each layer ofthe LCSO, as shown with circles in Fig. 1/H20849a/H20850. Therefore, it can be expected that specific oxygen atoms coordinating to aB-site cation in the unit cell are separately excited when the electron probe is placed on the B-site cation. Figure 1/H20849b/H20850 shows a typical atomic resolution HAADF image of LCSO,wherein B-site cations can be clearly distinguished owing to Z-contrast imaging. 27In order to avoid electron damage to the specimen, O K-edge ELNES spectra were acquired by scanning the electron probe on equivalent B-site columns along the caxis, as illustrated by the rectangular areas en- closed by white dotted lines in Fig. 1/H20849b/H20850. Another factor af-fecting the spatial resolution of the site-resolved EELS spec- tra is the propagation of the incident electron probe in thecrystal, the so-called electron channeling. 28The dynamical behavior of electrons in the crystal can be calculated by themultislice method. 29Figure 2shows two-dimensional maps of the electron wave function as a function of the specimenthickness calculated by the WIN HREM Version 3.5 software when the incident electron beam is placed on the B-site col- umns and the middle point of the Cu-Cu columns or Sn-Sncolumns, indicated by the arrows at the top row in the figure.When the electron probe is positioned on the B-site columns /H20851Figs. 2/H20849a/H20850and2/H20849c/H20850/H20852, most of the electrons propagate though theBsite and its nearest-neighbor O columns. Even when the electrons disperse with increasing specimen thickness,most effectively dechannel to the nearest-neighbor oxygenatoms forming each BO 6octahedron. When the incident electron probe is located on the middle point of Cu and Cu/H20851Fig.2/H20849b/H20850/H20852or Sn and Sn /H20851Fig.2/H20849d/H20850/H20852, very close to the oxygen atom shared with the same Bcations, most of the electrons effectively propagate near the oxygen atom, although it ap-pears that a small amount of electrons dechannel to La sites.Therefore, from the viewpoint of the electron channelingprocess, it can be expected that specific oxygen atoms coor-dinating to each B-site cation in the unit cell are separately excited when the electron probe is scanned on the sameB-site cation. III. RESULTS AND DISCUSSION Figure 3shows three O K-edge ELNES spectra acquired by scanning an electron probe over a whole unit cell ofLCSO, only from the Sn or the Cu site. These spectra clearlyhave different shapes; the peak labeled A is clearly observed FIG. 2. /H20849Color online /H20850Two-dimensional map of the electron wave function as a function of sample thickness when the electronprobe was positioned on /H20849a/H20850a Sn column, /H20849b/H20850the middle of Sn and Sn, /H20849c/H20850the Cu column, and /H20849d/H20850the middle of Cu and Cu. The position of electron probe is indicated by a yellow arrow in theprojected atomic map images.HARUTA et al. PHYSICAL REVIEW B 80, 165123 /H208492009 /H20850 165123-2in the spectrum from the Cu site, and the main peak consists of two peaks labeled B and C as observed in the spectra fromthe whole unit cell and the Sn site, while the spectrum fromthe Cu site only shows peak C. These characteristic featuressuggest that the chemically different oxygen atoms contrib-ute individually to the spectra of the Sn and Cu sites, al-though the spectrum from the whole unit cell contains theexcitation of all oxygen sites. As mentioned in Sec. I, there are eight nonequivalent oxygen sites in the unit cell, whichshould contribute to the O K-edge ELNES differently. There- fore, for simulating the ELNES spectrum, it is necessary tocalculate the partial density of states /H20849PDOS /H20850at each indi- vidual oxygen site. Figure 4shows the O K-edge ELNES calculated for each individual oxygen site /H20849from O1 to O8 /H20850in the LCSO. Here, we show the spectra calculated by including a core hole inthe 1 sstate of each oxygen atom /H20849solid lines /H20850together with those from ground state calculations /H20849broken lines /H20850without a core hole. The core-hole effect appears differently in eachspectrum, which will be discussed later. These spectra can be classified into three groups in relation to the nearest-neighborB-site atoms: the spectra of oxygen atoms binding only to a Sn atom /H20851Fig.4/H20849a/H20850/H20852or only to a Cu atom /H20851Fig.4/H20849b/H20850/H20852and the spectra of oxygen atoms linking both Sn and Cu atoms /H20851Fig. 4/H20849c/H20850/H20852. In order to simulate the O K-edge ELNES acquired over the whole unit cell, all the calculated spectra aresummed up by taking into account the occupancy of eachatomic site. Figures 5/H20849b/H20850and5/H20849c/H20850show the calculated spectra compared with the experimental spectrum /H20851Fig. 5/H20849a/H20850/H20852ac- quired from the whole unit cell. The calculated spectrumincluding a core hole is in better agreement with the experi-mental one. By comparing the spectral features with thePDOS, we find that peak A can be attributed to the transitionto the O 2 pstates hybridized with Cu 3 dstates near the Fermi level, while peaks B–D correspond mainly to the tran-sitions to the O 2 pstates hybridized with Sn 5 s,L a5 dand Cu 4 s , and Sn 5 pand/or Cu 4 sstates, respectively. Although this spectrum includes the contribution from all O sites, site-resolved spectra can discriminate individual contributions. Figure 6shows a comparison between the site-resolved spectra acquired experimentally from the Sn and Cu sites andthe corresponding calculated spectra including the core-holeeffect. Since the value of the spatial resolution of the presentEELS experiment is not very obvious because of delocaliza-tion of inelastic scattering /H20851Fig.1/H20849a/H20850/H20852and beam dechanneling /H20849Fig.2/H20850, two types of calculated spectra are shown in Fig. 6. One is constructed with all oxygen atoms forming the SnO 6 and CuO 6octahedrons /H20849a-2 and b-2 /H20850, while the other elimi- nates the contribution from the apex sites /H20849O4 and O5 /H20850of the octahedrons /H20849a-3 and b-3 /H20850. The experimental spectra /H20849a-1 and b-1 in Fig. 6/H20850acquired from the Sn and Cu sites agree well with the calculated spectra /H20849a-2 and b-2 in Figs. 6/H20850including FIG. 3. Experimental O K-edge ELNES from /H20849a/H20850the whole unit cell of LCSO, /H20849b/H20850Sn site, and /H20849c/H20850Cu site. FIG. 4. Calculated O K-edge ELNES for individual oxygen sites in LCSO. O KELNES of O sites /H20849a/H20850binding only to a Sn atom, /H20849b/H20850 binding only to a Cu atom, and /H20849c/H20850binding to both Sn and Cu atoms. Solid lines are spectra calculated by including the core-holeeffect and the dotted line by the ground state. FIG. 5. O K-edge ELNES of LCSO obtained from the whole unit cell. /H20849a/H20850Experimental results, /H20849b/H20850calculated with the O 1 score- hole state, and /H20849c/H20850calculated for the ground state. The calculated spectra were summed with a weight considering the number of sitescontained in the unit cell.SITE-RESOLVED OXYGEN K-EDGE ELNES OF THE … PHYSICAL REVIEW B 80, 165123 /H208492009 /H20850 165123-3the contribution from all oxygen atoms forming the SnO 6or CuO 6octahedrons, respectively. The clear difference in the OK-edge ELNES indicates that the experimental site- resolved spectra contain the local electronic structure aroundthe Sn or Cu separately, although the linking oxygen atoms/H20849O4 and O5 /H20850contribute to both experimental site-resolved spectra. The reason why O4 and O5 sites contribute to boththe spectra from Cu-O and Sn-O layers is attributed to thedelocalization of the inelastic scattering as discussed in Ref.21. The site-resolved O K-edge ELNES shows the following specific features. Peak A due to transitions to the hybridizedO2p-Cu 3 dstates appears only in the CuO 6spectrum /H20851Fig. 6/H20849b/H20850/H20852, while peak B attributed to the transition to the O 2 p states hybridized with Sn 5 sstates appears predominantly in the SnO 6spectrum /H20851Fig.6/H20849a/H20850/H20852. Peaks C and D appear in both spectra because these peaks mainly originate from transitionsto the hybridized O 2 p-La 5 dstates and O 2 p-Sn 5 pand/or Cu 4 sstates, respectively. It should be emphasized that prepeak A attributed to the transition to the O 2 pstate hybridized with the Cu 3 dstate is not observed in the SnO 6spectrum, although oxygen atoms /H20849O4 and O5 /H20850linking Sn and Cu also contribute to this spec- trum. This means that there are no unoccupied O 2 pstates hybridized with the Cu 3 dstate in the local DOS at O4 and O5 sites. This result contains important information on localelectronic structure arising from the Jahn-Teller distortion ofthe CuO 6octahedrons. Figure 7shows the structures of CuO 6and SnO 6octahedrons in the LCSO as determined by x-ray diffraction analysis.17The CuO 6octahedron in the LCSO is elongated in the zdirection by the Jahn-Teller ef- fect. Such a distortion from cubic symmetry /H20849Oh/H20850leads to a splitting of the t2gandegstates, as illustrated in Fig. 7. The crystal field of the Cu 3 dstates in the distorted CuO 6octa- hedron splits the t2gstates into b2g/H20849dxy/H20850andeg/H20849dyz,dzx/H20850states and the egstates into b1g/H20849dx2−y2/H20850and a1g/H20849d3z2−r2/H20850states.30 Since the b2g/H20849dxy/H20850and eg/H20849dyz,dzx/H20850states are occupied with Cu2+/H20849d9/H20850ion, there are two possibilities for the electron con- figuration at the top of the Cu 3 dstates: /H20849b1g/H208502/H20849a1g/H208501or/H20849a1g/H208502/H20849b1g/H208501. The b1g/H20849dx2−y2/H20850state is extended into the in plane, while the a1g/H20849d3z2−r2/H20850state is directed in the zdirection forming hybridized states at the O4 and O5 sites. This ex- perimental result clearly indicates that the Cu 3 dhole should be in the b1g/H20849dx2−y2/H20850state because there are no 3 dholes at the O4 and O5 sites linking both the Sn and Cu atoms, as pre- viously mentioned. Therefore, it is concluded that the elec-tron configuration should be /H20849a 1g/H208502/H20849b1g/H208501in the ground state. Such a configuration suggests that electrostatic repulsion be-tween Cu and O atoms at the O4 and O5 sites is strongerthan that for in-plane O atoms, which results in a slight ex-pansion of bond lengths between the Cu and the apex O sites/H20849O4 and O5 atoms /H20850compared to those of the in-plane O atoms. The elongated distortion along the zdirection is con- sistent with the structural analysis results, shown in Fig. 7. Consequently, it is demonstrated that the Jahn-Teller effectcan be detected by site-resolved EELS. Finally, we discuss core-hole effects in the material, which operate differently at the eight nonequivalent oxygensites, as shown in Fig. 4. The core-hole potential generally modifies the distribution of DOS in the bottom of the unoc-cupied band, 31–33leading to a change in ELNES. In the case of oxygen atoms around the Sn site, the spectrum shiftsdown as a whole to the Fermi level due to the core-holeeffect. Peak B corresponding to the transition to the hybrid-ized O 2 p-Sn 5 sstates is enhanced, while the intensity of peak D is considerably reduced due to transitions to the O 2 p states hybridized with Sn 5 pstates. Such a spectral modifi- cation is typically obtained as an excitonic effect in ionicinsulators; 34therefore, it appears that the character of chemi- cal bonding in the SnO 6octahedron is locally an ionic one. On the other hand, in the case of oxygen atoms around a Cusite, the intensity of peak A due to the transition to the O 2 p states hybridized with the Cu 3 dstates is significantly weak- ened by the core-hole effect, whereas the other peaks athigher energy are almost unchanged. A similar decrease inthe prepeak intensity owing to the core-hole effect has beenreported for O K-edge excitation spectra obtained from CuO /H20849Ref. 35/H20850and LaCoO 3.36This decrease in the prepeak inten- sity might indicate that some empty O 2 pstates lying at the bottom of the conduction band could be pulled below theFermi level to effectively screen the core-hole potential 37 because the bottom of the conduction band is mainly com- FIG. 6. Site-resolved spectra from the /H20849a/H20850Sn site and /H20849b/H20850Cu site, where spectrum 1 is the experimental spectrum, the calculated spec-trum 2 includes the contribution from all O sites surrounding eachB-site cation, and the calculated spectrum 3 eliminates the contri- bution from the apex O sites in octahedrons. FIG. 7. /H20849Color online /H20850/H20849a/H20850Local structure of CuO 6and SnO 6 octahedrons in the LCSO crystal. The CuO 6octahedron is distorted by the Jahn-Teller effect. The bond lengths shown in the CuO 6and SnO 6octahedrons are average values. /H20849b/H20850Crystal-field splitting of Cu 3 dstates owing to the Jahn-Teller distortion.HARUTA et al. PHYSICAL REVIEW B 80, 165123 /H208492009 /H20850 165123-4posed of Cu 3 dstates which are near the Fermi level com- pared to the Sn 5 spstates. IV. CONCLUSIONS In the present research, B-site-resolved oxygen K-edge ELNES was measured from LCSO using the STEM-EELSmethod. The experimental site-resolved spectra clearly havedifferent shapes reflecting the local electronic structurearound the SnO 6and CuO 6octahedrons. The prepeak attrib- uted to the transition to O 2 pstates hybridized with Cu 3 d states was not observed in the spectrum acquired from oxy-gen atoms around the Sn site, although oxygen atoms /H20849O4 and O5 /H20850linking Sn and Cu contribute to this spectrum, indi- cating that there are no unoccupied O 2 pstates hybridized with Cu 3 dstates at the O4 and O5 sites. From this result, the electron configuration at the top of the Cu 3 dstate was determined to be /H20849a 1g/H208502/H20849b1g/H208501caused by the Jahn-Teller dis- tortion of the CuO 6octahedrons. Consequently, it was dem-onstrated that the Jahn-Teller effect was detected only in the CuO 6octahedrons by site-resolved EELS. The experimental spectra agree well with the calculated spectra including the core-hole effect. The core-hole effectson any oxygen K-edge spectra have a tendency to shift the excitation energy a little lower. It was also found that thecore-hole effect appears differently around Sn and Cu sites,as shown in Fig. 4. In the case of O around the Sn site, the O2pstates hybridized with the Sn 5 sstates are enhanced by the core-hole effect. On the other hand, in the case of Oaround the Cu site, O 2 pstates hybridized with the Cu 3 d states are weakened by the core-hole effect. ACKNOWLEDGMENTS This work was partly supported by Grants-in-Aid for Sci- entific Research Grants No. 19GS0207 and No. 19310071and also by JSPS Fellows Grant No. 20-145 from the Min-istry of Education, Culture, Sports, Science and Technology,Japan. We acknowledge Kawasaki, Otsuka, and Nishimuraof TRC for preparing the cross-section samples. 1P. E. Batson, N. Dellby, and O. L. Krivanek, Nature /H20849London /H20850 418, 617 /H208492002 /H20850. 2P. D. Nellist, M. F. Chisholm, N. Dellby, O. L. Krivanek, M. F. 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PhysRevB.82.144110.pdf

Ground-state properties and high-pressure behavior of plutonium dioxide: Density functional theory calculations Ping Zhang,1Bao-Tian Wang,2and Xian-Geng Zhao1 1LCP , Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’ s Republic of China 2Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, People’ s Republic of China /H20849Received 6 May 2010; published 19 October 2010 /H20850 Plutonium dioxide is of high technological importance in nuclear fuel cycle and is particularly crucial in long-term storage of Pu-based radioactive waste. Using first-principles density-functional theory, in this paperwe systematically study the structural, electronic, mechanical, thermodynamic properties, and pressure-inducedstructural transition of PuO 2. To properly describe the strong correlation in Pu 5 felectrons, the local-density approximation /H20849LDA /H20850+Uand the generalized gradient approximation+ Utheoretical formalisms have been employed. We optimize Uparameter in calculating the total energy, lattice parameters, and bulk modulus at nonmagnetic, ferromagnetic, and antiferromagnetic configurations for both ground-state fluorite structure andhigh-pressure cotunnite structure. Best agreement with experiments is obtained by tuning the effective Hubbardparameter Uat around 4 eV within LDA+ Uapproach. After carefully testing the validity of the ground-state calculation, we further investigate the bonding nature, elastic constants, various moduli, Debye temperature,hardness, ideal tensile strength, and phonon dispersion for fluorite PuO 2. Some thermodynamic properties, e.g., Gibbs free energy, volume thermal expansion, and specific heat are also calculated. As for cotunnite phase,besides elastic constants, various moduli, and Debye temperature at 0 GPa, we have further presented ourcalculated electronic, structural, and magnetic properties for PuO 2under pressure up to 280 GPa. A metallic transition at around 133 GPa and an isostructural transition in pressure range of 75–133 GPa are predicted.Additionally, as an illustration on the valency trend and subsequent effect on the mechanical properties, thecalculated results for other actinide metal dioxides /H20849ThO 2,U O 2, and NpO 2/H20850are also presented. DOI: 10.1103/PhysRevB.82.144110 PACS number /H20849s/H20850: 71.27. /H11001a, 61.50.Ks, 62.20. /H11002x, 63.20.dk I. INTRODUCTION Actinide elements and compounds possess particular in- teresting physical behaviors due to the 5 fstates and have always attracted extensive attention because of their impor-tance in nuclear fuel cycle. 1–4Actinide dioxides, AO2/H20849A =Th, U, Np, or Pu /H20850, are universally used as advanced fuel materials for nuclear reactors and PuO 2also plays an impor- tant role in the plutonium reuse, separation, and/or long-termstorage. Recently, there has occurred in the literature a seriesof experimental reports 5–7on the strategies of storage of Pu- based waste. Exposure to air and moisture, metallic pluto-nium surface easily oxidizes to Pu 2O3and PuO 2. Although the existence of PuO 2+x/H20849x/H113490.27 /H20850was reported by Haschke et al. ,5recent photoemission study found that PuO 2was only covered by a chemisorbed layer of oxygen and can be easilydesorbed at elevated temperature, 7therefore, PuO 2is the sta- blest plutonium oxide. At ambient condition PuO 2crystal- lizes in a fluorite structure with space group Fm3¯m/H20849No. 225 /H20850. Its cubic unit cell is composed of four PuO 2formula units with plutonium atoms and oxygen atoms in 4 aand 8 c sites, respectively. By using the energy-dispersive x-ray dif-fraction method, Dancausse et al. 8reported that at 39 GPa, PuO 2undergoes a phase transition to an orthorhombic struc- ture of cotunnite type with space group Pnma /H20849No. 62 /H20850. As for the electronic-structure study of PuO 2, conven- tional density-functional theory /H20849DFT /H20850that applies the local- density approximation /H20849LDA /H20850or generalized gradient ap- proximation /H20849GGA /H20850underestimates the strong on-site Coulomb repulsion of the 5 felectrons and, consequently, describes PuO 2as incorrect ferromagnetic /H20849FM /H20850conductor9instead of antiferromagnetic /H20849AFM /H20850Mott insulator reported by experiment.10Same problems have been confirmed in studying other correlated materials within the pure LDA/GGA schemes. Fortunately, several approaches, theLDA /GGA+ U, 11–13the hybrid density functional of /H20849Heyd, Scuseria, and Enzerhof /H20850HSE,14and the self-interaction- corrected local spin density /H20849SIC-LSD /H20850,15have been devel- oped to correct these failures in calculations of actinide com-pounds. The effective modification of pure DFT byLDA /GGA+ Uformalisms has been confirmed widely in study of UO 2/H20849Refs. 11and13/H20850and PuO 2.16–20By tuning the effective Hubbard parameter in a reasonable range, the AFMMott insulator feature was correctly calculated and the struc-tural parameters as well as the electronic structure are well inaccord with experiments. Lattice dynamical properties ofUO 2and PuO 2and various contributions to their thermal conductivities were studied by using a combination of LDAand dynamical mean-field theory /H20849DMFT /H20850. 21However, de- spite that abundant researches on the structural, electronic,optical, and thermodynamic properties of plutonium dioxidehave been performed, relatively little is known regarding itschemical bonding, mechanical properties, and phonon dis-persion. In addition, the pressure-induced structural transi-tion has also not yet been studied by first-principles DFT+Ucalculations. In this work, we have systematically calculated the ground-state structural parameters, electronic, mechanical,thermodynamic properties, and pressure-induced structuraltransition of PuO 2by employing the LDA+ Uand GGA+ U schemes due to Dudarev et al.11–13The validity of the ground-state calculation is carefully tested. Our calculatedPHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 1098-0121/2010/82 /H2084914/H20850/144110 /H2084914/H20850 ©2010 The American Physical Society 144110-1lattice parameter and bulk modulus Bare well consistent with previous LDA+ Uresults.18The total energy, structural parameters, bulk modulus B, and pressure derivative of the bulk modulus B/H11032for AFM and FM phases calculated in wide range of effective Hubbard Uparameter are presented and our electronic spectrum reproduce briefly the main feature ofour previous study. 16In addition, the bonding nature of A-O bond in PuO 2, NpO 2,U O 2, and ThO 2involving its mixed ionic/covalent character is investigated by performing theBader analysis. 22,23We find that about 2.40, 2.48, 2.56, and 2.66 electrons transfer from each Pu, Np, U, or Th atom to Oatom, respectively. In study of the mechanical properties, we first calculate the elastic constants of both Fm3¯mandPnma phases and then the elastic moduli, Poisson’s ratio, and De-bye temperature are deduced from the calculated elastic con- stants. Hardness and ideal tensile strength of Fm3¯mPuO 2are also obtained by LDA+ Uapproach with one typical value of effective Uparameter. The hardness of PuO 2is equal to 26.6 GPa and the ideal tensile strengths are calculated to be 81.2,28.3, and 16.8 GPa for pulling in the /H20851001 /H20852,/H20851110 /H20852, and /H20851111 /H20852 directions, respectively. As for the thermodynamic study, thephonon dispersion illustrates the stability of PuO 2and we further predict the lattice vibration energy, thermal expan-sion, and specific heat by utilizing the quasiharmonic ap-proximation based on the first-principles phonon density ofstate /H20849DOS /H20850. One more aim of the present work is to extend the description of PuO 2to high pressures. The electronic, structural, and magnetic behavior of PuO 2under pressure up to 280 GPa have been evaluated. Results show that thereoccurs a metallic transition at around 133 GPa for Pnma phase. The isostructural transition, similar to UO 2/H20849Ref. 24/H20850 and ThO 2,25in pressure domain of 75–133 GPa is predicted. The rest of this paper is arranged as follows. In Sec. IIthe computational method is briefly described. In Sec. IIIwe present and discuss our results. In Sec. IVwe summarize the conclusions of this work. II. COMPUTATIONAL METHODS The DFT calculations are performed on the basis of the frozen-core projected augmented wave method of Blöchl26 encoded in Vienna ab initio simulation package /H20849VASP /H20850/H20849Ref. 27/H20850using the LDA and GGA.28,29For the plane-wave set, a cutoff energy of 500 eV is used. The k-point meshes in the full wedge of the Brillouin zone /H20849BZ /H20850are sampled by 9 /H110039 /H110039 and 9 /H1100315/H110039 grids according to the Monkhorst-Pack30 scheme for fluorite and cotunnite PuO 2, respectively, and all atoms are fully relaxed until the Hellmann-Feynmanforces become less than 0.02 eV /Å. The plutonium 6s 27s26p66d25f4and the oxygen 2 s22p4electrons are treated as valence electrons. The strong on-site Coulomb repulsionamong the localized Pu 5 felectrons is described by using the LDA /GGA+ Uformalisms formulated by Dudarev et al., 11–13where the double counting correction has already been included. As concluded in some previous studies ofactinide dioxides /H20849AO 2/H20850, although the pure LDA and GGA fail to depict the electronic structure, especially the insulat-ing nature and the occupied-state character of UO 2,11,24 NpO 2,18,31and PuO 2,16,17,19the LDA /GGA+ Uapproacheswill capture the Mott insulating properties of the strongly correlated U 5 f,N p 5 f, and Pu 5 felectrons in AO2ad- equately. In this paper the Coulomb Uis treated as a variable while the exchange energy is set to be a constant J =0.75 eV. This value of Jis same with our previous study of plutonium oxides.16,17Since only the difference between UandJis significant,12thus we will henceforth label them as one single parameter, for simplicity labeled as U, while keeping in mind that the nonzero Jhas been used during calculations. Both spin-unpolarized and spin-polarized calculations are performed in this study. Compared to FM and AFM phases,the nonmagnetic /H20849NM /H20850phase is not energetically favorable both in the LDA+ Uand GGA+ Uformalisms. Therefore, the results of NM are not presented in the following. The depen-dence of the total energy /H20849per formula unit at respective op- timum geometries /H20850onUfor both FM and AFM phases within the LDA+ Uand GGA+ Uformalisms is shown in Fig.1.A tU=0 and 1.0 eV, the total energy of the FM phase is lower than that of the AFM phase either in LDA+ U scheme or GGA+ Uscheme. However, as shown in Fig. 1,i t is clear that the total energy of the AFM phase decreases tobecome lower than that of the FM phase when increasing U. The total-energy differences /H20849E FM−EAFM /H20850within the LDA +Uand GGA+ UatU=4 eV are 0.705 eV and 0.651 eV, respectively. Both FM and AFM results will be presented inthe following analysis. Besides, while the spin-orbit coupling/H20849SOC /H20850is important for certain properties of heavy-metal compounds, it has been numerically found 9,32and physically analyzed14,33that the inclusion of the SOC has little effect on the bulk and one-electron properties of UO 2and PuO 2. Our test calculations also show that within LDA+ Uapproach with U=4 eV, inclusion of SOC will increase the optimum lattice constant by only 0.7% and the bulk modulus by about0.5 GPa. Therefore, in our following calculations of pluto-nium dioxide, the SOC is not included. In present work, the theoretical equilibrium volume, bulk modulus B, and pressure derivative of the bulk modulus B /H11032 are obtained by fitting the energy-volume data with the third- order Birch-Murnaghan equation of state /H20849EOS /H20850.34In order to calculate elastic constants, a small strain is applied ontothe structure. For small strain /H9280, Hooke’s law is valid and the crystal energy E/H20849V,/H9280/H20850can be expanded as a Taylor series,35FIG. 1. /H20849Color online /H20850Dependence of the total energies /H20849per formula unit /H20850onUfor FM and AFM PuO 2.ZHANG, WANG, AND ZHAO PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-2E/H20849V,/H9280/H20850=E/H20849V0,0/H20850+V0/H20858 i=16 /H9268iei+V0 2/H20858 i,j=16 Cijeiej+O/H20849/H20853ei3/H20854/H20850, /H208491/H20850 where E/H20849V0,0/H20850is the energy of the unstrained system with the equilibrium volume V0,/H9280is strain tensor which has ma- trix elements /H9255ij/H20849i,j=1, 2, and 3 /H20850defined by /H9255ij=/H20898e11 2e61 2e5 1 2e6e21 2e4 1 2e51 2e4e3/H20899, /H208492/H20850 andCijare the elastic constants. For cubic structures, there are three independent elastic constants /H20849C11,C12, and C44/H20850. So, the elastic constants for fluorite PuO 2can be calculated from three different strains listed as follows: /H92801=/H20849/H9254,/H9254,/H9254,0,0,0 /H20850,/H92802=/H20849/H9254,0,/H9254,0,0,0 /H20850,/H92803 =/H208490,0,0, /H9254,/H9254,/H9254/H20850. /H208493/H20850 The strain amplitude /H9254is varied in steps of 0.006 from /H9254= −0.036 to 0.036 and the total energies E/H20849V,/H9254/H20850at these strain steps are calculated. After obtaining elastic constants, we cancalculate bulk and shear moduli from the Voigt-Reuss-Hill/H20849VRH /H20850approximations. 36The Voigt /H20849Reuss /H20850bounds on the bulk modulus BV/H20849BR/H20850and shear modulus GV/H20849GR/H20850for this cubic crystal system are deduced from the formulas of elasticmoduli in Ref. 37. As for cotunnite PuO 2, the nine indepen- dent elastic constants /H20849C11,C12,C13,C22,C23,C33,C44,C55, andC66/H20850can be obtained from nine different strains listed in the following: /H92801=/H20849/H9254,0,0,0,0,0 /H20850,/H92802=/H208490,/H9254,0,0,0,0 /H20850,/H92803 =/H208490,0,/H9254,0,0,0 /H20850, /H92804=/H208490,0,0, /H9254,0,0 /H20850,/H92805=/H208490,0,0,0, /H9254,0/H20850,/H92806 =/H208490,0,0,0,0, /H9254/H20850, /H92807=/H20849/H9254,/H9254,0,0,0,0 /H20850,/H92808=/H208490,/H9254,/H9254,0,0,0 /H20850,/H92809=/H20849/H9254,0,/H9254,0,0,0 /H20850 /H208494/H20850 and the formulas of elastic moduli in VRH approximations36 are from Ref. 38. Based on the Hill approximation,36B =1 2/H20849BR+BV/H20850andG=1 2/H20849GR+GV/H20850. The Young’s modulus Eand Poisson’s ratio /H9271are given by the following formulas: E=9BG //H208493B+G/H20850,/H9271=/H208493B−2G/H20850//H208512/H208493B+G/H20850/H20852. /H208495/H20850 In addition, the elastic properties of a solid can also be related to thermodynamical parameters especially specificheat, thermal expansion, Debye temperature, melting point,and Grüneisen parameter. 39From this point of view, Debye temperature is one of fundamental parameters for solid ma-terials. Due to the fact that at low temperatures the vibra-tional excitations arise solely from acoustic vibrations, there-fore, the Debye temperature calculated from elastic constants is the same as that determined from specific-heat measure-ments. The relation between the Debye temperature /H20849 /H9258D/H20850and the average sound-wave velocity /H20849/H9271m/H20850is /H9258D=h kB/H208733n 4/H9266/H9024/H208741/3 /H9271m, /H208496/H20850 where handkBare Planck and Boltzmann constants, respec- tively, nis the number of atoms in the molecule and /H9024is molecular volume. The average wave velocity in the poly-crystalline materials is approximately given as /H9271m=/H208751 3/H208732 /H9271t3+1 /H9271l3/H20874/H20876−1 /3 , /H208497/H20850 where /H9271t=/H20881G//H9267/H20849/H9267is the density /H20850and/H9271l=/H20881/H208493B+4G/H20850/3/H9267are the transverse and longitudinal elastic wave velocities of thepolycrystalline materials, respectively. III. RESULTS A. Atomic and electronic structures of fluorite PuO 2 We report in Table Ithe lattice parameter /H20849a0/H20850, bulk modu- lus /H20849B/H20850, and pressure derivative of the bulk modulus /H20849B/H11032/H20850for AFM and FM PuO 2obtained in LDA+ Uand GGA+ U frameworks. All these values are determined by EOS fitting.For comparison, the experimental values of a 0/H20849Ref.5/H20850andB /H20849Ref. 40/H20850are also listed. In the overall view, the dependence of the lattice parameter a0of AFM PuO 2onUis well con- sistent with our previous study16either in LDA+ Uscheme or in GGA+ Uscheme. For the LDA /GGA+ Uapproaches, the calculated a0improves upon the pure LDA/GGA by steadily increasing its amplitude with U. Actually, at a typi- cal value U=4 eV, the LDA+ Ugives a0=5.362 Å, which is very close to the experimental value of 5.398 Å and theGGA+ Ugives a 0=5.466 Å. Note that recent /H20849Perdew- Burke-Ernzerhof /H20850PBE+ U/H20849Ref. 19/H20850and LDA+ U/H20849Ref. 18/H20850 calculations with U=4.0 eV and J=0.7 eV predicted the lattice parameter of AFM PuO 2to be 5.444 Å and 5.354 Å, respectively, and the HSE /H20849Ref. 14/H20850and SIC-LSD /H20849Ref. 15/H20850 calculations gave the values to be 5.385 Å and 5.44 Å, re-spectively. On the other hand, as shown in Table I, the ten- dency of a 0with Ufor FM PuO 2is similar to that of the AFM phase. At a typical value U=4 eV, the LDA /GGA +Ugives a0=5.338 and 5.452 Å. These values are slightly smaller than those of the AFM phase. Previous HSE /H20849Ref. 14/H20850calculation of FM PuO 2gave the lattice parameter to be 5.387 Å. As for the dependence of bulk modulus Bof AFM and FM PuO 2onU, it is clear that the LDA results are always higher than the GGA results, which is due to theoverbinding effect of the LDA approach. While the bulkmodulus Bof AFM phase increases steadily with increasing the amplitude of U, the bulk modulus Bof FM phase will decrease along with the increasing of Hubbard U. At a typi- cal value U=4 eV, the LDA+ Uand GGA+ Ugive B =225 GPa and 193 GPa for AFM phase, respectively, andB=218 GPa and 182 GPa for FM phase, respectively. Ap- parently, the GGA+ Uapproach gives more close values to the experimental data of B=178 GPa. 40In our present study,GROUND-STATE PROPERTIES AND HIGH-PRESSURE … PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-3as shown in Table II, the bulk modulus Bdeduced from elastic constants turns out to be very close to that obtained byEOS fitting. This indicates that our calculations are consis-tent and reliable. In addition, as listed in Table II, recent PBE+ U/H20849Ref. 19/H20850and LDA+ U/H20849Ref. 18/H20850calculations with U=4.0 eV and J=0.7 eV predicted the bulk modulus Bof AFM PuO 2to be 199 GPa and 226 GPa, respectively, and the HSE /H20849Ref. 14/H20850and SIC-LSD /H20849Ref. 15/H20850calculations gave the bulk modulus to be 221 GPa and 214 GPa, respectively. Asfor pressure derivative of the bulk modulus /H20849B /H11032/H20850, all our cal- culated values are approximately equal to 4.5. Overall, com-paring with the experimental data and recent theoretical re-sults, the accuracy of our atomic-structure prediction forAFM PuO 2is quite satisfactory by tuning the effective Hub- bard parameter Uin a range of 3–4 eV within theLDA /GGA+ Uapproaches, which supplies the safeguard for our following study of electronic structure and mechanicalproperties of PuO 2. The total DOS as well as the projected DOS for the Pu 5 f and O 2 porbitals within LDA, GGA, LDA+ U, and GGA +Uformalisms are shown in Fig. 2. Clearly, our results re- produce all the features included in our previous work.16In particular, we recover the main conclusion that although thepure LDA and GGA fail to depict the electronic structure,especially the insulating nature and the occupied-state char-acter of PuO 2, by tuning the effective Hubbard parameter in a reasonable range, the LDA /GGA+ Uapproaches can prominently improve upon the pure LDA/GGA calculationsand, thus, can provide a satisfactory qualitative electronic-structure description comparable with the photoemissionTABLE I. Calculated lattice parameters /H20849a0/H20850, bulk modulus /H20849B/H20850, and pressure derivative of the bulk modulus /H20849B/H11032/H20850for AFM and FM PuO 2 at 0 GPa. For comparison, experimental values are also listed. Magnetism Method Property U=0 U=1 U=2 U=3 U=4 U=5 U=6 Expt. AFM LDA+ Ua 0/H20849Å/H20850 5.275 5.313 5.338 5.351 5.362 5.371 5.378 5.398a B/H20849GPa /H20850 218 208 224 224 225 226 227 178b B/H11032 4.1 3.7 4.3 4.3 4.3 4.3 4.3 GGA+ Ua 0/H20849Å/H20850 5.396 5.433 5.446 5.457 5.466 5.473 5.480 B/H20849GPa /H20850 185 188 191 192 193 194 195 B/H11032 4.3 3.8 4.4 4.5 4.5 4.5 4.4 FM LDA+ Ua 0/H20849Å/H20850 5.270 5.290 5.309 5.325 5.338 5.350 5.361 B/H20849GPa /H20850 230 224 221 220 218 215 212 B/H11032 4.4 4.4 4.3 4.4 4.4 4.4 4.5 GGA+ Ua 0/H20849Å/H20850 5.384 5.405 5.424 5.439 5.452 5.464 5.476 B/H20849GPa /H20850 193 188 186 184 182 179 174 B/H11032 4.5 4.4 4.5 4.5 4.7 4.7 4.8 aReference 5. bReference 40. TABLE II. Calculated elastic constants, various moduli, Poisson’s ratio /H20849/H9271/H20850, spin moments /H20849/H9262mag /H20850, and energy band gap /H20849Eg/H20850forFm3¯m AFM PuO 2at 0 GPa. For comparison, experimental values and other theoretical results are also listed. MethodC11 /H20849GPa /H20850C12 /H20849GPa /H20850C44 /H20849GPa /H20850B /H20849GPa /H20850G /H20849GPa /H20850E /H20849GPa /H20850 /H9271/H9262mag /H20849/H9262B/H20850Eg /H20849eV/H20850 LDA+ U/H20849U=0/H20850 386.6 136.5 71.9 220 89.9 237.3 0.320 3.937 0.0 LDA+ U/H20849U=4/H20850 319.6 177.8 74.5 225 73.0 197.7 0.354 4.126 1.5 GGA+ U/H20849U=0/H20850 343.9 112.3 53.7 190 73.5 195.1 0.328 4.085 0.0 GGA+ U/H20849U=4/H20850 256.5 167.9 59.2 197 52.7 145.2 0.377 4.165 1.5 Expt. 178a1.8b PBE+ Uc199 3.89 2.2 LDA+ Ud226 1.7 HSEe221 3.4 SIC-LSDf214 1.2 aReference 40. bReference 10. cReference 19. dReference 18. eReference 14. fReference 15.ZHANG, WANG, AND ZHAO PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-4experiments.6,7In this study, the insulating energy band gap /H20849Eg/H20850is of 1.5 eV at U=4 eV within LDA /GGA+ Uap- proaches /H20849see Table II/H20850. In Table II, the experimental10value and previous theoretical calculations14,15,18,19results are also listed for comparison. Obviously, the HSE calculations resultin a larger E gby /H110111.9 eV /H20849Ref. 14/H20850and our calculated re- sults are well consistent with the measured value and othertheoretical results. The calculated amplitude of local spinmoment is /H110114.1 /H9262B/H20849per Pu atom /H20850for AFM PuO 2within the two DFT+ Uschemes and this value is comparable to previ- ous LDA /PBE+ U/H20849Ref. 19/H20850results of /H110113.9/H9262B. To understand the chemical-bonding characters of fluorite PuO 2, we present in Fig. 3the crystal structure of its cubic unit cell and the charge-density map of the /H20849110 /H20850plane cal- culated in LDA+ Uformalism with U=4 eV for AFM phase. Evidently, the charge density around Pu and O ionsare all near spherical distribution with slightly deformed to-ward the direction to their nearest-neighboring atoms andthere are clear covalent bridges between Pu and O ions. Inorder to describe the ionic/covalent character quantitativelyand more clearly, we plot the line charge-density distributionalong the nearest Pu-O bonds /H20849not shown /H20850. A minimum value of charge density /H208490.53 e/Å 3/H20850along the Pu-O bonds is ob- tained and is listed in Table III. For comparison, we have also calculated some corresponding properties of AFM UO 2 within the LDA+ Uformalism. Parameters of the Hubbard term are taken as U=4.51 eV and J=0.51 eV, which had been checked carefully by Dudarev et al.11–13In following study, results of UO 2are all calculated using these Hubbard parameters either for ground-state Fm3¯mphase or high-pressure Pnma phase. The lattice parameter a0=5.449 Å and bulk modulus B=220.0 GPa for Fm3¯mUO 2obtained by EOS fitting are in perfect agreement with results of recentLDA+ Ucalculation 18/H20849a0=5.448 Å and B=218 GPa /H20850and experiments40,41/H20849a0=5.47 Å and B=207 GPa /H20850. Charge- analysis results of Fm3¯mUO 2are listed in Table III. Clearly, the minimum values of charge density for PuO 2and UO 2, comparable to that along the Np-O bonds included in ourprevious study of NpO 2,31are prominently larger than that along the Th-O bonds /H208490.45 e/Å3/H20850in ThO 2.25This indicates that the Pu-O, U-O, and Np-O bonds have stronger cova-lency than the Th-O bonds. And this conclusion is in goodaccordance with previous HSE study of the covalency inRef. 2, in which significant orbital mixing and covalency in the intermediate region /H20849PuO 2-CmO 2/H20850of actinide dioxides was showed by the increasing 5 f-2porbital energy degen- eracy. Besides, the A-O /H20849A=Pu, Np, U, or Th /H20850bond distances calculated in present study and our previous works25,31are listed in Table III. Obviously, the Pu-O, Np-O, and U-O bond distances are all smaller than the Th-O bond distance. Thisfact illustrates that the covalency of AO 2has tight relation with their bond distances, thus influences their macroscopi-cal properties, such as hardness and elasticity. To see theionicity of AO 2, results from the Bader analysis22,23are pre- sented in Table III. The charge /H20849QB/H20850enclosed within the Bader volume /H20849VB/H20850is a good approximation to the total elec- TABLE III. Calculated charge and volumes according to Bader partitioning as well as the A-O distances and correlated minimum values of charge density along the A-O bonds for actinide dioxides. CompoundQB/H20849A/H20850 /H20849e/H20850QB/H20849O/H20850 /H20849e/H20850VB/H20849A/H20850 /H20849Å3/H20850VB/H20849O/H20850 /H20849Å3/H20850A-O /H20849Å/H20850Charge density min /H20849e/Å3/H20850 PuO 2 13.60 7.20 15.36 11.58 2.32 0.53 NpO 2 12.52 7.24 15.69 11.83 2.34a0.51a UO 2 11.44 7.28 16.35 12.05 2.36 0.52 ThO 2 9.34 7.33 17.67 13.35 2.43b0.45b aReference 31. bReference 25.FIG. 2. /H20849Color online /H20850The total DOS for the PuO 2AFM phase computed in the /H20849a/H20850LDA, /H20849b/H20850LDA+ U/H20849U=4/H20850,/H20849c/H20850GGA, and /H20849d/H20850 GGA+ U/H20849U=4/H20850formalisms. The projected DOSs for the Pu 5 fand O2porbitals are also shown. The Fermi energy level is set at zero. FIG. 3. /H20849Color online /H20850Cubic unit cell for PuO 2in space group Fm3¯m. The larger blue spheres represent plutonium atoms and the smaller white O atoms. A slice of the isosurfaces of the electrondensity for AFM phase in /H20849110 /H20850plane calculated in the LDA+ U formalisms with U=4 eV is also presented /H20849in unit of e/Å 3/H20850GROUND-STATE PROPERTIES AND HIGH-PRESSURE … PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-5tronic charge of an atom. Note that although we have in- cluded the core charge in charge density calculations, sincewe do not expect variations as far as the trends are con-cerned, only the valence charge are listed. Here, same withour previous study, we apply U=4.6 eV and J=0.6 eV for the Np 5 forbitals in calculations of NpO 2. Based on the data in Table IIIand considering the valence electron numbers of Pu, Np, U, Th, and O atoms /H2084916, 15, 14, 12, and 6, respec- tively /H20850, in our study of AO2, we find that about 2.40, 2.48, 2.56, and 2.66 electrons transfer from each Pu, Np, U, or Thatom to O atom, respectively. This indicates that the ionicityofAO 2shows decreasing trend with increasing Z. B. Mechanical properties 1. Elastic properties Our calculated elastic constants, various moduli, and Poisson’s ratio /H9271of the Fm3¯mPuO 2at 0 GPa are collected in Table IIand those of the Pnma phase are listed in Table IV. Obviously, the Fm3¯mphase of PuO 2is mechanically stable due to the fact that its elastic constants satisfy the followingmechanical stability criteria 35of cubic structure: C11/H110220, C44/H110220, C11/H11022/H20841C12/H20841, /H20849C11+2C12/H20850/H110220. /H208498/H20850 As for the high-pressure Pnma crystalline phase, we have optimized the structural parameters of its AFM phase at dif-ferent pressures within LDA+ Uformalism with the typical value of U=4 eV. To avoid the Pulay stress problem, we perform the structure-relaxation calculations at fixed vol-umes rather than constant pressures. Note that different from the structure optimization of the ground-state Fm3 ¯mphase, the coordinates of atoms and the cell shape of the Pnma phase are necessary to be optimized due to its internal de-grees of freedom. After fitting the energy-volume data to theEOS, we obtain the optimized structural lattice parameters a, b, and cfor the Pnma PuO 2AFM phase at 0 GPa to be 5.889 Å, 3.562 Å, and 6.821 Å, respectively, giving V =143.1 Å3. This volume is smaller than the equilibrium vol- ume of 154.2 Å3for the Fm3¯mphase. This feature implies that the Pnma phase will become stable under compression. The elastic constants listed in Table IVindicate that the Pnma PuO 2is also mechanically stable. Actually, they sat- isfy the following mechanical stability criteria35for the orthorhombic structure: C11/H110220, C22/H110220, C33/H110220, C44/H110220, C55/H110220, C66 /H110220,/H20851C11+C22+C33+2 /H20849C12+C13+C23/H20850/H20852/H110220, /H20849C11+C22−2C12/H20850/H110220,/H20849C11+C33−2C13/H20850/H110220, /H20849C22+C33−2C23/H20850/H110220. /H208499/H20850 One can see from Table IVthat the calculated C12,C23, and C13are largely smaller than C11,C22, and C33. Therefore, the mechanical-stability criteria is easily satisfied. For compari-son, we have also calculated the elastic properties of UO 2in its ground-state fluorite phase and high-pressure cotunnite phase within LDA+ Uformalism. For Fm3¯mphase, C11 =389.3 GPa, C12=138.9 GPa, C44=71.3 GPa, B =222.4 GPa, G=89.5 GPa, E=236.8 GPa, and Poisson’s ratio/H9271=0.323. Results for Pnma UO 2are presented in Table IV. Clearly, both fluorite and cotunnite phases of UO 2are mechanically stable. Moreover, comparing results of bulkmodulus B, shear modulus G, Young’s modulus E, and Pois- son’s ratio /H9271for fluorite phase and cotunnite phase, they are almost equal to each other for PuO 2. Particularly, the bulk modulus Bis only smaller by about 5 GPa for Pnma phase compared to that of Fm3¯mphase. As for UO 2, bulk modulus of cotunnite phase is smaller by approximately 9%, shearmodulus and Young’s modulus about 23%, compared tothose of fluorite structure. Besides, in our previous study ofThO 2/H20849Ref. 25/H20850we find that the bulk modulus, shear modu- lus, and Young’s modulus of cotunnite ThO 2are all smaller by approximately 25% compared to those of fluorite ThO 2. Therefore, after comparing the bulk modulus Bfor the two phases of ThO 2,U O 2, and PuO 2, we find that the difference is decreasing along with increasing of 5 felectrons. This trend is understandable. Although the softening in bulkmodulus upon phase transition for these three systems issimilar, the increase in 5 felectrons of actinide metal atoms will lead to more covalency and thus reduce the bulk modu-lus difference between the two phases of the actinide diox-ides across the series. To study the anisotropy of the linearbulk modulus for Pnma PuO 2and UO 2, we calculate the directional bulk modulus along the a,b, and caxes by the following equations:39 Ba=/H9011 1+/H9251+/H9252,Bb=Ba /H9251,Bc=Ba /H9252, /H2084910/H20850 where /H9011=C11+2C12/H9251+C22/H92512+2C13/H9252+C33/H92522+2C23/H9251/H9252. For orthorhombic crystals,TABLE IV. Calculated elastic constants, elastic moduli, pressure derivative of the bulk modulus B/H11032and Poisson’s ratio /H9271for cotunnite- type PuO 2and UO 2at 0 GPa. Except B/H11032and/H9271, all other values are in units of gigapascal. Compound C11 C22 C33 C44 C55 C66 C12 C23 C13 BB /H11032 Ba Bb Bc GE /H9271 PuO 2 355.6 327.4 336.3 36.0 81.5 96.7 140.5 199.0 141.2 219.9 5.7 598.5 669.5 720.3 73.0 197.2 0.351 UO 2 338.2 335.1 325.9 24.4 75.6 92.0 119.9 148.4 142.1 202.2 5.5 586.4 593.8 641.9 68.0 183.4 0.349ZHANG, WANG, AND ZHAO PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-6/H9251=/H20849C11−C12/H20850/H20849C33−C13/H20850−/H20849C23−C13/H20850/H20849C11−C13/H20850 /H20849C33−C13/H20850/H20849C22−C12/H20850−/H20849C13−C23/H20850/H20849C12−C23/H20850 /H2084911/H20850 and /H9252=/H20849C22−C12/H20850/H20849C11−C13/H20850−/H20849C11−C12/H20850/H20849C23−C12/H20850 /H20849C22−C12/H20850/H20849C33−C12/H20850−/H20849C12−C23/H20850/H20849C13−C23/H20850 /H2084912/H20850 are defined as the relative change in the bandcaxes as a function of the deformation of the aaxis. After calculation, we obtain values of /H9251and/H9252to be 0.8939 /H208490.9875 /H20850and 0.8309 /H208490.9135 /H20850for PuO 2/H20849UO 2/H20850, respectively. Results of Ba, Bb, and Bcare presented in Table IV. Clearly, results of the directional bulk moduli show that both Pnma PuO 2and UO 2 are easily compressed along aaxis at 0 GPa. The longest axis cis the hardest axis for these two actinide dioxides. Directional bulk moduli of PuO 2are all bigger than the cor- responding values of UO 2. Moreover, using results of elastic constants included in previous study,25the directional bulk moduli Ba,Bb, and BcofPnma ThO 2are calculated to be 528.3 GPa, 406.3 GPa, and 415.8 GPa, respectively. Thisillustrates that, in contrary to Pnma PuO 2and UO 2, the Pnma ThO 2is relatively harder to be compressed along a axis compared to other two axes. And all three directionalbulk moduli of Pnma ThO 2are apparently smaller than that ofPnma PuO 2and UO 2, which indicates relative weaker covalency of ThO 2compared with PuO 2and UO 2in their high-pressure phase. 2. Debye temperature In study of the sound velocities and Debye temperature, our calculated results of AFM PuO 2and AFM UO 2in their fluorite phase and high-pressure cotunnite phase are pre-sented in Table V. For comparison, results of NpO 2and ThO 2calculated by using the elastic data included in our previous studies25,31are also listed. As seen from Table V,i n their Fm3¯mstructure, Debye temperature and sound velocity of UO 2, NpO 2, and ThO 2are higher than that of PuO 2. Thisis interestingly associated with the fact that Debye tempera- ture /H20849/H9258D/H20850and Vickers hardness /H20849H/H20850of materials follow an empirical relationship:42 /H9258D/H11008H1/2/H90241/6M−1 /2, /H2084913/H20850 where Mis molar mass and /H9024is molecular volume. Al- though Fm3¯mPuO 2has close value of hardness compared with UO 2and NpO 2, it has relatively smaller molecular vol- ume and larger molar mass, as a consequence, has a lowerDebye temperature than that of UO 2and NpO 2. As for the Pnma structure, Debye temperature and sound velocity of PuO 2,U O 2, and ThO 2are lower than those of their Fm3¯m structure. This is since that the Pnma PuO 2,U O 2, and ThO 2 have smaller values of volume, bulk and shear moduli com- pared to their ground-state Fm3¯mstructure. 3. Hardness Hardness is also one fundamental physical quantity when considering the phase stability and mechanical properties.According to the hardness conditions, 43the hardness is closely related to interatomic distances, number of nearestneighbors, directional bonding, anisotropy, and the indenterorientation in the structures. To date there is still no availablecalculation method involving hardness anisotropy in differ-ent dimensions in the literature. In spite of that, however,recently a semiempirical approach of hardness was raised bySimunek and Vackar 44,45in terms of the atomistic bond- strength properties. This approach has been successfullytested by the authors on the more than 30 binary structureswith zinc blende, cubic fluorite, rocksalt crystals, and alsofor highly covalent crystals. 44,45There is no need for all those high-symmetry structures to consider the anisotropy.Also, this method has been applied to the crystals involvingcovalent and ionic bonding characters, such as in the com-pounds of transition metal and light atoms, and generalizedto the complex structure with more than two different bondstrengths. 44,45Moreover, in study of covalent crystals, the results of the method raised by Simunek and Vackar44,45 agree well with those of another hardness method of Gao et al.46Therefore, the hardness of optimized cubic fluorite structures of actinide dioxides can be calculated by theTABLE V. Calculated density /H20849in g /cm3/H20850, transverse /H20849/H9271t/H20850, longitudinal /H20849/H9271l/H20850, and average /H20849/H9271m/H20850sound velocities /H20849in m/s /H20850derived from bulk and shear modulus, and Debye temperature /H20849in K /H20850for actinide dioxides. Results of NpO 2and ThO 2are calculated by using the elastic data included in our previous studies /H20849Refs. 25 and31/H20850. Compound Phase /H9267 /H9271t /H9271l /H9271m /H9258D PuO 2 Fm3¯m 11.892 2477.6 5206.3 2787.0 354.5 PuO 2 Pnma 12.812 2387.0 4976.0 2683.9 350.0 NpO 2 Fm3¯m 11.347 2835.0 5566.5 3176.8 401.2 UO 2 Fm3¯m 11.084 2841.8 5552.7 3183.4 398.1 UO 2 Pnma 11.957 2384.2 4948.5 2680.2 343.7 ThO 2 Fm3¯m 9.880 2969.1 5575.5 3317.3 402.6 ThO 2 Pnma 10.505 2504.6 4738.4 2799.8 346.8GROUND-STATE PROPERTIES AND HIGH-PRESSURE … PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-7method of Simunek and Vackar.44,45In the case of two atoms 1 and 2 forming one bond of strength s12in a unit cell of volume /H9024, the expression for hardness has the form44 H=/H20849C//H9024/H20850b12s12e−/H9268f2, /H2084914/H20850 where s12=/H20881/H20849e1e2/H20850//H20849n1n2d12/H20850,ei=Zi/ri /H2084915/H20850 and f2=/H20873e1−e2 e1+e2/H208742 =1− /H20851/H20881/H20849e1e2/H20850//H20849e1+e2/H20850/H208522/H2084916/H20850 are the strength and ionicity of the chemical bond, respec- tively, and d12is the interatomic distance; C=1550 and /H9268 =4 are constants. The radius riis chosen to make sure that the sphere centered at atoms iin a crystal contains exactly the valence electronic charge Zi. For fluorite structure PuO 2, b12=32 counts the interatomic bonds between atoms Pu /H208491/H20850 and O /H208492/H20850in the unit cell, n1=8 and n2=4 are coordination numbers of atoms Pu and O, respectively, r1=1.698 /H20849Å/H20850and r2=1.005 /H20849Å/H20850are the atomic radii for Pu and O atoms, re- spectively, Z1=16 and Z2=6 are valence charge for Pu and O atoms, respectively, d12=2.32 /H20849Å/H20850is the interatomic dis- tance, and /H9024=154.16 /H20849Å3/H20850is the volume of unit cell. As for fluorite UO 2,b12=32, n1=8, n2=4, r1=1.737 /H20849Å/H20850,r2 =1.003 /H20849Å/H20850,Z1=14, Z2=6, d12=2.36 /H20849Å/H20850, and /H9024 =161.82 /H20849Å3/H20850. Using Eqs. /H2084914/H20850–/H2084916/H20850, we obtain s12=0.1010 andf2=0.0503 for PuO 2ands12=0.0919 and f2=0.0219 for UO 2. Therefore, the hardness of PuO 2at its ground-state fluorite structure can be described as H=26.6 /H20849GPa /H20850and for UO 2H=25.8 /H20849GPa /H20850. Clearly, the hardness of PuO 2, almost equal to the hardness of NpO 2/H2084926.5 GPa /H20850,31is slightly larger than that of UO 2. Besides, these three values of hardness are all larger than that of ThO 2/H2084922.4 GPa /H20850.25 4. Theoretical tensile strength Although many efforts have been paid on PuO 2, little is known on its theoretical tensile strength. The ideal strengthof materials is the stress that is required to force deformationor fracture at the elastic instability. Although the strength ofa real crystal can be changed by the existing cracks, disloca-tions, grain boundaries, and other microstructural features,its theoretical value can never be raised, i.e., the theoreticalstrength sets an upper bound on the attainable stress. Here,we employ a first-principles computational tensile test /H20849FPCTT /H20850/H20849Ref. 47/H20850to calculate the stress-strain relationship and obtain the ideal tensile strength by deforming the PuO 2 crystals to failure. The anisotropy of the tensile strength istested by pulling the initial fluorite structure along the /H20851001 /H20852, /H20851110 /H20852, and /H20851111 /H20852directions. As shown in Fig. 4, three geo- metric structures are constructed to investigate the tensilestrengths in the three typical crystallographic directions: Fig.4/H20849a/H20850shows a general fluorite structure of PuO 2with four Pu and eight O atoms; Fig. 4/H20849b/H20850a body-centered tetragonal unit cell with two Pu and four O; and Fig. 4/H20849c/H20850an orthorhombic unit cell with six Pu and 12 O. In FPCTT, the tensile stress iscalculated according to the Nielsen-Martin scheme 48/H9268/H9251/H9252 =1 /H9024/H11509Etotal /H11509/H9255/H9251/H9252, where /H9255/H9251/H9252is the strain tensor /H20849/H9251,/H9252=1,2,3 /H20850and/H9024 is the volume at the given tensile strain. Tensile process along the /H20851001 /H20852,/H20851110 /H20852, and /H20851111 /H20852directions is implemented by increasing the lattice constants along these three orienta-tions step by step. At each step, the structure is fully relaxeduntil all other five stress components vanished except that inthe tensile direction. The calculated total energy, stress, and spin moments as functions of uniaxial tensile strain for AFM PuO 2in the /H20851001 /H20852,/H20851110 /H20852, and /H20851111 /H20852directions are shown in Fig. 5. Clearly, all three energy-strain curves increase with increas-ing tensile strain, but one can easily find the inflexions byperforming differentiations. Actually, at strains of 0.36, 0.22,and 0.18, the stresses reach maxima 81.2, 28.3, and 16.8 GPafor pulling in the /H20851001 /H20852,/H20851110 /H20852, and /H20851111 /H20852directions, respec- tively. For clear comparison, all these maximum stresses/H20849i.e., the theoretical tensile strengths in the three typical crys- talline orientations /H20850and the corresponding strains are listed in Table VI. Our results indicate that the /H20851001 /H20852direction is the strongest tensile direction and /H20851111 /H20852the weakest. In fact, there are eight Pu-O bonds per formula unit for fluorite FIG. 4. /H20849Color online /H20850Schematic illustration of tension along /H20849a/H20850 /H20851001 /H20852,/H20849b/H20850/H20851110 /H20852, and /H20849c/H20850/H20851111 /H20852orientations. The AFM order is indi- cated by white arrows attached on Pu atoms. FIG. 5. /H20849Color online /H20850Dependence of the /H20849a/H20850total energy /H20849per formula unit /H20850,/H20849b/H20850stress, and /H20849c/H20850spin moments on tensile strain for AFM PuO 2in the /H20851001 /H20852,/H20851110 /H20852, and /H20851111 /H20852directions.ZHANG, WANG, AND ZHAO PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-8PuO 2. The angle of all eight bonds with respect to the pulling direction is 45° in /H20851001 /H20852direction. However, in /H20851110 /H20852direc- tion only four bonds make an angle of 45° with the pullingdirection. Four other bonds are vertical to the pulling direc-tion. In /H20851111 /H20852direction, two bonds are parallel to the pulling direction and six others make an angle of about 19.5° withthe pulling direction. The bonds vertical to the pulling direc-tion have no contributions on the tensile strength and thebonds parallel to the pulling direction are easy to fractureunder tensile deformation. Therefore, the fact that the tensilestrength along the /H20851001 /H20852direction is stronger than that along other two directions is understandable. Besides, we note thatthe stress in /H20851110 /H20852direction experiences an abrupt decrease process after strain up to 0.24. This is due to the fact that thecorresponding four Pu-O bonds /H20849make an angle of 45° with the pulling direction /H20850have been pulled to fracture. The frac- ture behaviors have been clarified by plotting valence-electron charge-density maps /H20849not shown /H20850. Under the same strain, the abrupt increase in spin moment can be clearly seen/H20851Fig.5/H20849c/H20850/H20852. While the spin moments in /H20851110 /H20852and /H20851111 /H20852direc- tions only increase from 4.13 to 4.23 /H9262Band 4.33 /H9262B, re- spectively, the spin moments in /H20851001 /H20852direction increase up to 5.27/H9262Bat the end of tensile deformation. In addition, the evolutions of the lattice parameters with strain in all threetensile processes are presented in Fig. 6. One can see from Fig.6that along with the increase in the lattice parameter in the pulling direction, other two lattice parameters vertical tothe pulling direction are decrease smoothly. In /H20851001 /H20852direc- tion, the evolutions of the lattice parameters along /H20851100 /H20852and /H20851010 /H20852directions are absolutely same due to the structural symmetry. For all three tensile deformations, no structuraltransition has been observed in our present FPCTT study. C. Phonon dispersion and thermodynamic properties of fluorite PuO 2 To our knowledge, no experimental phonon frequency re- sults have been published for PuO 2. In 2008, Yin andSavrasov21successfully obtained the phonon dispersions of both UO 2and PuO 2by employing the LDA+DMFT scheme. They found that the dispersive longitudinal optical /H20849LO /H20850 modes do not participate much in the heat transfer due totheir large anharmonicity and only longitudinal acousticmodes having large phonon group velocities are efficientheat carriers. In 2009, Minamoto et al. 49investigated the thermodynamic properties of PuO 2based on their calculated phonon dispersion within the pure GGA scheme. In presentwork, we have calculated the Born effective charges Z /H11569and dielectric constants /H9255/H11009of PuO 2before phonon-dispersion calculation due to their critical importance to correct theLO-TO splitting. For fluorite PuO 2, the effective charge ten- sors for both Pu and O are isotropic because of their positionsymmetry. After calculation, the Born effective charges of Pu and O ions for AFM PuO 2are found to be ZPu/H11569=5.12 and ZO/H11569=−2.56, respectively, within LDA+ Uformalism with the choice of U=4.0 eV. In addition, the macroscopic static di- electric tensor is also isotropic and our computed value ofdielectric constant /H9255 /H11009is 5.95 for the AFM phase. As for phonon dispersion, the Hellmann-Feynman theorem and thedirect method 50are employed to calculate the phonon curves along /H9003-X-K- /H9003-L-X-W-L directions in the BZ together with the phonon density of states. We use 2 /H110032/H110032 fcc supercell containing 96 atoms and 3 /H110033/H110033 Monkhorst-Pack k-point mesh for the BZ integration. In order to calculate theHellmann-Feynman forces, we displace four atoms /H20849two Pu and two O atoms /H20850from their equilibrium positions and the amplitude of all the displacements is 0.02 Å. The calculated phonon-dispersion curves are displayed in Fig.7for the AFM phase of PuO 2. In the fluorite primitive cell, there are only three atoms /H20849one Pu and two O atoms /H20850. Therefore, nine phonon modes exist in the dispersion rela-tions. One can see that the LO-TO splitting at /H9003point be- comes evident by the inclusion of polarization effects. Inaddition, due to the fact that plutonium atom is heavier thanoxygen atom, the vibration frequency of plutonium atom islower than that of oxygen atom. As a consequence, the pho-non density of states for PuO 2can be viewed as two parts. One is the part lower than 6.4 THz where the main contri-bution comes from the plutonium sublattice while the otherpart higher than 6.4 THz are dominated by the dynamics ofthe light oxygen atoms. Thermodynamic properties of AFM PuO 2calculated within LDA+ Uformalism with the choice of U=4.0 eV are determined by phonon calculation using the quasiharmonicapproximation, 51under which the Gibbs free energy G/H20849T,P/H20850 is written asTABLE VI. Calculated stress maxima and the corresponding strains in the tensile process. DirectionStress /H20849GPa /H20850 Strain /H20851001 /H20852 81.2 0.36 /H20851110 /H20852 28.3 0.22 /H20851111 /H20852 16.8 0.18 FIG. 6. /H20849Color online /H20850Dependence of the lattice parameters on tensile strain for AFM PuO 2in the /H20849a/H20850/H20851001 /H20852,/H20849b/H20850/H20851110 /H20852, and /H20849c/H20850/H20851111 /H20852 directions.GROUND-STATE PROPERTIES AND HIGH-PRESSURE … PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-9G/H20849T,P/H20850=F/H20849T,V/H20850+PV. /H2084917/H20850 Here, F/H20849T,V/H20850is the Helmholtz free energy at temperature T and volume Vand can be expressed as F/H20849T,V/H20850=E/H20849V/H20850+Fph/H20849T,V/H20850+Fel/H20849T,V/H20850, /H2084918/H20850 where E/H20849V/H20850is the ground-state total energy, Fph/H20849T,V/H20850is the phonon free energy, and Fel/H20849T,V/H20850is the thermal electronic contribution. In present study, we focus only on the contri-bution of atom vibrations. The F ph/H20849T,V/H20850can be calculated by Fph/H20849T,V/H20850=kBT/H20885 0/H11009 g/H20849/H9275/H20850ln/H208752 sinh/H20873/H6036/H9275 2kBT/H20874/H20876d/H9275, /H2084919/H20850 where /H9275=/H9275/H20849V/H20850represents the volume-dependent phonon fre- quencies and g/H20849/H9275/H20850is the phonon DOS. In calculation of F/H20849T,V/H20850, the ground-state total energy and phonon free energy should be calculated by constructingseveral 2 /H110032/H110032 fcc supercells. The temperature Tappears inF/H20849T,V/H20850via the phonon term only. Calculated free energy F/H20849T,V/H20850curves of PuO 2for temperature ranging from 0 up to 1500 K are shown in Fig. 8. The locus of the equilibrium lattice parameters at different temperature Tare also pre- sented. The equilibrium volume V/H20849T/H20850and the bulk modulus B/H20849T/H20850are obtained by EOS fitting. Figure 9shows the tem- perature dependence of the relative lattice parameter and thebulk modulus. Experimental results41are also plotted. Clearly, the lattice parameter increases approximately in thesame ratio as that in experiment. 41The bulk modulus B/H20849T/H20850is predicted to decrease along with the increase in temperature.This kind of temperature effect is very common for com-pounds and metals. Besides, the specific heat at constant vol-ume C Vcan be directly calculated through CV=kB/H20885 0/H11009 g/H20849/H9275/H20850/H20873/H6036/H9275 kBT/H208742exp/H6036/H9275 kBT /H20873exp/H6036/H9275 kBT−1/H208742d/H9275 /H2084920/H20850 while the specific heat at constant pressure CPcan be evalu- ated by the thermodynamic relationship CP−CV =/H9251V2/H20849T/H20850B/H20849T/H20850V/H20849T/H20850T, where the isobaric thermal-expansion co- efficient can be calculated according to the formula /H9251V/H20849T/H20850 =1 V/H20849/H11509V /H11509T/H20850P. Calculated CPand CVare presented in Fig. 10. Clearly, the calculated CVis in good agreement with experiment49in all investigating temperature domain while the predicted CPonly accords well with the corresponding experimental data52below 300 K due to the intrinsic fact thatFIG. 7. Phonon dispersion curves /H20849left panel /H20850and corresponding PDOS /H20849right panel /H20850for AFM PuO 2calculated within LDA+ Ufor- malism with U=4 eV. FIG. 8. Dependence of the free energy F/H20849T,V/H20850on crystal-lattice parameter afor a number of selected temperatures for AFM PuO 2 calculated within LDA+ Uformalism with U=4 eV.FIG. 9. /H20849Color online /H20850Temperature dependence of /H20849a/H20850relative lattice parameter /H20851a/H20849T/H20850−a/H20849300 /H20850/H20852/a/H20849300 /H20850, where a/H20849300 /H20850denotes the lattice parameter at 300 K and /H20849b/H20850bulk modulus B/H20849T/H20850of PuO 2. Experimental results from Ref. 41are also shown in panel /H20849a/H20850. FIG. 10. /H20849Color online /H20850Calculated specific heat at constant pres- sure /H20849CP/H20850and at constant volume /H20849CV/H20850. For comparison, previous experimental results from Refs. 49and52–54are also shown.ZHANG, WANG, AND ZHAO PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-10near zero-temperature CPand CVapproach to the same value. The disagreement of CPbetween theory and experiments53,54in high-temperature domain is believed to mainly originate from the quasiharmonic approximation weuse. D. High-pressure behavior of PuO 2 In the following, we will focus our attention on the be- havior of plutonium dioxide under hydrostatic compression. Two experimentally established structures, Fm3¯mandPnma phases, are investigated in detail. The relative energies /H20849per unit cell /H20850of the two phases at different volumes are calcu- lated and shown in Fig. 11. Obviously, the Fm3¯mphase is stable under ambient pressure while under high pressure the Pnma phase becomes stable. According to the rule of com- mon tangent of two energy curves, a phase transition at 24.3GPa is predicted by the slope shown in the inset of Fig. 11. Besides, we also determine the phase transition pressure bycomparing the Gibbs free energy as a function of pressure.At 0 K, the Gibbs free energy is equal to enthalpy H, ex- pressed as H=E+PV. The crossing between the two en- thalpy curves /H20849not shown /H20850also gives phase transition pres- sure of 24.3 GPa, which is fully consistent with above resultin terms of the common tangent rule. This value is smaller by/H1101115 GPa when compared to the experiment measurementby Dancausse et al. 8We notice that Li et al.55predicted a transition pressure of about 45 GPa employing the full-potential linear-muffin-tin-orbital method. However, theyonly considered the NM and FM phases in their calculations of the Fm3 ¯mandPnma PuO 2. No results were calculated for the AFM phase. In our present study, we first optimize the Pnma PuO 2at constant volume /H20849about 125.2 Å3/H20850of the ex- perimental value at 39 GPa.8Our calculated structural pa- rameters of the NM, FM, and AFM phases are tabulated inTable VII. Obviously, the AFM phase is the most stable phase and its structural parameters are well consistent withexperiment. As seen in Table VII, results of AFM phase at 49 GPa is also consistent with experiment. Therefore, we willonly consider the results of AFM phase in the followingdiscussion. In addition, obvious discrepancy between twoworks 8,40from the same experimental group involving the transition pressure of ThO 2was noticed in our previous study.25While the transition was reported first to start at 40 GPa,8later, through improving experimental measurement technique, which can capture the two phases cohabitationzone during transition process, Idiri et al. 40observed that the transition really begins around 33 GPa. Our previous study25 predicted that the phase transition of ThO 2started at around 26.5 GPa. Idiri et al.40also stated that the bulk modulus of PuO 2were largely overestimated in their previous work.8 Thus, we believe that their former report8also overestimated the transition pressure of PuO 2in some degree. So the pre- dicted transition pressure of PuO 2, which is very close to that of ThO 2,25is understandable. Figure 12compares the partial DOS /H20849PDOS /H20850of the Fm3¯m andPnma phases of PuO 2at a pressure of around 25 GPa, close to the transition pressure. One can see evident increasein the band gap from 1.51 eV in the fluorite phase to 1.65 eVin the cotunnite phase. In study of UO 2, Geng et al.24also predicted this similar behavior. However, our previous studyof ThO 2/H20849Ref. 25/H20850found that the band gap is reduced from Fm3¯mphase to Pnma phase. The reason is simply that ThO 2 is a charge-transfer insulator while PuO 2and UO 2are the Mott-type insulators. From Fig. 12, one can see that while O2sand Pu 6 pstates expand in the low bands, O 2 pand Pu 5 f/6dstates are mainly featured near the Fermi level and have prominent hybridization characters in a width of 6.3 /H208495.9/H20850eV for Fm3¯m/H20849Pnma /H20850phase. There is no evident dif- ference between the two phases in their 5 felectronic local- ization behavior. Therefore, our study of PuO 2supports ourFIG. 11. /H20849Color online /H20850Comparison of relative energy vs the cell volume for AFM PuO 2inFm3¯mandPnma phases. The total energy of Pnma phase at 0 GPa is set as zero. A phase transition at 24.3 GPa is predicted by the slope of the common tangent rule, asshown in the inset. TABLE VII. Calculated structural parameters /H20849in Å /H20850, pressure /H20849in GPa /H20850, and relative energy /H20849in eV /H20850with respect to the total energy at 0 GPa of the Pnma PuO 2at two constant volumes from experiment /H20849Ref.8/H20850. For comparison, the experimental values are also listed. Magnetism abc Pressure Relative energy NM 6.502 3.165 6.087 82.9 32.87 FM 7.478 3.088 5.424 124.2 31.80AFM 5.585 3.410 6.577 38.6 1.90Expt. 5.64 3.38 6.57 39Expt. 5.62 3.44 6.49 49AFM 5.492 3.398 6.539 48.8 2.80GROUND-STATE PROPERTIES AND HIGH-PRESSURE … PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-11previous25viewpoint: the phenomenon of volume collapse during high-pressure phase transition of the actinide dioxidesis mainly attributed to the ionic /H20849instead of electronic /H20850re- sponse to the external compression. The relative volume V/V 0evolution with pressure for PuO 2in both Fm3¯mandPnma phases are shown in Fig. 13. For comparison, the experimental8data are also shown in the figure. Clearly, our calculated P-Vequation of state is well consistent with the experimental measurement for the twophases of PuO 2. Specially, at the calculated transition pres- sure /H2084924.3 GPa /H20850, our result in Fig. 13gives that the volume collapse upon phase transition is 6.8%. This value is evi-dently underestimated compared with the experimental dataof 12%. 8The discrepancy between experiment and our cal- culation needs more experimental and theoretical works toexamine. After phase transition, we also find an isostructuraltransition occurring between 75 and 133 GPa for the Pnma phase. This isostructural transition of actinide dioxides wasfirst found in DFT calculations of UO 2by Geng et al.24and then observed in study of ThO 2.25The pressure dependence of the three lattice parameters /H20849with respect to their values at calculated transition pressure 24.3 GPa /H20850for the Pnma phase of PuO 2are plotted in Fig. 14. Similar to the studies of UO 2 and ThO 2,24,25in pressure region before 75 GPa, the re- sponses of the three relative lattice parameters to the com-pression are anisotropic in the sense that the compression ofthe middle axis ais most rapid compared to those of the long axis cand small axis b, which vary upon compression almost in the same tendency. When the pressure becomes higher tobe between 75 and 133 GPa, remarkably, it reveals in Fig. 14 that all the three relative lattice parameters undergo dramaticvariations by the fact that the small axis bhas a strong re- bound and the middle ais collapsed. When the pressure is beyond 133 GPa, then the variations in the three relativelattice parameters become smooth and approach isotropiccompression. This signifies a typical isostructural transitionfor the Pnma phase of PuO 2. Moreover, we also present in Fig. 14the evolution of the insulating band gap with pressure for Pnma phase of PuO 2. Apparently, the band gap behaves smooth in pressure regionof 25–92 GPa, then turns to decrease suddenly from about1.75 eV to zero under compression from 92 to 133 GPa. Thisclearly indicates that the Pnma phase will occur a metallic transition after external pressure exceeds 133 GPa. As Fig.15/H20849b/H20850shows, with increasing pressure in the crossover range between 92 and 133 GPa, the 5 felectrons in the cotunnite phase of PuO 2are more delocalized and the 5 fbands are largely broadened. As a result, the Mott-type band gap isnarrowed and even blurred, which is characterized by finiteoccupancies of O 2 pand Pu 5 forbitals at the Fermi level, by the increasing kinetic energy of 5 felectrons. In order to see the pressure behavior of electronic structure with differentvalues of Hubbard parameter Ufor cotunnite PuO 2, we have plotted in Fig. 15the total DOS and PDOS for the cotunnite PuO 2AFM phase calculated at 92, 133, and 248 GPa within LDA+ Uformalism with U=0, 4, and 6 eV. It clearly shows that /H20849i/H20850the pure LDA always produces incorrect electronic structure in the full pressure region we considered and /H20849ii/H20850it is not imperative to adjust Uas varying pressure by the fact revealed in Figs. 15/H20849b/H20850and15/H20849c/H20850that the electronic proper- ties and insulator-metal transition behavior at high pressuresshow the similar character for the two choices of Hubbardparameter U=4 eV and U=6 eV. Based on this observa- tion, in the above high-pressure calculations we have fixedthe value of Uto be 4 eV. It should be stressed that the metallic transition is not unique for PuO 2. Similar phenom- enon has also been observed in other actinide dioxides.24FIG. 12. /H20849Color online /H20850PDOS for /H20849a/H20850Fm3¯mphase and /H20849b/H20850 Pnma phase at around 25 GPa. The Fermi energy level is zero. The energy gaps of Fm3¯mphase and Pnma phase are 1.51 eV and 1.65 eV, respectively. FIG. 13. /H20849Color online /H20850Calculated compression curves of PuO 2 compared with experimental measurements. The volume collapses at our predicted phase transition point 24.3 GPa and experimental/H20849Ref. 8/H20850phase transition pressure 40 GPa are labeled.FIG. 14. /H20849Color online /H20850Pressure behavior of the relative lattice parameters of the Pnma phase, where the drastic change in the relative lattice constants /H20849region between dashed lines /H20850indicates an isostructural transition. Besides, the pressure behavior of the insu-lating band gap is also shown.ZHANG, WANG, AND ZHAO PHYSICAL REVIEW B 82, 144110 /H208492010 /H20850 144110-12Besides, the variation in the local magnetic moment of plu- tonium atoms is almost same for Fm3¯mand Pnma phases, implying that the magnetic property is insensitive to thestructure transition in PuO 2. Actually, the calculated ampli- tude of local spin moment varies from /H110114.1 to /H110113.8/H9262B/H20849per Pu atom /H20850for both fluorite and cotunnite phases in pressure range from 0 to 255 GPa. No paramagnetic transition for thismaterial has been observed in present study. IV . CONCLUSIONS In conclusion, the ground-state properties as well as the high-pressure behavior of PuO 2were studied by means of the first-principles DFT+ Umethod. By choosing the Hub- bard Uparameter around 4 eV within the LDA+ Uapproach, the electronic structure, lattice parameters, and bulk modulus were calculated for both the ambient Fm3¯mand the high- pressure Pnma phases of PuO 2and were shown to accord well with experiments. Results for UO 2were also presented for comparison. Based on these results, the Pu-O and U-Obonds were interpreted as displaying a mixed ionic/covalentcharacter by electronic-structure analysis. After comparingwith our previous calculations of NpO 2and ThO 2, we dem- onstrated that the Pu-O, U-O, and Np-O bonds have strongercovalency than the Th-O bond. The ionicity of Th-O bondwas found to be the largest among these four kinds of ac-tinide dioxides. In addition, the stability of the two phases atzero pressure was predicted through calculating elastic con-stants and phonon dispersion. The hardness, Debye tempera-ture, ideal tensile strength, and thermodynamic propertieswere calculated and analyzed to support the practical appli- cation of PuO 2. We showed that the hardness of Fm3¯mphase is/H1101127 GPa and the Debye temperatures of Fm3¯mand Pnma phases are 354.5 K and 350.0 K, respectively. ForFm3¯mPuO 2, the ideal tensile strengths are calculated within FPCTT to be 81.2 GPa, 28.3 GPa, and 16.8 GPa in tensiledeformations along the /H20851100 /H20852,/H20851110 /H20852, and /H20851111 /H20852directions, respectively. The volume thermal expansion and specific heatat constant volume curves are well consistent with availableexperiments. However, the discrepancy between measuredand our calculated specific heat at constant pressure in high-temperature domain is evident. This needs further theoreticalwork by including the anharmonic ionic contribution to de-crease this kind of discrepancy. In studying the pressure behavior of PuO 2, we showed that the Fm3¯m→Pnma transition occurs at 24.3 GPa. Al- though this value is large smaller than the experimental re-port, we believe that our calculated result is reasonable. Onereason is that the lattice parameters of Pnma PuO 2AFM phase calculated at around 39 and 49 GPa are precisely con-sistent with experiment. Another is the fact that the experi-ment needs improvement as having been indicated in studyof UO 2and ThO 2.40Furthermore, we extended the pressure up to 280 GPa for the two structures of PuO 2. A metallic transition at around 133 GPa and an isostructural transitionin pressure range of 75–133 GPa were predicted for the Pnma phase. Also, the calculated amplitude of local spin moment only varies from /H110114.1 to /H110113.8 /H9262B/H20849per Pu atom /H20850 for both fluorite and cotunnite phases in pressure range from0 to 255 GPa. No paramagnetic transition for this materialhas been observed. ACKNOWLEDGMENTS We gratefully thank G. H. Lu, H. B. Zhou, X. C. Li, and B. Sun for illustrating discussions. This work was supportedby the Foundations for Development of Science and Tech-nology of China Academy of Engineering Physics underGrant No. 2009B0301037.FIG. 15. /H20849Color online /H20850The total DOS for the cotunnite PuO 2AFM phase calculated at selected pressures within LDA+ Uformalism with /H20849a/H20850U=0 eV, /H20849b/H20850U=4 eV, and /H20849c/H20850U=6 eV. The projected DOSs for the Pu 5 fand O 2 porbitals are also shown. 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PhysRevB.97.155145.pdf

PHYSICAL REVIEW B 97, 155145 (2018) Chiral d-wave superconductivity in a triangular surface lattice mediated by long-range interaction Xiaodong Cao,1Thomas Ayral,2,3Zhicheng Zhong,1,4Olivier Parcollet,3Dirk Manske,1and Philipp Hansmann1,5 1Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany 2Physics and Astronomy Department, Rutgers University, Piscataway, New Jersey 08854, USA 3Institut de Physique Théorique (IPhT), CEA, CNRS, UMR 3681, 91191 Gif-sur-Yvette, France 4Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, 315201 Ningbo, China 5Institut für Theoretische Physik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen (Received 9 October 2017; revised manuscript received 18 December 2017; published 20 April 2018) Adatom systems on the Si(111) surface have recently attracted an increasing attention as strongly correlated systems with a rich phase diagram. We study these materials by a single band model on the triangularlattice, including 1 /rlong-range interaction. Employing the recently proposed TRILEX method, we find an unconventional superconducting phase of chiral d-wave symmetry in hole-doped systems. Contrary to usual scenarios where charge and spin fluctuations are seen to compete, here the superconductivity is drivensimultaneously by both charge andspin fluctuations and crucially relies on the presence of the long-range tail of the interaction. We provide an analysis of the relevant collective bosonic modes and predict how a cumulative charge and spin paring mechanism leads to superconductivity in doped silicon adatom materials. DOI: 10.1103/PhysRevB.97.155145 I. INTRODUCTION The search for materials with unconventional high temper- ature superconductivity (SC) has been one of the most activefields in correlated solid state physics since the discovery ofthe cuprate high T ccompounds. Sophisticated synthesis tech- nology nowadays allows for the construction of new materials like heterostructures or surface systems on an atomic length scale. Recently, many-body studies on experimentally well-controlled correlated adatom lattices X:Si(111) and X:Ge(111)with (X=Pb,Sn,C) led to interesting results [ 1–5] and allowed for the unification of the materials in a single phase diagram [ 3]. A first-principles derivation of the low-energy Hamiltonian ofthese systems [ 2,3] revealed sizable long-range interactions, explaining why the standard Hubbard model fails to capture the materials’ ground states or low-energy spin and charge fluctuations. Depending on the species of the adatom, some ofthe materials were shown to be in close vicinity to a triple pointbetween a Fermi liquid, a Mott insulator, and a charge-orderedinsulator. Sn:Si(111) and Pb:Si(111) in particular turned out tobe close to a charge-order Mott insulator phase transition withsizable charge fluctuations which, in the case of Sn:Si(111) were visible in core-level spectroscopy [ 5]. In complementary studies [ 4], the importance of spin fluctuations for Sn:Si(111) was emphasized. Silicon adatom systems with intrinsic long-range interactions are, hence, promising candidates to searchfor new physics like unconventional SC. For such systems, theoretical methods are needed which are capable to capture both local and non-local electroniccorrelations. Dynamical mean-field theory (DMFT) [ 6,7] has been proven to be a powerful approach to treat local cor- relations and Mott physics. If nonlocal interactions haveto be treated, extended DMFT (EDMFT) [ 8] captures their effects on the local self energy and charge polarization by aretarded onsite interaction. Local approximations like DMFTand EDMFT are, however, not sufficient when nonlocalfluctuations start to play an important role. To overcomethese shortcomings of DMFT, several extensions have been proposed [ 9,10]. Cluster extensions of DMFT in real and reciprocal space [ 10–13], e.g., are capable to treat nonlocal short-range fluctuations. Long-range fluctuations, on the otherhand, can be taken into account by DMFT +GW [ 14–16] or dual boson methods [ 17–20]. For our paper, we em- ploy the recently developed TRILEX approximation [ 21–24], which combines a balanced treatment of long-range spin andcharge fluctuations with comparatively little computationaleffort. In this paper, we show that the triangular lattice model for the adatom materials has a dome-shaped superconductingphase of chiral d-wave symmetry as a function of hole doping in realistic parameter regimes. The long-range interaction iskey for enhanced critical temperatures and distinguishes theadatom Hamiltonian from triangular Hubbard models [ 25– 32]. By analyzing spin- and charge-response functions (and their dependence on the materials’ long-range interaction), wefurther show that the pairing mechanism crosses over from acumulative spin/charge fluctuation character at small dopingsto a charge-dominated one at large doping. II. MODEL AND METHOD We make use of the extended Hubbard model on the triangular lattice with 1 /rinteractions, H=/summationdisplay i,j,σtijˆc† iσˆcjσ+1 2/summationdisplay i,jUijˆniˆnj−μ/summationdisplay iˆni, (1) to study the low-energy physics of adatom systems, following a first principles constrained random-phase approximation (cRPA) derivation [ 2] where ˆc† iσ(ˆciσ) are electron creation (annihilation) operators on site iwith spin σ=↑,↓.ˆni= ˆni↑+ˆni↓is the density operator on site i, and μis the chemical potential. tijandUijare the hopping integrals and long-range Coulomb interaction strength between sites iandj. 2469-9950/2018/97(15)/155145(5) 155145-1 ©2018 American Physical SocietyCAO, AYRAL, ZHONG, PARCOLLET, MANSKE, AND HANSMANN PHYSICAL REVIEW B 97, 155145 (2018) For translational invariant two-dimensional systems, the long- range Coulomb interaction, in momentum space, reads Uq= U0+V/summationtext i/negationslash=0eiq·Ri/|Ri|, where Riare real space coordinates, U0is the on-site interaction, and Vis the strength of the long-range interaction, respectively (Supplemental Material Ain Ref. [ 33] and Ref. [ 34]). More specifically, we adopt hop- ping parameters up to next-nearest-neighbors ( t=0.042 eV andt /prime=− 0.02 eV) from Refs. [ 2,3] derived from density- functional theory (DFT) for the Pb:Si(111) adatom system(closest to the triple point) and vary the interaction parametersin realistic regimes for the adatom materials found by cRPA [ 3]. TRILEX approximates the three-legged fermion-boson in- teraction vertex using a local self-consistent quantum im-purity model. For systems retaining SU(2) symmetry, theself-consistent TRILEX equations [ 21–24] for the fermionic single particle self-energy /Sigma1(k,iω n) and bosonic polarization in charge and spin channel Pc,s(q,iνn) can be rewritten as /Sigma1k,iωn=/Sigma1imp iωn−/summationdisplay η,q,iνnmη/tildewideGk+q,iωn+iνn/tildewideWη q,iνn/Lambda1imp,η iωn,iνn, Pη q,iνn=Pimp,η iνn+2/summationdisplay k,iωn/tildewideGk+q,iωn+iνn/tildewideGk,iωn/Lambda1imp,η iωn,iνn,(2) where the index η={c,s}corresponds to charge and spin channel, respectively, and ωnandνnare fermionic and bosonic Matsubara frequencies. Gk,iωnis the dressed Green’s function, andWc,s q,iνnare the fully screened interactions in the charge and spin channels, respectively. The local part of self-energyand polarization are replaced by their impurity counterparts /Sigma1imp iωnandPimp,η iνn, respectively, and for any quantity X,/tildewideXk,iωn= Xk,iωn−Xloc iωnwithXloc iωn=1 Nk/summationtext k∈B.Z.Xk,iωn.W ee m p l o yt h e Heisenberg decomposition of the interaction [ 22], for which we have mc=1,ms=3, and Wη q,iνn=Uη q[1−Uη qPη q,iνn]−1. Bare interactions in charge and spin channel are, hence, givenbyU c q=U0 2+vqandUs=−U0 6(for details see Supplemental Material A in Ref. [ 33]). This spin/charge ratio is a choice (dubbed “Fierz ambiguity” [ 22,24]). Moreover, in the param- eter range explored in this paper, we have observed (Fig. 2 and Supplemental Material B in Ref. [ 33] and Ref. [ 35]) that using /Lambda1imp,η iωn,iνn≈1i nE q .( 2) does not change our results qualitatively as it was also found in Ref. [ 23]. This simplified TRILEX version can be seen as a GW +EDMFT-like scheme which, however, can treat simultaneously both charge andspin fluctuations. The impurity problem was solved usingthe segment picture in the hybridization-expansion continuoustime quantum Monte-Carlo algorithm [ 36–40] implemented with the TRIQS library [ 41]. To probe SC instabilities, we solve the linearized gap equation with converged simplified TRILEX results as aninput [ 23]. For singlet d-wave pairing, the corresponding eigenvalue equation for the gap reads λ/Delta1 k,iωn=−/summationdisplay k/prime,iω/primen/vextendsingle/vextendsingleGk/prime,iω/primen/vextendsingle/vextendsingle2/Delta1k/prime,iω/primenVeff k−k/prime,iωn−iω/primen, (3) where the singlet pairing interaction is given by Veff q,iνn=mcWc q,iνn−msWs q,iνn, (4) and is therefore a combination of effective interaction in charge and spin channel. The SC instability occurs when the largest FIG. 1. Phase diagram of the Hamiltonian Eq. ( 1) as function of temperature (for T> 40 K) and doping for U0=0.7e V , V= 0.2 eV (circles) and V=0.3 eV (diamonds). Green/blue regions correspond to 1 /greaterorequalslantMax[−Ps(q,iνn=0)Us]/greaterorequalslant0.95 for q∈B.Z.Or- ange/red regions indicate chiral d-wave superconductivity. eigenvalue λ=1. The pairing symmetry is monitored by the kdependence of the gap function /Delta1k,iωn. III. RESULTS Emergence of d-wave SC. In Fig. 1, we plot the temperature– doping ( T–δ) phase diagram for V=0.2 eV and V=0.3e V for a fixed value of U0=0.7 eV in the simplified TRILEX approximation. At half-filling ( δ=0) we obtain a correlated Fermi liquid (Supplemental Material C in Ref. [ 33]) with strong magnetic fluctuations. The static spin-spin correlationfunction χ s(q,iνn=0) is very large at some qbut has not diverged yet, i.e., no phase transition has occurred. Moreprecisely, we use Max[ −P s(q,iνn=0)Us] with q∈B.Z., which reaches 1 at a second-order spin-ordering phase tran-sition to quantify the strength of the spin fluctuations andcolor code regions in the phase diagram for which 1 > Max[−P s(q,iνn=0)Us]/greaterorequalslant0.95 in green ( V=0.2 eV) and blue (V=0.3 eV). From this plot, we see that spin fluctuations are slightly enhanced by increasing V.F o rδ> 0.2 we observe the emergence of a dome-shaped superconducting phase (aplot of λin Eq. ( 3) as a function of temperature is shown in the Supplemental Material D in Ref. [ 33]). The pairing symmetry of the SC phase is of d-wave character and includes doubly degenerate d x2−y2- anddxy-wave pairing channels (see Supplemental Material E in Ref. [ 33] and Ref. [ 42]f o rap l o t of the gap function). The degeneracy of these two pairingsymmetries is protected by the C 6vpoint group of the triangular lattice, which then yields chiral d-wave symmetry below T cto maximize condensation energy. The predicted chiral SC phasedepends crucially on V:T cincreases from V=0.2e V( r e d circles) to V=0.3 eV (orange diamonds) as shown in Fig. 1. Moreover, for V=0.0 eV and V=0.1 eV (not shown here), we do not find a SC phase for T> 40 K. Impact of long-range interaction on susceptibilities and single particle spectra. The crucial impact of Von the SC instability is reflected in the effective singlet-pairing inter-action V eff q,iνn, which depends on fluctuations in both charge and spin channels. We analyze the respective susceptibilitiesχ c/s(q,iνn) with the data shown in Fig. 2: In the upper panels 155145-2CHIRAL d-W A VE SUPERCONDUCTIVITY IN A … PHYSICAL REVIEW B 97, 155145 (2018) 0.00.20.40.60.81.0 δ0510152025χc(qmax.,0)(eV−1)(a) 0.00.20.40.60.81.0 δ050100150200250300χs(qmax.,0)(eV−1)(b) 0.00 0.05 0.10 0.15 0.20 ω(eV)012345678Im [χc,s(qmax.,ω)] (eV−1) (c)χc,Λη=1 χc,Λη χs,Λη=1 χs,Λη 0.00.20.40.60.81.0 δ0.000.050.100.150.200.250.300.350.40ωc,s 0(eV)(d)ΓMK FIG. 2. Maximum values of the static charge (a) and spin (b) response functions versus hole doping. Color coding indicates the position of the maximum in the first Brillouin zone as definedin the inset. Data is shown for fixed U 0=0.7e V a n d T=40 K and nonlocal interaction strength V=0.3 eV (diamonds) and V= 0.2 eV (circles). (c) Charge- and spin-response functions on the real frequency axis (obtained by analytical continuation with the maximum entropy method [ 43]) at their maximum in momentum space ( qmax.) with (dashed) and without (solid) vertex corrections forT=116 K and δ=0.2. (d) Characteristic frequency of charge (filled symbols) and spin (open symbols) fluctuations with the same convention and parameters as (a) and (b). we show the maximum values of the static ( iνn=0) charge (left hand side) and spin (right hand side) susceptibilities asa function of hole doping. The corresponding position of themaximum in the first Brillouin zone is color coded (see inset). The charge fluctuations increase with hole doping to a maxi- mum value around δ=0.5 and, thereafter, decrease approach- ing the “empty” limit at δ=1. The spin fluctuations, instead, decrease monotonically as a function of δ. While χ c(q,iνn=0) always peaks at K, the maximum of χs(q,iνn=0) moves from MtoKwhen the system is slightly doped, and then follows K→M→/Gamma1when the system is further hole doped. The peak position of the charge response function as a functionof doping remains at the Kpoint since its momentum depen- dence is mainly determined by the doping-independent v(q), which energetically favors a 3 ×3 charge configuration in real space [ 3]. The momentum dependence of the spin response function, however, is mostly determined by the topology ofthe Fermi surface. Indeed, the Vdependence is much stronger for the charge response [compare diamond ( V=0.3 eV) and circle ( V=0.2 eV) symbols in Fig. 2]. There are, however, small effects of Vto the spin-response function, which can be understood by the V-dependent renormalization of the one-particle spectra as show in Fig. 3[44]. At fixed T=40 K andδ=0.2,Vis increased from 0 .0e Vt o0 .3 eV (subplots from left to right hand side). Upon increasing V, the bandwidth is effectively reduced and the spectral weight near to the Fermienergy is increased. Consequently, particle-hole excitationsthat contribute to the spin polarization P s(q,iνn) and the spin susceptibility are enhanced. ΓMK Γ−0.4−0.20.00.20.40.60.81.0ω−εF(eV)A(k,ω)(eV−1) V=0.0 eV MK ΓV=0.1 eV MK ΓV=0.2 eV MK ΓV=0.3 eV LowHigh FIG. 3. Single particle spectral function A(k,ω) along the path /Gamma1-M-K- /Gamma1(see inset of Fig. 2) for fixed doping δ=0.2,T=40 K, U0=0.7 eV and four values of V. We now extend these considerations to the frequency dependence of the bosonic fluctuations. In Fig. 2(c),w ep l o t the dynamic response functions at the q-points where they are maximal ( qmax.) for doping δ=0.2. The data clearly shows a peaked structure of the dynamic response functions.Moreover, we show in this plot the impact of the vertexcorrections (compare solid and dashed lines). which are onlyquantitative in the considered case as claimed in the intro-duction. Figure 2(d) shows the doping dependence of the characteristic frequency ω c,s 0(qmax.) defined by ωc,s 0(qmax.)=/integraltext∞ 0ωIm[χc,s(qmax.,ω)]dω//integraltext∞ 0Im[χc,s(qmax.,ω)]dωin both channels. Inside the superconducting region (indicated by thevertical red dashed lines), the characteristic frequency of thefluctuations are of the order of 100 −200 meV. Moreover, |ω c 0-ωs 0|is small and minimal for the region of maximum Tc.I n agreement with our discussion above, we see that an increase ofVyields even smaller |ω c 0-ωs 0|, which suggests that charge and spin contributions to the SC pairing mechanism are cumulative. Separating spin and charge channels in the pairing mech- anism. To disentangle the interplay between charge and spin degrees of freedom in gap Eq. ( 3), we solve for λ, including contributions from only spin ( λs) and only charge channel (λc), i.e., Veff q,iνn=− 3Ws q,iνnandVeff q,iνn=Wc q,iνn, respectively. First, we follow the phase boundary of the SC phase inthe underdoped regime for fixed V=0.3 eV, starting from (δ,T)=(0.2,40 K) up to ( δ,T)=(0.38,65 K). In Fig. 4(a)we plotλ,λ s, andλc: Since we are following the phase transition line,λ≈1.λcandλsare both smaller than λandλc+λs≈λ, indicating a cumulative charge and spin contribution for the chirald-wave pairing in the underdoped regime. This observa- tion is surprising and in stark contrast to the expectation thatspin and charge fluctuations drive competing instabilities. Thesame conclusion can be drawn when the λvalues are calculated at the critical doping δ c=0.2 as a function of the nonlocal interaction Vas depicted in Fig. 4(b). Our data indicates that overall both spin and charge fluc- tuations are important for the SC phase. As a function ofdoping, however, we observe that charge fluctuations becomeincreasingly dominant and λ sbecomes negligible. This effect is reflected in the Vdependence of the SC dome in Fig. 1, which is stronger at larger dopings. We arrive at the same conclusions 155145-3CAO, AYRAL, ZHONG, PARCOLLET, MANSKE, AND HANSMANN PHYSICAL REVIEW B 97, 155145 (2018) 0.20.220.260.30.38 δ0.00.20.40.60.81.01.2λ(a) 40K 45K 50K 55K 65K λc λs λ 0.00.10.20.30.4 V/U 0(b) FIG. 4. Eigenvalue λof the gap equation Eq. ( 3)(λ=1 signals SC transition) for full effective singlet pairing interaction Veff q,iνn(cyan) and charge/spin only channels (red/green). (a) Plot for V=0.3 eV along the SC phase boundary up to doping δ=0.38. (b) Plot as a function ofVfor fixed doping δ=0.2 and temperature T=40 K. when we analyze the dependence of λon the choice of the Fierz parameter that defines the charge-to-spin fluctuation ratio(Supplemental Material F in Ref. [ 33]). Let us stress two important points: (i) The true long- range character is crucial for the predicted SC in siliconadatom materials. If only short-range (i.e., nearest-neighbor)interactions are considered, charge ordering is overestimatedand, long before any SC emerges, the system turns into acharge-ordered insulator as proven by calculations shown inthe Supplemental Material G in Ref. [ 33]. (ii) The degeneracy of the d x2−y2- and dxy-wave-pairing state is important for the cumulative charge and spin interplay. Since the origin ofthis degeneracy is connected to the lattice symmetry group, adifferent behavior can be expected for the 2D square lattice (seeSupplemental Material H in Ref. [ 33]): in the square geometry with relatively large V/U 0,t h eqdependence of χc(q,iνn=0)favors dxy−pairing symmetry while χs(q,iνn=0) prefers dx2−y2−pairing symmetry, and the two channels compete with each other. IV . CONCLUSION We predict the existence of a dome-shaped unconventional chirald-wave superconducting phase for hole-doped triangular lattice systems with ∝1/rinteractions, which could be realized by hole-doping existing αphase Si(111) adatom materials. The analysis of spin and charge correlation functions revealsthat lattice geometry as well as the nonlocal interaction arenecessary conditions for the emergence of SC. The nature of thepairing undergoes a crossover from an unexpected combinedcharge/spin mechanism in the underdoped regime toward acharge fluctuation dominated one at higher doping. In futurestudies, high hole-doping levels will be considered in moredetail. Here, triplet f−wave pairing symmetry may begin to become important due to the appearance of a disconnectedFermi surface [ 45]. Note added. During the review process of this paper, we became aware of very recent experimental activities onstrongly hole-doped Sn:Si(111). F. Ming. and coworkers [ 46] were able to reach doping levels up to 10 −12% by using boron- doped silicon as a substrate without perturbing the symmetryof the adatom lattice. This first success proves the feasabilityof synthezising heavily doped adatom systems and motivatesfurther experimental work along the lines of our prediction. ACKNOWLEDGMENTS O.P. and T.A. are supported by the FP7/ERC under Grant Agreement No. 278472-MottMetals. We thank Yi Lu, Alessan-dro Toschi, Ciro Taranto, Thomas Schaefer, and Daniil Man-tadakis for helpful discussions. [1] S. Schuwalow, D. Grieger, and F. Lechermann, Phys. Rev. B 82, 035116 (2010 ). [2] P. 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PhysRevB.70.125319.pdf

Multiband k·ptheory of carrier escape from quantum wells V. V. Nikolaev Department of Electronics, University of York, Heslington, York, YO10 5DD, United Kingdom and A. F. Ioffe Physico-Technikal Institute, 26 Politechnicheskaya, St. Petersburg 194021, Russia E. A. Avrutin Department of Electronics, University of York, Heslington, York, YO10 5DD, United Kingdom (Received 23 March 2004; published 24 September 2004 ) A general theory of multichannel coherent particle escape introducing the escape matrix based on the scattering formalism has been developed and applied to the problem of hole escape from biased quantum wells.This approach incorporates band-mixing effects, such as valence-band nonparabolicity and heavy-to-light(light-to-heavy )hole transformations, in the theory of carrier escape from quantxum wells consistently with the principles of multiband effective-mass theory. The numerical calculations, made in the Luttinger approxima-tion, show significant influence of the band-mixing effects on the hole escape time and are in satisfactoryagreement with available experimental data. The effective decrease of the heavy-hole tunneling mass (“light- ening” )as a result of the light-hole admixture is found to be the major reason for the escape-time decrease, as compared to the simple parabolic-band effective-mass model.The concept of the escape matrix can be used forinvestigating multichannel coherent escape processes in different systems such as quantum wires, quantumdots, or photonic structures. DOI: 10.1103/PhysRevB.70.125319 PACS number (s): 73.21.Fg, 85.60.Bt, 42.79.Sz I. INTRODUCTION Understanding of particle escape processes is important for fundamental physics as well as for numerous deviceapplications. The escape of electrons and holes from low-dimensional semiconductor heterostructures plays a primaryrole in the functioning of devices such as solar cells, optical modulators, and photodetectors. In this respect, oneof the most important category of heterostructures isreversed-biased quantum wells (QW’s ), which are used, among others, as saturable absorbers in mode-locked lasers 1 and in recently proposed all-optical signal processingdevices. 2,3Electron and hole escape rates can determine such a critical parameter as the maximum operational frequencyof the device. Currently, the most widely used approach to the problem of modeling carrier escape from quantum wells is the onewhich considers two separate processes: thermionic emissionand tunneling. For thermionic emission, a simple expressionis usually employed, 4which was obtained using the three- dimensional (3D)density of states for particles with a para- bolic energy-momentum relationship. The tunneling is usu-ally assumed to take place exclusively from quasiboundstates described by the 2D density of states. 5–7This division is inherently approximate, since the biased QW is neither apurely 2D nor a 3D system. A significant advance in thetheory of carrier escape from QW’s was made by Lefebvreand Anwar 8,9who treated the above- (thermionic emission ) and below- (tunneling )barrier escape processes on the same footing. However, numerical complexity limits the useful-ness of their approach. Recently, we have developed a com-putationally efficient yet comparatively rigorous theorywhich does not employ any additional approximation exceptfor coherent tunneling, parabolic-band effective-mass, andquasiequilibrium approximations, 10thus representing an ad-vance compared to all previously used theories of carrier escape. It is well known that proper account of the valence-band structure is essential for a quantitative description of holes in QW’s. The band mixing leads to strong nonparabo-licity for subbands inside the QW. 11The results obtained using multiband effective-mass theory in the k·papproxima- tion show that the effects of heavy-to-light (light-to-heavy ) hole transformations play a significant role in the holetunneling process. 12–14Despite this, applying multiband effective-mass theory to the problem of carrier escapefrom biased QW’s has never been attempted before: to ourknowledge, all previously used theories of carrier escapefrom QW’s are based on a simple parabolic-band (effective mass )approach. In order to account for the nonparabolicity of hole subbands, previous methods used different effectivemasses for confined and above-barrier levels. 9,10,15Although such an approach can give realistic escape-time values, afirst-principles theory is preferable for detailed quantitativeanalysis. The purpose of this article is to develop a multichannel theory of coherent escape as well as to apply this theory toholes in QW’s. To do so, we introduce the concept of theescape matrix by analogy with the scattering matrix used in the theory of quantum conductance. 16We assume the tunnel- ing processes to be coherent and scattering to be elastic.Apart from that, the level of approximation would be deter-mined by the method used to find the elements of the escapematrix. In addition to the investigation of the carrier escapefrom quantum wells, the escape matrix approach can be usedto describe carrier escape from other nanostructures such asquantum wires or dots, as well as the photon escape fromcomplex photonic structures, such as microcavities and pho-tonic crystals.PHYSICAL REVIEW B 70, 125319 (2004 ) 1098-0121/2004/70 (12)/125319 (9)/$22.50 ©2004 The American Physical Society 70125319-1The article is organized as follows. In Sec. II A, we develop a general theory of multichannel escape. Inthe following section, we apply the escape matrix approachto the problem of hole escape from reverse-biased quantum-well structures using the Luttinger approximation. InSec. III, we compare the numerical results to experimentaldata and discuss the influence of band mixing on theescape time and escape current from GaAs/AlGaAsquantum wells. II. THEORY A. Escape matrix In this section, we introduce the concept of escape matrix, which stems from the same principles as the quantum-mechanical scattering matrix. 17–19We consider a topologi- cally one-dimensional system consisting of a finite objectsandwiched between two sufficiently large regions (see Fig. 1 ). Some particles are initially confined inside the object, and our task is to find the escape time. The largeregions on both sides of the object are considered as emptyreservoirs for escaping particles. In order to find the escape time, we shall look for the outside escape particle flux whichis caused by a predetermined occupation of quantum statesinside the structure. The particle flux can be easily found ifthe outside probability flux, associated with the quantumstates under consideration (i.e., theescape probability flux ), is known. Each reservoir is considered to be large enough so that it can be described by a quasicontinuum of propagatingeigenstates (waves ). Each wave carries a certain probability fluxf 0i, whereiis the complete set of relevant quantum numbers. By summing f0iover all quantum numbers, one can get a physical quantity suitable for the descriptionof the eigenstate structure of the reservoir, which isindependent of the reservoir size as long as the reservoiris sufficiently large so that this quantity is unaffected bythe presence of the finite-size object in contact with thereservoir. To make our proposal clearer, consider the example of a (quasi- )one-dimensional structure (a wire or a specified di- rection in a 3D system ).Ameaningful physical quantity here isthe energy density of the probability flux f e0=df0/de, wheredf0is the probability flux of the propagating states in an energy interval de. This quantity is given as a product ofthe group velocity vg="−1]e/]kand the density of states for particles moving in a particular direction ]n/]e. In the 1D case for particles with the spin degeneracy factor of 2, thisquantity is given as f e0=1 "]e ]k]n ]e=1 "]n ]k=1 p"s1d and, as expected, does not depend on the reservoir param- eters. The principle of independence of the reservoir states of the presence of the object is used in the derivation of thewell-known Buttiker-Landauer formula 20where Eq. (1)ap- pears as a factor in the expression for the conductance. If thecarriers in QW’s are described by simple parabolic disper-sion, where the in-plane longitudinal motion can be dealtwith separately, making the problem essentially one dimen-sional, this factor appears in the formula for the escapecurrent. 10 We propose that, in general, the escape probability flux can be found through the scattering formalism, taking thereservoir fluxes as source terms. To do that, we evoke theconcept of the scattering matrix for a finite-size structureunder consideration. Using an appropriate theoretical ap- proach one can calculate the amplitude transmission matrix tˆ of the structure, where ut jiu2gives the ratio of the particle flux transmitted into the propagating state (wave )jon the right to the incident particle flux of the wave ion the left. Similarly, the amplitude reflection matrix rˆdescribes reflection of par- ticle waves incident from the left. The scattering matrix Sˆ relates the amplitudes of the waves incident on the structure (represented by a vector I)to the amplitudes of the outgoing waves (vectorO)asO=SˆIand can be written in the follow- ing form: Sˆ=Srˆtˆ8 tˆrˆ8D. s2d Here, the prime denotes the substitution of “left” with “right” and vice versa in the definition of the matrices. In the topologically one-dimensional geometry, one can divide the structure in two, one to the left and the other to theright of some point z 0inside the structure. Each part can be assigned its own scattering matrix. One can express the totalscattering matrix of the combined system through the matri-ces of its parts using the following relations (cascading rules ): tˆ=tˆ 2fIˆ−rˆ18rˆ2g−1tˆ1, s3d rˆ=rˆ1+tˆ18rˆ2fIˆ−rˆ18rˆ2g−1tˆ1, s4d tˆ8=tˆ18hIˆ+rˆ2fIˆ−rˆ18rˆ18g−1rˆ1jtˆ28, s5d rˆ8=rˆ28+tˆ2fIˆ−rˆ18rˆ2g−1rˆ18tˆ28. s6d A simple consideration shows that each element of the total scattering matrix is a result of interference of all the differentways for a particle to get from one reservoir to another FIG. 1. Illustration of the idea for the escape matrix.The dashed arrow depicts penetration of the probability flux of reservoir statesinside the structure, the dot-dashed curves schematicly depict for-mation of the internal probability flux (backscattering ), and the solid arrows show the escape probability flux.V. V. NIKOLAEV AND E. A. AVRUTIN PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-2(transmission )or to enter and leave the structure on the same side (reflection ). Now let us suppose that z0is a point at which the particles are confined inside the structure (an arbitrary point in the quantum well, for example ).We propose that in this case, the outer probability flux generated by the inner states (escape probability flux )can be obtained by considering the interfer- ence of all different processes which account for penetrationof the incident flux of reservoir states into the structure andsubsequent escape from the structure into outgoing waves ofthe reservoirs. 10We will call the matrix with elements built to account for such processes the escape matrix Eˆ. It is clear that such a matrix will be closely connected to the scatteringmatrix of the structure. Indeed, the elements of the off-diagonal [or transmission; see Eq. (2)]blocks of the scatter- ing and escape matrices (which account for the penetration of the probability flux from one reservoir to another )are equivalent: Eˆ t;Sˆt;tˆ, s7d Eˆt8;Sˆt8;tˆ8, s8d which is due to the fact that in a topologically 1D structure, any process accounting for the probability flux penetrationfrom one side of the structure to another will involve trans-mission through the point z 0inside the structure. On the other hand, constructing the diagonal elements of the escapematrix one should exclude the paths which do not reach z 0 due to reflection. This approach gives the following off- diagonal escape matrix elements: Eˆr=tˆ18rˆ2fIˆ−rˆ18rˆ2g−1tˆ1, s9d Eˆr8=tˆ2fIˆ−rˆ18rˆ2g−1rˆ18tˆ28. s10d Here the subscripts denoting the escape matrix blocks indi- cate their position similarly to those of the scattering matrix,Eq.(2). The condition of flux conservation implies unitarity of the scattering matrix: SˆSˆ †=Iˆ. In addition, if the problem is in- variant under the operation of time reversal, the scattering matrix is symmetric:21Sˆ=SˆT. Although, obviously, the es- cape matrix is generally not unitary, it is easy to show thatthe time reversal symmetry of the problem leads to a sym-metric escape matrix Eˆ=Eˆ T. s11d To find the escape probability flux, one should take the reservoir probability flux as the source term for “incident”particle waves. Thus, uE iju2f0jwould give the escape prob- ability current carried by the outgoing ith wave in one of the reservoirs which is due tothe incident jth wave. One should note that in our problem there are no physical particles inci-dent on the structure, and the inclusion of the incident out-side waves in the formalism is in fact a form of boundaryconditions. The total escape particleflux, which is generated by the particles initially confined inside the structure, isfound by multiplying the escape probability flux by the oc- cupation number N iinof the particular state inside the struc- ture: J=o iout,iinuEioutiinu2f0iinNiin. s12d Here, the indices iinsioutdrun over all incident (outgoing ) states in both reservoirs. The direct correspondence between the incident outer waves and inner states (designated here by the subscript iinat the occupation number N)is a result of the energy conservation law (only elastic scattering is being con- sidered ). In the parabolic-band approximation (no band mix- ing)this approach reproduces the expressions of Ref. 10; in this case, it has been shown that the results of previous the-oretical methods based on the assumption of strong particlelocalization are reproduced by our theory in the correspond-ing limiting case. B. Hole escape in the Luttinger approximation In this section, we apply our general theory to the prob- lem of hole escape from quantum-wells. We consider aquantum-well structure of an arbitrary profile in an electricfield. The heterostructure is sandwiched between two bulklayers with a constant valence-band edge and no electric field(see, for example, Fig. 2 ). The structure with a complex valence-band profile between the two layers is discretized asa set of sufficiently narrow constant-band-offset layers.The most widely used theoretical method in the investigationof semiconductor heterostructures is the multiband effective-mass approximation. In this method, the carrier states inthe semiconductor layer with constant valence-band edge arerepresented by multiband envelope functions Fe ik·r, wherekis the wave vector. Although our method allows an arbitrary number of bands to be included in the formal-ism, in this work we use the Luttinger approach, whichconsiders only heavy- and light-hole bands. This is a goodapproximation for the GaAs/AlGaAs system. For some othermaterials, such as most III-V quaternaries, the influence ofthe spin-orbit split-off band cannot be ignored. In that case,one should include more bands in the escape matrix, in amanner similar to the scattering (or transfer )matrix formalism. 14 FIG. 2. Schematic valence-band-edge profile of a quantum well structure studied in Ref. 39.MULTIBAND k·pTHEORY OF CARRIER ESCAPE PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-3In the Luttinger approximation, the spin quantization is parallel to the growth (here, [001])direction. The top of the valence band in this case is described by the 4 34 Luttinger Hamiltonian.22As the hole in-plane anisotropy is not of par- ticular interest for this problem, it is convenient to use theaxial approximation, 23,24which averages the dependence of the Hamiltonian on the direction of the in-plane wave vector.In this approximation, one can perform the Broido-Shamtransformation 25to a new basis, in which the Luttinger Hamiltonian becomes block diagonalized:12,13,24,26 Hˆ=SHˆU0 0HˆLD, s13d where the upper and lower Hamiltonians are HˆU/L=SP±Q−VR− R+P7Q−VD. s14d Here P=G1skr2+kz2d,Q=G2skr2−2kz2d, and R±=˛3G¯kr2±i2˛3G3krkz, where Gj="2gj/s2med, G¯=sG2+G3d/2, andkr=skx2+ky2d1/2is the in-plane wave vec- tor,meis the free electron mass, g1,2,3are Luttinger param- eters, and Vis the valence-band offset. The Broido-Sham transformation allows one to deal with two separate prob-lems described by the 2 32 upper and lower Hamiltonians, Eq.(14), instead of the initial 4 34 problem, which makes for significant computational simplification. The eigenvalues of the both Hamiltonians are elhshhdskr,kzd−V=P±˛Q2+R+R−, s15d where the plus is for the light hole (LH)and the minus is for the heavy hole (HH). The normalized eigenvectors of these Hamiltonians can be written as13,27 F=1 V1/2suU1u2+uU2u2d1/2SU1 U2D, s16d where Vis the volume of the slab and the vectors U=sU1 U2d are UhhU=SP−Q−ehh+V −R+D,UlhU=S−R− P+Q−elh+VD, s17d for the upper Hamiltonian and UlhL=SP+Q−elh+V −R+D,UhhL=S−R− P−Q−ehh+VD s18d for the lower Hamiltonian. Thus the hole envelope wave function in the semiconductor layer can be represented as asum of the following envelope functions. For the fixed in-plane vector k rand energy ethere are two heavy-hole FhhUexps±ikhhz+ikr·rdand two light-hole FlhUexps±iklhz +ikr·rdmultiband plane waves for the upper Hamiltonian, whereFhhUandFlhUare given by Eqs. (16)and (17), and,similarly, two heavy-hole FhhLexps±ikhhz+ikr·rdand two light-hole FlhLexps±iklhz+ikr·rdplane waves for the lower Hamiltonian, with FhhLandFlhLgiven by Eqs. (16)and(18). The heavy-hole khhand light-hole klhzcomponents of the wave vector are obtained from Eq. (15)(see Ref. 12 for more details ). The probability flux in the zdirection corresponding to a multiband plane wave Feik·rcan be written as f0=ResF†JˆzFd. Here the current operator Jˆz, obtained by av- eraging the law of probability conservation over the unit cell of the material,13,28has the following form: Jˆ zU/L=1 "] ]kzHˆU/L=2 "SsG172G2dkz−i˛3G3kr i˛3G3kr sG1±2G2dkzD. s19d The bulk layers on both sides of the QW structure (see Fig. 2)can be considered as empty reservoirs, which are de- scribed in terms of propagating waves. For one of the fourtypes (bands )of hole waves propagating in the positive di- rection along zaxis in a bulk semiconductor slab one can write the differential probability flux as df 0=Re sF†JˆzFdd3k s2pd3=kr 4p2ResU†JˆzUd U†Udkzdkr,s20d where the independence of the probability flux on the direc- tion of the in-plane wave vector was used. One can use thisexpression as a source term in order to obtain the hole escapeprobability flux. This equation can be considered as a 2Dversion of the one-dimensional equation (1). As has been mentioned above, the complex valence-band energy profile between the two bulk layers (Fig. 2 )is ap- proximated as a series of small steps in which the valence-band edge is assumed to be constant. In each step the multi-band wave function can be represented as a sum of heavy-and light-hole plane waves, which propagate in the oppositedirections and “scatter” at the interfaces. The term “scatter-ing” here stands for the coherent reflection and transmissionprocesses. In order to describe the transmission and reflec-tion properties of the heterointerfaces between constant-band-edge steps one has to use appropriate boundary condi-tions. The problem of the boundary conditions at abruptinterfaces in the multiband effective-mass theory has beenunder detailed investigation recently. 29–31The latest research generated some new approaches, which introduce phenom-enological parameters in the boundary conditions. These pa-rameters are not well established for the semiconductor ma-terials under consideration yet. Furthermore, as one of thetasks of this paper is to compare our multiband hole escapetheory with the simple single-band approximation, it is ad-vantageous for us to use the boundary conditions which aredirectly related to the single-particle quantum-mechanicalboundary conditions. Therefore, we use here the conditionsof the component-by-component continuity of the envelopefunctionFand the normal component of the probability flux Jˆ zF, Eq. (19), across the interface, which have been most widely used for such problems up to now.11–14,27The second set of boundary conditions28can be derived directly from theV. V. NIKOLAEV AND E. A. AVRUTIN PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-4expressions for the Hamiltonians, Eq. (14), by integration of the multiband Schrödinger equation across the interface,32,33 similarly to the simple one-dimensional problem. It shouldbe noted that the level of approximation imposed by theseboundary conditions can be reduced with no major difficulty,provided the phenomenological material parameters for moreaccurate boundary conditions are known. The transmission tˆ j,tˆj8and reflection rˆj,rˆj8matrices for the jth interface are easily obtained using the condition of the incidence of the hole of a particular type from a particularside. 12There are two sets of the 2 32 transmission and re- flection matrices,12one set corresponding to the upper and another to the lower Hamiltonian. Apart from the materialparameters on both sides of the interface, these matrices de-pend only on the in-plane wave vector k rand the hole energy e, Eq. (15), which is a consequence of the coherence and elasticity of scattering.These reflection and transmission ma-trices form the 4 34 scattering matrix of the interface, as shown in Eq. (2). For the propagation across the constant- band-edge step only the diagonal elements of the transmis-sion matrices, representing the phase shifts of the heavy- andlight-hole waves traveling across the step, are nonzero. The scattering matrix for the part of the structure to the leftSˆ lskr,edand to the right Sˆrskr,edof a particular point inside the QW are obtained using the cascading rules, Eqs. (6). A similar approach based on the scattering matrix for- malism was used by Kumar et al.34for investigation of the hole tunneling problem. Their simulations showed that thescattering matrix approach is more numerically stable thanthe standard transfer-matrix method. 13,35Finally, the 4 34 upperEUskr,edand lower ELskr,edescape matrices are ob- tained using Eqs. (7)–(10). The 4 34 escape matrix, which is dependent on the in- plane wave vector krand the energy e, relates the heavy- or light-hole probability flux in one of the bulk layers on bothsides of the structure, Eq. (20), to the escape probability flux, carried away from the structure by the heavy or light holes inone of the two reservoirs. In order to get the escape currentone should multiply the outgoing flux of the confined state,characterized by the energy eon its occupation number N. Rigorous determination of the nonequilibrium carrier occu-pation number in quantum wells is a complex kinetic prob- lem(see, for example, Ref. 36 ). The interaction with optical phonons may, in general, play an important role for the cal-culation of the escape current. 37In this work, however, we assume that the carrier-carrier and carrier-phonon interac-tions are fast enough for the carrier levels in the QW to be inquasiequilibrium, 4,5,8–10,15and so the Fermi distribution func- tion can be used—i.e., NFsed=hexpfbse−mhdg+1j−1, where bis the inverse temperature and mhis the hole quasi-Fermi level. As the major contribution to the current is due to thenear-barrier-edge levels, this approximation is a good one aslong as the quantum wells are deep enough, and levels ofexcitation low enough, for the escape time to be much longerthan the carrier-phonon and carrier-carrier interaction times. The escape current can be expressed in the axial approxi- mation for the Luttinger Hamiltonian asJ= o X=U,L iout,iin=HHl,LHl,HHr,LHrE 0‘ dkiin 2pE 0‘ dkr 2pkruEioutiinXkr,eskr,kiindu2 3ResU†JˆzUdX U†UNFeskr,kiind, s21d where the subscript iin=HHlmeans that the source wave is of heavy-hole character and incident from the left,i out=LHrmeans that outgoing escape flux is carried out by the light-hole wave in the right region, and so on. In order toinvestigate band-mixing effects, it is more convenient towrite the escape current as the sum of the upper and lowerHamiltonian currents, J=J U+JL, where each current is de- composed into the sum of the “direct” and “mixed”currents—i.e., J X=JHH−HHX+JLH−LHX+JLH−HHX+JHH−LHX. s22d Here the subscripts denote the type of the escaping and “source” particles. As a consequence of the time invariancesymmetry (symmetricity of the escape matrix ), the “mixed” currents are equal, J LH-HHX=JHH-LHX. Equation (21)can be written in the form J=o X=U,L a,b=HH,LHJa−bX=o X=U,L a,b=HH,LHEEja−bXse,qdde2pqdq. s23d In the parabolic-band (effective-mass )approximation, the current “density” jdoes not depend on the in-plane wave vectorqand the index of the Hamiltonian X. By performing appropriate summation and integration one can in this caseobtain the single-band version of the escape current formula,Eq.(13)in Ref. 10. In order to find the escape time, one has to calculate the hole density n=on j, where the summation is performed over all quasibound bands in the biased QW. The density for a jth band is nj=e0‘dkrskrd/s2pdNF(eQBjskrd)where the (quasi )bound hole energies eQBmanifest themselves as minima of the following determinant: detsIˆ−rˆ1rˆ28d. s24d The escape time is given by the ratio t=nh/J. III. RESULTS AND DISCUSSION In order to investigate the influence of the band mixing on the hole escape time in a transparent and instructive manner,one should consider the contributions of band-mixing effectsto the escape current separately from the impact of the sub-band nonparabolicity on the hole density. Since a thoroughinvestigation of the subband structure by means of multibandeffective-mass theory in the k·papproximation has already been reported, 11,23,27,38we concentrate our attention on the change of escape current. The escape current calculations made using the full multi- bandk·pmodel are compared to those made within a simpleMULTIBAND k·pTHEORY OF CARRIER ESCAPE PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-5parabolic-band (effective-mass )approximation. In the latter model the escape matrix is constructed using diagonal reflec- tion srˆdand transmission stˆdmatrices, resulting in the two independent escape matrices for the heavy and light holes. This method is analogous to that used in Ref. 10. The subband nonparabolicity is accounted for by the use of the minima of Eq. (24)in the hole density calculation.The resulting density is applied for calculation of the hole escapetime for both mulitband and single-band methods, thus leav-ing only the alteration of the escape current as a source of thedifference between the two methods. In the single-band case,such an approach is similar to some previous calculations,where a larger in-plane mass was used for QW levels. 10,15,39 The use of an asymmetric QW structure, schematically depicted in Fig. 2, allowed Cavailles et al.39to perform, to our knowledge, the only available up-to-date separate mea-surements of the electron and hole escape.Their data, as wellas our calculations for the full multiband and parabolic-bandmodels, are plotted in Fig. 3. One can see that the inclusionof the band mixing into the theory gives a significant im-provement in the agreement with the experiment. For thelarge applied field, the escape time calculated using themultiband model is almost twice smaller than that given bythe parabolic-band approximation, which constitutes an al- most twofold increase of the escape current as a result of theband mixing. The impact of the band mixing on the escapetime is small at small applied fields and increases with anincrease of the applied field. The maximum of the escapetime, present in both calculated curves as well as in the ex-perimental data, occurs because of the fact that, as the biasincreases, the escape current in the directions opposite to theapplied field (to the left in Fig. 2 )is quenched, whereas the escape to the other side (right )is facilitated.The components of the calculated escape current [see Eq. (22)]are plotted in Fig. 4. One can see that for room tem- perature, the heavy-hole current is significantly larger thanthe light-hole current within the considered applied-field in-terval for both k·pand parabolic-band methods. At first glance, this contradicts the notion that light holes shouldleave the QW faster because of their smaller “tunneling”effective mass in the barrier and, hence, larger transmissioncoefficient. This contradiction is resolved if one presumesthat the escape process is dominated by the thermal activa-tion into the energy region near and above the barrier whereheavy holes have the advantage of the larger density ofstates. The effects of the band mixing on calculation resultsfor the escape current components (Fig. 4 )are as follows.As the bias increases, the “multiband” direct heavy-hole currentbecomes larger and light-hole current smaller than their“parabolic-band” counterparts. This effect will be explainedbelow. In addition, the results show the increase of the“mixed” LH-HH component with the applied field; for thefieldF=100 kV/cm, it becomes comparable with the direct LH component. As the majority of devices and experimental setups em- ploy structures with symmetric quantum wells, it is reason-able to study the effects of band mixing on the example of ausual symmetric QW structure. In what follows, we consider10-nm-wide GaAs/GaAlAs with 30% aluminum in the bar-riers. Figures 5 and 6 show calculation results of the escape current density [see Eq. (23)], integrated over energy, at room s300 K dand lowered s120 K dtemperature. There are several qualitative features which are present at both tem- peratures. First, the band mixing enhances the total escapecurrent in the whole range of the in-plane wave vector q (except at q=0). This is due to substantial “mixed” compo- nent LH-HH (HH-LH )and the increase of the direct HH component for certain values of q. Second, for both tempera- tures the LH-HH curve has a maximum and, third, the directLH component is decreased in the whole range of q. In the case of room temperature one can see clear connec- tion between the “mixed” component and the direct compo- FIG. 3. Experimentally measured and calculated hole escape time as a function of applied electric field (see the text for detail ). FIG. 4. Components of escape current as a function of applied electric field. Solid lines correspond to the multiband k·pmethod and dashed lines to the parabolic-band effective-mass method.V. V. NIKOLAEV AND E. A. AVRUTIN PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-6nents. In the region of qwhere the mixed component has a maximum, the direct HH component has a minimum, whichadds up to total enhancement of the HH current compared to“parabolic-band” case. One can see that decreasing temperature makes for sig- nificant quantitative changes. The high-temperature calcula-tions demonstrate significant HH current for both parabolic-band and multiband approximations, whereas the low-temperature “parabolic-band” current is dominated by lightholes. The reason for this is the fact that at low temperatures,thermal activation is quenched, and the escape takes placemainly through tunneling, where the light holes have an ad-vantage of a larger tunneling coefficient.As can be seen fromFig. 6, the admixture of the LH bands to the HH bandsincreases the HH current drastically. This shows that “light-ening” of the HH tunneling mass by the LH admixture playsa significant role in the escape process. On the other hand,the process of light holes becoming heavier while escapingthe QW (manifested as a decrease of the multiband LH-LH current in comparison with the parabolic-band model )is not crucial. Furthermore, the decrease of the direct LH current isovercompensated for by the appearance of the mixed LH-HHcomponent in the total LH current. All this makes for an increase of the total escape current and a decrease of theescape time, as compared to “parabolic-band” case. In Fig. 7 the escape current density at temperature of 120 K as a function of the hole energy at a particular in- plane wave vector qis shown. Confirming the above specu- lation on the enhanced role of tunneling at low temperature,the calculations show that in this situation, the main contri-bution to the escape current is due to the states below theenergy barrier. Here, in the “parabolic-band” model, the es-cape current is dominated by the single light-hole peak LH1,which corresponds to the LH quasibound state. In this model,the tunneling current from deep HH quasibound states (here HH1 )is negligible, and only the weakly bound heavy-hole state closer to the barrier edge (HH2 )gives an appreciable contribution to the total current. The LH1 and HH1 quasibound states, which have close energy positions at this particular value of q, are split into two quasibound states M1 and M2, as a result of band mix-ing. One can see that there are large peaks of direct light(LH-LH ), direct heavy (HH-HH )and mixed (LH-HH or HH- LH)escape currents at the energies corresponding to these mixed states. The direct HH tunneling current is drasticallyincreased as a result of the band mixing. In addition, one cansee that the “mixed” tunneling happens through the quasi-bound states. As the dispersion of weakly bound states isnearly parabolic, the peak M3 is near the correspondingheavy-hole peak HH2 in the parabolic-band approximation.One can see that even for a weakly bound state, sizableLH-LH and LH-HH current peaks appear in the k·papproxi- mation. In Fig. 8, we examine the room-temperature situation. Figure 5 shows the enhanced HH-HH current at large qin thek·papproach as compared to the parabolic-band model. Here, we take the in-plane wave vector of 0.6 nm −1to exam- ine this effect in detail. For such a wave vector, the direct FIG. 5. Integrated components of escape current density jint=ejUse,qddeas a function of in-plane wave vector qfor the upper Hamiltonian at 300 K. FIG. 6. Same as Fig. 5, but for temperature equal to 120 K. FIG. 7. The escape current density jUse,qdfor the upper Hamil- tonian [see Eq. (23)]as a function of the hole energy eat fixed in-plane wave vector q=0.15 nm−1. Temperature is 120 K. The en- ergy is measured from the barrier edge for heavy holes at a given q. The sharp peaks denoted by LH1, HH1, and HH2 correspond toquasibound light-hole and heavy-hole states in the parabolic-bandmodel, and M1, M2, and M3 denote mixed quasibound states in themultiband model.MULTIBAND k·pTHEORY OF CARRIER ESCAPE PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-7LH-LH current is negligible (see Fig. 5 ). The energy interval in which the escape current is significant shows that at roomtemperature, the majority of holes escape by thermally as-sisted tunneling and emission through the states near the bar-rier edge. The sharp peaks (HH1 and M1 )here are the weakly bound states, and the broad peaks (M2 and HH2 )are the precursors of the new quasibound states. 10As in the low- temperature situation, the “mixed” LH-HH component peaksat the energy position of the quasibound state. One can seethat the peak M1 in the multiband model is broader than thatof the parabolic-band model (HH1 ). Indeed, the integrated escape current from the M1 quasibound state is more than 3times larger than that for the HH1 state. The escape currentin the vicinity of the barrier edge (M2 and HH2 peaks )is as well enhanced by the mixing but with the more modest fac-tor of 1.32. This shows that, although the most drastic effectof the band mixing is in the increasing of heavy-hole tunnel-ing from well-localized states (which are especially impor- tant at lower temperatures ), the influence of mixing on the weakly localized near-barrier states is substantial. This effectis enhanced by the large HH density of states, which makesfor increasing of the HH escape current as the “penetration”of the outer HH states through the barrier is made easier bythe LH admixture. IV. CONCLUSION We have developed a general theory of multichannel co- herent escape and applied it to the problem of hole escapefrom quantum wells. The effects of the heavy-to-light (light- to-heavy )hole transformation and the subband nonparabolic- ity are included consistently with the multiband effective-mass theory in the Luttinger approximation. The agreementbetwen calculation results and the available experimentaldata is improved considerably compared to the parabolic-band effective-mass approximation. We found that for theGaAs/AlGaAs system, band mixing generally increases theescape current in comparison with the simple parabolic-bandeffective-mass approximation. This is due to the significantincrease of the direct heavy-hole current (heavy-hole light- ening as a results of the light-hole admixture )as well as the appearance of a new type of “mixed” component in the light-and heavy-hole total currents. 1E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, IEE Proc.-J: Opto- electron. 147, 251 (2000 ). 2I. Ogura, Y. Hashimoto, H. Kurita, T. Shimizu, and H. Yokoyama, IEEE Photonics Technol. Lett. 10, 603 (1998 ). 3C. Knöll, M. Golles, Z. Bakonyi, G. Onishchukov, and F. Led- erer, Opt. Commun. 187, 141 (2001 ). 4H. Schneider and K. v. Klitzing, Phys. Rev. B 38, 6160 (1988 ). 5A. Larson, P. A. Andrekson, S. T. Eng, and A. Yariv, IEEE J. Quantum Electron. 24, 787 (1988 ). 6D. Ahn and S. L. Chuang, Phys. Rev. B 34, 9034 (1986 ). 7D. C. Hutchings, Appl. Phys. Lett. 55, 1082 (1989 ). 8K. R. Lefebvre and A. F. M. Anwar, IEEE J. Quantum Electron. 33, 187 (1997 ). 9K. R. Lefebvre and A. F. M. Anwar, J. Appl. Phys. 80, 3595 (1996 ). 10V. V. Nikolaev and E. A. Avrutin, IEEE J. Quantum Electron. 39, 1653 (2003 ). 11L. C. Andreani, A. Pasquarello, and F. Bassani, Phys. Rev. B 36, 5887 (1987 ). 12S. L. Chuang, Phys. Rev. B 40, 10 379 (1989 ). 13C. Y.-P. Chao and S. L. Chuang, Phys. Rev. B 43, 7027 (1991 ). 14S. Ekbote, M. Cahay, and K. Roenker, Phys. Rev. B 58, 16 315 (1998 ). 15D. J. Moss, T. Ido, and H. Sano, IEEE J. Quantum Electron. 30,1015 (1994 ). 16P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, Phys. Rev. B 22, 3519 (1980 ). 17P. W. Anderson, Phys. Rev. B 23, 4828 (1981 ). 18B. Shapiro, Phys. Rev. B 35, 8256 (1987 ). 19M. Cahay, M. McLennan, and S. Datta, Phys. Rev. B 37, 10 125 (1988 ). 20M. Buttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985 ). 21R. G. Newton, Scattering Theory of Waves and Particles , 1st. ed. (McGraw-Hill, New York, 1966 ). 22J. M. Luttinger, Phys. Rev. 102, 1030 (1956 ). 23M. Altarelli, U. Ekenberg, and A. Fasolino, Phys. Rev. B 32, 5138 (1985 ). 24G. Fishman, Phys. Rev. B 52, 11 132 (1995 ). 25D. A. Broido and L. J. Sham, Phys. Rev. B 31, 888 (1985 ). 26A. Twardowski and C. Hermann, Phys. Rev. B 35, 8144 (1987 ). 27S. L. Chuang, Phys. Rev. B 43, 9649 (1991 ). 28M. Altarelli, Phys. Rev. B 28, 842 (1983 ). 29A. V. Rodina, A. Y. Alekseev, A. L. Efros, M. Rosen, and B. K. Meyer, Phys. Rev. B 65, 125302 (2002 ). 30E. L. Ivchenko, A. Y. Kaminski, and U. Rossler, Phys. Rev. B 54, 5852 (1996 ). 31M. V. Kisin, B. L. Gelmont, and S. Luryi, Phys. Rev. B 58, 4605 FIG. 8. Same as Fig. 7, but q=0.6 nm−1and temperature is 300 K.V. V. NIKOLAEV AND E. A. AVRUTIN PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-8(1998 ). 32G. Bastard, Phys. Rev. B 24, 5693 (1981 ). 33M. F. H. Schuurmans and G. W. ’t Hooft, Phys. Rev. B 31, 8041 (1985 ). 34T. Kumar, M. Cahay, and K. Roenker, Phys. Rev. B 56, 4836 (1997 ). 35R. Wessel and M. Altarelli, Phys. Rev. B 39, 12 802 (1989 ).36S. Khan-ngern and I. A. Larkin, Phys. Rev. B 64, 115313 (2001 ). 37A. F. M. Anwar and K. R. Lefebvre, Phys. Rev. B 57, 4584 (1998 ). 38R. Winkler and A. I. Nesvizhskii, Phys. Rev. B 53, 9984 (1996 ). 39J. A. Cavailles, D. A. B. Miller, J. E. Cunningham, P. L. KamWa, and A. Miller, IEEE J. Quantum Electron. 28, 2486 (1992 ).MULTIBAND k·pTHEORY OF CARRIER ESCAPE PHYSICAL REVIEW B 70, 125319 (2004 ) 125319-9

PhysRevB.80.153401.pdf

Aharonov-Bohm oscillations in the local density of states A. Cano1,*and I. Paul2,3,† 1European Synchrotron Radiation Facility, 6 rue Jules Horowitz, BP 220, 38043 Grenoble, France 2Institut Néel, CNRS/UJF, 25 Avenue des Martyrs, BP 166, 38042 Grenoble, France 3Institut Laue-Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble, France /H20849Received 16 June 2009; revised manuscript received 18 August 2009; published 1 October 2009 /H20850 The scattering of electrons with inhomogeneities produces modulations in the local density of states of a metal. We show that electron interference contributions to these modulations are affected by the magnetic fieldvia the Aharonov-Bohm effect. This can be exploited in a simple scanning-tunneling-microscopy setup thatserves as an Aharonov-Bohm interferometer at the nanometer scale. DOI: 10.1103/PhysRevB.80.153401 PACS number /H20849s/H20850: 73.20.At, 73.23. /H11002b, 73.40.Gk Scanning tunneling microscopy /H20849STM /H20850provides a power- ful tool to measure the density of states.1The scattering of electrons with inhomogeneities is known to modify the cor-responding density of states producing local modulations.These modulations were first probed by STM with atomicresolution in Ref. 2, where standing-wave patterns in the local density of states /H20849LDOS /H20850arising from scattering of electrons with impurities and step edges were observed onthe surface of Cu /H20849111/H20850. These patterns encode information about the electronic properties of the corresponding system.For example, in the case of a two-dimensional electronicsystem, a point defect produces oscillations in the LDOSwith wavevector 2 k Ffor very low-bias volatge.2The period of these oscillations changes as a function of the bias volt-age, from which one can infer the spectrum of the electrons.In the case of a superconductor, there can be additional fea-tures reflecting the symmetry of the superconducting gap. 3 Moreover, atomic manipulation permits the engineering ofsurfaces, making it possible to confine electrons into the so-called quantum corrals. 4This has attracted great attention since it permits the study of lifetime effects, Kondo physics,single-atom gating, etc. 5Very recently, open nanostructures have been shown to be amenable for quantum holographic encoding.6In the following we show that under suitable con- ditions, the LDOS exhibits oscillations due to the magneticfield that can be interpreted as due to the Aharonov-Bohmeffect. Therefore the STM setup can be designed to serve asan Aharonov-Bohm interferometer at the nanometer scale. For the sake of concreteness we consider the close-packed surface of a noble metal where the so-called Shockley sur-face states form a two-dimensional nearly free electron gas.Two atoms deposited on top of this surface can be modeledas two point scattering potentials for the surface electrons.This forms the simplest Aharonov-Bohm interferometer,where the role of the two different paths in conventionalAharonov-Bohm setups is played by the two scattering pathsshown in Fig. 1. In the presence of a magnetic field, elec- trons scattering along these loops pick up different phasesdepending on whether the scattering is clockwise or anti-clockwise /H20849the two paths being connected by time-reversal symmetry at zero field /H20850. This affects the interference contri- bution to the LDOS measured by the STM tip, which even-tually exhibits oscillations as a function of the magnetic fluxthat passes through area enclosed by the above paths. Thephysics of these LDOS oscillations is similar to the effect ofthe magnetic field on weak localization in disordered two- dimensional metals. 7 The dI/H20849r,V/H20850/dVmaps obtained experimentally are deter- mined by the LDOS of the sample N/H20849r,/H9275=eV/H20850, the tip den- sity of states, and the tunneling matrix elements.5,8In the Tersoff-Hamann approximation with a constant tip density ofstates dI/dVis proportional to N. 5,8However, the oscillations of the LDOS that we discuss are picked up by the experi-mental dI/dVeven if the above proportionality is lost due to voltage dependence of the tunneling matrix elements or dueto variation in the tip density of states with energy. TheLDOS of the sample can be obtained from the correspondingretarded Green’s function as N/H20849r, /H9275/H20850=−2 /H9266ImGR/H20849r,r;/H9275/H20850/H20849 1/H20850 /H20849the factor 2 is due to spin degeneracy /H20850. For a two- dimensional free-electron gas the Green’s function is GR/H20849r,r/H11032;/H9275/H20850=−i/H9266NH0/H208491/H20850/H20849k/H20849/H9275/H20850/H20841r−r/H11032/H20841/H20850. /H208492/H20850 Here N=m//H208492/H9266/H60362/H20850is the density of states of the electron gas per spin, H0/H208491/H20850is the zeroth-order Hankel function of the first FIG. 1. /H20849Color online /H20850STM interferometer. rrepresents the po- sition of the STM tip on the surface and r1andr2two impurities. The LDOS measured by the STM tip contains interference contri-butions due to electrons traveling along the two paths shown in thefigure. The magnetic field affects this interference via theAharonov-Bohm effect, producing oscillations in the LDOS that ismeasured by the tip.PHYSICAL REVIEW B 80, 153401 /H208492009 /H20850 1098-0121/2009/80 /H2084915/H20850/153401 /H208494/H20850 ©2009 The American Physical Society 153401-1kind and k/H20849/H9275/H20850is given by the dispersion relation k/H20849/H9275/H20850 =kF/H208491+/H9275//H9262/H208501/2, where /H9262=/H60362kF2//H208492m/H20850. For large distances /H20851r/H11271k−1/H20849/H9275/H20850/H20852we have GR/H20849r,/H9275/H20850/H11015−iN/H208732/H9266 k/H20849/H9275/H20850r/H208741/2 ei/H20851k/H20849/H9275/H20850r−/H9266/4/H20852. /H208493/H20850 The presence of impurities can be modeled by a term Himp=/H20885drU/H20849r/H20850/H9267/H20849r/H20850/H20849 4/H20850 in the Hamiltonian of the system, where Uis the scattering potential associated with the impurities and /H9267is the elec- tronic density. Following a perturbative approach5the Green’s function can be expressed as G=G0+/H9254G, where G0 is the Green’s function in the absence of impurities and /H9254G/H20849r,r/H11032/H20850=/H20885dr/H11033G0/H20849r−r/H11033/H20850U/H20849r/H11033/H20850G0/H20849r/H11033−r/H11032/H20850 +/H20885/H20885 dr/H11033dr/H11630G0/H20849r−r/H11033/H20850U/H20849r/H11033/H20850 /H11003G0/H20849r/H11033−r/H11630/H20850U/H20849r/H11630/H20850G0/H20849r/H11630−r/H11032/H20850+ ... /H208495/H20850 The dependence on the frequency is dropped since the scat- tering is assumed to be elastic. The latter quantity in Eq. /H208495/H20850 contains the interference contributions to the LDOS we areinterested in /H20849i.e., from terms second order and higher in the scattering potential /H20850. Two identical point impurities are described by the scat- tering potential U/H20849r/H20850=U 0/H20851/H9254/H20849r−r1/H20850+/H9254/H20849r−r2/H20850/H20852. /H208496/H20850 In this case the quantity /H9254G/H20849r,r/H20850naturally contains two types of terms. On one hand, there are /H20849additive /H20850terms in which the scattering with the impurities is produced separately. TheFourier transform of this contribution has quasiparticlepeaks, which contains information about the dispersion ofthe electrons /H20849see, e.g., Ref. 3/H20850. This contribution, however, plays no role in the physics that we intend to study. On theother hand, there are terms involving scattering with both theimpurities. Among the latter terms, there are processes inwhich the semiclassical scattering paths enclose a finite areaas in Fig. 1. It is important to note that such loops occur in pairs connected by time-reversal symmetry /H20849i.e., clockwise and anticlockwise /H20850. For example, to the lowest order in the impurity potential, the contribution to the Green’s functiondue to closed loops reads /H9254Gloop/H208492/H20850/H20849r,r/H20850=U02G0/H20849r−r1/H20850G0/H20849r1−r2/H20850G0/H20849r2−r/H20850+/H208491↔2/H20850. /H208497/H20850 In the absence of a magnetic field the two terms in this ex- pression are equivalent /H20849due to time-reversal symmetry /H20850, and therefore give the same contribution to the LDOS /H20849it can be said that the interference is constructive /H20850. In the presence of a magnetic field, however, time-reversal symmetry is broken.Then electrons traveling clockwise and anticlockwise alongthe above loop acquire different phases, so the subsequentinterference is affected. This results in Aharonov-Bohm os-cillations of the LDOS as we shall see explicitly below. Next we demonstrate that the simple geometrical picture above remains unchanged when higher order terms in theimpurity potential are taken into account. At higher order wehave to deal with the following additional ingredients: /H20849i/H20850 multiple scattering at the impurities and /H20849ii/H20850multiple scatter- ing where semiclassically the particle goes back and forthbetween the two impurities. The first type of multiple scat-tering can be easily taken into account by replacing each ofthe scattering potentials by their respective Tmatrices. That is, 9 U0→U˜0=U0 1−U0G0/H208490/H20850. /H208498/H20850 As regards the second point, we note that /H9254Gloop/H20849r,r/H20850is en- tirely due to processes where the path between the impuritiesis traversed an odd number of times /H20849otherwise the scattering path does not enclose a finite area /H20850. In the second order con- tribution Eq. /H208497/H20850, for example, the path between the impuri- ties is traversed once. At O/H20849U 04/H20850there are contributions where the path is traversed three times, and so on. Taking allthis into account, we obtain /H9254Gloop/H20849r,r/H20850=W2G0/H20849r−r1/H20850G0/H20849r1−r2/H20850G0/H20849r2−r/H20850+/H208491↔2/H20850, /H208499/H20850 where W2=U˜ 02 1−U˜ 02G0/H20849r1−r2/H20850G0/H20849r2−r1/H20850. /H2084910/H20850 It is worth noticing that in all the processes that finally give rise to Eq. /H208499/H20850we are actually dealing with the same area, since to go back and forth along the same line, for example,does not change the area of the resulting loop. This is thesimple reason why the nontrivial phase relation between theclockwise and the anticlockwise paths in the presence of amagnetic field is not washed out by multiple impurity scat-tering. Let us now consider explicitly the influence of the mag- netic field. In the low-field regime /H20849see below /H20850we can use the semiclassical approximation for the electron Green’sfunction: 7 G0/H20849r−r/H11032/H20850= exp/H20873i/H9266 /H90210/H20885 rr/H11032 A/H20849l/H20850·dl/H20874G00/H20849r−r/H11032/H20850./H2084911/H20850 Here G00/H20849r−r/H11032/H20850represents the Green’s function in the ab- sence of magnetic field, Ais the vector potential /H20849B=/H11612 /H11003A/H20850,/H90210=h//H208492e/H20850is the flux quantum, and the integral is along the straight line connecting rand r/H11032. The magnetic field then enters the interference contributions to the LDOSvia complex factors e /H11006i/H9266/H9021//H90210, where /H9021is the magnetic flux through the area enclosed by the corresponding scatteringpath /H20849the different signs of the phase corresponds to anti- clockwise and clockwise line integrals respectively /H20850.A sa result, the LDOS can be written asBRIEF REPORTS PHYSICAL REVIEW B 80, 153401 /H208492009 /H20850 153401-2N/H20849r,/H9275/H20850=NB=0/H20849r,/H9275/H20850+Nloop/H20849r,/H9275/H20850/H20851cos/H20849/H9266/H9021//H90210/H20850−1/H20852, /H2084912/H20850 where NB=0is the zero-field total LDOS and Nloop/H20849r,/H9275/H20850=−2 /H9266Im/H9254Gloop/H20849r,r;/H9275/H20850/H20849 13/H20850 represents the /H20849constructive /H20850interference contribution due to all closed paths in the absence of a magnetic field /H20851computed from Eq. /H208499/H20850with G00/H20852. This interference process picks up a nontrivial phase in the presence of a magnetic field, givingrise to oscillations in the LDOS as in Eq. /H2084912/H20850, which is a manifestation of the Aharonov-Bohm phenomenon. 10This can be revealed by varying either the magnetic field or therelative position between the STM tip and the impuritiessince/H9021changes in both cases. We note that, irrespective of the position of the STM tip, the magnitude of the correctionto the LDOS reduces with Bfor low magnetic fields. As in the case of negative magnetoresistance in weak localization,this is due to the fact that magnetic field induces destructiveinterference between the contribution of the two semiclassi-cal paths. Next we discuss the limitations of our calculations and the feasibility of our proposal to use the STM as an Aharonov-Bohm interferometer at a nanometer scale. The expression/H2084912/H20850for the LDOS has been derived within a semiclassical approach, so it holds as long as Eq. /H2084911/H20850can be used for describing the influence of the magnetic field on the electronsystem. This is possible if the magnetic field is such that theFermi wavelength is much smaller than the Landau orbits: /H9261 F/H11270aB=/H20873/H90210 /H9266B/H208741/2 . /H2084914/H20850 The magnetic field needed to observe a complete Aharonov- Bohm oscillation can be estimated as B2/H9266/H110114/H90210/d2, /H2084915/H20850 where dis the characteristic distance in the setup, i.e., the distance between the impurities and/or between the impuri-ties and the position of the tip. For d/H1101140−20 nm this field isB 2/H9266/H110115–20 T, which corresponds to Landau orbits aB /H1101111−6 nm. For Cu /H20849111/H20850and Ag /H20849111/H20850/H9261F=2.95 nm and 7.6 nm, respectively, and therefore the condition in Eq. /H2084914/H20850forthe semiclassical approximation is valid. Furthermore, in spite of the fact that the interference signal is long-range inthe sense that N loopdecays as 1 /din a power-law fashion /H20851see Eq. /H208493/H20850/H20852, in reality there are dephasing processes that introduce an extra attenuation /H20849thermal dephasing for ex- ample /H20850, and which, for simplicity, have not been taken into account in the current computation. Experimentally, in fact,the impurity-induced variations in the LDOS are typicallyobserved up to distances of the order of a few times theFermi wavelength, say 10 /H9261 F.11Therefore the characteristic distance in our setup must be d/H1135110/H9261F. For d/H1101140−20 nm we then need /H9261F/H114074−2 nm, which still can be smaller than aBat 5–20 T. Thus we find that the semiclassical interpreta- tion is good to describe the first few periods of the LDOSoscillations. The first corrections to our results will be due tothe curvature of the classical trajectories, which can still bedescribed within the semiclassical picture. 12To describe suf- ficiently high periods, however, one has to go beyond Eq./H2084911/H20850and consider the influence of the magnetic field within a Landau-level approach. These periods imply magnetic fieldsconsiderably high /H20849/H1102220 T /H20850, so we do not develop this latter approach here. The Aharonov-Bohm physics reveals also in the spatial variations of the LDOS for a fixed magnetic field. In oursetup/H9021varies only in the direction perpendicular to the line connecting the two impurities. In consequence, the STMscans along this direction will show a periodic envelope dueto the cosine factor in Eq. /H2084912/H20850. With two impurities sepa- rated 20 nm, fields of 5–20 T give rise to periods of 80 and20 nm, respectively, for such an envelope /H20849see Fig. 2/H20850. It is also worth mentioning that the Zeeman splitting, ne- glected so far, has a trivial influence on the Aharonov-Bohmoscillations if the spin is conserved in the scattering process.N loopin Eq. /H2084912/H20850actually results from contributions associ- ated with the two spin polarizations. It can be written as Nloop=1 2/H20849Nloop↑+Nloop↓/H20850if the spin is conserved. The eventual difference between Nloop↑andNloop↓due to the Zeeman split- ting can be probed by means of spin-polarized STM. How-ever this difference does not alter the oscillatory behavior ofthe LDOS described above. The situation is more subtle ifthe spin can flip during the scattering process as a result of,e.g., spin-orbit coupling. This can affect multiple scatteringin a nontrivial way, 13and can give rise to the analog of the antilocalization phenomenon if spin components are mea-sured separately. B=5T 20nm B=1 0T 20nm B=1 5T 20nm 20nmB=2 0T FIG. 2. /H20849Color online /H20850Expected STM patterns for two impurities 20 nm apart on the Ag /H20849111/H20850surface after subtraction of the B=0 signal. The horizontal stripes are produced by the Aharonov-Bohm effect in the LDOS, whereas the remaining elliptic features are due to theinterference contributions described by N loop /H20851with/H9261F=7.6 nm and real Wmatrices as defined in Eq. /H2084910/H20850/H20852.BRIEF REPORTS PHYSICAL REVIEW B 80, 153401 /H208492009 /H20850 153401-3In summary, we have shown that magnetic field affects electron interference contributions to the local density ofstates via the Aharonov-Bohm effect. This can be exploitedin building STM devices that serve as Aharonov-Bohm in-terferometers at the nanometer scale. We have illustrated thispossibility for the close-packed surface of a noble metal withtwo atoms adsorbed on top. The role of these atoms consistsin creating strong enough scattering potentials, which canalso be produced, for example, using additional STM tips.The implementation of this new functionality into the STM technique might broaden its applications notably, offeringnew perspectives for STM studies of the fundamental prop-erties of surfaces and underlying systems. We acknowledge I. Brihuega, P. Bruno, E. Kats, and R. Whitney for very fruitful discussions and M. Collignon forthe Fig. 1. *cano@esrf.fr †indranil.paul@grenoble.cnrs.fr 1C. J. Chen, Introduction to Scanning Tunneling Microscopy /H20849Ox- ford, New York, 1993 /H20850. 2M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature /H20849London /H20850 363, 524 /H208491993 /H20850; Y . Hasegawa and Ph. Avouris, Phys. Rev. Lett. 71, 1071 /H208491993 /H20850. 3J. M. Byers, M. E. Flatté, and D. J. Scalapino, Phys. Rev. Lett. 71, 3363 /H208491993 /H20850. 4M. F. Crommie, C. P. Lutz, and D. M. Eigler, Science 262, 218 /H208491993 /H20850. 5G. A. Fiete and E. J. Heller, Rev. Mod. Phys. 75, 933 /H208492003 /H20850. 6C. R. Moon et al. , Nat. Nanotechnol. 4, 167 /H208492009 /H20850. 7B. L. Altshuler, D. Khmelnitzkii, A. I. Larkin, and P. A. Lee, Phys. Rev. B 22, 5142 /H208491980 /H20850; G. Bergmann, Phys. Rep. 107,1 /H208491984 /H20850. 8J. Tersoff and D. R. Hamann, Phys. Rev. Lett. 50, 1998 /H208491983 /H20850; Phys. Rev. B 31, 805 /H208491985 /H20850; J. M. Blanco, F. Flores, and R. Perez, Prog. Surf. Sci. 81, 403 /H208492006 /H20850; O. Fischer et al. , Rev.Mod. Phys. 79, 353 /H208492007 /H20850. 9Expression /H208492/H20850gives a spurious divergence for the real part of G/H20849r,/H9275/H20850for r=0. This divergence is regularized by using Kramers-Kroning relations to define the real part of G/H208490,/H9275/H20850in expression /H208498/H20850as Re G/H208490,/H9275/H20850=−2 /H9266P/H20848ImG/H208490,/H9275/H11032/H20850 /H9275−/H9275/H11032d/H9275/H11032and by cutting off the integral with the corresponding electron bandwidth /H20851see, e.g., S. Doniach and E. H. Sondheimer, Green’ s Functions for Solid Physicists /H20849W.A. Benjamin Inc., London, 1974 /H20850,p .7 9 /H20852. 10Note that the experimental dI/dVmaps will show the Aharonov- Bohm oscillations even if dI/dVis not proportional to the local density of states of the sample. 11In the case of Cu /H20849111/H20850and Ag /H20849111/H20850the phase relaxation length can be as large as 66 and 60 nm, respectively, at low tempera-tures /H20851see, e.g., O. Jeandupeux, L. Burgi, A. Hirstein, H. Brune, and K. Kern, Phys. Rev. B 59, 15926 /H208491999 /H20850/H20852. 12T. A. Sedrakyan, E. G. Mishchenko, and M. E. Raikh, Phys. Rev. Lett. 99, 036401 /H208492007 /H20850. 13J. D. Walls and E. J. Heller, Nano Lett. 7, 3377 /H208492007 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 80, 153401 /H208492009 /H20850 153401-4

PhysRevB.78.075209.pdf

Systematic trends of first-principles electronic structure computations of Zn 1−xAxB diluted magnetic semiconductors R. D. McNorton, T. M. Schuler, and J. M. MacLaren Department of Physics, Tulane University, New Orleans, Louisiana 70118, USA R. A. Stern Seagate, 7801 Computer Avenue, Bloomington, Minnesota 55435, USA /H20849Received 27 March 2008; revised manuscript received 13 June 2008; published 26 August 2008 /H20850 This paper presents a study of the calculated electronic properties of the Zn-based II–VI dilute magnetic semiconductors in a Zn 22A2B24structure, where A=Cr, Mn, Fe, Co, and Ni, and B=S, Se, and Te. In this study we investigate the local densities of states of the magnetic ions and host semiconductor, the magnetic exchangeinteractions as a function of transition-metal ion separation and orientation, the origins of the magnetic cou-pling, the tendency for impurity atoms to cluster creating impurity rich regions—at least thermodynamically,and the local magnetic moments. The results show half-metallic behavior for Cr, Fe, and Ni impurities, and inthe case of Cr and Ni a ferromagnetic coupling consistent with the double-exchange mechanism. The Mn- andCo-doped materials are found to be semiconducting and couple antiferromagnetically, which can be explainedby the superexchange model. The Fe-doped materials show the sign of the coupling is dependent on theorientation and separation of the impurities. DOI: 10.1103/PhysRevB.78.075209 PACS number /H20849s/H20850: 75.30.Et, 75.50.Pp, 71.20. /H11002b, 31.15.A /H11002 I. INTRODUCTION Semiconductors containing small amounts of magnetic impurities, known as dilute magnetic semiconductors/H20849DMS /H20850, have been of interest to the physics and engineering communities for quite some time. 1–3The intriguing, and po- tentially technologically useful, property that arises out ofthese materials is the possibility of strongly spin polarizedcurrents. This is electronically possible when either the ma-jority or minority-carrier states dominate at the Fermi energy.In the extreme case of half-metallic materials, one spin chan-nel is conducting while the other spin channel is strictlyinsulating. 4It is this idea of spin polarized currents that would allow engineered devices that combine the propertiesof magnetism with the traditional semiconductors to createthe so-called spintronic devices. The electronic structure of the material depends on the particular host crystal structure and on the magnetic impurityatom. The particular DMS compounds studied here havecrystal structures in which the bonding interactions occurbetween the semiconductor s-porbitals and the magnetic ion’s dorbitals. The nature of this bonding determines the physical properties observed in our calculations. It has beensuggested 1,5that one of the variables determining the strength of the exchange interactions is the amount of sp-d hybridization between the host and impurity. To this end, wehave studied in detail the electronic structure of wide bandgap II–VI semiconductors ZnS, ZnSe, and ZnTe, which formin the zincblende structure and contain the magnetic impuri-ties Cr, Mn, Fe, Co, and Ni. Prior work has confirmed that the II–VI compounds doped with Cr are ferromagnetic 5–9/H20849FM/H20850even at room temperature, however they require a high concentration ofthe Cr impurity. 10–12Mn-doped systems have been studied extensively due to the giant Zeeman splitting of the d bands. This splitting results in Mn being easy to model andstudy due to a small number of exchange mechanisms being possible. Mn has been predicted and observed experimen-tally to have an antiferromagnetic /H20849AFM /H20850coupling via the superexchange. 2,13–15 Compounds containing Fe impurities, with their magneti- cally inactive singlet ground state /H20849GS/H20850, have been shown to be AFM,13FM,16and spin-glass7compounds. These differ- ing results indicate that the magnetism in Fe DMS com-pounds is frustrated which is manifested through a competi-tion between exchange mechanisms. Ni-doped systems havealso been shown to be FM /H20849Ref. 5/H20850and AFM /H20849Refs. 6and7/H20850 and Co-doped systems have been determined to be AFM innature. 5–7,17These studies show hybridization between the impurity’s dshell and the pshell of the near neighbor anion. The short range of the dorbital prevents any other significant hybridization between more distant neighbors to occur.15The hybridization between these orbitals is greatest in Cr, fol-lowed by Ni and Fe. All of these materials have been foundto be half-metallic. 5The results of a similar study for ZnCrTe are consistent with the data presented here, and suggest thatin determining the exchange properties of the DMS struc-tures one must take into account both the impurity separationand bonding direction between the impurities and theanions. 18This is to be expected since the mechanisms behind the magnetic coupling are the double-exchange /H20849DE/H20850and superexchange /H20849SE/H20850processes. In this paper we will further develop the results of a prior work5which found that Cr and Ni impurities in the semicon- ductor Zn X/H20849X=S, Se, Te /H20850result in a FM coupling, with Cr producing the stronger coupling of the two, while Mn and Coimpurities yield an AFM GS. The DE mechanism, presum-ably, is responsible for the ferromagnetism observed in theformer compounds. Compounds containing Fe impuritiesproduce a half-metallic state and thus are able to participateeffectively in a DE process via the conduction band /H20849CB/H20850, but do not exhibit a FM GS except at certain transition-metal/H20849TM/H20850separations and orientations.PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 1098-0121/2008/78 /H208497/H20850/075209 /H2084911/H20850 ©2008 The American Physical Society 075209-1It will be shown that the filling of the TM dstates and the orientation of the TMs to one another determines which d orbitals participate, the amount of p-dhybridization, and thus the exchange coupling. The ability or ease in whichelectrons can hop through the crystal given the particularatomic arrangement and density of states appears to be an-other contributing factor to the sign and strength of the ex-change coupling. While the SE mechanism is an indirectexchange mediated by the porbital of the group VI element, 2 we propose that in certain circumstances a DE coupling19 can occur which we suggest explains the observation of a FM alignment. We postulate that the two exchange mecha-nisms can be in competition and the one that is favored isdetermined by fundamental factors of the electronic struc-ture, the magnetic ion orbitals which are involved, and theseparation distance between the impurities. To gain an understanding of the effects of a random dis- tribution of impurities, which would occur in real samples,we have calculated the densities of states /H20849DOS /H20850and ex- change energies for a variety of impurity-impurity separa-tions and orientations. The calculation of the GS energies forvarious impurity separation distances suggests that there is atendency of these implanted ions to cluster together. Thespatial distribution of the magnetic impurities in the host isclearly important in determining the magnetic properties ofthese materials. Obtaining experimentally an atomic distribu-tion of magnetic impurities in a real sample is a difficult task,and one would expect that since there is a thermodynamictendency to cluster that there will be regions that are impu-rity rich as well as regions that are impurity poor. Such com-position fluctuations affect the electronic structure and mag-netic coupling of the system. The distribution of impuritieshas a significant effect in the cases where SE is the dominantmechanism. Anderson 20showed that the SE interaction was determined by the interatomic transfer integral between p-d wave functions 2J kbT=2b2 4S2U, /H208491/H20850 where Jis the interaction energy between magnetic ions, 2 S represents the number of unpaired electrons, Uis the energy required to add an extra electron to a neutral solid /H20849electron affinity /H20850, and bis the transfer integral. It has been previously shown that b/H11008d−8, where dis the spacing between impurities.21 A. Zincblende II–VI DMS The II–VI DMS materials have been studied extensively, in part owing to the fact that their band gap, lattice constant,and other band parameters can be tuned by varying the ioniccomposition. 2This property could make these II–VI’s poten- tially useful for a wide range of electronic applications andhas also been observed in III–V DMS materials. 22The struc- ture of these materials is that of zincblende for ZnS, ZnSe,and ZnTe when the impurity concentration is 4%, 2as was determined in this study. It has been suggested that someII–VI materials will form in the wurtzite structure at impurityconcentrations above 10%, 2however these concentrationsand structures are not considered in this work. The zincblende structure is derived from a diamond bravais lat-tice, with AandBatoms on adjacent sites. Thus each Aatom has four nearest-neighbor /H20849nn/H20850Batoms, and vice versa. This tetragonal symmetry at the atomic sites is important in deter-mining the exchange mechanism between magnetic impuri-ties. The crystal-field effect on a tetragonal site will split themagnetic ion dshell into two levels the e gsubshell, which is lower in energy and holds only two electron pairs, and the t2g subshell, which is higher in energy and holds three pairs of electrons. These materials also exhibit tetrahedral bonding at all lat- tice sites, which results in sp3bonding between constituents. This effect is well known to be maximized in the II–VI semi-conductor, owing to the complete filling of valence states. Ifwe introduce impurities into the host the amount of sp 3 bonding will depend on the impurity introduced. In a II–VI semiconductor doped with Mn this bonding strength is maxi-mized, as the Mn half-filled dshell can virtually act as a filled valence shell, similar to the cation II element. The Mnatom can thereby replace the cation and still remain virtuallyas stable as the host semiconductor. 2 II. COMPUTATIONAL DETAILS Supercell computations were performed with the Vienna Ab-initio Simulation Package /H20849V ASP /H20850,23using all-electron frozen-core projector augmented wave /H20849PAW /H20850potentials24,25 and the generalized gradient approximation /H20849GGA /H20850to the exchange-correlation energy.26The GGA functional was that of Perdew-Wang 1991 /H20849PW91 /H20850/H20849Ref. 26/H20850using the interpola- tion of V osko et al.27Potentials for all magnetic impurities, except Co, treated the 3 pand 4 ssemicore states as valence states, while the Co potential treated only 3 dstates as va- lence states. Energy cutoffs were calculated by convergingthe GS energy by varying the ENCUT parameter in V ASP.They were determined to be 319 eV for the Cr, Mn, and Cosystems, 345 eV for the Fe system, and 400 eV for the Nisystem. Lattice constants were converged in a manner simi-lar for the nonmagnetic semiconductor hosts. Their valueswere then used for the DMS computations as indicated inTable I. We did not consider any local lattice relaxations or spin-phonon interactions that may occur around the impurityatom. Calculated and experimental lattice constants and bandgaps for pure II–VI semiconductors are given in Table I. The computed lattice constants of the Zn Bcompounds are quite close to the experimental values, although the band gap issignificantly under estimated.TABLE I. Lattice constants and energy gaps of II–VI semicon- ductors in the zincblende structure, comparing the DFT calculationswith experimental /H20849Ref. 2/H20850measurements. Material L 0/H20849Å/H20850 Energy gap /H20849eV/H20850 calc. expt. calc. expt. ZnS 5.44 5.416 2.88 3.84 ZnSe 5.73 5.670 2.48 2.82 /H20849300 K /H20850 ZnTe 6.09 6.102 1.86 2.39 /H20849300 K /H20850MCNORTON et al. PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-2The DOS and exchange calculations of the magnetically doped materials were calculated via supercell computationsof the materials Zn 22A2B24, where B=S, Se, Te, and A=Cr, Mn, Fe, Co, Ni. This corresponds to a TM concentration ofroughly 4% for the entire crystal. Each unit cell consisted ofeight atoms, and the dimensions of the supercell were 1 unitcell in the xdirection, 2 unit cells in the ydirection, and 3 unit cells in the zdirection. These supercells were con- structed to yield either an FM or AFM GS. A 3 /H110033/H110031 grid for a total of four kpoints was used to reduce calculation time, while still maintaining a total-energy convergence ofbetter than 1.0 meV/atom for all calculations. Several differ-ent spacings of the magnetic impurity were modeled in thecalculations in order to represent the possible spacings of theimpurities allowed within the zincblende structure. Thesepossible spacings were found to be /H20849from least to greatest, in units of the lattice constant /H20851L o/H20852/H20850: .707 Lo, 1.000 Lo, 1.225 Lo, 1.414 Lo, 1.581 Lo, and 1.871 Lo. The actual ionic positions in the supercell are /H20849000 /H20850for the one impurity that remains fixed, and for the variable impurity the positions /H20849from least separation distance to greatest /H20850are/H208491/2 0 1/2 /H20850,/H20849100 /H20850,/H208491 1/2 1/2/H20850,/H20849101 /H20850,/H208490 1/2 3/2 /H20850, and /H208491 1/2 3/2 /H20850. For visualization, three of these configurations are portrayed in Fig. 1. III. ELECTRONIC STRUCTURE A. DMS DOS First we will summarize the electronic properties of the pure II–VI semiconductors. The total density of states forzincblende ZnS is shown in Fig. 2. The large sharp peaks at /H11011−6 eV are the Zn dstates which do not hybridize signifi- cantly with the group VI anion orbitals. The broad featurebetween the Zn dstates and the Fermi energy /H20849set to 0 eV /H20850is composed of Zn- sand S- pstates of bonding. The unoccupied states are also hybridized mixtures of Zn- sand S- pstates. The feature at /H11011−12 eV reflects the S- sstates.The DOS of these DMS materials can be used to study trends that emerge when the TM ions and their separationsare varied, and also determine whether the material is semi-conducting or half-metallic. In this study we focus on theeffect the TM separation has on the electronic and magneticproperties of the compound. The total and partial densities of states for different TM impurities at a constant spacing in ZnS are displayed in Fig.3. In the TM DOS /H20849solid black line /H20850we see that as the im- purity increases in atomic number from Cr to Ni the dpeaks of both the conduction and valence bands shift down in en-ergy such that the magnetic ion maintains an integer spinmoment. For Cr, Fe, and Ni impurities, the dpeaks of the TM coincide with the Fermi energy /H20849E F/H20850. For materials con- taining Mn and Co impurities there are no dstates at the Fermi energy. The II–VI materials that contain Cr, Fe, and Niare therefore determined to be half-metallic, while materials (a) (b) (c) FIG. 1. Atomic arrangement of impurities for three of the six possible supercell configurations. The separation of the impurities in the figure in terms of the lattice constant Loand are /H20849a/H20850=.707 Lo,/H20849b/H20850=1.225 Lo, and /H20849c/H20850=1.871 Lo, where Lorepresents the lattice constant of the undoped semiconducting compound. -12 -10 -8 -6 -4 -2 2 4 6-10-8-6-4-20246810Dens ity of States [ /atom /eV] Ener gy [eV]EF FIG. 2. Spin resolved DOS of zincblende ZnS, with the positive yaxis displaying spin-up states and the negative yaxis displaying spin-down states.SYSTEMATIC TRENDS OF FIRST-PRINCIPLES … PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-3doped with Mn and Co impurities are found to be semicon- ducting. Ni and Fe have spin minority carriers at the Fermi energy while Cr shows the presence of spin majority carriers. Whenthedorbitals of the TM impurity lie in the gap of the host semiconductor /H20849SC/H20850we can clearly see the crystal-field /H20849CF/H20850 splitting of the dband into the e gandt2gstates, which are ordered in the manner expected for an ion in a tetrahedralsite. However, as we progress through the TM series, thepopulated dstates decrease in energy and hybridize with the host valence band /H20849VB/H20850, forming a broad band from which thee gand t2gsuborbitals can no longer be clearly distin- guished. This behavior is seen in the DOS for all of theimpurities. By examining one particular impurity /H20849ZnCoS /H20850we can compare changes to the DOS as the impurity spacing is in-creased /H20849Fig. 4/H20850. In the FM ZnCoS DOS /H20849right column of Fig.4/H20850, we see that when the TM impurities get further apart the TM- dstates become more localized, as expected. This-2 -1 1 2 3-1.0-0.50.00.51.0Density of States [ /atom/eV] EF-0.50.00.51.0 Zn1-xNixS Energy[eV]-0.50.00.51.0 Zn1-xCoxS-0.50.00.51.0 Zn1-xFexS-0.50.00.51.0 Zn1-xMnxSZn1-xCrxS FIG. 3. DOS of Cr, Mn, Fe, Co, and Ni in ZnS, forming a FM DMS. The shaded region is the total DOS, the solid line representsthe TM- dstate, and the white line represents the anion- pstate.EFFM(3) EFFM(1) Energy[eV]-0.5 0.5-1.0-0.50.00.5AFM(6)-0.50.00.5AFM(3)-0.50.00.51.0AFM(1)Density ofStates [ /atom/eV] -0.5 0.5FM(6) FIG. 4. DOS of ZnCoS in the AFM state /H20849left column /H20850and FM state /H20849right column /H20850for the different crystal impurity spacings, with plots /H208491/H20850,/H208493/H20850, and /H208496/H20850corresponding to separations of 0.707 L0, 1.225 L0, and 1.871 L0, respectively. The shaded region is the total DOS, the solid line represents the TM- dstate, and the white line represents the anion- pstate.MCNORTON et al. PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-4results in a narrowing of the energy spectrum for the TM- d state which creates an increased band gap. This is to be ex-pected as when the impurities are placed further apart in thelattice there will be a reduction in d-dorbital interactions. In the AFM DOS /H20849left column of Fig. 4/H20850we see that the TM-dorbital structure resembles the d-orbital peaks in the FM DOS, though having slightly greater splitting. This isdue to the fact that in the AFM material, the d-dinteraction between like magnetic spins occurs at larger distances. Theonly noticeable change between the different impurity sepa-rations in this plot is the location of the AFM DOS peakswith respect to the top of the VB. A small repulsive interac- tion between the occupied dstates with the states at the top of the VB composed of hybridized porbitals of the anion and dorbitals of the TM. The consequence of separating the impurities is that it will further narrow the impurity band,which reduces the repulsive interaction and allows these VBstates to move slightly higher in energy, reducing the energyseparation between the impurity dand valence orbitals from 0.50 to 0.25 eV . This effect is seen in AFM compoundsdoped with Cr, Co, and Ni impurities, but not Mn or Feimpurities, and it is not observed in the FM case for any ofthe impurities because the energy separation between theTM-dand the spin-down VB states is large. Table IIlists the calculated band gaps for all of the DMS materials discussed in this work, and from this data we seethat the half-metals ZnCr B, ZnFe B, and ZnNi B/H20849B=S, Se, Te/H20850have a small gap between the valence and conduction bands within the semiconducting channel. The AFM orientedstates of Mn exhibit a band gap that ranges from 0.86 to 1.46eV . Likewise, the Co DMS have a range between 0.28 and0.412 eV . By comparing these values to the calculated bandgaps of the host semiconductors /H20849from Table I/H20850we see that, with the introduction of the impurity, the band gap generallydecreases as the number of occupied electron shells withinthe anion increases. This effect is evident when we plot theDOS of the TM d-majority-spin states for each potential an- ion of increasing atomic number, as is shown for Ni impuri-ties in ZnS, ZnSe, and ZnTe in Fig. 5. With the introduction of the greater number of occupied electronic shells associ-ated with there is a broadening of the VB orbitals resulting ina smaller band gap. Experimentally it is found that increas-ing the concentration of the impurity would result in an in-crease of the band gap, especially in Mn. 2One property of half-metallic materials is that the TM ion should exhibit integer spin moments, which is confirmed in all of our calculated half-metallic systems. In conjunctionwith the DOS, this would suggest that the hybridization be-tween the anion pshell and the TM dstates preserves the total integer moment of the TM, and the spin moment ispreserved because the exchange splitting of the TM is greaterthan the CF splitting, which ensures the filling of electronshells according to Hund’s rules. Though the CF splitting issufficient to separate the e gandt2gstates, the subbands show some of their own structure, particularly in the case of the t2g which shows apparent hybridization with the host states. One reason for the increased hybridization of the t2gstates is that these orbitals are oriented diagonally between the crystalaxes /H20849d xy,dzx, and dyz/H20850. The integer moments of each impurity correspond to the accepted free moments of the TM ions, which are 4 /H20849Cr/H20850, 5/H20849Mn/H20850,4/H20849Fe/H20850,3/H20849Co/H20850, and 2 /H20849Ni/H20850. The lone exception is that of ZnNiTe, where the Ni ion exhibits a maximum of 0.25 /H9262b reduction in the magnetic moment, representing a 12.5% re-TABLE II. Energy gap between the occupied and unoccupied orbitals are tabulated from the DOS of the FM alignment for the materials Zn /H20849Cr,Mn,Fe,Co,Ni /H20850/H20849S,Se,Te /H20850using the closest ionic spacing possible /H208490.707 L0/H20850. ZnCrS ZnMnS ZnFeS ZnCoS ZnNiS Spin ZnCrSe ZnMnSe ZnFeSe ZnCoSe ZnNiSe ZnCrTe ZnMnTe ZnFeTe ZnCoTe ZnNiTe 1.75 2.03 2.09 2.09 Majority 1.33 1.48 1.47 1.47 1.48 1.55 1.71 1.46 2.39 1.93 Minority 1.83 1.59 0.19 1.27 0.89 -2.0 -1.5 -1.0 -0.5 0.5 1. 00.00.20.40.6 Zn 1-xNi xSZn 1-xNi xSeDensity ofStates [ /atom/eV] Energy[eV]Zn 1-xNi xTe EF FIG. 5. DOS of the Ni- dVB for ZnNiS, ZnNiSe, and ZnNiTe are plotted to show the broadening of the VB that occurs as the sizeof the anion increases. This broadening is a product of the upper-most valence electrons shifting toward the Fermi energy /H208490e V /H20850as indicated by the two vertical lines. The larger electronic bands andincreased p-dhybridization associated with larger anion atoms cre- ate this effect.SYSTEMATIC TRENDS OF FIRST-PRINCIPLES … PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-5duction from the free atom. The moment calculated for the Ni ion in ZnNiTe for each ion separation in the order of leastto greatest is 1.75 /H9262b/H208491/H20850, 1.825 /H9262b/H208492/H20850, 1.85 /H9262b/H208493/H20850, 1.86/H9262b/H208494/H20850, 1.93 /H9262b/H208495/H20850, and t2g/H208496/H20850. The reason for this is apparent from the partial DOS of the Ni dstates in the VB presented in Fig. 5for our ZnNiS, ZnNiSe, and ZnNiTe sys- tems. As the anion changes from S →Se→Te, there is a definite broadening of the peaks at the locations of both thee gpeaks, which move from −0.60 to −0.30 eV with increas- ing anion size, and the t2gpeaks, which move from −0.40 to 0.00 eV with increasing anion size. Calculations of the par-tial charges and magnetizations of the Ni atoms in these ma-terials /H20849Table III/H20850show that the Ni ions are gaining charge from their nearest-neighbor anions, and that the amount ofcharge gained increases with the size of the anion. Thiswould suggest that the sp-dhybridization is greater for the ZnNiTe compound. This hybridization dictates the strengthof the SE so ZnNiTe should become more AFM orientedthan the S and Se counterparts. IV. SUPEREXCHANGE VERSUS DOUBLE EXCHANGE IN THE II–VI DMS Since the magnetic ions are not near neighbors the direct exchange is too weak to explain the size of the exchangecoupling observed. However, indirect exchange mechanismscan provide exchange couplings of the right size. In manyDMS compounds it is generally believed that Anderson’ssuperexchange 20,28and Zener’s DE mechanism are respon- sible for the magnetic coupling. One major difference be-tween these two exchange mechanisms is the role of themediator. In SE the role of the mediator is played by a pair of pelectrons of the host anion /H20849S, Se, Te /H20850. One pelectron is transferred to a neighboring TM while the other directly ex-change couples with another neighboring TM ion. This cou-pling normally reduces the kinetic energy and is FM for d shells that are less than half-filed, and AFM otherwise. TheDE process is mediated by the CB of the host crystal via aresonance state that mixes same spin dorbitals of the TM ions and extended CB states. Thus the DE coupling requiresan extended metallic state, and since this state couples wavefunctions of the same spin, the coupling has to be FM. For a SE mechanism to apply there needs to be a signifi- cant p-dhybridization between the anion- pand TM- dorbit- als to promote the electron transfer and a small relative en-ergy difference between the porbital and available dstates as would be expected from perturbation theory. Clearly, therequirements for the material to participate in a DE are thatthedelectron must be able to propagate through the CB and there must be an allowable empty dstate to propagate into. It has been often suggested that these two mechanisms are incompetition with one another. 17The following analysis will examine why and which DMS compounds exhibit this com-petition. The particular electronic structure of these DMS materials provides a qualitative explanation of which exchange mecha-nism might be involved. The DMS materials, studied here,doped with either Cr, Fe, or Ni are half-metallic and exhibita competition between these two exchange mechanismswhich can either manifest in a weak coupling /H20849Fe/H20850, strong coupling /H20849Cr/H20850, or moderate coupling /H20849Ni/H20850of the magnetic ions depending on the relative strengths of the two mecha-nisms. Those doped with Co and Mn are insulating and therefore expect predominantly exhibit an AFM SE coupling. Figures 6/H20849a/H20850–6/H20849e/H20850show how the DE mechanism operates. In Mn /H20851Fig. 6/H20849b/H20850/H20852the DE is prohibited and in Co /H20851Fig. 6/H20849d/H20850/H20852 is unfavorable. In both cases the compounds are insulating.Mn has no empty majority-spin dorbitals. Therefore the magnetism in Mn DMS compounds is strictly AFM andregulated by the SE. The Co DMS compounds are insulatingbut have some low lying unoccupied t 2gstates that could exhibit a weak FM DE. The three cases of Cr /H20851Fig. 6/H20849a/H20850/H20852,F e /H20851Fig. 6/H20849c/H20850/H20852, and Ni /H20851Fig.6/H20849e/H20850/H20852, the half-metallic nature of the electronic structure would permit a FM DE to occur. The Cr DMS compound ishalf-metallic in the spin-up direction while Fe and Ni arehalf-metallic in the spin-down direction. Being a half-metalpromotes considerable sp-dcoupling between the dorbital of the TM and sporbital of the CB. This coupling will fa- cilitate the transfer of spin between the two neighboringTM-dshells. The exchange splitting is large enough to en- sure that the transferred electron is accepted by the incom-plete t 2gshell of the neighboring TM. For Cr, Fe, and Ni the DE results in either a half-shell /H20849Cr/H20850, half-subshell /H20849Ni/H20850,o r both /H20849Fe/H20850electronic configurations. By maximizing Sthe ground-state energy is lowered due to guidelines of Hund’srules. The strength of the FM DE for each TM from thisanalysis in decreasing order is Cr, Ni, Fe, Co, and Mn. Figures 7/H20849a/H20850–7/H20849e/H20850show how the SE operates either by a FM or AFM coupling. The larger the energy difference be-tween the localized TM dpeaks and the porbitals /H20849VB/H20850and the narrower the dorbital is the weaker the SE is. In the case of Mn /H20851Fig.7/H20849b/H20850/H20852, the exchange splitting is large. The energy cost of promoting the pelectron to the TM- dshell is sub- stantial due to this large exchange splitting. Also the accept-ing orbital is an e gtype and is narrow in energy. These two factors will create a weak AFM SE coupling to exist in Mn.TABLE III. Calculated values of the partial charge /H20851units of electron charge /H20849e/H20850/H20852and magnetization /H20849/H9262b/H20850 on the Ni atoms in ZnS, ZnSe, and ZnTe compounds. Material s p d Total chg. mag. chg. mag. chg. mag. chg. mag. ZnNiS 0.742 0.006 6.903 0.085 8.832 1.208 16.477 1.299ZnNiSe 0.810 0.008 6.939 0.095 8.941 1.151 16.691 1.254ZnNiTe 0.927 0.008 7.091 0.103 9.136 0.898 17.154 1.009MCNORTON et al. PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-6FIG. 6. Schematic of the electronic configurations for TM- cation-TM pairs under the DE mechanism, which mediates the elec-tron transitions by requiring a FM alignment. The favorable ex-changes result in half-metallic compounds. FIG. 7. Schematic of the electronic configurations for the TM- cation-TM pairs under the SE mechanism. All of the compounds/H20849b/H20850–/H20849e/H20850are required to be antiferromagnetically aligned except for Cr/H20849a/H20850which is required to have its spins ferromagnetically aligned in order to mediate the electronic coupling.SYSTEMATIC TRENDS OF FIRST-PRINCIPLES … PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-7In Co /H20851Fig. 7/H20849d/H20850/H20852we see a shrinking exchange splitting. The promotion of the pelectron must be received by the wider TMt2gshell. The SE and AFM couplings are predicted to be stronger in the case of Co than in Mn given the broaderwidth of the t 2gsubband and energy cost of promoting the p electron. In the case of the DMS compounds containing Cr promot- ing the anion’s spin-up pelectron is favored by maximizing S. Since the exchange splitting is large the electron will be promoted to the t2gshell. The energy cost of this p-dpromo- tion is small. Hence we should expect strong FM couplingvia the SE for Cr. In the case of Fe, the energy cost ofpromoting the pelectron to the narrow TM e gshell is not as large as Mn, due to the smaller exchange splitting, but largerthan Cr. In Fe the DE and SE couplings should be compa-rable yet opposite in sign to one another and may lead to anoscillatory coupling. In Ni the exchange splitting has beenfurther reduced. In SE the pelectron is transferred to the neighboring TM /H20849left/H20850t 2gshell. The energy cost in promoting thispelectron is less than in any of the previous cases due to the close proximity of the TM- dpeak to the VB. However, given that Ni is simultaneously involved with a strong FMDE the overall strength and sign of the magnetic couplingfrom this analysis is inconclusive. The strength of the AFMSE for each TM from this analysis in decreasing order is Ni,Co, Cr, Fe, and Mn. We can see that the main difference in the p-dhybridiza- tion between the half-metallic elements Cr, Ni, and the FeDMS is the particular dorbital that is interacting with the p orbital of the anion. This acceptor orbital has to be energeti-cally able to acquire this electron and have significant over- lap with the donor orbital. The difference between the two d suborbitals lies in the bonding geometries /H20849Fig. 8/H20850. The d egorbitals /H20849dx2−y2and dz2/H20850point along the coordinate axes, while the dt2gorbitals /H20849dxy,dyz, and dxz/H20850point between the coordinate axes. These crystal geometries lead to differing amounts of suborbital overlap. Therefore, not only do weneed to take into account the proximity of the TM impuritiesto each other but also the bonding direction. Figure 8dem- onstrates how the degree of p-dsuborbital overlap is deter- mined by the bonding geometries between the TM impuritiesand nearest-neighbor anions. Fe with six valence electrons has electron transitions to and from the d egorbitals, and therefore the overlap describ- ing the p-dinteraction is between the px-dz2orbitals and results in a smaller overlap. The eight valence electrons of Ni provide a local moment from the electrons in the dt2gsubor- bital, therefore the p-dinteractions exist between the px-dxz suborbitals and is stronger than in the case of Fe. Lastly, Cr atoms have four valence electrons and one empty minority state in the dorbital, and the transitioning electron will travel to a dt2gorbital on the adjacent atom, as in the case of Ni. The difference in the strength of the ex- change between Ni and Cr is a consequence of the amount ofthed t2gordxzoverlap with the anion’s pshell. The d-shell radii of the elements have been calculated and show that as one fills up the dshell with electrons the radius of the shell shrinks.29The radius of a Cr dshell is roughly 30% larger than that of Ni, which explains Cr has a much larger ex-change energy than Ni.V. RESULTS A. Ground-state energies/exchange couplings To determine the preferred magnetic state of a compound calculations are performed for FM and AFM configurationsand the energies are compared. Intuitively, the larger the en-ergy difference between the two states the more probable thesystem will exist in that magnetic GS. The calculated GSenergies of the FM and AFM states for all of the DMS ma-terials discussed in this work are shown in Fig. 9as a func- tion of the impurity separation distance. From the trends dis-played in this figure, we can see that the only materials forwhich ferromagnetism is the favored state for all impurityseparation distances are those which contain Cr or Ni impu-rities /H20849except ZnNiTe /H20850, with Cr being more ferromagnetically stable. The compounds which contain Fe impurities exhibitan oscillatory coupling, favoring the AFM ordering at theshortest impurity spacing /H208490.707 L o/H20850and FM ordering at sub- sequent impurity spacings, exhibiting behavior which islikely caused by the competing SE and DE mechanisms. Thematerials containing Co and Mn impurities have AFM order-ing for all impurity spacings. The exchange energy is the change in GS energy between the two magnetic configurations. The FM DMS compounds (b)(a) FIG. 8. Bonding directions of the p,dt2g,/H20849left/H20850anddeg/H20849right /H20850 suborbitals at the closest TM spacing /H208490.707 Lo/H20850. The figure is meant to display the favoritism for certain bonding orbitals pertaining tothe SE mechanism.MCNORTON et al. PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-8are Cr and Ni. The maximum exchange energy for Cr is 0.335 eV and occurs in ZnSe. The maximum exchange en-ergy for Ni is 0.207 eV and occurs in ZnS. The exchangeenergy of the calculated half-metallic materials is shown inFig.10as a function of the impurity separation distance. The data displayed in this figure show that the closest impurityspacings /H208490.707 L o/H20850yields the greatest exchange energy, which then decreases dramatically as the separation betweenthe magnetic ions increases. This is due to a narrowing of thedbands as the ions are spaced further apart, and thus a smaller probability the electron will be transported in the DEand SE processes. In real samples, the statistical arrangementof the impurities would lead to a distribution of separationdistances between the impurities, making the actual ex- change energy a statistical averaging of the exchange distri-bution. The calculated exchange energies also show a dependence on the size of the anion in the semiconducting material, asshown in Fig. 11for ZnS, ZnSe, and ZnTe materials doped with either Cr, Mn, and Fe impurities. As the size of theanion increases so does the lattice constant. This increasesthe spacing between atoms. At first glance this might be ex-pected to reduce the strength of the coupling mediated by theSE. However, the valence porbitals extend out further for the larger anion. For example, the dorbitals of the TM series Ni→Pt→Pd broaden as you go down in the periodic table. The greater spatial extent of the porbitals creates an increase of the p-dhybridization and thus a strengthened SE cou- pling. We can verify this by examining the Cr and Mn com- pounds which couple strongly /H20849Cr/H20850and exclusively /H20849Mn/H20850by the SE. Figure 10shows that the exchange energy, as ex- pected, increases /H20849negatively /H20850for the Mn inset and increases /H20849positively /H20850for Cr with the increasing lattice constant. We investigate the effect anion size has on the strength of the DEcoupling. The strength of the DE is determined by thestrength of the Zener sp-dcoupling or the weight of the impurity dband at the Fermi energy. The band gap is known to decrease as the size of the anion increases for the II–VIsemiconductors; from Table I,S→Se→Te=2.88 eV →2.48 eV→1.86 eV thus the decreasing band gap should increase the weight of the impurity dband that coincides with the Fermi energy. We address the issue of which exchange mechanism is most sensitive to the lattice constant by examining the Fecompounds which have similar coupling strengths for the SE/H20849AFM /H20850and DE /H20849FM/H20850. In Fig. 11, the Fe inset shows that the exchange energy crosses the xaxis and becomes more nega- tive /H20849more AFM /H20850as the lattice constant increases. This be- havior suggests that SE reduces at a slower rate as the ionsspread apart than DE. Quantitatively the exchange energywill vary differently with the lattice spacing depending onthe impurity and the strength and preference of each ex-change coupling.0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-130.80-130.68-130.56 Zn1-xNixTeRelative Ground State Energy [eV] -133.92-133.84-133.76 Separation Distance [multiple of L0]Zn1-xCoxTe-136.24-136.20-136.16-136.12 Zn1-xFexTe-138.84-138.72-138.60-138.48-138.36 Zn1- xCrxSeZn1-xCrxS Zn1-xCrxTe Zn1-xMnxS Zn1-xMnxSe Zn1-xFexS Zn1-xFexSe Zn1-xCoxS Zn1- xCoxSe Zn1-xNixS Zn1-xNixSe-139.32-139.28-139.24 Zn1-xMnxTe FIG. 9. /H20849Color online /H20850Plots showing the trends in the calculated GS energies of the doped ZnTe materials with both FM /H20849closed circles /H20850and AFM /H20849open circles /H20850ordering as a function of the impu- rity separation distance. The smaller plots show the GS energies forthe ZnS and ZnSe materials in order to show that the response ofthe GS energy to the separation distance is similar for a family ofanions.5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1-0.02-0.010.000.010.021.000 L0 1.414 L0 1.871 L0 FeCr MnExchange Energy [eV] Lattice Constant [ L0]0.000.050.100.150.20 5 . 55 . 75 . 96 . 1-0.0 4-0.03-0.0 2-0.010.00 FIG. 10. /H20849Color online /H20850Plots showing the trends in the calcu- lated exchange energies as a function of the impurity separationdistance for the half-metallic impurities /H20849Cr, Fe, and Ni /H20850in ZnS, ZnSe, and ZnTe.SYSTEMATIC TRENDS OF FIRST-PRINCIPLES … PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-9In each of the compounds discussed in this work it was determined that the orientation which provided the lowestGS energy was the one in which the magnetic impuritieswere located closest to each other, suggesting a localizationof the impurities, and their magnetic moments, inside thecrystal upon its fabrication. This clustering effect may lowerthe Curie temperature in DMS structures, as has been shownin GaMnN systems. 30The orientation having the shortest im- purity separation distance was also found to yield the largestexchange energies of the potential orientations. A simple calculation provides a comparative measure of the tendency of impurities to cluster by comparing the dif-ference in the GS energy between the closest TM spacing/H208490.707 L 0/H20850and the spacing of one lattice constant /H20849L0/H20850. Larger values resulting from this calculation indicate a greater incli-nation of the magnetic ions to cluster together. From data presented in Fig. 12, we see that the half- metallic DMS materials, Cr and Ni, tend to cluster locallyand the preference to cluster for these impurities are in thedecreasing order Cr →Ni. This order of clustering strengths is similar to the strength of the magnetic exchange for theseimpurities. The compounds containing the impurities Cr andNi have significant clustering strengths, while Fe has a smallclustering strength indicative of its oscillation between AFM and FM phases from Fig. 10. The dashed lines in Fig. 12 represent linear fits of the data for each impurity in relationto the atomic number of the compounds anion. Extrapolationof the linear fit implies that the use of smaller anion atomsleads to a stronger attraction between impurities, and there-fore a more significant clustering effect of impurities in thecrystal. Owing to the smaller anion, this data fit suggests thatthe compounds ZnCrO and ZnNiO will have large attractionstrengths /H208490.270 and 0.170 eV , respectively /H20850, which may lead to some of the interesting characteristics discovered for ZnODMS. 13 This clustering effect has been previously studied for the Cr species of the ZnTe semiconductors.18Our results were comparable with this study on the aspect of the clusteringtrends that exist in these II–VI compounds. However, thatstudy assumes that the spatial inhomogeneity of the Cr ionswould automatically enhance the half-metallic property ofthe crystal. While that might be the case locally in the crystalthe overall effect on a bulk cluster of applicable size is not soclear cut. It should be noted that our study did not take intoconsideration the diffusion probability of the TM ions andassumes that the energy barrier is large, given the covalentbonding potential of the DMS material. VI. CONCLUSION The calculated densities of states presented in this study identify two metallic impurities /H20849Cr and Ni /H20850which may also produce half-metallic behavior in the Zn-based II–VI DMSstructures. The half-metallic behavior of Cr impurities insimilar systems has been shown previously, 6,7,11,31although the half-metallic property of Ni may require greater study.5,7 The other impurities studied /H20849Mn and Co /H20850show primarily AFM coupling, and therefore are eliminated from the possi-bility of being used as spin filters in spintronics. Fe-dopedsystems appear to exhibit a competition between the SE and0.6 0.8 1.0 1.2 1.4 1.6 1.8-0.2-0.10.00.10.20.3-0.10.00.10.20.3-0.10.00.10.20.3 Separation Distance [multiple of L0]Exc hange Energy [eV]Zn 1-xXX xS Zn 1-xXX xTeZn 1-xXX xSeCr Fe Ni FIG. 11. /H20849Color online /H20850Plot showing the trend of decreasing exchange energy values as the lattice constant increases for com-pounds doped with Fe /H20849left/H20850,C r /H20849top right /H20850, and Mn /H20849bottom right /H20850.5 1 01 52 02 53 03 54 04 55 05 56 06 50.060.080.100.120.140.160.180.200.220.240.26Cr Im purities Ni Impurities Te Se SCluster ing Strengt h[eV] Anion Atomic Number (A) FIG. 12. /H20849Color online /H20850Plot showing the calculated attraction strength /H20851/H9004E=Egs/H20849.707Lo/H20850−Egs/H208491.0Lo/H20850/H20852, for our Cr and Ni impurity systems as a function of the atomic number of the anion /H20849S, Se, or Te/H20850. The trend lines indicate a linear fit to the calculated data and can be used to determine the attraction strength for other possibleanions /H20849such as oxygen /H20850.MCNORTON et al. PHYSICAL REVIEW B 78, 075209 /H208492008 /H20850 075209-10DE mechanisms, which leads to an oscillatory magnetic cou- pling. Whether a particular Fe sample will demonstrate aFM, AFM, or spin-glass arrangement has a strong depen-dence on the spatial positioning of the impurities. So then theII–VI, Zn-based, DMS compounds doped with either Cr orNi have the greatest potential to be implemented successfullyinto spintronic devices The tendency of the impurities to cluster has been estab- lished in the materials for a concentration of 4%. This clus-tering effect is critical for Cr based DMSs since the local ferromagnetism mediated by the SE depends greatly on theTM crystal separation/orientation. The FM ordered materialsthat contain Cr and Ni display this tendency to cluster theions together. It must be considered that these impurity richregions in the crystal could result in the FM behavior beinglocal and not a bulk property of the crystal. Much of thiswork is verified by a similar study on ZnO DMS, 13which deduced similar results. 1J. K. Furdyna and J. Kossut, Diluted Magnetic Semiconductors /H20849Academic, New York, 1988 /H20850, V ol. 25. 2J. K. Furdyna, J. Appl. Phys. 64, R29 /H208491988 /H20850. 3J. K. Furdyna, J. Appl. Phys. 53, 7637 /H208491982 /H20850. 4C. M. Fang, G. A. Wijs, and R. A. Groot, J. Appl. Phys. 91, 8340 /H208492002 /H20850. 5R. A. Stern and T. M. Schuler, J. Appl. Phys. 95, 7468 /H208492004 /H20850. 6J. Blinowski, P. Kacman, and J. A. Majewski, Phys. Rev. B 53, 9524 /H208491996 /H20850. 7K. Sato, H. Katayama-Yoshida, S. Yamagata, Y . Suzuki, and K. Ando, Phys. Status Solidi B 229, 673 /H208492002 /H20850. 8H. Saito and W. Zayets, J. Appl. Phys. 91, 8085 /H208492002 /H20850. 9W. Mac, A. Twardowski, P. J. T. Eggenkamp, H. J. M. Swagten, Y . Shapira, and M. Demianiuk, Phys. Rev. B 50, 14144 /H208491994 /H20850. 10L. M. Sandratskii and P. Bruno, J. Phys.: Condens. Matter 15, 585 /H208492003 /H20850. 11H. Saito, V . Zayets, S. Yamagata, and K. Ando, J. Appl. Phys. 93, 6796 /H208492003 /H20850. 12T. M. Pekarek, D. J. Arenas, B. C. Crooker, I. Miotkowski, and A. K. Ramdas, J. Appl. Phys. 95, 7178 /H208492004 /H20850. 13L. M. Sandratskii and P. Bruno, Phys. Rev. B 73, 045203 /H208492006 /H20850. 14L. M. Sandratskii, Phys. Rev. B 68, 224432 /H208492003 /H20850. 15T. M. Schuler, R. A. Stern, R. McNorton, S. D. Willoughby, J. M. MacLaren, D. L. Ederer, V . Perez-Dieste, F. J. Himpsel, S. A.Lopez-Rivera, and T. A. Callcott, Phys. Rev. B 72, 045211 /H208492005 /H20850.16A. Twardowski, J. Appl. Phys. 67, 5108 /H208491990 /H20850. 17Y . Uspenskii, E. Kulatov, H. Mariette, H. Nakayama, and H. Ohta, J. Magn. Magn. Mater. 258, 248 /H208492003 /H20850. 18Y . Gou, J. Cao, X. S. Chen, and W. 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PhysRevB.99.125147.pdf

PHYSICAL REVIEW B 99, 125147 (2019) Universal dielectric response across a continuous metal-insulator transition Prosenjit Haldar,1,2,3,*M. S. Laad,2,3,†and S. R. Hassan2,3,‡ 1Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Sciences, Bangalore 560012, India 2Institute of Mathematical Sciences, Taramani, Chennai 600113, India 3Homi Bhabha National Institute Training School Complex, Anushakti Nagar, Mumbai 400085, India (Received 1 October 2018; revised manuscript received 18 March 2019; published 26 March 2019) A wide range of disordered materials, including disordered correlated systems, show universal dielectric response (UDR), followed by a superlinear power-law increase in their optical responses over exceptionallybroad frequency regimes. While extensively used in various contexts over the years, the microscopics underpin-ning UDR remains controversial. Here, we investigate the optical response of the simplest model of correlatedfermions, the Falicov-Kimball model, across the continuous metal-insulator transition (MIT) and analyze theassociated quantum criticality in detail using cluster extension of dynamical mean-field theory. Surprisingly,we find that UDR naturally emerges in the quantum critical region associated with the continuous MIT. Wetie the emergence of these novel features to a many-body orthogonality catastrophe accompanying the onset ofstrongly correlated electronic glassy dynamics close to the MIT, providing a microscopic realization of Jonscher’s time-honored proposal as well as a rationale for similarities in optical responses between correlated electronicmatter and canonical glass formers. DOI: 10.1103/PhysRevB.99.125147 I. INTRODUCTION Optical conductivity has long been used to characterize elementary excitations in condensed matter. The responseof matter to ac electromagnetic fields is usually encoded inthe complex conductivity [ 1],σ xx(ω)=σ/prime(ω)+iσ/prime/prime(ω), or the complex dielectric constant /epsilon1(ω), related to each other byσ/prime(ω)=ω/epsilon10/epsilon1/prime/prime(ω), where /epsilon1/prime/prime(ω) quantifies the dielectric loss and /epsilon10is the permittivity of free space. Optical studies have been especially valuable in strongly correlated electronicmatter [ 2] and, as a particular example, have led to insights into the breakdown of traditional concepts in cuprates [ 3]. Such studies have also led to much progress in understand- ing complex charge dynamics in disordered matter. In the1970s, the pioneering work of Jonscher [ 4,5] showed a univer- sal dielectric response (UDR) of disordered quantum matterto ac electromagnetic fields, wherein σ xx(ω)/similarequalωα, with α/lessorequalslant 1 in the subgigahertz regime. More recently, Lunkenheimerand Loidl [ 6] reported astonishingly similar responses in a wide class of disordered matter over a more extended energywindow: among others, doped, weakly, and strongly corre-lated semiconductors exhibit UDR, followed by a superlin-ear power-law increase in σ(ω), bridging the gap between classical dielectric and infrared regions. This behavior is alsocommon to dipolar and ionic liquids as well as to canonicalglass formers. Very recently, materials which belong to theelusive class of spin liquids [ 7] were also interpreted in terms of UDR: in this case, it is possible that intrinsic dis-order, arising from geometric frustration, is implicated in the *prosenjith@iisc.ac.in †mslaad@imsc.res.in ‡shassan@imsc.res.inemergence of UDR. This suggests involvement of a deeper,more fundamental and common element, related to the onsetof a possibly intrinsic, glassy dynamics in the emergence ofUDR. In the context of correlated quantum matter (such as the Mott insulator LaTiO 3and Pr 0.65(Ca 0.8Sr0.2)0.35MnO 0.35 (PCSMO) [ 6]), such unconventional “glassy” dynamics can emerge near the doping-induced metal-insulator transition(MIT) as a consequence of substitutional and /or intrinsic disorder due to inhomogeneous electronic phase(s) near theMIT. On the other hand, early on, Jonscher himself suggestedthe relevance of many-body processes akin to the seminalAnderson orthogonality catastrophe (AOC) for UDR. Thus,the link between AOC and an emergent, slow glassy dynamicsunderlying the electronic processes leading to UDR in dis-ordered, interacting electronic systems remains a challengingand largely unaddressed issue for theory, to the best of ourknowledge. Motivated thereby, we investigate these issues with a careful study of the optical response of the Falicov-Kimballmodel (FKM). The FKM is the simplest representative modelof correlated electrons on a lattice and possesses an exactsolution within both dynamical mean-field theory (DMFT) [ 8] and its cluster extensions (CDMFT) [ 9,10]. Remarkably, it can be solved almost analytically, even in CDMFT [ 10], leading to enormous computational simplifications in transport studies[11–13]. Across a critical U, the FKM is known to undergo aT=0 continuous MIT of the Hubbard band-splitting type [8]. As found earlier for transport properties, it turns out that precise computation of the optical response for the FKMwithin two-site cellular DMFT [ 14] is facilitated by the facts that (i) explicit closed-form expressions for the cluster propa-gators G(K,ω), with K=(0,0),(π,π ), greatly reduce com- putational cost, even in CDMFT, and (ii) the cluster-resolved 2469-9950/2019/99(12)/125147(7) 125147-1 ©2019 American Physical SocietyHALDAR, LAAD, AND HASSAN PHYSICAL REVIEW B 99, 125147 (2019) irreducible particle-hole vertex functions are negligibly small and we ignore them in the Bethe-Salpeter equations forall conductivities, thanks to an almost rigorous symmetryargument [ 15]: upon a cluster-to-orbital mapping (which is implicit in our mapping of the two-site CDMFT to two, “ S,P” channels [ 14]). In this “multiorbital” scenario, the irreducible vertex corrections entering the Bethe-Salpeter equation for theconductivity are still negligibly small [ 16]. Thus, the optical conductivity acquires a form similar to DMFT, but with anadditional sum over cluster momenta (or the two SandP channels on the cluster). The rest of this paper is organized as follows: In Sec. IIwe describe the model we study in this work and the calculationof optical conductivity using cluster DMFT formalism. InSec. IIIwe present our numerical results and analyze (i) Mott-like quantum criticality in the optical response usingCDMFT and (ii) universal dielectric response across the MIT.We then tie the UDR to an emergent many-body orthogonalitycatastrophe in the FKM within our CDMFT approach. InSec. IVwe discuss our findings in the context of real materials exhibiting UDR.II. GENERAL FORMULATION OF OPTICAL CONDUCTIVITY WITHIN CLUSTER DMFT The Hamiltonian of the spinless FKM model is HFKM=−t/summationdisplay /angbracketlefti,j/angbracketright(c† icj+H.c.)−μ/summationdisplay ini,c+U/summationdisplay ini,dni,c. (1) The Hamiltonian describes a band of dispersive fermions (c,c†) interacting locally via a “Hubbard”-type interaction with dispersionless dfermions. Since vi=Un i,dis a random (binary) potential in the symmetry-unbroken phases of theFKM, Eq. ( 1) can also be viewed (as has long been known [17]) as a model of fermions in a random binary alloy potential. In recent work [ 14], we used a cluster extension of DMFT to solve FKM using the Dyson-Schwinger equation of mo-tion technique. Remarkably, the cluster-local Green’s func-tion in two-site cluster DMFT is obtained analytically andreads ˆG=/bracketleftbigg G00(ω)Gα0(ω) Gα0(ω)G00(ω)/bracketrightbigg , where the matrix element Gij(ω), with i,j=0,α,i s Gij(ω)=/bracketleftbigg1−/angbracketleftn0d/angbracketright−/angbracketleft nαd/angbracketright+/angbracketleft n0dnαd/angbracketright ξ2(ω)+/angbracketleftn0d/angbracketright−/angbracketleft n0dnαd/angbracketright ξ2(ω)−U/bracketrightbigg/bracketleftbigg δij−F2(ω) [t−/Delta1α0(ω)](1−δij)/bracketrightbigg +/bracketleftbigg/angbracketleftnαd/angbracketright−/angbracketleft n0dnαd/angbracketright ξ1(ω)+/angbracketleftn0dnαd/angbracketright ξ1(ω)−U/bracketrightbigg/bracketleftbigg δij−F1(ω) [t−/Delta1α0(ω)](1−δij)/bracketrightbigg , (2) with ξ1(ω)=[ω−/Delta100(ω)−F1(ω)], ξ2(ω)=[ω− /Delta100(ω)−F2(ω)], and F1(ω)=(t−/Delta1α0)2 ω−U−/Delta100(ω),F2(ω)= (t−/Delta1α0)2 ω−/Delta100(ω), where the bath function ˆ/Delta1(ω) is related to the local Green’s function through a suitable self-consistency condition. The self-energy is given as ˆ/Sigma1(ω)=ˆG−1 0(ω)−ˆG−1(ω), (3) where ˆG0(ω) is the Weiss Green’s function, ˆG0(ω)=(ω+ μ)1−ˆ/Delta1(ω). We use the algorithm described in Ref. [ 14] to find the local Green’s function and self-energy. In thesymmetric basis (cluster momentum basis) we can write G S= (G00+Gα0) and GP=(G00−Gα0), where SandPare even and odd orbitals, respectively. The optical conductivity is evaluated using the Kubo- Greenwood formalism. In the near absence of vertex cor-rections, only the bare bubble, composed from the CDMFTpropagators, contributes. The explicit form of the opticalconductivity within cluster DMFT then reads σ /prime(ω)=σ0/summationdisplay K∈[S,P]/integraldisplay∞ −∞d/epsilon1v2(/epsilon1)ρK 0(/epsilon1) ×/integraldisplay∞ −∞d˜ωAK(/epsilon1,˜ω)AK(/epsilon1,˜ω+ω)f(˜ω)−f(˜ω+ω) ω, (4)with AK(/epsilon1,ω)=Im/bracketleftbigg1 ω+μ−/epsilon1−/Sigma1K(ω)/bracketrightbigg . (5) Here,ρK 0(/epsilon1) is the noninteracting spectral function of the even and odd orbitals, and f(ω) is the Fermi distribution. This simplification allows a comprehensive study of the opticalresponse of the FKM within CDMFT, which we now describe. III. RESULTS AND DISCUSSION We consider the Bethe lattice with the half bandwidth of the conduction electron ( cfermions) as unity (2 t=1). We de- fine the short-range order parameter f0αasf0α=/angbracketleftn0dnαd/angbracketright− /angbracketleftn0d/angbracketright/angbracketleftnαd/angbracketright. A. Quantum criticality near the MIT We exhibit the real part of the optical conductivity near and across the MIT (1 .6/lessorequalslantU/lessorequalslant2.0), computed from Eq. ( 4) as a function of Ufor (a) the completely disordered case (short-range order parameter f0α=0 in our earlier work [ 14]) in the top panel of Fig. 1and (b) the short-range ordered case (f0α/negationslash=0) in Fig. 2. Several features stand out clearly: in case (a),σ/prime(ω) shows an incoherent low-energy bump centered atω=0, whose weight decreases continuously as the MIT 125147-2UNIVERSAL DIELECTRIC RESPONSE ACROSS A … PHYSICAL REVIEW B 99, 125147 (2019) FIG. 1. Optical conductivity of the completely random ( f0α=0) FKM within two-site CDMFT, showing its evolution with Uat temperature T→0. The MIT occurs at Uc=1.8. Blue symbols show how an emergent scale /Omega10(U), associated with a smooth crossover between metallic and insulating states, collapses at the Mott transition ( U=1.8) as ( δU)νwithν=1.29, close to 4 /3 (see text). is approached (at U=1.8). It is important to note that (i) there is no low-energy Drude component in σ/prime(ω) since the CDMFT propagators have no pole structure [ 14] and (ii) as expected, low-energy spectral weight is continuously trans-ferred from the bad-metallic and midinfrared (MIR) regionsto high energies O(U) across the MIT. This is characteristic of a correlation-driven MIT, and the continuous depletion oflow-energy weight is a consequence of the continuous MITin the FKM driven by increasing U.I nF i g 2, we exhibit the effect of “antiferromagnetic alloy” (AF-A) short-rangeorder (SRO). Apart from the fact that the MIT now occursat (U)/similarequal1.35 [14], the above features seem to be reproduced in this case as well. Looking more closely, however, we seemarked changes in the low- and midinfrared energy range:the bad-metallic bump centered at ω=0 is suppressed by SRO, and σ /prime(ω) rises faster with ωin the MIR, showing the emergence of a low-energy pseudogap. These changes are tobe expected since AF-A SRO reduces the effective kineticenergy and increases the effective U, leading to a reduction of the low-energy spectral weight and a low-energy pseudogapin optics. A closer look at Fig. 1reveals very interesting features. We uncover a crossover scale /Omega1 0(U), separating “metallic” and “insulatorlike” behaviors in σ/prime(ω). As expected, it col- lapses at the MIT: interestingly, we find /Omega10(δU)/similarequal(δU)1.29,FIG. 2. The real part of the optical conductivity of the FKM with antiferromagnetic short-range order ( f0α=−0.15) within two-site CDMFT close to the MIT (1 <U<1.5). The critical curve at which the MIT occurs corresponds to ( U)c=1.35 (red dashed line). quite close to ν=4/3 found in earlier work [ 11]. This also motivates us to investigate underlying quantum criticality inthe optical response. In Fig. 3, we show that log 10[σc/σ/prime(ω)] plotted versus ω//Omega1 0(U) (the latter is taken from Fig. 1) indeed reveals clean quantum-critical scaling: the insulating ( I) and metallic ( M) data fall on two master curves, and the beautiful mirror symmetry relating the two testifies to the unambiguousmanifestation of the “Mott” quantum critical point (QCP) inthe optical response. Further, we also find that /Omega1 0(δU)/similarequal c|δU|η, withη=1.3/similarequal4/3, in excellent agreement with both Fig. 1and our previous study. Using our earlier result ξ/similarequal (U−Uc)−ν, with ν=4/3 and z=1, we thus expect that σ/prime(ω)/σcshould scale as y=|U−Uc|/Ucω=1/ωξ1/zν, i.e., FIG. 3. Clean quantum critical scaling of the optical conductivity across the Mott QCP, as shown by the fact that log10[σc/σ/prime(ω)] versus ω//Omega1 0(U) for the metal and insulator phases falls on two universal “master” curves. σcis the optical conductivity at the critical U, i.e., the separatrix. We estimate /Omega10(δU)/similarequal(δU)η, with η=1.31, in very good agreement with ν=4/3 from earlier work [ 11]. 125147-3HALDAR, LAAD, AND HASSAN PHYSICAL REVIEW B 99, 125147 (2019) FIG. 4. Clean quantum criticality as revealed in the scaling be- havior of the CDMFT optical conductivity, σ/prime(ω)/σcversus y= |U−Uc|/Ucω. Since the localization length ξ/similarequal|U−Uc|−4/3(see text), this implies that σ/prime(ω)/σ0=F(ωξ1/zν). This is a manifestation of the time-temperature superposition principle [ 5] following from Jonscher’s UDR. thatσ/prime(ω)/σc=F(ωξ1/zν). This is indeed adequately borne out in Fig. 4for both MandIphases. This is a manifes- tation of the “time-temperature superposition principle” [ 5], expressible as a scaling law,σ/prime(ω) σc=F(ω /Omega10), with Fbeing a T-independent scaling function and /Omega10(U) being a scaling parameter corresponding to the onset of conductivity disper-sion, precisely as found here. The variation of /Omega1 0with U reflects the nontrivial interplay between itinerancy (hopping)and Mott-like localization in the FKM. In analogy with the pa-rameter T 0(δU)/similarequalc(δU)zνfor the dc transport criticality [ 11], /Omega10also scales like ( δU)zν. Finally, the fact that /Omega10(δU)/similarequal (δU)zνinboth theMandIphases reflects the fact, alluded to in earlier work [ 11,18], that the basic electronic processes governing the Iphase are also relevant deep into the Mphase. B. Universal dielectric response Having shown a Mott-like quasilocal quantum criticality, we now turn to the UDR near the MIT. Since the FKM isisomorphic to the binary-alloy Anderson disorder problem,we inquire how CDMFT performs in the context of the re-markable universality in the dielectric response in disorderedquantum matter alluded to before [ 6]. In Figs. 5and6,w e show ln σ /prime(ω) and the dielectric loss, ln[ σ/prime(ω)/ω]v e r s u s ln(ω), as functions of Uto facilitate meaningful comparison with data of Lunkenheimer and Loidl. It is indeed quiteremarkable that allthe basic features reported for disordered quantum matter are faithfully reproduced by our CDMFTcalculation. Specifically, (i) for 1 .5<U<1.8, a “dc” con- ductivity regime at the lowest energy (up to 10 −4–10−3) smoothly goes over to a sublinear-in- ωregime (UDR, in the region 10−2–10−1) followed by a superlinear-in- ωregime (around 10−1), connecting up smoothly with the “boson” peak. These regimes are especially visible around U=1.8, precisely where the MIT occurs. (ii) Moreover, corroboratingFIG. 5. Real part of the optical conductivity versus frequency ω, plotted on a log-log scale to facilitate direct comparison with data for disordered and correlated electronic systems from Lunkenheimer and Loidl [ 6]. Very good agreement is clearly seen. More importantly, the crossover from the dc limit to UDR around ln( ω)/similarequal−3 close to the Mott QCP (for U/similarequal1.8) is also revealed, showing that UDR emerges in the quantum critical region associated with the continuous MIT. behavior is also clearly seen in Fig. 6, where we exhibit the dielectric loss function vs ωon a log-log scale. It is clearly seen that a shallow minimum separates the UDR and super-linear regimes at approximately ln( ω)=−0.8 (in the meV region) in the very bad metallic state close to the MIT. Thisis in excellent agreement with results for both LaTiO 3and PCSMO [ 6]. Moreover, the energy dependence of the optical conductivity also seems to be in good qualitative agreementwith data when we compare our results with Figs. 1,2, and 3 FIG. 6. The dielectric loss Re σxx(ω)/ω, plotted versus ωon a log-log scale to facilitate direct comparison with Lunkenheimer and Loidl [ 6]. In excellent agreement with data on correlated and disordered systems in Ref. [ 5], a shallow minimum separates the UDR from a superlinear power-law regime around ln ω/similarequal−1,−2a s Uincreases from 1.5 up to 1.8, the critical value for the Mott QCP. 125147-4UNIVERSAL DIELECTRIC RESPONSE ACROSS A … PHYSICAL REVIEW B 99, 125147 (2019) of Lunkenheimer and Loidl. In the right panel of Fig. 6,w e also show that short-range spatial correlations do not qualita-tively modify these conclusions, attesting to their robustnessagainst finite short-range order. Finally, precisely at the MIT(red curves in Fig. 6), we unearth a very interesting feature: Im/epsilon1(0,ω)/similarequalω −η, with η=0.8,0.75 for f0α=0.0,−0.15. Hence, apart from a “dielectric phase angle” cot( πη/2), the real part of the dielectric constant also varies like ω−ηas the MIT is approached from the metallic side. Remarkably, thisis a concrete manifestation of the dielectric (or polarization)catastrophe that is expected to occur at a QCP associated witha continuous MIT. This wide-ranging agreement with data is quite remarkable and begs a microscopic clarification in terms of basic elec-tronic processes at work near the MIT. A phenomenologicalway is to posit that the universality is linked to glassiness[6] as follows: (i) first, our finding [ 11]o fν=4/3 (and z= 1) is characteristic of percolative transport that is naturallyexpected to arise in glassy systems. (ii) It has also been shown[19] (for the disorder problem) that electronic glassy behavior precedes an insulating phase. Thanks to the mapping between the FKM and a binary-alloy “disorder” problem, we alsoexpect an intrinsic electronic glassy phase near the continuous MIT in the FKM. This suggests a close link between UDR andthe onset of electronic glassy dynamics near the continuousMIT in the FKM. On a more microscopic level, it is very interesting that Jonscher himself [ 5] hinted at the relevance of many-body processes at the heart of UDR. In particular, an explicitparallel with the seminal x-ray edgelike physics was already(phenomenologically) invoked to account for such features. Itis indeed remarkable that such x-ray edge physics naturallyfalls out in the DMFT and CDMFT solutions of the FKM in high D[20]: in DMFT, the mobile ( c-fermion) propagator exhibits a pseudogapped metal going over to an insulatoracross a critical U=U c, while the d-fermion propagator exhibits an “x-ray edge” singular behavior linked to theseminal Anderson orthogonality catastrophe (AOC). In ourCDMFT [ 14], we find similar behavior: (i) a clear correlation- induced pseudogapped metal without Landau quasiparticlesgoing over to an insulator at U c=1.8, with an anomalous self-energy Im /Sigma1(c) loc(ω)/similarequal|ω|1/3and density of states ρc(ω)/similarequal |ω|1/3at the Mott QCP and (ii) unusual power-law behavior of the dynamical charge susceptibility near Uc,a sw e l la s , most importantly, anomalous energy dependence of the d- fermion propagator at long times. In CDMFT, these featuresarise precisely from the fact that (a) the dispersive cfermions interact with the d ±=(d1±d2)/√ 2 dispersionless fermion modes at the intracluster level via U, while (b) the cfermions donothybridize at all with the d±localized mode at the single-fermion level. Physically, the origin of the AOC isthat the dynamical screening arising from strong intraclusterinteractions in the FKM is nontrivially affected by the hoppingmotion of the carriers: since the c ±fermions do nothybridize with the d±-fermion modes at the one-fermion level [there is no term of the form V(c† ±d±+H.c.) on the cluster], this screening is nontrivial and arises from the “slow” reaction ofthed ±modes to the “sudden” jumps of the c±fermions (the latter occurs on a much shorter timescale of ¯ h/tin the FKM). Due to the local symmetry implied by [ ni,d,HFKM]=0a teach i, a hopping carrier experiences a sudden change in the local potential on the cluster (from zero to Uand vice versa while hopping), now in the sense of a “sudden local quench.”The rigorous absence of c-d-fermion one-particle mixing in the FKM implies the lack of heavy-particle recoil in the “two-impurity” cluster problem, leading to “Kondo destruction”and generation of an AOC in our two-site CDMFT as above,in precise agreement with Jonscher’s original suggestion .W e present the origin of the many-body AOC using an analyticalapproach in the following section. C. Cluster orthogonality catastrophe in two-site CDMFT for the FKM Here, we present an analytic argument that exposes the venerated orthogonality catastrophe in our CDMFT approach.It turns out that this is most conveniently done by using theunderlying two-impurity problem of our CDMFT, which wenow describe. The two-impurity FKM is H FKM=t(c† 0cα+H.c.)+/summationdisplay k/epsilon1kc† kck +t/summationdisplay k,i=0,α(eik·Ric† ick+H.c.)+U/summationdisplay i=0,αni,cni,d. (6) Introducing the bonding-antibonding fermions, c±=(c0± cα)/√ 2,d±=(d0±dα)/√ 2. Then, HFKM=H12+Hcoupl+ Hband, with H12=t(nc,+−nc,−)+U 2/summationdisplay a,a/prime=±(nc,and,a/prime+c† aca/primed† ada/prime), (7) Hcoupl=√ 2t/summationdisplay k[cos( ka/2)c† +ck+isin(ka/2)c† −ck+H.c.], (8) and Hband=/summationdisplay k/epsilon1kc† kck. (9) This “cluster-to-orbital” mapping exposes the novel structure of the cluster-local correlations at the Mott QCP. Specifically,we observe that while the d ±fermions interact with the c± fermions via U, they do not hybridize with each other at the one-fermion level [i.e., there is no term of the form V±(c† ±d±+H.c.) in the FKM]. Thus, the two-impurity FKM maps rigorously onto a cluster version of the classic problemof recoilless, “infinite-mass” d ±scatterers in a “Fermi sea” of thec±, i.e., to the cluster version of the venerated x-ray edge problem. One now directly reads off that the spectral functionof the d ±fermions is infrared singular with an interaction- dependent power-law behavior [ 21]: ρd±(ω)/similarequalθ(ω) |ω|1−η±, (10) where η±=(δ±/π)2andπδ±=tan−1[Uρc±(0)π]i st h e scattering phase shift. There will be an additional contri-bution to the scattering phase shift arising from the term 125147-5HALDAR, LAAD, AND HASSAN PHYSICAL REVIEW B 99, 125147 (2019) U 2/summationtext a,a/primec† aca/primed† ada/prime, but this will not qualitatively change the power-law behavior above. This many-body orthogonality catastrophe will carry over into the self-consistently embed-ded two-site CDMFT solution of the FKM. Interestingly, we thus find that the orthogonality catastro- phe and the accompanying breakdown of adiabatic continuityalso hold for the case of spatially separated recoilless scat- terers on the length scale of the cluster. Using a different approach, this aspect was also studied previously [ 22]. Thus, incorporation of intersite correlations between spatially sep-arated scattering centers does not qualitatively modify theorthogonality catastrophe, an interesting result in itself. Inmodern parlance, this means that the vanishing fidelity as wellas the anomalous long-time behavior of the Loschmidt echo,both manifestations of the orthogonality catastrophe [ 23], also holds for spatially separated, sudden local quenches, a resultthat may have more widespread applications. Thus, the classic orthogonality catastrophe, arising from the sudden local but spatially correlated quenches in our two-impurity model, is a genuine feature in our CDMFT. Thisalso provides a concrete microscopic rationalization linkingthe Jonscher UDR to this many-body effect, as conjecturedearly on by Jonscher himself. IV . DISCUSSION AND CONCLUSION Our findings can profitably be utilized to interpret a wider range of data on dielectric responses of a wide range ofdisordered electronic matter, e.g., disordered semiconduc-tors, doped Mott insulators, p-njunction devices [ 5], etc. In reality, the optical response at low energy will nowbe Re σ xx(ω)=σdc+σ0ωα, with 0 <α< 1. This directly implies that Im σxx(ω)=tan(πα/2)σ0ωα+ω/epsilon10/epsilon1∞, with σdc being the dc conductivity and /epsilon1∞being the bare dielec- tric constant. The corresponding (dynamic) capacitance andimpedance read C(ω)=C ∞+(σ0/2π)tan(πα/2)ωα−1and Z∗(ω)/similarequal[iω/epsilon1∗(ω)]−1. Along with causality (Kramers-Krönig relations), UDR implies that the real and imaginary parts ofthe dielectric function (thus of the susceptibility) are relatedto each other by a dielectric phase angle, χ /prime(ω)/χ/prime/prime(ω)= cot(πα/2) [5],independent ofω, in stark contrast to Debye- like relaxation, where this ratio equals ωτ. Such forms have been widely used to analyze data in detail for a wide rangeof disordered matter [ 4,5,24] for a long time. Within CDMFT, our findings provide a microscopic rationale for use of theserelations.Theoretically, it is very interesting that such features ap- pear near a correlation-driven MIT in the FKM since this is aband-splitting-type Mott (rather than pure Anderson localiza-tion in a disorder model or a first-order Mott transition in thepure Hubbard model) transition. It supports views [ 11,25] that the disorder problem at strong coupling, where k Fl/similarequalO(1), is characterized by a different Mott-like quantum criticality, aview nicely supported by our earlier finding [ 11]o fβ(g)/similarequal ln(g) instead of β(g)/similarequal(d−2)−1/geven deep in the (bad) metallic phase. This is not unreasonable, as it has long beenknown [ 26] that the coherent potential approximation, the best mean-field theory for the Anderson disorder problem, isequivalent to the Hubbard III band-splitting view of the Motttransition (the latter becomes exact for the FKM in d=∞ [27]). (a) As concrete examples on the materials front, we note that various aspects of manganite physics have also been suc-cessfully modeled by an effective FKM, where the cfermions represent effectively spinless fermions (due to strong Hund’scoupling) strongly scattered by a disordered “liquid” of ef-fectively localized Jahn-Teller polarons [ 28]. In this context, it is also interesting to notice that a field-induced percolativeMIT has also long been known to occur in manganites [ 29]. Thus, our model can serve as the simplest effective modelfor PCSMO [ 6]. Application to LaTiO 3would require using a full Hubbard model very close to half filling, where intrinsic disorder due to inhomogeneous phases near the MIT wouldgenerally be expected to be relevant. (b) It is also veryinteresting that related features, namely, (i) the non-Landauquasiparticle (Drude) “strange” but infrared singular power- law optical response and (ii) anomalous optical phase angle,characterize the strange metallicity in near-optimally dopedcuprates [ 3]. One scenario, based on the hidden-Fermi-liquid idea, posits that an inverse orthogonality catastrophe also underlies [ 30] such non-Landau Fermi-liquid observations in cuprates. 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PhysRevB.73.014114.pdf

Gamma-ray spectra resulting from the annihilation of positrons with selected core levels of Cu, Ag, and Au S. Kim,1A. Eshed,1S. Goktepeli,2P. A. Sterne,3A. R. Koymen,1W. C. Chen,1and A. H. Weiss1 1Department of Physics, The University of Texas at Arlington, Arlington, Texas 76019, USA 2Motorola, 3501 Ed Bluestein Blvd., K10, Austin, Texas 78721, USA 3Lawrence Livermore National Laboratory, Livermore, California 94550, USA /H20849Received 22 July 2005; published 26 January 2006 /H20850 The energy spectra of /H9253rays resulting from positron annihilation with selected core levels of Cu, Ag, and Au were obtained separately from the total annihilation spectra. The separation was accomplished by measuringthe energy of /H9253rays detected in time coincidence with Auger electrons emitted consequent to the filling of holes resulting from the annihilation of core electrons. The results of these measurements are compared to thetotal annihilation spectra and with local-density approximation based theoretical calculations of the core con-tributions of the selected levels. Good agreement was found between calculated and measured values of thecore momentum densities with no adjustable parameters outside of the overall normalization. DOI: 10.1103/PhysRevB.73.014114 PACS number /H20849s/H20850: 61.72. /H11002y, 71.60. /H11001z, 78.70.Bj INTRODUCTION Spectroscopies based upon the detection and analysis of the/H9253rays emitted when a positron becomes trapped and annihilates in a defect are among the most sensitive probesof open volume or charged defects in metals andsemiconductors. 1–3In addition, the tendency of positrons to become trapped at open volumes in polymers, at surfaces, atinterfaces, and within nanoparticles has allowed positron-annihilation spectroscopy to be used as a highly selectiveprobe of these systems. 4–7The contributions to the Doppler- broadened annihilation spectra due to core electrons has be-come a subject of increased interest as the result of measure-ments demonstrating that it is possible to identify elements 3,8 from a chemically distinct spectral signature in the region ofthe /H9253spectra corresponding to core annihilation. In coinci- dence Doppler broadening /H20849CDB /H20850two Ge detectors are used to measure both the red and blue shifted annihilation /H9253rays in coincidence. The use of two detectors in coincidence hasmade it feasible to extract a statistically significant core an-nihilation contribution from the background resultingfrom the large valence contribution. 9The CDB tech- nique has been extensively applied in studies of vacancy-impurity complexes 1–3and quantum -dot nanoparticles and nanoprecipitates.5–7 In order to be confident in the elemental identification made using the coincidence-Doppler technique, it is impor-tant to understand the spectral contributions of the annihila-tion with core electrons in detail. However the core contri-butions constitute only a small fraction of the total spectrumdue to the fact that, typically, more than 90% of the annihi-lation events occur with the valence electrons due to therepulsion of positron from the positive core. This makes itimpossible, using only /H9253detection, to uniquely identify the core contributions to the spectra. Recently, Eshed et al.10 reported measurements of the Doppler-broadened /H9253-ray en- ergy spectra associated with the annihilation of a positronwith selected core levels of Cu and Ag using a technique inwhich /H9253rays are detected in coincidence with Auger elec-trons. In this paper we report research expanding on this previous work and report data on the Doppler-broadened /H9253-ray energy spectra associated with the annihilation of a positron with the core levels of Au. The core annihilation spectra for Au is compared to data obtained for Cu and Ag.An analysis applied to all three sets of data yielded improvedagreement with local-density approximation /H20849LDA /H20850calcula- tions of the core momentum densities by taking into accountthe contributions from deeper cores due to the Auger cascadeprocess. We also present a detailed discussion of the experi-mental system used in the /H9253-Auger coincidence measure- ments and the methods used in extracting the pair momen-tum densities from the Doppler broadened /H9253spectra. The Auger-coincidence methods, described in this paper, can beapplied in other types of momentum measurements includingangular correlation of annihilation radiation /H20849ACAR /H20850and to study the effects of adsorbates and reduced coordination onthe core level momentum densities of atoms at the surface. The experimental measurements of the annihilation /H9253 spectra for core levels reported in this paper provide a strin-gent test of theoretical calculations of core annihilation mo-mentum distributions, and a guide to the construction of im-proved descriptions of the electron-positron correlationeffects as they pertain to annihilation with core electrons.The addition of the Au data to that obtained for Cu and Aghas allowed us to test if LDA-based theory adequately ac-counts for the ratio of high momentum to low momentumspectral weight in the extracted momentum densities forcores with increasing principal quantum number. It wasfound that excellent agreement could be obtained betweentheory and experiment for Au and Cu when the contributionsof all core annihilations events that lead to Auger electronemission in the appropriate energy range are included. Theagreement between the data, which were obtained from theannihilation of surface state positrons, and the theory, whichwas calculated for annihilation of a positron in a sphericalstate around the atom, indicate that positron wave functioneffects do not cause serious problems in calculating momen-tum densities for core electrons.PHYSICAL REVIEW B 73, 014114 /H208492006 /H20850 1098-0121/2006/73 /H208491/H20850/014114 /H208497/H20850/$23.00 ©2006 The American Physical Society 014114-1BACKGROUND Positrons in solids annihilate predominantly into two /H9253 rays. In the center of mass frame these /H9253rays are emitted equal in energy and opposite in propagation direction. In thelaboratory frame, the center of mass motion of the positron-electron pair results in a Doppler shift of magnitude,/H20849P Lc/H20850/2, yielding energies, E/H92531andE/H92532for the two annihi- lation/H9253rays: E/H92531=m0c2−EB/2 + /H20849PLc/H20850/2, E/H92532=m0c2−EB/2 − /H20849PLc/H20850/2. /H208491/H20850 Where m0is the rest mass of the electron /H20849positron /H20850,cis the speed of light, EBis the binding energy of the electron, and PLis the component of the center of mass momentum of the electron-positron pair along the direction of the /H9253-ray emis- sion. Equation /H208491/H20850can be inverted to obtain the momentum of the electron-positron pair at the time of annihilation in thedirection of the HPGe /H9253detector15–17 PL=2E/H92531−2m0c2+EB c. /H208492/H20850 As a consequence, a histogram of the energy of detected annihilation /H9253’s can be used to obtain a one-dimensional pro- jection of the momentum distribution of annihilatingelectron-positron pairs. This distribution can be modeled byappropriate two-dimensional integration of a calculated mo-mentum distribution given by /H9267/H20849p/H20850=/H9266r02c/H20858 i/H20879/H20885dreip.r/H9023+/H20849r/H20850/H9023i−/H20849r/H20850/H20851/H9003i/H20849p,r/H20850/H208521/2/H208792 ,/H208493/H20850 where r0is the classical electron radius, pis the total mo- mentum of the annihilation pair, and /H9023+/H20849r/H20850is the positron wave function /H9023i−/H20849r/H20850is the wave function for the ith electron and/H9003i/H20849p,r/H20850is a weighting function that models “enhance- ment” i.e., electron-positron correlation affects which lead to an annihilation rate higher than that predicted in the indepen-dent particle approximation. 11 Calculations of the annihilation /H9253spectra with sufficient accuracy to extract chemical information in positron defectstudies require a detailed understanding of the enhancementfactor, /H9003 i/H20849p,r/H20850, for core levels. Conventional measurements /H20849including those using /H9253-/H9253coincidence techniques3,8,9/H20850probe the total momentum density of the system including bothcore and valence electrons. In modeling this data, Eq. /H208493/H20850 must be summed over all occupied electron states. Thus con-ventional spectra must be compared with calculations ofsums of individual level momentum densities weighted bymomentum dependent enhancement factors that introduceuncertainties that seriously limit the reliability of the com-parison. The /H9253spectra obtained using the /H9253-Auger coincidence technique make it possible to compare the measured andmodel momentum distributions of selected core levels andprovide a unique means to test theoretical efforts to go be-yond the local density approximation /H20849LDA /H20850,/H20849which can be expected to break down for the core levels because of theirwide range of momenta and large electron density gradients 12/H20850, such as the generalized gradient approximation /H20849GGA /H20850/H20849Ref. 13 /H20850and explicitly non-local treatments12such as the weighted density approximation /H20849WDA /H20850. In addition, the/H9253-Auger coincidence measurements provide the only means available, to date, of measuring the low momentumpart of the annihilation spectra for cores /H20849the low momentum contribution of the cores is swamped by valence band anni-hilation using /H9253spectroscopy alone /H20850. The measurement of the low momentum part of the spectra of the cores makes itpossible to determine the ratio of the high momentum to lowmomentum contributions providing a test of attempts tomodel the momentum dependence of /H9003. 14 EXPERIMENT The method of selecting /H9253rays associated with the anni- hilation of core electrons relies on the fact that an energeticcore hole left by the annihilation event can decay via thealmost simultaneous emission of one or more Auger elec-tron/H20849s/H20850with energies characteristic of the core level /H20849s/H20850. 15,16 For the outer core levels /H20849the levels of most relevance to Doppler broadening measurements /H20850almost all the core hole excitations decay via Auger processes.17,18For example, in a core-valence-valence Auger process, a valence electron car-ries off the energy made available when another valenceelectron fills the core hole left by annihilation. Previous mea-surements have demonstrated that it is possible to detectannihilation-induced Auger electrons with high efficiencyand with an energy resolution sufficient to infer the energylevels of the initial core holes. 15,16As a result, /H9253spectra associated with positron annihilation with electrons in se-lected core levels can be obtained by measuring the energiesof /H9253’s detected in coincidence with annihilation-induced Auger electrons of the appropriate energy. The/H9253-Auger coincidence data were collected using a magnetically guided positron beam system describedpreviously. 19The measurements were performed using a pos- itron beam energy of 12 eV and a flux of /H110112/H11003104positrons/ second. The beam system is equipped with a trochoidal en-ergy analyzer which is used for Positron annihilation-induced Auger spectroscopy, an ion-sputter gun and aconventional electron stimulated Auger system /H20849PHI–1100 /H20850 /H20849the later two systems are operated with the magnetic field of the positron beam turned off /H20850. The previous configuration of the beam system was augmented with the addition of aHPGe detector /H20849ORTEC-GEM-30185P, 58.6 mm diam. /H1100354.8 mm, relative efficiency 32% at 1.33 MeV /H20850, which was mounted perpendicular to the positron beam, 0.058 mfrom the sample, and behind a 0.0016 m stainless steelvacuum window. The full width at half maximum /H20849FWHM /H20850 of the detector resolution was measured to be 1.23 keV at 514 keV using a 85Sr calibration source. The samples were cut to a size of 20 mm /H1100320 mm from pure Ag, Cu, and Au foils, etched in a 48% solution of hy-drofluoric acid and rinsed in acetone and ethyl alcohol beforeloading into the vacuum chamber which was evacuated andS. KIM et al. PHYSICAL REVIEW B 73, 014114 /H208492006 /H20850 014114-2baked to obtain UHV conditions. The samples were initially cleaned by repeated sputter-anneal cycles /H2084930 min sputtering by neon /H20849Ag/H20850or argon /H20849Cu, Ag, Au /H20850ions followed by anneal- ing at /H11011150 °C. The sample was maintained under UHV conditions and sputtered for 3 h two times a week during theperiod of data acquisition /H20849/H1101120 days per sample /H20850. Surface cleanliness was monitored throughout the /H1101120 day period required for data accumulation by conventional electronstimulated Auger spectroscopy /H20849EAES /H20850and contamination levels were observed to be below 1% except for C and O forwhich the surface concentration stayed below 10% duringthe data collection period. We note that the spectral weight inthe energy range of interest from the low energy tails of theannihilation-induced C /H20849O/H20850Auger lines for 100% C /H2084950% O /H20850 surfaces are only a few percent of the positron annihilationinduced Auger intensities of Cu, Ag, and Au. 20Consequently, we estimate that less than 0.2% of the /H9253-Auger coincidence signal is from the C and O cores. The/H9253-Auger coincidence spectra /H20849containing contribu- tions only from core annihilation /H20850were obtained by gating the input from the HPGe detector into the multichannel ana-lyzer /H20849MCA /H20850with a pulse resulting from the detection of electron in the selected energy range within 600 ns of thedetection of the /H9253-ray /H20849see Fig. 1 /H20850. Conventional “non- coincidence” /H9253-spectra /H20849containing contributions from both core and valence annihilation /H20850were obtained by setting the gate input high allowing all of the HPGe pulses into theMCA. The annihilation /H9253spectra of the core levels /H20849levels with binding energies greater than or equal to that of the 3 plevel /H20850 of Cu were obtained by requiring coincidence with electronsin the energy range 57–59 eV. This range spans the peak ofthe energy distribution of the M 23VV Auger transition. Simi- larly the annihilation /H9253spectra of the core levels /H20849levels with binding energies greater than or equal to that of the 4 plevel /H20850 of Ag were obtained by requiring coincidence with electronsin the energy range 35–38 eV /H20849corresponding to the Ag N 23VV Auger transition /H20850and the annihilation /H9253spectra of the core levels /H20849levels with binding energies greater than or equal to that of the 5 plevel /H20850for Au were obtained by requir-ing coincidence with electrons in the range 38–40 eV for Au /H20849corresponding to the Au O 23VV Auger transition /H20850. A small background /H20849accounting for 5.4% of the total in- tensity for Cu, 11.6% for Ag, and 5.5% for Au /H20850due to acci- dental coincidences between the /H9253signal and uncorrelated pulses from the microchannel plate /H20849MCP /H20850was determined directly from a measurement of the integrated intensity of the /H9253signal taken in coincidence with electrons detected in an energy range where no true coincidences are present /H2084920 eV above the annihilation-induced Auger peak /H20850. The accidental contribution was then removed by subtracting a high statis-tics, noncoincidence /H9253spectra scaled to match the measured intensity of the accidental contribution to the spectra. Figure2 shows a comparison of the /H9253-Auger coincidence data as- sociated with the annihilation of positrons with Cu 3 pelec- trons before and after subtraction of the accidental back-ground. We note that the kinetic energy of the positrons hitting the surface at 12 eV, was below the impact-ionization thresholdfor all of the core levels studied. This was important to en- FIG. 1. Details of the experimental setup. A low energy positron beam is usedto place positrons at the sample surface.The signal from the preamp of the HPGe /H9253 detector is amplified and connected to theADC of a multichannel analyzer /H20849MCA /H20850. The Auger electrons were energy selectedusing an E/H11003Bfilter and detected using a microchannel plate /H20849MCP /H20850. The Doppler- broadened /H9253spectrum for a selected core level was acquired by using a fast coinci-dence circuit to open the gate to the MCAonly after the simultaneous detection of a /H9253ray and an electron from the selected Auger transition. Conventional Doppler-broadened spectra were acquired by hold-ing the MCA gate open. FIG. 2. Comparison of /H9253-ray -Auger electron coincidence data from a Cu surface as collected and the same data with the accidentalbackground subtracted. The /H9253spectra were obtained by impinging 12 eV positrons on a Cu surface and taken in coincidence with thedetection of electrons in the Cu M 23VV Auger peak.GAMMA-RAY SPECTRA RESULTING FROM ... PHYSICAL REVIEW B 73, 014114 /H208492006 /H20850 014114-3sure that the Auger electrons detected resulted from annihi- lation with core electron and not from Auger electrons result-ing from impact ionization. If a positron beam-energy higher than the core ionization energy was to be used, it wouldexcite Auger electrons both by positron annihilation withcore electrons and by impact ionization. The use of too higha positron-beam energy would also result in positron-inducedsecondary electrons with energies in the range of the Augerelectrons. 21Since the positrons that cause impact ionization or impact-induced secondaries are free to annihilate with va-lence electrons after the impact, the presence of Auger elec-trons /H20849or secondary electrons /H20850in the Auger energy range from impact excitation would result in an undesirable va-lence background in the coincidence measurements. Notealso that a large fraction of the positrons injected into thesamples at 12 eV diffuse back to the surface and becometrapped in an image correlation well before they annihilate.This greatly increases the escape probability of the Augerelectrons and implies that the /H9253-Auger coincidence technique predominantly samples atoms in the topmost atomic layer. THEORETICAL CALCULATIONS The calculations were based on an atomic code using a local-density form for the electron-positron correlationfunction. 22The calculations include appropriate integration of the three-dimensional /H208493D/H20850radial momentum distribution to correspond to the 1D Doppler measurements. We use anapproach in which the momentum integration is performedanalytically using a /H9254-function identity, thereby reducing the expression for the 1D momentum density to real-space inte-grals over well-behaved radial functions. As a result of theintegration, nodes in the radial momentum distributions re-sult in breaks in momentum that appear as shoulders in the1D momentum distribution. The calculations use a generalized-gradient approxima- tion /H20849GGA /H20850for the electron-positron enhancement. 23Sepa- rate calculations were performed using state-dependent andr-dependent enhancement. State-dependent enhancement uses a constant, momentum-independent enhancement factorequal to the average enhancement of the individual atomicstate, corresponding to /H9003 i/H20849p,r/H20850=/H9253iin Eq. /H208493/H20850.I nr-dependent enhancement, the enhancement factor in /H9003i/H20849p,r/H20850Eq. /H208493/H20850is replaced by a density-dependent enhancement factor, /H9253/H20851n/H20849r/H20850/H20852 prior to performing the radial integrations to produce the 1D momentum density, resulting in a momentum-dependent en-hancement function. The electron-positron enhancement function /H9253/H20851n/H20849r/H20850/H20852be- comes very large when the density becomes small. In our atomic calculation, the electron charge density drops off rap-idly at large radii, while in a real solid the charge densityfrom the neighboring atoms would maintain a much largercharge density. For this reason, we limit the enhancement toa value determined by the interstitial charge density in eachof the elements we consider here, with r svalues of 2.67, 3.02, and 3.0, respectively, for Cu, Ag, and Au. RESULTS AND DISCUSSION Figure 3 /H20849a/H20850shows a comparison of the energy distribution of annihilation /H9253rays from positrons incident on Cu mea-sured in coincidence with an Auger electron emitted as a result of filling the 3 pcore hole in Cu with a spectra ob- tained without the requirement of coincidence. Similarly,panels 3 /H20849b/H20850and 3 /H20849c/H20850show comparisons of the energy spectra FIG. 3. Comparison of the core + valence and core annihilation /H9253-ray energy spectra resulting from the bombardment of polycrys- talline Cu /H20849a/H20850,A g /H20849b/H20850, and Au /H20849c/H20850foils with a 12 eV positron beam. The core + valence spectra /H20849open symbols /H20850were acquired without a coincidence requirement. The core spectra /H20849solid symbols /H20850were ac- quired in time coincidence with the detection of an electron in therange of the peaks of the energy distribution of Auger electronsemitted as a result of transitions involving an initial hole in the Cu3p/H20849M 23/H20850,A g4 p/H20849N23/H20850, and Au 5p /H20849O23/H20850levels for panels /H20849a/H20850,/H20849b/H20850, and /H20849c/H20850, respectively.S. KIM et al. PHYSICAL REVIEW B 73, 014114 /H208492006 /H20850 014114-4of annihilation /H9253rays obtained with and without the require- ment for coincidence with Auger electrons emitted as a resultof filling the 4 plevel in Ag and the 5 plevel in Au, respec- tively. All the data sets shown in Fig. 3 were scaled such thatthey coincide at the peak. In all three cases the widths of the coincidence spectra are significantly larger than those of the non-coincidence spectra.The full width at half maximum /H20849FWHM /H20850of the noncoinci- dence spectra are 2.24, 2.73, and 2.38 keV from Cu, Ag, andAu, respectively, while the corresponding FWHM of the co-incidence spectra are 5.5, 4.6, and 4.4 keV, respectively. Thisis consistent with the fact that the noncoincidence data aredominated by /H9253’s resulting from annihilation with the rela- tively low momentum valence electrons and the coincidencedata characterizes the energy spectra of /H9253rays emitted as a result of annihilation with relatively high momentum coreelectrons. A qualitative understanding of the spectral widths of the noncoincidence spectra can be obtained by estimating thewidth of the momentum distribution of the valence electronsalone. To the lowest approximation this contribution can bemodeled by a parabola representing annihilation with freeconduction electrons. The parabola cuts off at an energy/H9004E max /H9004Emax=m0c2VF 2c/H110151 KeV /H208494/H20850 where VFis the Fermi velocity of an electron /H20849/H11011106ms−1/H20850.15–17 The larger width of the coincidence spectra is due to the relatively large Doppler shift associated with the core elec-trons which are the sole contributors to the coincidence spec-tra. As noted above, the FWHM of the coincidence spectraare 5.5, 4.6, and 4.4 keV from the Cu 3 p,A g4 p, and Au 5 p levels, respectively. The ratios of these widths, 1.25:1.05:1,correspond approximately to the ratios of the square root ofthe binding energies of the p 3/2levels /H20849a rough estimate of the ratios of the magnitudes of momentum of these levels /H20850 1.15:1.01:1. The fact that the Au 5 pis less tightly bound than the Ag 4 pwhich in turn is less tightly bound than the Cu 3 p implies that the Au 5 pis wider in real space than the Ag 4 p which is again wider than the Cu 3 pand hence their widths in momentum space are reversed. We note that the use of /H9253-Auger coincidence, like the use of/H9253-/H9253coincidence eliminates background due to Ps, nuclear decay and cosmic ray /H9253’s, etc. However, only /H9253-Auger coin- cidence is capable of separating the core part of the annihi-lation spectra from the much larger /H2084920 times at the peak /H20850 valence contribution. Figure 4 shows a comparison between measurements made using the Auger- /H9253coincidence technique and first prin- ciples calculations of the one-dimensional projection of themomentum distribution of annihilating electron-positronpairs of core electrons determined using the methods de-scribed above with state-dependent enhancement. The mo-mentum is expressed in dimensionless atomic units, where q is the wave vector and a 0is the Bohr radius. The calculated values are the result of summing the intensities of the mo-mentum distribution for all core levels with binding energies greater than or equal to those of the 3 pin Cu, the 4 pin Ag, and the 5 pin Au. We note that we have modified the analysis used previously in Ref. 10, in which only the contributionform the outer core was included, by including the contribu-tion from the deeper core levels. The previous analysis as-sumed that all the /H20849MVV /H20850Auger transitions in Cu and FIG. 4. Momentum distributions of the positron-electron pairs involved in the annihilation of positrons with the core levels of Cu/H20849a/H20850,A g /H20849b/H20850, and Au /H20849c/H20850. The experimental distributions /H20849solid squares /H20850were extracted from the Doppler-broadened spectra by us- ing Eq. /H208492/H20850. The measured distributions are compared to LDA-based calculations normalized to have the same value at the peaks.GAMMA-RAY SPECTRA RESULTING FROM ... PHYSICAL REVIEW B 73, 014114 /H208492006 /H20850 014114-5/H20849NVV /H20850Auger transitions in Ag were preceded by annihila- tion with M shell electrons in Cu and N shell electrons in Ag,respectively. This assumption neglects the contribution to an-nihilation spectra from events in which /H9253rays emitted from deeper core levels are detected in coincidence with Augerelectrons emitted in a cascade process. 24For example, the annihilation of an electron in the L shell of Cu can result ina MVV Auger transition via the cascade process in which aLMV or LMM transition is followed by a MVV Auger tran-sition. Although such processes account for only a smallfraction of the electrons emitted in the energy range of theCu MVV peak /H20849which are overwhelmingly electrons emitted in a direct Auger transitions in which the positron annihilateswith electrons in the M shell /H20850, the contribution of coinci- dence events involving cascade processes is significant in thehigh momentum region of the /H9253spectra. This follows from the fact that the high momentum region of the total annihi-lation spectra is dominated by contributions from annihila- tion with the deeper cores due to the high momentum of thecore electrons even though the probability for such annihila-tion events is low due to core repulsion together with the factthat holes in the deep core level are converted into holes inthe outer cores with high efficiency. 24Consequently the spec- tral contribution of annihilation events involving the deeperholes needs to be included in order to accurately model thecoincidence data in the high momentum region. We note thatcontributions from individual core levels could be obtainedby comparing a sequence of /H9253spectra taken in coincidence with higher and higher energy Auger peaks. The agreement between theory and experiment is remark- able given the complexities of both the experiment andtheory and the fact that the calculations were done indepen-dently with no adjustable parameters aside from the overallnormalization. Referring to Fig. 4 panels 4 /H20849a/H20850and 4 /H20849c/H20850,i t may be seen that the agreement between theory and experi-ment is within the limits of statistical uncertainty for Cu andAu. However, referring to panel 4 /H20849b/H20850it may be seen that there are differences between theory and experiment for Agthat are outside of the statistical uncertainties of the experi-ment. Specifically, when both theory and experiment are setto be equal at the peak, the calculation for Ag lies consis-tently below the experimental values for qa 0/H110223. Because the normalization procedure made the measured and theoret-ical values coincide in the low momentum part of the spec-tra, some care should be taken in assuming the comparisonindicates that the discrepancy between theory and experi-ment is only at high momentum. There are a number of possible explanations for the dis- crepancies between theory and experiment in Ag: /H208491/H20850. Auger cascade processes lead to a larger relative contribution ofannihilation events with deeper cores in Ag than in the caseof Cu and Au. We note however that modeling of the dataindicates that in order to get better agreement, the contribu-tions from coincidence events involving annihilation withdeeper cores would need to be much larger for Ag than forCu or Au, while the Auger cascade processes should havesimilar efficiencies for all three metals. /H208492/H20850. The LDA-based calculation of the core electron momentum distribution mayunderestimate the high momentum tails. However, while theLDA is known to give the wrong core level binding energies,the charge densities from which the momentum distributions are calculated are believed to be accurate. /H208493/H20850. The discrep- ancies may be due to the level of approximation used inmodeling the positron wave function in which only a s-like state was included. While sstates have appeared to be ad- equate for approximating the positron state in bulk calcula-tions, it is likely that a mixed s-pstate may be more appro- priate for the overlap of a positron in a surface state with asurface atom. We note however, that the observed discrepan-cies appear to be in the high momentum region in which thepositron’s contribution to the total pair momentum could beexpected to be small due to the fact that, on the average, thepositrons are at thermal energies at the time of annihilation./H208494/H20850. The discrepancy may reflect inadequacies in the treat- ment of electron-positron correlation effects and the need foran enhancement term with an explicit momentum depen-dence that increases at higher momentum. We note, however,that current treatments of the momentum dependence of theLDA based theories predict the opposite momentum depen-dence, and our r-dependent enhancement factor calculations, which introduce a momentum-dependent enhancement fac-tor, in fact showed a preferential enhancement of the low-momentum electrons, further worsening the agreement withexperiment for Ag. CONCLUSIONS The data presented in this paper represents the results of the first measurements of the Doppler-broadened /H9253-ray spec- tra resulting from the annihilation of positrons with indi-vidual core levels. Annihilation /H9253spectra from selected core levels in Cu, Ag, and Au were obtained by measuring theenergies of /H9253-rays time coincident with Auger electrons emitted as a result of positrons annihilating with a core level.A comparison with calculations of the annihilation spectrafor the core levels shows excellent qualitative agreementwith no adjustable parameters aside from the overall normal-ization. However, differences with theory are well outside ofthe statistical uncertainties for Ag. We note that our measurements directly separate the low momentum contributions of the core from the much largersignal from annihilation with valence electrons. The methodof using coincidence with the detection of Auger electrons toselect core annihilation events, while used in this study tomeasure the Doppler broadened /H9253spectra, is of general ap- plicability in studies of core annihilation. Future /H9253-Auger coincidence measurements could be used to measure the corespectra of impurity atoms at the surface. The core signatures of impurities thus obtained could then be used to provideconfirmation of the signatures of vacancy-impurity com-plexes in the bulk as seen in Doppler spectra obtained using /H9253-/H9253coincidence. The Auger coincidence technique can also be used in conjunction with the measurement of the angularcorrelation of annihilation radiation, /H20849ACAR /H20850, in high- resolution fundamental studies of core electron momentumdistributions.S. KIM et al. PHYSICAL REVIEW B 73, 014114 /H208492006 /H20850 014114-6ACKNOWLEDGMENTS This work was supported in part by the NSF DMR- 9812628 and The Welch Foundation Grant Nos. Y-1100 andY-1215. Part of this work was performed under the auspices of the U.S. Department of Energy by the University of Cali- fornia, Lawrence Livermore National Laboratory under Con-tract No. W-7405-Eng-48. 1M. Rummukainen, I. Makkonen, V. 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Lett. 77, 2097 /H208491996 /H20850. 9K. G. Lynn, J. E. Dickman, W. L. Brown, M. F. Robbins, and E. Bonderup, Phys. Rev. B 20, 3566 /H208491979 /H20850. 10A. Eshed, S. Goktepeli, A. R. Koymen, S. Kim, W. C. Chen, D. J. O’Kelly, P. A. Sterne, and A. H. Weiss, Phys. Rev. Lett. 89, 075503 /H208492002 /H20850. 11M. Alatalo, B. Barbiellini, M. Hakala, H. Kauppinen, T. Kor- honen, M. J. Puska, K. Saarinen, P. Hautojärvi, and R. M. Ni-eminen, Phys. Rev. B 54, 2397 /H208491996 /H20850. 12A. Rubaszek, Z. Szotek, and W. M. Temmerman, Phys. Rev. B58, 11285 /H208491998 /H20850. 13B. Barbiellini, M. Hakala, M. J. Puska, R. M. Nieminen, and A. A. Manuel, Phys. Rev. B 56, 7136 /H208491997 /H20850. 14B. B. J. Hede and J. P. Carbotte, J. Phys. Chem. Solids 33, 727 /H208491972 /H20850. 15A. Weiss, R. Mayer, M. Jibaly, C. Lei, D. Mehl, and K. G. Lynn, Phys. Rev. Lett. 61, 2245 /H208491988 /H20850. 16A. H. Weiss and P. G. Coleman, in Positron Beams and Their Applications , edited by Paul Coleman /H20849World Scientific, Sin- gapore, 2000 /H20850, pp. 152–165. 17E. J. McGuire, Phys. Rev. A 5, 1052 /H208491972 /H20850. 18D. Briggs and M. P. Seah, Practical Surface Analysis /H20849Wiley, New York, 1994 /H20850,p .9 2 . 19C. Lei, D. Mehl, A. R. Koymen, F. Gotwald, M. Jibaly, and A. H. Weiss, Rev. Sci. Instrum. 60, 3656 /H208491989 /H20850. 20J. G. Zhu, M. P. Nadesalingam, A. H. Weiss, and M. Tao, J. Appl. Phys. 97, 103510 /H208492005 /H20850; Shupping Xie, Ph.D. dissertation, University of Texas at Arlington, /H208492002 /H20850. 21A. Weiss, D. Mehl, A. R. Koymen, K. H. Lee, and Chun Lei, J. Vac. Sci. Technol. A 8, 2517 /H208491990 /H20850. 22P. A. Sterne, P. Asoka-Kumar, and R. H. Howell, Appl. Surf. Sci. 194,7 1 /H208492002 /H20850. 23B. Barbiellini, M. J. Puska, T. Torsti, and R. M. Nieminen, Phys. Rev. B 51, 7341 /H208491995 /H20850; B. Barbiellini, M. J. Puska, T. Kor- honen, A. Harju, T. 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PhysRevB.100.174303.pdf

PHYSICAL REVIEW B 100, 174303 (2019) Scaling behavior of the stationary states arising from dissipation at continuous quantum transitions Davide Rossini and Ettore Vicari Dipartimento di Fisica dell’Università di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy (Received 21 July 2019; revised manuscript received 5 November 2019; published 20 November 2019) We study the critical behavior of the nonequilibrium dynamics and of the steady states emerging from the competition between coherent and dissipative dynamics close to quantum phase transitions. The latter is inducedby the coupling of the system with a Markovian bath, such that the evolution of the system’s density matrixcan be effectively described by a Lindblad master equation. We devise general scaling behaviors for the out-of-equilibrium evolution and the stationary states emerging in the large-time limit for generic initial conditionsin terms of the parameters of the Hamiltonian providing the coherent driving and those associated with thedissipative interactions with the environment. Our framework is supported by numerical results for the dynamicsof a one-dimensional lattice fermion gas undergoing a quantum Ising transition in the presence of dissipativemechanisms which include local pumping and decay of particles. DOI: 10.1103/PhysRevB.100.174303 I. INTRODUCTION One of the major challenges of current experimental and theoretical investigations in the field of quantum statisticalmechanics and condensed-matter physics is the understand-ing of the out-of-equilibrium dynamics of open many-body systems, arising from coherent Hamiltonian drivings and dis- sipative mechanisms. Recent technological breakthroughs inthe manipulation of atomic and quantum optical systems arepaving the way to a careful study of the interplay between thecoherent quantum dynamics and the dissipative effects due tothe interaction with the environment [ 1–4]. The competition between these two mechanisms may lead to stationary states which are not related to a thermalization process, i.e., whose properties are not describable in terms of thermal Gibbs distri-butions. In particular, novel phenomena may emerge close to aquantum phase transition [ 5], where the low-energy properties of the system are particularly sensitive to variations of theexternal conditions. In this paper we investigate the dynamics of a many-body system in the proximity of a continuous quantum transition inthe presence of dissipation arising from the interaction withthe environment. We focus on a class of dissipative mech-anisms whose dynamics can be reliably described througha Lindblad master equation governing the time evolutionof the system’s density matrix [ 6,7]. This framework is of experimental interest; indeed, the conditions for its validityare typically realized in quantum optical and circuit quantumelectrodynamics (c-QED) implementations [ 1,2,8]. We argue that, in the presence of homogeneous dissipators, the compe-tition between coherent and dissipative drivings develops dy-namic scaling laws involving some relevant parameters of thetwo mechanisms. This occurs within a low-dissipation regime,where the decay rate of the dissipation is comparable to thegap of the Hamiltonian [ 9]. General scaling behaviors are put forward which are expected to be developed along the timeevolution described by the Lindblad equation, especially inthe large-time limit where stationary states set in. Analogousto the scaling laws of closed systems at quantum transitions, the dynamic scaling behavior in the presence of dissipationis expected to be universal, i.e., largely independent of themicroscopic details. To verify the emerging scaling scenario, we found it con- venient to consider, as an example, the paradigmatic Kitaev quantum wire [ 10]. Namely, we study its dynamic behavior close to the quantum transition in the presence of dissipationdue to local incoherent pumping or decay. Our numericalresults support the general dynamic scaling theory, addressingthe mutual interplay between coherent dynamics and dissipa-tion at a continuous quantum transition. Some features arising from the competition of coherent and dissipative dynamics close to quantum transitions wererecently analyzed within a dynamic finite-size scaling frame-work (see Ref. [ 9]), with specific emphasis on finite systems of linear size Land the dynamic behavior for relatively small times t∼L z(where z>0 is the dynamic critical exponent associated with the quantum transition), thus not including thelarge-time stationary regime. In this paper we focus on somecomplementary regimes. Indeed, we consider infinite-sizesystems, for which we derive dynamic scaling laws extendingto the large-time limit of the evolution described by theLindblad equation, thus valid for the corresponding stationarystates. We shall emphasize that, at this stage, our dynamicalscaling theory for quantum dissipative systems should beconsidered as a conjecture arising from phenomenologicalscaling arguments. Therefore, the scaling behaviors devisedfor the two mentioned distinct situations are not triviallyrelated, and careful numerical checks are crucial to validateour general framework. We finally mention that dynamic scaling behaviors have also been put forward and numerically checked for opencritical systems when the environment is constituted by asingle qubit homogeneously coupled to the whole many-bodysystem [ 11]. This paper is organized as follows. In Sec. IIwe present the general setup of our dynamic problem, recalling the scaling 2469-9950/2019/100(17)/174303(9) 174303-1 ©2019 American Physical SocietyDA VIDE ROSSINI AND ETTORE VICARI PHYSICAL REVIEW B 100, 174303 (2019) laws developed by closed systems at quantum transitions, and the main features of the dissipative mechanisms describedby the Lindblad equation. Then, in Sec. IIIwe put forward the scaling laws describing the competition of coherent anddissipative drivings as described by the Lindblad equation,extending the scaling laws of closed systems at quantum tran-sitions to incorporate the effects of the dissipation. In Sec. IV we introduce the one-dimensional Kitaev Fermi model in thepresence of dissipation arising from local pumping and decayand define the observables that we are going to consider inorder to characterize the time evolution of the system, inparticular of the stationary states. Section Vcontains and discusses the outcomes of our numerical results, which def-initely support the dynamic scaling theory in the presence ofdissipation. Finally, in Sec. VIwe summarize and draw our conclusions. II. CRITICAL SYSTEMS IN THE PRESENCE OF DISSIPATION A. Many-body systems at a quantum transition We start by summarizing the general scaling features of a many-body system at a continuous quantum transition.Consider a generic d-dimensional many-body system with Hamiltonian ˆH, close to a zero-temperature transition driven by quantum fluctuations [ 5,12]. A quantum transition is gener- ally characterized by few relevant perturbations, whose tuninggives rise to quantum critical behaviors, characterized by adiverging length scale, and universal power laws. Let us assume that the system Hamiltonian has one relevant parameter μ, whose tuning toward the point μ cdevelops a quantum critical behavior. The critical power laws are gen-erally characterized by the renormalization-group dimensiony μassociated with the relevant parameter μand the dynamic exponent z, so that the diverging length scale of the critical modes behaves as ξ∼λ≡|¯μ|−ν,¯μ≡μ−μc,ν=y−1 μ, (1) and the suppression of the gap (that is, the difference of the two lowest energy levels) behaves as /Delta1∼ξ−z. (2) Moreover, the correlation function of generic local operators ˆO(x), G(x;¯μ)≡/angbracketleft0μ|ˆO(x)ˆO(0)|0μ/angbracketright, (3) where |0μ/angbracketrightis the ground state associated with the parameter μ, obeys an asymptotic scaling law of the form [ 12,13] G(x;¯μ)≈b−2yoG(x/b,¯μbyμ), (4) where bdenotes an arbitrary positive number and yois the renormalization-group dimension of ˆO(x). The above scaling equation neglects further dependences on other irrelevantHamiltonian parameters, which are supposed to be suppressedin the large- blimit. Then, if we fix the arbitrary parameter b by requiring |¯μ|b yμ=1( 5 )and introduce the variable λas in Eq. ( 1), we obtain the asymptotic scaling behavior G(x;¯μ)≈λ−2yoG(x/λ)( 6 ) around μc. One may also consider equilibrium states at finite temperature T, related to a Gibbs distribution of the quantum states. For sufficiently small temperatures, the dependence onTis taken into account [ 5] by adding a further dependence on Tb zin Eq. ( 4), turning into a dependence on Tλzin the scaling equation ( 6), that is, G(x;¯μ,T)≈λ−2yoG(x/λ,Tλz). (7) These asymptotic scaling behaviors are expected to be ob- served in the limit λ→∞ . Their approach is generally char- acterized by power-law scaling corrections, which may comefrom different sources [ 13], such as irrelevant perturbations at the fixed point describing the quantum transition and analyticcorrections in the nonlinear scaling fields entering the scalinglaws. One may also study out-of-equilibrium evolutions close to a quantum transition, for example, arising from a suddenquench of the Hamiltonian parameter μatt=0 from an initial value μ itoμ/negationslash=μi, starting from the ground state at μi. The corresponding time dependence after the quench can betaken into account within the scaling laws as well by addinga further dependence on tb −zin Eq. ( 4). More precisely, the fixed-time correlation functions are expected to behave as[14,15] G(x,t;¯μ i,¯μ)=/angbracketleft/Psi1(t)|ˆO(x)ˆO(0)|/Psi1(t)/angbracketright ≈λ−2yoG(x/λ,tλ−z,¯μi/¯μ), (8) where |/Psi1(t)/angbracketrightis the quantum state after a time tfrom the quench and ¯ μi≡μi−μc. The above scaling behaviors have been shown to develop for closed states around the critical point μc. In the following we study the effects of the presence of dissipation in systemsdescribed by critical Hamiltonians. B. Dissipative interactions Suppose that the many-body system is also subject to dissipative interactions with the environment, so that thetime dependence of its density matrix ρis described by the Lindblad master equation [ 6] ∂ρ ∂t=−i ¯h[ˆH,ρ]+uD[ρ], (9) where the first term provides the coherent driving, while the second term accounts for the coupling to the environment,characterized by a global coupling constant u>0. We restrict ourselves to homogeneous dissipation mech- anisms, preserving translational invariance, as sketched, forexample, in Fig. 1. In the case of systems weakly coupled to Markovian baths, the trace-preserving superoperator can bewritten as a sum of local terms, such as [ 16,17] D[ρ]=/summationdisplay oDo[ρ], (10) Do[ρ]=ˆLoρˆL† o−1 2(ρˆL† oˆLo+ˆL† oˆLoρ), (11) 174303-2SCALING BEHA VIOR OF THE STATIONARY STATES … PHYSICAL REVIEW B 100, 174303 (2019) BBBBBBBBBB FIG. 1. Sketch of a quantum system on a one-dimensional lattice (black squares), in which different sites may undergo a coherent and uniform nearest-neighbor coupling. Each site is supposed to be homogeneously and weakly coupled to an external and independent bath B(blue boxes), whose effect is to introduce local incoherent mechanisms. where ˆLois the Lindblad jump operator associated with the local system-bath coupling scheme and odenotes an appropri- ate spatial coordinate. Dissipation mechanisms described bylocal Lindblad operators ˆL ohave been considered in various contexts (see, e.g., Refs. [ 1,2,8,18,19]). In several quantum optical devices, in particular in systems with photon leakageor with some qubit relaxation or dephasing, the conditionsleading to Eqs. ( 9)–(11) are typically satisfied [ 1,2,8]; there- fore, this formalism constitutes the standard choice for theirtheoretical investigation. Other interesting implementations inthis respect may be provided by hybrid light-matter systems,where atoms are strongly coupled to cavities, thus mediatingphoton-photon interactions (see, e.g., Ref. [ 19]). In the next section, we will address the formation of large- time stationary states arising from the competition of the co-herent and dissipative drivings, as described by the Lindbladequation ( 9). As we shall see below, while the coupling with a bath generally drives the system to a noncritical behavior, evenwhen the Hamiltonian parameters are critical, it is, however,possible to identify a peculiar low-dissipation regime wherethe steady state may develop the above-mentioned compe-tition and display a critical behavior. To this purpose, wewill put forward general scaling behaviors for the stationarystates in the low-dissipation regime when the many-bodyHamiltonian is within the quantum critical regime, somehowextending the scaling laws reported in Sec. II A. III. SCALING BEHA VIOR OF THE STATIONARY STATES Let us assume that, without loss of generality, the quantum many-body system is initialized at t=0 into the ground state of the Hamiltonian ˆH(μi) for a given parameter μi. The time evolution for t>0 is dictated by the Lindblad equation ( 9) with the Hamiltonian ˆH(μ), where μmay differ from μi, thus realizing a sudden quench, and the system-bath couplingstrength is fixed by the coupling u. The dissipator D[ρ]m a y drive the system to a steady state, which is generally non-critical, even when the Hamiltonian parameters are critical.In the case of dissipation leading to a unique stationary state,the choice of the initial state (that is, the initial state of thetime evolution, which is eventually fixed by the parameterμ i) is not relevant for the long-time properties of the system; nonetheless, it may determine the initial and intermediate timedependence. We now argue that it is possible to identify a low- dissipation regime where the dissipation is sufficiently smallto compete with the coherent evolution driven by the (critical)Hamiltonian. This leads to a late-time stationary state whichcan present a critical behavior, depending also on the strength of the system-bath coupling. The effects of a small dissipationare taken into account by adding a further dependence on ascaling variable associated with uin the out-of-equilibrium scaling laws, i.e., ub ζ, where ζis a suitable exponent which ensures the substantial balance (thus competition) with thecritical coherent driving. Since dissipation is predicted togive rise to a relevant perturbation at the quantum transition,we expect ζ> 0. Thus, the peculiar low-dissipation regime outlined above should be characterized by u∼λ −ζ. As already argued in Ref. [ 9], the exponent ζshould gen- erally coincide with the dynamic exponent z. This is expected by noting that the parameter uin Eq. ( 9) plays the role of a decay rate, i.e., of an inverse relaxation time for the associateddissipative process [ 6], and any relevant timescale t sat a quantum transition behaves as ts∼/Delta1−1[14]. In other words, to observe a nontrivial competition between critical coherentdynamics and dissipation, one should consider a sufficientlysmall coupling u∼λ −z, so that its size is comparable with the energy difference /Delta1∼λ−zof the lowest energy levels of the Hamiltonian. An analogous conjecture was put forward todescribe the approach to thermalization of some specific opensystems close to a quantum transition [ 20]. Here we extend it to more general situations, even when the final stationary stateis not thermal. On the basis of such scaling arguments, generic fixed- time correlation functions in the large-volume limit are thusexpected to behave as G(x,t;¯μ i,¯μ,u)=Tr [ρ(t)ˆO(x)ˆO(0)] ≈λ−2yoG(x/λ,tλ−z,¯μi/¯μ,uλz),(12) where ρ(t) is the time-dependent density matrix of the system. We conjecture that this scaling ansatz describes the low-dissipation regime of quenching protocols for many-bodysystems at quantum transitions. The main features of such adynamic critical regime are expected to depend only on theuniversality class of the transition and the general propertiesof the dissipative mechanism. In order to derive the scaling laws for the asymptotic stationary states, we need to consider the large-time limitof the scaling equation ( 12). Hereafter we shall assume that the asymptotic stationary state is unique, i.e., independent ofthe initial conditions of the protocol that we consider, as isthe case for several classes of dissipators [ 21–24]. Such a stationary state should appear when t/greatermuchλ z. Therefore, we conjecture the scaling law Gs(x;¯μ,u)≡G(x,t→∞ ;¯μi,¯μ,u) ≈λ−2yo/Gamma1(x/λ,uλz), (13) where in the definition of Gswe assumed that the dependence on the initial parameter ¯ μidrops under an assumption of unicity of the stationary state. It is also possible to define a correlation length ξsfrom the exponential large-distance decay of the correlation functionG s. Indeed, since we expect that the large-distance behavior Gs∼e−x/ξs, we may define ξ−1 s(¯μ,u)=−lim x→∞lnGs(x;¯μ,u) x. (14) 174303-3DA VIDE ROSSINI AND ETTORE VICARI PHYSICAL REVIEW B 100, 174303 (2019) Using the scaling laws derived for Gs, we obtain ξs≈λL1(uλz), (15) or, alternatively, ξs≈u−1/zL2(uλz). (16) Note that this scaling equation is formally analogous to that obtained at the thermodynamic equilibrium after replacing u with the temperature T[see Sec. II A, in particular Eq. ( 7)]. However, we stress that our arguments extend to cases wherethe asymptotic stationary state does not coincide with a Gibbsequilibrium state. This will be, indeed, the case for the Kitaevmodel subject to incoherent particle decay or pumping [seeEq. ( 20) and the discussion below]. We finally stress that the strictly local (on-site) nature of the Lindblad operators is not essential for the dynamic scalingbehavior put forward here. The important point is that suchinteractions induce only finite-size correlations, whose lengthscale becomes negligible in the quantum critical limit, wherecritical quantum correlations develop a diverging length scalein the system. IV . THE KITAEV QUANTUM WIRE SUBJECT TO DISSIPATION To provide evidence of the scaling laws put forward in Sec. III, we consider a Kitaev quantum wire defined by the Hamiltonian [ 10] ˆHK=− tL/summationdisplay j=1(ˆc† jˆcj+1+δˆc† jˆc† j+1+H.c.)−μL/summationdisplay j=1ˆnj,(17) where ˆ cjis the fermionic annihilation operator on the jth site of the chain, ˆ nj≡ˆc† jˆcjis the density operator, and δ>0. We set ¯h=1 and t=1 as the energy scale. The Hamiltonian ( 17) can be mapped into a spin-1 /2 XY chain by means of a Jordan-Wigner transformation. Itundergoes a continuous quantum transition at μ=μ c=−2 independent of δbetween a disordered quantum phase (μ<μ c) and an ordered quantum phase ( |μ|<|μc|). This transition belongs to the two-dimensional Ising universalityclass [ 5], characterized by the length-scale critical expo- nentν=1, related to the renormalizaton-group dimension y μ=1/ν=1 of the Hamiltonian parameter μ(more pre- cisely, the difference ¯ μ≡μ−μc). The dynamic exponent associated with the unitary quantum dynamics is z=1. More- over, the renormalization-group dimension of the fermionicoperators ˆ c jand ˆ c† jisyc=1/2, and that of the density operator ˆ njisyn=1[5]. In the following we fix δ=1, such that the corresponding spin model is the quantum Ising chain ˆHIs=−/summationdisplay jˆσ(3) jˆσ(3) j+1−g/summationdisplay jˆσ(1) j, (18) with ˆσ(k) jbeing the Pauli matrices and g=−μ/2. In the following we prefer to stick with the Kitaev quantum wirebecause the dissipation that we consider is more naturallydefined for Fermi lattice gases. We focus on the dynamic behavior of the Fermi lattice gas ( 17) close to its quantum transition in the presence ofhomogeneous dissipation mechanisms following the Lindblad equation ( 9). The dissipator D[ρ] is defined as a sum of local (single-site) terms of the form D j[ρ]=ˆLjρˆL† j−1 2(ρˆL† jˆLj+ˆL† jˆLjρ), (19) where ˆLjdenotes the Lindblad jump operator associated with the system-bath coupling scheme and the index jcorresponds to a lattice site [thus replacing the index oin Eqs. ( 10) and (11)]. The on-site Lindblad operators ˆLjdescribe the coupling of each site with an independent bath B(Fig. 1). We consider dissipation mechanisms associated with either particle losses(l) or pumping (p) [ 25,26]: ˆL l,j=ˆcj,ˆLp,j=ˆc† j. (20) The uniqueness of the eventual steady state has been proven for the above decay and pumping operators [ 21–24]. The choice of such dissipators turns out to be particularly conve-nient for the numerical analysis, allowing us to obtain resultsfor a quite large Kitaev model (see below). Our protocol starts from the ground state of ˆH Kfor a generic ¯ μi≡μi−μc, sufficiently small to stay within the critical regime. To address the competition between coherentand dissipative dynamics, we study the evolution after aquench of the Hamiltonian parameter to ¯ μand a simultaneous sudden turning on of the dissipation coupling u. To character- ize the dynamic properties of the evolution described by theLindblad equation, in particular the corresponding asymptoticlarge-time behavior, we consider the fixed-time correlations G p(x,t)=Tr[ρ(t)(ˆc† jˆc† j+x+ˆcj+xˆcj)], (21a) Gc(x,t)=Tr[ρ(t)(ˆc† jˆcj+x+ˆc† j+xˆcj)], (21b) Gn(x,t)=Tr[ρ(t)ˆnjˆnj+x]−Tr[ρ(t)ˆnj]T r [ρ(t)ˆnj+x], (21c) where we used the space translation invariance of the system. According to the scaling arguments reported in Sec. III for systems close to a continuous quantum transition, inparticular for λ≡|¯μ| −1→∞ , these correlation functions are expected to develop the scaling laws given in Eqs. ( 12) and (13). The approach to the asymptotic behavior is foreseen to be controlled by power-law corrections. Relying on theanalysis of closed systems undergoing quantum transitionbelonging to the two-dimensional Ising universality class (seeRef. [ 13]), we generally expect O(λ −1) corrections, generally arising from analytic corrections to the nonlinear scaling fields[while scaling corrections arising from the leading irrelevantperturbation are more suppressed in the two-dimensional Isinguniversality class, as O(λ −2)]. The numerical results reported in the next section nicely support these predictions. V . NUMERICAL RESULTS In this section we report numerical evidence of the scaling laws put forward in Sec. IIIfor quantum wires in the presence of dissipation arising from incoherent particle losses andpumping. The choice of the dissipators ( 20) allows us to numerically solve the dynamic problem for systems with upto thousands of sites [ 27,28]. Indeed, the dynamics of the dissipative fermionic Kitaev chain (Fig. 1) can be written in 174303-4SCALING BEHA VIOR OF THE STATIONARY STATES … PHYSICAL REVIEW B 100, 174303 (2019) 10-1100101t / λ0.120.160.20.24λ Gpλ = 10 λ = 20 λ = 40 λ = 80 0 0.05 0.1 1/λ0.2350.2360.2370.2380.239λ Gp 0 0.05 0.1 1/λ0.150.15250.1550.15750.16 0 0.05 0.1 1/λ0.1340.1360.1380.140.142(a) (b1)( b2)( b3) FIG. 2. (a) The correlation function Gp(x,t)i nE q .( 21a), mul- tiplied by λ, versus the rescaled time t/λ. Different curves are for various values of λ, as indicated in the legend. Here we fix x/λ=1. The dissipation is induced by incoherent particle losses, and its rescaled strength is kept constant, such that uλ=1. We also fix ¯μi=¯μ(no Hamiltonian quenches) and take ¯ μ< 0 (i.e., approaching the critical point from the left side, μ<μ c), which vary for each curve as |¯μ|=λ−1[see Eq. ( 1)]. The curves approach a scaling function with increasing λ, thus supporting the scaling equation (12). (b1)–(b3) The convergence with λof the various curves in (a) for fixed values of the rescaled time, t/λ=0.1, 2, and 25, respectively [arrows in (a)]. Straight lines are fits ∝λ−1of numerical data (circles), whose extrapolation to λ→+ ∞ produces the values 0.2385, 0.1585, and 0.1429, respectively. terms of coupled linear differential equations, whose number scales linearly with the size L. We employ a fourth-order Runge-Kutta method in order to integrate them. Further de-tails on the computation of the time trajectories from theLindblad equation ( 9) are reported in Ref. [ 9]. In our analysis we always consider antiperiodic bound- ary conditions, such that ˆ c L+1=− ˆc1, which turn out to be technically convenient for the calculations. However, sincewe always consider sufficiently large chains to be effectivelyin the thermodynamic limit (more precisely, L/greatermuchλ, where λ=¯μ −1), the results that we present are not affected by this particular choice. The realization of the infinite-size limit iseasily verified by comparing the numerics at fixed λand u, with increasing L. All the data presented here should be considered in the infinite-size limit with great accuracy (finite-size effects are invisible on the scales of all the figures shownbelow). Let us start by presenting some numerical outcomes for the correlation function G p(x,t)[ s e eE q .( 21a)] in the presence of dissipation arising from incoherent particle losses. The curvesshown in Fig. 2were obtained for the protocol with ¯ μ i=¯μ (no Hamiltonian quenches) and ¯ μ< 0 (thus approaching the critical point from the left side: μ<μ c), keeping the rescaled dissipation strength constant, such that uλz=uλ=1. The results in Fig. 2(a) clearly support the asymptotic dynamic10-1100101t / λ-0.12-0.08-0.04λ Gc λ = 10 λ = 20 λ = 40 λ = 80 0 0.05 0.1 1/λ-0.061-0.06-0.059-0.058-0.057λ Gc 0 0.05 0.1 1/λ-0.087-0.086-0.085-0.084-0.083 0 0.05 0.1 1/λ-0.091-0.09-0.089-0.088-0.087(a) (b1)( b2)( b3) FIG. 3. Same as in Fig. 2, but for the correlation function Gc(x,t) in Eq. ( 21b). All the various parameters are the same as in Fig. 2, except for the value of x/λ, which here is fixed and taken to be equal to 2. The asymptotic values of the various curves in (a) for λ→+ ∞ , extrapolated from a linear fit in λ−1, are (b1) −0.0606 ( t/λ=0.1), (b2)−0.0869 ( t/λ=2), and (b3) −0.0914 ( t/λ=25). scaling behavior put forward in Eq. ( 12); that is, for μi=μ, Gp(x,t;μ,u)≈λ−1Gp(x/λ,t/λ,uλ). (22) Indeed, the data for λGp(x,t)v e r s u s t/λat fixed x/λ=1 appear to converge toward an asymptotic curve exhibitinga nontrivial behavior when increasing λ. Notice also that the long-time limit of such a curve for G pis different from zero, thus signaling the approach to a nontrivial stationarystate. Moreover, as expected (see the end of Sec. IV), such a convergence is characterized by O(λ −1) corrections, which is visible from the bottom panels of Fig. 2, which focus on three different values of t/λ: 0.1 [Fig. 2(b1) ], 2 [Fig. 2(b2) ], and 25 [Fig. 2(b3) ]. We have extensively verified that the same scaling behavior as in Eq. ( 22) can be observed for any value of the ratio x/λ and of the rescaled dissipation strength uλ. Furthermore, an analogous dynamic scaling is developed by the correlationfunction G cdefined in Eq. ( 21b), as explicitly shown in Fig. 3. For the sake of clarity in our presentation, in Fig. 3we adopt the same framework and conventions as in Fig. 2and use the same set of parameters, with the exception of x/λ, which is set equal to 2. Also in that situation we notice a remarkable agreement with the predicted O(λ−1) scaling for finite-λcorrections (see bottom panels of Fig. 3). As a further check of the dynamic scaling theory, Fig. 4 displays results for the density-density correlation functionG n(x,t)[ s e eE q .( 21c)] for two different values of rescaled dissipation strength uλ. Again, they support the dynamic scaling behavior put forward in Sec. III[see, in particular, Eq. ( 12)], taking into account that the renormalization-group dimension of the density operator is now yn=1. Therefore, for ¯μi=¯μ, Gn(x,t;μ,u)≈λ−2Gn(x/λ,t/λ,uλ); (23) 174303-5DA VIDE ROSSINI AND ETTORE VICARI PHYSICAL REVIEW B 100, 174303 (2019) -0.400.40.81.2102λ2Gn λ = 10 λ = 20 λ = 40 λ = 80 10-1100101 t / λ-0.400.40.81.2102λ2Gn0 0.05 0.11/λ0.1280.130.132102λ2 Gn 0 0.05 0.11/λ0.1550.15750.16103λ2 Gnuλ = 1 uλ = 2(a) (b) FIG. 4. The correlation function Gn(x,t)i nE q .( 21c), multiplied byλ2, versus the rescaled time t/λ. (a) is with dissipation strength uλ=1, while (b) is with uλ=2. Here we fix x/λ=1, ¯μi=¯μ and take ¯ μ=−1/λ. The horizontal dashed line indicates the zero value, which is reached asymptotically at long times, and is plottedas a guideline. The two insets show the convergence with λof the curves at t/λ=2 (arrow in the two main panels). The rather complex behavior of the large- λconvergence reflects the oscillatory behavior of the curves as a function of t/λ, which becomes more evident on the scale of the main frames for 0 .3/lessorsimilart/λ/lessorsimilar1. The dashed lines of the insets are drawn to guide the eyes (they suggest that the large- λ limit approached by the data is ≈0.127 for uλ=1a n d≈0.155 for uλ=2). that is, the product λ2Gn(x,t) is expected to approach a nontrivial asymptotic large- λscaling behavior in terms of the scaling variables x/λ,t/λ, and uλ, which eventually vanishes fort/λ→∞ . The data in the two insets, which focus on a fixed rescaled time t/λ=2, reveal that the 1 /λcorrections to the scaling are, for this observable, superimposed to fluctua-tions which witness the complex oscillatory behavior of thescaling functions (compare the different scales on the yaxes of the main panels and of the insets). Analogous scaling results can be obtained when approach- ing the critical point from the right side: μ>μ c; thus, ¯ μ> 0, as shown by the curves reported in Fig. 5for the three considered correlation functions. Comparing the results forthe asymptotic stationary states with those for μ<μ cat equal values of |¯μ|[see, in particular, Figs. 2(a),3(a), and 4], we no- tice that while the absolute values of the rescaled correlationfunctions are the same when approaching the critical pointfrom either side, the respective signs for the correlation G c are exchanged. Let us now discuss in more detail the behavior of stationary states, which are approached in the large-time limit. Figure 6 shows that the quantities λGp,sandλGc,s, obtained by taking thet→∞ limit of GpandGc, approach an asymptotic large- λscaling form as a function of x/λand uλ[cf. Eq. ( 13)].-0.2-0.100.10.20.3λ Gpλ = 10 λ = 20 λ = 40 λ = 80 -0.2-0.100.10.2λ Gc 10-1100101 t / λ00.511.5102λ2Gn(a) (b)(c) FIG. 5. Results for (a) λGp(x,t), (b) λGc(x,t), and (c)λ2Gn(x,t) versus the rescaled time t/λ. The various system parameters are set as in Fig. 4(a), with the exception of the Hamiltonian control parameter ¯ μi=¯μ, with |¯μ|=λ−1, which here has been chosen to be positive (i.e., approaching the critical point from the right side: μ>μ c). Indeed, they show that Gp/c,s(x;μi,μ,u)≈λ−1/Gamma1p/c(x/λ,uλ), (24) as put forward in Eq. ( 13). Moreover, they appear to decay exponentially for sufficiently large distances, i.e., λGp/c,s∼exp[−x/ξs]=exp/bracketleftbigg −x/λ ξs/λ/bracketrightbigg , (25) where ξsdefines a correlation length which depends on the dissipation strength [see Eq. ( 14)]. Therefore, Eq. ( 25)a l s o implies the scaling behavior ξs∼λ≡|¯μ|−ν, (26) as predicted by the scaling equation ( 15). All the results presented so far have been obtained for μi=μ, thus in the absence of any Hamiltonian quench. Relaxing this assumption, we found that the stationary valuesapproached by the various correlators in the long-time limitare independent of the initial condition of the protocol, inparticular of the value of μ i/negationslash=μ, and thus obey the same asymptotic scaling behaviors discussed above. This is explic-itly shown in Fig. 7for the correlation function G p.T h i s fact is consistent with the observation that, for our choice ofdissipators, the asymptotic stationary states are indeed unique. We conclude our analysis by mentioning that completely analogous results can be obtained in the case the dissipativemechanism is related to a uniform local pumping, associatedwith the Lindblad operator ˆL p,jin Eq. ( 20) (not shown). 174303-6SCALING BEHA VIOR OF THE STATIONARY STATES … PHYSICAL REVIEW B 100, 174303 (2019) 0.050.10.2λ Gp,s λ = 20 λ = 40 λ = 80 λ = +∞ 0 0.5 1 1.5 2 x / λ0.050.10.20.3λ Gc,suλ = 1 uλ = 2(a) (b) FIG. 6. Results for (a) λGp,s(x)a n d( b ) λGc,s(x)i nt h el a r g e - time limit (checked numerically with great accuracy) as a function of the rescaled variable x/λ. Smaller and partially filled symbols stand for three specific values of λ, according to the legend, while fully filled symbols denote the values extrapolated for λ→+ ∞ , that is, by extrapolating to zero the data at finite λ−1[see, e.g., Figs. 2(b1) –2(b3) and3(b1) –3(b3) ]. Black data sets and circles are for a dissipation strength uλ=1, while red data sets and squares are for uλ=2. Here we always choose ¯ μ< 0. Straight lines are exponential fits Gs∼e−(x/λ)/(ξ/λ)of the extrapolated numerical data forλ→+ ∞ . For both kinds of correlation functions, the obtained decay rates are ξ/λ=1.789 (for uλ=1) and ξ/λ=1.414 (for uλ=2), with a relative discrepancy smaller than 10−4, in support of the scaling law ( 15). VI. SUMMARY AND CONCLUSION We have investigated the effects of dissipation on the dy- namics of many-body systems close to a continuous quantumphase transition, arising from the interaction with the envi-ronment, as sketched in Fig. 1. The latter is modeled through a class of dissipative mechanisms that can be effectivelydescribed by Lindblad equations for the density matrix of thesystem [ 6,7], with local and homogenous Lindblad operators, such as those reported in Eqs. ( 9)–(11). This framework is of experimental interest; indeed, the conditions for its validityare typically realized in quantum optical and c-QED imple-mentations [ 1,2,8]. We have analyzed how homogeneous dissipative mecha- nisms change the scaling laws of closed systems at quantumtransitions. For this purpose, we have considered a relativelysimple dynamic protocol: the quantum many-body system isinitialized at t=0 into the ground state of the Hamiltonian ˆH(μ i) for a given parameter μi; then for t>0 the system evolves according to the Lindblad equation ( 9), where the coherent driving is provided by the Hamiltonian ˆH(μ), and10-1100101 t / λ-0.4-0.200.20.4λ Gp μi = -0.01 μi = +0.01 μi = -0.025 μi = +0.025 μi = -0.1 μi = +0.110 15 20t / λ-2×10-5-1×10-501×10-5δ (λ Gp) FIG. 7. The correlation function Gp(x,t)i nE q .( 21a), mul- tiplied by λ, versus the rescaled time t/λ.H e r ew efi x x/λ=1,uλ=1, and ¯ μ=−0.01 (corresponding to λ=100). The various curves correspond to different initial conditions, determined by the ground states of the Kitaev Hamiltonian with changing ¯ μi, being either negative (i.e., μi<μ c) or positive ( μi>μ c; see the legend). The inset provides a magnification of the numerical data for 10/lessorequalslantt/λ/lessorequalslant20 (box in the main frame) after subtracting the asymp- totic value for t/λ→∞ . the dissipation arising from the system-bath interaction is effectively described by the dissipator D[ρ] with a fixed coupling strength u. The large-time stationary state is usually noncritical, even when the Hamiltonian parameters are critical. However, weidentified a low-dissipation regime where the dissipation issufficiently small to compete with the coherent evolutiondriven by the critical Hamiltonian, leading to stationarystates which present critical behaviors depending also on thestrength of the dissipation coupling. The above-mentionedregime is generally realized when the dissipation parameteruscales as the gap /Delta1of the Hamiltonian of the many-body system, i.e., u∼/Delta1. Therefore, it is a low-dissipation regime in that the gap gets suppressed at the quantum transition, as/Delta1∼ξ −z, where ξis the large length scale developed by the critical correlation functions. This reflects the fact that at aquantum transition the perturbation arising from dissipationis always relevant, such as the temperature at equilibrium[5,12,13]. This also means that, when u/greatermuch/Delta1, critical fluc- tuations do not survive to the dissipation. We argue that, under such low-dissipation conditions, open many-body systems develop dynamic scaling laws,which apply to the time evolution described by the Lind-blad equation ( 9), in particular to the stationary states aris- ing in the large-time limit (see Sec. III). Analogous to the scaling laws of closed systems at quantum transitions,the dynamic scaling behavior in the presence of dissipa-tion is expected to be largely independent of the micro-scopic details of both coherent and dissipative drivings,like for the critical behavior of closed systems, whichdepends only on the universality class of the quantum 174303-7DA VIDE ROSSINI AND ETTORE VICARI PHYSICAL REVIEW B 100, 174303 (2019) transition. Further investigation is called for to assess the actual extension of the universality of the dynamic scalingfunctions with respect to the properties of the dissipativemechanisms. The dynamic scaling laws obtained in this paper apply to complementary regimes with respect to those recentlyreported in Ref. [ 9], where finite systems of linear size Land dynamic behavior for relatively small times t∼L zwere con- sidered, thus not including the large-time stationary regime.On the other hand, here we focused on infinite-size systems,for which we managed to derive scaling laws extending to thelarge-time limit of the evolution described by the Lindbladequation, thus valid for the corresponding stationary states. The dynamic scaling scenario has been checked within fermion wires [see Eq. ( 17)] in the presence of dissipation due to local incoherent pumping and decay, which are describedby the Lindblad operators reported in Eq. ( 20). Our numerical analysis supports our general, phenomenological, dynamicscaling theory addressing the competition between coherentdynamics and dissipation at a continuous quantum transition.Further checks of the dynamic scaling behaviors may turn out to be interesting for other many-body systems at quan-tum transitions, possibly belonging to different universalityclasses, and /or dissipation mechanisms, including nonlocal ones [ 29–31]. The arguments leading to this scenario are quite gen- eral. We believe that analogous phenomena should developin any homogeneous d-dimensional many-body system at a continuous quantum transition whose Markovian interactionwith the environment can be described by local or extendeddissipators within the Lindblad equation ( 9). These arguments should also apply to non-Markovian system-bath couplings[32] (not described by Lindblad equations), replacing uwith the parameter controlling the decay rate. We finally mention that some experimental breakthroughs were recently achieved in the control of dissipative quan-tum many-body dynamics through different platforms, suchas Rydberg atoms and c-QED technology. Quantum criticalbehaviors in such an out-of-equilibrium context were reportedin Refs. [ 19,33,34]. [1] A. A. Houck, H. E. Türeci, and J. Koch, On-chip quantum simulation with superconducting circuits, Nat. Phys. 8,292 (2012 ). [2] M. Müller, S. Diehl, G. Pupillo, and P. Zoller, Engineered open systems and quantum simulations with atoms and ions, Adv. At. Mol. Opt. Phys. 61,1(2012 ). [3] I. Carusotto and C. Ciuti, Quantum fluids of light, Rev. Mod. Phys. 85,299(2013 ). [4] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86,1391 (2014 ). [5] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999). [6] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002). 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PhysRevB.92.180509.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 92, 180509(R) (2015) Magnetization of underdoped YBa 2Cu 3Oyabove the irreversibility field Jing Fei Yu,1,*B. J. Ramshaw,2I. Kokanovi ´c,3,4K. A. Modic,2N. Harrison,2James Day,5Ruixing Liang,5,6 W. N. Hardy,5,6D. A. Bonn,5,6A. McCollam,7S. R. Julian,1,6and J. R. Cooper3 1Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada 2Los Alamos National Laboratory, Mail Stop E536, Los Alamos, New Mexico 87545, USA 3Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom 4Department of Physics, Faculty of Science, University of Zagreb, P .O. Box 331, Zagreb, Croatia 5Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada 6Canadian Institute for Advanced Research, 180 Dundas St. W, Toronto, Ontario M5S 1Z8, Canada 7High Field Magnet Laboratory (HFML-EMFL), Radboud University, 6525 ED Nijmegen, The Netherlands (Received 18 September 2015; revised manuscript received 1 November 2015; published 23 November 2015) Torque magnetization measurements on YBa 2Cu3Oy(YBCO) at doping y=6.67 (p=0.12), in dc fields ( B) up to 33 T and temperatures down to 4.5 K, show that weak diamagnetism persists above the extrapolatedirreversibility field H irr(T=0)≈24 T. The differential susceptibility dM/dB , however, is more rapidly suppressed for B/greaterorsimilar16 T than expected from the properties of the low field superconducting state, and saturates at a low value for fields B/greaterorsimilar24 T. In addition, torque measurements on a p=0.11 YBCO crystal in pulsed field up to 65 T and temperatures down to 8 K show similar behavior, with no additional features at higher fields. We offertwo candidate scenarios to explain these observations: (a) superconductivity survives but is heavily suppressed athigh field by competition with charge-density-wave (CDW) order; (b) static superconductivity disappears near24 T and is followed by a region of fluctuating superconductivity, which causes dM/dB to saturate at high field. The diamagnetic signal observed above 50 T for the p=0.11 crystal at 40 K and below may be caused by changes in the normal state susceptibility rather than bulk or fluctuating superconductivity. There will beorbital (Landau) diamagnetism from electron pockets and possibly a reduction in spin susceptibility caused bythe stronger three-dimensional ordered CDW. DOI: 10.1103/PhysRevB.92.180509 PACS number(s): 74 .72.Gh,74.25.Bt,74.25.Ha,74.25.Op The possible existence of bulk superconductivity as T→ 0 K above the irreversibility field ( Hirr)[1] in the cuprates has been a long-standing question. Not only is this problemimportant for our understanding of the cuprates, but alsobecause there is still debate [ 2,3] about whether Cooper pairs persist in the region of the field-temperature plane wherequantum oscillations are seen [ 4]. Many experimental efforts have been made to address this issue [ 5–8]. Diamagnetism has consistently been reported using torque magnetometry at high fields in many familiesof cuprates and it is argued that this observation shows thepersistence of Cooper pairs above H irr[5]. For YBa 2Cu3Oy (YBCO), resistivity measurements have established Hirr(T= 0) to be below 30 T for fields along the caxis for dopings between p=0.11 (OII) and p=0.12 (OVIII) [ 9]. Moreover, x-ray [ 10–12], NMR [ 13], and sound velocity measurements [14] have demonstrated the existence of static charge-density- wave (CDW) order that competes with superconductivity:Reference [ 12] shows a distinct long-range three-dimensional (3D) order that emerges at high field and continues to grow at28 T for an OVIII crystal, consistent with that first observed inNMR studies [ 13]. The CDW is strongest and the suppression ofH c2is largest at p=0.125 for YBCO [ 11,15]. Recent thermal conductivity measurements by Grisson- nanche et al. [7] show a sharp transition precisely at the extrapolated Hirr(T=0)/similarequal22 T for OII YBCO. They have interpreted this feature (henceforth referred to as HK)a sa *jfeiyu@physics.utoronto.casignature of Hc2, arguing that the end of the rapid rise in thermal conductivity at 22 T reflects a corresponding increasein the mean free path as a result of the sudden disappearance ofvortices at H c2. On a crystal with same doping, Marcenat et al. [16] show that the specific heat saturates at a field Hcp.Hcp(T) lies above Hirr(T), but they extrapolate to the same value at T=0K[ 16]. In contrast, torque measurements by Yu et al. [6] on a crystal with the same doping suggested diamagnetism persisting to fields much higher than HK. The debate is thus still open. To resolve this problem, we conducted torque magnetom- etry measurements of magnetization ( M)o nt w o p=0.12 (OVIII, Tc=65 K) crystals in dc fields and one p=0.11 (OII, Tc=60 K) crystal in pulsed fields. The crystals were mounted on piezoresistive cantilevers and placed on a rotating platform,with the CuO 2planes parallel to the surface of the lever. dc field sweeps, first from 0 to 10 T and later from 0 to 33 T,were performed with the caxis of the OVIII crystal at a small angleθfrom the field. The magnetoresistance of the levers was eliminated by subtracting data from the complementary angle(−θ) (see Supplemental Material for raw data [ 17]). Similar procedures were used for the OII crystal in pulsed magneticfields up to 65 T. For strongly anisotropic superconductors,where out-of-plane screening currents can be neglected, thetorque τper unit volume Vat an angle θfrom field Bis given by [ 18]τ/V= 1 2χD(T)B2sin 2θ+McBsinθ.H e r e χD(T)=χc(T)−χab(T) is the anisotropic susceptibility in the normal state and Mcis the magnetization from in-plane screening currents for a field of Bcosθalong the caxis. This is a good approximation when Mc/greatermuchχDBor when the superconducting gap and Mcare both small. 1098-0121/2015/92(18)/180509(5) 180509-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS JING FEI YU et al. PHYSICAL REVIEW B 92, 180509(R) (2015) 0 50 100 150 200 250 300810121416 Temperature (K)χD (A/m/T) FIG. 1. (Color online) Black dots: high temperature anisotropic susceptibility χD(T) of the OVIII crystal at 10 T. Blue solid line: fit to this data above 120 K using Eq. ( 1). The parameters A= 11.09 A/m/T,TF=680 K are taken from Ref. [ 19], while the fit givesχVV=5.84 A/m/T,T∗=330 K and χR(0)=1.26 A/m/T; Red dashed line: Linear fit with χ(T)=1.22×10−2×(T+948), following Ref. [ 6] but with different parameters. Note that 1 × 10−4emu/mol=9.73 A/m/T. A key challenge with magnetization measurements in the cuprates is the separation of the normal state from the super-conducting contributions, because superconducting fluctua-tions are thought to contribute to χ(T) even at temperatures far above T c[18], while χnormalis temperature dependent to well below Tc. We follow the procedure outlined in Refs. [ 18,19] and interpret χD(T) in the normal state of underdoped YBCO as arising from the pseudogap and g-factor anisotropy, plus a superconducting fluctuation term that sets in below 120 K.Neglecting isotropic Curie and core susceptibility terms, whichdo not contribute to τ, the total normal state contribution to χ D(T)i s[19] χnormal D (T)=χPG D(T)+χVV D+χR D(T), (1) where χVV Dis the T-independent Van Vleck susceptibility, χPG D(T) is the pseudogap contribution assuming a V-shaped density of states [ 20], and χR D(T) is thought to arise from an electron pocket or Fermi arcs in the region 0 .0184<p< 0.135. Specifically, χPG D=A{1−y−1ln[cosh( y)]}, where A=N0μ2 B,y=Eg/2kBT,Eg=kBT∗andT∗is the pseu- dogap temperature, and χR D(T)=χR(0)[1−exp(−TF/T)] where TFis the Fermi temperature. The fit is shown in Fig. 1, along with a linear model for the normal state χused in Ref. [ 6]. Both fits agree well with the data for T/greaterorequalslant120 K. Our background is almost twice as small as that of the linearfit atT=0 K. Subtraction of the background magnetization using this nonlinear model should thus give a significantlyweaker diamagnetic signal at T→0 K than the linear model would (by about 160 A /ma t3 0T ) . In Figs. 2(a)and3(a),w es h o w M cvsBzcurves at selected temperatures for the OVIII and OII crystals, obtained bysubtracting M BG=χBGBz, where χBGis the blue line in Fig. 1, andBzis the field projected along the crystalline c axis. For the OVIII crystal, at T=103 K, we see that Mc is almost zero. At 58 K, just below Tc, we see significant diamagnetism that gradually tends to about −130 A /ma t high field. Figure 2(a) shows that the crystal remains weakly0 5 10 15 20 25 30−3000−2500−2000−1500−1000−5000500 Bz (T)Mc (A/m)4.5K103K 58K 35K 20.5K 16K10K7K27K MBG(T=4.5) 12 14 16 18 20 22 24 26 28 30 32050100150200250300 Bz (T)dM/dB (A/m/T) 27K20.5K10K 4.5K 103K16Kκ = 41 κ = 50 κ = 150(a) (b) FIG. 2. (Color online) (a) Magnetization ( Mc) of the OVIII crystal vs Bz, the field parallel to the caxis. Here Mc(T,H )= Mobs(T,H )−MBG(T,H ), where MBG=χDBzandχDis the blue line in Fig. 1. Dashed line: MBGat 4.5 K. Diamagnetism is present even at our highest field of 33 T. (b) Differential susceptibility dM/dB of the OVIII crystal vs Bzat selected temperatures. The lines are guides to the eye. We call the characteristic field at which dM/dB departs from linearity Hd. Red: calculated mean field dM/dB near Hc2withκ=50, with κ=41 (purple) and with κ=150 (blue). diamagnetic down to 4.5 K in fields up to 33 T. Similar behavior was found for the OII crystal in pulsed fields. Asshown in Fig. 3(a),M cis still diamagnetic at the highest field Bz=63 T, but has a small value – about −90 A/ma t8K . Our results differ from those of Yu et al. [6]: Our normal state susceptibility is larger than theirs by approximately 8A/m/T, and after background subtraction, at 10 K and 20 T we find M cto be up to four times larger (see Supplemental Material for details on the calibration procedure); at 30 T wefind about −200 A/m for OII and OVIII rather than their value of−75 A/m. Our estimated uncertainty in χ D(0) corresponds to±32 A/mi nMcat 33 T and ±61 A/ma t6 3T . Although the weak diamagnetic signal persists to higher fields, we are able to see a signature in our differentialsusceptibility dM/dB at fields comparable to H K(22 T) found by thermal conductivity [ 7]. In each curve of Figs. 2(b) and3(b),dM/dB decreases linearly, up to a field we call Hd(T), before saturating to a small but nonzero value. At the lowest temperatures for both OVIII and OII crystals, we findH d≈24 T, which is close to the extrapolated Hirr(T=0). This is consistent with the feature at HKfound by thermal 180509-2RAPID COMMUNICATIONS MAGNETIZATION OF UNDERDOPED YBa 2Cu3O . . . PHYSICAL REVIEW B 92, 180509(R) (2015) 0 10 20 30 40 50 60 70−3000−2500−2000−1500−1000−5000 Bz(T)Mc (A/m) 8 K 10 K 15 K 20 K 30 K 40 K 50 K 60 K 80 K40 50 60−200−150−100−50050 10 20 30 40 50 60 70050100150200 Bz(T)dM/dB(A/m/T) 8 K 10 K 15 K 20 K 30 K 40 K 50 K 60 K 80 K(dM/dB)MF at Hc2 with κ = 50(a) (b) FIG. 3. (Color online) (a) Magnetization ( Mc)o ft h eO I Ic r y s t a l measured in pulsed magnetic field up to Bz=63 T, where Mc= Mobs−MBG,MBG=χDBz,a n dχDis the blue line in Fig. 1.F o r clarity only the falling-field sweeps are shown. Diamagnetism is present though extremely weak at high field (inset). The small offset inMcbetween the T/lessorequalslant40 K and T/greaterorequalslant50 K curves may be due to the transition to long-range CDW order near 40 K in high fields as observed in both sound velocity [ 14]a n dN M R[ 13]. (b) Differential susceptibility for the OII crystal in pulsed field. dM/dB is seen to be small and constant up to the highest field of 63 T. Blue: calculated mean field dM/dB nearHc2withκ=50. conductivity [ 7], though unlike HK,Hddoes not correspond to a sharp transition. Hdvaries very little with temperature forT< 10 K, a result that is consistent with the findings of Ref. [ 7], though the Tdependence at high temperatures is not consistent with that found by Refs. [ 6,16]. Surprisingly, we do not observe in any of our crystals the broad peak in dM/dB reported by Ref. [ 6]. In highly anisotropic type-II superconductors, the magne- tization calculated using mean field (MF) Ginzburg-Landau(GL) theory for an s-wave superconductor, which we use in the absence of a d-wave theory, yields logarithmic behavior at low field (in cgs units), −4πM=αφ 0/(8πλ2)l n (βHc2/H) for 0.02<H / H c2<0.3, where αandβare numbers of order 1,φ0is the flux quantum for Cooper pairs, and λis the London penetration depth [ 21].μSR at low fields has shown a√ H0 10 20 30 40 50 60 70051015202530 Temperature (K)Bz (T)fit to OII Hirr OII H irr fit to OVIII Hirr OVIII Hirr OVIII Hd OII Hd OVIII GL Hc2 FIG. 4. (Color online) Hdfor both OII and OVIII crystals show similar temperature dependencies. Exponential fits to Hirrof OII [23.2e x p (−T/13.5)] and OVIII [23 .7e x p (−T/20.5)] give extrap- olated values Hirr(0)=23.2 and 23.7 T. These values are close to the low temperature Hdfor both crystals. Note that Hc2from GL fits (see main text) connects smoothly to Hd. field dependence [ 22]f o rλ(T=0), but results of tunneling experiments on Bi-2212 imply thermally induced pair breakingnear the nodes [ 23], indicating a weaker field dependence at higher T. Thus, for simplicity, we assume a negligible field dependence of λ.W ea l s oa s s u m e[ 21]α=0.77 and β=1.44 for 0.02<H / H c2<0.3, in reasonable agreement with later works [ 24,25], and we fit the low field magnetization and obtain an estimate of Hc2(T), shown in Fig. 4. Since our GL values of Hc2join smoothly to Hd, it is possible to interpret Hdas the low temperature GL-type Hc2. When H/H c2>0.3, and again using cgs units for an s-wave superconductor, the magnetization is expected to obey 4 πM=(H−Hc2)/[(2κ2−1)βA], where κis the GL parameter and βA=1.16 is the Abrikosov parameter [ 26,27]. Figures 2(b) and 3(b) show that for B> 28 T, dM/dB has the mean field property of saturating toward a constantvalue, but this is very small and requires κ/similarequal150, a value far greater than κ=50 given in Ref. [ 7]. This means that our high field dM/dB is nearly ten times smaller than would be expected. This may be due to the field dependentcharge-density-wave (CDW) order within the vortex liquidregion [ 11,12]. The CDW competes with superconductivity and is partially suppressed at low field. As increasing fieldsuppresses superconductivity, the CDW order is graduallyrestored [ 14]. The presence of a relatively strong CDW would increase λand thus increase κ, as illustrated in Fig. 5. A linear region in M c(B) can also be seen in Fig. 2(a),f o rT=20 K andT=16 K and B/lessorequalslant17 T, with κ=41, and in Fig. 3(a),f o r T=20 K and B/lessorequalslant17 T, with κ=50. These linear regions are not present above 20 K, where Mc(B) is likely to be smeared out by thermal fluctuations. As shown in Fig. 5,f o rt h eO V I I I crystal, the low field Mcextrapolates to zero around 24 T, consistent with our GL-type Hc2. To summarize, clear linear regions, with slopes corresponding to the expected values ofκ, have been observed for both doping levels below 20 K. The value of H c2(0)≈Hd≈24 T obtained from these GL analyses may refer to a low field, unreconstructed Fermi 180509-3RAPID COMMUNICATIONS JING FEI YU et al. PHYSICAL REVIEW B 92, 180509(R) (2015) 0 5 10 15 20 25 30−3000−2500−2000−1500−1000−5000 Bz(T)Magnetization (A/m)κ = 145 κ = 41T = 16K3D CDW sets in above 15 T κ increases FIG. 5. (Color online) Magnetization data of the OVIII crystal at 16 K. The blue dashed line shows the MF behavior near Hc2for an s-wave superconductor with κ=41. The stronger (3D) CDW sets in above 15 T for OVIII YBCO. At higher fields the data are consistent withκ=145 (solid line). surface. For fields greater than 24 T, we may be observing MF behavior of weak superconductivity arising from thesmall electron pockets [ 4,28] resulting from the appearance of CDW order. The GL-type theory we applied assumes s-wave superconductivity and we cannot rule out possible d-wave effects on the determination of H c2. An obvious possibility is the V olovik effect whereby the Cooper pairs near the nodeson the Fermi surface are broken up, and consequently, λand κwould increase. Alternatively, the diamagnetism that we observe above 24 T could be caused by superconducting fluctuations. The OIIdata in the inset of Fig. 3(a) show that it is ∼−100 A /m between 35 and 63 T. This is five times smaller and fallsmore quickly with field than predicted by theory [ 29]f o ra two-dimensional s-wave superconductor at low temperatures and high fields. This is a robust statement because in theclean limit all parameters in the theoretical expression [ 29] forM c(B) above Hc2are known. Nernst data [ 30]f o rO V I I I crystal show saturation near 30 T to the negative value expectedfor an electron pocket. This does not necessarily rule out bulksuperconductivity above 30 T because in the presence of aCDW, the vortex core entropy – which dominates the Nernsteffect – could be reduced. However, at a qualitative level,the Nernst data between 24 and 30 T may be more consistentwith superconducting fluctuations. Since torque magnetizationis sensitive to superconducting fluctuations, while thermalconductivity sees only the normal quasiparticles which arethe only source of entropy, this may explain why we do notobserve the sharp transition at H Kseen in Ref. [ 7]. Finally, the diamagnetism of −90 A/m observed at 63 T might arise from orbital (Landau) diamagnetism of the electronpockets [ 31] possibly combined with a suppression of spin susceptibility [ 32] associated with the stronger (3D) CDWorder that sets in above 15 T [ 12]. The change required would be 1.36 A/m/Ti nχ D(0). This is consistent with the significant decrease in diamagnetism between 40 and 50 K shown in theinset of Fig. 3(a), the region where the 3D CDW seen at high fields goes away [ 12]. The above discussion highlights the importance of compet- ing CDW and superconductivity instabilities [ 11,33]. Little is known about the size of the CDW energy gap, or theMF behavior expected for a d-wave superconductor just below H c2asT→0 K. Therefore the linear Hdependence ofdM/dB we observe below Hdmight be a fundamental property of a d-wave superconductor. In other words, because of V olovik-type pair breaking effects, the MF transition atH c2could have a discontinuity in d2M/dB2, rather than in dM/dB , which is the standard MF result for the second order transition in a conventional s-wave superconductor. In summary, we observe diamagnetism in OVIII YBCO at fields up to 33 T and OII YBCO at fields up to 65 Tusing torque magnetometry. The analysis uses a differentmodel for the high temperature normal state susceptibility thatgives a smaller correction at low temperature compared withearlier models [ 6]. We also find that dM/dB departs from a linear lower field behavior at fields H d≈Hirr(0)≈24 T, and approaches a constant value at higher fields. We proposetwo candidate scenarios: a competing order scenario where afully fledged CDW at high field mostly suppresses the super- conductivity so that the diamagnetism at high field could be attributed to bulk superconductivity; or a fluctuation picture inwhich for H>H d, the system crosses over to superconducting fluctuation behavior. The diamagnetism at 65 T for the OIIcrystal could arise from the orbital susceptibility of carrierpockets and a reduction in spin susceptibility associated withthe stronger 3D CDW order. It would be of interest to developd-wave expressions for the MF magnetization and for the fluctuation contribution in the low temperature, high fieldregime, for comparison with our data. This could settle thedebate over the existence of the high field vortex liquid region. We thank G. Grissonnanche for useful discussions. This work was generously supported by NSERC under Grant No.RGPGP 170825-13 and No. RGPIN-2014-04554 and CIFARof Canada, Canada Research Chair, EPSRC (UK) under GrantNo. EP/K016709/1, Croatian Science Foundation project (No.6216), and the Croatian Research Council, MZOS NEWFEL-PRO project No. 19. We thank HFML-RU, a member of theEuropean Magnetic Field Laboratory. The work at LANL wasfunded by the Department of Energy Basic Energy Sciencesprogram “Science at 100 T.” The NHMFL facility is fundedby the Department of Energy, the State of Florida, and theNational Science Foundation (NSF) Cooperative AgreementNo. DMR-1157490. [1] For ease of comparison with Refs. 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PhysRevB.83.195120.pdf

PHYSICAL REVIEW B 83, 195120 (2011) Ultrafast insulator-metal phase transition in VO 2studied by multiterahertz spectroscopy A. Pashkin,1C. K ¨ubler,1H. Ehrke,1,*R. Lopez,2,3A. Halabica,3R. F. Haglund Jr.,3R. Huber,1and A. Leitenstorfer1 1Department of Physics and Center for Applied Photonics, University of Konstanz, D-78457 Konstanz, Germany 2Department of Physics and Astronomy and Institute of Advanced Materials, Nanoscience and Technology, University of North Carolina, Chapel Hill, North Carolina 27599, USA 3Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA (Received 22 December 2010; revised manuscript received 4 April 2011; published 12 May 2011) The ultrafast photoinduced insulator-metal transition in VO 2is studied at different temperatures and excitation fluences using multi-THz probe pulses. The spectrally resolved midinfrared response allows us to trace separatelythe dynamics of lattice and electronic degrees of freedom with a time resolution of 40 fs. The critical fluence of theoptical pump pulse, which drives the system into a long-lived metallic state, is found to increase with decreasingtemperature. Under all measurement conditions, we observe a modulation of the eigenfrequencies of the opticalphonon modes induced by their anharmonic coupling to the coherent wave-packet motion of V-V dimers at6.1 THz. Furthermore, we find a weak quadratic coupling of the electronic response to the coherent dimeroscillation resulting in a modulation of the electronic conductivity at twice the frequency of the wave-packetmotion. The findings are discussed in the framework of a qualitative model based on an approximation of localphotoexcitation of the vanadium dimers from the insulating state. DOI: 10.1103/PhysRevB.83.195120 PACS number(s): 71 .30.+h, 72.80.Ga, 78 .47.J− I. INTRODUCTION The insulator-metal transition in transition-metal oxides, first reported by Morin,1is characterized by a sharp decrease in resistivity by several orders of magnitude when the compoundis heated above a critical temperature T c. Later, a large number of compounds were found to exhibit this type ofphase transition. Extensive experimental information has beencollected and the physical mechanisms responsible for thetransition have been relatively well understood in many cases. 2 Nevertheless, the origin of the phase transformation is stillunder debate in some systems. VO 2is a classical example that has attracted considerable interest for fundamental reasons aswell as for possible applications, since its critical temperatureT c=340 K (67◦C) is located in the vicinity of ambient conditions. The change of the electronic properties in VO 2is accom- panied by a remarkable modification of the structure fromthe highly symmetric rutile to the low-temperature monoclinicphase where tilted V-V pairs are formed. 1The phase transition is of first order as confirmed by the divergence of the heatcapacity and the presence of latent heat. 3The high-temperature rutile phase of VO 2(space group D14 4h) can be visualized in terms of a body-centered-tetragonal lattice formed by thevanadium atoms as depicted in Fig. 1(a), with each metal atom surrounded by an oxygen octahedron. 4Each octahedron has virtually cubic symmetry with very small orthorhombicdistortions. Symmetry lowering during the transition into themonoclinic phase (space group C 5 2h) leads to the doubling of the unit cell and the formation of V-V pairs5,6as illustrated in Fig. 1(a). The soft mode of the rutile lattice corresponding to this structural transformation is located at the point R= (1/2,0,1/2) of the Brillouin zone.7–9Correspondingly, the monoclinic phase can be mapped onto the rutile lattice bytwo normal vibrational modes of vanadium atoms located atthe/Gamma1point as a result of the Brillouin zone folding in the monoclinic phase. 10Working out the exact details of the electronic band structure of VO 2poses a true challenge to modern many- body theories. In particular, the correct prediction of theinsulating gap remains a difficult task because the hierarchyof the microscopic degrees of freedom (spin, charge, orbital,and lattice) contributing to the phase transition is not clear.However, the phenomenological band scheme proposed byGoodenough 11,12provides a quite realistic description by using simple arguments based on molecular orbitals. It serves asa starting point for the discussion of the electronic propertiesof VO 2. In metallic VO 2, hybridization between the oxygen 2 pand vanadium 3 dorbitals leads to σ- andπ-type overlap.13The filledσandπstates will be primarily of O 2 pcharacter, whereas the corresponding antibonding bands are dominatedby the V 3 dorbitals [Fig. 1(b)]. The overlap of vanadium orbitals along the tetragonal c Raxis parallel to the vanadium chains results in the formation of a narrow band commonlylabeled d ||. The Fermi level is located within all of the three t2gbands, which are still partially overlapping despite being no longer degenerate, as shown in Fig. 1(b). Upon entering the low-temperature insulating phase, the experimentally observed structural changes bring the vana-dium atoms closer to the apex of the barely modified oxygencage, thus increasing the hybridization. Now only the d ||band is occupied and the dimensionality of the electronic systemis effectively reduced to unity. Moreover, photoemissionexperiments report a splitting of the d ||band14,15by 2.5 eV that eventuates in the formation of a band gap of 0.6–0.7 eVbetween the lower d ||band and the egband and the transition to the insulating state.16 Several theoretical models have been proposed based on Peierls12,17and Mott-Hubbard18,19scenarios. The former supposes a lattice instability caused by electron-phononinteraction within a system of reduced dimensionality. Thelatter is based on strong electron-electron correlation effects.Recent ab initio calculations on the basis of cluster dynamical 195120-1 1098-0121/2011/83(19)/195120(9) ©2011 American Physical SocietyA. PASHKIN et al. PHYSICAL REVIEW B 83, 195120 (2011) cR eg ** V3d O2 peg egE-E (eV)f 22 -2 -846 d||(a) (b)TT<c T>Tc cMaM bM d||d|| egaRbR 0 FIG. 1. (Color online) (a) Vanadium sublattice of the rutile (left) and monoclinic (right) crystal structure of VO 2. (b) Schematic band structure according to Refs. 12and15. The red arrow denotes the photoexcitation by an optical pump pulse with an energy of 1.5 eV . mean-field theory20have demonstrated that the interplay of strong electronic interactions and the dimerization ofvanadium atoms in the low-temperature phase play a crucialrole in forming the energy gap. In this scenario, which may bedescribed as a “many-body Peierls insulator,” strong Coulombcorrelations in VO 2lead to the formation of dynamical V-V singlet pairs that are necessary to trigger a Peierls transition.The calculated electronic structure and optical conductivity inboth phases of VO 2agree well with experimental results.15,21,22 The delicate balance of interactions drives transition-metal oxides, such as VO 2, into a critical regime that is ruled by phase competition and that reacts exceedingly sensitively toexternal stimuli. When ultrafast photoexcitation favors one ofthe competing phases via the interaction of a photoexcitedstate with lattice, spin, or charge degrees of freedom, adramatic phase conversion may occur. These phenomena arehighly cooperative so that the structural relaxation processesof the electronic excited states are not independent, as inconventional dilute excitonic or photochemical absorption,but entail a photoinduced phase transformation toward anew lattice structure and electronic order. 23,24This scenario opens the way for a light pulse to induce symmetry breakingfrom a stable phase and so to establish a new long-rangeorder. In this context, ultrafast time-resolved techniques areemerging as a tool to study phase transitions in complexmaterials. 25Time is introduced as an additional parameter. It promotes the possibility to unravel the contributing degreesof freedom and advance our understanding of sophisticatedphysical processes. Evidently, the time-resolved technique ofchoice should be sensitive to the relevant degrees of freedom.Interest in the insulator-metal transition of VO 2was rejuve- nated after Becker et al.26discovered that the metallic state of VO 2may not only be induced thermally but also optically on an ultrafast time scale. Later, x-ray diffraction27and optical28 pump-probe experiments proved the nonthermal character ofthe transition and suggested the limiting time for switching tobe less than 100 fs. A large transient pump-induced increaseof the conductivity in the THz frequency range was recentlyreported by several groups. 29–31Therefore, VO 2has attracted considerable attention due to potential applications such asultrafast control of light in VO 2-based photonic crystals,32 optical switches,33–35and optical storage devices.36Efficient harvesting of this large technological potential demands athorough understanding of the microscopic physical processes. Recently, we reported measurements of the insulator-metal transition in VO 2induced by 12-fs optical pulses and probed by ultrabroadband multi-THz transients.30These experiments made it possible to discriminate spectrally between theexcitation dynamics of electronic and lattice degrees offreedom, revealing their fundamentally different character.Based on these observations, a novel qualitative model ofthe photoinduced insulator-metal transition in VO 2has been suggested. The present paper extends the previous Brief Report by addressing such important aspects as a temperature depen-dence of the photoinduced changes and a detailed insight intothe coherent oscillation dynamics. In this paper, we presenta systematic study of the photoinduced transition at differenttemperatures and excitation fluences. The minimum energyof the pump pulse required to induce the cooperative non-thermal transition into the metallic state is found to becomparable to the thermodynamic energy difference betweenthe monoclinic and rutile phases. This fact indicates anultrafast switching of the lattice structure that instantaneouslyfollows the electronic excitation of vanadium dimers. As acharacteristic fingerprint of this process, a modulation of theeigenfrequencies of the optical phonon modes induced by theiranharmonic coupling to the coherent wave-packet motion ofV-V dimers is observed under all measurement conditions.In addition, a modulation of the electronic response at twicethe frequency of the wave-packet motion gives evidenceof a quadratic coupling to the structural order parameter.The observed phenomena allow a deepened insight into theinterplay of electronic and lattice degrees of freedom duringthe insulator-metal phase transition in VO 2. The qualitative model of the ultrafast phase transition proposed previously isdiscussed in view of the additional results. II. EXPERIMENT The sample is a 120 nm thin film of polycrystalline VO 2grown by pulsed laser deposition on a chemical vapor deposition (CVD) diamond substrate.37A vanadium-metal target is ablated in 250 mTorr of oxygen by a KrF excimer laser(repetition rate 25 Hz) at a wavelength of 248 nm and a fluenceof 4 mJ /cm 2. The as-grown films are nonstoichiometric with an approximate composition close to VO 1.7. Therefore, the film is subsequently oxidized at 450◦C under 7 ×10−3mbar of oxygen pressure. After oxidizing to VO 2, the stoichiometry is verified via Rutherford backscattering and observation of 195120-2ULTRAFAST INSULATOR-METAL PHASE TRANSITION IN ... PHYSICAL REVIEW B 83, 195120 (2011) Cryost atEmitterFilter PolarizerDetector polarization sensitive detectionTi:sapphire laser system tp center= 12 fs, = 800 nm rep. rate: 400 kHz ... 4 MHz pulse energy: up to 0.5 J FIG. 2. (Color online) Schematic of the optical pump–multi-THz probe setup. The output of the amplified Ti:sapphire laser system is split into three branches: (1) The optical pump pulse with variable delayτ; (2) the THz probe pulse with variable delay T; and (3) the gating pulse for field-resolved electro-optic detection. The sample is mounted in a liquid helium cryostat equipped with CVD diamond windows. electrical and optical switching, while crystallinity and phase homogeneity are confirmed by x-ray diffraction. Our multi-THz setup is based on a home-built low-noise Ti:sapphire amplifier system for intense 12-fs light pulsescentered at a photon energy of 1 .55 eV (800 nm wavelength). 38 Pulse energies up to 0.5 μJ are generated at variable repetition rates of up to 4 MHz. The simplified schematic of the opticalpump–multi-THz probe setup is depicted in Fig. 2. Part of the laser output denoted as branch 1 is used for optical pumpingof the sample. The beam in branch 2 generates phase-lockedmulti-THz probe transients by optical rectification in a 50- μm- thick GaSe crystal. 39A pair of parabolic mirrors focuses the THz beam to the spot excited by the pump beam. Themulti-THz transients are focused onto the sample, which isheld at a preset substrate temperature T L. The transmitted radiation is refocused by another set of two parabolic mirrorsonto the 50 μm GaSe electro-optic sensor. 40By varying the delay T of the gating pulse (branch 3) with respect to thetransient, the time evolution of the electric field can be tracedfor selected delay times τof the pump pulse. Fourier analysis of the time-domain data yields two-dimensional (2D) mapsof both absolute amplitude and phase of the pump-inducedtransmission change. The time resolution along the τaxis in our setup is determined by the detector bandwidth and set tobe approximately 40 fs. The real and imaginary parts of therefractive index of the VO 2thin film are extracted from the measured equilibrium transmission and the pump-inducedtransmission change by numerical solution of a correspondingFresnel equation. 41Any other optical constants, e.g., optical conductivity σ(ω,τ) or permittivity /epsilon1(ω,τ), are available from these data through basic electrodynamic relations.42 The spectrally integrated dynamics of the optical conduc- tivity is measured if the delay T is fixed at the maximum of themulti-THz transient and only the delay of the pump pulse τis scanned. Since the total spectral intensity of the probe pulse isproportional to the peak value of the electric field, a measuredpump-induced change of its amplitude characterizes changes3 2 03 3 03 4 03 5 03 6 00.20.40.60.81.0ΔTc = 6.5 KTransmittance Temperature (K)50 60 70 80 90Temperature (oC) 10 15 20 25 40 60 80 10001812 363 K 294 K Energy (meV)σ1 (100 Ω-1cm-1) Frequency (THz) (a) (b) FIG. 3. (Color online) (a) Temperature dependence of the nor- malized transmittance through a VO 2thin film upon heating (red guide to the eye) and cooling (blue guide to the eye). (b) Real part of the optical conductivity of the VO 2sample below and above Tc. in absorption or optical conductivity of a sample in the whole spectral range covered by the multi-THz transient. III. RESULTS A. Equilibrium optical response of the VO 2sample When the sample temperature is varied across the phase transition, the transmission in the THz range shows a pro-nounced hysteresis illustrated in Fig. 3(a). The insulator-metal transition occurs at T L=339.5±1 K upon heating and at 333±1 K when cooling down. The resulting narrow hysteresis width of 6.5 K attests to the high quality of the thin-filmsample. 37The hysteretic behavior confirms the first-order character of the phase transition in VO 2. Previous calorimetric studies demonstrated the presence of a latent heat of 241J/cm 3at the transition point.3All measurements presented in this paper were acquired on the heating part of the hysteresiscycle. Figure 3(b) shows the real part of the optical conductivity σ 1(ω) for two selected sample temperatures below and above Tc. Although σ1of insulating VO 2is expected to vanish, at TL =295 K it exhibits pronounced maxima at ¯ hω=50, 62, and 74 meV (13, 15, and 18 THz). These peaks correspond to thetransverse optical phonon resonances of the monoclinic latticerelated to vibrations of the oxygen cages surrounding the Vatoms. 9,43Vanadium-dominated normal modes, in contrast, are known to oscillate in the low-frequency regime between2a n d6T H z . 9,28The spectral region above 85 meV is free of infrared-active resonances and displays a low conductivityin the insulating phase. The spectral positions and weightsof the phonon resonances in the monoclinic phase are ingood agreement with previous infrared studies on VO 2single crystals.9,43The observed phonon resonances can be identified by comparing their eigenfrequencies and oscillator strengthswith data reported by Barker et al. 43Two strong resonances at 62 and 74 meV are assigned to the oxygen vibrationsperpendicular to the monoclinic a Maxis (collinear with cR of the rutile phase). The weaker resonance around 50 meV is related to the vibration polarized along the aMaxis. As expected for polycrystalline thin films, the phonon linewidthsof about 10 meV observed in our sample are about twice 195120-3A. PASHKIN et al. PHYSICAL REVIEW B 83, 195120 (2011) as large as those of single crystalline samples (4–5 meV).43 First, our thin-film samples are polycrystalline and have reduced long-range order, which causes a broadening of theobserved lattice modes. Second, the experiments performedwith polycrystalline thin films always average over differentcrystal orientations. The optical conductivity in the metallic phase increases monotonically with temperature. However, it does not increasewith decreasing energy as expected for a Drude responseof conventional metals [Fig. 3(b)]. The peculiar spectral shape arises from the coexistence of insulating and metallicdomains typical of a first-order transition. Hence, in a dcresistivity measurement, a sample appears metallic as soonas a percolation path connecting metallic domains is formed.On the other hand, the optical response averaged over manydomains is determined by an effective dielectric functionthat depends on the volume fractions of the monoclinic andrutile phase. A good description of experimental infraredspectra using Bruggeman’s effective-medium theory 44has been demonstrated.45–47Based on this success, recent studies rely on the effective-medium theory for a quantitative analysisof the optical spectra. 29,48–50 B. Time-resolved spectrally integrated multi-THz response of photoexcited VO 2 Now we consider the nonequilibrium response of VO 2 induced by the optical pump pulse. The pump photons at 1.55 eV promote electrons from the split-off d||band below the Fermi level to the conduction-band πstates [see Fig. 1(b)]. The optically generated free electron-hole pairs, in turn, give riseto a finite photoconductivity. Figure 4(a) depicts the ultrafast dynamics of the spectrally integrated conductivity change/Delta1σ 1(τ), for a series of pump fluences /Phi1recorded at a sample temperature of TL=295 K. The initial increase of the infrared conductivity peaks within a resolution-limited time intervalof 60 fs, marking the completion of the excitation process.At low excitation densities, a sharp onset of the pump-inducedsignal is followed by a nonexponential sub-ps decay. At higherfluences, an increasing background of long-lived conductivityappears that is constant within a time window of 10 ps. Thedensity of directly excited photocarriers contributing to theinitial conductivity depends linearly on the fluence. Thus,we expect the quasi-instantaneous signal to scale linearlywith/Phi1, as well. Extracting the values of the pump-induced conductivity at τ=60 fs confirms this expectation, as shown in Fig. 4(b). In contrast, the corresponding value at τ=1p s vanishes for small fluences while it increases rapidly above athreshold /Phi1 c(295 K) =4.6 mJ /cm2, which is determined by extrapolating the linear part of the latter graph to zero [insetof Fig. 4(b)]. The threshold in excitation density separates two regimes: below /Phi1 cthe directly photogenerated electron-hole pairs populate delocalized states and thus give rise to the initialphotoconductivity. However, the lifetime of these states islimited by a very effective trapping or relaxation mechanismthat results in the rapid sub-ps decay of the quasi-instantaneoussignal. Increasing the excitation density above the threshold ofonly one absorbed photon per approximately 10 V-V dimerstriggers a cooperative transition from the insulating to themetallic state that renders the relaxation pathway inoperative.250 300 3502460.0 0.5 1.002468 3 mJ/cm2 6 mJ/cm210 mJ/cm2 8 mJ/cm2 Delay time τ (ps)Δσ1(arb.units)TL = 295 K 02468 1 002468 )b( )a( 60 fs Fluence Φ(mJ/cm2) Δσ1(arb. units) 1 ps 02460123Δσ1(arb.units)320 K 295 K Fluence Φ(mJ/cm2)(d) (c) Φc (mJ/cm2) Temperature TL (K)Tc FIG. 4. (Color online) (a) Spectrally integrated transient change of the THz conductivity after excitation by a 12-fs near-infrared laser pulse at TL=295 K for various pump fluences. (b) Fluence dependence of /Delta1σ 1(τ)a tτ=60 fs (blue crosses) and 1 ps (red triangles). (c) Extrapolation of /Delta1σ 1(1 ps) curves (red triangles: 295 K, magenta circles: 320 K) to a critical fluence of /Phi1c(295 K) = 4.6 mJ /cm2and/Phi1c(320 K) =3.5 mJ /cm2, respectively. (d) Dependence of threshold fluence /Phi1con lattice temperature TL. The solid line is the thermodynamic energy difference between themetallic state and the insulating state at a given temperature calculated according to Ref. 3. The dashed line marks the phase-transition temperature T c. The subsequent persistence of the photoconductivity indicates the transition to the metallic phase. However, the initialinsulating state is restored before the arrival of the nextpump pulse due to heat dissipation to the diamond substrate(the substrate temperature T Lwas kept below the hysteresis region). Raising the sample temperature closer to Tcleads to a decrease of the threshold fluence /Phi1c[see Fig. 4(c)] and the photoinduced conductivity becomes more long-lived ascompared to room temperature. On the other hand, thecooperative transition may be suppressed when the latticeis cooled to cryogenic temperatures, although the excitationdensity exceeds its critical room-temperature value (seeSec. III C). Indeed, as it is shown in Fig. 4(d), the fluence threshold experiences a significant reduction as the criticaltemperature is approached from below. Since the thresholdfluence corresponds to the minimal energy of the pump pulsenecessary to induce the cooperative transition into the metallicphase, it is instructive to compare it to the difference ofthe thermal energy of the VO 2lattice between the metallic and insulating phases. This difference can be calculated asthe integral of the lattice heat capacity between a giventemperature and T cplus the latent heat of the insulator-metal transition. The resulting curve calculated using the data from 195120-4ULTRAFAST INSULATOR-METAL PHASE TRANSITION IN ... PHYSICAL REVIEW B 83, 195120 (2011) Ref. 3is shown in Fig. 4(d) as a solid line. To enable a direct comparison with pump fluence, the thermal energy density isrecalculated in terms of a surface energy density assuming a100-nm-thick excited layer. The agreement between both datasets is remarkable. Since the electronic contribution to the specific heat in the studied temperature range is negligible compared to thelattice contribution, we can conclude that optical pumpingabove /Phi1 c(T) should induce a structural transformation of VO 2 into the rutile phase on a time scale shorter than 1 ps after the excitation that we have chosen for the estimate of the excitationthreshold. Indeed, the dynamics of the optical conductivity inFig.4(a)confirms this assumption. It should be noted that the agreement between optical pump and thermal energies [Fig. 4(c)] does not imply a thermal character of the photoinduced transition in VO 2. Clearly, the sub-ps switching time would be untypically shortfor a thermally driven process. As a matter of fact, bothcoherent (optical switching) and incoherent (thermal heating)mechanisms are observed, but they are well separated in time.For example, after photoexcitation at 295 K, the pump-probesignal starts rising again after several tens of ps and reachesits maximum level after 300 ps, independent of the excitationfluence (not shown). The phase growth in the thin film is drivenby hot phonons propagating at the speed of sound and proceedsincoherently at spatially separated sites. The conductivitydynamics of this thermal phase transition activated by laserheating was studied in detail by Hilton et al. 29and is not a subject of this paper. A closer look at the decay dynamics of the THz conductivity reveals a fast oscillation superimposed on the monotonicallydecreasing signal [Fig. 5(a)]. The presence of this feature in the pump-probe signal is a first substantial signature of 04 0 0 8 0 0 1 2 0 0-0.10.01234 oscillatory component cosine fit(c)(b)Δσ1 (arb.units) Delay time τ (fs) pump-probe response exponential fit(a) 01 56701THz probe Amplitude (arb.units) Frequency (THz)optical probe Raman FIG. 5. (Color online) Decomposition of the decay dynamics. (a) Spectrally integrated dynamics of the THz conductivity afterexcitation at fluence /Phi1=14 mJ /cm 2and temperature TL=4K (black circles). Subtracting the biexponential decay component (red line) reveals a coherent oscillation (blue circles) to which a cosinefunction with a frequency of ω/2π=6.1 THz (green line) was fitted. (b) Spectrum of the oscillatory component (blue circles) and fit by a Lorentzian function (blue line). (c) The spectrum of the oscillatorymodulation observed in a degenerate pump-probe experiment 28at 1.55 eV (red line) and the unpolarized spontaneous Raman spectrum51 (gray line).the participation of lattice modes in the phase transition. Figure 5(a) shows how subtraction of a biexponential decay from the pump-probe signal reveals a coherent cosine-likeoscillation with a frequency of 6.1 THz. Apparently, the firstcycle of the observed coherent oscillation is notably strongerand cannot be well described by a simple cosine function.This points out a strongly nonequilibrium character of themulti-THz response during the first oscillation period of 160 fswhen the lattice distortion is accompanied by a strong changein electronic structure. The following harmonic oscillationsdescribe a pure lattice vibration around a new quasiequilibriumconfiguration. It should be noted that coherent oscillations ata constant frequency of 6.1 THz are observed at all latticetemperatures below T cindependent of the excitation fluence. Pumping the metallic phase ( T> T c) reduces the conductivity, and the sign of the spectrally integrated response is inverted.Thus, for delay times τ> 100 fs, the observed behavior is typical of a hot electron gas. The Fourier transform of the oscillatory component ex- cluding the first cycle is shown in Fig. 5(b).I ti sw e l lfi t t e d by a single Lorentzian term. In contrast, all-optical ultrafastreflectivity experiments 28have revealed impulsive excitation of two vibrational modes at 5.85 and 6.75 THz, which coincidewith totally symmetric A gRaman modes of the monoclinic lattice51,52[Fig. 5(c)]. These phonon modes are critical to the metal-insulator transition in VO 2and describe stretching and tilting of vanadium dimers, which map the monoclinic ontothe rutile lattice. 10Thus, the corresponding resonances are only observed in the ground insulating state of VO 2probed by spontaneous Raman spectroscopy. The observation of asingle oscillation frequency instead of the two frequencies inthe dimerized monoclinic phase indicates the higher latticesymmetry of the excited state probed in our experiment. In general, pump-probe techniques utilizing an impulsive excitation may be sensitive to ground-state as well as toexcited-state vibrations. 53,54This can be clearly seen in Fig. 5(c), where the experimental data from a degenerate femtosecond pump-probe experiment at a photon energy of1.55 eV are shown. 28Besides two AgRaman modes of the ground state of VO 2, an additional shoulder appears between both maxima, exactly at the frequency of 6.1 THz where thecoherent oscillation in the multi-THz probe occurs (see thevertical dashed line in Fig. 5). In contrast, the multi-THz probe obviously is not sensitive to the coherent wave-packetdynamics in the ground state of VO 2and probes almost exclusively the coherent oscillation in the excited electronicstate. The reason for such selectivity cannot be understoodbased solely on the spectrally integrated data, which do notunravel the multiple participating degrees of freedom. Thislimitation is overcome in a full 2D optical pump–multi-THzprobe experiment. C. Spectrally resolved dynamics The spectrally resolved pump-probe response is obtained by measuring the complete time profiles of multi-THz transientsas well as their pump-induced changes at varying pump-probedelay times τand subsequent Fourier analysis as described in Sec. II. The resulting 2D plots shown in Fig. 6depict the spectral profiles of the conductivity changes /Delta1σ 1(ω,τ) 195120-5A. PASHKIN et al. PHYSICAL REVIEW B 83, 195120 (2011) (b) 4 K, 7.5 mJ/cm²-1000100200300400500 (fs)1 (c m )-1 -1 050100150 050100150 -1000100200300400500 (fs) 20406080(c) 50 60 70 80 90 100 (meV)1 (cm)-1 -1 050100150(d)250 K, 3 mJ/cm² 250 K, 7.5 mJ/cm²1 (c m )-1 -1 -1000100200300400500 (fs)(a) FIG. 6. (Color online) 2D optical pump–multi-THz probe data: (a) Equilibrium conductivity of insulating VO 2at 295 K. Color plots of the pump-induced changes of the conductivity /Delta1σ 1(ω,τ): (b) at TL=4Ka n da ni n c i d e n tfl u e n c eo f /Phi1=7.5 mJ /cm2;a tTL=250 K and pump fluence (c) /Phi1=3m J/cm2and (d) /Phi1=7.5 mJ /cm2. The broken vertical lines indicate the frequency positions of crosssections reproduced in Fig. 7. as a function of time delay τat different temperatures and pump fluences. The equilibrium optical conductivity spectrumof the insulating phase of VO 2is depicted in Fig. 6(a) for comparison. Figures 6(b) and6(c)show data for an excitation fluence /Phi1below the transition threshold /Phi1c. As discussed in Sec. III B, the threshold fluence depends on temperature TL.I n particular, excitation densities that completely switch the VO 2 to the metallic phase at room temperature are not sufficientto do so if the lattice is cooled down to 4 K. Performingthe 2D measurement with a fluence of /Phi1=7.5 mJ /cm 2,a t TL=4 K, results in the data displayed in Fig. 6(b).T h e observed dynamics are essentially identical to below-thresholdexcitation with the fluence /Phi1=3m J/cm 2atTL=250 K shown in Fig. 6(c). Within our time resolution of 40 fs, ultrafast photodoping induces a quasi-instantaneous onset of conductivity due todirectly injected mobile carriers. For pump fluences belowthe threshold, the electronic part of the optical conductivity/Delta1σ 1(ω,τ) decays promptly within approximately 400 fs. In contrast, the pump-induced changes in the phonon resonancesare more long-lived. While the onset of the phononic responsevaries for the three modes, we find intriguing common features.0 100 200 300 400-20020-20020 Δσ1 (Ω-1cm-1) Delay Time τ (fs) Δσ1 (Ω-1cm-1) 050100150 (b)Δσ1 (Ω-1cm-1)(a) 4 K, 7.5 mJ/cm2 250 K, 3 mJ/cm2 x 2.4 250 K, 7.5 mJ/cm2 (d)(c) 0 200 4000100200Δσ1 (Ω-1cm-1) Delay Time τ (fs) FIG. 7. (Color online) Cross sections of Fig. 6along the time axis τfor a photon energy of (a) ¯ hω=60 meV and (b) ¯ hω=100 meV . The curves taken at /Phi1=3m J/cm2are scaled up by a factor of 2.4. (c) and (d) The oscillating components of the cross sections throughthe 2D scan in Fig. 6(d) for a photon energy of (c) ¯ hω=60 meV and (d) ¯hω=92 meV . Green solid lines: (c) fit of the oscillating component by a cosine function with frequency of 6.1 THz, (d) the fitting functionshown in panel (c) squared with subtracted constant background. The oscillation occurs at a doubled frequency of 12.2 THz. Photoexcitation induces increased polarizability on the low- frequency side of each phonon resonance, while minimalchange is seen on the blue wing: the resonance frequenciesare redshifted. For all modes, the change in frequency is su-perimposed on a remarkable coherent oscillation of /Delta1σ 1(ω,τ) along the pump-probe delay axis τ. This phenomenon is most notable at a THz photon energy of 60 meV marked by a verticalbroken line in Fig. 6. The corresponding cross sections of the 2D maps are shown in Fig. 7(a). After subtraction of a slowly varying background, the coherent oscillation is well fitted by a cosine function witha frequency of 6.1 THz, as shown in Fig. 7(c). Thus, the true origin of the periodic conductivity modulation that appearedalready in the spectrally integrated data of Fig. 5is related to the periodic change in the phononic response. The analysis showsthat the failure of the simple decomposition in explaining thespectrally integrated conductivity dynamics for τ< 400 fs [first oscillation cycle in Fig. 5(a)] arises from the presence of the quickly decaying electronic contribution in the totalresponse. As argued in Sec. III B, the coherent oscillations at the single frequency of 6.1 THz observed by multi-THz probingin contrast to the all-optical experiment 28(Fig. 5) indicate that the utilized multi-THz probe is more sensitive to thecoherent dynamics of the excited electronic state of VO 2.T h e 2D maps clearly demonstrate that the origin of the coherentdynamics traced by the multi-THz probe originates in theanharmonic coupling between the high-frequency oxygen and low-frequency vanadium lattice vibrations. Since themulti-THz response is dominated by the vanadium vibrationinherent to the rutile phase, we assume that the anharmonicityof the VO 2lattice in the excited electronic state is strongly enhanced as compared to the insulating ground state. The spectral response of VO 2and its dynamics shown in Figs. 6(b) and 6(c) are very similar. This fact convincingly 195120-6ULTRAFAST INSULATOR-METAL PHASE TRANSITION IN ... PHYSICAL REVIEW B 83, 195120 (2011) demonstrates once more the interchangeability of temperature and fluence. We may thus conclude that the observed interplayof electronic and phononic degrees of freedom that eventuallyleads to the emergence of the metallic phase is not unique tothe optically driven case but plays an important role in thethermal transition as well. Figure 6(d) shows the 2D map measured with an excitation fluence of /Phi1=7.5 mJ /cm 2exceeding a threshold fluence /Phi1c=5.3 mJ /cm2at a temperature TL=250 K. The spectral response of the optical phonon resonances is qualitativelysimilar to Figs. 6(b) and 6(c) with the same dynamics characterized by the coherent oscillations [see Fig. 7(a)]. In contrast, the dynamics of the electronic photoconductivity dif-fers profoundly: After a resolution-limited onset of /Delta1σ 1(ω,τ) due to nearly instantaneous photodoping, the THz conductivitylevels off at a high value, indicating the final transition into ametallic phase. This behavior is seen also in Fig. 7(b), where the cross sections of the 2D maps at the energy of 100 meVare shown. Remarkably, at this energy there are no pronounced oscillations imprinted on the electronic conductivity, incontrast to the phononic response. Nevertheless, carefulinspection reveals a periodic modulation of the multi-THzresponse around 90 meV . This energy range is well above thehighest phonon resonance and related to the electronic partof the conductivity. Most clearly, this effect is seen in thecross section of Fig. 6(d) made at an energy of 92 meV . The oscillating component in this cross section shown in Fig. 7(d) occurs exactly at twice the V-V stretching frequency of6.1 THz. Moreover, the first three cycles of the oscillationshown in Fig. 7(d) are well described by squaring the fitting function of the 6.1 THz oscillation in Fig. 7(c) (and subtracting an offset). For later delay times τ, the modulation amplitude rapidly decreases and a relation to the coherentwave-packet motion cannot be traced anymore. The doubledfrequency of the oscillation indicates that the dominant termin the electron-phonon coupling of the 6.1 THz phonon modeis quadratic with respect to the phonon normal coordinate. The redshift of the oxygen modes is equal in all three investigated cases and hence depends neither on fluence nortemperature. The same holds for the period of the modulation,which always consistently matches the vanadium 6.1 THzmode, as displayed in Fig. 7(a). Interestingly, the amplitude of the modulation scales with fluence but hardly depends ontemperature at all. IV . DISCUSSION A qualitative understanding of the experimental observa- tions may be reached using a recent model that describes theelectronic structure in thermal equilibrium within a clusterdynamical mean-field theory. 20In this picture, the low- temperature insulating phase is regarded as a molecular solid ofvanadium dimers embedded in a matrix of oxygen octahedra.To first order, the correlated electronic state of each dimermay be described by bonding and anti-bonding Heitler-Londonorbitals. The energy dependence of these states on the nuclearpositions (such as V-V separation) is depicted schematicallyin Fig. 8. The minimum of the lower-energy bonding state defines the atomic position in the monoclinic phase. FIG. 8. (Color online) Schematic energy surfaces as a function of a dimer coordinate Q. The energy minima of the potential surfaces QMandQRcorrespond to spatial configurations of the V-V dimers in the monoclinic and rutile phases, respectively. Vertical dashed arrows denote Franck-Condon-type photoexcitation of spin singlets into a conductive state. (a) Structural relaxation and coherent vibrationsabout the new energy minimum below the pumping threshold /Phi1< /Phi1 c. (b) Intense excitation above the threshold /Phi1>/Phi1 cleads to a structurally assisted collapse of the mobility edge. Absorption of a near-infrared photon removes an electron from the bonding orbital, destabilizing the dimer, while the lat-tice site is left in a vibrationally excited Franck-Condon state.Symmetry requires that the energy minimum of antibondingorbitals should be located near the rutile configuration.Ultrafast photoexcitation thus launches a coherent structuraldeformation of excited dimers followed by oscillations of A g symmetry around the new potential minimum (Fig. 8). The appreciable lattice deformations impose strong distortions onthe surrounding oxygen octahedra and shift their resonancefrequencies. A similar effect has been reported in carbonnanotubes. 55Remarkably, the frequency shift is very pro- nounced for the oxygen vibrations polarized perpendicularto the a Maxis (modes at 62 and 74 meV) and relatively weak for the phonon mode at 50 meV polarized along the aMaxis. This fact attests to an anisotropic character of the anharmoniccoupling between normal modes of the vanadium and oxygensublattices. Since the oscillation period of the oxygen-related phonon modes is shorter by a factor of 3 as compared to the inversefrequency of the A gvibration, the eigenfrequencies may follow the structural changes of the vanadium dimer adiabatically. Ourexperiment directly demonstrates the influence of a coherentlattice motion on other phonon resonances: An overall redshiftof the oxygen modes attests to a modified average structure ofthe vanadium dimers, while the coherent modulation at 6 THzreflects large-amplitude oscillations about the new potentialminimum in the excited state. Although additional shifts ofinfrared-active phonon modes may be expected from screeningvia delocalized charge carriers, the distinctly different decaytimes of electronic and phononic conductivity prove that thiseffect is not dominant (Fig. 7). The density of excited dimers scales linearly with the laser fluence and so does the amplitudeof the observed coherent modulation. On the other hand,the lattice temperature has no measurable influence on thecoherent modulation of the oxygen modes. The model also provides an instructive explanation of the transient electronic conductivity. Photodoping leaves thedimers initially in a higher-energy state with delocalizedelectronic character (Fig. 8) generating a quasi-instantaneous onset of electronic conductivity. Structural distortion drives thedimers into a new energy minimum. The rapid nonexponential 195120-7A. PASHKIN et al. PHYSICAL REVIEW B 83, 195120 (2011) decay of the electronic conductivity seen in Fig. 7(b) indicates that the structural deformation shifts the energy of exciteddimers below a mobility edge for extended electronic states[Fig. 8(a)]. This process is analogous to a self-trapping of excitons considered by Dexheimer et al. 54Although self- trapped states do not contribute to the electronic conductivity,their structural distortions manifest themselves by the an-harmonic shift of oxygen-related phonon modes that persistfor at least 1 ps [see Fig. 5(a)]. On the other hand, at high enough pump fluences electronic conductivity does notdecay rapidly. We suggest long-range cooperative effects toaccount for this finding: Strong distortions that locally maponto the rutile lattice are expected to adiabatically renormalizethe electronic bands 56and lower the mobility edge. Due to the enhanced critical fluctuations at elevated temperatures, thesystem is more susceptible to this renormalization in accordwith experimentally observed lowering of the critical fluenceupon approaching T c. Instead of the persistent coherent modulation at the frequency of the Agvibration, a weak modulation of the electronic conductivity at the doubled frequency is observed.The amplitude of this oscillation rapidly vanishes after aboutthree oscillation cycles [Fig. 5(d)]. This behavior suggests that the lowest-order term in the coupling between the A g mode and the electronic conductivity band of the photoexcited state is proportional to /Delta1Q2, where /Delta1Q=Q−QRdenotes the deviation of the dimer coordinate from the metastablerutile structure (see Fig. 8). Since, in a first approxima- tion, this coupling is rather weak, the electronic systemmay be considered as decoupled from the ionic motion inthe nonequilibrium regime after the photoinduced insulator-metal transition. Consequently, the long-lived high value ofelectronic conductivity cannot originate from the continuingmotion of the V-V dimers. Instead, we propose to explain thisfeature by the collapse of the mobility edge 57below the Fermi energy [Fig. 8(b)]. In this picture, the critical pump fluence /Phi1c determines whether self-trapping or cooperative delocalization of electron-hole pairs prevails.V . CONCLUSION The ultrabroadband THz experiments presented here allow us to directly trace the the temporal evolution of the electronicconductivity and the optical phonon resonances during thetransient formation of the metallic phase in VO 2. This study reveals fingerprints of a coherent V-V intradimer wave-packetmotion at 6.1 THz, which couples anharmonically to infrared-active phonon modes. The coherent response of the crystallattice is explained within a model description of local V-Vdimers photoexcited into an antibonding state. The electronicconductivity shows a weak quadratic coupling to the coherentwave-packet motion. A threshold fluence of the excitationpulse for the photoinduced transition is found to decrease as theequilibrium lattice temperature T Lapproaches the transition temperature Tc. Below threshold, the electronic correlations remain undisturbed and the midinfrared conductivity vanisheson a subpicosecond time scale. Above a critical excitationdensity, a cooperative insulator-metal transition occurs on atime scale set by the first half of the oscillation cycle ofthe V-V coherent wave-packet motion in the excited state.So far, the data indicate a strong structural component ofthe phase-transition mechanism. However, right after thetransition to the metallic phase, the electronic system isrendered insensitive to the continuing coherent wave-packetmotion. 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PhysRevB.88.235129.pdf

PHYSICAL REVIEW B 88, 235129 (2013) Pressure study of nematicity and quantum criticality in Sr 3Ru 2O7for an in-plane field Dan Sun,1W. Wu,1S. A. Grigera,2R. S. Perry,3A. P. Mackenzie,2,4,5and S. R. Julian1,4 1Department of Physics, University of Toronto, 60 St. George Street, Toronto M5S 1A7, Canada 2Scottish Universities Physics Alliance, School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, United Kingdom 3Center for Science at Extreme Conditions, School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland 4Canadian Institute for Advanced Research, 180 Dundas Street West, Suite 1400, Toronto M5G 1Z8, Canada 5Max Planck Institute for Chemical Physics of Solids, Noethnitzerstraße 40, Dresden 01187, Germany (Received 12 September 2013; published 27 December 2013) We study the relationship between the nematic phases of Sr 3Ru2O7and quantum criticality. At ambient pressure, one nematic phase is associated with a metamagnetic quantum critical end point (QCEP) when theapplied magnetic field is near the caxis. We show, however, that this metamagnetic transition does not produce the same nematic signatures when the QCEP is reached by hydrostatic pressure with the field applied in the ab plane. Moreover, a second nematic phase, that is seen for field applied in the abplane close to, but not right at, a second metamagnetic anomaly, persists with minimal change to the highest applied pressure, 16.55 kbar. Takentogether our results suggest that metamagnetic quantum criticality may not be necessary for the formation of anematic phase in Sr 3Ru2O7. DOI: 10.1103/PhysRevB.88.235129 PACS number(s): 75 .30.Kz,71.27.+a,75.20.En I. INTRODUCTION Electronic nematic phases are electronic analogs of nematic liquid crystals. They are characterized by spontaneouslybroken rotational symmetry in their transport properties. Suchnematic phases, which have a symmetry level between that of aFermi liquid and a Wigner solid, 1have been unveiled in several strongly correlated electron systems, including quantum Hallsystems, 2and iron-pnictide3and cuprate superconductors.4,5 A prominent example is Sr 3Ru2O7,6which is a clean system for nematicity in the sense that accompanying the nematicphase transition, there is no magnetic ordering or charge den-sity wave formation, although an area-preserving symmetry-breaking lattice distortion of order 10 −6occurs within the nematic phase.7 Sr3Ru2O7is the bilayer member of the Ruddlesden-Popper series of layered perovskite ruthenates. It has orthorhombicBbcb symmetry, arising from ordered rotations of the RuO 6 octahedra, but is nearly tetragonal in terms of the lattice parameter: the 5 parts in 104difference between the a andblattice parameters cannot be detected by Laue x-ray diffraction.8The resistivity is isotropic in the two in-plane principal axis directions in the absence of an external field. There are two distinct nematic regions that have been found in the magnetic field vs field angle phase diagram ofSr 3Ru2O7, as shown in Fig. 1.12These nematic phases can only be seen in very pure samples, at temperatures below1 K. The nematic phase 1 extends between 0 ◦and 40◦from thecaxis and is bounded by first-order transitions, which are demonstrated by peaks in the imaginary part of the ac susceptibility χ/prime/prime(H).6,9,13Nematic phase 2 is found from 60◦ to 90◦from the caxis, i.e., adjacent to the abplane. It is bounded on each side by double peaks in the real part ofthe susceptibility χ /prime(H), which was obtained by taking the derivative, dM/dB , of the dc magnetization.14χ/prime/prime(H)s h o w s a small peak at the lower-field boundary of nematic phase 2at ambient pressure at the abplane. 15There is no observable feature in χ/prime/prime(H) at the upper-field boundary of the phase.The two nematic phases are closely related to metamagnetic features that survive to much higher temperature and that arenot sensitive to sample purity. Metamagnetism is defined asa sudden increase in magnetization within a narrow range offieldH. This is illustrated for Sr 3Ru2O7in Fig. 2, which shows low excitation frequency ( ν=14 Hz) susceptibility data χ(H) (red line) taken at 70 mK with the external field between 4 and8 T applied in the abplane, together with the magnetization M(H) obtained by integrating χ(H), as described in the Appendix. The weak H 3metamagnetic transition marks the upper boundary of nematic phase 2 and, in contrast to the H1 andH2metamagnetic transitions, it is not robust, disappearing rapidly with increasing T. The dependence of the H1andH2 metamagnetic features on field angle is shown in Fig. 1. TheH1metamagnetic transition exhibits an interesting kind of quantum criticality. This transition is a first-order jumpat low temperature for magnetic field in the abplane, as illustrated in Fig. 2atH 1. When the field is rotated towards thecaxis, the jump decreases in magnitude, vanishing at a quantum critical end point (QCEP) approximately 10◦from thecaxis in samples where purity is not high enough to see the nematic phase.16 The fact that nematic phase 1, in very high purity samples, emerges near a QCEP has caused speculation of a deepconnection. A connection at some level is plausible, becausethe first-order metamagnetic jump and nematicity are mutuallyexclusive ways to avoid a van Hove singularity at the Fermienergy ε F.6,17Puetter et al. ,18using a tight-binding model, identified two van Hove singularities close to εF.A sa sweeping magnetic field progressively spin-splits the energybands, the two van Hove singularities will consecutively passthrough ε F, producing peaks in the density of states, g(εF). In mean-field theories for itinerant systems, the tendency to orderincreases as Ig(ε F), where Iis an electron-electron interaction strength. In the case of Sr 3Ru2O7, an intraorbital Hubbard U interaction will drive a metamagnetic transition,19while on- site interorbital electron repulsion10,20and nearest-neighbor repulsion18,21drive different forms of nematic order. The key 1098-0121/2013/88(23)/235129(6) 235129-1 ©2013 American Physical SocietySUN, WU, GRIGERA, PERRY , MACKENZIE, AND JULIAN PHYSICAL REVIEW B 88, 235129 (2013) 8.0 7.0 6.0 5.0020406080 H (T)Angle(degrees)H1 H2 Nematic phase 2 Nematic phase 1H3 FIG. 1. (Color online) Phase diagram at ambient pressure in the (H,θ)p l a n ea t T/lessmuch1 K, from Refs. 9–11.0◦is thecaxis, while 90◦is theabplane. The two metamagnetic transitions follow the H1andH2 lines. Where the lines are solid the metamagnetic transitions are first order, while dotted lines show crossovers. The green and blue shaded regions show nematic phase 1 and 2, respectively. Nematic phase 1 is bounded by first-order transitions. For the H2metamagnetic transition, and nematic phase 2, first-order behavior has only been observed at H2at the abplane. H3indicates the end of nematic phase 2. difference between the first-order metamagnetic transition and nematicity is that in the former all four symmetrically placedcopies of a van Hove singularity in the Brillouin zone arejumped over at once, while in the latter they are jumpedover two at a time. In the nematic phase, when only twoof the four singularities have been jumped over, the Fermisurface is strongly distorted. The point is that because first-order metamagnetism and nematicity are mutually exclusive,weakening the strong first-order metamagnetic jump as itapproaches its QCEP would naturally be a precondition forthe appearance of the nematic phase. 4 5 6 7 8 H (T)00.10.20.30.4M (μB/Ru mol) 00.10.20.30.4 χ (a.u.)H1H2 Nematic phase 2H3 FIG. 2. (Color online) Low-temperature susceptibility χ(H) (Ref. 15) (red) and integrated susceptibility M(H) (black) at near- ambient pressure. The two metamagnetic transitions H1andH2are indicated by dashed lines. A first-order jump in M(H)a tH1is added by hand (see Appendix). The weak H3anomaly appears only below ∼0.4 K.Beyond this mutual connection to an underlying van Hove singularity the nematic phase 1 gives the appearance ofscreening the metamagnetic QCEP, just as quantum criticalsuperconductivity is often found to screen antiferromagneticquantum critical points. 22 We tried to investigate this connection in a previous high- pressure study, by measuring the susceptibility. We succeededin inducing the QCEP for the H 1metamagnetic transition withH/bardblabplane, at 13.6 ±0.2 kbar, but we did not see the bifurcation of the peak in χ/prime(H) that is associated with the appearance of the nematic phase when H/bardblc.15This is in contrast to the theoretical prediction of Ref. 10that the nematic phase 1 should extend to the abplane when it is not preempted by a first-order metamagnetic transition. In contrast to nematic phase 1, nematic phase 2 has not been considered theoretically. It has very similar transportsignatures to nematic phase 1, and it is similarly sensitiveto impurities and disappears above 0.4 K. 14It is not obvious however that it screens a metamagnetic QCEP. The metam-agnetic feature at H 2is weak; moreover the nematic phase occurs not at but beside the metamagnetic transition fieldH 2, and it extends to fields that are quite a lot higher (see Fig. 2). However, in heat capacity measurements at widely spaced fields, C/T increased logarithmically with decreasing temperature near H2, suggesting quantum critical behavior. But it is not necessarily metamagnetic quantum criticality: recent theoretical work23suggests that a nematic phase would have a non-Fermi liquid normal state regardless of proximityto a metamagnetic QCEP. The motivation for the present study was to examine more deeply the connection between nematic phases and quantumcriticality. The plan was to use resistivity anisotropy to searchagain for nematicity at the H 1QCEP and, second, to see the effect of pressure on nematic phase 2. Pressure should reducethe peak in g(ε F) at the van Hove singularity, weakening metamagnetism, so if there is a connection to quantumcritical metamagnetism nematic phase 2 should weaken withincreasing pressure. Our main results are that we find no evidence of a nematic phase at the QCEP of the H 1metamagnetic transition, while nematic phase 2 is robust against pressure up to 16.55 kbar. II. EXPERIMENT We simultaneously measured the resistivity of two samples, ρ/bardblwith current parallel to the field and ρ⊥with current perpendicular to the field, in the same clamp-type pressurecell. Daphne oil 7373 was used as the pressure mediumand the pressure at low temperature was determined bythe calibrated pressure dependence of the superconductingtransition temperature of tin. The field and the current wereboth applied in the abplane. The current directions for ρ /bardbl andρ⊥were 8.5◦and 15.5◦from the closest in-plane principal axis, respectively (the aandbaxes are indistinguishable under Laue x-ray diffraction). At eight different pressures rangingfrom 1.87 kbar to 16.55 kbar, we carried out field sweepscrossing both of the metamagnetic anomalies, at temperaturesfrom 100 mK to 2.5 K. At each of these pressures we alsocarried out 20 or so temperature sweeps at fixed fields, toextract the Acoefficient in ρ=ρ 0+AT2. These temperature 235129-2PRESSURE STUDY OF NEMATICITY AND QUANTUM . . . PHYSICAL REVIEW B 88, 235129 (2013) 5 6 7 8 H (T)11.52ρ (a.u.)5 6 7 82.5 K 2.0 K 1.5 K 1.1 K 0.7 K 0.5 K 0.3 K 0.1 K 8 1 01 21 41.522.5 8 1 01 21 41.87 kbar(a) 1.87 kbar(b) 13.71 kbar 13.71 kbar(c) (d)ρ|| ρ||ρ⊥ ρ⊥ FIG. 3. (Color online) In-plane magnetoresistivity ρ(H) with currents parallel ρ/bardbl(left) and perpendicular ρ⊥(right) to the applied field at different temperatures. The upper two panels (a) and (b) show data at 1.87 kbar and the lower two (c) and (d) show data at 13.71 kbar. ρ/bardblandρ⊥are normalized according to the geometry of the samples and the amplification gain. sweeps went from 100 mK to 700 mK. The samples were cut from ultrapure single crystals ( ρ0<0.4μ/Omega1cm) grown at St. Andrews University, UK. III. RESULTS Typical in-plane magnetoresistivity data at different tem- peratures and pressures are shown in Fig. 3, while Fig. 4 focuses on the lowest temperature data at 1.87 kbar. Thetwo metamagnetic anomalies are clearly visible. At hightemperatures, these anomalies are two overlapping peaks.At low temperature [see Fig. 4(a)], the H 1metamagnetic transition is a cusplike feature in both ρ/bardblandρ⊥, while the H2 transition is a clear peak in ρ⊥, but a weak shoulder in ρ/bardbl. As the pressure increases, the H1andH2transitions shift to higher fields, roughly linearly with increasing pressure, asshown in Fig. 4(b). Figure 4(b) also shows H 1(crosses) and H2 (stars) from our previous susceptibility measurements.15By fitting the susceptibility points, we obtained linear functionsH 1(p) andH2(p). The features from resistivity and susceptibil- ity align well, although the agreement is not perfect. In Fig. 5, for the lowest temperature data at each pressure, we have usedthese H 1(p) and H2(p) functions to rescale the horizontal axis using H→[H−H1(p)]/[H2(p)−H1(p)]. IfH1and H2from our resistivity data agreed perfectly with the fit of our susceptibility results then H1would be at 0 and H2at 1 for every curve. Some scatter is however apparent, particularly atthe lowest pressures; for example in ρ /bardblat 1.87 kbar the H1 peak is clearly above 0.0 5 10 15 Pressure (kbar)51015H (T)H1 of χ H2 of χ H1 of ρ||4 5 6 7 89 H (T)0.60.70.8ρ (a.u.) H1H2ρ|| ρ⊥ H3 H2 of ρ|| H1 of ρ⊥H2 of ρ⊥H3 of ρNematic phase 2 (a) (b) FIG. 4. (Color online) (a) ρ/bardblandρ⊥at 100 mK at 1.87 kbar with a linear background subtracted. The curves are normalized according to the geometry of the samples and the circuit gain and then shifted for clarity. The positions of H1,H2,H3and nematic phase 2 are indicated in the figure. (b) The pressure dependence of the H1andH2 metamagnetic anomalies and the upper boundary of nematic phase 2, H3. The green crosses and the blue stars show the results from earlier susceptibility measurements, and the two lines H1(p)a n dH2(p)a r e linear fits to these data. The other five sets of data are from the present resistivity measurements, as shown in the legend. The H3points were obtained by averaging the estimated upper phase boundaries from ρ/bardbl andρ⊥, with error bars showing the uncertainty. We now turn to the pressure dependence of ρ/bardbland ρ⊥. At a given pressure, ρ/bardblandρ⊥behave similarly at high temperatures. At low temperatures, however, they showqualitative differences that grow with increasing pressure. Forexample, as noted above, the behavior at H 2is different [see Fig. 4(a)]. Moreover, while ρ⊥only shows mild changes in shape with increasing pressure, the low-temperature curves forρ /bardblchange from concave upwards to concave downwards in the two regions: H<H 1andH1<H<H 2.F o rH>H 2,ρ/bardblis concave downwards at both low and high pressures, but thecurvature is more pronounced at high pressures [see Fig. 5(a)]. The small bump located just above H 2corresponds to ne- matic phase 2, discussed in the introduction9,14[see Fig. 4(a)]. The bumps can be seen in both ρ/bardblandρ⊥, but the signal in ρ/bardbl is stronger. The bump can only be seen at temperatures lower than∼0.5 K, but it does not seem to show a strong dependence on pressure (see Fig. 6). TheAcoefficient (see Fig. 7), which is proportional to the square of the effective mass of the quasiparticles, is obtainedby fitting the temperature-dependent magnetoresistivity. AtH 1,t h eAcoefficient is enhanced, but not divergent even near the the critical pressure. The peak at H2and a shoulder-like feature above H2are evident. 235129-3SUN, WU, GRIGERA, PERRY , MACKENZIE, AND JULIAN PHYSICAL REVIEW B 88, 235129 (2013) -1 0 1 21.21.41.61.8ρ (a.u.) -1 0 1 23 (H - H1(p))/(H2(p) - H1(p))16.55 kbar 14.53 kbar 13.71 kbar 12.55 kbar 11.21 kbar 8.01 kbar 4.45 kbar1.87 kbar16.55 kbar 14.53 kbar 13.71 kbar 12.55 kbar 11.21 kbar 8.01 kbar 4.45 kbar 1.87 kbarρ||ρ⊥ (a) (b)Η1Η2Η1Η2 FIG. 5. (Color online) (a) and (b) The magnetoresistivity, ρ/bardbland ρ⊥, at 100 mK at all pressures. Panels are scaled in the field as H→[H−H1(p)]/[H2(p)−H1(p)] so that the two metamagnetic transitions align, where H1(p)a n dH2(p) are the pressure dependence of the metamagnetic transitions extracted by fitting earlier suscepti- bility measurements (Ref. 15). The two dashed lines are guides to the eye for H1andH2. A linear background has been subtracted at 1.87 kbar so that the resistivity returns to its original values outside the region of interest and the same background is also subtracted from the data of all the other pressures. The curves have been shifted vertically for clarity. The green arrows indicate the position of nematic phase 2at each of the pressures. IV . DISCUSSION The pressure-dependent metamagnetic transition fields H1(p) andH2(p)o fρ/bardblandρ⊥line up well with each other and with our susceptibility measurement,15demonstrating hydrostatic pressure inside the cell. This is important be-cause, although hydrostatic pressure tunes Sr 3Ru2O7away from ferromagnetism, uniaxial stress components can induceferromagnetism. 24The extrapolation to zero pressure also agrees well with measurements at ambient pressure for H/bardblab, showing that both samples were well aligned with the field intheabplane. Although this does not ensure that the current was properly aligned parallel or perpendicular to the field, thegeometry of the pressure cell constrains the current to be nearlyparallel to the field for the ρ /bardblsample. We cannot rule out that theρ⊥signal may have a small component of current parallel to the field. Our motivation in measuring ρ/bardblandρ⊥was to search for signs of nematicity; however this turned out to be more compli-cated than we expected. In their study of nematic phase 1 withthe field applied near thecaxis, Borzi et al. 9defined a nematic order parameter as the anisotropy ( ρ/bardbl−ρ⊥)/(ρ/bardbl+ρ⊥), where ρ/bardblandρ⊥referred to whether the current was parallel or perpendicular to the small in-plane component of the magnetic0.75 1.00 1.25 1.50 (H−H1(p))/(H2(p)−H1(p))1.041.081.121.161.201.241.281.32ρ(a.u.) 0.75 1.00 1.25 1.500.000.020.040.060.080.100.120.14 Δρ (a.u.) 1.87 1.87 kbar8.018.01 kbar11.2111.21 kbar12.5512.55 kbar13.7113.71 kbar 14.5314.53 kbar 16.5516.55 kbar 0.0 5.0 10.0 15.0 P(kbar)0.40.60.81.0area (a .u.) (c)(a) ρ (b) FIG. 6. (Color online) Pressure dependence of nematic phase 2. (a) The magnetoresistivity near H2. In order to estimate the pressure dependence of the nematic bump, we interpolated a background across the nematic phase as follows: The magnetoresistance was fittedby polynomials in two regions, above and below H 2(indicated by the lowest arrow in the 12.5 kbar data). In fitting the region above H2, the data in the nematic region were ignored (the region between thetwo upper arrows in the 12.5 kbar data). The data on panel (b) were obtained by subtracting the resulting fitted lines from the measured data. This procedure was adopted purely to remove a backgroundsignal, and we do not assert that the parameters of the polynomials have any particular physical significance. (c) This figure shows the integrated area of the resulting nematic bump. The error bars wereestimated by expanding and contracting the ignored region in the fit. field. In their configuration, because the ∼8 T field was near thecaxis, both ρ/bardblandρ⊥were predominantly measuring the transverse magnetoresistance and, outside of the nematicphase, they agreed well with each other. In our measurement,there are no field ranges where ρ /bardblandρ⊥agree well, and the disagreement grows as the pressure increases. [Compare forexample the lowest temperature curves in Figs. 3(c) and3(d).] That is, ( ρ /bardbl−ρ⊥)/(ρ/bardbl+ρ⊥) is nonzero everywhere. This may be caused by the fact that our measurement configurationdiffers in two significant ways from that of Borzi et al. First, they measured ρ /bardblandρ⊥in the same sample. We are unable to tilt our pressure cell by 90◦, so we measured ρ/bardblandρ⊥ in different samples. We attempted to minimize the effect of this by using samples from the same batch, in the same 235129-4PRESSURE STUDY OF NEMATICITY AND QUANTUM . . . PHYSICAL REVIEW B 88, 235129 (2013) 6 8 10 12 14 16 H(T)00.51.01.52A (a.u.)4.45kbar 8.01kbar 11.21kbar 12.55kbar 13.71kbar 14.53kbar 16.55kbar -1 0 1 23 (H-H1(p))/(H2(p)-H1(p))012345A(a.u.)(a) (b) FIG. 7. (Color online) The T2coefficient of resistivity. Resistiv- ity vs temperature data for our ρ/bardblsample between 100 mK and 500 mK were fitted with a form ρ(T)=ρ◦+AT2.( a )AvsHat each pressure. (b) AvsHreplotted with the field axis rescaled as in Figs. 5and6, and with each plot shifted vertically for clarity. pressure cell, but we cannot rule out a sample-dependent effect, e.g., the currents are at different angles to the principalaxis of the samples. Second, in our experiment the magneticfield is purely in the abplane, so ρ /bardblandρ⊥measure the longitudinal and transverse magnetoresistance, respectively.This too could contribute to the different shape of ρ /bardblandρ⊥at low temperature. Beyond the conventional magnetoresistivity,the applied magnetic field breaks the symmetry in the planeand could also induce “metanematic” anisotropy. Whateverthe explanation, nonzero ( ρ /bardbl−ρ⊥)/(ρ/bardbl+ρ⊥) is probably not, in our measurement configuration, a reliable signatureof nematicity. Nevertheless, a reliable signature of a nematicphase could be an abrupt increase in (ρ /bardbl−ρ⊥)/(ρ/bardbl+ρ⊥)o n entering a nematic phase,10,25as we see at the boundaries of nematic phase 2. It is therefore significant that no dramaticchange in anisotropy is seen at H 1, even near the critical pressure, 13.6 kbar of the QCEP [see Figs. 5(a) and 5(b)]. Thus our magnetoresistance measurement provides no evidence ofa nematic phase at the pressure-induced QCEP of H 1. The bump in the resistivity just above H2was shown by Borzi et al.9to correspond to a nematic phase with a strong (ρ/bardbl−ρ⊥)/(ρ/bardbl+ρ⊥) signature at ambient pressure. We find that this bump is robust against pressure, as seen in Figs. 5 and 6: surprisingly, after subtraction of the background by interpolating across the bump, the size of the peak does notchange with pressure [Figs. 6(b) and 6(c)], within the error. In our earlier susceptibility study, 15we found the peaks in the real part of χ(H) that mark the boundary of this phase. The peak at the lower boundary had the double feature observedby Perry et al. 14(see Fig. 8) and its amplitude depended only4.5 5 5.5 6 6.5 7 H (T)00.511.5χ′ (a.u.)70 mK 1400 mK FIG. 8. (Color online) Real part of susceptibility χ(H)a td i f f e r - ent temperatures. The second and third peaks in 70 mK data mark theentrance and exit of nematic phase 2, respectively, with the second peak being weakly split. weakly on pressure,15which is consistent with this nematic phase being relatively unaffected by pressure. Nematic phase 2 may have different physics from the better- studied nematic phase 1. Nematic phase 1 is associated withthe field-angle-tuned QCEP of H 1.H2at ambient pressure may be close to its QCEP, since at ambient pressure χ/prime/primeshows weak first-order behavior, while susceptibility measurement at0.59 kbar and higher pressures did not observe any χ /prime/primesignal atH2. Regardless of this, a key point is that nematic phase 2 does not screen a metamagnetic QCEP, because it is locatedbeside H 2(see Figs. 1,3,4, and 5). Moreover, our finding that the nematic bump is relatively unaffected by pressure alsoseems to rule out fine tuning to a QCEP as being necessary forformation of this nematic phase. An obvious explanation for the robustness of nematic phase 2, in terms of a symmetry-breaking Fermi surfacereconstruction, would be that the in-plane applied magneticfield already breaks fourfold symmetry via coupling of theelectron momentum to the applied field 25as well as through magnetoelastic coupling, which normally stretches a crystal parallel to /vectorHand shrinks it perpendicular to /vectorH. The resulting distortion of the energy bands would lower the degeneracy ofany van Hove singularity in the Brillouin zone from one set offourfold degenerate to a pair of twofold degenerate van Hovesingularities, increasing the tendency for the Fermi surface toreconstruct in a two-stage process. The double peaks in χ /prime(H) atH2at low temperatures may arise from such a splitting of the van Hove singularity. However, it is quite clear from thenarrowness of the H 2metamagnetic transition compared with the width of nematic phase 2 region that any such splitting istiny compared with the width nematic phase 2, and that thevan Hove singularity remains below the lower boundary ofnematic phase 2. The physics of nematic phase 2 remains amystery. There is evidence that the resistivity in nematic phase2 is affected by the angle between the current and the principalaxis. More extended experiments on the magnetoresistancewith different angles between the current and the principalaxis are needed in order to explore this behavior. Moreover,quantum oscillation measurements under pressure would helpto show how the Fermi surface changes across this nematic 235129-5SUN, WU, GRIGERA, PERRY , MACKENZIE, AND JULIAN PHYSICAL REVIEW B 88, 235129 (2013) phase, which would should also enhance understanding of the underlying physics. V . SUMMARY In conclusion, the relation of nematicity and quantum crit- icality has been studied in Sr 3Ru2O7by applying hydrostatic pressure when the magnetic field is in the abplane. There is no evidence of a nematic phase at H1when there is a QCEP induced by hydrostatic pressure. This is in contrastwith the appearance of the nematic when the QCEP is obtainedby field-angle tuning. Another nematic phase, persistent withpressure, does not occur at a metamagnetic quantum criticalpoint. These two findings suggest that, in contrast to quantumcritical superconductors, the nematic phase is not driven byquantum criticality. ACKNOWLEDGMENTS We are grateful to H. Y . Kee for helpful discussions. This research was supported by the NSERC (Canada) and EPSRC(UK). S.A.G. was partially supported by the Royal Society(UK), CONICET, and ANPCyT (Argentina), and A.P.M.acknowledges support from a Royal Society Wolfson ResearchMerit Award. APPENDIX In our previous pressure experiment,15acmagnetic sus- ceptibility χac(H,ω ) was measured at low frequency. There are different possible regimes for χacin the low-frequency limit depending on the value of ωcompared with τ−1,t h e inverse characteristic time of the system: if ω/lessmuchτ−1,χac tends to the isothermal susceptibility, χT=(∂M ∂H)T.O nt h eopposite limit, ω/greatermuchτ−1, the system has no time to exchange energy with its surroundings and what is measured is the adiabatic susceptibility χS, which is usually smaller in size. A good rule of thumb, based on the analysis by Casimir anddu Pr ´e, 26is to work in the very low frequency regime where the imaginary part of χac(H,ω ) is negligible. In this regime, χac(H,ω )≈(∂M ∂H)Tand the magnetization M(H,T ) can, in principle, be obtained by simple integration over the field H. In practice, in addition to keeping a low frequency, this isdifficult to do because the filling factor of the pickup coil andthe overall gain of the system are not known with sufficientaccuracy. The integrated susceptibility in Fig. 2was therefore calculated by the following equation: M(H)=/integraldisplay (aχ /prime ac(H)+b)dH, (A1) where aandbwere chosen so that M(H) matches magnetiza- tion measurements by Perry et al. at 70 mK.14 In addition to the constants aandb, there is another adjustment to be made since the rise in M(H)a tH1is of first order. It can be seen that the first peak is much largernear the critical temperature of 1550 mK (see Fig. 8). At lower temperatures the dynamical response is affected by thephysics of a first-order metamagnetic transition, in particularby domain wall pinning, with the consequent growth of thecharacteristic time, τ. In this region, the imaginary part of χ acis no longer negligible, and the real part of χacdecreases towards χS. Thus at the lowest temperature, 70 mK, while the size of the second peak is appreciable, the first peak hasbecome very small. In order to compensate for the signal lossand achieve an agreement with the measured M(H), we add a first-order jump at H 1in Fig. 2. 1E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and A. P. Mackenzie, Annu. Rev. Condens. Matter Phys. 1,153(2010 ). 2M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 82,394(1999 ). 3J. Chu et al. ,Science 329,824(2010 ). 4Y . Ando, K. Segawa, S. Komiya, and A. N. Lavrov, Phys. Rev. Lett. 88,137005 (2002 ). 5V. H i n k ov et al. ,Science 319,597(2008 ). 6S. A. Grigera et al. ,Science 306,1154 (2004 ). 7C .S t i n g l ,R .S .P e r r y ,Y .M a e n o ,a n dP .G e g e n w a r t , Phys. Rev. Lett. 107,026404 (2011 ). 8H. Shaked, J. Jorgensen, O. Chmaissem, S. Ikeda, and Y . Maeno, J. Solid State Chem. 154,361(2000 ). 9R. A. Borzi et al. ,Science 315,214(2007 ). 10S. Raghu et al. ,P h y s .R e v .B 79,214402 (2009 ). 11A. P. Mackenzie, J. A. N. Bruin, R. A. Borzi, A. Rost, and S. A. Grigera, Physica C 481,207(2012 ). 12Strictly speaking, whenever the field angle is not 0◦, these phases are “metanematic” (Ref. 25) since the applied field already breaks therotational symmetry of the crystal, but we use the term “nematic”throughout this paper.13A. W. Rost, R. S. Perry, J.-F. Mercure, A. P. Mackenzie, and S. A.Grigera, Science 325,1360 (2009 ). 14R. S. Perry et al. ,J. Phys. Soc. Jpn. 74,1270 (2005 ). 15W. Wu et al. ,P h y s .R e v .B 83,045106 (2011 ). 16S. Grigera et al. ,P h y s .R e v .B 67,214427 (2003 ). 17H.-Y . Kee and Y . B. Kim, Phys. Rev. B 71,184402 (2005 ). 18C. M. Puetter, J. G. Rau, and H.-Y . Kee, P h y s .R e v .B 81,081105 (2010 ). 19E. P. Wohlfarth and P. Rhodes, Philos. Mag. 7,1817 (1962 ). 20W.-C. Lee and C. Wu, Phys. Rev. B 80,104438 (2009 ). 21C. M. Puetter, S. D. Swiecicki, and H.-Y . Kee, New J. Phys. 14, 053027 (2012 ). 22N. Mathur et al. ,Nature (London) 394,39(1998 ). 23P. R. Ponte, D. Cabra, and N. Grandi, E u r .P h y s .J .B 86,85 (2013 ). 24H. Yaguchi, R. S. Perry, and Y . Maeno, AIP Conf. Proc. 850,1203 (2006 ). 25C. Puetter, H. Doh, and H.-Y . Kee, P h y s .R e v .B 76,235112 (2007 ). 26H. Casimir and F. Du Pr ´e,Physica 5,507(1938 ). 235129-6

PhysRevB.98.245402.pdf

PHYSICAL REVIEW B 98, 245402 (2018) Distribution of waiting times between electron cotunneling events Samuel L. Rudge and Daniel S. Kosov College of Science and Engineering, James Cook University, Townsville, QLD, 4811, Australia (Received 21 June 2018; revised manuscript received 12 November 2018; published 4 December 2018) In the resonant tunneling regime, sequential processes dominate single-electron transport through quantum dots or molecules that are weakly coupled to macroscopic electrodes. In the Coulomb blockade regime, however,cotunneling processes dominate. Cotunneling is an inherently quantum phenomenon and thus gives rise tointeresting observations, such as an increase in the current shot noise. Since cotunneling processes are inherentlyfast compared to the sequential processes, it is of interest to examine the short time behavior of systems wherecotunneling plays a role, and whether these systems display nonrenewal statistics. We consider three questionsin this paper. Given that an electron has tunneled from the source to the drain via a cotunneling or sequentialprocess, what is the waiting time until another electron cotunnels from the source to the drain? What are thestatistical properties of these waiting time intervals? How does cotunneling affect the statistical properties of asystem with strong inelastic electron-electron interactions? In answering these questions, we extend the existingformalism for waiting time distributions in single-electron transport to include cotunneling processes via ann-resolved Markovian master equation. We demonstrate that for a single resonant level, the analytic waiting time distribution including cotunneling processes yields information on individual tunneling amplitudes. Forboth a SRL and an Anderson impurity deep in the Coulomb blockade, there is a nonzero probability for twoelectrons to cotunnel to the drain with zero waiting time in between. Furthermore, we show that at high voltages,cotunneling processes slightly modify the nonrenewal behavior of an Anderson impurity with a strong inelasticelectron-electron interaction. DOI: 10.1103/PhysRevB.98.245402 I. INTRODUCTION With the ever-present search for smaller transistors and the advent of modern technologies such as quantum computing,the world in recent years has turned its gaze inward to probeelectron transport through nanoscale devices, where a funda-mental understanding of quantum dynamics is required. Thishas yielded intriguing experimental and theoretical results:for example, single-molecule transistors, quantum heat en- gines, and spintronics [ 1–3]. Of particular interest in quantum nanoscale systems is the potential for encountering micro-scopic current fluctuations and phenomena that are classi-cally forbidden, such as the existence of electron transportthrough virtual quantum states that temporarily violate energyconservation laws; both of which form the focus of thispaper. Electron transport through quantum systems can display a phenomena known as cotunneling . Inelastic cotunneling was first proposed theoretically by Averin and Odintsov [ 4] and confirmed experimentally shortly after the theoreticalprediction by Geerligs et al. [5] with the introduction of the modern combined inelastic and elastic theory detailedsimultaneously by Averin and Nazarov [ 6]. In contrast to sequential tunneling , which describes single-electron tunnel- ing events and can essentially be described classically, co-tunneling is a coherent quantum process that involves thetunneling of an electron from the source to the drain (or viceversa) through an intermediate “virtual” state, which mayor may not be classically forbidden [ 7–9]. Elastic cotunnel- ing leaves the system with the same energy, while inelasticcotunneling leaves the intermediate quantum system in an excited state. The common explanation is that cotunneling isan example of the uncertainty principle /Delta1t/Delta1E∼¯h; energy conservation can be violated only if the electron spends a suf-ficiently short time in the intermediate virtual state, althoughin recent years this notion has been challenged by Romito andGefen [ 10]. Cotunneling processes dominate transport in the Coulomb blockade regime, as the electronic energy levels are pushedoutside the voltage bias window and sequential tunnelingis exponentially suppressed. Hence, cotunneling manifestsexperimentally as a small current in the Coulomb blockade regime, and as a small correction to the sequential current in the resonant tunneling regime [ 5,11]. Theoretical research into cotunneling has investigated its effect on transport in sys-tems with inelastic scatterings, such as electron-electron andelectron-phonon interactions [ 12–14]. Additional cotunneling research has focused on heat conductance [ 15,16], transport in double quantum dots [ 17,18], and inelastic cotunneling spectroscopy [ 19,20]. Recently, multiple authors have studied the noise and full counting statistics (FCS) of cotunnelingphenomena in an attempt to explore its effect on currentfluctuations [ 21–31]. Such investigations have demonstrated that inelastic cotunneling transport induces super-Poissonianshot noise for a variety of systems, which is in agreement withexperimental measurements [ 32–34]. Alongside the zero-frequency noise and FCS, the waiting time distribution (WTD) has been shown to be a useful tool fordescribing current fluctuations in quantum nanoscale systems,as it contains information complementary to that found in 2469-9950/2018/98(24)/245402(13) 245402-1 ©2018 American Physical SocietySAMUEL L. RUDGE AND DANIEL S. KOSOV PHYSICAL REVIEW B 98, 245402 (2018) other statistics [ 35–44]. In contrast to current cumulants, which require theoretical calculations over long-time inter-vals, WTDs can reveal interesting short-time physics that mayotherwise be inaccessible. Of particular interest is observing aviolation of renewal statistics, where the assumption is thatw(τ 1,τ2)=w(τ1)w(τ2). Nonrenewal statistics is character- ized by short-time correlations between subsequent waitingtimes, and is thus invisible in the current cumulants. Perhapsthe recent interest has been spurred onward in part by thedevelopment of real-time single electron detection techniques,which have enabled experimental measurement of micro-scopic current fluctuations for many different quantum sys-tems [ 45–48]. However, there remain experimental difficulties in measuring electron tunnelings via virtual processes dueto the collapse of the intermediate state [ 49,50]. A possible method for experimentally accessing waiting times includingquantum processes is the reconstruction of the WTD fromlow-order charge correlation functions [ 51]. Although there are multiple definitions of the WTD in statistics [ 52], in the context of quantum transport it is the conditional probabilitydensity that, given an extra electron was counted in the drainelectrode at time t, another extra electron was counted in the drain at time t+τ, where no intermediate tunneling events to the drain are allowed. Historically, WTDs have been extensively used in quan- tum optics as a statistical tool [ 36,37] and they were in- troduced to mesoscopic quantum transport by Brandes, whocalculated WTDs by defining jump operators from a quan-tum master equation [ 35]. The master equation method for calculating WTDs has since been applied to a diverserange of scenarios, such as systems with electron-electroninteractions, electron-phonon interactions, coherent internaltransport, non-Markovian quantum transport, and spintronics[35,38–41,53–58]. Alongside the master equation approach, there exist various techniques for calculating WTDs in meso-copic transport. For example, Albert et al. [44] developed a scattering matrix approach suitable for fully coherent trans-port, and described single channel and multichannel transport[44,59], transport through superconducting junctions [ 54,60], and transport of electron pulses [ 43,61]. Despite this success, the scattering matrix approach is unable to calculate WTDsoutside of the steady state; thus, nonequilibrium Green’sfunctions are used to describe coherent transport in the tran-sient regime [ 62–64]. However, so far WTDs have not been used to study the statistical properties of electron cotunnelingevents for systems with strong electron-electron interactions.Although the scattering matrix approach and nonequilibriumGreen’s functions are applicable to coherent transport, andthus seem tailor-made for describing cotunneling, they strug-gle to include strong inelastic scatterings in the quantum sys-tem; thus, in this paper we use the master equation technique. We study the WTD in an Anderson impurity for successive tunnelings to the drain, including cotunneling, and compareit to the WTD for successive tunnelings to the drain for onlysequential tunneling processes. We first develop a systematicmethod for extending the current master equation approachfor WTDs developed by Brandes [ 35] to include cotunneling processes, and then demonstrate its use for transport throughan Anderson impurity, as well as the limiting case of strongCoulomb repulsion and no level splitting when the systembehaves as a single resonant level (SRL). In doing so, we examine the relationship between inelastic scatterings and theinherently coherent quantum cotunneling process, as well asthe effect cotunneling has on nonrenewal statistics. The master equation approach to quantum transport is a powerful method for analyzing quantum electron transportthrough mesoscopic systems [ 65–69]. Although the full mas- ter equation is useful for describing quantum effects suchas interference [ 70], decoherence between double quantum dots [ 71], electron transport through quantum dot attached to superconducting leads [ 72,73], and driven quantum transport [74], in many cases the transport is incoherent and thus is effectively described by rate equations [ 75– 77]; this is the approach taken in this paper. To connect this formalism towaiting times, we will in fact have to work with the n-resolved master equation [ 66,76,78]. The transition rates in the master equation are calculated using the T-matrix approach: a perturbation expansion aroundthe tunneling coupling H T. Sequential tunneling corresponds to the lowest order of this expansion, and cotunneling pro-cesses correspond to next-to-leading order in H T: first and second order in the tunneling coupling strength γ, respec- tively. Cotunneling rates developed from a purely second-order perturbative expansion about H Tare well-known to formally diverge due to higher-order tunneling effects notbeing taken into account. To overcome this we follow theapproach first developed by Averin [ 79], and extended to the T-matrix context by Turek and Matveev [ 14] and Koch et al. , [12,13] of introducing a finite width to the energy of the intermediate virtual state. Specifically, we closely follow themethodology of Koch et al. and obtain similar analytic results, although we note that we focus on the WTDs associated withan electron-electron interaction whereas Koch et al. focus on an electron-vibration interaction in the limit U→∞ . Once the rates are defined, and similar to Thomas and Flindt’sapproach, we start with an n-resolved master equation, then derive the WTD from the idle time probability and showthat for forward tunneling only it reduces to the methodintroduced by Brandes, albeit with a nonintuitive Liouvilliansplitting [ 38]. We demonstrate that, likewise to the WTD for sequential tunneling through a SRL, the WTD including cotunnelingoffers information on the individual electrode coupling pa-rameters [ 35]. Furthermore, for an Anderson impurity, cotun- neling processes slightly increase the nonrenewal behavior;this is evident in the comparison of the correlation betweensubsequent waiting times, which is largely controlled by thestrength of the Coulomb repulsion. However, the use of themethod presents difficulties in two key areas: when the level isinside the voltage bias window and when backward tunnelingprocesses are included. The paper is organized as follows. Section IIoutlines the construction of the master equation and the derivation ofthe WTD including cotunneling. Section IIIdetails analytic results for cotunneling through an Anderson impurity and aSRL. Section IVoutlines the main results and discusses future work. The Appendix details calculations and derivations usedthroughout the paper. Throughout this paper we use natural units: ¯ h=k e= e=1. 245402-2DISTRIBUTION OF WAITING TIMES BETWEEN … PHYSICAL REVIEW B 98, 245402 (2018) II. METHODS A. Quantum rates for cotunneling processes In this paper, we examine the transport of electrons, mod- eled as fermions with spin. Let us consider a nanoscalequantum system weakly coupled to two macroscopic metalelectrodes: the source and drain. The source and drain areheld at different chemical potentials to cause a voltage biasacross the system and induce a nonequilibrium state. For sucha setup, the Hamiltonian is H=H S+HD+HM+HT. (1) The source and drain are modeled as a sea of noninteracting electrons with the Hamiltonians HS=/summationdisplay s,σεs,σa† s,σas,σ andHD=/summationdisplay d,σεd,σa† d,σad,σ.(2) The operators a† s/d,σ(as/d,σ) represent creation (annihilation) of an electron in the single-particle state s/dwith spin σand free energy εs/d. We examine transport through an Anderson impurity, which is described by the Hamiltonian HM=/summationdisplay σεσa† σaσ+Ua† ↑a↑a† ↓a↓, (3) where the operator a† σ(aσ) creates (annihilates) an electron with spin σon the single-particle level with energy εσ, and Uis the Coulomb repulsion. When U→∞ and there are no spin split energy levels, the system can be modeled by a SRL: HM=εa†a. (4) The interaction between the nanoscale quantum system and the macroscopic electrodes is described by the Hamiltonian HT=tS/summationdisplay s,σ(a† s,σaσ+a† σas,σ)+tD/summationdisplay d,σ(a† d,σaσ+a† σad,σ), (5) where tSandtDare tunneling amplitudes between the molecule and source and drain electrode, respectively. The quantum system has four states; it can either be empty (/angbracketleft0|), occupied by a single spin up electron ( /angbracketleft↑ |), occupied by a single spin down electron ( /angbracketleft↓ |), or occupied by a spin-up and spin-down electron ( /angbracketleft2|). These states have the associated probabilities P0=/angbracketleft0|ρ|0/angbracketright,P↑=/angbracketleft ↑ |ρ|↑/angbracketright,P↓=/angbracketleft ↓ |ρ|↓/angbracketright, andP2=/angbracketleft2|ρ|2/angbracketright, where ρis the reduced density matrix of the Anderson impurity. The dynamics of the system is defined by a quantum master equation, which is constructed from quantum ratesassociated with electron tunneling processes. The rate oftransforming from reduced system state mto reduced system statenis denoted /Gamma1 nm. To calculate the /Gamma1nm,w eu s et h e T-matrix approach, which is suitable as it provides a directmethod for calculating transition rates between eigenstatesof quantum many-body systems. Cotunneling has previouslybeen explored via a comprehensive real-time diagrammaticmethod [ 26,30,80,81]; however, the T-matrix approach is a suitable approximation for this more rigorous method whenthe dynamics does not exhibit non-Markovian phenomena[24] This occurs for large temperatures k BT/greatermuchγalongside a large gap between the Fermi energies of the baths and theenergy levels participating in the transport δ=|eV−ε|/greatermuchγ, a condition that is met in the Coulomb blockade regime.Our calculations are all performed with γ=0.5k BT, which falls within this regime. Furthermore, the construction ofthe WTD requires that backscattering from the drain is notincluded, and so in the tunneling regime we necessarily haveγ/lessmuchk BT/lessmuchδ. Finally, using the rate equation requires the secular approximation; coherences in the off-diagonals of thefull density matrix are ignored. For the sake of self-completeness and to introduce relevant notations used throughout the paper, below we explicitly de-rive the sequential and cotunneling rates used in the quantummaster equation. Here, we briefly summarize the methodoutlined by Bruus and Flensberg [ 8,82]. First, the Hamiltonian is reformulated as H(t)=H S+HD+HM+HTeηt, (6) where the time-independent part H0=HS+HD+HMhas a trivial but fast time-evolution e−iH0t, and the complex but slow time-evolution is due to the interaction HTeηt, which is treated as a perturbation. The time factor eηtensures that the perturbation is turned on adiabatically at t=− ∞ by assuming that ηis an infinitesimal positive real number. The starting point for the T-matrix approach is the prob- ability Pf(t) that the system is in state |f/angbracketrightat time tgiven that time t=0 it was in state |i/angbracketright, which is just the square of their overlap; and from here, the transition rate between thetwo states is the time derivative of P f(t): /Gamma1fi=d dt/vextendsingle/vextendsingle/angbracketleftf|i(t)/angbracketright/vextendsingle/vextendsingle2. (7) Using the interaction picture, Eq.( 7) is transformed to /Gamma1fi=2π|/angbracketleftf|T|i/angbracketright|2δ(Ei−Ef), (8) where the T-matrix is T=HT+HT1 Ei−H0+iηHT +HT1 Ei−H0+iηHT1 Ei−H0+iηHT+.... (9) The sequential tunneling regime corresponds to second order in HTin the transition rates: the first linear term in the T-matrix. So the sequential rates are /Gamma1fi=2π|/angbracketleftf|HT|i/angbracketright|2δ(Ei−Ef), (10) which is just the standard Fermi’s Golden Rule. In the many-body configuration in the full Fock space, the initial and final states are tensor products of the discretesystem states (molecular or quantum dot) and continuous elec-trode states: |m/angbracketright⊗ |i S/D/angbracketrightand|n/angbracketright⊗ |fS/D/angbracketright, with eigenen- ergies Em+εis/dandEn+εfs/drespectively. Consequently, there are multiple final and initial states that correspond toa system state of |m/n/angbracketright; they must be summed over, and the initial states weighted with a thermal distribution function WS/D im: /Gamma1S/D nm=2π/summationdisplay fS/D,iS/D|/angbracketleftfS/D|/angbracketleftn|HS/D T|m/angbracketright|iS/D/angbracketright|2WS/D im ×δ(Em−En+εiS/D−εfS/D). (11) 245402-3SAMUEL L. RUDGE AND DANIEL S. KOSOV PHYSICAL REVIEW B 98, 245402 (2018) At this point, we can now calculate the sequential rates for electron tunneling between the electrodes and the system: /Gamma1S/D σ0=γS/DnF(εσ−μS/D), (12) /Gamma1S/D 0σ=γS/D(1−nF(εσ−μS/D)), (13) /Gamma1S/D σ2=γS/D(1−nF(εσ+U−μS/D)), and (14) /Gamma1S/D 2σ=γS/D(εσ+U−μS/D), (15) where γS/D=2π|tS/D|2ρ(εS/D) andρ(εS/D) is the density of states for the source and drain electrodes, which is assumed tobe constant. In the limiting case of a SRL, the rates reduce to /Gamma1S/D 10=γS/DnF(ε−μS/D) and (16) /Gamma1S/D 01=γS/D(1−nF(ε−μS/D)). (17) Throughout the paper we use a symmetric coupling, such thatγS=γD=γ 2.T h e nF(ε−μS/D) are the Fermi-Dirac distributions for the source and drain electrodes: nF(ε−μS/D)=1 1+e(ε−μS/D)β, (18) where β=1 kBT. When the electronic level is within the bias window and in the limit of infinite source-drain bias, which isachieved by making the voltage μ S−μDlarge, the configura- tion undergoes forward tunneling only: that is, from the sourceto the molecule or from the molecule to the drain. However,in the Coulomb blockade regime the electronic levels areoutside the bias window, regardless of the large voltage. Toreconcile the two scenarios, we note that their combinedprocesses are tunneling from the source to the molecule, fromthe molecule to the source, and from the molecule to thedrain. In effect, the total sequential rates for an Andersonimpurity reduce to /Gamma1 σ0=/Gamma1S σ0,/Gamma10σ=/Gamma1S 0σ+/Gamma1D 0σ,/Gamma12σ=/Gamma1S 2σ, and/Gamma1σ2=/Gamma1S σ2+/Gamma1D σ2. Similarly, the total sequential rates for aS R La r e /Gamma110=/Gamma1S 10and/Gamma101=/Gamma1S 01+/Gamma1D,w h e r ew eh a v e adopted the shorthand /Gamma1D=/Gamma1D 01. The next-to-leading term in the T-matrix expansion is sec- ond order in the tunneling coupling γ, which is fourth order in HTin the rate expression, and describes cotunneling effects. For an Anderson impurity in the infinite bias limit, there aremultiple cotunneling pathways, which can be categorized aseither inelastic or elastic. Elastic cotunneling processes leave the system in the same energetic state; for example, an electron tunnels into an emptysystem from the source and another electron tunnels out tothe drain in the same process, leaving the molecule empty andwith an extra electron in the drain. We denote the transitionrate of this process /Gamma1 SD 00, where SDimplies that an electron is moved from the source to the drain. Note that this processcan occur for ↑or↓electrons, so there are actually two pathways contained within the rate /Gamma1 SD 00. Similarly, we define /Gamma1SD 22as the rate of elastically tunneling through an originally doubly occupied system from the source to the drain. Inthis scenario, the first process must be an electron tunnelingfrom the molecule to the drain, which is then replaced by anelectron from the source. Again, the process can occur for either ↑or↓electrons, so the rate contains contributions from both pathways. Finally, we define /Gamma1 SD σσas the rate of elastically cotunneling from the source to the drain through an originallyσoccupied system. This process can occur via an original tunneling of a ¯ σelectron from the source to the molecule followed by a subsequent tunneling of a ¯ σelectron from the molecule to the drain, or by the σelectron tunneling to the drain first; so it too has two contributions to the rate. Sincethe system is experiencing infinite bias voltage, cotunnelingprocesses that move an electron from the drain to the sourcedo not contribute to the transport. Inelastic cotunneling processes leave the system occupied by the same number of electrons, but in a different energystate. For an Anderson impurity, the only inelastic cotunnelingprocesses are those that transform the system from beingoccupied by a single σelectron to being occupied by a single ¯σelectron. The rate of moving an electron from the source to the drain and changing the system occupation from σfrom ¯σis then /Gamma1 SD ¯σσ. This rate has two processes as well; either a σ electron tunnels from the molecule to the drain and is replacedfrom the source by a ¯ σelectron, or a ¯ σelectron tunnels from the source to the molecule and then a σelectron tunnels from the molecule to the drain. We also define inelasticcotunneling processes involving the same electrode: /Gamma1 SS ¯σσand /Gamma1DD ¯σσ. Although these processes do not move electrons across the system, they affect the occupation probabilities of theimpurity and thus are included in the transport description. The wide variety of cotunneling rates involved in transport through an Anderson impurity are all derived by going tofourth order in H T, so that Eq. ( 8) becomes /Gamma1αβ n/primen=2πlim η→0+/summationdisplay α,β=S,D/summationdisplay i,f/vextendsingle/vextendsingle/vextendsingle/angbracketleftf|/angbracketleftn/prime|Hβ T1 Ei,n−H0+iη ×Hα T|n/angbracketright|i/angbracketright/vextendsingle/vextendsingle/vextendsingle2 Wα i,nWβ i,n×δ(εi−εf), (19) where n=n/primefor elastic cotunneling processes and n/negationslash=n/primefor inelastic cotunneling processes, and the notation recognizesthe fact that cotunneling always leaves the system occupiedby the same number of electrons as it was before the process. It is assumed that the thermal probabilities for the source and drain are independent and so can be factored: W S inWD in. Additionally, we assume weak coupling, so that the elec-trode thermal probabilites are independent of the state of thequantum system at time t=t 0. The imaginary component iηin Eq. ( 19) ensures that, due to tunneling processes not included in a second-order expansion, the intermediate energyof the intermediate virtual state has a finite width, and withits inclusion divergent integrals in the rate are avoided. Theinclusion of iη, and the assumption that it is O(γ), forms the first part of a regularization procedure necessary to calculatecotunneling rates. The second component of regularization isremoving any parts of the rate that are O(γ), as they corre- spond to a sequential tunneling. These appear because anycotunneling process can also be achieved via two sequentialtunneling processes. The regularization procedure we followis that detailed by Koch et al. [12,13], which is equivalent to the method outlined by Turek and Matveev [ 14], where the finite energy width was first noted by Averin [ 79]. Evaluating 245402-4DISTRIBUTION OF WAITING TIMES BETWEEN … PHYSICAL REVIEW B 98, 245402 (2018) Eq. ( 19), applying the regularization procedure and taking the appropriate limits, one obtains the general form of the elastic cotunneling rates for an Anderson impurity as /Gamma1SD nn=γSγDnB(μD−μS)/bracketleftbiggβ 4π2/Ifractur/braceleftbigg ψ(1)/parenleftbigg1 2+iβ 2π(μD−E1)/parenrightbigg −ψ(1)/parenleftbigg1 2+iβ 2π(μS−E1)/parenrightbigg +ψ(1)/parenleftbigg1 2+iβ 2π(μD−E2)/parenrightbigg −ψ(1)/parenleftbigg1 2+iβ 2π(μS−E2)/parenrightbigg/bracerightbigg ±1 π(E1−E2)/Rfractur/braceleftbigg ψ/parenleftbigg1 2−iβ 2π(μS−E2)/parenrightbigg −ψ/parenleftbigg1 2−iβ 2π(μS−E1)/parenrightbigg −ψ/parenleftbigg1 2−iβ 2π(μD−E2)/parenrightbigg +ψ/parenleftbigg1 2−iβ 2π(μD−E1)/parenrightbigg/bracerightbigg/bracketrightbigg , (20) where E1andE2refer to the energies of the tunneling pathways involved in the process and the ±is negative only for /Gamma1SD σσ. Furthermore, the transition rates defined in Eq. ( 20) use the digamma ψ(x) and trigamma ψ(1)(x) functions, as well as the Bose-Einstein distribution function nB(μD−μS): nB(μD−μS)=1 e(μD−μS)β−1. (21) The inelastic cotunneling rates are similarly defined: /Gamma1αβ ¯σσ=γαγβnB(μβ−μα−εσ+ε¯σ)/bracketleftbiggβ 4π2/Ifractur/braceleftbigg ψ(1)/parenleftbigg1 2+iβ 2π(μβ−(εσ+U))/parenrightbigg −ψ(1)/parenleftbigg1 2+iβ 2π(μα−(ε¯σ+U))/parenrightbigg +ψ(1)/parenleftbigg1 2+iβ 2π(μβ−εσ)/parenrightbigg −ψ(1)/parenleftbigg1 2+iβ 2π(μα−ε¯σ)/parenrightbigg/bracerightbigg −1 πU/Rfractur/braceleftbigg ψ/parenleftbigg1 2−iβ 2π(μα−ε¯σ)/parenrightbigg −ψ/parenleftbigg1 2−iβ 2π(μα−(ε¯σ+U))/parenrightbigg −ψ/parenleftbigg1 2−iβ 2π(μβ−εσ)/parenrightbigg +ψ/parenleftbigg1 2−iβ 2π(μβ−(εσ+U))/parenrightbigg/bracerightbigg/bracketrightbigg . (22) For a SRL, the number of cotunneling processes is much more limited; either an electron tunnels into the empty levelfrom the source and another electron tunnels out to the drainin the same quantum process, or an electron tunnels out fromthe level into the drain and is replaced by an electron fromthe source in the same quantum process. The two processes have transition rates /Gamma1 (2) 00and/Gamma1(2) 11, respectively, and one can show that /Gamma1(2) 00=/Gamma1(2) 11=/Gamma1(2). Since the same molecu- lar energy level is filled and emptied, both processes areelastic: /Gamma1 (2)=β/Gamma1S/Gamma1D 4π2nB(μD−μS)/Ifractur/braceleftbigg ψ(1)/parenleftbigg1 2+iβ 2π(ε−μS)/parenrightbigg −ψ(1)/parenleftbigg1 2+iβ 2π(ε−μD)/parenrightbigg/bracerightbigg . (23) The details of the derivations for Eqs. ( 20), (22), and ( 23) are in the Appendix. From here it is tempting to construct the standard rate equa- tion for occupation probabilities of the impurity. However,since elastic cotunneling rates do not change the state of thequantum system, they do not contribute to the rate equationfor the system state probabilities. Instead, one must considerthen-resolved system state probabilities. B.n-resolved master equation The master equation can be resolved upon the number of electrons transferred to the drain; so P0(n,t) is the probability that the system is empty at time tand that nelectrons weretransferred to the drain in the interval [0 ,t], and similarly forPσ(n,t) and P2(n,t). For the infinite bias regime n= 0,1,2,3,...,+∞. Thus, the total probability that nelectrons were transferred by time tis P(n,t)=(I,P(n,t)) (24) =P0(n,t)+P↑(n,t)+P↓(n,t)+P2(n,t),(25) where Iis the identity vector, I=/bracketleftbig1111/bracketrightbig , (26) andP(n,t) is the probability vector, P(n,t)=⎡ ⎢⎢⎢⎣P 0(n,t) P↑(n,t) P↓(n,t) P2(n,t)⎤ ⎥⎥⎥⎦. (27) Then-resolved Markovian master equation follows the general form, ˙P(n,t)=/summationdisplay n/primeL(n−n/prime)P(n,t). (28) For the tunneling interaction defined in Eq. ( 5), each nis connected only to its neighboring values n/prime=n,n±1 and for an Anderson impurity in the infinite bias regime, includ-ing cotunneling processes, the n-resolved master equation is 245402-5SAMUEL L. RUDGE AND DANIEL S. KOSOV PHYSICAL REVIEW B 98, 245402 (2018) intuitively ˙P(n,t)=⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣−/parenleftbig /Gamma1 S ↑0+/Gamma1S ↓0+/Gamma1SD 00/parenrightbig /Gamma1S 0↑ /Gamma1S 0↓ 0 /Gamma1S ↑0 −/parenleftbig /Gamma10↑+/Gamma1S 2↑+/Gamma1SD ↑↑+/Gamma1(2) ↓↑/parenrightbig /Gamma1SS ↑↓+/Gamma1DD ↑↓ /Gamma1S ↑2 /Gamma1S ↓0 /Gamma1SS ↓↑+/Gamma1DD ↓↑ −/parenleftbig /Gamma10↓+/Gamma1S 2↓+/Gamma1SD ↓↓+/Gamma1(2) ↑↓/parenrightbig /Gamma1S ↓2 0 /Gamma1S 2↑ /Gamma1S 2↓ −/parenleftbig /Gamma1↑2+/Gamma1↓2+/Gamma1SD 22/parenrightbig⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦P(n,t) +⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣/Gamma1 SD 00/Gamma1D 0↑/Gamma1D 0↓ 0 0/Gamma1SD ↑↑/Gamma1SD ↑↓/Gamma1D ↑2 0/Gamma1SD ↓↑/Gamma1SD ↓↓/Gamma1D ↓2 000 /Gamma1SD 22⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦P(n−1,t). (29) Here, we have excluded those rates that involve back-tunneling processes from the drain, as they have a negligible contribution in the infinite bias regime. Additionally, we use the notation /Gamma1 (2) ¯σσ=/Gamma1SS ¯σσ+/Gamma1DD ¯σσ+/Gamma1SD ¯σσ. (30) Evidently, the n-resolved master equation is an infinite set of coupled equations since n=0,1,2,...,+∞.T os o l v e ,w e use the elegant idea, proposed first by Nazarov and extended to master equations by Bagrets and Nazarov, of introducing acontinuous counting field χ, with 0 /greaterorequalslantχ/greaterorequalslant2π[83,84]: P(χ,t)=/summationdisplay neinχP(n,t), and (31) P(n,t)=1 2π/integraldisplay2π 0e−inχP(χ,t)dχ. (32) Multiplying Eq. ( 29)b yeinχand transforming/summationtext neinχP(n−1,t)→/summationtext mei(m+1)χP(m,t), one obtains the n-resolved master in χ-space in the form ˙P(χ,t)=L(χ)P(χ,t): d dt⎡ ⎢⎢⎢⎣P 0(χ,t) P↑(χ,t) P↓(χ,t) P2(χ,t)⎤ ⎥⎥⎥⎦=⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−/parenleftbig /Gamma1 S ↑0+/Gamma1S ↓0/parenrightbig +/Gamma1SD 00(eiχ−1)/Gamma1S 0↑+/Gamma1D 0↑eiχ/Gamma1S 0↓+/Gamma1D 0↓eiχ0 /Gamma1S ↑0−/parenleftbig /Gamma10↑+/Gamma1S 2↑+/Gamma1(2) ↓↑/parenrightbig +/Gamma1SD ↑↑(eiχ−1)/Gamma1SS ↑↓+/Gamma1DD ↑↓+/Gamma1SD ↑↓eiχ/Gamma1S ↑2+/Gamma1D ↑2eiχ /Gamma1S ↓0 /Gamma1SS ↓↑+/Gamma1DD ↓↑+/Gamma1SD ↓↑eiχ−/parenleftbig /Gamma10↓+/Gamma1S 2↓+/Gamma1(2) ↑↓/parenrightbig +/Gamma1SD ↓↓(eiχ−1)/Gamma1S ↓2+/Gamma1D ↓2eiχ 0 /Gamma1S 2↑ /Gamma1S 2↓−/parenleftbig /Gamma1↑2+/Gamma1↓2/parenrightbig +/Gamma1SD 22(eiχ−1)⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡ ⎢⎢⎢⎣P 0(χ,t) P↑(χ,t) P↓(χ,t) P2(χ,t)⎤ ⎥⎥⎥⎦.(33) Equation ( 33) has the formal solution: P(χ,t)=eL(χ)tP(χ,0), (34) where the inital condition is P(χ,0)=P(n=0,0), since it is assumed that electron counts are monitored after t=0. We also assume that the system was prepared in the steady stateatt=0, so that P(n=0,0)=¯Pwith ¯Pbeing a null vector of the standard Liouvillian: L(0)¯P=0. (35)Then, the probability that nelectrons have been transferred to the drain by time tis P(n,t)=1 2π/integraldisplay2π 0e−inχ(I,eL(χ)t¯P)dχ. (36) At this point, one could define a moment-generating function M(χ,t)=(I,eL(χ)t¯P) and derive the moments of transferred charge /angbracketleftnk/angbracketright=(−i)k∂k ∂χkM(χ,t)|χ=0to obtain the FCS. How- ever, we are interested in the WTD. 245402-6DISTRIBUTION OF WAITING TIMES BETWEEN … PHYSICAL REVIEW B 98, 245402 (2018) C. WTD definition Based on the ideas from quantum optics single photon counting theories [ 36,37], Brandes first introduced the con- cept of a WTD to electron transport with a formalism thatused “jump” operators defined from the master equation ofthe system [ 35]. To include cotunneling rates, however, we will start with the conditional WTD defined in terms of theidle time probability [ 38,52]: w(τ)=1 p∂2 ∂τ2/Pi1(τ). (37) The idle time probability /Pi1(τ) is the probability that there were no electron tunnelings to the drain in the measurementtimeτ. Here, pis the initial probability of observing an electron tunneling to the drain, and can be defined in terms of/Pi1(τ)a sw e l l : p=− ∂ ∂τ/Pi1(τ)|τ=0. The key relation is that the idle time probability is the probability for no electrons to betransferred to the drain between time t=0 and time t=τ,s o that when forward tunneling only is included /Pi1(τ)=P(0,τ) [38]. The moment-generating function can be written as M(χ,τ)=P(0,τ)+∞/summationdisplay n=1einχP(n,τ); (38)hence, in the infinite bias regime the idle time distribution is /Pi1(τ)=lim χ→i∞(I,eL(χ)τ¯P). (39) Combining with the definition of the WTD from Eq. ( 37), we get w(τ)=− lim χ→i∞(I,L(χ)eL(χ)τL(χ)¯P) (I,L(χ)¯P), (40) and in Laplace space, ˜w(z)=− lim χ→i∞(I,L(χ)(z−L(χ))−1L(χ)¯P) (I,L(χ)¯P). (41) Similar to the sequential tunneling case, L(χ) is formally split into a quantum jump part J(χ)=Jeiχ, containing the χ-dependence, and the χ-independent L0: L(χ)=⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣−/parenleftbig /Gamma1 S ↑0+/Gamma1S ↓0+/Gamma1SD 00/parenrightbig /Gamma1S 0↑ /Gamma1S 0↓ 0 /Gamma1S ↑0 −/parenleftbig /Gamma10↑+/Gamma1S 2↑+/Gamma1SD ↑↑+/Gamma1(2) ↓↑/parenrightbig /Gamma1SS ↑↓+/Gamma1DD ↑↓ /Gamma1S ↑2 /Gamma1S ↓0 /Gamma1SS ↓↑+/Gamma1DD ↓↑ −(/Gamma10↓+/Gamma1S 2↓+/Gamma1SD ↓↓+/Gamma1(2) ↑↓) /Gamma1S ↓2 0 /Gamma1S 2↑ /Gamma1S 2↓ −/parenleftbig /Gamma1↑2+/Gamma1↓2+/Gamma1SD 22/parenrightbig⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ +⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣/Gamma1 SD 00/Gamma1D 0↑/Gamma1D 0↓ 0 0/Gamma1SD ↑↑/Gamma1SD ↑↓/Gamma1D ↑2 0/Gamma1SD ↓↑/Gamma1SD ↓↓/Gamma1D ↓2 000 /Gamma1SD 22⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦e iχ =L0+Jeiχ. (42) The splitting is similarly defined for a SRL: L(χ)=/bracketleftBigg −(/Gamma1S 10+/Gamma1(2)) /Gamma1S 01 /Gamma1S 10 −(/Gamma1D+/Gamma1S 01+/Gamma1(2))/bracketrightBigg +/bracketleftBigg /Gamma1(2)/Gamma1D 0/Gamma1(2)/bracketrightBigg eiχ. (43) Using the splittings in Eqs. ( 42) and ( 43) the WTD be- comes w(τ)=− lim χ→i∞(I,(L0+Jeiχ)e(L0+Jeiχ)τ(L0+Jeiχ)¯P) (I,(L0+Jeiχ)¯P), (44) which is w(τ)=−(I,L0eL0τL0¯P) (I,L0¯P). (45) Noting that L0=L(0)−J,L(0)¯P=0 and ( I,L(0)A)=0 for arbitrary A, we obtain the standard expressions for theWTD in the time domain: w(τ)=(I,JeL0τJ¯P) (I,J¯P), (46) which in Laplace space becomes ˜w(z)=(I,J(z−L0)−1J¯P) (I,J¯P). (47) Here, we see that in the case of forward tunneling the WTD reduces to the one calculated using Brandes’ method[35]. Despite this, the n-resolved master equation is still 245402-7SAMUEL L. RUDGE AND DANIEL S. KOSOV PHYSICAL REVIEW B 98, 245402 (2018) necessary as it tells us how to construct L0from the quantum jump operator J. We notice that the method breaks down if backward tunneling processes are included, as their factore −iχwill diverge in the limit χ→i∞. This is a serious limitation of the approach, and it is not yet clear how toresolve it. Although the single WTD is itself an interesting quan- tity, to compute higher-order expectation values and analyzemicroscopic fluctuations, we must also generalize it to twoor more consecutive waiting times. For example, the WTDfor two waiting times, w 2(τ2,τ1), is defined as the joint probability distribution that the first electron waits time τ1and the next electron waits time τ2before tunneling to the drain [58,85]: w2(τ2,τ1)=(I,JeL0τ2JeL0τ1J¯P) (I,J¯P). (48) Moments of the single WTD are easily calculable by introducing a moment-generating function over τ: K(x)=/integraldisplay∞ 0dτeixτw(τ)=(I,JG(x1)J¯P) (I,J¯P), (49) where xis a real number and G(x)=(L0+ix)−1. (50) We obtain all possible moments by direct differentiation with respect to x, such that /angbracketleftτn/angbracketright=/integraldisplay∞ 0dτ τnw(τ) =n!(−1)n+1(I,JG(0)n+1J¯P) (I,J¯P). (51) The second-order expectation value is calculated similarly: /angbracketleftτ2τ1/angbracketright=/integraldisplay∞ 0dτ1/integraldisplay∞ 0dτ2τ1τ2w2(τ2,τ1) =(I,JG(0)2JG(0)2J¯P) (I,J¯P). (52) III. RESULTS In this section, we analytically and numerically investigate statistics of waiting time intervals between successive elec-tron cotunneling events, for both the SRL and an Andersonimpurity. The WTD for a SRL in Laplace space is obtained via Eq. ( 47) using the splitting from Eq. ( 43): ˜w(z)=a+bz (z+z+)(z+z−), (53) and the corresponding WTD in the time domain is w(τ)=a−bz− z+−z−e−z−τ−a−bz+ z+−z−e−z+τ, (54)where the coefficients of the linear function in the numera- torare a=/braceleftbig/parenleftbig /Gamma1D/Gamma1S 10/parenrightbig2+/Gamma1(2)/bracketleftbig /Gamma1D+/Gamma1S 10+/Gamma1S 01/bracketrightbig/bracketleftbig (/Gamma1(2))2+2/Gamma1D/Gamma1S 10/bracketrightbig +(/Gamma1(2))2/bracketleftbig (/Gamma1D)2+/parenleftbig /Gamma1S 10+/Gamma1S 01/parenrightbig2+/Gamma1D/parenleftbig 2/Gamma1S 01+3/Gamma1S 10/parenrightbig/bracketrightbig/bracerightbig ×/slashbig/braceleftbig /Gamma1D/Gamma1S 10+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 10+/Gamma1S 01/parenrightbig/bracerightbig and (55) b=/Gamma1(2)/parenleftbig 2/Gamma1D/Gamma1S 10+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 10+/Gamma1S 01/parenrightbig/parenrightbig /Gamma1D/Gamma1S 10+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 10+/Gamma1S 01/parenrightbig. (56) The poles of the Laplace space WTD, which are also the exponents in the time-space WTD, are z±=1 2/parenleftbig 2/Gamma1(2)+/Gamma1D+/Gamma1S 10+/Gamma1S 01 ±/radicalBig (/Gamma1D)2+2/Gamma1D/parenleftbig /Gamma1S 01−/Gamma1S 10/parenrightbig +/parenleftbig /Gamma1S 01+/Gamma1S 10/parenrightbig2/parenrightbig .(57) Interestingly, the position of the poles yield information on the individual source-drain couplings, similar to the resultsBrandes found for sequential tunneling through a SRL. The moments of the WTD can be derived analytically for a single resonant-level model, using Eq. ( 49). The average waiting time is /angbracketleftτ/angbracketright=/Gamma1 D+/Gamma1S 01+/Gamma1S 10 /Gamma1D/Gamma1S 10+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 10/parenrightbig=1 /angbracketleftI/angbracketright(2), (58) where /angbracketleftI/angbracketright(2)is the forward current including cotunneling processes. The short time behavior of the WTD is evident from Eq. ( 54): w(0)=/Gamma1(2)/parenleftbig 2/Gamma1D/Gamma1S 10+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 10+/Gamma1S 01/parenrightbig/parenrightbig /Gamma1D/Gamma1S 10+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 10+/Gamma1S 01/parenrightbig. (59) For sequential tunneling only, a SRL is a single reset system; that is, after an electron tunneling to the drain thesystem is always left empty. Consequently, in such a regimethe probability density at τ=0 is zero as two electrons cannot be detected in the drain right after one another. In contrast,Eq. ( 59) shows that when cotunneling processes are included w(0)/negationslash=0, which implies that it is now a multiple reset system. Physically, this is plausible as the cotunneling processes thatmove electrons from the source to the drain occur regardlessof the SRL occupancy. The short-time behavior is further characterized by the Pearson correlation coefficient: p=/angbracketleftτ 1τ2/angbracketright−/angbracketleftτ/angbracketright2 /angbracketleftτ2/angbracketright−/angbracketleftτ/angbracketright2, (60) where τ1andτ2are subsequent waiting times. For sequen- tial tunneling through a SRL p=0, such that w(τ1,τ2)= w(τ1)w(τ2) and waiting times between subsequent tunnelings to the drain are completely uncorrelated. Consequently, insuch a regime the renewal assumption is satisfied. Whencotunneling processes are included, however, the Pearsoncorrelation coefficient is nonzero: p=−A 2 B·C, (61) 245402-8DISTRIBUTION OF WAITING TIMES BETWEEN … PHYSICAL REVIEW B 98, 245402 (2018) 01234500.511.5210-7 Sequential Cotunneling 0 50 100 150 200 2500.511.5210-8 Sequential Cotunneling (a) (b) FIG. 1. Sequential and cotunneling WTDs for two voltages; in (a) the level is in the Coulomb blockade regime and in (b) the level is in the tunneling regime. The yaxis represents wseq(τ)a n dwco(τ) for the sequential and cotunneling WTDs, respectively. The energies of the spin split electronic levels are ε↑=0.5 meV and ε↓=1.5 meV, the Coulomb repulsion is U=4m e V ,k BT=75μeV, and γ=0.5k BT. Parameters for each plot are (a) μS=−μD=0.25 meV, /angbracketleftτ/angbracketrightseq=2.04ns, and /angbracketleftτ/angbracketrightco=1.06 ns; (b) μS=−μD=5meV, /angbracketleftτ/angbracketrightseq=46.81 ps, and/angbracketleftτ/angbracketrightco=46.43 ps. where the components are A=/Gamma1(2)/Gamma1D/Gamma1S 10, (62) B=(/Gamma1(2))2+/Gamma1D/Gamma1S 10+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 01+/Gamma1S 10/parenrightbig ,and (63) C=/parenleftbig /Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 01+/Gamma1S 10/parenrightbig/parenrightbig2+/Gamma1(2)/parenleftbig /Gamma1D+/Gamma1S 01+/Gamma1S 10/parenrightbig3 +/Gamma1D/Gamma1S 01/parenleftbig (/Gamma1D)2+2/Gamma1D/Gamma1S 01+/parenleftbig /Gamma1S 01+/Gamma1S 10/parenrightbig2/parenrightbig . (64) Equation ( 61) shows that, contrary to sequential tunneling, electron waiting times for cotunneling through a SRL arenegatively correlated, since A 2,B, andCare all positive. However, the correlation is negligibly small, as expected fromthe small perturbative changes that cotunneling brings. Turning now to the Anderson impurity, when neither of the spin split levels are in the voltage window, we expect thesequential current to be negligible. Consequently, in such acase, we also expect the average sequential waiting time /angbracketleftτ/angbracketright seq to be large; that is, on average it takes a long time for electrons to be transferred from the source to the drain. In contrast,for an Anderson impurity experiencing Coulomb blockade,cotunneling provides a quantum pathway for electrons totunnel through the system that is not visible in the sequentialphysics. This is evident in Fig. 1(a), where /angbracketleftτ/angbracketright cois double /angbracketleftτ/angbracketrightseq; whereas in the tunneling regime, shown in Fig. 1(b), sequential processes dominate and /angbracketleftτ/angbracketrightcois comparable to /angbracketleftτ/angbracketrightseq. At high voltages, an Anderson impurity behaves as a mul- tiple reset system, since an electron tunneling to the drain canleave the system singly occupied or empty, which is shownin Fig. 1(b) asw(0)/negationslash=0 for both sequential tunneling and cotunneling. In comparison, at low voltages double occupancyis energetically denied and it behaves as a single reset sys-tem, which is shown in Fig. 1(a) asw(0)=0 for sequentialtunneling. Again, however, when cotunneling processes are included, the WTD displays multiple reset behavior at shorttimes, as cotunneling processes can leave the system eithersingly occupied or empty. Sequential tunneling through an Anderson impurity dis- plays nonrenewal statistics in the high voltage regime, whichis seen in Fig. 2(b). Here, due to the strong inelastic electron- electron interaction when a spin-up and spin-down electronare occupying the impurity, the correlation between subse-quent waiting times is negative; a short waiting time is morelikely to be followed by a long waiting time and vice versa.We note that the Coulomb repulsion is an order of magnitudegreater than the electronic single-particle energies, so that ifthe system is doubly occupied it is likely for both electronsto subsequently tunnel out, which is a short waiting time,and then for the system to fill and empty again, which isa long waiting time. Thus, the nonrenewal behavior doesnot arise from non-Markovian behavior, as we work underthe Markovian assumption, but rather from the multiple tun-neling processes contained in the drain jump operator [ 58]. Importantly, even though sequential processes dominate inthis regime cotunneling still has an effect on the nonrenewalstatisticis, slightly increasing the strength of the negativecorrelation between subsequent waiting times. Multiple authors have shown that when the renewal as- sumption is satisfied there is a direct link between the cumu-lants of the WTD and the current cumulants [ 43,86]. Here, we focus on the Fano factor, which is the ratio of the zero-frequency noise to the average current: F=S(0) 2e/angbracketleftI/angbracketright(65) =/angbracketleftn2/angbracketright−/angbracketleftn/angbracketright2 /angbracketleftn/angbracketright. (66) 245402-9SAMUEL L. RUDGE AND DANIEL S. KOSOV PHYSICAL REVIEW B 98, 245402 (2018) 0.2 0.4 0.6 0.8 VSD (meV)-5-4-3-2-101p10-5 Sequential Cotunneling 8 9 10 11 12 VSD (meV)-0.1-0.08-0.06-0.04-0.020pSequential Cotunneling (a) (b) FIG. 2. Pearson correlation coefficient/angbracketleftτ1τ2/angbracketright−/angbracketleftτ/angbracketright2 /angbracketleftτ2/angbracketright−/angbracketleftτ/angbracketright2over a range of voltages in (a) the Coulomb blockade regime and (b) the tunneling regime. The energies of the spin split electronic levels are ε↑=0.5 meV and ε↓=1.5 meV, the Coulomb repulsion is U=4m e V ,k BT=75μeV, andγ=0.5kBT. The Fano factor in terms of waiting times is given by the randomness parameter [ 43,58,86]: R=/angbracketleftτ2/angbracketright−/angbracketleftτ/angbracketright2 /angbracketleftτ/angbracketright2. (67) If the renewal assumption holds, then F=R. Indeed, in Fig. 3(b) one can see that the two parameters diverge at the same voltage that the sequential correlation coefficientbecomes nonzero, and that the difference between the Fand Rincreases as the correlation increases. Furthermore, when cotunneling processes are included, FandRdiverge at a later voltage, following the behavior of the cotunneling correlation. Since multiple cotunneling rates appear in the drain jump operator, one might expect that nonrenewal behavior couldbe observed even in the Coulomb blockade regime when the strong Coulomb repulsion does not play a part in the transport.Figure 2(a) shows that for small voltages the correlation is nonzero but as with a SRL the magnitude of the correlationsare negligibly small. This is apparent in Fig. 3(a); the pres- ence of cotunneling changes the Fano factor and randomnessparameter from their sequential values, but they still are notvisibly different. Note that the divergence of the Fano factorat zero voltage is due to the complete suppression of thePoissonian shot noise in comparison to the thermal noise. So far, we have shown plots that are either deep in the Coulomb blockade regime or well in the tunneling regime.This is because for certain voltage ranges between thesetwo extremes the approach produces unphysically negative 0.2 0.4 0.6 0.8 VSD (meV)0.80.911.11.2F,RSequential Cotunneling 8 9 10 11 12 VSD (meV)0.50.520.540.560.580.60.62F,R Sequential Cotunneling (a) (b) FIG. 3. (Color online.) Exact Fano factor Fand its prediction from waiting times under the renewal assumption Rover a range of voltages in (a) the Coulomb blockade regime and (b) the tunneling regime. The energies of the spin split electronic levels are ε↑=0.5 meV and ε↓=1.5 meV, the Coulomb repulsion is U=4m e V ,k BT=75μeV, and γ=0.5k BT. 245402-10DISTRIBUTION OF WAITING TIMES BETWEEN … PHYSICAL REVIEW B 98, 245402 (2018) probability densities for small waiting times. For a SRL, these unphysical WTDs clearly occur when ε<eV and second- order contributions actually reduce the total current, whichamounts to negative regularized cotunneling rates. From the point of view of the theory, the total transition rate /Gamma1 00= γ 2nF(ε−μS)(1−nF(ε−μD))+/Gamma1(2)is still positive, but /Gamma1(2)can be negative [ 12,13]. In such a regime and for small τ (∼103fs), the WTD for a SRL is negative, which is shown in Eq. ( 59) when /Gamma1(2)<0. For an Anderson impurity, the situation is more complex; it appears that when at voltages where cotunneling processesdramatically decrease the sequential current, the WTD isnegative for small τ. It is not yet clear how to resolve this interesting pathology; evidently there should be a well-definedWTD for all voltage ranges. This positivity violation could bean artifact of only going to second-order perturbation theory;on the other hand, it may not be physically correct to includenegative rates at all in the definition of the jump operators. IV . CONCLUSION In this paper, we have extended the Markovian master equation technique for calculating WTDs in quantum electrontransport to include cotunneling effects, and demonstratedthe method for transport through a SRL and an Andersonimpurity. Additionally, we have demonstrated that, similarto the WTD for sequential tunneling through a SRL, thecotunneling WTD in Laplace space provides information onthe individual source-drain couplings. Of particular interestis how cotunneling processes affect the nonrenwal statisticsalready present in the Anderson impurity, where electrons experience strong inelastic electron-electron interactions. Wehave shown that for large voltages, cotunneling increases themagnitude of the nonnegligible negative correlation betweenwaiting times of subsequent electron tunnelings to the drain,which is caused by a strong electron-electron interaction,and thus increases the nonrenewal behavior shown by thedifference in the Fano factor and the randomness parameter.However, in the Coulomb blockade regime where cotunnel-ing processes dominate, the correlation between subsequentwaiting times is negligible and the system displays renewalbehavior. ACKNOWLEDGMENTS We thank the two anonymous reviewers whose insightful comments and suggestions helped significantly improve thispaper. This work was supported by an Australian GovernmentResearch Training Program (RTP) Scholarship to SLR. APPENDIX: COTUNNELING RATES In this Appendix, we derive the cotunneling rates shown in Eqs. ( 20), (22), and ( 23), from the starting point of Eq. ( 19). We note that the derivation generally follows the regulariza-tion procedure detailed by Koch et al. [12,13]. As an example, consider the case of elastic tunneling through an initiallyempty dot. The cotunneling rate from the source to the drainis /Gamma1SD 00=2πlim η→0+/summationdisplay σ/summationdisplay i,f/vextendsingle/vextendsingle/angbracketleftf|/angbracketleft0|HD,σ T1 Ei,0−H0+iηHS,σ T|0/angbracketright|i/angbracketright/vextendsingle/vextendsingle2WS i,0WD i,0×δ(εi,s−εf,d), (A1) where we have summed over σto account for tunneling through either the ↑or↓level. The initial state of the dot is |0/angbracketright⊗ |i/angbracketright and the final state is aσa† σ|0/angbracketright⊗a† νDaνS|i/angbracketright. To span the possible configurations after the cotunneling, the rate is summed over the electrode states νSandνD. Additionally, it is assumed that the metal electrodes’ density of states is constant. With these assumptions, the rate reduces to /Gamma1SD 00=γSγD 2πlim η→0+/integraldisplay dε/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 ε−ε↑+iη+1 ε−ε↓+iη/vextendsingle/vextendsingle/vextendsingle/vextendsinglen F(ε−μS)[1−nF(ε−μD)]. (A2) In general then, elastic cotunneling rates have the form /Gamma1SD nn=γSγD 2πlim η→0+/integraldisplay dε/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 ε−E1+iη±1 ε−E2+iη/vextendsingle/vextendsingle/vextendsingle/vextendsinglen F(ε−μS)[1−nF(ε−μD)], (A3) where E1andE2are derived from the cotunneling pathways involved in the rate, and the ±is only positive for elastic tunneling through an initially empty or initially doubly occupied system. Similarly, inelastic cotunneling rates have the general form /Gamma1αβ ¯σσ=γαγβ 2πlim η→0+/integraldisplay dε/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 ε−ε¯σ−U+iη−1 ε−ε¯σ−iη/vextendsingle/vextendsingle/vextendsingle/vextendsinglenF(ε−μS)[1−nF(ε−μD+εσ−ε¯σ)]. (A4) The expanded form of either Eq. ( A3)o rE q .( A3) consists of two square terms and the real component of the cross term; for example, /Gamma1SD nn=γSγD 2πlim η→0+/bracketleftbigg/integraldisplay dε1 (ε−E1)2+η2nF(ε−μS)[1−nF(ε−μD)]+/integraldisplay dε1 (ε−E2)2+η2nF(ε−μS)[1−nF(ε−μD)] ±2/Rfractur/integraldisplay dε1 ε−E1+iη·1 ε−E2−iηnF(ε−μS)[1−nF(ε−μD)/bracketrightbigg . (A5) 245402-11SAMUEL L. RUDGE AND DANIEL S. KOSOV PHYSICAL REVIEW B 98, 245402 (2018) It has been noted in the literature that the intermediate virtual state in the dot has a finite width, which is propor-tional to the coupling strength η∼γ, and so a divergence is avoided in the denominator of the integrand [ 12–14,79]. Additionally, the two square terms in the overall rate includenot only the rate of that particular event from cotunneling,but also the contribution to that process from sequentialtunneling, as a cotunneling event can be mimicked by twosequential tunneling events. Thus, it is necessary to removethis sequential overcounting by expanding the integrands ofthe first two integrals in Eq. ( A5) in a power series about η, and discarding the η −1term as with the γSγDprefactor it is overall O(γ). For a simple system such as a SRL, some groups choose to remove the overcounting and then computethe rate numerically with Cauchy’s principal value [ 14]. How- ever, we follow the approach of Koch et al. and evaluate the integral analytically by transforming it to a contour integralover a semicircle in the upper half-plane of complex spaceand using residue theory. The final expressions for the elasticand inelastic rates are given in Eqs. ( 20) and ( 22), where the trigamma functions ψ (1)(x) come from the squared terms in the rate, the digamma functions ψ(x) come from the cross term, and both originate from the complex poles of the Fermi-Dirac distributions n F(ε−μS/D), known as the Matsubara frequencies. 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PhysRevB.89.195142.pdf

PHYSICAL REVIEW B 89, 195142 (2014) Concomitant charge-density-wave and unit-cell-doubling structural transitions in Dy 5Ir4Si10 M. H. Lee,1,2C. H. Chen,1,2C. M. Tseng,3C. S. Lue,4Y . K. Kuo,5H. D. Yang,6and M.-W. Chu2,7,* 1Department of Physics, National Taiwan University, Taipei 106, Taiwan 2Center for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan 3Institute of Physics, Academia Sinica, Taipei 115, Taiwan 4Department of Physics, National Cheng Kung University, Tainan 701, Taiwan 5Department of Physics, National Dong Hwa University, Hualian 974, Taiwan 6Department of Physics, National Sun-Yat-Sen University, Kaohsiung 804, Taiwan 7Taiwan Consortium of Emergent Crystalline Materials, Ministry of Science and Technology, Taipei 106, Taiwan (Received 17 December 2013; revised manuscript received 14 April 2014; published 29 May 2014) The tetragonal rare-earth transition-metal silicide system with three-dimensional crystallographic structure R5T4Si10,w h e r e Ris Dy, Ho, Er, Tm, and Lu, and T=Ir and Rh, has been shown to exhibit fascinating charge- density-wave (CDW) phase transitions, a phenomenon largely found in otherwise low-dimensional systems. Inthis study, we report the investigations of CDW in Dy 5Ir4Si10at different temperatures using transmission electron microscopy techniques including electron diffraction and dark-field imaging. Incommensurate superlattice spotsalong the caxis were observed in the electron-diffraction patterns when the sample was cooled below the CDW transition temperature at ∼208 K. CDW becomes commensurate with further cooling and configurations of CDW dislocations convincingly show that the CDW phase transition is accompanied by a concomitant cell-doublingcrystallographic structural phase transition. Intriguingly, the cell-doubling transition is featured by a brokeninversion symmetry along the caxis and a disparity in the CDW-modulation vectors with opposite signs, which gives rise to two sets of CDW domains with reversed contrasts. The profound physics underlining this notabledomain-contrast behavior is discussed. DOI: 10.1103/PhysRevB.89.195142 PACS number(s): 71 .45.Lr,64.70.Rh,68.37.Lp,61.44.Fw I. INTRODUCTION The series of ternary rare-earth–transition-metal silicides system R5T4Si10, where R=Dy, Ho, Er, Tm, and Lu, and T=Ir and Rh, have been shown to exhibit various interesting phase transitions observable in the thermal and electricaltransport measurements, ranging from superconductivity andlong-range magnetic ordering at low temperatures (below10 K), to charge-density waves (CDWs) at higher temperatures[1–15]. Indeed, CDW transitions are known to occur largely in low-dimensional solids where it is feasible to achieve goodnesting of Fermi surfaces leading to electronic instability. Therather unexpected presence of CDW phase transitions in thisclass of three-dimensional R 5T4Si10has drawn much attention in recent years [ 1–15]. The compound Dy 5Ir4Si10adopts a tetragonal crystal structure (space group, P4/mbm ) at room temperature with a=12.577 ˚A and c=4.237 ˚A[2]. It is generally thought that the rare-earth ions of the crystal form quasi-one-dimensionalchains, which could lead to a CDW instability. Accordingly,the rare-earth network around the central site can be viewed asa channel of conductivity parallel to the chain. However, thefollowing observations argue against the above conjectures:(1) The spacing of the rare-earth ions along the “chains” is4.2˚A, which is too large to be considered as one dimensional. (2) The anisotropy in electrical resistivity is much smaller thantypical one-dimensional systems such as NbSe 3[16]. It has been suggested that precursory or concomitant structural phase transitions might facilitate the formation ofthe CDW state in this class of materials due to modified band *Corresponding author: chumingwen@ntu.edu.twstructures, which might improve the nesting conditions at theFermi surface [ 5,8]. In fact, it has also been substantiated that CDW transitions in this case are often accompanied by acell-doubling structural transition [ 14,17]. The presence of CDW in Dy 5Ir4Si10was hinted in thermal conductivity and thermoelectric power (TEP) measurementswhere anomalies at 208 and 165 K were found [ 10]. However, there was no direct evidence regarding the presence ofsuperlattice reflections characteristic of CDW formation inDy 5Ir4Si10using diffraction techniques [ 10]. In this work, we report on the CDW phase transitions in Dy 5Ir4Si10by selected area electron diffraction (SAED), convergent beam electrondiffraction (CBED), and direct observation in real spaceusing superlattice dark-field imaging technique in transmissionelectron microscopy (TEM). The dark-field images obtainedfrom the superlattice reflections associated with the CDWphase transitions clearly show that the CDW incommensuratephase should be accurately described as discommensurate withsmall commensurate domains separated by sharp discommen-surations (DCs) [ 14]. Moreover, careful SAED investigations indicate that a crystallographic cell-doubling lattice distor-tion occurs simultaneously with the CDW phase transition,pointing to a common occurrence in this class of materials[5]. Most surprisingly, we demonstrated by CBED that the cell-doubling transition is accompanied with the breaking ofthe primitive inversion symmetry along the caxis, which is closely correlated with the observation of two different sets ofCDW domains with reversed contrasts. II. EXPERIMENT The polycrystalline sample Dy 5Ir4Si10was prepared by arc-melting high-purity elements under argon atmosphere. The 1098-0121/2014/89(19)/195142(7) 195142-1 ©2014 American Physical SocietyLEE, CHEN, TSENG, LUE, KUO, Y ANG, AND CHU PHYSICAL REVIEW B 89, 195142 (2014) detailed preparation and characterization of the samples have been reported elsewhere [ 2,10]. Thin specimens for TEM studies were prepared by mechanical polishing followed byion milling at liquid nitrogen temperature and more than threesamples of the Dy 5Ir4Si10compound were measured to check experimental reproducibility. A JEOL 2000 FX transmissionelectron microscope operating at 200 kV and equipped with alow-temperature sample stage was used for the present study.The temperature accuracy of the stage has been carefullycalibrated using the transition temperature of a known ma-terial, Lu 2Ir3Si5, and the temperature difference between the stage and the thermometer reading was found to be within3K[ 14,17]. III. RESULTS AND DISCUSSION TEM examinations show that typical grain size is larger than a few micrometers, which can be treated as single crystalsin the present case, since SAED patterns require only an areamuch smaller than 1 μm in size. Firstly, the electron-diffraction studies at room temperature [e.g., inset of Fig. 1(a)]are consistent with the primitive tetragonal structure with spacegroup P4/mbm [4,18]. Further electron-diffraction evidence of the CDW formation in Dy 5Ir4Si10is shown in Figs. 1(a) and 1(b), which represent the [100]zone-axis SAED patterns taken at 100 and 190 K, respectively. The two temperatures werechosen to be below the two respective transitions at 165 and208 K indicated in the thermal transport measurements [ 10]. The presence of systematic superlattice peaks in addition to thefundamental Bragg peaks is obvious, with Figs. 1(a) and1(b) showing commensurate and incommensurate modulations,respectively. These superlattice spots are characterized bymodulation wave vectors q=(0,0, 1 4±δ), where δis the temperature-dependent incommensurability with δ=0i n Fig. 1(a) andδ∼0.03 in Fig. 1(b). It is noted in Fig. 1that one observes (0, k,l) reflections with k=odd and l=all integers, which are symmetry forbidden to the room-temperature P4/mbm structure. In addition to the CDW superlattice peaks characterized by qin Fig. 1, we also find commensurate spots systemically located at(0,k, 1 2) with k=all integers and these reflections can be interpreted as different fundamental Bragg spots due to astructural phase transition involving a cell-doubling along thecaxis. From the concomitant appearance of qmodulations and symmetry-forbidden reflections in Fig. 1, the CDW and structural transitions in Dy 5Ir4Si10should take place at the same temperature. This is different from the situation inthe CDW transition of Lu 2Ir3Si5, in which the cell-doubling structural transition sets in at a slightly higher temperaturethan the CDW transition [ 17,19,20]. We first focus on the direct observation of CDW domains and domain walls usingthe superlattice dark-field imaging in TEM. This rather uniqueand powerful imaging technique has been shown to reveal themost direct details and the temperature-dependent evolutionof CDW phase transitions [ 14,21–25]. Figures 2(a) and2(b) show the dark-field images obtained from the commensurate CDW-modulation wave vectors q += (0, 0,1 4) andq−=(0, 0,−1 4) as shown in Fig. 1(a), respectively, and reveal many large dark (or otherwise bright) domainsseparated by sharp boundaries. Surprisingly, the domaincontrast in Fig. 2(a) is reversed to that in Fig. 2(b). Considering that the order parameter of charge density in CDW systemsis a pure scalar quantity, the CDW modulation generallydescribed by the formulation of ρ=cos(qx+φ)(ρ, charge FIG. 1. (a) Electron-diffraction pattern along the [100]zone axis obtained at 100 K, which shows the commensurate superlattice spots of CDW modulations characterized by q+=(0, 0,1 4)a n dq−=(0, 0,−1 4). The length between center to (001) Bragg peak corresponds to 1.48 ˚A−1. The room-temperature pattern along the same projection is also shown (inset). (b) [100]zone-axis diffraction pattern taken at the same sample region as (a) at otherwise 190 K, showing the emergence of incommensurate superlattice peaks as indicated by the white circle. In (a) and (b), the systematic presence of (0, 0,1 2)-type reflections indicate the cell doubling. The pair of the superlattice spots close to the (0, 0,1 2)-type reflections with an incommensurability of 2 δ, rather than δcharacteristic of the q-modulation spots, is a result of dynamical electron-multiple scattering as confirmed by their systematic disappearance upon tilting the grain slightly away from the [100]zone. By contrast, theq-modulation spots indicated by the white circle are persistent upon the tilting. Note that the patterns taken at 100 K, 190 K, and room temperature have been subject to slightly different exposure times from one to another considering the finite dynamical range of the cameraand the possible saturation of the intense central beam. 195142-2CONCOMITANT CHARGE-DENSITY-W A VE AND UNIT- . . . PHYSICAL REVIEW B 89, 195142 (2014) FIG. 2. (a), (b) The superlattice dark-field TEM images obtained from the respective commensurate q+=(0, 0,1 4)a n dq−=(0, 0,−1 4) vectors in Fig. 1(a). The reversed contrasts of one to the other clearly indicate that the modulation vectors, q+andq−, actually come from different domains as a result of the symmetry breaking along the caxis upon the CDW modulation. density; q, wave vector; x, position; ϕ, phase) should remain invariant upon the sign reversal of qandx[26–28]. Herein, the modulation vectors of q+andq−should thus be equivalent and the dark-field imaging using either q+orq−would then lead to the same contrast characteristics. However, the contrastreversal shown in Fig. 2clearly indicates that q +andq−are inequivalent and q+andq−arise from different domains. Note that neither chemical inhomogeneity nor impurity canbe found between the two sets of domains using chemicalmicroanalysis by energy-dispersive x-ray spectroscopy withan electron-probe size of ∼30 nm, which is also the beam diameter used in the following CBED studies (Fig. 3). The disparity in q +andq−observed (Fig. 2) must come from a structural origin with intriguing coupling to the primaryorder parameter of charge density. We examined this structuralaspect by exploiting CBED (Fig. 3), which is known to bepowerful in revealing structural microdistortion and subtle symmetry breaking that cannot be easily resolved by SAED. Figure 3(a) shows a CBED pattern taken along the [100] direction at room temperature, with high-order Laue zone(HOLZ) observed as rings centering around the enlargedzero-order Laue zone (ZOLZ) inset. A close inspection ofFig.3(a) indicates that the characteristic features in (0, k,l) and (0,k,−l) reflections are identical (see the white arrowheads), pointing to the existence of a mirror symmetry perpendicularto thecaxis. In Fig. 3(a), a mirror plane perpendicular to the b axis can also be observed and these mirror-symmetry elementsalong the [100]projection correspond to a symmetry class of 2mm. Classically, the 4 /mmm point group characteristic of P4/mbm w o u l dg i v er i s et oa [100]-zone CBED symmetry of 2mm [29], indeed in agreement with our observations in Fig. 3(a). The CBED pattern taken at 98 K along the same FIG. 3. CBED patterns along the [100]zone axis. (a) The reflection characteristics in the HOLZ rings and enlarged ZOLZ inset reveal the presence of two respective mirror planes, perpendicular to the bandcaxes. (b) The CBED pattern acquired at low temperature, unveiling dynamical absences along the caxis as indicated by the FOLZ-reflection features in the white-block areas (for clarity, these reflections having been enlarged and shown by the reversed contrasts). These reflection features also reveal the breaking of the primitive mirror symmetryperpendicular to the caxis at low temperature, while the otherwise mirror-symmetry element perpendicular to the baxis persists. 195142-3LEE, CHEN, TSENG, LUE, KUO, Y ANG, AND CHU PHYSICAL REVIEW B 89, 195142 (2014) zone axis is further shown in Fig. 3(b). Notably, only the mirror plane perpendicular to the baxis remains and the other mirror plane perpendicular to the caxis has now disappeared, as indicated by the distinct loss of mirror symmetry in thefirst-order Laue zone (FOLZ) features (see the rectangularwhite blocks and their blowups shown in reversed contrasts forclarity). In the low-temperature CDW state [Fig. 3(b)], such a symmetry-element change indicates that the concomitantcell-doubling structural transition is effectively entangled withthe breaking of the primitive inversion symmetry along the c axis (4 /m)[29]. In Fig. 3(b), the adjoining reflections of the (1,±k,±11) type are strongly excited, while the (1 ,0,±11) reflections appear to have zero intensity on the mirror linerunning through them (see the insets) as a result of thecharacteristic dynamical absence [ 23]. Due to the symmetry breaking along the caxis in this CDW state, q +andq−become inequivalent and the dynamical-scattering nature of CBED renders the excitationintensities of associated (1 ,±1,−11) and (1 ,±1,11) readily different, with (1 ,±1,11) reflections much stronger than (1,±1,−11) reflections as shown in the insets of Fig. 3(b) [23,29]. Otherwise, Friedel’s law, characterized by equal intensities between ( h,k,l )-and (−h,−k,−l)-type reflections in the framework of kinematical scattering, would forbid theintensity disparity between q +andq−[29,30]. Indeed, such a disparity in reflection intensities has been well establishedin ferroelectrics, which features the breaking of inversioncenters in the unit cell, and the associated observation ofcontrast reversals of the 180 °ferroelectric domains in BaTiO 3 represents a classical example [ 30]. Upon the careful setup of dynamical-scattering condition in BaTiO 3, it has been firmly shown that the inversion-symmetry breaking of theorder parameter (spontaneous polarization) along the caxis can result in the breakdown of Friedel’s law [ 30]. Using the associated qvectors with opposite signs for TEM dark-field imaging as in Fig. 2, a direct observation of 180 °domainswith characteristic reversed contrasts can be straightforwardly achieved [ 30]. What we observed in Figs. 2and3altogether, withq +andq−contributed by two separate sets of domains showing contrast reversals, is remarkably similar to the BaTiO 3 exemplification [ 30]. Although a microstructural model of lattice distortions is not available at the present time, itis clear that the CDW lattice modulation in Dy 5Ir4Si10 breaks the spatial-invariance principle and produces thedomain-contrast reversal in resemblance to the 180 °-twin domains in ferroelectrics [ 26–28,30,31]. We note that similar domain-contrast reversal has also been observed in othercommensurate CDW states such as 2 H-TaSe 2and 1T-VSe 2, but detailed lattice distortions and related origin were notclearly addressed therein [ 21–23,32]. Later in Fig. 5,t h e dark-field imaging near the commensurate-to-incommensuratetransition also revealed a similar contrast reversal, reaffirmingthe loss of inversion symmetry along the caxis in the CDW transition. Upon cooling from room temperature, the incommensurate CDW of Dy 5Ir4Si10[Fig. 1(b)]locks into a commensurate state[Fig. 1(a)]at the so-called lock-in temperature of 165 K [10]. However, detailed domain structure observations and the change of CDW superlattice reflections as a function oftemperature discussed below unveil a slight difference in thelock-in transition temperature [ 10]. Figure 4(a) illustrates the development of the incom- mensurate peaks beginning at 208 K and merging intoa commensurate q=(0,0, 1 4) single peak at temperatures below ∼143 K, with the corresponding temperature-dependent incommensurability δplotted in Fig. 4(b). It is interesting to note in Fig. 4(b) that the lock-in temperature observed by TEM, ∼143 K, is significantly different from the previous thermal transport measurements, 165 K [ 10], and this phenomenon could be associated with variations in local environment, suchas the possible presence of strains, in the thin TEM samples[14,17,21,22]. FIG. 4. (Color online) (a) The Gaussian-fitted intensity profiles across the superlattices along the c-modulation direction at various temperatures, showing the commensurate-to-incommensurate transition. (b) The value of incommensurability, δ, is varying from ∼0.03 (incommensurate) to 0 (commensurate) with decreasing temperatures. 195142-4CONCOMITANT CHARGE-DENSITY-W A VE AND UNIT- . . . PHYSICAL REVIEW B 89, 195142 (2014) FIG. 5. Superlattice dark-field TEM images taken at various temperatures from the same area, which show the residual DC domain walls in the commensurate phase (a) and the presence and movement of the CDW dislocations in (b)–(d). All the images were taken from the satellite peaks located at (4, 0,1 4±δ). It is noted that two DCs are always required to form a CDW dislocation. A few corresponding dislocations marked by arrows in (a) and (b) seem to move at a slower speed than the others. Upon further warming, the density of the DCs has rapidly increasedas shown in (d). Inset in (c) shows the dark-field imaging of the dotted region using the qvector with an otherwise opposite sign. The contrast in the inset is reversed to that using q +. Figures 5(a)–5(d) show a series of dark-field images obtained from the CDW satellite peaks at several different tem- peratures near the commensurate-to-incommensurate CDWtransition of ∼143 K on warming and the dark lines therein are DC domain walls, where the CDW phase changes rapidly[14,17]. As the temperature approach the transition temper- ature, DCs are found to nucleate within the commensuratedomain [black arrows, Fig. 5(a)]and the nearly incommensu- r a t es t a t ei nF i g . 5(d) is characterized by the presence of small commensurate domains separated by DCs, which has beenknown as a discommensurate phase [ 14,17]. It is well known that nucleation, growth, and annihilation of DCs play themost critical role for the commensurate-to-incommensuratetransition and the physics of the transition has been welldescribed based on the Landau theory in the early literature[26,31,33]. Furthermore, it is obvious that the DC domain walls start to appear near the edge of the sample at ∼136 K. It deserves mentioning that there exists a discernible changein the TEP results of the material at ∼136 K [ 10]. Indeed, the TEP measurements are known to be very sensitive to thechanges in the Fermi-level density of states (DOS) [ 10], and this would suggest that the commensurate-to-incommensurate transition observed by TEM is accompanied by a notablechange of DOS near the Fermi surface. Further experimentswill be required to elucidate the ∼136 K transition, which is a separate issue on its own. It should be mentioned thata close inspection of the SAED pattern corresponding toFig. 5reveals a nearly vanishing incommensurability, δ,f o r the otherwise broad superlattice spots [see Fig. 4(a), 133 K, for instance ], although nonzero δwould be generally expected for CDW phases with appreciable DCs as in Fig. 5.T h e vanishing δherein, in effect, arises from the strong intensity of the (0 ,0, 1 4)-type reflections of the commensurate domains, which buries the subtle incommensurability in the intensecommensurate superlattices and leads to the broad width ofthe modulation peaks. In addition, nodes at which DCs join together are known as CDW dislocations and many CDW dislocations are evidentin Figs. 5(a)–5(d) (black arrows). It is clear that every CDW dislocation involves only two DCs. The fact that two DCs 195142-5LEE, CHEN, TSENG, LUE, KUO, Y ANG, AND CHU PHYSICAL REVIEW B 89, 195142 (2014) are always needed in the present case to form a CDW dislocation demonstrates that the phase shift across a DCisπand the modulation wave vector should be of the type q=(0,0, 1 2) instead of q=(0,0,1 4), which would otherwise require four DCs to merge at a CDW dislocation node.Thec-axis cell-doubling structural phase transition, which simultaneously occurs with CDW, would automatically lower the order of commensurability ( n)f r o m n=4t on=2 and, hence, requires only two DCs to form a CDW dislocation.Nucleation and growth of the CDW dislocations are usuallythe most fascinating aspect of the entire evolution process ofvarious CDW phases. With increasing temperature [Figs. 5(a)– 5(d)], while new CDW dislocations are being nucleated, the existing CDW dislocations continue to move and result in further lengthening and increasing density of DCs. The CDW dislocations are found to move with different speeds, whichcould reflect the local variations of strains and pinning strengthfluctuations along the path they encounter. A few CDWsmarked by arrows in Fig. 5(b) seem to move much more slowly than others and are still visible in Fig. 5(c). It is noted that the original residual DCs observed at lower temperatures as shown in Fig. 5(a) remain largely motionless and unchanged in the initial warming process, which signifies the strong pinningeffect of the CDWs. We note also that annihilation of the CDWdislocations also plays an important role during the thermalevolution of the phase transition [ 14,17]. The evolution of the commensurate-to-incommensurate phase transition is entirelycontrolled by the process of nucleation, growth, and annihi- lation of the CDW dislocations and the commensurate phase can be satisfactorily understood as a transition underlined bythe disappearance of DCs from the incommensurate phase[26,31,33]. Following this latter established conception, the dark-field imaging of the incommensurate phase using q − should then lead to the same contrast-reversal phenomenon as in Fig. 2.I nF i g . 5(c), the exploitation of q−for the imaging (inset) indeed results in a contrast reversal compared to q+.W e show in the present work that detailed investigation in TEMdark-field technique can reveal profound details about CDW. Moreover, it deserves mentioning that the recent study of two-dimensional 1 T-TiSe 2using scanning tunneling mi- croscopy revealed intricate variations in the intensity ofcharge-density superlattice reflections, with the respectiveintensities of the three characteristic commensurate qmod- ulations diminishing in the order of a clockwise or counter-clockwise pattern [ 27]. Such a helical-like intensity variation breaks the parity in qand further implies the emergence of chiral CDW along the common projected vector of the q’s [27,28]. In Fig. 1(b), we also observed a subtle intensity fluctuation of the superlattice spots and Fig. 5(c) unveils the accompanied disparity in q. It seems plausible that the subtle disparity of superlattice reflections in Dy 5Ir4Si10could arise from the handedness of the CDW modulations, resultingnaturally in two types of domains as we have discussed.Dy 5Ir4Si10features, however, only one single set of qand whether CDW with a handedness could emerge along this q vector would require additional theoretical and experimental verifications in the future, such as the polarization-dependent transient reflectivity also exploited in the study of 1 T-TiSe 2 [27]. Indeed, the subject of chiral CDW in three-dimensional crystalline systems represents an intriguing and largely unex-plored area. IV . CONCLUSION The CDW transition and domain characteristics of Dy5Ir4Si10were thoroughly investigated on the basis of electron-diffraction studies. Using superlattice dark-fieldimaging, configurations of the corresponding CDW disloca-tions, where two, instead of four, DCs were found to forma dislocation node, convincingly establish that the CDWtransition is accompanied with a concomitant cell-doublingstructural transition. The careful CBED investigations of thislow-temperature CDW phase revealed a broken symmetryalong the caxis, leading to the observation of the intriguing phenomenon of two different sets of CDW domains withreversed contrasts in this class of materials. Moreover, theCDW dislocations appear to occur at ∼136 K rather than the generally accepted lock-in transition temperature of 165 K,suggesting an associated change in DOS near the Fermisurface. Indeed, an electronic anomaly at ∼136 K has been previously pointed out using macroscopic thermoelectricmeasurements without knowing the profound details andthis work provides useful microscopic details for the futureelaboration of the problem. [1] R. N. Shelton, L. S. 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PhysRevB.84.235411.pdf

PHYSICAL REVIEW B 84, 235411 (2011) Linear response theory of interacting topological insulators Dimitrie Culcer ICQD, Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, Anhui, China and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA (Received 26 July 2011; revised manuscript received 17 November 2011; published 1 December 2011) Chiral surface states in topological insulators are robust against interactions, nonmagnetic disorder, and localization, yet topology does not yield protection in transport. This work presents a theory of interactingtopological insulators in an external electric field, starting from the quantum Liouville equation for the many-bodydensity matrix. Out of equilibrium, topological insulators acquire a current-induced spin polarization. Electron-electron interactions renormalize the nonequilibrium spin polarization and charge conductivity, and disorderin turn enhances this renormalization by a factor of 2. Topological insulator phenomenology remains intact inthe presence of interactions out of equilibrium, and an exact correspondence exists between the mathematicalframeworks necessary for the understanding of the interacting and noninteracting problems. DOI: 10.1103/PhysRevB.84.235411 PACS number(s): 73 .23.−b, 71.10.Pm, 72 .25.Dc, 73 .20.At I. INTRODUCTION The understanding of insulating behavior has been revolu- tionized by the landmark discovery of topological insulators (TIs),1–4which are bulk band insulators with spin-orbit induced conducting states on the surface [three dimensional (3D)] or edge [two dimensional (2D)]. These states are a manifestation of Z2topological order: topology guarantees the existence in equilibrium of a crossing of bands connecting time-reversal invariant momenta, which is robust againstsmooth time-reversal invariant perturbations such as nonmag- netic disorder and electron-electron interactions. The surface states of 3D TIs are described by a Rashba Hamiltonian 5with Dirac-cone-like dispersion, and are gapless and chiral, with a well-defined spin texture (spin-momentum locking.) Theycarry a πBerry phase, which protects against backscattering and thus localization, and is associated with Klein tunneling, a half quantized anomalous Hall effect 6and a giant Kerr effect.7 Nontrivial topology makes TI a platform for the observation ofMajorana fermions 8and for topological quantum computing.9 The rise of topological insulators is following a close parallel to the rise of graphene a short time ago. Three- dimensional topological insulators have grown from nonexis- tence to a vastly developed mature field involving hundreds of researchers practically overnight. Within this time span, chiralsurface states started out as a mere theoretical concept, were predicted to exist in several materials, and were subsequently imaged. 2,3Unlike graphene, the Hamiltonian of topological insulators is a function of the real spin, rather than a sublattice pseudospin degree of freedom. This implies that spin dynamics is qualitatively different from graphene. Moreover, the twofold valley degeneracy of graphene is not present in topological insulators. Despite the apparent similarities, the study oftopological insulators is thus not a simple matter of translating results known from graphene. Due to the dominant spin- orbit interaction, topological insulators are also qualitatively different from ordinary two-dimensional spin-orbit coupled semiconductors. The topological order present in TIs is a result of one- particle physics. In light of this, we recall that electron-electroninteractions modify the effective mass, heat capacity, andground-state energy of solids, as well as the response of solids to external magnetic fields. 10In fact, electron-electron interactions can lead to spontaneous magnetism in itinerantelectron systems. The best-known example of this effect isPauli paramagnetism in interacting electron systems. It isknown from Fermi-liquid theory that the Pauli paramagneticsusceptibility is enhanced by electron-electron interactions.This can be derived rigorously using various types of linear-response formalisms, such as diagrammatic Kubo linear-response theory, the Keldysh kinetic equation formalism, ordensity-matrix formalisms based on the Liouville equation.Interaction effects in systems with strong spin-orbit inter- actions have been studied in 2D TI s 11,12and 3D TIs,13–19 and previously in spin-orbit coupled semiconductors.20–25In topological insulators the focus has been on phenomena inequilibrium and in the quantum Hall regime. [Remarkably,Ostrovsky et al. , Phys. Rev. Lett. 105, 036803 (2010), showed that interactions can fully localize surface states in strongly disordered TI.] In the meantime, transport in topological insulators has made enormous strides recently. 26In initial experimental efforts, it appeared impossible to identify any signatures what-soever of the elusive surface states. Yet lately experimentalwork on transport in topological insulators has begun toadvance at a brisk pace, and is without doubt entering itsheyday, in the way angle-resolved photoemission spectroscopy(ARPES) and scanning tunneling microscopy (STM) work didtwo years ago. A beautiful experiment 27recently detected the topological surface states of Bi 2Se3, in which Sb was partially substituted for Bi to reduce the bulk carrier density to 1016 cm−3. At large magnetic fields the surface states were clearly seen, with Shubnikov–de Haas oscillations depending onlyon the perpendicular magnetic field, and oscillatory featuresgrowing with increasing field. Another work showed thatcarrier densities can be tuned over a wide range with a backgate. 28A more recent experimental breakthrough29investi- gated surface transport in thin films of Bi 2Se3of thickness ≈10 nm, observing Landau levels that evolve continuously from electronlike to holelike. In another breakthrough, Kimet al. 30studied Bi 2Se3surfaces in samples with thicknesses of <10 nm, using a gate electrode to remove bulk carriers entirely 235411-1 1098-0121/2011/84(23)/235411(7) ©2011 American Physical SocietyDIMITRIE CULCER PHYSICAL REVIEW B 84, 235411 (2011) and take both surfaces through the Dirac point simultaneously. Ambipolar transport was observed with well-defined pandn regions, together with a minimum conductivity of the order ofe 2/h, reflecting the presence of electron and hole puddles. Exciting developments in HgTe transport have also beenreported. 31,32 Due to spin-momentum locking, the charge current flow- ing on the surface of a TI is intimately linked to itsspin polarization. 33First, it is evident that an out-of-plane spin polarization can be generated by a magnetic field ormagnetization. However, an in-plane magnetic field cannot generate an in-plane spin polarization for a Dirac cone: itsimply shifts the origin of the cone and can be removedby a gauge transformation. 34On the other hand, the com- bination of spin-momentum locking plus an electric fieldcan be understood as a net effective magnetic field, whichis in the plane of the TI, and generates an in-plane spinpolarization. This paper presents a study of the role of electron-electron interactions in topological insulators in an electric field, andtheir effect on the spin polarization generated electrically in the plane of the TI. A fundamental question is whether basicTI phenomenology survives interactions out of equilibrium . It is known that in transport topology only protects againstbackscattering . Topological protection stems from time- reversal symmetry, whereas transport is inherently irreversible.Therefore robustness against electron-electron interactions inequilibrium does not translate into the same robustness intransport. I will demonstrate that the effect of interactionscan be absorbed by a renormalization of the noninteractingcharge conductivity and spin polarization, and the response isqualitatively the same. Topological insulator phenomenologytherefore remains unchanged by electron-electron interactionsin the steady state. A multiband matrix formulation is imperative to capture in- terband dynamics and disorder effects, which give a nontrivialfactor of 2 to the renormalization factor appearing in the chargecurrent and spin polarization. This paper uses an alternativematrix formulation of linear response theory, which containsthe same physics as conventional approaches and is potentiallymore transparent, relying on the quantum Liouville equationin order to derive a kinetic equation for the density matrix.This theory was first discussed for graphene monolayers 35and bilayers,36and was recently extended to topological insulators including the full scattering term to linear order in the impuritydensity. 33Peculiarities of topological insulators, such as the absence of backscattering, which reflects the πBerry phase and leads to Klein tunneling, are built into this theory in atransparent fashion. In this work, the formalism of Ref. 33 is extended to account for electron-electron interactions via amean-field approach. Since transport in noninteracting systemswas studied in that work, only minimal overlaps required forconsistency have been retained in this paper. It is assumedthatT=0 so that electron-electron scattering is absent. The theory assumes ε Fτ/¯h/greatermuch1, where εFis the Fermi energy, located in the surface conduction band, and τis the momentum relaxation time. The physics considered here isdistinct from spin-Coulomb drag, 24,37which requires electron- electron scattering , and from previous work on transport in noninteracting TIs.33Electron-electron interaction effects have also been studied in graphene transport.38The interaction physics discussed here is to be distinguished from that of graphene, since, as statedabove, graphene is a multivalley system, and its Hamiltonianis a function of the pseudospin, due to the sublattice degree offreedom, rather than the real spin. It is also important to realizethat the mean-field Hartree-Fock treatment of interactions isparticularly advantageous in topological insulators, becauseformulating a large- Nrenormalization group expansion is a challenging task. This is because, whereas in graphenethe spin and valley degeneracies yield N=g sgv=4, but in topological insulators N=gsgv=1. The outline of this paper is as follows. In Sec. IIa general effective Hamiltonian for interacting systems is introduced.The dynamics of the density matrix in interacting systems arediscussed in a mean-field formulation in Sec. III. Following that, the effective one-particle kinetic equation is derived inSec. IV. This is then solved so as to obtain the correction to the conductivity and its renormalization due to disorder,followed by a brief discussion of current TIs, a summary, andconclusions. II. EFFECTIVE HAMILTONIAN FOR INTERACTING SYSTEMS The many-body Hamiltonian is H=/summationdisplay αβHαβc† αcβ+1 2/summationdisplay αβγ δVee αβγ δc† αc† βcγcδ =H1e+Vee. (1) The two-particle matrix element Vee αβγ δ in a general basis {φα(r)}is given by Vee αβγ δ=/integraldisplay d3r/integraldisplay d3r/primeφ∗ α(r)φ∗ β(r/prime)Vee(r−r/prime)φδ(r)φγ(r/prime). (2) Hermiticity implies Vαβγ δ=V∗ γδα β and identity of electrons Vαβγ δ=Vβαδγ. The antisymmetrized form is ˜Vαβγ δ=1 2(Vαβγ δ−Vαβδγ). (3) I will consider henceforth the crystal momentum represen- tation, where α≡ks. The electron-electron interaction is taken to be explicitly of the Coulomb form. The many-bodyHamiltonian is written as H 1e+Vee, where H1e=/summationdisplay ksHkk/primess/primec† ksck/primes/prime, (4) Vee=1 2/summationdisplay qv(q)/summationdisplay kk/primess/primec† k+q,sc† k/prime−q,s/primeck/primes/primecks. The one-particle matrix element Hkk/primess/primeincludes band- structure terms and disorder. The matrix element v(q)=v(q) is given by ( /epsilon1ris the relative permittivity) v(q)=e2 2/epsilon10/epsilon1rq. (5) 235411-2LINEAR RESPONSE THEORY OF INTERACTING ... PHYSICAL REVIEW B 84, 235411 (2011) The real v(q) arises from Coulomb interaction matrix elements between plane waves, Vk1s1,k2s2,k3s3,k4s4=δs1s4δs2s3δk1+k2,k3+k4v(k2−k3).(6) The term with k2=k3is canceled by the positive background of the lattice, so v(0)=0. In TIs in the random-phase approximation (RPA), abbre- viating q=k/prime−k, one replaces v(q)≡v(q)→v(q)/ε(q), where ε(q) is the dielectric function. The polarization function is obtained by summing the lowest bubble diagram. At T=0 the long-wavelength limit of the dielectric function is39 /epsilon1(q)=1+e2 4π/epsilon10/epsilon1rA/parenleftbiggkF q/parenrightbigg . (7) The polarization function was also calculated in Ref. 19.T h e screened electron-electron Coulomb potential has the form v(q) ε(q)=e2 2/epsilon10/epsilon1r1/radicalbig k2+k/prime2−2kk/primecosγkk/prime+rskF 2. (8) The Wigner-Seitz radius rs(alternatively, the effective fine- structure constant), which parametrizes the relative strengthof the kinetic energy and electron-electron interactions, is aconstant for the Rashba-Dirac Hamiltonian, given by r s= e2/(2π/epsilon10/epsilon1rA). In addition to the electron-electron Coulomb potential, the matrix element ¯Ukk/primeof a screened Coulomb potential between plane waves, which will be relevant intransport below, is given by ¯U kk/prime=Ze2 2/epsilon10/epsilon1r1 |k−k/prime|+kTF, (9) where Zis the ionic charge (which I will assume for simplicity to beZ=1) and kTF=kFrs/2 is the Thomas-Fermi wave vector, with kFthe Fermi wave vector. III. DENSITY MATRIX The many-particle density matrix Fobeys40 dF dt+i ¯h[H,F ]=0. (10) The one-particle reduced density matrix ρis the trace ρζη=Tr (c† ηcζF)≡/angbracketleftc† ηcζ/angbracketright≡/angbracketleftF/angbracketright1e. (11) The reduced density matrix satisfies dρζη dt+i ¯h[H1e,ρ]ζη−i ¯h/angbracketleft[Vee,c† ηcζ]/angbracketright=0. (12) In terms of the antisymmetric Coulomb matrix element ˜Vαβγ δ defined above, the last term on the left-hand side (LHS) is [Vee,c† ηcζ]=/summationdisplay αβγ[˜Vαβγ ηc† αc† βcγcζ+˜Vβγαζc† ηc† αcβcγ].(13) The many-electron average is evaluated as follows: /angbracketleft[Vee,c† ηcζ]/angbracketright=/summationdisplay αβγ/angbracketleft˜Vαβγ ηc† αc† βcγcζ+˜Vβγαζc† ηc† αcβcγ/angbracketright (14a) /angbracketleftc† αc† βcγcδ/angbracketright=/angbracketleftc† αcδ/angbracketright/angbracketleftc† βcγ/angbracketright−/angbracketleftc† αcγ/angbracketright/angbracketleftc† βcδ/angbracketright+Gαβγ δ. (14b)The focus of this work is on the first two terms on the right- hand side (RHS) of Eq. ( 14b), which represent the Hartree- Fock mean-field part of the electron-electron interaction. Theremainder, G αβγ δ, gives the electron-electron scattering term in the kinetic equation,40is second-order in the interaction, and vanishes at T=0. I will treat the case of zero temperature and reserve electron-electron scattering for a forthcomingpublication. To evaluate the Hartree-Fock factorization, notethat ˜V αβγ ηργβcancels, and the remainder becomes /angbracketleft[Vee,c† ηcζ]/angbracketright=/summationdisplay αβγ[˜Vαβγ η (ρζαργβ−ργαρζβ) +˜Vβγαζ (ργηρβα−ρβηργα)]. (15) I will introduce two mean-field terms by letting ˜Vαβγ ηργβ= AMF αηand ˜Vαβγ ηργα=BMF βη, then /angbracketleft[Vee,c† ηcζ]/angbracketright=[ρ,AMF−BMF]ζη. (16) The effective kinetic equation becomes dρ dt+i ¯h[H1e,ρ]+i ¯h[AMF−BMF,ρ]=0. (17) The one-particle Hamiltonian is renormalized by Heff ee=AMF−BMF. (18) I emphasize that in the final analysis one is interested only in the impurity average of ρin the crystal-momentum repre- sentation. In general one may always write ρkk/prime=fkδkk/prime+ gkk/prime,33where the k-off-diagonal part, gkk/prime, is eventually integrated out to yield the scattering term in any desiredapproximation. In the impurity average of Eq. ( 21), out of the commutator /angbracketleft[H eff ee(ρ),ρ]/angbracketrightζηonly the terms [ Heff ee(f),f] and [Heff ee(g),g] survive, where to first order in the electric fieldHeff ee(ρ)ρ≡Heff ee(ρ0)ρE+Heff ee(ρE)ρ0. This implies that Heff ee(g)g≡Heff ee(g0)gE+Heff ee(gE)g0. In linear response g0= 0, and we are left with Heff ee(f)f. Specializing to this term, spin indices are omitted and fkis treated henceforth as a 2 ×2 matrix in spin space. To determine Heff ee(f), we evaluate the two mean-field terms. Beginning with AMF, with summation implied over repeated indices, AMF αη=˜Vαβγ ηfγβ≡˜Vk1s1,k2s2,k2s3,k4s4fk2s3s2 =v(k2−k3)δs1s4δs2s3δk1+k2,k3+k4fk2s3s2(19) =v(0)δs1s4δk1k4trfk2 ∴→0,since v(0)=0. Therefore AMFvanishes in the most general case. Next, BMF is given by BMF βη=˜Vαβγ ηργα≡/summationdisplay k/primev(k−k/prime)fk/primesβsη. (20) Note that BMFcan be interpreted as an effective magnetic field due to the Hartree-Fock mean-field electron-electron interac-tion. This result reproduces the correct exchange energy, 10and yields exchange enhancement of Zeeman field-induced spinpolarizations, as found in Fermi-liquid theory. It is similar inspirit to the treatment of Ref. 41. 235411-3DIMITRIE CULCER PHYSICAL REVIEW B 84, 235411 (2011) Equation ( 12) is reduced to dρ dt+i ¯h[H1e,ρ]=i ¯h[BMF,ρ]. (21) The single-particle Hamiltonian is renormalized by BMF(f), which is itself a function of the single-particle density matrix. At this stage one may include explicitly disorder and driving electric fields in the one-particle Hamiltonian and write H1e kk/prime= H0kδkk/prime+HEkk/prime+Ukk/prime, where H0kis the band Hamiltonian, HEkk/primeis the electrostatic potential due to the driving electric field, and Ukk/primeis the disorder potential. The effective single- particle kinetic equation takes the form dfk dt+i ¯h[H0k,fk]+ˆJ(fk)=−i ¯h/bracketleftbig HEk,fk/bracketrightbig +i ¯h/bracketleftbig BMF k,fk/bracketrightbig . (22) One writes fk=f0k+fEk+fee k, where f0kis the equilib- rium density matrix, fEkis induced by the electric field, andfee kby electron-electron interactions. (The interaction correction to the energy contributes a diagonal term to thekinetic equation, which drops out of the Hamiltonian and doesnot contribute to f 0k.). Equation ( 22) is solved iteratively in BMF. Let the bare driving term Dk=−i ¯h[HEk,fk]. The approach is to solve the kinetic equation first with Dkas the source term. This will give a spin polarization. The spin polarization will givea nonzero B MF k, which in turn will give an additional source term, referred to as Dee kin the next section. Then one solves the kinetic equation again with Dee kas the source term, dfEk dt+i ¯h[H0k,fEk]+ˆJ(fEk)=−i ¯h[HEk,f0k], (23)dfee k dt+i ¯h/bracketleftbig H0k,fee k/bracketrightbig +ˆJ/parenleftbig fee k/parenrightbig =i ¯h/bracketleftbig BMF k,f0k/bracketrightbig . On the RHS of the second equation only the equilibrium density matrix f0kappears because BMF kis first order in the electric field. The iteration is continued to all orders in theWigner-Seitz radius r s(that is, to all orders in the effective fine-structure constant). We recall that electron-electron and electron-impurity potentials are screened, with screening treated in the random-phase approximation. The density-matrix formalism usedhere is thus equivalent to the GW approximation. In thenonequilibrium diagram technique, the correction discussed inthis work is obtained by including the real part of the Green’sfunction due to electron-electron interactions. 41 IV . KINETIC EQUATION FOR INTERACTING TI Henceforth I specialize to TIs. The band Hamiltonian H0k=¯h 2σ·/Omega1k, where /Omega1k=−2Ak ¯hˆθ, with ˆθthe tangential unit vector in polar coordinates in reciprocal space. Interactionwith the electric field is given by H E,kk/prime=(eE·ˆr)kk/prime1= ieE·∂ ∂kδ(k−k/prime)1, with 1the identity matrix in spin space. Uncorrelated impurities located at RIare represented by Ukk/prime=¯Ukk/prime/summationtext Iei(k−k/prime)·RI, with ¯Ukk/primethe Fourier transform of the potential of a single impurity. I will write fk=nk1+Sk, withnkthe number density and Sk=1 2Sk·σthe spin density. One decomposes Sk=Sk/bardbl+Sk⊥, where [ H0k,Sk/bardbl]=0 and Sk/bardblis the fraction of carriers in eigenstates of H0k, while Sk⊥represents interband dynamics, i.e., Zitterbewegung. Further, Sk/bardbl=(1/2)sk/bardblσk/bardblandSk⊥=(1/2)sk⊥σk⊥, with the matrices σk/bardbl=−σ·ˆθandσk⊥=σ·ˆk. A. Single-particle kinetic equation The general single-particle kinetic equation is dSk/bardbl dt+P/bardblˆJ(Sk)=Dk/bardbl, (24a) dSk⊥ dt+i ¯h[Hk,Sk⊥]+P⊥ˆJ(Sk)=Dk⊥, (24b) where the driving term Dk=eE ¯h·∂ρ0k ∂k, andρ0kis the equilib- rium density matrix. This equation is solved as an expansion inthe small parameter ¯ h/(ε Fτ), where the momentum relaxation timeτis defined below. The leading-order term in this expansion is ∝[¯h/(εFτ)](−1)and is found from P/bardblˆJ(Sk/bardbl)=Dk/bardbl. (25) The solution to Eq. ( 28) requires certain approximations. With respect to the scattering potential one expands in thesmall parameter ¯ h/(ε Fτ). In the steady state in the Born approximation the leading term in the solution to the kineticequation is ∝[¯h/(ε Fτ)](−1). It is trivial to check that at finite doping the next term in the expansion, i.e., ∝[¯h/(εFτ)](0), vanishes identically, which was demonstrated in Ref. 33.A term∝[¯h/(εFτ)](0)would appear in the weak localization regime, yet this correction is not relevant in the regimeε Fτ/¯h/greatermuch1 considered in this work. The Born-approximation scattering term has the form ˆJ(fk)=1 ¯h2/angbracketleftbigg/angbracketleftbigg/integraldisplay∞ 0dt/prime[ˆU,e−iˆHt/prime/¯h[ˆU,ˆf]eiˆHt/prime/¯h]/angbracketrightbigg/angbracketrightbigg kk,(26) withγ=θ/prime−θthe angle between the incident and scattered wave vectors, kandk/primerespectively, and /angbracketleft/angbracketleft/angbracketright/angbracketright denoting the average over impurity configurations. The projections of ˆJ(fk) needed in this work have been determined before,33 P/bardblˆJ(Sk/bardbl)=kniσk/bardbl 8¯hπA/integraldisplay dγ|¯Ukk/prime|2(sk/bardbl−sk/prime/bardbl)(1+cosγ), P⊥ˆJ(Sk/bardbl)=kniσk⊥ 8¯hπA/integraldisplay dγ|¯Ukk/prime|2(sk/bardbl−sk/prime/bardbl)s i nγ, (27) P/bardblˆJ(Sk⊥)=kniσk/bardbl 8¯hπA/integraldisplay dγ|¯Ukk/prime|2(sk⊥+sk/prime⊥)s i nγ, where γ=θ/prime−θis the angle between the incident and scattered wave vectors. The small sk/bardblandsk⊥are scalars, sk/bardbl=−Sk·ˆθandsk⊥=Sk·ˆk. The scattering terms contain factors of (1 +cosγ)( r e - flecting the πBerry phase) or sin γ, both of which prohibit backscattering and give rise to Klein tunneling. Since thecurrent operator ∝σ, only S kis needed. In the absence of scalar terms in the Hamiltonian, ˆJ(fk) does not couple nkwith Sk, and Eq. ( 22) makes evident the fact that the interaction term does not couple nkandSk, thusnkmay be dispensed with for the remainder of this work. The equation satisfied by Skis dSk dt+i ¯h[H0k,Sk]+ˆJ(Sk) =−i ¯h/bracketleftbig HEk,Sk/bracketrightbig +i ¯h/bracketleftbig BMF, (1) k,Sk/bracketrightbig . (28) 235411-4LINEAR RESPONSE THEORY OF INTERACTING ... PHYSICAL REVIEW B 84, 235411 (2011) With respect to the electron-electron interaction one also needs to define a perturbation expansion in order to solve Eq. ( 28), which is done in what follows. Within the approximationsused in this paper, this expansion can be summed exactly . The method of solution is summarized as follows. The kineticequation first with B MF kset to zero. This solution is already known33and gives a spin polarization, which in turn generates a nonzero BMF k, which itself yields a new driving term, and so forth. The full solution is found as a perturbation expansionin the electron-electron interaction, which can be summedexactly. To obtain the solution in the interacting case, it is therefore first necessary to solve the noninteracting problem. In theabsence of interactions 33the steady-state solution to the density matrix in the Born approximation is33 SEk/bardbl=τeE·ˆk 4¯h∂f0+ ∂kσk/bardbl, (29)1 τ=kni 4¯hA/integraldisplaydγ 2π|¯Ukk/prime|2sin2γ. Above niis the impurity density, while the factor of sin2γ represents the product (1 +cosγ)(1−cosγ). The first term in this product is characteristic of TI and ensures backscatteringis suppressed, while the second term is characteristic oftransport, eliminating the effect of small-angle scattering. Innoninteracting TI the Zitterbewegung contribution to the con- ductivity/spin density (i.e., due to S ee,(0) Ek⊥) vanishes identically at finite doping. But in the interacting case it is necessary toconsider both the electron and the hole bands to capture the spindynamics. The solution obtained, S Ek≡SEk/bardbl, is fed into BMF k, which in turn generates a new driving term in the kinetic equation. Each term in this expansion by the index α, thus BMF, (α) k. The solution found in Eq. ( 29) corresponds to α=0, that is, in the noninteracting case SEk/bardbl≡See,(0) Ek/bardbl. The driving term due to electron-electron interactions is generically denoted Dee,(α) k. The decomposition Dee,(α) k⊥=(1/2)dee,(α) k⊥σk⊥is also used. The solution to the spin part of the density matrix to order αis denoted by See,(α) k. The driving term Dee,(α) kis always orthogonal to H0k, therefore Dee,(α) k≡Dee,(α) k⊥.T h e kinetic equation for the solution See kin the presence of electron-electron interactions can be written for each orderas dSee,(α) k⊥ dt+i ¯h/bracketleftbig Hk,See,(α) k⊥/bracketrightbig =Dee,(α) k⊥, (30a) P/bardblˆJ/parenleftbig See,(α) k/parenrightbig =−P/bardblˆJ/parenleftbig See,(α) k⊥/parenrightbig , (30b) where in Eq. ( 30b) it is understood that the RHS, found from Eq. ( 30a), acts as the source for the LHS. The scattering term does not appear in Eq. ( 30a) since, as was argued above, Dee,(α) k/bardbl=0 always. I will dwell first on the solution of Eq. ( 22) due to BMF, (1) k, i.e., first order in the interaction, which requires See,(0) Ek/bardbl.F r o mEq. ( 29), BMF, (1) k=e2kF 4π/epsilon10/epsilon1r/integraldisplay1 0dl/primel/prime/integraldisplay2π 0 ×dγ 2πSee,(0) Ek/prime/radicalbig l2+l/prime2−2ll/primecosγ+rs 2, (31) where l=(k/kF). The term in BMF, (1) kin which Sk/prime→S0k/prime gives a vanishing contribution. For E/bardblˆx BMF, (1) k=e3Exτ 16π¯h/epsilon10/epsilon1r/bracketleftbig I(1) ccosθσk/bardbl−I(1) eesinθσk⊥/bracketrightbig , (32) I(1) ee(l,rs)=1 2/integraldisplaydγ 2π(cos 2γ−1) (/radicalbig 1+l2−2lcosγ+rs 2. InI(1) c(l,rs) the sign of the cosine term is flipped. Although BMF, (1) kitself has a part ∝σk/bardbl, this part drops out of the driving term in Eq. ( 33), because one is working to first order in the electric field and the commutator [ BMF, (1) k,Sk]→ [BMF, (1) k,S0k], and S0k∝σk/bardbl. The effective electron-electron interaction Hamiltonian BMF, (1) ktherefore contributes a driving term orthogonal to H0k, yielding a correction See,(1) Ek⊥to the density matrix. The scattering term does not appear in the equation for See,(1) k⊥. Equation ( 28) takes the simple form dSee,(1) Ek⊥ dt+i ¯h/bracketleftbig H0,See,(1) Ek⊥/bracketrightbig =i ¯h/bracketleftbig BMF, (1) k,S0k/bracketrightbig . (33) This equation is solved using the time evolution operator See,(1) Ek⊥=eExrsτI(1) ee(l,rs) 16¯hkf0+sinθσ·ˆk. (34) Another contribution stems from the projection P/bardblˆJ/bracketleftbig See,(1) k/bardbl/bracketrightbig =−P/bardblˆJ/bracketleftbig See,(1) k⊥/bracketrightbig . (35) It is understood that the RHS, found from Eq. ( 34), acts as the source for the LHS. Straightforwardly, See,(1) k/bardbl=eExrsτI(1) ee(l,rs) 16¯hkf0+cosθσ·ˆθ. (36) See,(1) Ek⊥andSee,(1) k/bardblcontribute equally to the charge current determined below. In effect, scattering from See,(1) Ek⊥intoSee,(1) k/bardbl doubles the contribution to the electrical conductivity due to See,(1) Ek⊥. The longitudinal current density operator jx=eA ¯hσy:t h e current density is equivalent to a spin polarization. The conduc- tivityσ0 xxof the noninteracting system is σ0 xx=(e2 h)(AkFτ 4¯h).33 The first-order conductivity correction in the electron-electron interaction is σee,(1) xx=/parenleftbigge2 h/parenrightbigg/parenleftbiggAkFτ 4¯h/parenrightbiggrsI(1) ee 2≡σ0 xx/parenleftbiggrsI(1) ee 2/parenrightbigg , (37) where I(1) ee(rs)=/integraltext1 0dlI(1) ee(l,rs). The electrical current and nonequilibrium spin polarization are renormalized (reduced)by electron-electron interactions. Equation ( 37) has been obtained to first order in the (screened) interaction. The source term due to dee,(1) Ekcon- tains only e±iθ, identical in structure to the noninteracting 235411-5DIMITRIE CULCER PHYSICAL REVIEW B 84, 235411 (2011) problem.33One solves for all higher terms in Hee kby iterating steps ( 31)–(36), obtaining the exact result for the conductivity (and spin polarization), σxx σ0xx=1+rs 2/bracketleftbigg I(1) ee+rs 4I(2) ee+/parenleftbiggrs 4/parenrightbigg2 I(3) ee+···/bracketrightbigg .(38) The general formula for the dimensionless integral I(n) eefor n> 1i s I(n) ee=/Pi1i=n i=1/integraldisplay1 0dli/integraldisplay2π 0dγi 2π⎛ ⎝1 rs 2+/radicalBig l2 1+l2 2−2l1l2cosγ1⎞ ⎠ ×⎛ ⎝1 rs 2+/radicalBig l2 2+l2 3−2l2l3cosγ2⎞ ⎠ ···/parenleftBigg (−1)nsin2γn rs 2+/radicalbig 1+l2n−2lncosγn/parenrightBigg . (39) In two dimensions v(q)∝1/q, while in TI rsis density independent, and the screened Coulomb potential v(q)//epsilon1(q)∝ 1/qalso. Thus Hee kdoes not introduce density dependence: at larger densities the Coulomb interaction is weaker. SolvingforS ee Ek⊥introduces a factor of 1 //Omega1 k, which is canceled bykin the 2D volume element. Thus 2D physics and TI linear dispersion combine to ensure that the renormalizationis density independent. The renormalization reflects the interplay of spin- momentum locking and many-body correlations. A spin at kfeels the effect of two competing interactions. The Coulomb interaction between Bloch electrons with kandk /primetends to align as p i na t kwith the spin at k/prime, equivalent to a ˆ zrotation—hence the driving term in Eq. ( 33)i s∝σz. The total mean-field interaction tends to align the spin at kwith the sum of all spins at all k/prime, andHee kencapsulates the amount by which the spin at kis tilted as a result of the mean-field interaction with all other spins on the Fermi surface. The effective field /Omega1k tends to align the spin with itself. As a result of this latter fact, out of equilibrium, an electrically induced spin polarization isalready found in the noninteracting system. 33Given that the spins at kandk/primeare in the plane, interactions tilt the spin atkin the direction of the spin at k/prime. Thus far the argument helps one understand why, if there is no spin polarization tostart with, electron-electron interactions do not give rise toa spin polarization. The mean-field result is zero, so thereis no overall tilt on any one spin due to the spins of theremaining electrons. Interactions tend to align electron spins inthe direction opposite to that of the existing polarization. Theeffective ˆ zrotation explains the counterintuitive observation that the renormalization is related to interband dynamics,originating as it does in S k⊥. Many-body interactions give an effective k-dependent magnetic field /bardblˆz, such that for E/bardblˆx the spins syand−syare rotated in opposite directions. Due to spin-momentum locking, a tilt in the spin becomes a tilt in thewave vector: spin dynamics create a feedback effect on chargetransport, renormalizing the conductivity. This feedback effectis even clearer in the fact that the projection −P /bardblˆJ(Sk⊥) doubles the renormalization. This doubling is valid for any elastic scattering. FIG. 1. Fractional change in the conductivity |δσee xx/σ0 xx|for 0/lessorequalslantrs/lessorequalslant1. The current generation of topological insulators has rs/lessmuch1, so the theory presented in this work is applicable to these materials. I will discuss next the magnitude of this renormalization in currently known topological insulators. Several materials werepredicted to be topological insulators in three dimensions. Thefirst was the alloy Bi 1−xSbx,42,43followed by the tetradymite semiconductors Bi 2Se3,B i 2Te3, and Sb 2Te3.44These materials have a rhombohedral structure composed of quintuple layersoriented perpendicular to the trigonal caxis. The covalent bonding within each quintuple layer is much stronger thanweak van der Waals forces bonding neighboring layers. Thesemiconducting gap is approximately 0.3 eV , and the TIstates are present along the (111) direction. In particularBi 2Se3and Bi 2Te3have long been known from thermoelectric transport as displaying sizable Peltier and Seebeck effects, andtheir high quality has ensured their place at the forefront ofexperimental attention. 2Initial predictions of the existence of chiral surface states were confirmed by first-principles studiesof Bi 2Se3,B i 2Te3, and Sb 2Te3.45In the current generation of topological insulators, rsis small. Currently /epsilon1rranges between 30 and 100 (200 for Bi 2Te3), making rsbetween 0.14 and 0.46. The theoretical treatment adopted in this workis therefore applicable, and interactions provide a correction tothe steady-state response. A plot of |δσ ee xx/σ0 xx|for 0/lessorequalslantrs/lessorequalslant1 is shown in Fig. 1, from which it emerges that interactions may account for up to ≈15% of the observed conductivity of surface states in the regime studied. I note that Heusler alloyswere recently predicted to have topological surface states, 46as well as chalcopyrites,47yet more work is needed to establish the size of rsin these materials. At this stage in topological insulator research, the results found in this work are interesting for conceptual reasons, sincethey demonstrate that TI phenomenology is unchanged by in-teractions. The electrical conductivity/spin polarization has thesame form as in the noninteracting case, with a renormalizationthat can be incorporated into a redefinition of the spin-orbitconstant, or alternatively of the Fermi velocity, and thus thedensity of states. For large r sa nonperturbative treatment that goes beyond the random-phase approximation is necessary, yetsuch a theory must await materials progress. In this context,I would like to note that the growth of new materials is anontrivial issue, and obtaining high-quality samples whereonly the surface electrons can be accessed in transport hasproved to be a difficult task. It is especially important to 235411-6LINEAR RESPONSE THEORY OF INTERACTING ... PHYSICAL REVIEW B 84, 235411 (2011) recall that future work may initially be hampered by factors such as roughness and dirt inherent in solid-state interfaces.In addition, it remains true that all current TI materials areeffectively bulk metals because of their large unintentional doping—at present, bulk carriers are only removed temporarilyby gating. Discussing TI surface transport in such bulk-dopedTI materials retains some ambiguity, since it necessarilyinvolves complex data fitting and a series of assumptionsrequired by the necessity of distinguishing bulk versus surfacetransport contributions. Real progress is expected when surfaceTI transport can be carried out unambiguously, withoutany complications arising from the (more dominant) bulktransport channel. The immediate tasks facing experimen-talists are getting the chemical potential in the gap withoutthe aid of a gate, further experimental studies confirmingambipolar transport, and the measurement of a spin-polarizedcurrent.V . CONCLUSIONS I have demonstrated that, from the point of view of the nonequilibrium spin polarizations and charge current, TIbehavior remains intact in the presence of interactions withonly quantitative modifications. The conductivity and spinpolarization are renormalized by electron-electron interactionsentering through a combination of interband dynamics andscattering. ACKNOWLEDGMENTS I am greatly indebted to S. Das Sarma, A. H. MacDonald, Yafis Barlas, W. K. Tse, Stephen Powell, Junren Shi, JeilJung, Dagim Tilahun, and Arun Paramekanti. This work wassupported in part by the Chinese Academy of Sciences and inpart by the National Science Foundation under Grant No. NSFPHY05-51164. 1C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 2M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 3X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). 4J. E. Moore and L. Balents, P h y s .R e v .B 75, 121306 (2007). 5Y . A. Bychkov and E. I. Rashba, JETP Lett. 39, 66 (1984). 6J. Zang and N. Nagaosa, Phys. Rev. B 81, 245125 (2010). 7W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105, 057401 (2010). 8L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008). 9C. Nayak, S. H. Simon, A. 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PhysRevB.99.155152.pdf

PHYSICAL REVIEW B 99, 155152 (2019) Torsional response and Liouville anomaly in Weyl semimetals with dislocations Ze-Min Huang,1,2Longyue Li,1Jianhui Zhou,3,*and Hong-Hao Zhang1,† 1School of Physics, Sun Yat-Sen University, Guangzhou 510275, China 2Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA 3Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, Anhui, China (Received 24 July 2018; published 29 April 2019) Weyl nodes in three-dimensional Weyl semimetals break the Liouville equation, leading to the Liouville anomaly. Here we present an approach to derive the semiclassical action and equations of motion for Weylfermions in the presence of electromagnetic fields and torsions from the quantum field theory: combining theWigner transformation with band projection operation. It has been shown that the Liouville anomaly, including apure torsion anomaly term, a mixing term between the electromagnetic fields and torsions, and the conventionalchiral anomaly, entirely differs from the counterpart of axial gauge fields. We find various torsional responsesand reproduce the chiral vortical effect and the torsional chiral magnetic effect. A torsion-modified anomalousHall effect due to the mixing term in the Liouville anomaly is predicted, and its implementation is also discussed.Therefore, our work not only provides insights into the torsional responses for Weyl fermions but also acts as astarting point to investigate their topological responses. DOI: 10.1103/PhysRevB.99.155152 I. INTRODUCTION Quantum anomalies, the breaking of classical symmetries by quantum fluctuations, have attracted much attention in condensed-matter physics due to the deep connection withtopology [ 1,2]. Many exotic responses of topological phases of matter can be understood in the language of quantumanomalies, including topological insulators [ 3,4] and topo- logical semimetals [ 5–7]. Recently, the chiral anomaly in the context of three-dimensional Weyl and Dirac semimetals hasled to rich physical phenomena [ 8–11], such as the chiral magnetic effect (CME) [ 5,12–20], the negative longitudinal magnetoresistance [ 21], the nonlocal transport [ 22], the giant planar Hall effect [ 23,24], and the unconventional collective excitations [ 25–29], some of which have been observed ex- perimentally [ 30–36]. Historically, the chiral anomaly was first derived by use of the perturbation method [ 37,38] and later by Fujikawa’s path integral method [ 1] and from the transport of the chiral zeroth Landau level [ 39]. Recently, the Berry-curvature-modified semiclassical equations of motion were used to derive theequation of the chiral anomaly [ 40]. Within the framework of the semiclassical equations of motion [ 41], the chiral anomaly equation manifests itself as breaking the conservation of thephase-space current or the Liouville equation, which is alsodubbed the Liouville anomaly [ 40,42,43]. Compared with Fujikawa’ s method, the absence of an ultraviolet cutoff in theLiouville anomaly can be traced back to the charge pumpingbetween Weyl nodes with opposite chiralities. In the presenceof dislocations or temperature gradients, gravity with torsion *jhzhou@hmfl.ac.cn †zhh98@mail.sysu.edu.cnwill emerge. A nonvanishing torsion contributes the Nieh-Yanterm to the chiral anomaly equation [ 44], leading to novel geometrical responses for Weyl fermions [ 45,46]. In fact, the construction of the semiclassical equations of motion with torsions is highly nontrivial. Both the wave-packet approach [ 47] and the chiral kinetic theory [ 40]a r e based on the Hamiltonian mechanics, while the Hamiltonianfrom the curved-space-time Dirac equation is hindered by theHermiticity problem [ 48]. Although a Hermitian Hamiltonian can be obtained from some careful manipulations [ 49], it would be too cumbersome for our purposes. In addition,semiclassical chiral kinetic theory can also be derived fromquantum field theory, but it remains valid only in the homo-geneous limit [ 50]. Therefore, a new method for deriving the semiclassical equations of motion with torsions is highly de-sirable and crucial to investigations of the related topologicalresponses. In this paper, we develop a formalism to derive the semi- classical action and equations of motion for Weyl fermions in the presence of electromagnetic fields and torsions from the quantum field theory: combining the Wigner transforma-tion with band projection operation. The relevant Liouvilleanomaly consists of a pure torsion anomaly term and a termmixing the electromagnetic fields and torsions, in additionto the conventional chiral anomaly. Various responses areobtained, such as the chiral vortical effect and the proposed torsional chiral magnetic effect. Meanwhile, we find a torsion- modified anomalous Hall effect (AHE) from the mixed termin the Liouville anomaly and discuss its implementation inWeyl semimetals with broken time-reversal symmetry such asCo 3Sn2S2. In addition, we find that the chiral zero modes lo- calized in dislocations can be canceled by the compensation ofthose states from the bulk via the Callan-Harvey mechanism. 2469-9950/2019/99(15)/155152(11) 155152-1 ©2019 American Physical SocietyHUANG, LI, ZHOU, AND ZHANG PHYSICAL REVIEW B 99, 155152 (2019) The rest of this paper is organized as follows. In Sec. II, the Lagrangian density for Weyl fermions in the presence oftorsions is introduced. In Sec. III, we derive the one-band effective Green’s function by combining the band projectionwith Wigner’s transformation. In Sec. IV, the semiclassical dynamics in the presence of both torsions and electromagneticfields are derived. In Sec. V, we consider torsional responses and derive the anomaly equations within the semiclassicalformalism. We also predict a torsion-modified AHE and dis-cuss its experimental implementation. In Sec. VI,t h em a i n results of this paper are summarized. Finally, we give detailedcalculations in the Appendices. II. MODEL In the presence of dislocations, the corresponding defor- mation of media is described by the displacement vectorsu a(x), where the superscript a=0,1,2,3 denotes locally flat space-time coordinates with the metric tensor ηab= diag(1 ,−1,−1,−1) [51]. That is, under a lattice defor- mation, lattice coordinates are shifted, i.e., x→x+u.S o ∂μ=ea μ∂a, where ea μis the vielbein, i.e., ea μ=δa μ+∂μua, and μ=0,1,2,3 denotes the curved space-time coordinates (or laboratory coordinates) with the metric tensor gμν=ea μηabebν. The action has the form [ 52,53] S=1 2/integraldisplay d4x/vextendsingle/vextendsingledetea μ/vextendsingle/vextendsingle/bracketleftbig¯/Psi1eμ aγa(i∂μ/Psi1)−(i∂μ¯/Psi1)eμ aγa/Psi1/bracketrightbig , (1) where γaare the 4 by 4 gamma matrices and eμ ais the inverse ofea μ. The action is written in this way to ensure Hermiticity locally. The torsions or torsional electromagnetic fields aredefined as T a μν=∂μea ν−∂νea μ. III. BAND-PROJECTED GREEN’S FUNCTION AND WIGNER’S TRANSFORMATION The Green’s function for the right-handed Weyl fermions can be read off from the action above directly. One can seethat this Green’s function depends on both momentum andposition. It is well known that in quantum physics, the positionoperator and the momentum operator do not commute witheach other. To develop a semiclassical theory described byboth momentum and position, one needs to utilize the Wignertransformation and has the Wigner-transformed Green’s func-tion (see Appendix A) i˜G −1=(1+w)pμδμ aσa−wμ apμσa+O(u2), (2) where σa=(1,σi),wμ a≡δρ a∂ρubδμ b, and w≡δa μwμ a.T h e factor (1 +w) comes from the determinant |detea μ|. One can see that, up to the linear-order terms in u, this Green’s function is equivalent to i˜G−1≈(1+w)[pμδμ a−(pμwμ a)]σa. Thus, wμ acouples to Weyl fermions with a coupling charge pin a way similar to the electromagnetic gauge fields. In order to derive the semiclassical action, one must project the two-band Green’s function in Eq. ( 2) onto its positive- energy bands, i.e., ˜G−1 ++=/angbracketleftu+|∗(˜G−1)∗|u+/angbracketright,where |u+/angbracketrightis the positive-energy eigenstates: p·σ|u+/angbracketright=|p||u+/angbracketright[54]. The Moyal star product, ∗=exp [−i 2(←−∂qμ−→∂pμ−←−∂pμ−→∂qμ)], is from the Wigner transformation. After lengthy calculations,one gets the projected Green’s function (see Appendix A) i˜G−1 ++=iG−1−ap·∂qiG−1+ξviel, (3) where iG−1=(1+w)paδa νˆpν+pawa νˆpνand ai p= /angbracketleftu+|i∂pi|u+/angbracketrightis the Berry connection for electrons in the con- duction band. The bold alphabet here is used for vectors in Eu-clidean space; for example, q μ=(q0,−qi), and ∂i q=∂ ∂qiis the derivative with respect to coordinates q.I nE q .( 3), the first two terms can be regarded as first-order Taylor’s expansionofG −1(q−ap). Hence, compared to electromagnetic fields, the Berry connection is like gauge fields in the momentumspace. In addition, G −1is the next-lowest-order expansion of (det ea μ)paea μˆpμ.ˆpμoriginates from eμ b/angbracketleftu+|σb|u+/angbracketright, where eμ blinks the locally flat space-time to the laboratory coordi- nates. Because, for right-handed Weyl fermions, the velocityoperator is v a=∂H/∂pa=σa,ˆpμwill link to velocity in the laboratory coordinate, as we shall show later. Finally,the energy correction ξ viel=/epsilon10αβσ ˆpσ 2|p|(1 2pbTb αβ) describes the coupling between the orbital magnetic moment ˆp/2|p|and the spatial components of the torsion tensor Ta ij. The torsional magnetic field ˜Tais defined as the Hodge dual of the torsion tensor ( Ta)jk=∂j q(ea)k−∂k q(ea)j, i.e., ( ˜Ta)i=1 2/epsilon1ijk(Ta)jk. IV . SEMICLASSICAL ACTION AND EQUATIONS OF MOTION In this section, we construct the semiclassical dynamics based on the band-projected Green’s function above. Thedispersion relation for the positive-energy particles can beobtained by solving the equation ˜G −1 ++=0. That is, the on- shell particles are located at poles of the Green’s function. Bykeeping terms up to order uand restoring the electromagnetic fields, one can straightforwardly find the solution to Eq. ( 3), leading to the semiclassical action (see Appendix A) L=k·˙q−(|k|−ξ viel−ξem) +[(wa)μka−Aμ]˙qμ−ak·˙k, (4) where ka=(|k|,−ki), ˙qμ=(1,˙qi),Aμ=(φ,−Ai)i s the electromagnetic gauge potential, and ξem=/epsilon10αβσ 1 2|p| (1 2Fαβ)ˆpσstems from the orbital magnetic moment of elec- trons. It is clear that dislocations modify the semiclassicalaction through two ways: the shift of the gauge potential andthe correction of the energy dispersion, which implies that (w a)μdoes behave like the electromagnetic gauge fields Aμ but with a coupling charge ka. Note that we have changed the variable from canonical momentum pto mechanical momen- tumk=p+(w0)|p|+(wj)pj−A. The corresponding equations of motion can be derived from the semiclassical action in Eq. ( 4), D˙qi=/braceleftbig/bracketleftbig 1−∂n k(kawa)n/bracketrightbig δij+∂i k(kawa)j/bracerightbig vj −/epsilon1ijk/bracketleftbig˜/Omega1k−˜/Omega1m∂m k(kawa)k/bracketrightbig/parenleftbig ∂j qE−Tj ele/parenrightbig −(˜/Omega1·v)Ti mag (5) 155152-2TORSIONAL RESPONSE AND LIOUVILLE ANOMALY IN … PHYSICAL REVIEW B 99, 155152 (2019) and D˙ki=−/braceleftbig/bracketleftbig 1−∂n k(kawa)n/bracketrightbig δij+∂j k(kawa)i/bracerightbig/parenleftbig ∂j qE−Tj ele/parenrightbig +/epsilon1ijkvj/bracketleftbig −δkm+∂k k(kawa)m/bracketrightbig Tm mag +˜/Omega1i[Tmag·(∂qE−Tele)], (6) where vi=∂i k(E−wa 0ka) is the velocity, with ∂i k=∂/∂ ki being the derivative with respect to momentum k.˜/Omega1i≡ 1 2/epsilon1ijk/Omega1jkis the Hodge dual of the Berry curvature /Omega1ij≡ ∂i kaj k−∂j kai k. In addition to the torsional magnetic fields, the torsional electric fields ( Ta 0)i=−Ta 0i, which link to the thermal transport [ 55–58], also exist. Due to the common role played by torsions and the electromagnetic fields, we could define Ti mag=−Bi+ka(˜Ta)iandTi ele=Ei+ka(Ta 0)i. The modified density of states is given as D/(2π)3, with D= 1−˜/Omega1jTj mag+˜/Omega1l∂l k(kawa)iTi mag. With no torsions, Dwill reduce to 1 +˜/Omega1·B, which is well known in semiclassical physics [ 59–65]. Interestingly, the torsion coupling charge k leads to an extra term in D. Equations ( 4)–(6) are part of the main results in this work. Let us now turn to the physics encoded in Eq. ( 5). The terms in the first line show that the velocity is modified by tor-sions. The terms in the second line correspond to the anoma-lous Hall effect. Because the temperature gradient can bedefined as T 0 0i[58] with e0 μcoupling with the energy current, the anomalous thermoelectric effect is also included. The termin the last line contains both the CME and the torsional CME(see Appendix B). To be more specific, the current caused by torsional magnetic fields is −/integraltext d3k (2π)3fn(˜/Omega1n·vn)(˜Taka),where index ndenotes bands and chirality. fnis the Fermi-Dirac distribution function. Because fn(˜/Omega1n·vn)(˜Tiki)i sa no d d function of momentum, this current should vanish unless thereis a pair of Weyl nodes with opposite chiralities located atdifferent positions in energy-momentum space, which can beimplemented through breaking either time-reversal symmetryor inversion symmetry. Meanwhile, the terms in the first line of Eq. ( 6)a r et h e electric force and the counterpart from torsions. Similarly,those in the second line are the Lorentz force and its torsionalcounterpart. One can clearly find that the torsional magneticfields behave expectedly like the conventional magnetic fields.The last term closely relates to the Liouville anomaly. The anomaly term on the right-hand side in Eq. ( 6) contains a mixing term between the electromagnetic fields and tor-sions. But the Berry curvature always leads to a Dirac δfunc- tion in the anomaly equation. Since the coupling charge oftorsions is momentum, nontrivial results require Weyl nodesdeviate from the origin. Hence, we assume that Weyl nodeswith chirality s(s=±1) are located at λ saand then have an extra term in Eq. ( 1), i.e., /Sigma1s/integraltext |detea μ|¯/Psi1(−λsaPs)γa/Psi1, with Ps=1 2(1+sγ5) andλsa=(λs0,−λi s), where we have restored the chirality index. The corresponding semiclassicalaction becomes L s=k·˙q−t(|k−λs|+λs0−ξviel−ξem) +(wa)μka˙qμ−Aμ˙qμ−aks·˙k, (7)where aks=aks(k−λs) is a function of ( k−λs). The energy correction from the orbital magnetic moment term becomes ξviel+ξem=¯h/epsilon10αβσ1 2(kbTb αβ+Fαβ)/hatwidek−λsσ 2|k−λs|. Note that, up to the low- est order in external fields, the dispersion relation is k0= |k−λs|+λs0. V . TORSIONAL RESPONSES AND LIOUVILLE ANOMALY In this section, we turn to consider the torsional responses and derive the anomaly equations within the semiclassicalformalism. We also predict a torsion-modified AHE anddiscuss its experimental implementation. The Callan-Harveymechanism is used to discuss the local charge conservation. From the equation of motion in Eq. ( 5), one finds the current stemming from ( ˜/Omega1·v)T a magkaandξvielas j=/Lambda1λi 2π2˜Ti+μλi 2π2/parenleftbigg 1+1 3/parenrightbigg ˜Ti +μ π2/bracketleftbigg μ5/parenleftbigg1 2+1 3/parenrightbigg −λ0 6/bracketrightbigg ˜T0, (8) where /Lambda1is the energy cutoff rather than the momentum cutoff and is actually from the distribution function for negative-energy particles: f s−={exp [β(−|k|+λs0−μs)]+1}−1. The chemical potential for sWeyl fermions is μs=μ+sμ5, andμ5is the chiral chemical potential induced by the chiral anomaly. In addition, we assume λsμ=sλμhereafter. The first term on the right-hand side of Eq. ( 8)i st h e torsional CME (see Appendix B). The relevant current is proportional to the energy cutoff /Lambda1, which is actually from the distribution function for negative-energy particles. Notethat the /Lambda1-dependent current was tested numerically in a tight-binding model [ 66]. The coefficients of 1 and 1 /3i n the second term come from ( ˜/Omega1·v)T i magandξviel, respectively. The second term means that the torsional magnetic fields caninduce currents proportional to the chemical potential ratherthan the chiral chemical potential. Physically, in the presenceof an external magnetic field, Weyl fermions with differentchiralities would move oppositely, and the net current is thusproportional to the chiral chemical potential, which gives riseto the CME [ 39]. On the other hand, for torsional magnetic fields, λ siprovides an extra minus sign, so the current turns out to be proportional to the chemical potential μ.A sw e shall show later, the current from ( ˜/Omega1·v)Ti magis closely related to the Liouville anomaly in Eq. ( 7). Compared with the chiral pseudomagnetic effect [ 16,67–69], both currents are proportional to the chemical potential. However, the extraminus sign in the chiral pseudomagnetic effect comes fromthe opposite coupling between the axial gauge fields andthe right-handed or left-handed Weyl fermions. For the lastterm, the coefficient of μ 5/2 comes from ( ˜/Omega1·v)T0 magk0; both the coefficients μ5/3 and λ0/6 come from ξviel. Because ˜T0 links to the background rotation, μμ 5˜T0/2π2corresponds to the chiral vortical effect [ 12,40,70]. In analogy to the dynamical CME [ 71–74],−λ0μ˜T0/6π2can be regarded as the dynamical chiral vortical effect, which stems from theorbital magnetic moment of electrons on the Fermi surfaceas well. 155152-3HUANG, LI, ZHOU, AND ZHANG PHYSICAL REVIEW B 99, 155152 (2019) The Liouville theorem states that the phase-space volume does not change under evolution. If we define /Omega1Las the volume form in the extended phase space (position, mo-mentum, and time), then the Liouville theorem is equivalenttoL V/Omega1L=0.V=˙qi∂ ∂qi+˙ki∂ ∂ki+∂ ∂tis a vector related to translation along time: for an arbitrary function g(q,k,t), Vg=d dtg.LVis the Lie derivative along vector V. Then, in the presence of the Berry connection, it can be shown thatL V/Omega1L∝d/Omega1×(···)[43], where /Omega1jkis the Berry curvature, d/Omega1=∂ki/Omega1jkdki∧dkj∧dkk. That is, the Liouville equation no longer holds because of singularities of the Berry curvatureat the Weyl nodes. Hence, the Liouville anomaly originatesfrom the infrared physics. But the Nieh-Yan term states that∂ μj5μ=/Lambda12 r 16π2/epsilon1μνρσ∂μea ν∂ρebσ, where /Lambda1ris the energy- momentum cutoff. So it is aware of the cutoff and thusconflicts with the picture from semiclassical physics. From theLiouville equation in the collisionless limit, one reaches theanomaly equation in the presence of both the electromagneticfields and torsions (see Appendix C), ∂ μjμ s=−s/epsilon1μνρσ 32π2/parenleftbig FμνFρσ+λsaλsbTa μνTb ρσ−2λsaFμνTa ρσ/parenrightbig , (9) where jμ s=(j0 s,js) is the current of Weyl fermions with chirality s. This Liouville anomaly is another main result in our work. It is clear that the last two terms explicitly dependon the positions of Weyl nodes in energy-momentum spaceand thus significantly differ from the counterparts of axialgauge fields [ 69]. Although these axial gauge fields from crystal deformations do not change our main results, theymay become significant out of the weak-displacement-fieldregime in real materials. According to Ref. [ 75], the coupling charge between the torsional electric fields (or temperaturegradient) and particles is k 0−μsrather than k0, where terms proportional to temperature are neglected for simplicity. Thisshift of the coupling charge leads to some extra terms inEq. ( 9), one of which is proportional to −sμs 4π2[(T0 0)iBi] and can provide an intuitive explanation for the recent negativemagnetothermal resistance in the Weyl semimetal NbP [ 76]. It is straightforward to derive the axial current ∂ μj5μ=−/epsilon1μνρσ 16π2/parenleftbig FμνFρσ+λaλbTa μνTb ρσ/parenrightbig (10) and the continuity equations for Weyl fermions, ∂μjμ=/epsilon1μνρσ 8π2/parenleftbig λaFμνTa ρσ/parenrightbig , (11) which can be understood from the chiral zeroth Landau level (see Appendix D). Assuming the screw dislocations are along thezaxis, the Weyl nodes with chirality slocate at sλμ, with s=±1. For simplicity, we set λμ=(0,0,0,−λz) and λz>0. The displacement vector is assumed to be along the zaxis, so only one component of the vielbeins survives, e3 μ=1 2(0,−˜Ty,˜Tx,0). Note that the surface density of the Burgers vector fields rather than the field itself is constant.Consequently, the zeroth Landau levels near p z=sλzare pz−sλz. Turning on an electric field along the zaxis, charges are pumped up from the Dirac sea, and extra particles are “produced.” Specifically, the level degeneracy is roughlyλz˜T 2πin the vicinity of pz=sλz, and the variation of momentum is/trianglepz=E/trianglet. Hence, the total variation of charge density is given as ∂tj0=1 2π2λz˜TE, whose covariant form is Eq. ( 11). The first term in Eq. ( 10) corresponds to the conventional chiral anomaly [ 37,38]. Unlike the Nieh-Yan term, the second term specifically depends on the locations of Weyl nodesbut is independent of the cutoff. A finite chiral chemicalpotential could be developed by a time-dependent dislocationeven without any external electromagnetic fields and wouldbe crucial to the anomalous transport phenomena for Weylsemimetals [ 77]. From the anomaly equation in Eq. ( 11), one finds the following solution [ 78]: j=−λ a 2π2/parenleftbig wa×E−wa 0B/parenrightbig . (12) In this work, we mainly focus on the static dislocations and would like to neglect the term proportional to wa 0.H e r e j= −λa 2π2wa×Ecan be dubbed the torsion-modified AHE. This resulting anomalous Hall current is still perpendicular to theelectric field but can be parallel to the momentum spacingbetween the two Weyl nodes with opposite chirality λ.T o show this point, let us set both screw dislocations and λ along the zaxis, i.e., u a=(0,0,0,u3(x,y)). In addition, the only nonvanishing waisw3=((w3)1,(w3)2,0).When the electric field is along the xaxis, the response current is along thezaxis, i.e., j3=λ3 2π2(w3)2E1. One can see that the ratio between the torsion-modified AHE and the intrinsic AHE isroughly equal to |u|/a, with abeing the crystal constant. For the Weyl semimetal Co 3Sn2S2, the giant intrinsic AHE is about 103/Omega1−1cm−1[79]. The torsion-modified AHE can thus reach tens of /Omega1−1cm−1(|u|/ais from 0.01 to 0.1.). Note that the mixing term in Eq. ( 11) also underlies the terms of μλi 2π2˜Tiin Eq. ( 8). In the thermal field theory, the chemical potential would couple to j0, i.e.,/integraltext μj0. So by keeping terms to leading order in μ, the effective action from this mixing term is/epsilon10ijk 4π2/integraltext μAi(λaTa jk). Thus, the response current is given asj=μ 2π2λi˜Ti[80]. Interestingly, Eq. ( 11) involves a mixing term between the electromagnetic fields and torsions, which seems to violatethe charge conservation. However, one can understand thisanomaly equation from the Callan-Harvey mechanism [ 81]. Namely, anomalies due to the chiral zero modes localizedin defects are canceled by the compensation of those statesfrom the bulk. The gauge invariance is thus locally preserved,leading to the local conservation of electric charges. It hasbeen shown that, in Weyl semimetals with a dislocation,the chiral zero modes with opposite chiralities are localizedin the dislocation and the boundary [ 82]. In addition, the chiral zero mode in the dislocation or antidislocation was alsonumerically found in Weyl semimetals [ 66]. To specifically demonstrate the cancellation of the anomaly due to the chiralzero mode in the dislocation by the bulk states through thetorsion-modified AHE, we consider a dislocation along the z direction located at x=y=0( s e eF i g . 1). Because of the chiral zero modes localized at the dislocation (black arrows),under external electric fields along the zdirection ( E 3), par- ticles are pumped up from the Dirac sea, which implies aone-dimensional anomaly ∂ μjμ∼1 2πE3in the dislocation. 155152-4TORSIONAL RESPONSE AND LIOUVILLE ANOMALY IN … PHYSICAL REVIEW B 99, 155152 (2019) FIG. 1. Schematic picture of the Callan-Harvey mechanism for the cancellation of the anomaly due to the chiral zero modes in thedislocation by the bulk states through the torsion-modified AHE. The dashed vertical line refers to the dislocation along the zdirection. The black arrows refer to the chiral anomalous current along thedislocation, and blue arrows refer to the torsion-modified anomalous Hall current under an external electric field along the zdirection. Note that ( wa)1=−ba 2πy (x2+y2)and ( wa)2=+ba 2πx (x2+y2)are solutions to the equation Ta μν=−baδ(x)δ(y). So the corre- sponding torsion-modified anomalous Hall currents are j1= −λaba 2π2x (x2+y2)E3andj2=−λaba 2π2y (x2+y2)E3(blue arrows), which means that this current flows toward (outward) the disloca- tion for λaba>0(λaba<0). Consequently, the chirality of localized chiral zero modes should depend on the sign ofλ aba, and the extra charges in the dislocation are compensated by the bulk states. Therefore, the electric charges are locallyconserved. Note that the torsion-modified AHE here plays arole similar to the AHE in the cancellation of the anomalyof the one-dimensional domain wall embedded in the (2 +1)- dimensional massive Dirac fermion system [ 83]. VI. CONCLUSIONS AND DISCUSSION In summary, we have presented a formalism to construct the semiclassical action and equations of motion for Weylfermions in the presence of both electromagnetic fields andtorsions from the quantum field theory. It has been shownthat the torsional electromagnetic fields make the Liouville anomaly equation essentially different from the counterpartof axial gauge fields. Our results could give rise to varioustorsional responses and reproduce the torsional CME and thechiral vortical effect. In addition, a torsion-modified AHEoriginating from the mixing term in the Liouville anomaly waspredicted, and its implementation in Weyl semimetals lackingtime-reversal symmetry was discussed as well. Recent progress on material realization of Weyl semimetals with no time-reversal symmetry in Co 3Sn2S2could facilitate the experimental investigation of the torsional responses asso-ciated with dislocations. ACKNOWLEDGMENTS The authors are grateful to Professor M. Stone for pointing out the Callan-Harvey mechanism and to L. Dong, B. Han,T. Qin, and C. Xu for insightful discussions. Z.-M.H., L.L.,and H.-H.Z. were supported by the National Natural ScienceFoundation of China under Grant No. 11875327. J.Z. was sup-ported by the 100 Talents Program of the Chinese Academyof Sciences. APPENDIX A: WIGNER TRANSFORMATION, BAND PROJECTION, AND SEMICLASSICAL ACTION In this Appendix, we provide the detailed derivations of the Wigner-transformed and band-projected Green’s function.For clarity, we focus on the case in which the torsional elec-tromagnetic fields and the electromagnetic fields are parallel. The action for Weyl semimetals in the presence of disloca- tions is given as [ 52,53] S=/integraldisplay d 4x/radicalbig −detg1 2/bracketleftbig¯/Psi1eμ aγa(i∂μ/Psi1)−(i∂μ¯/Psi1)eμ aγa/Psi1/bracketrightbig , (A1) where eμ ais the frame fields, ηabis flat space-time coordi- nates with indices a,b=0,1,2,3, and gμν=ea μebνηabis the curved space-time metric with indices μ, ν=0,1,2,3. The torsion is defined as Ta μν=∂μea ν−∂νea μ. If the lattice displacement is ua(x), then ea μ=δa μ+∂μua, and eμ a=δμ a− δρ a∂ρubδμ b+O(u2). Then, by keeping terms up to order O(u), the action for right-handed Weyl fermions becomes SR=/integraldisplay ddx/braceleftbigg /Psi1† R/bracketleftbigg σaδμ ai∂μ+wσμ(i∂μ)+1 2i(∂μw)σμ−wμ aσai∂μ−1 2/parenleftbig i∂μwμ a/parenrightbig σa/bracketrightbigg /Psi1R/bracerightbigg , (A2) where wμ a=δν aδμ b∂νubandw=δμ a∂μua. The corresponding Green’s function is iG−1=σaδμ ai∂μ+/bracketleftbig wα b(iσμ∂ν)+1 2∂νwα biσμ/bracketrightbig/parenleftbig ηb αην μ−ην αηb μ/parenrightbig . (A3) Performing the Wigner transformation [ 84] leads to i˜G−1=pμσaδμ a+/parenleftbig ηb αην μ−ην αηb μ/parenrightbig/bracketleftbig wα b∗pνσμ+i1 2∂νwα bσμ/bracketrightbig =pμσaδμ a+/parenleftbig ηb αην μ−ην αηb μ/parenrightbig/parenleftbig wα bpνσμ/parenrightbig , (A4) where ∗=exp [−i 2¯h(←−∂qμ−→∂pμ−←−∂pμ−→∂qμ)] is Moyal’s product from the Wigner transformation. Although we have used the natural unit, for heuristic purposes, ¯ hin Moyal’s product will not be set to 1 hereafter. Because we are most interested in the semiclassical limit, keeping ¯ hin Moyal’s product enables us to keep track of this. 155152-5HUANG, LI, ZHOU, AND ZHANG PHYSICAL REVIEW B 99, 155152 (2019) In addition, we project ˜Gonto the space spanned by its positive eigenstates to obtain semiclassical action, e.g., |u±/angbracketrightwith p·σ|u±/angbracketright=± | p||u±/angbracketright. For simplicity, we focus on the positive-energy band, and the counterpart for the negative-energy band is similar. The projected Green’s function becomes i˜G−1 ++=/angbracketleftu+|∗˜G−1∗|u+/angbracketright =/angbracketleftu+|/bracketleftbigg 1−i¯h 2/parenleftbig←−∂qμ−→∂pμ−←−∂pμ−→∂qμ/parenrightbig/bracketrightbigg/bracketleftbig pμσaδμ a+(ηb αην μ−ην αηb μ)/parenleftbig wα bpνσμ/parenrightbig/bracketrightbig ×/bracketleftbigg 1−i¯h 2/parenleftbig←−∂qμ−→∂pμ−←−∂pμ−→∂qμ/parenrightbig/bracketrightbigg |u+/angbracketright+O(¯h2) =(p0−|p|)(1+w)−/bracketleftbig w0 0p0+/parenleftbig w0 ip0ˆpi+wi 0pi/parenrightbig +wj ipjˆpi/bracketrightbig +/bracketleftbig p0(∂qμw)−∂qμw0 0p0−/parenleftbig ∂qμwi 0/parenrightbig pi/bracketrightbig/braceleftbiggi 2¯h/bracketleftbig/parenleftbig ∂pμ/angbracketleftu+|/parenrightbig |u+/angbracketright−/angbracketleft u+|/parenleftbig ∂pμ|u+/angbracketright/parenrightbig/bracketrightbig/bracerightbigg +/bracketleftbig −pi(∂qμw)−/parenleftbig ∂qμw0 i/parenrightbig p0−/parenleftbig ∂qμwj i/parenrightbig pj/bracketrightbig/braceleftbiggi 2¯h/bracketleftbig/parenleftbig ∂pμ/angbracketleftu+|/parenrightbig σi|u+/angbracketright−/angbracketleft u+|σi/parenleftbig ∂pμ|u+/angbracketright/parenrightbig/bracketrightbig/bracerightbigg +O(¯h2). (A5) where pμ=(p0,−p). We keep terms only up to order ¯ hin the second line. In the fourth line, we use /angbracketleftu+|σi|u+/angbracketright= ˆpi. The Berry connection is defined as /angbracketleftu+(p)|∂pμ|u+(p)/angbracketright=− iAμ p=−i(0,ap). (A6) Because /angbracketleftu+(p)|σi∂pμ|u+(p)/angbracketright =/angbracketleftu+(p)|σi/parenleftbigg|u+(p)/angbracketright−|u+(p−/trianglep)/angbracketright /trianglepμ/parenrightbigg ,(A7) by using the modified Gordon’s identity in Appendix E, one finds Im/angbracketleftu+|σi/parenleftbig ∂pμ|u+/angbracketright/parenrightbig =− iAμ pˆpi+i/epsilon1iμkˆpk 2|p|/vextendsingle/vextendsingle/vextendsingle/vextendsingle μ/negationslash=0+O(/trianglep) (A8) and Im/parenleftbig ∂pμ/angbracketleftu+|/parenrightbig σi|u+/angbracketright=iAμ pˆpi−i/epsilon1iμkˆpk 2|p|/vextendsingle/vextendsingle/vextendsingle/vextendsingle μ/negationslash=0+O(/trianglep). (A9) Thus, the projected Green’s function becomes i˜G−1 ++=iG−1 0(q)−¯hAμ p∂qμiG−1 0 −¯h/epsilon1imkˆpk 2|p|/bracketleftbig/parenleftbig ∂qmw0 ip0+∂qmwj ipj/parenrightbig/bracketrightbig ,(A10) where iG−1 0(q)=(p0−|p|)(1+w)−wμ apμˆpaand ˆ pa= (1,ˆp). Because ˆ pais from /angbracketleftu+|σa|u+/angbracketright, it is supposed to link to velocity. In reality, the response current is measured in laboratory coordinates with the index μ, so we shall change the indices of ˆ p, which leads to i˜G−1 ++=iG−1(q)−¯hAμ p∂qμiG−1(q)+ξviel, (A11) with iG−1(q)=(1+w)paδa νˆpν+pawa νˆpν(A12)and ξviel=¯h/epsilon10αβσ/parenleftbig1 2pbTb αβ/parenrightbig ˆpσ 2|p|. (A13) Because G−1(q)−¯hAμ p∂qμG−1(q)/similarequalG−1/parenleftbig q−¯hAμ p/parenrightbig +O(¯h2), (A14) the Berry connection acts as gauge fields in momentum space. Then, the dispersion relation can be determined by the polesof the Green’s function, p 0/similarequal|p|−wa μpaˆpμ−ξviel+ap·˙p /similarequal|p+(w0)|p|+(wj)pj|−[(w0)0|p|+(wi)0pi] −ξviel+ap·˙p, (A15) where we have neglected terms of order O(u2) and O(¯h2). After changing the variable from ptok=p+(w0)|p|+ (wj)pj, one gets the action L=k·˙q−(|k|−ξviel)+wa μka˙qμ−ak·˙k,(A16) with kμ=(|k|,−k), ˙qμ=(1,˙q), and ξviel=¯h/epsilon10αβσ/parenleftbig1 2kbTb αβ/parenrightbigˆkσ 2|k|. (A17) APPENDIX B: TORSIONAL CHIRAL MAGNETIC EFFECT In this Appendix, both the torsional chiral magnetic effect and chiral magnetic effect are derived from equations ofmotion with a careful treatment of the cutoff. For simplicity,we consider the zero-temperature limit. 155152-6TORSIONAL RESPONSE AND LIOUVILLE ANOMALY IN … PHYSICAL REVIEW B 99, 155152 (2019) The torsional chiral magnetic effect comes from ( ˜/Omega1·v)˜Tiin the equations of motion. For a=i, it becomes j(1) s=−/integraldisplayd3k (2π)3[f+ s(˜/Omega1+ s·v+)(˜Tiki)+f− s(˜/Omega1− s·v−)(˜Tiki)] =−/integraldisplayd3k (2π)3/braceleftbigg1 exp[β(|k|+λs0−μs)]+1+1 exp[β(−|k|+λs0−μs)]+1/bracerightbigg (˜/Omega1s·v)(˜Tiλsi) =s 4π2/braceleftbigg/integraldisplay/Lambda1 λs0d/epsilon1+1 exp[β(/epsilon1+−μs)]+1+/integraldisplay/Lambda1 −λs0d/epsilon1−/bracketleftbigg 1−1 exp[β(/epsilon1−+μs)]+1/bracketrightbigg/bracerightbigg (˜Tiλsi)=s(˜Tiλsi) 4π2(μs+/Lambda1),(B1) where plus and minus symbols denote positive- and negative-energy bands. In the second line, we have shifted variables kto k+λs./epsilon1±is defined as /epsilon1±=|k|±λs0./Lambda1refers to the cutoff for energy rather than momentum, i.e., λs0<|k|+λs0</Lambda1 and −/Lambda1<−|k|+λs0<λ s0. So the energy ranges from −/Lambda1to/Lambda1.f±are the distribution functions for positive- and negative-energy particles, respectively. Note that /Lambda1now plays the role of the energy reference [ 19]. Ifa=0, this current is j(2) s=−/integraldisplayd3k (2π)3[f+ s(˜/Omega1+ s·v+)(˜T0k+ 0)+f− s(˜/Omega1− s·v−)(˜T0k− 0)] =−/integraldisplayd3k (2π)3(˜/Omega1s·v)/braceleftbigg1 exp[β(|k−λs|+λs0−μs)]+1(˜T0/epsilon1+)+1 exp[β(−|k−λs|−λs0−μs)]+1(−˜T0/epsilon1−)/bracerightbigg =s 4π2/braceleftbigg/integraldisplay/Lambda1 λs0d/epsilon1+˜T0/epsilon1+ exp[β(/epsilon1+−μs)]+1+/integraldisplay/Lambda1 −λs0d/epsilon1−/bracketleftbigg 1−1 exp[β(/epsilon1−+μs)]+1/bracketrightbigg (−˜T0/epsilon1−)/bracerightbigg =s 8π2/parenleftbig μ2 s−/Lambda12/parenrightbig˜T0,(B2) where k± 0=±/epsilon1±. We have changed the variable in the third line: k→k+λs. Above all, if μ±=μ±μ5andλ±μ= ±λμ, then this current is j=/Lambda1λi 2π2˜Ti+μλi 2π2˜Ti+μμ 5 2π2˜T0, (B3) where the first term is the torsional chiral magnetic effect proposed in Ref. [ 66]. Let us turn to currents from the orbital moment. For a= 1,2,3, this current in the zero-temperature limit is j/prime(1) s=s/integraldisplayd3k (2π)3f+∂k(ki˜Ti·/hatwiderk−λs) 2|k−λs| =s(μs−λs0) 12π2(λsi˜Ti), (B4) where the contribution from the negative-energy band is zero and we have performed a partial integral and the variablechange k→k+λ sin the second line. Fora=0, we can find j/prime(2) s=s/integraldisplayd3k (2π)3f+∂k(k+ 0˜T0·/hatwiderk−λs) 2|k−λs| =sμs(μs−λs0) 12π2(˜T0). (B5) Similarly, if μ±=μ±μ5andλ±μ=±λμ, this current be- comes j/prime s=μ 6π2(λi˜Ti)+μμ 5 3π2˜T0−μλ0 6π2˜T0. (B6) APPENDIX C: LIOUVILLE ANOMALY The Liouville equation says that the phase-space current is conserved. However, the Berry curvature is singular at Weylnodes and thus breaks the Liouville equation, which is calledthe Liouville anomaly. In this Appendix, we derive the Liou- ville equation in the language of the differential form [ 43]. In addition, for clarity, we here use the four-vector notation,with the metric diag(1 ,−1,−1,−1) and neglect both ξ viel andξem. We define the one-form −ηH=kidqi+(|k−λs|+λs0)dt −wa μkadqμ−aμ skdkμ+Aμdqμ, (C1) and the two-form ωH=dηH, i.e., −ωH=δi jdki∧dqj−/hatwidek−λi sdki∧dt −/parenleftbig kaTa+wi μdki∧dqμ−w0 μ/hatwiderk−λsidki∧dqμ/parenrightbig −/Omega1s+F, (C2) where s=±1 for the chirality of Weyl fermions and k0= |k−λs|+λs0./Omega1s=1 2/Omega1ij sdki∧dkj,F, and Taare the Berry curvature, electromagnetic tensor, and torsion, respectively.Because ω H=dηH, one would naively expect that dωH=0. However, this is not true. To appreciate this point, let uscalculate dω H, −dωH=−dTaka−∂νwi μdqν∧dki∧dqμ +∂νw0 μ/hatwiderk−λsidqν∧dki∧dqμ −(−/hatwiderk−λsnT0dkn+Tidki)−d/Omega1s+dF =− (dTaka+d/Omega1s−dF), (C3) where dF=dTa=0.d/Omega1is not necessarily zero but relates to monopole charges. We then define a vector V=˙qi∂ ∂qi+ ˙ki∂ ∂ki+∂ ∂t, which is about translation along time: for an arbi- trary function g(q,k,t),Vg=d dtg. 155152-7HUANG, LI, ZHOU, AND ZHANG PHYSICAL REVIEW B 99, 155152 (2019) Then, the Liouville equation is LV/Omega1L=1 2!d/Omega1s∧/bracketleftbig/parenleftbig dki∧dqi−ˆkidki∧dt−wi μdki∧dqμ+w0 μˆkidki∧dqμ/parenrightbig −kaTa−/Omega1+F/bracketrightbig2 =1 2!d/Omega1(kaTa−F)∧(kbTb−F), (C4) where /Omega1L=1 3!ω3 H∧dtis the phase-space volume form and LVis the Lie derivative of vector V.LV/Omega1Lis now a top form, so for convenience, we employ the Hodge star operator to transform it to a scalar function, i.e., ⋆LV/Omega1L=⋆/bracketleftBigg/parenleftbigg1 2!/parenrightbigg4/parenleftbigg∂ ∂kl/Omega1ij sdkl∧dki∧dkj/parenrightbigg/parenleftbig kaTa μν−Fμν/parenrightbig/parenleftbig kaTa ρσ−Fρσ/parenrightbig dqμ∧dqν∧dqρ∧dqσ/bracketrightBigg =/bracketleftBigg/parenleftbigg1 2!/parenrightbigg4 /epsilon1lij/parenleftbigg∂ ∂kl/Omega1ij s/parenrightbigg /epsilon1μνρσ/parenleftbig kaTa μν−Fμν/parenrightbig/parenleftbig kaTa ρσ−Fρσ/parenrightbig/bracketrightBigg =1 8(∂k˜/Omega1s)/epsilon1μνρσ/parenleftbig kaTa μν−Fμν/parenrightbig/parenleftbig kbTb ρσ−Fρσ/parenrightbig , (C5) with (∂k˜/Omega1s)=−2πsδ(k−λs). Thus, the Liouville equation is ∂D ∂t+∂D˙q ∂q+∂D˙k ∂k=−πs/epsilon1μνρσ 4δ3(k−λs)/parenleftbig kaTa μν−Fμν/parenrightbig/parenleftbig kbTb ρσ−Fρσ/parenrightbig . (C6) Because of k0=|k−λs|+λs0,t h i sδfunction implies k0=λs0. In addition, by inserting the distribution function back into it, the equation above becomes ∂μjμ s=−s/epsilon1μνρσ 32π2/parenleftbig λsaTa μν−Fμν/parenrightbig/parenleftbig λsbTb ρσ−Fρσ/parenrightbig . (C7) APPENDIX D: LANDAU LEVELS INDUCED BY SCREW DISLOCATIONS In this Appendix, we derive the “Landau levels” for Weyl fermions under screw dislocations. From the action in Eq. ( 1)i n the main text, one obtains the corresponding equation of motion, 1 2/bracketleftbig/parenleftbig deteb β/parenrightbig γaeμ a(i∂μ/Psi1)+γai∂μ/parenleftbig deteb βeμa/Psi1/parenrightbig/bracketrightbig −/parenleftbig deteb β/parenrightbig λaγaγ5/Psi1=0. (D1) For simplicity, we consider screw dislocations with the displacement vector along the zaxis, ea μ=δa μ+wa μ, (D2) withw3 μ=1 2(0,−˜Ty,˜Tx,0) and wa μ=0f o r a/negationslash=3. That is, the torsional magnetic field ˜Tis along the zaxis. The Hamiltonians for the right-handed fermions ( HR) and the left-handed fermions ( HL) are thus given as HR=/parenleftBigg (pz−λz)/parenleftbig px+1 2˜Typ z/parenrightbig −i/parenleftbig py−1 2˜Txp z/parenrightbig /parenleftbig px+1 2˜Typ z/parenrightbig +i/parenleftbig py−1 2˜Txp z/parenrightbig −(pz−λz)/parenrightBigg (D3) and HL=−/parenleftBigg (pz+λz)/parenleftbig px+1 2˜Typ z/parenrightbig −i/parenleftbig py−1 2˜Txp z/parenrightbig /parenleftbig px+1 2˜Typ z/parenrightbig +i/parenleftbig py−1 2˜Txp z/parenrightbig −(pz+λz)/parenrightBigg . (D4) This implies that Weyl fermions under a momentum-dependent magnetic field ˜Tpzentirely differ from the cases of magnetic fields and axial magnetic fields. Because the Hamiltonian commutes with ˆ pz, the quantum number pzcan be used to label eigenstates, and the Hamiltonian can be recast as HR=⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩/parenleftBigg (p z−λz)/radicalbig |2˜Tpz|ˆA† /radicalbig |2˜Tpz|ˆA−(pz−λz)/parenrightBigg /parenleftBigg (pz−λz)/radicalbig |2˜Tpz|ˆA /radicalbig |2˜Tpz|ˆA†−(pz−λz)/parenrightBiggfor ˜Tpz>0, for ˜Tpz<0,(D5) 155152-8TORSIONAL RESPONSE AND LIOUVILLE ANOMALY IN … PHYSICAL REVIEW B 99, 155152 (2019) where ˆbx=−i∂x−i1 2˜Txp z,ˆby=−i∂y−i1 2˜Typ z, ˆax(y)=⎧ ⎨ ⎩1√|˜Tpz|ˆbx(y) 1√|˜Tpz|ˆb† x(y)for ˜Tpz>0, for ˜Tpz<0,(D6) and ˆA=ˆax+iˆay√ 2. (D7) It is straightforward to verify that [ ˆA,ˆA†]=1 and [ ˆA,ˆA]=0. The square of the Hamiltonian is H2 R/vextendsingle/vextendsingle˜Tpz>0=/parenleftBigg (pz−λz)2+|2˜Tpz|ˆA†ˆA 0 0 |2˜Tpz|(ˆA†ˆA+1)+(pz−λz)2/parenrightBigg (D8) and H2 R/vextendsingle/vextendsingle˜Tpz<0=/parenleftBigg (pz−λz)2+|2˜Tpz|(ˆA†ˆA+1) 0 0 |2˜Tpz|ˆA†ˆA+(pz−λz)2/parenrightBigg . (D9) Thus, the dispersion relation for the right-handed fermions can be calculated as ER=⎧ ⎪⎪⎨ ⎪⎪⎩/braceleftBigg p z−λz −(pz−λz)for ˜Tpz>0, for ˜Tpz<0, ±/radicalbig (pz−λz)2+2|n˜Tpz|n=0, n2/greaterorequalslant1,(D10) where nis an integer. Similarly, for the left-handed fermions, one has EL=⎧ ⎪⎪⎨ ⎪⎪⎩/braceleftBigg −(p z+λz) (pz+λz)for ˜Tpz>0, for ˜Tpz<0, ±/radicalbig (pz+λz)2+2|n˜Tpz|n=0, n2/greaterorequalslant1.(D11) One clearly finds a chiral zeroth Landau level for the left- or right-handed Weyl fermions, as shown in Fig. 2. APPENDIX E: MODIFIED GORDON’s IDENTITY The well-known Gordon’s decomposition is valid for mas- sive Dirac fermions. Therefore, in this Appendix, we derived FIG. 2. Schematic picture of the Landau levels due to screw dislocations. The red lines refer to the zeroth chiral Landau level for the right-handed Weyl fermions near pz=λzand the left-handed Weyl fermions near pz=−λz.a modified Gordon’s identity, which also holds for massless Weyl fermions. Assume |u+(p)/angbracketrightis the eigenfunction of Weyl’s equation, p·σ|u+(p)/angbracketright=+|p||u+(p)/angbracketright. (E1) From the identity for Pauli’s matrices [σi,σj]=2σiσj−2δij=−2σjσi+2δij, (E2) one can thus obtain 1 2/angbracketleftu+(p/prime)|[σi,σj](p/prime−p)j|u+(p)/angbracketright =/angbracketleftu+(p/prime)|[(−σjσi+δij)p/prime j−(σiσj−δij)pj]|u+(p)/angbracketright =/angbracketleftu+(p/prime)|[−σjσip/prime j−σiσjpj+δij(p/prime j+pj)]|u+(p)/angbracketright =/angbracketleftu+(p/prime)|(|p/prime|+|p|)σi|u+(p)/angbracketright −(p/prime+p)i/angbracketleftu+(p/prime)|u+(p)/angbracketright, with pμ=(p0,−p). Thus, the modified Gordon’s identity for Weyl fermions is of the form /angbracketleftu+(p/prime)|σi|u+(p)/angbracketright =1 (|p/prime|+|p|)[−i/epsilon1ijk/angbracketleftu+(p/prime)|(p/prime−p)j ×σk|u+(p)/angbracketright+(p/prime+p)i/angbracketleftu+(p/prime)|u+(p)/angbracketright].(E3) That is, for the right-handed Weyl fermions, we have found /angbracketleftu+(p)|σi|u+(p−/trianglep)/angbracketright =−i/epsilon1ijk[/angbracketleftu+(p)|σk|u+(p−/trianglep)/angbracketright](/trianglep)j (|p|+|p−/trianglep|) +/angbracketleftu+(p)|(2p−/trianglep)i (|p|+|p−/trianglep|)|u+(p−/trianglep)/angbracketright, (E4) 155152-9HUANG, LI, ZHOU, AND ZHANG PHYSICAL REVIEW B 99, 155152 (2019) which leads to the following equation by iterating /angbracketleftu+(p)|σi|u+(p−/trianglep)/angbracketright=/angbracketleft u+(p)|u+(p−/trianglep)/angbracketright/bracketleftbigg(2p−/trianglep)i (|p|+|p−/trianglep|)−i/epsilon1ijk (2p−/trianglep)k (|p|+|p−/trianglep|)(/trianglep)j (|p|+|p−/trianglep|)/bracketrightbigg +O(/trianglep2). 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PhysRevB.94.245145.pdf

PHYSICAL REVIEW B 94, 245145 (2016) Incommensurate spiral magnetic order on anisotropic triangular lattice: Dynamical mean-field study in a spin-rotating frame Shimpei Goto,1,*Susumu Kurihara,1and Daisuke Yamamoto2 1Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan 2Department of Physics and Mathematics, Aoyama-Gakuin University, Sagamihara, Kanagawa 252-5258, Japan (Received 16 August 2016; revised manuscript received 25 November 2016; published 28 December 2016) We study the ground-state magnetism of the half-filled Hubbard model on the anisotropic triangular lattice, where two out of three bonds have hopping tand the third one has t/primein a unit triangle. Working in a spin-rotating frame and using the density matrix renormalization group method as an impurity solver, we provide a properdescription of incommensurate magnetizations at zero temperature in the framework of the dynamical mean-fieldtheory (DMFT). It is shown that the incommensurate spiral magnetic order for t /prime/t/greaterorsimilar0.7 survives the dynamical fluctuations of itinerant electrons in the Hubbard interaction range from the strong-coupling (localized-spin) limitdown to the insulator-to-metal transition. We also find that when the anisotropy parameter t /prime/tincreases from the N´eel-to-spiral transition, the magnitude of the magnetic moment exhibits a maximum at the isotropic triangular lattice point t/prime/t=1 and then rapidly decreases in the range of larger t/prime/t. This work gives a solid foundation for further extension of the study including nonlocal correlation effects neglected at the standard DMFT level. DOI: 10.1103/PhysRevB.94.245145 I. INTRODUCTION The interplay of geometrical frustration and quantum fluctuations of itinerant electrons has drawn much at-tention because of its essential role for the realizationof spin liquid (SL) states in organic compounds suchasκ-(BEDT-TTF) 2Cu2(CN) 3,E t M e 3Sb[Pd(dmit) 2]2, and κ-H3(Cat-EDT-TTF) 2[1–3]. In these compounds, dimerized molecules form layered anisotropic triangular lattices spacedby insulating nonmagnetic layers. Many theoretical effortsaimed at understanding the quantum magnetism of anisotropictriangular-lattice systems have been made with the Heisenbergmodel of localized spins in both semianalytical [ 4–13] and numerical [ 14–23] manners. These studies have shown that the spatial anisotropy in spin exchange interactions gives riseto an incommensurate spiral magnetic order. It has been alsofound that strong quantum fluctuations are induced in theanisotropy parameter range where the competition betweenthe commensurate N ´eel and incommensurate spiral orders takes place or where the low dimensionality is enhanced bylarge anisotropy. These strong fluctuation effects could lead toquantum nonmagnetic states including SLs, although differentapproaches have given different conclusions [ 5–21] about the anisotropy parameter range where the SL states appear. The Hubbard model describes additional fluctuation effects that come from the itinerancy of electrons, which may alsoplay an important role on the magnetism of the organiccompounds and other strongly correlated electron systems.However, the theoretical studies on the anisotropic triangularHubbard model [ 24–38] remain far from consensus due to the difficulty in dealing with itinerant electron systems withfrustration. In order to reach full understanding of the itinerantfrustrated magnetism on the anisotropic triangular lattice, it iscrucial to properly treat the strong fluctuation effects betweenitinerant electrons and the incommensurability of magneticorders [ 7–24,39]. Furthermore, the consistency with the known *goto@kh.phys.waseda.ac.jpresults for the Heisenberg model of localized spins has to be achieved in the large Hubbard-interaction limit. In this paper we study the magnetic properties, including the incommensurability of magnetic orders, of the half-filledHubbard model on the anisotropic triangular lattice by meansof the dynamical mean-field theory (DMFT) [ 40–42]. The DMFT treats local correlation effects between electrons in anonperturbative fashion by mapping the original many-bodyproblem onto an effective impurity model, which becomesexact in the limit of lattices with an infinite coordination.Therefore, the spirit of the approximation is similar to those ofthe Weiss molecular field theory for localized spins [ 43] and the Gutzwiller approximation for lattice bosons [ 44,45]. These “single-site” approximations have offered a good startingpoint for understanding the role of fluctuations in quantummany-body systems. Based on the single-site approximations,the neglected nonlocal correlations can be taken into accountby, e.g., their cluster extensions [ 46–52] and perturbative expansions with collective-mode excitations (such as thespin-wave theory [ 53]). Although several cluster extensions of the DMFT and the related approaches [ 32–38] have been already applied to the Hubbard model on the anisotropic triangular lattice, theincommensurate magnetic order has not been properly treatedin those studies. Here we describe fully incommensurateorders by applying a local gauge transformation on the spinspace of the electron operators. Dealing with an effectiveimpurity model in the spin-rotating frame by means of a solverbased on the density matrix renormalization group (DMRG)[54–56], we study the effects of dynamical fluctuations on the incommensurate spin spiral states in the framework of theDMFT. The zero-temperature phase diagram determined byour DMFT shows that the incommensurate magnetic orderin insulating states survives the dynamical fluctuations ofelectrons in the interaction range from the strong-coupling(localized-spin) limit down to the insulator-to-metal transition.This indicates that it is crucial for the study of anisotropictriangular lattice to properly treat the incommensurability ofthe magnetic order. The role of the local, dynamic fluctuations 2469-9950/2016/94(24)/245145(10) 245145-1 ©2016 American Physical SocietySHIMPEI GOTO, SUSUMU KURIHARA, AND DAISUKE Y AMAMOTO PHYSICAL REVIEW B 94, 245145 (2016) FIG. 1. Square-lattice geometry that is topologically equivalent to the triangular lattice with spatially anisotropic hoppings −t(solid bonds) and −t/prime(dashed bonds). in realizing quantum SL states in strongly correlated electron systems [ 1,2] will be also discussed. This paper is organized as follows. In Sec. IIwe introduce the Hamiltonian of the model considered here and provide theprocedure of the DMFT calculations in the spin-rotating frame.In Sec. IIIwe present the phase diagram of the model and show the behaviors of the magnetic moment and the ordering wavevector as a function of the system parameters. The role ofthe dynamical fluctuations in realizing the SL state is alsodiscussed. Conclusions are given in Sec. IV. II. DYNAMICAL MEAN-FIELD THEORY FOR INCOMMENSURATE SPIRAL ORDERS A. Model Hamiltonian and the strong-coupling limit We study the half-filled Hubbard model on a spatially anisotropic triangular lattice: H=/summationdisplay ijσtijc† iσcjσ+U/summationdisplay ini↑ni↓−μ/summationdisplay iσniσ, (1) where ciσis an annihilation operator of an electron at site i with spin σ,Uis the on-site Hubbard interaction, μis the chemical potential, and niσ=c† iσciσ. The spin index σtakes two values, ↑=1/2o r↓= − 1/2. The spatially anisotropic triangular lattice is equivalent to the square lattice with oneadditional set of diagonal bonds (see Fig. 1). We assume the hopping integral t ijas tij=⎧ ⎨ ⎩−t<0( rj−ri=±e1,±e2), −t/prime/lessorequalslant0[rj−ri=± (e1+e2)], 0 (otherwise) ,(2) withe1=(1,0),e2=(0,1), and ribeing a position vector of site i. The geometry of the lattice can be viewed as an interpolation between the square lattice and the one-dimensional chain by varying t /prime/tfrom 0 to ∞through the isotropic triangular lattice at t/prime/t=1. In the strong coupling limit of U/greatermucht,t/primeat half-filling, the charge degrees of freedom are frozen out, and the Hubbardmodel is mapped onto the Heisenberg model with exchangecouplings J=4t 2/UandJ/prime=4(t/prime)2/Ufor solid and dashed bonds in Fig. 1, respectively. The classical-spin analysis on the anisotropic triangular Heisenberg model has shown thatthe local spins form a magnetic order with the ordering vectorQ=(q,q) where [ 7,8] q=/braceleftbigg arccos( −J/2J /prime)(J/prime/J > 1/2), π (J/prime/J/lessorequalslant1/2).(3) Increasing the value of J/prime/Jfrom 0 leads to a commensurate- incommensurate transition occurs at J/prime/J=1/2(t/prime/t= 1/√ 2≈0.707) from the N ´eel to incommensurate spiral state. When J/prime/Jis increased further, the ordering vector takes (2π/3,2π/3), which corresponds to a commensurate 120◦ order, at J/prime/J=1 and approaches ( π/2,π/2) in the one- dimensional limit of J/prime/J→∞ . B. Dynamical mean-field theory Let us now turn to the discussions away from the strong coupling limit to consider the effects of charge degrees offreedom on the magnetic orders. In order to deal with the N ´eel and spin spiral orders within the framework of DMFT, werotate the local phase of the electron operators as ˜c iσ=ciσeiσ(Q·ri+φ), (4) where φis an arbitrary phase shift. Under this local gauge transformation, the Hamiltonian becomes ˜HQ=/summationdisplay ijσtijeiσ[Q·(ri−rj)]˜c† iσ˜cjσ+U/summationdisplay ini↑ni↓−μ/summationdisplay iσniσ, (5) where niσ=c† iσciσ=˜c† iσ˜ciσ. Each component of spin opera- tor is transformed as Sx i=1 2(c† i↑ci↓+c† i↓ci↑) =1 2(˜c† i↑˜ci↓ei(Q·ri+φ)+˜c† i↓˜ci↑e−i(Q·ri+φ)) ≡˜Sx icos(Q·ri+φ)−˜Sy isin(Q·ri+φ), (6) Sy i=−i 2(c† i↑ci↓−c† i↓ci↑) =−i 2(˜c† i↑˜ci↓ei(Q·ri+φ)−˜c† i↓˜ci↑e−i(Q·ri+φ)) ≡˜Sx isin(Q·ri+φ)+˜Sy icos(Q·ri+φ), (7) Sz i=1 2(c† i↑ci↑−c† i↓ci↓)=1 2(˜c† i↑˜ci↑−˜c† i↓˜ci↓)≡˜Sz i.(8) Therefore a magnetically ordered spiral state in the xyplain can be described by a uniform magnetization /angbracketleft˜Sx i/angbracketright=M and/angbracketleft˜Sy i/angbracketright=/angbracketleft ˜Sz i/angbracketright=0 in the spin-rotating frame, which is convenient for the DMFT formulation of the system withincommensurate spiral orders [ 57,58]. It is expected for finite Uthat due to the charge fluctuation effects, the magnetization Mis reduced and the ordering vector Q=(q x,qy) is shifted from the classical-spin result in Eq. ( 3). The local Green’s function for ˜ciσis given by G(ω)=/parenleftBigg /angbracketleft/angbracketleft˜ci↑;˜c† i↑/angbracketright/angbracketrightω/angbracketleft/angbracketleft˜ci↑;˜c† i↓/angbracketright/angbracketrightω /angbracketleft/angbracketleft˜ci↓;˜c† i↑/angbracketright/angbracketrightω/angbracketleft/angbracketleft˜ci↓;˜c† i↓/angbracketright/angbracketrightω/parenrightBigg =1 N/summationdisplay k1 (ω+μ)1−εQ(k)−/Sigma1(k,ω), (9) 245145-2INCOMMENSURATE SPIRAL MAGNETIC ORDER ON . . . PHYSICAL REVIEW B 94, 245145 (2016) where Nis the number of lattice sites and εQ(k) is a diagonal matrix whose component εQσσ(k)= −2t/summationtext ν=x,ycos (kν+σqν)−2t/primecos [/summationtext ν=x,y(kν+σqν)] is the single-particle dispersion of ˜ciσ. The effects of spatial and dynamical fluctuations induced by the interactions U are taken into account through the momentum k=(kx,ky) and frequency ωdependencies of the self-energy /Sigma1(k,ω). In the simple DMFT, the self-energy is approximated as/Sigma1(k,ω)≈/Sigma1(ω) to study the local fluctuation effects. Under the approximation, the problem is mapped onto the singleimpurity Anderson model (SIAM) [ 42], whose Hamiltonian is given by H SIAM=Un↑n↓−μ/summationdisplay σnσ+Nb/summationdisplay lσσ/prime(Vlσσ/primea† lσ˜cσ/prime+H.c.) +Nb/summationdisplay lσ/epsilon1la† lσalσ, (10) where ˜cσis an annihilation operator of an electron at impurity site with spin σ,nσ=˜c† σ˜cσ,alσis an annihilation operator of an electron at lth bath orbital with spin σ, andNbis the number of bath orbitals. The bath parameters Vlσσ/primeandεlshould be optimized so that the impurity Green’s function Gimp(ω)=1 (ω+μ)1−/Gamma1(ω)−/Sigma1(ω)(11) is equal to the local Green’s function G(ω) of the original lattice problem [Eq. ( 9)] with the replacement of /Sigma1(k,ω)b y /Sigma1(ω). Here the hybridization function /Gamma1(ω)i sg i v e nb y /Gamma1(ω)=/summationdisplay lVlV† l ω−/epsilon1l, (12) where Vlis a two-by-two matrix whose component is Vlσσ/prime. The spin-flip couplings Vl↑↓andVl↓↑are required to describe the in-plane magnetization M=/angbracketleft ˜Sx/angbracketright. In order to compute the impurity Green’s function Gimp(ω), we employ the imaginary-time matrix product state solver[59] based on the DMRG technique, which can treat dozens of bath orbitals and access zero temperature. In the DMRGcalculations, which provide the ground state of the system,the SIAM Hamiltonian is arranged in the star geometry[60], and the truncation error is set to lower than 10 −8.T h e imaginary-time Green’s function Gimp(τ) can be computed from a one-electron (one-hole) excited state [ 59], which is obtained by applying a creation (annihilation) operator to theground state. For an efficient Fourier transformation of theGreen’s function with respect to τ, we perform the fitting of each component of G imp(τ)i nt h ef o r m/summationtext iαie−βiτwith the matrix pencil method [ 61]. This procedure gives the impurity Green’s function Gimp(ω) on the imaginary axis for a given set of the bath parameters Vlσσ/primeandεl. The details of the optimization of the bath parameters under the condition Gimp(ω)=G(ω) are given in Appendix A. In addition to the self-consistent optimization of the bath parameters, one has to determine spin spiral ordering vector Q so that the energy of the system can be minimized with respecttoQ. The energy of the system E(Q) as a function of Qisgiven by the Galitskii-Migdal formula [ 62] E(Q)=1 N/summationdisplay k/integraldisplay Cdω 2πiTr/bracketleftbigg/parenleftbigg εQ(k)+1 2/Sigma1(ω)/parenrightbigg Glatt(k,ω)/bracketrightbigg . (13) HereCdenotes a contour which surrounds the negative real axis counterclockwise and Glatt(k,ω) is the lattice Green’s function of the DMFT which is given by Glatt(k,ω)=1 (ω+μ)1−εQ(k)−/Sigma1(ω). (14) This contour integration can be transformed into an integration over the positive imaginary axis [ 63]. Note that the minimiza- tion of the energy function E(Q) with respect to Qcan be also obtained by the stability condition ∂ ∂qν/angbracketleft˜HQ/angbracketright=/angbracketleftjνQ/angbracketright=0(ν=x,y), (15) where jνQ≡∂˜HQ ∂qν=/summationtext ijσiσqνtij(νi−νj)˜c† iσ˜cjσis the spin current operator in the νdirection. Here νiis theνcomponent of the vector ri=(xi,yi). The local quantities including the filling/summationtext σ/angbracketleftnσ/angbracketrightand the spin moments /angbracketleft˜S/angbracketrightcan be directly calculated from the local Green’s function G(ω) with the optimized values of the bath parameters and the ordering vector Q. In order to consider the half-filled case, the chemical potential μhas to be numerically tuned so that/summationtext σ/angbracketleftnσ/angbracketright=1 since the system for t,t/prime/negationslash=0 does not possess the particle-hole symmetry. Using the above-mentioned DMFT procedure in the spin- rotating frame, one can describe the insulating state with anincommensurate spiral magnetic order and the commensurateN´eel and 120 ◦antiferromagnetic states, as well as metallic states. In the following, we will mainly discuss the chargefluctuation effects on the magnetic properties of the insulatingstates in the region of large but finite values of U.T h e possibility of the d-wave superconducting state [ 32]f o r intermediate U/t is out of the scope of this paper since spatial correlations are neglected. III. MAGNETIC ORDERS AND METAL-INSULATOR TRANSITIONS A. Magnetic phase diagram In Fig. 2we show the ground-state phase diagram obtained by the DMFT calculations in the spin-rotatingframe. The phase diagram consists of three phases: theN´eel-antiferromagnetic and spin-spiral insulators as well as a nonmagnetic-metal phase. The magnetic orders of the formertwo are characterized by the ordering vector Q=(π,π) and Q=(q,q) withπ/2<q<π , respectively. In Fig. 3we show the chemical potential dependence of the filling/summationtext σ/angbracketleftnσ/angbracketrightfor a typical spin-spiral insulator and metallic states. It can be seenthat the slope is zero in a finite range of μin the spin spiral state, which indicates the opening of a charge gap. Figure 4shows how the anisotropy t /prime/taffects the magnetic orders in the insulator phases at strong interactions. Whent /prime/t=0, the system is reduced to the simple square-lattice Hubbard model, which is well known to exhibit a robust 245145-3SHIMPEI GOTO, SUSUMU KURIHARA, AND DAISUKE Y AMAMOTO PHYSICAL REVIEW B 94, 245145 (2016) FIG. 2. Magnetic phase diagram of the half-filled Hubbard model on the anisotropic triangular lattice. The line with blue circles (greensquares) represents a first-order (second-order) transition boundary. The spin spiral phase has an incommensurate magnetic order except att /prime/t=1 (dashed line) N´eel order due to the perfect nesting of the itinerant electron Fermi surface. As shown in the lower panel of Fig. 4,e v e ni f the lattice geometry is changed by finite t/prime/t,t h eN ´eel order with commensurate wave vector ( π,π) persists up to a certain critical value ( t/prime/t)c.F o rt/prime/t > (t/prime/t)c, the minimum of the energy function E(Q) is shifted from ( π,π) to an incommen- surate momentum ( q,q) as shown in Fig. 5, which indicates a transition to a state with an incommensurate magnetic order. Ast /prime/tincreases, the value of qcontinuously moves away from πand reaches 2 π/3 at the isotropic triangular-lattice point t/prime/t=1. The wave vector Q=(2π/3,2π/3) corresponds to a commensurate (three-sublattice) 120◦order expected in triangular-lattice antiferromagnetic systems [ 64–67]. For a dominant diagonal hopping t/prime>t,t h ev a l u eo f qfurther decreases and approaches π/2 in the one-dimensional limit of t/prime>t→∞ . This behavior of magnetic order as a function of the anisotropy t/prime/tfor large U/t is consistent with the classical- spin analysis of the antiferromagnetic Heisenberg model on FIG. 3. Chemical potential μdependencies of filling/summationtext σ/angbracketleftnσ/angbracketrightat t/prime/t=0.9. The line with blue circles (green squares) corresponds to spin spiral (metal) phase at U/t=9.0(U/t=8.0). Here μhalfis the value of the chemical potential when/summationtext σ/angbracketleftnσ/angbracketright=1.2/3 FIG. 4. Upper panel: Diagonal hopping t/primedependence of energy per site at U/t=10. Middle panel: The hopping dependence of magnetic moment M. Lower panel: The hopping dependence of ordering vector parameter q. the anisotropic triangular lattice [ 7,8,11]. In fact, the ordering vector ( q,q) and the magnetic moment Mapproach the classical-spin results Eq. ( 3) with J/prime/J=(t/prime/t)2andM= S=1/2 in the limit of the infinite Hubbard interaction U/t→ ∞. This agreement is not surprising since the DMFT neglects the spatial fluctuations (the kdependence) in the self-energy /Sigma1(k,ω) as in classical-spin systems. Therefore, the reduction of the magnetic moment Mshown in the middle panel of Fig. 4is purely the result of the local, dynamical fluctuations that stem from the itinerant charge degrees of freedom. Themagnetic moment Mexhibits a dip at the transition point between the commensurate N ´eel and incommensurate spiral phases, although the reduction from M=Sis at most only ∼10%. In the spiral phase, the curve of Mshows a peak (at t /prime/t∼0.8 in the case of Fig. 4), and then decreases as t/prime/t increases. FIG. 5. The ordering vector dependence of the energy function E(q,q)a tU/t=10.0a n d t/prime/t=0.75. The bath parameters are optimized for each value of q. 245145-4INCOMMENSURATE SPIRAL MAGNETIC ORDER ON . . . PHYSICAL REVIEW B 94, 245145 (2016) FIG. 6. Upper panel: Hubbard interaction Udependence of energy per site at t/prime/t=0.9. The vertical dashed line represents the first-order transition point. Middle panel: The interaction dependence of magnetic moment M. Lower panel: The interaction dependence of ordering vector parameter q. On decreasing the interaction U/t, the system with t/prime/t > 0 undergoes a first-order transition from a magnetic insulator toa metallic state at a certain value of U/t. This is because the perfect nesting condition of the half-filled square latticeis violated for t /prime/t/negationslash=0, and finite U/t is required to stabilize magnetic orders. As shown in Fig. 6, the magnetic moment Msuddenly vanishes at the metal-insulator transition point. In our DMFT analysis, no magnetic metal state is foundbetween the magnetic insulator and nonmagnetic metal phasesin the parameter range of the phase diagram in Fig. 2.T h i si s consistent with the previous studies in Refs. [ 25–35,37,38,68], although several works including the Hartree-Fock mean-field analysis [ 24] and the variational cluster approach [ 36] have predicted the existence of magnetic metal phases forintermediate interactions. B. Dynamical property: Density of states By fitting the self-energy on the imaginary axis, we can derive the lattice Green’s functions on the real axis (seeAppendix B), which allow for accessing dynamical properties of the system, e.g., density of states (DOS). Figure 7(a) shows theU/t dependence of the ground-state DOS for t /prime/t=1( t h e isotopic triangular lattice). Reflecting the band structure of thetriangular lattice, the single-particle ( U/t=0) DOS does not possess the particle-hole symmetry even at half-filling and onecan see a van Hove singularity in the particle excitations. Whenthe Hubbard interaction U/t is finite but still small, the DOS has a finite value at the Fermi level, which indicates that thesystems remains in the nonmagnetic metallic phase. Once theFIG. 7. (a) Density of states at the triangular point t/prime/t=1.0f o r different values of the Hubbard interaction U/t=0 (free electrons), U/t=4.0 (nonmagnetic metal), U/t=9.0 (120◦antiferromagnetic insulator), and U/t=20.0 (120◦antiferromagnetic insulator). (b) Density of states at U/t=9.0 for different values of the anisotropic parameter t/prime/t=0( N ´eel insulator), t/prime/t=0.5( N ´eel insulator), t/prime/t=0.8 (incommensurate spiral insulator), and t/prime/t=1.2 (incom- mensurate spiral insulator). interaction gets strong enough to induce the phase transition to the anitiferromagnetic insulating phase, the spectral gapopens at the Fermi level and the weights around ω=±U/2 get larger. The enhancement of the weights around ω=±U/2 becomes more pronounced for U/t/greatermuch1 [see the bottom panel of Fig. 7(a)], which indicates the formation of upper and lower Hubbard bands. In Fig. 7(b) we show the change in the DOS shape as the value of t /prime/tincreases when the system is in magnetic insulating phase at U/t=9. For the isotropic square lattice (t/prime/t=0), the DOS possesses the particle-hole symmetry and has large weights at the edge of the spectral gap reflecting thevan Hove singularity in the bare DOS at the Fermi level. Oncethe diagonal hopping element t /primeis introduced, the system loses the particle-hole symmetry since the lattice structure becomesno longer bipartite. As the value of t /prime/tincreases from 0, the position of the van Hove singularity in the bare DOS shiftstoward the ω> 0 side, and, as a result, the DOS weights of particle excitations become higher and thinner compared tothe hole-excitation ( ω< 0) side. C. Possible spin liquid: Spatial and dynamical fluctuations The possibility of quantum SLs on anisotropic triangular lattice has been discussed in both localized-spin systems[5–21] and insulating yet barely itinerant electrons [ 25–38,68]. In those strongly correlated electron systems, two types ofquantum fluctuation effects play a key role for “quantummelting” of conventional magnetic long-range order: strongspatial fluctuations due to the frustrated lattice geometry anddynamical charge and spin fluctuations due to the itinerancyof electrons. The former effects have been studied in terms of the Heisenberg model of localized spins with anisotropic exchange 245145-5SHIMPEI GOTO, SUSUMU KURIHARA, AND DAISUKE Y AMAMOTO PHYSICAL REVIEW B 94, 245145 (2016) JandJ/prime(or the half-filled Hubbard model in the large U/t limit) [ 4–21,23]. Of particular interest is the anisotropy range where the classical spin configuration changes fromthe commensurate N ´eel to incommensurate spiral phase. The linear spin-wave theory has shown that the spin-wavevelocity along the ( k,k) direction vanishes at the N ´eel-spiral transition [ 7,8], which indicates that the magnetic order is destroyed by long-wavelength excitations. However, differentapproximations including several types of spin-wave theories[9], Schwinger-boson mean-field method [ 11], and series- expansion approach [ 23] have led to different conclusions on the search of SL phases in this anisotropy region, andmore sophisticated numerical studies [ 14,18] have been very limited. In the region where the anisotropic triangular latticecan be regarded as weakly coupled chains ( J /prime/J > 1), the fate of the classical spiral state under the influence ofquantum fluctuations has been examined by various numericalcalculations, which have suggested the emergence of nontrivialground states including essentially one-dimensional (gapless)SLs [ 5,6,16–20], a gapped SL close to the isotropic point [16,17], and a collinear antiferromagnetically ordered state [4,12]. On the other hand, the effects of the local, dynamical fluctuations unique to itinerant electrons has been discussedseparately from the spatial fluctuations in our present DMFTanalysis on the Hubbard model with finite values of U/t.F o r the Hubbard model, the previous study with a cellular DMFT[32] has shown that a nonmagnetic SL state may appear in a wide range of the anisotropy parameter, 0 .9/lessorsimilart /prime/t < 1.2, for largeU/t. However, it should be noted that such real-space cluster-based approximations [ 32,34–36] can describe only a commensurate magnetic order allowed by the size of theassumed cluster (four sites in Ref. [ 32]). The phase diagram obtained by our DMFT in the spin-rotating frame (Fig. 2) shows that the incommensurate spiral phase persists until itundergoes a first-order transition to the metallic phase, and noSL phase is formed only by the local quantum fluctuations dueto the itinerant change degrees of freedom. It is noted that increasing the parameter t /prime/tgives a larger bandwidth as well as bigger anisotropy. In order to extractonly the anisotropy effects, here we renormalize the Hubbardinteraction Uby the bandwidth W=/braceleftbigg8t (t /prime/t < 1/2), (2t/prime+t)2/t/prime(t/prime/t/greaterorequalslant1/2).(16) In Fig. 8we present the same magnetic phase diagram as Fig. 2 but as a function of U/W . It can be seen that the metal-insulator transition requires the strongest Hubbard interaction (in unitsofW) at the isotropic triangular-lattice point. However, it should be noticed that this does not mean that the magneticlong-range order is most fragile at t /prime/t=1 in the spiral phase. Figure 9shows the t/prime/tdependence of the magnetization M when U/W is fixed. One can see that as t/prime/tincreases, the magnetization Mshows a shallow dip around t/prime/t≈0.5 and then exhibits a maximum at the isotropic triangular-latticepoint in the spiral phase. A similar suppression of fluctuationsat the isotropic triangular-lattice point has also been reportedin the studies of the Heisenberg model. The linear spin-waveanalysis [ 7,8], the Schwinger boson mean-field approach [ 11],FIG. 8. The same as Fig. 2but with the vertical axis in units of the bandwidth W. and the coupled cluster method [ 14]h a v ea l ls h o w nt h a t the spatial spin fluctuations on the spiral magnetic orderis most suppressed at the isotropic triangular-lattice point.Surprisingly, Fig. 9shows that the value of Mfor the triangular lattice is even larger than that for the square lattice. Basically,mean-field order in the large- Uregime is more robust for the lattice with a larger coordination number z. Nevertheless, it is known that the spatial spin fluctuation on the antiferromagneticorder is much stronger for the triangular lattice ( z=6) than that for the square lattice ( z=4) since the effects of strong geometric frustrations enhance the spatial fluctuations on the120 ◦order of the triangular lattice. On the other hand, the DMFT results shown in Fig. 9might indicate that this is not the case for the local dynamic fluctuations of itinerantelectrons since nonlocal correlation effects are omitted fromthe self-energy /Sigma1(ω). In order to reach the final conclusion on the ground-state magnetic property of the Hubbard model for generic valuesoft /prime/t, it is required to take into consideration the interplay of both the spatial and dynamical fluctuations and comparethe energies of incommensurate spiral state and SL (or theother candidate) states. Given our DMFT results and the FIG. 9. Magnetization Mas a function of t/prime/tforU/W =2.0 (upper panel) and U/W =1.0 (lower panel). 245145-6INCOMMENSURATE SPIRAL MAGNETIC ORDER ON . . . PHYSICAL REVIEW B 94, 245145 (2016) previous studies on the Heisenberg model, we can conclude that the commensurate 120◦magnetic order at the isotropic triangular-lattice point is most robust, compared to the otherincommensurate spiral states, against both the dynamic andspatial fluctuations. Nevertheless, as seen in the phase diagramof Fig. 8, it is most unstable against the transition into metallic phase when the interaction strength U/W decreases. These facts indicate that the realization of quantum SLs in theHubbard model is most unlikely when t /prime/t=1i ns p i r a l phases, although the above discussion cannot rule out thepossibility of a SL ground state [ 25–27,30–36] induced by some synergistic effects of dynamic and spatial quantumfluctuations. On the other hand, our DMFT calculation shows that the local, dynamic fluctuations strongly reduce the magnetizationMwhen the anisotropy parameter increases from t /prime/t=1, which is attributed to the enhancement of low dimensionality.In addition, the curve of Mis slightly suppressed in the vicinity of the transition point between the N ´eel and spiral phases (see a dip around t /prime/t∼0.5i nF i g . 9), although the reduction effect is very small. The magnetization reduction dueto the dynamic fluctuations in these two anisotropy parameterregions could help the emergence of the SLs expected in thestudies on the same parameter regions of the Heisenberg model[5–11,16–20]. IV . CONCLUSION In this paper we studied the effects of the itinerant electron degrees of freedom on the magnetic properties of the systemson the anisotropic triangular lattice that interpolates from thesquare lattice ( t /prime/t=0) to decoupled one-dimensional chains (t/prime/t→∞ ) via the isotropic triangular lattice ( t/prime/t=1). We performed a local gauge transformation that rotated thespin-quantization axis into the direction of the magneticmoment at each site to properly describe an incommensuratespin spiral order. Working in the spin-rotating frame and usingthe imaginary-time matrix product state solver [ 59] based on the DMRG, we determined the magnetic phase diagram ofthe half-filled anisotropic-triangular Hubbard model at zerotemperature in the framework of the DMFT. It was foundthat the metal-insulator transition for t /prime/t/negationslash=0 takes place at a nonzero value of U/t due to the lack of perfect nesting, and in a discontinuous (first-order) fashion. When the anisotropyparameter t /prime/tincreases from 0 in the insulating state at a fixed value of U/t, the ordering vector of the magnetic long-range order changes from the commensurate value ( π,π) to an incommensurate one ( q,q)a tt/prime/t∼0.7, and gradually goes to ( π/2,π/2) ast/prime/t→∞ . In the vicinity of the transition between the commensurate N´eel and incommensurate spiral states, the magnetic moment reduction caused by the fluctuation effects is slightly pro-nounced. Moreover, for large values of t /prime/t, the magnetic moment decreases rapidly with t/prime/tdue to the enhancement of low dimensionality. Although no nonmagnetic insulating statewas formed only by the local, dynamic fluctuations consideredin the DMFT, these fluctuation effects could support theemergence of quantum SL ground states in the Hubbard modelwhen taking into consideration the strong spatial fluctuationsin the same anisotropy parameter regions [ 5–11,16–20]. Itis noteworthy that in the isotropic triangular lattice the 120 ◦ magnetic insulator undergoes a sudden first-order transition into nonmagnetic metal with relatively large interaction U/W (compared to that of the other spiral magnetic states att /prime/t/negationslash=1), whereas the magnetic long-range order just before the transition is still robust. This fact is disadvantageous to theemergence of nonmagnetic insulating states in the isotropictriangular-lattice Hubbard systems. The inclusion of nonlocal fluctuation effects has been partially carried out by real-space cluster extensions of DMFT[32,33], which have, however, treated only commensurate magnetic orders allowed within the assumed cluster shape.As was pointed out in the present study, incommensurabilityof magnetic order is essential for the magnetic propertyof the anisotropic triangular-lattice systems, and moreover,long-wavelength fluctuations are important for the breakingof long-range magnetic orders according to linear spin-wavepredictions [ 7,8]. Our present DMFT calculations in the spin-rotating frame provide a solid physical and mathematicalbasis for further study in this direction, e.g., with diagrammaticextensions of DMFT [ 69,70], which can include the effects of long-range quantum correlations through diagrammaticcorrection, or reciprocal space cluster extensions of DMFTsuch as the dynamical cluster approximation [ 71], which does not break the periodicity of original lattices. ACKNOWLEDGMENTS The DMRG calculations in this paper are performed using ITensor library, http://itensor.org . This paper is partially supported by KAKENHI Grants from Japan Society for thePromotion of Science No. 26800200 (D.Y .), and a part of theoutcome of research performed under a Waseda UniversityGrant for Special Research Projects No. 2015S-100 (S.G.). APPENDIX A: HOW TO OPTIMIZE BATH PARAMETERS From Eqs. ( 11) and ( 12), the self-consistent condition of the DMFT, Gimp(ω)=G(ω), can be rewritten as /summationdisplay lVlV† l ω−/epsilon1l=ω+μ−/Sigma1(ω)−G−1(ω). (A1) Using Eq. ( A1) we adjust the bath parameters Vlσσ/primeandεlin an iterative manner: First, the SIAM in Eq. ( 10) is solved by the DMRG technique given in Sec. II B, and the self-energy /Sigma1(ω) is extracted by the calculated Gimp(ω)v i aE q .( 11). Substituting /Sigma1(ω), one can evaluate the right-hand side of Eq. ( A1). [Note that the self-energy /Sigma1(k,ω)i nG(ω) should be replaced by /Sigma1(ω) in the DMFT.] Then a new set of Vlσσ/primeandεlis given by fitting the evaluated right-hand-side value in the form ofthe left-hand side as a function of ω. Using the updated bath parameters we solve again the SIAM by the DMRG technique,and the procedure is repeated until convergence is reached. Theconvergence criterion used in this study is /summationdisplay ω/bardbl/Gamma1(ω)−/Gamma1/prime(ω)/bardbl<5×10−3t, (A2) where /Gamma1(ω) and /Gamma1/prime(ω) are the hybridization function/summationtext l(VlV† l)/(ω−/epsilon1l) with the bath parameters before and after 245145-7SHIMPEI GOTO, SUSUMU KURIHARA, AND DAISUKE Y AMAMOTO PHYSICAL REVIEW B 94, 245145 (2016) a single step of the DMFT iteration. Here we take the summa- tion over a set of 200 sample points ω=(0.1it,0.2it,..., 20it) on the imaginary axis. The fitting of both sides of Eq. ( A1) for updating the bath parameters is performed by minimizing the distance function d=/summationdisplay ω/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay lVlV† l ω−/epsilon1l−[ω+μ−/Sigma1(ω)−G−1(ω)]/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble.(A3) Since the distance function is nonconvex, the minimization by ordinary gradient methods is practically difficult (see theSupplemental Material of Ref. [ 72]). Thus, to perform the minimization in an efficient way, we use the vector fitting(VF) method [ 73,74], which gives a fitting of the numerical data for the right-hand side of Eq. ( A1) with a rational expression/summationtext lAl/(ω−/epsilon1l). The matrix Vlcan be obtained by the Cholesky decomposition of the matrix Al. It should be noted that if the number of the bath orbitals Nb(the number of the bath parameters) is too large, the VF method may providea nonpositive definite matrix A l, which cannot be decomposed by the Cholesky decomposition, and/or a complex value for /epsilon1l due to “overfitting.” To avoid it, we try the fittings with different Nb(typically up to Nb∼25 in the present study), and choose the best fitting out of them. The value of the distance functiondfor the 200 ωpoints is smaller than 10 −4tthroughout the calculations. APPENDIX B: LATTICE GREEN’S FUNCTIONS ON THE REAL FREQUENCY AXIS In order to numerically calculate the lattice Green’s func- tions on the real frequency axis, we first fit the self-energy ofthe DMFT /Sigma1(ω) in the form /Sigma1(ω)=n/summationdisplay iθiθ† i ω−λi+/Sigma1∞. (B1) Hereθiis a two-by-two matrix, λiis a real pole of the self- energy, and /Sigma1∞denotes the self-energy in the high-frequency limit, which is given by [ 75] /Sigma1∞=/parenleftBigg U/angbracketleftn↓/angbracketright− U/angbracketleft˜c† ↓˜c↑/angbracketright −U/angbracketleft˜c† ↑˜c↓/angbracketrightU/angbracketleftn↑/angbracketright/parenrightBigg . (B2) Fitting the numerical data for /Sigma1(ω) with the parameters θiand λiis performed in the similar way to that for the hybridization function in Appendix A. From the self-energy in the form of Eq. ( B1), we can obtain the Lehmann representation of the lattice Green’s function onthe real frequency axis as follows [ 76]. We introduce a matrix A=/bracketleftBigg (ω+μ)1−/Sigma1∞−n/summationdisplay iθiθ† i ω−λi/bracketrightBigg−1 . (B3) The matrix Ais also given as the upper left block of a block matrix ⎡ ⎢⎢⎢⎢⎢⎢⎣ω1−⎛ ⎜⎜⎜⎜⎜⎜⎝/Sigma1 ∞−μ1θ1θ2··· θn θ† 1 λ11 0··· 0 θ† 2 0λ21...... ............ 0 θ† n 0··· 0λn1⎞ ⎟⎟⎟⎟⎟⎟⎠⎤ ⎥⎥⎥⎥⎥⎥⎦−1 ≡[ω1−U/Lambda1U†]−1 =U1 ω1−/Lambda1U†, (B4) where Uand/Lambda1are a unitary matrix and a diagonal matrix, respectively. Therefore, the matrix Acan be expressed as A=θ/prime1 ω1−/Lambda1θ/prime†, (B5) withθ/primebeing the first two rows of U. From the definition of Ain Eq. ( B3), the lattice Green’s function is given in the Lehmann representation as Glatt(ω,k)=1 (ω+μ)1−εQ(k)−/Sigma1(ω) =1 A−1−εQ(k) =1 /parenleftbig θ/prime1 ω1−/Lambda1θ/prime†/parenrightbig−1−εQ(k) =θ/prime 1 ω1−(/Lambda1+θ/prime†εQ(k)θ/prime)θ/prime† =θ/prime 1 ω1−U/prime k/Lambda1/prime(k)U/prime† kθ/prime† =θ/primeU/prime k1 ω1−/Lambda1/prime(k)U/prime† kθ/prime†, (B6) where U/primeand/Lambda1/primeare a unitary matrix and a diagonal matrix, respectively. 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PhysRevB.89.165121.pdf

PHYSICAL REVIEW B 89, 165121 (2014) Dot-bound and dispersive states in graphene quantum dot superlattices A. Pieper,1R. L. Heinisch,1G. Wellein,2and H. Fehske1 1Institut f ¨ur Physik, Ernst-Moritz-Arndt-Universit ¨at Greifswald, 17487 Greifswald, Germany 2Regionales Rechenzentrum Erlangen, Universit ¨at Erlangen-N ¨urnberg, 91058 Erlangen, Germany (Received 13 February 2014; revised manuscript received 8 April 2014; published 18 April 2014) We consider a square lattice configuration of circular gate-defined quantum dots in an unbiased graphene sheet and calculate the electronic, particularly spectral properties of finite albeit actual sample sized systems by meansof a numerically exact kernel polynomial expansion technique. Analyzing the local density of states and themomentum resolved photoemission spectrum we find clear evidence for a series of quasibound states at the dots,which can be probed by optical measurements. We further analyze the interplay of the superlattice structure withdot-localized modes on the electron energy dispersion. Effects of disordered dot lattices are discussed too. DOI: 10.1103/PhysRevB.89.165121 PACS number(s): 72 .80.Vp,73.21.Cd,73.21.La I. INTRODUCTION Graphene, in which the carbon atoms are condensed in a strictly two-dimensional honeycomb lattice due to their sp2 hybridization, constitutes a unique form of quantum matter, in- teresting for both fundamental science and applications [ 1–5]. Specifically the electronic properties of graphene are extraor- dinary. Graphene is a topological material where the quasi-particles (low-energy excitations) near the so-called Diracnodal points behave as massless relativistic Dirac fermionspossessing a linear energy dispersion. In neutral graphene theFermi energy crosses exactly the Dirac points. Hence, having avanishing density of states at the Fermi energy but no gap in the excitation spectrum, the system is a hybrid between a metal and a semiconductor. These features lead to many unusualand sometimes counterintuitive transport phenomena such asa finite universal dc conductivity at the neutrality point, Kleintunneling, or an anomalous quantum Hall effect [ 2]. From an application-technological point of view, the tunability of graphene’s electronic and optical properties by external electric and magnetic fields is of particular impor- tance [ 6]. Amongst others, this provides unique possibilities to modify—for instance, by gating—the properties of finite areas in graphene-based structures. Exploiting the parallels between optics and (Dirac) electronics, this has led to the proposal of potential steps such as Veselago lenses for propagating electron beams [ 7] or the experimental implementation of the counterpart of optical fiber cables [ 8]. For a circular gated region, refraction at the boundary leads to two coalescing caustics that focus the electron density in the dot [ 9]. Similarly, circular gated regions have also been studied in bilayer graphene [ 10] and in monolayer graphene with spin-orbit coupling which induces birefringence [ 11]. Interestingly, circular dots in unbiased graphene allow electrostatic electron confinement in spite of Klein tunneling [ 12]. For biased graphene, long-living temporary bound states appear [ 13]. These modes entail a peculiar angular scattering characteristic: In particular, forward scattering and Klein tunneling can bealmost switched off by a Fano resonance phenomenon [ 14,15]. While quantum dots on etched graphene have been studied as potential hosts for spin qubits [ 16–19], single gate-defined dots [ 20] and multiple dots arranged in corrals [ 21]h a v e been used to model the scattering of Dirac electron waves by impurities or metallic islands placed on a graphene sheet. Ifthere is more than a single graphene quantum dot, interdot coupling—realized, e.g., via direct tunneling between the dots or through the continuum states of graphene—gains in importance [ 22]. In this contribution, we investigate the electronic properties of graphene gate-controlled quantum dots arranged in a regularsquare lattice configuration. Graphene superlattices offer anexciting prospect to tailor the charge carrier behavior througha renormalization of the group velocity at the Dirac point orthe emergence of higher order Dirac points [ 23–27]. Using the kernel polynomial method (KPM) [ 28,29] in order to obtain unbiased numerical results, we monitor the (local) density ofstates (LDOS/DOS), the optical conductivity of the sample,and the single-particle excitation spectrum. Thereby, we firstdiscuss the existence of quasilocalized states at the quantumdot regions with energies near the normal modes of an isolatedfreestanding graphene dot. Then we consider the interplay ofthe additional flat bands stemming from the normal modes ofthe dots with sublattice effects such as higher order Dirac conesand group velocity renormalization. Finally, we demonstratehow different types of disorder destroy or preserve superlatticeand normal mode induced spectral signatures. II. MODEL AND METHOD We model the electronic structure by a tight-binding Hamiltonian, H=/summationdisplay iVic† ici−t/summationdisplay /angbracketleftij/angbracketright(c† icj+H.c.), (1) where c(†) iis a fermionic annihilation (creation) operator acting on lattice site iof the honeycomb lattice with Lsites. The nearest-neighbor hopping amplitude is t/similarequal3 eV . The setup we consider (see Fig. 1) is a graphene sheet on a gated substrate with circular regions where an additional external potential Vi=V/summationdisplay (n,m)/Theta1(R−| /vectorri−/vectorr(n,m)|)( 2 ) is applied. 1098-0121/2014/89(16)/165121(6) 165121-1 ©2014 American Physical SocietyA. PIEPER, R. L. HEINISCH, G. WELLEIN, AND H. FEHSKE PHYSICAL REVIEW B 89, 165121 (2014) () FIG. 1. (Color online) Left: Graphene quantum dot array used in this work. The dots are defined electrostatically, by applying a constant bias V. The dot radius is R; the square dot superlattice constant is D. To ensure the gate potential to be smooth on the scale of the lattice spacing a, we adopt a linear interpolation of Viwithin a small range R±0.01R. The underlying graphene honeycomb lattice structure is shown in the lower left corner; we have zigzag(armchair) edges in x(y) direction with N(M) dots. Periodic boundary conditions (PBCs) were used at the edges of the sample. Right: Sketch of Dirac electron scattering at a single quantum dot.ForE<V , where the dot embodies an n-pjunction, the incident (ψ i) and reflected ( ψr) electron waves reside in the conduction band, while the transmitted ( ψ(qb) t) wave inside the dot corresponds to a state in the valence band. Owing to the double-cone dispersion (near KandK/prime) nonevanescent waves can exist in the dot, i.e., ψ(qb) tmight give rise to a quasibound state. The local electronic properties of this graphene quantum dot superlattice are reflected in the LDOS, ρi(E)=L/summationdisplay l=1|/angbracketlefti|l/angbracketright|2δ(E−El), (3) where |i/angbracketright=c† i|0/angbracketright, and |l/angbracketrightis a single-electron eigenstate of Hwith energy El. The LDOS can be directly probed by scanning tunneling microscopy [ 30]. For the noninteracting system ( 1),ρi(E) can be determined to, de facto , arbitrary precision by the KPM, which is based on an expansion ofthe (rescaled) Hamiltonian into a finite series of Chebyshevpolynomials [ 28,29]. The mean DOS follows as ρ(E)=/summationtext L i=1ρi(E). The momentum-resolved single-particle spectral function at zero temperature, A(/vectork,E)=L/summationdisplay n=1|/angbracketleftl|ψ(/vectork)/angbracketright|2δ(E−El), (4) is easily accessible by the KPM as well [ 28,29]. Here |ψ(/vectork)/angbracketright= L−1/2/summationtext iexp(i/vectork/vectorri)c† i|0/angbracketright. Note that |ψ(/vectork)/angbracketrightis not a Bloch eigenstate of infinite graphene due to its sublattice structure. Within our KPM scheme, we also have access to the real part of the optical conductivity [ 28,29]: σ(ω)=π/planckover2pi1 ω/Omega1/summationdisplay l,l/prime|/angbracketleftl|Jx|l/prime/angbracketright|2[f(El)−f(El/prime)]δ(ω+El−El/prime) (5) withJx=− (iet//planckover2pi1)/summationtext /angbracketlefti,j/angbracketright(rj,x−ri,x)c† icjthexcomponent of the current operator. In ( 5),f(E)=[e(E−μ)/T+1]−1denotes the Fermi function containing the temperature Tand thechemical potential μ. Moreover, /Omega1=33/2La2/4, where a/similarequal 1.42˚A is the carbon-carbon distance. III. NUMERICAL RESULTS AND DISCUSSION We begin our discussion with the signatures of localized modes for a graphene sample with a regular array of quantumdots. To identify dot-induced features, we compare withresults for a single circular quantum dot with sharp boundaryin an infinite graphene sheet treated within the continuumDirac-equation approximation [ 12–14,31,32]. In this case the electronic states in the dot are resonances with finite trappingtime. Due to interference effects the trapping time may evenbecome infinite in unbiased graphene for particular parametersof the sharp circular confinement potential [ 12–14]. The quasibound states for an isolated dot, a m, can be classified according to their angular momentum. The mode amis made up of states with total angular momentum j=± (m+1/2) (composed of orbital momentum mand pseudospin ±1/2). The modes amare fourfold degenerate: twice with respect to±jand twice with respect to valley degrees of freedom KandK/prime. For small energies the mode a0is relatively broad. While contributing significantly to electron scatteringit does not evolve into a true bound state for E→0. For higher modes, however, the electron is strongly localized atthe dot, overcoming Klein tunneling. For unbiased graphenedot-localized modes appear for the “dot parameter” η= VR/ v F=jm,s, where vF=3at/2/planckover2pi1is the Fermi velocity in pristine graphene and jm,sis thesth zero of the Bessel function Jm[12]. We note that electron confinement of such kind can persist for relatively small dots, even taking the latticediscreteness into account [ 15,32]. If we consider an array of gate-defined quantum dots, for very large interdot distances,D/greatermuchR, all dots have the same energy spectrum. When D comes up to R’s order of magnitude, the interdot coupling results in a splitting of the degenerate energy levels. This hasbeen demonstrated for a periodic chain of quantum dots [ 22]. We now compare the DOS of samples with and without quantum dot superlattice (see Fig. 2). If all V i=0, the DOS of the nearest-neighbor πelectron tight-binding model ( 1), describing pure graphene in that case, can be calculatedanalytically in terms of a complete elliptic integral of the firstkind with energy-dependent prefactors [ 33]. Most notably, close to the Dirac points KandK /prime,ρ(E) is proportional to|E|/v2 F. Figure 2gives the DOS near the Dirac point of a finite graphene system with PBCs, where 10 ×10 [panels (a)–(c)], respectively, 20 ×20 [panel (d)] quantum dots were arranged periodically in a square (super-) lattice configuration.Contributions emanating from quasibound modes a mare superimposed on the DOS of pristine graphene. They form narrow energy bands, except for the broad a0mode. In panel (a) the voltage V/t=0.085 46 was chosen to fix the lowest a1mode—originally located in the lower Dirac cone—at zero energy. If we further increase the gate potential the energyladder of dot-bound states is shifted upwards. Accordingly, inpanel (b), where V/t=0.170 92, the first a 4related band has reached the Fermi energy, whereas states assigned to the firstand second a 1, as well as to the first a2anda3resonances passed the Dirac point already. We note that bands belongingtoa mstates with larger mare less spread in energy. Panel (c) 165121-2DOT-BOUND AND DISPERSIVE STATES IN GRAPHENE . . . PHYSICAL REVIEW B 89, 165121 (2014) 00.020.040.06 00.020.040.06 00.020.040.06 -0.1 -0.05 0 0.05 0.1 E/t00.020.040.06a4a3 a5a6a1a2a3 a1a2 a1a2a3R=9.55nm, V/t=0.08546, R=4.775nm, V/t=0.17092,R=9.55nm, V/t=0.17092, R=4.775nm, V/t=0.17092,D=38.2nm, 10x10 dots D=38.2nm, 10x10 dots D=38.2nm, 10x10 dots D=19.1nm, 20x20 dotsgraphene sheet: 3104zzx1792ac = (381.7x381.7)nm2 a1a2a3DOS(a) a3(b) (c) (d) FIG. 2. (Color online) DOS of the graphene quantum dot super- lattice in dependence on R,D,a n dV/t. Peaks related to quasibound dot modes are designated by am. Note that the dot parameter ηis the same in panels (a), (c), and (d). The DOS is calculated by the KPM on a lattice with 3104 (1792) sites in zigzag (armchair) edgedirection, using 16 384 Chebyshev moments. To identify effects due to the finiteness of the sample, the PBCs, and the resolution of the KPM—leading to small deviations from the strictly linear increase of the DOS near E=0—we included the DOS obtained numerically for the case V i=0∀i(dashed lines). The total DOS, of course, fulfills the sum rule/integraltext∞ −∞ρ(E)dE=1. shows that reducing the size of the quantum dots the different quasibound states become more separated energetically. Finitedot-size effects provoke the splitting of some a mbands. A larger number of dots will of course enhance the weight of thequasibound states [compare panels (c) and (d)]. To confirm the spatial localization of the states associated with the dot normal modes, we depict in Fig. 3the LDOS for the four representative energies indicated in the lower DOSpanel. Here the first three energies fit to the corresponding dotmodes, the fourth energy bgives an account of the situation in bulk graphene away from the resonances. For the energiesclose to the dot-bound modes the LDOS profile is reflectiveof the quantum dot superlattice structure. Within the dotregions the intensity of the LDOS is enhanced in a ring shape.Since the KPM has a finite energy resolution the LDOS as-sembles contributions from several eigenstates in the energeticvicinity of the target energy E[34]. We like to emphasize, however, that these dot states are not strictly localized in realspace. They are in superposition with each other (and alsowith bulk graphene states), leading to coherent transport but on a strongly reduced energy scale. For energies far off the resonances the LDOS is almost uniformly distributed (seepanel energy b). Here the quantum dot superlattice behaves like a pristine graphene sample. We next demonstrate that transitions between the dot-bound states could be induced optically. To this end we havecalculated the optical response of the graphene quantum dotarray. Figure 4gives the optical conductivity for the system studied in Figs. 2(d) and3. Besides the Drude peak at ω=0, noticeable absorption is only found for the transition fromthea 1to thea2band, which corroborates the optical selection 0 0.02 0.04 0.06 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 E/ta1 ba2a3 DOS 0 0.1 0.2a1LDOS b 10nm 0 0.1 0.2a2LDOS a3 FIG. 3. (Color online) LDOS intensity plots for the central part of a (larger) square quantum dot superlattice with 8 ×8 dots (upper panels). The LDOS is given for the energies indicated in the lowerpanel, showing the mean DOS of the whole panel. To diminish finite- size effects the LDOS was calculated using 4096 Chebyshev moments only; therefore the splitting of the a mbands is not resolved here (recall that the resolution of the KPM scales with the inverse square root of the number of Chebyshev moments [ 28]). System parameters are R=4.775 nm, D=19.1 nm, and V/t=0.170 92 [as in Fig. 2(d)]. ruleam→am±1obtained within the Dirac approximation [ 15]. Tuning μ, different optical transitions might be singled out. As a matter of course the gap in the optical absorption spectrumfills with spectral weight at higher temperatures. To investigate the interplay of the dot-bound modes with superlattice effects we calculate the energy-momentum dependence of the single-particle spectral function A(/vectork,E). 0 0.05 0.1 ω/t0123456σ/σ0 T=300K T=30Kμ/t=-0.02 FIG. 4. (Color online) Optical conductivity (in units of σ0= e2/8/planckover2pi1)f o ra2 0 ×20 graphene quantum dot superlattice with V/t=0.170 92 and R=4.775 nm, D=19.1 nm. We chose μ/t= −0.02<0 so that the Drude peak appears at ω=0 and the transition between the occupied a2and the unoccupied a1mode appears at ω/t/similarequal0.045 (note that the transition a3toa2does not appear as both modes are occupied). 165121-3A. PIEPER, R. L. HEINISCH, G. WELLEIN, AND H. FEHSKE PHYSICAL REVIEW B 89, 165121 (2014) (a) (c)(b) (d) (e) (f) (g) (h) (i) FIG. 5. (Color online) Single-particle spectral function along the /Gamma1K direction (horizontal; left-hand panels) and parallel to the /Gamma1M/prime direction (vertical, right-hand panels) through the Dirac Kpoint, as indicated in the upper central figure (b), showing graphene’s Brillouin zone. Panels (a) and (c) give A(/vectork,E) for a finite sample of pristine graphene ( V/t=0) with PBC. Below, results for a 20 ×20 dot superlattice with R=4.775 nm and D=19.1n ma r es h o w n . In (d) and (e) V/t=0.107 27 (mode a0falls on E=0), in (f) and (g) V/t=0.170 92 (mode a1onE=0), and in (h) and (i) V/t=0.229 11 (mode a2onE=0). The green marker (circle) traces the energy shift of the nodal point for pristine graphene when Vis increased. In view of the transfer of spectral weight to other nodalpoints it should, however, no longer be identified as the “genuine” Dirac point. Note also that for mode a 1atE=0 the Dirac point of pristine graphene evolves into the nodal point at E=0 when the dot spacing Dis reduced [compare panel (e) and Fig. 6(b)]. Within the KPM 8192 moments were used. Reflected in angle-resolved photoemission spectroscopy, this quantity gives insight into the electronic band structure.Figure 5displays the results in the vicinity of the Dirac point. In comparison to the perfectly linear energy bands of pure graphene [where V i≡0; see panels (a) and (c)], we observe for the graphene quantum dot array, on the one hand, a “ladder”of nearly dispersionsless bands formed by the quasibound dotstates, and on the other hand, a sequence of dispersive bandsdisplaced against each other by reciprocal superlattice vectors.The latter bands will collapse if D→∞ (compare with Fig. 6 showing results for a larger D). Note also the emergence of secondary nodal points due to intersection of energy bands atthe edges of the superlattice Brillouin zone. To assess how the dot-bound modes affect the dispersive bands of the superlattice we show A(/vectork,E)i nF i g . 5for -0.2 -0.1 0 0.1 0.2 ky 3a-0.1-0.0500.050.1E/t(a) -0.2 -0.1 0 0.1 0.2 ky 3a(b) FIG. 6. (Color online) Single-particle spectral function parallel to the /Gamma1M/primedirection through the Dirac Kpoint for a superlattice of 10×10 dots with R=4.775 nm and D=38.2 nm. In (a), V/t= 0.107 27 so that the mode a0falls on E=0 [as in Figs. 5(d) and5(e)]. In (b), V/t=0.170 92 so that mode a1falls on E=0 [as in Figs. 5(f) and5(g)]. different applied voltages at the dots. In panels (d) and (e) V/t=0.107 27 so that the first a0mode is at E=0. This mode is not localized at the dots and no dispersionless bandis formed at E=0. Instead, the mode a 0hybridizes with the extended states outside the dot. This leads to a shift of theoriginal Dirac cone to higher energy as propagating states alsoreside in the dots where the potential is higher. Furthermore,the group velocity at this nodal point is reduced by about26% which is even larger than the reduction by 19% obtainedin second order perturbation theory [ 23]. In panels (f) and (g) V/t=0.170 92 so that the mode a 1falls on E=0. This mode is very sharp and shows only negligible hybridization with thepropagating states outside the dot. Hence, the dispersionlessband originating from the a 1mode is superimposed at E=0 on a Dirac cone which is only marginally affected by the dots.The renormalization of the group velocity at the higher Diracpoint amounts to 51% in agreement with 48% in second orderperturbation theory. The different behavior between the modesa 0anda1atE=0 is also reflected by Fig. 6where results for a larger Dare shown. For a0hybridization between electronic states inside and outside the dot transfers spectral weight to the -0.1 -0.05 0 0.05 0.1 E/t00.020.040.06DOSa1a2a3 R=4.775nm, V/t=0.17092, D=19.1nm, 20x20 dots FIG. 7. (Color online) DOS for a 20 ×20 superlattice sample of graphene quantum dots with random radii. Shown are the results for a single, typical realization, where all the dot radii were drawn from a uniform distribution of radii with mean value R=4.775 nm andR(n,m)/R∈[0.975,1.025] (red dashed line), [0.95,1.05] (blue dashed-dotted line), [0.925,1.075] (green double-dot-dashed line), and [0.9,1.1] (violet dotted line). The black line gives the DOS withoutdisorder. Again, we have D=19.1n ma n d V/t=0.170 92. 165121-4DOT-BOUND AND DISPERSIVE STATES IN GRAPHENE . . . PHYSICAL REVIEW B 89, 165121 (2014) -0.2 -0.1 0 0.1 0.2 kx -0.1-0.0500.050.1E/t(a) -0.2 -0.1 0 0.1 0.2 ky (b) FIG. 8. (Color online) Single-particle spectral function through the Dirac Kpoint for a 20 ×20 graphene quantum dot superlattice with random R(n,m)/R∈[0.9,1.1]. The parameters R,D,a n dVare chosen as in Figs. 5(f)and5(g) so that the mode a0falls on E=0. newly emerging nodal points and the original Dirac point at E=0 vanishes. For the dot-localized a1mode, which couples negligibly to the extended states outside the dots, the originalDirac cone is preserved. Panels (h) and (i) of Fig. 5show the case of mode a 2atE=0. This mode overlaps with the broad second a0mode so that the nodal point is shifted to higher E. We conclude that if the dots support only one very sharp modea m>0atE=0 the dot superlattice leaves the conical energy dispersion of pristine graphene close to E=0 intact and the dispersionless dot band is merely superposed. This situation isbest realized for the mode a 1. Finally, we address the question of how disorder of a certain kind will affect the results shown so far. Intrinsicdisorder, for instance, leads to the formation of electron-holepuddles [ 35] characterized by potential variations of typically less than 50 meV ( ≈0.017t), which is small compared to the dot potential in our study. Hence, we expect our resultsto be relatively robust against intrinsic disorder and focus inthe following on variations of the radii and the spacing of thegate-defined quantum dots as these should be uncertain to someextent in experiments. Therefore we study, on the one hand,a square superlattice of quantum dots with radii uniformlydistributed around a mean value. Figure 7presents the DOS of typical samples of such a random system. Obviously thepeaks stemming from the quasibound states still exist butare considerably washed out if the disorder increases. The samehappens to the optical absorption (not shown). Looking at thesingle-particle excitation spectrum, we realize that the almostdispersionless bands, originating from the small overlap ofthe dot quasibound states, were destroyed (see Fig. 8). That means, quasibound dot states are still there but their coherenceis lost. The dispersive graphene states, on the contrary, arerather insusceptible against the randomness induced by thedifferent size of the quantum dots, at least close to the Diracpoints KandK /prime. If, on the other hand, the dots are randomly shifted away from their superlattice sites (see Fig. 9), the -0.1-0.0500.050.1E/t(a) (b) -0.2 -0.1 0 0.1 0.2 kx-0.1-0.0500.050.1E/t(c) -0.2 -0.1 0 0.1 0.2 ky(d) FIG. 9. (Color online) Single-particle spectral function through the Dirac Kpoint for a 20 ×20 graphene quantum dot superlattice where the dots are displaced from their superlattice sites by r(n,m)∈ [0,/Delta1r] in random direction. In panels (a) and (b) /Delta1r=0.1; in panels (c) and (d) /Delta1r=0.4. The parameters R,D,a n dVare chosen as in Figs. 5(f)and5(g) so that the mode a0falls on E=0. displaced dispersive bands become much weaker while the central cone as well as the dot-induced dispersionless bandspersist. We have also considered superlattices with elliptic dots(not shown). As the electron confinement is optimal only forcircular dots, noncircular dots lead to a significant broadeningof the flat bands. To conclude, superlattices of gate-defined quantum dots in graphene show clear indications of dot-bound modes inthe (local) density of states, the optical conductivity, and thesingle-particle spectral function. For superlattices with onlyone sharp localized mode at the charge neutrality point adispersionless dot band emerges while the conical energydispersion is preserved and pinned to E=0. For other choices of the dot potential the group velocity at the Dirac coneis significantly renormalized. 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PhysRevB.99.060406.pdf

PHYSICAL REVIEW B 99, 060406(R) (2019) Rapid Communications Magnetoelectric effect and orbital magnetization in skyrmion crystals: Detection and characterization of skyrmions Börge Göbel,1,*Alexander Mook,1Jürgen Henk,2and Ingrid Mertig1,2 1Max-Planck-Institut für Mikrostrukturphysik, D-06120 Halle (Saale), Germany 2Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany (Received 26 October 2017; revised manuscript received 16 May 2018; published 13 February 2019) Skyrmions are small magnetic quasiparticles, which are uniquely characterized by their topological charge and their helicity. In this Rapid Communication, we show via calculations how both properties can be determinedwithout relying on real-space imaging. The orbital magnetization and topological Hall conductivity measure the arising magnetization due to the circulation of electrons in the bulk and the occurrence of topologically protected edge channels due to the emergent field of a skyrmion crystal. Both observables quantify the topologicalHall effect and distinguish skyrmions from antiskyrmions by sign. Additionally, we predict a magnetoelectriceffect in skyrmion crystals, which is the generation of a magnetization (polarization) by application of anelectric (magnetic) field. This effect is quantified by spin toroidization and magnetoelectric polarizability. Thedependence of the transverse magnetoelectric effect on the skyrmion helicity fits that of the classical toroidalmoment of the spin texture and allows one to differentiate skyrmion helicities: It is largest for Bloch skyrmionsand zero for Néel skyrmions. We predict distinct features of the four observables that can be used to detect andcharacterize skyrmions in experiments. DOI: 10.1103/PhysRevB.99.060406 Introduction. Skyrmionics has attracted enormous interest over the recent years, as skyrmions [ 1–5]—small magnetic quasiparticles that are topologically protected—are aspirantsto be “bits” in future data storage devices [ 6–16]. The integral of the local spin chirality n Sk(r)=s(r)·/parenleftbigg∂s(r) ∂x×∂s(r) ∂y/parenrightbigg (1) of a skyrmion with magnetic texture s(r) tells the skyrmion number NSk=±1[17,18], that is, the topological invariant which characterizes skyrmions and antiskyrmions [ 19–22], respectively. On top of this, nSk(r) induces a topological Hall effect (THE) [ 23–34], which is an additional contribution to the Hall effect [ 35] of electrons in skyrmion crystals (SkXs, a periodic array of skyrmions; Fig. 1). Another quantity related to the magnetic texture is the orbital magnetization, which is explained in a semiclassicalpicture by the circulation of conduction electrons in the pres- ence of spin-orbit coupling (SOC) [ 36–41]. Recently, it has been shown that spin chirality, for example, in SkXs, canas well induce an orbital magnetization, even without SOC[42–44]. In this Rapid Communication, we establish a complete scheme (Fig. 1) for identifying the type of SkX in an exper- iment, without reverting to real-space imaging (e.g., Lorentzmicroscopy [ 45]). The TH conductivity and the orbital mag- netization describe the THE and are proportional to N Sk; therefore they differentiate skyrmions from antiskyrmions.Furthermore, we predict a magnetoelectric effect in SkXs,which is within experimental reach; the magnetoelectric po- *Corresponding author: bgoebel@mpi-halle.mpg.delarizability [ 46–48] and the spin toroidization [ 49,50]a l - low one to determine the skyrmion helicity, by which Néelskyrmions are differentiated from Bloch skyrmions. While theTHE quantities are based on reciprocal space Berry curvature,the magnetoelectric effect is characterized by the mixed Berrycurvature analogs (Fig. 1). Model and methods. We consider a two-dimensional square lattice with a fixed skyrmion texture {s i}(unit length, ilattice site). The resulting skyrmions and antiskyrmions can havevarious helicities [cf. Figs. 1(c)–1(e)]. The electrons in the SkX are described by a tight-binding Hamiltonian H=/summationdisplay ijtc† icj+m/summationdisplay isi·(c† iσci)( 2 ) (c† iandcjcreation and annihilation operators, respectively), with Hund’s rule coupling. The electron spins interact withthe magnetic texture ( mcoupling energy; s iunit vector; σ vector of Pauli matrices), which could be created by localizeddelectrons that are not explicitly featured in this one-orbital Hamiltonian. From the eigenvalues E n(k) and eigenvectors |un(k)/angbracketrightof the Hamiltonian ( 2) we calculate the kspace and the mixed Berry curvature for band n, /Omega1(ij) n(k)=−2I m/angbracketleftbig ∂kiun(k)/vextendsingle/vextendsingle∂kjun(k)/angbracketrightbig , (3a) D(ij) n(k)=−2I m/angbracketleftbig ∂kiun(k)/vextendsingle/vextendsingle1 m∂sjun(k)/angbracketrightbig , (3b) respectively. With v(j) nl(k)≡/angbracketleftun(k)|∂kjH(k)|ul(k)/angbracketright, (4a) s(j) nl(k)≡/angbracketleftun(k)|σj|ul(k)/angbracketright (4b) 2469-9950/2019/99(6)/060406(6) 060406-1 ©2019 American Physical SocietyGÖBEL, MOOK, HENK, AND MERTIG PHYSICAL REVIEW B 99, 060406(R) (2019) FIG. 1. Core message of this Rapid Communication. (a) Skyrmions and antiskyrmions are distinguished by the topologicalHall effect. (b) The helicity of skyrmions (e.g., Bloch and Néel skyrmions) is differentiated by the magnetoelectric effect. With these quantities, (c) Bloch skyrmions, (d) Néel skyrmions, and (e) anti-skyrmions can be distinguished. The color scale in (c)–(e) indicates the in-plane orientation of the spins (arrows). (j=x,y,z), we arrive at /Omega1(ij) n(k)=−2I m/summationdisplay l/negationslash=nv(i) nl(k)v(j) ln(k) [En(k)−El(k)]2, (5a) D(ij) n(k)=−2I m/summationdisplay l/negationslash=nv(i) nl(k)s(j) ln(k) [En(k)−El(k)]2. (5b) Integration over the occupied states [shorthand notation/integraltext occ(·)≡/summationtext n/integraltext (·)/Theta1(En(k)−EF)d2kwith EFFermi energy and/Theta1Fermi distribution at zero temperature] yields the conductivity σij[51] and the magnetoelectric polarizability αij[47–49], σij(EF)=−e2 h1 2π/integraldisplay occ/Omega1(ij) n(k), (6a) αij(EF)=gμBe (2π)2/integraldisplay occD(ij) n(k). (6b) From the orbital magnetic moment [ 36,37] mn(k)=−e 2¯hIm/summationdisplay l/negationslash=nvnl(k)×vln(k) En(k)−El(k), (7) we calculate the orbital magnetization [ 39], Mz(EF)=1 (2π)2/integraldisplay occm(z) n(k)+e ¯h1 (2π)2 ×/integraldisplay occ/Omega1(xy) n(k)−/Omega1(yx) n(k) 2[EF−En(k)]; (8) likewise, from the spin toroidal moment, tn(k)=gμB 2Im/summationdisplay l/negationslash=nvnl(k)×sln En(k)−El(k), (9)the spin toroidization, as recently shown by Gao et al. [49], Tz(EF)=1 (2π)2/integraldisplay occt(z) n(k)−gμB1 (2π)2 ×/integraldisplay occD(xy) n(k)−D(yx) n(k) 2[EF−En(k)].(10) The terms with m(z) nandt(z) ncapture the intrinsic contributions of each Bloch electron, while the other terms account for theBerry curvatures /Omega1(ij) nandD(ij) n, which modify the density of states [ 39]. Topological Hall effect as a quantum Hall effect. Before discussing the novel results concerning the energy-dependentorbital magnetization, magnetoelectric polarizability, and spintoroidization, a sketch of the band formation and the THconductivity is adequate; cf. Refs. [ 30,31]. Form=0 in the Hamiltonian ( 2), the so-called zero-field band structure is spin degenerate because there is neither spin-orbit coupling nor coupling to the SkX magnetic texture. Ifmis turned on, the spin degeneracy is lifted and the electron spins tend to align locally parallel or antiparallel withthe magnetic texture. At m≈5tthe spin alignment is almost complete and two blocks with n b(number of sites forming a SkX unit cell) bands each are formed: one for parallel (higherenergies) and one for antiparallel alignment (lower energies);see Fig. 2(a). In the limit m→∞ the alignment is perfect and the electron spins follow the skyrmion texture adiabatically. Bothblocks are identical but shifted in energy. Roughly speaking,besides the rigid shift by ±m, the nontrivial Zeeman term leads to a “condensation” of bands [identified as Landau levels(LLs) in what follows]. The perfect alignment for m→∞ motivates the transfor- mation of the Hamiltonian ( 2): A local spin rotation diagonal- izes the Zeeman term [ 29–31,52] and alters the hopping term (the hopping strengths t ijbecome complex 2 ×2 matrices). Since the system can be viewed as consisting of two (uncou-pled) spin species, it is sufficient to consider only one species.The resulting Hamiltonian describes a spin-polarized versionof the quantum Hall effect (QHE). Since we discuss chargeconductivities, the diagonal Zeeman term is dropped and wearrive at the Hamiltonian H /bardbl=/summationdisplay ijt(eff) ij˜c† i˜cj (11) of a quantum Hall (QH) system (spinless electrons on a lattice) [ 53–59]. The effective hopping strengths t(eff) ij de- scribe the coupling of the electron charges with a collinearinhomogeneous magnetic field, B (z) em(r)∝nSk(r). (12) This emergent field [ 17,18] is given by the spin chirality ( 1), that is, the real-space Berry curvature in the continuous limit[17,18]. The parallel (antiparallel) alignment of the electron spins, corresponding to the upper (lower) block in the bandstructure for m→∞ , manifests itself in the sign of the nonzero average of B em. For finite m, the mapping of the THE onto the QHE—and the one-to-one identification of bands and LLs—is reasonableas long as the band blocks are separated, i.e., for m/greaterorequalslant4t[60]. 060406-2MAGNETOELECTRIC EFFECT AND ORBITAL … PHYSICAL REVIEW B 99, 060406(R) (2019) FIG. 2. Properties of a skyrmion crystal. Parameters read nb=36 (sites in the skyrmion unit cell), coupling m=5t.AB l o c h( NSk=+1, γ=π/2; topological charge and helicity), a Néel ( NSk=+1,γ=0), and an antiskyrmion ( NSk=−1,γ=0) are compared. (a) Band structure, (b) TH conductivity σxy, (c) orbital magnetization Mz, (d) magnetoelectric polarizability αxy, and (e) spin toroidization Tzare separated into blocks in which the electron spins are aligned parallel [red in (a)] or antiparallel (blue) with the skyrmion magnetic texture ( σ0≡e2/h, M0≡te/¯h,α0≡gμBe/at,a n d T0≡gμB/a;ais the lattice constant). The band structure is identical for all skyrmion types. In (d) results for an intermediate skyrmion with γ=π/6 are shown in addition (green). Colored dots refer to Fig. 3. (f)αxyin the strong-coupling limit m=900t and (g) for larger skyrmions nb=48 on a different lattice (triangular). The LL character of the bands arises in Chern numbers and in the TH conductivity [Fig. 2(b)]. Bands of the upper (lower) block carry Chern numbers of +1(−1) due to the positive (negative) average emergent field. As a result, theTH conductivity is quantized in steps of e 2/h.A tt h es h i f t e d van Hove singularity (VHS) EVHS=±m, the TH conductivity changes sign in a narrow energy window. The quantization and the sign change are closely related to the zero-field band structure [ 30,31,61]. At the VHS the char- acter of the Fermi lines changes from electron- to holelike.The bands close to the VHS are simultaneously formed fromelectron- and holelike states, leading to a large Chern numberthat causes this jump. Having sketched the influences of the zero-field band structure on the THE, we derive consequences for the orbitalmagnetization. Orbital magnetization. The block separation manifests it- self in the orbital magnetization ( 8) as well. Its energy depen- dence within the lower block is similar to that in the upper onebut with opposite sign [Fig. 2(c)]; the latter is explained by the alignment of the electron spin with the magnetic texture. M z(EF) shows rapid oscillations with zero crossings within the band gaps, which is explicated as follows. The emergentfield leads to a rotation of an electron wave packet aroundits center of mass. The first term in Eq. ( 8), given by m n(k), changes continuously within the bands but is constant withinthe band gaps. In contrast, the phase-space correction due tothe Berry curvature [ 39] (second term) varies continuously in energy. Its slope in the band gaps is determined by the THconductivity, ∂ ∂EFMz(EF)=1 2e[σyx(EF)−σxy(EF)]. (13)Both gauge-invariant contributions are similar in absolute value but differ in sign. Consequently, their small differenceleads to one oscillation per band. For a better understanding we relate the orbital magnetiza- tion in a SkX to that of the associated QH system with (al-most) dispersionless bands [ 62–64]. Besides the oscillations we identify a continuous envelope function [ 62] (Fig. S1 in the Supplemental Material [ 65]). In the SkX this envelope is “deformed” due to the inhomogeneity of the emergent field.Nevertheless, the spectrum of the QH system is quite similarto that of the SkX. The influences of the two terms in Eq. ( 8) show up “undis- torted” in the QH system. The orbital magnetic moment perband (entering the first term) decreases (increases) stepwise atenergies below (above) the shifted zero-field VHSs at E VHS≡ ±m[cf. Fig. S1(c)]. There is no sign change at the VHSs, in contrast to the TH conductivity. Still, a zero-field explanationholds as Mis also based on the k-space Berry curvature. At energies below a VHS, LLs are formed from electronlikeorbits with a fixed common circular direction. At energiesabove a VHS, holelike orbits are formed in addition. Sincethese exhibit the opposite circular direction, they contributewith opposite sign. Both contributions result in an extremumat the VHS. The size and shape of the orbits dictate the magnitude of the contributions of each band. Therefore, on one hand, the os-cillation amplitudes in Figs. 2(c)and S1(b) [corresponding to the step heights in Fig. S1(c)] increase with increasing energydistance of the Fermi energy and band edges. On the otherhand, the oscillation amplitudes vanish at the VHS. Recall thatthe Fermi lines have zero curvature at this particular energy. When exchanging skyrmions with antiskyrmions the sign of the emergent field changes and so does the sign of both theTH conductivity and the orbital magnetization in Figs. 2(b) and 2(c), as both characterize the THE. These quantities 060406-3GÖBEL, MOOK, HENK, AND MERTIG PHYSICAL REVIEW B 99, 060406(R) (2019) distinguish skyrmions from antiskyrmions but cannot distin- guish Bloch and Néel skyrmions. Magnetoelectric polarizability. The independence of all above quantities on the skyrmions’ helicity calls for furthercharacterization: This is met by the magnetoelectric effect de-scribed by magnetoelectric polarizability and spin toroidiza-tion. Both quantities are derived from the mixed Berry cur- vature D (ij) n. If the Fermi energy lies between two Landau levels, the system is insulating. In this case, the transversemagnetoelectric polarizability α xy=∂My ∂Ex/vextendsingle/vextendsingle/vextendsingle/vextendsingle B=0=∂Px ∂By/vextendsingle/vextendsingle/vextendsingle/vextendsingle E=0(14) quantifies the magnetoelectric coupling [ 46] to in-plane fields that are applied to a sample in the SkX phase: An in-plane magnetization M(polarization P) can be modified by an orthogonal in-plane electric field E(magnetic field B[66]). If the Fermi energy lies within a Landau level, the systemis metallic and cannot exhibit a polarization. Nevertheless, an in-plane magnetization can be produced by perpendicular in-plane currents that are brought about by an applied electricfield. This so-called magnetoelectric effect in metals is equiva-lent to an intrinsic Edelstein effect [ 67] and was predicted [ 48] and confirmed experimentally for UNi 4B[68], which shows a coplanar toroidal order. The Onsager reciprocal effect is the inverse Edelstein effect: the generation of a current via theinjection of a nonequilibrium spin polarization. For a Bloch SkX, the spectrum of the magnetoelectric polarizability α xy(EF), Eq. ( 6b), shows a sign reversal of the two separated blocks [Fig. 2(d)]. Although αxyexhibits plateaus, it is not quantized. Around the VHS the curve showsa sharp peak (circle). Form/greatermuchtthe spectrum of each block becomes symmetric [Fig. 2(f)]. Within a block the sign of α xymostly remains, in contrast to σxy. The monotonicity, however, is reversed above the VHS, the reason being the exchange of vlnandslnin Eqs. ( 5a) and ( 5b). While the sign of the velocity is given by the electron or hole character, the spin is aligned withthe magnetic texture, irrespective of the electronic characterof band l. The mixing of electron and hole states in a small energy window about E VHSleads to a collapse of αxywith a reversed sign for this small energy region. This energywindow corresponds to the jump in σ xy. Spin toroidization. As the magnetoelectric polarizability is related to the TH conductivity, the spin toroidization ( 10)i s related to the orbital magnetization. It comprises two terms:one given by the spin toroidal moments t n, and the other by the phase-space correction due to the mixed Berry curvature.In analogy to Eq. ( 13), its slope ∂ ∂EFTz(EF)=1 2e[αyx(EF)−αxy(EF)] is given by the magnetoelectric polarizability in the band gap [49]. Tz(EF) oscillates rapidly for the Bloch SkX [Fig. 2(e)]. In the strong-coupling limit m/greatermuchtthe shape of the oscillations becomes more pronounced. Relation to skyrmion helicity. Changing continuously the skyrmion helicity, from Bloch to Néel skyrmions, αxy and Tzare reduced by a Fermi-energy- independent factortzSkyrmion tzAntiskyrmion xy EF=−4.59t xy EF=−4.91t xy EF=−5.49t Ne −3/4Bl −1/4Ne 1/4 Bl 3/4Ne−15−10−5051015 −0.3−0.2−0.100.10.20.3 Helicity ()Classical toroidal moment tz(t0) Magnetoelectric polarizability xy(0) FIG. 3. Dependence of the classical toroidal moment tz(blue, red) and the magnetoelectric polarizability αxyon the helicity of a skyrmion for selected Fermi energies EF[distinguished by color, as indicated; also marked in Fig. 2(d)].tzis proportional to αxy, with the proportionality factor depending on EF.t0≡gμBa,α0≡gμBe/at. [Fig. 3and green curve in Fig. 2(d)]; both quantities vanish for Néel SkXs by symmetry. We find that this factor is quantifiedby the classical toroidal moment [ 48] t=gμ B 2/summationdisplay iri×si∝sin(γ)ez (15) (riposition of spin siwith respect to the skyrmion center). t is a pure real-space quantity given by the skyrmion helicity γ (blue line in Fig. 3). This easily accessible quantity success- fully reproduces the functional dependence of αxy(and also of Tz) on the helicity but fails to reproduce the proportionality factor because it does not depend on EF. Being a classical quantity, tcannot explain the shape of αxy(EF) and Tz(EF). For a Bloch skyrmion ( γ=π/2) the full αtensor is anti- symmetric ( αxy=−αyx) and has no longitudinal components. A Néel skyrmion ( γ=0) exhibits only a longitudinal effect αxx=αyyidentical to αxyof the Bloch skyrmion, since all spins are rotated by π/2 around the zaxis [ 69]. For anti- skyrmions Eq. ( 15) always gives zero. This is why the αtensor is symmetric and Tzis zero in this case. Rotation of the sample always allows one to diagonalize the tensor for antiskyrmioncrystals since γmerely orients the two principal axes of an antiskyrmion, for which the texture points into oppositedirections giving opposite longitudinal effects α xx=−αyy. The full tensor of the texture-induced magnetoelectric po- larizability for a structural square lattice reads α(EF)=αBloch xy(EF)/parenleftbigg cos(γ)s i n ( γ) −NSksin(γ)NSkcos(γ)/parenrightbigg . The measurement of all tensor elements allows one to de- termine topological charge NSkand helicity γof an unknown skyrmion. Conclusion. In this Rapid Communication, we established a complete scheme for the characterization of the skyrmioncrystals’ topological charge and helicity (Fig. 1). Our findings on the topological Hall effect and the magnetoelectric effectare explained by quite simple pictures: a quantum Hall 060406-4MAGNETOELECTRIC EFFECT AND ORBITAL … PHYSICAL REVIEW B 99, 060406(R) (2019) system and the classical toroidal moment of a spin texture, respectively. Our prediction of the helicity-dependent magnetoelectric effect allows one to discriminate Néel and Bloch skyrmions,without reverting to real-space imaging of their magnetictexture (which is in particular difficult for skyrmions arising atinterfaces). For an electric field of 10 8V/m an additional in- plane magnetic moment of one-hundredth of gμBis induced per atom [ 70]. The collapse of αxynear van Hove singularities is a significant feature and could establish a new hallmark ofthe SkX phase: It is observable by shifting the Fermi energy(e.g., by a gate voltage or by chemical doping). As shown in Figs. 2(f)and2(g) as well as in the Supple- mental Material [ 65], the main claims of this Rapid Com- munication depend qualitatively neither on skyrmion size,strength of the exchange interaction, nor on the lattice ge-ometry. The established scheme for discrimination (Fig. 1) is a general result, which is not limited to specific materi-als. All presented quantities arise solely due to coupling of“spinful” electrons with the skyrmion texture and vanish in the absence of skyrmions. The skyrmion-induced contributions are distinguishable from the corresponding nonskyrmionic counterparts, e.g., the anomalous Hall effect in the pres-ence of spin-orbit coupling, the “conventional” magnetization,and the “conventional” magnetoelectric effect in multiferroicmaterials [ 14,71–73]). An experimental proof of the predicted magnetoelectric effect can be done simplest for a nonmultiferroic materialwith a crystal symmetry that allows only for Bloch skyrmions(e.g., MnSi) [ 74]. 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PhysRevB.83.205405.pdf

PHYSICAL REVIEW B 83, 205405 (2011) Surface defects and conduction in polar oxide heterostructures N. C. Bristowe,1,2P. B. Littlewood,1and Emilio Artacho2,3 1Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom 2Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom 3Donostia International Physics Centre, Universidad del Pais Vasco, E-20080 San Sebastian, Spain (Received 27 August 2010; revised manuscript received 28 March 2011; published 17 May 2011) The polar interface between LaAlO 3and SrTiO 3has shown promise as a field effect transistor, with reduced (nanoscale) feature sizes and potentially added functionality over conventional semiconductor systems. However,the mobility of the interfacial two-dimensional electron gas (2DEG) is lower than desirable. Therefore to progress,the highly debated origin of the 2DEG must be understood. Here we present a case for surface redox reactions asthe origin of the 2DEG, in particular surface O vacancies, using a model supported by first-principles calculationsthat describes the redox formation. In agreement with recent spectroscopic and transport measurements, wepredict a stabilization of such redox processes (and hence Ti 3 doccupation) with film thickness beyond a critical value, which can be smaller than the critical thickness for 2D electronic conduction, since the surface defectsgenerate trapping potentials that will affect the interface electron mobility. Several other recent experimentalresults, such as lack of core-level broadening and shifts, find a natural explanation. Pristine systems will likelyrequire changed growth conditions or modified materials with a higher vacancy free energy. DOI: 10.1103/PhysRevB.83.205405 PACS number(s): 73 .20.Hb, 73 .40.Lq, 73 .50.Bk I. INTRODUCTION Complex oxides offer the potential to replace conventional semiconductors in a range of devices due to reduced featuresizes and added functionality (see, e.g., Refs. 1and 2). The polar interface between LaAlO 3(LAO) and SrTiO 3(STO) (Ref. 3) has shown promise as a field effect transistor.4,5One problem hindering its development is the low mobility of theinterface two-dimensional electron gas (2DEG). To progress,the origin of the 2DEG must be understood. Explanationsproposed to date can be classed into three categories: (i) elec-tron transfer countering the “polar catastrophe,” 6(ii) doping through O vacancies in the film5,7–9or in the substrate,10–12 and (iii) cation intermixing at the interface.6,13,14Growth conditions have been shown to affect the observed behavior,particularly O 2partial pressure and post-annealing treatments, affecting carrier density and confinement to the interface (asopposed to conduction through the STO substrate). 4,11 The polar catastrophe6arises from a polarization dis- continuity between the nonpolar STO substrate and polarLAO film. 15In LAO films grown on TiO 2terminated STO substrates this polar discontinuity induces a divergence of the electric displacement field ( /vector∇·/vectorD) in the pristine system equal to considering the effective polarization response andnet effective charges σ cof precisely +e/2 and −e/2 per formula unit at the interface and surface, respectively.15–17 The electrostatic potential then builds up across the LAO film, accumulating electrostatic energy. To counter it a chargetransfer between the interface and the surface is required. Amechanism intrinsic to the pristine system is given by thetransfer of electrons from the surface to the interface. Fromband bending arguments, electrons are transferred from the O2pat the top of the valence band at the LAO surface to the Ti 3dconduction band at the interface, once the potential drop across the LAO layer reaches the effective band gap, whichis calculated to happen for a LAO thickness of five unit cells(see Ref. 18and references within). However, the absence ofa 2D hole gas at the surface and the observation of populated Ti 3dstates for films as thin as one or two bilayers 19–21raises doubts about this mechanism. Two important points should be kept in mind. First, the samples are generally not kept or measured in vacuum. The surface can thus be covered by adsorbants like water, and it will likely be far from an ideal surface termination, possiblyincluding chemical alterations such as the hydroxylation seenon many wet oxides. The second point is more central,however: whatever the chemistry, the relevant electrostaticsacross the film can only be affected by processes in whichcharge is altered at each side of the film, either by charge transport across it, or by charges arriving at either side from external reservoirs. If there are no such external sourcesand the chemical processes are confined to the surface, theremaining possibility is that of surface redox processes. Theytransform surface bound charge into free-carrier charge, theelectrons or holes then being free to move to the buriedinterface. The clearest and quite relevant example is that of oxygen vacancy formation whereby surface O 2−anions transform into O 2molecules, releasing two electrons to the n interface, as illustrated in Fig. 1.5,7–9Surface protonation22,23 is an analogous process in which the surface is also reduced by the oxidation of O2−into O 2, although in this case the process depends on the presence of water, H 2O→1/2 O2+2e−+2H+, and both the energetics and kinetics will be different from the previous one.22The protons attach to the surface O atoms, while the electrons are again free to go totheninterface (note that the non-redox hydroxylation, H 2O→ OH−+H+, does not affect the electrostatics across the film). In Sec. IIwe present a model for the formation of surface redox processes. We then apply the model to theLAO/STO system in Sec. IIIusing parameters obtained from first-principles calculations (see the Appendix). We show that,for such a surface redox process, (i) the process is favored byfilm thickness through the effect of the electric field across 205405-1 1098-0121/2011/83(20)/205405(6) ©2011 American Physical SocietyN. C. BRISTOWE, P. B. LITTLEWOOD, AND EMILIO ARTACHO PHYSICAL REVIEW B 83, 205405 (2011) the film; (ii) the growth of the density of the related surface defects with film thickness is predicted, showing a minimumcritical thickness; (iii) the model is in excellent agreement withfirst-principles calculations and reasonable agreement withwhat is observed experimentally for Ti 3 doccupation; (iv) the potential drop across the film does not change substantiallywith thickness once vacancies start to appear; (v) for thinfilms the carriers at the interface are still trapped by theelectrostatic potential generated by the vacancies in deepdouble-donor states: a strongly disordered two-dimensionalelectron system; (vi) as thickness increases the levels becomeshallower and the number density increases, closing the gapto the conduction band; (vii) the conduction onset is thereforeat a higher film thickness than the one for interfacial carrierpopulation. Finally we note that the model can also be appliedtop-type interfaces and ultra-thin ferroelectric films, where alteration of the surface chemistry has also been proposed asa possible screening mechanism (see, for example, Ref. 24). II. MODEL We consider a pristine polar thin film on a nonpolar substrate defining a n-type interface (see Fig. 1), and a surface reduction process (the model can be trivially extended top-type interfaces and surface oxidizing reactions, see below, and to ferroelectric thin films). The introduction of a surfacedefect via a redox reaction produces a donor level in the gap.The defect provides Zelectrons (two for an O vacancy) that can transfer to the interface (see Fig. 1) thus contributing to the screening of the polarization or compositional charge σ cat either side of the polar film ( ±0.5e/f.u. in LAO15–17). Considering this electron migration, we can model the formation energy of one such surface defect, Ef,i nt h e presence of an area density nof surface defects as Ef(n)=C+EE(n)+αn, (1) where we have separated an electrostatic term associated with the internal electric field in the film, EE(n), from a surface/interface chemistry term Cand a term accounting for defect-defect interactions other than electrostatic in amean-field sense. It can be seen as arising from E f(n)=E0 f−Z(W−ECD)+EE(n)+others/summationdisplay iJiri, (2) where E0 fis the formation energy of an isolated surface defect in the absence of a field across the film, and where Wand ECDare defined in Fig. 1. [A valence-band offset has been omitted for clarity since it is known to be small, but canalso be included alongside the conduction-band offset withoutaffecting the model.] E 0 f(and thus C) depends on the particular surface chemical process, and the reference chemical potentialfor the relevant redox counterpart species in the environment,e.g.,μ O2(P,T ), which depends on experimental conditions. Taking Eq. ( 1), the surface excess energy for a given area density of surface defects is then /Omega1(n)=/integraldisplayn 0Ef(n/prime)dn/prime=Cn+/Omega1E(n)+1 2αn2. (3)FIG. 1. (Color online) Schematic band diagram of an interface between a polar film and a nonpolar substrate along the normaldirection z. (a) The pristine system under the critical film thickness. (b) The creation of a donor state at the surface via a redox reaction and subsequent electron transfer. (c) The reconstruction reduces the film’s electric field. The key of the proposed energy decomposition is that the /Omega1E term is simply the energy gain of partly discharging a capacitor, which is /Omega1E(n)=d 2/epsilon1/bracketleftbig (σc−σv)2−σ2 c/bracketrightbig , (4) where dis the film thickness, /epsilon1is the LAO dielectric constant, σcis the compositional charge, and σv=n(Ze) is the charge density of the carriers confined to the interface (note thatthese electrons may not all be mobile, as discussed below).Equation ( 4) assumes no screening by electronic recon- struction, which is right if the onset for defect stabilizationhappens earlier than the one for electronic reconstruction. Acomplete description of all possible regimes will be presentedelsewhere. 25We limit ourselves to the regime given by Eq. ( 4) since a wider discussion of the model is irrelevant here. The equilibrium defect density is determined by finding the value that minimizes /Omega1. Taking Eqs. ( 3) and ( 4), n=dZ eσ c−C/epsilon1 (Ze)2d+α/epsilon1. (5) A critical thickness arises for defect stabilization, dc=C/epsilon1/(Zeσc), (6) 205405-2SURFACE DEFECTS AND CONDUCTION IN POLAR OXIDE ... PHYSICAL REVIEW B 83, 205405 (2011) ntending to σc/Ze for large d, which is the value required to completely screen the film’s intrinsic polarization. III. DISCUSSION A. Interface carrier density We now consider the specific case of O vacancy formation at the surface as the most prominent candidate redox process.5,7,9 Figure 2(a) shows quantitative agreement between the model’s ¯Ef=1 n/Omega1(n)=1 n/integraltextn 0Efdn/primeand first-principles calculations of the surface vacancy formation energies in Ref. 7on the full LAO/STO structure. ¯Efis the right magnitude to compare to first-principles results since it accounts for the energydifference between the system with a given concentrationof surface defects ( n) and the pristine system ( n=0), per surface defect, thus /Omega1(n)/n. The physical constants used in the model were determined independently by separate density-functional theory (DFT) calculations (see the Appendix) andthen compared to DFT results for films of varying thickness[Fig. 2(a)]. The predicted behavior of n(andσ v)i nL A O / S T O is shown in Fig. 2(b), where it is compared with Ti 3 d occupation (both trapped and mobile) as measured with hardx-ray photoelectron spectroscopy (HAXPES). 19Two bands are plotted: the colored one uses a range in C, the striped one a range in αand/epsilon1(see the Appendix). The bulk dielectric constant of LAO is about 24.26It may be substantially different for a strained ultra-thin film (see Ref. 18and references within) and so the range 21 </epsilon1< 46 has been considered. The agreement in Fig. 2(b) between model and experiment is only qualitative given the ambiguities in some of the magni-tudes of key parameters defining the problem, most notablythe chemical potential of O 2in experimental conditions. Despite this, the model predicts a critical thickness forthe appearance of carriers at the interface for a LAO filmthickness below the five unit cells predicted by the purelyelectronic mechanism. Other qualitative features observed butnot understood in this system also find a natural explanation(below). B. Electric field in LAO: Pinning of potential drop The electrostatic potential drop across the LAO film is V=(σc−σv)d//epsilon1. (7) Substituting σv, the drop is essentially independent of thick- ness,V≈C/(Ze), when the vacancy-vacancy interaction is small, α/lessmuch(Ze)2d//epsilon1. Using the parameters for LAO/STO the difference in potential drop per LAO layer added isbetween 0.0 and 0.2 eV/f.u., much smaller than the predictedthickness dependence for the electronic screening model,and consistent with recent x-ray photoemission spectroscopy(XPS) measurements, which show no core-level broadeningwith film thickness. 27The pinning of Vis also consistent with reduced cation-anion relative displacement with increasingLAO thickness as measured by SXRD. 28 C. Onset of conduction: Electron trapping The redox processes proposed above explain the absence of hole-mediated transport at the surface, while electrons allow FIG. 2. (Color online) (a) Defect formation energy ¯Ef(see definition in text) versus LAO film thickness dfor various vacancy densities n. The model (lines) is compared with the DFT calculations (circles) of Ref. 7of the surface vacancy formation energy on the full LAO/STO structure (see the Appendix for the determination of the model parameters). (b) Equilibrium area density of interface carriers σvversus d. The red (gray) band is the model prediction for 2.1 eV <C< 5.0e V ,/epsilon1=25, and α=0.8e V/(vac/f.u.)2. The striped band is forC=3.6e V ,2 1 </epsilon1< 46, and 0 <α< 8e V/(vac/f.u.)2.T h e circles indicate the Ti 3 doccupation as measured with HAXPES in Ref. 19. Open circle indicates the sample was not annealed. The crosses indicate the carrier density from Hall measurements in Ref. 4. 2D conduction at the interface. An important observation that remains unexplained, however, is the fact that the onset ofinterfacial Ti 3 doccupation, as measured with HAXPES , 19–21 happens at lower film thickness than the onset for interfacial 2D conduction.4It has been suggested that the 2DEG lies in several Ti 3 dsubbands, some of which are not mobile due to Anderson localization.29Whether Anderson localization occurs on an energy scale as high as room temperaturedepends on the energy scale of the disorder distribution. Thesurface defects associated to the redox processes representpoint sources of effective charge, very much as a dopant ina semiconductor, 17e.g.,+2efor an O vacancy. They then generate trapping potentials for the carriers at the interface plane of the form Vtrap=Ze2//epsilon1/radicalbig ρ2+d2,in atomic units, where dis the film thickness and ρ2=x2+y2corresponds to the radial variable in the plane. This potential is sketchedin Fig. 3for several dvalues. Its depth decays with thickness as 1/d. Figure 3shows estimates of the ground-state electron level associated to the double donor state arising at the interfacedue to an O vacancy at the surface. These trapped interface 205405-3N. C. BRISTOWE, P. B. LITTLEWOOD, AND EMILIO ARTACHO PHYSICAL REVIEW B 83, 205405 (2011) FIG. 3. (Color online) Trapping potential Vcreated by a surface O vacancy as seen by interface electrons versus distance within the interface plane x(x=0 is directly below the vacancy) for film thickness d=a(deepest), 2 a,3a,4a(shallowest), with harmonic estimates of corresponding donor ground states [taking meff=3me (Ref. 32)]. Inset: Sketch of range and density of trapped states. levels may be the “in-gap states” seen in a recent spectroscopic study.30For a thin film the traps are deep and few, but as it grows thicker, the donor states become shallower andthe area density of traps grows, as illustrated in the inset of Fig. 3. A transition from insulating to conducting behavior is thusexpected at a larger film thickness than the critical thicknessfor surface defect stabilization. With growing thickness, notonly do dopant levels tend to overlap as in a degeneratesemiconductor, but the doping-level band is pushed toward theconduction band. For Z=2, as for O vacancies, the physics of this transition is that of band overlap and disorder, since alldopant states are doubly occupied. If the mechanism involvesZ=1 defects, as in the hydroxylation case, the transition will be rather Mott-Anderson, as each dopant state is singlyoccupied. The different phenomenologies could be used toascertain on the mechanism. The surface potential distributionfrom charged defects is consistent with a recent Kelvin probeforce microscopy study. 31 IV . FINAL REMARKS AND CONCLUSIONS A recent atomic-force microscopy (AFM) study of LAO/STO has proposed the mechanism for conductivityswitching 4,5as the writing of surface charge.33Applying a biased tip to the surface alters the field across the LAOfilm, which either increases or decreases the stability ofvacancies (and hence σ v) depending on the sign of the bias. An implication of this observation is that the kinetics for theseredox processes is accessible at room temperature as used inthese experiments, not only the much higher Tused for growth. The model proposed can also be used for pinterfaces, holes, acceptor levels, and surface oxidation processes. Thiswould be the case for LAO grown on SrO terminated STO. Itis less symmetric than it seems, however, since, in additionto different energetics and chemical potentials, the largeconduction-band offset at the interface ( Win Fig. 1) favors the situation for electrons toward ninterfaces much more than the much smaller valence-band offset for holes and pinterfaces. For thin-film ferroelectrics with outward (inward) polarizationon metallic substrates, the important alignment becomes theacceptor (donor) level with the metal Fermi level. This could bebehind the stability of switchable ultrathin ferroelectric films under open circuit conditions (see, e.g., Refs. 34and24). We conclude that in LAO/STO, the onset of electrostatic modulation doping is precluded by the thermodynamic cre-ation of surface defects and thus carrier mobilities producedby this method will be much lower than at a pristine interface.Intrinsic systems will likely require changed growth conditionsor modified materials with a higher vacancy free energy. ACKNOWLEDGMENTS We acknowledge H. Hwang, M. Pruneda, and M. Stengel for valuable discussions, the support of EPSRC and thecomputing resources of CamGRID at Cambridge Universityand the Spanish Supercomputer Network (RES). APPENDIX: MODEL PARAMETERS Here we describe the determination of the parameters, α,/epsilon1, and Cused in the model in Fig. 2of the main paper. The parameters used in the model were independentlydetermined from first principles in appropriate LAO-basedsystems (see below), and then the model was checked againstDFT calculations of vacancy formation energies in the fullLAO/STO system as a function of thickness [Fig. 2(a)]. When comparing the model with experiment [Fig. 2(b)], we account for inaccuracies of DFT and the ambiguity of the experimentalchemical potential in the determination of each of theseparameters. A. Vacancy-vacancy interaction term α The defect-defect term was defined in Eq. ( 1) to include interactions other than electrostatic. Therefore to determine α we performed first-principles calculations (see Ref. 16for the method) of the charge neutral defect, i.e., oxygen vacancies in“bulk” LAO, which include the double donor electrons. Oneoxygen vacancy was placed in a simulation cell of 1 ×1×8, 2×2×8, and 3 ×3×8 unit cells of LAO to approximate 2D arrays of vacancies of various area densities. From thedifference in formation energy per vacancy between the threecalculations, αwas found to be 0 .8e V/(vac/f.u.) 2, which was used in Fig. 2(a).αis formally defined as the interaction between vacancies at the film surface, however, we believethis bulk value to be a good estimate. For the comparisonwith experiment, to account for any error associated with thisdetermination we choose the range 0 <α< 8e V/(vac/f.u.) 2 in Fig. 2(b) of the paper. B. Dielectric constant /epsilon1 The dielectric constant /epsilon1consists of lattice and electronic contributions. For LAO, we take /epsilon1=28 as the total for Fig. 2(a), as is consistent with Ref. 35. When comparing with experiment, we note the error and inconstancy of DFTcalculations of /epsilon1, and additionally the effect of strain as highlighted in Ref. 18. Due to this we choose the range 21</epsilon1< 46 for Fig. 2(b). 205405-4SURFACE DEFECTS AND CONDUCTION IN POLAR OXIDE ... PHYSICAL REVIEW B 83, 205405 (2011) C. Surface/interface chemistry term C Cconsists of three terms: C=E0 f,μ=0+μ+2(ECV−W). (A1) From the electronic structure presented by Li et al. ,t h e last term is found to be approximately −1.2 eV . From first- principles calculations of Ref. 36the formation energy of an isolated oxygen vacancy at the surface of LAO (in the absenceof a field) with reference to oxygen in an isolated molecule(1/2E[O 2]),E0 f,μ=0, is approximately 6.0 eV . We define the zero of chemical potential relative to this reference state, whichis appropriate for the DFT comparison and hence the value ofCused in Fig. 2(a) is taken as 4.8 eV . The DFT underestimation of the band gap requires correc- tions to both E 0 f,μ=0(see Ref. 37) andWfor the comparison with experiment in Fig. 2(b). At this point we note the difficulties and variation in first-principles determination offormation energies of donor/acceptor states (see, for example,Ref. 38). From Ref. 37, the formation energy correction of a donor defect, required due to DFT band-gap underestimation, issimply /Delta1E 0 f=Z/Delta1E c, (A2) where /Delta1Ecis the change in conduction-band edge between LDA and experiment (or corrected DFT). By comparing theelectronic structure presented in Li et al. and Ref. 39,t h i s correction could be as large as 1.0 eV . Therefore we take6.0e V<E 0 f,μ=0<7.0e V . From experimental band alignment40and theoretical calcu- lations determining the gap states,39the third term in Eq. ( A1) is approximately 2.0 eV (not 1.2 eV). Correcting for these DFTerrors we take 4.0e V+μ<C< 5.0e V+μ (A3) The chemical potential of oxygen in the growth conditions used in Ref. 19(T=1073 K and p=2.0×10 −8atm) relative to the zero reference defined above is calculated to be −1.9e V assuming the environment acts as an ideal gas-like reservoir.The effect of post-annealing and cooling to room temperatureand pressure is to shift the chemical potential toward zero.With these limits on the chemical potential and the inequalityin Eq. ( A3), the range of Cbecomes 2 .1e V<C< 5.0e V , as used in Fig. 2(b). 1J. Mannhart and D. Schlom, Science 327, 1607 (2010). 2H. Takagi and H. Hwang, Science 327, 1601 (2010). 3A. Ohtomo and H. Hwang, Nature (London) 427, 423 (2004). 4S. Thiel, G. Hammerl, A. Schmehl, C. W. Schneider, and J. Mannhart, Science 313, 1942 (2006). 5C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E. Andersen, C. S. Hellberg, J. Mannhart, and J. Levy, Nat. Mater. 7, 298 (2008). 6N. Nakagawa, H. Hwang, and D. Muller, Nat. 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PhysRevB.72.075118.pdf

Double-exchange model for noninteracting electron spins coupled to a lattice of classical spins: Phase diagram at zero temperature David Pekker,1Swagatam Mukhopadhyay,1Nandini Trivedi,2and Paul M. Goldbart1 1Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801-3080, USA 2Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, Ohio 43210, USA /H20849Received 25 February 2005; revised manuscript received 23 May 2005; published 11 August 2005 /H20850 The analytical zero-temperature phase diagram of the double exchange model for classical background spins as a function of the carrier density and Hund’s coupling in the entire range of these parameters is presented. Byconstructing a continuum field theory we explore the possibility of a continuous phase transition from ferro-magnetic state to a gently varying textured state. We find such a transition in and below two dimensions andshow that the emerging stable state is a spin-spiral which survives the tendency towards phase separation intocommonly considered phases, and is also energetically favored to the canted state, for low carrier density. DOI: 10.1103/PhysRevB.72.075118 PACS number /H20849s/H20850: 75.30.Et, 75.47.Lx I. INTRODUCTION The double-exchange model /H20849DEM /H20850describes the motion of noninteracting itinerant electrons through a lattice of clas-sical spins to which the electron spins are coupled. Thismodel is relevant, e.g., to the manganites, which show colos-sal magnetoresistance. Although electron-phonon interac-tions play an important role in the manganites through aJahn-Teller effect, 1some of the key features of the strongly coupled spin, charge, and lattice degrees of freedom are cap-tured by a DEM with lattice-distortion effects. Another classof materials for which the DEM is relevant are the dilutemagnetic semiconductors, such a GaAs doped with Mn,which are important for spintronics applications. 2In these materials, the S=5/2 local moments on the Mn sites are exchange coupled to the hole carriers generated by substitut-ing Mn for Ga. While a more realistic model would includethe effects of disorder 3,4arising from the random substitution of the Mn atoms, as well as spin-orbit coupling effects,5,6a careful study of the DEM is a necessary first step. Thus, the DEM is a paradigm for a wide class of materials that show a strong coupling between the charge and spindegrees of freedom. Pioneering work on this model focusedon the ferromagnetic and canted antiferromagnetic phases atnonzero temperature. 7Over the past few years, the DEM has received renewed attention, stimulated in part by the numeri-cal studies reported in Refs. 8 and 9, where the zero-temperature phase diagram of the simple DEM has been ex-plored. More elaborate extensions of the DEM—intended tomove it closer to real systems by augmenting it with physicalprocesses such as the super-exchange and Coulomb interac-tions, disorder, and Jahn-Teller distortions, etc.—have alsobeen investigated over the past few years. For example, thestability of a spin spiral state in one extension of the modelwas examined in Ref. 10 and was contrasted with the stabil-ity of the canted state. The ground state of a model aug-mented with Coulomb interactions and large Hund couplingwas addressed in Ref. 11. The instability of the homogeneouscanted state with respect to phase separation, for large Hundcoupling in a model that includes tunable super-exchangeinteraction, was studied in Ref. 12. The phase diagram of athree-dimensional model in the infinitely large Hund- coupling limit was studied in Ref. 13, and near the Curietemperature an instability with respect to spontaneous trans-lational symmetry breaking was proposed, as was the possi- bility of phase separation. Recently, Ref. 14 addressed thestability of the spin spiral state as the ground state of theDEM in the large- Slimit. In spite of the considerable amount of theoretical and nu- merical work on the DEM, the continuous phase transitionfrom the ferromagnetic phase to a spin-textured phase re-mains incompletely understood. Considerable attention hasbeen paid to the model in the infinite Hund-coupling regime,motivated by the fact that in the manganites, the Hund cou-pling is large, compared with the electronic bandwidth. Thezero-temperature phase diagram has been recently studied inRef. 15 within dynamical mean field theory /H20849DMFT /H20850for the entire range of carrier concentrations and Hund coupling. Itwas shown that the stable ground state in different regions ofparameter space is either a ferromagnet, or a commensurateantiferromagnet, or some incommensurate phase with an in-termediate wave vector. Moreover, a second-order phasetransition /H20849from the ferromagnetic to the incommensurate phase /H20850and a first-order transition /H20849from the antiferromag- netic phase to a region of phase separation /H20850were identified. In this paper, we address the magnetic ordering of the single-band DEM with classical background spins at zerotemperature /H20849ignoring orbital or charge ordering /H20850. In contrast to previous work, which involved either a numericalapproach 8or the DMFA,15,16we use a continuum field theory and a gradient expansion to determine the critical line ofcontinuous phase transitions separating the ferromagneticand textured phases in the parameter space of electron den-sity and Hund coupling. This approach allows us to considerall possible long wavelength textures that can emerge from the DEM. We argue that the spin spiral is the energeticallyfavored one from amongst this class of textures. As is wellknown, the ground state of the DEM is phase separated atlarge values of the Hund coupling. However, by explicitlycomparing the energetics of phase separation for commonlyconsidered textures, we argue that there is a region of phasespace in which the continuous transition survives the ten-PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 1098-0121/2005/72 /H208497/H20850/075118 /H208499/H20850/$23.00 ©2005 The American Physical Society 075118-1dency towards phase separation. We also show that the spin spiral state is favored, energetically, over the canted state,thereby making precise the nature of the emergent “incom-mensurate state” found in earlier works. 8,15,16We present the phase diagram of the model, after including the commonlyconsidered candidate phases for a preemptive phase separa-tion at high Hund coupling. Our phase diagram looks quali-tativly identical to the numerical and DMFA diagrams. 8,15,16 The one-dimensional DEM with quantum background spins/H20849S=1/2 /H20850was analyzed via DMRG, and was also found to have a similar, but somewhat more complicated, phase dia- gram, which also includes a spin spiral phase. 17In systems for which the Hund coupling is small /H20849e.g., the cobaltates, diluted magnetic semiconductors, etc. /H20850and the disorder is low, the approach used here may be of relevance. The present paper is organized as follows. In Sec. II a continuum version of the DEM is derived. The symmetries of this continuum version are considered and the electronicdegrees of freedom are integrated out, yielding an effectiveHamiltonian for the background spins within a gradient ex-pansion. A line of continuous phase transitions is determined,and it is shown that the spin-spiral state is the emergentstable state. In Sec. III the issue of whether the phase tran-sition from the ferromagnetic states to the spin spiral state ispreempted by another transition /H20849to either an antiferro/ ferrmagnetic microphase-separated state or to a canted ferro-magnetic state /H20850is considered. It is shown that there is a re- gion of the phase diagram in which the spin spiral state is astable state. II. DOUBLE-EXCHANGE MODEL; ANALYTICAL STRATEGY The double-exchange model describes noninteracting itin- erant electrons moving on a lattice of static “background”spins whose moments are typically large compared to that ofthe electron spins, and hence may be treated classically. TheHamiltonian is taken to beH=−t /H20858 /H20855i,j/H20856a/H20849cia†cja+ h.c. /H20850−JH S/H20858 jabSj·cja†/H9268abcjb +JAF S2/H20858 /H20855i,j/H20856Si·Sj, /H208492.1/H20850 where the first, second, and third terms respectively describe electronic hopping, double-exchange, and super-exchangecouplings. Here, S iis the background spin at lattice site i, which we approximate as a classical vector, JH/Sis the strength of the ferromagnetic Hund coupling, and JAF/S2is the strength of the antiferromagnetic coupling between thelocalized spins. Aspects of the microscopic origin of thisHamiltonian are discussed, e.g., in Refs. 5, 18, and 19. Weshall explore the J Hvs. electron density phase-diagram, con- sidering JAFto be vanishingly small. However, for technical reasons to be explained later, we shall need JAFto be non- zero. Earlier work has mostly focused on the JAFvs. electron density phase diagram of the model in the regime in whichJ His very large, so that the electrons are aligned with the background spins. It is convenient to transform to a new spin basis at each lattice site, so that the local part of the Hamiltonian /H20849i.e., the part that dominates at large JH/H20850is diagonal. To this end, we rewrite Eq. /H208492.1/H20850in terms of the new operators /H20853di†,di/H20854and /H20853ei†,ei/H20854which are, respectively, the creation and annihilation operators associated with the spin basis aligned and anti- aligned with the background spin direction at site j: /H20873c↑j c↓j/H20874=/H20849/H9253j/H9253j/H11036/H20850/H20873dj ej/H20874, /H208492.2/H20850 where the spinors /H9253jand/H9253j/H11036are defined in terms of the the polar and azimuthal angles of the background spin directionat site j, i.e., /H9258iand/H9278i, and are given by /H9253j=e+i/H9273j/H20898e−i/H9278j/2cos/H9258j 2 e+i/H9278j/2sin/H9258j 2/H20899, /H208492.3a /H20850 FIG. 1. Lowest-order Feynman diagrams /H208491 and 2 /H20850: solid lines represent aligned electron propagators, dashed lines represent anti-aligned electron propagators, curly lines represent /H9004 /H9251, and dotted lines represent /H11509/H9251nˆ·/H11509/H9251nˆ. FIG. 2. Ferromagnetic stiffness in dimension dford=1.5 /H20849dot- ted/H20850,d=2 /H20849solid /H20850, and d=3 /H20849dashed /H20850. Here, JH=1 and JAF=0. Note that for d=2, the stiffness becomes zero at /H9262=JHbut does not become negative.PEKKER et al. PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-2/H9253j/H11036=e−i/H9273j/H20898−e−i/H9278j/2sin/H9258j 2 e+i/H9278j/2cos/H9258j 2/H20899. /H208492.3b /H20850 /H9253and/H9253/H11036form an orthonormal local basis corresponding to the aligned and antialigned spin states. This mapping is de-fined up to a phase factor /H9273j, which is a gauge freedom.21 Note that in the aligned antialigned spin basis, the Hund’sterm is diagonal. The kinetic energy term in this basis is −t/H20858 /H20855i,j/H20856/H20849di†/H9253i*·/H9253jdj+ei†/H9253i/H11036*·/H9253j/H11036ej+di†/H9253i*·/H9253j/H11036ej +ei†/H9253i/H11036*·/H9253jdj+ h.c. /H20850. /H208492.4/H20850 We derive the continuum limit of the Hamiltonian,23defined on a hypercubic lattice, by expanding the kinetic energy termup to second order in gradients of the angles /H9258,/H9278, and/H9273and the electronic operators dande. In the continuum limit, the Hamiltonian is /H20849up to the corresponding gauge transforma- tion /H20850given by H=−/H20885dx/H20875/H9274†/H20849x/H20850/H20873/H20849/H11509/H9251−iA/H9251/H208502+1 4/H20849/H11509/H9252nˆ·/H11509/H9252nˆ/H20850−JH/H20874/H9274/H20849x/H20850 −/H9272†/H20849x/H20850/H20873/H20849/H11509/H9251+iA/H9251/H208502/H9272/H20849x/H20850+1 4/H20849/H11509/H9252nˆ·/H11509/H9252nˆ/H20850+JH/H20874/H9272/H20849x/H20850 +/H20873/H9274†/H20849x/H20850/H9004/H9251/H20849x/H20850/H11509/H9251/H9272/H20849x/H20850+1 2/H9274†/H20849x/H20850/H20849/H11509/H9251/H9004/H9251/H20849x/H20850/H20850/H9272/H20849x/H20850+ h.c./H20874 −JAF/H20849/H11509/H9252nˆ·/H11509/H9252nˆ/H20850/H20876, /H208492.5/H20850 where /H9274/H20849x/H20850and/H9272/H20849x/H20850are, respectively, the field /H20849annihilation /H20850 operators describing locally aligned and antialigned elec- trons, xis the position vector, nˆ/H20849x/H20850is the unit vector along the background spin direction, and Aand/H9004are vector po- tentials originating from a Berry phase and are defined by A/H9251/H20849x/H20850=1 2cos/H9258/H20849x/H20850/H11509/H9251/H9278/H20849x/H20850, /H208492.6a /H20850 /H9004/H9251/H20849x/H20850=−/H11509/H9251/H9258/H20849x/H20850+isin/H9258/H20849x/H20850/H11509/H9251/H9278/H20849x/H20850. /H208492.6b /H20850 Note that /H9004/H9251*/H9004/H9251=/H11509/H9252nˆ·/H11509/H9252nˆ. Also note that in passing to the continuum limit we have retained only the lowest-orderterms in the gradient expansion of the field operators. As aresult, the electron kinetic energy, and hence the band struc-ture that corresponds to it, is of a simple parabolic form.A. Symmetries of the continuum Hamiltonian 1. Local gauge invariance The mapping from a vector /H20849representing a classical spin /H20850 in three-dimensional space to a spinor in SU/H208492/H20850is defined up to an angle /H9273/H20849x/H20850, an overall phase factor, which is a U/H208491/H20850 gauge freedom,21i.e., under the simultaneous transforma- tions A/H20849x/H20850→A/H20849x/H20850+/H11612/H9273/H20849x/H20850/H9004/H20849x/H20850→e2i/H9273/H20849x/H20850/H9004/H20849x/H20850, /H9274/H20849x/H20850→ei/H9273/H20849x/H20850/H9274/H20849x/H20850,/H9278/H20849x/H20850→e−i/H9273/H20849x/H20850/H9278/H20849x/H20850, /H208492.7/H20850 the Hamiltonian is invariant. 2. Global spin rotation invariance The Hamiltonian is also invariant under rotation of all the background spins by a global angle. One can show that theeffect of a global rotation is identical to a gauge transforma-tion. For example, let us consider rotations about the xaxis by a small angle /H9275. The unit vector nˆ/H20849x/H20850transforms to nˆ/H11032/H20849x/H20850 where /H9258/H11032/H20849x/H20850/H11015/H9258/H20849x/H20850−/H9275sin/H9278/H20849x/H20850 and /H9278/H11032/H20849x/H20850/H11015/H9278/H20849x/H20850 −/H9275cot/H9258/H20849x/H20850cos/H9278/H20849x/H20850. Notice that the above transformation leaves /H11509/H9252nˆ·/H11509/H9252nˆinvariant, but the vectors A/H20849x/H20850and/H9004/H20849x/H20850 transform, up to first order in /H9275, in the following manner: A/H9251→A/H9251−/H9275 2/H11509/H9251/H20873cos/H9278 sin/H9258/H20874, /H208492.8a /H20850 FIG. 3. Arrangements of background spins on the 2 /H110032 plaquettes that are considered in the text. FIG. 4. /H20849Color online /H20850Dependence of the ground state of the double exchange model on the carrier density and the Hund cou-pling at zero antiferromagnetic coupling. At low carrier densitiesand high Hund couplings ferromagnetism is favored. In certainother regions, distinct homogeneous states are favored, includingspin spiral /H20849see Fig. 6 /H20850, A- and G-type antiferromagnetism /H20849see Fig. 3/H20850. In yet other regions several types of FM/AFM microphase- separated states are favored. As explained in the text, the boundariesindicated for microphase-separated states are in fact stability limits;the true boundaries must lie within these stability limits. Only inundotted region can one be certain that the spin-spiral state is thefavored state.DOUBLE-EXCHANGE MODEL FOR NONINTERACTING … PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-3/H9004/H9251→exp/H20849−i/H9275cos/H9278/sin/H9258/H20850/H9004/H9251. /H208492.8b /H20850 On identifying the factor − /H9275cos/H9278/2 sin /H9258with the gauge parameter /H9273/H20851see Eq. /H208492.7/H20850/H20852, we see that Eq. /H208492.8a /H20850describes a gauge transformation corresponding to rotations. A similaranalysis can be carried out for rotations about the yaxis; for rotations about the zaxis the transformation is trivial. There- fore, global rotational invariance corresponds to the gaugefreedom in the model. B. Effective Hamiltonian In this section we derive an effective Hamiltonian Heff governing the spatially dependent background spin orienta- tion /H20849texture /H20850, in the limit that the texture varies on length scales much bigger than the inverse Fermi wave vector. Todo this, we integrate out the electronic degrees of freedom,assuming that the zeroth-level description corresponds to asystem in the presence of a spatially uniform texture. Thecontribution due to any inhomogeneity of the texture is thentreated as a perturbation via a gradient expansion. The de-merit of this continuum approach is that background spinconfigurations that change abruptly from one site to another/H20849e.g., canted states or antiferromagnetic state /H20850are excluded from consideration. The effective Hamiltonian that results from this approach is a functional of A/H20849x/H20850,/H9004/H20849x/H20850, and nˆ/H20849x/H20850and their derivatives. Working at fixed chemical potential /H9262, the effective Hamil- tonian is defined as exp/H20849−/H9252Heff/H20851A,/H9004,nˆ,/H9262/H20852/H20850 /H11013/H20885D/H9274D/H9272exp/H20853−/H9252/H20849H/H20851/H9274,/H9272,A,/H9004,nˆ/H20852−/H9262N/H20850/H20854 =/H20885D/H9274D/H9272exp/H20853H0/H20851/H9274,/H9272,/H9262/H20852+H1/H20851/H9274,/H9272,A,/H9004,nˆ/H20852/H20854, /H208492.9/H20850 where N/H11013/H20848dx/H20851/H9274†/H20849x/H20850/H9274/H20849x/H20850+/H9272†/H20849x/H20850/H9272/H20849x/H20850/H20852,H0is the free Fermi gas Hamiltonian, and the perturbation H1is a functional of A,/H9004, and /H11509nˆ, each of which has one spatial derivative /H20851see Eq. /H208492.6a /H20850and /H208492.6b /H20850/H20852, and therefore is small in the sense of our approximation scheme. The form of Heffis constrained by gauge invariance. Thus, keeping allowed terms to quarticorder in gradients, but for now setting J AFto zero, we find the following form, arranged in increasing order:Heff/H20851A/H20849x/H20850,nˆ/H20849x/H20850,/H9004/H20849x/H20850/H20852=/H20885dx/H20851a/H20849/H9262/H20850/H11509/H9251nˆ·/H11509/H9251nˆ +b/H20849/H9262/H20850/H20849/H11509/H9251nˆ·/H11509/H9251nˆ/H208502+c/H20849/H9262/H20850F/H9251/H9252F/H9251/H9252 +d/H20849/H9262/H20850/H20841D/H9251/H9004/H9251/H208412+e/H20849/H9262/H20850/H20841D/H9251/H9004/H9252/H208412/H20852 +¯, /H208492.10 /H20850 where the coefficients a/H20849/H9262/H20850,b/H20849/H9262/H20850,c/H20849/H9262/H20850,d/H20849/H9262/H20850, and e/H20849/H9262/H20850are evaluated by computing the corresponding Feynman dia- grams. Not surprisingly, the results are compactly expressedin terms of the following quantities: F /H9251/H9252/H11013/H11509/H9251A/H9252/H20849x/H20850 −/H11509/H9252A/H9251/H20849x/H20850andD/H9251=/H11509/H9251−2iA/H9251; note that the combination D/H9251/H9004/H9251 is gauge invariant. The Feynman diagrams contributing to the lowest-order terms in the gradient expansion are shown in Fig. 1. Theamplitudes of these diagrams are both proportional to /H20849 /H11509/H9251nˆ/H208502. Therefore, their coefficients, /H92671and/H92672, add to give the stiff- ness of the ferromagnetic state /H9267; when /H9267goes negative, the ferromagnetic state becomes linearly unstable. The depen-dence of the corresponding limit of stability on d, /H9262, and JH is among the central results of this paper. The contributions from these two diagrams compete with one another. The firstgives a positive contribution to the energy, because a spatialvariation of the background spins decreases the hopping am-plitude, via the Anderson-Hasegawa mechanism: t →tcos/H20849/H9052 ij/2/H20850where /H9052ijis the angle between the nearest- neighbor spins iandjin the discrete version of this model. The second diagram gives a negative contribution; spatialvariations in the background spin orientation allow for mix-ing of aligned and anti-aligned bands, thereby lowering theenergy. The contributions to the stiffness /H20851i.e., the coefficient of/H20849 /H11509/H9251nˆ/H208502/H20852are /H92671=/H20849/H9262+JH/H20850d/2+/H20849/H9262−JH/H20850d/2/H9052/H20849/H9262−JH/H20850 21+d/H9266d/2d/H9003/H20849d/2/H20850, /H208492.11a /H20850 /H92672=/H20849/H9262+JH/H208501+d/2−/H20849/H9262−JH/H208501+d/2/H9052/H20849/H9262−JH/H20850 2d+1/H9266d/2/H208492+d/H20850JH/H9003/H208491+d/2/H20850, /H208492.11b /H20850 where /H9003/H20851·/H20852is the gamma function, /H9052/H20851·/H20852is the Heaviside step function, and dis the dimension of space /H20849see Appendix B /H20850. By examining the stiffness as a function of /H9262andd/H20849see Fig. 2 /H20850we observe that d=2 is a threshold dimension, in the sense that the instability occurs for dimensions less than two/H20849but not for dimensions greater than two /H20850. We emphasize that FIG. 5. Background spin configuration in the canted ferromag- netic state. FIG. 6. Spin spiral configuration of background spins, with cone angle /H9273and wavevector /H9251.PEKKER et al. PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-4the precise location of the transition, as well as the threshold dimension, depends on the form of the bare electronic bandstructure, which we have taken to be parabolic. Correctionsto the parabolic electronic dispersion relation would alterboth the location of the transition and the threshold dimen-sion. In particular, in two dimensions the contributions combine to give the stiffness J H2−/H92622 32/H9266JH. /H208492.12 /H20850 This shows that in two dimensions and at zero temperature there is a critical chemical potential /H9262c=JHabove which the ferromagnetic phase loses stability and, as we shall see, un-dergoes a transition to a “textured” phase. This criticalchemical potential coincides with the bottom of the anti-aligned electron band. To investigate how the instability ofthe ferromagnetic state is resolved, it would therefore be nec-essary to raise the chemical potential above its critical value,which would begin populating this band. However, as the gradient expansion is an expansion powers of q texture /kFupper,i t would not converge, as kFupperwould be very small. We ap- proach this dilemma by noting that these results were ob-tained in the absence of an antiferromagnetic term in theoriginal Hamiltonian 2.1, i.e., for J AF=0. The precise loca- tion/H9262c/H20849JH,d/H20850of the instability of the ferromagnetic state is perturbed, and in general shifted to lower value, in the pres- ence of a positive JAF, thus creating a region in the /H20849/H9262,JAF/H20850 plane in which the ferromagnetic state has become unstable and yet /H9262is still smaller than JAF, so that the upper band remains unoccupied. This scheme opens up a region of thephase diagram in which our gradient expansion remainsvalid and, at the same time, a textured state is preferred. In order to investigate the form of the /H20849stable /H20850textured state that replaces the /H20849unstable /H20850ferromagnetic state at chemical potentials immediately greater than the critical one/H20849given in two dimensions by /H9262=JH/H20850, we expand the effective Hamiltonian density to fourth order in gradients of nˆand minimize it with respect to all textures that vary only onlength scales longer than the Fermi wavelength. Via the ex-tension to quartic order of the diagrammatic expansion de-scribed in the present section, and for the case of two dimen- sions, we find the effective Hamiltonian to be H eff=JH2−/H92622 32/H9266JH/H20849/H11509/H9251nˆ/H208502+/H9262/H20849JH2+/H92622/H20850 256/H9266JH3/H20849/H11509/H9251nˆ/H208504 +3JH2/H20849JH+/H9262/H20850−/H92623 48/H9266JH3F/H9251/H9252F/H9251/H9252 +/H20849JH+/H9262/H208502/H208492JH−/H9262/H20850 192/H9266JH3/H20841D/H9251/H9004/H9252/H208412−JH3−/H92623 96/H9266JH3/H20841D/H9251/H9004/H9251/H208412 −JAF/H20849/H11509/H9251nˆ/H208502, /H208492.13 /H20850 where the terms associated with JAFarise from the antiferro- magnetic term in Eq. /H208492.1/H20850. The details of making this extension to quartic order are straightforward, and follow along the lines of Appendix B.For /H9262larger than its critical value, the coefficient of the first term in Heff/H20849e.g., the ferromagnetic stifness /H20850is negative, and therefore it is favorable for the ground state to have a non-uniform texture. We now show that the remaining terms,which are fourth order in gradients and serve to restabilizethe textured state, are all positive definite whenever 0 /H11021 /H9262 /H11021JH. The coefficients of /H20849/H11509/H9251nˆ/H208504andF/H9251/H9252F/H9251/H9252are positive for 0/H11021/H9262/H11021JH, ensuring that these terms are indeed positive definite. If we neglect the surface terms, the forth and fifthterms can be recast in the following form: /H9262/H20849JH2−/H92622/H20850 64/H9266JH3/H20841D/H9251/H9004/H9252/H208412+JH3+/H92623 48/H9266JH3sin2/H9258/H20841/H11612/H9258/H11003/H11612/H9278/H208412. /H208492.14 /H20850 In this form, the coefficients of each of these terms is posi- tive for 0 /H11021/H9262/H11021JH, and therefore all the fourth-order terms are indeed positive definite. Keeping only the first two terms,it is easy to check from the differential equation for theground state, which follows from varying H effwith respect to the fields /H9258and/H9278, that a spin spiral state /H20849e.g.,/H9258=/H9266/2 and /H9278=q·x, where qis a suitably chosen wave vector /H20850mini- mizes the energy. The third, forth, and fifth terms vanish forthe spiral state. Hence, the spiral state is a local minimum ofthe energy, as small perturbations around it would certainlyincrease the contribution from the first two terms, and theremaining terms can only give a positive contribution to theenergy /H20849as they are positive definite and zero to begin with /H20850. The implication of this analysis is as follows: in the double-exchange model there is a region of the zero-temperature/H20849 /H9262,JH/H20850phase diagram in which a spin spiral state is /H20849at least locally /H20850a stable ground state. This state emerges on the high- /H9262/H20849or, equivalently, low- JH/H20850side of the continuous phase transition line, on the other side of which the ferromagneticstate is the stable state. By using the fourth-order terms torestabilize the instability caused by the negative stiffness andincluding the effects of J AF, we find that the wave vector /H9251of the spiral is given by FIG. 7. Band structure in the presence /H20849solid line /H20850and absence /H20849dashed line /H20850of the spin-spiral magnetic state in two dimensions. Notice that the Fermi surface just touches the bottom of the anti-aligned band for the optimal spin-spiral state.DOUBLE-EXCHANGE MODEL FOR NONINTERACTING … PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-5/H92512=4JH2/H20849/H92622−JH2+3 2/H9266JHJAF/H20850 /H9262/H20849/H92622+JH2/H20850/H110154/H20849/H9262−/H9262c/H20850 1−1 6 /H9266/H20849JAF/JH/H20850, /H208492.15 /H20850 where /H9262c2=JH2/H208491−32/H9266JHJAF/H20850. The approximate form holds for/H9262/H11407/H9262c. Here, we note that if JAFis greater than zero, then the spin-spiral would not persist to arbitrary small carrier den-sity, as the antiferromagnetic state becomes stable. Is such acontinuous transition preempted by a first-order transitioninto a microphase-separated state? We explore this possibil-ity in the next section. III. PHASE SEPARATED AND CANTED STATES A. Collinear magnetic states In this section we compare the energies of several com- monly studied types of microphase-separated states22in double-exchange magnetic systems, with the aim of compar-ing their stability relative to the spin spiral state. By mi- crophase separated we mean states that have mesoscalestructure /H20849magnetic and/or electronic /H20850controlled by a com- petition between long-range interactions and interfacial ener-gies. Prior work 9,20,22has focused on the competition be- tween the super-exchange and double-exchange couplingstrengths, and has commonly assumed the latter to be infi-nite, or at least very large. In the present setting, we areconcerned with the entire range of double-exchange coupling strengths, but only with small super-exchange couplingstrengths. In order to have the coexistence of the ferromagnetic and antiferromagnetic microphases associated with the mi-crophase separation that we are considering, their thermody-namic and chemical potentials should coincide with oneanother. 19These condition are necessary for microphase co- existence, but they are not sufficient in settings involvinglong-range interactions, such as those due to distinct charge-densities in the coexisting microphases, or interface energiesassociated with regions separating microphases. However, byexamining the complement of the regions of the phase dia-gram that satisfy the aforementioned necessary conditions orare antiferromagnetic, we can locate the regions in which thehomoegenous ferromagnetic state or the textured state have achance of being stable. /H20849As we shall be limiting our consid- eration to the various types of antiferromagnetic orderinglisted in Fig. 3, we may fail to exclude some regions of thephase diagram that we shall be calling ferromagnetic or tex-tured. /H20850 We proceed by locating those regions of the /H20849J H,n/H20850phase diagram in which either the conditions for microphase coex- istance are satisfied or there is an antiferromagnetic state oflower energy than the ferromagnetic state. The single-electron band structure of the DE Hamiltonian /H208492.1/H20850for fer- romagnet /H20849F/H20850, G-type /H20849G/H20850, and A-type /H20849A/H20850antiferromagnet are given by /H20849F/H20850 /H9280k=±JH±2tcoskx±2tcosky−/H9262, /H208493.1a /H20850 /H20849G/H20850/H9280k=±/H20881JH2+4t2/H20849coskx± cos ky/H208502−/H9262,/H208493.1b /H20850 /H20849A/H20850/H9280k=± 2 tcoskx±/H20881JH2+4t2cos2ky−/H9262./H208493.1c /H20850 For both antiferromagnetic arrangements /H20849G and A /H20850, all the eigenvalues are doubly degenerate. The energy density andthe electron density at zero temperature are given by E=/H20885 BZd2k 4/H92662/H9052/H20849/H9280k/H20850/H9280k, /H208493.2a /H20850 N=/H20885 BZd2k 4/H92662/H9052/H20849/H9280k/H20850, /H208493.2b /H20850 where for the antiferromagentic cases the Brillouin zone should be halved /H20849in each direction /H20850, relative to the ferrmo- agnetic case. From these ingredients we find numerically the lines in the(/H9262,/H20849JH/H20850)plane at which the A-AFM/FM and G-AFM/FM phase transitions occur. On these phase boundaries, for each FIG. 8. Feynman diagrams associated with the Hamiltonian /H208492.5/H20850, contributing to the free energy up to fourth order in gradients of the background spin texture. Solid lines represent aligned elec-tron propagators; dashed lines represent anti-aligned electron propa-gators; the remaining lines are labeled.PEKKER et al. PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-6pair of competing states we calculate a pair of lines nc/H20849JH/H20850 corresponding to the density of each state. On the /H20849JH,n/H20850 phase diagram these lines bound the regions depicted in Fig. 4 inside which the two competing microphases have achance of coexisting. B. Uniform canted magnetic states Next we consider the instability of the ferromagnetic state with respect to the canted ferromagnet state, in order to de-termine the phase boundary between them. Repeating theproceedure of the previous subsection, we find the single-electron band structure of Eq. /H208492.1/H20850 /H9280k=−2 tcosky −/H9262±/H20881JH2+2t2/H208491 + cos 2 kx/H20850±4JHtcoskxcos/H9258, /H208493.3/H20850 where /H9258is the canting angle of the background spin with respect to the zaxis, as shown in Fig. 5. To find the phase transition line in the /H20849/H9262,JH/H20850diagram, we determine when the stiffness of the ferromagnetic state with respect to canting becomes zero. The resulting phase-boundary line is shown inFig. 4. From the diagram, it appears that the canted state isalways preempted by a microphase-separated state. IV. CONCLUDING REMARKS To study smoothly varying textures of the double- exchange model, we have followed the familiar program ofexpanding the free energy of the model in powers of gradi-ents of the background spin texture and tracing out the elec-tronic degrees of freedom. This program has wide applica-bility to the study of long wavelength patterns in latticemodels such as the double-exchange model. The main resultof our paper is the phase diagram 4 for the double-exchangemodel, which we have obtained analytically as a function ofthe carrier density and the Hund coupling J Hbetween the carrier spins and the lattice of classical backgropund spins.Through the application of this program, we find that thespin-spiral state is indeed a stable state for low carrier-densities and has a continuously varying wave vector /H9251 /H11011/H20841/H9262−/H9262c/H208411/2. By direct diagonalization we also find that the transition from the ferromagnetic state to the canted state isessentially preempted by phase separation into differenttypes of antiferromagnetic states. ACKNOWLEDGMENTS We acknowledge helpful discussions with S. L. Cooper, D. I. Golosov, R. M. Martin, H. R. Krishnamurthy, T. V .Ramakrishnan, and M. B. Salamon. We are especially grate-ful to B. H. Lee for sharing with us his unpublished numeri-cal work on the DEM, which prompted us to consider theissues addressed in the present paper. This work was sup-ported by the U.S. Department of Energy, Division of Mate-rials Sciences under Award No. DEFG02-91ER45439,through the Frederick Seitz Materials Research Laboratory,and the National Science Foundation under Grant No. DMR-9976550, and through the Materials Computation Center atthe University of Illinois at Urbana-Champaign. APPENDIX A: SIMPLE MODEL OF SPIN SPIRAL TEXTURE IN TWO DIMENSIONS In this appendix, we solve for the exact ground-state en- ergy of the DEM for the spin spiral state as the backgroundspin texture. We confirm that in two dimensions the criticalline for the transition lies at /H9262=JH, as stated in Sec. II B. The integrating-out of the electronic degrees of freedom, whichwe have carried out in the main text to determine the ground-state energy, becomes more transparent through this ex-ample, in which the process is nonperturbative. The simpli-fication ensues when on restricts attention to a specific classof background spin configurations, viz., spin spirals, whichhave the form /H9258/H20849x/H20850=/H9273;/H9278/H20849x/H20850=/H9251·x, /H20849A1/H20850 where /H9251is a constant wave vector /H20849Fig. 6 /H20850. With this choice, A/H20849x/H20850=/H9251/2 cos /H9273,/H9004/H20849x/H20850=i/H9251sin/H9273, and /H20849/H11509/H9262nˆ/H208502=/H92512sin2/H9273. The Hamiltonian /H208492.5/H20850then reduces to H2D=/H20885„/H9274†/H20849k/H20850/H9272†/H20849k/H20850…/H20873k2+/H9251·kcos/H9273+/H92512/4 −JH−/H9262 −/H9251·ksin/H9273 −/H9251·ksin/H9273 k2−/H9251·kcos/H9273+/H92512/4 +JH−/H9262/H20874/H20873/H9274/H20849k/H20850 /H9272/H20849k/H20850/H20874d2k 4/H92662. /H20849A2/H20850 On diagonalizing this Hamiltonian and calculating the effec- tive energy for the spin spiral state, we find that the criticalchemical potential is equal to the Hund coupling. The re-sponse of the band structure to the spin-spiral state is shownin Fig. 7. When /H9262is larger than JHthe ferromagnetic state is unstable with respect to the formation of a spin spiral state.For /H9273=/H9266/2 and /H9262/H11021JHthe energy has the formEeff=−/H20849JH+/H9262/H208502 8/H9266+/H92512JH2−/H92622 32/H9266JH+/H92514/H9262/H20849JH2+/H92622/H20850 256/H9266JH3./H20849A3/H20850 For/H9262/H11022JH, minimizing Eeffwith respect to /H9251gives /H92512=4JH2/H20849/H92622−JH2/H20850 /H9262/H20849/H92622+JH2/H20850, /H20849A4/H20850 which determines the pitch of the stabilizing spin spiral.DOUBLE-EXCHANGE MODEL FOR NONINTERACTING … PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-7APPENDIX B: EVALUATION OF FEYNMAN DIAGRAMS In this appendix we present a sample diagram calculation by considering the two diagrams of lowest /H20849i.e., second /H20850or- der in the gradient expansion that determine the stiffness/H20851i.e.,a/H20849 /H9262/H20850in Eq. /H208492.10 /H20850; see Sec. II B /H20852. We have computed the higher order diagrams in the same fashion, but specializ- ing to two spatial dimentions. The Matsubara Green functions for aligned /H20849/H9274/H20850and anti- aligned /H20849/H9272/H20850electrons are G/H9274/H20849p,ipn/H20850=1 ipn−/H20849p2−/H9262−JH/H20850, /H20849B1/H20850 G/H9272/H20849p,ipn/H20850=1 ipn−/H20849p2−/H9262+JH/H20850. /H20849B2/H20850 The two diagrams that contribute to the ferromagnetic stiff- ness are diagrams 1 and 2 in Fig. 8. The amplitude corre-sponding to the first diagram is 1 /H9252/H20858 iqn/H20885d–q/H20851G/H9274/H20849q,iqn/H20850+G/H9272/H20849q,iqn/H20850/H208521 4/H20849/H11509/H9252nˆ·/H11509/H9252nˆ/H20850,/H20849B3/H20850 where d–qstands for ddq//H208492/H9266/H20850d. The integral over the internal momentum results in the following expression, which is es-sentially the the sum of the volumes of two d-dimensional spheres of respective radii /H20881/H9262−JHand/H20881/H9262+JH: 2−1−d/H9266−d/2/H9262d/2 d/H9003/H20849d/2/H20850/H20851/H9052/H20849/H9262+JH/H20850/H20849/H9262+JH/H20850d/2 +/H9052/H20849/H9262−JH/H20850/H20849/H9262−JH/H20850d/2/H20852/H20885dx/H20849/H11509/H9252nˆ·/H11509/H9252nˆ/H20850. /H20849B4/H20850 The amplitude corresponding to the second diagram is 1 /H9252/H20858 iqn/H20885d–qd–kG/H9272/H20849q+k,iqn/H20850G/H9274/H20849q,iqn/H20850 /H11003/H9004/H9262/H20849k/H20850/H9004/H9263*/H20849k/H20850/H20849q+k/2/H20850/H9262/H20849q+k/2/H20850/H9263 /H20849B5/H20850 =1 /H9252/H20858 iqn/H20885d–qd–k /H11003/H9004/H9262/H20849k/H20850/H9004/H9263*/H20849k/H20850/H20849q+k/2/H20850/H9262/H20849q+k/2/H20850/H9263 /H20851iqn−/H20849q2−/H9262−JH/H20850/H20852/H20851iqn−/H20849/H20841k+q/H208412−/H9262+JH/H20850/H20852. /H20849B6/H20850 On applying the standard Feynman trick and simplifying, the previous amplitude becomes 1 /H9252/H20858 iqn/H20885d–qd–k/H20885 01 dz/H9004/H9262/H20849k/H20850/H9004/H9263*/H20849k/H20850/H20849q+k/2/H20850/H9262/H20849q+k/2/H20850/H9263 /H20851/H208491−z/H20850/H20851iqn−/H20849q2−/H9262−JH/H20850/H20852+z/H20849iqn−/H20849/H20841k+q/H208412−/H9262+JH/H20850/H20850/H208522/H20849B7/H20850 /H110151 /H9252/H20858 iqn/H20885d–qd–k/H20885 01 dz/H9004/H9262/H20849k/H20850/H9004/H9263*/H20849k/H20850q/H9262q/H9263 /H20851iqn−q2+/H9262−k2z+k2z2+JH/H208491−2 z/H20850/H208522, /H20849B8/H20850 where in the final step we have dropped terms of higher order in k, as they do not contribute to the stiffness. This follows from the obsevation that /H9004/H9251*/H9004/H9251=/H11509/H9252nˆ·/H11509/H9252nˆ. Reversing the order of summation and integration, and simplifying further by noting that the denominator sums to a /H9254function in the zero-temperature limit, we obtain /H20885d–qd–k/H20885 01 dz/H9004/H9262/H20849k/H20850/H9004/H9263*/H20849k/H20850/H20849q/H9262q/H9263/H20850/H9254/H20851−q2+/H9262−k2z+k2z2+JH/H208491−2 z/H20850/H20852 /H20849 B9/H20850 =−/H9052/H20849/H9262−JH/H20850/H20849/H9262−JH/H208501+d/2+/H9052/H20849JH+/H9262/H20850/H20849JH+/H9262/H208501+d/2 22+d/H9266d/2JH/H9003/H208492+d/2/H20850 /H20885dx/H20849/H11509/H9252nˆ·/H11509/H9252nˆ/H20850, /H20849B10 /H20850 which simplifies to the expressions /H208492.11 /H20850in the text. 1T. V . Ramakrishnan, H. R. Krisdhnamurthy, S. R. Hassan, and G. Venkateswara Pai, cond-mat/0308396 /H20849unpublished /H20850;S .K u m a r and P. Majumdar, cond-mat/0406085 /H20849unpublished /H20850. 2H. Ohno, A. Shen, F. Marsukara, A. Oiwa, A. Endo, S. Katsumo, and Y . Iye, Science 281, 951 /H208491988 /H20850;H .O h n o et al. , Appl. Phys. Lett. 69, 363 /H208491996 /H20850; T. Dietl, Semicond. Sci. Technol. 17, 377 /H208492002 /H20850; H. Akai, Phys. Rev. Lett. 81, 3002 /H208491998 /H20850. 3M. P. Kennett, M. Berciu, and R. N. Bhatt, Phys. Rev. B 66,045207 /H208492002 /H20850. 4S. Kumar and P. Majumdar, Phys. Rev. Lett. 91, 246602 /H208492003 /H20850. 5C. Timm and A. H. MacDonald, Phys. Rev. B 71, 155206 /H208492005 /H20850. 6See the review by J. Konig, J. Schliemann, T. Jungwirth, and A. H. MacDonald, in Electronic Structure and Magnetism of Com-plex Materials, edited by J. D. Singh and D. A. Papaconstanto-poulos /H20849Springer, Berlin 2003 /H20850; cond-mat/0111314 /H20849unpub- lished /H20850; see also B. H. Lee, X. Cartoixa, N. Trivedi, and R. M.PEKKER et al. PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-8Martin, cond-mat/0410051 /H20849unpublished /H20850. 7C. Zener, Phys. Rev. 82, 403 /H208491951 /H20850; P. W. Anderson and H. Hasegawa, ibid. 100, 675 /H208491955 /H20850; P.-G. de Gennes, ibid. 118, 141 /H208491960 /H20850; see also D. P. Arovas and F. Guinea, Phys. Rev. B 58, 9150 /H208491998 /H20850. 8S. Yunoki, J. Hu, A. L. Malvezzi, A. Moreo, N. Furukawa, and E. Dagotto, Phys. Rev. Lett. 80, 845 /H208491998 /H20850; E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344,1/H208492001 /H20850. 9E. Dagotto, S. Yunoki, A. L. Malvezzi, A. Moreo, J. Hu, S. Cap- poni, D. Poilblanc, and N. Furukawa, Phys. Rev. B 58, 6414 /H208491998 /H20850. 10M. Hamada and H. Shimahara, Phys. Rev. B 51, 3027 /H208491995 /H20850;J . Inoue and S. Maekawa, Phys. Rev. Lett. 74, 3407 /H208491995 /H20850. 11L.-J. Zou, Q.-Q. Zheng, and H. Q. Lin, Phys. Rev. B 56, 13669 /H208491997 /H20850. 12M. Yu. Kagan, D. I. Khomskii, and M. V . Mostovoy, Eur. Phys. J. B12, 217 /H208491999 /H20850. 13L. Sheng, H. Y . Teng, and C. S. Ting, Phys. Rev. B 58, 8186 /H208491998 /H20850. 14L. Yin, Phys. Rev. B 68, 104433 /H208492003 /H20850. 15A. Chattopadhyay, A. J. Millis, and S. Das Sarma, Phys. Rev. B 64, 012416 /H208492001 /H20850.16E. Kogan, M. Auslender, and E. Dgani, cond-mat/0206018 /H20849un- published /H20850. 17D. J. Garcia, K. Hallberg, B. Alascio, and M. Avignon, Phys. Rev. Lett. 93, 177204 /H208492004 /H20850. 18P. M. Krstajic, V . A. Ivanov, F. M. Peeters, V . Fleurov, and K. Kikoin, Europhys. Lett. 61, 235 /H208492003 /H20850; P. M. Krstajic, F. M. Peeters, V . A. Ivanov, V . Fleurov, and K. Kikoin, Phys. Rev. B 70, 195215 /H208492004 /H20850. 19See, e.g., E. L. Nagaev, Colossal Magnetoresistance and Phase Separation in Magnetic Semiconductors /H20849Imperial College Press, London, 2002 /H20850. 20J. L. Alonso, J. A. Capitan, L. A. Fernandez, F. Guinea, and V . Martin-Mayor, Phys. Rev. B 64, 054408 /H208492001 /H20850. 21See, e.g., A. Auerbach, Interacting Electrons and Quantum Mag- netism /H20849Springer-Verlag, New York, 1998 /H20850,p .7 2 . 22D. I. Golosov, J. Appl. Phys. 91, 7508 /H208492002 /H20850; H. Aliaga, B. Normand, K. Hallberg, M. Avignon, and B. Alascio, Phys. Rev.B64, 024422 /H208492001 /H20850. 23In deriving the continuum limit we assume that the density of impurities is larger then the density of fermions. Thus, our app-proach only applies to DMS when the magnetic impurities arenot very dilute. Coincidently, in this limit, the spin-orbit scatter-ing is found to be negligible /H20849Ref. 5 /H20850.DOUBLE-EXCHANGE MODEL FOR NONINTERACTING … PHYSICAL REVIEW B 72, 075118 /H208492005 /H20850 075118-9

PhysRevB.64.165103.pdf

Effects of magnetic-ion dilution in Kondo insulators Tetsuya Mutou RIKEN (The Institute of Physical and Chemical Research), Wako, Saitama 351-0198, Japan ~Received 13 March 2001; revised manuscript received 25 May 2001; published 28 September 2001 ! On the basis of the periodicAnderson model with randomly distributed impurity sites without felectrons, the effects of magnetic-ion dilution in Kondo insulators are investigated. Using a new scheme in the framework ofthe dynamical mean-field theory with the coherent potential approximation, the density of states, the opticalconductivity, the resistivity, and their concentration dependence for impurities are calculated. The impurityconcentration dependence of the optical conductivity obtained in the present study is qualitatively consistentwith the recent experimental result of an optical conductivity measurement for Yb 12xLuxB12(0<x<1). DOI: 10.1103/PhysRevB.64.165103 PACS number ~s!: 71.10.Fd, 71.27. 1a In many studies of heavy-fermion compounds, several types of ground state have been reported so far. In particular,for some compounds which have an insulating ~or semicon- ducting !ground state, the origin of a tiny gap in those com- pounds has attracted interest, and intensive experimental andtheoretical studies have been carried out in the recentdecade; 1,2those heavy-fermion compounds which have the insulating ~semiconducting !ground state are referred to as Kondo insulators ~semiconductors !. For general heavy- fermion compounds, there have been some experimentalstudies in which nonmagnetic ions are substituted for mag-netic ions, and the relation between heavy-fermion systemsand single-Kondo-impurity systems has been investigated. 3 Especially for Kondo insulators ~semiconductors !, such a magnetic-ion dilution experiment plays an important role instudying the origin of the energy gap. Since the dilution ofmagnetic ions affects the electronic coherence among forbit- als at different sites with magnetic ions, it can be expectedthat the insulating ground state changes in these diluted com-pounds.Actually, there have been several magnetic-ion dilu-tion experiments for Ce-based Kondo insulators, and it hasbeen observed that the insulating behavior of the resistivity isweakened with increasing concentration of nonmagnetic Laions. 4 In the recent study,5Okamura et al.carried out a measure- ment of the optical conductivity for Yb-based compounds Yb12xLuxB12(0<x<1).6Since the ground state of the compound YbB 12is semiconducting, it is classified into Kondo semiconductors and Yb 12xLuxB12is called a diluted Kondo semiconductor.5Okamura et al.observed that the en- ergy gap in the spectrum is rapidly filled in with increasing x, while the shoulder of the gap edge remains up to x51/2, and suggested that the characteristic energy for the gap remainsunchanged in a wide range of x. This is the first systematic experimental study, and there have not been theoretical stud-ies of the optical conductivity for diluted Kondo insulators sofar. To study the characteristic xdependence of the optical- conductivity spectrum observed in Yb 12xLuxB12,i ti si m - portant to investigate some effects of magnetic-ion dilutionin the theoretical model. In the present paper, we treat a theoretical model for Yb 12xLuxB12and calculate physical quantities including the optical conductivity. In order to treat diluted heavy-fermion systems theoreti- cally, it is necessary to consider two important effects: therandomness effect of the magnetic-ion distribution and the strong-correlation effect between felectrons in magnetic ions. For general diluted heavy-fermion compounds, a theo-retical model was suggested by Yoshimori and Kasai ~YK!, 7 and they treated the randomness effect with the coherent po- tential approximation ~CPA!. The insufficiency that the strong-correlation effect was not treated self-consistently inYK was later improved by using the slave-boson mean-fieldapproximation ~SBMFA !. 8The scheme with the CPAand the SBMFA was also applied to a similar model for Ce-baseddiluted Kondo insulators. 9,10However, the SBMFA applied in the above studies cannot treat the many-body correlationeffect sufficiently; it is insufficient to describe the high-energy excitation due to the strong correlation. We use the dynamical mean-field theory ~DMFT !in the present study to treat the many-body correlation effect be-yond the Hartree-Fock approximation. 11In the DMFTframe- work, one can reduce a lattice problem to an impurity prob-lem embedded in an effective medium, and treat the many-body correlation effect more correctly by solving theeffective impurity problem appropriately. Concerning therandomness effect, we also apply the CPA as in the above-mentioned studies. The physical idea of the DMFT frame-work is the same as that of the CPA scheme, 12and it is expected that one can construct the effective scheme consist-ing of the DMFT and the CPA to treat the diluted heavy-fermion systems. Although both the DMFT and the CPA are exact in the infinite-dimensional ( d5‘) system since they are based on the mean-field theory, 12,13it should be noted that the spatial correlation of electronic states is not takeninto account beyond the mean-field treatment when one ap-plies them to the finite-dimensional system. Nevertheless, itcan be expected that the present scheme is applicable to thediluted heavy-fermion systems because forbitals in magnetic ions are rather localized. The purpose of the present study isto clarify the impurity concentration dependence of physicalquantities such as the optical conductivity with using the theoretical model for Yb 12xLuxB12by the scheme mentioned above. In the present study, we treat the periodicAnderson model ~PAM!with randomly distributed sites which have no f electrons.14The model is same as that used in studies for Ce-based diluted Kondo insulators, and it is a quite generalmodel for diluted Kondo insulators. One can expect that thePHYSICAL REVIEW B, VOLUME 64, 165103 0163-1829/2001/64 ~16!/165103 ~5!/$20.00 ©2001 The American Physical Society 64165103-1model is suitable to understand qualitative properties of Yb12xLuxB12though quantitative details of the materials may not be described by the simple model. We refer to siteswith and without felectrons as a host site ~denoted by H !and an impurity site ~denoted by I !, respectively. Host and impu- rity sites correspond to the ions Yb 31and Lu31in Yb12xLuxB12. The model Hamiltonian is defined as follows: H5( k,s«kcks†cks1V( i,s~cis†fis1H.c.! 1( i,sH~12ji!U 2fis†fis@fi2s†fi2s21#1jiEIfis†fisJ [H01Hf, where Hfdenotes the third term and H0includes the other two terms. A random variable jiis defined as ji51(iPI) andji50(iPH). The impurity concentration xis expressed asx5(iji/N;Ndenotes the number of lattice sites. On impurity sites corresponding to nonmagnetic ions, felectrons are excluded by taking the limit EI!‘in the calculation.7–10 The density of electrons is as nf1nc522x. Other notation in the Hamiltonian are conventional. The present system hasparticle-hole symmetry, and the hybridization gap opens in the homogeneous case ( ji50 for all sites !; the model with particle-hole symmetry has been used as a theoretical modelof Kondo insulators. 15–18Since substituting impurity sites for host sites is expected to break the coherence of the insulatingground state, the Hamiltonian can be a model of Yb 12xLuxB12. We assume that the wave-number dependence of the self- energy can be neglected ~the local approximation !and apply the CPAto the alloy system.According toYK, the usual CPAprocedure is applied to the present model. We start with theeffective one-body Hamiltonian H eff5H01( i,s$~12ji!Ss~ivn!1jiEI%fis†fis, where Ss(ivn) denotes the self-energy ~from which the Har- tree term is subtracted !due to the felectron correlation. The CPA Hamiltonian is defined as HCPA5H01( i,sSs~ivn!fis†fis, whereSs(ivn) denotes the coherent potential. We obtain the CPA condition by equating the random average of the site- dependent Green’s function Gi(ivn)[$ivn2(HCPA 1vi)%21with the CPA Green’s function GCPA(ivn)[(ivn 2HCPA)21. The site-dependent potential viis defined as ( ivi[Heff2HCPA. The CPA condition in the limit EI!‘can be expressed by the following equation:7 $S~ivn!2S~ivn!%Gff~ivn!5x~xÞ1!, ~1!where we have omitted the spin index because we deal with only the paramagnetic state. The fcomponent of the CPA ~or averaged !Green’s function is denoted by Gff(ivn): Gff~ivn!5Ednr0~n!Gff~ivn;n! 5Ednr0~n! ivn2S~ivn!2V2 ivn2n, ~2! where the density of states r0(n)5(kd(n2«k)/Nhas been introduced. In this study, we assume the half-elliptic form of r0(n):r0(n)52A12(n/D)2/(pD) for unu<Dandr0(n) 50 for unu.D. Hereafter we take Das the unit of energy: D51. Next step of the procedure is to determine the self-energy S(ivn). YK suggested that S(ivn) should be calculated in terms of the fcomponent of the site-dependent Green’s func- tionGiPHff(ivn) on the hostsite.7If one can calculate an exact self-energy from GiPHff(ivn), the above procedure by YK would be the complete one. In fact, however, one needs an appropriate procedure with approximations by which thestrong-correlation effect can be treated properly. In the present study, we calculate S(i vn) by the iterated perturba- tion theory ~IPT!in the DMFT framework. Namely, we in- troduce the so-called Weiss function G(ivn) in the DMFT ~Ref. 11 !: @G~ivn!#21[@Gff~ivn!#211S~ivn!. ~3! Using this Weiss function, we can express GiPHff(ivn)a s @GiPHff(ivn)#215@G(ivn)#212S(ivn). Following the pro- cedure of the IPT, we calculate S(ivn) in terms of G(ivn) by second-order perturbation so that the scheme is consistentwith that for the usual ~homogeneous !PAM in the case of x50;S(i vn) is equal to S(ivn) forx50. We symbolically express S(ivn) calculated using G(ivn) as the functional of G(ivn): S~ivn!5S@G~ivn!#. ~4! Now we have four equations ~1!,~2!,~3!, and ~4!for four functions Gff,G,S, and S. We determine these functions self-consistently from the above equations. Once self-consistent solutions of Green’s functions are found, the one-particle density of states ~DOS!can be ob- tained from the Green’s function as rf(c)~v!521 pImGff(cc)~v1id!, whereGcc(ivn) is defined similarly to Gff(ivn). Using the one-particle Green’s function, we can also calculate the op-tical conductivity in the present scheme with the approach fromd5‘.Assuming that the expression of the optical con- ductivity formulated on the hypercubic lattice in d5‘can be also applied to the case with the general lattice, we obtainTETSUYA MUTOU PHYSICAL REVIEW B 64165103 165103-2the following expression of the optical conductivity.19With omitting some constant factors, we define the reduced opticalconductivity 20 s˜~v![Ednr0~n!Edercc~e1v;n!rcc~e;n! 3f~e!2f~e1v! v, where rcc(e;n)52ImGcc(e1id;n)/pandf(e) denotes the Fermi distribution function. Note that the above expressionof the optical conductivity does not include so-called vertexparts because the contribution to the conductivity due to ver- tex corrections vanishes with the approach from d5‘~Ref. 21!and s˜(v) is expressed by the convolution of one-particle Green’s functions. The resistivity r˜(T) is defined as r˜(T) 51/s˜(v50). First, we show the results for xdependence of the DOS in Fig. 1. The hybridization gap opens at the Fermi level ~the center of the spectrum !forx50; it corresponds to the usual ~homogeneous !Kondo insulator. The value of the DOS gap DDOSis about 8.2 31022for the present parameters: U53 andV50.5.22In high-energy regions one can see that there are side peaks due to the on-site Coulomb interaction be-tweenfelectrons. Note that these high-energy structures re- lated to the charge excitation can be obtained by the presentscheme with the DMFT; these cannot be described by theSBMFA ~Fig. 2 !. In the SBMFA scheme, the many-body ef- fect is treated by the mean-field theory and it affects only thenarrowing of the bandwidth ~or the hybridization gap !. Namely, the SBMFA is insufficient to describe such a high-energy excitation. For finite x, the spectrum has a structure in the gap, and the structure grows with the impurity concentration @the inset of Fig. 1 ~a!#. This structure inside the gap corresponds to an impurity band. 9,10The gap edge of the spectrum is almost unchanged for lower concentration. Since the Fermi level sitsat the center of the growing structure, the ground state of thesystem is expected to be metallic. The impurity concentra-tion becomes much higher, the gap is filled up, and it turns out to be a pseudogap structure. Finally, for x50.8, there is a sharp peak instead of the pseudogap structure at the center @Fig. 1 ~a!#, and the spectrum rf(v) has a three-peak struc- ture. Considering rf(v) per one host site, rf(v)/(12x), the side-peak structures in high-energy regions are almost un-changed for higher concentrations ~which corresponds to di- lute concentrations for magnetic ions !, while the central structure changes drastically. It is suggested that the structurein high-energy regions corresponds to the local excitationand it is not so affected by the substitution. On the contrary,in low-energy regions, the structure is sensitive to the substi-tution; the coherence is destroyed by a small amount of im- purities. In the DOS spectrum, rc(v) for conduction elec- trons, the gap is filled up and side-peak structures becomesmall with increasing x. Then, for higher concentrations, the spectrum of rc(v) becomes similar to that of r0(v) as ex- pected. Figure 3 shows the optical conductivity spectra s˜(v) for several values of x. One can see that the spectrum for x.0 has a sharp large peak. In particular, for x50~not shown !,17,18the optical conductivity spectrum has the gap structure caused by the hybridization, and at its gap edge there is a sharp peak. The gap-edge peak for x50 corre- sponds to the above-mentioned large peak for x.0.23Once the impurity concentration becomes finite, charge excitationfrom the Fermi level can occur, and then the gap structure isfilled up rapidly. For low concentrations, one can see a dip structure at v.0.1, which corresponds to DDOS. The value FIG. 1. Impurity concentration dependence of DOS spectra for ~a!f-electron component rf(v)/(12x) per a host site and ~b! conduction-electron component rc(v) forU53 andV50.5 atT 50.The inset of the panel ~a!shows spectra rf(v)/(12x) near the Fermi level ( v50) for low concentrations: x50.00, 0.05, 0.10, 0.15, and 0.20. FIG. 2. Comparison of DOS spectra rf(v)/(12x) calculated by the present scheme with those by the SBMFA ~inset!.EFFECTS OF MAGNETIC-ION DILUTION IN KONDO. . . PHYSICAL REVIEW B 64165103 165103-3ofDDOSis determined by parameters proper to the system, and it does not depend on x. Since the DOS gap is renormal- ized~narrowed !by the strong correlation, the position of the dip structure in s˜(v) also shifts to lower energy with in- creasingU~the inset in Fig. 3 !; one can see that the peak structure grows for larger values of U. With increasing the impurity concentration, the intensity in lower-energy regionsincreases, and the peak structure shifts toward lower energy. The spectrum for x50.9 has a single large peak at v50. We show the temperature dependence of the resistivity for several values of xin Fig. 4. For x.0, the resistivity in- creases as the temperature decreases. Such a semiconductor-like increasing behavior of the resistivity is suppressed formuch higher concentration of impurity sites, and the resistiv-ity reaches a finite value at the lowest temperature. The re-sidual resistivity decreases with increasing x. This behavior is reasonable because the gap originates from the coherenthybridization in the present system and substituting host sitesfor impurity sites lowers the electronic coherence. One cansee more clearly the xdependence of the residual resistivityin the inset of Fig. 4. The residual resistivity diverges at x 50; the system becomes insulating, while its value normal- ized by the host-site concentration (1 2x) reaches a finite value in the limit of x!1. For the present model treated by the formulation we use, one can obtain an analytical form of thexdependence of r˜(T50): r˜~T50!5p2D2 412x2 x. In the inset of Fig. 4, the analytical curve of r˜(T50)/(1 2x) is also shown. The value of r˜(T50)/(1 2x) in the limit x!1i sp2D2/2, and this should correspond to the residual resistivity in the unitarity limit for the so-called Kondo impurity system. In summary, we have treated the inhomogeneous PAM by a scheme consisting of the CPA and the DMFT in order to study effects of magnetic-ion dilution in Yb 12xLuxB12. Im- purity ~nonmagnetic ion !concentration dependences of the DOS, the optical conductivity, and the resistivity are inves-tigated. Introducing impurities produces an impurity band ina hybridization gap, and the system becomes metallic. How-ever, the magnitude of the gap remains unchanged for lowerimpurity concentration. For much higher concentration thegap itself becomes a pseudogap. Concerning the optical con-ductivity spectrum, the position of the dip structure corre-sponding to the gap edge of the DOS almost remains un-changed, while the gap is filled up with increasing impurityconcentration. These results show that the gap edge itself is related to local parameters ( UandV!proper to the original system with host sites and is not dependent on impurity con-centration, though the electron coherence is destroyed by in-troducing impurity sites. The present result for the impurityconcentration dependence is qualitatively consistent with therecent result obtained by the optical conductivity measure- ment for Yb 12xLuxB12.5 For high concentrations, the resistivity at the lowest tem- perature reaches the finite value corresponding to the unitar-ity limit of the Kondo impurity system. One can obtain theanalytical form of the xdependence of the residual resistivity in the present model and formulation. In fact, we can also write down the xdependence of the felectron DOS rf(v 50) at the Fermi level as rf(v50)/(1 2x) 5D/(2pV2)Ax. In this expression, one can confirm that rf(v50)/(1 2x) in the limit of x!1 is equal to the DOS of the Kondo impurity system at the Fermi level. Therefore wecan obtain properties of the coherent Kondo insulator and theKondo impurity system on the same footing with the presentscheme. ACKNOWLEDGMENTS We would like to thank H. Okamura for informative dis- cussions on recent experimental results of Yb 12xLuxB12. The author was supported by the Special Postdoctoral Re-searchers Program from RIKEN. A main part of the numeri-cal calculations was performed on the supercomputerVPP700E of RIKEN. FIG. 3. Impurity concentration dependence of optical conduc- tivity spectra s˜(v) forU53 andV50.5 atT50.001. The enlarge- ment of the lower parts of the spectra for several values of Uis shown in the inset ( U51, 2, and 3; V50.5). FIG. 4. Temperature dependence of the resistivity r˜(T)/(1 2x) normalized by the host-site concentration (1 2x) forx50.1, 0.2, 0.4, 0.6, and 0.8. The inset shows the numerical result of the x dependence of the residual resistivity r˜(T50)/(1 2x) and its ana- lytical form ( p2D2/4)(1 1x)/x~see text !.TETSUYA MUTOU PHYSICAL REVIEW B 64165103 165103-41For a few reviews of experimental studies, see G. Aeppli and Z. Fisk, Comments Condens. Matter Phys. 16, 155 ~1992!;L .D e - giorgi, Rev. Mod. Phys. 71, 687 ~1999!. 2For a few reviews of theoretical studies, see P. Schlottmann, in Current Problem in Condensed Matter: Theory and Experiment , edited by J. L. Moran-Lopez ~Plenum, New York, 1998 !;P . Riseborough, Adv. Phys. 49, 257 ~2000!. 3For example, A. Sumiyama, Y. Oda, H. Nagano, Y. O ¯nuki, K. Shibutani, and T. Komatsubara, J. Phys. Soc. Jpn. 55, 1294 ~1986!. 4For example, M.F. Hundley, P.C. Canfield, J.D.Thompson, and Z. Fisk, Phys. Rev. B 50, 18142 ~1994!. 5H. Okamura, M. Matsunami, T. Inaoka, T. Nanba, S. Kimura, F. Iga, S. Hiura, J. Klijn, and T. Takabatake, Phys. Rev. B 62, R13 265 ~2000!. 6F. Iga, S. Hiura, J. Klijn, N. Shimizu, T. Takabatake, M. Ito, Y. Matsumoto, F. Masaki, T. Suzuki, and T. Fujita, Physica B 259- 261, 312 ~1999!. 7A.Yoshimori and H. Kasai, Solid State Commun. 58, 259 ~1986!. 8Z. Li and Y. Qiu, Phys. Rev. B 43,1 29 0 6 ~1991!. 9P. Schlottmann, Phys. Rev. B 46, 998 ~1992!. 10R. Shiina, J. Phys. Soc. Jpn. 64, 702 ~1995!. 11A. Georges, G. Kotliar, W. Krauth, and M.J. Rozenberg, Rev. Mod. Phys. 68,1 3~1996!.12For a review of the infinite-dimensional approach, see D. Voll- hardt, in Correlated Electron Systems , edited by V.J. Emery ~World Scientific, Singapore, 1992 !,p .5 7 . 13R. Vlaming and D. Vollhardt, Phys. Rev. B 45, 4637 ~1992!. 14Magnetic ions Yb31and Ce31have anf holeand anf electron in Yb- and Ce-based compounds, respectively. Here, however, werefer to both of them as f electrons ; we do not distinguish the above compounds in the present theoretical model. 15M. Jarrell, H. Akhlaghpour, and Th. Pruschke, Phys. Rev. Lett. 70, 1670 ~1993!. 16D.S. Hirashima and T. Mutou, Physica B 199&200 , 206 ~1994!. 17T. Mutou and D.S. Hirashima, J. Phys. Soc. Jpn. 63, 4475 ~1994!. 18M. Jarrell, Phys. Rev. B 51, 7429 ~1995!. 19Strictly speaking, the conductivity disappears in the infinite- dimensional limit. Note that the expression of the optical con-ductivity in the text is the leading term of O(1/d). 20T. Mutou, Phys. Rev. B 60, 2268 ~1999!. 21A. Khurana, Phys. Rev. Lett. 64, 1990 ~1990!. 22We define DDOSas the energy difference between the Fermi level and the lower edge of the upper part ( v.0) of the DOS. 23In some studies for the homogeneous PAM ( x50), it was sug- gested that this peak position to the direct excitation gap ~Refs. 18 and 11 !.EFFECTS OF MAGNETIC-ION DILUTION IN KONDO. . . PHYSICAL REVIEW B 64165103 165103-5

PhysRevB.84.121202.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 84, 121202(R) (2011) Electronic structure of Ga 1−xMn xAs probed by four-wave mixing spectroscopy M. Yildirim,1S. March,1R. Mathew,1A. Gamouras,1X. Liu,2M. Dobrowolska,2J. K. Furdyna,2and K. C. Hall1 1Department of Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia B3H1Z9, Canada 2Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA (Received 13 July 2011; published 28 September 2011) Four-wave mixing experiments on GaMnAs indicate an increase in the optical response near the band gap with increasing Mn concentration. These findings are attributed to ( s,p)-dhybridization, which leads to an enhancement in the density of states in the valence band. Our experiments show that the nonlinearity of thefour-wave mixing technique provides a highly sensitive probe for Mn-related changes in the electronic structureof GaMnAs. DOI: 10.1103/PhysRevB.84.121202 PACS number(s): 75 .50.Pp, 78 .47.J−,7 8.47.nj The III-V diluted magnetic semiconductors (DMS) exhibit a unique combination of semiconducting and ferromagneticproperties, offering the ability to control magnetic charac-teristics through modification of the carrier density. 1–6This feature makes DMS materials of interest for developingsemiconductor-based magneto-sensitive electronic and pho-tonic devices that would exploit the spin degree of freedom. 7–16 Of the various DMS systems under study, GaMnAs has become the prototype material, with a considerable body ofwork over the past decade providing the foundation for ageneral picture of ferromagnetism in the III-V DMS. Theability to tailor the growth conditions and optimize post-growthannealing procedures has led to continual improvements inT cfor GaMnAs, reaching close to 200 K for Mn densities ∼10%.17–19 Despite these experimental advances, the theoretical under- standing of the band structure and the underlying ferromag-netic interactions is still evolving, due in part to the complexity of treating ( s,p)-dhybridization, localization effects, and Coulomb interactions in the presence of disorder. 20–26An ongoing debate concerning the location of the Fermi level,which may lie within the p-type valence band or within an impurity band with primarily dcharacter, has critical implications regarding the nature of transport and ferromag-netic order in this system. 22,24,25,27–32Results of linear optical spectroscopy experiments have provided crucial input into thisdebate, as well as illuminating the fundamental propertiesof GaMnAs. 21,25,27–35For instance, an absorption feature around 200 meV has provided a possible means to distinguishbetween valence band and impurity band models, 29although sample-to-sample variations have complicated the interpreta-tion of experimental results. 25Magnetic circular dichroism and spectroscopic ellipsometry studies have also provided information regarding the sign (antiferromagnetic) and wave-vector dependence of the p-dexchange interaction. 27,30,31,33–37 Four-wave mixing spectroscopy provides another tool to investigate the electronic structure of GaMnAs. In this nonlin- earoptical technique, two incident optical pulses with wave vectors k1andk2together generate a third-order nonlinear optical polarization that emits light with wave vector 2 k2- k1. The spectral content of this emission, together with its dependence on the time delay between the two exciting laserpulses, provides information regarding the density and natureof the electronic transitions, interactions within the excitedcarrier system, and the coherence decay time. 38–43Four-wave mixing experiments are expected to provide a sensitive probeof Mn-related features in the electronic structure of III-Mn-Vsemiconductors, including the prevalence of defect-to-bandtransitions 27,33and the influence of ( s,p)-dhybridization on the valence states.23,28This sensitivity is tied to the nonlinearity of the technique, as the four-wave mixing polarization isproportional to the cube of the energy-dependent dipole matrixelement. For example, four-wave mixing experiments haverevealed the relative strength of exciton and band-to-bandcontributions in nonmagnetic Ge, GaAs, and InP. 39,40,43 Here we report measurements of the four-wave mixing response of a family of Ga 1−xMnxAs structures with xin the range 0 to 0.1%. The properties of GaMnAs in the low xlimit have been the subject of several recent works as this regimeprovides a window into the electronic properties of Mn pointdefects and the effects of ( s,p)-dhybridization. 22,23,44–51Our experiments indicate a strong increase in the optical responseabove the fundamental band gap with the addition of Mn withxas low as 0.005%. We attribute this observation to an increase in the density of states near the valence band edge caused byhybridization between the dlevels of the Mn Gaand the p states of the host GaAs crystal, as predicted theoretically.23 Mn-induced states near the valence band maximum wereseen in angle-resolved photoemission, 52however the energy resolution was relatively low ( ∼100 meV). A broad increase in the optical response of GaMnAs in the vicinity of theband gap with the addition of Mn has also been observedin absorption and ellipsometry experiments, 27,33although a strong absorption tail characteristic of low-temperaturegrown GaAs and GaMnAs masks any Mn-related featuresnear the fundamental band gap in linear spectroscopy. 53Our experiments indicate that the absorption tail provides only aweak contribution to the four-wave mixing spectrum due tothe difference in the dipole transitions strengths for defect-to-band and band-to-band absorption channels. This sensitivity,which is afforded by the nonlinearity of the technique usedhere, has allowed us to observe clear optical signatures of(s,p)-dhybridization, providing new insight into the electronic structure of GaMnAs. The samples for this work were grown by molecular beam epitaxy and consist of 800 nm of Ga 1−xMnxAs grown on top of a 175 nm Al 0.27Ga0.73As stop etch layer on a semi-insulating GaAs substrate. The samples were glued face 121202-1 1098-0121/2011/84(12)/121202(5) ©2011 American Physical SocietyRAPID COMMUNICATIONS M. YILDIRIM et al. PHYSICAL REVIEW B 84, 121202(R) (2011) TABLE I. Summary of characteristics of Ga 1−xMnxAs samples for four-wave mixing experiments. HT (LT) indicates growth at high (low) substrate temperature. The Mn concentration was determined by secondary ion mass spectrometry.54 Sample Growth temperature x(%) HT GaAs 600 0 LT GaAs 250 0LT GaMnAs 250 0 .005 LT GaMnAs 250 0 .045 LT GaMnAs 250 0 .100 HT GaMnAs a600 0 .100 aThe Mn content in the HT GaMnAs sample is an upper estimate only, as discussed in the text. down to c-cut sapphire windows, and the substrates were removed using mechanical polishing followed by wet etchingin order to enable experiments in the transmission geometry.The substrate growth temperature and Mn concentration forthe samples studied here are listed in Table I.T h eM n content for the single GaMnAs sample grown at 600 ◦C is only an estimate as the Mn will exist in a combina-tion of Mn point defects and precipitates. The thicknessof the GaMnAs layer was chosen to provide a sufficientfour-wave mixing signal without introducing propagationeffects. The linear absorption spectra for the samples described in Table Iare shown in Fig. 1(a). These measurements were made using a continuous-wave white light source (Ocean OpticsLS-1). The absorption coefficient was extracted from the rawtransmission data using a self-consistent model incorporatingmultiple reflections in the sample layer. Weak residual FabryPerot oscillations persist for low photon energies, but do notobscure the dominant trends in Fig. 1(a). The two samples grown at 600 ◦C exhibit a clear excitonic peak and a sharp band edge response, as expected since the elevated temperature ofthe substrate during growth results in a low density of defects.The exciton absorption peak in HT GaMnAs is observedto be red shifted by 2.5 meV relative to HT GaAs. Weattribute this shift to the dominant emission of excitons boundto Mn-related defects. In contrast, no exciton is visible inany of the low-temperature-grown samples, and there is asubstantial absorption below the band gap. Smearing of theband edge features with low temperature growth is due toexcess arsenic, an effect that has been well studied in LTGaAs. 53,55Excess arsenic leads to the formation of band tail states associated with local potential fluctuations55,56and a deep donor band due to As Gaantisite defects,28,53both of which lead to absorption pathways for photon energies belowthe band gap of HT GaAs. The absorption tail is largest in theLT GaAs sample, as expected because the As Gamid-gap donor band will be occupied, leading to strong linear absorptionfrom the mid-gap band to the conduction band. In the LTGaMnAs samples, compensation due to the Mn Gaacceptors reduces this absorption channel relative to LT GaAs.28For increasing Mn concentration, the absorption tail increases inthe LT GaMnAs samples, reflecting the increasing degree ofMn-related local potential fluctuations and associated bandtailing. FIG. 1. (Color online) Results of linear transmission experiments at 10 K. The laser spectrum for the 20-fs pulses used in the four-wavemixing experiments is indicated by the filled circles. (b) Schematic diagram of the four-wave mixing apparatus. Spectrally-resolved four-wave mixing experiments were carried out in the two-pulse self-diffraction geometry, as shownschematically in Fig. 1(b). The optical source is a mode-locked Ti:Sapphire oscillator, producing pulses with a center photonenergy of 1.55 eV . A prism compressor was integrated into theoptical setup to compensate for dispersion, providing a pulseduration at the sample position of 20 fs, detected using thezero background autocorrelation signal at an equivalent focus.The laser pulse spectrum, which is broad enough to overlapthe band edge region as well as the absorption tail, is shownin Fig. 1(a) by the filled circles. The four-wave mixing signal was spectrally resolved using a monochromator and detectedusing a photomultiplier tube. The spectral sensitivity of theentire setup was calibrated using the white light source. Thepulse delay ( τ) was varied using the rapid scan approach, with a retroreflector mounted to the cone of a woofer speaker.The zero delay position was determined by simultaneouslydetecting the four-wave mixing signal along 2 k 1-k2and 2k2-k1, which form a mirror image about τ=0.40The density of electron-hole pairs excited by the laser pulse was estimatedusing the incident and transmitted powers through the sampleto be 6 ×10 16cm−3. The optical density of the samples is 0.2, indicating that propagation effects may be neglected.57 The sample was held at 10 K in a liquid helium flow-throughcryostat for all measurements. The results of four-wave mixing measurements on the HT GaAs reference sample are shown in Fig. 2(a). The signal is strongly dominated by the optical responseof the exciton, which exhibits a spectrally narrow peakcentered at zero time delay. This signal decays very rapidly 121202-2RAPID COMMUNICATIONS ELECTRONIC STRUCTURE OF Ga 1−xMnxAs... PHYSICAL REVIEW B 84, 121202(R) (2011) FIG. 2. (Color online) Results of spectrally-resolved four-wave mixing experiments at 10 K on the samples described in Table I.T h e contour scale indicates the amplitude of the four-wave mixing signal, which is equal for all LT samples within the experimental uncertainty. The maximum contour scale for the two HT samples is much larger: by a factor of 20 for HT GaAs in (a) and by a factor of 12 for HTGaMnAs in (b). versus delay, with a decay time that is comparable to the laser pulse duration. A weak response from the valence toconduction band transitions is also observed on the highenergy side of the exciton. The continuum response peaksat positive time delay and decays more slowly than theexciton response. These general characteristics were observedin earlier experiments on high-temperature-grown GaAs, Ge,and InP. 40,58,59The interband response is characteristic of an inhomogeneously-broadened transition,60,61while the exciton signal is dominated by an interaction-induced nonlinearityrelated to exciton-carrier scattering. 41–43As seen in Fig. 2(b), HT GaMnAs exhibits a similar four-wave mixing responseto HT GaAs, reflecting the clean band structure in thissample. Faster dephasing kinetics in the HT GaMnAs samplecauses the interband contribution to peak closer to zerodelay than in HT GaAs, 60likely reflecting carrier scattering with the Mn impurities.62The signal characteristics change qualitatively when the structure is grown at low temperature:For all of the samples in Figs. 2(c)–2(f), an additional signal component appears below the band gap. With increasing x, the four-wave mixing response of LT-GaMnAs progressivelybroadens to the high energy side. Measurements for a range of FIG. 3. (Color online) (a) Four-wave mixing spectra at 10 K taken at zero pulse delay. (b) The laser excitation spectrum. excited carrier density indicate some evidence of saturation at the exciton in the HT-GaAs response, likely reflectingscreening of the exciton-carrier scattering coefficient. 41For the continuum response of HT GaAs at higher photon energies,and for data at all energies in all other samples, the powerdependence indicates that the nonlinear response is in the χ (3) regime. These trends may be seen more clearly in Fig. 3, which shows the four-wave mixing spectrum at zero delay. The sharppeak at the exciton is dominant in the optical response forHT GaAs and HT GaMnAs. A 2.5 meV red shift is observedfor the HT GaMnAs exciton response, consistent with thelinear absorption data in Fig. 1(a). The exciton peak in HT GaMnAs is also broader than that in HT GaAs (11 meVcompared to 8 meV in HT GaAs), consistent with an increaseddephasing rate due to scattering with Mn-related defects. Sincethe peak of the signal due to the interband transitions occurscloser to zero delay in the HT GaMnAs sample than in HTGaAs, it may be seen in the zero delay spectrum for HTGaMnAs as the small peak on the high energy side of theexciton. A low energy shoulder appears in all of the samplesgrown at low temperatures, with a similar spectral shape andstrength across the samples. The four-wave mixing responseremains peaked at the exciton energy for both the LT GaAssample and the sample with the lowest Mn concentration(x=0.005%), although the latter signal is considerably broader and assymetric to the high-energy side. For higherx, the peak broadens further, and the peak center shifts to higher photon energies. As the low energy shoulder is present for all samples grown at low temperature, it is not tied to the Mn doping. We attributethis shoulder to the nonlinear optical response of band tailstates induced by local potential fluctuations, as the associatedoptical transitions are expected to be similar in strength tothe interband transitions. 55Optical transitions between the AsGamid-gap donor band and the conduction band provide 121202-3RAPID COMMUNICATIONS M. YILDIRIM et al. PHYSICAL REVIEW B 84, 121202(R) (2011) an additional absorption channel for the photon energies investigated here, however these transitions are expected togenerate a broad, featureless response, with a spectral shapesimilar to the laser spectrum, in contrast to the observations inFig. 3. The absence of a significant four-wave mixing response associated with these defect-to-band transitions indicates thatthe matrix element of the dipole operator associated withthese transitions is small compared to that for the interbandtransitions. This conclusion is consistent with findings inp-type GaAs, where no response associated with the acceptor- deionization band was observed even up to 120 K. 63Differ- ences in transition strength for various signal contributionsare amplified in nonlinear optical experiments like four-wavemixing spectroscopy because the source polarization in theexperiments is proportional to the cube of the dipole matrixelement for the transition. The 3dorbitals of the Mn Gasubstitutional impurity hy- bridize with the porbitals of the As atoms. This ( s,p)-d hybridization is the origin of the strong exchange coupling be-tween the hole and the Mn local moments responsible for ferro-magnetic order. The bound Mn acceptor level located 110 meVabove the valence band edge has been well characterized usinginfrared absorption spectroscopy. 46,64Much less is known about the influence of ( s,p)-dhybridization on the delocalized valence states. In angle-resolved photoemission experiments,an increase in the emission intensity was observed for energiesbetween the Fermi energy and 0.5 eV below. 52This was attributed to states induced by Mn near the valence bandmaximum, although the energy resolution in those experimentswas limited to ∼100 meV . Using a local tight-binding model, Tang and Flatt ´e predicted a strong enhancement in the valence band density of states due to hybridization extending fromthe band edge to more than 1 eV below. 23The blue shift and broadening we observe is consistent with an increased opticalresponse associated with the interband transitions inducedby an increase in the valence band density of states. Wetherefore attribute these observations to signatures of ( s,p)-d hybridization. No such signatures have been observed inlinear absorption experiments due to the dominant absorptioncontributions of band tail states and transitions involving theAs Gaimpurity band.33Band edge smearing prevented even the observation of a critical point at the band gap in GaMnAsin spectroscopic ellipsometry experiments. 27The contrast in sensitivity of linear and nonlinear spectroscopy is evidentfrom a comparison of the evolution of the spectrum with x in Figs. 1(a) and 3: The four-wave mixing signal exhibits a gradual evolution as xincreases, whereas no qualitative difference is observed in the linear absorption data for anyof the low-temperature grown samples. Our ability to observeclear effects of ( s,p)-dhybridization in four-wave mixing experiments is tied to the nonlinearity of the technique andthe associated enhancement in sensitivity to fine features inthe optical joint density of states. In conclusion, we have studied the nonlinear optical response of GaMnAs using femtosecond four-wave mixingspectroscopy. Our experiments indicate a distinct reshapingof the four-wave mixing spectrum with the addition ofMn, corresponding to an increase in the optical responseabove the band gap. We attribute these observations to theeffects of ( s,p)-dhybridization on the valence band states. Our results show that four-wave mixing techniques provideincreased sensitivity to Mn-related changes in the electronicstructure for states in the vicinity of the band gap whencompared to traditional linear spectroscopy experiments. Ourfindings provide new insight into the fundamental propertiesof GaMnAs. This research is supported by the Canada Founda- tion for Innovation, the Natural Sciences and EngineeringResearch Council of Canada, Lockheed Martin Corporation,the Canada Research Chairs Program, and the NationalScience Foundation (Grant No. DMR10-05851). 1H. Ohno, D. Chiba, F. Matsukura, T. 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PhysRevB.73.035209.pdf

Ab initio simulation of polyamorphic phase transition in hydrogenated silicon Murat Durandurdu Department of Physics, University of Texas at El Paso, El Paso, Texas 79968, USA /H20849Received 19 June 2005; revised manuscript received 14 September 2005; published 17 January 2006 /H20850 We study the pressure-induced phase transition of a hydrogenated amorphous silicon model with a void, using a constant pressure ab initio technique within the generalized gradient approximation. With the applica- tion of pressure, the 223 atom model of hydrogenated amorphous silicon with a 5-atom void undergoes a firstorder phase transition from a semiconducting low density amorphous state to a metallic high density amor-phous state with a discontinuous volume change of about 16%. The transition is accompanied by a coordina-tion change from fourfold to about sixfold. DOI: 10.1103/PhysRevB.73.035209 PACS number /H20849s/H20850: 61.43.Dq, 64.70.Kb, 64.70.Dv I. INTRODUCTION Silicon /H20849Si/H20850systems have a wide range of applications in electronic and photoelectronic devices, and hence they havebeen studied extensively for decades. Yet their structures andproperties are still being discovered. Under hydrostatic pres-sure, crystalline silicon /H20849c-Si/H20850undergoes a series of polymor- phic transitions. Diamond → /H9252-Sn→Imma→simple hexago- nal /H20849SH /H20850→Si/H20849VI/H20850→hexagonal close packed transitions have been synthesized1–4as its density is increased. c-Si subjected to nanoindentations also forms the /H9252-Sn structure5similar to that found in hydrostatic pressure experiments. The ex- panded phase of Si, clathrate, with remarkable electronic properties, converts to /H9252-Sn on compression as well.6 The formation of /H9252-Sn under pressure is not always a characteristic of Si. In nanocrystalline Si, a direct transitionto a SH is observed depending on its size. 7Recently Deb et al.8have reported pressure-induced high density amorphiza- tion of the crystalline portions in a porous Si sample. In thecase of amorphous silicon /H20849a-Si/H20850, experiments 9,10have found thata-Si shows a unique response to pressure and undergoes a transition from amorphous semiconductor-to-amorphousmetal with a sharp resistivity drop. In the study, it is arguedthat the transition is first order and the metallic amorphousstate is probably similar to liquidlike or disordered /H9252-Sn structures. However, this anomalous transition of Si is notfully clarified because of the limitations in the experiments.In the previous simulations using a constant-pressure ab ini- tiotechnique, 11we have provided a clear atomic-level pic- ture of the amorphous to amorphous phase transition in Siand found that the transition is indeed first order with achange in density of about 19%. Furthermore, the metallicamorphous phase is found to be resemblance to liquid-Si/H20849l-Si/H20850. Recently, Morishita 12has proposed a fivefold coordi- nated high-density amorphous /H20849HDA /H20850phase of Si and made a connection between the HDA phase and its liquid state.This prediction is particularly important because it indicatesdistinct high-density amorphous forms of Si as seen in water.A Raman scattering investigation of a-Si provides additional support for the amorphous to amorphous phase transition inSi. 13All these studies clearly suggest the existence of distinct amorphous states of Si with different densities and bondingenvironments. This phenomenon is called polyamorphism 14and has been observed in a wide variety of amorphous or liquid systems to date.15–18The polyamorphic phase transi- tion of Si has revived the idea of an underlying liquid-liquidphase transition, indicating the existence of a low-densityliquid and its glass transition to the amorphous solid. 8,13In- deed, the very fast time scale of ion hammering experimentsprovides the first evidence for a low-density liquid state ofSi. 19In a recent experiment, however, Kim et al.20have questioned the existence of a liquid-to-liquid transition in Si. For hydrogenated amorphous silicon /H20849a-Si:H /H20850,t oo u r knowledge, no pressure-induced phase transition has been reported so far. In this paper, we extend our study for ana-Si:H model with a void and explore its response to pres- sure using an ab initio technique within the generalized gra- dient approximation. A first order polyamorphic phase tran-sition is observed for the a-Si:H model through the simulation and the nature of semiconductor-to-metal transi-tion is characterized. Our findings probably provide informa-tion on silicon’s phase changes and electronic structure athigh pressure. II. COMPUTATIONAL METHOD Thea-Si:H model is due to Narkhmanson and Drabold.21 We have chosen this configuration because it consists of a void which allows us to better understand its effects to thebehavior of a-Si at high pressure. In order to generate a hydrogenated network, a cluster of Si atoms was removedfrom a perfectly coordinated configuration 22and then every dangling bond was terminated by a hydrogen atom. The hy-drogenated topology is, therefore, a passivated network. Thesupercell consists of 223 atoms /H20849211 Si and 12 H atoms /H20850with periodic boundary conditions. The model at zero pressurewas relaxed by the SIESTA code 23which employs the first principles pseudopotential method within the density-functional formalism and the generalized gradient approxi-mation of Perdew-Burke and Ernzerhof for the exchange-correlation energy. 24The ab initio technique uses a linear combination of atomic orbitals as the basis set, and norm-conserving Troullier-Martins pseudopotentials. 25A split- valence single- /H9256basis set was employed. A uniform mesh with a plane wave cutoff of 40 Ry was used to present theelectron density, the local part of the pseudopotentials, andPHYSICAL REVIEW B 73, 035209 /H208492006 /H20850 1098-0121/2006/73 /H208493/H20850/035209 /H208495/H20850/$23.00 ©2006 The American Physical Society 035209-1the Hartree and the exchange-correlation potential. We used /H9003-point sampling for the supercell’s Brillouin zone integra- tion. The results were checked with four special k-points but this made no variation in the energy and the volume differ-ences, and the atomic coordinates at zero pressure. Zero-point energies were omitted in the calculation because theychanged by less than 0.05 eV. 26The omitting zero-point en- ergies does not affect accuracy of the present simulationsince the relative energy difference between a-Si and a HDA phase was about 0.29 eV/atom. Pressure was increased in anincrement of 3 GPa up to 15 GPa and after this pressure, a1 GPa increase was used to predict accurately the transitionpressure and the topology of high pressure phase. For eachvalue of the applied pressure, the lattice vectors were opti-mized together with the atomic coordinates until the stresstolerance was less than 0.25 GPa and the maximum atomicforce was smaller than 0.01 eV/Å. The optimization wasperformed using a conjugate-gradient technique. III. RESULTS AND DISCUSSION In order to determine the nature of the phase transition in the configuration, we first plot its pressure-volume curve inFig. 1. Accordingly, the volume monotonically decreaseswith increasing pressure and the low-density amorphousphase is still preserved to 16 GPa. As the pressure is in-creased from 16 GPa to 17 GPa, the structural phase transi-tion begins and is accompanied by a dramatic volume drop,which is a characteristic of a first order phase transition.Owing to the transformation, the low-density a-Si:H net- work transforms into a high density amorphous state /H20849in the rest of the paper we refer to this high-pressure phase asHDA:H /H20850. The initial and HDA:H phases are illustrated in Fig. 2. These observations are particularly important becausethey again provide direct evidence for a discontinuous tran-sition between two distinct amorphous forms of Si and lendfurther support to an underlying transition between a high-density and a low-density liquid phase in supercooled Si. The volume reduction due to the phase change is found to be about 16%. This value is apparently smaller than 19%obtained for a-Si. Moreover, a-Si:H transforms to a less dense structure, compared to a-Si /H20849see Fig. 1 /H20850. This findingindicates that the HDA:H and HDA forms of Si have differ- ent bonding environments. The pressure-volume relation is fit to the third order Birch-Murhanham equation of state. The fitting yields thebulk modulus B=61 GPa, which is considerably less than that of a-Si, B=86 GPa calculated by an ab initio technique 11andB=87 GPa computed using a tight-binding method.27Such a result is expected because the existence of the void gives additional freedom in compressibility mecha-nism of the model, but note that the void does not change thethermodynamic aspect of the amorphous-to-amorphousphase transition in Si. It is also worth mentioning that during the compression of thea-Si:H model the simulation box remains almost cubic until the phase transition occurs. At 17 GPa, however, one ofthe simulation cell angles changes dramatically from 90° to97°. This means that the transformation into HDA:H in-volves clearly shear deformations. We next analyze the pressure-induced changes at the ato- mistic level in terms of real space pair distribution functions.Figure 3 shows the Si-Si and Si-H and H-H partials as afunction of the applied pressure. We find that the Si-Si bondlengths are gradually shifted to lower distances by increasingpressure and they are abruptly elongated at 17 GPa. Thisobservation reflects a transition into a higher-coordinatedstructure at this pressure. The second and the third peaks ofthe Si-Si correlation are also monotonically decreased underpressure. The simulation also reveals that the Si-H bondlengths remain almost invariant up to the transition pressureat which point the Si-H separations are dramatically ex- FIG. 1. Pressure-volume curve of the a-Si:H and a-Si models. The pressure-volume relation of a-Si is adopted from Ref. 11. FIG. 2. /H20849Color online /H20850/H20849a/H20850Low-density a-Si:H model at zero pressure. /H20849b/H20850The high-density amorphous phase predicted at 17 GPa.MURAT DURANDURDU PHYSICAL REVIEW B 73, 035209 /H208492006 /H20850 035209-2tended, similar to that seen in the Si-Si bonding. Moreover, we observe the formation of H-H bond /H20849s/H20850in HDA:H as clearly indicated by a peak around 1.0 Å in the H-H partialdistribution. In order to investigate the structural changes through the transition further, we plot the angle distribution function ofthe model at several pressures in Fig. 4. At zero pressure, asexpected, the function has single peak near the tetrahedralangle. As the configuration is compressed, the distribution issmoothly broadened with a decrease in its intensity. Thisimplies that the tetrahedral angles are slightly distorted up to17 GPa. As the system undergoes the phase transition, thefunction becomes rather broad with three main peaks around60° and 90° and 160°. The angles at 60° and 90° are a clearindication of metallic bonding in the network and a charac-teristic of the SH structure. The small peak near 160° is closeto the value of 149° presented in /H9252-Sn. These results suggest that the HDA:H phase partially consists of /H9252-Sn-like and SH-like configurations. Also it should be noted that the gen-eral shape of the bond angle distribution function fairly re-sembles that of l-Si. Table I presents the structural properties of the model athigh pressures. The initial compression shortens the Si-Si bond lengths and narrows the tetrahedral angles but has al-most no effect on the Si-H bond lengths because of the void.The configuration still remains fourfold coordinated up to17 GPa. Owing to the phase transition, 25% of H atomsforms two bonds with Si atoms. Each H atom has a shortbond length at 1.650–1.720 Å and a slightly larger one at1.733–1.791 Å. The Si-H-Si angles range from 139° and171°. These twofold-coordinated H topologies might be con-sidered as body centered H clusters. The body centered Hhas been studied extensively because of its possible rel-evance to H migration in a-Si:H. The bond lengths and the bond angles of the twofold coordinated H clusters in theHDA:H phase are found to be comparable with those of thebody centered H configurations studied in a-Si:H. 28,29We also see the formation of a H-H bond at 17 GPa. MolecularH is also not an unusual configuration in a-Si:H because recent studies have argued that at least some of gooda-Si:H samples consist of substantial amounts of molecular H. 30,31The rest of the H atoms are still onefold coordinated and the Si-H bond lengths are about 1.597–1.773 Å. Weshould note that the Si-H separations in HDA:H are notice-ably larger than those of the low density Si:H model andhence they are weakly bound to the Si atoms. Further relax-ing the model at 17 GPa until the force is smaller than0.004 eV/Å did not change this feature of the H clusters.Therefore, these weakly bonded H configurations might berelated to the topology of the high density phase. Indeed, anab initio simulation 31has revealed that H forms weak bonds with tetrahedrally coordinated Si. This indicates that the co-ordination number of Si can be a factor for determining thenature of Si-H bonds. Additionally, when a system trans-forms into a higher coordinated state, bond lengths are gen-erally enlarged, and hence one expects a similar tendency forSi-H separations as well. Therefore, we believe that the for-mation of weakly bonded H is due to higher coordination ofthe HAD:H phase. Furthermore, these H clusters are prob-ably stable configurations since their bond lengths are com-parable with those of the body centered H clusters consid-ered in a-Si:H. 28,29 As for the phase transition, the average coordination of Si increases to about 6.2. This coordination is significantly dif-TABLE I. Structural parameters of the a-Si:H model. The av- erage Si-Si and Si-H bond lengths and the percentage of Si and Hcoordination. Pressure /H20849GPa /H20850 0 9 16 17 Si-Si /H20849Å/H20850 2.45 2.37 2.33 2.55 Si-H /H20849Å/H20850 1.59 1.58 1.58 1.69 C 4-Si 100.0 100.0 100.0 1.89 C5-Si 0.0 0.0 0.0 15.16 C6-Si 0.0 0.0 0.0 55.9 C7-Si 0.0 0.0 0.0 21.8 C8-Si 0.0 0.0 0.0 5.24 C1-H 100.0 100.0 100.0 75 C2-H 0.0 0.0 0.0 25 FIG. 3. The partial pair distribution functions under pressure. FIG. 4. Pressure dependence of the bond angle distribution function.AB INITIO SIMULATION OF POLYAMORPHIC PHASE … PHYSICAL REVIEW B 73, 035209 /H208492006 /H20850 035209-3ferent from that of a-Si reported in Refs. 11 and 12, but close to an octahedrally coordinated HDA phase predicted in the Raman study.13It is worth mentioning here that Morishita12 obtained the distinct HDA forms of Si by compression on thefivefold coordinated HDA phase and proposed that the amor-phous phases of Si nearly correspond to those of the crystal-line forms /H20849the diamond, /H9252-Sn and SH structures /H20850. Therefore, one might see different HDA phases of Si in simulationsdepending on initial geometry of models, loading conditions,pressure-control techniques, and temperature. It is also noteworthy that the HDA:H phase consists of differently bounded domains as seen in l-Si. The formation of distinct geometries in amorphous systems can be ex-plained in terms of inhomogeneous stress distributions: Un-like crystalline structures, bond lengths and angles of disor-dered materials are not uniform and hence they have strainedtopologies even at zero pressure. When they are subjected topressure, the local stress varies from site to site. Therefore,some parts of disordered networks are expected to be morecompressed than the other parts. Furthermore such a featureproduces nonuniform nucleation in the networks. Actually,we find that the more compressed parts of the model due tothe distortions behave as nucleation centers in the model andform higher coordinated domains. These behaviors differ forcrystalline states in which stress is uniformly distributed onall positions because all bonds and angles have the samevalues, and nucleation occurs homogeneously across entirelattice structures to preserve the transition symmetry. We finally examine the pressure-induced changes in the electronic properties of the model because the applications ofSi systems in technology strongly depend on the nature ofelectron states near the Fermi level. Figure 5 shows thechange of the highest occupied molecular orbital /H20849HOMO /H20850, the lowest unoccupied molecular orbital /H20849LUMO /H20850, and the band gap energy E g/H20849eV/H20850as a function of the applied pres- sure. With increasing pressure, both HOMO and LUMO states gradually shift to higher energies but the shift of theLUMO state is larger than that of the HOMO state. Such atrend results to an increase of the band gap energy. At17 GPa the band gap is abruptly closed, indicating apressure-induced semiconductor-to-metal transition in thesystem. The pressure derivative of the band gap from a linear fit in the pressure ranges from 0 GPa to 12 GPa is calculatedto be +1.4 meV/kbar. This value is a factor of 6 larger thanthe experimental result of +0.25 meV/kbar of a-Si. 32 IV. CONCLUSIONS Anab initio constant-pressure technique is applied to study the pressure-induced phase transition of an a-Si:H model with a void. For the first time, the simulation revealsall characteristic of a transformation from a low-densityamorphous phase to a HDA:H phase in a-Si:H through a discontinuous transition. Therefore, the existence of a void does not change the nature of phase transition in a-Si:H. Our findings are important for understanding of amorphous Siand its relations to liquid-Si. ACKNOWLEDGMENTS The author is grateful to Professor D. A. Drabold for read- ing the paper and the amorphous model and the SIESTAcommittee for providing the ab initio MD code. Most of the work was done on the AIX-IBM clusters at the High-Performance Computer Center, the University of Texas atEl Paso. 1M. C. Gupta and A. L. Ruoff, J. Appl. Phys. 51, 1072 /H208491980 /H20850. 2H. Olijnyk, S. K. Sikka, and W. B. Holzapfel, Phys. Lett. 103A , 137 /H208491984 /H20850. 3J. Z. Hu, L. D. Merkle, C. S. Menoni, and I. L. Spain, Phys. Rev. B34, 4679 /H208491986 /H20850; J. Z. Hu and I. L. Spain, Solid State Commun. 51, 263 /H208491984 /H20850. 4M. I. McMahon and R. J. Nelmes, Phys. Rev. B 47, 8337 /H208491993 /H20850; M. I. McMahon, R. J. Nelmes, N. G. Wright, and D. R. Allan,ibid. 50, 739 /H208491994 /H20850. 5V . Domnich and Y . Gogotsi, Rev. Adv. Mater. Sci. 3,1 /H208492002 /H20850;V . Domnich, Y . Gogotsi, and M. Trenary, Mater. Res. Soc. Symp.Proc. Q8.9.1 ,6 4 /H208492001 /H20850. 6G. Ramachadran, P. F. McMillan, S. K. Deb, M. Somayazulu, J. Gryko, Jianjun Dong, and O. T. Sankey, J. Phys.: Condens.Matter 12, 4013 /H208492000 /H20850. 7S. H. Tolbert, A. B. Herhold, L. E. Brus, and A. P. Alivisatos, Phys. Rev. Lett. 76, 4384 /H208491996 /H20850. 8S. K. Deb, M. Wilding, M. Somayazulu, and P. F. McMillan, Nature /H20849London /H20850414, 528 /H208492001 /H20850. 9O. Shimomura, S. Minomura, N. Sakai, K. Asaumi, K. Tamura, J. Fukushima, and H. Endo, Philos. Mag. 29, 547 /H208491974 /H20850. 10S. Minomura, High Pressure and Low-Temperature Physics /H20849Ple- num, New York, 1978 /H20850, p. 483. 11M. Durandurdu and D. A. Drabold, Phys. Rev. B 64, 014101 /H208492001 /H20850. 12T. Morishita, Phys. Rev. Lett. 93, 055503 /H208492004 /H20850. 13P. F. McMillan, J. Mater. Chem. 14, 1506 /H208492004 /H20850. 14P. H. Poole, T. Grande, F. Sciortino, H. E. Stanley, and C. A. FIG. 5. The HOMO and LUMO states and the band gap energy Eg/H20849eV/H20850as a function of pressure.MURAT DURANDURDU PHYSICAL REVIEW B 73, 035209 /H208492006 /H20850 035209-4Angle, Comput. Mater. Sci. 4, 373 /H208491995 /H20850. 15O. Mishima, L. D. Calvert, and W. Whalley, Nature /H20849London /H20850 314,7 6 /H208491985 /H20850. 16D. J. Lacks, Phys. Rev. Lett. 84, 4629 /H208492000 /H20850;80, 5385 /H208491998 /H20850. 17P. F. McMillan, B. Piriou, and R. Couty, J. Chem. Phys. 81, 4234 /H208491984 /H20850. 18O. B. Tsiok, V . V . Brazhkin, A. G. Lyapin, and L. G. Khvostant- sev, Phys. Rev. Lett. 80, 999 /H208491998 /H20850. 19A. Hedler, S. L. Klaumunzer, and W. Wesch, Nat. Mater. 3, 804 /H208492004 /H20850. 20T. H. Kim, G. W. Lee, B. Sieve, A. K. Gangopadhyay, R. W. Hyers, T. J. Rathz, J. R. Rogers, D. S. Robinson, K. F. Kelton,and A. I. Goldman, Phys. Rev. Lett. 95, 085501 /H208492005 /H20850. 21S. M. Narkhmanson and D. A. Drabold, Phys. Rev. B 58, 15325 /H208491998 /H20850. 22B. R. Djordjevic, M. F. Thorpe, and F. Wooten, Phys. Rev. B 52, 5685 /H208491995 /H20850.23P. Ordejón, E. Artacho, and J. M. Soler, Phys. Rev. B 53, R10441 /H208491996 /H20850; D. Sánchez-Portal, P. Ordejón, E. Artacho, and J. M. Soler, Int. J. Quantum Chem. 65, 453 /H208491997 /H20850. 24J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 /H208491996 /H20850. 25N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 /H208491991 /H20850. 26C. G. Van de Walle, P. J. H. Denteneer, Y . Bar-Yam, and S. T. Pantelides, Phys. Rev. B 39, 10791 /H208491989 /H20850. 27J. L. Feldman, N. Bernstein, D. A. Papaconstantopoulos, and M. J. Mehl, Phys. Rev. B 70, 165201 /H208492004 /H20850. 28P. A. Fedders, Phys. Rev. B 64, 165206 /H208492001 /H20850. 29B. Tuttle and J. B. Adams, Phys. Rev. B 57, 12859 /H208491998 /H20850. 30P. A. Fedders, D. J. Leopold, P. H. Chan, R. Borzi, and R. E. Norberg, Phys. Rev. Lett. 85, 401 /H208492000 /H20850. 31P. A. Fedders, Phys. Rev. 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PhysRevB.87.174512.pdf

PHYSICAL REVIEW B 87, 174512 (2013) Superconductivity induced by electron doping in La 1−xMxOBiS 2(M= Ti, Zr, Hf, Th) D. Yazici,1,2K. Huang,1,2,3B. D. White,1,2I. Jeon,1,2,3V . W. Burnett,1,2A. J. Friedman,1,2I. K. Lum,1,2,3 M. Nallaiyan,4S. Spagna,4and M. B. Maple1,2,3,* 1Department of Physics, University of California, San Diego, La Jolla, California 92093, USA 2Center for Advanced Nanoscience, University of California, San Diego, La Jolla, California 92093, USA 3Materials Science and Engineering Program, University of California, San Diego, La Jolla, California 92093, USA 4Quantum Design, 6325 Lusk Boulevard, San Diego, California 92121, USA (Received 26 March 2013; revised manuscript received 26 April 2013; published 16 May 2013) We report a strategy to induce superconductivity in the BiS 2-based compound LaOBiS 2. Instead of substituting F for O, we increase the charge-carrier density (electron dope) via substitution of tetravalent Th+4,H f+4,Z r+4,a n d Ti+4for trivalent La+3. It is found that both the LaOBiS 2and ThOBiS 2parent compounds are bad metals and that superconductivity is induced by electron doping with Tcvalues of up to 2.85 K. The superconducting and normal states were characterized by electrical resistivity, magnetic susceptibility, and heat capacity measurements. Wealso demonstrate that reducing the charge-carrier density (hole doping) via substitution of divalent Sr +2for La+3 does not induce superconductivity. DOI: 10.1103/PhysRevB.87.174512 PACS number(s): 74 .70.Dd, 74 .25.F−,7 4.25.Op I. INTRODUCTION Superconductivity with Tc=8.6 K has recently been reported in the layered compound Bi 4O4S3.1,2Following this report, several other BiS 2-based superconductors, including LnO1−xFxBiS 2(Ln=La, Ce, Pr, Nd, Yb) with Tcas high as 10 K, have been synthesized and studied.3–12These materials have a layered crystal structure composed of superconducting BiS 2layers and blocking layers of Bi 4O4(SO 4)1−xfor Bi 4O4S3 andLnOf o r LnO1−xFxBiS 2(Ln=La, Ce, Pr, Nd, and Yb). This structural configuration is similar to the situation encountered in the high- Tclayered cuprate and Fe-pnictide superconductors in which superconductivity primarily resides in CuO 2planes and Fe-pnictide layers, respectively.13–17Even though BiS 2-based superconductors share a similar crys- tal structure with Fe-pnictide superconductors, they exhibit some important differences. The undoped parent compounds,LnFeAsO, display a spin density wave (SDW) or a structural instability near 150 K. 15–17Superconductivity emerges when the SDW is suppressed towards zero temperature eitherthrough charge carrier doping or application of pressure. 18Tcis raised as high as 55 K by replacement of La by other rare-earth elements such as Sm.15,16In contrast, the phosphorus-based analogues LnFePO do not show a SDW transition or structural instability but still exhibit superconductivity.19,20For LaFePO, values of Tcthat range from 3 to 7 K,20,21without charge carrier doping, have been reported. Both LnFeAsO and LnFePO remain metallic to low temperatures, while the undoped parentcompounds LnOBiS 2are bad metals. It has been suggested that superconductivity emerges in close proximity to an insulating normal state for the optimal superconducting sample.11 Several distinct examples of chemical substitution have been found to induce superconductivity in LnFeAsO com- pounds including substituting F for O,11,22–27Co for Fe,28Sr for La,29Th for Gd,30and also the introduction of oxygen vacancies.31,32Substituting F for O induces superconductiv- ity in BiS 2-based superconductors.3,4,6–12,33–36To determine whether superconductivity might emerge under other condi-tions, we chose to dope electrons via chemical substitutionon the La site in LaOBiS 2. In this study, we demonstratethat substitution of tetravalent Th+4,H f+4,Z r+4, and Ti+4 for trivalent La+3in LaOBiS 2induces superconductivity. We also observed that substitution of divalent Sr+2for La+3(hole doping) does not induce superconductivity. II. EXPERIMENTAL METHODS Polycrystalline samples of La 1−xThxOBiS 2(0/lessorequalslantx/lessorequalslant1), La1−xHfxOBiS 2(0/lessorequalslantx/lessorequalslant0.4), La 1−xZrxOBiS 2(0/lessorequalslantx/lessorequalslant0.3), La1−xTixOBiS 2(0/lessorequalslantx/lessorequalslant0.3), and La 1−xSrxOBiS 2 (0/lessorequalslantx/lessorequalslant0.3) were prepared by a two-step solid state reaction method using high-purity starting materials. Initially,the Bi 2S3precursor powders were prepared by reacting Bi and S grains together at 500◦C in an evacuated quartz tube for 10 hours. Starting materials of La, La 2O3,S ,a n dB i 2S3 powders and either M=Th or Ti chunks, Hf granules, or Zr foil, were weighed in stoichiometric ratios based on nominalconcentrations La 1−xMxBiS 2(M=Th, Hf, Zr, Ti). Next, they were thoroughly mixed, pressed into pellets, sealed inevacuated quartz tubes, and annealed at 865 ◦C for 72 hours. Additional regrinding and sintering at 700◦C for 3 days was performed to promote phase homogeneity. The crystalstructure was verified by means of x-ray powder diffraction(XRD) using a Bruker D8 Discover x-ray diffractometer withCu-K αradiation. The resulting XRD patterns were fitted via Rietveld refinement37using the GSAS +EXPGUI software package.38,39The chemical composition was investigated by means of energy dispersive x rays (EDX) using a FEI Quanta600 scanning electron microscope equipped with an INCAEDX detector from Oxford instruments. Electrical resistivitymeasurements were performed using a home-built probe in aliquid 4He Dewar for temperatures 1 K /lessorequalslantT/lessorequalslant300 K by means of a standard four-wire technique using a Linear ResearchLR700 ac resistance bridge. Magnetization measurementswere made for 2 K /lessorequalslantT/lessorequalslant300 K and in magnetic fields H=5 Oe using a Quantum Design MPMS. Alternating current magnetic susceptibility was measured down to ∼1K in a liquid 4He Dewar. Specific heat measurements and electrical resistivity measurements under applied magnetic 174512-1 1098-0121/2013/87(17)/174512(8) ©2013 American Physical SocietyD. YAZICI et al. PHYSICAL REVIEW B 87, 174512 (2013) FIG. 1. (Color online) XRD patterns for selected concentrations of La 1−xThxOBiS 2where values of xare explicitly labeled. The arrow indicates a Bi and/or Bi 2S3impurity. field for La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2samples were performed for 0.36 K /lessorequalslantT/lessorequalslant30 K in a Quantum Design PPMS DynaCool with a3He insert using a thermal relaxation technique. III. RESULTS A. Crystal structure and sample quality Figure 1shows the powder XRD patterns of LaOBiS 2 and ThOBiS 2samples. Overall, the main diffraction peaks of these two samples can be well indexed to a tetragonalstructure with space group P4/nmm . The crystal structure has previously been designated as having the ZrCuSiAs typestructure, 3–5,8however, the CeOBiS 2type structure may be a more appropriate structure assignment.11While the crystal structure of LaOBiS 2and ThOBiS 2has a different Wyckoff sequence than that of ZrCuSiAs, it has the same Wyckoffsequence and positions, and similar unit cell parameter ratios(c/a) as the crystal structure for CeOBiS 2.40The crystallo- graphic data for ThOBiS 2obtained from Reitveld refinement of the powder XRD pattern given in Fig. 1are tabulated in Table I. The inset of Fig. 2(a), displays the crystal structure of LaOBiS 2. The structure is composed of stacked La 2O2 layers and two BiS 2layers in each unit cell. To dope electrons into the BiS 2conduction layers, we substituted tetravalent Th+4,H f+4,Z r+4, and Ti+4for La+3and to dope holes into the BiS 2conduction layers, we substituted divalent Sr+2for trivalent La+3. We find that samples for the entire range of Th substitutions below the solubility limit, Hf substitutions inthe range 0 .1/lessorequalslantx/lessorequalslant0.4, and Zr, Ti, and Sr substitutions in the range 0 .1/lessorequalslantx/lessorequalslant0.3 can be described by the same spaceTABLE I. Crystallographic data for the compound ThOBiS 2, which was obtained from Rietveld refinement of a powder x-ray diffraction pattern collected at room temperature. Chemical Formula ThOBiS 2 Space Group P4/nmm (No. 129) a(˚A) 3.9623 (8) c(˚A) 13.5062 (5) V(˚A3) 212.041 (6) Z 2 Atom site xy z Th 2c 0.25 0.25 0.1017(5) Bi 2c 0.25 0.25 0.6013(2) S1 2c 0.25 0.25 0.3080(17) S2 2c 0.25 0.25 0.8382(12) O 2a 0.75 0.25 0 group. As shown in Fig. 2(b), the lattice parameters aandc for La 1−xThxOBiS 2(0/lessorequalslantx/lessorequalslant1) decrease with increasing Th concentration, although the relative decrease of ais much smaller than that of c. With Th doping, the positions of Bragg reflections shift systematically. For example, the (004)reflections shift to higher angles, consistent with a decreaseofc. This result indicates that Th is really incorporated into the lattice; although, some amounts of elemental Bi and/orBi 2S3separate from the lattice, indicated by arrows in Figs. 1 xxxxa c x FIG. 2. (Color online) (a) X-ray diffraction (XRD) pattern for LaOBiS 2. Black circles represent data and the red solid line is the result of Rietveld refinement of the data. The arrows indicatea Bi and/or Bi 2S3impurity. (b) Lattice parameters aandcvs nominal Th concentration x. (c) Unit cell volume Vvs nominal Th concentration x. 174512-2SUPERCONDUCTIVITY INDUCED BY ELECTRON DOPING ... PHYSICAL REVIEW B 87, 174512 (2013) and2(a). We could not always distinguish between whether the impurity phase was Bi or Bi 2S3, since their Bragg reflections occur at very similar positions. If the impurity is assumedto be Bi, the amount is between 3% to 15% by mass and ifthe impurity is assumed to be Bi 2S3, the amount is between 5% to 18% by mass. In either case, the ratio increases withincreasing Th concentration. If the impurity is Bi, there shouldbe other La-S-O based impurities, but we could not detectany elemental S or La impurities or binary compounds madefrom La-S-O. The impurity is therefore likely Bi 2S3, which should be balanced with a La 2O3impurity. Unfortunately, the Bragg reflections of La 2O3overlap with the main phase peak positions, which makes it difficult to estimate the impurityratio of La 2O3using Rietveld refinement. For 0 .5/lessorequalslantx/lessorequalslant0.9, the lattice parameters (not shown) are close to the latticeparameters of ThOBiS 2and do not change appreciably with x. The impurity amount increases with xand other impurity peaks begin to appear in this region, indicating the probableexistence of a solubility limit of La/Th near x=0.45. More direct evidence of Th incorporation into the lattice comes from chemical composition measurements by energy-dispersive x-ray (EDX) microanalysis. The EDX spectrumwas collected from a single grain. Quantitative analysis forselected samples gives La:Th ratios of 0.08, 0.12, 0.18, and0.24 for the samples with nominal ratios 0.1, 0.15, 0.20, and0.25, respectively. This result demonstrates that most of theTh was successfully doped into the samples. For the Hf, Zr, and Ti substitutions, the powder XRD patterns and lattice parameters aandcare given for La 1−xMxOBiS 2with M= Hf, Zr, Ti, Sr and x=0.2i nac FIG. 3. (Color online) (a) XRD patterns and (b) lattice parameters aandcfor La 1−xMxOBiS 2with M= Hf, Zr, Ti, Sr and x=0,0.2. The arrows indicate Bi and/or Bi 2S3impurities.01234500.30.60.91.2 0 50 100 150 200 250 3000123450.51.01.52.02.50 50 100 150 200 250 300 (c) (T) / (5K) x= 0.15 x= 0.20 x= 0.25 x= 0.30 x= 0.35 x= 0.40 x= 0.45 T (K)La1-xThxOBiS2 (b)La1-xThxOBiS2 x= 0 x= 0. 1 x= 0. 15 x= 0.20 x= 0.25 x= 0.30 x= 0.35 x= 0.40 x= 0.45 x= 1.00(T) / (290K) T (K)T (K) LaOBiS2 ThOBiS2(T) / (290K) (a) FIG. 4. (Color online) (a) Electrical resistivity ρ, normalized by its value at 290 K, vs temperature Tfor LaOBiS 2and ThOBiS 2.( b ) ρ, normalized by its value at 290 K, vs Tfor La 1−xThxOBiS 2(0/lessorequalslant x/lessorequalslant1). (c) ρ, normalized to its value in the normal state at 5 K, vs T emphasizing the transition into the superconducting state. Figs. 3(a)and3(b), respectively. The lattice parameters aand cexhibit behavior similar to Th substitution, wherein the c lattice parameter decreases with increasing Hf, Zr, and Ticoncentration and the alattice parameter does not show an appreciable concentration dependence. For Sr substitution, thealattice parameter increases slightly, while cdecreases with increasing Sr concentration. Even though the ionic size of Th isbigger than the others (Hf, Zr, and Ti), the clattice parameters are very close to each other. The decrease of the caxis may be attributed to a strengthening of the interlayer bonding as aconsequence of doping. B. Electrical resistivity Electrical resistivity ρ(T) data, normalized by the value of ρat 290 K, are shown in Fig. 4for the pure LaOBiS 2and ThOBiS 2samples. For both compounds, ρinitially decreases with decreasing temperature, exhibits a minimum at T=220 and 206 K, respectively, and then shows semiconductor-likebehavior down to the lowest temperatures measured. For theLa 1−xThxOBiS 2system, the minimum in ρis suppressed forx=0.1, and ρ(T) exhibits relatively strong temperature dependence and an inflection point (indicated by an arrow) asshown in Fig. 4(b).F o rx=0.15, this feature disappears and ρdrops to zero below T c=2.85 K. ρ(T) becomes less tem- perature dependent for 0 .15/lessorequalslantx/lessorequalslant0.45 and increases with decreasing temperature until the onset of superconductivity.First-principles calculations have suggested that there may bea charge density wave instability or enhanced correlations inthe LaO 1−xFxBiS 2system.33We are unable to unambiguously observe such an instability from the electrical resistivitymeasurements in this study. However, the inflection point couldbe related to such an effect. Electrical resistivity measurements for La 1−xHfxOBiS 2, La1−xZrxOBiS 2, and La 1−xTixOBiS 2samples are shown in Figs. 5–7, respectively. Resistive superconducting transition curves for these systems are indicated in the right inset 174512-3D. YAZICI et al. PHYSICAL REVIEW B 87, 174512 (2013) 01234500.30.60.91.2 0 50 100 150 200 250 30000.20.40.60.8 0 50 100 150 200 250 30000.20.40.60.8 (T) / (5K) T (K) La0.8Hf0.2OBiS2 La0.7Hf0.3OBiS2 La0.6Hf0.4OBiS2 (Ω cm) T (K) La0.85Hf0.15OBiS2 (Ω cm) T (K) FIG. 5. (Color online) Electrical resistivity ρvs temperature Tfor La1−xHfxOBiS 2. The left inset shows ρvsTfor La 0.85Hf0.15OBiS 2. The right inset displays the superconducting transition curves forsamples with concentrations 0 .2/lessorequalslantx/lessorequalslant0.4. of Figs. 5–6and in Fig. 7(c). All of these doping studies show similar characteristics such as an inflection pointanomaly in the normal state with low concentration andinduced superconductivity for concentrations starting withx=0.15 for La 1−xThxOBiS 2andx=0.2f o rL a 1−xHfxOBiS 2, La1−xZrxOBiS 2, and La 1−xTixOBiS 2. Furthermore, all of these systems show semiconductor-like behavior in the nor-mal state. The inflection point in ρof samples with low concentration is emphasized in the left inset of Figs. 5 and6, and in Fig. 7(b) where the anomaly is indicated by an arrow. These anomalies all appear to be present at acommon temperature of roughly 120 K in concentrationsxjust below those where superconductivity is induced. To compare the superconducting transition temperatures ( T c), we consider data for 20% substitution of La by Th, Hf,Zr, and Ti in Fig. 8(a). The superconducting transition temperatures, as shown in Fig. 8(b), are characterized by the temperatures where the electrical resistivity drops to 50% ofthe normal state resistivity, and the width of the transition is FIG. 6. Electrical resistivity ρvs temperature T, plotted for La0.8Zr0.2OBiS 2. The left inset shows ρvsTfor La 0.9Zr0.1OBiS 2. The right inset displays the superconducting transition for La0.8Zr0.2OBiS 2.01234500.20.40.60.81.0 0 50 100 150 200 250 300010203040500510150 50 100 150 200 250 300 (c) La0.8Ti0.2OBiS2 La0. 7Ti0.3OBiS2 (T) / (5K) T (K)(b) La0.9Ti0.1OBiS2 (mΩ cm) T (K)T (K) La0.8Ti0.2OBiS2 La0.7Ti0.3OBiS2 (mΩ cm) (a) FIG. 7. (Color online) (a) Electrical resistivity ρvs tempera- tureT, plotted for La 1−xTixOBiS 2(x=0.2–0.3). (b) ρvsTfor La0.9Ti0.1OBiS 2. (c) The superconducting transition curves for x= 0.2–0.3. determined by the temperatures where the resistivity drops to 90% and 10% of the normal state resistivity. Electrondoping clearly induces superconductivity in LaOBiS 2.T h e Tc’s are quite similar to one another, but the transition width is sharper for La 0.8Hf0.2OBiS 2and La 0.8Ti0.2OBiS 2than for La0.8Zr0.2OBiS 2and La 0.8Th0.2OBiS 2.T h el o w e s t Tcis seen in La 0.8Zr0.2OBiS 2. There does not appear to be a clear correlation between Tcand the lattice parameters. Meanwhile, Fig.9shows ρ(T) measurements for La 1−xSrxOBiS 2wherein no evidence of a superconducting transition is observed downto∼1 K in the range 0 .1/lessorequalslantx/lessorequalslant0.3. This result suggests that hole doping is not sufficient to induce superconductivity. Itis, however, interesting to note that the magnitude of ρat low temperatures increases with increasing Sr concentration,which is similar to the behavior observed with Th, Hf, Zr, andTi doping. xx x FIG. 8. (Color online) (a) The superconducting transition for 20% concentration of Th, Hf, Zr, and Ti. (b) Superconducting transition temperatures for La 1−xMxOBiS 2with M=Th, Hf, Zr, Ti and x=0.2. 174512-4SUPERCONDUCTIVITY INDUCED BY ELECTRON DOPING ... PHYSICAL REVIEW B 87, 174512 (2013) 0 50 100 150 200 250 3000.010.1110100 (Ω cm) T (K) LaOBiS2 La0.9Sr0.1OBiS2 La0.8Sr0.2OBiS2 La0.7Sr0.3OBiS2 FIG. 9. (Color online) Electrical resistivity ρvs temperature T for La 1−xSrxOBiS 2(0/lessorequalslantx/lessorequalslant0.3), plotted on a semilogarithmic scale. The temperature dependence of ρ, normalized by its value at 5 K for La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2samples, is shown in Figs. 10(a) and10(b) , respectively, under several ap- plied magnetic fields ( H=0, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 1, and 9 T) down to 0.36 K. Both samples undergo relativelysharp superconducting transitions at T c=2.85 and 2.40 K for La 0.8Th0.2OBiS 2and La 0.8Hf0.2OBiS 2, respectively. With increasing magnetic field, the transition width broadens andthe onset of superconductivity gradually shifts to lower tem-peratures. Similar broadening of the transition was observed in the high- T clayered cuprate and Fe-pnictide superconductors and attributed to the vortex-liquid state.34,41Figures 10(c) and 10(d) show the upper critical field Hcversus Tfor 01200.30.60.91.20123 00.30.60.91.2 00.30.60.91.20123 012300.30.60.91.2 (d)Hc(T) 90%n 50%n 10%nHc2(0) = 1.12 T HirrHc2La0.8Hf0.2OBiS2 T (K)(b) (T) / (5 K)H = 9 T La0.8Hf0.2OBiS2 H = 0 0.01 T 0.05 T 0.1 T 0.2 T 0.3 T 0.4 T 0.5 T 1 T 9 TT (K) T (K) H = 0La0.85Th0.15OBiS2 (T) / (5 K)H = 9 T (a) (c)Hc2(0) = 1.09 T HirrHc2La0.85Th0.15OBiS2 90%n 50%n 10%nHc(T) T (K) FIG. 10. (Color online) (a) and (b) Resistive superconducting transition curves for La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2samples, respectively, measured under several different applied magnetic fields ( H=0, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 1, 9 T). (c) and (d) The temperature dependence of the upper critical field Hc2, andHirr, determined from the 90% and 10% normal state ρfor La0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2samples, respectively. The temperature corresponding to the 50% normal state ρis also shown.0123-1.0-0.8-0.6-0.4-0.20-1.6-1.2-0.8-0.40012345 -4-20012345 0123-0.2-0.10 (b)Tc ~ 2.05 KLa0.85Th0.15OBiS2(arb. units) T (K)T (K) Tc ~ 2.51 KH = 5 Oe La0.85Th0.15OBiS2ZFC FC(10-3 emu/g) (a) (c)T (K) La0.8Hf0.2OBiS2Tc ~ 2.81 KZFC FC(10-3 emu/g) H = 5 Oe (d) T (K)Tc ~ 1.42 KLa0.8Hf0.2OBiS2dc dcac (arb. units)ac FIG. 11. (Color online) (a) and (c) dc magnetic susceptibility χdcvs temperature Tfor La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2, respectively, measured in field cooled (FC) and zero field cooled(ZFC) conditions with a 5 Oe applied magnetic field. (b) and (d) Real part of the ac magnetic susceptibility χ acvsTfor La 0.85Th0.15OBiS 2 and La 0.8Hf0.2OBiS 2, respectively. The superconducting critical temperature Tcis indicated and labeled explicitly. La0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2samples, correspond- ing to the temperatures where the resistivity drops to 90% ofthe normal state resistivity ρ n(T,H )(Tc,onset), 50% of ρn(T,H ) (Tc), and 10% of ρn(T,H )(Tc,zero) in applied magnetic fields. Using the conventional one-band Werthamer-Helfand-Hohenberg (WHH) theory, 42the orbital critical fields Hc2(0) for La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2compounds were inferred from their initial slopes of Hcwith respect to T, yielding values of 1.09 and 1.12 T, respectively. These valuesofH c2(0) are very close to the values seen in Sr 1−xLaxFBiS 2 (1.04 T)43and LaO 0.5F0.5BiS 2(1.9 T),8suggesting that the superconducting state in BiS 2-based superconductors probably shares a common character. C. Magnetization To determine whether superconductivity is a bulk phe- nomenon in La 1−xMxOBiS 2, zero field cooled (ZFC) and field cooled (FC) measurements of the magnetic suscepti-bility χ dc(T) were made in a magnetic field of 5 Oe for the La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2samples. These measurements are plotted in Figs. 11(a) and 11(c) .Z F C measurements yield diamagnetic screening signals with Tc onset values that are lower than values obtained from ρ(T) data, while FC measurements reveal evidence for strongvortex pinning. Alternating current magnetic susceptibilitymeasurements for La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2 samples are shown in Figs. 11(b) and 11(d) , respectively. These data exhibit signatures of SC with slightly lower Tc’s than those observed in ρandχdcmeasurements. It is clear that, even though the superconducting transition is incompleteat 1.3 K, the volume fractions are ∼67% and ∼16% for La 0.85Th0.15OBiS 2and La 0.8Hf0.2OBiS 2samples, respectively, according to the ac magnetic susceptibility measurements. 174512-5D. YAZICI et al. PHYSICAL REVIEW B 87, 174512 (2013) FIG. 12. (Color online) Specific heat Cdivided by temperature T,C/T ,v sTfor La 0.85Th0.15OBiS 2.C/T vsT2i ss h o w ni nt h e inset in the upper left hand part of the figure. The red line is a fit of the expression C(T)=γT+βT3to the data that yields γ=0.58 mJ/mol f.u. K2and/Theta1D=220 K. The inset in the lower right part of the figure shows a plot of Ce/TvsT,w h e r e Ce/Tis the electronic contribution to the specific heat, in the vicinity of the superconducting transition. An idealized entropy conserving construction yields Tc= 2.5Ka n d /Delta1C/γT c=0.91. D. Specific heat Specific heat data for La 0.85Th0.15OBiS 2and La0.8Hf0.2OBiS 2samples are displayed in Figs. 12and13, respectively. The data for La 0.85Th0.15OBiS 2cover the temperature range 0.36 to 30 K. A fit of the expressionC(T)=γT+βT 3to the data in the normal state, where γis the electronic specific heat coefficient and βis the FIG. 13. (Color online) Specific heat Cdivided by temperature T,C/T ,v sTfor La 0.8Hf0.2OBiS 2. A plot of C/T vsT2is shown in the inset in the upper left-hand part of the figure. The red line is afit of the expression C(T)=γT+βT 3to the data that yields γ= 2.12 mJ/mol f.u. K2and/Theta1D=228 K. A plot of Ce/TvsT,i nt h e vicinity of the superconducting transition, is displayed in the inset inthe lower right hand side of the figure. The superconducting transition temperature T c, obtained from the electrical resistivity measurements (Tc=2.36 K), is indicated by an arrow.0 0.1 0.2 0.3 0.40123 Tc from (T) Tc from χac Tc from χdcTc (K) XSCLa1-xThxOBiS2 FIG. 14. (Color online) (a) Superconducting transition tempera- tureTcvs Th concentration xphase diagram for the La 1−xThxOBiS 2 system. The phase diagram is constructed from electrical resistivity ρ(filled triangles), dc magnetic susceptibility χdc(filled diamonds), and ac magnetic susceptibility χac(filled circles) measurements. The Tc-xphase boundary (dark gray area) separates the superconducting phase (SC) from the normal phase. The value of Tcfromρ(T)w a s defined by the temperature where ρdrops to 50% of its normal state value and vertical bars indicate the temperatures where the electrical resistivity drops to 90% and 10% of its normal state value. Values of Tcfromχacandχdcwere defined by the onset of the superconducting transition. The dashed line indicates another possible upper bound for the SC region. coefficient of the lattice contribution, is plotted as a function ofT2in the inset in the upper left-hand side of Fig. 12. From the best fit, which is explicitly indicated by a line inthe inset, we obtain values of γ=0.58 mJ /mol f.u. K 2and β=0.91 mJ /m o lf . u .K4;t h ev a l u eo f βcorresponds to a Debye temperature of /Theta1D=220 K. In the inset in the lower right-hand side of Fig. 12,Ce/Tversus Tis plotted. A clear feature is observed between 1 and 3 K. If this feature isrelated to the transition into the superconducting state, we canestimate T c=2.5 K from an idealized entropy conserving construction. This value of Tcis close to the temperature obtained from the electrical resistivity ( Tc=2.85 K). The presence of the feature may suggest that superconductivityis a bulk phenomenon in this compound. The ratio of thespecific heat jump to γT c,/Delta1C/γT c=0.91, was calculated u s i n gaj u m pi n Ce/Tof 0.53 mJ /mol f.u. K2, extracted from the entropy conserving construction as seen in the inset in thelower right-hand side of the figure. This value is less than thevalue of 1.43 predicted by the BCS theory, but is similar tothat seen in LaO 0.5F0.5BiS 2.12 The specific heat data for La 0.8Hf0.2OBiS 2are displayed between 0.36 and 30 K in Fig. 13. The upper inset of Fig.13highlights the linear fit to the C/T data plotted versus T2, from which we extracted γ=2.12 mJ /mol f.u. K2and β=0.82 mJ/mol f.u. K4and calculated /Theta1D=228 K. In the inset in the lower right-hand side of the figure, the electroniccontribution to the specific heat C e/Tversus Tis shown, which has been obtained by subtracting the lattice contributionβT 3fromC(T). Absent from these data is any clear evidence for a jump at the Tcobtained from either the electrical resistivity or dc and ac magnetic susceptibility ( Tc=2.36, 2.81, and 1.42 K, respectively) measurements. However, there 174512-6SUPERCONDUCTIVITY INDUCED BY ELECTRON DOPING ... PHYSICAL REVIEW B 87, 174512 (2013) 00 . 1 0 . 2 0 . 3 0 . 400.40.81.21.62.02.42.83.2 La1-xThxOBiS2 La1-xHfxOBiS2 La1-xZrxOBiS2 La1-xTixOBiS2Tc (K) X FIG. 15. (Color online) Superconducting transition temperature Tcvs concentration xphase diagram for the La 1−xMxOBiS 2system (M=Th, Hf, Zr, Ti), constructed from electrical resistivity [ ρ(T)] measurements. The value of Tcwas defined by the temperatures where ρdrops to 50% of its normal state value and vertical bars are defined by the temperatures where ρdrops to 90% and of 10% of its normal state value. is a small feature around the Tcobtained from the electrical resistivity ( Tc=2.36 K) as indicated by an arrow in the lower right-hand side of the figure. The absence of a well-definedsuperconducting jump at T cis probably a consequence of the superconducting transition being spread out in temperaturedue to sample inhomogeneity. There is an upturn in specific heat below roughly 1.2 and 0.9 K for the La 0.8Hf0.2OBiS 2and La 0.85Th0.15OBiS 2 samples, respectively. The same kind of upturn is also seen in Sr 0.5La0.5FBiS 2at a similar temperature.43This upturn may be a contribution to specific heat from a Schottky anomalyor may be indicative of a second superconducting phase inthis compound. However, measurements must be made attemperatures below 0.36 K to unambiguously clarify the natureof this feature. IV . DISCUSSION The results from ρ(T),χdc, and χacmeasurements are summarized in a phase diagram of Tcvs. Th concentration xshown in Fig. 14.Tc(x) decreases with xfrom 2.85 K atx=0.15 to 2.05 K at x=0.20 and exhibits roughly concentration-independent behavior at higher concentration.The superconducting region may be defined by the dark grayregion in Fig. 14, and apparently lacks a domelike character typically seen for both the high- T clayered cuprate andFe-pnictide superconductors. Domelike superconductor re- gions are also seen in LaO 1−xFxBiS 2and NdO 1−xFxBiS 2.6,9 At concentrations below the SC domes in these com- pounds, electrical resistivity measurements reveal bad metal orsemiconducting-like behavior. These results are in agreementwith first-principles calculations, 33which suggest that the density of states at the Fermi level increases with increasingelectron doping, such that the insulating parent compoundbecomes metallic. This effect is expected to be maximal athalf-filling ( x=0.5). The LaO 1−xFxBiS 2system6shows a maximum Tcforx=0.5, while the NdO 1−xFxBiS 29system exhibits its highest Tcatx=0.3. Other scenarios may be possible depending on how we define the SC region because ofthe broadness of the superconducting transitions. For example,a dashed line is also shown in Fig. 14that mostly resides within the ranges characterized by the transition. In order to betterdefine the phase boundary, studies on samples with sharpertransitions would be beneficial. A similar result can be seen in Fig. 15where the highest T c(x) is observed at x=0.2 for Hf, Zr, and Ti doping. Tc(x) decreases initially with increasing xand then becomes concentration independent. The character of Tc(x) observed for La1−xMxOBiS 2and displayed in the phase diagram suggests that the superconducting state is similar for these systems.Considering other examples of inducing superconductivityby electron doping such as in LaO 1−xFxBiS 24,6,8–10,12and Sr1−xLaxFBiS 2,43our results suggest that electron doping is a viable approach to induce superconductivity in BiS 2-based compounds. V . CONCLUDING REMARKS In summary, we have synthesized polycrystalline samples of La 1−xMxOBiS 2(M=Th, Hf, Zr, Ti, and Sr) with the CeOBiS 2crystal structure. Electrical resistivity, dc and ac magnetic susceptibility, and specific heat measurementswere performed on selected samples. Electron doping viasubstitution of tetravalent Th +4,H f+4,Z r+4,T i+4for triva- lent La+3induces superconductivity while hole doping via substitution of divalent Sr+2for La+3does not. These results strongly suggest that electron doping by almost any meansmay be sufficient to induce superconductivity in BiS 2-based compounds. ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the US Air Force Office of Scientific Research - MultidisciplinaryUniversity Research Initiative under Grant No. FA9550-09-1-0603 and the US Department of Energy under Grant No.DE-FG02-04-ER46105. IKL was supported by a gift fromQuantum Design, Inc. *Corresponding Author: mbmaple@ucsd.edu 1Y . Mizuguchi, H. Fujihisa, Y . Gotoh, K. Suzuki, H. Usui, K. Kuroki, S. Demura, Y . Takano, H. Izawa, and O. Miura, P h y s .R e v .B 86, 220510(R) (2012).2S. K. Singh, A. Kumar, B. Gahtori, S. Kirtan, G. Sharma,S .P a t n a i k ,a n dV .P .S .A w a n a , J. Am. Chem. Soc. 134, 16504 (2012). 3B. Li, Z. W. 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PhysRevB.84.235439.pdf

PHYSICAL REVIEW B 84, 235439 (2011) Band symmetries and singularities in twisted multilayer graphene E. J. Mele* Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Received 12 October 2011; revised manuscript received 9 December 2011; published 27 December 2011) The electronic spectra of rotationally faulted graphene bilayers are calculated using a continuum formulation for small fault angles that identifies two distinct electronic states of the coupled system. The low-energy spectraof one state features a Fermi velocity reduction, which ultimately leads to pairwise annihilation and regenerationof its low-energy Dirac nodes. The physics in the complementary state is controlled by pseudospin selectionrules that prevent a Fermi velocity renormalization and produce second generation symmetry-protected Diracsingularities in the spectrum. These results are compared with previous theoretical analyses and with experimentaldata. DOI: 10.1103/PhysRevB.84.235439 PACS number(s): 73 .22.Pr, 77.55.Px, 73 .20.−r I. INTRODUCTION The variation of the electronic properties of few-layer graphenes (FLGs) with their layer stacking is receiving increasing attention. FLGs represent a family of materials that bridge the pseudorelativistic properties of single-layergraphene with the more conventional semimetallic behavior of bulk graphite. The atomic registry of neighboring layers and stacking sequence are structural parameters that determine their electronic properties. 1–5In twisted FLGs where the crystallographic axes of neighboring layers are misaligned by a rotation angle θ/negationslash=nπ/3, the interlayer interactions produce remarkably rich physics that is being actively studied.6–22 This paper presents a continuum theory of the low-energy electronic physics in twisted bilayer graphenes for small rotation angles, as illustrated in Fig. 1. Our approach reveals the existence of two distinct electronic states in this system that present quite different electronic properties. The behavior of one state is identified with the situation described by a frequently adopted continuum formulation of this problem:8,16 the interlayer coupling renormalizes the Fermi velocities of the individual layers and hybridizes their Dirac cones in the spectral region where they merge. In the complementary state, we find that the Fermi velocity renormalization is nearly completely prevented by a pseudospin selection rule and the interlayer hybridization inherits a novel momentum space geometry producing a set of second generation Dirac singularities. The behavior in this latter family agrees well with properties experimentally observed for rotationally faulted FLGs thermally grown on SiC (000 ¯1),9,11,15suggesting that this physics is realized in this form of FLG. We briefly discuss the relation of our new results to prior theoretical and to experimental studies of these systems. The physics described below is identified by consideration of the effects of the lattice symmetry on the low-energy elec-tronic spectra. We show that the geometrical structure of thelow spectrum is determined by a symmetry-allowed threefoldanisotropy in the interlayer coupling amplitudes which, thoughabsent from conventional two-center tight-binding models,occur in empirical models of interlayer coupling in graphite.We find that the sign of this anisotropy distinguishes two quitedifferent electronic states of this system.II. SPATIALLY MODULATED HOPPING The coupling between the two sublattices in the two layers can be represented by a (position-dependent) 2 ×2m a t r i x operator ˆT12(/vectorr). As shown in Fig. 2, for small angle faults the registry between layers in the unit cell evolves smoothly fromregions locally characterized by AB (region α),BA (β), and AA (γ). The smoothest possible supercell-periodic matrix- valued expression for ˆT(/vectorr) is given by the expansion ˆT 12(/vectorr)=ˆt0+6/summationdisplay n=1ei/vectorGn·/vectorrˆtn, (1) with constant matrix coefficients ˆtnand where /vectorGnare the six elements of the first star of reciprocal lattice vectors dual tothe superlattice translations /vectorT 1and/vectorT2. The matrix coefficients ˆtn(n=1,6) can be determined from the couplings in the locally registered regions; for example, in the geometry ofFig. 2the even elements of the first star have coefficients ˆt neven=tG/parenleftbigg e−i/vectorGn·/vectorrγe−i/vectorGn·/vectorrα e−i/vectorGn·/vectorrβe−i/vectorGn·/vectorrγ/parenrightbigg =tG/parenleftbigg z1 z∗z/parenrightbigg , (2) where z=e2πi/3and the coefficients for the odd elements are tnodd=t∗ neven. The constant matrix has the form ˆt0=/parenleftbigg caacab cbacbb/parenrightbigg , (3) with real coefficients satisfying caa=cbbandcab=cba.T h e interlayer operator of Eqs. ( 2) and ( 3) is thus parametrized by three real constants tG,caa, and cab. We choose these coefficients so that the interlayer matrix ˆT(/vectorrα) matches the Slonczewski-Weiss-McClure (SWMcC) interlayer parametersγ 1,γ3, andγ4for Bernal stacked graphite shown in the inset of Fig. 2,24with the results in Table I. We note that the γ3 parameter (hopping between unaligned sublattices in the two layers) is comparable to γ1and that the γ4parameter (hopping between aligned and unaligned sublattice sites) is relativelyweak. The conventional continuum description of twisted bilayer graphene 8,16can be derived from the constant matrix ˆt0. The low-energy Hamiltonian is a long-wavelength expansionaround the zone corner points in each layer; in this Diracbasis, the matrix elements in Eq. ( 1) acquire the phases 235439-1 1098-0121/2011/84(23)/235439(5) ©2011 American Physical SocietyE. J. MELE PHYSICAL REVIEW B 84, 235439 (2011) FIG. 1. (Color online) Lattice structures of twisted graphene bilayers rotated away from AA stacking by angles θ=3.89◦(top) and its commensuration partner14withθ=56.11◦(bottom). The insets show schematically the dispersions of two nearby Dirac cones in these structures in the absence of their interlayer coupling. exp[i(/vectorG/prime·/vectorτ/prime i−/vectorG·/vectorτj)], where /vectorG(/vectorG/prime) are reciprocal lattice vectors in the two separate layers and /vectorτj(/vectorτ/prime i) are sublattice positions. Boosts by a triad of ( /vectorG,/vectorG/prime) pairs translate the Hamiltonian to three pairs of zone corner points that areseparated by /Delta1/vectorKand its ±2π/3-rotated counterparts. This generates three possible constant coupling matrices indexed by the momentum differences /Delta1/vectorK i. With a conventional choice of origin,8,16where the Asublattice site of one layer is aligned FIG. 2. (Color online) Lattice structure for a segment of twisted bilayer at rotation angle 3 .89◦, with superlattice translation vectors T1andT2. The points labeled α,β,a n dγare high symmetry points in the unit cell. The inset23illustrates three hopping processes in the interlayer Hamiltonian.TABLE I. Fourier coefficients (meV units) for the interlayer hopping operator Eq. ( 2), fitted to the Slonczewski-Weiss-McClure parametrization for Bernal stacked layers. Model I: γ1=390 meV; γ3=γ4=0. Model II: γ1=390 meV , γ3= 315 meV , and γ4= 44 meV . Coefficient Parametrization I II tG (γ1−γ3)/94 3 .38 .3 caa γ4+(γ1−γ3)/3 130 .06 9 .0 cab (γ1+2γ3)/3 130 .0 340 .0 with the Bsublattice of the other, the matrices are ˆT1=/parenleftbigg 10 01/parenrightbigg ˆt0/parenleftbigg 10 01/parenrightbigg =/parenleftbigg caacab cbacbb/parenrightbigg , ˆT2=/parenleftbigg 10 0z/parenrightbigg ˆt0/parenleftbigg z0 01/parenrightbigg =/parenleftbigg zcaacab z∗cbazcbb/parenrightbigg , (4) ˆT3=/parenleftbigg 10 0z∗/parenrightbigg ˆt0/parenleftbigg z∗0 01/parenrightbigg =/parenleftbigg z∗caacab zcbaz∗cbb/parenrightbigg . In one of these valleys, the Hamiltonian for the coupled bilayer with a momentum offset /Delta1/vectorKis H=/parenleftbigg ¯hvFσ·(−i∇) ˆT† 1ˆT1 ¯hvFσθ·(−i∇−/Delta1/vectorK)/parenrightbigg , (5) where σθare Pauli matrices resolved along the axes of the θ-rotated layer. The problem can be written in dimensionless form by scaling all momenta by the offset |/Delta1/vectorK|and energies byEθ=¯hvF|/Delta1/vectorK|. The scaled coupling coefficients are ˜c= c/Eθ=3ac/[8π¯hvFsin(θ/2)] (where ais the single-layer graphene lattice constant), which increase with decreasingrotation angle. III. PARAMETRIZATIONS Model I (Table I) is an isotropic interlayer model with γ3=γ4=0. For an isotropic coupling model caa=cbb=w and the interlayer matrices are ˆT1=w/parenleftbigg 11 11/parenrightbigg ,ˆT2=w/parenleftbigg z1 z∗z/parenrightbigg , (6) ˆT3=w/parenleftbigg z∗1 zz∗/parenrightbigg , withw=130 meV. The form of these matrices and their prefactor agree with the estimates ( w≈110 meV) obtained from tight-binding calculations.8,16Our construction shows that these terms project the q=0 term of the interlayer potential into the Dirac K-point (pseudospin) basis, thereby coupling the electronic states in the two layers with identicalcrystal momenta. Since only the q=0 term in the coupling is retained, it does not depend on a relative lateral translationof the two layers, in agreement with earlier work 16and physically reasonable, since for small twist angle a rigid-layer translation produces insignificant changes to the Moiresuperlattice. Thus model I reproduces the existing continuumtheoretic phenomenology of the coupled system, and thecalculation leading to Eq. ( 6) provides an alternate (and compact) derivation of the effective Hamiltonian used in these 235439-2BAND SYMMETRIES AND SINGULARITIES IN TWISTED ... PHYSICAL REVIEW B 84, 235439 (2011) FIG. 3. (Color online) Electronic spectra for twisted bilayers using the interaction parameters (a) ˆt0=˜c(I+σx),˜c=0.21, (b) ˆt0=˜cσx,˜c= 0.55, and (c) ˆt0=˜cI,˜c=0.55.QparandQperpare momenta in units of the offset |/Delta1/vectorK|and the ordinate is the scaled energy E/E θ=E/(¯hvF/Delta1K). In (b), Dirac cones with opposite helicity are coupled; in (c), Dirac cones with the same helicity are coupled. The insets give the locations of singular points in the spectra describing the annihilation and regeneration of Dirac nodes (red diamond) in the compensated case (b) and theappearance of a singular point of degeneracy ( C) for the uncompensated case (c). The point Cis a second generation Dirac point singularity in the coupled spectrum. earlier studies.8,16The left panel of Fig. 3shows the bilayer spectra computed in this model, which shows the expected(θ-dependent) reduction of the Dirac cone velocities and a hybridization of the two branches in the spectral region wherethey merge. We now consider a refinement of the interlayer coupling matrices using the parametrization of model II. The salient properties of the SWMcC parametrization are the introductionof the interlayer amplitudes γ 3andγ4withγ3comparable to γ1andγ4significantly smaller. Note that γ3andγ4represent interlayer hopping processes at the same range, but in differentdirections with respect to the layer crystallographic axes.The asymmetry between γ 3andγ4thus reflects an intrinsic threefold lattice anisotropy in the interlayer amplitudes which, though symmetry-allowed, does not occur in the isotropictwo-center tight-binding approximation. Significantly, theseadditional terms break the symmetry between the pseudospin-diagonal and off-diagonal terms in ˆt 0(Table I), so that the coupling matrix is dominated by its off-diagonal amplitudes.An instructive limit considers ˆt 0∝σxfor which the Fig. 3(b) shows the spectrum calculated for a θ=3.89◦rotation away from Bernal stacking. Here the two Dirac cones have mergedat low energy, producing two composite low-energy singularpoints. Note that the linear low-energy dispersion is replacedby an approximately quadratic form near the center ofsymmetry of these spectra and that the momentum offsetbetween the singular points in the spectrum is along the Q perp axis, i.e., π/2-rotated with respect to the original Dirac cone offset /Delta1/vectorK.These spectral changes reflect the proximity to a critical point that occurs at ˜c=1/2 in this theory. This can be understood by considering a single-layer sublattice exchangeoperation implemented by the gauge transformation ˜H=/parenleftbigg I 0 0ˆσ x/parenrightbigg/parenleftbiggˆHK(/vectorq) ˜cˆσx ˜cˆσxˆHK(/vectorq−/Delta1/vectorK)/parenrightbigg/parenleftbigg I 0 0ˆσx/parenrightbigg =/parenleftbiggˆHK(/vectorq) ˜cI ˜cI ˆσx·ˆHK(/vectorq−/Delta1/vectorK)·ˆσx/parenrightbigg , (7) demonstrating that this system has a scalar coupling between Dirac cones with compensating helicities (Berry’s phase ±π). Increasing the control parameter ˜c(by decreasing θ)d r a w s the nodes together until they become coincident at a criticalcoupling strength ˜c=1/2 and annihilate [Fig. 3(b) inset]. For ˜c> 1/2, new singularities emerge at E=0 separated by /Delta1/vectorQ directed perpendicular to the original offset /Delta1/vectorK.U s i n gt h e parameters listed in Table I,˜c(θ=3.89 ◦)=0.55, i.e., just on the strong-coupling side of this transition. The residualcurvature in the low-energy spectrum and the associated reorientation of /Delta1/vectorQare both clearly evident in Fig. 3(b).I ti s noteworthy that the momentum separation between the zeroenergy contact points is not determined purely geometricallyby the rotation angle, as is generally assumed, but insteadis modified by the interlayer coupling. This occurs becausethe interactions between layers produces an effective gaugefield seen within each layer that shifts the momentum of itszero-energy states. The π/2 rotation of the momentum offset 235439-3E. J. MELE PHYSICAL REVIEW B 84, 235439 (2011) that bridges the contact points on the strong-coupling side of the transition is a striking consequence of this gauge coupling. Reversing the sign of the threefold anisotropy in the interlayer matrix ˆt0produces a distinct electronic state. The complementary behavior is understood by considering thelimit ˆt 0∝I, which describes the coupling of Dirac cones with the same helicity, preventing annihilation of the Dirac nodesand leading to a qualitatively different geometry in the bilayerspectrum [Fig. 3(c)]. The dispersing bands from the uncoupled cones are degenerate everywhere along the line that bisects /Delta1/vectorK. However, along the line that connects the Dirac nodes the pseudospins of the intersecting branches are orthogonaland the interlayer coupling is symmetry-forbidden, turningon linearly as a function of the transverse momentum Q perp. Thus the coupled system retains a twofold point degeneracymidway between the displaced Dirac nodes. 25The cancellation of the interlayer coupling at this critical point is the bilayeranalog of the “absence of backscattering” due to the π Berry’s phase in single-layer graphene. In the vicinity of thiscritical point, interlayer coupling is allowed and proportionalto the transverse momentum. Thus this system exhibits secondgeneration Dirac singularities in its coupled-layer spectrum asshown in Fig. 3(c): hybridization of the two layers is symmetry forbidden at a discrete critical crossing point. We refer to thiscomplementary state as the uncompensated bilayer state. IV. FERMI VELOCITY The relative helicity of the two Dirac cones also controls the renormalization of their Fermi velocities, further distinguish-ing these two states. For Dirac cones of opposite helicities,perturbation theory on the Hamiltonian in Eq. ( 7) for small ˜c modifies the velocity operators, ˆv +=vFσ+→vF(1−˜c2)σ+, (8) ˆv−=vFσ−→vF(1−˜c2)σ−, which symmetrically reduces both vxandvy; summation over the triad of offset momenta /Delta1/vectorKiyields the renormalized velocity v∗ F=vF(1−9˜c2) exactly as found in earlier work.8,16 By contrast, for coupling between nearby cones of the same helicity, perturbation theory yields ˆv+=vFσ+→vF(σ+−˜c2σ−), (9) ˆv−=vFσ−→vF(σ−−˜c2σ+), so that in this class the corrections to the velocity are weaker, ∝˜c4. Moreover, they have a twofold cos(2 φ) anisotropy, so they vanish by symmetry after summing over the threefold symmetric triad of /Delta1/vectorKi. Thus the Fermi velocity is unchanged by the interlayer coupling in this class of bilayers. Physically,this can be understood by observing that the bands dispersingfrom the Dirac nodes are connected smoothly to the second generation points of degneracy at /Delta1/vectorK/2. V. DISCUSSION The distinction between the compensated and uncompen- sated states in the small-angle limit reflects a lattice-scaleproperty that determines the matrix structure of the long-wavelength coupling in Eq. ( 1). This should be distinguished from the different mechanism by which sublattice exchangesymmetry determines the direct coupling between the Diracnodes. 14The latter requires finite momentum umklapp inter- layer hopping processes which, though significant for low-order rational commensurate rotations, are negligible in thesmall-angle limit considered here. For example, note thatsublattice exchange “even” and “odd” commensurations arerelated by a rigid sublattice translation of one layer at a fixedrotation angle. In the small angle regime, this translationsimply permutes regions of the bilayer that are locally inAB,BA, and AA registry as shown in Fig. 1, but it does not change ˆt 0, which determines the spectrum. Thus sublattice exchange even and odd structures become indistinguishablein the small-angle limit. Note also that bilayers at rotationangles θand ¯θ=π/3−θare commensuration pairs that can be distinguished by their sublattice exchange parity. 14Even- and odd-parity commensurations are, respectively, inflatedgeneralizations of the primitive AAandAB stacked bilayers. This symmetry ultimately determines the valley structureof the interlayer amplitudes that directly couple the Diracnodes of neighboring layers. This interlayer umklapp couplingderives from the finite momentum terms in the interlayerHamilonian in contrast to the q=0 terms that control the physics for small-angle rotations. The spectra for these two classes are ultimately determined by the pseudospin asymmetry in ˆt 0. The conventional SWMcC model selects the class that couples cones with compensatinghelicities. In this model, the spectral transition illustrated inFig. 3occurs for rotation angles near 4 ◦, i.e., in a range that is frequently studied experimentally.17,18The physics of the uncompensated class occurs for caa>cab, which requires γ4>γ 3. Although this is excluded by the conventional SWMcC parametrization, it is important to note that thisparametrization is designed to fit data for Bernal stacking,and it likely does not properly represent the matrix structure ofthe coupling in AA registered regions. In particular, using the parametrization of Table I, the spatial dependence of Eq. ( 1) shows that strong interlayer coupling in AA stacked regions requires γ 4>γ 3. Microscopically, this originates from interlayer tunneling processes along the edges of eclipsedhexagons in the aligned AA structure, a motif which does not appear at all for Bernal stacking. In the spirit of theSWMcC theory, it is therefore appropriate to retain γ 3andγ4 as parameters that can be determined from the experimentally observed properties of twisted graphenes. In fact, the phenomenology of the uncompensated class pro- vides a striking explanation for many of the puzzling observedspectral properties for rotationally faulted graphenes thermally grown on SiC(000 ¯1). 11,12,15Landau-level spectroscopy shows a negligible renormalization of the Fermi velocity in thesestructures 12and, furthermore, angle-resolved photoemission finds no evidence for a hybridization-induced avoided crossingof the intersecting Dirac cones, despite a careful search. 15This is completely consistent with the existence of a node in theinterlayer coupling at the midpoint between offset Dirac conescharacteristic of the uncompensated class. This assignment canbe confirmed definitively by measurements of the quasiparticledispersion along an azimuth passing through the midpoint between the displaced Dirac cones, but perpendicular to /Delta1/vectorK; 235439-4BAND SYMMETRIES AND SINGULARITIES IN TWISTED ... PHYSICAL REVIEW B 84, 235439 (2011) these should show a band splitting linear in the transverse momentum around the point of degeneracy. Alternatively, if these bilayers exist in the compensated class, photoemissionshould be able to detect the annihilation and reemergence oftheir singular contact points along with the band curvature intheir spectra in the crossover regime as illustrated in Fig. 3(b). By contrast, experiments on rotationally faulted chemical vapor deposition (CVD)-grown graphenes have observedphenomena that have been associated with the spectralproperties of the compensated class. 18,20Features due to the van Hove singularities arising from the avoided crossingof hybridized Dirac cones 18and a θ-dependent low-energy velocity renormalization have both been reported.20These features are at least qualitatively consistent with the predictedbehavior of the compensated class and have been analyzedwithin a theoretical model representative of this class. 8We note that these measurements study samples at small rotation angle,where the proximity to the merger of the Dirac singularities(Fig. 3) should be manifest in these data, though their effects have not yet been considered in the analysis. It is interestingthat these samples exhibit a large periodic height modulation≈1˚A in the superlattice unit cell peaked in the AA-registered zones. 26It is tempting to speculate that these CVD samples are grown as rippled structures that partially delaminate inthese regions, thereby locally weakening their contributionto the q=0 coupling coefficients. In this scenario, the strongly coupled regions would maintain Bernal registry asdescribed by the conventional SWMcC parametrization andidentify these samples as members of the compensated bilayer family. The distinction between the two complementary states is controlled by an important threefold anisotropy in theinterlayer tunneling amplitudes. This physics is not capturedby an isotropic two-center tight-binding theory, which in-evitably leads one to the coupling model in Eq. ( 6), which happens to occur at a crossover between two rather differentelectronic models for the system. The effects of the threefoldanisotropy are accessible in density-functional calculations ofthese structures, but for practical reasons these have beenrestricted to short period superlattices that do not addressthe small-angle regime where the continuum theory is mostappropriate. For short-period commensurate structures, theFermi velocities found in these calculations are consistent withthe values for single-layer graphene. This could arise from thesmall value of ˜cin the large-angle regime, the intrinsic behavior of the uncompensated class, or an interlayer mass term, whichis important for short period superlattices. 14 ACKNOWLEDGMENTS I thank P. First, C. Kane, M. Kindermann, S. Zaheer, and F. Zhang for their comments on the manuscript andE. Andrei for communication of unpublished data. This workwas supported by the Department of Energy, Office of BasicEnergy Sciences under Contract No. DE-FG02-ER45118. *mele@physics.upenn.edu 1E .M c C a n na n dV .I .F a l ’ k o , P h y s .R e v .L e t t . 96, 086805 (2006). 2T. Ohta et al. ,Science 313, 951 (2006). 3F. Guinea, A. H. Castro Neto, and N. M. R. Peres, Phys. Rev. B 73, 245426 (2006). 4H. Min and A. H. MacDonald, Phys. Rev. B 77, 155416 (2008). 5M. Koshino and E. McCann, P h y s .R e v .B 80, 165409 (2009). 6C. Berger et al. ,Science 312, 1191 (2006). 7S. Latil, V . Meunier, and L. Henrard, P h y s .R e v .B 76, 201402(R) (2007). 8J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto,P h y s .R e v .L e t t . 99, 256802 (2007). 9J. Hass, F. Varchon, J. E. Mill ´an-Otoya, M. Sprinkle, N. Sharma, W. A. de Heer, C. Berger, P. N. First, L. Magaud, and E. H. Conrad,P h y s .R e v .L e t t . 100, 125504 (2008). 10S. Shallcross, S. Sharma, and O. A. Pankratov, Phys. Rev. Lett. 101, 056803 (2008). 11M. Sprinkle, D. Siegel, Y . Hu, J. Hicks, A. Tejeda, A. Taleb- Ibrahimi, P. Le F `evre, F. Bertran, S. Vizzini, H. Enriquez, S. Chiang, P. Soukiassian, C. Berger, W. A. de Heer, A. Lanzara, and E. H.Conrad, P h y s .R e v .L e t t . 103, 226803 (2009). 12D. M. Miller, K. D. Kubista, G. M. Rutter, W. A. de Heer, P. N. First, and J. A. Stroscio, Science 324, 9242 (2009). 13G. T. Laissardiere, D. Mayou, and L. Magaud, Nano Lett. 10, 804 (2010).14E. J. Mele, Phys. Rev. B 81, 161405 (2010). 15J. Hicks, M. Sprinkle, K. Shepperd, F. Wang, A. Tejeda, A. Taleb- I b r a h i m i ,F .B e r t r a n ,P .L eF `evre, W. A. de Heer, C. Berger, and E. H. Conrad, P h y s .R e v .B 83, 205403 (2011). 16R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci. USA 108, 12233 (2011). 17G. Li, A. Luican, and E. Y . Andrei, P h y s .R e v .L e t t . 102, 176804 (2009). 18G. Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Castro Neto,A. Reina, J. Kong, and E. Y . Andrei, Nature Phys. 6, 109 (2010). 19M. Kindermann and P. N. First, Phys. Rev. B 83, 045425 (2011). 20A. Luican, G. Li, A. Reina, J. Kong, R. R. Nair, K. S. Novoselov,A. K. Geim, and E. Y . Andrei, Phys. Rev. Lett. 106, 126802 (2011). 21R. de Gail, M. O. Goerbig, F. Guinea, G. Montambaux, and A. H. Castro Neto, P h y s .R e v .B 84, 045436 (2011). 22M-Y . Choi, Y-H. Hyun, and Y . Kim, Phys. Rev. B 84, 195437 (2011). 23M. Freitag, Nature Phys. 7, 596 (2011). 24M. S. Dresselhaus and G. Dresselhaus, Adv. Phys. 51,6 9 (2002). 25Because of the rotation of σθin Eq. ( 6), this point degeneracy is slightly shifted along the line that bisects /Delta1/vectorK. This does not significantly change the physics. 26E. Y . Andrei (unpublished). 235439-5

PhysRevB.71.195331.pdf

Atomic structure of Si-rich 6H-SiC0001¯-2ˆ2 surface Y. Hoshino,1,3,*R. Fukuyama,1Y. Matsubara,1T. Nishimura,1S. Tanaka,2M. Kohyama,2and Y. Kido1 1Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan 2Materials Science Research Group, Research Institute for Ubiquitous Energy Devices, National Institute of Advanced Industrial Science and Technology, Ikeda, Osaka 563-8577, Japan 3Department of Electronic Science and Engineering, Kyoto University, Nishikyo-ku, Kyoto 610-8510, Japan sReceived 19 December 2004; revised manuscript received 3 March 2005; published 27 May 2005 d The atomic structure of the Si-rich 6H-SiC s0001¯d-232 surface has been determined by high-resolution medium energy ion scattering sMEIS dand photoelectron spectroscopy using synchrotron-radiation-light sSR-PES d. The MEIS analysis reveals the fact that Si adatoms s0.2–0.3 ML doverlie a Si adlayer s0.8–0.9 ML d sitting on the bulk-truncated surface s1 ML=1.22 31015atoms/cm2d. In fact, we observed two surface-related components in the Si 2 pcore level spectra corresponding to the adatom and the adlayer Si atoms and the intensity ratio of the former to the latter was ,1/3. On the other hand, the C 1 score level observed has the bulk component only. We propose an adatom-adlayer model satisfying the 2 32 surface reconstruction, three- fold symmetry and the results obtained by MEIS and SR-PES. Further MEIS analysis using focusing/blockingeffect clearly shows that the Si adatoms take a Hsite.The surface structure predicted by the ab initiomolecular dynamics calculation coincides with the above structure model except for slight lateral displacements of the Siadlayer and reproduces well the present experimental results of MEIS and SR-PES. DOI: 10.1103/PhysRevB.71.195331 PACS number ssd: 68.49. 2h, 68.47.Fg, 61.18.Bn, 79.60.Bm I. INTRODUCTION Silicon carbide has attracted much attention as the best candidates for high-temperature, high-power, and high-frequency electronic devices. In the device fabrication, theatomic structure of crystal surfaces influences the quality ofepitaxial layers and the reaction processes with various gasesand metal films. So far, a lot of investigations 1–5have been done for the Si-terminated s0001 dsurfaces mainly from the practical aspect of its low interfacial defect densities andgood quality of the epilayers. In contrast, we have only a few reports on the C-terminated s0001¯dsurfaces because of its large defect levels at oxide/semiconductor interfaces and dif- ficulties in controlling dopant concentrations. Recently,Fukudaet al. 6succeeded in the considerable reduction of the interfacial levels by a pyrogenic oxidation followed by an-nealing under hydrogen ambient. In addition, the oxidationrate for the C-terminated SiC surface is much larger than thatfor the Si-terminated one. Therefore, now it is strongly re-quired to characterize the oxidation and the metal/SiC con-tact formation for the C-terminated surface. It is known that the SiC s0001 ¯dsurface takes several sur- face reconstructions dependent on sample preparation. Bern- hardtet al.reported that a s333dsurface reconstruction ap- peared by heating at 1050 °C for 15 min in an ultra-high vacuum sUHV dand prolonged heating at 1075 °C led to a Si-depleted 2 32 surface sf232gCd.7,8These two phases tend to coexist presumably due to nearly equal surface ener- gies of the two structures. Scanning tunneling microscopesSTM dand quantitative low-energy electron diffraction sLEED danalyses showed that the f232g Csurface consisted of Si adatoms of 0.25 ML s1 ML=1.22 31015atoms/cm2d taking aH3site. Heating at temperatures above 1150 °C led to as131dgraphitized surface. On the other hand, annealingat 1150 °C in Si flux of about 1 ML/min for 20 min formed a Si-enriched SiC s0001¯d-232 surface sf232gSid. Johansson et al.9also prepared the 3 33 reconstructed surface by heat- ing at 1050 °C and concluded from C 1 score level analysis that the surface consisted of at least two carbon overlayers. In this work, we first prepared the Si-enriched SiCs0001¯d-232 surface by heating at 950 °C for 5 min in UHV after predeposition of Si s3M L d. It is shown that this 232 surface is the Si-enriched f232gSisurface whose exis- tence was reported by Bernhardt et al.7,8Unfortunately, however, the surface structure has not been clarified yet.High-resolution medium energy ion scattering sMEIS dcom- bined with photoelectron spectroscopy using synchrotron-radiation-light sSR-PES dis a powerful tool to determine the atomic configuration of surfaces and interfaces. The struc-tural analysis was carried out in situat the beamline named SORIS allowing both MEIS and SR-PES.We also performedtheab initio calculations based on the density-functional theory sDFT dusing the pseudopotential method. Finally, an energetically stable and most probable surface structure sat-isfying the experimental results is proposed. II. EXPERIMENT We purchased N-doped 6H-SiC s0001¯dwafers from CREE Inc. and cut them into small pieces with a typical size of 10310 mm2. After cleaning the surface by a modified RCA method,10the sample was introduced into an UHV chamber and degassed at 600 °C for 5 h. Then cooling it down toroom temperature sRTd, a small amount of Si s3M L dwas deposited by molecular beam epitaxy sMBE dand after that the sample was annealed at 950 °C for 5 min. Reflectionhigh-energy electron diffraction sRHEED dshowed a sharpPHYSICAL REVIEW B 71, 195331 s2005 d 1098-0121/2005/71 s19d/195331 s5d/$23.00 ©2005 The American Physical Society 195331-1232 pattern with strong Kikuchi lines. This surface is stable without coexistence of other surface phases. The experiment was performed at beam line 8 constructed at Ritsumeikan SR Center, which combined MEIS with SR-PES. All the systems were working under UHV conditions.The samples were heated by infrared radiation and the tem-perature was monitored with a Pt-Rh thermocouple set about1 mm above the sample. A well collimated He +beam was incident on a sample and backscattered He+ions were de- tected by a toroidal electrostatic analyzer sESA dwith an ex- cellent energy resolution sDE/Edof 9.0 310−4. On the other hand, emitted photoelectrons were analyzed by a hemispheri- cal ESA with an energy resolution of about 10 meV at atypical pass energy of 2.95 eV. The total energy resolutionwas estimated to be 100–150 meV including contributionsfrom a Doppler broadening and energy spreads of incidentphotons. The details of the experimental setup are describedin the previous reports. 11,12The present MEIS analysis deter- mined the elemental depth profiles and the atomic configu-ration using the ion blocking and focusing effects. Comple- mentally, the information on the chemical bonding and theelectronic structure was obtained by SR-PES analysis. It must be noted that the toroidal ESA detected only He + ions. In order to determine the absolute amount of atoms ofinterest, we need the scattered He +fractions as a function of scattered He+velocity. For He ions scattered from low Z atoms sZ,,20dlocated near a top surface, the He+fraction does not reach an equilibrium.13So we measured in advance the surface peaks for amorphous-Si and graphite targets andobtained the He +fractions, which was enhanced by 120% and 200%, respectively compared with the equilibrium He+ fractions which were derived from the scattering yields fromthe deeper layers. III.AB INITIO CALCULATIONS Theab initiomolecular dynamics sMDdcalculations were performed by the ab initiopseudopotential method based on the DFTwithin the local density approximation.14The norm- conserved pseudopotentials developed by Troullier-Martins15 were employed. In order to get the electronic ground state,we used the residual minimization method and direct inver-sion in the iterative subspace 16,17sRMM-DIIS dand the conjugate-gradient method18coupled with the efficient charge-mixing scheme.19,20Here, a plane-wave cutoff energy of 40 Ry was selected based on the tests of total energyconvergence. We developed an appropriate supercell of a 6H-SiC s0001¯d-232 surface. The supercell consists of a Si adatom, three Si-adlayer atoms, six C-Si bilayers with a sur- face C layer, and four hydrogen atoms for termination ofdangling bond of the backside surface C atoms. In addition,the supercell includes six vacuum layers, which sufficientlyseparates each 2 32 surface slab. Three and six sampling k points for self-consistent calculations were used. Thus weobtain stable atomic configurations through relaxation pro-cesses according to Hellmann-Feynman forces, which con-verged within 0.001 eV/Å per each atom. The bottom C-Sibilayer was fixed in the bulk configuration of the theoreticallattice constants sa=3.08 Å, c=15.08 Å d. The present scheme was successfully applied previously to adhesive en-ergies and Schottky barrier-heights of 3C-SiC/Ti interfaces 21,22and 3C-SiC/Al interfaces.23,24 IV. RESULTS AND DISCUSSION Figure 1 shows the MEIS spectrum observed for 120 keV He+ions incident at 54.7° and scattered to 85.1° in the s112¯0dplane with respect to surface normal. Such a grazing emergence geometry makes it possible to improve the depth resolution and thus to separate the scattering componentsfrom each atomic layer. The surface peak from Si consists oftwo components, from a Si adlayer and the top C-Si bilayer.The C front edge shifts by ,1 keV to the lower energy side, indicating the existence of a Si adlayer of about 1 MLon thetop C-Si bilayer. The absolute amount of the Si adlayer isderived to be 1.1±0.1 ML from the area of the deconvolutedsurface peak and from the knowledge of the integrated beamcurrent, the solid angle subtended by the toroidal ESA, andthe He +fraction. Taking a further glancing emergent geom- etrys88.0° dclearly resolves the previous Si adlayer into two scattering components, Si adatoms of ,0.25 ML and an un- derlying Si adlayer of ,0.95 ML ssee Fig. 2 d. Figure 3 shows the Si 2 pcore level spectra observed for incident photon energy of 140 and 280 eVat emission anglesof 0° and 60° with respect to surface normal. The spectra aredecomposed into three components, bulk and two surface-related sS1 andS2dones, assuming Gaussian shapes, the singlet/triplet branching ratio of 1/2, and the energy intervalof 0.60 eV. From the bottom to top, the spectra are moresurface sensitive. The binding energy of the bulk componentmeasured from the Fermi level was determined precisely bytaking higher photon energy of 280 and 420 eV which givesa larger escape depth. The surface-related components de-noted by S1 andS2 have lower binding energies of FIG. 1. MEIS spectrum observed for 120 keV He+ions incident on the 6H-SiC s0001¯d-232 surface at an angles of 54.7° and scat- tered to 85.1° in the s112¯0dplane. Open circles indicate the ob- served one and the solid curves are the best-fitted spectrum assum-ing a Si adlayer of 1.1 ML sitting on the top C-Si bilayer. The boldand thin curves correspond to the total and the scattering componentfrom each atomic layer, respectively.HOSHINO et al. PHYSICAL REVIEW B 71, 195331 s2005 d 195331-2−1.21±0.1 and −0.41±0.1 eV, respectively, relative to that of the bulk. From the relative intensity ratios dependent onphoton energy and emission angle, the components S1 and S2 are assigned to the Si adatoms and the Si adlayer, respec- tively. The intensity ratio of S1/S2 at normal emission is estimated to be about 1/3. This is consistent with the previ-ous MEIS result. We also observed the C 1 sspectra at a photon energy of 420 eV and found the bulk componentonly. This also supports the surface structure taking Si ada-toms on a Si adlayer overlying the top C-Si bilayer. Now we propose a probable surface structure satisfying the above MEIS and PES results considering the 2 32 recon- struction with threefold symmetry, as shown in Fig. 4. Theshaded area indicates the 2 32 unit cell. In this model, three Si atoms of the Si adlayer making a trimer bonded to one Siadatom and the amounts of the Si adatoms and Si adlayer are0.25 and 0.75 ML, respectively. There are two possibilities concerning the location of the Si adatoms, a Tsitesleft side of Fig. 4 dor aHsitesright side of Fig. 4 d. TheTsiteson-top or terminal site dgenerally means the location on top of the C or Si atoms in the first bilayer. On the other hand, the Hsite shollow site dindicates the center of a hexagon consisting of C and Si atoms in the first bilayer. According to this model,the amount of 0.25 ML of C atoms of the top C-Si bilayer isvisible from the surface normal direction. To confirm thissituation, we measured the MEIS spectrum at normal inci-dence.As a result, it was found that almost all C atoms of thetop C-Si bilayer are visible from the normal direction. Thissuggests a significant lateral distortion of the Si adlayer. Considering the above structural model and the C 3vsym- metry, we performed the ab initio MD calculations. We de- termined the atomic configurations minimizing the total en-ergy for three specific terminations of the stacking sequence,i.e.,S-1sCACBAB d,S-2sBCACBA d, andS-3sABCACB d against the bulk stacking sequence ABCACB of 6H-SiC. 25 The most stable surface structure is shown in Fig. 5. Thesurface structure does not depend on the termination of thestacking sequence as the previous calculation of silicate ad- layers on ˛33˛3 surface,25and no significant preference is seen for HorTsites of the Si adatoms. The Si adlayer consists of two types of trimers, PQR and PQ 8R8, as indi- cated in Fig. 5, holding C 3vsymmetry and the larger one sPQR dis bonded to one Si adatom sSin Fig. 5, where Si adatoms take a Hsited. The presence of the two types of trimers on the Si adlayer means the lateral displacement ofeach Si atom above the C atom, which is consistent with theMEIS result obtained at normal incidence. It is interestingthat the triangular bonding of Si occurs stably at the smallertrimer sPQ 8R8dwith enough bond charges. In contrast, the Si adatoms take a perfectly symmetric position with respect to the bulk crystal structure. The bond length between the adlayer Si atom and the C atom is 1.88 Å, which is close to the bulk SiC bond length,1.87 Å in theoretical calculations, and 1.89 Å in experiments.The interlayer distance between the Si adlayer and the topC-Si bilayer is slightly smaller than this Si-C bond length, FIG. 2. MEIS spectrum observed for 120 keV He+ions incident at incident and detection angles of 54.7° and 88.0°, respectively in thes11¯00dplane. The bright and deep gray areas correspond to the scattering components from the Si adtoms and the Si adlayer,respectively. F I G .3 .S i2 pcore level spectra observed at photon energy of 140 eV with emission angles of 60° sadand 0° sbdand at 280 eV with an emission angle of 0° scd. The components of S1 andS2 comes from the Si adatoms and the Si adlayer, respectively. FIG. 4. Top and side views of a probable surface structure for the 6H-SiC s0001¯d-232. Open and closed circles denote Si and C atoms, respectively. The larger the circles’size, the upper the latticepositions. The Si adatoms take a Tsitesleft side dand anHsite sright side d.ATOMIC STRUCTURE OF Si-RICH 6H-SiC s0001¯d-232 PHYSICAL REVIEW B 71, 195331 s2005 d 195331-3because of the lateral displacement of each Si-adlayer atom, 0.35 Å, as shown in Fig. 5. The bond length between the adlayer Si atoms is 2.48 Å, which is slightly larger than thatof 2.41 Å between the adlayer Si atom and the Si adatom.These bond lengths are larger than the bulk Si bond length of2.33 Å in theoretical calculations and 2.35 Å in experiments.The angle between the two adatom-adlayer bonds is 99.7°,which is about 9% smaller than the tetrahedral angle of thebulk Si. The origin of this distortion may have some relationwith the electronic structure of the dangling bond of the ada-tom. In any case, the present structure minimizes the number of dangling bonds and also minimizes the total energy, althoughthere exist bond length and bond angle distortions associatedwith peculiar bonding network. There remain two kinds ofdangling bonds at the top C-Si bilayer and the Si adatom,which should cause some surface electronic states. It is in-teresting to compare the surface structure of the f232g Si with that of the f232gC. According to Bernhardt et al.,7the f232gCsurface consists of a single Si adatoms in a H3site, i.e., located above the center of the Si-C hexagons of the topmost C-Si bilayer. The number of dangling bonds for theSi-depleted f232g Csurface is quite the same as that for the present Si-enriched f232gSi. Experimentally, we have re- cently observed the valence-band spectra and found nondis- persive two surface states with energies of 1.5 and 2.2 eVbelow the Fermi level probably originating from the dan-gling bonds of the Si adatoms and of the top C-Si bilayer visible from the normal direction. 26The detailed theoretical and experimental results of the electronic structure will begiven in the near future. In order to determine which site the Si adatoms take pref- erentially, the HorTsite, we performed an azimuth scan for the scattering component from Si of the top C-Si bilayer atfixed incident and emergent angles of 54.7° and 85.1° mea-sured from surface normal, respectively. Such a grazing-angle emission condition enhances the focusing and blockingeffects due to the contribution from a large number of atomslying in a string not from a single atom only. If the Si ada-toms take a Hsite, the He ions scattered from Si of the top C-Si bilayer would undergo a pronounced focusing effect by the Si adatoms along the f112 ¯0gazimuth ssee the side views of Figs. 4 and 5 d. On the other hand, a T-site location gives a blocking effect in this scattering geometry. At the f11¯00g-azimuth, both HandTsites block the He ions scat- tered from Si of the top C-Si bilayer and lead to a reduction of the scattering yield. Figure 6 shows the observed azimuthscan profile for the scattering component mainly from Si ofthe top C-Si bilayer. Here, 0° and 30° correspond to the f11 ¯00g- and f112¯0g-azimuth angle, respectively.We also per- formed Monte Carlo simulations of ion trajectories assuming the atomic configurations predicted by the ab initio calcula- tions. Apparently, the observed profile has a pronounced fo- cusing effect at the f112¯0g-azimuth s30°dand thus supports preferential location of the Hsite rather than the Tsite. The reason why the H-site location is dominant is not clear at the present. The present ab initio calculations show that the en- ergy for the Hsite is slightly lower than that for the Tsite but the energy difference is less than 10 mRyd/cell. Thus ener- FIG. 5. Top and side views of the surface structure predicted by theab initio calculations. Open and closed circles show Si and C atoms, respectively.The adlayer Si atoms form two types of trimers,PQR and PQ 8R8underlying the Si adatoms denoted by S. FIG. 6. sColor online dAzimuth scan profile observed for the scattering component mainly from Si of the top C-Si bilayer sopen circles d. The incident and detection angles were fixed at 54.7° and 85.1° with respect to surface normal. The azimuth angles of 0° and 30° correspond to the crystal axes of f11¯00gandf112¯0g, respec- tively. The black curve connects the observed data points and theblue and red ones denote the simulated profiles obtained from MCsimulations assuming the T- andH-site location of the Si adatoms, respectively predicted by the ab initio calculations.HOSHINO et al. PHYSICAL REVIEW B 71, 195331 s2005 d 195331-4getically there is no significant difference between the Hand Tsite locations. However, taking the Hsite may be kinemati- cally favorable during the surface cleaning procedure of theSi predeposition followed by annealing at 950 °C. V. CONCLUSION The 6H-SiC s0001¯d-232 surface was prepared by heating at 950 °C in UHV after Si deposition of 3 ML. This surface corresponds to just the Si-rich f232gsurface sf232gSid which was reported by Bernhardt et al.7,8Our high- resolution MEIS and SR-PES analyses have revealed the factthat there exist the Si adatoms s,1/3ML don the Si adlayer s,0.8–0.9 ML dsitting on the top C-Si bilayer. The present MEIS and PES results predict the surface structure consist- ing of the Si adlayer which forms hexagons in the unit of atrimer bonded to a Si adatom sHorTsited. However, the amount of the C atoms visible from the normal direction wassignificantly larger than that expected from the above struc-ture model, suggesting a slight lateral distortion of the Siadlayer. So, we performed the ab initio MD calculations us- ing the plane waves as a basis function and employing thepseudopotential method. The surface atomic configurationminimizing the total energy basically supports the Si-adatom/Si-adlayer model but the bonding between the Sitrimer and Si adatom is symmetrically elongated in the lat-eral plane. This structure model explains completely theMEIS and SR-PES results. However, there are no significantdifferences between the total energies and also the structuresof the Si adlayer calculated assuming the H- andT-site loca- tion of the Si adatoms. The azimuth scan for the scatteringcomponent from Si of the top C-Si bilayer under a grazingemission condition clearly shows preference of the H-site location. ACKNOWLEDGMENTS The authors would like to thank Professor H. Namba and Dr. K. Ogawa for maintaining the SR-PES system of BL-8 atRitsumeikan SR Center. Special thanks are also due to Dr. T.Okazawa for his help in the MEIS experiment. *Corresponding author. Email address: yht23389@se.ritsumei.ac.ip 1J. R. Waldrop, J. Appl. Phys. 75, 4548 s1994 d. 2L. I. Johansson, F. Owman, and P. Mårtensson, Phys. Rev. B 53, 13 793 s1996 d. 3H. Matsunami and T. Kimoto, Mater. Sci. Eng., R. 20, 125 s1997 d. 4Y. Hoshino, T. Nishimura, T. Yoneda, K. Ogawa, H. Namba, and Y. Kido, Surf. Sci. 505, 234 s2002 d. 5Y. Hoshino, S. Matsumoto, and Y. Kido, Phys. Rev. B 69, 155303 s2004 d. 6K. Fukuda, S. Suzuki, T. Tanaka, and K. Arai, Appl. Phys. Lett. 76, 1585 s2000 d. 7J. Bernhardt, M. Nerdling, U. Starke, and K. Heinz, Mater. Sci. Eng., B61, 207 s1999 d. 8J. Bernhardt, A. Seubert, M. Nerdling, U. Starke, and K. Heinz, Mater. Sci. Forum, 338–342, 345 s2000 d. 9L. I. Johansson, P.-A. Glans, and N. Hellgren, Surf. Sci. 405, 288 s1998 d. 10W. Kern and D. A. Poutinen, RCA Rev. 31, 187 s1970 d. 11Y. Kido, H. Namba, T. Nishimura, A. Ikeda, Y. Yan, and A. Yag- ishita, Nucl. Instrum. Methods Phys. Res. B 135–138, 798s1998 d. 12T. Nishimura, A. Ikeda, and Y. Kido, Rev. Sci. Instrum. 69, 1671 s1998 d. 13Y. Kido, T. Nishimura, and F. Fukumura, Phys. Rev. Lett. 82, 3352 s1999 d. 14J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 s1981 d. 15N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 s1991 d. 16P. Pulay, Chem. Phys. Lett. 73, 393 s1980 d. 17G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11 169 s1996 d. 18D. M. Bylander, L. Kleinman, and S. Lee, Phys. Rev. B 42, 1394 s1990 d. 19P. Pulay, J. Comput. Chem. 3, 556 s1982 d. 20G. P. Kerker, Phys. Rev. B 23, 3082 s1981 d. 21S. Tanaka and M. Kohyama, Phys. Rev. B 64, 235308 s2001 d. 22M. Kohyama and J. Hoekstra, Phys. Rev. B 61, 2672 s2000 d. 23S. Tanaka and M. Kohyama, Appl. Surf. Sci. 216, 471 s2003 d. 24J. Hoekstra and M. Kohyama, Phys. Rev. B 57, 2334 s1998 d. 25W. Lu, P. Krüger, and J. Pollmann, Phys. Rev. B 61, 13 737 s2000 d. 26Y. Hoshino, R. Fukuyama and Y. Kido, Phys. Rev. B 70, 165303 s2004 d.ATOMIC STRUCTURE OF Si-RICH 6H-SiC s0001¯d-232 PHYSICAL REVIEW B 71, 195331 s2005 d 195331-5

PhysRevB.95.064517.pdf

PHYSICAL REVIEW B 95, 064517 (2017) Supercurrent generation by spin injection in an s-wave superconductor–Rashba metal bilayer A. G. Mal’shukov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia (Received 25 October 2016; revised manuscript received 19 January 2017; published 28 February 2017) The spin-galvanic (inverse Edelstein) and inverse spin-Hall effects are calculated for a hybrid system that combines thin superconductor and Rashba metal layers. These effects are produced by a nonequilibrium spinpolarization that is injected into the normal metal layer. This polarization gives rise to an electric potential thatrelaxes within some characteristic length, which is determined by Andreev reflection. Within this length, thedissipative electric current of quasiparticles in the normal layer converts into the supercurrent. This processinvolves only subgap states, and at low temperature the inelastic electron-phonon interactions are not important.It is discussed how such a hybrid system can be integrated into a SQUID, where it produces an effect similar toa magnetic flux. DOI: 10.1103/PhysRevB.95.064517 I. INTRODUCTION An interplay of spin-orbit coupling (SOC), magnetism, and superconducting correlations in some solids and theirinterfaces leads to a number of electron transport phenom-ena, which have attracted recent interest in connection withpotential spintronic applications. At the heart of the unusualtransport properties of such systems lie the direct and inversespin-galvanic effects (SGEs). The former produces the electric current by polarized spins. The inverse to the SGE is also called the Edelstein effect. These phenomena were predicteda long time ago for normal systems [ 1–4]. The electric current induced by polarized spins was first observed in asemiconductor quantum well (QW) in Ref. [ 5], where spin polarization was created by optical excitation. Closely relatedto the SGE are direct and inverse spin-Hall effects (SHEs),which convert the charge current into perpendicular spincurrent and back again. For a review of the SHE in normalsystems, see Ref. [ 6]. There is a fundamental difference between these effects in superconducting and normal systems. For example, insuperconductors the spin-charge conversion can occur inthermodynamic equilibrium conditions. Thus, a spontaneoussupercurrent may be produced by an equilibrium spin po- larization induced by a static Zeeman field [ 7]. That is impossible in a normal metal. This effect, however, cannotbe observed in spatially uniform systems, because in a weakuniform Zeeman field the so called helix phase with aninhomogeneous order parameter is formed [ 7–13]. In such a superconducting state, the electric current is absent. Onthe other hand, the supercurrent may be induced in thepresence of an inhomogeneous Zeeman field [ 14–16]. The equilibrium SGE, as well as the equilibrium analog of theinverse spin-Hall effect, were also predicted in the so calledphi-0 Josephson junctions [ 17–23]. The supercurrent may also be produced by subgap light illumination of a hybridsuperconductor–semiconductor system [ 24]. The inverse SGE was considered for two-dimensional (2D) superconductorsand normal metal–superconductor hybrid systems, where the supercurrent gives rise to an equilibrium magnetization by polarizing spins of triplet Cooper pairs [ 7,23,25–28]. Another group of spin-charge conversion effects involves nonequilibrium spin polarization, as well as spin currentpumped into a superconducting system by some external source. As was shown, for SOC caused by spin-orbit impu-rities, such a nonequilibrium spin distribution can generatethe electric current and the electric potential in supercon-ductors [ 29,30]. This nonequilibrium situation bears a strong resemblance to the analogous effects in normal systems.In superconductors, however, the electric and spin transportparameters are determined by quasiparticle characteristics thatare strongly renormalized by the gap in the electron energyspectrum [ 29]. In addition, there are typical charge imbalance effects for superconductors that have not been discussed yet inthis context. In this paper, the spin-charge conversion effect will be considered for a bilayer system consisting of a normal metallayer with strong Rashba SOC and an s-wave superconducting layer. Both layers are coupled through a tunneling barrier. The nonequilibrium spin polarization is injected into the normal layer, as shown in Fig. 1. One advantage of such a system is that it combines the strong spin-orbit coupled electronsof the normal metal and the correlated Cooper pairs of thesuperconductor. There are good candidates for the former,such as narrow gap semiconductor quantum wells (QWs) andsome insulator interfaces [ 31,32]. For example, high-quality epitaxy-grown hybrid semiconductor–superconductor systems have been reported recently [ 33,34]. In addition, niobium or aluminum films can be employed as the superconductinglayer. The spin polarization can be injected by passingthe electric current through a ferromagnetic-paramagneticinterface [ 29,35,36]. Due to Rashba SOC, in such a bilayer system the injected spin polarization gives rise to the electric current inside thenormal layer. The mechanisms for such a transformation arethe SGE and the inverse SHE. Note that in bounded Rashbasystems, such as strips of finite width, it is impossible todistinguish between these two effects. The electric currentcreated by these effects is dissipative and is carried byquasiparticles whose energies are below the energy gap ofthe superconducting layer and above the proximity-inducedminigap in the spectrum of the normal metal. Then, thiscurrent converts into condensate supercurrent through Andreevreflection, so that at large enough distances from the point ofinjection the current is carried by the condensate. The electric 2469-9950/2017/95(6)/064517(11) 064517-1 ©2017 American Physical SocietyA. G. MAL’SHUKOV PHYSICAL REVIEW B 95, 064517 (2017) FIG. 1. The bilayer system consists of a Rashba metal Nand a superconducting layer S. They are coupled through a tunneling barrier. The spin polarization penetrates into the normal layer froma nonmagnetic lead. In this lead, the polarization can be created by spin injection from the ferromagnetic lead (not shown), or otherwise. The superconducting layer may be connected to a superconductingcircuit, including, e.g., a flux qubit. The spin galvanic effect in the Rashba metal gives rise to the electric current of quasiparticles above the minigap. Due to Andreev reflection, this current transforms in theSlayer into a current of Cooper pairs potential, which is associated with the quasiparticle’s current, also vanishes at large distances together with this current.Such a mechanism of charge imbalance relaxation is the mostrelevant mechanism in the considered low-temperature regime.It should be noted that in the nonequilibrium system consideredherein, quasiparticle spins play a major role in spin-chargeconversion. At the same time, in a normal Rashba metalthat comes in contact with a superconductor, the proximityeffect gives rise to triplet Cooper correlations [ 25,37], so that total spins of correlated electron pairs may potentiallycontribute to the spin-charge conversion effects [ 26,27]. The electric current, however, can be produced only if thesespins are polarized. In the absence of a Zeeman field, theycould get some polarization from polarized quasiparticle spinsthrough the electron-electron exchange interaction [ 38]. This presumably weak interaction will be ignored below, althoughit can be important in systems with strong exchange effects. The problem will be considered within the semiclassical approximation. For a dirty system, the corresponding Usadelequations will be employed for the electron Green functions. Itis important that, within the main semiclassical approximation,the standard Usadel equations miss the charge-spin coupling,which determines the spin-charge conversion effects. There-fore, such a term will be derived separately as a nonclassicalcorrection to the Usadel equations that is linear in h F/μ/lessmuch1, where hFis the spin-orbit splitting of the electron energy at the Fermi level, EF/similarequalμ, andμis the chemical potential. In this way, the expression for the generated electric current, as well ascoupled differential equations for the order-parameter phaseand the quasiparticle distribution function, will be obtained.In some important limiting cases, these equations will beanalyzed analytically. It is important to emphasize that although the strong SOC favors the spin-charge conversion effects in a normal metallayer, it should not be too strong in the considered case ofa disordered system. To reach an injected spin polarizationthat is high enough, a quasiparticle’s spins must survive manycollisions with impurities. The corresponding regime of slowD’yakonov-Perel [ 39] spin relaxation may be achieved ifthe elastic scattering rate is much larger than the spin-orbit splitting of electron energies. This regime cannot be realized,for example, for Dirac electrons on the surface of three-dimensional topological insulators, where SOC is comparableto the Fermi energy, and the spin relaxation time coincides withthe elastic scattering time, because the spin is locked to theelectron momentum. For such a material, the nonequilibriumSGE could be considered in the clean limit. However, that isbeyond the scope of this work. The paper is organized as follows: In Sec. II, the Usadel equation for the bilayer system, which accounts phenomeno-logically for the coupling of the injected spin-dependentdistribution to the spin-independent Green function, will bederived. In Sec. III, such a nonclassical spin-charge coupling term will be calculated in the Usadel equation and in theelectric current expression. Also, charge-imbalance relaxationwill be analyzed, and the differential equation for the phaseof the order parameter will be obtained. We shall considerthe electric current generated in a closed loop and evaluatethe effective electromotive force that is produced by the spininjection. II. SEMICLASSICAL EQUATIONS A. Hamiltonian, semiclassical Green functions, and self-energies of a bilayer system One of the most convenient tools for an analysis of electron transport in the range of characteristic energies /lessmuchμand lengths /greatermuch1/kF, where kFis the Fermi wave vector, is a formalism of semiclassical equations for the energy-integratedGreen functions [ 40,41]. This method operates with the three functions G r(X1,X2),Ga(X1,X2), and GK(X1,X2), where X1=(r1,t1) and X2=(r2,t2) denote space-time variables. Gr(X1,X2) and Ga(X1,X2) are, respectively, retarded and advanced Green functions, while GK(X1,X2) is the so called Keldysh function. The former carries information about theenergy spectrum and wave functions of an electron system,while G K(X1,X2) depends on its statistical properties. It is convenient to combine these functions in the 2 ×2m a t r i x ˆG, such that G11=Gr,G22=Ga,G12=GK, andG21=0. This function satisfies the Dyson equation that can be writtenin either of two forms, namely ( iτ 3∂t1−Hτ 3−ˆ/Sigma1)ˆG=ˆ1o r −i∂t2ˆGτ3−ˆG(Hτ 3+ˆ/Sigma1)=ˆ1, where His the one-particle Hamiltonian and ˆ/Sigma1≡ˆ/Sigma1(X,X/prime) is the self-energy matrix. An integration over intermediate space-time coordinates isimplied in the products ˆ/Sigma1ˆGand ˆGˆ/Sigma1. The Pauli matrices τ 1, τ2, andτ3operate in Nambu space. The Hamiltonians of the superconducting and normal layers HSandHNhave the form HSτ3=(/epsilon1Sˆk−μS)+eφS(r)−iRe/Delta1(r)τ2+iIm/Delta1(r)τ1, HNτ3=(/epsilon1Nˆk−μN)+eφN(r)+hˆkσ, (1) where /epsilon1Sˆk=ˆk2/2m,/epsilon1Nˆk=ˆk2/2m∗, and ˆk=−i∇r.μSand μNare the chemical potentials of electron gases in two layers (the band offset is included). In the case when the normallayer is a two-dimensional electron gas, the vector kin/epsilon1 Nk has only xandycomponents, which are parallel to the interface. SOC is represented by the third term in HN, where σ=(σx,σy,σz) is the vector of Pauli matrices. The spin-orbit field hk=−h−kis assumed to be a linear function of k. 064517-2SUPERCURRENT GENERATION BY SPIN INJECTION IN . . . PHYSICAL REVIEW B 95, 064517 (2017) This situation takes place if hkis represented by the Rashba field [ 42]hk=α(ez×k), or by the linear Dresselhaus [ 43] fieldhx=βkx,hy=−βky, as well as by their combination. Below, for simplicity we assume the Rashba SOC. The electricpotentials φ SandφNappear in Eqs. ( 1) due to charge imbalance, which is caused by a conversion of the injectedspin polarization into the electric current of quasiparticles. Itshould be noted that the spin injection explicitly enters onlyin the quasiparticle’s distribution function, while in Eq. ( 1) it is represented implicitly through the electric potential and/Delta1. In the unperturbed state, we assume Im /Delta1(r)=0 and Re/Delta1(r)=/Delta1 0. In principle, the injected spin polarization might enter into Eqs. ( 1) as an effective Zeeman field that is produced by polarized electrons via the electron-electronexchange interaction, as was discussed in Ref. [ 38]. To evaluate this field, let us assume that the injected spins are in aquasiequilibrium state that is characterized by the differenceδμbetween chemical potentials of two spin projections. In this case, the effective Zeeman field is Z∼Gδμ , where G is the Landau-Fermi liquid exchange parameter. For simplemetals, |G|/lessmuch1. It is even less in semiconductors, where the Coulomb interaction effects are weaker. In such a case, theeffect of the Zeeman field may be ignored, because it cannotcompete with the much stronger effect of quasiparticle spins,which is determined by δμ. In the semiclassical regime, ˆGvaries slowly in both layers as a function of the center of gravity, r=(r 1+r2)/2. At the same time, as a function of r1−r2it oscillates fast, within the Fermi wavelength. Therefore, it is convenient to Fouriertransform ˆGwith respect to ( r 1−r2) and retain intact its dependence on r. Also, in the considered stationary regime, ˆG depends only on the time difference t1−t2, and hence it can be Fourier-transformed to the frequency variable ω. Accordingly, let us introduce the Green function as ˆGk(r,ω)=/integraldisplay dn(r1−r2)e−ik(r1−r2)ˆG(r1,r2,ω),(2) where nis a dimension of the electron gas in a film (the labels NandSare omitted for a while). The self-energy may be represented in a similar way. The semiclassical Green function gν(ω) is defined by integrating Eq. ( 2) over ξ=/epsilon1k−μat a fixed direction of ν=k/kon the Fermi surface. Hence, we have ˆgν(r,ω)=i π/integraldisplay dξˆGk(r,ω). (3) The so defined function is normalized, such that ˆg2 ν=1. A procedure for obtaining the semiclassical equations for thisfunction is well described in the literature [ 41,44,45]. For each layer, these so called Eilenberger [ 40] equations can be written in the compact form iv F∇ˆgν+[ωτ3−Hτ 3−ˆ/Sigma1ν,ˆgν]=0, (4) where vFis the Fermi velocity. The nonclassical term associated with the spin-orbit part of the velocity operator∇ k(hkσ) has been neglected in Eq. ( 4), because it is as small ashk/EF. It will be included together with other nonclassical terms in a correction to the Usadel equations in Sec. III.T h e right-hand side of Eq. ( 4) should contain the inelastic scatteringterm. For the situation considered herein, this scattering is not important. Therefore, it was skipped. Let us consider the self-energy term in more detail. First of all, it contains a contribution from electron collisionswith impurities. In the Born approximation for a short-rangeisotropic scattering amplitude, the corresponding self-energycan be written as [ 41,44,45] ˆ/Sigma1(r,ω)=−i 2τscˆg(r,ω), (5) where τscis the elastic scattering time and ˆg(r,ω) is the angular average of ˆgν(r,ω). Other contributions to the self-energy describe couplings of 2D normal electrons to the supercon-ductor layer and the spin injector. They will be denotedas/Sigma1 NSand/Sigma1NM, respectively. Within the semiclassical approach, these self-energies are presented only in layerscarrying a two-dimensional electron gas, for example in asemiconductor QW. At the same time, in a bulk layer, whosethickness is much larger than k −1 F, the coupling between layers may be taken into account with the help of boundaryconditions for ˆg ν. The self-energies /Sigma1NSand/Sigma1NMare determined by virtual electron tunneling from the normal layerto an adjacent layer and back. Let us assume the normalmetal–superconductor tunneling Hamiltonian in the form/summationtext k,k/prime{tNS k,k/primeexp[i(k−k/prime)rNS]cNkc† Sk/prime+H.c.}, where rNSis the interface position in the zdirection. In the xandydirections, the interface is homogeneous, so that the parallel wave vector isconserving. One can easily write the corresponding self-energyin the form ˆ/Sigma1 NS k(r)=/summationdisplay k/primez,qz/vextendsingle/vextendsingletNS k,k/prime/vextendsingle/vextendsingle2/integraldisplay dzˆGSk/prime(r,z)eiq(z−zNS), (6) where the vectors randkare directed along the interface, qandzare perpendicular to it, and k/prime=k+kz, with kz denoting a vector that is perpendicular to the interface. Within the semiclassical approximation, the qdependence of tNS k,k/primewas neglected, because qis small in comparison with kandk/prime, which are approximately equal to the electron Fermi wavevector. Then, one may set k=k FNandk/prime=kFSin|tNS k,k/prime|2, where kFNandkFSare the Fermi wave vectors of the normal metal and the superconductor, respectively. By integrating G in Eq. ( 6) over energy, we arrive at the simple expression ˆ/Sigma1NS ν(r,ω)=−iTNSˆgSν/prime(r,z=zNS,ω), (7) where TNS=(m/2kFS)|tNS k,k/prime|2 k=kFN,k/prime=kFS(cosθ0)−1and the polar angle of k/primeandν/primeis fixed at θ0given by |sinθ0|= kFN/kFS. The self-energy ˆ/Sigma1NM ν, which is associated with a contact to the spin injector, has the same form as Eq. ( 7), with ˆgSνandTNSsubstituted for ˆgMνandTNM. The tunnel coupling with the injector is not zero only in the part of thebilayer system where the normal layer contacts to the injector. B. Usadel equations for a bilayer system Equation ( 4) can be simplified considerably in dirty systems where vFτsc/lessmuchvF//Delta1,v F/hkFand other length scales that characterize spatial variations of the Green functions ˆgS(N)ν(r,ω). In this case, these functions are almost isotropic, and it is possible to obtain closed equations for their isotropicparts, ˆg S(N)(r,ω)[46]. The corresponding formalism can be 064517-3A. G. MAL’SHUKOV PHYSICAL REVIEW B 95, 064517 (2017) found in Refs. [ 41,44,45]. These so called Usadel equations in NandSlayers can be written in the form DS∇ˆgS∇ˆgS+i[ωτ3+i/Delta1τ,ˆgS]=0, (8) DN˜∇ˆgN˜∇ˆgN+i[ωτ3+iTNSˆgS+iTNMˆgM,ˆgN]=0,(9) where /Delta1τ=Re/Delta1(r)τ2−Im/Delta1(r)τ1,˜∇∗=∇∗−i[A,∗], and the gauge-field vector components are Ax=−αmσ yand Ay=αmσ xfor Rashba SOC [ 23,47,48]. The parameters DSandDNdenote the electron diffusion coefficients in the superconductor and normal layers, respectively. The firstequation is a standard equation of an s-wave superconductor. The second equation contains the spin-orbit effects that arerepresented by the gauge field A. This equation is written for a 2D electron gas in the normal-metal film. If a 3D gasoccupies the film, the self-energies ˆ/Sigma1 NSand ˆ/Sigma1NMare absent. Instead, a contact with the superconductor and injector canbe taken into account with the help of boundary conditions(BCs). On the interface between two dirty systems iandj, where i,j=S,N,M , the conventional form of the boundary condition is [ 49] D iˆgi∇zˆgi=γij[ˆgi,ˆgj], (10) where the zaxis is directed from itoj, and γijcan be expressed in terms of the interface resistance. It is expectedthat Eq. ( 10) may be modified in the presence of Rashba SOC. In the main semiclassical approximation, the modified BCmay be obtained in the same way as in Ref. [ 49]. As a result, ∇ zis substituted for ˜∇zon the left-hand side of Eq. ( 10). Both derivatives, however, are equal to each other, becausethe gauge field A z=0. The so obtained BC can, in principle, contain nonclassical corrections ∼hkF/EF, as was shown for transparent interfaces in clean systems [ 27]. In the case of a low transparent interface considered herein, there is no reasonto consider such small terms, and the nonclassical correctionswill be ignored. The Usadel equation for ˆg Smay be further simplified [50] by assuming that Green functions vary slowly across a thin film, whose thickness dSis much less than the superconductor’s coherence length√DS/|/Delta1|. By integrating Eq. ( 8) over zand taking into account BC Eq. ( 10), we arrive at the equation for ˆgS(r)=(1/dS)/integraltext dzˆgS(r,z), DS∇ˆgS∇ˆgS+i[ωτ3+i/Delta1τ+iTSNˆgN,ˆgS]=0, (11) where TSN=DSγSN/dS. If the normal metal is a 3D film, one may perform the same manipulation with a corresponding3D equation. It should be taken into account that, accordingto the chosen model, ˜∇ z=∇ z. That results in the equation of the same form as Eq. ( 9), with TNS=DNγNS/dN.T h e parameters TNSandTSNare related to each other through the equation dNNFNTNS=dSNFSTSN, where NFNandNFSare 3D state densities at the Fermi level in the normal metal andsuperconductor (in the normal state). The same relation will beassumed for T NSentering into the self-energy Eq. ( 7)o ft h e2 D electron gas, with dNNFNsubstituted for 2D density. Such a relation is necessary for the conservation of the charge currentthrough the NS interface. It should be noted that the aboverelation between T NSandTSNmeans that the latter is much smaller than TNS, when the Nfilm is a semiconductor, but thesuperconducting layer is a usual metal, whose state density exceeds that of the semiconductor by orders of magnitude. Furthermore, we will focus on some properties of Eqs. ( 9) and ( 11) that allow us to simplify the considered problem dramatically. First of all, we note that in the case of aspin injector represented by a massive nonmagnetic film, towhich the spin polarization is pumped from ferromagneticleads, its retarded and advanced Green functions are simply g r(a) M=±τ3. This means that they are scalars in spin space. Hence, the retarded and advanced projections of Eq. ( 9)a r e scalars, except for terms with the gauge field A. Such terms, however, appear only in the form of commutators of Awith spin scalars gr(a) N. Therefore, they vanish. By representing ˆgN(S)in the form ˆgN(S)=ˆg0N(S)+ˆgN(S)σ, one can see from Eqs. ( 9) and ( 11) that gr(a) N=gr(a) S=0. Furthermore, let us consider the Keldysh projection of Eqs. ( 9) and ( 11). In contrast to the retarded and advanced functions, the functiong K Mis spin-dependent, because it is determined by the spin- dependent distribution function of the injector. Therefore,g 0K Nas well as gK Nare finite. At the same time, one can see from Eqs. ( 9) and ( 11) that at gr(a) N=gr(a) S=0, the equations for g0K NandgK Nare decoupled from each other. The equations for the three functions gK x,gK y, andgK zdescribe the energy-dependent spin diffusion, the D’yakonov-Perel spinrelaxation, and the spin precession associated with the Rashbainteraction. These processes are essentially the same, and theyare described by the same parameters as in normal metalsin the absence of superconducting proximity effects. In otherwords, the proximity to the superconductor results in the samerenormalization factor for all these effects. We will return tothe spin transport in Sec. II D. The independence of the charge transport on the spin injection is the main reason why onecannot consider the spin-charge conversion effects within themain semiclassical approximation. C. Retarded and advanced Green functions For spin-independent retarded and advanced functions gr(a) 0N, one may substitute ˜∇→∇in Eq. ( 9), so that the spin-charge coupling is only implicitly represented through the phase χ(r) of the order parameter /Delta1(r). Let us first consider the Usadel equations for the retarded functions by neglecting a contactwith the spin injector. We also neglect for a moment the termscontaining ∇χ, which are small, because they are proportional to the weak spin-charge coupling. In such a case, g r(a) 0N(S)do not depend on coordinates. As a result, for the retarded functions,Eqs. ( 9) and ( 11) reduce to /bracketleftbig ωτ 3+i/Delta1τ+iTSNgr N,gr S/bracketrightbig =0, /bracketleftbig ωτ3+iTNSgr S,gr N/bracketrightbig =0, (12) where /Delta1τ=/Delta10[cosχ(r)τ2+sinχ(r)τ1]. A similar equation can be written for the advanced functions. For TNS,TSNand ω/lessmuch/Delta10, the solution of Eq. ( 12) is given by gr(a) S=1 /Delta10/parenleftbigg −iωτ 3+/Delta1τ+TSNωτ3/radicalbig (ω±iδ)2−/Delta12m/parenrightbigg , gr(a) N=ωτ3+iTNS(/Delta1τ//Delta10)/radicalbig (ω±iδ)2−/Delta12m. (13) 064517-4SUPERCURRENT GENERATION BY SPIN INJECTION IN . . . PHYSICAL REVIEW B 95, 064517 (2017) The small terms ∼TSN//Delta10andω//Delta1 0have been taken into account in gr(a) Sbecause they are important in the effects associated with the Andreev reflection. One can see that inthe quasiparticle spectrum of the normal layer, the minigap/Delta1 m=|TNS|/lessmuch/Delta10opens, which is a common property of SN bilayer systems [ 50]. Since /Delta1has a phase that is varying in space, in some cases it is necessary to take into account corresponding correctionsto Eqs. ( 13). For Green functions of the superconductor, these corrections can be easily obtained in the range of highenergies ω>|/Delta1 0|by ignoring a contact with the normal layer, whose effect is weak in this frequency range. Itis convenient to perform the unitary transformation g→ exp(iτ 3χ/2)˜gexp(−iτ3χ/2). Furthermore, by keeping only the terms in Eq. ( 11) that are linear in ∇χ,w ea r r i v ea tt h e Fourier-transformed function ˜gr(a) Sin the form ˜gr(a) S=ωτ3+iτ2/Delta10 /Omega1δq,0+τ1Dq2χq/Delta10 Dq2/Omega1+2i/Omega12, (14) where /Omega1=/radicalBig (ω±iδ)2−/Delta12 0, and qis the wave vector. As will be seen below, this correction is important for calculating the spin injection effect on the order parameter. Let us now consider a contact with the injector, as a small correction δgr(a)to the functions given by Eqs. ( 13). Let us assume that in Fig. 1, the length bof the injector–normal metal contact in the xdirection is small in comparison with the diffusion length lN=(DN/2/radicalbig ω2−/Delta12m)1/2(ω> |/Delta1m|). Then, one can represent TNM(x)i nE q .( 9)i nt h e formTNM(x)=bTNMδ(x). For the massive injector film, the unperturbed value gr(a) M=±τ3is also assumed. By linearizing Eq. ( 9) with respect to δgr(a) N,w ea r r i v ea t δgr(a) N=∓blN 2l2 NMexp−(1+i)|x|√ 2lNgr(a) N/bracketleftbig gr(a) N,τ3/bracketrightbig , (15) where l2 NM=DN/TNM. Therefore, this correction is small at blN/l2 NM/lessmuch1, which will be assumed in the following. D. Distribution functions In this section, we will consider Eqs. ( 9) and ( 11)f o r Keldysh functions. These equations can be transformed tokinetic equations for the distribution function f(r,ω), which is defined by the equation [ 41,45] g K=grf−fga. (16) The function gKexpressed in this way satisfies the proper normalization condition grgK+gKga=0, which is a non- diagonal projection of the general condition ˆg2=1. The distribution function, in turn, can be represented as f(r,ω)= f0(r,ω)+f(r,ω)σ. As was noted above, the spin and charge variables are decoupled in Eqs. ( 9) and ( 11). This means that we have separate equations for the scalar ( f0) and triplet ( f) parts of the distribution function. 1. Spin-distribution function Let us first consider the spin-dependent triplet part. We assume that the spin injector is a normal metal, where the spinpolarization is creating by electric current passing througha normal metal–ferromagnet interface [ 35,36], or by other means. The thermodynamic equilibrium will be assumed forboth spin projections. Hence, the spin-distribution function inthe injector is given by f M=s 2/parenleftbigg tanhω+μs kBT−tanhω−μs kBT/parenrightbigg , (17) where sdenotes the unit vector that is parallel to the spin polarization, and 2 μsis the difference between chemical potentials of spin distributions corresponding to two spinprojections. It will be assumed that μ s/lessmuch/Delta10. Since fMis a scalar in Nambu space, one can expect that fNandfSare also scalar functions. Equations for these functions are obtainedby substituting Eqs. ( 16) and ( 17) into Eqs. ( 9) and ( 11) and taking the trace over Nambu variables. The terms containing∇χhave been neglected. In this way, the equations for f Nare obtained in the form 0=(−˜DN∇2+˜/Gamma1s)fN/bardbl+4αm∗˜DN∇fNz +˜T(1) NS(fN/bardbl−fS/bardbl)+˜T(1) NM(fN/bardbl−fM/bardbl), (18) 0=(−˜DN∇2+2˜/Gamma1s)fNz−4αm∗˜DN∇fN/bardbl +˜T(1) NS(fNz−fSz)+˜T(1) NM(fNz−fMz), where the labels /bardblandzdenote projections of the vector fonto thex,yplane and the zaxis, respectively. Apart from tunneling terms, these equations resemble well-known spin-diffusionequations [ 51–54], where spin-charge coupling effects have been neglected. However, in Eq. ( 18) the spin-diffusion and D’yakonov-Perel spin-relaxation coefficients ˜D Nand ˜/Gamma1s, respectively, are renormalized by the superconductor proximity effect. The renormalization factor is the samefor both transport parameters, such that ˜D N/DN=˜/Gamma1s//Gamma1s= (1/4)Tr[1 −gr Nga N], where DN=v2 Fτsc/2 and /Gamma1s=2h2 kFτsc. The couplings to the superconductor and injector layers aregiven by ˜T(1) NS=TNS 4Tr/bracketleftbig/parenleftbig gr N−ga N/parenrightbig/parenleftbig gr S−ga S/parenrightbig/bracketrightbig , ˜T(1) NM=TNM 4Tr/bracketleftbig/parenleftbig gr N−ga N/parenrightbig/parenleftbig gr M−ga M/parenrightbig/bracketrightbig , (19) where gr M−ga M=2τ3.I tf o l l o w sf r o mE q s .( 17)–(19) that the injector spin-distribution function plays the role of a sourcein Eqs. ( 18). At low temperatures, the spectral power of this source is distributed in the energy range ω/lessorsimilarμ s/lessmuch|/Delta1|.A t these energies, the tunnel coupling between the normal and superconductor layers is weak because gr S−ga Sin˜T(1) NSis finite only due to subgap quasiparticle states. The contribution of these states is given by the small third term of gr(a) Sin Eq. ( 13). In bounded systems, Eqs. ( 18) must be appended by boundary conditions. For example, in Fig. 1one needs BCs at the edges, y=±w/2, where wis the width of the bilayer. A generalization of BCs that takes into account Rashba SOChas been discussed in Sec. II B. It can be achieved by the substitution ∇ y→˜∇yin Eq. ( 10). By setting at the edges γ=0 and multiplying Eq. ( 10)b y ˆgi, we get ˜∇yˆgi=0. Since they-independent retarded and advanced functions are scalars in spin space, the latter equation may be reduced, with the helpof Eq. ( 16), to a set of equations for vector components of the 064517-5A. G. MAL’SHUKOV PHYSICAL REVIEW B 95, 064517 (2017) spin distribution function. These equations have the form ∇yfz+2αm∗fy=∇ yfy−2αm∗fz=∇ yfx=0,(20) where all functions are taken at y=±w/2. The same equations take place for the spin density in normal systemsif the diffusive spin current parallel to the yaxis becomes zero at the hard-wall boundary [ 55–58]. Therefore, Eqs. ( 20)f o r the distribution function seem reasonable, although this issuedeserves a separate study. Due to spin precession in the Rashba field, the boundary conditions Eq. ( 20) always mix various spin components. For example, if the injected spin polarization in Eq. ( 17) is initially oriented parallel to the yaxis and homogeneous in the y direction, it will rotate toward the zaxis during propagation along the strip. Therefore, it is impossible to observe apure spin-galvanic effect in bounded systems, because in thepresence of z-polarized spins the inverse spin-Hall effect also takes place. When the strip width is much larger than the spindiffusion/precession length, l so=1/αm∗, one may neglect the boundary conditions. In this case, the solution of Eq. ( 18)i s fNx=fNz=0, and fNydepends only on the xcoordinate if fMis chosen in the form of Eq. ( 17) with sparallel to the y axis. By assuming in Eq. ( 18)2T(1) NM(x)=T(1) NM[θ(x+b/2)+ θ(x−b/2)], where θ(x) is the step function, this solution can be obtained in the form fNy=Aθ/parenleftbiggb 2−|x|/parenrightbigg (1−Bcosh 2 xκ/prime) +ABθ/parenleftbigg |x|−b 2/parenrightbiggκ/prime κsinhbκ/primee−2xκ, (21) where A=˜T(1) NMfMy/(˜/Gamma1s+˜T(1) NM),B=κ(κcoshbκ/prime+ κ/primesinhbκ/prime)−1,κ=1/lso, and 4 κ/prime2=4κ2+(˜T(1) NM/˜DN). We neglected in Eq. ( 21) a leakage of the spin polarization into the superconductor. Also, it was assumed that ˜/Gamma1s/greatermuch˜T(1) NS.T h e opposite case of a narrow strip with w/lessmuchlsois considered in Sec. III B 2 . 2. Particle distribution function In Eq. ( 16), the spin-independent scalar function f0can be represented in the form of a diagonal matrix in Nambu space [45]. Accordingly, we have f0=f(1) 0+τ3f(2) 0. Let us first consider the Usadel equation for f(1) 0. It is important that the spin injector pumps into the system not only nonequilibriumspins, but also particles that are out of thermodynamic equi-librium. Indeed, the spin-independent distribution function inthe injector is f (1) 0M=1 2/parenleftbigg tanhω+μs kBT+tanhω−μs kBT/parenrightbigg . (22) This function differs from the equilibrium distribution. Such sort of a distribution function was considered in Ref. [ 59]. Let us look at what happens in the bilayer geometry shown in Fig. 1.F r o mE q s .( 9) and ( 11), the Usadel equations for f(1) 0N andf(1) 0Scan be obtained in the form 0=− ˜DN∇2f(1) 0N+˜T(1) NS/parenleftbig f(1) 0N−f(1) 0S/parenrightbig +˜T(1) NM/parenleftbig f(1) 0N−f(1) 0M/parenrightbig , 0=− ˜DS∇2f(1) 0S+˜T(1) SN/parenleftbig f(1) 0S−f(1) 0N/parenrightbig . (23)The renormalization factor for ˜T(1) SNis the same as for ˜T(1) NS in Eqs. ( 19). The diffusion constant in the superconductor is given by ˜DS=(DS/4)Tr[1 −gr Sga S]. The solution of Eqs. ( 23) isf(1) 0S=f(1) 0N=f(1) 0M. This solution is valid as long as the inelastic scattering was ignored. If inelastic relaxationprocesses are taken into account, the distribution functionsin both layers will relax to the thermal equilibrium at a largedistance from the injection point. We will assume that thetemperature is small enough such that this distance is muchlarger than the spin-orbit relaxation/precession length l soand other characteristic lengths that determine the spin-chargeconversion. Since the quasiparticle’s energy distribution inthe superconductor’s layer is different from the equilibriumone, the gap will decrease slightly. The distribution functionin the form of Eq. ( 22) produces a weak effect at k BT/lessmuch /Delta1andμs/lessmuch/Delta1[59]. We will assume that this effect is included in the gap. Such a gap depends slightly on xand relaxes together with f(1) 0Sto its unperturbed value at large distances. The functions f(2) 0Sandf(2) 0Ncontrol the kinetics of the spin-charge conversion. Within the considered model, such a function is zero in the injector, i.e., f(2) 0M=0. Therefore, according to Eqs. ( 8) and ( 9),f(2)must be zero in the entire system. On the other hand, these equations miss thespin-charge coupling terms that are responsible for the directand inverse spin-Hall and spin-galvanic effects. These termsplay the role of nondiagonal elements that couple two setsof Usadel equations for spin-independent and spin-dependentGreen functions, g 0andg, respectively. As will be shown in the next section, in the transport equations for the spin-independent function g K 0Nthe spin-charge coupling appears as a term that is proportional to gK. The latter, in turn, can be expressed through the spin-distribution function fthat was considered in Sec. II D 1 . As a result, the effective ω-dependent electromotive force E, which is given by Eq. ( 33), appears in the equation for f(2) 0N. By substituting Eq. ( 16) into the scalar projection of Eq. ( 9) and adding the spin-charge coupling E, the transport equations for f(2)take the form 0=− ˜D(2) N∇2f(2) 0N−jN∇f(1) 0N−eDN∇E +˜T(2) NS/parenleftbig f(2) 0N−f(2) 0S/parenrightbig +˜T(2) NMf(2) 0N, (24) 0=− ˜D(2) S∇2f(2) 0S−jS∇f(1) 0S−Rf(2) 0S +˜T(2) SN/parenleftbig f(2) 0S−f(2) 0N/parenrightbig , where the energy-dependent transport parameters are [ 60] jN(S)=DN(S) 4Tr/bracketleftbig/parenleftbig gr N(S)∇gr N(S)−ga N(S)∇ga N(S)/parenrightbig τ3/bracketrightbig , ˜D(2) N(S)=DN(S) 4Tr/bracketleftbig 1−τ3gr N(S)τ3ga N(S)/bracketrightbig , R=DS 4Tr/bracketleftbig/parenleftbig gr S+ga S/parenrightbig /Delta1τ/bracketrightbig . (25) The tunneling parameters are given by ˜T(2) AB=(TAB/4)Tr/bracketleftbig/parenleftbig τ3gr A−ga Aτ3/parenrightbig/parenleftbig gr Bτ3−τ3ga B/parenrightbig/bracketrightbig ,(26) where AandBtake the values N,S,o rM.I nE q s .( 24), the terms with the spectral supercurrents jN(S)are small because 064517-6SUPERCURRENT GENERATION BY SPIN INJECTION IN . . . PHYSICAL REVIEW B 95, 064517 (2017) ∇f(1) 0N(S)are inversely proportional to the large inelastic relaxation length, as follows from the above analysis. Thespectral supercurrents j N(S)are also small because they are proportional to ∇χ. The latter is determined by the weak spin-charge coupling. Therefore, these terms will be neglected.Furthermore, the coupling to the injector may also be neglected because the function f(2) 0Nchanges its sign in the range of the injector, which makes a leakage of f(2) 0Ninto the injector at b/lessmuchlNMvery inefficient. The transport parameters in Eqs. ( 24) can be calculated by using Eqs. ( 13) and ( 25). By taking into account only the leading terms at ω/lessmuch|/Delta10|, we obtain ˜D(2) N=DNω2 ω2−|/Delta1m|2,˜D(2) S=DS,R=2/Delta10expiχ, ˜T(2) NS TNS=˜T(2) SN TSN=2TSNω2 /Delta10(ω2−|/Delta1m|2). (27) With these parameters, the kinetics of the charge imbalance relaxation, which is controlled by Eq. ( 24), becomes clear. Indeed, in the normal metal layer the electromotive force E generates the nonzero f(2) 0N. The latter, in turn, is related to the electric potential according to the equation [ 45] eφ=−1 8/integraldisplay dωTr/bracketleftbig τ3/parenleftbig gr N−ga N/parenrightbig/bracketrightbig f(2) 0N. (28) This potential implies the presence of a charge imbalance that relaxes through electron tunneling into the adjacent superconducting layer, where f(2) 0Sbecomes zero relatively fast due to the quasiparticle’s absorption by the condensate.According to Eqs. ( 24) and ( 27), the characteristic relaxation length of f (2)in the superconductor is the smallest one of lR andlSN, which are given by lSN=/radicaltp/radicalvertex/radicalvertex/radicalbt˜D(2) S/vextendsingle/vextendsingle˜T(2) SN/vextendsingle/vextendsingle=1√ 2ω|TSN|/radicalBig |/Delta10|DS/parenleftbig ω2−/Delta12m/parenrightbig , lR=/radicalBigg ˜D(2) S |R|=/radicalBigg DS 2/Delta10. (29) It is seen from this equation that, except for a narrow region of energies close to /Delta1m, the distribution function f(2) 0Svanishes within the superconductor’s coherence length lR.A tt h es a m e time,f(2) 0Ndecreases in space only due to a slow leak into the superconductor, within the length lNS=(˜D(2) N/˜T(2) NS)1/2= (DN/Delta10/TSNTNS)1/2, which is assumed much larger than lR. Therefore, lNSdetermines the charge-imbalance relaxation in the whole system. This mechanism is different from the well-known charge-imbalance relaxation at superconductor–normalmetal interfaces, which involves inelastic electron-phononscattering [ 61–63]. A relaxation of the electric potential is accompanied by a transformation of the electric currentof quasiparticles into the supercurrent. This issue will bediscussed in more detail in Sec. III B .III. CALCULATION OF THE SPIN-CHARGE COUPLING TERM IN USADEL EQUATIONS A. Quantum corrections to the Usadel equations and electric current As was shown in the previous section, the Usadel equa- tions, which were obtained within the main semiclassicalapproximation, are decoupled into two independent sets ofequations for spin-singlet and spin-triplet Green functions. Inthis section, the quantum correction, which leads to a mixingof these two sets, will be calculated in first order with respecttoα/v F∼hkF/μ. It follows from Sec. IIthat, as long as the Fermi liquid effects are ignored, the retarded and advancedfunctions stay scalar in spin space. Therefore, let us focus onthe Keldysh function. We start from the Dyson equation (ωτ 3−HNτ3)GK N=/Sigma1r N◦GK N+/Sigma1K N◦Ga N. (30) In this equation, the Green function and self-energy depend on two spatial coordinates, and “ ◦” denotes the integration over an intermediate coordinate. Equation ( 30) can be simplified by taking into account that the self-energies Eqs. ( 5) and (6) are local functions of rand can be expressed through the semiclassical angular averaged Green functions. Also, one should take into account that Gr(a) Ndepend weakly on coordinates, as was discussed in Sec. II. By combining the first term on the right-hand side of Eq. ( 30) with the expression on the left, one may express the perturbed part of the Keldyshfunction in the form G K=Gr◦/Sigma1K◦Ga. Furthermore, by transforming GKto the mixed representation Eq. ( 2) and integrating it over k, the Fourier-transformed spin-independent part of gK(q) can be expressed as gK 0N(q)=1 2/summationdisplay kTrσ/bracketleftbig Gr Nk+q 2/Sigma1K N(q)Ga Nk+q 2/bracketrightbig . (31) One may obtain the Usadel equation ( 9) for the Keldysh function by expanding Gr(a) Nk±q 2inq,ω, andhkand performing the integration over kwithin the main semiclassical approx- imation. That means that all slowly varying entries in theintegral are evaluated at k=k F. Since we are interested in quantum corrections, such terms have to be expanded neark F. The task, however, is not so vast, because the goal is to calculate only the terms that couple the spin-independentand spin-dependent functions g K 0NandgK N. Therefore, only the spin-dependent part of /Sigma1K N(q) will be taken into account in Eq. ( 31). Furthermore, contributions to /Sigma1K N(q) that are caused by electron tunnelings to the injector and superconductor aremuch smaller than the self-energy associated with the elasticimpurity scattering. Therefore, they will be ignored below. At the small frequency ω/lessmuch/Delta1 0, the retarded and advanced functions are obtained from Eqs. ( 1) and ( 5) in the form Gr(a) Nk=1 4/summationdisplay σ=±1/bracketleftbigggr(a) N+1 λ/Omega1−ξσ+gr(a) N−1 λ/Omega1+ξσ/bracketrightbigg [1+σ(nσ)],(32) where ξσ=ξ+σhk,n=hk/hk,/Omega1=/radicalbig (ω±iδ)2−(/Delta1m)2, and the factor λis given by λ/Omega1=/Omega1+iω/2τsc|ω|.T h e functions gr(a) Nin this equation are given by Eq. ( 13), where the phase factor in the order parameter is ignored, so that/Delta1τ=/Delta1 0τ2. This simplification is dictated by the accuracy of 064517-7A. G. MAL’SHUKOV PHYSICAL REVIEW B 95, 064517 (2017) nonclassical corrections, which must be linear in hkF/μ. Since the phase is small on this parameter, one must neglect it in gr(a) N. For the same reason, the electric potential φNshould also be neglected. In view of the relatively large scatteringrate 1/τ sc, the calculation of the integral over ξin Eq. ( 31) can be performed by expanding the denominators in Eq. ( 32) with respect to hk,/Omega1, andq.A l s o , hk±q/2,nk±q/2, andξk±q/2 should be expanded with respect to q. It is also crucial to take into account the kdependence of hknear the poles of Eq. ( 32). Some details of such a calculation may be found in Ref. [ 14]. A lengthy algebra yields the leading term for the quantum correction of the order of ( q/kF)(hkFτsc)3. In turn, the “electromotive” force Ein Eq. ( 24)i sg i v e nb y eE=h2 kFτ2 sc 2DN∇kTr/bracketleftbig τ3hkgK N/bracketrightbig . (33) A similar expression controls the spin-galvanic effect in Josephson junctions [ 20] and normal systems [ 64]. In the latter case, gK Nshould be substituted for the spin density. In addition to the Usadel equations, the nonclassical corrections also appear in the electric current. The currentis expressed from Eqs. ( 30) in the form J(q)=ie 4/integraldisplaydω 2π/summationdisplay kTr/bracketleftbig vkτ3GK Nk(q)/bracketrightbig =ie 4/integraldisplaydω 2π/summationdisplay kTr/bracketleftbig vkτ3Gr Nk+q 2/Sigma1K N(q)Ga Nk+q 2/bracketrightbig ,(34) where vk=∇k(/epsilon1Nk+hkσ). The sought-after nonclassical correction Jnccan be obtained from the spin-dependent part of/Sigma1K N(q), similar to the above calculation of the correction to the Usadel equation. In the coordinate representation, it takesthe form J nc(r)=eτscNFN 4/integraldisplay dωTr/bracketleftbig /Gamma1s∇k/parenleftbig hkτ3gK N/parenrightbig −2DNα2m∗(ez×∇r)τ3gK Nz/bracketrightbig . (35) The first term in the integrand represents the spin-galvanic effect. In the case of Rashba SOC, this term gives rise tothe electric current in the xdirection if spins are polarized parallel to the yaxis. The corresponding example of spin injection was considered in Sec. II D. The second contribution to the current stems from spins polarized in the zdirection. This is the inverse spin-Hall effect. It has been noted abovethat in bounded systems, whose size is comparable to l so,i ti s difficult to distinguish these two effects, because gK NzandgK Ny(x) are coupled to each other via the boundary conditions. This situation will be analyzed in more detail in the next section. The total current J(q) consists of Jncand the usual diffusion currents [ 41,44,45] of quasiparticles in the normal metal and superconductor, as well as supercurrents due to the order-parameter phase gradient in both layers. We thus obtain forthe current density in the bilayer J=J nc+e/integraldisplay dω/parenleftbig NFN˜D(2) N∇f(2) 0N +dSNFS˜D(2) S∇f(2) 0S/parenrightbig +/parenleftbiggenS 2m+enN 2m∗/parenrightbigg ∇χ, (36)where nS=2πmD SdSNFS/Delta10tanh(/Delta10/2kBT) and nN= 2πm∗DNNFN/Delta1mtanh(/Delta1m/2kBT) are 2D densities of super- conducting electrons in the normal and superconducting layers[45]. Note that N FNis the state density of a 2D gas. The 2D density of states in the superconductor film is given instead byd SNFS.I nE q .( 36), the supercurrent in the normal layer may be neglected, because /Delta1m/lessmuch/Delta10. Moreover, one should expect thatdSNFS/greatermuchNFNif the superconducting film is thick enough, or if the normal system is a 2D electron gas in a semiconductorquantum well. The phase of the order parameter can be foundfrom the continuity equation ∇J=0. When the operator ∇ is applied to Eq. ( 36), one should take into account Eq. ( 24), Eq. ( 26), and the relations between the tunneling parameters T NSandTSN, which were discussed below Eq. ( 11). Also, Eqs. ( 33) and ( 35)g i v e ∇Jnc=σN 2/integraldisplay dω∇E, (37) where σN=2e2DNNFNis the conductivity of the normal metal. In this way, the equation for the phase can be obtainedin the form ed SNFS/integraldisplay dωRf(2) 0S−enS 2m∇2χ=0. (38) The above equation has been obtained from the charge conservation. It is instructive to derive it in a different way,such as directly from the gap equation. The latter has the form /Delta1 λ=1 8/integraldisplay dωTrτ/bracketleftbig τgK 0S/bracketrightbig =1 8/integraldisplay dωTrτ/bracketleftbig τ/parenleftbig gr S−ga S/parenrightbig f(1) 0S+τ/parenleftbig gr S+ga S/parenrightbig f(2) 0S/bracketrightbig ,(39) where λis the electron-electron pairing constant. The first term in the integrand is determined by quasiparticle energiesabove the gap where, as shown in Sec. II D 2 ,a tk BT/lessmuch/Delta10 andμs</Delta1 0the distribution function f(1) 0S=tanh(ω/2kBT), while the retarded and advanced Green functions are givenby Eq. ( 14) (in the rotated representation). The integral of the unperturbed function, which is given by the first term inEq. ( 14), cancels with the left-hand side of Eq. ( 39), while the second term in Eq. ( 14)g i v e s( 1 /d S/Delta10)(enS/2m)∇2χ.B y taking into account Eq. ( 25), it is easy to see that Eq. ( 39) coincides with Eq. ( 38). B. Electric current induced by spin injection 1. Current in a wide strip Let us consider a simple situation of a wide enough strip, such that boundary effects at y=±w/2 may be neglected. Also, the injected spin polarization will be assumed uniform in the ydirection. In the case of Rashba SOC, this example was considered in Sec. II D 1 with spins polarized in the y direction. Hence, in Eq. ( 35) only the first term in the integrand contributes to Jnc.F r o mE q .( 33) the current density Jncin the xdirection can be expressed in terms of the “electromotive force,” Jnc=σN 2/integraldisplay dωE. (40) 064517-8SUPERCURRENT GENERATION BY SPIN INJECTION IN . . . PHYSICAL REVIEW B 95, 064517 (2017) As seen from Eq. ( 33), Eq. ( 21), and Sec. II D 1 , spatial variations of Jncare determined by the larger of the Dyakonov- Perel spin-relaxation length lsoand the size of the injector. In turn, both lengths have been assumed much smaller than othercharacteristic lengths of the system, such as l NS,lSN, andlR. Let us consider a solution of Eqs. ( 24) that vanishes at large x. This takes place in a situation when the length of the strip in Fig.1is larger than all characteristic lengths, and an external bias is absent. By substituting this solution into the secondterm of Eq. ( 36) and combining it with J nc, we obtain the dissipative quasiparticle current Jdin the form Jd=σN 2/integraldisplay dωUir1r2 r1−r2/parenleftbiggei√r1|x| √r1−ei√r2|x| √r2/parenrightbigg , (41) where U=/integraltext dxE, andr1andr2are given by r1=i l2 R/parenleftbig l2 R+il2 NS−l2 NSl2 Rl−2 SN/parenrightbig /parenleftbig l2 R+il2 NS/parenrightbig , r2=−1 l2 NS/parenleftbig l2 R+il2 NS+l2 NSl2 Rl−2 SN/parenrightbig /parenleftbig l2 R+il2 NS/parenrightbig . (42) In Eq. ( 41), the signs of√r1(2) are chosen such that Im(√r1(2))>0. Since lR/lessmuchlNSandlSN,w eh a v e r1/similarequali/l2 R andr2/similarequali/l2 NS, so that r1/greatermuchr2. Hence, at a large distance the quasiparticle current is given by the second term in theparentheses in Eq. ( 41), which in turn is determined by the Andreev reflection. To determine the total current, one needs a boundary condition for the order parameter at a large distance. For this,let us assume that the wire has the form of a closed loop withlength L. A change of the phase χon this length is 2 πn, where nis a whole number. By integrating the current density Eq. ( 36) over the strip area and taking into account that the total current Iis constant and the distribution functions are periodic, we obtain nwπen S m+/integraldisplay dxdyJ nc=IL, (43) where the integral of Jnccan be obtained from Eq. ( 40). It should be noted that Eq. ( 43) is valid at an arbitrary relation between Land the relaxation length lNSof the quasiparticle current. If L/greatermuchlNS, most of the current is formed by the condensate, while in the opposite case the current is producedby quasiparticles. In the former case, the second term inEq. (43) plays the role of an effective magnetic flux through the loop, similar to the equilibrium magnetoelectric effect inducedby the Zeeman field [ 14]. In the latter case, the electric current has a mostly dissipative nature, and it is more reasonable todescribe the effect in terms of an effective electromotive force,which is associated with a nonequilibrium spin polarization,as in the case of the spin-galvanic effect in normal metals.The number nin Eq. ( 43) must be chosen to minimize the energy of the moving condensate, similar to the Little-Parkseffect [ 65]. It is likely that the strong enough spin-galvanic effect might cause a sort of Little-Parks oscillations in the caseofL/greatermuchl NS. A more realistic possibility, given a weakness of the effect, might be a measuring of a shift in Little-Parksoscillations produced by an external magnetic field. Also, thetime-modulated spin injection can significantly affect a fluxFIG. 2. The effective electric voltage induced by the spin injection as a function of the chemical potential difference of two injected spinprojections. The voltage is given in units of the effective voltage, which is calculated for a normal metal without the superconducting proximity effect (see the text). The chemical potential is measured inunits of the minigap in the spectrum of the normal layer. The curves (from top to bottom) are calculated at 2 T NM//Gamma1s=0.005, 0.01, 0.05, and 0.1. qubit when the qubit’s resonance frequency coincides with the modulation frequency. In this case, the spin-injection effectwill be similar to an oscillating magnetic flux [ 66]. Note that in superconducting systems, the spin-galvanic effect is very similar to the equilibrium magnetoelectric effect,which is produced by a Zeeman field. In particular, in bothcases they result in an effective magnetic flux [ 14]. There is, however, a fundamental difference. The Zeeman field gives riseto triplet Cooper pairs, whose dynamics in the Rashba field andconversion to singlet pairs lead to the magnetoelectric effect.In contrast, the spin injection does not produce any changesin the condensate wave function. It modifies the quasiparticledistribution function only. The spin-injection effect becomes stronger at large SOC. Within the considered theory, a strength of this coupling isrestricted only by the smallness of the semiclassical parameterα/v F. Also, the effect increases with the larger contact size bof the injector. Therefore, the most interesting case correspondstob/greatermuchl soin Eq. ( 21). In Fig. 2, the electromotive voltage U, which is defined as U=/integraltext Edxdω , is shown as a function of the injection strength μsat various TNM//Gamma1s. In the normal layer at small temperature, this voltage is a linear function ofμ s, namely UN=4(b/l so)(TNM/μN)μs.I nF i g . 2, this value is used as a normalization factor. The nonlinear dependence of U onμsin Fig. 2is associated with the presence of the minigap in the quasiparticle spectrum. By taking b/l so=5,TNM/μN= 10−3, andμs=10−4eV, we obtain U∼UN∼10−6V. The above evaluation of Uis mostly restricted by limitations of the theory, which does not allow us to take larger TNM,μs, andα. One cannot exclude the possibility that larger Umay be reached within a more general theory. 064517-9A. G. MAL’SHUKOV PHYSICAL REVIEW B 95, 064517 (2017) 2. Current in a narrow strip As noted in Sec. II D 1 , in bounded systems SOC may strongly modify the spatial distribution of the injected spinpolarization. In the case of a narrow strip, whose width is lessthan the spin-relaxation length, one must take into accountthe boundary conditions Eq. ( 20). The importance of such an analysis becomes evident from Eq. ( 35) for the nonclassical electric current. Indeed, in the case of Rashba SOC this currentcan be written in the form J x nc=eτscDNNNF 2m∗l2so/integraldisplay dωTr/bracketleftbig 2αm∗τ3gK Ny+∇yτ3gK Nz/bracketrightbig .(44) Since, according to Eq. ( 16),gK/bardblf, one can apply the boundary conditions Eq. ( 20) to the integrand of Eq. ( 44). As a result, Jx ncvanishes at the strip boundaries y=±w/2, because just the expression 2 αmf Ny+∇yfNzenters in the integrand of Eq. ( 44). Hence, in the case of a narrow strip, whose width w/lessmuchlso=1/αm∗, one could expect that Jx ncis small inside the strip. On the other hand, the Dyakonov-Perel’spin-relaxation time increases dramatically in narrow wires[51,67]. Therefore, the spin accumulation in the strip increases and may compensate for a cancellation of the two terms inEq. ( 44). To check that such a compensation indeed takes place, let us assume that the injector width bis small ( b/lessmuchl so). Also, in the case of a weak metal-injector coupling, one mayneglect the back flow of spins from the normal layer. Thismeans that only f Mmust be retained in the tunnel coupling term ˜T(1) NM(fN−fM)i nE q s .( 18). A weak coupling to the superconductor will also be neglected. It is easy to calculate thearea integral of J x nc, which enters in Eq. ( 43). After integration of Eqs. ( 18) over x, the remaining equations may be solved by expanding fyNandfzNin power series in y, while fxN=0. In this way, we obtain /integraldisplay dxdy (2αmf Ny+∇yfNz)=−2bαm∗˜T(1) NM /Gamma1sfyM.(45) Let us compare this result with the similar integral obtained in the case of a wide strip. For such a strip, fNzmay be neglected,whilefNyis given by Eq. ( 21). By expanding this expression with respect to small b/l so, we obtain the same result as Eq. ( 45). Therefore, there is no difference between wide and narrow wires. On the other hand, in narrow wires the slowerspin relaxation causes an enhanced leak of the spin polarizationinto the injector and the superconductor. Consequently, suchan effect may become important at larger bandT NM. IV . CONCLUSION It has been shown that the spin-galvanic effect in a hybrid superconductor–Rashba metal bilayer system also hasa hybrid character. The injected spin polarization induces botha dissipative quasiparticle current in the normal layer and asupercurrent in the superconducting layer. It depends on thesize of the system, and which of the two effects dominates.There is some characteristic length where a conversion of the quasiparticle’s current into the supercurrent through the Andreev reflection occurs. In either of the two cases,the current of quasiparticles is strongly influenced by theproximity-induced minigap in the electron spectrum. 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PhysRevB.92.165433.pdf

PHYSICAL REVIEW B 92, 165433 (2015) Nonlocal transport and the hydrodynamic shear viscosity in graphene Iacopo Torre,1,2,*Andrea Tomadin,3Andre K. Geim,4and Marco Polini2 1NEST, Scuola Normale Superiore, I-56126 Pisa, Italy 2Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy 3NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy 4School of Physics, and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom (Received 3 August 2015; published 30 October 2015) Motivated by recent experimental progress in preparing encapsulated graphene sheets with ultrahigh mobilities up to room temperature, we present a theoretical study of dc transport in doped graphene in the hydrodynamicregime. By using the continuity and Navier-Stokes equations, we demonstrate analytically that measurements ofnonlocal resistances in multiterminal Hall bar devices can be used to extract the hydrodynamic shear viscosityof the two-dimensional (2D) electron liquid in graphene. We also discuss how to probe the viscosity-dominatedhydrodynamic transport regime by scanning probe potentiometry and magnetometry. Our approach enablesmeasurements of the viscosity of any 2D electron liquid in the hydrodynamic transport regime. DOI: 10.1103/PhysRevB.92.165433 PACS number(s): 72 .10.−d,72.80.Vp I. INTRODUCTION Transport in systems with many particles (such as gases and liquids) undergoing very frequent interparticle collisionshas been studied for more than two centuries and is describedby the theory of hydrodynamics [ 1–3]. In the hydrodynamic regime, transport is described by [ 1–3] three nonlinear partial-differential equations—the continuity, Navier-Stokes,and energy-transport equations—reflecting the conservationof mass, momentum, and energy, respectively. The Navier-Stokes equation contains two transport coefficients [ 1–3]: The shear viscosity η, which describes friction between adjacent layers of fluid moving with different velocities, and the bulkviscosity ζ, which describes dissipation arising in a liquid due to homogeneous compressionlike deformations. The energy-transport equation contains the thermal conductivity κ, which describes dissipative heat flow between regions with differenttemperatures. These coefficients quantify the tendency of theliquid to restore a homogeneous state in response to a velocityor thermal gradient: They therefore control the magnitude ofnonlocal contributions to the linear-response functions of theliquid. Viscous flow of dilute classical gases attracted the attention of Maxwell, who theoretically discovered [ 3] a puzzling property of the shear viscosity of a dilute gas. Using amolecular approach [ 3], he found that the shear viscosity of a dilute gas is independent of density n(and depends on temperature Taccording to η∝T 1/2), a counterintuitive result that he felt needed immediate experimental testing [ 4]. The importance of ηin the hydrodynamic behavior of dilute gases and liquids stems from the fact that this parameter controlsthe term of the Navier-Stokes equation that opposes turbulentflow [ 1–3]. Recent years have witnessed a tremendous interdisciplinary interest in the hydrodynamic flow of strongly interacting quan-tum fluids. This interest was sparked by a series of results [ 5], which were obtained via the anti-de Sitter/conformal fieldtheory (AdS/CFT) correspondence, for the shear viscosity *iacopo.torre@sns.itof a large class of strongly interacting thermal quantumfield theories. These efforts culminated in 2005 when it wasconjectured [ 6] that all quantum fluids obey the following universal lower bound: η/s/greaterorequalslant/planckover2pi1/(4πk B), where sis the entropy density. Note that this bound does not contain the speedof light, thereby explaining why the conjecture was extendedalso to nonrelativistic quantum field theories. Fluids thatsaturate this bound have been dubbed “nearly perfect fluids”(NPFs) [ 7], i.e., fluids that dissipate the smallest possible amount of energy and satisfy the laws of hydrodynamics atdistances as short as the interparticle spacing. Currently, twolaboratory systems come closest to saturating the AdS/CFTbound: (i) the quark-gluon plasma [ 8], which is created at Brookhaven’s Relativistic Heavy Ion Collider and at CERN’sLarge Hadron Collider by bashing heavy (e.g., gold and lead)ions together, and (ii) ultracold atomic Fermi gases [ 9,10] (such as 6Li) close to a Feshbach resonance. Although mathematical counterexamples have appeared in the literature [ 11], there are no known experimental violations of the AdS/CFT bound. The present paper is motivated by the following questions: Do electron liquids display hydrodynamic behavior? If so,how can it be experimentally proven that the electron systemhas entered the hydrodynamic transport regime? Once in thehydrodynamic regime, how can the shear viscosity of anelectron liquid be measured in a solid-state device? Can anelectron liquid in a solid-state device be a NPF? Hydrodynamics has been used for a long time to describe transport of electrons in solid-state devices [ 12–25]. However, since the (Bloch) momentum of an electron in a solid is apoorly conserved quantity due to collisions against impurities,phonons, and structural defects in the crystal, experimentalsignatures of hydrodynamic electron flow are expected only inultraclean crystals, at sufficiently low temperatures. Second,electron-electron (e-e) interactions need to be sufficientlystrong to ensure that the mean-free path /lscript eefor e-e collisions is the shortest length scale in the problem, i.e., /lscriptee/lessmuch/lscript,W,v F/ω. Here, vFis the Fermi velocity, /lscriptis the mean-free path for momentum-non-conserving collisions, Wis the sample size, andωis the frequency of the external perturbation. Unfortunately, the low-temperature requirement (needed to mitigate the breakdown of momentum conservation in 1098-0121/2015/92(16)/165433(11) 165433-1 ©2015 American Physical SocietyTORRE, TOMADIN, GEIM, AND POLINI PHYSICAL REVIEW B 92, 165433 (2015) a solid-state environment) and the strong e-e interaction requirement are conflicting: At low temperatures, where /lscriptis large, the mean-free path /lscripteefor e-e collisions is also very large due to Pauli blocking. Indeed, normal Fermi liquidsat low temperatures [ 26–28] display very large values of /lscript ee, i.e.,/lscriptee∝(TF/T)2for temperatures T/lessmuchTFwithTFas the Fermi temperature. (In two spatial dimensions there isa very well-known [ 29] logarithmic correction that has been dropped.) These severe restrictions, imposed by a rigid Fermisurface on the phase space for two-body e-e collisions, canbe relieved by increasing temperature. Indeed, /lscript eequickly decreases for increasing T. Short e-e mean-free paths therefore require operating at sufficiently elevated temperatures. At suchtemperatures, strong scattering of electrons against opticalphonons (e.g., in polar crystals, such as GaAs) often leads tothe unwanted inequality /lscript</lscript ee. Realizing hydrodynamic flow at “high” temperatures therefore requires not only ultracleancrystals, but also crystals where electron-phonon couplingis extremely weak. We note that it is in principle easierto reach the hydrodynamic transport regime in nonpolarcrystals, such as graphene, where the dominating mechanismat high temperatures is scattering of electrons against acousticphonons. In this case /lscriptdecays as 1 /T, i.e., slower than /lscript ee. For these reasons, evidence of hydrodynamic transport in solid-state devices is, to the best of our knowledge, limited toearly work carried out by Molenkamp and de Jong [ 30] and de Jong and Molenkamp [ 31] in electrostatically defined wires in GaAs/AlGaAs heterostructures. These authors measured anonmonotonic dependence of the four-point longitudinal resis-tivityρ xxon the electronic temperature T, which is increased above the lattice temperature by using a large dc heatingcurrent. The decrease in ρ xxwith increasing temperature above a certain value of Twas attributed to e-e collisions [ 30,31]. This is the so-called “Gurzhi effect” and will be discussedextensively in this article. More recently, indirect evidenceof hydrodynamic flow comes from an explanation [ 32]o f Coulomb drag between two neutral graphene sheets [ 33], which differs from that offered by the authors of Ref. [ 33]. Recent experimental progress [ 34–39], however, has made it possible to fabricate samples with ultrahigh carrier transportmobilities up to room temperature . These are graphene sheets encapsulated between thin hexagonal boron nitride(hBN) slabs, which display ultrahigh mobilities reaching10 5cm2V−1s−1in a wide range of temperatures up to room temperature. These values can be achieved in GaAs/AlGaAsheterostructures only below 100 K because of polar phononscattering [ 40]. In addition, the finite electron mass and moderate doping required to achieve high mobilities limitsthe Fermi energy to values below a few tens of meV ,which makes the Fermi-level smearing important even atliquid-nitrogen temperatures. Encapsulated samples [ 33–39] are special because electrons roaming in graphene suffervery weak scattering against acoustic phonons [ 41–44] and because hBN provides an exceptionally clean and flat dielectricenvironment for graphene [ 45]. Furthermore, microscopic calculations based on many-body diagrammatic perturbationtheory [ 46–48] indicate that the e-e mean-free path /lscript eein graphene is shorter than 400 nm in a wide range of carrier den-sities and temperatures T/greaterorequalslant150 K. We therefore conclude that hBN/graphene/hBN stacks are ideal samples where the long-sought hydrodynamic regime can be unveiled and explored. Indeed, recent nonlocal transport measurements [ 49] carried out in high-quality encapsulated single-layer graphene (SLG)and bilayer graphene (BLG) samples have demonstrated thatthis is the case. The authors of Ref. [ 49] have reported evidence of hydrodynamic transport, showing that doped grapheneexhibits an anomalous (negative) voltage drop near currentinjection points, which has been attributed to the formationof whirlpools in the electron flow. From measurements ofnonlocal signals, Bandurin et al. [49] extracted the viscosity of graphene’s electron liquid and found it to be in quantitativeagreement with many-body theory calculations [ 48]. In this article, we present a fully analytical theoretical study of nonlocal dc transport in the two-dimensional (2D)electron liquid in a graphene sheet in the hydrodynamicregime. In Sec. IIwe present the theoretical framework that was used in Ref. [ 49] to interpret the experimental results, i.e., a linearized steady-state hydrodynamic approach based on thecontinuity and Navier-Stokes equations. Suitable boundaryconditions for these hydrodynamic equations are discussedin Sec. II A. In Sec. IIIwe present analytical solutions for longitudinal transport in a rectangular Hall bar, and we discussthe dependence of the solutions on the boundary conditions,providing details on the Gurzhi effect. In Sec. IVwe present analytical results for the spatial dependence of the 2D electricalpotential, nonlocal resistance, and current-induced magnetic field as obtained by solving the hydrodynamic equations with the free-surface boundary conditions, which we believe to bethe appropriate boundary conditions for the linear-responseregime. Finally, in Sec. Vwe present a summary of our main results and offer some perspectives. We remark that, in the present paper, we focus only on doped SLG and BLG sheets where the applicability of theFermi-liquid theory is granted. However, it is believed thatthe hydrodynamic behavior of the semimetals is particularlyinteresting [ 16,17] when these are in the charge neutral state. In this case the Fermi surface shrinks to a point, and Fermi-liquidtheory is not applicable. For example, the authors of Ref. [ 17] have found that the ratio η/sfor the 2D electron liquid in a graphene sheet at the charge neutrality point (CNP) comesclose to saturating the AdS/CFT bound. In nearly neutralsemimetals /lscript eeis also short due to frequent collisions between thermally excited carriers ( T/greatermuchTF). It is, however, well known that any theory at the CNP must take into account: (i)the spatially inhomogeneous pattern of electron-hole puddlescreated by disorder [ 50–53] and (ii) coupling between charge and energy flow [ 32,54,55]. In the regime of doping we consider, we can safely ignore both effects. For this reason,our theoretical predictions below cannot be extrapolated downto the CNP. II. LINEARIZED STEADY-STATE HYDRODYNAMIC THEORY We consider a two-dimensional electron liquid in a doped SLG or BLG sheet, deep in the hydrodynamic transport regime(/lscript ee/lessmuch/lscript,W). For the sake of definiteness, we consider the Hall bar geometry sketched in Fig. 1. Since the energy-momentum dispersion of electrons in these systems is particle-holesymmetric [ 56], we assume, without loss of generality, that 165433-2NONLOCAL TRANSPORT AND THE HYDRODYNAMIC SHEAR . . . PHYSICAL REVIEW B 92, 165433 (2015) FIG. 1. (Color online) Schematic of the nonlocal transport setup analyzed in this paper. A dc current Iis injected (red arrow) into an encapsulated graphene Hall bar of width W. Current injection occurs at a lateral contact located at x=x0andy=−W/2. The same current is drained (blue arrow) at a contact located at x=−x0 andy=−W/2. Measurements of voltage drops /Delta1Vnear the current injection region are sensitive to the kinematic viscosity νof the two- dimensional massless Dirac fermion liquid. The notion of “vicinity” between voltage probe and current injector is defined by a crucial length scale, i.e., the vorticity diffusion length Dν=√ντ. Here τ (exceeding 1 ps in high-quality encapsulated devices) represents a phenomenological scattering time due to momentum-non-conserving collisions of a fluid element (and notof single electrons). the sample hosts a back-gate-controlled equilibrium electron density equal to ¯n. (The charge density is −e¯n,−ebeing the electron charge.) We neglect thermally excited carriers andcoupling between charge and heat flow [ 54], which is strong only at the charge neutrality point. Finally, we consider thelinear-response regime and steady-state transport. In this framework of approximations, the hydrodynamic transport equations [ 23,24] for the 2D electron liquid greatly simplify and reduce to ∇·J(r)=0, (1) and ¯ne m∇φ(r)+ν∇2J(r)=J(r) τ. (2) In Eqs. ( 1) and ( 2) we have introduced the linearized steady- state particle current density J(r)=¯nv(r), where v(r)i st h e linearized steady-state fluid-element velocity. Equation ( 1) is the continuity equation, whereas Eq. ( 2)i s the Navier-Stokes equation. The latter contains three forcesacting on a fluid element: (i) the electric force −eE(r)= e∇φ(r), written in terms of the electric potential φ(r)o nt h e 2D plane where electrons are moving, which is generated bythe steady-state charge distribution n(r) in response to the drive current I, (ii) the internal force due to the shear viscosity η=η(¯n,T) of the 2D electron liquid, here written in terms of the kinematic viscosity [ 1–3], ν=η m¯n, (3) and (iii) friction exerted on a fluid element by agents external to the electron liquid, such as phonons and impurities, whichdissipate the fluid-element momentum at a rate of τ −1=1/τ(¯n,T). The latter is a phenomenological parameter, which depends on ¯nandT, and is commonly used in modeling transport in semiconductor devices [ 57]. In Eqs. ( 2) and ( 3)mis a suitable effective mass defined by m=/braceleftbiggmc for SLG , 0.03mefor BLG ,(4) where mc=/planckover2pi1kF/vFis the 2D massless Dirac fermion cy- clotron mass [ 56],kF=√π¯nbeing the Fermi wave number, vF∼106m s is the Fermi velocity, and meis the bare electron mass in vacuum. Multiplying both members of Eq. ( 2)b yτ, we obtain σ0 e∇φ(r)+D2 ν∇2J(r)=J(r). (5) In Eq. ( 5) we have introduced the following characteristic length scale of the problem: Dν≡√ντ. (6) Forτ=1 ps (as in high-quality hBN/graphene/hBN samples) andν=0.1m2s (see Ref. [ 48]) we obtain Dν≈0.3μm. The physical significance of Dνcan be understood as follows. We first note that we can rewrite ∇2J(r)b yu s i n g the following identity: ∇2J(r)=∇[∇·J(r)]−∇×[∇×J(r)]. (7) Because of ( 1), we can drop the first term on the right-hand side of Eq. ( 7). The second term is finite and related to the vorticity [ 1,2], ω(r)≡1 ¯n∇×J(r)=ω(r)ˆz, (8) which in 2D is oriented along the ˆ zaxis. We can then rewrite Eq. ( 5)a sf o l l o w s : σ0 e∇φ(r)−¯nD2 ν∇×ω(r)=J(r). (9) Taking the curl of Eq. ( 9) and using the identities ∇×∇φ(r)= 0 and∇·ω(r)=0 (the latter being valid because ω’s only nonvanishing component is along ˆ z, whereas ∇acts only on the 2D ˆx-ˆyplane), we obtain a damped-diffusion equation for the vorticity, D2 ν∇2ω(r)=ω(r). (10) We therefore see that Dνplays the role of a diffusion length forω(r). In Eq. ( 5) we have also introduced a “Drude-like” conduc- tivity, σ0≡e2¯nτ m. (11) Since we are in the hydrodynamic regime, σ0should not be confused [ 24] with the ordinary dc conductivity in the dif- fusive transport regime: Once again, τ=τ(¯n,T) represents a phenomenological parameter that should be fit to experimentaldata as we discuss below in Sec. III. This naive description of momentum-non-conserving collisions in the hydrodynamictransport regime can be relaxed by following arguments similarto those in Ref. [ 19]: This is however well beyond the scope of the present article and will be the topic of future studies. In 165433-3TORRE, TOMADIN, GEIM, AND POLINI PHYSICAL REVIEW B 92, 165433 (2015) the absence of viscosity, Eq. ( 2) reduces to a local version of Ohm’s law, i.e., J(r)=σ0∇φ(r)/e. Finally, we note that taking the divergence of Eq. ( 5) and making use of Eq. ( 1) we obtain the Laplace equation ∇2φ(r)=0 for the electric potential φ(r) on the 2D plane. This should not be confused with the usual three-dimensional(3D) Poisson equation for the 3D electrostatic potential /Phi1(r,z), /parenleftbigg ∇ 2+∂2 ∂z2/parenrightbigg /Phi1(r,z)=4πen(r)δ(z). (12) The 2D potential in Eq. ( 2)i sφ(r)=/Phi1(r,z=0). On the right- hand side of Eq. ( 12) we note the steady-state charge-density distribution −en(r) which occurs in the sample in response to the drive current I. Equation ( 12) needs to be solved in 3D space with suitable boundary conditions—depending on thedielectric environment, gates, etc., surrounding the graphenesheet—if one is interested in determining n(r). In this article we will focus our attention on J(r) andφ(r). Equations ( 1) and ( 5) will be used to describe transport in the Hall bar geometry pictorially represented in Fig. 1. Mathematically, it is convenient to work in a Hall bar of infinitelength in the longitudinal direction ˆxsince this allows us to use the Fourier transform to solve the equations of motion—seeSec. IV. The width Wof the Hall bar will be kept finite. In the next section we will describe a crucially important ingredientof the theory: boundary conditions. A. Boundary conditions In order to find φ(r) and J(r) in the Hall bar geometry depicted in Fig. 1, we need to solve Eqs. ( 1) and ( 5)i nt h e rectangle ( −∞,∞)[−W/2,W/2] with appropriate boundary conditions (BCs) at the edges, i.e., at y=±W/2. Lateral electrodes acting as current injectors/collectors are described through BCs on the component of the currentperpendicular to the edges, J y(x,y=±W/2)=J±(x). (13) HereJ±(x) is a function that describes a distribution of current injectors and collectors on the upper (lower) edge of themultiterminal Hall bar. It is through Eq. ( 13) that the total drive current Iinjected into the system at the boundaries enters the problem. Following Abanin et al. [58], we model the electrodes as pointlike (i.e., δfunction) sources and sinks. (A more realistic modeling of electrodes has been carried out in Ref. [ 49] where finite-width effects and metallic boundary conditionsat extended electrodes have been taken into account in afully numerical solution of Eqs. ( 1) and ( 5). Such details have essentially no impact on the physics we are goingto highlight below.) For example, for the setup depicted inFig.1with a current injector at x=x 0, a current collector at x=−x0, and no injectors/collectors on the upper edge, we will use J−(x)=−I eδ(x−x0)+I eδ(x+x0), (14) andJ+(x)=0. In the presence of a finite shear viscosity ν, we need an additional BC on the tangential component of the current atthe top ( y=+W/2) and bottom ( y=−W/2) edges of the Hall bar. We use the following BC: [∂yJx(x,y)+∂xJy(x,y)]y=±W/2=∓Jx(x,y=±W/2) lb, (15) where lbis a “boundary slip length,” i.e., a length scale describing friction at the physical boundaries of the sample.This BC can be explained as follows. The left-hand sideof Eq. ( 15) is proportional to the off-diagonal component of the stress tensor [ 1], calculated at the edges of the Hall bar. It represents the tangential component of thefrictional force exerted by the boundaries of the Hall baron the 2D electron liquid [ 1]. This force depends on the tangential velocity of the 2D electron liquid and boundaryroughness: In the linear-response regime, it is natural toreplace such unknown dependence with a linear law char-acterized by the single parameter l bas on the right-hand side of Eq. ( 15). In the description of transport of molecular liquids in constrained geometries, such as water in a pipe, where theinteractions between the molecules of the fluid and the wallsof the container are of the same nature as of those betweenmolecules of the fluid, the most used BCs are the so-called“no-slip” BCs [ 1] in which the component of the current tangential to the boundary vanishes. The no-slip BCs can beobtained from Eq. ( 15) by taking the limit l b→0. In the opposite limit of a free-surface geometry, such as the surfaceof water in an open bucket, the tangential force applied fromthe boundary to the fluid element vanishes at the boundary.These free-surface BCs [ 1,24] can be obtained from Eq. ( 15) by taking the limit l b→+ ∞ . Which of these BCs should be used to model the experi- ments in Ref. [ 49] will become clear at the end of Sec. III. B. Applicability of the linearized theory The validity of the linearized Navier-Stokes Eq. ( 5) relies on the smallness of the Reynolds number [ 1–3]RW.T h i s is a dimensionless parameter (which depends on the samplegeometry) that controls the smallness of the nonlinear term[v(r,t)·∇]v(r,t) in the convective derivative with respect to the viscous term. In our case we can define the Reynoldsnumber as follows: /vextendsingle/vextendsingle/vextendsingle/vextendsingle[v(r,t)·∇]v(r,t) ν∇2v(r,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/similarequal¯vW ν=I e¯nν≡RW, (16) where ¯ vis the typical value of the fluid-element velocity. For an injected current [ 49]I=2×10−7A, a Hall bar width ofW=1μm and an equilibrium density of ¯n=1012cm−2, we obtain ¯ v∼I/(e¯nW)≈104cm s. We note that ¯ vis much smaller than the graphene Fermi velocity of vF∼106m s and the flow is therefore nonrelativistic. The corresponding valueof the Reynolds number is R W∼10−3/lessmuch1, obtained by using a kinematic viscosity ν∼103cm2s of the 2D electron liquid in graphene [ 48]. Our linearized theory in Eqs. ( 1) and ( 5)i s therefore fully justified. 165433-4NONLOCAL TRANSPORT AND THE HYDRODYNAMIC SHEAR . . . PHYSICAL REVIEW B 92, 165433 (2015) III. LONGITUDINAL TRANSPORT AND THE GURZHI EFFECT We first consider the situation in which no current is injected or extracted laterally at the Hall bar edges, i.e., J±(x)=0. In this case the local current J(r) does not depend on the longitudinal coordinate x, and all the spatial derivatives with respect to xin Eqs. ( 1), (5), and ( 15) vanish. The continuity equation implies that Jydoes not depend on yand vanishes identically because of Eq. ( 13). Therefore the ycomponent of the electric field must also vanish. The xcomponent of the current respects the following equation: Jx(y)−D2 ν∂2 yJx(y)= −σ0Ex/e, where E=−∇φ(r) is the electric field. Note that Excannot depend on ybecause Eyvanishes and ∇×E=0. The solution of this equation that fulfills the BC ( 15)i s Jx(y)=−σ0 eEx/bracketleftbigg 1−Dν ξcosh/parenleftbiggy Dν/parenrightbigg/bracketrightbigg , (17) where we have introduced the length, ξ≡lbsinh/parenleftbiggW 2Dν/parenrightbigg +Dνcosh/parenleftbiggW 2Dν/parenrightbigg . (18) We can calculate the total longitudinal current Icarried by the flow by integrating Eq. ( 17) in the transverse direction, i.e., I=−e/integraldisplayW/2 −W/2dy J x(y)=σ0WEx(1−F), (19) where we have defined the dimensionless quantity, F≡2D2 ν Wξsinh/parenleftbiggW 2Dν/parenrightbigg . (20) Measuring the longitudinal potential drop /Delta1V between two lateral contacts at positions xandx+Lyields a four-point longitudinal conductivity σxxof the form σxx≡I /Delta1VL W=σ0(1−F). (21) Equation ( 21) is the most important result of this section. In the limit lb→∞ (i.e., free-surface BCs) F→0. For this choice of BC the longitudinal conductivity σxxdepends only on the rate of momentum-non-conserving collisions τ−1 (through σ0) and is independent of ν. On the other hand, in the limit lb→0 (i.e., no-slip BCs) Eq. ( 21) reduces to σxx=σ0/bracketleftbigg 1−2Dν Wtanh/parenleftbiggW 2Dν/parenrightbigg/bracketrightbigg . (22) We can easily understand two asymptotic limits of Eq. ( 22). In the limit Dν/lessmuchWEq. ( 22) yields σxx=σ0(1−2Dν/W): The small correction to the Drude-like conductivity σ0is due to a reduction of the fluid-element velocity in a thin regionof width D νnear the top and bottom edges of the Hall bar. In the opposite limit Dν/greatermuchW, we obtain σxx=σ0W2/(12D2 ν)= e2¯nW2/(12mν). In this limit the problem is equivalent to that of Poiseuille flow in a pipe [ 1,2] with a velocity profile vx(y) that depends quadratically on the transverse coordinate yand a resistance that is entirely due to viscosity. A summary of our main results for longitudinal electron transport in the presence of a finite viscosity is reported inFig.2.0.0 0.5 1.0 1.5 2.0 Dν/W0.00.20.40.60.81.01(a) (b).2σxx/σ0 −W/2 0 W/2 y0.00.20.40.60.81.01.21.41.6−eJxW/I FIG. 2. Panel (a) The longitudinal conductivity ( 21) (in units of σ0) is plotted as a function of the ratio Dν/Wfor different values of the boundary scattering length lb:lb=∞ (thick solid line), lb=10W (dashed line), lb=W(dotted line), lb=0.1W(dashed-dotted line), andlb=0 (thin solid line). Panel (b) The current-density profile −eJx(y), normalized by the total current I/W , is plotted as a function ofyforlb=0 (no-slip BCs) and different values of Dν:Dν=0 (thick solid line), Dν=0.01W(dashed line), and Dν=0.1W(dotted line). The case of Dν/greatermuchW, corresponding to Poiseuille flow, is represented by a thin solid line. The Gurzhi effect We now would like to make a remark on the temperature dependence of σxxin Eq. ( 21). For the sake of simplicity, we assume that lbdoes not depend on temperature. We observe that the derivative of σxxwith respect to T, dσxx dT=dσ0 dT(1−F)−σ0dF dDνdDν dT(23) is the sum of two contributions with opposite signs. The first term on the right-hand side of Eq. ( 23) is negative because F<1 and dσ0/dT < 0. The latter inequality holds because the scattering rate τ−1describing momentum nonconserving collisions is a monotonically increasing function of tempera-ture [ 49]. On the contrary, the second term is positive because dF/dD ν>0 anddDν/dT < 0. The vorticity diffusion length Dνdecreases with increasing temperature because both ν andτare decreasing functions of T. We therefore conclude that due to viscosity σxx(ρxx) can increase (decrease) upon increasing temperature. This is the so-called Gurzhi effect [ 12]. 165433-5TORRE, TOMADIN, GEIM, AND POLINI PHYSICAL REVIEW B 92, 165433 (2015) The existence of this effect relies crucially on the nature of BCs that are used to solve the hydrodynamic equations. Inparticular, it disappears for free-surface BCs. All previousexperimental studies of transport in graphene and other 2Delectron liquids we are aware of have reported monotonictemperature dependencies (i.e., no evidence of the Gurzhieffect) in the ordinary longitudinal geometry in the linear-response regime. We therefore conclude that free-surface BCsare the most appropriate for a weak driving current I.I nt h i s case,σ xxdepends only on the unknown damping rate τ−1, which can therefore be determined from an ordinary four-pointlongitudinal transport measurement at every value of ¯nandT, i.e.,τ −1=e2¯n/(mσxx). In the next section, we will discuss another hydrodynamic phenomenon occurring in 2D electron liquids, i.e., the forma-tion of whirlpools in electron flow [ 49], yielding a clear-cut experimental signal of hydrodynamic transport in weaklynonlocal linear-response transport measurements. Since theexperimental data in Ref. [ 49] do not show any Gurzhi effect in the linear-response regime, in the next section we willutilize only the free-surface BCs ( l b→∞ ). Whirlpools inhydrodynamic electron flow, however, do exist also when no- s l i pB C sa r eu s e d[ 49]. In this sense whirlpools are a much more robust phenomenon that the Gurzhi effect in longitudinaltransport. Whirlpools are also more dramatic in experimentalappearance. IV . NONLOCAL TRANSPORT AND THE IMPACT OF VISCOSITY We now present the solution of the problem posed by Eqs. ( 1), (5), (13), and ( 15) in the rectangle (−∞,∞)[−W/2,W/2]. We use free-surface boundary con- ditions, corresponding to lb→∞ . To this end, we introduce the Fourier transform with respect to the longitudinal coordinate xin the two equations ( 1) and ( 5) (the latter for the two components JxandJy). The Fourier transform of a function f(x,y) will be denoted by ˆf(k,y). These equations can be grouped into a linear system of three second-order ordinary differential equations (ODEs)with respect to the independent variable y. It is convenient to rewrite this system in terms of four first-order ODEs. We find ∂y⎛ ⎜⎜⎜⎝kˆJ x(k,y) kˆJy(k,y) ∂yˆJx(k,y) k2σ0ˆφ(k,y)/e⎞ ⎟⎟⎟⎠=k⎛ ⎜⎜⎝00 1 0 −i 000 1+1/(kD ν)200 −i/(kDν)2 01 +(kDν)2i(kDν)20⎞ ⎟⎟⎠⎛ ⎜⎜⎜⎝kˆJ x(k,y) kˆJy(k,y) ∂yˆJx(k,y) k2σ0ˆφ(k,y)/e⎞ ⎟⎟⎟⎠. (24) Equation ( 24) can be solved by diagonalizing the 4 ×4m a t r i x on the right-hand side, which has four distinct eigenval- ues:±1 and ±/radicalbig 1+1/(kDν)2. The general solution will therefore be a linear combination of exponentials of theform/summationtext iaiviexp(λiky) where vi(λi) are the eigenvectors (eigenvalues) of the matrix. The four unknown coefficientsa ican be found by enforcing the desired BCs. These are found by taking the Fourier transform of Eqs. ( 13) and ( 15) with respect to x, ˆJy(k,y=±W/2)=ˆJ±(k), (25) and [∂yˆJx(k,y)+ikˆJy(k,y)]y=±W/2=0. (26) The solution reads as follows: ˆφ(k,y)=/summationdisplay α=±eˆJα(k)W σ0/bracketleftbigg ˆF1α/parenleftBig kW,y W/parenrightBig +2D2 ν W2ˆF2α/parenleftBig kW,y W/parenrightBig/bracketrightbigg , (27) ˆJx(k,y)=/summationdisplay α=±ˆJα(k)W/braceleftbigg ik/bracketleftbigg ˆF1α/parenleftBig kW,y W/parenrightBig +2D2 ν W2ˆF2α/parenleftBig kW,y W/parenrightBig/bracketrightbigg −2D2 ν W2∂yˆF3α/parenleftbigg kW,y W,Dν W/parenrightbigg/bracerightbigg , (28)and ˆJy(k,y) =/summationdisplay α=±ˆJα(k)W/braceleftbigg ∂y/bracketleftbigg ˆF1α/parenleftBig kW,y W/parenrightBig +2D2 ν W2ˆF2α/parenleftBig kW,y W/parenrightBig/bracketrightbigg +2D2 ν W2ikˆF3α/parenleftbigg kW,y W,Dν W/parenrightbigg/bracerightbigg . (29) In writing Eqs. ( 27)–(29) we have introduced the following functions of dimensionless arguments: ˆF1±(˜k,˜y)=1 2/bracketleftbiggsinh( ˜k˜y) ˜kcosh( ˜k/2)±cosh( ˜k˜y) ˜ksinh( ˜k/2)/bracketrightbigg ,(30) ˆF2±(˜k,˜y)=˜k 2/bracketleftbiggsinh( ˜k˜y) cosh( ˜k/2)±cosh( ˜k˜y) sinh( ˜k/2)/bracketrightbigg , (31) and ˆF3±(˜k,˜y,λ)=i˜k 2/bracketleftBigg cosh( ˜y/radicalbig˜k2+λ−2) cosh(1 /2/radicalbig˜k2+λ−2) ±sinh( ˜y/radicalbig˜k2+λ−2) sinh(1 /2/radicalbig˜k2+λ−2)/bracketrightBigg . (32) Equations ( 27)–(32) are the most important results of this article. In general, it is not an easy task to inverse Fourier transform Eqs. ( 27)–(29) to real space after the functions ˆJ±(k)h a v e been specified. Indeed, this requires calculating a convolution 165433-6NONLOCAL TRANSPORT AND THE HYDRODYNAMIC SHEAR . . . PHYSICAL REVIEW B 92, 165433 (2015) which involves the BCs and the functions F1±(˜x,˜y),F3±(˜x,˜y), andF3±(˜x,˜y,λ) in real space. We now introduce the inverse Fourier transforms of the functions ˆF1±(˜k,˜y),ˆF2±(˜k,˜y), and ˆF3±(˜k,˜y,λ), which read F1±(˜x,˜y)=1 4πln/bracketleftbigg1+e−2π|˜x|+2s i n (π˜y)e−π|˜x| 1+e−2π|˜x|−2s i n (π˜y)e−π|˜x|/bracketrightbigg ∓1 4πln[1+e−4π|˜x|+2 cos(2 π˜y)e−2π|˜x|] ∓|˜x| 2, (33) and F2±(˜x,˜y)={ −πsin(π˜y)e−π|˜x|(1+e−2π|˜x|) ×{1+e−4π|˜x|−2[cos(2 π˜y)+2]e−2π|˜x|} ±2πe−2π|˜x|[cos(2 π˜y)(1+e−4π|˜x|)+2e−2π|˜x|]} ×[1+e−4π|˜x|+2 cos(2 π˜y)e−2π|˜x|]−2. (34) The functions F3±(˜x,˜y,λ) do not have simple expressions in terms of elementary functions but can be cast in the form ofan exponentially converging series, F 3±(˜x,˜y,λ)=−πsgn(˜x)/braceleftBigg∞/summationdisplay /lscript=0(2/lscript+1)(−1)/lscriptcos[(2 /lscript+1)π˜y] ×exp[−|˜x|/radicalbig λ−2+π2(2/lscript+1)2] ∓∞/summationdisplay /lscript=1(2/lscript)(−1)/lscriptsin(2/lscriptπ˜y) ×exp[−|˜x|/radicalbig λ−2+π2(2/lscript)2]/bracerightBigg . (35) In the following we will make use of a number of asymptotic behaviors of the functions F1±(˜x,˜y),F2±(˜x,˜y), andF3±(˜x,˜y,λ), which have been listed for the sake of convenience in Table I. The task of calculating the potential and currents in real space simplifies substantially if the currents J±(x)i nE q .( 13) can be represented by the sum of a finite number of δfunctions in real space. Let us focus on the geometry of Fig. 1where the Fourier transform of the BCs ( 14) reads ˆJ−(k)=−I(eikx 0− e−ikx 0)/eand ˆJ+(k)=0. In this case, we find that the steady- state current pattern can be written as J(r)=σ0 e∇φ(r)−¯nD2 ν∇×ω(r), (36)where the potential φ(r) and vorticity ω(r)≡ˆzω(r)a r eg i v e n by φ(r)=−I σ0/braceleftbigg F1−/parenleftbiggx− W,y W/parenrightbigg −F1−/parenleftbiggx+ W,y W/parenrightbigg +2D2 ν W2/bracketleftbigg F2−/parenleftbiggx− W,y W/parenrightbigg −F2−/parenleftbiggx+ W,y W/parenrightbigg/bracketrightbigg/bracerightbigg ,(37) and ω(r)=−2I e¯nW2/bracketleftbigg F3−/parenleftbiggx− W,y W,Dν W/parenrightbigg −F3−/parenleftbiggx+ W,y W,Dν W/parenrightbigg/bracketrightbigg . (38) In Eqs. ( 37) and ( 38) we have introduced the shorthand x±≡x±x0withx+(x−) representing the lateral separation between the observation point and the collector (injector). From Eq. ( 36) we clearly notice an important feature of the solution, i.e., for vanishing viscosity the current flow isirrotational. More precisely, the viscosity plays a twofold role:It modifies the irrotational contribution due to the electricpotential φ(r)andintroduces a finite vorticity. It is noteworthy that these effects yield independent experimental signatures:The modification of the electrical potential can be detectedby monitoring the resistances in a nonlocal configuration (orby carrying out scanning probe potentiometry), whereas thevorticity generates a magnetic field, which can be detected byscanning probe magnetometry. These two effects are discussedin detail in the following sections. A. Spatial dependence of the 2D electrical potential, charge current, and nonlocal resistances Illustrative results for the spatial map of the 2D electrical potential φ(r)—Eq. ( 37)—and the charge current pattern −eJ(r)—Eq. ( 36)—are shown in Fig. 3. For typical values of the drive current Iand conductivity, i.e., I=20μA through a submicron constriction and σ0=20 mS, we find that the scale over which the 2D electrical potential changesisφ 0≡I/σ 0=1 mV. We clearly see that in the case of ν/negationslash=0—panels (b) and (c) in Fig. 3—whirlpools with a spatial extension ∼Dνdevelop in the spatial current pattern −eJ(r) to the right of the current injector and to the left of the currentcollector. Once again, the spatial variations of the 2D electricalpotential φ(r) are amenable to experimental studies based on scanning probe potentiometry. In passing, we note that near the current injector at x=x 0 the potential is dominated by the singular parts of the functions TABLE I. Explicit expressions of the functions Fm−defined in the main text, evaluated at ˜y=∓1/2. We also summarize useful asymptotic behaviors in the limits |˜x|/lessmuch1a n d|˜x|/greatermuch1. Similar expressions can also be obtained for the quantities Fm+(˜x,˜y) by noting that Fm+(˜x,˜y)= Fm−(˜x,−˜y). |˜x|/lessmuch1 |˜x|/greatermuch1 ˜y=∓1/2 ˜y=−1/2 ˜y=1/2 ˜y=∓1/2 F1− π−1ln(1∓e−π|˜x|)+|˜x|/2 π−1ln(π|˜x|) π−1ln(2) |˜x|/2∓π−1e−π|˜x| F2− ±πe−π|˜x|/(1∓e−π|˜x|)2(π˜x2)−1−π/4 ±πe−π|˜x| F3− (1±1)δ/prime(˜x)/2 δ/prime(˜x)0 0 165433-7TORRE, TOMADIN, GEIM, AND POLINI PHYSICAL REVIEW B 92, 165433 (2015) −x0 0 x0−W/20W/2 −φ00φ0 −x0 0 x0−W/20W/2 −φ00φ0 −x0 0(a) (b) (c)x0−W/20W/2 −φ00φ0 FIG. 3. (Color online) Steady-state spatial map of the 2D elec- trical potential φ(r) (in units of φ0≡I/σ 0) and charge current streamlines −eJ(r) in a Hall bar device, such as the one depicted in Fig. 1withx0=W. Different panels refer to different values of the vorticity diffusion length Dν:Dν=0 [panel (a)], Dν=0.5W [panel (b)], and Dν=W[panel (c)]. Whirlpools are clearly seen in the bottom right and bottom left of panels (b) and (c). No whirlpools occur in the absence of viscosity as in panel (a). In each panel, thecurrent streamlines change color from white (high current density) to black (low current density). Fm−. Keeping only the leading terms for |x−x0|/lessmuchx0,Wand |y+W/2|/lessmuchx0,Win Eq. ( 37) we find an extremely simple expression for the potential near the injector, φ(r/prime,θ)=−I πσ0/bracketleftbigg ln/parenleftbiggr/prime R/parenrightbigg −2D2 νcos(2θ) r/prime2/bracketrightbigg , (39) where r/primeis the distance from the injection point, θis the angle measured from the injection direction ˆy, and Ris a length determined by BCs far from the contact. Note that: (i)changing Ris equivalent to changing φby an arbitrary additive constant, and (ii) the limit of a very large internal viscous force(compared to the frictional force exerted on a fluid element byagents external to the electron liquid) can be taken in Eq. ( 39) by letting τ→∞ . The end result of this limit is lim τ→∞φ(r/prime,θ)=2mIν π¯ne2cos(2θ) r/prime2. (40) Equation ( 40) explains the negative lobes of the electrical potential near the injector that are present in panel (c) of Fig. 3. If dc transport is to be used as the main tool to detect hydrodynamic electron flow, it is pivotal to understand thespatial dependence of the nonlocal resistance R NL, which we define in the following way: RNL(x,y)≡φ(x,y)−φ(x→+ ∞ ,y) I, (41) where the quantity φ(x→+ ∞ ,y) does not depend on y. Because of ( 37), we find that, at each point in space, RNL(x,y)is a quadratic function of Dν, RNL(x,y)σ0=a(x,y)D2 ν+b(x,y), (42) where a(x,y)=2 W2/bracketleftBig F2−/parenleftBigx+ W,y W/parenrightBig −F2−/parenleftBigx− W,y W/parenrightBig/bracketrightBig , (43) and b(x,y)=F1−/parenleftBigx+ W,y W/parenrightBig −F1−/parenleftBigx− W,y W/parenrightBig −x0 W. (44) To make contact with Ref. [ 49], we now introduce the vicinity resistance, which is the nonlocal resistance measured on theedge where the current is injected at a distance /Delta1xfrom the current injector, R V(/Delta1x)≡RNL(x0+/Delta1x,−W/2). (45) Using Eqs. ( 42)–(44), the asymptotic results in Table I, and taking the limit x0/greatermuchW, we find RV(/Delta1x)σ0=−2πe−π|/Delta1x|/W W2(1−e−π|/Delta1x|/W)2D2 ν +/bracketleftbigg −1 πln(1−e−π|/Delta1x|/W)+/Delta1x /Theta1 (−/Delta1x)/bracketrightbigg . (46) Here/Theta1(x) is the Heaviside step function. For positive /Delta1xthe two terms in Eq. ( 46) have opposite signs. For this reason, RV(/Delta1x) is expected to change sign as a function of Dν.T h e change in sign of the vicinity resistance is a key signature ofthe viscous contribution to the electric potential. Maximumsensitivity to viscosity is achieved when slightly nonlocalor vicinity voltage drops are measured outside the region[−x 0;x0] where the current flux is maximum. The vicinity resistance ( 46) rapidly decays for |/Delta1x|/greatermuchW/π : It is therefore pivotal to measure [ 49] the potential φ(x,y) for a lateral separation /Delta1xfrom the current injection point which is on the order of the vorticity diffusion length Dν. Once the rate of momentum-non-conserving collisions τ−1(¯n,T) is measured from an ordinary four-point longitudinal measurement of σxx(as explained in Sec. III), a measurement of the vicinity resistance RV(/Delta1x) yields a map ν=ν(¯n,T)o f the kinematic viscosity of the 2D electron liquid. We hastento stress that our all-electrical nonlocal protocol to measurethe kinematic viscosity ν=ν(¯n,T) applies to any 2D electron liquid driven into the hydrodynamic transport regime (and notonly to doped SLG and BLG). For the sake of completeness, we note that the authors of Ref. [ 24] have proposed an ac Corbino disk viscometer, which allows a determination of the hydrodynamic shear viscosityfrom the dc potential difference that arises between the innerand the outer edges of the disk in response to an oscillating magnetic flux. B. Spatial dependence of the current-induced magnetic field Because the steady-state current −eJ(r) generates a mag- netic field in the proximity of 2D electron system, whirlpoolsand viscosity-dominated hydrodynamic transport can also bedetected by scanning probe magnetometry (see, for example,Refs. [ 59–61]). 165433-8NONLOCAL TRANSPORT AND THE HYDRODYNAMIC SHEAR . . . PHYSICAL REVIEW B 92, 165433 (2015) −x0 0(a) (b)x0−W/20W/2 −B00B0 −x0 0 x0−W/20W/2 −B00B0 FIG. 4. (Color online) Spatial map of the ˆ zcomponent of the magnetic field Bz(r,z) (in units of B0≡μ0Id/W2), generated by the 2D steady-state current pattern J(r) and calculated immediately above the graphene sheet, i.e., for 0 <z/lessmuchW,D ν. These results have been obtained for the same parameters as in Fig. 3. Different panels refer to different values of Dν:Dν=0.5W[panel (a)] and Dν=W [panel (b)]. In this figure we have not shown results for Dν=0: In the absence of viscosity Bz(r,z) is identically zero. As shown in the Appendix, in a sample in which a backgate is placed at a distance z=−dbelow the graphene sheet with d/lessmuchW,D ν, a local relation exists between the ˆ zcomponent Bz(r,z > 0) of the magnetic field and the vorticity ω(r). In Syst`eme International (SI) units, this relation reads as follows: Bz(r,z > 0)=−eμ0¯ndω (r), (47) where μ0=4π×10−7NA2is the magnetic constant and the vorticity has been introduced earlier in Eq. ( 38). A typical 2D spatial map of Bz(r,z > 0) is shown in Fig. 4 for different values of the vorticity diffusion constant Dν. In this figure the magnetic field is plotted in units of B0≡ μ0Id/W2.F o rI=200μA,W=1μm, and d=80 nm, we findB0=20μT. This value is well within reach of current technology [ 59–61]. V . SUMMARY AND FUTURE PERSPECTIVES To summarize, we have presented a theoretical study of dc transport in graphene driven into the hydrodynamic regime.As highlighted in Ref. [ 49], this regime occurs in a wide range of temperatures and carrier concentrations. Our theory, which applies only to the doped regime, relies on the continuity ( 1) and Navier-Stokes ( 5) equations, augmented by suitable boundary conditions at Hall bar edges.We have demonstrated analytically that a combination ofordinary four-point longitudinal transport measurements andmeasurements of nonlocal resistances in Hall bar devices canbe used to extract the hydrodynamic shear viscosity of thetwo-dimensional electron liquid in graphene [ 49]. We have also discussed how to probe the viscosity- dominated hydrodynamic transport regime by scanning probemethods. Indeed, we believe that it possible to observehydrodynamic electron flow with spatial resolution by usingavailable scanning probe potentiometry and magnetometrysetups. Spatial maps of the two-dimensional electrical poten-tialφ(r) and current-induced magnetic field B z(r,z > 0) for experimentally relevant parameters are shown in Figs. 3and4. We wish to emphasize that our theoretical approach is immediately applicable to any 2D electron liquid in thehydrodynamic transport regime. In particular, it would beinteresting to revisit the hydrodynamic electron flow of 2Dparabolic-band electron liquids in high-quality GaAs/AlGaAssemiconductor heterostructures by employing the nonlocalmeasurement geometry proposed in Ref. [ 49] and used in this paper. As discussed in Ref. [ 14], in such heterostructures the hydrodynamic regime is expected to occur around 30 K. In the future, we plan to extend our theoretical approach to semimetals with linear and quadratic band touchings atthe charge neutrality point where viscosity is expected to bevery low [ 17] and the Reynolds number ( 16) is expected to be very large, possibly enabling the observation of electronicturbulence. This will require dealing with thermally excitedcarriers, coupling between charge and energy flow, and non-linear terms in the Navier-Stokes equation. We also would liketo gain a fully microscopic understanding of momentum-non-conserving collisions in the hydrodynamic transport regimeby treating smooth scalar andvector potentials due to disorder (strain [ 62], charged impurities [ 53], etc.) along the lines of what was done by the authors of Ref. [ 19] for the case of a smooth scalar potential. Last, but not least, we believe that hydrodynamic flow and the shear viscosity of 2D electron liquids can also be ac- cessed [ 63] by scattering-type near-field optical spectroscopy (see, for example, Ref. [ 38] and references therein to earlier work) in the terahertz spectral range since this techniquemeasures the nonlocal conductivity σ(q,ω). Terahertz radi- ation is required: (a) to make sure that the hydrodynamicinequality ωτ ee/lessmuch1 is satisfied (with τee=/lscriptee/vF) and (b) to have measurable nonlocal effects due to viscosity since thelatter decreases quickly [ 48] as a function of the external probe frequency ω. ACKNOWLEDGMENTS This work was partially supported by MIUR through the program “Progetti Premiali 2012”—Project “ABNAN-OTECH.” Free software ( www.gnu.org ,www.python.org ) was used. We thank M. F. Crommie, L. S. Levitov, L.A. Ponomarenko, G. Vignale, and A. Yacoby for fruitfuldiscussions. After this paper was submitted, we became awareof related work [ 64,65] on the impact of viscosity on graphene transport. APPENDIX A: LOCAL INDUCTANCE APPROXIMATION In this Appendix we present a derivation of Eq. ( 47). We use SI units and the Coulomb gauge ∇·A=0 for the vector potential. We assume that a bottom gate positioned at z=−d exists below the graphene sheet. This will be treated as aperfect conductor. The vector potential is related to the steady-state current pattern by the following 3D Poisson equation: /parenleftbigg ∇ 2+∂2 ∂z2/parenrightbigg A(r,z)=μ0eJ(r)δ(z). (A1) 165433-9TORRE, TOMADIN, GEIM, AND POLINI PHYSICAL REVIEW B 92, 165433 (2015) This is similar to the Poisson equation in Eq. ( 12) for the scalar potential /Phi1(r,z). Fourier transforming Eq. ( A1) with respect to the in-plane coordinate rwe find /parenleftbigg −q2+∂2 ∂z2/parenrightbigg ˆA(q,z)=μ0eˆJ(q)δ(z), (A2) where q·ˆJ(q)=0 because of the continuity equation Eq. ( 1). The general solution of Eq. ( A2)i sa sf o l l o w s : ˆA(q,z)=−μ0eˆJ(q)e−q|z| 2q+ˆa+(q)eqz+ˆa−(q)e−qz.(A3) The quantities ˆa±(q) must obey the condition ˆa±z(q)= ∓iq·ˆa±(q)/qto enforce the Coulomb gauge and must be determined from BCs. Requiring ˆA(q,z→+ ∞ )=0 implies thatˆa+(q) must vanish.The corresponding zcomponent of the magnetic field is given by ˆBz(q,z)=[iq׈A(q,z)]z. (A4) SinceBzmust vanish at the gate position, i.e., for z=−d,w e find: ˆa−(q)=μ0eˆJ(q)e−2qd/(2q). In deriving the previous result we have assumed that ˆa−(q)·q=0. The Fourier transform of the vector potential is therefore given by ˆA(q,z)=−μ0eˆJ(q)e−q|z|−e−q(2d+z) 2q. (A5) Now, if dand|z|are small with respect to the lateral length scales WandDνover which the steady-state current pattern J(r) changes in the sample, i.e., d,|z|/lessmuchW,D ν, the above formula can be approximated for z>0a s ˆA(q,z > 0)≈−eμ0dˆJ(q). (A6) Making use of Eq. 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PhysRevB.99.035416.pdf

PHYSICAL REVIEW B 99, 035416 (2019) Monolayer transition metal dichalcogenides in strong magnetic fields: Validating the Wannier model using a microscopic calculation J. Have,1,2,*G. Catarina,3T. G. Pedersen,1,4and N. M. R. Peres5,6 1Department of Materials and Production, Aalborg University, DK-9220 Aalborg East, Denmark 2Department of Mathematical Sciences, Aalborg University, DK-9220 Aalborg East, Denmark 3QuantaLab, International Iberian Nanotechnology Laboratory (INL), 4715-330 Braga, Portugal 4Center for Nanostructured Graphene (CNG), DK-9220 Aalborg East, Denmark 5International Iberian Nanotechnology Laboratory (INL), 4715-330 Braga, Portugal 6Center and Department of Physics, and QuantaLab, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal (Received 16 October 2018; revised manuscript received 27 November 2018; published 9 January 2019) Using an equation of motion (EOM) approach, we calculate excitonic properties of monolayer transition metal dichalcogenides perturbed by an external magnetic field. We compare our findings to the widely usedWannier model for excitons in two-dimensional materials and to recent experimental results. We find goodagreement between the calculated excitonic transition energies and the experimental results. In addition, we findthat the exciton energies calculated using the EOM approach are slightly lower than the ones calculated usingthe Wannier model. Finally, we also show that the effect of the dielectric environment on the magnetoexcitontransition energy is minimal due to counteracting changes in the exciton energy and the exchange self-energycorrection. DOI: 10.1103/PhysRevB.99.035416 I. INTRODUCTION The first use of an external magnetic field to study exci- tons and the electronic structure in thin-film transition metaldichalcogenides (TMDs) was published in 1978 [ 1]. Since then, the study of magnetoexcitons has been an active fieldof research. With the recent emergence of monolayer TMDs,research in this area has undergone a rapid development, duein part to the interesting electronic and optical propertiesof monolayer TMDs [ 2–4], including large exciton binding energies on the order of 0.5–1 eV [ 5–7]. Additionally, ex- citing magneto-optical phenomena of monolayer TMDs [ 8– 10] have inspired novel applications, for which a detailed understanding of the effect of a magnetic field on the excitonsis necessary. These phenomena include the valley Zeemaneffect, a magnetic field assisted lifting of the degeneracy ofthe inequivalent KandK /primevalleys [ 11–13]. This control of the degeneracy could prove useful in the area of valleytronics[14]. Another phenomenon lending itself to possible optical applications is Faraday rotation [ 15], which has also been observed in monolayer TMDs perturbed by a magnetic field[16,17]. In addition to potential applications, perturbation by an external magnetic field provides experimental insight into theproperties of excitons, such as their spatial extent [ 18,19] and the effect of the dielectric environment [ 20]. Using strong magnetic fields of up to 65 T, the Zeeman valley effect anddiamagnetic shift of the excitonic states have been measuredfor the four most common monolayer TMDs: MoS 2[21,22], *jh@nano.aau.dkMoSe 2[19,21,23], WS 2[24,25], and WSe 2[20,26]. The analysis of such experimental results would benefit froma thorough theoretical study of the effect of an externalmagnetic field on excitons. But while there is a plethora of ex-perimental results on magnetoexcitons, there have been fewertheoretical studies. The difficulties related to a theoreticaldescription of magnetoexcitons in two-dimensional materialsis, in part, due to the magnetic field breaking the translationsymmetry. In one-dimensional systems, translation symmetrycan be retained by choosing a suitable gauge for the magneticvector potential [ 27], but in two- and three-dimensional systems that option is not available. The standard theoretical approach has been to use an effective mass model such as the Wannier model [ 28], where the effective mass is calculated from the band structure of theunperturbed system. Using this approach, results regardingthe binding energy of excitons, trions, and biexcitons inmonolayer TMDs perturbed by a magnetic field were recentlypublished in Ref. [ 29]. But with no other theoretical models for magnetoexcitons in 2D materials, it can be difficult tovalidate the effective mass model. In addition, the effectivemass model does not take into account the unique Landaulevel structure of monolayer TMDs [ 8,10], which affects the magneto-optical response. In this paper, we provide analternative approach for describing magnetoexcitons, whichdoes not depend on the effective mass approximation. Theapproach is an extension of the equation of motion (EOM)method in Ref. [ 7] to the case in which the TMDs are per- turbed by an external magnetic field. This model has severaladvantages, which include accounting for the Landau levelstructure of TMDs, allowing coupling between distinct bandsand valleys, and providing a more self-contained theoretical 2469-9950/2019/99(3)/035416(13) 035416-1 ©2019 American Physical SocietyHA VE, CATARINA, PEDERSEN, AND PERES PHYSICAL REVIEW B 99, 035416 (2019) framework. The EOM approach can also be used to calcu- late the optical response and was previously used to includesecond-order effects in the electric field in Ref. [ 30]. The present paper is structured as follows: In Sec. II,w e introduce the single-particle Hamiltonian, which will serveas the outset for our study. In Sec. III, the EOM approach is briefly introduced. Section IVcontains the definition of the electron-electron interaction Hamiltonian, as well as thederivation of the EOM for the excitonic problem. Section V serves to introduce the Wannier model, which we will use forcomparison with the results obtained in the EOM approach.Finally, in Sec. VIour results are presented and compared to recent experiments. II. SINGLE-PARTICLE HAMILTONIAN In this section, we present the system and the single- particle Hamiltonian, which is the outset for our study ofmagnetoexcitons. The system is illustrated in Fig. 1. A mono- layer TMD material, possibly deposited on some dielectricsubstrate with relative dielectric constant κ aand capped by a dielectric with relative dielectric constant κb, is perturbed by a uniform static magnetic field perpendicular to the TMD.Under absorption of an incident photon with energy ¯ hωan exciton is generated. The properties of the exciton, i.e., sizeand energy, are affected by the magnetic field. To describe magnetoexcitons in monolayer TMDs, we need an accurate description of the single-particle propertiesof unperturbed TMDs. For that purpose, we apply the effec-tive Hamiltonian from Ref. [ 3]. This effective Hamiltonian describes a massive Dirac system, and has been found toreproduce the band structure of monolayer TMDs in the low-energy range around the direct band gaps in the KandK /prime valleys, including the spin-orbit splitting of the bands. For a monolayer in the xyplane the Hamiltonian is given by ˆH0=vF(τσxpx+σypy)+/Delta1τ,sσz+ξτ,sI, (1) where vFis the Fermi velocity, τ=± 1 is the valley index (+1f o rt h e Kvalley and −1f o rt h e K/primevalley), σiare the Pauli matrices with i∈{x,y,z },pxandpyare the canonical momentum operators, Iis the 2 ×2 identity matrix, and /Delta1τ,s andξτ,sare the valley- and spin-dependent mass and on-site Incident light - ħω B FIG. 1. Sketch of the system under consideration: Excitons in a monolayer TMD material perturbed by a uniform static magnetic field perpendicular to the monolayer. The monolayer may be encap- sulated between a dielectric substrate and a capping material.TABLE I. Parameters of the effective Hamiltonian for the four common types of TMDs. The mass parameters and the Fermi ve- locities are taken from Ref. [ 2] and the spin-orbit couplings are f r o mR e f .[ 31]; both sets of parameters were calculated from first principles. An alternative set of parameters is provided in Ref. [ 4]. /Delta1(eV) ¯ hvF(eV Å−1)/Delta1V soc(eV) /Delta1C soc(eV) MoS 2 0.797 2.76 0.149 −0.003 MoSe 2 0.648 2.53 0.186 −0.022 WS 2 0.90 4.38 0.430 0.029 WSe 2 0.80 3.94 0.466 0.036 energy, respectively. The mass and on-site energy are given by /Delta1τ,s=/Delta1−τs/Lambda11 2,ξ τ,s=τs/Lambda12 2, (2) where s=± 1(+1 for the spin up and −1 for spin down), /Lambda11=(/Delta1V soc−/Delta1C soc)/2, and /Lambda12=(/Delta1V soc+/Delta1C soc)/2. The pa- rameters vF,/Delta1,/Delta1V soc, and/Delta1C socare material dependent, and found by fitting to first-principles band structure calculation[2,31]. The material parameters used in this paper are pro- vided in Table I. The single-particle energy bands are the eigenvalues ε τ,sofˆH0, which are given by ετ,s=±/radicalBig ¯h2v2 F|k|2+/Delta12τ,s+ξτ,s. (3) Note that the eigenvalues only depend on the product τs=± 1, and not on τandsas individual parameters. The eigenvalues of MoS 2are plotted as dashed lines in Fig. 2.W e observe that the energy dispersion shows spin-orbit splittingof both valence and conduction bands and that the KandK /prime valleys are inequivalent due to spin. FIG. 2. Single-particle spectrum at the KandK/primevalleys of MoS 2 with (solid lines) and without (dashed lines) magnetic field. Red and blue indicate spin up and spin down, respectively. The Landau levelspectrum is plotted for a very high magnetic field (600 T) to make it possible to distinguish the individual Landau levels. Qualitatively similar features are found at lower magnetic field strengths. 035416-2MONOLAYER TRANSITION METAL DICHALCOGENIDES IN … PHYSICAL REVIEW B 99, 035416 (2019) The next step is the inclusion of a perpendicular magnetic field B. The magnetic field is introduced using the minimal coupling substitution p/mapsto→p+eA, where pis the momentum operator, −eis the electron charge, and Ais the magnetic vector potential, related to the magnetic field by ∇×A=B. Using the Landau gauge, A=Bxˆy, the effective perturbed Hamiltonian is ˆHB=vF[τσxpx+σy(py+eBx)]+/Delta1τ,sσz+ξτ,sI.(4) The eigenvalues and eigenfunctions of ˆHBcan be found by expressing ˆHBin terms of creation and annihilation operators [8,32], and then expanding the eigenfunctions in a basis of harmonic oscillator eigenfunctions. We find that the eigenval-ues and the normalized eigenfunctions are given by E n,λ τ,s=λ/radicalBig /Delta12τ,s+n(¯hωc)2+ξτ,s, (5) /Psi1n,λ τ,s,k y(r)=eikyy /radicalbigLy/Phi1n,λ τ,s(˜x). (6) Here,n/greaterorequalslant(1+τλ)/2 is the integer Landau level (LL) index, λ=± indicates the type of LLs ( +for conduction type LLs and −for valence type LLs), ¯ hωc=√ 2¯hvF/lBis the cyclotron energy, lB=√¯h/(eB) is the magnetic length, Ly is the length of the system in the ydirection, and the spinor wave function is /Phi1n,λ τ,s(˜x)=1√ 2/parenleftBigg Bn,λ τ,sφn−(τ+1)/2(˜x) Cn,λ τ,sφn+(τ−1)/2(˜x)/parenrightBigg . (7) Here, ˜x=x+l2 Bky,φn(˜x) are the usual harmonic oscillator eigenstates, and Bn,λ τ,sandCn,λ τ,sare normalization constants given by Bn,λ τ,s=λ/radicalBig 1+λαnτ,s,Cn,λ τ,s=/radicalBig 1−λαnτ,s, (8) where αn τ,s=/Delta1τ,s/√ /Delta12 τ,s+n(¯hωc)2. The harmonic oscilla- tor eigenstates are given by φn(˜x)=1√ 2nn!/parenleftbigg1 πl2 B/parenrightbigg1 4 e−˜x2 2l2 BHn/parenleftbigg˜x lB/parenrightbigg , (9) where Hnare the physicist’s Hermite polynomials, which are defined by Hn(x)=(−1)nex2dn dxne−x2. (10) Note that the energies En,λ τ,sdefine a discrete set of LLs that have a degeneracy corresponding to the number of distinct ky values. The Landau level spectrum of MoS 2is plotted (solid lines) in Fig. 2.F r o mF i g . 2and the allowed values of n, w es e et h a taL Lw i t h n=0 is only allowed when τ/negationslash=λ. This gives rise to a magnetic-field-dependent increase of theband gap. Finally, the valley Zeeman splitting [ 12] is not included in the effective Hamiltonian ˆH B. It could have been by adding additional terms to ˆHB[33], but since the focus of the present paper is on the excitonic effects, it is ignored forsimplicity.A. Dipole matrix elements In this section, the dipole matrix elements for the single- particle wave functions are calculated. In addition to beingnecessary for calculating the optical response, the dipole ma-trix elements provide information about the optical selectionrules, which can be used to exclude some dark transitions fromour excitonic calculations. This speeds up the numerical stud-ies performed below by a significant factor. The interactionof the system with the incident light is included, within thedipole approximation, via the interaction Hamiltonian H I=−d·E(t)=er·E(t). (11) Here, d=−eris the dipole moment operator and E(t)t h e time-dependent electric field of the light. By construction,transitions between different valleys and different spins arenot allowed. We introduce some notation to simplify theexpressions. Let αbe shorthand for {n,λ,k y}andηfor {τ,s}; then the dipole matrix elements are written as dα→α/prime η= /angbracketleft/Psi1n,λ τ,s,k y|d|/Psi1n/prime,λ/prime τ/primes/prime,k/primey/angbracketright, where /Psi1n,λ τ,s,k yare the single-particle eigen- states of ˆHB. For the dipole matrix elements in the xdirection, we find dα→α/prime η,x=−eδky,k/primey/angbracketleftbig /Phi1n,λ τ,s/vextendsingle/vextendsinglex/vextendsingle/vextendsingle/Phi1n/prime,λ/prime τ,s/angbracketrightbig =−eδky,k/primey/angbracketleftbig /Phi1n,λ τ,s/vextendsingle/vextendsingle[ˆHB,x]/vextendsingle/vextendsingle/Phi1n/prime,λ/prime τ,s/angbracketrightbig En,λ τ,s−En/prime,λ/prime τ,s. (12) The commutator is simply [ ˆHB,x]=−i¯hvFτσx. A similar expression holds for the commutator with y. Consequently, the dipole matrix elements are found to be dα→α/prime η=e¯hvFδky,k/primey 2/Delta1En,λ n/prime,λ/prime/bracketleftbigg Bλ,n τ,sCn/prime,λ/prime τ,s/parenleftbigg−iτ 1/parenrightbigg δn−τ,n/prime −Bn/prime,λ/prime τ,sCn,λ τ,s/parenleftbiggiτ 1/parenrightbigg δn+τ,n/prime/bracketrightbigg . (13) Here,/Delta1En,λ n/prime,λ/prime:=En,λ τ,s−En/prime,λ/prime τ,s. The nonzero dipole matrix elements correspond to the bright interband transitions.Equation ( 13) shows that the allowed interband transitions from a LL with index nare to LLs with index n /prime=n±1 and at the same kypoints. III. EQUATION OF MOTION APPROACH The excitonic properties will be calculated using an EOM approach similar to that of Ref. [ 7], which is an extension of the method introduced to describe the magneto-optics ofgraphene in a cavity in Ref. [ 34]. The approach relies primar- ily on writing and solving Heisenberg’s equation of motion,which is given by −i¯h∂ˆρ ∂t=[ˆH,ˆρ]. (14) Here ˆH, is the full Hamiltonian including ˆHI, and ˆρis the density matrix for the states of ˆHB. To compute the density matrix, we introduce the cre- ation and annihilation operators ˆc† α,η(t) and ˆcα,η(t), which, respectively, create or annihilate an electron in state /Psi1η α≡ /Psi1n,λ τ,s,k y(recall that αis short for {n,λ,k y}andηis short 035416-3HA VE, CATARINA, PEDERSEN, AND PERES PHYSICAL REVIEW B 99, 035416 (2019) for{τ,s}). The creation and annihilation operators obey the usual anticommutator relations. Using these operators, we canexpress the single-particle Hamiltonian and the light-matterinteraction Hamiltonian as ˆH B(t)=/summationdisplay α,ηEη αˆρη α,α(t), (15) ˆHI(t)=−E(t)·/summationdisplay α,α/prime,ηdα→α/prime η ˆρη α,α/prime(t), (16) where ˆ ρη α,α/prime(t)=ˆc† α,η(t)ˆcα/prime,η(t) are elements of the density matrix in a basis of the eigenstates of ˆHB. Note that only a few of the terms in the sum over α/primegive nonzero contributions toˆHIdue to the optical selection rules from Sec. II. Solving Heisenberg’s EOM exactly as expressed in Eq. ( 14) is not possible. Consequently, we take the expectation value on both sides of Eq. ( 14) with respect to the equilibrium state, and get the following EOM for the expectation value: −i¯h∂ ∂tpη α,α/prime=/angbracketleftbig/bracketleftbigˆH,ˆρη α,α/prime/bracketrightbig/angbracketrightbig , (17) withpη α,α/prime=/angbracketleftρη α,α/prime/angbracketright. Note that the diagonal elements α=α/prime define a new electron distribution. The commutators of ˆHB and ˆHIwith the density matrix are calculated in Appendix A and can be used to calculate the single-particle optical re-sponse as in Ref. [ 35]. We now turn to the problem of including electron-electron interactions in the Hamiltonianand then find the excitonic states by solving Eq. ( 17). IV . ELECTRON-ELECTRON INTERACTIONS From this point on, we consider the full Hamiltonian given by ˆH=ˆHB+ˆHI+ˆHee, where the electron-electron interaction Hamiltonian is defined by ˆHee=1 2/integraldisplay dr1dr2ˆψ†(r1)ˆψ†(r2)U(r1−r2)ˆψ(r2)ˆψ(r1). (18) Here, the integrals also cover spin, U(r) is the electron- electron interaction potential defined below, and ˆψ(r)i st h e field operator, given by ˆψ(r)=/summationdisplay α,ηˆcα,η/Psi1η α(r). (19) Here and in the following, we drop the explicit time depen- dence of ˆcα,η(t) and ˆρη α,α/prime(t) to simplify notation. Although monolayer TMDs are not strictly 2D materials, the electrons are effectively confined to move in two dimen-sions by the negligible thickness of the layer [ 6,7]. Conse- quently, instead of the usual Coulomb potential, we model theelectron-electron interaction U(r) by the Keldysh potential [36], which is valid for strict 2D systems. In momentum spaceTABLE II. Parameters used in the calculation of the excitonic properties for the four common types of TMDs. The first and second columns contain the reduced exciton masses for the spin-up and spin- down bands, respectively. The third column is the in-plane screeninglength, and is taken from Ref. [ 6]. μτ,+1(me) μτ,−1(me) r0(Å) MoS 2 0.380 0.418 41.4 MoSe 2 0.355 0.417 51.7 WS 2 0.159 0.199 37.9 WSe 2 0.170 0.223 45.1 the Keldysh potential has the following simple form [ 36–38]: U(q)=e2 2ε01 q(κ+r0q), (20) where q=|q|,ε0is the vacuum permittivity, r0is a material- dependent in-plane screening length, and κ=(κa+κb)/2i s the average of the relative dielectric constant of the substrateand the capping material. The in-plane screening lengths arerelated to the in-plane polarizability, and can be calculatedfrom a first-principles band structure. The parameters usedin this paper were calculated in Ref. [ 6] and are listed in Table II. It is worth mentioning that the Keldysh potential previously has been used successfully to describe variousexcitonic properties of TMDs [ 6,7,46,47]. Before calculating the commutator of ˆH eewith the density matrix and solving the Heisenberg EOM, we will rewrite ˆHeeslightly. Assuming that the electron-electron coupling between different valleys is negligible, the ˆHeecan be written as ˆHee=1 2/summationdisplay τ,s,s/prime α1,α2 α3,α4Uτ,s,s/prime α1α4,α2α3ˆc† α1,τ,sˆc† α2,τ,s/primeˆcα3,τ,s/primeˆcα4,τ,s, (21) where two of the summations over spin cancel because of the spin integrals in Eq. ( 18), and the so-called Coulomb integrals are Uτ,s,s/prime α1α4,α2α3=1 4π2/integraldisplay d2qU(q)Fτ,s α1,α4(q)Fτ,s/prime α2,α3(−q).(22) Here,Fτ,s α,α/prime(q) are structure factors defined as Fτ,s α,α/prime(q)=/integraldisplay d2reiq·r/bracketleftbig /Psi1α τ,s(r)/bracketrightbig∗/Psi1α/prime τ,s(r). (23) An explicit expression for the structure factors is provided in Appendix B.U s i n gE q .( 21), we calculate the commutator of the full Hamiltonian with the density matrix in Appendix A and find that the EOM in Eq. ( 17) can be written as /parenleftbigg Eη α/prime−Eη α−i¯h∂ ∂t/parenrightbigg pη α,α/prime=/summationdisplay α1,α2 α3pη α1,α3/parenleftbig Uτ,s,s α/primeα3,α1α2pη α,α 2−Uτ,s,s α1α,α 2α3pη α2,α/prime/parenrightbig −E(t)·/summationdisplay α/prime/prime/parenleftbig dα/prime/prime→α ηpη α/prime/prime,α/prime−dα/prime→α/prime/prime ηpη α,α/prime/prime/parenrightbig .(24) 035416-4MONOLAYER TRANSITION METAL DICHALCOGENIDES IN … PHYSICAL REVIEW B 99, 035416 (2019) Here, Eη α≡Eτ,s n,λand the expectation value of the four-body operator in ˆHeehas been truncated at the random phase approximation (RPA) level [ 39]. Comparing the EOM to what was found in Ref. [ 7], we see that the general form of the equation is equivalent to the expression for a system with anarbitrary number of bands. In the following subsections, wekeep only the terms of Eq. ( 24), which are of first order in the electric field and collect the terms corresponding to the ex- change self-energy corrections and electron-hole interactions. A. Exchange self-energy corrections In this section, we briefly touch upon the exchange self- energy corrections caused by the electron-electron interac-tions. The term exchange should be understood in the senseof the Hartree-Fock approximation, where there are twocorrections to self-energy: the Hartree correction, which iscanceled by the interaction with the positive background (seeAppendix A), and the exchange correction. Although exchange self-energy corrections are not the main focus of this work, it is still important to include them ifwe hope to accurately describe the transition energy of the ex-citons. This is because the self-energy correction has a strongimpact on the value of the single-particle gap. In Appendix A, the first-order terms that result in a renormalization of the LLsare collected. It is found that the self-energy-renormalizedLLs, ˜E η α,a r eg i v e nb y ˜Eη α=Eη α−/Sigma1η α,/Sigma1η α=/summationdisplay α/primef/parenleftbig Eη α/prime/parenrightbig Uτ,s,s α/primeα,αα/prime. (25) Here,/Sigma1η αis the exchange self-energy correction and f(E) is the Fermi-Dirac distribution. We calculate the exchangeself-energy correction using the structure factors fromAppendix B. Converting the sum over k yto an integral, the exchange self-energy can be written as /summationdisplay α/primeUτ,s,s α/primeα,αα/primef/parenleftbig Eη α/prime/parenrightbig =/summationdisplay n/prime,λ/primef/parenleftbig Eη n/prime,λ/prime/parenrightbig Iη λ/primen/prime,λn, (26) where the integrals are defined as Iη λn,λ/primen/prime=1 16π2/integraldisplay d2qU(q)e−l2 Bq2 2/vextendsingle/vextendsingleJη λn,λ/primen/prime(q)/vextendsingle/vextendsingle2. (27) Here,Jη λn,λ/primen/primeis the function defined in Eq. ( B4). The integral in Eq. ( 27) is simplified by the fact that U(q) and|Jη λn,λ/primen/prime(q)|2 only depend on q=|q|, meaning that the angular integral simply gives a factor of 2 π. In the remainder of the paper, we assume that the system is undoped, i.e., the Fermi levelis in the band gap, and that T=0 K. This implies that the sum in Eq. ( 26) only runs over the valence-type LLs, which simplifies the numerical calculations. For graphene described in the Dirac approximation, the exchange self-energy correction has been found to divergelogarithmically when summing over an infinite number of va-lence LLs [ 40]. We have observed the same type of divergence numerically for the expression in Eq. ( 26). Consequently, a cutoff N cutof the summation over LLs has to be introduced. In Ref. [ 41]( s e ea l s oR e f .[ 42]), the cutoff was calculated for graphene by equating the concentration of electrons in Ncut LLs to that in the filled valence band. The same approach canbe used for TMDs and we find a cutoff equal to Ncut=πl2 B /Omega10, (28) with/Omega10=√ 3a2/2 the area of the primitive unit cell of the TMD. Taking a=3.2 Å for all four TMDs [ 43], we get a cutoff equal to Ncut≈2.33×104/BT. B. Excitonic effects Finally, using the exchange self-energy corrected LLs, we proceed to calculating the excitonic effects of TMDsperturbed by an external magnetic field. As shown inAppendix A, the excitonic states can be found by solving the first-order equation /parenleftbigg ˜E η α/prime−˜Eη α−i¯h∂ ∂t/parenrightbigg pη,1 α,α/prime =/parenleftBigg/summationdisplay α1,α2Uτ,s,s α/primeα2,α1αpη,1 α1,α2−E(t)·dα/prime→α η/parenrightBigg /Delta1fη α/prime,α,(29) where/Delta1fη α/prime,α=f(Eη α/prime)−f(Eη α). As in Ref. [ 7] the excitonic transition energies can be calculated by solving the homoge-neous equation, i.e., setting E(t)=0. Changing from time to frequency domain, we get the homogeneous equation /parenleftbig˜Eη α/prime−˜Eη α−E/parenrightbig pη α,α/prime=/summationdisplay α1,α2Uτ,s,s α/primeα2,α1αpη α1,α2/Delta1fη α/prime,α.(30) Here, pη α,α/primeshould be understood as the Fourier transform ofpη,1 α,α/primeandEis the exciton transition energy for a fixed combination of spin and valley. The excitonic states are theinterband solutions of Eq. ( 30), i.e., where αandα /primecorre- spond to valence and conduction states, respectively. Thus,we assume that to be the case. Additionally, the sum over α 1 andα2can be split into two contributions: one where α1and α2are valence and conduction states, respectively, and one where the converse holds. We denote these cases the resonantcontribution and the nonresonant contribution, respectively. In the following, we keep only the resonant contribution. It has been shown in Ref. [ 7] that this is a valid approximation. To clearly distinguish the valence and conduction states, we write α vandαcforα1andα2in Eq. ( 30), respectively. Setting ky=k/prime y(which corresponds to ignoring the dark non- vertical transitions; see Sec. II A), we simplify the right-hand side of Eq. ( 30) by writing /summationdisplay αv,αcUτ,s,s α/primeαc,αvαpη αv,αc≈/summationdisplay nv,nc/integraldisplay∞ −∞dqyKτ,s,s n/primenc,nvn(qy−ky) ×pη nv,nc(qy). (31) Here, we write the approximate sign to indicate the approxi- mations discussed above, and we denote pη αv,αcbypη nv,nc(ky) for the case in which the kyvalues associated with αcandαv are equal. The different λparameters are fixed by the previous assumptions and are not written explicitly. The electron-holeinteraction kernel K τ,s,s n/primenc,nvnis calculated using the structure 035416-5HA VE, CATARINA, PEDERSEN, AND PERES PHYSICAL REVIEW B 99, 035416 (2019) FIG. 3. Electron-hole interaction kernels plotted for MoS 2in a magnetic field of 100 T. The kernels are plotted for the Kvalley, spin up, and ( n, n/prime,nv)=(0,1,0). factors and is found to be Kτ,s,s n/primenc,nvn(qy)=1 16π2/integraldisplay∞ −∞dqxU(q)e−l2 B|q|2 2 ×Jτ,s +n/prime,+nc(q)Jτ,s −nv,−n(−q), (32) where the integral over qxmust be performed numerically. This finally implies a homogeneous first-order equation givenby /parenleftbig˜Eη α/prime−˜Eη α−E/parenrightbig pη n,n/prime(ky) =/summationdisplay nv,nc/integraldisplay∞ −∞dqyKτ,s,s n/primenc,nvn(qy−ky)pη nv,nc(qy). (33) Equation ( 33) corresponds to the Bethe-Salpeter equation for electron-hole pairs [ 44], and it can be written as an eigenvalue problem with eigenvalues Eby discretizing the integral over qy. The size of the eigenvalue problem scales as NkNcNv, where Nkis the number of points used to discretize the integral, and where NcandNvare the number of conduction and valence LLs, respectively. It is clear that only if theelectron-hole kernel decays sufficiently fast with increasing n c andnvcan we hope to solve Eq. ( 33), since that would imply that the sums over ncandnvcan be truncated. Fortunately, the kernel does decay quite fast in ncandnv, as illustrated for ncin Fig. 3. In the next section, we turn our attention to an alternative (and nonmicroscopic) description of the excitonicproperties of TMDs. V . WANNIER MODEL In this section, we briefly introduce the Wannier model [28] for excitons. The Wannier model is based on the ef- fective mass approximation for a single pair of valence andconduction bands. For a two-dimensional semiconductor in aperpendicular magnetic field (using the symmetric gauge forthe magnetic vector potential), the operator describing zeroangular momentum excitons, i.e., s-type states, is [ 45] ˆH ex=−¯h2 2μ∇2+e2B2 8μr2−U(r). (34) Here, μis the reduced effective mass, ∇2is the 2D Laplace operator, ris the relative electron-hole distance, and U(r) is the electron-hole interaction potential given as thereal-space representation of Eq. ( 20). Taking the inverse Fourier transform of Eq. ( 20), we find U(r)=e2 8ε0r0/bracketleftbigg H0/parenleftbiggκr r0/parenrightbigg −Y0/parenleftbiggκr r0/parenrightbigg/bracketrightbigg , (35) withr=|r|,H0the Struve function, and Y0a Bessel function of the second kind. For a direct comparison of the Wannier model with the solutions to Eq. ( 33), we want to use the same parameters in both models. Thus, we calculate the effective mass from the eigenvalues of the unperturbed single-particle operator ˆH0. Expanding the eigenvalues in Eq. ( 3) around |k|=0, we find that the effective mass of an electron or hole in the τvalley and with spin sis m∗ τ,s=|/Delta1τ,s| v2 F. (36) The effective masses of electrons and holes are equal due to the symmetric conduction and valence bands. The reducedeffective mass is then μ τ,s=m∗ τ,s/2, for which the values for the four common TMDs are given in Table II. Thes-type excitons, corresponding to bright excitons [ 46], can be found by solving the eigenvalue problem ˆHexψ(r)= Eexcψ(r), where Eexcis the exciton energy. We solve it by expanding ψ(r) in a basis of Bessel functions, more specif- ically the basis φi(r)=J0(λir/R), where λiis the ith zero of the Bessel J0function and r/lessorequalslantR. This basis corresponds to introducing an infinite barrier at r=R, but this should not affect the results as long as Ris sufficiently large. The same basis was recently used to describe the Stark shift of excitonsin monolayer TMDs [ 46,47]. VI. RESULTS In this section, our results are presented and discussed. In addition, we devote some attention to the computationalapproaches applied. All results were obtained using the pa-rameters in Tables Iand II. Evaluating the integrals in the exchange self-energy correction, i.e., Eq. ( 27), is done using an adaptive quadrature and a numerical high-precision library[48]. This approach, although computationally expensive, is found to provide accurate results for the rapidly oscillatingintegrands that occur when nandn /primeare large. In contrast, since the sum in Eq. ( 33) can be truncated at reasonably low values of ncandnv, as illustrated by Fig. 3, the integral in the electron-hole kernel can be evaluated using the Gauss-Hermite quadrature. For the calculation of excitonic energiesusing the Wannier model, we use 400 basis functions and fixRatR=20 nm. The kinetic and magnetic matrix elements can be calculated analytically in this basis, while the potentialmatrix elements are computed numerically using a Gauss-Legendre quadrature. First, we consider the exchange corrections. We denote the exchange self-energy corrected and the uncorrected band gapsas˜E τs gandEτs g, respectively. The ˜Eτs gandEτs gband gaps are plotted in Fig. 4as a function of magnetic field for τs=+ 1, i.e., spin up at the Kvalley or spin down at the K/primevalley. The results show that the self-energy correction gives rise toan opening of the band gap on the order of 0.8 to 1.0 eV .Similar values hold for the τs=− 1 gaps. We find smaller 035416-6MONOLAYER TRANSITION METAL DICHALCOGENIDES IN … PHYSICAL REVIEW B 99, 035416 (2019) FIG. 4. Plot of τs=+ 1 band gaps of suspended monolayer TMDs, i.e., taking κ=1. The uncorrected (black) and exchange self-energy corrected (red) band gaps are shown as a function of magnetic field. In addition, the exchange self-energy correction to the band gaps, /Delta1/Sigma1τs≡˜Eτs g−Eτs g, is plotted (blue). The blue lines refer to the blue axes, while the rest refer to the black axes. exchange self-energy corrections than those of Ref. [ 7]f o rt h e case of unperturbed monolayer TMDs. The explanation forthis discrepancy is twofold: First, we use a different parameterset. Second, the cutoffs that are used are different. But, aswill be shown later, our approach results in exciton transitionenergies that match experiments quite well. Considering the magnetic field dependence of the band gaps, we see that the uncorrected band gaps calculated usingthe LL energies in Eq. ( 5) vary linearly with magnetic field for the field range in Fig. 4. We also find a linear magnetic field dependence of the exchange self-energy correction to theband gap with slopes of 5 .57μeV/Tf o rM o S 2,7.76μeV/T for MoSe 2,2 0.0μeV/Tf o rW S 2, and 19 .3μeV/T for WSe 2. The slopes are for τs=+ 1 states, but similar slopes hold for theτs=− 1 states. This apparent linear behavior of /Delta1/Sigma1τs= ˜Eτs g−Eτs gcan be explained by studying the expression in Eq. ( 26). For small B, it can be shown using Eqs. ( 27) and (B4) that the integrals Iη λn,λ/primen/primeare proportional to√ B,f o r allλ,λ/prime,n, andn/prime. If we can show that Iη −0,−n/prime−Iη +1,−n/primeis proportional to ( n/prime+1)−3/2as a function of n/prime, the result is a linear behavior of /Delta1/Sigma1τssince /Delta1/Sigma1τs∝√ BNcut/summationdisplay n/prime(n/prime+1)−3 2≈√ B/integraldisplayNcut+1 1dn/primen/prime−3 2 ≈2√ KB. (37) Here, the last approximation holds for a cutoff of the type Ncut=K/B , with Ksome constant, and for small B.T h einset in Fig. 5shows Iη −0,−n/prime−Iη +1,−n/primeon a log-log scale for MoS 2, with B=100 T and τs=+ 1. Fitting with a linear function, we find a power of q=− 1.33±0.03 covering the range from 20 T to 100 T. Thus, an approximately linearbehavior of the exchange self-energy correction is expected. FIG. 5. Convergence of the transition energy of the Aexciton in MoS 2in a 100 T field. The black line refers to the situation where all LLs up to a cutoff Nv=Ncare included and the red line refers to the situation where only significant transitions are included, i.e., of thetypen vtonc∈[nv−1,nv+3]. The dashed blue line is the exciton transition energy calculated. Finally, the inset shows the integrals Iη −0,−n/prime−Iη +1,−n/primeon a log-log scale for τs=+ 1. 035416-7HA VE, CATARINA, PEDERSEN, AND PERES PHYSICAL REVIEW B 99, 035416 (2019)0 1 234 5 6 7 8912345678910 00.20.40.60.81 FIG. 6. Plot of the squared eigenvector of the Aexciton in MoS 2in an external field of 100 T at ky=0. The elements of the eigenvector have been normalized, such that the largest norm is unity. The plot shows that only a few transitions are significant, and that they are centered around transitions allowed by the optical selectionrules. In photoluminescence and spectroscopy experiments, it is typically the exciton transition energy and not the ex-change self-energy corrected band gap that is measured. Butdemonstrating that the exchange self-energy correction isapproximately linearly in the magnetic field is important ifthe diamagnetic shift of the exciton transition energy is usedto estimate the exciton size, as was done in Refs. [ 18,20,22]. Any finite quadratic dependence of the exchange self-energycorrection would result in errors in the estimates of the excitonsizes. Although the results presented here do not exclude finitequadratic terms in the exchange self-energy correction, theyappear to be small enough that any error in the estimation ofthe exciton size should be negligible. Turning our attention to the exciton states, we note that it is difficult to separate the bright and dark exciton statescalculated in the EOM approach, since Eq. ( 33) mixes dark and bright transitions. This difficulty might be resolved bywriting the magnetic vector potential in the symmetric gaugeinˆH Band repeating the derivations in Sec. IV, but this study is left for future work. At the present time, we will instead focuson the exciton with the lowest transition energy, also calledthe ground state exciton. We follow convention and denotethe spin up and down ground state excitons at the Kvalley as AandB, respectively. Similarly, we have A /primeandB/primeexcitons in the K/primevalley. In the absence of valley Zeeman splitting, theAandA/primeexcitons are energetically degenerate and the same holds for the BandB/primeexcitons. Consequently, in the following, only the AandBexcitons are considered. In Fig. 6, the squared eigenvector of the Aexciton in MoS 2is plotted for ky=0. The plot shows that the significant transitions between LLs are where nvcouples to nc=nv+1, which coincides exactly the bright transitions according to Sec. II A.W ea l s o find that the same holds for the Bexciton. Consequently, the AandBexcitons must be bright.When solving Eq. ( 33), discretizing the integral over qy using a Gauss-Hermite quadrature with Nk=300 nodes has been found to result in good convergence. If we then includethe first 15 valence and conduction LLs in the summationin Eq. ( 33), the resulting matrix has size 67 500 ×67 500 and is at the limit of what we can handle numerically. Butfor these values the exciton transition energy has not yetconverged, as illustrated for the Aexciton in MoS 2by the black line in Fig. 5. Alternatively, we can utilize that only a few transitions are significant in the exciton ground state,as was demonstrated in Fig. 6. In fact, calculating the norm of the eigenvector where only transitions of the type n vto nc∈[nv−1,nv+3] have been included, we find that the squared overlap is only 2% less than unity. Including onlythese significant transitions allows us to include more valenceLLs and, as illustrated by the red line in Fig. 5, obtain a better convergence. The cost is a small error on the orderof a few meV . The numerical difficulties associated withincluding a high number of LLs in the excitonic calculationsresult in a restriction on the magnetic field strength usedhence: as the magnetic field strength decreases, more LLsneed to be included in the calculations to secure sufficientlyconverged results. Eventually, the current computational re-strictions limit us to magnetic fields above 100 T. Turning to the exciton transition energies, it has been shown in Refs. [ 19,22] that for magnetic fields in the range considered here the transition energies E τcan be approxi- mated by Eτ=E0+μgB+τμZB+σdiaB2, (38) withτthe valley index, E0the zero-field exciton transition energy, μgBthe field-dependent change in band gap, τμZB the valley Zeeman shift, and finally σdiaB2the diamagnetic shift. Since the valley Zeeman shift is not included in oursingle-particle Hamiltonian, the transition energies found bysolving Eq. ( 33) can be approximated by E=E 0+μgB+ σdiaB2. To allow for comparisons between the theoretical and the experimentally measured exciton transition energies,we average the experimentally measured exciton transitionenergies from the KandK /primevalleys to remove the valley Zeeman splitting, i.e., use E=(E+1+E−1)/2. The exciton transition energies of the AandBexcitons are presented in Table III. In columns three and four, we show the theoretical transition energies, which were calcu-lated by solving Eq. ( 33). Columns five and six contain the experimental exciton transition energies when there is noexternal magnetic field. In columns seven and eight, we showthe experimental exciton transition energies at approximately65 T. Comparing the zero-field transition energies with theexperimental transition energies in columns seven and eight,we see that the exciton transition energies exhibit a minimaldependence on the magnetic field. In fact, experiments predictthat the quadratic diamagnetic shift is on the order of only afew meV [ 18,22] for a magnetic field of 100 T. Consequently, we can compare the calculated transition energies to themeasured transition energies in a system with no magneticfield. Table IIIshows that the transition energies of MoS 2, WS 2, and WSe 2are very well captured by our model, with differences on the order of 10 meV . The calculated resultsfor MoSe 2differ more from the experimental results, with 035416-8MONOLAYER TRANSITION METAL DICHALCOGENIDES IN … PHYSICAL REVIEW B 99, 035416 (2019) TABLE III. Theoretical and experimental transition and exciton energies for AandBexcitons in TMDs with different dielectric environments. All theoretical energies are computed at 100 T. Transition energies Exciton energies EOM Experimental, B=0 T Experimental, B≈65 T EOM Wannier TMD κAB A B A B A B A B MoS 21.00 1.918 2.076 −0.620 −0.632 −0.617 −0.632 1.55 1.907 2.066 1.895 [ 22], 1.948 [ 21] 2.042 [ 22], 2.092 [ 21] 1.896 [ 22], 1.948 [ 21] 2.044 [ 22], 2.094 [ 21]−0.491 −0.504 −0.489 −0.503 MoSe 21.00 1.516 1.735 −0.526 −0.542 −0.513 −0.533 1.55 1.512 1.730 1.660 [ 13] −0.419 −0.434 −0.409 −0.428 WS 2 1.00 2.042 2.467 −0.559 −0.584 −0.520 −0.555 1.55 2.030 2.453 2.039 [ 25], 2.045 [ 22] 2.442 [ 25], 2.453 [ 22] 2.040 [ 25], 2.046 [ 22] 2.442 [ 25], 2.454 [ 22]−0.426 −0.450 −0.392 −0.424 WSe 21.00 1.761 2.216 −0.511 −0.535 −0.468 −0.505 1.55 1.755 2.209 1.744 [ 12] −0.393 −0.417 −0.357 −0.391 3.30 1.721 2.173 1.732 [ 20] 1.733 [ 20] −0.229 −0.247 −0.197 −0.224 4.50 1.700 2.152 1.723 [ 18] 1.724 [ 18] −0.177 −0.192 −0.144 −0.168 the calculated transition energy being approximately 150 meV below the experimental transition energy. This discrepancyindicates a problem with the material parameters used and notthe method, as the results agree well for the three other typesof materials. In the final four columns of Table III, the exciton energies calculated using the EOM approach and the Wannier modelare presented. For the EOM method, the exciton energies arefound from E exc=E−˜Eg, where Eis the exciton transition energy found by solving Eq. ( 33) and ˜Egis the exchange self- energy corrected band gap. Comparing the results, we see thatall the exciton energies calculated using the EOM approachare below the Wannier results. That is to be expected sincethe EOM approach relies on less strict approximations. Thedifferences between the calculated energies are quite smalland vary from a few meV to 50 meV . Thus, if errors in thisrange are acceptable, the Wannier model provides a usefulmodel for excitons in monolayer TMDs. Finally, we also consider the effect of changing the di- electric environment of the TMDs, i.e., varying the screeningparameter κin the potentials in Eqs. ( 20) and ( 35). The effect is illustrated in Fig. 7for MoS 2in a magnetic field of 100 T. The figure shows that the exchange self-energy correctedband gap decreases while the exciton energy increases as afunction of κ. These two counteracting effects result in exciton transition energies, which only exhibit minimal dependenceon the dielectric environment, as illustrated by the blue lineand green squares in Fig. 7. This effect has previously been demonstrated in TMDs with no external magnetic field [ 49], but Fig. 7illustrates that it still holds for systems in the presence of a perpendicular magnetic field. This phenomenonfurther underlines the importance of including the exchangeself-energy corrections in a self-contained model. We find thatsimilar results hold for the other TMDs. Comparing the EOM method and the Wannier model, we see that both have advantages and disadvantages. The EOMmethod provides a self-contained framework, including theunique LL structure and a higher accuracy of the excitonenergies. The disadvantage is that the numerical computationsare demanding and, as a consequence, small magnetic fieldscannot be considered. For the Wannier method, the numericalcalculations are relatively simple and arbitrary magnetic field strengths can be considered. The disadvantages are that forsome systems the accuracy is lower than the EOM method andthat only the excitonic properties are described. The Wanniermodel provides no information about the unique LL structure,the band gap, or the field-dependent change of the bandgap. Consequently, the choice between the EOM method andthe Wannier method depends on the application, and whichaspects are deemed important. VII. SUMMARY In summary, starting from a Dirac-type Hamiltonian de- scribing the band structure of monolayer TMDs around the K andK/primepoints, we have introduced an external magnetic field and then included electron-electron interactions to accountfor the exchange self-energy corrections and excitons. In this 1234-0.75-0.50-0.250.000.252.002.252.502.75 FIG. 7. Plot of the corrected band gap (red line), the exciton transition energy (blue line and green diamonds), and the exciton energy (black line) as a function of the relative dielectric constant of the surrounding medium for MoS 2, with B=100 T and τs=+ 1. The exciton transition energy calculated from the Wannier results (blue line) is the sum of exciton energy (black line) and the corrected band gap (red line), i.e., E=˜Eg+Eexc. 035416-9HA VE, CATARINA, PEDERSEN, AND PERES PHYSICAL REVIEW B 99, 035416 (2019) setup, we used the EOM approach to find the low-energy A andBexcitons. Our results were compared to the popular Wannier model for excitons and recent experimental results. When comparing with the Wannier model, we found that theAandBexciton energies match quite well. Conse- quently, the EOM method validates the Wannier model in thiscase. The exciton energies only exhibit a small dependenceon the magnetic field (up to a few meV for realistic fieldstrengths), but the optical properties are expected to changesignificantly. These changes include optical transitions be-tween discrete LLs, which depend strongly on the magneticfield, and a finite optical Hall conductivity giving rise toFaraday rotation in TMDs. Thus, we will focus on the opticalproperties of magnetoexcitons in future projects. We also ex-pect to see more pronounced differences between the opticalresponse calculated using the EOM approach and the Wanniermodel. Comparing the calculated transition energies with the ex- perimental values, we also found a very good agreement.This shows that the exchange self-energy correction is centralif accurate theoretical calculations of the exciton transitionenergies are needed. Finally, we considered the effect of thedielectric environment on the exciton transition energy. Wefound that increasing the dielectric constant of the environ-ment causes a decrease in the corrected band gap and anincrease in the exciton energy. These two counteracting effectscause a minimal dependence of the exciton transition energieson the dielectric environment. This holds for both the EOMmethod results and transition energies calculated from theWannier model results. ACKNOWLEDGMENTS J.H. and T.G.P. gratefully acknowledge financial support by the QUSCOPE Center, sponsored by the Villum Foun-dation. Additionally, T.G.P. is supported by the Center forNanostructured Graphene (CNG), which is sponsored by theDanish National Research Foundation, Project No. DNRF103. G.C. acknowledges financial support from FCT for the P2020-PTDC/FIS-NAN/4662/2014 project. N.M.R.P. acknowledgessupport from the European Commission through the project“Graphene-Driven Revolutions in ICT and Beyond” (Ref.No. 785219), and the Portuguese Foundation for Science andTechnology (FCT) in the framework of the Strategic Financ-ing UID/FIS/04650/2013. Additionally, N.M.R.P. acknowl-edges COMPETE2020, PORTUGAL2020, FEDER, and thePortuguese Foundation for Science and Technology (FCT)through Project No. PTDC/FIS-NAN/3668/201. APPENDIX A: COMMUTATOR RELATIONS AND THE EQUATION OF MOTION In this section, we present the commutator relations be- tween ˆH=ˆHB+ˆHI+ˆHeeand the density matrix, as well as the relevant equation of motion. First, we calculate thecommutator relations using the following relation: /bracketleftbig ˆρ η α1,α2,ˆρη/prime α3,α4/bracketrightbig =ˆρη α1,α4δα2,α3δη,η/prime−ˆρη α3,α2δα1,α4δη,η/prime.(A1) Applying this relation to the first two terms of the commutator [ˆH,ρη α,α/prime], we find /bracketleftbigˆHB,ˆρη α,α/prime/bracketrightbig =/summationdisplay α/prime/prime,η/primeEη/prime α/prime/prime/bracketleftbig ˆρη/prime α/prime/prime,α/prime/prime,ˆρη α,α/prime/bracketrightbig (A2) =/parenleftbig Eη α−Eη α/prime/parenrightbig ˆρη α,α/prime (A3) and /bracketleftbigˆHI,ˆρη α,α/prime/bracketrightbig =−E(t)·/summationdisplay α1,α2,η/primedα1→α2 η/prime/bracketleftbig ˆρη/prime α1,α2,ˆρη α,α/prime/bracketrightbig (A4) =−E(t)·/summationdisplay α/prime/prime/parenleftbig dα/prime/prime→α η ˆρη α/prime/prime,α/prime−dα/prime→α/prime/prime η ˆρη α,α/prime/prime/parenrightbig .(A5) In the commutator relation between the electron-electron in- teraction Hamiltonian and the density matrix, the following commutator relation is useful: /bracketleftbigˆc† α1,τ,s/primeˆc† α2,τ,s/prime/primeˆcα3,τ,s/prime/primeˆcα4,τ,s/prime,ˆc† α,τ/prime,sˆcα/prime,τ/prime,s/bracketrightbig =δτ,τ/prime/parenleftbigˆc† α1,τ,sˆc† α2,τ,s/prime/primeˆcα3,τ,s/prime/primeˆcα/prime,τ,sδα,α 4δs,s/prime+ˆc† α1,τ,s/primeˆc† α2,τ,sˆcα/prime,τ,sˆcα4,τ,s/primeδα,α 3δs,s/prime/prime −ˆc† α,τ,sˆc† α2,τ,s/prime/primeˆcα3,τ,s/prime/primeˆcα4,τ,sδα/prime,α1δs,s/prime−ˆc† α1,τ,s/primeˆc† α,τ,sˆcα3,τ,sˆcα4,τ,s/primeδα/prime,α2δs,s/prime/prime/parenrightbig .(A6) Applying Eq. ( A6)t ot h e[ ˆHee,ˆρη α,α/prime] commutator, we find /bracketleftbig Hee,ˆρη α,α/prime/bracketrightbig =/summationdisplay s/prime,α1 α2,α3/braceleftbig Uτ,s,s/prime α1α,α 2α3ˆc† α1,τ,sˆc† α2,τ,s/primeˆcα3,τ,s/primeˆcα/prime,τ,s−Uτ,s,s/prime α/primeα1,α2α3ˆc† α,τ,sˆc† α2,τ,s/primeˆcα3,τ,s/primeˆcα1,τ,s/bracerightbig , (A7) where we also used the relation Uτ,s,s/prime α1α4,α2α3=Uτ,s/prime,s α2α3,α1α4. (A8) Collecting the terms in Eqs. ( A3), (A5), and ( A7), we can now write Heisenberg’s equation of motion for the full Hamiltonian including electron-electron interactions. To write Eq. ( 17), we compute the expectation value of the commutator relations keeping terms that are of first order in the electric field. While the expectation values of Eqs. ( A3) and ( A5) are found by straightforward calculation, we apply the random phase approximation (RPA) [ 39] to find /angbracketleftbig/bracketleftbig Hee,ˆρτ,s α,α/prime/bracketrightbig/angbracketrightbig =/summationdisplay s/prime,α1 α2,α3/braceleftbig Uτ,s,s/prime α1α,α 2α3/parenleftbig pτ,s/prime α2,α3pτ,s α1,α/prime−δs,s/primepτ,s α1,α3pτ,s α2,α/prime/parenrightbig −Uτ,s,s/prime α/primeα1,α2α3/parenleftbig pτ,s/prime α2,α3pτ,s α,α 1−δs,s/primepτ,s α2,α1pτ,s α,α 3/parenrightbig/bracerightbig , (A9) 035416-10MONOLAYER TRANSITION METAL DICHALCOGENIDES IN … PHYSICAL REVIEW B 99, 035416 (2019) where pτ,s α,α/prime=/angbracketleftˆρτ,s α,α/prime/angbracketright. Terms allowing mixing of spins correspond to the Hartree terms in Hartree-Fock theory. They are canceled by the interaction with the positive background [ 50] and, as a result, the expectation value has the following form: /angbracketleftbig/bracketleftbig Hee,ˆρτ,s α,α/prime/bracketrightbig/angbracketrightbig =/summationdisplay α1,α3pα1,α3/summationdisplay α2/parenleftbig Uτ,s,s α/primeα3,α1α2pτ,s α,α 2−Uτ,s,s α1α,α 2α3pτ,s α2,α/prime/parenrightbig . (A10) This gives the following EOM for the expectation value: /parenleftbigg Eη α/prime−Eη α−i¯h∂ ∂t/parenrightbigg pη α,α/prime=/summationdisplay α1,α2 α3pη α1,α3/parenleftbig Uτ,s,s α/primeα3,α1α2pη α,α 2−Uτ,s,s α1α,α 2α3pη α2,α/prime/parenrightbig −E(t)·/summationdisplay α/prime/prime/parenleftbig dα/prime/prime→α ηpη α/prime/prime,α/prime−dα/prime→α/prime/prime ηpη α,α/prime/prime/parenrightbig .(A11) The final step is to expand the expectation values in orders of the electric field and collect first-order terms in Eq. ( A11). The zeroth order of the expectation value can be expressed using the Fermi-Dirac distribution pη,0 α,α/prime=f/parenleftbig Eη α/parenrightbig δα,α/prime, (A12) where f(E) is the Fermi-Dirac distribution. Consequently, the first-order equation is /parenleftbigg Eη α/prime−Eη α−i¯h∂ ∂t/parenrightbigg pη,1 α,α/prime=/parenleftBigg/summationdisplay α1,α2Uτ,s,s α/primeα2,α1αpη,1 α1,α2−E(t)·dα/prime→α η/parenrightBigg /Delta1fη α/prime,α+/summationdisplay α1,α2f/parenleftbig Eη α1/parenrightbig/parenleftbig Uτ,s,s α/primeα1,α1,α2pη,1 α,α 2−Uτ,s,s α1α,α 2α1pη,1 α2,α/prime/parenrightbig , (A13) where/Delta1fη α/prime,α=f(Eη α/prime)−f(Eη α) and pη,1 α,α/primeis the first-order term of the expectation value. We rewrite the last term on the right-hand side to isolate the exchange self-energy correction /summationdisplay α1,α2f/parenleftbig Eη α1/parenrightbig/parenleftbig Uτ,s,s α/primeα1,α1,α2pη,1 α,α 2−Uτ,s,s α1α,α 2α1pη,1 α2,α/prime/parenrightbig =/Sigma1η α/prime−/Sigma1η α+/summationdisplay α1f/parenleftbig Eη α1/parenrightbig⎛ ⎝/summationdisplay α2/negationslash=α/primeUτ,s,s α/primeα1,α1α2pη,1 α,α 2−/summationdisplay α2/negationslash=αUτ,s,s α1α,α 2α1pη,1 α2,α/prime⎞ ⎠, (A14) where/Sigma1η αis the exchange self-energy correction given by /Sigma1η α=/summationdisplay α1f/parenleftbig Eη α1/parenrightbig Uτ,s,s α1α,αα 1. (A15) The remaining terms in Eq. ( A14) correspond to density terms and will be disregarded in this work. Thus, the first-order EOM for the expectation value of the density matrix reads /parenleftbigg ˜Eη α/prime−˜Eη α−i¯h∂ ∂t/parenrightbigg pη,1 α,α/prime=/parenleftBigg/summationdisplay α1,α2Uτ,s,s α/primeα2,α1αpη,1 α1,α2−E(t)·dα/prime→α η/parenrightBigg /Delta1fη α/prime,α, (A16) with ˜Eη α=Eη α−/Sigma1η α. The interband solutions to the system of first-order differential equations in Eq. ( A16) give the excitonic states. APPENDIX B: STRUCTURE FACTORS In this section, we find an explicit expression for the structure factors Fτ,s α,α/primedefined in Eq. ( 23). The explicit expression allows for a numerical evaluation of the Coulomb integrals in Eq. ( 22). Inserting the expression for the single-particle wave function, Eq. ( 6), in the structure factors, we find Fτ,s α,α/prime=/integraldisplay d2rei(qy−ky+k/prime y)y Lyeiqxx/bracketleftbig Bn,λ τ,sBn/prime,λ/prime τ,sφnτ,−(˜x)φn/prime τ,−(˜x/prime)+Cn,λ τ,sCn/prime,λ/prime τ,sφnτ,+(˜x)φn/prime τ,+(˜x/prime)/bracketrightbig , (B1) where the notation is ˜x=x+l2 Bky,˜x/prime=x+l2 Bk/prime y,nτ,−=n−(τ+1)/2, and nτ,+=n+(τ−1)/2. For each term of Eq. ( B1), we calculate an integral of the type /integraldisplay dxeiqxxφn(˜x)φn/prime(˜x/prime)=exp/parenleftbigg −l2 B(ky−k/prime y)2+l2 Bq2 x 4+iqxl2 B 2(ky+k/prime y)/parenrightbigg/radicalBigg n<! n>!/parenleftbiggilBqx+lBsgn(n−n/prime)(ky−k/prime y) √ 2/parenrightbiggn>−n< ×Ln>−n< n</parenleftbiggl2 Bq2 x+l2 B(ky−k/prime y)2 2/parenrightbigg , (B2) 035416-11HA VE, CATARINA, PEDERSEN, AND PERES PHYSICAL REVIEW B 99, 035416 (2019) where n>=max{n,n/prime},n<=min{n,n/prime}, andLm nare associated Laguerre polynomials. The detailed calculation of the integral in Eq. ( B2) was provided in Ref. [ 51]. The previous expression allows us to write the structure factors as Fτ,s α,α/prime(q)=πδ(qy−ky+k/prime y) Lyexp/parenleftbigg −l2 B|q|2 4+iqxl2 B 2(ky+k/prime y)/parenrightbigg Jτ,s λn,λ/primen/prime(q), (B3) where the function Jη λn,λ/primen/primeis defined as Jτ,s λn,λ/primen/prime(q)=/parenleftbiggilBqx+lBsgn(n−n/prime)qy√ 2/parenrightbiggn>−n</bracketleftBigg/radicalBigg [n<−(τ+1)/2]! [n>−(τ+1)/2]!Bn,λ τ,sBn/prime,λ/prime τ,sLn>−n< n<−(τ+1)/2/parenleftbiggl2 B|q|2 2/parenrightbigg +/radicalBigg [n<+(τ−1)/2]! [n>+(τ−1)/2]!Cn,λ τ,sCn/prime,λ/prime τ,sLn>−n< n<+(τ−1)/2/parenleftbiggl2 B|q|2 2/parenrightbigg/bracketrightBigg . (B4) The expression for the structure factors in Eq. ( B3) is used to calculate both the excitonic properties and the exchange self-energy corrections. [1] M. Tanaka, H. Fukutani, and G. Kuwabara, J. Phys. Soc. Jpn. 45,1899 (1978 ). [2] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108,196802 (2012 ). [3] A. Kormányos, V . Zólyomi, N. D. 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PhysRevB.72.073205.pdf

Electronic behavior of rare-earth dopants in AlN: A density-functional study S. Petit *and R. Jones School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom M. J. Shaw and P. R. Briddon School of Natural Sciences, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, United Kingdom B. Hourahine and T. Frauenheim Theoretische Physik, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany /H20849Received 20 January 2005; revised manuscript received 15 April 2005; published 11 August 2005 /H20850 Local density functional calculations are carried out on Er, Eu, and Tm rare-earth /H20849RE/H20850dopants in hexagonal AlN. We find that the isolated impurities prefer to substitute for Al and, in contrast with isolated RE dopantsin GaAs and GaN, RE Aldefects are electrically active and introduce deep donor levels around Ev+0.5 eV. RE complexes with oxygen and vacancies are discussed; some of these have deep levels in the upper third of thegap and could account for a threshold excitation energy around 4 eV observed for intra- ftransitions at 465 and 478 nm in AlN:Tm. DOI: 10.1103/PhysRevB.72.073205 PACS number /H20849s/H20850: 71.55.Eq, 68.55.Ln, 61.72.Ji, 71.15.Mb I. INTRODUCTION Rare-earth /H20849RE/H20850doped semiconductors exhibit sharp intra- foptical transitions which have long been of interest for displays. Recently, it has become apparent that in wide-band materials such as GaN and SiC, and in contrast with Siand GaAs, the luminescence is not quenched at room tem- perature. As far as we are aware, all confirmed optical transitions in doped semiconductors are due to the RE defect in a trivalentoxidation state RE 3+. Thus, irrespective of doping and mate- rial, the fshell of Er, for example, contains a fixed number of 11 electrons. The localized nature of the fshell ensures that the influence of the crystal field of the host on the RE isslight and confined to splitting the degenerate multiplet statesof the RE and relaxing the selection rules for dipole-allowedtransitions. However, it is by no means obvious that the REdefect does not seriously perturb the electronic structure ofthe host. The size of the RE ion, the number of valenceelectrons, and its electronegativity may differ from the host;thus, it may be expected that the RE could induce one ormore gap levels occupied by valence electrons and not thoseof the fshell. This is particularly true for group IV semicon- ductors, where the RE has a different valence from the host. 1 However, we have previously shown that RE impurities, sub-stituting Ga in two III–V semiconductors GaAs and GaN, donot introduce any gap levels. 2,3Consequently, the RE defect is unable to bind photogenerated electrons or holes, and ex-citation of the fshell is inefficient. This suggests that com- plexes with other defects are required to act as exciton orcarrier traps which survive room temperature and which re-combine through an excitation of the fshell. This immedi- ately explains why large doping concentrations of the RE/H20849/H110111%/H20850are required, 4and that high-resolution photolumines- cence /H20849PL/H20850and photoluminescent excitation spectroscopy /H20849PLE /H20850reveal5the presence of many RE optical defects with low symmetry. In GaAs, it appears that RE oxygen defectsare the most important, 2while in GaN the RE-V Ndefect hasbeen suggested to be a dominant trap.3Nevertheless, it is no means obvious a priori that substitutional RE defects do not bind carriers in all III–V materials, and this is less likely inwider band gap materials. Indeed, we show here that Er, Tm,and Eu dopants in hexagonal AlN, having a gap of 6.12 eV,introduce a deep donor level around 0.5 eV above the va-lence band. This result points to a unique property of REdopants in AlN. Several RE optical transitions have been reported in AlN. The 1.54 /H9262m transition due to Er3+has been closely studied.6,7PLE studies show a broad band with excitation energies above about 2 eV; superimposed on this band aresharp spikes. This indicates at least two classes of Er 3+cen- ters. The broad band is attributed to optically excited Er-related defects possessing gap levels and the spikes to directintra- fexcitation. 6PL studies of AlN:Tm reveal intense 465 and 478 nm blue lines when excited above 4.3 and 4 eV,respectively, possibly relating to two different defects. 8RE doping of AlN and AlGaN alloys seems to result in moreefficient and temperature-stable luminescence than GaN. 7,6 Moreover, the PL intensity of Al xGa1−xN:Tb /H20849Ref. 9 /H20850and the CL intensity of Al xGa1−xN:Eu /H20849Ref. 10 /H20850increase dramati- cally with xup to 15%. Lattice location studies demonstrate that implanted Er,11Yb, and Tm /H20849Ref. 12 /H20850primarily lie at substitutional Al sites, although more recent RBS data13sug- gest some displacement away from the Al site. We investigate here the structure and electrical properties of RE dopants in AlN. In Sec. II we describe the theoreticalmethod that is used. In Sec. III, we apply the method to studynon-RE defects, namely the Si Al,OAl, and the nitrogen va- cancy V Nto assess the errors likely to be encountered. In Sec. IV we analyze the structure and electronic behavior ofEr, Eu, and Tm defects in AlN, and conclude in Sec. V. II. METHOD We use a spin-polarized local density functional code, AIMPRO , with localized basis sets of Gaussian s,p,d, and fPHYSICAL REVIEW B 72, 073205 /H208492005 /H20850 1098-0121/2005/72 /H208497/H20850/073205 /H208494/H20850/$23.00 ©2005 The American Physical Society 073205-1orbitals. Hartwigsen-Goedecker-Hutter14pseudopotentials are also used to eliminate core electrons. These pseudopoten-tials were developed for Gaussian basis sets. Following ear-lier studies, 3thefshell for the RE defects is frozen and treated as part of the core. All atoms were relaxed in 72 atomsupercells formed with lattice vectors 3 a,3b, and 2 c, where a,b, and care the unit vectors of bulk AlN. A Monkhorst- Pack 3 3sampling scheme15was used for these supercells. We found lattice parameters of bulk AlN to be a=3.076 Å and c=4.930 Å, which are within 1% of experiment,16and a bulk modulus Bequal to 201 GPa and consistent with experimen- tal values of 185–212 GPa.16The calculated heat of forma- tion of AlN, 3.38 eV, is found from the energies of bulk AlN,Al, and N 2gas,17and close to the experimental value of 3.30 eV.16The band structure is similar to that found in a previous study.18 LetEq/H20851R/H20852denote the energy of a defective supercell in charge state qwith structure defined by Rand made up of ni atoms of species iwith chemical potential /H9262i.I fR/H20849q/H20850denotes the most stable structure for the charge state q, then the de- fect formation energy is given by Ef=Eq/H20851R/H20849q/H20850/H20852−/H20858ini/H9262i −q/H20849EF−Ev/H20850, where qis the excess electron charge, EFandEv the Fermi energy and valence band maximum, respectively. The chemical potentials for Al and N are assumed to be forstoichiometric growth conditions. The chemical potential forO is derived from the O 2molecule, while that of the RE is found from hexagonal RE-nitride.3The formation energy Ef is related to the equilibrium concentration of impurities and of intrinsic defects. These concentrations are given bygNexp/H20849−E f/kT/H20850, where Nis the density of lattice locations for the defect and gis the orientational degeneracy. Experi- ence with other defects indicates less than perfect quantita-tive agreement with solubility data, but they give a guide tothe expected equilibrium defect concentration. The electrical levels of the defect are related to the elec- tron affinities and ionization energies or differences in for-mation energies between charged defects. However, the the-oretical band gap evaluated from E −/H20851R/H20849−/H20850/H20852+E+/H20851R/H20849+/H20850/H20852 −2E0/H20851R/H208490/H20850/H20852is 5.27 and 0.8 eV below the experimental gap. The Kohn-Sham energy gap derived from the band structure is even smaller at /H110114.2 eV. These well-known underesti- mates are a consequence of local density functional theory,and cause difficulties in relating the levels to the band edges.As an alternative, the levels can be found by comparing the defect ionization energy with that of the host or preferablyanother defect with similar levels. 2In this last method, we first evaluate E−/H20851R/H20849−/H20850/H20852−E0/H20851R/H208490/H20850/H20852and E0/H20851R/H208490/H20850/H20852−E+/H20851R/H20849+/H20850/H20852. The relaxation resulting in differences in the structures R/H208490/H20850 andR/H20849−/H20850is taken into account in these expressions. To ex- tract the electrical levels, these energies are compared with similar ones for a marker defect with known levels; hence,the relative levels can be found. Such a method eliminatessystematic errors in the calculation of the energies. In par-ticular, Makov-Payne correction terms would largely vanishif the two defects had similar charge distributions. Themethod works best when the defect levels are close to thoseof the standard. Here, we choose Si Alto be the standard defect which has donor and acceptor levels19found experi- mentally at Ev+6.06 eV and Ec−0.32 eV. This donor level isclose to a PL line attributed to a Si bound exciton at 6.024 eV.20 If there are large lattice relaxations in one or more charge states of the defect, then optical transitions may not be di-rectly related to the donor or acceptor levels. For example,the energy for a vertical Franck-Condon transition, taking anelectron from the top of the valence band and adding it tothe neutral defect, is the acceptor level referenced to thevalence band, together with the relaxation energy E −/H20851R/H208490/H20850/H20852 −E−/H20851R/H20849−/H20850/H20852 /H20849see Fig. 1 /H20850. The relaxation energy may be con- siderable and leads to a broad PL spectrum typically seen in large band gap semiconductors. RE defects are an exception,as the internal excitation will the 4 fshell will not lead to any appreciable structural change. We examined the convergence in the structure and ener- gies of some defects using a larger cell of 192 atoms. Forexample, in the defect Er Al−/H20849ON/H208502, the EruO bond lengths along the caxis and in the basal plane are 2.36 and 2.10 Å, respectively, in the 72 atom cell, and 2.36 and 2.09 Å in thelarger cell. Clearly, the structure of the defects is well con-verged. Energy differences are more sensitive to cell size.However, E −/H20851R/H20849−/H20850/H20852−E0/H20851R/H208490/H20850/H20852and E0/H20851R/H208490/H20850/H20852−E+/H20851R/H20849+/H20850/H20852are −10.397 and 9.282 eV in the 72 atom cells, and −10.288 and 9.467 eV in the 192 atom cell, respectively. Thus, these en-ergy differences are converged to about 0.2 eV. III. SILICON, OXYGEN, AND VACANCY-NITROGEN DEFECTS We find both the neutral and the positively charged Si Al defect to lie on-site but, in the negative charge state, the SiuN bond along c/H20849Ref. 21 /H20850breaks, rather than one of the SiuN bonds lying near the basal plane.22,23Using the bulk ionization energy and electron affinity as a marker, the cal-culated Si donor and acceptor levels are E v+5.22 eV and Ec−0.53 eV, respectively. These values are deeper than the FIG. 1. Schematics of the energy configurations describing the capture of an electron by a neutral defect resulting in a verticaloptical transition. The Franck-Condon absorption energy /H20849FC/H20850is E −/H20851R/H208490/H20850/H20852−E0/H20851R/H208490/H20852/H20850−Ev. This energy is numerically the sum of the acceptor level of the defect referenced to the valence band,E −/H20851R/H20849−/H20850/H20852−E0/H20851R/H208490/H20850/H20852−Ev, together with the relaxation energy /H20849R.E. /H20850, which is E−/H20851R/H208490/H20850/H20852−E−/H20851R/H20849−/H20850/H20852.BRIEF REPORTS PHYSICAL REVIEW B 72, 073205 /H208492005 /H20850 073205-2experimental values of Ev+6.06 eV and Ec−0.32 eV men- tioned above. The underestimate of the donor level is almostthe same as the underestimate of the gap. The theoreticallevels indicate a positive- Ubehavior, as the donor level at E v+5.22= Ec−0.9 eV lies below the Ec−0.53 eV acceptor level. This is in conflict with the experimental donor andacceptor levels at E c−0.06 and Ec−0.32 eV. However, the difference is relatively small. The optical /H20849Franck-Condon /H20850 ionization energy of the negative defect, resulting in the for-mation of a neutral defect and an electron in the conductionband, is E c+E0/H20851R/H20849−/H20850/H20852−E−/H20851R/H20849−/H20850/H20852. This can be written as Ec −/H20849E−/H20851R/H20849−/H20850/H20852−E0/H20851R/H20849−/H20850/H20852/H20850or the depth of the acceptor level, Ec−/H20849E−/H20851R/H20849−/H20850/H20852−E0/H20851R/H208490/H20850/H20852/H20850, together with the relaxation en- ergy E0/H20851R/H20849−/H20850/H20852−E0/H20851R/H208490/H20850/H20852, and is found to be 2.09 eV. Experi- mentally, a broad photoconductivity spectrum with a peak around 2.0 eV and a threshold around 1.5 eV is found.19 The formation energy of the substitutional oxygen defect, ON, is −1.1 eV. This is 2.3 eV lower than that of interstitial oxygen, demonstrating that the interstitial species would be aminority one. The negative value of the formation energy ofO Nindicates that AlN would be readily oxidized, and mate- rial with a large concentration of oxygen would be antici-pated. In the neutral and positive charge states, the oxygenatom stays on-site but, in the negative charge state, one ofthe three Al uO Nlying in the basal plane breaks22rather than the bond Al uONalong c.23ONhas a donor level at 0.21 eV deeper than Si and, using Si as the marker, is placedatE v+5.85 eV or Ec−0.27 eV. An acceptor level is also evaluated 0.75 eV below that of Si and is then placed at Ec −1.06 eV. Like Si Al,ONis a DX center, in agreement with previous work.22,23The optical /H20849Franck-Condon /H20850ionization energy for ON−is found to be 2.25 eV, and close to a 2.8–2.9 eV optical absorption peak reported in AlN:O.24The low formation energy for O Alin AlN suggests that high con- centrations can be expected and larger oxygen aggregates arereadily formed. A close-by pair of O Ndefects is the most stable among defects with two oxygen atoms, and this pos-sesses a donor level at E v+4.54 eV. The depth of this level from Ecshows a possible link with an oxygen-related PL band at 2.05 eV.25As the acceptor level is close to that of Si, atEc−0.53 eV, the O NuONcomplex is not a DX center in contrast with O N. Other defects investigated also have deep levels. VN+has C3vsymmetry, but a small distortion of the four Al atoms around the vacant site lowers the symmetry of the neutraland negative charge states from C 3vtoC1h. The donor and acceptor levels lie around Ev+4.64 and Ec−1.36 eV, respec- tively, showing that V Nis a positive- Udefect26in contrast with an earlier report.27The high formation energy of 4.98 eV of the neutral defect indicates a low equilibriumconcentration, although the defect could be readily formedduring growth or implantation. IV. RARE-EARTH DEFECTS We now turn to the properties of RE defects in AlN. Dif- ferent sites for the RE atom have been investigated: the RE Al and RE Nsubstitutional sites, as well as the corresponding interstitial sites. We find RE Alis the most stable in agreementwith site-location studies.11,12With respect to solid RE ni- tride, the formation energies of the substitutional Er, Eu, andTm defects lie between 1.6 and 2.2 eV. These positive ener-gies reveal that the isolated defects are less stable than acorresponding RE-nitride precipitate. RE Aldefects have C3v symmetry with three equal RE–N bonds of lengths 2.10, 2.08, and 2.15 Å for, respectively, Er, Tm, and Eu. Thefourth RE–N bond along cis/H110110.05 Å longer. In contrast with GaN /H20849Ref. 3 /H20850and GaAs /H20849Ref. 2 /H20850, we find that substitutional Er, Eu, and Tm defects in AlN are deepdonors. Comparing the ionization energy of the defect withbulk AlN reveals that donor levels lie around E v+0.5 eV. Thus, photoionization of an isolated RE center, with radia-tion with an energy greater than about 5.6 eV, should lead toRE-related CL, EL, and PL spectra. We next studied RE-defect complexes starting with RE Al -VN. There are two inequivalent types of Al atoms bordering a nitrogen vacancy. Their replacement by RE leads to eitheraC 3vorC1hdefect. The latter is more stable by about 0.36 eV. Despite a binding energy between the isolated va-cancy and the RE impurity of more than 1.3 eV, the forma- tion energy of RE Al-VNcomplexes is very large /H20849/H110225.2 eV /H20850 and the equilibrium concentration of this defect negligible. This is a consequence of the high formation energy of thevacancy. However, it might be introduced during growth orion implantation. Deep single donor and acceptor levels liearound E v+4.4 eV and Ec−1.5 eV, respectively. Second lev- els might also exist but have not been considered here. Cap-ture of an electron excited from the valence band by theneutral RE Al-VNdefect requires about 5.0 eV. This threshold would be reduced for the excitation of a RE Al-VN–RE Alpair separated by, say, 5 Å. The reduction is 0.5 eV due to thedonor level of the RE Aland /H110111 eV due to the electrostatic energy of the charged defects. The resulting threshold iscomparable with the reported one for efficient excitation ofthe 478 nm blue line of Tm /H208494e V /H20850. 8 As oxygen impurities seem to be unavoidable in AlN, we have investigated the RE Al-complex with one oxygen atom. In the stable C1hconfiguration, RE Alis bound to O Nlying in the basal plane with an energy of 0.9 eV and similar to thebinding energy of Er Alwith O Asin GaAs /H20849Ref. 2 /H20850but greater than found in GaN.3The formation energies of these RE Al -ONcomplexes are around −0.1 eV. In the negative charge state, the oxygen atom is displaced from its site. As the de-fects possess a donor level around E v+5.7 eV and an accep- tor level about Ec−0.8 eV, they are negative- Ucenters. Elec- tron capture by the neutral defect requires around 6 eV butoptical ionization of the DX-negative charge state requires2.0 eV. In GaAs, complexes of Er with two oxygen atomsare found in EXAFS studies 28and may play a part in RE luminescence.2Similarly in AlN, RE Al-/H20849ON/H208502defects have very low formation energies around −3 eV and would im- pede precipitation of the RE. These defects have a deep do-nor level around E v+4.3 eV and an acceptor level around Ec−0.5 eV. Electron capture by the neutral defect then re- quires around 5.9 eV. Such thresholds could be reduced bythe involvement of close-by RE Aldefects as discussed above, and these defects could then be involved in the enhancementof the 465 nm blue line of Tm for excitation above 4.2 eV. 8 Several RE Al-ONuOidefects, which possess very similarBRIEF REPORTS PHYSICAL REVIEW B 72, 073205 /H208492005 /H20850 073205-3formation energies around −2 eV, exhibit acceptor levels be- tween Ev+1.8 eV and Ev+2.8 eV. These levels are much lower than other defects. Such complexes might be involvedin a broad PLE band with an onset around 2 eV detected inAlN:Er. 6 V. CONCLUSION In summary, we have performed density-functional calcu- lations to investigate Er, Eu, and Tm dopants in hexagonalAlN. We find that these impurities substitute for Al atomsand, in contrast with isolated rare-earth /H20849RE/H20850dopants in GaAs and GaN, introduce deep donor levels /H20849E v+0.5 eV /H20850 which may be involved in the luminescence mechanism, es- pecially during electro- and cathdoluminescence. Theycould, for example, account for the enhanced CL intensityobserved in Al xGa1−xN:Eu for x/H110220.15.10However, substi- tutional RE defects are less stable than RE-N precipitates.On the contrary, complexes with two substitutional oxygenatoms are more stable than RE-N precipitate, and act to pre-vent precipitation. The RE Al-ON,R E Al-/H20849ON/H208502and RE Al-VN defects possess acceptor and donor levels in the upper third of the gap. Excitations of these defects, especially whenpaired with deep donors like RE Al, could account for the threshold excitation energy around 4 eV observed8for prominent intra- ftransitions for AlN:Tm. 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PhysRevB.102.075429.pdf

PHYSICAL REVIEW B 102, 075429 (2020) Quantum dots in AA-stacked bilayer graphene H. S. Qasem,1H. M. Abdullah ,2,*M. A. Shukri,1H. Bahlouli,3and U. Schwingenschlögl2,† 1Department of Physics, Faculty of Science, Sanaa University, Sanaa 009671, Yemen 2Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia 3Department of Physics, King Fahd University of Petroleum and Minerals, 31261 Dhahran, Saudi Arabia (Received 16 March 2020; revised 28 July 2020; accepted 29 July 2020; published 13 August 2020) While electrostatic confinement in single-layer graphene and AA-stacked bilayer graphene is precluded by Klein tunneling and the gapless energy spectrum, we theoretically show that a circular domain wall that separatesdomains of single-layer graphene and AA-stacked bilayer graphene can provide bound states. Solving the Dirac-Weyl equation in the presence of a global mass potential and a local electrostatic potential, we obtain the energyspectrum of these states and the corresponding radial probability densities. Depending on the mass potentialprofile, regular bound states can exist inside the quantum dot and topological bound states at the domain wall.Controlling the electrostatic potential inside the quantum dot enables the simultaneous presence of both typesof states. We find that the number of nodes of the radial wave function of the regular bound states inside thequantum dot is equal to the radial quantum number. The energy spectra of the bound states display anticrossings,reflecting coupling of electron- and holelike states. DOI: 10.1103/PhysRevB.102.075429 I. INTRODUCTION Quantum dots (QDs) are often referred to as artificial atoms with confined electrons in quantized states. Realizing electro-static QDs in single-layer graphene (SLG) and AA-stackedbilayer graphene (AA-BLG) is hampered by Klein tunneling[1–5] and the lack of an energy gap [ 6,7]. Only in quasi- QDs [ 5,8–11] can electrons be confined to quasibound states [12,13] for a finite time, which hinders deployment of QDs in nanoelectronics. Various proposals have been advanced toopen an energy gap and overcome this bottleneck, for exam-ple, using Li +ions as a dopant [ 14,15] or introducing a mass potential through a substrate [ 16,17]. Rigorous control of the charge carriers is indispensable in electronic devices, whichrequires perfect confinement as well as external tunability. BLG exists in configurations with AA and AB stacking [18,19], with AA-BLG having a linear energy spectrum sim- ilar to SLG at low energy. It is known that domain walls(DWs) in BLG can give rise to one-dimensional topologicallyprotected current [ 20–22]. Such DWs may be established by considering potentials of kink and antikink shapes [ 23], delamination of one or both layers [ 24,25], and switching of the stacking order from AB to BA [ 26,27]. Despite the fact that AB-BLG is energetically favorable over AA-BLG,stable samples of AA-BLG have been realized experimentally[28–31], triggering theoretical investigations of the material properties [ 19,32–40]. In AA-BLG the interlayer coupling is symmetric; that is, an electrostatic potential cannot openan energy gap, making confinement a great challenge. Fewmechanisms have been proposed to achieve bound states in *hasan.abdullah@kaust.edu.sa †udo.schwingenschlogl@kaust.edu.saAA-BLG, such as combining an electrostatic potential witha strong magnetic field [ 41] and using finite-size flakes [ 42]. However, these mechanisms would limit the applicability dueto the complications that a strong magnetic field brings alongin a nanostructured system (e.g., formation of Landau levels)or the fact that small flakes are hard to control in experimentsand their edges affect the electronic transport. We study QDs in AA-BLG in the presence of a global mass potential (a method that very recently led to the prediction oftopological bound states in SLG [ 43] with subsequent experi- mental confirmation [ 44]) and a local electrostatic potential. The mass potential (which can be realized by embeddingthe QD into dielectric materials [ 45]) introduces the required energy gap that prevents electrons from leaving the QD,while the electrostatic potential provides rigorous control ofthe doping level inside the QD [ 46]. A homogeneous mass potential can support regular bound states localized inside theQD, and an inhomogeneous mass potential can additionallysupport topological bound states localized at the DW. This paper is organized as follows. In Sec. IIwe introduce the electronic model and describe the considered mass poten-tial profiles as well as the implemented boundary conditions.The numerical results and discussion are presented in Sec. III, while Sec. IVis devoted to final remarks about the main findings of our work. II. ELECTRONIC MODEL AND ENERGY SPECTRUM The unit cell of SLG is hexagonal and comprises two atoms, AandB, with an interatomic distance of a=0.142 nm and an intralayer coupling of γ0=3e V[ 47]. AA-BLG is created by stacking two sheets of SLG on top of each othersuch that atoms A 2and B2in the top layer (layer 2) are located directly above atoms A1andB1in the bottom layer 2469-9950/2020/102(7)/075429(8) 075429-1 ©2020 American Physical SocietyH. S. QASEM et al. PHYSICAL REVIEW B 102, 075429 (2020) (a) (b) FIG. 1. Low-energy spectrum of (a) SLG and (b) AA-BLG. Solid black and dashed red lines correspond to /Delta11=/Delta12=0a n d /Delta11= −/Delta12=0.5γ1, respectively. (layer 1) with a direct interlayer coupling of γ1=0.2e V [48]. At low energy the electron dynamics can be described in the continuum approximation. In the vicinity of the Diracpoints ( KandK /primepoints of the Brillouin zone) and in the presence of a mass potential the Hamiltonian of AA-BLG inthe basis ( φ A1,φB1,φB2,φA2)T, with the elements referring to the sublattices, takes the form [ 34] Hτ AA=/parenleftbigg Hτ 1γ1σx γ1σxHτ 2/parenrightbigg , (1) where Hτ i=v0I+vF[pxσx+(−1)ipyσy]+τ(−1)i/Delta1iσzde- scribes the ith layer, τ=+1(−1) is the index of the K (K/prime) valley, v0is the electrostatic potential, Iis the 2 ×2 identity matrix, vF∼106m/s is the Fermi velocity of SLG [7],px,y=−i¯h∂x,y,σx,y,zare the Pauli matrices, and /Delta1iis the mass potential in the ith layer. The energy spectrum and transport properties of SLG and AA-BLG [ 49] can be significantly altered by a mass potential [see Figs. 1(a)and1(b), where the solid black and dashed red lines represent the energy spectrum without and with the masspotential, respectively]. Both materials have linear bands, withthose of AA-BLG being copies of those of SLG shifted up anddown by 2 γ 1. By breaking the sublattice and layer symmetries the mass potential opens an energy gap [ 19,35]. The structures of the studied QDs with a circular DW are illustrated in Figs. 2(a) and 2(b). The bottom layer is infinite, and the top layer terminates at the DW such thatthe region inside the QD consists either of SLG (QD1) orof AA-BLG (QD2). These QDs can be realized in terms ofAA-BLG samples with a SLG hole and SLG samples withan AA-BLG island, respectively [ 50,51]. For each QD we consider two mass potential profiles: The mass potential inlayer 1 is either the same inside and outside the QD or hasthe same magnitude with opposite signs [see the dashed blue (a) (c) (e) (f)(d)(b) FIG. 2. (a) and (b) Schematic pictures of the studied QDs. A circular DW with radius Rseparates domains of SLG and AA-BLG. Mass potential profiles considered for layers 1 and 2 in the case of (c) and (e) QD1 and (d) and (f) QD2. Yellow and gray regions referto SLG and AA-BLG. line in Figs. 2(c)–2(f)]. The mass potential in layer 2 remains unchanged [see the dashed red line in Figs. 2(c)–2(f)]. Note that an abruptly changing mass potential is justified by the factthat in the vicinity of the Dirac points the Fermi wavelength ismuch larger than the characteristic width of the DW. To calculate the energy spectrum we need to solve the Schrodinger equation H/Phi1=E/Phi1and match the so- lutions at the DW through appropriate boundary condi-tions. The momentum operator π ±=px±ipyreads, in polar coordinates [ 24], π±=¯h ie±iθ/bracketleftbigg∂ ∂ρ±i ρ∂ ∂θ/bracketrightbigg , (2) where ρis the radial distance to the center of the QD and θis the azimuthal angle. Since the properties of the QDs depend on the interlayer coupling γ1and length l=¯hvF/γ1∼ 3.3 nm, we scale the energy and spatial coordinate in the following with respect to γ1andl, respectively. A. AA-stacked BLG The orbital angular momentum Lz=−i¯h∂θdoes not com- mute with the Hamiltonian in Eq. ( 1) and thus is not a conserved quantity. However, the operator L=Lz+¯h 2/bracketleftbigg/parenleftbigg −σz+I 0 0 σz+I/parenrightbigg/bracketrightbigg , (3) with eigenvalue mand eigenstate /Phi1τ(r)=⎛ ⎜⎜⎝φτ A1(ρ)eimθ iφτ B1(ρ)ei(m−1)θ iφτ B2(ρ)ei(m−1)θ φτ A2(ρ)eimθ⎞ ⎟⎟⎠, (4) 075429-2QUANTUM DOTS IN AA-STACKED BILAYER GRAPHENE PHYSICAL REVIEW B 102, 075429 (2020) commutes with the Hamiltonian. Note that the angular parts of connected sublattices are the same. In AA-BLG the connectedsublattices are A1↔A2 and B1↔B2. The radial dependence of the components of /Phi1 τ(r)i s described by the (dimensionless, due to our scaling) fourcoupled first-order differential equations /bracketleftbiggd dρ−m−1 ρ/bracketrightbigg φτ B1=[E−v0−τ/Omega1+τ/Delta1 0]φτ A1−φτ A2, (5a) /bracketleftbiggd dρ+m ρ/bracketrightbigg φτ A1=−[E−v0+τ/Omega1−τ/Delta1 0]φτ B1+φτ B2, (5b) /bracketleftbiggd dρ+m ρ/bracketrightbigg φτ A2=−[E−v0−τ/Omega1−τ/Delta1 0]φτ B2+φτ B1, (5c) /bracketleftbiggd dρ−m−1 ρ/bracketrightbigg φτ B2=[E−v0+τ/Omega1+τ/Delta1 0]φτ A2−φτ A1, (5d) where /Omega1=(/Delta12−/Delta11)/2 and /Delta10=(/Delta12+/Delta11)/2. For sim- plicity, we set /Delta11=−/Delta12, i.e.,/Delta10=0. Equations ( 5a)–(5d) can be written as a single fourth-order differential equationwith two sets of orthogonal solutions given by the second-order differential equation /bracketleftbiggd 2 dρ2+1 ρd dρ−/parenleftbiggm2 ρ2+(αξ ±)2/parenrightbigg/bracketrightbigg φτ,ξ ±,A1(ρ)=0,(6) withφτ,ξ ±,A1(ρ)=φτ ±,A1(√ξρ), (αξ ±)2={ −ξ[(E−v0)2−/Omega12+1]±2/radicalbig (E−v0)2−/Omega12}, (7) andξ=−1(+1) for QD1 (QD2). For QD1 (AA-BLG out- side) the solutions must be regular in the limit ρ→∞ , which is satisfied by the modified Bessel functions of the second kindK m(α+1 ±ρ). On the other hand, for QD2 (AA-BLG inside) the solutions must be defined at the origin, which is guaranteedby the Bessel functions of the first kind J m(α−1 ±ρ). In the case of QD1 the square root of Eq. ( 7) is complex; that is, the solu- tions of Eq. ( 6) are Bessel functions with a complex argument [52]. Notice that α+1 +=(α+1 −)∗; that is, the two independent solutions of Eq. ( 6) can be written as superposition of the real and imaginary parts of Bessel functions. Using Eqs. ( 5a)–(5d), we find φτ,ξ A1(ρ)=Cτ,ξ 1Re/bracketleftbig Bξ m(αξ +ρ)/bracketrightbig +Cτ,ξ 2Im/bracketleftbig Bξ m(αξ −ρ)/bracketrightbig , (8a) φτ,ξ B1(ρ)=Cτ,ξ 1Re/bracketleftbig aτ,ξ +bτ,ξ +Bξ m−1(αξ +ρ)/bracketrightbig +Cτ,ξ 2Im/bracketleftbig aτ,ξ −bτ,ξ −Bξ m−1(αξ −ρ)/bracketrightbig , (8b) φτ,ξ B2(ρ)=Cτ,ξ 1Re/bracketleftbig bτ,ξ +Bξ m−1(αξ +ρ)/bracketrightbig +Cτ,ξ 2Im/bracketleftbig bτ,ξ −Bξ m−1(αξ −ρ)/bracketrightbig , (8c) φτ,ξ A2(ρ)=Cτ,ξ 1Re/bracketleftbig aτ,ξ +Bξ m(αξ +ρ)/bracketrightbig +Cτ,ξ 2Im/bracketleftbig aτ,ξ −Bξ m(αξ −ρ)/bracketrightbig , (8d)with Bξ m(x)=/braceleftbigg Km(x)f o rξ=+1, Jm(x)f o rξ=−1,(9) aτ,ξ ±=±ξ/radicalbig (E−v0)2−/Omega12/[(E−v0)+τ/Omega1],bτ,ξ ±=αξ ±[ξ±/radicalbig (E−v0)2−/Omega12]/[(E−v0)2−/Omega12−1], and Cτ,ξ 1,Cτ,ξ 2∈C. B. SLG SLG (layer 1) appears inside QD1 (confinement region) and outside QD2. The Hamiltonian of SLG can be written as Hτ 1=/parenleftbigg v0−τ/Delta1 1 vFπ+ vFπ− v0+τ/Delta1 1/parenrightbigg . (10) The wave function is given by the first two components of the vector in Eq. ( 4). Inserting Eq. ( 10) and this wave function into the Schrodinger equation yields /bracketleftbiggd dρ−m−1 ρ/bracketrightbigg φτ,ξ B1=[E−v0+τ/Delta1 1]φτ,ξ A1, (11a) /bracketleftbiggd dρ+m ρ/bracketrightbigg φτ,ξ A1=−[E−v0−τ/Delta1 1]φτ,ξ B1.(11b) These coupled first-order differential equations can be reduced to the second-order differential equation /bracketleftbiggd2 dρ2+1 ρd dρ−/parenleftbiggm2 ρ2+(ηξ)2/parenrightbigg/bracketrightbigg φτ,ξ A1(ρ)=0, (12) with (ηξ)2=−ξ[(E−v0)2−/Delta12 1]. (13) The solutions are φτ,ξ A1(ρ)=Dτ,ξ/braceleftbig Re/bracketleftbig Bξ m(ηξρ)/bracketrightbig +Im/bracketleftbig Bξ m(ηξρ)/bracketrightbig/bracerightbig , (14a) φτ,ξ B1(ρ)=−Dτ,ξξηξ/braceleftbig Re/bracketleftbig Bξ m−1(ηξρ)/bracketrightbig +Im/bracketleftbig Bξ m−1(ηξρ)/bracketrightbig/bracerightbig E−v0−τ/Delta1 1, (14b) with Dτ,ξ∈C. We are now in a position to calculate the en- ergy spectrum by imposing appropriate boundary conditionsin the next section. C. Boundary conditions and energy levels There are two possible terminations of AA-BLG, called zigzag (ZZ) and armchair (AC) edges. A ZZ edge can be established in two ways, namely, ZZ1 and ZZ2, where φτ,ξ B2 andφτ,ξ A2, respectively, are set to be zero at ρ=R. For layer 1 the boundary conditions are established by equating thespinor components inside and outside the QD at ρ=R. The single-valley approximation is no longer valid for theAC edge; that is, the boundary conditions for layer 2 areintervalley mixed [ 53], φ +1,ξ A2−φ−1,ξ A2=0,φ+1,ξ B2+φ−1,ξ B2=0. (15) As the AC edge therefore requires more involved calculation than the ZZ edge, we focus here on the ZZ edge. Despitethe symmetric interlayer coupling in AA-BLG, the ZZ1 andZZ2 edges are not equivalent due to the broken sublattice 075429-3H. S. QASEM et al. PHYSICAL REVIEW B 102, 075429 (2020) symmetry as a result of the mass potential. However, their energy levels are connected through specific symmetries.Hence, we consider here only the ZZ1 edge and then explainthe connection to the ZZ2 edge. For the ZZ1 edge we obtain a set of three equations for three unknowns that can be writtenin matrix form as Mτ,ξ⎛ ⎝Dτ,ξ Cτ,ξ 1 Cτ,ξ 2⎞ ⎠=0,Mτ,ξ=⎛ ⎜⎝Bξ m(ηξR) −Re/bracketleftbig Bξ m(αξ +R)/bracketrightbig −Im/bracketleftbig Bξ m(αξ −R)/bracketrightbig Bξ m−1(ηξR)−Re/bracketleftbig aτ,ξ +bτ,ξ +Bξ m−1(αξ +R)/bracketrightbig −Im/bracketleftbig aτ,ξ −bτ,ξ −Bξ m−1(αξ −R)/bracketrightbig 0 −Re/bracketleftbig bτ,ξ +Bξ m−1(αξ +R)/bracketrightbig −Im/bracketleftbig bτ,ξ −Bξ m−1(αξ −R)/bracketrightbig⎞ ⎟⎠. (16) The energy levels Eτ,ξ n,m(R)f o rt h eD Wa t ρ=Rare derived through the roots of the determinant of Mτ,ξ. Here, nis the radial quantum number, corresponding to |n|modes that emerge with increasing R. Subsequently, we obtain the corresponding wave function /Phi1τ,ξ n,m(r) by solving, at given Eτ,ξ n,m(R) and R, for the constants Dτ,ξ,Cτ,ξ 1, and Cτ,ξ 2. Then the radial probability density (RPD) is given by ρ|/Phi1τ,ξ n,m(r)|2, obeying the normalization /integraldisplay∞ 0/integraldisplay2π 0ρ/vextendsingle/vextendsingle/Phi1τ,ξ n,m(r)/vextendsingle/vextendsingle2dθdρ=1. (17) III. RESULTS AND DISCUSSION The parameters controlling the electronic states are the mass potentials of the two layers, /Delta11and/Delta12, and the elec- trostatic potential v0. As we assume /Delta12=−/Delta11, the induced energy gap is 2 /Delta11. To confine the bound states, we set the electrostatic potential to v< 0=/Delta11andv> 0=0, respectively, using the symbols <and>to refer to the regions inside and outside the QD. In the following we discuss results for theZZ1 edge and afterwards address the relation to the ZZ2 edge. A. QD1 In Fig. 3we show the energy spectrum of the bound states in QD1 as a function of the DW radius Rfor angular momentum m=1 and the mass potential profiles of Figs. 2(c) and 2(e). Brown and black curves refer to the Kand K/prime valleys, respectively. To thoroughly understand these results we show the corresponding band structure in Fig. 4.W e observe the same energy gap of size 2 /Delta11inside and outside the QD, as expected. The band edges inside the QD are locatedat an energy of v < 0±/Delta11, and those outside the QD are located at±/Delta11. While regular bound states generally can exist only within the energy gap outside the QD, i.e., the yellow and blue regions in Fig. 4, we find them only in the blue region, as they require propagating states to be available inside theQD. Note that the valley symmetry is broken, while the boundstates in both valleys are connected through E +1,+1 n,m(/Delta11)= E−1,+1 n,m(−/Delta11) according to Eq. ( 1). To ensure that the states are confined inside the QD (in layer 1), we plot on the left side of Fig. 5the RPDs of the two layers for the states marked by colored circles in Fig. 3(a). Interestingly, the number of peaks inside the QD equals ex-actly the radial quantum number |n τ|. This is a manifestation of the nodes associated with the two radial components of theDirac spinor of SLG. Generally, node features of the radialcomponent of the wave function in the nonrelativistic case aredistinct from those in the relativistic case. In the case of the Dirac equation the number of nodes can be the same for stateswith the same Rbut different energies, while this does not apply to nonrelativistic states. Under specific conditions oneradial component of the Dirac spinor can behave exactly as thenonrelativistic wave function, where |n τ|counts the number of nodes [ 54]. In our case, we find that the number of nodes νB1 inside the QD associated with φτ,ξ B1(ρ)i s|nτ|, while φτ,ξ A1(ρ) possesses νA1=νB1−1 nodes that almost coincide with the extrema of φτ,ξ B1(ρ). This behavior of the radial components of the Dirac spinors results in the peak pattern of the RPD shownon the left side of Fig. 5. Note that we show the RPD only for theK /primevalley, as for the Kvalley it is almost the same. When the sign of the mass potential in layer 1 is opposite inside and outside the QD [see Fig. 3(b)], we find, in addition to the regular bound states, two topological bound stateswith∼1/Rdependence which can exist in the yellow and blue regions of Fig. 4and to which we assign n τ=0. For small Rthere are only topological bound states with Eτ,+1 0,m(R), (a) (b) FIG. 3. Energy spectrum of the bound states in QD1 [see Fig. 2(a)] as a function of the DW radius Rform=1,/Delta11= 0.5γ1,v< 0=0.5γ1,v> 0=0, and the mass potential profiles of (a) Fig. 2(c) and (b) Fig. 2(e). Dashed orange lines mark the energy gap±0.5γ1outside the QD, and the dashed green line marks the top of the valence band v< 0−0.5γ1inside the QD. 075429-4QUANTUM DOTS IN AA-STACKED BILAYER GRAPHENE PHYSICAL REVIEW B 102, 075429 (2020) (a) (b) FIG. 4. Band structure (a) inside and (b) outside QD1 with the parameters of Fig. 3. Note that the band structure is the same for the mass potential profiles of Figs. 2(c)and2(e). pertaining to different valleys and conducting at the DW in opposite directions. The nature of the regular bound stateswith E τ,+1 n>0,m(R) in the energy range [ −0.5γ1,0] is exactly the same as in the case of Fig. 3(a). In particular, the number of peaks in the RPD associated with these states is equalto|n τ|; see states 1, 2, and 3 in Fig. 5. On the other hand, FIG. 5. RPD for the labeled states in Fig. 3(QD1). The dashed green line indicates the radius of the QD. 20E = -0.263 γ1 E = -0.026 γ1 E = 0.278 γ1 E = -0.378 γ1 E = -0.147 γ1 E = 0.220 γ1 20 20MaxLocal density of states 0.0y/l x/l FIG. 6. Spatial distributions of the local density of states for the bound states with energy Eτ,+1 n,1(10l) in Fig. 3(b). The black dashed circles represent the QD with radius R=10l. The bottom and top rows refer to the KandK/primevalleys, respectively. state 4 is localized at the DW in layer 1 and decays quickly away from the DW. We notice that the RPDs in layer 2 areexactly zero at the DW but finite in the classically forbiddenregion just outside the QD (which is enhanced for states nearthe continuum spectrum). Imaging of the topological boundstates at a BLG DW has been achieved experimentally [ 20], making these states accessible to scanning probe techniquessuch as scanning tunneling microscopy, the signal of whichis proportional to the local density of states. In Fig. 6we therefore show typical spatial distributions of the local density of states,/summationtext m,nδ(E−Em,n)|/Phi1τ,ξ m,n(r)|2, for the states along the vertical red dashed line in Fig. 3(b). We find that the regular bound states are localized inside the QD, while the topologicalbound states are localized at the DW. B. QD2 In Fig. 7we show the energy spectrum of the bound states in QD2 as a function of the DW radius Rfor angular momen- tum m=1. For the mass potential profile of Fig. 2(d) only regular bound states appear [see Fig. 7(a)], while topological bound states exist as well for the mass potential profile ofFig. 2(f) [see Fig. 7(b)]. For the same parameters as in the case of QD1 the energy levels of QD2 are very different. Par-ticularly, they exhibit anticrossings, which manifest couplingof electron- and holelike states. To understand the essence ofthe anticrossings, let us consider an AA-BLG nanodisk withthe ZZ1 edge (both layers) and the same parameters as used inFig.7(a). The obtained energy spectrum is shown in Fig. 7(c) as blue dashed curves with the energy spectrum of the Kvalley from Fig. 7(a) superimposed as brown curves. We observe for the AA-BLG nanodisk no anticrossings of the boundstates but distinct crossing points close to the anticrossingsof QD2, implying that both the anticrossings and crossingpoints are due to coupling of electron- and holelike states.When we expand the bottom layer of the AA-BLG nanodiskto infinity, we also break the layer symmetry (remove thedegeneracy), and anticossings appear (due to the presence ofcoupled electron- and holelike states in AA-BLG). A similaranticrossing behavior is expected for the topological bound 075429-5H. S. QASEM et al. PHYSICAL REVIEW B 102, 075429 (2020) (a) (c)(b) FIG. 7. Energy spectrum of the topological and bound states in QD2 [see Fig. 2(b)] as a function of the DW radius Rform=1, /Delta11=0.5γ1,v< 0=0.5γ1,v> 0=0, and the mass potential profile of (a) Fig. 2(d) and (b) Fig. 2(f). Dashed orange lines mark the energy gap±0.5γ1outside the QD, and the dashed green line marks the top of the valence band v< 0−0.5γ1inside the QD, as indicated in Fig. 4 for switched bands inside and outside the QD and the same v0.I n (c) the solid brown lines are the energy levels of the Kvalley from (b), and the blue dashed lines are the energy levels of an AA-BLG nanodisk with ZZ1 edge for m=1. states in Fig. 7(b). We notice that the number of bound states is larger for QD2 than for QD1 at the same R, which agrees with the larger density of states of AA-BLG compared to SLGat a specific energy. Figure 8shows the RPDs of the states labeled in Figs. 7(a) and7(b). For layer 1 the RPDs are continuous at the DW, while for layer 2 they terminate. The left column of Fig. 8 shows the same pattern as in the case of QD1 with the statescloser to the continuum spectrum localized closer to the DWin layer 1. Due to the ZZ1 edge there are dangling bonds atthe DW in layer 2, which results in a finite RPD attributed toFIG. 8. RPD for the labeled states in Fig. 7(QD2). The dashed green line indicates the radius of the QD. |φτ,−1 A2(ρ)|2. At the anticrossings the RPDs of the bound states show maxima at the DW in layer 2, as can be deduced fromstates 1 and 3. The topological bound state 4 is localized at theDW in layer 1, while the RPDs are strictly zero at the DW inlayer 2. This is a direct consequence of the sign change of themass potential at the DW. We have considered so far the ZZ1 edge and results for m=1. Corresponding results for the ZZ2 edge and different angular momenta are directly related. The energy spectrumof QD1 for a larger angular momentum is similar to that form=1 but for a larger size of the QD. Since in the case of QD1 the bound states are confined in layer 1, we expect thatthe edge type in layer 2 has a negligible effect when Ris much larger than the localization length ¯ hv F//Delta11, i.e., the bound states become similar to those of a SLG QD [ 55]. Calculating the spectrum for the ZZ2 edge accordingly yields /parenleftbig Eτ,+1 n,m/parenrightbig ZZ1=/parenleftbig Eτ,+1 n,−m+1/parenrightbig ZZ2. (18) On the other hand, for QD2 it is difficult to anticipate the connection between the results for the ZZ1 and ZZ2 edges dueto the distribution of the bound states over layers 1 and 2 aswell as the broken sublattice and layer symmetries. By explicitcalculation we confirm the symmetry given by Eq. ( 18). IV . CONCLUSION In conclusion, we have proposed AA-BLG QDs as a platform to realize electron confinement in a graphene-basedmaterial. In the presence of a global mass potential and a localelectrostatic potential, we have assessed the properties of thebound states using the four-band continuum model. We havedemonstrated that AA-BLG QDs can support both regular and 075429-6QUANTUM DOTS IN AA-STACKED BILAYER GRAPHENE PHYSICAL REVIEW B 102, 075429 (2020) topological bound states and that their radial quantum number (number of nodes) coincides with the number of peaks in theRPD. For QD1 these states are localized in layer 1 and thusare robust against the edge type (ZZ1 or ZZ2) of layer 2. Incontrast, for QD2 they have a finite RPD in both layers as aresult of the interlayer coupling in AA-BLG. This makes QD2more sensitive to the edge type. Due to coupling of electron-and holelike states and the broken layer symmetry, the energyspectrum of the bound states of QD2 displays anticrossingsthat can be mapped onto the energy spectrum of an AA-BLGnanodisk. Results for the ZZ2 edge follow from those of theZZ1 edge directly by the given symmetries. Formation ofregular bound states, topological bound states, or both can becontrolled by the electrostatic potential. For the AC edge the calculations are more involved as a result of valley mixing. However, due to the symmetric inter-layer coupling in AA-BLG we expect that the properties of the bound states are not significantly altered. Experimentally,the DW of a QD most likely will be a mixture of ZZ andAC edges. The energy spectrum of the bound states then willbe intermediate between those of the ZZ and AC edges. Weexpect that our results can be elegantly verified by scanningtunneling microscopy, as precise control of the stacking order[56] and creation of DWs [ 50,57] is possible. ACKNOWLEDGMENTS H.B. acknowledges the support of the Saudi Center for Theoretical Physics (SCTP) and of KFUPM under physicsresearch group Project No. RG181001. 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PhysRevB.92.134408.pdf

PHYSICAL REVIEW B 92, 134408 (2015) Localized itinerant electrons and unique magnetic properties of SrRu 2O6 S. Streltsov,1,2I. I. Mazin,3and K. Foyevtsova4 1M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620137, Ekaterinburg, Russia 2Theoretical Physics and Applied Mathematics Department, Ural Federal University, Mira St. 19, 620002 Ekaterinburg, Russia 3Code 6393, Naval Research Laboratory, Washington, DC 20375, USA 4Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4 (Received 11 August 2015; published 12 October 2015) SrRu 2O6has unique magnetic properties. It is characterized by a very high N ´eel temperature, despite its quasi-two-dimensional structure, and has a magnetic moment more than twice reduced compared to the formalionic count. First-principles calculations show that only an ideal N ´eel ordering in the Ru plane is possible, with no other metastable magnetic solutions, and, highly unusually, yield dielectric gaps for both antiferromagneticand nonmagnetic states. We demonstrate that this strange behavior is the result of the formation of very specificelectronic objects, recently suggested for a geometrically similar Na 2IrO 3compound, whereby each electron is well localized on a particular Ru 6hexagon, and completely delocalized over the corresponding six Ru sites, thus making the compound both strongly localized and highly itinerant. DOI: 10.1103/PhysRevB.92.134408 PACS number(s): 75 .50.Ee,71.15.Mb,75.30.Et,71.27.+a The recently discovered [ 1]S r R u 2O6has attracted consid- erable attention because, despite being a very two-dimensional(2D) material, it shows an exceptionally high N ´eel temperature of∼560 K [ 2,3]. As we will argue in this paper, this is by far not the only, and maybe not even the most intriguing, propertyof this material. Ru 5+has a half-filled t2gelectronic shell, and exhibits insulating behavior. Naturally, it was interpreted as aSlater insulator (maybe Mott-enhanced), with Ru in the highspin state S=3/2.However, the experimentally measured ordered magnetic moment is only M=1.3–1.4μ B[2,3], 2.3 times smaller than expected for S=3/2(M=3μB). This was ascribed to hybridization with oxygen [ 2–4], but it should be noted that such strong suppression of magnetic moment in agood insulator is unheard of. Even in the metallic SrRuO 3the hybridization suppresses the total magnetic moment of Ru4+ only from 2 to 1.7 μB,and in Sr 2YRuO 6Ru5+has essentially exactly 3 μB,with basically the same Ru-O distances as in SrRu 2O6[5]. Hiley et al. [2] mention the case of Li 3RuO 4[6], where a suppression down to M=2.0μBwas reported for the same oxidation state, which is, however, still twice smallera reduction compared to SrRu 2O6,and the material might actually be a metal (no transport data have been published). Electronic structure calculations [ 3,4] so far have not resolved the mystery, but have only added to the confusion. Itwas found that only the ideal N ´eel state can be stabilized in the calculations, even though ions with S=3/2 are usually very stable, and while they disorder with temperature, neverlose their magnetic moment completely. At the same time themoment found in the calculations matches the experimentallymeasured one within 8%, suggesting that the role of Coulombcorrelations beyond the standard density functional theory(DFT) is negligible [ 7]. The instability of the ferromagnetic (FM) state was traced down to the presence of a dielectric gapin nonmagnetic calculations [ 4], but that essentially translates one mystery into another: why does a highly symmetricRu sublattice, with no dimerization or clusterization, with ahalf-filled t 2gband, show a sizable nonmagnetic gap? Singh mentions [ 4] that the gap is allowed by symmetry, since the unit cell includes two Ru atoms that can, in principle, forma bonding and an antibonding bands, but does not elaborate about how a structure with each Ru having three equivalentbonds manages to develop a bonding-antibonding splitting. Similarly, it was pointed out that, even though SrRu 2O6 is extremely 2D magnetically, there is still some residualinterlayer coupling, J ⊥M2≈1.5 meV , as well as a single-ion magnetic anisotropy, estimated to be ≈1.4m e V /Ru [4]. It was suggested that the anisotropy [ 4] or interlayer coupling [ 3]a r e responsible for the large TN,implying that the (unknown) mean field transition temperature is extremely high. Tianet al. [3] attempted to describe this system by a three nearest- neighbor Heisenberg model with parameters derived within theperturbation theory in the limit of a Hubbard Umuch larger than the hopping, U/greatermucht. However, the fact that ferromagnetic arrangement is completely unstable (in fact, as we show below,no parallel nearest-neighbor moments are stable), indicatesthat the system is strongly non-Heisenberg, casting very strongdoubt on the relevance of such models. Additionally, thefact that the system is very weakly correlated makes sucha perturbation theory unphysical. Similarly, Hiley et al. [2] used a hybrid functional that overestimates the equilibriummagnetic moment and thus the exchange parameters [ 8], as well as yields a very large band gap of 2.15 eV , totallyinconsistent with the observed weak temperature dependenceof the resistivity. An explanation of all these oddities can be consistently found in the so-called molecular orbitals (MO) picture, whichwas first brought up in connection to Na 2IrO 3[9] and later found also in RuCl 3[10]. Basically, this picture is based on the idea that for ideal 90◦Ru-O-Ru bond angles (the actual angles are 101◦) the O-assisted Ru-Ru hopping is only allowed for one particular pair of the t2gorbitals for each hexagonal bond, denoted t/prime 1in Ref. [ 9]. If all other hoppings are neglected, it leads to a curious situation where every electronic state isfully delocalized over a particular hexagon, but never leavesthis hexagon. One can say that the electrons are fully localized(form nondispersive levels) and fully delocalized (each stateis an equal weight combination of six orbitals belonging to sixdifferent sites). If a direct overlap of the t 2gorbitals (which 1098-0121/2015/92(13)/134408(5) 134408-1 ©2015 American Physical SocietyS. STRELTSOV , I. I. MAZIN, AND K. FOYEVTSOV A PHYSICAL REVIEW B 92, 134408 (2015) FIG. 1. (Color online) Density of states (DOS) projected on molecular orbitals of different symmetries in nonmagnetic GGA calculations (WIEN2k results). The Fermi energy is set to zero. always exists in the common edge geometry) is included, as well as deviations of the angle from 90◦, two more hoppings emerge: one between the same orbitals on the neighboringsitest 1,and the other an O-assisted second neighbor hopping between unlike orbitals t/prime 2.As long as t/prime 1is dominant, the MO model still applies, and can be readily solved. Thesolution entails six bands, A 1g,E2u,E1g, and B1u(theE bands being double degenerate in each spin channel), whosedispersion is controlled by t 1,and whose centers are located at 2(t/prime 1+t/prime 2),(t/prime 1−t/prime 2),−(t/prime 1+t/prime 2), and −2(t/prime 1−t/prime 2),respectively. In Na 2IrO 3t/prime 1≈−3t/prime 2,so that the A1gandE2upractically merge. This accidental degeneracy also leads to much strongerspin-orbit effects than would have been possible had the MObands remained well separated, and to considerable destructionof the MO picture in the relativistic case. On the other hand, thehopping parameters for SrRu 2O6,as calculated in Ref. [ 11], a r es i m i l a rt ot h o s ei nN a 2IrO 3,in the sense that again t/prime 1=300 meV is by far the largest hopping, and the only other sizable hoppings are t1=160 meV and −t/prime 2≈100–110 meV . Note that here |t/prime 2|is again about 1 /3o ft/prime 1. Thus, the A1g andE2ubands merge, while E1gandB1uremain separated, as one can see in Fig. 1. Projecting the density of states onto MOs, we observe that the predicted characters are very wellreproduced. The distance between the centers of the E 2uand E1gbands is about 0.8 eV , and their width is about 0.6–0.7 eV , thus providing for a small gap of ≈50 meV . It is instructive to compare SrRu 2O6with Na 2IrO 3and with Li 2RuO 3.All these compounds share the same crystallo- graphic motif, but feature a different number of delectrons: 5, 4, or 3. In the iridate, a single hole in the upper A1gsinglet is prone to both strong correlations and, due to near degeneracybetween A 1gandE2u,to spin-orbit interaction. As a result, as one increases the spin-orbit coupling, the A1gsinglet is gradually transformed into the jeff=1/2 singlet [ 9]. Either way, a half-filled singlet triggers Mott physics even if theHubbard Uis small. This transformation controls most of the interesting physics in this compound. Li 2RuO 3has two dholes, providing it with an opportunity to form strongly bound covalent dimers. This is exactly what happens, and theMO on the hexagons transforms to an MO on the Ru dimersresulting in the spin singlet ground state [ 12]. Neither Mott nor spin-orbit physics is relevant on the background of thestrong covalent bonding in dimers. Finally, SrRu 2O6has the six MO bands half-filled, and the gap is formed between thelower and the upper MO triads. Similar to Li 2RuO 3,both Mott and spin-orbit effects are of minor importance, and thegap structure inherent to the MO picture gives rise to uniquemagnetic properties. Let us now turn to the energetics of the material. First, we have confirmed, using the WIEN2k package [ 13,14], the numbers published by Singh [ 4] regarding the interplanar coupling, single-site anisotropy, and Ru magnetic moment.We also confirmed that the ferromagnetic structure cannotbe stabilized. Moreover, the so-called stripy and zigzagmagnetic patterns [ 9], where one or two out of three bonds are ferromagnetic, and the net moment is zero, cannot bestabilized. This indicates that besides the obvious influenceof the nonmagnetic gap there are other factors stronglydisfavoring ferromagnetic bonds. In fact, given that the gapvalue is ten times smaller than Ru Stoner factor [ 5], and the calculated magnetic moment in the N ´eel state is ∼1.3μ B,i ti s surprising that the ferromagnetic bonds do not stabilize with afinite moment. In order to gain more insight into the problem, we turned to the V ASP code [ 14,15], which is faster and has the capability to restrict magnetic moments to a certain direction, or toboth a direction and a magnitude (we confirmed that theenergies of collinear magnetic states agree with those foundin WIEN2k). First, we computed the total energy for a cantedantiferromagnet (AFM), restricting the angle with the zaxis to be±φfor the two Ru’s in the cell. The results are shown in Fig. 2. Note that for the largest canting angle we were able to converge, 35 ◦, the energy of the magnetic state is already higher than that of the nonmagnetic one. Also note how softthe magnetic moments are: despite the sizable equilibriummoment, the energy cost of total suppression of magnetismis less than 80 meV , only 50% larger than the transitiontemperature. This is, again, an indication of the great role ofitinerancy, and specifically, delocalization over Ru 6hexagons. Interestingly, suppression of magnetism with canting can- not be described by a naive combination of a local Hamiltonianfor itinerant magnets [ 16],E=/summationtext i/greaterorequalslant0aiM2i, where Mis the magnetization, with a Heisenberg term. While the totalenergies at a fixed canting angle φ/lessorsimilar35 ◦can be very well described by this Hamiltonian with just three terms, -90-80-70-60-50-40-30-20-10 0 10 0 5 10 15 20 25 30 35 1.22 1.24 1.26 1.28 1.3 1.32 1.34 1.36magnetic energy (meV/Ru) magnetic moment ( /Ru) µB canting angle (deg) FIG. 2. Magnetic energy (squares) and magnetic moments (cir- cles) as a function of the canting angle of spins, starting from the N ´eel antiferromagnetic structure. Results are from V ASP calculations. 134408-2LOCALIZED ITINERANT ELECTRONS AND UNIQUE . . . PHYSICAL REVIEW B 92, 134408 (2015) E(M)−E(0)=a1M2+a2M4+a3M6,not only does the first coefficient show appreciable angular dependence (as in the Heisenberg model), but also the second, and, to a lesserdegree, the third. Instead, a good fit could be obtained with thefollowing formula: E=−81.3M 2+16.9M4+2.0M6+359.2M2sin2(φ) −165.8M4sin2(φ)+27.6M6sin2(φ), (1) in meV /Ru. Note that the angle between the moments is θ= π−2φ,and that there are 1.5 times more bonds than sites. Thus, the proposed Hamiltonian looks as follows: H=/summationdisplay sites{98.3M2−66.0M4+15.8M6} +/summationdisplay n.n. bonds{179.6(M·M/prime)−82.9|M||M/prime|(M·M/prime) +13.8|M|2|M/prime|2(M·M/prime)}. (2) The Heisenberg term is extremely strong [ JM2= ∂H/∂ cos (θ)≈1600 K], and, without it, local magnetic moments fail to form. To this Hamiltonian one needs to add a small interlayer term/summationtextJ⊥Mi·Mi/prime,where iandi/primebelong to the neighboring planes, and the magnetic anisotropy/summationtextDM2 z,where J⊥≈0.9m e V , andD≈0.8m e V . In principle, at this point one would need to perform a Monte Carlo simulation using this Hamiltonian and determine thetransition temperature. However, it is notoriously difficult todistinguish a Kosterlitz-Thouless phase in a quasi-2D systemfrom the true long range order, so that one should be very skep-tical of any Monte Carlo simulation that claims to establish aN´eel temperature T Nwithout first showing that in the isotropic 2D limit TNtruly vanishes. The softness of the moment, expressed via Eq. ( 2), additionally complicates the simulation. We leave this daunting task to more experienced MonteCarlo simulators, but mention that the numbers that we havededuced are in the right ballpark. For instance, Costa and Piresshowed [ 17] that for the square lattice T N/TMF≈0.8(D/J )0.2. For three neighbors, the mean field transition temperatureT MF≈JM2≈1600 K, which together with DM2≈1.4m e V results in TN∼500 K. On the other hand, for the cubic quasi-2D model with J⊥/negationslash=0,D=0,Yasuda et al. [18] found that TN≈4.27JM2/[3.12+log(J/J⊥)],which for our parameters translates into 900 K. Thus, we conclude that (a) theMermin-Wagner theorem is mainly lifted via the interplanarcoupling [ 3], and not via the single site anisotropy [ 4], and (b) the softness of the magnetic moment, i.e., longitudinalfluctuations, plays an important role, suppressing T Nby up to a factor of 2. Let us now discuss how and why MOs support a N ´eel antiferromagnetism in SrRu 2O6. In the nonmagnetic state, the three lower MO bands, B1uandE1g,are fully occupied. Imposing uniform spin polarization does not change the occu-pancy of these states, unless the induced exchange splittingis larger than the gap, and this is why the ferromagneticorder is unstable. On the contrary, imposing the staggeredmagnetic field of ±/Delta1does not break the MO band structure, but rather increases the gap between E 1gandE2u(in the lowest 0 5 10 15 20 25 30 35 40 45 -1.5 -1 -0.5 0 0.5 1N(E) (states/f.u.) Energy (eV) M=0 M=0.7 M=1.3 M=1.7 FIG. 3. (Color online) Total density of states (DOS) calculated for several values of Ru moments Min the fixed-spin-moment procedure for the N ´eel AFM. The black lines illustrate the fact that the band gap is approximately quadratic in M. Results of the V ASP calculations. order in /Delta1,by/Delta12/t/prime 1). In the same order we can calculate the change of the occupancies and find that the spin-up sitesacquire magnetization of 5 /Delta1/2t /prime 1μB, and the spin-down sites −5/Delta1/2t/prime 1μB.The signs are consistent with the assumed signs of /Delta1,which tells us that with sufficiently large Hund’s rule coupling the system will become unstable against sucha staggered magnetization (but will resist any ferromagneticcomponent); of course, quantitative analysis is impossible onthis level of simplification. Obviously, the equilibrium momentcan be anything between 0 and 3 μ B.It is not “suppressed” from the putative S=3/2 state, but is set by the interplay between the Hund’s rule coupling on Ru and the details of the density ofstates of MOs. A corollary from the above arguments is that thedielectric gap depends quadratically on the Ru moment; Fig. 3 illustrates that this is indeed the case, to a reasonable accuracy. Let us emphasize that the molecular orbitals are not just another way to describe the electronic structure ofSrRu 2O6, but have profound physical meaning. It is instructive to compare it with another recently investigated high- TN material, SrTcO 3, where the transition metal also has a 4 d3 configuration and S=3/2. It was argued [ 19] that TNis so high because SrTcO 3is in an intermediate regime between itinerancy and localization, which is optimal for magneticinteractions. Indeed, LDA +DMFT calculations, well suited to this regime, have been performed by Mravlje et al. [19], who found T N≈2200 K. The experimental number is about 1100 K. To compare this result with SrRu 2O6,we have also performed LDA +DMFT calculations with the AMULET code [ 20], using an effective Hamiltonian constructed for Rut2gorbitals and interaction parameters U=2.7 and J= 0.3 eV as calculated in Ref. [ 3] (parameters for Tcare very similar). The corresponding temperature dependence of themagnetic moment is shown in Fig. 4. Not surprisingly, we found about the same N ´eel temperature (2000 K) as Mravlje et al. [19], and an even larger magnetic moment ( M≈2.7 vs 2.5 μ B). The difference, however, is that experimentally in SrRu 2O6bothTNandMare about twice smaller than in 134408-3S. STRELTSOV , I. I. MAZIN, AND K. FOYEVTSOV A PHYSICAL REVIEW B 92, 134408 (2015) 0 500 1000 1500 2000 2500 3000 Temperature (K)00.511.522.53Magnetic moment per Ru ( µB) FIG. 4. Magnetic moment as calculated in the LDA +DMFT approach. The continues-time quantum Monte Carlo (CT-QMC)solver [ 22] was used in these calculations. SrTcO 3.Mravlje et al. ascribed their overestimation of TN to nonlocal fluctuations, missing in the DMFT, but observed no reduction in the ordered moment at all, while in ourcase the reduction in bothT NandM2is of the same order, about a factor of 4. This clearly indicates that there is a fundamental difference between the two compounds, going much beyond just the difference in dimensionality, which isrelated to the presence of MOs in one and their absence in theother compound. A proper account of the molecular orbitalswithin DMFT can only be done in the cluster extension of thismethod [ 21], which could shed more light on this compound. Another interesting question that arises in connection with this material is what would happen if it were doped with, forinstance, a rare earth element. To address this scenario, wesimulated doping by adding electrons to the system (with acompensating constant background). The energy differencebetween the FM and N ´eel AFM states decreases upon electron doping, as seen from Fig. 5. The FM configuration immediately becomes metastable, whereby all doped electrons go intoone spin subband, rendering the material is half-metallic.The ground state remains antiferromagnetic, but its energyadvantage is gradually decreasing. Thus, one expects that thecritical angle φ(which was ∼35 ◦in undoped case) will grow with doping, and the Hamiltonian ( 2) will be correspondingly modified; this may result in a rapid change of magneticproperties with doping, which deserves further theoretical andexperimental investigation. To summarize, we have found that:(i) The electronic structure of SrRu 2O6is dominated by molecular orbitals. Each electron is, to a good approximation,localized on a particular Ru 6hexagon, and completely delo- calized over the corresponding six Ru sites. (ii) This structure sports an excitation gap that prevents for- mation of ferromagnetic bonds, but is consistent with nearest-neighbor antiferromagnetism. The corresponding magneticinteractions cannot be mapped onto a localized spin model,be it Heisenberg or biquadratic Hamiltonian with arbitrarylong range. Neither can it be described as purely itinerantmagnetism, but features interesting elements of both. This0.2 0.4 0.6 0.8 1 1.2 1.4 00.20.40.60.811.21.41.6Magnetic moment ( µB)Ru moment, FM Ru moment, AFM Total moment, FM 0.2 0.4 0.6 0.8 1 1.2 1.4 Electron doping (x)050100150200EFM - EAFM (meV/f.u) FIG. 5. (Color online) Electron doping dependence of magnetic moments (per Ru and total) and total energy difference between FM and N ´eel AFM states on the electron doping. duality reflects the dual character of the electronic structure, where electrons are simultaneously completely delocalizedand strongly localized on the Ru hexagons. A corollary isthat any deviation from the collinear N ´eel order is severely punished by kinetic energy, which, in turn, provides for theanomalously large transition temperature. (iii) The gaps in the nonmagnetic and antiferromagnetic states have the same nature, and one is continuously trans-formed into the other as the magnetization increases. Onthe contrary, the ionic picture assigning the moment of3μ Bto each Ru and associating the gap in magnetic states with spin-up/spin-down splitting is qualitatively incorrect.The observed and calculated magnetic moment of 1.3 μ Bis a manifestation of the molecular orbital nature of electronicstates, and should not be viewed as a spin S=3/2 reduced by hybridization. (iv) The magnetic properties of doped SrRu 2O6(e.g., by Na or La) are expected to be very different from the stoichiometriccase. One may anticipate interesting and very different physicsemerging, which can be a subject of forthcoming research. S.S. and I.M. are grateful to R. Valenti and University of Frankfurt (where this work was started) for the hospitalityand to A. Ruban, S. Khmelevskii, A. Poteryaev, D. Khomskii,and K. Belashchenko for useful discussions. This work wassupported by Civil Research and Development Foundation 134408-4LOCALIZED ITINERANT ELECTRONS AND UNIQUE . . . PHYSICAL REVIEW B 92, 134408 (2015) via program FSCX-14-61025-0, the Russian Foundation of Basic Research via Grant No. 13-02-00374, Ural branch ofRussian academy of Science via program 15-8-2-4 and FASO(theme Electron No. 01201463326). I.M. is supported by ONR through the NRL basic research program. [1] C. I. Hiley, M. R. Lees, J. M. Fisher, D. Thompsett, S. Agrestini, R. I. Smith, and R. I. Walton, Angew. Chem. Int. Ed. 53,4423 (2014 ). [2] C. I. Hiley, D. O. Scanlon, A. A. Sokol, S. M. Woodley, A. M. Ganose, S. Sangiao, J. M. De Teresa, P. Manuel, D. D.Khalyavin, M. Walker, M. R. Lees, and R. I. Walton, Phys. Rev. B92,104413 (2015 ). [3] W. Tian, C. Svoboda, M. Ochi, M. Matsuda, H. B. Cao, J.-G. Cheng, B. C. Sales, D. G. Mandrus, R. Arita, N. Trivedi, andJ.-Q. Yan, P h y s .R e v .B 92,100404(R) (2015 ). [4] D. J. Singh, P h y s .R e v .B 91,214420 (2015 ). [5] I. I. Mazin and D. J. Singh, P h y s .R e v .B 56,2556 (1997 ). [6] P. Manuel, D. T. Adroja, P.-A. Lindgard, A. D. Hillier, P. D. Battle, W.-J. Son, and M.-H. Whangbo, P h y s .R e v .B 84,174430 (2011 ). [7] One may think of possible cancellation of errors, as for instance in FeTe, where the local moment is underestimated becauseof strong correlations, yet suppression of the ordered momentis also underestimated because of strong spin fluctuation.However, extremely high T Nindicates that fluctuations at low temperatures are weak, and cannot appreciably suppress theordered moment. Indeed, calculations reported in Ref. [ 2], using a hybrid method enhancing on-site correlations, overestimate the equilibrium moment by nearly 50% (R. Walton, privatecommunication). [8] These authors report Monte Carlo simulations for the calcu- lated set of parameters, and claim a good agreement withthe experiment; however, their nearest-neighbor only simu-lations yielded an unphysical result of T N=JM2,which is the mean field temperature, rather than TN=0,as re- quired by the Mermin-Wagner theorem, indicating that thermalfluctuations have not been properly accounted for in these simulations. [9] K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii, and R. Valenti, Phys. Rev. B 88,035107 (2013 ). [10] R. D. Johnson, S. Williams, A. A. Haghighirad, J. Singleton, V . Zapf, P. Manuel, I. I. Mazin, Y . Li, H. O. Jeschke, R. Valenti,and R. Coldea, arXiv:1509.02670 . [11] D. Wang, W.-S. Wang, and Q.-H. Wang, Phys. Rev. B 92,075112 (2015 ). [12] S. A. J. Kimber, I. I. Mazin, J. Shen, H. O. Jeschke, S. V . Streltsov, D. N. Argyriou, R. Valenti, and D. I. Khomskii, Phys. Rev. B 89,081408(R) (2014 ). [13] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k, An Augmented Plane Wave +Local Orbitals Program for Calculating Crystal Properties (Techn. UniversitatWien, Wien, 2001). [14] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.92.134408 for computational details. [15] G. Kresse and J. Furthm ¨uller, Phys. Rev. B 54,11169 (1996 ). [16] T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer, Berlin, 2012). [17] B. V . Costa and A. S. T. Pires, J. Magn. Magn. Mater. 262,316 (2003 ). [18] C. Yasuda, S. Todo, K. Hukushima, F. Alet, M. Keller, M. Troyer, and H. Takayama, Phys. Rev. Lett. 94,217201 (2005 ). [19] J. Mravlje, M. Aichhorn, and A. Georges, Phys. Rev. Lett. 108, 197202 (2012 ). [20] A. I. Poteryaev et al. ,http://amulet-code.org . [21] G. Biroli and G. Kotliar, P h y s .R e v .B 65,155112 (2002 ). [22] P. Werner, A. Comanac, L. de’Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 97,076405 (2006 ). 134408-5

PhysRevB.81.201302.pdf

Quantum oscillations in the microwave magnetoabsorption of a two-dimensional electron gas O. M. Fedorych,1M. Potemski,1S. A. Studenikin,2J. A. Gupta,2Z. R. Wasilewski,2and I. A. Dmitriev3,* 1Grenoble High Magnetic Field Laboratory, CNRS, Grenoble, France 2Institute for Microstructural Sciences, NRC, Ottawa, Ontario, Canada K1A-0R6 3Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany /H20849Received 22 February 2010; published 4 May 2010; corrected 3 June 2010 /H20850 We report on the experimental observation of the quantum oscillations in microwave magnetoabsorption of a high-mobility two-dimensional electron gas induced by Landau quantization. Using original resonance-cavitytechnique, we observe two kinds of oscillations in the magnetoabsorption originating from inter-Landau-leveland intra-Landau-level transitions. The experimental observations are in full accordance with theoretical pre-dictions. Presented theory also explains why similar quantum oscillations are not observed in transmission andreflection experiments on high-mobility structures despite of very strong effect of microwaves on the dcresistance in the same samples. DOI: 10.1103/PhysRevB.81.201302 PACS number /H20849s/H20850: 73.50.Mx, 73.50.Jt, 78.20.Ls, 78.67.De Quantum oscillations in absorption /H20849QMA /H20850by a two- dimensional electron gas /H208492DEG /H20850in perpendicular magnetic field B, governed by the ratio /H9275//H9275cof the wave frequency /H9275=2/H9266fmwof external electromagnetic wave and the cyclo- tron frequency /H9275c=eB /mc, were predicted long ago by Ando1and observed experimentally2in the IR absorption on a low-mobility and high-density 2DEG in a Si-inversionlayer. Recently, similar /H9275//H9275coscillations were discovered in the dc resistance of a high-mobility 2DEG irradiated bymicrowaves. 3Particularly intriguing are zero-resistance states4which develop in the minima of these microwave- induced resistance oscillations /H20849MIRO /H20850. Theoretically, both MIRO /H20849Refs. 5–7/H20850and QMA /H20849Refs. 1 and5–8/H20850stem from microwave-assisted transitions between disorder-broadened Landau levels /H20849LLs/H20850. However, in experi- ments on high-mobility samples no QMA were observed sofar despite strong MIRO showing up in the same experimen-tal conditions. Several attempts to measure microwave re-flection or transmission simultaneously with MIRO reportedeither single cyclotron resonance /H20849CR/H20850peak 9–11or more complex structure dominated by confined magnetoplasmons/H20849CMPs /H20850. 12–14 Using original resonance-cavity technique, in this work we observe well-pronounced QMA in a high-mobility GaAs/AlGaAs sample which also reveals strong MIRO in dc trans-port measurements. For both the T-independent QMA and dynamic Shubnikov-de Haas oscillations /H20849SdHO /H20850, the experi- mental results fully agree with the theoretical predictions ofRef. 5, which generalizes theory 1of Ando for the case of smooth disorder potential appropriate for high-mobilitystructures. In addition, we explain the failure to observe suchquantum oscillations in transmission and reflection experi-ments. We start with a summary of relevant theoretical results which includes QMA theory 5for dynamic conductivity at high LLs and nonlinear relation between the absorption anddynamic conductivities 8,10,15specific for high-mobility 2DEG samples. Consider a plane wave normally incident tothe 2DEG at the interface z=0 between two dielectrics with permittivity /H92801/H20849z/H110210/H20850and/H92802/H20849z/H110220/H20850. The electric field Re El of external /H20849l=e/H20850, reflected /H20849l=r/H20850, and transmitted /H20849l=t/H20850waves is a real part of El=Elexp/H20849iklz−i/H9275t/H20850/H20858/H11006s/H11006/H20849l/H20850e/H11006, where the wave numbers ke//H20881/H92801=−kr//H20881/H92801=kt//H20881/H92802=/H9275/c, coeffi- cients s/H11006/H20849l/H20850describe the polarization, /H20858/H11006/H20841s/H11006/H20849l/H20850/H208412=1, and/H208812e/H11006=ex/H11006iey. According to the Maxwell equations, boundary conditions at z=0 read Et=Er+Eeand /H11509z/H20849Et−Er−Ee/H20850=/H208494/H9266/c2/H20850/H11509t/H9268ˆEt. It follows that: /H20881/H92801Ees/H11006/H20849e/H20850/Ets/H11006/H20849t/H20850=/H20881/H9280eff+2/H9266/H9268/H11006/c, /H208491/H20850 where 2 /H20881/H9280eff=/H20881/H92801+/H20881/H92802. Further, /H9268/H11006=/H9268xx/H11006i/H9268yxare the ei- genvalues of the complex conductivity tensor /H9268ˆhaving the symmetries /H9268xx=/H9268yyand/H9268xy=−/H9268yx, namely, /H9268ˆe/H11006=/H9268/H11006e/H11006. Equation /H208491/H20850yields the absorption A, transmission T, and reflection Rcoefficients /H20849see also Refs. 8,10, and 15/H20850, A=/H20858 /H11006/H20881/H92801/H20841s/H11006/H20849e/H20850/H208412 /H20841/H20881/H9280eff+2/H9266/H9268/H11006/c/H208412Re4/H9266/H9268/H11006 c, /H208492/H20850 T=/H20858 /H11006/H20881/H92801/H92802/H20841s/H11006/H20849e/H20850/H208412 /H20841/H20881/H9280eff+2/H9266/H9268/H11006/c/H208412, /H208493/H20850 R=/H20858 /H11006/H20841s/H11006/H20849e/H20850/H208412/H20879/H20881/H92801−/H20881/H92802−4/H9266/H9268/H11006/c /H20881/H92801+/H20881/H92802+4/H9266/H9268/H11006/c/H208792 . /H208494/H20850 It is important to mention that the dynamic conductivity /H9268/H11006, which is the focus of present study, is the response to thescreened electric field acting on 2D electrons. By contrast,coefficients in Eqs. /H208492/H20850–/H208494/H20850represent a response to the /H20849un- screened /H20850electric component of incoming wave and, there- fore, measure both single-particle /H20849transport /H20850and collective /H20849screening /H20850properties of 2DEG. The dynamical screening, represented by the denomina- tors in Eqs. /H208492/H20850–/H208494/H20850, becomes particularly strong in high- mobility structures where the ratio /H208412 /H9266/H9268/H11006/c/H20841reaches values much larger than unity. Indeed, in the absence of Landau quantization the conductivity /H9268/H11006=/H9268/H11006Dis given by the Drude formulaPHYSICAL REVIEW B 81, 201302 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 1098-0121/2010/81 /H2084920/H20850/201302 /H208494/H20850 ©2010 The American Physical Society 201302-1/H9268/H11006D=ne2/m /H9270tr−1−i/H20849/H9275/H11006/H9275c/H20850, /H208495/H20850 where /H9270tris the momentum relaxation time. The absorption, Eq. /H208492/H20850, takes the form AD=/H20881/H92801 /H9280eff/H20858 /H11006/H20841s/H11006/H20849e/H20850/H208412/H9024/H9270tr−1 /H20849/H9024+/H9270tr−1/H208502+/H20849/H9275/H11006/H9275c/H208502, /H208496/H20850 where /H6036/H9024=2/H9251/H9255F//H20881/H9280eff,/H9251=e2//H6036c/H112291/137 is the fine- structure constant, and /H9255Fis the Fermi energy of 2DEG. In high-mobility 2DEG /H9024/H9270tr/H112711 and the width of the cyclotron peak in Eqs. /H208492/H20850–/H208494/H20850and /H208496/H20850is dominated by strong reflection of microwaves. In the region /H20841/H9275−/H9275c/H20841/H11351/H9024, where /H208412/H9266/H9268−/H20841/H11271c, the collective effects are pronounced. In this region, a specialcare should be taken to avoid the finite-size magnetoplasmoneffects. 12–14 According to Ref. 5/H20849which generalizes the results of Ref. 1for the relevant case of smooth disorder potential, see also Refs. 7and8/H20850, Landau quantization at high LLs modifies Drude formula to the form which we call the quantum Drudeformula /H20849QDF /H20850in what follows: Re /H9268/H11006=ne2 /H9275m/H20885d/H9255/H20849f/H9255−f/H9255+/H9275/H20850/H9263˜/H20849/H9255/H20850/H9270tr,B−1/H20849/H9255+/H9275/H20850 /H20851/H9270tr,B−2/H20849/H9255/H20850+/H9270tr,B−2/H20849/H9255+/H9275/H20850/H20852/2+/H20849/H9275/H11006/H9275c/H208502, /H208497/H20850 where f/H9255is the Fermi distribution function. The Landau quantization leads to the oscillatory density of states /H20849DOS /H20850, /H9263/H20849/H9255/H20850=/H9263/H20849/H9255+/H9275c/H20850/H11013/H92630/H9263˜/H20849/H9255/H20850, and to renormalization of the trans- port relaxation time, /H9270tr,B/H20849/H9255/H20850/H11013/H9270tr//H9263˜/H20849/H9255/H20850, where /H92630=m/2/H9266/H60362is the zero- BDOS per spin orientation. In the limit of strongly overlapping LLs, /H9275c/H9270q/H112701, where /H9270qis the quantum relax- ation time, the DOS is weakly modulated by magnetic field /H9263˜/H20849/H9255/H20850=1−2 /H9254cos2/H9266/H9255 /H9275c,/H9254=e−/H9266//H9275c/H9270q/H112701. /H208498/H20850 In the opposite limit /H9275c/H9270q/H112711, LLs become separated, /H9263˜/H20849/H9255/H20850=/H9270qRe/H20881/H90032−/H20849/H9255−/H9255n/H208502, where /H9003=/H208812/H9275c//H9266/H9270q/H11021/H9275c/2, and /H9255nmarks the position of the nearest LL. At the CR /H9275=/H9275c, QDF /H20851Eq. /H208497/H20850/H20852reads /H9268−/H20841/H9275c=/H9275=/H9268−D/H20841/H9275c=/H9275/H20885d/H9255/H9008/H20851/H9263/H20849/H9255/H20850/H20852f/H9255−f/H9255+/H9275 /H9275, /H208499/H20850 where the integration is over the regions with /H9263/H20849/H9255/H20850/H110220. In the case of overlapping LLs, this produces the classical Drude result/H9268−=/H9268−Dmeaning that quantum effects in the vicinity of the resonance are absent. In the case of separated LLs, /H9275c/H11271/H9003, the conductivity is reduced, /H9268−/H20841/H9275c=/H9275=/H208492/H9003//H9275c/H20850/H9268−D.A t the same time, the CR width increases from /H9270tr−1to /H9270tr−1/H9275c//H9003.5,16 In what follows we consider the region /H20841/H9275−/H9275c/H20841/H11271/H9270tr−1 where one can safely neglect /H9270tr−1in the denominator of Eq. /H208496/H20850and QDF, Eq. /H208497/H20850, which givesA/AD=/H20885d/H9255f/H9255−f/H9255+/H9275 /H9275/H9263˜/H20849/H9255/H20850/H9263˜/H20849/H9255+/H9275/H20850. /H2084910/H20850 Equation /H2084910/H20850is the key theoretical result for our study. It expresses the imbalance between the rates of absorption andemission of the microwave quanta, both proportional to theproduct of initial and final densities of states. The screening properties of a 2DEG at /H20841 /H9275/H11006/H9275c/H20841/H11271/H9270tr−1are solely determined by the nondissipative part of conductivity,Im /H9268/H11006/H11229ne2/m/H20849/H9275/H11006/H9275c/H20850/H11271Re/H9268/H11006, which remains finite at /H9270tr−1→0. That is why the transmission and reflection coeffi- cients, Eqs. /H208493/H20850and /H208494/H20850, in high-mobility structures are al- most independent of disorder scattering making it very diffi-cult to observe QMA in direct transmission and reflectionexperiments. 9–13 Magnetoabsorption at two temperatures T=2 and 0.5 K as given by Eqs. /H208496/H20850and /H2084910/H20850is illustrated in Fig. 1for high- mobility 2DEG similar to best structures used for MIROmeasurements. The broadening of the envelope Drude peak/H20851Eq. /H208496/H20850/H20852is dominated by strong reflection of microwaves near the CR since in our example /H9024/H11011 /H9275/H11271/H9270tr−1. The Drude peak in Fig. 1is modulated by the dynamic SdHO /H20849at/H9275c/H11407/H9275and T=0.5 K /H20850and by the temperature- independent QMA with maxima at the CR harmonics /H9275=N/H9275c, which we discuss below. At low temperature, XT=2/H92662T//H6036/H9275c/H113511, the absorption /H20851Eq. /H2084910/H20850/H20852manifests the dynamic SdHO, which in the case of overlapping LLs /H20851Eq. /H208498/H20850/H20852are given by A/AD=1−4XT/H9254 sinh XT/H9275c 2/H9266/H9275sin2/H9266/H9275 /H9275ccos2/H9266/H9255 /H9275c. /H2084911/H20850 The dynamic SdHO are exponentially suppressed at XT/H112711 similar to SdHO in the dc resistance. In the inset it is clearlyseen that the dynamic SdHO are periodically modulated ac-cording to sin /H208492 /H9266/H9275 //H9275c/H20850with nodes at the CR harmonics. Ate−XT/H11270/H9254, i.e., T/H11271TD=/H6036/2/H9266/H9270q, SdHO /H20851Eq. /H2084911/H20850/H20852be- come exponentially smaller than T-independent /H9275//H9275coscil- lations of second order O/H20849/H92542/H20850which represent QMA,FIG. 1. /H20849Color online /H20850Magnetoabsorption calculated using Eqs. /H208496/H20850and /H2084910/H20850for two temperatures T=2 and 0.5 K. Frequency fmw=60 GHz, density n=1.8/H110031011cm−2, mobility /H9262=107cm2/V s, and /H9270q=15 ps. The difference between traces at T=0.5 and 2 K in the inset shows the T-dependent dynamic SdHO.FEDORYCH et al. PHYSICAL REVIEW B 81, 201302 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 201302-2A/AD/H11229/H20855/H9263˜/H20849/H9255/H20850/H9263˜/H20849/H9255+/H9275/H20850/H20856=2/H92542cos2/H9266/H9275 /H9275c. /H2084912/H20850 Here the angular brackets denote /H9255—averaging over the pe- riod/H9275c. Maxima of QMA seen in Fig. 1appear at integer harmonics of the CR /H9275//H9275c=N. The amplitude of QMA /H20851Eq. /H2084912/H20850/H20852becomes of order unity when the DOS modulation is pronounced, i.e., /H9254/H110111.17 Since QMA lie in the microscopic origin of MIRO,5it is instructive to compare the expression /H2084912/H20850to results for MIRO in the same regime. For inelastic mechanism ofMIRO, 5,6the photoresistivity /H9267phin terms of Drude resistiv- ity/H9267Dreads /H9267ph /H9267D=1+2 /H92542−4/H9270inPD /H92752/H92630/H925422/H9266/H9275 /H9275csin2/H9266/H9275 /H9275c, /H2084913/H20850 where the microwave power dissipated in the absence of Landau quantization is PD=AD/H20881/H92801cEe2/4/H9266and/H9270inis the in- elastic relaxation time. Below we use this equation to deter-mine /H9270qfrom dc measurement of MIRO. The known value of /H9270qenables a direct comparison of the measured and calcu- lated QMA /H20851Eq. /H2084912/H20850/H20852without fitting parameters. Experiment. Two rectangular samples were cleaved from a single molecular beam epitaxy-grown wafer of a high-mobility GaAs/AlGaAs heterostructure V0050. Afterillumination with red light the electron concentrationwas n=3.6/H1100310 11cm−2and mobility /H9262=5/H11003106cm2/Vs . Sample 1 was prepared for dc transport measurements. In-SnOhmic contacts were made by rapid annealing in reducingatmosphere of argon bubbled through hydrochloric acid. Thesample was placed in a helium cryostat equipped with a su-perconducting magnet. Similar to Refs. 10and12, micro- waves from an HP source, model E8257D, were delivered tothe sample using Cu-Be coaxial cable terminated with a3 mm antenna. Inset in Fig. 2shows dc resistance R xxmeasured in sample 1 at fmw=70 GHz and T=2 K which displays well- resolved MIRO. Main panel in Fig. 2presents the depen- dence of log 10/H20849/H9004Rxx/H9275c//H9275/H20850vs/H9275//H9275c, where /H9004Rxxis the am- plitude of MIRO measured between adjacent peaks and dips.In accordance with Eq. /H2084913/H20850containing /H92542=exp /H20849−2/H9266//H9275c/H9270q/H20850, this dependence is linear. The slope gives the quantum relax-ation time /H9270q=9.1 ps which we use for calculation of QMA in sample 2. Sample 2 was cleaved from the same wafer in vicinity to sample 1. In order to reduce the CMP effects,14,15,18which can obscure weak QMA oscillations under the investigation,we cleaved a narrow 0.5 /H110031.5 mm 2rectangular stripe. The magnetoabsorption experiment was performed using ahome-built microwave cavity setup at liquid-heliumtemperatures. 19The cavity with a tunable resonance fre- quency had a cylindrical shape with 8 mm diameter and theheight between 3 and 8 mm adjustable with a movableplunger. The cavity operated in TE 011mode, where /H20853011/H20854are the numbers of half-cycle variations in the angular, radial,and longitudinal directions, respectively. The 2DEG stripewas placed at the bottom of the cavity with the externalmagnetic field normal to the 2DEG plane and the microwaveelectric field of the TE 011mode oriented along the short side of the rectangular sample. The sample was placed in a “faceup” fashion such that the active 2DEG layer is separatedfrom the plunger surface by the substrate. This geometry hashigher sensitivity to weak-absorption signals such as QMAbut is not appropriate to study, i.e., CMPs in the vicinity ofthe CR due to cavity over-coupling effects. To further im-prove sensitivity we measure the differential signal with re-spect to magnetic field. The top curve in Fig. 3presents Bdependence of the second derivative of the measured absorption for f mw=58.44 GHz whereas the bottom curve shows the sec- ondBderivative of the absorption coefficient A, Eqs. /H208496/H20850and /H2084910/H20850, calculated without fitting parameters using /H9270q=9.1 ps determined from MIRO measurements on sample 1 /H20849Fig.2/H20850. Both curves demonstrate well-resolved QMA with maximaat harmonics of the CR and dynamic SdHO with maximadetermined by the position of the chemical potential withrespect to LLs. Clearly, the theory reproduces the experimen-tal trace in Fig. 3quite well everywhere except for the shaded region around the CR where the signal is distorteddue to the CMP absorption. The dimensions of sample 2FIG. 2. /H20849Color online /H20850MIRO measured on sample 1 for fmw=67 GHz at T=2 K /H20849inset /H20850, and the amplitude of MIRO vs /H9275//H9275cin semilog scale /H20849main panel /H20850. The linear fit yields the quantum-scattering time /H9270q=9.1 ps, see Eqs. /H208498/H20850and /H2084913/H20850.FIG. 3. /H20849Color online /H20850Magnetoabsorption measured on sample 2a tT=2 K for fmw=58.44 GHz /H20849top curve /H20850and absorption coef- ficient A/H20851Eqs. /H208496/H20850and /H2084910/H20850, bottom curve /H20852calculated without fitting parameters using /H9270q=9.1 ps determined from MIRO measurements on sample 1 /H20849Fig.2/H20850.QUANTUM OSCILLATIONS IN THE MICROWAVE … PHYSICAL REVIEW B 81, 201302 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 201302-3were chosen to allow only one CMP mode at B=0.092 T. The absence of higher modes for fmw/H1102163 GHz enabled the possibility to observe clear quantum oscillations on bothsides of the shaded region where finite-size effects 14,15,18are not essential. The low-field traces /H20849/H9275c//H9275/H110211/2/H20850of the magnetoabsorp- tion are shown in Fig. 4for several microwave frequencies together with the function /H92542cos/H208492/H9266/H9275 //H9275c/H20850. The phase and B damping of the observed QMA follow well the theoreticaldependence, Eq. /H2084912/H20850, without fitting parameters. We, there- fore, believe that the observed oscillations are indeed QMApredicted in Refs. 1and5, which provides an important ex- perimental evidence supporting the theory of MIRO/H20849Refs. 5–7/H20850based on inter-LL transitions. In our sample, QMA are strongly damped /H20851 /H92542=0.02 at /H9275c=/H9275/2, /H9275/2/H9266=fmw=58.44 GHz, and /H9270q=9.1 ps, see Eq. /H2084912/H20850/H20852 which makes their observation difficult. We expect that muchstronger QMA as well as dynamic SdHO can be observed onsamples with higher mobility /H20849longer /H9270q/H20850, as simulated in Fig. 1, provided the CMP effects are avoided or sufficiently reduced. In summary, we have observed quantum magneto- oscillations in the microwave absorption and dynamic SdHO in a high-mobility 2DEG. For this purpose we used a sensi-tive high- Qcavity technique and developed a special setup to avoid undesirable magnetoplasmon effects masking thequantum oscillations. Using the quantum Drude formula 5 and the quantum relaxation time extracted from the MIROmeasurements on the same wafer we were able to reproducethe experimental results for absorption without fitting param-eters, which provides a strong experimental support to thetheory of MIRO and QMA based on inter-LL transitions. We are thankful to A. D. Mirlin, D. G. Polyakov, B. I. Shklovskii, S. A. Vitkalov, and M. A. Zudov for fruitful dis-cussions. This work was supported by the DFG, by the DFG-CFN, by Rosnauka under Grant No. 02.740.11.5072, by theRFBR, and by the NRC-CNRS project. *Also at Ioffe Physical Technical Institute, 194021 St. Petersburg, Russia. 1T. Ando, J. Phys. Soc. Jpn. 38, 989 /H208491975 /H20850. 2G. Abstreiter, J. P. Kotthaus, J. F. Koch, and G. Dorda, Phys. Rev. B 14, 2480 /H208491976 /H20850. 3M. A. Zudov, R. R. Du, J. A. Simmons, and J. L. Reno, Phys. Rev. B 64, 201311 /H20849R/H20850/H208492001 /H20850;P .D .Y e et al. ,Appl. Phys. Lett. 79, 2193 /H208492001 /H20850. 4R. G. Mani et al. ,Nature /H20849London /H20850420, 646 /H208492002 /H20850;M .A . Zudov, R. R. Du, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 90, 046807 /H208492003 /H20850. 5I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. Lett. 91, 226802 /H208492003 /H20850. 6I. A. Dmitriev, M. G. Vavilov, I. L. Aleiner, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. B 71, 115316 /H208492005 /H20850. 7M. G. Vavilov and I. L. Aleiner, Phys. Rev. B 69, 035303 /H208492004 /H20850. 8O. E. Raichev, Phys. Rev. B 78, 125304 /H208492008 /H20850. 9J. H. Smet, B. Gorshunov, C. Jiang, L. Pfeiffer, K. West, V. Umansky, M. Dressel, R. Meisels, F. Kuchar, and K. von Klitz-ing,Phys. Rev. Lett. 95, 116804 /H208492005 /H20850. 10S. A. Studenikin, M. Potemski, A. Sachrajda, M. Hilke, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 71, 245313 /H208492005 /H20850. 11L. C. Tung et al. ,Solid State Commun. 149, 1531 /H208492009 /H20850.12S. A. Studenikin et al. ,IEEE Trans. Nanotechnol. 4, 124 /H208492005 /H20850. 13A. Wirthmann, B. D. McCombe, D. Heitmann, S. Holland, K. J. Friedland, and C. M. Hu, Phys. Rev. B 76, 195315 /H208492007 /H20850. 14S. A. Studenikin, A. S. Sachrajda, J. A. Gupta, Z. R. Wasilewski, O. M. Fedorych, M. Byszewski, D. K. Maude, M. Potemski, M.Hilke, K. W. West, and L. N. Pfeiffer, Phys. Rev. B 76, 165321 /H208492007 /H20850. 15K. W. Chiu, T. K. Lee, and J. J. Quinn, Surf. Sci. 58, 182 /H208491976 /H20850. 16M. M. Fogler and B. I. Shklovskii, Phys. Rev. Lett. 80, 4749 /H208491998 /H20850. 17Resonant features at overtones of CR can be induced by the quasiclassical memory effects even in the absence of Landauquantization, /H9254→0/H20851see I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. B 70, 165305 /H208492004 /H20850/H20852.These classical oscillations originate from the non-Markovian memory effects inthe presence of both long-range and short-range randomness inthe static impurity potential. The predicted phase of classicaloscillations in the dynamic conductivity /H20849and therefore in the absorption coefficient /H20850is opposite to the phase of quantum os- cillations /H20851Eq. /H2084912/H20850/H20852while the amplitude can be comparable to the value of Drude conductivity. 18O. M. Fedorych et al. ,Int. J. Mod. Phys. B 23, 2698 /H208492009 /H20850. 19M. Seck and P. Wyder, Rev. Sci. Instrum. 69, 1817 /H208491998 /H20850.FIG. 4. /H20849Color online /H20850QMA measured at different frequencies compared to the theory /H20851Eq. /H2084912/H20850, parameters as in Fig. 3/H20852.FEDORYCH et al. PHYSICAL REVIEW B 81, 201302 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 201302-4

PhysRevB.84.201305.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 84, 201305(R) (2011) Anomalous mass enhancement in strongly correlated quantum wells Satoshi Okamoto* Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Received 10 September 2011; published 14 November 2011) Using dynamical-mean-field theory, we investigate the electronic properties of quantum wells consisting of at1 2g-electron system with strong correlations. The special focus is on the subband structure of such quantum wells. The effective mass is found to increase with increase in the value of the bottom of the subband, i.e.,decrease in the subband occupation number. This is due to the combination of Coulomb repulsion, whose effectis enhanced on surface layers, and longer-range hoppings. We discuss the implication of these results for therecent angle-resolved photoemission experiment on SrVO 3thin films. DOI: 10.1103/PhysRevB.84.201305 PACS number(s): 73 .21.−b, 71.10.−w Two-dimensional electron gases (2DEGs) realized in a variety of oxide interfaces have been attracting significantinterest. 1,2In particular, electronics utilizing oxides with strong correlations would benefit from their rich phase behaviors.3 For example, control of the band structure of 2DEGs intransition-metal oxides has been proposed as a way to createnoncuprate high- T csuperconductivity.4Yet, the realization of metallic behavior in few-unit-cell-thick oxides remainschallenging. 5,6 Two-dimensional metallic behavior in confined geometry, i.e., in quantum wells (QWs), has been studied for conventional metals. Reconstructed band structures or subband dispersion relations in QWs of Ag thin films have been confirmedusing photoemission spectroscopy. 7,8The subband dispersion of 2DEGs realized on the surface of a band insulator SrTiO 3was also observed using angle-resolved photoemission spectroscopy (ARPES).9,10More recently, Yoshimatsu and co-workers have performed ARPES measurements on QWs in thin films of the correlated metal SrVO 3.11The subband structures realized in such QWs can be explained reasonablywell using a simple tight-binding-type description. However, the effective mass of such subbands was found to increase with decreasing binding energy of the subband. Since this trend isopposite to what we expect based on the bulk behavior, i.e., theeffective mass is reduced with decreasing binding energy and decreasing band occupancy, the origin of such an anomalous mass enhancement remains to be understood. In this Rapid Communication, we analyze model QWs consisting of a t 1 2g-electron system as experimentally con- sidered by Yoshimatsu et al. We employ layer dynamical- mean-field theory (DMFT) with the exact-diagonalizationimpurity solver. 12,13In correlated QWs, a smaller coordination number on surface layers induces larger mass enhancementthan in the bulk region. 14–18This brings about the anomalous subband-dependent mass enhancement; the effective massbecomes larger with decreasing subband binding energy ordepopulation of the subband. With the additional effect comingfrom the longer-range hopping, the subband-dependent massenhancement was found to increase dramatically. We arguethat the anomalous mass enhancement reported for thin filmsof SrVO 3is due to strong correlations and long-range transfer integral. We consider the three-band Hubbard model involving t2g electrons, H=Hband+/summationtext rHloc(r). The first term describesthe noninteracting part of the system as Hband=−/summationdisplay τ,σ/summationdisplay r,r/primetτ rr/primed† rτσdr/primeτσ, (1) where drτσstands for the annihilation operator for an electron at site rin orbital τwith spin σ, andtτ rr/primeis the hopping integral between orbitals τat sites randr/prime. For the orbital τ=xy,w e take the nearest-neighbor hoppings tτ rr/prime=tπforr/prime=r±ˆx(ˆy) andtτ rr/prime=tδforr/prime=r±ˆz, and the second-neighbor hopping tτ rr/prime=tσ/primeforr/prime=r±ˆx±ˆy. Here, ˆx(ˆy,ˆz) is the unit vector along the x(y, z) direction. The hopping parameters for the other orbitals are given by interchanging the coordinates x,y, andzaccordingly. Parameter values are taken from density- functional theory results as tπ=0.281,tδ=0.033, and tσ/prime= 0.096 (all in eV).19Hlocdescribes the local interaction as Hloc=1 2/summationdisplay τ,τ/prime,τ/prime/prime τ/prime/prime/prime,σ,σ/primeUττ/primeτ/prime/primeτ/prime/prime/primed† τσd† τ/primeσ/primedτ/prime/prime/primeσ/primedτ/prime/primeσ−μ/summationdisplay τ,σd† τσdτσ. (2) Here, the site index ris suppressed for simplicity, and μis the chemical potential. Since we consider t2g-electron systems, we assume the relation U=U/prime+2J, where U=Uττττ (intraorbital Coulomb), U/prime=Uττ/primeττ/prime(interorbital Coulomb), J=Uττ/primeτ/primeτ(interorbital exchange) =Uτττ/primeτ/prime(interorbital pair transfer) for τ/negationslash=τ/prime, and other components are absent.20As in Ref. 11, we consider QWs in which a finite number of correlated layers stack along the zdirection with the open-boundary condition and the periodic-boundary conditionalong the xandydirections. Before going into the detailed analysis taking into account the correlation effects, let us first discuss the low-energyelectronic behavior, focusing on quasiparticle bands. For thispurpose, we consider the following effective one-dimensionalSchr ¨odinger equation: 21 /bracketleftbig Zτ z/braceleftbig ˜ετ k−μ+Re/Sigma1τ z(0)/bracerightbig δz,z/prime+/radicalBig ZτzZτ z/primetτ kδz,z/prime±1/bracketrightbig ϕτα z/primek =Eτα kϕτα zk, (3) where /Sigma1τ z(ω) is the electron self-energy at orbital τin layer z computed in layer DMFT, and Zτ zis the layer-dependent quasi- particle weight defined by Zτ z={1−Re∂ω/Sigma1τ z(ω)|ω=0}−1.˜ετ k is the in-plane dispersion and tτ kis the out-of-plane hopping element for orbital τwith in-plane momentum k=(kx,ky). 201305-1 1098-0121/2011/84(20)/201305(5) ©2011 American Physical SocietyRAPID COMMUNICATIONS SATOSHI OKAMOTO PHYSICAL REVIEW B 84, 201305(R) (2011) FIG. 1. (Color online) (a) Dispersion relation as a function of momentum for a noninteracting 5-ML-thick quantum well. Crosses indicate Eyzα k=0(binding energy at k=0times −1). (b) Three- dimensional plot of subband quasiparticle weight /tildewideZyz α,Eyzα k=0,a n dQ W thickness. For each QW, /tildewideZyz αis defined at Eyzα k=0<0. When projected on the left (right) vertical plane, /tildewideZyz αis given as a function of Eyzα k=0 (QW thickness). Forτ=yz, these are explicitly given by ˜ εyz k=− 2tπcosky− 2tσ/primecoskxandtyz k=−tπ−2tδcosky.αlabels the subband with the eigenfunction ϕτα zkin increasing order of the subband energy Eτα k. As an example, the energy eigenvalue Eτα k for a 5-ML-thick noninteracting ( Zτ z=1) QW is plotted in Fig. 1(a) (ML indicates monolayer). We notice that subbands originating from yz(xz) orbitals are not parallel, while xy subbands are. This is because the second-neighbor hoppingbetween neighboring layers induces kdependence in the out-of-plane hopping tyz(xz) k . For orbital yz(xz),tτ k=−tπ− 2tδcosky(x)and, therefore, the subband separation becomes large when ky(x)approaches 0. As a result, the Fermi velocity of high-energy (less-populated) bands becomes small, as if the effective mass is enhanced . The low-energy electronic behaviors of correlated QWs are governed by the quasiparticle subbands. The correlationeffects enter as the quasiparticle weight of the subband. Usingthe solution of Eq. ( 3), the subband-dependent quasiparticle weight is given by 21,22 Zτ α=/summationdisplay zZτ z/vextendsingle/vextendsingleϕτα zk=kα F/vextendsingle/vextendsingle2. (4) From Eq. ( 4), it is clear that the subband quasiparticle weight becomes unity in the absence of correlations, i.e., Zτ z=1 leads to/summationtext z|ϕτα zk=kα F|2=1 (normalization of the quasiparticleeigenfunction). Another important quantity is the effective quasiparticle weight defined by /tildewideZτ α=∂kEτα k ∂kετ k0/vextendsingle/vextendsingle/vextendsingle/vextendsingle k=kτα F. (5) Here, kτα Fis the Fermi momentum for the αth subband, and ετ kkzis the bulk dispersion. For τ=yz,w eh a v e εyz kkz=˜εyz k+ 2tyz k−2tπcoskz. Thus, /tildewideZτ αmeasures the change in the Fermi velocity with respect to its bulk value. In Ref. 11,/tildewideZτ αwas used to discuss the mass enhancement. Because of the kdependence of the interlayer hopping matrix tτ k,/tildewideZcan be smaller than unity even without correla- tions. Figure 1(b) summarizes the results for Eyzα k=0and 1//tildewideZfor noninteracting QWs with thickness varied from 4 to 10. As theQW becomes thin, Eyzα k=0increases and the number of occupied subbands is reduced (see the basal plane). For a 5-ML-thickQW, 1 //tildewideZyz αis projected in the left vertical plane and thus is s h o w na saf u n c t i o no f Eyzα k=0(binding energy times −1). As Eyzα k=0approaches 0, 1 //tildewideZyz αis increased. This trend can be seen in all QWs studied (see the projection of 1 //tildewideZyz αon the right vertical plane). A similar trend was reported experimentally.However, 1 //tildewideZ τ αis enhanced from ∼1t o∼1.7, so is at most 70%. Therefore, the band effect alone does not account forthe large mass enhancement reported in Ref. 11, where 1 //tildewideZ τ α varies from ∼1.7 to ∼4.5. The experimental enhancement in 1//tildewideZτ αis nearly 300%, and 1 //tildewideZτ αat the largest binding energy is already ∼70% larger than the band mass. These results indicate the influence of the correlation effects. In order to see the effect of correlations rather quantita- tively, here we employ layer DMFT, whose self-consistencycondition is closed by 14–16,21 Gτ z(ω)=/integraldisplayd2k (2π)2Gτ zz(k,ω). (6) Here,Gτ zis the local Green’s function on layer z, and the lattice Green’s function matrix on the right-hand side is given as afunction of kand the z-axis coordinate as ˆG(k,ω)=[(ω+ μ)1−ˆH band(k)−ˆ/Sigma1(k,ω)]−1. The hopping matrix ˆHband(k) is given by an in-plane Fourier transformation of Hband as ˆHband(k)=(˜ετ kδz,z/prime+tτ kδz,z/prime±1)δτ,τ/prime. The self-energy matrix is approximated as ˆ/Sigma1(k,ω)=/Sigma1τ z(ω)δz,z/primeδτ,τ/prime. The local self- energy is obtained by solving the effective impurity modeldefined by the local interaction term coupled with an effectivemedium. In this study, we use the exact diagonalizationimpurity solver with the Arnoldi algorithm. 23,24Here, the effective medium is approximated as a finite number of bathsites, and the impurity Hamiltonian is given by H imp=Hloc+/summationdisplay i,τ,σεiτc† iτσciτσ+/summationdisplay i,τ,σ(viτc† iτσdτσ+H.c.). (7) ciτσis the annihilation operator of an electron at the ith bath site with potential εiτand hybridization strength with the impurity orbital τdenoted by viτ. Because of the exponentially growing Hilbert space with respect to the numbers of orbitalsand electrons, we consider two bath sites per correlated orbital,i.e.,i=1,2. In our numerical simulations, we use temperature T=10 −2eV to retain low-energy states with Boltzmann 201305-2RAPID COMMUNICATIONS ANOMALOUS MASS ENHANCEMENT IN STRONGLY ... PHYSICAL REVIEW B 84, 201305(R) (2011) FIG. 2. (Color online) (a) Orbitally resolved spectral function as a function of momentum and frequency for the interacting 5-ML- thick QW with U=4a n d J=0.5 eV . Dotted lines indicate the solution of Eq. ( 3),Eτα k, obtained using the layer DMFT result for the quasiparticle weight Zτ z. Note that yzandxzbands are symmetric with respect to X↔Y. (b) Subband quasiparticle weights Zyz αand /tildewideZyz αas functions of Eyzα k=0.( c )Eyzα k, (d) quasiparticle weight Zyz α,a n d (e) effective quasiparticle weight /tildewideZyz αas functions of the thickness of QWs. Plots (b)–(e) are generated using the method displayed inFig. 1(b). See also Ref. 25(Fig. S1). factors larger than 10−6and consider only paramagnetic solutions. Figure 2(a) shows the results for the orbitally resolved spectral function Aτ(k,ω)=−1 π/summationtext zImGτ zz(k,ω)a sw e l la s Eτα kas dotted lines for a 5-ML-thick interacting QW with U=4 andJ=0.5 (both in eV). For Aτ(k,ω), the self-energy is extrapolated to the real axis using the Pad ´e approximation.24 In comparison with the noninteracting case, the bandwidth is reduced by about 50% due to correlations. We notice that Eτα k reproduces the peak positions of Aτ(k,ω) fairly well. There are five subbands for both yzandxy, but those in the latter are indistinguishable because all subbands are located within therange of 2 t δ∼0.07 eV . Thus, we focus on yzsubbands in the following discussion. Using the same procedure as in Fig. 1(b), we analyze Eyzα k=0and the mass enhancements 1 /Zyz αand 1//tildewideZyz α. Figure 2(b) shows plots of 1 /Zyz αand 1//tildewideZyz αas functions of Eyzα k=0for a 5-ML-thick QW. At the largest binding energy, both 1/Zyz αand 1//tildewideZyz αare about 2, the mass enhancement expected in the bulk region. Figures 2(c),2(d), and 2(e)summarize Eyzα k=0, 1/Zyz α, and 1 //tildewideZyz α, respectively, for interacting QWs with thickness varied from 4 to 10. Although 1 /Zyz αshows an increase with increasing Eyzα k=0,i ti so n l yf r o m ∼2t o∼2.2. On the other hand, 1 //tildewideZyz αshows a rather steep increase from ∼2t o∼3.8, as reported experimentally.11FIG. 3. (Color online) (a) Orbitally resolved occupation number as a function of position zfor a noninteracting 10-ML-thick QW. (b) Same as (a) for the interacting model, and (c) the local quasiparticle weight as a function of z. Layer DMFT with U=4 andJ=0.5 eV is used for (b) and (c). Inset: Quasiparticle wave functions ϕyzα zk=kyzα F. Aside from the quantitative difference, noninteracting QWs and interacting QWs behave quite similarly. As a small butclear difference, some of interacting QWs have a larger numberof occupied yzsubbands, 7- and 9-ML-thick QWs. This is caused by the different orbital polarization. As shown inFigs. 3(a) and 3(b), noninteracting QWs have larger orbital polarization with smaller occupancy in the yzandxzorbitals on surface layers than interacting QWs. This is because thesebands have a quasi-one-dimensional character on the surfacelayers, with a reduced effective bandwidth. The averagecharge density also shows Friedel-type oscillatory behaviorwith respect to z. On the other hand, in the interacting case, the orbital polarization and the charge redistributionare significantly suppressed because the charge susceptibilityis suppressed near integer fillings. This behavior was foundto be insensitive to the choice of the interaction strength asshown in Ref. 25(Fig. S2). Therefore, the effect of charge relaxation is expected to be small, in contrast to LaTiO 3/SrTiO 3 heterostructures. Figure 3(c) shows the position-dependent quasiparticle weight Zτ z, and its inset the quasiparticle eigenfunctions for ayzelectron at the Fermi level. Strong mass renormalization takes place in surface layers where the coordination numberis smaller. 14–18In the current case, Zτ zis∼0.43 (0.51) on surface layers (in the bulk region at z=5); thus there is about 15% stronger mass renormalization on the surface.This small difference comes from the fact that SrVO 3is not so strongly correlated. With increasing U,Zτ zon surfaces are more strongly renormalized [see Ref. 25,F i g .S 3(a)]. Since an eigenfunction with larger αhas larger weight on the 201305-3RAPID COMMUNICATIONS SATOSHI OKAMOTO PHYSICAL REVIEW B 84, 201305(R) (2011) FIG. 4. (Color online) Subband mass enhancement. (a) Effective mass enhancement 1 //tildewideZyz αfor the noninteracting model as a function of Eyzα k=0. Mass enhancement (b) 1 /Zyz αand (c) 1 //tildewideZyz αfor the interacting model with U=4a n d J=0.5e V .( d )1 //tildewideZyz αfor the interacting model with the parameter values indicated. Gray bold lines are guides to the eye. surface layers, the effective mass of such a subband is more strongly renormalized. But the renormalization of Zyz α,u pt o ∼10%, is smaller than that of Zτ zbecause of the interlayer hybridization. The additional enhancement in 1 //tildewideZyz αis caused by the momentum-dependent interlayer hoppings, as discussedearlier. Figure 4summarizes the mass enhancement as a function of the position of the bottom of the subband. The enhance-ment in 1 //tildewideZ yz αis rather small for noninteracting QWs because it comes from the small hopping parameter tσ/prime.T h em a s s enhancement 1 /Zyz αoriginating purely from the correlation effects also shows rather small dependence on Eyzα k=0.O nt h eother hand, 1 //tildewideZyz αshows strong Eyzα k=0dependence because both the band and correlation effects are included. Theeffective mass enhancement 1 //tildewideZyz αsomewhat depends on the correlation strength, as shown in Fig. 4(d). Comparison of 1/Zyz αfor different interaction strengths is presented in Ref. 25[Figs. S 3(b)–S3(d)]. With a reasonable parameter set, the experimentally reported mass enhancement can besemiquantitatively reproduced. We notice that the number of subbands is overestimated by∼1 for interacting QWs compared with the experimental observation. 11A possible explanation for this discrepancy is that, in the experiment of Ref. 11, the surface layer is made of VO 2, so that the symmetry and the valence state of surface V ions greatly deviate from those in the bulk. Also, we cannotexclude the possibility of surface lattice relaxation by whichconduction electrons are strongly localized on the surfacelayer. In these cases, the surface V sites would not contributeto the ARPES spectrum near the Fermi level as do those in thebulk. Detailed study including these effects would be necessaryto fully understand the nature of SrVO 3QWs, including the dimensional crossover and the metal-insulator transition.5,11 Yet the present study provides a reasonable account for the anomalous mass enhancement reported for SrVO 3thin films. Summarizing, using dynamical-mean-field theory, we in- vestigated the electronic properties of correlated quantumwells consisting of a t 1 2g-electron system. The special focus is on the subband structure of such quantum wells. The subbandeffective mass was found to increase with decreasing bandoccupancy as reported for SrVO 3thin films. The present theory provides a reasonable account for this observationas the combined effect of Coulomb repulsion, whose effectis enhanced on surface layers, and longer-range hoppings.Inclusion of these two effects is essential to correctly interpretexperimental observations. The author thanks H. Kumigashira, K. Yoshimatsu, and A. Fujimori for valuable discussions and for sharing experi-mental data prior to publication. The author is grateful to V .R. Cooper for discussion and to C. G. Baker for his advice oncoding. This work was supported by the US Department ofEnergy, Office of Basic Energy Sciences, Materials Sciencesand Engineering Division. *okapon@ornl.gov 1A. Ohtomo, D. A. Muller, J. L. Grazul, and H. Y . Hwang, Nature (London) 419, 378 (2002). 2A. Ohtomo and H. Y . Hwang, Nature (London) 427, 423 (2004). 3H. Takagi and H. Y . Hwang, Science 327, 1601 (2010); J. Mannhart a n dD .G .S c h l o m , ibid.327, 1607 (2010). 4J. Chaloupka and G. Khaliullin, P h y s .R e v .L e t t . 100, 016404 (2008). 5K. Yoshimatsu, T. Okabe, H. Kumigashira, S. Okamoto, S. Aizaki,A. Fujimori, and M. Oshima, P h y s .R e v .L e t t . 104, 147601 (2010). 6J. Liu, S. Okamoto, M. van Veenendaal, M. Kareev, B. Gray,P. Ryan, J. W. Freeland, and J. Chakhalian, P h y s .R e v .B 83, 161102(R) (2011).7D. A. Evans, M. Alonso, R. Cimino, and K. Horn, P h y s .R e v .L e t t . 70, 3483 (1993). 8I. Matsuda, T. Ohta, and H. W. Yeom, P h y s .R e v .B 65, 085327 (2002). 9A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailh `es, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi,P. Le F `evre, G. Herranz, M. Bibes, N. Reyren, Y . Apertet, P. Lecoeur, A. Barth ´el`emy, and M. J. Rozenberg, Nature (London) 469, 189 (2011). 10W. Meevasana, P. D. C. King, R. H. He, S.-K. Mo, M. Hashimoto,A. Tamai, P. Songsiriritthigul, F. Baumberger, and Z.-X. Shen, Nat. Mater. 9, 114 (2011). 11K. Yoshimatsu, K. Horibe, H. Kumigashira, T. Yoshida, A. Fujimori, and M. Oshima, Science 333, 319 (2011). 201305-4RAPID COMMUNICATIONS ANOMALOUS MASS ENHANCEMENT IN STRONGLY ... PHYSICAL REVIEW B 84, 201305(R) (2011) 12A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 13M. Caffarel and W. Krauth, P h y s .R e v .L e t t . 72, 1545 (1994). 14M. Potthoff and W. Nolting, P h y s .R e v .B 59, 2549 (1999). 15M. Potthoff and W. Nolting, P h y s .R e v .B 60, 7834 (1999). 16S. Schwieger, M. Potthoff, and W. Nolting, Phys. Rev. B 67, 165408 (2003). 17A. Liebsch, Phys. Rev. Lett. 90, 096401 (2003). 18H. Ishida, D. Wortmann, and A. Liebsch, Phys. Rev. B 73, 245421 (2006) 19E. Pavarini, A. Yamasaki, J. Nuss, and O. K. Andersen, New J. Phys. 7, 188 (2005).20S. Sugano, Y . Tanabe, and H. Kamimura, Multiplets of Transition-Metal Ions in Crystals (Academic, New York, 1970). 21S. Okamoto and A. J. Millis, P h y s .R e v .B 70, 241104(R) (2004). 22A. R ¨uegg, S. Pilgram, and M. Sigrist, Phys. Rev. B 75, 195117 (2007). 23R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide (SIAM, Philadelphia, 1997). 24C. A. Perroni, H. Ishida, and A. Liebsch, Phys. Rev. B 75, 045125 (2007). 25See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.84.201305 for more information. 201305-5

PhysRevB.62.130.pdf

C28: A possible room temperature organic superconductor N. Breda Dipartimento di Fisica, Universita `di Milano, Via Celoria 16, I-20133 Milano, Italy and INFM, Unita di Milano, Milano, Italy R. A. Broglia Dipartimento di Fisica, Universita `di Milano, Via Celoria 16, I-20133 Milano, Italy; INFN, Sezione di Milano, Milano, Italy; and The Niels Bohr Institute, University of Copenhagen, D-2100 Copenhagen, Denmark G. Colo` Dipartimento di Fisica, Universita `di Milano, Via Celoria 16, I-20133 Milano, Italy and INFN, Sezione di Milano, Milano, Italy G. Onida Dipartimento di Fisica, Universita `di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italy and INFM, Unita `di Roma Tor Vergata, Roma, Italy D. Provasi Dipartimento di Fisica, Universita `di Milano, Via Celoria 16, I-20133 Milano, Italy E. Vigezzi INFN, Sezione di Milano, Milano, Italy ~Received 11 January 2000 ! The electron-phonon coupling in fullerene C28has been calculated from first principles. The value of the associated coupling constant l/N(0) is found to be a factor 3.4 larger than that associated with C60. Assuming similar values of the density of levels at the Fermi surface N(0) and of the Coulomb pseudopotential m*for C28-based solids as those associated with alkali doped fullerides A3C60, one obtains Tc(C28)’8Tc(C60). The valence properties of small fullerenes,1in particular, of the smallest fullerene yet observed C 28, is a fascinating question at the fundamental level as well as in terms of itspotential applications for the synthesis of new materials. 2–7 In supersonic cluster beams obtained from laser vaporiza- tion, C 28is the smallest even-numbered cluster, and thus the fullerene displaying the largest curvature, which is formedwith special abundance. In fact, under suitable conditions, C 28is almost as abundant as C 60.3At variance with its most famous family member C 60,C28is expected to form a cova- lent crystal ~like C36, Refs. 8–10 !, and not a van der Waals solid.11However, similarly to C 60, fullerene C 28maintains most of its intrinsic characteristics when placed inside aninfinite crystalline lattice. 2The transport properties of the associated metal doped fullerides, in particular superconduc-tivity, can thus be calculated in terms of the electron-phonon coupling strength lof the isolated molecule, and of the den- sity of states of the solid. 12,13In keeping with the fact that curvature–induced hybridization of the graphite sheet por- bitals, seems to be the mechanism explaining ~cf. Refs. 12–15 and references therein !the large increase in Tcin going from graphite intercalated compounds ( Tc’5K!~Ref. 16!to alkali-doped C 60fullerides ( Tc’30–40 K !,17–19 fullerene C 28is a promising candidate with which to form a high-Tcmaterial. These observations call for an accurate, first-principle investigation of the electronic and vibrationalproperties, as well as of the electron-phonon coupling strength of this system. In the present work we present theresults of such a study, carried out within ab initio density- functional theory ~DFT!in the local spin-density approxima- tion~LSDA !. Our findings are that the associated value of l/N(0) is a factor 3.4 and 1.2 larger than that associated with C 60~Ref. 13 !and C36~Ref. 9 !, respectively. Under simi- lar assumptions for the density of levels at the Fermi energy N(0) and for the Coulomb pseudopotential m*as those as- sociated with alkali-doped fullerides A3C60, one will thus expectTc(C28)’8Tc(C60), opening the possibility for C28-based fullerides which are superconducting at, or close to, room temperature. The equilibrium geometry of C 28obtained in the present calculation is similar to that proposed by Kroto and co-workers,20and has the full Tdpoint-group symmetry. All atoms are threefold coordinated, arranged in 12 pentagonsand 4 hexagons. The large ratio of pentagons to hexagons makes the orbital hybridization in C 28more of sp3type rather than sp2, the typical bonding of graphite and C 60. The sp3-like hybridization is responsible for a series of remark- able properties displayed by small fullerenes in general and by C28in particular. Some of these properties are: ~i!the presence of dangling bonds, which renders C 28a strongly reactive molecule, ~ii!the fact that C 28can be effectively stabilized @becoming a closed-shell system displaying a largePHYSICAL REVIEW B 1 JULY 2000-I VOLUME 62, NUMBER 1 PRB 62 0163-1829/2000/62 ~1!/130~4!/$15.00 130 ©2000 The American Physical Societyhighest occupied molecular orbital–lowest unoccupied mo- lecular orbital ~HOMO-LUMO !energy gap #by passivating the four tetrahedral vertices either from the outside (C 28H4) or from the inside ~U@C28).3It also displays a number of hidden valences: in fact, C 28H10,C28H16,C28H22, and C28H28are essentially as stable as C 28H4~all displaying HOMO-LUMO energy gap of the order of 1.5 eV !,1in keep- ing with the validity of the free-electron picture of pelec- trons which includes, as a particular case, the tetravalentchemist picture, ~iii!while typical values of the matrix ele- ments of the deformation potential involving the LUMOstate range between 10 and 100 meV, the large number ofphonons which couple to the LUMO state produces a totalelectron-phonon matrix element of the order of 1 eV ~cf. Table I !, as large as the Coulomb repulsion between two electrons in C 28. This result @remember that the correspond- ing electron-phonon matrix element is ;0.1 eV and the typi- cal Coulomb repulsion is ;0.5–1 eV for C 60~Ref. 13 !#tes- tifies to the fact that one should expect unusual properties for both the normal and the superconducting state of C 28-based fullerides, where the criticisms leveled off against standard theories of high Tcof fullerenes ~cf., e.g., Refs. 13 and 21–24 and references therein !will be much in place. In Fig. 1 ~a!, we report the electronic structure of C 28 ccomputed within the local spin-density approximation, asobtained from a Car-Parrinello25molecular-dynamics scheme.26,27Near the Fermi level we find three electrons in a t2orbital, and one in a a1orbital, all with the same spin, in agreement with the results of Ref. 3. The situation is notaltered, aside from a slight removal of the degeneracy, when the negative anion, C 282, is considered @see Fig. 1 ~b!#. In this case, the additional electron goes into the t2state, and has a spin opposite to that of the four valence electrons of neutral C28. The wave numbers, symmetries, and zero-point ampli- tudes of the phonons of C 28are displayed in Table I, together with the matrix elements of the deformation potential defin-ing the electron-phonon coupling with the LUMO state. Thetotal matrix element summed over all phonons is equal to 710 meV. The partial electron-phonon coupling constantsl a/N(0),also shown in Table I, sum up to 214 meV. This value is a factor 2.5 larger than that observed in C 60,13and aTABLE I. Phonon wave numbers, symmetries, and zero-point amplitudes @Ga[(\/2Mva)1/2#~columns 1, 2, and 3 !of the phonons of C28which couple to the LUMO state. In columns 4 and 5 the corresponding electron-phonon matrix elements gaand partial coupling constants la/N(0) are displayed. In the last row we show the corresponding summed values. 1/l@cm21#symm. Ga(1023Å!Matrix element ga@meV#la/N(0) @meV# 351 E 63.3 7.9 1.0 391 T2 59.9 10.7 2.4 524 T2 51.8 49.7 38.0 565 A1 49.9 12.9 0.8 570 E 49.6 37.0 12.9 607 E 48.1 55.7 27.5 707 T2 44.6 42.5 20.6 724 T2 44.1 42.8 20.4 763 A1 42.9 46.2 7.5 771 T2 42.7 12.4 1.6 791 T2 42.1 0.9 0.0 976 E 37.9 43.6 10.5 983 T2 37.8 15.2 1.9 1093 T2 35.9 3.4 1.0 1101 A1 35.7 45.2 50.0 1116 E 35.5 68.9 22.8 1171 A1 34.6 6.4 0.1 1191 T2 34.3 43.6 12.9 1220 A1 33.9 30.5 2.0 1260 T2 33.4 21.2 2.9 1306 E 32.8 57.5 13.6 1381 T2 31.9 6.7 0.3 1414 E 31.5 49.2 9.2 Total: 710 214 FIG. 1. Kohn-Sham levels of the neutral ~a!and negatively charged ~b!C28cluster calculated within the LSD approximation. a andblabel the zprojection of the electron spin and arrows repre- sent the valence electrons. FIG. 2. Calculated electron-phonon coupling constant l/N(0) for C70~Ref. 30 !,C60~Ref. 13 !,C36~Ref. 9 !,C28~cf. Table I !.PRB 62 131 BRIEF REPORTSfactor 1.2 larger than the value recently predicted for C 36.9 In Fig. 2 we display the values of l/N(0) for C 70,C60,C36, and C28,9,30–32which testify to the central role the sp3cur- vature induced hybridization has in boosting the strengthwith which electrons couple to phonons in fullerenes. 12–15 In keeping with the simple estimates of Tccarried out in Refs. 13 and 9 for C 60and C36based solids, we transform the value of l/N(0) of Table I into a critical temperature by making use of McMillan’s solution of Eliashbergequations, 33,34 Tc5vln 1.2expF21.04~11l! l2m*~110.62l!G, ~1! where vlnis a typical phonon frequency ~logarithmic aver- age!,lis the electron-phonon coupling and m*is the Cou- lomb pseudopotential, describing the effects of the repulsive Coulomb interaction. Typical values of vlnfor the fullerenes under discussion is vln’103K~cf., e.g., Refs. 35 and 36 !. Values of N(0) obtained from nuclear magnetic resonance lead to values of 8.1 and 12 states/eV spin for K 3C60and Rb3C60, respectively ~cf. Ref. 13 and references therein !. Similar values for N(0) are expected for C 36.9Making useof these values of N(0) for all C n-based solids ( n570, 60, 36, and 28 !, one obtains 0.1 <l<3 for the range of values of the associated parameter l. The other parameter entering Eq. ~1!, namely m*and which is as important as lin determin- ingTcis not accurately known. For C 60,m*is estimated to be’0.25.13Using this value of m*, and choosing N(0) so thatTc’19.5 K for C 60, as experimentally observed for K3C60,13one obtains Tc(C28)’8Tc(C60) andTc(C28) ’1.3Tc(C36).37 We conclude that C 28-fullerene displays such large electron-phonon coupling matrix elements as compared tothe repulsion between two electrons in the same molecule, that it qualifies as a particular promising high- T csupercon- ductor. From this vantage point of view one can only specu-late concerning the transport properties which a conductor constructed making use of the other fullerene C 20as a build- ing block, can display.40In fact, this molecule is made en- tirely out of 12 pentagons with no hexagons, being the small-est fullerene which can exist according to Euler theorem forpolyhedra, and thus displaying the largest curvature a carboncage can have. Calculations have been performed on the T3E Cray com- puter at CINECA, Bologna. 1C. Milani, C. Giambelli, H.E. Roman, F. Alasia, G. Benedek, R.A. Broglia, S. Sanguinetti, and Y. Yabana, Chem. Phys. Lett.258, 554 ~1996!. 2E. Kaxiras, L.M. Zeger, A. Antonelli, and Yu-min Juan, Phys. Rev. B49, 8446 ~1994!. 3T. Guo, M.D. Diener, Yan Chai, M.J. Alford, R.E. Haufler, S.M. McClure, T. Ohno, J.H. Weaver, G.E. Scuseria, and R.E. Smal-ley, Science 257, 1661 ~1992!. 4D.M. Bylander and L. Kleinman, Phys. Rev. B 47,1 09 6 7 ~1993!. 5B.I. Dunlop, O. Ha ¨berben, and N. Ro ¨sch, J. Phys. Chem. 96, 9095 ~1992!. 6M.R. Pederson and N. Laouini, Phys. Rev. B 48, 2733 ~1993!. 7A. Canning, G. Galli, and J. Kim, Phys. Rev. Lett. 78, 4442 ~1997!. 8J.C. Grossman, M. Cote ´, S.G. Louie, and M.L. Cohen, Chem. Phys. Lett. 284, 344 ~1998!. 9M. Cote´, J.C. Grossman, M.L. Cohen, and S.G. Louie, Phys. Rev. Lett.81, 697 ~1998!. 10C. Piskoti, J. Yager, and A. Zettl, Nature ~London !373, 771 ~1998!. 11M.S. Dresselhaus, G. Dresselhaus, and P.C. Eklund, Science of Fullerenes and Carbon Nanotubes ~Academic Press, New York, 1996!. 12M. Schlu¨ter, M. Lannoo, M. Needels, G.A. Baraff, and D. To- manek, Phys. Rev. Lett. 68, 526 ~1992!. 13O. Gunnarsson, Rev. Mod. Phys. 69, 575 ~1997!. 14A. Devos and M. Lannoo, Phys. Rev. B 58, 8236 ~1998!. 15V.H. Crespi, Phys. Rev. B 60, 100 ~1999!. 16I.T. Belash, A.D. Bronnikov, O.V. Zharikov, and A.V. Pal’nichenko, Synth. Met. 36, 283 ~1990!. 17K. Tanigaki, T.W. Ebbesen, S. Saito, J. Mizuki, J.S. Tsai, Y. Kubo, and S. Kuroshima, Nature ~London !352, 222 ~1991!. 18T.T.M. Palstra, O. Zhou, Y. Iwasa, P.E. Sulewski, R.M. Fleming,and B.R. Zegarski, Solid State Commun. 93, 327 ~1995!. 19T.T.M. Palstra, A.F. Hebard, R.C. Haddon, and P.B. Littlewood, Phys. Rev. B 50, 3462 ~1994!. 20H. Kroto, Nature ~London !329, 529 ~1987!. 21P. Anderson ~unpublished !. 22L. Pietronero and S. Stra ¨ssler, Europhys. Lett. 18, 627 ~1992!. 23L. Pietronero, S. Stra ¨ssler, and C. Grimaldi, Phys. Rev. B 52, 10 516 ~1995!. 24C. Grimaldi, L. Pietronero, and S. Stra ¨ssler, Phys. Rev. B 52,1 0 530~1995!. 25R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 ~1985!. 26J. Hutter et al., MPI Fu¨r Festko¨rperforschung, Stuttgart, and IBM research, 1990–1997. The code has been partially modified tocalculate the matrix elements of the deformation potential. 27The whole calculation ~i.e., geometry optimization of the cluster, Kohn-Sham levels, phonons and deformation potential !has been carried out by setting C28in a fcc supercell with lattice constant a526 a.u. A norm-conserving Trouiller-Martins ~Refs. 28 and 29!pseudopotential has been employed in the calculation, with a cutoff of 40 Ry. 28N. Troullier and J.L. Martins, Phys. Rev. B 43, 1993 ~1991!. 29M. Fuchs, and M. Scheffler, Comput. Phys. Commun. 119,6 7 ~1999!. 30D. Provasi, N. Breda, R.A. Broglia, G. Colo `, H.E. Roman, and G. Onida, Phys. Rev. B 61, 7775 ~2000!. 31O. Gunnarsson, Phys. Rev. B 51, 3493 ~1995!. 32N. Breda, R.A. Broglia, G. Colo `, H.E. Roman, F. Alasia, G. Onida, V. Ponomarev, and E. Vigezzi, Chem. Phys. Lett. 286, 350~1998!. 33W.C. McMillan, Phys. Rev. 167, 331 ~1968!. 34G.M. Eliashberg, Zh. E´ksp. Teor. Fiz. 38, 966 ~1960!@Sov. Phys. JETP11, 696 ~1960!#. 35D.S. Bethune, G. Meijer, W.C. Tang, H.J. Rosen, W.G. Golden,132 PRB 62 BRIEF REPORTSH. Seki, C.A. Brown, and M.S. de Vries, Chem. Phys. Lett. 179, 181~1991!. 36Z.H. Wang, M.S. Dresselhaus, G. Dresselhaus, and P.C. Eklund, Phys. Rev. B 48, 16881 ~1993!. 37Similar values of Tcare obtained using Allen’s solution ~Refs. 38 and 39 !of Eliashberg equations. 38P.B. Allen and R.C. Dynes, Phys. Rev. B 12, 905 ~1975!. 39P.B. Allen and B. Mitrovic ˇ,Solid State Physics , edited by H.Ehrenreich, F. Seitz, and D. Turnbull ~Academic Press, New York, 1982 !, Vol. 37, p. 1. 40While no clear evidences have been found in carbon cluster beams for a particularly abundant bare C20cluster, the fully hydrogenated C20H20molecule, dodecahedrane, turns out to be stable ~Ref. 41 !. 41L.A. Paquette, R.J. Ternansky, D.W. Balogh, and G.J. Kentgen, J. Am. Chem. Soc. 105, 5446 ~1983!.PRB 62 133 BRIEF REPORTS

PhysRevB.39.10028.pdf

PHYSICAL REVIEW B VOLUME 39,NUMBER 14 15MAY1989- Electronic dampingofadsorbate fundamental andovertone vibrations atmetalsurfaces Z.Y.Zhang Department ofPhysicsandAstronomy, Rutgers Th—eStateUniversity ofPewJersey,Piscataway, NewJersey088550-849 DavidC.Langreth Department ofPhysicsandAstronomy, Rutgers The—StateUniversity ofXewJersey,Piscataway,¹wJersey088550849- andInstituteofTheoretical Physics, Chalmers University ofTechnology, S-41296Goteborg, Sweden (Received 7October 1988) Amodelcalculation ofthevibrational lineshapeofadsorbates atmetalsurfaces damped bythe generation ofelectron-hole pairsispresented. Twovibrational excitations areconsidered, onefun- damental andoneovertone. Tocalculate thetotalabsorption ofaradiation fieldbythesystem, we usetheformalism developed inthepreceding papertoincludecontributions, a&andu~~,fromthe respective E-fieldcomponents perpendicular andparallel tothesurface. Thegeneralized asym- metricFanoline-shape formula isshowntoapplytoboththefundamental andtheovertone. For eachcase,theisotopeeffectontheline-shape parameters isobtained bothinthelimitsofweakand strongbreakdown ofadiabaticity, wheretheweak-orstrong-breakdown limitismodeled byassum- ingaRatspectral densityneartheFermilevel,orasharpstructure inthatregionontopoftheOat part.Itisfoundthatastrongbreakdown ofadiabaticity weakens theisotopedependence ofthe asymmetry factorandthelinewidth, whiletheisotopeeffectfortheexcitation strength remains the sameasintheweak-breakdown limit.Themagnitude oftheexcitation strength ismodified asthe strong-breakdown limitisreached. Theseconclusions remainvalidwhether thevibration contrib- utestoa&ortoo.~~.Whenappliedtothewagovertone ofhydrogen onW(100), reasonably good agreement isfoundbetween thetheoryinthestrong-breakdown limitandexperiment assuming that theovertone absorption isgivenbya&,theperpendicular response. Wealsodiscussotherpossibili- tiesforthemeasured absorption, including thecontribution fromtheovertone bya~~,thetangential response, andfromotherexcitations nearbyinfrequency. I.INTRODUCTION Studiesofvibrational spectraofatomsormolecules ad- sorbedatsurfaces havebeenoneoftheveryactiveareas insurfacescienceoverthelastdecade,boththeoretically andexperimentally.'Suchstudiesareexpected toyield important information onthebonding geometry ofada- tomsandontheunderlying physical principles ofthe dynamical processes whichoccur.Among manyofthe concerns ofexperimentalists arethefrequency shifts,the linewidths, thelineshapes, andexcitation strengths. Thesequantities characterizing alineareusually ob- tainedandstudiedatdifferent coverages andtempera- tures,withisotopesubstitutions orwiththecoadsorption ofanother species.Yet,atpresent, itisstilloftendifficult toidentify whichmechanism ispredominately involved foragivensystem,notonlybecause generally morethan onemechanism coexists andcompetes, butalsobecause westillneedabettertheoretical understanding ofthe characteristic differences inthespectra caused by different mechanisms. Foradsorbates onametalsurface, oneoftheeffective broadening mechanisms istheenergyrelaxation dueto excitations ofelectron-hole pairsinsidethemetal. Suchamechanism becomes important because the electron-hole-pair continuum ofthemetalsubstrate ex- tendsdowntozeroenergy. Inanidealized case,'an adsorbed molecule contributes anemptyleveltotheadsorbate-substrate systemrightabovetheFermilevel. Astheadsorbate vibrates, thislevelalsomovesupand downrelative totheFermilevelasaresultofthe response ofthescreening electrons. Thechanging occu- pationofthisadsorbate-induced resonant stateleadsto thegeneration ofelectron-hole pairsinthesubstrate, thus enhancing thedamping ofthevibration. Aboutonede- cadeago,BrivioandGrimley madeamicroscopic- theoretic firstestimateofthelinewidth yforsuchaprob- lem.Subsequently, Persson andHellsing'"performed first-principles calculations ofyforanatomonajellium surface andamodepolarized perpendicular tothesur- face.Theirvalues,typically afewmeV,compare favor- ablywithexperimental observations. Theyalsopredicted the1/Misotopeeffectiny,aswellasitsnegligible tern- perature dependence. Indeed, intheadiabatic approxi- mationtherewouldbenocontribution tothelinewidth fromelectron-hole pairexcitation.'The1/Misotope dependence, alsoderived byPersson andRyberg,'as- sumesthatthebreakdown ofadiabaticity issmall,andis obtained byretaining onlytheleading nonvanishing term inanexpansion ofthewidthinpowersofco.Inaddition, theweaktemperature dependence ofyhasoftenbeen usedtosuggestthatthebroadening mechanism mightbe electron-hole —pairgenerations.'' Recently, noticing theimportance ofthephasedelayin theelectronic partofthedynamic dipolemomentofthe adsorbed molecule relativetotheionicpart,oneofthe 3910028 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10029 present authors(D.C.L.)predicted thataninescapable consequence oftheelectron-hole-pair decaymechanism isthatthelineshapehasanasymmetric Fano-like' form, aconclusion laterarrivedatbySorbello.'Suchan asymmetry inlineshapewasobserved byChabal''in infrared reflectance studiesofHonW(100)atsaturation coverage. Langreth'originally considered asinglead- sorbedmolecule vibrating perpendicular tothesurface in thefundamental ofthestretchmode.Morerecently, CrljenandLangreth generalized thetreatment tothe casewheredipole-dipole interactions between themole- culesoftheadsorbed layerwereincluded, andagainthe asymmetric lineshapewasobtained, butwithrenormal- izedparameters. Whentheirtheorywasappliedtothe C—0stretchvibration onCu(100),'verygoodagree- mentwasobtained. Inalloftheseprevious investiga- tionsthevibration wasassumed tobeafundamental (not anovertone) vibrating perpendicularly tothesurfaceand excitedbyafieldperpendicular tothesurface. Forasaturation coverage ofHonW(100),''and morerecently HonMo(100),''twospectral features wereobserved whichwereassigned tobethefundamental ofthesymmetric stretchmodeandtheovertone ofthe wagmode;onlythelattershowedastrongasymmetry in itslineshape.Theovertone ofthewagprovides anex- amplewherethevibration isparalleltothesurface.For bothsystems, theasymmetric lineshapecouldbefitted verywellbythelineshapederived byLangreth, suggest- ingthattheelectron-hole —pairenergy-relaxation mecha- nismisoperative. Ontheotherhand,themeasured iso- topedependences ofthecharacteristic parameters ofthe lineshape,theasymmetry factor,andthelinewidth, are consistently weakerthanwhatispredicted forafunda- mentalinthelimitofweakbreakdown ofadiabaticity. In addition, following theGrstreportonthise-hpair damped asymmetric lineshapeofanovertone,'anexten- siverelatedstudyhasbeencarriedoutbyReutt,Chabal, andChristman,'suggesting thattheovertoneofthewag couldbeexcitedbyanelectric fieldparalleltothesurface viasurface-state excitations. Thehighasymmetry and thestrongexcitation strength measured doinfactindi- cateastrongcoupling between thevibration andtheelec- tronicstates. Inthepresent investigation, wegeneralize theformal- ismfortheelectron-hole —pairmechanism developed be- foreinseveralaspects. First,weincludethepossibility of exciting bothafundamental andanovertone, andobtain theisotopeeffectsinthelimitofweakbreakdown ofadia- baticity. Ourcalculations predictthattheasymmetry andwidthhavethesameisotopic dependence forthe overtone astheydoforthefundamental. Onlytheexci- tationstrengths exhibitasmalldifference inisotope dependence. Nextweconsider thelimitofstrongbreakdown ofadi- abaticity. Heretheexpansion inpowersofcoisnolonger valid.Weassumethattheelectron-hole —paircreation occursviatheexcitation ofanelectron outof(orinto)a sharpstructure inthespectral density, whichisbetween theFermilevelandtheresonant vibrational level.We thusobtain newisotope effectsforthelinewidth and asymmetry, whicharesystematically weakerthanthoseinthenearlyadiabatic limit.Inparticular, theisotope eff'ectsareinmuchbetteragreement withexperiment for thewagovertone forHonW(100), suggesting thatthere isindeedastrongbreakdown oftheadiabatic approxima- tionforthissystem. Thisconclusion isconsistent with thelargeobserved asymmetry ofthismode.Inaddition, thestrong-breakdown limitalsoproduces anenhance- mentintheexcitation strength compared toresultsinthe weak-breakdown limit. Wealsoconsider thepossibility ofexciting vibrational modesbyanelectric-field component paralleltothesur- face.Asintheperpendicular case,weusethegeneral theorydeveloped inRef.22.Experimentally onemight determine whichexcitation method isoperative byangu- larand/orpolarization dependence, provided thatone hasenough sensitivity togoawayfromnearglancing in- cidence. IntheH/W(100) orH/Mo(100) case,itwould seemthatiftheE~Icoupling isimportant, thenuncertain- tymayexistonthemodeassignments. However, 'the agreement oftheobserved isotope dependence ofthe asymmetry withourprediction strongly suggests acou- plingviatheelectrons, andastrongbreakdown ofthe adiabatic approximation. Thispaperisorganized asfollows. Inthenextsection wedescribe ourHamiltonian andtheformalism usedto obtainthevibrational lineshapeofthesystem. Section IIIcontains ourcalculations andresultsintheweak- breakdown limit.InSec.IVwederivetheisotopeeffects inthestrong-breakdown limit.Comparisons withexperi- mentanddiscussions aremadeinSec.V.Therewecon- cludebysummarizing thesuccesses ofthismodeland stressing theremaining controversies anddiscrepancies. II.DESCRIPTION QFTHEPROBLEM Inthissection, wedescribe theformalism andthe Hamiltonian usedtostudyadsorbate vibrations. Wecon- siderawell-defined faceofsomesubstrate solidwithcu- bicsymmetry withthex,y,andzaxesalongcrystalline directions, asassumed inthepreceding paper. Ontop ofthissurface, takentolieinthexyplane,thereisanor- deredlayerofchemisorbed atomsormolecules. Wefur- therassume thattheadsorbate-substrate system also possesses highsymmetry, sothatallofthenormalmode vibrations arestrictly alongthecrystalline directions. TheH(1X1)/W(100) andtheH(1X1)/Mo(100) systems studied inRef.21andalsoanalyzed herewhencontact withexperiment ismade,satisfytheserequirements. p- polarized incident electromagnetic radiation onsucha system willingeneralproduce anelectric fieldintheper- pendicular zdirection(Ej),aswellasacomponent (E~~) paralleltothesurfacetakentobealongthexaxis.Fora systemwithadielectric constantoflargemagnitude, asis typicalofmetalsintheinfrared range,thezcomponent ofthefieldismuchlargeroutside thesample, butis screened toaverysmallvaluewithinafewangstroms of thesurface; theparallel component, although typically muchsmalleroutsideandinthescreening layer,eventu- allybecomes theonlysizable Geldleft,sincetolowestor- deritisnotscreened.'Foragivensystem,theproper- 10030 Z.Y.ZHANG ANDDAVIDC.LANGRETH 39 tiesordistributions ofthesefieldswillpartially determine theexcitation strengths ofthevibrational modesofthe adatoms; theotherfactordetermining theexcitation strengths isthedynamic dipolemoments ofthosemodes alongthefielddirections. Inmostcasesthelfieldwillbe muchmoreimportant (seeRef.22andlaterdiscussions), butinsomespecialcasestheroleofthe ~~fieldmaybe- comemoreimportant, aswassuggested tobetrueforthe excitation ofthewagovertone forH(1X1)/W(100) and theH(1X1)/Mo(1OO) systems.'Weintendtoconsider a generalpicture inthepresentpaper,ratherthanassum- ingatthebeginning thatonlytheeffectsofl(or ~~)field arepredominant. Onceexcited, theadatom vibrations candecaybygen- erationofelectron-hole pairsviatheCoulomb interaction V,,(R—r)between theionandelectrons insurfacestates and/orintheconduction bandofthemetal.Weconsider thecasewheretheionicvibrations haveafrequency muchhigherthanthemaximum bulkphonon frequency, sothatgeneration ofe-hpairsistheonlyeffective energy-relaxation mechanism. Theincident radiation canalsodirectly excite electron-hole pairsinthecombined systemofadsorbate andsubstrate. Depending onthespecific symmetry con- ditions, theseelectron-hole pairscanbeexcitedbyeither EgE[~orboth.Aspointed outbefore,'theinterfer- encebetween thedirectabsorption bytheelectrons and thedirectabsorption bytheioncangiverisetoanasym- metriccontribution tothelineshape.Infact,thepossi- bilityfortheexternal fieldtoexcitetheelectronic system directly isessential fortheasymmetry toappear. The present paperfocuses onthecaseofanovertone excita- tionparalleltothesurface. Inapurelyharmonic approx- imation, suchamodecannotbeexcitedbythedirectcou- plingofEitotheionicmotion. Therefore, thedirectex- citationofeithervirtualorrealelectron-hole pairs,some ofwhichhaveasymmetry whichallowstheirrecombina- tiontocreateovertone vibrations, becomes theonly channel leftforabsorption attheovertone frequency. Eveninthemoregeneral casewheretheeffectofthe parallel fieldEt~isincluded, thewagovertone atthefre- quency 2vz(wefollowthenotationofRefs.17and21for thefrequencies) isstilluncoupled totheionicmotion un- lessstronganharmonicity isassumed. Thismeansthat assistance fromelectron-hole pairexcitations (whichmay havedifferent symmetry fromthoseexcited byEi)is againneededforastrongabsorption at2+2.Incompar- ingtheirinfrared differential reflectivity resultswithdi- poleelectron-energy-loss spectroscopy (EELS) resultsof othersonthesamesystems, Reutt,Chabal, andChrist- man'arguethatthestrongabsorption peakatfrequency 2vzforH/W(100) andH/Mo(100) atsaturation coverage isduetotheinterference ofthismodewithelectron- hole—pairexcitations insurface statesprobably ofd„ symmetry; thiswouldenablethemtobestrongly coupled withtheovertone atZv2,tobeexcitable byE~~,andnotto bedetectable byspecularEELS.Asanapplication ofthe presenttheory, wewillfirstexamine thispossibility. We findthatthesymmetry ofbothH/W(100) and H/Mo(100) tendstoeliminate thepossibility ofexciting 2v2byE~~unlessthereAection symmetry inthexyplaneisRFresnel16mqIm[(nui)sin 8—(nal)/e],2 cosO(2.1) whereAR=R—RF„,„,&andRF„,„,&istheFresnel value ofR,0istheincident angle,qisthewavevector,andeis themacroscopic dielectric function ofthesubstrate sys- tem.Thes-polarization analogue of(2.1)is ARncxll =16mqcosOIme—1(2.2) Fresnel In(2.1)and(2.2),thesurfacepolarizability perunitarea n~~orna~~is,asmentioned before,uniquely givenbythe samemicroscopic expression (afterdropping thebulk term):lostduetothepresenceofimpurities anddefects. We wi11alsoexamine otherpossible contributions tothisab- sorption peak,suchastheparticipation ofelectron-hole pairsexcitable byE~andcoupledtotheovertone at2v2, andtheabsorption duetothenearlydegenerate asym- metricstretchvibration (v3). Thelattermodecould infactonlybeexcitedbyE~tifthereflection symmetry is assumed. Theeffectofincident radiation istobuildupanoscil- latingelectric fieldtowhichthesystemresponds. The systems underconsideration havebulkdielectric con- stantsthatareverylargeinmagnitude, sothatatany essentially nonzero angleofincidence, ap-polarized in- cidentbeamwi11produce afieldwithalargezcomponent andamuchsmallerxcomponent outsidethesample (we takethesurfacenormaltobeinthezdirection andthe incident wavevectortobeinthexzplane);asonepasses through thescreening-surface absorption layer,thislarge zcomponent isscreened downtoamuchsmaller value, leaving theessentially unscreened xcomponent asthe predominant fieldcomponent afteroneisseveral angstroms deepinthesolid.Inthecasesofinterest, the ultimate exponential extinction ofthesebulkfieldsoccurs oversolongadistance (—10A)thatitcanbeneglected. Arigorous expression forthereQectivity involving the microscopic surface polarizabilities perpendicular and paralleltothesurface, andwhichtakestheabovefield distributions intoaccount exactly, wasderived inthe preceding paper, andwillbeusedasthebasisofthecal- culations here. Although theparallel (~~)andperpendicular (J.)polari- zabilities entertheexpression forthereAectivity in different ways,theformalexpressions forthepolariza- bilitiesthemselves haveidentical forms,withoneinvolv- ingtheparallel dipoleoperator andtheotherinvolving theperpendicular dipoleoperator. Thus,exceptfor selection rulescoming fromsymmetry considerations, andthewayinwhichscreening istreated, thecalculation ofoneproceeds inthesamewayasthatfortheother. Wewilltherefore notrepeatthesamealgebratwice,and willpicktheperpendicular caseasaworking example. Withsomedifferences tobediscussed later,themaincon- clusions extracted fromtheformalcalculation oftheper- pendicular polarizability canbeappliedtotheparallel caseaswell. Forap-polarized incident beam,theformula forthy change inreAectivity, assuming ~e~cos0))1,is 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10031 [na„(a))]= —iAfdte'"'([p,'~(t),p,'~(0)]), 0 (2.3) wherep'„~iseitherthelorthe ~~component ofthedi- poleoperator perunitarea(thetotalareabeingA),and whichincludes bothionicandelectronic contributions. Nowwespecialize tothecasewheretheelectric fieldis alongz.Equation (2.3)canthenberewritten as [nai(co)]=—fdrfdr'rE~(r,r',co)r'Ei,(2.4) whereyisthesurfacechargeresponse function ofthe system,randr'runoverallrelevant electronic andionic coordinates withinaunitarea,andEj=zisaunitvector alongz. Withtheseformulas inhand,themainworkleftisto evaluate themicroscopic response function forthesystem described. TodothisweneedtospecifytheHarniltonian oftheadsorbate-substrate system. Weconfine ourselves tothecasewhereanydirectcoupling between adjacent adatoms isveryweak,andcanthusbeneglected. Ifthere isaneedtoincludethedipolecoupling amongtheadsor- bates,theformalism usedbyCrljenandLangreth, and veryrecently byPersson, shouldbeappropriate. Inthe casewheresuchcoupling canbeneglected, asforin- stanceforHonW(100) andHonMo(100), insteadofa layerwecanjustfocusonasingleion(apurenucleusor anucleus withitsinner-shell electrons), chemisorbed ona metalsurface, andplacedattheoriginofcoordinates. Quantities inrealmeasurements foranadsorbated layer will,toagoodapproximation, beproportional tothose calculated forasingleadsorbate. TheHamiltonian for thissystemcanbewrittenastionthattheaccuracy intreating thisscreening shouldbe inaccordance withtheaccuracy usedtoobtainother relevant quantities, suchasthepurelyelectronic partof thesurface charge response functiong,appearing in (3.19),aquantitytobedefinedlater. Formally, theHarniltonian usedherehasthemeritof treating equallytheelectron-hole —pairexcitations associ- atedwiththeNewns-Anderson —typechargetransfer be- tweentheadsorbate andthesubstrate, andthosesolely restricted withintheconduction and/or surface-state bandsofthemetalbythelong-range dipolefieldofthevi- bratingion.Itcanbesimplified easilytotreataspecial caseifnecessary. Inparticular, forthe2v2modeof H/W(100), asimpleNewns-Anderson modelseemstobe inadequate, astheelectronic statesmediating theexcita- tionoftheovertone aresuggested tobedifferent from bonding orantibonding states,''although insome casesthechargeoscillations involving somebonding or antibonding stateswithinaNewns-Anderson modelcan bemuchmoreeffective togivedamping ofanadatom vi- bration thanthelong-range dipoleGeld.'Inthecase wherechargeoscillations donotdominate, theeffectof thedipolefieldcanbemuchmoreimportant thanexpect- ed,especially whenoneisinthestrong-breakdown limit. Theionicmotion ismuchsmaller inmagnitude com- paredwithelectronic motion, becauseofitsverylarge masscompared withm„takenheretobe1.Wecan therefore expandV;,(R—r)in(2.7)intermsoftheionic displacement vectorRaroundtheoriginofcoordinates, takentobetheequilibrium positionoftheion.Usingthe identityV&V;,(R—r)=—V,V;,(R—r)weget Hintgaiaj H=~O+~In~+~COUiamb where(2&) (i~(—RV,)"[V;,(R—r)]o~j)/n!,(2.9) n=1 Ho=+A bb+pe;a;a; . (2.6) In(2.6),thefirsttermdescribes thevibrational statesof theion,withb(b„)theraising(lowering) operator for modemoffrequencyQ,andthesecondtermrefersto electronic motions, wherea,(a;)isthecreation (destruc- tion)operator forelectronic statep,(r)(wetakeatomic units irt=m,=e=1).Thesecondtermin(2.5)isgivenby H;„,=pa;aj(i~ V;,(R—r)~j), (2.7) where (i~V;,(R—r)~j)=fdrqr,'(r)V,,(R—r)y(r)(2.8) isthetransition matrixbetween twodiferent electronic states ~i)and ~j),V;,(R—r)istheinteraction potential between anelectronofcoordinates randtheionofcoor- dinatesR,wheretheionisinsomevibrationally excited state.Thesimplest formofV,,(R—r)isabareCoulomb potential, orthatforavibrating dipole,butinpractice it mustbescreened bytheCoulomb partHc,„&,bfromthe restoftheelectrons. Weintendtogivemorediscussions lateronthescreening ofV,,(R—r);hereweonlymen-1 2Mn where g'isthepolarization vectorofthevibrational statem,andMistheeffective massassociated withthe normal modeoffrequency0.ForHonWorHon Mo,theeffective massforeachmodeofHvibration can beverywellapproximated bythemassofthehydrogen atom. Finally, inrelationtothederivation ofthegeneralized Fanolineshapebyotherauthors,''wemention thatwherethesubscript 0ofV;,(R—r)meansthattheionic coordinate Rshouldbesettozeroafterthederivative is taken.Noticeagainthatthestaticn=0term,already included in(2.6),isexcluded herefrom(2.9),sincebyas- sumption H;„,describes thoseprocesses ofthesystem originating fromthebreakdown ofadiabaticity in electron-ion interaction, whichbydefinition shouldin- volvetransitions between electronic states,andthus necessarily alsoinvolves themotionoftheion,or,more strictly, thetransition between theionicstates.In(2.9),a smallionicdisplacement Rmaybewritten intermsof phononcreation anddestruction operators bandbas 1/2 g.(b.'+b.), (2.10) 10032 Z.Y.ZHANG ANDDAVIDC.LANGRETH 39 theexternal fieldenterstheproblem viaitscoupling to thedynamic dipolemoments ofthesystemassociated withelectronic andalsoionicmotion. Fromthiscou- plingonearrivesatformulas (2.3)inlinear-response theory, whichforalfieldproduces (2.4).' Usingtherather general Hamiltonian andthe reAection formulas givenabove,wecaninprinciple cal- culatetheabsorption byanadsorbate-substrate system withtheexcitation ofanyvibrations (fundamental and/or overtone) involved. InSec.IIIwewillcalculateain severalspecialcases:firstweoutlinethemainresultsal- readyobtained beforeforafundamental stretch mode perpendicular tothesurface, givingexplicit isotope dependences fortheparameters characterizing theline shapeforsystems whichhaveaAatdensityofstatesnear theFermilevel;thenwestudythecasewherethesecond harmonic ofthevibration parallel tothesurface is present. III.CALCULATIONS ANDRESULTS INTHELIMITOFAWEAKBREAKDOWN OFADIABATICY A.Thefundamental ofastretchvibration Thevibrational linewidths andlineshapesassociated withanadsorbate stretchvibration offundamental fre- quencyperpendicular tothesurfacecoupledtotheelec- tronshavebeenstudied quitethoroughly inprevious theoretical investigations. Actually, nearlyallprevious workconsiders thiscaseasaworking example,"''ex- ceptforaveryrecentpaperbyPersson, inwhichafun- damental modeisstillconsidered, butvibrating parallel tothesurface.Foravibration alongz,onlynaj(orE~) wasconsidered andwasassumed tobedominant, which isindeedthecaseformanysystems. Inthissection, we stillrestrictourselves tothiscase.Thatistosay,wewill againconsider thecontribution, toaz,ofastretchmode vibrating perpendicularly tothesurfaceatthefundamen- talfrequency. Butoneshouldunderstand thatthefor- malism developed foritwouldgoexactlythesamewayif (1)thevibration isparalleltothesurfaceexcited bythe fieldE~,(2)thevibration isalongzexcited byE~~(anex- ampleofthiskindisstilllacking), and(3)boththevibra- tionandthefieldareparallel tothesurface.'The difference liesonlyinphysical considerations. Firstly, different electronic stateswillparticipate fordifferent sit- uations. Secondly, withintheharmonic approximation, theionicemotionalong,say,thezaxisdoesnotdirectly coupletothefieldEII.Therefore, quantities depending on thiscoupling (suchasa,;;defined below) willvanish. Thirdly, thedetailed mechanisms fortheexcitation of electron-hole pairscanbedifferent fordifferent direc- tions:forthelfield,thepresence ofthesurfaceactsas thesourceofmomentum neededfortheexcitation, while forthe ~~field,periodicity isstillassumed andtherefore eitherthelatticemustprovide momentum equaltoasur- facereciprocal latticevector,orelsea"free-carrier" scattering-type mechanism mustbeoperative. Finally, wenotethat(2.1)contains notermwhichaccounts for theinterference between thecontribution ofthejandthewhereeandistandforadirectcoupling oftheelectronic andionicmotions tothefieldE~atthesurface, respec- tively.Forsimplicity, wehavedropped thecommon fac- torninfrontofeachain(3.1),aswearenowconsidering onlyasingleadsorbate. Wehavealsosuppressed the subscriptsj,sinceonlythiscaseisconsidered here. Whencontact withexperiment ismade,propersubscripts shouldbeused,andthesingle-adsorbate quantities de- rivedaretobesubstituted backintotheexpression for thereAectivity ofarealsystem. Incalculating a(co)forafundamental, onlythefirst derivative oftheinteraction potential V~,(R—r)with respecttoZneedstobetaken,andg'=zin(2.10).Adi- agrammatic representation foreachpieceofthepolariza- bilityisgiveninFig.1,'whereadouble-dashed line represents thenormalized propagator D,(co)ofthevibra- tionalmode,andthetotalelectroriic chargeresponse functiong,isrepresented byasphere.TheDysonequa- tionforD,(co)isshowninFig.2,wherethesingle-dashed lineisabarevibrational propagator definedby 2Q,oD,o(~)— Q)Qgo+ ltJ(32) Q,oistheunperturbed frequency, andq=0+an (c) FIG.1.Diagrams ofthetotalpolarizability a,forthestretch vibration offundamental frequency, whichisseparated into fourpieces: {a)u,;;,(b)a,;„(c)a,,;,(d)a,,,Adouble- dashedlinerepresents afullynormalized propagator fortheion (seeFig.2).Thesphererepresents thetotalelectronic charge responseg,(seeFig.3).~~fieldcomponents. Suchinterference wasneglected in thederivation of(2.1)inRef.22,because theelectronic stateswhichcanbestrongly coupledtoonefieldcom- ponent usuallycannotbecoupled efFectively totheother fieldcomponent. Sincethemainframework hasalready beengivenbe- foreforaperpendicular stretchmodeoffundamental fre- quencycontributing toca~,weonlyneedtobrieAysum- marizetheresults. Meanwhile, whenever possible, more attention willbepaidtothederivation oftheisotope dependences fortheparameters characterizing thevibra- tionallineshape. Following theconvention usedbefore,'wedividethe response functiongappearing in(2.4),andhencetheto- talpolarizability a,intofourpieces: (3.1) 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10033 a...(co)=p—,,D,(co)p,; 2Q„ (33) FIG.2.TheDysonequation forthepropagator ofthevibra- tionalmode.Thesingle-dashed lineistheunperturbed one. infinitesimal. Thenormalized phononpropagator D,(co),determined fromFig.2,willbeakeyexpression inthederivation since,asshowninFig.1,D,(co)enterseverypieceofthe polarizability. Firstforthei-ipartwehave,fromFig. 1(a),wheretherenormalized frequency Q,„=Q,O+5Q„5Q, isthefrequency shifttobeobtained fromitsdefinition andtherelationQ,„=Q~~O+2Q, ORe[m(Q, o)],n(Q,o)be- ingtheusualself-energy ofthevibrational propagator givenbelow. Oneveryimportant quantity appearing in (3.3)isthevibrational linewidth y,arising fromthe creationofelectron-hole pairs(weuseasubscript ztodis- tinguish quantities forvibrations alongzfromthose alongaparalleldirection xconsidered inSec.III8).By definition, y,isrelatedtotheimaginary partofthepho- nonself-energy, m(co),as y,=—21mil(co) 2A 2MQ,Oco(3.4) XP,(r)P;(r)P, (r')P;(r') . (3.5) Inmatrixnotation, thetotalelectronic response function isrelatedtoitsirreducible partby,fromtheDysonequa- tiongiveninFig.3(a), +eQ (a)whereg,;.(co)isthefrequency-dependent factorcon- tainedinageneral termoftheirreducible partofthe electronic response function,g,(r,r';co),asascreened electron-ion interaction potentialV;(R—r)isused simultaneously.''Therelation ofg,(r,r',co)to y,(r,r',co),showninFig.3(a),isgivenbyanintegral equation, andwithintherandom-phase approximation (RPA)[seeFig.3(b)]g,(r,r',co)canbewrittenas f(s;)—f(E,)g,(r,r';co)= ,,e;—s—(co+i')e1—(3.6) where VisthebareCoulomb potential givenby V(r,r')=1/~r—r'~.Thelastquantity tobeexplained in (3.4)isthedynamic dipolemomentp,;,defined by ][[,,=e(2MQ,„)'~,wheree,'istheefFective chargeas- sociated withtheioncore.Weassumethate;*isrealand static.Forthecaseofhydrogen adsorption itissimply equaltoone. In(3.3)and(3.4)wehavepurposely written quantities inawaythatanexpansion intermsofthefrequency co canbemadeeasily.Thisexpansion, whichwillbemade later,isjustified because cuissmallcompared withelec- tronicenergies. Foradetailed discussion onthefrequen- cydependence ofthesusceptibility g„orofthelinewidth y„and itsrelationtotheadiabatic Born-Oppenheimer approximation, seeLangreth.' From(3.4)wecanderivetheisotope (andtegnperature) dependences ofthelinewidth. Todothisweneedtoun- derstand thefrequency andthemassdependences ofthe summations overelectronic statesshownin(3.4).Forthe benefitoflaterdiscussions, wewillstudythefollowing moregeneral typeofsummation, wh&chcanbespecial- izedtoaparticular casewhenever necessary: ~RPA (E;)—(s,)J(co)—=gF,~'~c,;—s~(co+i')—(3.7) FIG.3.(a)Theintegral equation fory,.(b)Thelowest-order diagrams fortheirreducible partg,.Thefirsttermgivesthe RPAresult.whereF,isanysmoothfunctionofI',andj,andisoftena multiplication ofseveralinteraction and/orexcitation matrices. In(3.7)wehaveimplicitly assumed thatany frequency dependence ofF;canbeneglected. Thisas- sumption iswelljustified whenweconsider theweak- breakdown limit.'Todemonstrate thisjustification, we noticethatinmostcasesI';hascodependence because V,';(R—r)generally has;whileintheweak-breakdown 10034 Z.Y.ZHANG ANDDAVIDC.LANGRETH 39 (3.9) whered0isanelementofsolidangle. Throughout thissection weassumethattherelevant densityofstatesoftheelectrons isaveryslowlyvarying function withintheregiona~+co-c~+Q, whereQisthe vibrational frequency beingstudied. Suchacasecorre- spondstoaweakbreakdown ofadiabaticity, andwepost- poneadiscussion onastrongbreakdown tothenextsec- tion.Inthisweak-breakdown limit,ithasbeenshown thatthetemperature dependence ofthevibrational line shapeatasurfacecanbetotallyneglected ifthevibration isdamped byelectron-hole —pairexcitations. Therefore, wecantaketheT=0limitin(3.8)andfind ImJ(co)=~fdE—p(s)fds'p(E')F(e, E') 0 0 X[1—f(s)]f(e') X5(s—E—co) EF+co vrfdEp(e—)p(eco)F(e,—Eco).— Ep(3.10)limit,thedominant contribution tothescreening of V,,(R—r),represented bythedielectric function E(co)of thesystem(thespatialdependences ofearesuppressed here),canwellbeobtained fromthestaticlimitofe(co),' whichisrealandbydefinition alsofrequency indepen- dent.Forsimplicity, inthenextsection wewillstill neglect anycodependence ofF;.whenweconsider the strong-breakdown limit,corresponding tothecasewhere asharppeakexistsinthespectral density withinthe range0oftheFermilevel,lyingontopofaAatpart. Equivalently, wewillassumethatthecontribution tothe screening involving virtualorrealtransitions between the sharppeakandtheHatpartisstillmuchsmallerthanthe staticlimitofthecontribution fromallotherpossible vir- tualorrealtransitions wherethesharppeakisnotin- volved,although itwillbeshowninSec.IVthatineither therealorimaginary partofthecontribution toe(co) fromthepeak-to-fiat orAat-to-peak alone,anco- dependent, dynamic termhastobekept. Ingeneral, astatedenoted byihasanenergyandsome angular variables: ~i)—+~E,Q).Whenthesummations defined in(3.7)arereplaced bytheircorresponding in- tegrals, wecansingleouttheintegrals overenergyexplic- itlybyintroducing aneffective densityofstatesp(E)for theelectrons. Byeffective wemeanthosestateswhich arecapableofmediating acoupling between theexternal fieldandthevibrational modebeingstudied. This effective spectral density mayresemble thedensityof statesmapped outinphotoemission andinversephoto- emission measurements, butisnotnecessarily thesame. Intermsofp(E)wecanthenrewrite(3.7)as J(co)=fdep(s)fde'p(e') (3.8) Es'(co—+i—g) whereF(sE)i,su'suallyassumed' tobeasmoothly vary- ingfunction definedbyWenowexpandImJ(co) incoandkeeponlytheleading contribution. Sincebyassumption bothF(E,E+co)and p(E)arefiatfunctions ofE,wecansimplyreplace them bytheircorresponding valuesattheFermisurface, and (3.10)isthensimplified to ImJ(co)=~F—(E~,e~)p(s~)p(E~)co . (3.11) 6D, COde)P—ED~'—67 b EDM 2ED+coIn ED+co(3.12) wherePindicates theprincipal-value integral, andE~is acutoffparameter intheexcitation spectrum of electron-hale pairs.SincewealwayshaveED))co-Q, thelogarithmic termcanbedropped, andweendupwith theconclusion thatReJ(co)cansimplybeobtained from itsstaticlimitbysetting~=0.Thisconclusion istrueas longaswehaveaBatspectral distribution inarange cz+Q,nornatterhowrapidthevariation ofthespectral density isawayfromthisrange.Infact,thefirsttermin (3.12),whichwasobtained byassuming afiatspectral densityforelectron-hole —pairexcitations overthewhole range2ED,isbynomeansaccurate, andshouldinreality bereplaced byatermotherthanthatgivenin(3.12);but, thekeyfeatures thatitisfrequency independent andis muchlargerthanthefrequency-dependent secondterm in(3.12)remainunchanged. Therefore, itwouldbestill validtoneglect anyfrequency dependence inReJ(co). Oneshouldalsonoticethattheconclusions madehere aboutthefrequency (andthusionicmasswhenever appl- icable)dependences oftherealandimaginary partsof J(co),whichwerearrivedatwithintheRPA,willstill holdifsomemoreaccurate approximation ismadefor theelectronic chargeresponse function. Foreitherthefundamental considered hereorthe overtone tobediscussed inthenextsubsection, wewill encounter threephysically different kindsofsummations. Eachofthemcanbeformally represented bythetypeof summation J(co)discussed above,withdifferent expres- sionsforF(s,8+co) Clearly the.linewidth y,givenby (3.4)contains onesuchsummation; theothertwo,which willappearshortly, arecontained intheelectronic com- ponentofthedynamic dipolemoment defined byEq. (3.14)andthebackground electronic absorption givenby Eq.(3.19),respectively. Wenowextract theisotopic dependence oftheTherefore ImJ(co)=bco,wherebisaconstant determined fromtheFermi-surface electronic properties ofthesys- ternandhenceindependent oftheionicmass. Laterwewillneedthefrequency dependence ofthe realpartofsumsofthetypeJ(co).Wepresenthereonly acrudeevaluation whichextrapolates theassumedfat- nessofthespectral densityoverthewholeband,butthe basicconclusion arrivedatisessentially correct. Using theKramers-Kronig relation, andEq.(3.11),wehaveap- proximately fortherealpartofJ(co) +QO 1ReJ(co)=—fdco'ImJ(co')P 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10035 linewidth. FromEqs.(3.4)and(3.11),y,=—ImJ(co)/ (Mco)= b—/M.Therefore wehavey,ccM',animpor- tantresultfirstderived byPersson andHellsing. Nextwecontinue ourcalculation ofthetotalpolariza- bility.FromFigs.1(b)and1(c)wehave a...(co)=a...(co)=p,—,D,(co)p,,(co), (3.13) (3.14) andp,,=e,*/(2MB,„)'isthesameasdefinedbefore, e,,zstandsforthe"effective" electronic chargescreening thevibrating ion,whoserealpartisrelatedtothefre- quently usednotation e,*ofthedynamic dipoleas e,.~+Re(e,,z)=e,+e,*,=e,*,anditsimaginary partpro- videsinformation ontheasymmetry ofthelineshape derivable fromthetotalpolarizability a„'g,;.(co)has beenseenbefore[see(3.5)].Hereweemphasize again thatp,,from(3.14)hasbothrealandimaginary parts becauseoftheinterference between thedirectexcitation bytheexternal fieldoftheelectron-hole pairsontheone hand,represented bythecoupling matrices—(i ~rEi~j), andofthevibration ontheotherhand,represented by p,;.Theinterference ismediated bytheelectron-ion in- teraction, whichinturnisscreened bythepresence of otherelectrons. Forthisinterference tobeeffective, itis essential thatwehaveelectronic stateswhichpossessthe rightsymmetry tobedirectly excitable bytheexternal field,andalsotocouplestrongly totheionicmotion. Fi- nally,wenoticethatfortwoextended electronic statesof ~i)and ~j)belonging toaconduction band,formally the matrixelement—(i~rEi~j)in(3.14)wouldbedivergent ifwelookatthematrixalone.Suchadivergence is prevented bythesummations overiandj,whichweas- sumearedonebeforeimplied spatialintegrals. Thefourthpieceofthepolarizability canbewritten downinasimilarwayfromFig.1(d): a...(co)=p„(co)D, (co)p,—,(co). Adding up(3.3),(3.13),and(3.15)yields a,(co)=—[p,;+p,,(co)]'D, (co).(3.15) (3.16) Nowwerewrite(3.16)inamorefamiliar form.Letting p,;+p,,:—p,,+ip,2,anddefi.ning A,,=p,z/p,„weget a,(co)=—p„s(1+i A,), Q7Qzz+lCO+z(3.17) wherep,,s-=e,'/(2MB,„)'~.Finallyforthelineshape wehaveI.(co)=—21ma,(co),andwhen(3.17)isused,an asymmetric Pano-like formula isproduced. Using(3.14),wecannowdiscusstheisotope depen- dencesoftheparameters otherthany,whichcharacter-wherep,,istheelectronic component ofthedynamic di- poledefinedas p,,(co)=—p,;e„s(co), =—y(ilrEilJ)X,,;,(~)izethelineshape.Asalready mentioned (3.14)contains another summation whosefrequency dependence canbe equally represented bythesummation J(co)defined in (3.7).Herewemaymaketheidentification thatJ(co)=e,,s(co).Ifafiatspectral densityisagainassumed neartheFermienergy,thesummation appearing in(3.14) shouldthenhavethesamefrequency dependence asthe summation in(3.4),whichisgivenbyafrequency- independent realpart,andanimaginary partlinearincu. Thatis,inbothcaseswehaveJ(co)=a+ibco,andforthe electronic contribution tothedynamic dipolemoment we maywrite,fromEq.(3.14),p,,(co)=(a+ibco)p,;,where aandbareconstants independent ofthefrequency. Of course,themagnitudes ofaandbwillbedifferent for different summations. Thedifference between thetwo summations appearing in(3.4)andin(3.14)isthatthe transition matrices iV',R—roj between thestates ~i)and ~j)coupled bythefirstderiva- tiveoftheelectron-ion interaction potential V,';(R—r)in (3.4)arereplaced in(3.14)bythetransition matrices—(i~rEi ~j)between thesamestates ~i)and ~j),but nowcoupled bythescreened potential oftheexternal ra- diation. Sinceneither—(i~r-Ei~j)nor contain anydependence onthemassofthevibrating ion, weseethattheconstants aandbarealsoindependent of themassoftheion.'Therefore, inthelimitofweak breakdown ofadiabaticity, theratio A.,=(b/a)co intro- ducedhere,calledtheasymmetry factoranddenoted by previously, hasitsisotope dependence given onlybyitslineardependence on~.Sincenearthereso- nancewecanalwaysreplaceubythevibrational fre- quency0,„,wegetfinally (3.18) Next,theprefactor p,,sappearing in(3.17),which givesthestrength oftheexcitation (assuming A,,((I, otherwise amorecomplicated formhastobeused)hasa simpleisotope effectp,,sccp,,o:I/(2MB,„)a:M Finally, wemention thatanequivalent wayofexpressing theexcitation strength isgivenbytheeffective chargee* ofthedynamic dipole moment, defined by p,s=e*/(2M&„)', whichforthefundamental ofthe stretchmodehasnodependence onM.Alltheseisotope effectsarecollected inTableI. Beforeweturntotheovertone, weshouldpointout thatatermproportional totheimaginary partofao(co) shouldbeaddedtothelineshapeobtained from(3.17), whichcorresponds tothebackground absorption. This background isduepurelytoelectronic excitations, corre- sponding tothecasewherethevibrational modeis frozen. Anexpression forao(co)canbewritten down simply: 10036 Z.Y.ZHANG ANDDAVIDC.LANGRETH 39 TABLEI.Predicted isotopeeffectsforaweakbreakdown ofadiabaticity. Mode Asymmetry Linewidth Strength 2Pea' M—I/2Effective charge ao(co)=—g(i~—r.Ei Ij)y,;,(~)(jl—r'Eili&.(3.19) Towritedownanexpression fortheimaginary part,we assumeImp,«Reg„so that Imp,= Im(g, )1 1 1—I'X, Thenwehave Imao( co)=—g(i ~@"(r)~j)(3.20) XI',;(co)(j~C&"(r')i),(3.21) where4"isproportional tothescreened versionofthe external potential: Eo+(r)=—rEi,Eobeingthecon- stantmagnitude ofthefield. Usingagaintheproperty ofthesummation ofthetype defined in(3.7),weseethatthebackground absorption is linearincoforaOatspectral densityofelectronic excita- tions.Therelative magnitudes ofthebackground ab- sorption andthevibration-involved absorption varywith systems. Although onetendstothinkthatthevibration- assisted absorption shouldbemoredistinctive asoneis calculating Uibrational lineshapes,therearecaseswhere thebackground absorption seemstobeactually much stronger thanthevibrational part.ifso,theinterference between thesetwokindsofabsorption mayproduce an unexpected enhancement attheabsorption peakofthevi- brational mode.Suchacaseissuggested byReutt,Cha- bal,andChristman'forH(1X1)/W(100) andon H(1X1)/Mo(100). Wewillcomebacktothispointin ourdiscussion section. B.Theovertone ofawagvibration Nowweconsider thefirstovertone ofadsorbate vibra- tionsparalleltothesurface. Atlowtemperature, this represents atransition fromthegroundtothesecondex- citedstateoftheoscillator. Weconsider thecasewhere theexternal disturbance buildsupadominant electric fieldintheperpendicular zdirection inthesurfaceregion relevant forexciting thevibration, sothatonlyca~willbe calculated. Similartothecaseofthefundamental dis- cussedabove,thetheorydeveloped belowcanbeequally appliedtothecalculation of+~I,nomatterwhether the overtone vibration isparallelorperpendicular tothesur- face.Ofcourse, tomeetsymmetry requirements, different situations willinvokediferent selection rules. Thescreening willalsobedi6'erent fordi8'erent situa- tions.Consideration ofex~Ibecomes important ifaspecial experimental technique, suchasshinings-polarized radia-tiononthesurface, produces adominant electric-field distribution inthe ~~direction, or,alternatively, ifoneis inaregioninsidethesurfacescreening layerwherethe ~~ fieldismuchlarger. Beforewespecialize toanovertone, westressonegen- eralaspectofthecontributions toa~fromvibrations paralleltothesurface. Withinthepictureofdipolecou- pling,onlythosevibrations intheparallel direction whichpossessadynamic dipolemoment alongzcancon- tribute. Therefore, withintheharmonic approximation forthebareionicpotential, thedynamic dipolemoment associated withthestrictlyparallel vibration canonlybe provided bythemotionofelectrons screening theion. Thismeansthatp„;=0,because(n„~z~0) isstrictlyzero ifthevibrations denoted by~n„)areharmonic, withno mixtureofdisplacement alongz.Ontheotherhand,p mightnotbeequaltozero.Onesuchexample where p,;=0butp,&0isthe0—Ofundamental stretch vi- brationof02chemisorbed onPt(111) withthe02lying paralleltothesurface. Asthemolecule vibrates, the shortening andelongating motionsofthemolecule away fromitsequilibrium bondlengthdisturb theelectron cloudsurrounding theadsorbate di6'erently, resul:ing ina netdynamic dipolemoment alongzoriginating fromthe electronic response. Suchasystem wasanalyzed quite successfully byPersson usingtheelectron-hole —pair damping mechanism combined withtheinclusion ofdi- polecoupling. %'ecandiscusstheexcitation ofaparallel modebased ontheanharmonicity inthevibrational potential. EfFectively, thisanharmonicity arisesfortwodi5'erent reasons. Thefirstistheusualanharmonic termsinthe adiabatic ionicpotential, suchasaterm~ZX,which wouldgiveacoupling between, say,afundamental along zandanovertone alongx.TheefFectofthiskindofcou- plingiscapableofcreating anindirect excitation ofa ~~ modeviathedirectexcitation ofalvibration first;butin general thisisaveryweakchannelfortheexcitation ofa parallel mode,unlessthetwovibrations involved are nearlydegenerate infrequency, aspecialcaseknownas theFermiresonance. Thenovelfeatures fromthisad- ditional interference between twovibrations ofanada- tom,duetoaFermiresonance, willnotbediscussed here. Thesecondcaseofanharmonicity ismoreelectronic in nature,arisingfromthenonadiabatic interaction between theelectrons andtheion.Thatis,incalculating a~, p,„,&0fora ~~vibration meansthatthereisaneff'ective anharmonic z-dependent perturbation inthetotalHamil- tonianfortheionicmotioncontributed bytheelectron- ioninteraction. Sincethereexistelectronic transitions thatcanbeexcitedbythefieldalongz,andhaveenergies 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10037 equalornearlyequaltothevibrational frequency, the effectofthiselectron-contributed anharmonicity canbe quitestrong, yielding astrongabsorption atthe ~~mode frequency ina~. Nowwenarrowourselves downtoaparticular classof parallel vibration, anexampleofwhichisthewagmotion ofHchemisorbed onW(100),''''oron Mo(100),''atsaturation coverage. Forthesesys- tems,thehydrogen atomisinatwofold bridgeposition, givingthesystems reflection symmetry aboutthe(x,z)or (y,z)plane.Toahighaccuracy, suchreAection symme- trycanbeassumed topersistfortherealsystems studied experimentally, eventhough somedisorder (impurities, steps,etc.)onthesurface isunavoidable. Therefore, if theatomvibrates inthefundamental ofthewagorthe asymmetric stretch, thereisnonetdynamical dipolemo- mentperpendicular tothesurface evenduetoelectronic motions: thecontributions top„,frompositivexand negativexcanceleachother,andtheresultantp„,=0. Thissituation istobecontrasted withthatforOzon Pt(111). Ontheotherhand,thereisafinitedipolemo- mentassociated withthefirstovertone ofthewag motion, asthesecond-order derivative oftheelectron- ioninteraction potential isneededtoexcitethisovertone, whichsymmetrizes theeffective excitation matrix with respecttox.Theinvolvement ofanovertone makesthe example moreinteresting toconsider, asitseemstobe thefirsttheoretical discussion onthisaspectofanover- tone. Foranovertone vibration, theionicdisplacement coor- dinate,unlikethatofaproperfundamental, hasnonet coupling toanelectric fieldinthedipoleapproximation withintheharmonic approximation, nomatterwhatthe fielddirection is.Thismeansthatquantities likea;,.and a;,areidentically zero,eveninthecasewherethevibra- tionandthefieldarealongthesamedirection. Theonly possibility ofexcitation leftisfrom0,,„which foral fieldcanbefinite. Tocalculate thevibrational lineshapeforsuchan overtone derivable fromthepolarizability a„,,(co),we notethat,intheharmonic approximation, theexcitation(a) (b) FIG.4.I',a)Thelowest-order diagram whichgivestheexcita- tionoftheovertone. (b)Thefullynormalized diagram, in whichtheovertone isbroadened. 2Q„D„(co)= Q~y+lco'Vx where(3.22)ofthefirstwagovertone isequivalent totheexcitation of twoidentical vibrational quanta, eachwithafundamen- talwagfrequency0p.Ofcourse,higher-order anhar- monicinteractions notincluded herewouldchangethe excitation energyslightly awayfrom2Qp.Thelowest- orderdiagram considered forthisprocess isshownby Fig.4(a),whichbyinspection produces anabsorption peakedat2Qpofinfinitesimal width. Where line- broadening processes areconcerned, animportant point tonoticeisthat,although neitherofthetwoquanta(each ofenergyQ„o)canbeexcited individually bythefieldEi andtheymustbecreatedtogether, onceexcited,eachone ofthemcandecayviatheexcitation ofelectron-hole pairs,aslongasthereexistelectronic stateswithin c~+Q p,whosesymmetry property allowsthemtobe strongly perturbed bythewagging vibrations ofthefun- damental. ForH/W(100) atsaturation coverage, these statesarejustavailable, asanalyzed byChabal.''A fulldiagram fortheexcitation oftheovertone, including decayprocesses, isgivenbyFig.4(b),whereadouble- dashed lineisidentically givenbyFig.2,butnow represents thenormalized propagator D„(co)forthewag mode.FromSec.IIIAwecanimmediately writedown anexpression forD„(co): 20p s)mxj[V,';(R—r)]Diy,;(rr)i[V,';(R—r)]rjl. (3.23) Thisisidentical informwith(3.4),exceptforthedifferent partialderivatives. Needless tosay,theelectronic states ~i) and ~j)entering in(3.23)and(3.4)arealsodifferent, bothinsymmetry properties andinenergy. Ontheotherhand, thescreening totheelectron-ion interaction isexpected tobesimilarinbothcases.Discussions ontheisotopedepen- denceofydefined in(3.23)willbepostponed untilweobtainanexpression forapseudopropagator representing the overtone. Nextwecalculate thediagram shownbyFig.4(b),whichcanbewrittenas a,(~):a...(~)=—]t],,(~)1(~—)]M,(~), wherep,isgiveninanalogyto(3.14),(3.24) (3.25) 10038 Z.Y.ZHANG ANDDAVEDC.LANGRETH 39 andI(co)atfinitetemperature isasummation overcom- plexfrequencies:wheren(co)=(e~" —1)'isthePlanckfunction. When T~O, n(co)becomes aunitstepfunction, andweobtain, fortheimaginary part,I(ice„)=—gD„(ice„)D (iso„—iso). (3.26) B(co') D„(iso)=~ ~27Tlco—co(3.27) wehave,aftersumming overv,In(3.24),acombinational factor2hasbeenassociated withthediagram (wesuppress anypossible spinindices, astheyareirrelevant toanydynamic process described here),andco,=2vmlP, vbeinganinteger. I3'=Tisthe temperature. Thesummation in(3.26)canbeevaluated byastan- dardmethod. Expressing D(ice)intermsofthespec- traldensityB(co),ImI(co)=—fdao'B(co')B„(n) co—').(3.29) B(co)=—21mD„(co)Nextwerecallthatweareinterested intherangeofco closeto20,„.Atco-20,„,B(co')ispeakedat+0„„, whileB(coco')is—peakedat0„,and30,„.Therefore, inthepresentcaseofy«Q„,toaverygoodapproxi- mation wecanextendtherangeofintegration in(3.29) overtheentire co'axis.Now,onceagainusingthefact that 2A, Ct)QXq+lQ)P2Q„, co0„lQ)p(3.30) X[n(co")n(——co')] CO COM+E'g(3.28)substituting (3.30)into(3.29),andcarrying outthecon- tourintegral, weobtain 8Q„ (co—4'0„„—5'y„+40„y„)+iso(2y )(2'—40„„)(3.31) Wemayalsowritedownanexpression forthecorre- sponding realpartofI(co).ButReI(co)canbeobtained moreeasilyifwesimplify(3.31)further. First,wenotice that,whensetequaltozero,therealpartofthedenomi- natorinthelargeparentheses definesaresonant frequen- cy: n'„=4@'„+—"y'Xr 2X (3.32) =4Q„, (3.33) whereineachstepwehaveusedthecondition that y«Q„.Inaddition, forconottoofarawayfromthe resonant valueof20„,wecansimplify (3.31)as ro(2y„)—ImI(co)=2(20 „) [co—(20„„)]+co(2y„) (3.34) Following (3.34),wecanimmediately writedownanex- pression forReI(co),andfurtherforI(co): 2(20„)I(co)=zco—(20„„)+iso(2y„)(3.35) ThefinalformofEq.(3.35),incomparison withEq. (3.22),suggests thatwecandefineapproximately apseu- dopropagator fortheovertone vibration, givenbyD'(co):I(co),which ha—saresonant frequency0„—=20„.Inaddition, Eq.(3.35)alsoindicates thattheI lineshapefortheovertone, obtained byconvoluting a lineshapeofwidthywithanother oneofidentical form, willhaveatotalwidthy=2@„,whereyisgivenbefore byEq.(3.23). Afterobtaining anexpression forthelinewidth ofthe overtone, wecannoweasilydiscussitsisotope depen- dence.From(3.23),weseethaty„contains asummation representable againbythegeneralsummationJ(co)intro- ducedinSec.IIIA.Therefore, thefrequency cuinthe denominator of(3.23)willcancelwiththefrequency co appearing inImJ(co), whichhastheformImJ(co)=bco atsmallco,whereagaintheconstant bhasnoisotope dependence. Thismeansthatthelinewidth y„forthe wagfundamental, likey,forthestretch fundamental, hasanM'isotope dependence. Thisresulthasnow beenshowntobevalidevenforthefirstovertone ofthe wagmode;thatis,wealsohavey„*~M Nowwegobacktothepolarizability defined in(3.24) todiscussotherquantities. Wefirstexpressp„,(co)given by(3.25)inafamiliar way.Asforthestretchfundamen- tal,weseparatep,(co)intoitsrealandimaginary parts: p«=p»+ipx2,anddefinekx=pxz/px &.Equation (3.24)thenbecomes 2(20, )a„(co)=—p,,(1+iA„), co—(20„)+ice(2y ) (3.36) whichisidentical informto(3.17)fora,(co),showing 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10039 thattheovertone alsohasanasymmetric lineshapeof theFanotype. Theisotopedependences oftherestoftheparameters canbeobtained similarly fortheovertone excitation. From(3.24),(3.25),and(3.36),andusingonceagainthe frequency dependence ofthesummation J(co),weagain have A,~M',asfortheasymmetry factorofthe stretchfundamental. Thisgeneralisotopedependence in theasymmetry factorcaninfactbeappliedtoanyexcita- tion,notjustafundamental orthefirstovertone, aslong IasthesystemhasaRatelectron densityofstateswithin cF+0,.Asimilarobservation canalsobemadeforthe isotopedependence ofthelinewidth, wherethecommon exponent is—1.Incontrast, theexcitation strength for anovertone hasastronger isotopedependence thanfora fundamental: wehave, from(3.25),p,ir——p„,cc(MQ„)~M'.Ifwestilldefineaneffective charge e„*fortheovertone wagbyp„,s=e„*/[2M(2Q„„)]' thenwehavetheexpression fore„*: ,zzReg(irHJj)j, ;(ru)j[V;(R—r'i]0il, (3.37) showing thate*hasaweakisotope dependence:e*~M'.Forabettercomparison ofisotope effects between thetwomodes,seeTableI. Sofarwehaveconsidered onlytheleading diagram Fig.4(b)whichcontributes toa(co).Therearealso higher-order contributing diagrams, corresponding tothe casewherethetwovibrational quantadecaysimultane- ously.Onesuchdiagram isshowninFig.5.Itiseasyto showthattheeffectofincluding thesehigher-order dia- gramsistogiveanadditional frequency shift,andto broadeny*further byanamount equalto5y„*,where 5y/y„*~M',showing thatthecontribution ishigher inorderinexpansion ofelectron massoverionmass. Thesetermsarecompletely negligible whenm,«Mas assumed here. IV.STRQNG BREAKDOWN ClpADIABATICITY InSec.III,wehavederived formulas forthepolariza- bility a(co)[andthelineshape given by L(co)=—21ma(co)] ofanadsorbate-substrate systemin thepresenceofadsorbate vibrations. Theseformulas are validwithcertaingenerality, andshouldbeapplicable to avarietyofsystems. Whenwediscussed theisotope dependences oftheparameters characterizing theline shape,wespecialized tosystems whichhada+atdensity ofstatesoftheelectrons aroundtheFermilevel.Thatis, wehavestudiedtheproblem intheregimeofweakbreak- downofadiabaticity. Inthissection weconsider theoth- erregime, whichcorresponds tothecasewherethe effective spectral densityvariesrapidlynearcF.Arapid- lychanging p(e)nearE~invalidates thesmall-co expan- sionusedinevaluating thegeneral summation J(co) defined by(3.7).Therefore, newresultsbearing onthep,(E)=p(e~)[1+cEF5(EEF+6)],—(4.1) whereasubscript sisusedtoindicate thestrong- breakdown limit.Thefirstterminthesquarebrackets stillrepresents theAatpartinthespectral density, while thesecondrepresents thesharply peakedpartwhichlies ontopofit;cisadimensionless constant givingthe weightofthepeak,and6&0isapositive parameter de6ning thepositionofthepeak.Asitturnsout,both termsarerequired forastrongbreakdown tobeopera- tional.Thefiatpartin(4.1)provides electrons ifthepeak isabovetheFermilevel,andreceives electrons otherwise. Inaddition, asharppeakeitherbeloworabovetheFermi levelisnecessary tocauseastrongbreakdown. Fora givensystem, onlyonesignneedstobekeptinfrontofA. Itcanbeshownthattheresultsobtained usingEq.(4.1) wouldbethesameforarealistic situation withapeakof finitewidthI,aslongasI«6&(co—I)-(0—I). Separation oftheeffective spectral densityintoaAat partandasharply peakedpartsuggests thatasimilar separation canalsobemadefortheelectronic partofthe chargeresponse functiony„but byadifferent criterion, whichistobeshownbelow.Firstweseparate theirre- duciblepartoftheelectronic response function,g„into XeXe,g+Xe,sI isotopeeffectsshouldbeexpected. Todemonstrate howthingschange, wetaketheex- tremecasewherep(c,)hasasharppeak,whichiscloser toc~than0,andapproximate thispeakbya5function. Theexistence ofsuchasharppeakinsomesystems is conceivable, andmayoriginate fromanadsorbate- contributed surfacestatebroadened byahopping term between theelectron inthislevelandtheconduction electrons, asintheNewns-Anderson model. Weformally writethespectral densityp(e)as FIG.5.Oneofthehigher-order diagrams whichisnegligible whencompared withthediagramofFig.4(b).In(4.2)weusesubscripts gandstoindicate diff'erent con- tributions totheresponse function fromdifferent elec- tronicstates.Byswemeanthecontribution fromsome specialelectronic states;theseelectronic statesarecap- ableofmediating acoupling between theexternal field andthevibrational modebeingstated,andarethuscon- tainedintheeff'ective spectral densityp,(c,)in(4.1).That 10040 Z.Y.ZHANG ANDDAVIDC.LANGRETH 39 is,g,,expresses thecontribution fromvirtualorreal transitions between statesinthesharppeakandstatesin thefIatpart,orbetween statesintheAatpartonly.In contrast, thefirstterm,y,,represents thecontribution fromallothergeneral, virtual,orrealtransitions between statesintheconduction bandorothersurface-state bands.Noneofthelattertransitions cancouplebothto theexternal fieldandtothevibrational mode;therefore, electronic statescontributing toy,areexcluded from theeflectiue spectral densityp,(s),definedfortheexcita- tionofthevibration. Ontheotherhand,theseelectronic statescontributing tog,aregenerally themaincontri- butors inscreening theexternal field.Ifweassume))y,„wehave,from(3.6)and(4.2),andtofirstor- derin/,„ 1+e+e,g r7e,g1 1 1—y,V'1—Vyg+e,s (4.3) Equation (4.3)showsthatthetotalelectronic response hasalsobeenseparated intotwoparts.Laterinthissec- tionwhenanexpression forthedynamic dipolemoment ofthevibration isdesired, onlythecontribution fromthe secondtermin(4.3)needstobeconsidered. Thefirst termgivesnocontribution atall. Toobtainthecorresponding changes inisotopedepen- dencesofthelineshape,J(co)mustbecalculated using thenewdensityofstatesdefined by(4.1).Theevaluation ofthesummation J(co)inSec.IIIuptothefirstlineof Eq.(3.10)isstillvalidinthepresentcase.Beyond this point,wearefacedwiththeproblem that,ifthepeakis soclosetoE~thatb,iscomparable toorlowerthanP thenthermal excitation ofelectrons cannotbeneglected. Becauseoftherestriction fromthederivativelike feature oftheFermidistribution function, aweak,butnoticeable, temperature dependence inthelineshapewouldbeex- pected. Hereweexclude thispossibility, andrestrictthe parameter6tobesuchthatP'«b&Q.Withthisre- striction wehave,fromthelastequalityof(3.10)and (4.1), ImJ,(ro)=bIco+cEF[+6(+co+ b,)+6(+co+b,)]I, (4.4) whereb=mF(EF,e~)—[p(EF)] isaconstant, which,like itsanalogue bdefined inthelimitofweakbreakdown of adiabaticity, isindependent oftheionicmass.Thecorre- sponding realpartis b EDcoReJ,(ro)=—2E+coinED+co ED2~2 +eEln (4.5) CO Onceagain,thediscussion abouttheaccuracy ofthefirst termintheparentheses in(3.12)shouldalsoapplyhere forthefirsttermintheparentheses in(4.5). Nextwediscusstheimplications ofthesetwoexpres- sions.Interesting information iscontained inboththe realandtheimaginary partsofJ,(oi).Firstwelookat therealpart.Whencompared withthefirstterm,the second terminthelargeparentheses in(4.5)canbePx,ea +„(EF,E~)[P(E)]2 xr ED Q)X2ED+ccFln22(4.6) where ~a2(4.7) with ~i)~~AE). Thefirstpointtonoticein(4.6)isthatthesecondterm inthelargeparentheses givesafrequency-dependent con- tribution. Inaddition, thissecond termispositive definite andcomparable tothefirstterm,producing a largeenhancement inthemagnitude ofdynamic dipole momentifmeasured bythecontribution purelyfromthe Aatpart.Ontheotherhand,ifthepositionofthesharp peakissuchthat6«co=0, wehave ED Q7 EDln22=2ln &21n (4.8) Thetermproportional to21n(ED/b, ),obtained from In~(ED—co)/(b,—co)~bysettinga~=0,corresponds to anequivalent contribution, fromapeakcontained inthe spectral densityofagivensystem, intypical self- consistent all-electron, butfrozenphonon, calculations. - Equation (4.8)showsthatalthough astrongbreakdown ofadiabaticity enhances theexcitation strength com- paredwiththeresults intheweak-breakdown limit (whereonlythecontribution fromthefiatpartisinclud- ed),itdoesnothelptoexplainpossible discrepancies be- tweenthelargeexcitation strength measured experimen- tally,andamuchsmallerstrength calculated numerically usingadiabatic procedures (withthecontribution from thepeakalsoincluded). Itmaywellbethathigher- ordertermsinexpansion ofy,ing,,areimportant.dropped againasintheweak-breakdown case,whilethe thirdtermcaneasilybecome comparable tothefirstterm inthepresentcaseevenforaverysmallvalueofc(such asc=10'),andmustbekept.Therefore, eventhough therealpartdoesnotgivenoticeable changes inisotope effectssincethereisonlyalogarithmically frequency- dependent terminit,themagnitude ofReJ,(co)willbe modified awayfromthatobtained usingaAatdensityof states. Assuming theAatpartremains thesame,the modification isalwaysanenhancement. Physically, theincrease inthemagnitude ofReJ,(co) impliesthat(a)alargerfrequency shiftforthenormal modeunderconsideration willbeobserved, and(b)a stronger excitation strength willbeexpected. Thisistrue because boththeself-energy ~(co)ofthephononpropaga- torandtheeffective dipolemoment operator containa summation ofthetypegivenbyJ,(co).Todemonstrate thispointmoreexplicitly, wereferthelattercasetothe wagovertone, andhave,fromEqs.(3.7),(3.9),(3.37), (4.3),and(4.5),andusing therelations 4(1—y,V)'=4"and(1—Vy,)'V,,=V,';, 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10041 Thesewouldtendtoweaken thescreening andhence strengthen theeffective coupling overwhatitisinthe adiabatic approximation. Wewillcomebacktothis pointwhenwemakecontact withexperiment. Fortheimaginary part,itisquitepossible thatthe secondterminthelargeparentheses of(4.4)couldstill giveacontribution comparable tothefirstterm,sincefor everysystemofinterestczisnormally about 1orderof magnitude largerthanQ.Thissituation canexistevenif thesharppeakcarriesonlyasmallfractionofthetotal spectral density (i.e.,c«1).Insuchasituation, represented bycu-Q=ccz, thecorresponding frequency (andthusisotope) dependence ofaline-shape parameter willbechanged fromapowerlawtoamorecomplicated polynomial form,aslongasthisline-shape parameter is proportional totheimaginary partgivenin(4.4),suchas thelinewidth ortheasymmetry factor.If0«ccFfora givensystem, wecanevendropthefirsttermin(4.4),and haveanImJ,whichcontains nocudependence atall;for suchacasewewillagainhavesimplepower-law isotope dependences fortheline-shape parameters, butnewex- ponents aretobeexpected. Wenowspecialize ourresultstovariousvibrational ex- citations ofasystem whosespectral density hasthe correctpeakdesiredbothinposition(5&Ll)andalsoin weight(Q«cEF). First,forthestretchvibration with fundamental frequency alongz,onlyoneenergybound- arylimiting thepositionofstrongpeakinspectral densi- tycomesintoplay,thatistheunperturbed 0,0,andwe musthavetherestriction6&0,o.Fromtheanalyses givenabove,wecaneasilyconclude thattheisotopeeffect inboththelinewidth y,andtheasymmetry factor A,, shouldbeweakerherebyafactorofM'relativeto theweak-breakdown limit,becauseImJ(co) contains one morefactorofcothanImJ,(co).Noticealsothattheex- pression forthebackground absorption duepurelyto electronic excitations, another quantity containing asum- mationofthetypeJ,(co),becomes alargerconstant in thepresentcase,whileitwassmaller andlinearincoin thelimitofweakbreakdown ofadiabaticity considered before. Ontheotherhand,thereisnochange intheiso- topedependence oftheexcitation strength, whichispro- videdbytherealpartofJ,(co). Nextwespecialize tothecaseoftheovertone ofthe wagmode.Inordertoapplytheformalism developed heretotheovertone, onefacestwopossibilities, referring tothepositionofthesharppeak.Thefirstisdefined by6&Qxo,andthesecond isdefined byQ„z&4butwith 5&2A„O. WerecallthatQomaybehave asanenergy boundary tothepositionofpeakinspectral densitybe- causeeachofthetwopropagators oftheovertone vibra- tiondecaysindividually. If4&Ao,thechanges iniso-topeeffectsofthelinewidth andtheasymmetry factor shouldbemadeinexactlythesamewayasforthefunda- mentalofthezstretchmode,thatis,hfactorM'isto betakenawayfromtheasymmetry factorandthe linewidth. Ifb,&20„0, butb.)00,thenonlytheasym- metryfactor A,„oischanged fromM'toM,whilethe linewidth yoremains thesameasfortheweak- breakdown case.Inaddition, thepurelyelectronic back- groundabsorption becomes afrequency-independent con- stant,relatively largerthanitsweak-breakdown analo- gue,aslongasthelocationofthispeaksatisfies thecon- dition(b,&20,0)andthepeakhassufficient weightthat fLo«ccF. Finally, likethecaseof-thefundamental, the excitation strength oftheovertone haspractically no change inisotope effectcompared withtheweak- breakdown case.Alltheseisotopeeffectsarecollected in TableII. V.DISCUSSION ANDCOMPARISON WITHEXPERIMENT Wedevotethissectiontoadiscussion ofthetheoretical resultsderived aboveandinthepreceding paper, as theyrelatetorecentexperimental investigations. Inaseriesofpapers,"''''''Chabalandhis collaborators measured andinterpreted thesurface in- fraredreAectance spectraforhydrogens adsorbed onthe (100)surfacesofWandMo.Herewefocusonlyonthe H(1X1) phaseatthesaturation coverage, asthisisthe phasewhereastrongderivativelike asymmetric vibra- tionalpeakisobserved forbothsystems, andtheFano line-shape formula foranelectron-hole —pairmechanism seemstoapply. Atthesaturation coverage, theH(1X1)/W(100) and theH(1X1)/Mo(100) surfaces exhibit twoinfrared ab- sorptiori bandsinthe800—4000-cm regionstudied. Thesetwopeakswereidentified asbeingduetothesym- metricstretchvibration (v,)andthefirstovertone ofthe wagmotion(2v2).Thefirstpeakcanbewellrepresented byaLorentzian form.Forthesecondpeak,theasym- metricFanoline-shape formula givesaverygoodfitto theexperimental data,'suggesting thattheelectron- hole—pairenergy relaxation mechanism isoperative. TableIinRef.21summarizes thesituation quantitative- ly. Forthesymmetric stretchmode,energyrelaxation via generation ofelectron-hole pairsisveryunlikelytobethe dominant broadening mechanism, asinitially concluded byChabal.'Theabsenceofline-shape asymmetry was usedasthestrongest evidence toruleoutthispossibility. Otherfacts,suchasthenarrowing oftheHlinewidth upon0dilution, thestrongtemperature dependence in itslinewidth,'andthelackofelectronic statescapable TABLEII.Predicted isotopeeffectsforastrongbreakdown ofadiabaticity (assuming6&Q„o). Mode Asymmetry Linewidth Strength 2PeaEffective charge Vl 2V2 10042 Z.Y.ZHANG ANDDAVIDC.LANGRETH 39 ofmediating theenergytransfer viatheirstrongcoupling tov&,alltendtoargueagainst theoperation ofan electron-hole —pairmechanism asthedominant onefor thebroadening ofthismode. Ontheotherhand,thereshouldbelittledoubtthatthe sharpasymmetric vibrational peakatthefrequency 2v2is duetoelectron-hole —pairenergyrelaxation. Anyenergy relaxation viaphonon emission, orphaserelaxation byphonon modes''orelectron-hole pairs, wouldbe strongly temperature dependent, whileexperimentally ei- therveryweakornonexistent temperature dependence wasobserved fortheline-shape parameters. Inaddition, noneoftheseprocesses willproduce astrongasymmetry inthelineshape. Broadening bydisorder, whosetem- perature dependence isstillinquestion,'mayinprinci- pleproduce anasymmetric lineshape.'However, as pointed outbyChabal,'thecharacteristic Fanoline shape,witha"negative" partdigging intothecontinu- um,unambiguously rulesoutsuchadisorder-induced broadening asthedominant mechanism fortheabsorp- tionat2vz. Important information canalsobeprovided bythesign oftheasymmetry factork.Werecallthat A,isdefinedby, following Eq.(3.16),p«,=p;+p,—=p,+ipz=p, tt(1+A,), where A,=@2/p,.Sincep;contains noimaginary part, thenp,~:—p,=p;+Rep„ from which follows A,=(Imp,/Rep,)[Re@,/(p;+Rep,)].The ratio Imp,/Rep,~ImJ/ReJ ispositive definite, asseenfrom (3.11)and(3.12).Nowassume wehaveasystemwhere thedirectcoupling oftheexternal fieldtotheiongives theleadingcontribution, suchthatthesignofp,+Rep, isthesameasthesignofp;.Then A.ispositiveornega- tive,depending onwhether theextraelectronic contribu- tiontothedynamic dipolemoment isparallelorantipar- alleltothecontribution oftheionicpart.Ontheother hand,ifthedirectcoupling totheionisnegligible, asfor thecaseof0—0stretchvibration onPt(111), orthe 2v2modeofH(1X1)/W(100) andH(1X1)/Mo(100), then A,=Imp,/Rep„which isnowpositive definite,cor- responding tothecasewherethereisalwaysalongtail onthelowerfrequency sideoftheabsorption peak. Indeed, positive valuesofA,wereobserved forthe2vz modeofH(lX1)onbothW(100) andMo(100), asalso expected fromourtheorydeveloped hereforasinglead- sorbate. Nextweanalyze theisotope dependence fortheline-shapeparameters intheirexperiment. Dataforthispur- poseareavailable onlyontheW(100)surface,''which werereproduced inTableIII.Thelastthreerowscom- paretheindicesofthepowerlawdefined byMforiso- topedependence ofagivenparameter. Noticethat,since thesignal-to-noise ratioisquitelowwhentheadsorbate isdeuterium,'thedataontheisotopedependence ofthe intensity seemtobelessaccurate. Butthekeymessage presented, thattheisotope effectsmeasured aremuch weaker thanthosegiveninTableIintheweak- breakdown limit,andareinconsiderably betteragree- mentwiththepredictions shown inTableIIforthe strong-breakdown limit,shouldbereliable. Inparticular, excellent agreement isobtained intheisotopedependence oftheasymmetry factor.Theasymmetry isuniquetothe electron-hole —pairmechanism: thevalueoftheasyrn- metryfactorproduced byanelectron-hole —pairmecha- nismremains thesameevenifsomeotherbroadening process producing asymmetric lineshapecoexists. In contrast, otherparameters willbestrongly infiuenced by eitherthecoexistence ofotherbroadening mechanisms (aswiththelinewidth), ortheuseofanother phaseasa reference (aswiththestrength). Thismeansthatthebe- havioroftheasymmetry factoristhemostinformative abouttheelectron-hole —pairmechanism, andcon- clusions basedonitwouldbethemostreliable. There- fore,theexcellent agreement intheisotopedependence of theasymmetry factorbetween theexperiment andthe prediction inthelimitofstrongbreakdown ofadiabatici- tysuggests theexistence ofasharppeakintheeffective spectral densityofthesystemneartheFermilevel.We emphasize thattheagreement madeherefortheovertone ofthewagmodedoesnotdependonthedetailsofhow themodeisexcited.Forexample, thesameconclusion is madenomatterwhether thefieldexciting theovertone is parallelorperpendicular tothesurface. Thereareotherquestions remaining tobeanswered. Inparticular, wewishtounderstand (a)whereasharp peakinspectral density mightoriginate, (b)howthe electronic statesbelonging toasharppeakwouldmediate theovertone excitation at2vz,and(c)bywhichfieldcom- ponent thesestatesthemselves areexcited. Theseare closelyrelatedandimportant questions. Whatweintend todonextistobrieflyreviewvarious possibilities pro- posedpreviously whichmaygiveananswertooneques- tionoranother, andwherever possible, suggest newpos- TABLEIII.Isotopeetfectsforthewagovertone ofH/W(100). Quantities inthefirsttworowsare KtakenfromRef.21.Thethirdrowisobtained fromthefirsttwoandthedefinitions, e.g.,X~M"P'. ThelasttworowsaretakenfromTablesIandII,respectively. Frequency0,(cm') 1270 915Asymmetry 0.44 0.46Linewidth y(cm') 26 22Strength 2Pea 0.438 0.142 +expt Kq—0.473—0.5—0.5Exponent ofM 0.064 0.0—0.5—0.241—0.5—1.0—1.625—1.0—1.0 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10043 sibilities andfutureinvestigations. Inthesurfaceregion,sincep-polarized external radia- tionatgrazing incidence buildsupanelectric field predominately perpendicular tothesurface withmuch smallerparallel fieldcomponent,'itismostnaturalto lookforthepossibility ofexciting theovertone bythe perpendicular fieldE~first,asfirstdonebyChabalfor H/W(100).''Chabalalsosuggested thattheelectronic stateswhichcouplestrongly tothewagovertone aresur- facestatesofdsymmetry, basedonresultsfromfirst- principles calculations ofRichter andWilkins. Morerecently, Reutt,Chabal, andChristman'sug- gestedanewpossibility whicharosefromthecomparison oftheirresultswiththosefromelectron-energy-loss spec- troscopy (EELS) measurements onthesamesystems. Heretheysuggested thatthecoupling between theradia- tionfieldandthewagovertone isagainmediated byelec- tronsinsurfacestatesofdsymmetry, butthefieldcom- ponent istheoneparalleltothesurface. Thisnewpossi- bilityisveryintriguing, because itwouldnotonlypro- videananswertoessentially eachofthequestions listed above,butitwouldalsoresolvethepuzzleregarding a difference betweenEELSandirfindings. Inthepast,an extensive studyofthevibrational properties ofHon, W(100)atsaturation coverage hadbeencarriedoutusing EELS.''Although therehadbeensomecontro- versiesaboutmodeassignments between different experi- mental groups,''atleastoneareaofagreement seemstohaveemerged: thisistheabsenceofthefirst overtone ofthewagmotion (2v2)inallspecularEELS measurements.'Thisovertone excitation wasalsoab- sentinspecularEELSforHonMo.Sincetheir,which strictlysatisfies dipoleselection rules,andthedipolepart ofEELS,whichcorresponds toexcitations bythefieldE~,areexpected toyieldsimilarspectra, theabsenceof 2v2indipoleEELS,anditsstrongabsorption inirmea- surements, ledReutt,Chabal, andChristman tosuggest thattheelectronic statesresponsible fortheabsorption at the2v2frequency couldonlybestrongly coupledtothe ~~ field,nottothelfield.Ifthisisthecase,theabsenceof absorption at2v2wouldbecome thenquitenatural, asfor dipoleEELStheresultant fieldisperpendicular tothe surfaceandnoparallel fieldcomponent exists. Thesur- facestates,possible ofd„symmetry, werefurther identified tobeabout0.15eVbelowcz,witharelatively narrow width (=0.25eV).Theyalsosuggested thatelec- tronicexcitations areintraband, sincetheovertone ener- gyisgenerally toosmallforelectronic interband transi- tions. Suppose thewagovertone contributes toa~~,asReutt, Chabal, andChristman havesuggested. Thatwould meanthatwehaveasystemwhosereAection symmetry waslostduetothepresence ofdisorder, steps,etc.,or duetothepresence ofthermal phonons. Suchdisorder maycontribute negligibly tothelinewidth, butitspres- enceisnecessary, orelsetheovertone excitation isforbid- denfromthefollowing symmetry arguments. Consider anelectric fieldparalleltothesurface, chosentobealong x,whosespatialdependence isneglected. Thediagram fortheexcitation ofthe2v2modebythisparallel fieldis showninFig.4(b).Nowweshowthattheeff'ective ver- texforexciting the2v2mode,mediated bythedirectcou- plingofE~~toelectronic states,indicated bythedotted blockinFig.4(b),andexpressed byamatrixelementof thedipolemoment, p„'„iszeroifthesystempossesses reAection symmetry inthexyplane.Inanalogy tothe caseofaperpendicular field,wehave[compare with (3.25)] (5.1) wherep~~istheparallel component ofthepolarization operator fortheelectrons, whichisdefined inRef.22. Underanoperation ofareAection alongtheyzplane,the innerproductofp~~withE~~behaves liketheelectronic coordinate x.Therefore, inorderfor(i~— p~~E~~~j)tobe nonzero, ineachpairtheinitialandfinalstates ~i)and ~j)musthaveopposite parities inthexyplane.Onthe otherhand,itiseasytoseethat8/3X[V;(R—r')]o behaves likethesumofevenpowersofx'.Thismatrix element isidentically zerowhensandwiched inbetween theabovepairofelectronic states ~i)and ~j),showing thatthe2v2modecannotbeexcited byaparallel field. Therefore, weeitherhaveasystem whichhasno reAection symmetry inthexyplane,orelsethewagover- tonecannotcontribute toa~~.Iftheformercaseistrue, eitherintraband transition withinthesurface bandvia free-carrier scattering orinterband transition havetoex- ist.Insuchcasesthetheorydeveloped beforeremains essentially unchanged. Inparticular, isotope depen-dencesidentical tothoseobtained beforewouldbeob- tainedagainforeitherthestretch fundamental orthe wagovertone ifthesystem isagaininthestrong- breakdown limit.Weemphasize thatfortheexcitation of thewagovertone byafieldcomponent alongz,the difficulty relatedtosymmetry arguments doesnotappear, becausethereisnoreAection symmetry alongthezaxis. Sofareverything seemsallrightforthewagovertone tocontribute toaI~,withtheassumption oftheexistence ofdisorder onthesurfaces. Buttherearealsosomeun- certainties. Whatkindofdisorder isit?Atwhatconcen- trationandwithwhatspatialdistribution canithelpto excitethewagovertone butnottobroaden it significantly?'''IfthereAection symmetry inthexy planeislost,thereshouldalsobeafiniteeffective dipole moment associated withthefundamental oftheasym- metricstretch(v3),contributing toai.Furthermore, as wi11beshownlater,afiniteeffective dipolemoment isex- pectedforv3,contributing toa~~,evenifthereAection 10044 Z.Y.ZHANG ANDDAVIDC.LANGRETH 39 symmetry inthexyplaneisnotlost.Thenwhyisv3not detected intheexperiment? Isittooweaktobe identified? Another difficult question concerns theexcitation strength at2v2.itisnotweakcompared withthe strength atvi,andismuchstronger thancommonly ex- pectedfromatypicalovertone excitation. Toshowthe absorption at2vzandvi,acommon absorption curveob- tainedexperimentally fromadifferent phaseofthe H/W(100) system(notatthesaturation coverage) was usedasareference.'Therefore, theexperimentally mea- suredabsorption ofcomparable magnitude atthetwofre- quencies indicates thattheproductoftheeffective dy- namicdipolemoment foronevibration withthemagni- tudeofthecorresponding fieldexciting itshouldbecom- parabletotheproductofthetwoquantities fortheother vibration. Sincethedynamic dipolemoment associated withthefundamental islargerthanthatassociated with theovertone,'thenwewouldrequirethattheE~~exciting theovertone shouldnotbesmallerthanE~exciting the fundamental. Inthislastrequirement liesthemain difficulty. Asstressed inthelastsectionofRef.22,and alsoshownby(2.1)inthispaper,theparallelcontribution na~~isreduced inthescreening layerbyalargedielectric constant e,whilena~isnot.Therefore, inmostcases no,'~shouldgivethedominant contribution, evenifthere maybeonlyaverysmallportionofelectronic statesin thespectral density whichcancouple withboththe external radiation andthevibrational modeundercon- sideration. Onlydeepinthesubstrate, pastthescreening layer,wouldthelcontribution bereduced byafactore,becoming muchsmaller thanthe ~~contribution, whichisstillreduced bythesamefactore'andremains unchanged. Butitishardtoimagine thatanysurface statescouldliesodeeplyinsideandstillbeinAuenced strongly bytheadsorbate (recallthatthesurface-state bandneededtomediate thecoupling between theexter- nalfieldandthewagovertone issensitive totheadsorp- tionofhydrogen atoms,beginning toformonlywhensat- urationcoverage isapproached ').Inthisregard, weem- phasize againthatthedifferent screening ofdifferent field components alsoexplains whyitismorenaturaltoiden- tifytheabsorption atthefrequency 2v2asavibration contributing toaz,asdonebyChabal' initially. Indeed, forH(1X1)/W(100), theexistence oftheelectronic states capableofcoupling withbothE~and2v2isindicated by thenumerical workofBiswasandHamman. Theycal- culatedchanges inworkfunction asafunctionofthewag coordinate squared(X),fromwhichafinitevaluecanbe estimated fortheeffective chargeofthewagovertone. Ontheotherhand,thenumerical resultofBiswasand Hamman, withtheadiabatice*=0.014,issmaller than thatmeasured byafactorofnearly3.Thisdiscrepancy remains apuzzleinthepicturewherethe2vzmodecon- tributestoO.z,despitethefollowing qualitative arguments orspeculations. InEq.(4.8),wehaveshownthatif b«0,theexcitation strength obtained hereinthe strong-breakdown limitisevensmaller thanBiswasand Hamman's value,because therealpartoftheelectronic susceptibility function, Rey„entering thedynamic di-polemomentp,zissmalleratfinitefrequency thanits zero-frequency limit.Sinceasimilardecrease alsoexists intherealpartofthedielectric function, onemighthope thatanoverallenhancement couldbeobtained inp,&,as thedielectric function appears twiceinthedenominator inEq.(4.7).Actually thisdoesnothappen inthepresent treatment, because byassumption, thesharppeakcon- tributes negligibly tothescreening oftheexternal fieldor theelectron-ion interaction, asexpressed byEq.(4.3). Therefore, togetanenhancement inp,&,weshouldlook forcaseswherethemagnitude ofReg,appearing inthe numerator islargeratfinitefrequency thanatzerofre- quency. Wecertainly cannot putthepositionofthe sharppeaktoocloseto0inordertoget In(EJi—fl)/(b,—0)~&21n(ED/b, ),fortworeasons. Firstly,ifthiswerethecase,themethod usedinthe strong-breakdown limitwouldnolongerbevalid.And secondly, ifsuchaspecialcaseheldfor,say,Dvibration, itcouldnotholdforHvibration because0,wouldshift by=50%and6wouldremainunchanged. Experimen- tally,however, strongabsorption wasobserved at2v2for bothHandDvibration.'Enhancement couldalso occurwhen ~b,—0 ~(5,withb,nottooclosetofl. However, aquantitative estimate onthesizeofsucha possible enhancement cannotbemadewithout morein- formation aboutthestructure intherelevant spectral density. Finally, wemention thatsincethecalculated effective chargeforthesymmetric stretch isalsosmaller totheexperimental valuebynearlyafactorof2,itis possible thatthecalculations maysystematically underes- timatetheexcitation strength. Basedonthediscussions givenbeforeinthissection, it isclearthatonehastodetermine whichfieldcomponent isreallyresponsible fortheexcitation atfrequency 2vz. Atheoretically idealexperiment wouldbetouses- polarized incident radiation, whichhasnolfieldatall. Unfortunately, inthiscasethesignalappearstooweakto bemeasured.' Attention paidtotheparallel fieldcomponent also bringsinanother possibility, involving theexcitation ofa different modeentirely. Thatis,ifafieldE~~canbegen- eratedinthevicinityofthesurface, thenthefundamental oftheasymmetric stretch, v3,wouldalsobecome active toit,eventhoughitisnotactivetoafieldalongzifthe reAection symmetry inthexyplaneisassumed, orhas veryweakstrength whenthereAection symmetry inthe xyplaneislost.Inaddition, thisv3modemaybeexcited evenbythedirectcouplingoftheionicmotiontoE~~.We wouldthenhaveacontribution directly froma,,too, whichmightgivetheexcitation strength alargemagni- tude.Similarly, wewouldalsohavecontributions from e;„o,,;,andu,„which together wouldyieldtheob- servedasymmetry inthelineshape.Forallthistohap- pen,wewouldneedtohaveelectronic stateswhichcould couplestrongly tobothv3andE~~.Thebestcandidates arethosewithd22symmetry, whoseexistence wouldx—y needfuturejustification inthetwosystemsofH/W(100) andH/Mo(100). Inaddition, stateswithdsymmetry couldalsobeinvolved, although theycoupletov3much lessstrongly thanthed22stateswould,andalsolessx—y 39 ELECTRONIC DAMPING OFADSORBATE FUNDAMENTAL AND... 10045 strongly thanto2v2.Asforintraband versusinterband transitions oftheelectrons fortheexcitation ofv3,we againneedtoinvokeafree-carrier scattering mecha- nisms,possibly involving statesofd»symmetry inthe dband.Inaddition, finiteexcitation strength can,in principle, alsobeobtained ifthesystems possess refiection symmetry intheir(x,y)planes.Buttosatisfy thesymmetry requirements, interband transitions be- tweenthesurface-state bandandsomecontinuum band wouldberequired, wherethecontinuum behaves asthe fiatpartinthespectral density defined in(4.1). Thepossibility oftheasymmetric stretchmodev3con- tributing toa~~,orlesslikely,toaj,createssomedifhculty inmodeassignments, arising fromthefactthatthe mode's frequency (1296cm'inEELS measure- ments'')isveryclosetothatofthewagovertone (=1270cm')inirmeasurements.'''IntypicalEELS measurements theresolution isaboutseveral tensof cm',avalueseveraltimeslargerthantheknownsepa- rationbetween thetwovibrations. Inaddition, theob- servedwidthatthewagovertone frequency inirissome 20cm',whichisnotnarrower thantheseparation. Therefore, itispossible thatthemeasured absorption at thewagovertone frequency mightbeactually duetoa fundamental oftheasymmetric stretch. Atleastitseems tousthatnotenoughevidence hasbeenestablished sofar tosafelyexclude thispossibility. Lastly,wemention thatanelectric fieldE~~inthevicin- ityofthesurfacecouldalsoexcitethewagfundamental (vz),inamanner similartotheexcitation oftheasym- metricstretch(v3),andwithastrength roughlyofthe sameorderasforv3butmuchgreaterthanthatexpected to2v2.Ifitexists,itsdetection wouldhelpnotonlyin confirming thatwereallyhaveanefficient parallel fie$d E~~,butalsoinassigning therightmodetotheabsorption measured atthefrequency 2vz.Unfortunately, thiswag fundamental liesbelowthefrequency rangecovered byir. Ontheotherhand,moreinformation fromfirst-principles calculations, withresolved spectral densitiesof states,wouldproveveryhelpfulinunderstanding theex- periments. Toclose,westressthattheexperimental resultsonthe wagovertone ofH/W(100) andH/Mo(100) canbeat leastpartially understood successfully usingtheelectron- hole—pairmechanism inthelimitofstrongbreakdown of adiabaticity developed here,indicating thatsharpstruc- tureintheeffective spectral density exists.However, puzzles remainastotheexcitation mechanism. Compet- ingpossibilities include (a)theovertone ofthewagmode 2v2,excited byEi;(b)theovertone ofthewagmode2v2, excited byEl(requires disorder); (c)thefundamental of theasymmetric stretch modev3,byE~~.Thereisone morepossibility whichislessprobable: (d)theasym- metricstretch v&,excited byEi[refiection symmetry bro- keninthe(x,y)plane]. Wesuggestthatthesesystems arestillripeforfutureexperimental andtheoretical work. ACKNOWLEDGMENTS Weacknowledge numerous discussions withand stimulating suggestions fromY.J.Chabal. %"ethank himalsoforacriticalreadingofthemanuscript. 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PhysRevB.97.075402.pdf

PHYSICAL REVIEW B 97, 075402 (2018) Quantum model of gain in phonon-polariton lasers M. Franckié,*C. Ndebeka-Bandou, K. Ohtani, and J. Faist Institute for Quantum Electronics, ETH Zurich, Auguste-Piccard-Hof 1, 8093 Zurich, Switzerland (Received 2 November 2017; revised manuscript received 18 January 2018; published 1 February 2018) We develop a quantum model for the calculation of the gain of phonon-polariton intersubband lasers. The polaritonic gain arises from the interaction between electrons confined in a quantum well structure and phononsconfined in one layer of the material. Our theoretical approach is based on expressing the resonant matterexcitations (intersubband electrons and phonons) in terms of polarization densities in second quantization, andtreating all nonresonant polarizations with an effective dielectric function. The interaction between the electronicand phononic polarizations is treated perturbatively, and gives rise to stimulated emission of polaritons in the caseof inverted subbands. Our model provides a complete physical insight of the system and allows us to determinethe phonon and photon fraction of the laser gain. Moreover, it can be applied and extended to any type of designsand material systems, offering a wide set of possibilities for the optimization of future phonon-polariton lasers. DOI: 10.1103/PhysRevB.97.075402 I. INTRODUCTION A polariton [ 1] is a composite excitation arising from the coupling of light with a material excitation. As such,polaritons are exhibiting properties that are inherited fromtheir two original constituents and can be tailored over alarge range through the strength of the light-matter coupling.Polaritons can be seen as forced to interact through theirmatter part, while the photons will carry the imprint ofthe coherence properties, enabling their measurement in thefar field using photodetectors. For well-chosen experimen-tal parameters, the polaritons exhibit features of a quan-tum fluid whose properties have attracted a lot of attentionrecently [ 2]. While much attention has been given to exciton-polaritons and their properties in the visible [ 1,3–6], the study of polaritons in the mid-infrared portion of the spectrum hasalso some unique features [ 7–9]. The coupling between an intersubband electronic system and longitudinal optical (LO)phonons was described recently as an intersubband polaron[10], and the coupling between an intersubband system and light, called an intersubband (or cavity) polariton [ 11,12], was theoretically investigated in the Power-Zienau-Woolley (PZW)gauge [ 13,14] by Todorov et al. in Ref. [ 15]. In addition, the light can resonantly couple to transverse optical (TO) phononsforming phonon-polaritons [ 16], which have mostly been stud- ied at the surface of polarizable materials [ 17,18], and recently also in the bulk using classical theory [ 19–21]. The strong coupling properties have also been observed experimentallyand described using a dielectric function approach [ 22]. As shown schematically in Fig. 1, when the cavity andthe intersubband transitions are chosen to be energetically resonantwith a mechanical resonance of the semiconductor lattice, aunique tripartite coupling can be achieved. For large electron *martin.franckie@phys.ethz.chconcentrations and in thermal equilibrium, the resulting po-laritonic dispersion arises due to the coupling of light to bothexcitations. An interesting feature of the intersubband systemis that it can be electrically excited, providing optical gain.Solid state phonon lasers were proposed [ 23] and analyzed using either a pure phononic gain [ 24] or using an electronic Raman approach [ 25]. In contrast, we present a fully quantized model that treats both the photons and phonons, as well asthe intersubband system on an equal footing, in the PZWgauge. This allows us to account for the spatial variation ofthe material optical response, as the phonon polarization isspatially confined. We thus fully account for the tripartitecoupling, albeit using a basis of phonon-polaritons, since thephoton-phonon coupling is the stronger one. In this basis,the rate of stimulated emission of phonon-polaritons fromintersubband excitations, is derived in first order perturbation FIG. 1. Scheme of the three interactions involved in the lasing process of a phonon-polariton laser. The red arrow symbolizes thestrong coupling between the cavity modes and the TO phonon modes that creates the phonon-polariton. The green arrow symbolizes the weak interaction between the phonon-polariton and the ISBtransitions that generates the laser gain. 2469-9950/2018/97(7)/075402(9) 075402-1 ©2018 American Physical SocietyFRANCKIÉ, NDEBEKA-BANDOU, OHTANI, AND FAIST PHYSICAL REVIEW B 97, 075402 (2018) theory. We also provide computational examples for a resonant tunneling diode (RTD) and a quantum cascade laser (QCL),where the phonon-polaritons are confined to potential barriersin the conduction band profile. However, the theory can easilybe expanded to account for arbitrary 2D heterostructures, aswell as other material excitations provided their quantizedpolarization. This paper is organized as follows: In Sec. IIwe derive the classical Hamiltonian for oscillating polarization densities inthe presence of a time-dependent electromagnetic field as thestarting point of our quantum formulation. Then we quantizethis Hamiltonian in Sec. IIIby introducing the polarization density operators for the intersubband system (Sec. IV) and the relevant phonon excitations (Sec. V), in a second quan- tization. In Sec. VIwe derive the interaction Hamiltonian for the phonon and photon fields, which is then diagonalizedto give the phonon-polariton creation-annihilation operatorsand dispersion relation. Finally, in Sec. VII we describe the polariton-ISB interaction responsible for the polaritongain and provide computational examples of the model inSec. VIII. II. CLASSICAL FORMULATION The starting point for our model is the Lagrangian density [ 26] L=ε0 2(˙A+∇φ)2−1 2μ0(∇×A)2 +/summationdisplay i/parenleftbigg1 2χi˙P2 i−ω2 i 2χiP2 i−Pi(˙A+∇φ)/parenrightbigg (1) for a polarization Piin vacuum, represented by a sum of harmonic oscillators with eigenfrequency ωiand “mass” χi, under the application of an electromagnetic field with vectorand scalar potentials Aandφ. The last term accounts for the electric potential energy stored in the “springs” of the oscillators. This leads to the Hamiltonian H=/integraldisplay d 3r⎡ ⎢⎢⎢⎣1 2ε0D2+1 2μ0(∇×A)2+1 2/summationdisplay i/negationslash=jPiPj +/summationdisplay i⎛ ⎜⎜⎜⎝/parenleftbiggω2 i 2χi+1 2ε0/parenrightbigg /bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright ω/prime i/2χiP2 i+1 2χi˙P2 i−1 ε0D·Pi⎞ ⎟⎟⎟⎠⎤ ⎥⎥⎥⎦,(2) where D=−ε0(˙A+∇φ)+/summationtext iPiis the electric displace- ment field satisfying ∇·D=0. Here the frequencies ( ω/prime i)2= ω2 i+ω2 P,iof the oscillators are shifted by the plasma frequency ω2 P,i=χi ε0with respect to the bare mechanical frequency of Eq. ( 1). We shall consider only a few polarization terms, namely those coming from TO phonons and the electronsconfined in the conduction band. We thus treat all nonresonantoscillators with an effective dielectric constant defined by D=ε 0E+/summationdisplay i∈b.Pi+/summationdisplay i∈e,LPi≡εrε0E+/summationdisplay i∈e,LPi,(3)where the first sum is over the background (b), and the second over the ISB (e) and the resonant lattice ( L) polarizations. Assuming Pi∈boscillate at the cavity frequency ω, εr=1+/summationdisplay i∈bχi ε0/parenleftbig ω2 i−ω2/parenrightbig. (4) Considering only the background, the Hamiltonian reads Hb=/integraldisplay d3r/parenleftbigg1 2E·D+1 2B·H/parenrightbigg +Hmat (5) =/integraldisplay d3r/parenleftbigg1 2ε0εr(z)D2+μ0 2H2/parenrightbigg +Hmat, (6) where ∇×A=B=μ0Hand we assumed in Eq. ( 1) that there are no magnetic moments in the system. Hmatcontains all terms of Eq. ( 2) which contain the background matter polarizations Pionly. Physically, this term contains the energy contribution of all the crystal ions, and will not affect the following theorywhere the we treat the conduction band electrons in theenvelope function approximation. We will thus suppress thisterm from now on. Adding the special polarizations/summationtext i∈e,LPi, not included in εr, we find H=/integraldisplay d3r/parenleftbigg1 2ε0εrDD2+μ0 2H2/parenrightbigg +1 ε0εr/integraldisplay d3r⎛ ⎝−/summationdisplay i∈e,LD·Pi+1 2/summationdisplay (ij)∈e,LPiPj⎞ ⎠ +/integraldisplay d3r/summationdisplay i∈e,L1 2χi/parenleftbig ω2 iP2 i+˙P2 i/parenrightbig . (7) This Hamiltonian resembles the one of Ref. [ 15]. We will use Eq. ( 7) and diagonalize the terms in the quantized Hamiltonian containing PLonly in Sec. VI. We also note that in a het- erostructure, εr=εr(z,ω) will acquire a zdependence which in principle has to be considered when performing the volumeintegral. The background dielectric function ε rresults from both interatomic polarizations and bound electrons. In addition,electrons in quantum states spatially separated from the res-onant phonon polarization may give a small contribution. Wewill assume that ε ris close to the bulk values of the constituent materials, why we will later set εr=ε∞of the bulk well material. III. QUANTUM FORMULATION In order to quantize the system, we write down the quantized version of the Hamiltonian ( 7)a s ˆH=ˆHrad+ˆHL+ˆHe+ˆHint, (8) where the radiation in the cavity is written in second quantiza- tion as ˆHrad=/summationdisplay q¯hωcav,q/parenleftbigg a† qaq+1 2/parenrightbigg . (9) 075402-2QUANTUM MODEL OF GAIN IN PHONON-POLARITON LASERS PHYSICAL REVIEW B 97, 075402 (2018) For the cavity modes, we use the quantized displacement field of the TM mode in the PZW gauge (see, e.g., Ref. [ 27]) ˆDz(R)=i/summationdisplay q/radicalBigg εrε0¯hωopt,q 2SLcaveiq·rgq(z)(aq−a† −q),(10) where SandLcavis the surface area and length of the cavity, respectively, and gq(z) is the mode profile normalized as /integraldisplay∞ −∞g2 q(z)dz=Lcav. (11) These are the only modes than can propagate in 2D heterostruc- tures. Still neglecting magnetic interactions, the light-matterinteraction in Eq. ( 7) leads to an interaction Hamiltonian having the form ˆH int=/integraldisplay d3r1 ε0εr/parenleftbigg −ˆD·ˆPmat+1 2ˆP2 mat/parenrightbigg , (12) where the sum in Eq. ( 7) runs over the intersubband transitions and the lattice contributions ˆPmat=ˆPe+ˆPL, respectively. The formalism developed so far does not take into account any dissipative couplings for the phonons, electrons, or pho-tons. In the first part of this paper, we will completely neglectthese coupling terms. Later, when we calculate the polaritongain, however, we need to include the phonon and photondecays via an effective decay rate into acoustic phonons andcavity losses, respectively. For the ISB system, dissipation dueto optical and acoustic phonons, as well as elastic scatteringwith ion impurities, alloy disorder, and interface roughness,can be included in the transport calculations providing the self-consistent populations of the ISB levels undergoing stimulatedpolariton emission. Since we are interested in a situation where the electronic system provides optical gain but will remain in the weakcoupling with radiation, we split the Hamiltonian ˆH=ˆH rad+ˆHL+/integraldisplay d3r1 ε0εr/parenleftbigg −ˆD·ˆPL+1 2ˆP2 L/parenrightbigg /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright Diagonalize ˆHP +/integraldisplay d3r1 ε0εr/parenleftbigg −ˆD·ˆPe+1 2ˆPL·ˆPe/parenrightbigg /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright Perturbation +/integraldisplay d3r1 ε0εr1 2ˆP2 e /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright Neglect+ˆHe (13) into three parts. The first part ˆHPcontains the lattice-radiation coupling and will lead to our polaritonic basis after diago-nalization. Amplification or attenuation of these polaritonsthrough their interaction with the intersubband system willbe computed using Fermi’s golden rule applied to the secondpart of ˆH. Finally, as we are dealing with a low electron population, we can safely neglect the intersubband polarizationself-energy, which accounts for the depolarization shift. IV . ELECTRON POLARIZATION The electronic subband states nare defined by their ener- giesEn(k)=En+¯h2k2 2m∗and their wave functions /angbracketleftr,z|n,k/angbracketright=1√ Seik·rχn(z), where Sis the sample area, m∗is the carrier effective mass, and kand r=(x,y) are the in-plane wave vector and the in-plane coordinate, respectively. Starting fromthe initial level |i,k/angbracketright(where the carriers are either residing or electrically injected), each possible ISB transition |i,k/angbracketright→ |n,k /prime/angbracketrightis labeled by the index jand occurs at a frequency ωj=ωi−ωn. The intersubband polarization is [ 15] ˆPe(r)=¯he 2Sm∗/summationdisplay j,q/radicalbig/Delta1Njξj(z) ωjeiqr[b† j,−q+bj,q],(14) where e=− |e|is the electron charge, /Delta1Nj=Ni−Nnis the population inversion, and is expressed as a function of thebright mode creation operators [assuming that the transitionsare vertical in the kspace ( k≈k /prime)] b† j,q=1/radicalbig/Delta1Nj/summationdisplay kc† n,kci,k≡b† j. (15) Herec† n,kandci,kare the creation and annihilation operators for the one-electron ISB states |n,k/angbracketright. The microcurrents for the transition between state are defined from the wavefunctions as ξ j(z)=χi(z)∂zχn(z)−χn(z)∂zχi(z). (16) In this formalism, the electron Hamiltonian is expressed as ˆHe=/summationdisplay jb† jbj¯hωj. (17) V . PHONON POLARIZATION The only lattice vibrations interacting with a light field are the transverse optical (TO) phonons, which are assumed tobe dispersionless and have a mechanical frequency ω TO.W e will assume that the vibrations are localized in layers in thex-yplane, with z-coordinate z i. The phonon polarization (see Appendix A) ˆPL,z(R)=¯he 2SM/summationdisplay q,iξL,i(z) ωTOeiq·r(d† i,−q+di,q) (18) is similar to the electronic one, with Mbeing the vibrational mass and d† i,−qcreating a TO phonon with in-plane momentum −q. The phonon microcurrent is (see Appendix A) ξL,i=/radicalbigg 2 π1 σ2e−(z−zi)2/σ2(19) andσis related to the standard deviation from the equilibrium position zi. These phonons are represented globally by the Hamiltonian ˆHL=/summationdisplay qi¯hωTOd† qidqi, (20) provided the phonon plasma frequency is ω2 P,i=¯he2/integraltext ξ2 L,i(z)dz 2SM2/epsilon10/epsilon1rωTO. (21) The operator d† i,qcan be thought of as creating a phonon ex- citation confined in one monolayer, traveling with momentum qinside this layer. The polarization operator ( 18) depends on 075402-3FRANCKIÉ, NDEBEKA-BANDOU, OHTANI, AND FAIST PHYSICAL REVIEW B 97, 075402 (2018) the density of oscillators via the plasma frequency, which fixes the oscillator “mass” M. The index ican either represents physical atomic layers, or spatially separated thin layers ofthe bulk material (so thin that the lattice ions oscillate in phaseand the phonon microcurrent can be represented by a Gaussianfunction). VI. HAMILTONIAN OF THE PHONON-POLARITON The diagonalization of ˆHPin thermal equilibrium, assum- ing all the carriers are in the ground state, yields polaritonsthat combine lattice and electronic excitations [ 1], and have been observed experimentally [ 22]. In this section we will diagonalize ˆH Pin two steps and find the polariton eigenstates. First, we will incorporate the phonon polarization self-energy 1 2ε0εrˆP2 Linto the bare phonon Hamiltonian ˆHL, which will lead to an energy renormalization similar to the depolarization shiftof the intersubband system. The following calculation will besignificantly lightened by neglecting the terms of P 2 Lmixing ξP,iwith different layer indices i, as motivated in Appendix A. Thus, we diagonalize ˆH/prime L≡ˆHL+/integraldisplay d3r1 2ε0/epsilon1rˆP2 L=¯hωTO/summationdisplay iqd† i,qdi,q +/summationdisplay iq¯h/Theta1i(d† i,q+di,−q)(d† i,−q+di,q), (22) where /Theta1i=e2¯h 8SM2ε0/epsilon1rω2 TO/integraldisplay ξ2 L,i(z)dz (23) (here we assumed that the background dielectric constant does not vary on the scale of the phonon layers), with the newoperators p i,q=ω/prime i−ωTO 2/radicalbig ω/prime iωTOd† i,−q+ω/prime i+ωTO 2/radicalbig ω/prime iωTOdi,q. (24) The eigenvalue is ( ω/prime i)2=(ωTO)2+4ωTO/Theta1i, and interpreting this as the longitudinal optical (LO) phonon frequency, wededuce the plasma frequency ω 2 P,i=4ωTO/Theta1i. Expressing ˆHP in second quantization format now leads to the expression ˆHP=/summationdisplay q¯hωcav,q/parenleftbigg a† qaq+1 2/parenrightbigg +/summationdisplay q,i¯hω/prime ip† i,qpi,q +i/summationdisplay q,i¯h/Lambda1i,q(a† q−a−q)(p† i,−q+pi,q), (25) with /Lambda1i,q=ωP,i 2/radicalbiggωopt,q ω/prime ifP,i, (26) where fL,i=⎛ ⎝/integraltext gq(z)ξL,i(z)dz/radicalBig Lcav/integraltext ξ2 L,i(z)dz⎞ ⎠2 . (27) The factor fL,imeasures the filling of the cavity by the mechanical oscillators and is equal 1 for the bulk material.Proceeding with the second step of the diagonalization of ˆHP, the Hamiltonian ( 25) can be exactly diagonalized through a Bogoliubov transformation and by the introduction of the polariton operator /Pi1q=xqaq+yqa† −q+zqpq+tqp† −q.T h e two real solutions ωq,±of the eigenvalue equation [ /Pi1q,ˆHP]= ¯hω q/Pi1qare the frequencies of the two polaritonic branches and are readily obtained by ωq,±=1√ 2/radicalBig ω/prime2+ω2 opt,q±√ /Delta1, (28) /Delta1=ω/prime4+4ω2 cω2 opt,q−2ω/prime2ω2 opt,q+ω4 opt,q, (29) with ω2 c=fpω2 P. The two polaritonic branches have the asymptotes ωq→0,+=√ ω2 TO+ω2 P≡ω/primeandωq→∞,−=√ ω2 TO+ω2 P(1−fP)≡ω/prime/prime. Additionally, the diagonalization of (25) in the polaritonic basis allows us to determine the mixing fractions of the phonon-polariton, namely its photonic(h l,q=|xq|2−|yq|2) and phononic ( hp,q=|zq|2−|tq|2) fractions. For instance, in the upper branch ( ωq=ωq,+), we have the fractions h+ l,q=ω2 q,+−ω2 TO ω2 q,+−ω2 q,−,h+ p,q=1−h+ l,q. (30) In the lower branch, the mixing fractions are simply obtained byh− l,q=1−h+ l,q. While the limits when ωq,+→ω/primeand ωq,+→ωTOcorrespond to mostly phonon states, in the vicin- ity of the anticrossing the mixing fractions reach a value of 0.5,indicating a maximum phonon-photon admixture. A suitabledesign of the active region enables us to achieve a lasing emis-sion at frequencies close to this maximum admixture point,where a nonvanishing phononic gain is therefore expected. If the filling factor f Lis small (as, e.g., for a thin-layer structure such at those in Figs. 2and 4) the equivalent Rabi frequency at resonance /Lambda1R=ωP√fL/2 becomes small compared to the bare cavity and TO phonon frequencies. Inthis regime of weakly coupled oscillators ( /Lambda1 R/ωTO/lessmuch1), the mixing fractions can be approximated by hl≈|xq|2andhp≈ |zq|2as well as the polariton operator /Pi1q≈xqaq+zqpq= /Pi1l,q+/Pi1p,q[12]. VII. POLARITON-ISB INTERACTION The second step of our approach consists of describing the interaction between the phonon-polariton and the ISBtransitions. We express the full quantum Hamiltonian thatdescribes the phonon-polariton-ISB system as follows: ˆH p-ISB=/summationdisplay q¯hω q,p/Pi1† q/Pi1q+/summationdisplay j¯hωjb† jbj +i/summationdisplay q,j¯h/Omega1j,q(a−q−a† q)(b† j+bj) +/summationdisplay q,j¯h/Xi1j,q(pq+p† −q)(b† j+bj). (31) The first term in Eq. ( 31) is the polaritonic part of the Hamil- tonian with the approximate polariton operator /Pi1qalready defined above, in either the upper or lower polariton branch.The second term contains the ISB part. Similarly to Eq. ( 26), 075402-4QUANTUM MODEL OF GAIN IN PHONON-POLARITON LASERS PHYSICAL REVIEW B 97, 075402 (2018) FIG. 2. Transport scheme of a phonon-polariton resonant tunnel- ing diode (RTD). The red oscillators represent the confining barrier of the TO phonon modes which at the same time serves as the tunneling barrier. The electronic wave functions of the upper and the lowerlasing states are plotted thick yellow and green lines, respectively, and overlap the TO phonon modes such that an additional gain arising from the TO phonon-ISB transition coupling is expected. Under the lasingbias, the energy spacing between the two lasing states is resonant with the TO phonon energy. The layer sequence of the structure in ˚Ai s 250/100/12/120/12/50/5000, where boldface denotes AlGaSb barriers, italic face denotes the AlInAs barrier, and the underlined layers are doped to 2 ×1016cm−3. the two last terms in Eq. ( 31) are the interaction components with respective coupling frequencies /Omega1j,q=ωPj 2/radicalbiggωopt,q ωj/integraltext gq(z)ξj(z)dz/radicalBig Lcav/integraltext ξ2 j(z)dz, (32) /Xi1j,q=ωPjωP 4/radicalbigω/primeωj/integraltext ξL(z)ξj(z)dz/radicalBig/integraltext ξ2 j(z)dz/integraltext ξ2 L(z)dz, (33) where ωPjis the ISB plasma frequency proportional to the injected carrier density [ 15]. Since the phonon-polariton mode and the ISB transitions are in the weak coupling regime, Fermi’s golden rule can beapplied to compute the emission rate, i.e., the gain cross sectionof the phonon-polariton-ISB system. In a cavity containing N q phonon-polaritons, we consider all the transitions |ul,N q/angbracketright→ |n,N q+1/angbracketrightwith an electron initially in the upper laser state ( ul) and finally in a lower energy level n, which lead to the emission of a phonon-polariton, and we calculate the total emission rate.By Fermi’s golden rule, then the emission rate becomes g(ω q)=2π ¯h/summationdisplay j|/angbracketleftn,N q+1|[i¯h/Omega1j,q(a−q−a† q)(b† j−bj) +¯h/Xi1j,q(pq+p† −q)(b† j−bj)]|ul,N q/angbracketright|2δ(ω−ωj). (34) Retaining only the terms in Eq. ( 34) that describe an emission process (the ones that are proportional to a† qbjandp† qbj), we find the expression of the total gain cross section g(ωq)=2π¯h(Nq+1)/summationdisplay j|z∗ q/Xi1j,q−ix∗ q/Omega1j,q|2δ(ωj−ωq) =2π¯h(Nq+1)/summationdisplay j||zq|/Xi1j,q+|xq|/Omega1j,q|2δ(ωj−ωq) as shown in Appendix B. Here special care needs to be taken to the phase between xqandzq, since this is evidently crucial to the role of the mixed terms in Eq. ( 35). If we write xq= |xq|eiϕ, then zq=|zq|ei(ϕ+π/2)leading to the second line of Eq. ( 35). Thus, the mixed terms contribute constructively to the emission rate, if the couplings /Omega1j,qand/Xi1j,qhave the same sign. The δfunction can be replaced by a Lorentzian function of characteristic width γ:δ(ωj−ωq)→γ/π (ωj−ωq)2+γ2. Along the same lines, the optical losses of the device can be estimated from the respective photon ( τcav) and phonon ( τp) lifetimes in the cavity. Accounting again for the mixed natureof the polariton, the loss rate is written as α(ω q)=hl(q) τcav+hp(q) τp, (35) where, from experimental studies values of phonon lifetimes, τpof 3.5 ps at 300 K and 7.8 ps at 77 K were determined [ 28] andτcavcan be readily estimated from the cavity losses. In the following section we will employ the developed theory to compute the phonon-polariton dispersion and gainin experimentally realizable 2D systems. VIII. COMPUTATIONAL EXAMPLES A. InGaAs-based resonant tunneling diode As a first example, we consider a resonant tunneling diode (RTD) structure. The benefit of such a structure for emissionin the THz region, is that it can be heavily doped and thushave a large inversion, in addition to easily tunable emissionfrequency by changing the layer widths. In addition, thesimple layer structure provides an excellent starting pointfor a theoretical analysis of polariton gain in heterostructure.However, such devices typically have population inversion inregions of negative differential conductance (NDC), and thuscannot operate in a serial configuration. In addition, in thestructure shown in Fig. 2, a 4 monolayer thick InAlAs barrier serves as both the injection barrier of the RTD, giving riseto inversion between the level indicated by a thick yellowline and the two semibound states of the subsequent quantumwell, as well as the confining layer for the AlAs phonons.In the following computations we treat the four monolayersas one effective layer with σ=0.48 nm and ω P=17 meV . The computed optical loss for this structure is ∼410−1with a Au/Au double-metal waveguide. In comparison, we calculate amaximum optical gain of ∼800 cm −1, using a nonequilibrium Green’s function model [ 29]. From Eq. ( 28) we compute the phonon-polariton dispersion which is shown in Fig. 3(a). Due to the small filling factor fL= 1.67×10−2, this structure exhibits a much smaller polaritonic gap than the one of bulk AlAs. Figure 3(b) shows the contributions to the gain rate of Eq. (35) from the photon, phonon, and mixed parts, as functions 075402-5FRANCKIÉ, NDEBEKA-BANDOU, OHTANI, AND FAIST PHYSICAL REVIEW B 97, 075402 (2018) (a) (b) FIG. 3. (a) Calculated dispersion of light ωq±for the phonon- polariton RTD in Fig. 2. The polaritonic phonon ( hp) and photon ( hl) mixing fractions of as functions of the energy in the upper branch, arealso shown with thin lines. The dashed line shows the bare cavity mode withω= ck√/epsilon1r. (b) Gain fraction of the different gain components arising from the photon (green), phonon (blue), and mixed terms(orange) in ( 35). The right axis shows the ratio of the gain to the losses in the UP branch. of the energy in the upper polariton branch. For low energies, close to the polariton gap, the phonon fraction is maximal anddecreases rapidly with increasing ω q. Reversely, the photonic gain vanishes when ω→ω/prime, but dominates at high frequencies as the photon fraction increases. Due to the small fillingfactor, the coupling ratio /Xi1 q//Omega1 q/lessmuch1 and the total gain is mostly dominated by the photonic gain. However, a maximumnonphotonic gain of 20% is already achieved at the frequencywhere gain overcomes the losses, despite a phonon extensionof only a few monolayers. Figure 3(b) also shows the ratio g/α as a function of the energy in the UP branch. For the bias considered here, the bare optical gain is peaked at a frequencyof ¯hω=48.05 meV . The loss rate being energy dependent through the mixing fractions, dividing the gain by the losses,shifts its maximum by 0.1 meV , which corresponds to the lasingenergy of the device. While the phonon gain fraction at thisenergy is only 1 .2×10 −4, the nonphotonic contribution to the gain is still 2% of the total gain. In addition, this structureFIG. 4. Band structure and eigenstates of the proposed phonon- polariton QCL for an applied electric field of 18 kV /cm. The marked AlInAs barrier hosts the polaritons and overlaps the gaintransition, from the upper laser state ( ul) to the lower laser state (ll). The electrons are depopulated from the llinto the ulof the next period via cascading down the potential wells through coherenttunneling, as well as incoherent transport. The layer sequence in˚A is, starting from the rightmost barrier, 48/54/3/86/7.5/ 82/7.5/81/8.5/71/11.2/61/16/ 64/30/72, where boldface denotes GaAsSb barriers, italic font denotes the InAlAs barrier, and the underlined well is doped to 4 .1×1017cm−3. has relatively low optical losses, and the contribution of the phonon part of the polariton is expected to be more importantfor structures where the optical losses are higher. B. InGaAs-based quantum cascade laser Our second example is a quantum cascade laser (QCL) [ 30] where the TO phonons are provided by a barrier close to theinverted ISB transition. In contrast to RTDs, QCLs are reliablesources of coherent radiation in the THz frequency region, witha well proven growth and fabrication technique. In addition, op-erating at a bias of positive differential resistance, one QCL pe-riod can be repeated hundreds of times in a several micrometer-thick structure, potentially allowing significantly more opticalpower to be extracted than from a single period RTD structure. In the structure in Fig. 4, a monolayer-thick AlInAs barrier plays the role of the phonon barrier in a InGaAs/GaAsSbactive region [ 31]. This barrier is placed where the the inverted subbands ulandllhave significant overlap, thus emitting phonon-polaritons via stimulated emission. In this bound-to-continuum design, the carriers are extracted from llin a cascade ending on the black state of lowest energy in Fig. 4, where it is subsequently injected into the ulstate of the next period of the QCL. For this structure, the electron transport is calculated ina density matrix approach [ 32]. The calculated dispersion and mixing fractions are shown in Fig. 5(a). In this structure, f L=4.3×10−3is even smaller than for the RTD. However the design frequency is adjustedto be close to the maximum splitting between the branches,where the fraction of the phonon to photon Hopfield coefficientis close to 50%. For this and slightly higher frequencies, thedesign has a phonon fraction of about 3 ×10 −4of the total 075402-6QUANTUM MODEL OF GAIN IN PHONON-POLARITON LASERS PHYSICAL REVIEW B 97, 075402 (2018) (a) (b) FIG. 5. (a)Dispersion relation of the proposed phonon-polariton QCL, where the design frequency belongs to the upper branch, as well as phonon and phonon hop field coefficients (thin lines). The dashed line shows the bare cavity mode with ω=ck√/epsilon1r. (b) Gain fraction of the different gain components arising from the photonic part (green), the phononic part (blue), and the mixed terms (orange) in ( 35). The right axis shows the ratio of the gain to the losses of the UP polariton mode. gain, while the total nonphotonic gain accounts for ∼3%, as seen in Fig. 5(b). In this figure we also show the ratio of the calculated gain to the losses from Eq. ( 35), and we find the maximum value at an energy slightly blueshifted from thedesign frequency. The second, lower, peak at 62 meV , arisesdue to emission to a lower electronic state which has lessoverlap with the upper laser level. Despite the fact that the gain of these devices remains mainly dominated by the standard dipole coupling, there isroom for increasing the phononic contribution. An optimizeddesign with a suitable location and thickness of the phononlayer could lead to larger overlaps between the phonon and ISBmicrocurrents. The choice of a material with a larger polaritongap (with larger ω P), such as ZnO/ZnMgO or GaN/AlN, could dramatically increase the phonon part of the gain. Inparticular for structures with large optical losses compared tooptical gain, the phonon contribution to the polaritonic gaincan then increase the stimulated emission rate and thereby theconversion efficiency of electrical power into power radiatedin the electric field.IX. CONCLUSION In conclusion, we have developed a quantum approach for the description of gain in phonon-polariton lasers. Comparedto an effective dielectric model [ 33], this formalism is more appropriate for the description of confined modes in thinlayers and has the advantage to provide a complete physicalinsight of the system, especially by directly giving the phononand photon fractions of the lasing modes that are the keyparameters for the gain computation. Our model can be appliedto a wide variety of designs and material systems, offeringa wide set of possibilities for the optimization of futurephonon-polariton QCLs. As a demonstration of the flexibilityof the model, we have proposed and simulated resonanttunneling diodes and quantum cascade lasers made from theconventional InGaAs/InAlAs/InGaSb material system, as wellas the less explored ZnO material system. While the former twostructures show a small nonphotonic contribution to the gainof∼10%, this number can be increased by employing more phonon material to increase the filling factor, or using othermaterial systems with larger phonon plasma frequency, suchas ZnO/ZnMgO or GaN/AlN. ACKNOWLEDGMENTS This work is partially funded by the ERC Advanced grant Quantum Metamaterials in the Ultra Strong Coupling Regime(MUSiC) with the ERC Grant 340975. The authors also ac-knowledge financial support from the Swiss National ScienceFoundation (SNF) through the National Centre of Competencein Research Quantum Science and Technology (NCCR QSIT).J.F. thank A. Vasanelli, S. De Liberato, and J. B. Khurgin forvery fruitful discussions. APPENDIX A: PHONON POLARIZATION IN SECOND QUANTIZATION For modeling the lattice vibrations in thin layers at z=zi, we expand the collective lattice vibrations in the basis of vibra-tional harmonic oscillator modes /Psi1 α,ij(R)=ψα,ij(z)χα,ij(r), where jlabels the in-plane coordinate and αis the harmonic oscillator excitation: ˆ/Psi1†=1√ S/summationdisplay α,ijd† α,ijψα,i,j(z)χα,i,j(r), (A1) where Sis the sample area and d† α,ijis the creation operator for a lattice vibration excitation. Due to the rotational symmetryin the x−yplane, we write the total wave function /Psi1(R) as a product between the harmonic oscillator wave functionψ α(z), and the in-plane (periodic) wave function χα,j(r). As in Ref. [ 15], the polarization is found via its relation to the current density operator ˆJz(R)=1 i¯h[ˆPL,z,H], (A2) defined by ˆJz(R)=i¯he 2M/bracketleftbigg ˆ/Psi1†∂ ∂zˆ/Psi1−/parenleftbigg∂ ∂zˆ/Psi1†/parenrightbigg ˆ/Psi1/bracketrightbigg =i¯he 2SM/summationdisplay αβ/summationdisplay ii/prime,jj/primeξii/prime αβ(z)χ∗ αjχβj/primed† α,ijdβ,i/primej/prime,(A3) 075402-7FRANCKIÉ, NDEBEKA-BANDOU, OHTANI, AND FAIST PHYSICAL REVIEW B 97, 075402 (2018) where Mis parametrizing the inertia of the ions. Here the phonon microcurrent is defined as ξii/prime αβ(z)=φα,i(z)∂ ∂zφ∗ β,i/prime(z)−φ∗ β,i/prime(z)∂ ∂zφα,i(z). (A4) The only allowed transitions of the harmonic oscillators are those with |α−β|=1, and we will consider only the lowest excitation with ( α,β)∈{0,1}. Then, the current density oper- ator becomes ˆJz(R)=i¯he 2SM/summationdisplay i,jj/primeξii 10(z)(χ∗ 1,ijχ0,ij/primed† 1,ijd0,ij/prime −χ1,ij/primeχ∗ 0,ijd0,ijd† 1,ij/prime). (A5) Here we neglected the mixing of different layers i/negationslash=i/prime, which will give a very small contribution if the layers are separatedby a few standard deviations σ. The terms in the parentheses are periodic in the plane with period a x/y=2π qx/y, since we are interested in solutions where the polarization is a traveling wave with momentum q, and the first term is Fourier expanded to (the second term is just the complex conjugate of the first one)/summationdisplay jj/primeχ∗ 1,ijχ0,ij/primed† 1,ijd0,ij/prime≡/summationdisplay qd† i,qe−iqr. (A6) Thus, the current density becomes ˆJz(R)=i¯he 2SM/summationdisplay i,qξL(z)eiqr(d† i,−q−di,q), (A7) where ξL(z)≡ξii 10(z). Using the commutation relations [d† iq,ˆH]=− ¯hωTOd† iqand [diq,ˆH]=+ ¯hωTOdiqtogether with Eq. ( A2), we find the polarization density operator of Eq. ( 18). The similar form of the phonon polarization to the electronic one, prompts us to write the Hamiltonian for the TO phononas H L=/summationdisplay iq¯hωTOd† i,qdi,q. (A8) Inserting Eq. ( 18) in place of the classical polarization in the classical Hamiltonian [the last line of Eq. ( 7)] gives HL=/integraldisplay d3r1 2χL/parenleftbig ω2 TOP2 L+˙P2 L/parenrightbig (A9) =1 2χL¯h2e2 SM2/integraldisplay ξ2 L(z)dz/summationdisplay iqd† i,qdi,q (A10) (apart from the constant vacuum energy shift which is irrele- vant here). This is equal to Eq. ( A8) if we identify the phonon plasma frequency ω2 P=χL//epsilon10/epsilon1ras in Eq. ( 21). APPENDIX B: PHONON-POLARITON EIGENSTATES In order to compute the polariton scattering rates, we need to express the polariton states in terms of its phonon and photonconstituents. The polariton states can be calculated by repeated application of the polariton creation operator /Pi1† qon the vacuum state|0/angbracketrightas |Nq/angbracketright=CNq(/Pi1† q)Nq|0/angbracketright, (B1) with a normalization constant CNq, to be determined. From now on we suppress the index qfor ease of notation. Equation ( B1) can be rewritten by noting that ( /Pi1†)N=(x∗a†+z∗p†)Nandusing the binomial formula as |N/angbracketright=CNN/summationdisplay k=0N! k!(N−k)!(x∗)k(z∗)N−k(a†)k(p†)N−k|0/angbracketright =CNN/summationdisplay k=0N!√k!(N−k)!(x∗)k(z∗)N−k|k,N−k/angbracketright,(B2) where |n,m/angbracketright≡|n/angbracketrightphot⊗|m/angbracketrightphonspan the Hilbert space of the phonon-polariton. Here we used the approximation of smallcoupling strength /Lambda1 R/ωTO/lessmuch1. However, the resulting |N/angbracketright will also be the same without this approximation, since terms likea−qa† q|0/angbracketright=0. The normalization constant is found by solving 1=/angbracketleftN|N/angbracketright =|CN|2N/summationdisplay k,k/primeN!N!xk/prime(x∗)kzN−k/prime(z∗)N−k √k!(N−k)!√k/prime!(N−k/prime)! ×/angbracketleftk/prime,N−k/prime|k,N−k/angbracketright. (B3) The bra-ket gives δkk/prime, and again using the binomial formula, we find |CN|2/summationdisplay kN!N! k!(N−k)!|x|2k|z|2N−2k=|CN|2N!(|x|2+|z|2)N. (B4)By definition, |x|2+|z|2=1, and we readily find that CN= 1/√ N!, and |N/angbracketright=N/summationdisplay k=0/radicalBigg N! k!(N−k)!(x∗)k(z∗)N−k|k,N−k/angbracketright. (B5) We can easily check that the number operator gives the correct result by using [ a,(a†)N]=N(a†)N−1: /Pi1†/Pi1|N/angbracketright=/Pi1†/Pi1(/Pi1†)NCN|0/angbracketright =/Pi1†N(/Pi1†)N−1CN|0/angbracketright=N|N/angbracketright. (B6) Now lets calculate the emission rate /Gamma1em. For this we need to compute terms with /angbracketleftN+1|a†|N/angbracketright=N/summationdisplay k=0(x∗)2k+1(z∗)2N−2k k!(n−k)!/radicalbig N!(N+1)! ≡x∗√ N+1N/summationdisplay k=0C2 k,N, (B7) where we defined |N/angbracketright≡/summationtext kCk,N|k,N−k/angbracketright. Similarly, we find that /angbracketleftN+1|p†|N/angbracketright=z∗√ N+1N/summationdisplay k=0C2 k,N. (B8) The normalization of |N/angbracketrightmeans that/summationtextN k=0C2 k,N=1 and so /angbracketleftN+1|a†|N/angbracketright=/angbracketleftN+1|/Pi1† l|N/angbracketright x∗=x∗√ N+1,(B9) /angbracketleftN+1|p†|N/angbracketright=/angbracketleftN+1|/Pi1† p|N/angbracketright z∗=z∗√ N+1.(B10) Equations ( B9) and ( B10) inserted into Eq. ( 34)g i v et h e emission rate in Eq. ( 35). 075402-8QUANTUM MODEL OF GAIN IN PHONON-POLARITON LASERS PHYSICAL REVIEW B 97, 075402 (2018) [1] J. J. Hopfield, Phys. Rev. 112,1555 (1958 ). [2] I. Carusotto and C. Ciuti, Rev. Mod. Phys. 85,299(2013 ). [3] H. Deng, G. Weihs, C. Santori, J. 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PhysRevB.86.085424.pdf

PHYSICAL REVIEW B 86, 085424 (2012) Temperature dependence of the dynamics of the first image-potential state on Ag(111) S. S. Tsirkin,1,2S. V . Eremeev,1,3E. V . Chulkov,2,4M. Marks,5K. Schubert,5J. G¨udde,5and U. H ¨ofer2,5 1Tomsk State University, Tomsk, 634050, Russia 2Donostia International Physics Center (DIPC), 20018 San Sebasti ´an/Donostia, Basque Country, Spain 3Institute of Strength Physics and Materials Science, pr. Academicheskii 2/4, Tomsk, 634021, Russia 4Departamento de F ´ısica de Materiales UPV/EHU, Centro de F ´ısica de Materiales CFM–MPC and Centro Mixto CSIC-UPV/EHU, 20080 San Sebasti ´an/Donostia, Basque Country, Spain 5Fachbereich Physik und Zentrum f ¨ur Materialwissenschaften, Philipps-Universit ¨at, D-35032 Marburg, Germany (Received 14 June 2012; published 14 August 2012) The temperature dependence of the dynamics of electrons in the n=1 image potential state on the Ag(111) surface has been investigated by means of time-resolved two-photon photoemission spectroscopy and many-bodycalculations. We show that the decay rate of electrons in this state grows linearly with temperature. The thermalshortening of the lifetime is caused by the increase of the electron-electron scattering rate, due to deeperpenetration of the image state wave function into the bulk metal at higher temperature. The electron-phononscattering in this state is found to be small. DOI: 10.1103/PhysRevB.86.085424 PACS number(s): 73 .20.−r, 72.10.−d, 79.60.Bm, 78 .47.J− I. INTRODUCTION Lifetimes of electronic excitations are crucial for many physical and chemical phenomena, such as charge andenergy transport, 1interaction of atoms and molecules with the surface,2and catalytic reactions.3At metal surfaces, in addition to bulk electronic states, single-particle excitationsoccur in intrinsic surface states 4,5and in image-potential states.6–8Surface states arise from the breaking of the crystal symmetry along the surface normal and are localized near theoutermost atomic layer. Image-potential states are localizedon the vacuum side of the surface. They are formed by theimage potential, originating from the attractive interactionof the electron with the polarization charge. Far from thesurface plane this potential converges asymptotically to theclassical image potential, which is inversely proportional tothe distance from the image plane. In the vicinity of the /Gamma1 point of the surface Brillouin zone image-potential states forma Rydberg-like series, converging to the vacuum level E vac:9,10 En(k/bardbl)=Evac−0.85 eV (n+a)2+¯h2k2 /bardbl 2m∗n, (1) where n=1,2,... is the principal quantum number and ais the quantum defect. The dispersion of image-potential stateswith the wave vector k /bardblin the surface plane is free-electron- like with effective masses m∗ nclose to the free-electron mass m0. Due to their well-defined energy spectrum, which weakly depends on the microscopic details of the surface underconsideration, image-potential states have been widely usedas a model system for both theoretical and experimentalinvestigations of the dynamics of electronic excitations atmetal surfaces. 11 The total decay rate (or linewidth) /Gamma1=¯h/τ (τbeing the lifetime) of an electronic excitation is determined by the contri-butions of four different processes: inelastic electron-electronscattering ( /Gamma1 e-e), electron-phonon ( e-ph) interaction ( /Gamma1e-ph), electron-defect scattering ( /Gamma1e-d), and energy-conserving res- onant one-electron tunneling into the bulk ( /Gamma11e e-e). The latter process contributes to the decay rate only when the initialelectronic state is degenerate with projected bands of the bulk metal.12–14 The temperature dependence of the dynamics of electrons and holes in the surface states has been extensively stud-ied both theoretically and experimentally on various metalsurfaces, 15–33but only few works considered the temperature dependence of the image state lifetimes.20,34–36 The thermal change of the decay rate is usually attributed to thee-ph interaction15,22,27,28,32,35–38and scattering on thermally activated defects,19,24while the electron-electron scattering contribution is usually assumed to be independent of temper-ature. This approximation is justified for the surface states,as far as e-ph interaction is rather strong in these states and /Gamma1 e-phoften exceeds /Gamma1e-eat room temperature. However, e-ph scattering in image-potential states has been predicted to bequite weak. 34–36The theoretical e-ph coupling parameter is typically λe-ph<10−2, which cannot account for the thermal decrease of the n=1 image state lifetime from 22 ±3f s at 25 K to 14 ±3 fs at 350 K observed in two-photon photoemission for a Cu(111) surface.20Hence, in order to investigate theoretically the temperature dependence of thedecay rate of electrons in image-potential states, it is necessaryto take into account the temperature dependence of the inelasticelectron-electron contribution. In this article we present a comprehensive study of the temperature dependence of the dynamics of electrons in the(n=1) image-potential state on Ag(111) by means of time- resolved two-photon photoemission spectroscopy (2PPE) andtheoretical many-body calculations. Photoemission experi-ments reveal a temperature dependence of the energy and dy-namics of the first image-potential state. The image-potentialstate shifts to lower energies with increasing temperature.In addition, a decrease of the inelastic lifetime is observed.We perform many-body calculations of the electron-electronand electron-phonon contributions to the decay rate of the(n=1) state in the temperature range from 0 to 300 K. We account for the temperature dependence of /Gamma1 e-eby taking into consideration the experimentally measured temperaturedependence of the band structure. 085424-1 1098-0121/2012/86(8)/085424(7) ©2012 American Physical SocietyS. S. TSIRKIN et al. PHYSICAL REVIEW B 86, 085424 (2012) In Sec. IIwe outline the theoretical methods used to cal- culate the decay rate. Experimental techniques are describedin Sec. III. The results of the experiments and calculations are presented and discussed in Sec. IV. The article is summarized in Sec. V. Unless otherwise explicitly stated, atomic units are used in the equations, i.e., ¯ h=m 0=e=1. II. THEORY A. Inelastic electron-electron scattering The calculations of the inelastic electron-electron scattering are performed in the self-energy formalism of many-body the-ory using the GW approximation. 39The method is described in detail elsewhere,11and here we give just a brief overview. Within this formalism the inelastic electron-electron scatteringcontribution to the decay rate /Gamma1 e-eof the electronic state in bandiwith wave function /Psi1i,ki, energy Ei,ki, and wave vector kiis obtained as the projection of the imaginary part of the self-energy operator /Sigma1onto this state: /Gamma1e-e=−2/angbracketleftbig /Psi1i,ki/vextendsingle/vextendsingleIm/Sigma1/vextendsingle/vextendsingle/Psi1i,ki/angbracketrightbig =−2EF<Ef,kf<Ei,ki/summationdisplay f,kf/integraldisplay/integraldisplay/bracketleftbig /Psi1∗ i,ki(r)/Psi1f,kf(r) ×ImW/parenleftbig r,r/prime,/vextendsingle/vextendsingleEi,ki−Ef,kf/vextendsingle/vextendsingle/parenrightbig /Psi1∗ f,kf(r/prime)/Psi1i,ki(r/prime)/bracketrightbig drdr/prime (2) Here the self-energy is represented by the first term of the expansion in terms of the screened Coulomb interaction W, which is calculated within the random phase approximation.The summation is carried out over all final electronic states(f,k f) with energies between the Fermi energy EFand the energy of the initial state Ei,ki. Thus, the many-body decay rate is determined by three main factors: (i) the phase spaceof the final states ( f,k f), (ii) the overlap between the wave functions of the initial and final states, and (iii) the magnitudeof the imaginary part of the screened Coulomb interactionImW. The latter is given in linear response theory by W(r,r /prime;ω)=V(r−r/prime)+/integraldisplay/integraldisplay [V(r−r1) ×χ(r1,r2;ω)V(r2−r/prime)]dr1dr2, (3) where V(r−r/prime) is the bare Coulomb interaction and χ(r1,r2;ω) is the density-density response function of the interacting electron system, which is evaluated from theequation χ(r 1,r2;ω)=χ0(r1,r2;ω)+/integraldisplay/integraldisplay [χ0(r1,r3;ω) ×V(r3−r4)χ(r4,r2;ω)]dr3dr4.(4) Hereχ0(r1,r2;ω) is the density-density response function of a noninteracting electron system: χ0(r1,r2;ω)=2/summationdisplay α,kα/summationdisplay β,kβθ/parenleftbig EF−Eαkα/parenrightbig −θ/parenleftbig EF−Eβkβ/parenrightbig Eαkα−Eβkβ+ω+iη ×/Psi1αkα(r1)/Psi1∗ βkβ(r1)/Psi1βkβ(r2)/Psi1∗ αkα(r2).(5)In this equation αandβindicate the band numbers, kαand kβare wave vectors, and ηis an infinitesimally small positive constant. The formalism outlined above does not contain temperature explicitly. However, the contributions to the decay rate /Gamma1e-eare determined by electronic energies En,kand wave functions /Psi1n,k. Thus, in order to study the thermal change of /Gamma1e-e, we take into account the temperature dependence of theelectronic structure using a procedure, described below inSec. II C. B. Electron-phonon scattering The contribution of the electron-phonon ( e-ph) interaction to the decay rate /Gamma1e-phis expressed in terms of the Eliashberg function40α2F(ω) which accounts for phonon emission ( E) and absorption ( A) processes in scattering of electronic excitations: /Gamma1e-ph=2π/integraldisplayωmax 0/braceleftbig α2FA i,ki(ω)/bracketleftbig n(ω)+f/parenleftbig Ei,ki+ω/parenrightbig/bracketrightbig +α2FE i,ki(ω)/bracketleftbig 1+n(ω)−f/parenleftbig Ei,ki−ω/parenrightbig/bracketrightbig/bracerightbig dω, (6) where n(ω) and f(/epsilon1) are the Bose and Fermi functions, respectively, and ωmaxis the maximum phonon frequency. In the high-temperature limit the e-ph decay rate grows linearly with temperature ( T), /Gamma1e-ph=2πλi,kiT, (7) where λi,kiis thee-ph coupling parameter defined as λi,ki=/integraldisplayωmax 0α2FE i,ki(ω)+α2FA i,ki(ω) ωdω. (8) The Eliashberg spectral function of an electron or hole is defined as α2FA(E) i,ki(ω)=1 (2π)2/integraldisplay d2q/summationdisplay ν,fδ(ω−ωq,ν)|gi,f(ki,kf,q,ν)|2 ×δ/parenleftbig Ei,ki−Ef,kf±ωq,ν/parenrightbig , (9) where gi,f(ki,kf,q,ν)i st h e e-ph matrix element reproducing the probability of a transition from an initial state ( i,ki)t o a final state ( f,kf) by emission or absorption of a phonon with a frequency ωq,νand momentum q, which satisfies the relation ±(kf−ki)=q+G/bardbl. (10) Here “ +” and ‘‘ −” signs correspond to phonon absorption and emission processes, respectively. G/bardblis a two-dimensional reciprocal lattice vector. The summation in (9)is carried out over all possible final states /Psi1kfand all phonon modes. The matrix element is defined by gi,f(ki,kf,q,ν)=/parenleftbigg1 2Mω q,ν/parenrightbigg1/2/angbracketleftbig /Psi1i,ki/vextendsingle/vextendsingleˆεq,ν·∇RVe-i/vextendsingle/vextendsingle/Psi1f,kf/angbracketrightbig , (11) withMbeing the mass of an atom, ˆ εq,νthe phonon polar- ization vector, and ∇RVe-ithe gradient of an electron-ionic pseudopotential with respect to atomic positions R. To calculate the matrix element (11) we use the method developed in Refs. 22and37. This method consists of three 085424-2TEMPERATURE DEPENDENCE OF THE DYNAMICS OF THE ... PHYSICAL REVIEW B 86, 085424 (2012) Evac 4 3 2 1 0 –0.5 0 0.5 kII (–1)E – EF (eV)M Γ M Ag(111)SSn=1n=2 5.8 6.0 6.2 6.4 Final State Energy E–EF (eV)2PPE signal (arb. units)300 K 151 K 30 KSS 5.8 6.0 6.2 6.4n=1 –100 0 100 200 30010–310–210–1100101n=1 300 K 151 K 30 K Pump–probe delay (fs)2PPE signal (arb. units) –100 0 100 200 300EFEvac ωUV ωvis SSn=1 0 100 200 300–0.30–0.26–0.22 EedgeElower T (K)0.00 –0.04 –0.08E – EF (eV) SS3.763.803.84n=13.883.923.96 EedgeEupper(a) (b) (c) (d) FIG. 1. (Color online) (a) Electronic surface band structure of Ag(111). Gray areas indicate projected bulk states. The dashed and solid lines represent the dispersion of the Shockley surface state (SS), the ( n=1) image-potential state, and the ( n=2) image-potential resonances, respectively. EFandEvacare Fermi and vacuum level, respectively. (b) Normalized 2PPE spectra as functions of the final-state energy with respect to the Fermi energy EF. The spectra were recorded at the indicated sample temperatures and analyzed for normal emission. The inset depicts the excitation schemes for the ( n=1) image-potential state and the Shockley surface state (SS). (c) Time-resolved pump-probe traces at the energies of the ( n=1) image-potential state for different sample temperatures. The solid lines are fits of a rate equation model. The dashed line represents the cross correlation of the laser pulses. (d) Temperature-dependent energy of the Shockley surface state (SS), the ( n=1) image-potential state, as well as of the lower Elower edgeand the upper Eupper edge edges of the projected bulk gap. independent approximations: (i) the electron-ionic potential Ve-iis taken to be the Ashkroft empty-core pseudopotential, screened in Thomas-Fermi approximation, (ii) phonon spectraare calculated on the basis of the embedded atom methodderived interatomic potentials, 41and (iii) the electronic struc- ture of the surface is calculated within a one-dimensionalpotential, 42described in Sec. II C. It should be noted, that the electronic energies and wave functions depend on temperature,and hence the matrix element (11), the Eliashberg function (9), and the e-ph coupling constant (8)are also functions of temperature. C. Model for the description of electronic states In order to describe the electronic energies En,kand wave functions /Psi1n,kwe utilize a model pseudopotential,42,43which varies only in the direction zperpendicular to the surface and remains constant in the plane of the surface. The parameters ofthe pseudopotential are adjusted to reproduce the main featuresof the surface band structure at the /Gamma1point: the energies of the lower ( Elower edge) and the upper ( Eupper edge) edges of the projected bulk band gap, the Shockley surface state ESS, and the ( n=1) image-potential state En=1. This method has been widely used for calculations of the lifetimes of excitations in surface statesand image-potential states on close-packed surfaces of variousmetals. 44 The energies of the Shockley surface state and the ( n=1) image-potential state are measured in the present experimentat different temperatures in the range from 30 to 300 K. Tocalculate the temperature-dependent edges of the energy gapwe utilize the multiple-reflection model. 9,45This model allows one to calculate the positions of the Shockley state and the(n=1) image state from the positions of the edges of theenergy gap. 9,45In this work we solve the converse problem (analogous to Ref. 20) to calculate the energies of the gap edges from the energies of the Shockley state and the firstimage-potential state. Some estimations of the accuracy ofsuch an approach should be made. The model was used inRef. 46to reproduce the experimentally measured temperature dependence of the energy E SS(T) of the Shockley states on the (111) surfaces of noble metals. It was found, that this modelaccurately reproduces the slope dE SS/dT, while the calculated dependence ESS(T) is shifted from the experimental data by ∼50 meV . Thus we may suppose that the multiple-reflection model reproduces the positions of the gap edges correctly withan accuracy of ∼50 meV . The resulting values E lower edge,Eupper edge,ESS, and En=1are presented in Fig. 1(d). In order to account for the temperature dependence of the band structure, we adjust different sets ofparameters of the pseudopotential for different temperaturesin the range from 0 to 300 K. Thus, the values E lower edge,Eupper edge, ESS, and En=1calculated with the temperature-dependent pseudopotential coincide with those presented in Fig. 1(d). III. EXPERIMENT The experiments were performed in an ultrahigh vacuum chamber with a base pressure of 5 ×10−11mbar.47,48A Ti:sapphire laser amplifier system was used to pump an opticalparametric amplifier which provided laser pulses with a photonenergy of ¯ hω vis=2.13 eV and a pulse duration of around 55 fs. The output was split into two parts. One part wasfrequency doubled in order to obtain ¯ hω UV=4.27 eV/55 fs laser pulses, while the second part was guided over a motor-driven delay stage. Then the p-polarized laser pulses were 085424-3S. S. TSIRKIN et al. PHYSICAL REVIEW B 86, 085424 (2012) aligned collinearly and focused onto the sample at an angle of incidence of 72◦relative to the surface normal. The photoemitted electrons were analyzed with respect to theirkinetic energy and emission angle using a hemisphericalelectron analyzer with an angle-resolved lens mode (SpecsPhoibos 150) and detected with a two-dimensional charge-coupled-device detector. 49The overall energy resolution of the 2PPE experiment was /Delta1E/lessorsimilar30 meV . The Ag(111) crystal was prepared by repeated sputtering and annealing cycles asdescribed in Ref. 14. The surface quality was checked by x-ray photoemission spectroscopy and low-energy electrondiffraction. IV . RESULTS AND DISCUSSION The schematic plot of the Ag(111) band structure is presented in Fig. 1(a). The Shockley surface state (SS) and the (n=1) image-potential state are located in the projected bulk gap near the /Gamma1point. Image-potential states with quantum numbers n/greaterorequalslant2 are degenerate with projected bulk states, and hence they form image-potential resonances. Figure 1(b) shows three 2PPE spectra at /Gamma1in the energy region of the ( n=1) state and the Shockley state signals for sample temperatures of 300, 151, and 30 K. The 2PPE spectrawere normalized and scaled differently in the respectiveregions for better comparison. With increasing sample temper-ature, the signal of the ( n=1) state shifts to lower final-state energies. Simultaneously, a shift to higher energies is observedfor the 2PPE signal of the Shockley state. The inset in Fig. 1 depicts the applied excitation scheme for the ( n=1) state and the Shockley state. Electrons are excited into the ( n=1) state by the UV pulses (¯ hω UV) and subsequently photoemitted by absorption of photons ¯ hωvis. Electrons from the Shockley state are photoemitted in a direct, nonresonant 2PPE process withabsorption of one photon of each pulse (¯ hω UV+¯hωvis). Taking into account the different photoemission pathways, the bindingenergies of the ( n=1) state and the Shockley state with respect to E Fare shown as functions of sample temperatures in Fig. 1(d). The temperature dependence of the energy of the Shockley surface state ESShas a plateau at T< 100 K which is in good agreement with low-temperature measurementsby means of scanning tunneling spectroscopy [ −65 meV at T=5K( R e f . 50) and at T=50 K (Ref. 51)]. The energy calibration of the 2PPE spectra was matched in such a way thatthe Shockley state is found at E−E F=−0.065 meV for low temperatures in accordance with results from photoemissionspectroscopy [ −60 meV at T=5K( R e f . 46)]. For T> 100 K E SSgrows linearly with temperature. This dependence is also in agreement with data reported in Ref. 46. The energy of then=1 image state En=1decreases linearly from 3.83 eV atT=0 K to 3.80 eV at T=300 K. This is in agreement with the previously reported value En=1=Evac−0.77 eV = 3.79 eV (Ref. 52) for room temperature. [The work function Evac−EF=4.56 eV (Ref. 52) is used.] The temperature-dependent dynamics of electrons excited into the ( n=1) state were investigated by time-resolved 2PPE. In the pump-probe traces that are shown in Fig. 1(c)the (n=1) state intensity is plotted as function of the relative delaybetween the laser pulses. For negative pump-probe delaysthe photoemission pulses (¯ hω vis) arrive at the sample beforeFIG. 2. (Color online) Probability distribution of the first image- potential state at T=28 K (solid line), T=151 K (dotted line), and T=300 K (dashed line), and the surface state at T=300 K (light gray line), integrated over directions parallel to the surface. The zaxis is perpendicular to the surface; z=0 corresponds to the outermost atomic layer. the UV excitation, while positive delay times correspond to delayed photoemission pulses. The lifetimes τof the exponential decay of the excited ( n=1) state population has been extracted from the pump-probe traces by a fit using arate equation model. This model includes the cross correlationof the laser pulses and an exponential population decay. Theresults of the fits are shown as thin solid lines in Fig. 1(c).T h e experimentally determined population decay rate is obtainedfrom/Gamma1=¯h/τand corresponds to the inelastic decay rate of the excited electrons. The experimental data show that not only thebinding energy of the ( n=1) state is modified with increasing temperature, but also the decay rate increases. The signalwhich is observed for negative pump-probe delays originatesfrom hot electrons. The nonthermal distribution is excitedby the intense ¯ hω vispulse and electrons are photoemitted by absorbing UV photons.53 Using the multiple-reflection model, we calculate the temperature dependence of the positions of the edges ofthe projected bulk gap at the /Gamma1point.9,20,45The zero- temperature limits Eupper edge(T→0)=3.96 eV and Elower edge(T→ 0)=−0.30 eV are not far from the result of first-principles calculations Eupper edge=3.9 eV and Elower edge=−0.40 eV .42 When the temperature is raised, the upper edge of the projected gap and the ( n=1) state move downward with the rates dEupper edge/dT=−0.26 meV /K and dEn=1/dT= −0.10 meV /K, respectively. Thus, the ( n=1) state ap- proaches the gap edge, and therefore its wave function ismodified. As demonstrated in Fig. 2, at higher values of Tthe main probability peak of the ( n=1) state is slightly decreased, while the weight inside the crystal is increased. 085424-4TEMPERATURE DEPENDENCE OF THE DYNAMICS OF THE ... PHYSICAL REVIEW B 86, 085424 (2012) (a) (b) (c) FIG. 3. (Color online) (a) Electron-electron scattering contri- bution /Gamma1e-eto the decay rate of the ( n=1) image-potential state (filled squares) and contributions from scattering to bulk ( /Gamma1B e-e,fi l l e d circles) and surface ( /Gamma1S e-e, filled triangles) states; (b) dependence of the bulk contribution /Gamma1B e-efrom the penetration integral Pn=1 (empty circles); and (c) penetration integral Pn=1(filled diamonds) as functions of temperature. Dashed lines — linear approximation of the calculated data; dotted line in b— proportional approximation /Gamma1B e-e=Pn=1×130 meV . The inelastic electron-electron contribution to the decay rate of the ( n=1) image-potential state contains contributions from scattering into bulk electronic states /Gamma1B e-eand into the surface state /Gamma1S e-e[Fig. 3(a)]./Gamma1S e-econtributes 30% to the total decay rate at T=0 and slightly grows with temperature: d/Gamma1S e-e/dT=0.006 meV /K./Gamma1B e-eincreases considerably faster with temperature ( d/Gamma1B e-e/dT=0.022 meV /K) and mostly determines the linear growth of the total decay rate /Gamma1e-e. This is the direct consequence of the change of the wave function of the ( n=1) state. As follows from Eq. (2),/Gamma1e-e is roughly proportional to the overlap between the charge densities of the initial state |/Psi1n=1|2and of the final states |/Psi1f,kf|2. In the case of transitions into bulk states this overlap may be estimated by the penetration integral Pn=1=/integraldisplayas/2 −∞|/Psi1n=1(z)|2dz, (12) as=2.35˚A being the interlayer spacing of the silver in the [111] direction. Figure 3(b) demonstrates the proportionality between Pn=1and/Gamma1B e-e. As shown in Fig. 3(c),Pn=1grows linearly with temperature, resulting in the linear growthof/Gamma1 B e-e. As can be seen in Fig. 2,t h e(n=1) wave function in the vicinity of z=0 (where the surface state is mostly localized) practically does not depend on temperature, and hence /Gamma1S e-eis weakly modified with temperature. This behavior is similar tothat observed in the study of the wave-vector dependence of thedecay rates of image states on Cu(111) and Ag(111): 48,54Theapproach of the ( n=1) state to the gap edge results in deeper penetration of the ( n=1) wave function into the bulk and enhances the scattering into bulk states, while the scatteringrate into the Shockley state remains practically constant. We find that the electron-phonon scattering of electrons in the ( n=1) image-potential state is rather weak. The e-ph coupling constant (8)varies from λ=4×10 −3atT=0t o λ=5×10−3atT=300 K for Ag(111). These values are the same order of magnitude as for the ( n=1) image state on Pd(111)36(λ=2×10−3), Cu(100)34(λ=10−2), Ag(100)34 (λ=5×10−3), and the first two image states on Pd(110)35 (λ=2×10−3). The resulting contribution to the decay rate /Gamma1e-phdoes not exceed 1 meV up to room temperature, and one may expect that the contribution of the e-ph scattering to the dephasing rate is also very small. However, a significantly larger value of the electron-phonon coupling constant λ=6×10−2, was reported for the ( n=1) image-potential state of Cu(111) by Knoesel et al.20Given the similar band structure of Cu(111) and Ag(111) surfaces thislarge difference between e-ph coupling strengths of Cu(111) and Ag(111) is surprising and deserves further attention.Therefore, we also calculated the e-ph coupling constant for the (n=1) image state on Cu(111). We obtained the value λ=7×10 −3, which is the same order of magnitude as for Ag(111), as well as for the other surfaces listed above. Thereare several reasons why the experiment of Ref. 20might have overestimated the e-ph coupling constant λ. First of all, Knoesel et al. extracted λfrom the temperature dependence of the pure dephasing rate T ∗ 2which was determined from a comparison of 2PPE decay rates and linewidths.20For small T∗ 2 times, however, this method is generally less straightforward than, e.g., quantum-beat spectroscopy or direct methods ofdetermining quasielastic electron-scattering rates. 55–57More- over, the increase of the pure dephasing rate with temperaturecan have other origins than coupling to phonons. Many defectsare known to scatter image-potential electrons efficiently. 58 Stronger penetration of the image state wave function intothe bulk at higher temperature, discussed above, is expectedto lead to an increase of the defect scattering rate withtemperature. Also thermally activated defects 19,24contribute to the temperature dependence of the scattering rate. The temperature dependence of the calculated ( /Gamma1Th= /Gamma1e-e+/Gamma1e-ph) and measured ( /Gamma1Ex) decay rates of the image- potential state is shown in Fig. 4. Both, theory and experiment show linear growth of the decay rate with temperature. The cal-culated slope d/Gamma1 Th/dT=0.030 meV /K is also in agreement with the value d/Gamma1Ex/dT=0.034 meV /K, obtained by linear fitting the experimental data. However, /Gamma1This approximately 16 meV larger than /Gamma1Exover the whole temperature range considered. There may be two reasons for such discrepancy. First, /Gamma1Thmay be overestimated because of neglecting thedelectrons in the present calculations. The inclusion ofdelectrons decreases the surface plasmon energy below the energy of ( n=1) state on Ag(111).59,60Although this opens an extra inelastic decay channel for electrons in theimage-potential state, it has been shown 59,60that the lifetimes of image states are increased due to the strongly nonlocalcharacter of the self-energy near the surface. For example, thesurface plasmon channel reduces the decay rate of the ( n=1) image-potential state on Ag(111) by 8 meV . 60 085424-5S. S. TSIRKIN et al. PHYSICAL REVIEW B 86, 085424 (2012) 2015 25 30 35 40 τ(fs) 0 100 200 300203040 T (K)/τ ( meV)Theory Experiment FIG. 4. The measured (open dots) and calculated (filled squares) decay rate of the n=1 image-potential state on Ag(111) as a function of temperature. Dashed (dotted) lines represent linear interpolationsof the experimental (theoretical) data. The slopes are 0.034 meV /K (0.030 meV /K). Second, the overestimation of /Gamma1Thmay be attributed to the uncertainty of the determination of the upper gap edge position.According to our calculations, the main reason for the thermalshortening of the lifetimes is the change of the penetrationof the ( n=1) wave function into the bulk metal due to the approach of the image state to the upper gap edge. The decayrate depends linearly on the energy difference E upper edge−En=1, while the derivative d/Gamma1Th/d(Eupper edge−En=1)=−0.18. Thus, a 50 meV inaccuracy in the determination of Eupper edge−En=1may cause an inaccuracy ∼10 meV in the calculated decay rate of the ( n=1) image state. However, such an inaccuracy only shifts the /Gamma1Th(T) dependence by a certain value, while leaving the slope d/Gamma1Th/dT unchanged. V . SUMMARY In summary, we have investigated the temperature de- pendence of the lifetime of electrons in the n=1 image potential state on the Ag(111) surface by means of time-resolved two-photon photoemission spectroscopy and many-body calculations. 2PPE experiments show that the decayrate/Gamma1of the ( n=1) image-potential state grows linearly with temperature. Theoretical investigations confirm this resultand reveal the origin of such behavior: the shortening of thelifetime is caused by the increase of the electron-electronscattering rate, due to deeper penetration of the image statewave function into the bulk metal at higher temperature.The contribution of electron-phonon scattering to the decayrate was found to be very small. The calculated derivatived/Gamma1 Th/dT=0.030 meV /K is close to the experimental value d/Gamma1Ex/dT=0.034 meV /K. ACKNOWLEDGMENTS We thank P. M. 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PhysRevB.101.235448.pdf

PHYSICAL REVIEW B 101, 235448 (2020) First-principles calculation of shift current bulk photovoltaic effect in two-dimensional α-In2Se3 Rajender Prasad Tiwari , Balaji Birajdar , and Ram Krishna Ghosh* Special Center for Nano Sciences, Jawaharlal Nehru University, New Delhi 110067, India (Received 24 February 2020; revised manuscript received 13 May 2020; accepted 4 June 2020; published 30 June 2020) Shift current is the dominant dc-current response in the bulk photovoltaic effect (BPVE), which is the conversion of solar energy into electricity in the materials with broken inversion symmetry. While the guidingprinciple of BPVE is a lack of inversion symmetry in a material which also results in ferroelectricity, it istherefore, expected that a significantly large shift current is achieved in ferroelectric materials. In this work,we calculate shift current using first principles in two-dimensional α-In 2Se3which has both in-plane and out-of-plane polarization at room temperature. To understand the implications of in-plane and out-of-planepolarization on shift current BPVE, mono- and bilayer structures of 3 Rand 2 Hα-In 2Se3are considered in our calculations. It suggests that the in-plane polarization doesn’t affect the shift current response of this material.In monolayer, a dominant shift current response of magnitude 750 μA/V 2is obtained along the direction of out-of-plane polarization under uniform illumination of zz-polarized light at a photon energy of 4.16 eV. The doping engineering is further implemented to tune the shift current response to visible light. Bismuth (Bi)is used to substitute the indium element at the tetrahedral site thereby introducing more energy levels in theconduction band which importantly are contributed by porbitals of Bi and take part in the transition process. Consequently, a giant shift current of 1200 μA/V 2is obtained at a photon energy of 2.98 eV. The present study would be an instrumental in understanding the shift current BPVE and would pave the path for designing efficientphotovoltaic devices based on α-In 2Se3and similar material systems. DOI: 10.1103/PhysRevB.101.235448 I. INTRODUCTION The conversion of solar energy into electric current in the materials with broken inversion symmetry is called bulk pho-tovoltaic effect (BPVE) [ 1]. It is a technologically important phenomenon as it can generate a photovoltage more than theband gap of the material [ 2]. Moreover, its photoconversion efficiency is not limited by the Shockley-Queisser limit unlikethe conventional p-njunction solar cells [ 3–5]. Therefore, BPVE is viewed as one of the most efficient alternativesources of green energy. Because of its prospective scope insolar energy harvesting applications, intense research is beingdone to understand or formulate a mechanism to improveits efficiency. Though the necessary ingredient to optimizethe BPVE is still not fully understood, a material with theband gap in visible range (1.1–3.1 eV) [ 5,6], large electronic densities of states near the band edge, and high anisotropy arebeneficial [ 5,7,8]. The latter ensures a preferential direction of current (called “shift current”) flow through such material-based devices under uniform illumination. The shift current BPVE is a nonlinear optical process that arises from second-order interaction with monochromaticlight. Photoexcitation of electrons from one band to anotheraccompanying a coordinate shift allows a net current flowfrom the asymmetry of the potential resulting in a shift current[3–5]. The shift current is a dominant dc-current response *Author to whom correspondence should be addressed: ramki.phys@gmail.comin BPVE [ 7] therefore, the shift current and BPVE are used interchangeably in this paper. The shift current tensor is givenas [8–11] σ abc(0;ω,−ω) =−iπe3 2¯h2/integraldisplay [dk]/summationdisplay n,mfnm/parenleftbig rb mnrc nm;a+rc mnrb nm;a/parenrightbig ×δ(ωmn−ω), (1) where a,b,care Cartesian indices, nand mare the band indices, fnm=fn−fmis the Fermi-Dirac occupation number, ωnm=ωn−ωmis the band energy difference, rb mnis the dipole matrix elements, rb nm;arepresents generalized deriva- tives, the integral is over the first Brillouin zone with [ dk]= dkd/(2π)dinddimensions, and kis the wave vector. The dipole matrix elements are defined as rb mn≡Ab mnwhen n/negationslash=m or zero otherwise [ 7]. The term Ab mnis the Berry connection matrix given as Ab mn=i/angbracketleftum|∂kb|un/angbracketright, where |un/angbracketrightdenotes the cell periodic part of a Bloch eigenstate. The generalizedderivative is defined as r b nm;a=∂karb nm−i(Aa nn−Aa mm)rb nm. Under a monochromatic light of form Eb(t)= Eb(ω)eiωt+Eb(−ω)e−iωtwith frequency ωand linearly polarized along the direction b, the dc-photocurrent density (J) from the linear BPVE is given as [ 7,10] Ja shift(ω)=2/summationdisplay bσabb(0;ω,−ω)Eb(ω)Eb(−ω) (2) 2469-9950/2020/101(23)/235448(9) 235448-1 ©2020 American Physical SocietyTIWARI, BIRAJDAR, AND GHOSH PHYSICAL REVIEW B 101, 235448 (2020) and the second-order response function for the shift current becomes σabb(0;ω,−ω)=πe3 ¯h2/integraldisplay [dk]/summationdisplay n,mfnmRa,b nm ×rb nmrb mn×δ(ωmn−ω), (3) where Ra,b nmis a shift vector [ 9,10] defined as Ra,b nm=∂kaφb nm−Aa mm+Aa nn;( 4 ) hereφb nmis the phase of rb nm=|rb nm|e−iφb nm. The shift vector Ra,b nmhas a unit of length and can be physically interpreted as, on average, the displacement of coherent carriers duringtheir lifetimes when they are pumped from band mto band n. According to Fermi’s golden rule, the product r b nmrb mn× δ(ωnm−ω)=|rb nm|2δ(ωnm−ω)i nE q .( 3) can be interpreted as a transition rate from band mto band n. Therefore, the shift current response is expressed as the shift vector multiplied bythe transition rate. BPVE has long been studied intensively in conventional ABO 3ferroelectric oxides because the fundamental require- ment of broken inversion symmetry is well satisfied in suchmaterials. However, in most of such material systems, thebands involved in the transition process are strongly con-tributed by the localized dorbitals which tend to decrease the shift current [ 12]. Moreover, studies show that the shift current can be improved by choosing the material systems with a largejoint density of states where the band edge is closely alignedwith the peak of the solar spectrum [ 5,12]. Since the band edge always induces a Van Hove singularity in the density of states,the requirement of a large peak in the photoresponse can benaturally better satisfied by low dimensional materials, whichgenerically present stronger singularities [ 5,7,12]. Therefore, the recent development of a number of two-dimensional(2D) materials with broken inversion symmetry has openedup a new paradigm to the BPVE. Recently, Rangel et al. reported a large shift current ( ∼100μA/V 2) in single-layer GeS monochalcogenide [ 8]. Interestingly, the reported shift current in single-layer GeS (thickness ∼2.6 Å) is far larger than that in the most studied ferroelectric perovskite oxidessuch as BaTiO 3(30μA/V2), and PbTiO 3(50μA/V2)i nb u l k [4], suggesting that the 2D van der Waals (vdW) layered materials are promising for photovoltaic applications at anultimate scaling thickness. The polar 2D vdW materials, especially, are more interest- ing because a large shift current BPVE is reported in such ma-terial systems [ 12]. Although polarization and shift current are not linked in any specific way despite the fact that both orig-inate from inversion symmetry breaking, it is reported that amaterial with larger polarization would significantly enhancethe bulk photovoltaic response [ 6,13–15]. The shift current response has been studied in many material systems includ-ing perovskite oxides [ 3,4], organometal-halide perovskites [16], and 2D material system [ 8,10,11], etc., however, to our knowledge it has been studied in the material systems in whichpolarization exists in one direction (i.e., either in plane or outof plane). But a recent development in the 2D material systemwith broken inversion symmetry leads to the emergence of thegroup III 2−VI3vdW materials which are reported to possess a robust in-plane as well as out-of-plane polarization [ 17–19].Understanding the implications of coexisting in-plane and out-of-plane polarization on shift current is, therefore, vitalin down selecting the material and designing the photovoltaicdevices with improved efficiency. Therefore, in this study,we choose α-In 2Se3, a crystalline family member of In 2Se3, which is a prototypical 2D layered material system havingboth in-plane and out-of-plane polarization. In 2Se3is an intriguing III −VI binary chalcogenide with diverse electronic properties and crystalline polymorphismexisting in at least five different crystalline phases, viz.,α,β,γ,δ, and κ[20,21]. At room temperature, α-In 2Se3 has a hexagonal crystal structure and is a multidirectional ferroelectric layered semiconductor with a robust in-planeand out-of-plane polarization [ 17,21]. It has been widely studied theoretically and experimentally with the view of itsapplications in two-dimensional memory and optoelectronicssuch as photodetectors [ 22–25]. However, the photovoltaic application of α-In 2Se3is still not well explored. Therefore, here we theoretically study its shift current BPVE applicationand implications of in-plane and out-of-plane polarization onit. We find that a large BPVE shift current of 750 μA/V 2 is obtained along the direction of out-of-plane polarization and a negligible shift current is obtained along the direc-tion of in-plane polarization direction when illuminated by azz-polarized and xx/yy-polarized light, respectively. Further- more, the highest shift current response lies in the ultravioletrange ( ∼4.16 eV) which can be tuned effectively by doping. Wisely choosing the dopant and doping site, an enhanced shiftcurrent ( ∼1200μA/V 2) is obtained in single-layer InBiSe 3 system that importantly occurs in the visible spectrum range. II. METHODOLOGY Density functional theory (DFT) calculations were per- formed using the QUANTUM ESPRESSO package [ 26] with the ultrasoft pseudopotential [ 27] as parameterized by the Perdew-Zunger exchange correlation [ 28]. The plane-wave basis set with an energy cutoff of 70 Ry and 8 ×8×1 Monkhorst-Pack k-point grid was found to optimally con- verge self-consistent energy up to 10−5Ry using Davidson diagonalization algorithm [ 29]. Fermi-Dirac distribution with a width of kBT=0.005 Ry was used for smearing the oc- cupation of numbers of electronic states. A vacuum layer of20 Å along the Zaxis was used to minimize the interac- tions between the periodic images of the layers. In order toaccount for vdW interaction, we tested many vdW correc-tion functionals [ 30,31] and the vdW-df-obk8 functional was used as it defined the lattice constants of this material moreaccurately. The calculations were sufficiently converged toallow the atomic structures to be optimized until the residualforces on the atoms are less than 10 −4Ry/bohr. Uniform k-point grid of 8 ×8×1 and 30 ×30×1 was used for self- consistent and non-self-consistent calculations, respectively.Polarization was calculated by the quantum theory of po-larization approach [ 32] using Berry’s phase method. The hybrid functional Heyd-Scuseria-Ernzerhof-06 (HSE06) [ 33], as implemented in QuantumATK [ 34], was used to correctly describe the electronic band structures. Based on HSE06band-structure calculations, Hubbard Uwas further optimized using electronic property calculation [ 6] and used as a basis to 235448-2FIRST-PRINCIPLES CALCULATION OF SHIFT … PHYSICAL REVIEW B 101, 235448 (2020) FIG. 1. Showing the side view of an α-In2Se3quintuple (a); quintuple layers and their stacking in rhombohedral (3 R) and hexagonal (2 H) arrangement (b); variation of in-plane ( Pin) and out-of-plane ( Pout) polarization with the number of layers in these stacking arrangements (c). In a quintuple, the In atoms, In ocand In trare coordinated octahedrally and tetrahedrally, respectively, while the Se atoms at the top (Se tg1)a n d bottom layer (Se tg2) are coordinated trigonally, and the central Se tratom (encircled) is coordinated tetrahedrally via which the PinandPoutin a quintuple layer (QL) are interlinked. The alignment of in-plane polarization ( P/prime in) and out-of-plane polarization ( P/prime out) of the individual layer are also indicated (green arrow). take into account the correlation effect in Wannier functions inQUANTUM ESPRESSO . Since electronic bands near the Fermi level of In 2Se3are greatly influenced by porbitals of In and Se, therefore, in order to treat the large self-interaction errororiginating from porbitals, an optimum Hubbard Uof 4.6 eV was applied on the porbitals of both In and Se. Considering the optimized local-density approximation plus Hubbard U(USPP/LDA +Uor simply LDA +Uunless until specified) Kohn-Sham potential, maximally localizedWannier functions (MLWFs) were created using the Wan-nier90 code [ 35] in the slab geometry as obtained after orthog- onalizing the hexagonal α-In 2Se3. On obtaining MLWFs, the Wannier interpolation method was used to calculate the shiftcurrent conductivity using a refined kgrid of 200 ×200×1 and a boarding factor of 0.04 eV. The kgrid of 200 ×200×1 was chosen after performing the shift current convergence testwith the kpoints. Furthermore, to check the stability of band structures and the shift current response against the change ofpseudopotentials in α-In 2Se3, along with the Hubbard U,w e used ultrasoft pseudopotential with Perdew, Burke, and Ernz-erhof exchange correlation (USPP/GGA +U), and projector- augmented wave parametrized by Perdew-Zunger exchange-correlation functional (PAW/LDA +U). The optimized Hub- bard Uof 4.8 and 5.0 eV were used with USPP/GGA and PAW/LDA, respectively applied on the porbitals of both In and Se. III. RESULTS AND DISCUSSION Structure of α-In2Se3— Among the five ( α,β,γ,δ, and κ) known crystalline polymorphisms of In 2Se3,t h eα-In2Se3is a layered 2D semiconductor at room temperature and is a technologically important material system [ 19–21,24]. Using mechanical or chemical exfoliation, mono to a few layers ofα-In 2Se3is easily obtained [ 21,36]. The building block of α-In2Se3is a quintuple consisting of Se-In-Se-In-Se atomic sequence connected by the strong covalent bond vertically[Fig. 1(a)]. Similar quintuple layers (QLs) can be stacked stable by the weak out-of-plane vdW interactions. Dependingupon the geometrical orientation of stacked quintuple layers,theα-In 2Se3may adopt two different crystal structures: rhom- bohedral (3 R) or hexagonal (2 H)[21], as shown in Fig. 1(b). The 3 Rstructure has a three-layered cell unit comprising of identical QLs with a constant translation along the xyplane, belonging to the R3m (no. 160) space group, whereas the 2 H structure has a two-layered cell unit with each successive QLrotated by 180° in the xyplane to its preceding QL and it belongs to the P6 3/mc(no. 186) space group. The 3 Rand 2Hstructures are easily transferred into each other by turning and moving the layers along the xyplane. An optimized in-plane lattice constant of 4.0751 Å in 3 Rand 4.0761 Å in the 2Hstructure is obtained which matches closely the previous reports [ 17,21] (a lattice constant calculated using different pseudopotentials is shown in Table S1 in the SupplementalMaterial [ 37]). In addition to the enigma of QL stacking in α-In 2Se3, the Se and In atoms have different coordination polyhedra.In a quintuple, the Se atoms at the top (Se tg1) and bottom layer (Se tg2) are coordinated trigonally with In ocand In tr, respectively (see Fig. S1 in the Supplemental Material [ 37]). The In ocon the other hand is coordinated octahedrally by six Se atoms, and the In tris coordinated tetrahedrally by 235448-3TIWARI, BIRAJDAR, AND GHOSH PHYSICAL REVIEW B 101, 235448 (2020) FIG. 2. (a) Electronic band structure of single-layer α-In2Se3as obtained with LDA +U. It is an indirect band semiconductor with conduction-band minimum (CBM) at the /Gamma1point and the valance-band maximum (VBM) lies between /Gamma1andMpoints as indicated by an arrow; (b) fat band of single-layer α-In2Se3showing the contribution of sandporbitals of In and Se. four Se atoms, whereas the central Se tratom is tetrahedrally sandwiched by In ocand In tratoms. The Se tratom plays a vital role in persuading the distinguished attribute of coexistenceof in-plane and out-of-plane polarization in α-In 2Se3. In its low-energy symmetry, the Se tratom [Fig. 1(a)] undergoes a vertical movement along the Zaxis in addition to the lateral movement along the xyplane. Due to its vertical movement, there is a dramatic difference in the bond length betweenSe tr-Inocand Se tr-Intr(∼0.36 Å) leading to the breaking of out-of-plane centrosymmetry and hence an out-of-planepolarization ( P out) along the Zaxis, whereas the lateral move- ment of Se trleads to the breaking of in-plane centrosymmetry, resulting in an in-plane polarization ( Pin)i nt h e xyplane; therefore, PinandPoutare interlinked in this material system. In a QL of thickness 6.8 Å, we calculate a robust Pinand Poutof 0.53 nC /m and 0.023 nC /m, respectively which is quite larger than the previously reported in-plane polarizationinγ-SbP (0.38 nC /m) [38] and out-of-plane polarization in γ-LiAlTe 2[39]. Therefore, α-In2Se3can provide a robust multidirectional polarization at an ultimate scaling length ofa few angstroms. Interestingly, in a multilayer α-In 2Se3system, the strength of polarization depends upon the stacking style of the QL (i.e., 3 Ror 2H). Variation of Poutand Pinin 3Rand 2 H α-In2Se3with layers is shown in Fig. 1(c). If the in-plane and out-of-plane polarization of individual layer is represented asP /prime inandP/prime outas indicated in Fig. 1(b), then in both 3 Rand 2 H stacking arrangements, the P/prime outof each layer is always along the−z(or+z) axis, therefore, the total out-of-plane polariza- tionPoutgradually increases with the number of layers [ 40]. The strength of the net in-plane polarization Pincomponent, however, depends upon the QL stacking style. In 3 Rarrange- ment, it increases up to the trilayer system and thereafter itgradually weakens and eventually becomes smaller than P out after the quad layer. On the other hand, in the 2 Hsystem, thePinvanishes in an even number of layer structures and is nonzero in an odd number of layer structures because in thisstacking arrangement each successive QL is a mirror image ofits preceding QL in the xyplane [see Fig. 1(b)], therefore, P /prime incomponents cancel themselves in the even number of QLs. Our theoretical calculation of polarization in single andthe layered 3 Rand 2 Hstructures of α-In 2Se3are in goodagreement with the results reported earlier. For instance, a similar trend of polarization in the layered 3 Rstructure was earlier reported by Dai et al. [40] and Ding et al. [17]a s calculated using Berry’s phase method, while Cui et al. [25] reported an odd-even nature of in-plane polarization in thelayered 2 Hstructure. Band structure and shift current — It is seen above that the ferroelectric properties of α-In 2Se3are sensitive to the nature of stacking of QLs, however, the electronic band profileof these structures is essentially similar, suggesting that theinterlayer interaction is quite weak in nature [ 22,41]. A repre- sentative band structure of a single-layer α-In 2Se3is shown in Fig.2(a). Single-layer α-In2Se3is an indirect band gap semi- conductor with conduction-band minimum (CBM) at the /Gamma1 point and the valance-band maximum (VBM) lies between the/Gamma1andMpoints. The fat band of a QL [Fig. 2(b)] reveals that thep x/yorbitals of In oc(at octahedra site) and pz/xorbitals of Intr(at the tetrahedral site) form a degenerated orbital which combines with the porbitals of Se tg1(at the top layer) and the pzorbital of Se tr(at the tetrahedral site) to form the VBM, whereas the CBM is composed of the sorbital of In tr(at the tetrahedral site) hybridized with the porbitals of Se tg2(at the bottom layer) and the pzorbital of In tr(at the tetrahedral site). We calculate the band gap with different methods (Table I) which match closely with the previous reports [ 17–19,43]. The USPP/LDA +U(UIn=4.6e V , USe=4.6 eV) band gap ofα-In2Se3matches very well with the band gap calcu- lated using hybrid functional HSE06. The band structuresof single-layer α-In 2Se3, as calculated using LDA +Uand HSE06, are shown in Fig. S2 in the Supplemental Material[37]. Therefore, using the optimized LDA +UKohn Sham potentials, the MLWFs are further obtained. To perform theWannierization, a frozen window of 14 eV is chosen to containthe low-energy region, whereas the outer window extends upto 24 eV to capture the manifold of 50 bands. For initialprojections, we choose sandptrial orbitals of each atom. The LDA+Uand Wannier-interpolated energy bands are shown in Fig. 3. The Wannier interpolation is further extended to cal- culate shift current response in α-In 2Se3in the postprocessing step. Following Ref. [ 11], we report 3D-like shift current re- sponse ( σ) which is obtained by rescaling the response of the 235448-4FIRST-PRINCIPLES CALCULATION OF SHIFT … PHYSICAL REVIEW B 101, 235448 (2020) TABLE I. Electronic band gaps of single-layer 3 Rα-In2Se3calculated with different approaches. Results are also compared with previous reports. Band gap USPP/LDA +U USPP/GGA +U PAW/LDA +U HSE06 Previous reports Indirect (eV) 1.404 1.402 1.407 1.404 1.46a[17] Direct (eV) 1.416 1.414 1.415 1.467 1.55b[42] aObtained with HSE06. bElectronic band gap measured using high-resolution electron energy-loss spectroscopy (HR-EELS). slab as σ=(slab thickness /layer thickness) σslab.T h es h i f t current is calculated with the optimized k-point interpolation mesh obtained after performing the convergence test. With thekgrid of 30 ×30×1 in the non-self-consistent calculations, a well converged shift current spectrum of α-In 2Se3can be obtained by using the k-point interpolation mesh of 200 × 200×1. On further increasing the k-point interpolation mesh, the shift current changes insignificantly (see Fig. S3 in theSupplemental Material [ 37]). Furthermore, the stability of shift current against different pseudopotentials is also exam-ined (see Fig. S4 in the Supplemental Material [ 37]). Different pseudopotentials, USPP/LDA, USPP/GGA, and PAW/LDA,yield similar shift current response (with some qualitativedifferences mostly at the higher energy regions) with almostequal magnitude. Therefore, here we present the shift currentresponse calculated using USPP/LDA. Besides, we success-fully reproduce the previously reported shift current responseof bulk GaAs and single-layer GeS [ 11] by the Wannier90 code which indicates the robustness of our calculations (seeFigs. S5 and S6 in the Supplemental Material [ 37]). Theα-In 2Se3belongs to C3vsymmetry and has a mirror plane perpendicular to ˆ x, leading to five independent non- vanishing shift current response tensors [ 44]:Zxx=Zyy, Yy y=−Yxx=−Xxy=Xyx,Zzz,Xxz=Yy z, and Xzx= Yz y, where the upper case letter represents the direction of shift current susceptibility and the last two lower caseletters represent light polarization. For a monochromaticlight linearly polarized along a particular direction, we cal-culate longitudinal ( σ Zzz,σYy y) and transverse ( σZyy)s h i f t current responses. The longitudinal component, σZzz(σYy y), FIG. 3. LDA +Uand Wannier-interpolated energy bands of monolayer α-In2Se3.represents the shift current response along the Z(Y) axis, i.e., along the direction of out-of-plane (in-plane) polarization duetozz(yy) polarized light, whereas the transverse component σ Zyyrepresents the shift current response along the Zaxis due toyypolarized light. The longitudinal and transverse shift current responses in a single-layer α-In2Se3are shown in Fig. 4(a). Surprisingly, the shift current response σZzzis the only dominant current FIG. 4. (a) Longitudinal ( σZzz,σYy y) and transverse ( σZyy)s h i f t current responses in single-layer α-In2Se3. (b) Absorption of zz, yy,a n d xxpolarized light in α-In2Se3. A dominant shift current response σZzzalong Poutunder the illumination of zz-polarized light is indeed consistent with the larger absorption of zz-polarized light inα-In2Se3. 235448-5TIWARI, BIRAJDAR, AND GHOSH PHYSICAL REVIEW B 101, 235448 (2020) response in this material. In a QL of α-In2Se3with a thickness of 6.8 Å, the σZzzis strikingly large and is of the order of 750μA/V2at a photon energy of 4.16 eV. Previously, the shift current of ∼50μA/V2and∼30μA/V2was reported in the prototypical ferroelectrics PbTiO 3and BaTiO 3, respectively [4], and that of ∼40μA/V2was reported in the bulk GaAs [11] (see Fig. S5 in the Supplemental Material [ 37]). Addi- tionally, in 2D materials, the shift current of ∼100μA/V2is reported in single-layer GeS [ 8] (see Fig. S6 in the Supple- mental Material [ 37]). Therefore, the shift current in α-In2Se3 is quite larger than that in the previously reported 3D bulk and 2D materials. It is interesting to note in Fig. 4(a)that the σZzzis only a dominant current response and σYy yis nearly zero despite the fact that Pout<Pinin single-layer α-In2Se3. This observation raises two questions: (i) Why is σZzztheonly dominant current response and the σYy ycomponent of the shift current is negligible? (ii) Is the shift current not influenced by Pin in this material? In order to understand the first question, we calculate the dielectric absorptive spectrum (imaginarydielectric constant, ε im bb) which is given as [ 11] εim bb(ω)=iπe2 ¯h/integraldisplay [dk]/summationdisplay n,mfnmrb mnrb nm×δ(ωmn−ω).(5) The product of the last two terms in the right-hand side of Eq. (5),rb nmrb mn×δ(ωmn−ω)=|rb nm|2δ(ωmn−ω),represents the transition rate from band mto band n.F r o mE q s .( 5) and (3) it is clear that the shift current depends upon the transition rate, therefore, it inherits most of its features from the absorp-tion spectrum [ 4,8,11]. The absorption spectrum, shown in Fig.4(b), indicates the highly anisotropic nature of α-In 2Se3 which agrees well with the previous reports [ 36,45]. Higher absorption of zz-polarized light than the xx/yy -polarized light in single-layer α-In2Se3is indeed consistent with the large σZzztensor response in this material. Previously, a similar trend was also reported in the single-layer GeS 2D materialsystem [ 8,11]. It is to be noted in Fig. 4that the maximum absorption of zz-polarized light and the corresponding shift current response along the Zaxis,σ Zzz,occur at ∼4.16 eV which approximately equals the energy separation betweenthe upper valance band due to the Se porbitals and the lower conduction band due to the In sorbital (see Fig. S7 in the Supplemental Material [ 37]). Clearly, the peak of shift current response lies well outside the visible spectrum [liesin the ultraviolet (UV) range]. However, for efficient solarconversion, the peak of shift current response in the visiblespectrum is highly desired. Therefore, in the later section, wewill discuss a mechanism by which the shift current responseinα-In 2Se3can be brought inside the visible spectrum. Now, coming back to understand the question (ii) that if the shift current is not influenced by Pinin single-layer α-In2Se3, it is interesting to note that the Pinis greater than Pout,b u ta negligible shift current is obtained along the direction of Pin. Although the strength and direction of polarization are notdirectly related to the shift current in any obvious ways, themaximum shift current response has been reported along thedirection of polarization in many material systems (leavingfew exceptions, e.g., BiFeO 3[3]), including 2D materials. For example, in prototypical ferroelectrics, BaTiO 3, and PbTiO 3, FIG. 5. Shift current in the bilayer of 3 Rand 2 Hα-In2Se3.T h e bilayer of 3 Rhas both PoutandPinwhereas 2 Hhasonly P outnonzero component. The σZzzisonlya dominant shift current in 3 Rand 2 Hα- In2Se3along Poutunder zz-polarized light, also σZzzin 3Rand 2 Hare equal in magnitude; this confirms that Pindoesn’t affect shift current BPVE in α-In2Se3. the bulk polarization and shift current are reported along the same direction [ 4]. Similar reports are also available for perovskite halide CH 3NH 3PbI 3[16] and 2D material like GeS [ 8]. The α-In2Se3material system, however, is one of its kind in which Pinand Poutcoexist. The dominant shift current σZzzalong the direction of Pout[Fig. 4(a)] suggests that in-plane polarization has either no or less significance tothe shift current in this material. In order to further confirmthe influence of P inon shift current, we calculate the shift current in bilayer 3 Rand 2 Hα-In2Se3structures where in the former case there exist both Pinand Poutwhile in the latter structure there exists only P outas explained above in Fig. 1. Remarkably, in both 3 Rand 2 Hα-In2Se3bilayer structures, the σZzzisonly a dominant shift current response with almost equal magnitude as shown in Fig. 5.I ti st o be noted that to eliminate the spurious interlayer screening[46], the shift current in the bilayer is calculated using a vacuum level of 40 Å. The trend of shift current in mono- andbilayer structures confirms that the particular value of in-planepolarization ( P in) is insignificant in BPVE shift current in single-layer α-In2Se3; rather the symmetry breaking along with higher absorption spectrum is leading to the large shiftcurrent. Shift current engineering via doping — It is clear from the above discussion that in single-layer α-In 2Se3the maximum BPVE shift current response can be achieved along the Zaxis when illuminated by zz-polarized light. But the peak of shift current response lies well outside the visible spectrum in theultraviolet range [Fig. 4(a)]. The peak of shift current response in the visible spectrum is highly desired for efficient solarenergy conversion. Therefore, here in this section, we discussa prototypical doping strategy by which the shift current re-sponse is tuned effectively for photovoltaic application undervisible solar light. 235448-6FIRST-PRINCIPLES CALCULATION OF SHIFT … PHYSICAL REVIEW B 101, 235448 (2020) FIG. 6. Demonstrating doping engineering in a QL of α-In2Se3.( a )I n trat the tetrahedral site is substituted by Bi to obtain a QL of InBiSe 3 with Pin=0.391 nC /m, and Pout=3.088 nC /m; (b) fat band of QL of InBiSe 3shows that the Bi substitution introduces more bands in CBM which importantly composed of porbitals of Bi and Se. (c) Dominant shift current response in InBiSe 3is along the direction of Poutunder zz-polarized light. Two sharp peaks, one in the visible spectrum and another in the ultraviolet range, which is much higher than that in α-In2Se3, is obtained. (d) Proposed device model based on α-In2Se3and InBiSe 3material system. It is well known that doping engineering is a key mecha- nism to modulate electronic [ 47], optical [ 48], magnetic [ 49], and many other exotic properties of 2D materials for differentapplications [ 41]. The ultrathin nature of 2D materials not only allows traditional substitutional doping strategies by im-plantation or diffusion processes, but also enables new meth-ods such as surface charge transfer, intercalation, and field-effect modulation methods. Here we theoretically demonstratethe substitutional doping of the bismuth (Bi) element in aQL of α-In 2Se3which results in improving the shift current response under visible light in the resulting QL of InBiSe 3. Notably, the substitution of a foreign atom for any particularelement in α-In 2Se3is a bit tricky in the sense that atoms are differently coordinated and substitution at a particular sitemay lead to an attribute suitable for a particular application. As seen above in Fig. 4(a), the maximum shift current response σ Zzzfor the QL of α-In2Se3occurs at 4.16 eV which approximately equals the energy separation between the uppervalance band due to Se porbitals and the lower conduction band due to In sorbital (see Fig. S7 in the Supplemental Ma- terial [ 37]). Therefore, increasing the density of states (DOS) near the CBM would enhance the transition rates under visiblelight and thereby modulate the shift current response. How-ever, the increment of DOS should not be on account of theintroduction of localized dorbitals as it may reduce the shift current response [ 12]. Therefore, we choose the bismuth (Bi)element as it has half filled porbitals ([Xe]4 f 145d106s26p3) which contribute to bands taking part in the transition processnear the Fermi level. It is worth noting here that the CBM ofα-In 2Se3is primarily contributed by the sorbital of In tr(at the tetrahedral site, Fig. 1); therefore, we substitute this indium element by Bi to obtain a QL of InBiSe 3, as shown in Fig. 6(a). Although the doping percentage considered in this study maybe difficult to obtain experimentally, it serves a good examplefor demonstrating the doping engineering of shift current inα-In 2Se3. Single-layer InBiSe 3is an indirect semiconductor with band gap 1.28 eV as obtained using the hybrid functionalHSE06. The CBM lies at Mand the VBM lies between /Gamma1 andM. Substitution of In trwith Bi introduces more bands in the CBM as shown in the fat bands of InBiSe 3in Fig. 6(b). The CBM comprises Bi- pand Se tg2-porbitals and the VBM is not affected by Bi substitution. When illuminated under thezz-polarized light, a giant shift current ( σ Zzz) response of mag- nitude ∼1200μV/cm2is obtained, which is almost double that of the QL of α-In2Se3along the Zaxis in the visible spectrum (at 2.98 eV) as shown in Fig. 6(c) (the transverse and longitudinal components of the shift current are shownin Fig. S8 in the Supplemental Material [ 37]). Thus, using doping engineering, the shift current response of α-In 2Se3 can be modulated effectively to occur in the visible spectrum. Notably, the shift current response of InBiSe 3contains two 235448-7TIWARI, BIRAJDAR, AND GHOSH PHYSICAL REVIEW B 101, 235448 (2020) sharp peaks, one at 2.98 eV (in the visible spectrum) and another at 3.89 eV (in the ultraviolet spectrum) suggesting that InBiSe 3can generate a larger photocurrent in visible as well as ultraviolet light. It is worth mentioning here thatwe used both longitudinal and transverse polarized lights,however σ Zzzis the dominant shift current response in the QL of InBiSe 3, indicating that a vertical device would be more efficient than a lateral device. Such a device modelis schematically shown in Fig. 6(c) which can be realized easily using exfoliation [ 19] or chemical vapor deposition [ 20] techniques on a substrate (such as Si/SiO 2) with a conducting layer acting as a back contact. A transparent conducting layer(such as of indium tin oxide) can be deposited at the top whichwould allow the solar light to fall on to the InBiSe 3layer. The maximum photogenerated charge carriers are collectedvia the transparent conducting layer at the top along the Z axis, thereby the maximum solar energy conversion efficiency can be obtained with such devices in this material system. IV . CONCLUSION α-In2Se3is a 2D van der Waals layered material with in-plane and out-of-plane polarizations coexisting at roomtemperature. We calculate the shift current in this materialto understand the implication of polarizations on BPVE. Ina single layer, a larger shift current of 750 μA/V 2at the photon energy of 4.16 eV is obtained along the direction ofout-of-plane polarization under the illumination of zz- polarized light whereas a negligible shift current is obtainedalong the direction of in-plane polarization under yy-polarized light. Furthermore, in the bilayer, a dominant shift currentalong the direction of out-of-plane polarization is obtainedin both 3 Rand 2 Harrangement, confirming that in-plane polarization doesn’t affect the shift current in α-In 2Se3. Moreover, we have found that the maximum shift current response occurs in the ultraviolet energy range in this material.Therefore, for efficient solar energy conversion, the shiftcurrent is tuned to the visible spectrum by doping engineering.The indium element at the tetrahedral site is substituted bybismuth, thereby a giant shift current response of 1200 μA/V 2 is obtained at 2.98 eV in the resulting composition, InBiSe 3, under zz-polarized visible light along the direction of out-of- plane polarization. This study would be instrumental in under-standing the BPVE in α-In 2Se3and similar material systems. A device model is also proposed which would guide design ofan efficient solar energy converter based on this material. ACKNOWLEDGMENTS R.K.G. thanks the Department of Science and Technology, Government of India, for DST INSPIRE Faculty Grant No.IFA17-ENG206 for financial support. R.P.T. thanks UGC forfinancial assistance through UGC-SRF. [1] V. M. Fridkin, Crystallogr. 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PhysRevB.69.235306.pdf

Carrier tunneling in nanocrystalline silicon–silicon dioxide superlattices: A weak coupling model B. V. Kamenev,1G. F. Grom,2D. J. Lockwood,3J. P. McCafrey,3B. Laikhtman,4and L. Tsybeskov1 1Department of Electrical and Computer Engineering, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102, USA 2Agere Systems, Alhambra, California 91030, USA 3Institute for Microstructural Sciences, National Research Council, Ottawa, Canada K1A 0R6 4Racah Institute of Physics, Hebrew University, Jerusalem, Israel (Received 15 July 2003; revised manuscript received 8 September 2003; published 8 June 2004 ) Differential conductivity measurements in partially disordered nanocrystalline Si–amorphous SiO 2superlat- tices reveal a double-peak structure associated with tunneling via energy levels of light and heavy holes. Thetheoretical model and numerical simulations presented here show good agreement with experiment and predictthat this system does not have stable solutions for an injected carrier concentration greater than 10 17cm−3. Similar to a weakly coupled superlattice, a larger carrier concentration results in current instabilities. Theseinstabilities have been observed and can be partially suppressed by using pulsed carrier photoinjection. DOI: 10.1103/PhysRevB.69.235306 PACS number (s): 73.21.Ac, 73.21.La Attempts to develop a system of coupled nanoscale ob- jects in one, two, or three dimensions have attracted consid-erable attention recently due to their predicted unique physi-cal properties and wide range of potential applications. 1–6 This task in group IV semiconductors, specifically for Si, isquite complex because of the absence of matching materialsfor the construction of a high-quality heterostructure. Con-trolled growth of Si/SiO 2quantum structures is even more complicated due to the physical nature of the materials [crys- talline Si (c-Si)and amorphous SiO 2(a-SiO2)]and the dif- ference in their thermal-expansion properties. Nevertheless,the high quality of the Si/SiO 2interface and the large poten- tial barriers of Si/SiO 2could be utilized in systems with efficient carrier confinement and for novel quantum devicedevelopment. It has been demonstrated previously that a pe-riodic structure consisting of layers of Si nanocrystals sepa-rated by a-SiO 2layers possesses abrupt and nearly-defect- free interfaces.7The observations of acoustic phonon Brillouin-zone folding8and a preliminary indication of the resonant nature in carrier tunneling in these structures9have suggested that nanocrystalline snc-Si d/a-SiO2superlattices (SL’s)can be used in electron quantum devices.These results are surprising, because so far only epitaxially grown semi-conductor superlattices have consistently demonstrated suchphenomena as resonant carrier tunneling via quantized en-ergy levels, the formation of high-electric-field domains, andthe generation of sustained current oscillations. 10Here we show that sequential resonant tunneling via quantized energylevels and related phenomena can be quantitatively modeled in partially disordered nc-Si/ a-SiO 2SL’s with the assump- tion of sequential carrier resonant tunneling. Our numericalsimulations show that at carrier concentration higher than10 17cm−3this process may lead to the formation of electric- field domains and sustained current oscillations. These simu-lations together with experiments demonstrate that pulsedcarrier injection can suppress the current instabilities. Theresults reported here demonstrate that partially disorderednc-Si/a-SiO 2SL’s exhibit phenomena similar to that ob- served previously in epitaxially grown weakly coupledsuperlattices10and indicate considerable promise for their device applications. The nc-Si/ a-SiO2SL’s were prepared by the technique of controlled solid-phase crystallization, as described in Ref.7–9. This nc-Si and a-SiO 2layers were deposited onto an n-type 10 k Vcmc-Si substrate, forming an eight period SL with Si nanocrystals of an elliptical shape measuring 4 and6 nm in the vertical and lateral dimensions, respectively.These layers of Si nanocrystals were separated by2-nm-thick SiO 2spacers (extracted from transmission elec- tron microscopy data, not shown ). A top Al contact with an area of ,1m m2was defined and fabricated using photoli- thography and wet etching techniques. To achieve precise control over broad range of carrier concentrations at low temperature, the fabricated structureswere illuminated by low-power s!1m W dcw radiation from a He-Ne laser sl=632.8 nm d. Since the absorption coeffi- cient inc-Si at that wavelength is ,10 4cm−1and the entire superlattice is only 52-nm thick, the radiation is totally ab-sorbed in the c-Si substrate. Under these experimental con- ditions, photogenerated carriers are subsequently collected inthe substrate or accumulated in the substrate depletion regionnear the c-Si/SiO 2interface, depending on the sign of the applied bias. Therefore, the experimental conditions can bemodeled as carrier injection from a substrate and, dependingon the sign of the applied bias, may provide hole or electroninjection. Here, only experimental results related to holetransport will be discussed. Differential photoconductance asa function of the applied bias was measured as a low-signalac conductivity using an ac signal of 10 mV at 88 Hz super-imposed on the slowly varying s10 mV/s ddc voltage. The measurements were performed at T <10 K. Differential photoconductivity in a nc-Si/ a-SiO 2SL as a function of the applied bias is shown in Fig. 1 for variousintensities of the incident laser beam. At the lowest level ofexcitation s100 mWc m−2d, two well-separated peaks are clearly observed in the differential conductivity, with almost an order of magnitude ratio between their maximum andminimum. As the intensity of laser excitation increases, thePHYSICAL REVIEW B 69, 235306 (2004 ) 0163-1829/2004/69 (23)/235306 (5)/$22.50 ©2004 The American Physical Society 69235306-1peak full width at half maximum (FWHM )increases, peaks shift to higher voltage, and the ratio between the maximumand minimum decreases. Since the experimental conditions are chosen to provide a monopolar hole injection from the c-Si substrate into the nc-Si/a-SiO 2SL, we attribute the two peaks to hole tunnel- ing via energy levels of heavy (HH)and light (LH)holes. Our choice is motivated by earlier calcultions, which indi-cated that the HH2 energy level is higher than the LH1 en-ergy level. 11,12In conventional superlattices, tunneling from the HH level to the LH level is not very efficient because ofspin conservation in pure vertical carrier transport. However,this selection rule is not so strong here due to the LH-HHhole mixing for nonzero in-plane momentum and the pos-sible lateral carrier transport component in nc-Si/ a-SiO 2 SL’s.13An additional perturbation of the wave-function sym- metry due to the applied electric field must also be taken intoaccount. Therefore, in a simplified model of sequential tun-neling, the first peak in the photoconductance dependence onthe applied dc bias may be associated with HH-HH tunnelingand the second peak with HH-LH tunneling. The model isillustrated in Fig. 2, where after tunneling to the LH level (1) carriers relax down to the HH level (2). This process of carrier relaxation and tunneling is not fast and requires a sufficiently long residence time for the chargecarrier within a quasi well before the next tunneling eventoccurs. Our model is consistent with the anticipated ex-tremely low vertical carrier mobility in nc-S/SiO 2SL’s due to a weak coupling between the adjacent Si nanocrystals: theobserved sample time constant t=RCvaries, depending on the excitation conditions, from 1 to 100 ms. Therefore, ourassumption of the carrier tunneling process comprising se-quences of tunneling/relaxation/tunneling is a reasonableone. A more complete model needs to include the carrier tun- neling dependence on the gradient of the hole concentrationand the electric-field distribution across the nc-Si/ a-SiO 2 SL. The presence of carriers in the layers of Si nanocrystalsinduces a redistribution of the applied electric field over the entire SL.This process is responsible for a local modificationof the resonance conditions between the adjacent wells indifferent parts of the SL, tuning the system in and out of theresonance [Fig. 2 ]. To model the electric-field distribution in a nc-Si/a-SiO 2SL, the balance equations for the carrier con- centration in each layer of Si nanocrystals and the Poissonequation have to be solved self-consistently. This equationsystem can be written as ]p1 ]t=psTin−p1T1−p1Tin+p2T1, ]p2 ]t=p1T1−p2T2−p2T1+p3T2, As 1d ]pn ]t=pn−1Tn−1−pnTout−pnTn−1, jn+1=jn+e «pndB, jn=−wn+1−wn dSLs2d with the following boundary condition: Swn+wk=V s3d Herepsis the hole concentration in the substrate, pnis the hole concentration in the nth well, jnandwnare the electric field and potential drop in the nth well,dBanddSLare the thicknesses of the dioxide barrier and quantum well, Tnis the tunneling coefficient between nth and sn+1dth wells,eis the free-electron charge, «is the dielectric constant, Vis the external bias, and Tin,Toutare the SL in and out contact FIG. 1. Differential photoconductivity at T =10 K as a function of the external bias underdifferent levels of 632.8-nm excitation.B. V. KAMENEV et al. PHYSICAL REVIEW B 69, 235306 (2004 ) 235306-2tunneling coefficients, respectively. The first and third terms are the rates between the nth and sn+1dth well, while the second and fourth are between the nth and sn+1dth well. The energy barriers snc-Si/a-SiO2dbetween adjacent quasiwells (Si nanocrystals )are very high and the energy dependence of the tunneling coefficients Tncomes mainly from the density of states in the quasiwells. A dispersion ofnc-Si sizes and inhomogeneous broadening of the density ofstates is modeled by a Gaussian function. The tunneling co-efficientT ndepends on the potential drop wnacross the nth barrier as Tn,Ah-he−s−ewn/dEhd2+Ah-le−fsDE−ewnd/dElg2, s4d whereAh-handAh-ldescribe tunneling between HH and LH energy levels, respectively, and contain the correspondingtunneling matrix element squared; dEh,dElare the widths of the heavy and light hole Gaussian-shape density of states;andDEis the separation between light and heavy hole en- ergy levels. In our numerical simulations, dEh,dEl, and DE are used as adjustable parameters. It is significant that thevalues of dEh,dElobtained from our calculation are ,20 meV. Simple estimations using different methods14,15show that for a Si nanocrystal diameter of ,4 nm an energy dispersion of ,20 meV corresponds to ,10% dispersion in the Si nanocrystal vertical dimension. An additional contri-bution is expected from small variations in Si nanocrystalshape and crystallographic orientation. Coefficients A h-handAh-lare estimated from a simple relationship for tunneling through a square potential barrier, T=" mSL*dSL2e−2dB˛2mB*U/", s5d where "is the Planck constant, mSL*andmB*are the effective masses of holes in the nc-Si quantum well and SiO 2barrier, andUis the barrier height. Results of our numerical simulations are shown in Fig. 3. In accordance with our experiment (Fig. 1 ), the increase in carrier concentration results in an increase of the FWHM indifferential conductivity peaks, a decrease in their peak-to-valley ratio, and shift to a higher voltage. More importantly,our simulations predict that the system of Eqs. (1)–(4)with the chosen boundary conditions does not have a stable solu-tion for an injected carrier density into the silicon substrate(i.e., injecting contact ).10 17cm−3. What happens when the carrier concentration exceeds 1017cm−3? Figure 4 shows the experimentally observed sustained current oscillations undera much greater cw laser excitation intensity of100 mW cm −2. By varying the carrier concentration and the applied bias, the oscillation time domain can be changedfrom milliseconds to hundreds of nanoseconds, most likelydue to limitations in the sample RC. A quantitative understanding of carrier tunneling in nc -Si/a-SiO 2SL’s can immediately be applied to resolve a long-standing issue concerning the demonstration of NDC inSi-based nanostructures. Since the first reported observationof resonant tunneling and NDC in semiconductor SL’s byEsakiet al., 16,17the importance of a uniform electric field applied across a SL has been pointed out and is well under-stood. Our simulations show that in order to achieve such a FIG. 2. Valence-band energy diagrams of a SL with (a)homo- geneous and (b)nonhomogeneous electric-field distributions. The indicated processes are (1)resonance tunneling of holes via energy levels of heavy (HH)and light (LH)holes; (2)relaxation of holes from LH down to HH levels; (3)destroyed resonance conditions in a SL with a high carrier concentration. FandMdenote the Fermi level of the c-Si substrate and metal contact, respectively. FIG. 3. The calculated dependencies of differential photocon- ductivity as a function of the external bias for different levels ofcarrier concentrations.CARRIER TUNNELING IN NANOCRYSTALLINE PHYSICAL REVIEW B 69, 235306 (2004 ) 235306-3uniform field, the carrier concentration in a nc-Si/SiO 2SL should be as low as possible. Partial screening of the appliedelectric field by carriers localized within nc-Si quasiwellsdestroys the tunneling resonances. However, an effective car-rier injection can be produced by a relatively short pulse oflight, mostly absorbed in the substrate. At a fixed appliedbias, carriers (holes )accumulate at the substrate/SL interface and are entirely immobile until the resonant tunneling con-ditions occur. When resonance is achieved, carrier mobilityincreases dramatically and the peak photocurrent as a func-tion of the applied bias may show more pronounced features.A similar technique of transient carrier injection has beenapplied to study carrier transport in weakly coupled III-Vsemiconductor superlattices. 18,19Here this technique is used to detect differential photoconductance under chopped pho-toexcitation with a low duty cycle. Figure 5 show results ofexperiments with 632.8-nm excitation at a peak intensity of1Wc m −2and a duty cycle of ,1:10. (Note that under the chosen excitation conditions, the pulse duration of 1–10 msis less than the relaxation time t=RC.)As a result, a very sharp peak and NDC are observed, indicating that the screen-ing of the applied electric field is less significant. The shapeof the photoconductance as a function of the applied bias and the remnant of instabilities indicate that the system is notcompletely stable. In conclusion, the observed double-peak feature in photo- conductance of nc-Si/ a-SiO 2SL’s is attributed to hole se- quential resonant tunneling and quantitatively modeled tak-ing into account a nonuniform distribution of the appliedelectric field due to screening by photogenerated carriers.Asan extreme case, this process generates current instabilitiesand sustained oscillations, similar to phenomena observedpreviously in epitaxially grwon III-V superlattices. These in-stabilities can be partially suppressed using pulsed photoin-jection of carriers, and thus a better peak-to-valley ratio canbe achieved at higher carrier densities. The demonstratedquantitative understanding of carrier resonant tunneling inpartially disordered Si nanostructures now permits quantita-tive modeling and construction of practical electron devices. This work was supported by Semiconductor Research Corporation, NSF,ARO, Motorola,AMD, and Foundation atNJIT. 1L. J. Lauhon, M. S. Gudiksen, D. Wang, and C. M. Lieber, Nature (London )420,5 7 (2002 ). 2B. A. Korgel, Phys. Rev. Lett. 86, 127 (2001 ). 3T. M. Wallis, N. Nilius, and W. Ho, Phys. Rev. Lett. 89, 236802 (2002 ). 4S. Sun and C. B. Murray, J. Appl. Phys. 85, 4325 (1999 ). 5A. J. Williamson, J. C. Grossman, R. Q. Hood, A. Puzder, and G. Galli, Phys. Rev. Lett. 89, 196803 (2002 ). 6S. A. Crooker, J. A. Hollingsworth, S. Tretiak, and V. I. Klimov, Phys. Rev. Lett. 89, 186802 (2002 ).7L. Tsybeskov, K. D. Hirschman, S. P. Duttagupta, M. Zacharias, P. M. Fauchet, J. P. McCaffrey, and D. J. Lockwood, Appl. Phys.Lett.72,4 3 (1998 ). 8G. F. Grom, D. J. Lockwood, J. P. McCaffrey, H. J. Labbe, L. Tsybeskov, P. M. Fauchet, B. White, J. Diener, H. Heckler, D.Kovalev, and F. Koch, Nature (London )407, 358 (2000 ). 9L. Tsybeskov, G. F. Grom, R. Krishnan, L. Montes, P. M. Fauchet, D. Kovalev, J. Diener, V. Timoshenko, F. Koch, J. P.McCaffrey, H. J. Labbe, G. I. Sproule, D. J. Lockwood, Y. M.Niquet, C. Delerue, and G. Allan, Europhys. Lett. 55, 552 FIG. 4. Experimentally observed current instabilities and sus- tained oscillations under a cw photoexcitation of ,100 mW/cm2. Arrows show faster, unresolved components of the oscillations. FIG. 5. Differential photoconductivity as a function of the ex- ternal bias under pulsed photexcitation with a duration of ,10 ms and a peak intensity of ,1W/cm2. The pulses are provided by modulation of a cw He-Ne laser beam with a duty cycle of 1:10.The arrows shows the NDC at ,1.3 V followed by the partially suppressed instabilities.B. V. KAMENEV et al. PHYSICAL REVIEW B 69, 235306 (2004 ) 235306-4(2001 ). 10See, for example, Semiconductor Superlattices: Growth and Elec- tronic Properties , edited by H. T. Grahm (World Scientific, Sin- gapore, 1995 ), p. 250. 11A. V. Chaplik and L. D. Shvartsman, Phys. Chem. Mech. Surf. 2, 73(1982 ). 12B. Laikhtman, R. A. Kiehl, and D. J. Frank, J. Appl. Phys. 70, 1531 (1991 ). 13D. L. Miller and B. Laikhtman, Phys. Rev. B 54, 10 669 (1996 ). 14C. Delerue, M. Lannoo, and G. Allan, Phys. Rev. Lett. 84, 2457(2000 ). 15S. Ögüt, J. R. Chelikowsky, and S. G. Louie, Phys. Rev. Lett. 79, 1770 (1997 ). 16L. Esaki and L. L. Chang, Phys. Rev. Lett. 33, 495 (1974 ). 17R. Tsu, L. L. Chang, G. A. Sai-Halasz, and L. Esaki, Phys. Rev. Lett.34, 1509 (1975 ). 18S. Tarucha, K. Ploog, and K. von Klitzing, Phys. Rev. B 36, 4558 (1987 ). 19W. Muller, D. Bertran, H. T. Grahn, K. von Klitzing, and K. Ploog, Phys. Rev. B 50, 10 998 (1994 ).CARRIER TUNNELING IN NANOCRYSTALLINE PHYSICAL REVIEW B 69, 235306 (2004 ) 235306-5

PhysRevB.92.115119.pdf

PHYSICAL REVIEW B 92, 115119 (2015) Topological nature of the FeSe 0.5Te0.5superconductor Zhijun Wang,1,2P. Zhang,1Gang Xu,1,3L. K. Zeng,1H. Miao,1Xiaoyan Xu,1T. Qian,1Hongming Weng,1,4P. Richard,1 A. V . Fedorov,5H. Ding,1,4,*Xi Dai,1,4,†and Zhong Fang1,4,‡ 1Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 3Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA 4Collaborative Innovation Center of Quantum Matter, Beijing, China 5Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 23 June 2015; revised manuscript received 18 August 2015; published 8 September 2015) We demonstrate, using first-principles calculations, that the electronic structure of FeSe 1−xTex(x=0.5) is topologically nontrivial and characterized by an odd Z2invariant and Dirac cone type surface states, in sharp contrast to the end member FeSe ( x=0). This topological state is induced by the enhanced three-dimensionality and spin-orbit coupling due to Te substitution (compared to FeSe), and characterized by a band inversion at theZpoint of the Brillouin zone, which is confirmed by our ARPES measurements. The results suggest that the surface of FeSe 0.5Te0.5may support a nontrivial superconducting channel in proximity to the bulk. DOI: 10.1103/PhysRevB.92.115119 PACS number(s): 74 .25.Jb,73.43.−f,74.70.Xa,79.60.−i I. INTRODUCTION Among the Fe-based superconductors, the FeSe 1−xTex family of compounds [ 1–4] is of particular interest. First, it has the simplest PbO structure (space group P4/nmm ) with Se (or Te) atoms forming distorted tetrahedra aroundFe [see Fig. 1(b)] similar to the structure of FeAs planes in the families of FeAs-based high- T csuperconductors [ 5]. Second, the internal parameters can be systematically tuned by thesubstitution of Se by Te [ 6–8], which provides us a platform for in-depth study of possible superconducting mechanismsand topological characters. Third, superconductivity has beenobserved for a wide range of composition x[2–4], and the transition temperature T ccan be further enhanced by pressure [ 9–11]. More recently, superconductivity with Tc higher than 77 K was suggested for single unit cell FeSe films [ 12] epitaxially grown on SrTiO 3substrates. Despite these interesting properties though, the particular- ities of the system have still not been fully explored. Earlierstudies, both theoretical and experimental, suggest the similar-ity of the electronic structures of the Fe chalcogenides (FeSe,FeTe) [ 13–15] and the FeAs-based [ 16–18] superconductors. Indeed, the low-energy physics around the Fermi level is dom-inated by the Fe-3 dstates, and the morphology of the Fermi surfaces is similar. On the other hand, a surprisingly stable(no splitting under external magnetic field) zero-energy boundstate (ZBS) at randomly distributed interstitial excess Fe siteswas observed in very recent scanning tunneling microscopy(STM) measurements on the surface of superconductingFe(Te,Se) [ 19], suggesting a possible topological feature of its electronic structure. Obviously, the 5 porbitals of Te are more extended and have stronger spin-orbit coupling (SOC)than the 4 porbitals of Se. The consequences of Te substitution, particularly for the bulk topological character of FeSe 1−xTex, have been largely ignored in the literature and will be the *dingh@aphy.iphy.ac.cn †daix@aphy.iphy.ac.cn ‡zfang@aphy.iphy.ac.cnmain purpose of the present paper. Based on first-principlescalculations combined with angle-resolved photoemissionspectroscopy (ARPES) measurements, here we report that theelectronic structure of FeSe 0.5Te0.5is topologically nontrivial, in sharp contrast to its end member FeSe. The topologicalproperties of FeSe 0.5Te0.5can be characterized by an odd Z2number, and the existence of Dirac cone type surface states, in proximity to bulk superconductivity, should favortopologically superconducting surface states, as suggested byFu and Kane [ 20]. After introducing our methodology in Sec. II, we present the DFT band structures and main results in Sec. III, and provide a section on the ARPES experimental data in Sec. IV, which is followed by our conclusion in Sec. V. II. METHODOLOGY The electronic structures of FeSe and FeSe 0.5Te0.5are calculated with SOC included. The calculations are performedbased on the density functional theory (DFT) [ 21,22] and the generalized gradient approximation (GGA) for the exchange-correlation potential [ 23], as implemented in the plane-wave pseudopotential based BSTATE (Beijing Simulation Tool ofAtomic TEchnology) package [ 24]. The experimental lattice parameters [ 6,25] are used in the calculations. Maximally localized Wannier functions (MLWF) [ 26] are constructed from bulk calculations, and then used to study the surfacestates in the semi-infinite system. To treat the substitutionproperly, we have calculated FeSe 0.5Te0.5by using both the virtual crystal approximation and the two-formula cell withordered Se and Te sites. Both calculations give convergingresults. III. RESULTS AND DISCUSSION A. Electronic structures without SOC and band inversion We first neglect the SOC and concentrate on the comparison between the electronic structures of FeSe and FeSe 0.5Te0.5[as shown in Figs. 2(a) and2(b). The band structure of FeSe is very similar to that of LaOFeAs as reported before [ 17]. At the/Gamma1point, the valence band top is not occupied, leading to 1098-0121/2015/92(11)/115119(7) 115119-1 ©2015 American Physical SocietyZHIJUN W ANG et al. PHYSICAL REVIEW B 92, 115119 (2015) FIG. 1. (Color online) Unit cell of Fe X(X=Se0.5Te0.5), with the xaxis pointing along Fe nearest neighbors. (a) Top view. (b) Crystal structure. Gray and green balls represent Fe and Xatoms, respectively. (c) Schematic plot of the hybridization along the zaxis between the combined orbitals D− xyandP− z(see details in Appendix B), consisting of intralayer pdbonding and interlayer ppbonding. the well known hole pockets of Fermi surfaces around /Gamma1.T h e three topmost states at /Gamma1can be labeled as twofold degenerate /Gamma1+ 5states ( dyz/dxzorbitals) and nondegenerate /Gamma1+ 4state (dxyorbital), respectively. There is a clear band gap larger than 0.6 eV above the valence band top at the /Gamma1point. All the d-d antibonding states (with negative parity) are located abovethe gap except the d z2orbital that has a weaker antibonding state. Among them, the most interesting state is the secondhighest one with remarkable red circles, which belongs tothe/Gamma1 − 2representation and comes from the antibonding dxy orbitals of Fe and pzorbital of chalcogen. The weight of the pzcomponent is illustrated by the size of red circles, which suggest that the /Gamma1− 2state can be affected by hybridizing with thedxyandpzorbitals. Looking along the /Gamma1-Zdirection, the band dispersion of the /Gamma1− 2state is the strongest among all d states. Nevertheless, since it is energetically high, the bandstructure around the Zpoint of the Brillouin zone (BZ) is only slightly affected and remains similar to that around /Gamma1.T h e result suggests that FeSe is quite two-dimensional. Immediate differences can be seen in comparing these results with the band structure of FeSe 0.5Te0.5shown in Fig. 2(b):( 1 )t h e /Gamma1− 2state in FeSe 0.5Te0.5is significantly pushed down and almost touches the valence band top, andthe band gap at /Gamma1(above the valence and top) is nearly closed; (2) the band dispersion of this /Gamma1 − 2state along the /Gamma1-Zdirection is strongly enhanced. As a result, a band inversion occurs attheZpoint, which implies a change of topological property. Compared to FeSe, the caxis and the experimental intralayer distance d zof FeSe 0.5Te0.5are enlarged by 7.8% and 9.7%, respectively (see Appendix A), while the aaxis and the interlayer distance change only very little. As a result,the intralayer hybridizations, especially the pdbonding as shown in Fig. 1(c), are seriously weakened, and the /Gamma1 − 2band Γ Z-0.100.10.20.30.4 Γ7+ Γ7+ Γ6+Γ6+Γ6− Γ6−Γ7+Γ7+ Λ6Λ7 Λ7 Λ6Γ6+Γ6+(a) (b) (c)(d)M Γ XR Z A ΓZΓ2− Γ2−Γ4+Γ5+ Γ2− Γ2−Γ2− Γ2−Γ2− Γ2− } dz2 M Γ XR Z A ΓZ-2-1012Energy (eV)M Γ XR Z A ΓZ-2-1012Enerygy (eV)Γ2− Γ2− Γ5+ Γ4+Γ2− Γ2−Γ2− Γ2−Γ2− Γ2− FIG. 2. (Color online) DFT electronic band structures. (a) Band structure of FeSe with internal parameter zX=0.2345 without SOC. (b) Band structure of FeSe 0.5Te0.5(zX=0.2719) without SOC. (c) Band structure of FeSe 0.5Te0.5with SOC. (d) Zoom-in view of the solid red box area in (c). The size of the red circles in (a) and (b) indicates the weight of the pzcomponent of the chalcogen atoms. The two dz2bands are indicated below EFin (b). The original /Lambda15bands split into /Lambda16and/Lambda17. Along the /Gamma1-Zline, two /Lambda16bands cross and hybridize to open a SOC gap of about 10 meV . The red dashed line corresponds to a Fermi curve across the gap. 115119-2TOPOLOGICAL NATURE OF THE FeSe 0.5Te0.5. . . PHYSICAL REVIEW B 92, 115119 (2015) center is lowered very much and becomes close to EF,a s shown in Fig. 2(b). On the other hand, because the Te 5 p orbitals are much more extended than the Se 4 porbitals, the interlayer hybridization through the ppbonding becomes stronger. Therefore, the Te substitution enhances the hoppingbetween layers and gives rise to a larger dispersion for /Gamma1 − 2in FeSe 0.5Te0.5. These two observations are in good agreement with the band structure of FeSe 0.5Te0.5shown in Fig. 2(b). Therefore, the band inversion in FeSe 0.5Te0.5is a consequence of the weakened intralayer hopping and enhanced interlayerhopping originating from the Te substitution. B. Electronic structures with SOC and nontrivial Z2invariant Once SOC is included, the band structure of FeSe 0.5Te0.5 opens a direct SOC gap and a nontrivial Z2invariant can be defined by assuming a “curved chemical potential”—thered dashed line in Fig. 2(c)—lying between the 10th and 11th bands (neglecting the spin-doublet degeneracy of thebands). Generally, the doubly degenerate /Gamma1 + 5states split into /Gamma1+ 6and/Gamma1+ 7, and the odd /Gamma1− 2state turns into /Gamma1− 6(see details in Appendix B). Along the /Gamma1-Zhigh-symmetry line, two /Lambda16 bands under C4vsymmetry hybridize and open a gap of about 10 meV , which is clearly shown in Fig. 2(d). When defining a Fermi curve through the SOC gap, the Z2invariant is easily calculated from the parity criterion, which comes out to 1. [Theparities at all time-reversal-invariant momenta (TRIM) arepresented in Appendix B.] This nonzero Z 2invariant indicates that FeSe 0.5Te0.5is in a topological phase that can support the nontrivial surface states (SS). Due to the substantial SOC ofTe, increasing the content xenlarges the SOC gap, which is beneficial for the detection of the SS in the gap. Moreover,we also performed dynamical mean-field theory (DMFT)calculations to confirm the band inversion and identify thestrong band renormalization due to electronic correlations(see Appendix Cfor details). The correlation effects do not change the detail of the electronic bands, but simply reducethe bandwidth. C. Spin-resolved Fermi surface Next, we analyze the spin-resolved Fermi surfaces around the Dirac point of the semi-infinite system formed by the SSto identify the nontrivial topology. With the surface Green’sfunction calculated from the modified effective Hamiltonianconsidering DMFT on-site modification, the spin-filter surfacestates and the corresponding Fermi surfaces can be obtaineddirectly. From the dispersion of the SS shown in Fig. 3(a), the protected SS emerges in the SOC gap around E Fdue to the nontrivial topological nature of the bulk system. TheFermi surface of the SS ( E F=50 meV) is illustrated as a bright yellow circle in Fig. 3(b), and the spin orientation for the SS around the Fermi surface is marked by green arrows,which form a πBerry phase enclosed. The magnitude of the zspin component is very small compared to the in-plane component. The πBerry phase signifies the topological nontrivial properties of the bulk. By inducing an s-wave superconducting gap, the chiral SS can play an important rolein producing the Majorana zero energy mode. FIG. 3. (Color online) Topological nontrivial 001 surface states and spin-resolved Fermi surface of these states. (a) Surface LDOS on the (001) surface for FeSe 0.5Te0.5considering DMFT on-site modification. (b) Fermi surface of the topological state (bright yellowcircle beside the projected bulk states). The in-plane spin orientation is indicated by green arrows. IV . ARPES MEASUREMENTS We performed ARPES measurements in order to demon- strate experimentally the existence of the /Gamma1− 2band with strong pzorbital character crossing EFalong the /Gamma1-Zdirection. Large single crystals of FeSe 0.45Te0.55were grown using the self-flux method, and LiFeAs with FeAs flux method. ARPESmeasurements were performed at the Advanced Light Sourceand at Synchrotron Radiation Center, using a VG-Scientaelectron analyzer. The K source used for evaporation is made ofa SAES K dispenser. In the experiments, the largest coverageis less than one monolayer. All data were recorded with linearhorizontal polarized photons with a vertical analyzer slit ( σ geometry). Under this configuration, odd orbitals with respectto the emission plane are visible, such as the d xyanddyz orbitals, while the dxzband should not be detected. In addition, our experimental setup leads to a zcomponent of the light polarization, and thus orbitals with mainly zcomponents such asdz2andpzorbitals are also observable [ 27]. The comparison of the DFT band calculations on FeSe and FeSe 0.5Te0.5, shown in Figs. 2(a) and2(b), suggests that while it is pushed far above EFat/Gamma1in FeSe, the /Gamma1− 2band in FeSe 0.5Te0.5forms an electron band just above EF. To prove its existence, we raised the chemical potential by performingin situ K doping, and we measured ARPES spectra along /Gamma1-M athν=30 eV , which coincides with k z=0[28]. The results before and after evaporation (6 minutes) are shown in Figs. 4(a) 115119-3ZHIJUN W ANG et al. PHYSICAL REVIEW B 92, 115119 (2015) FIG. 4. (Color online) ARPES spectra along the /Gamma1-Mdirection in the σ-polarization geometry. (a) FeSe 0.5Te0.5before surface evaporation. (b) FeSe 0.5Te0.5after 6 minutes of K surface doping. (c), (d) MDCs corresponding to data from the dashed boxes in panels (a) and (b), respectively. and4(b), respectively. As expected, the hole band is sinking further below EFafter evaporation. Interestingly, an additional electron band is observed, as clearly shown by contrastingthe momentum distribution curves (MDCs) obtained before[Fig. 4(c)] and after [Fig. 4(d)] evaporation. This band, locating about 30 meV above the top of the valence band in FeSe 0.5Te0.5, is very similar to the small 3D electron pocket reported in(Tl,Rb) yFe2−xSe2[27], which mainly has a pzcomponent. In the presence of SOC, the DFT calculations indicate that the/Gamma1+ 6,/Gamma1+ 7, and/Gamma1+ 7states at the /Gamma1point mostly come from thedxz,dyz, anddxyorbitals, respectively, while the /Gamma1− 6state (labeled as /Gamma1− 2without SOC) have an important pzcomponent besides the dxyorbital. As shown in Fig. 5(a), while the pz band locates above EFat/Gamma1, it is shifted below EFupon moving along /Gamma1-Zand it reaches its minimum at the Zpoint. On its way down, the pzband crosses the dxy,dyz, anddxz bands, opening a SOC gap with the dxzband. This situation is illustrated schematically in Fig. 5(a). We now ask how the hybridization of the pzband with the dxzband should affect the ARPES measurements. Around the /Gamma1point, the pz band forms an electron-like band located above EFalong the /Gamma1-Mdirection. Therefore, it should not be observed in our experimental geometry. Due to selection rules, the dxzband should not be observed either and only the dxyanddyzband can possibly be seen, as shown schematically in Fig. 5(b). However, because it hybridizes with the pzband, one should expect to be able to detect the dxzband near the crossing point below EF, as illustrated schematically in Fig. 5(c). Finally, our calculations indicate that the pzband has a hole-like dispersion near the Zpoint, and we should be able to observe it. On the other hand, the dxzband no longer hybridizes strongly with FIG. 5. (Color online) Normal-emission ARPES spectra. First row: Schematic band dispersions. The main feature is the pzband crossing the dxy,dyz,a n ddxzbands, opening a SOC gap with the dxz band. Second row: ARPES data on FeSe 0.5Te0.5. Third row: ARPES data on LiFeAs. First column: Dispersion at kx=0 along the /Gamma1-Z direction. Second, third, and fourth columns: Electronic dispersionsalong cuts indicated in the first column. thepzband and its intensity should be significantly suppressed again in the σpolarization, as described in Fig. 5(d). In order to confirm this dispersion, we compare in Fig. 5(e) ARPES spectra recorded on FeSe 0.45Te0.55with different photon energies. We see some intensity between the /Gamma1and Zpoints that we assign to the pzband sinking down from /Gamma1 toZ.A tt h e Zpoint, this band has merged with the strong dz2band and it is thus undistinguishable. We display in Figs. 5(f)–5(h) three ARPES intensity cuts along /Gamma1-Mrecorded on FeTe 0.55Se0.45and corresponding respectively to kzvalues around /Gamma1[cut 1, Fig. 5(f)], the hybridization between the pzanddxzbands [cut 2, Fig. 5(g)], and around Z[cut 3, Fig. 5(h)]. As the dxyband heavily renormalizes compared to the dxz/dyzbands, it is shallower and hardly resolved with weaker intensity [ 29]. In cut 1 [Fig. 5(f)], as expected, none of the pzanddxzbands are detected near kz=0. In contrast, not only the dyzband is observed in cut 2 [Fig. 5(g)], but the dxzband as well due to its hybridization with the pzband. At theZpoint, away from hybridization, the dxzband disappears from the ARPES spectrum [Fig. 5(h)]. Unfortunately, the pz band is too close to the strong dz2band to be distinguished unambiguously. As shown in Figs. 5(i)to5(l), the strong kzdispersion of thepzband is also observed in LiFeAs, which exhibits very clear spectral features. Because its bottom is located awayfrom the d z2band, the dispersion of the pzband along /Gamma1-Zcan be identified very clearly, as illustrated in Fig. 5(i). As with FeSe 0.45Te0.55, the intensity of the dxzband is the strongest along cut 5 [Fig. 5(k)], where it hybridizes with the pzband. Figures 5(k) and5(l)also show clearly that the pzband has a hole-like dispersion below EF, as expected theoretically. 115119-4TOPOLOGICAL NATURE OF THE FeSe 0.5Te0.5. . . PHYSICAL REVIEW B 92, 115119 (2015) TABLE I. Structural parameters of PbO-structure Fe X. The lattice parameters are from experimental data [ 6,25], and both optimized (Opt.) and experimental (Exp.) internal chalcogen positions zXare shown. dzis the Cartesian distance in the zdirection from the Xplane to the Fe plane. a(˚A) c(˚A) zX(Opt.)/dz(˚A) zX(Exp.) /dz(˚A) FeSe [ 25] 3.7724 5.5217 0.2345 /1.2948 0.2673 /1.4759 Fe1.068Te [6] 3.8123 6.2517 0.2496 /1.5604 0.2829 /1.7686 FeSe 0.493Te0.507[6] 3.7933 5.9552 0.2476 /1.4745 0.2719 /1.6192 V . CONCLUSION In conclusion, we have presented both theoretical and experimental evidence for a topologically nontrivial phasein FeSe 0.5Te0.5. From the DFT calculations, we show that the band topology is sensitive to the intralayer and interlayerhopping terms, which can be tuned by the Te substitution. TheTe substitution content xstrongly affects the structural, elec- tronic, and topological properties of these materials. We haveidentified the topologically nontrivial electronic band structureof FeSe 0.5Te0.5with a band inversion, characterized by an odd Z2invariant and spin-moment-locked SS. Our ARPES data strongly support that the /Gamma1− 2band forms a band inversion at theZpoint. Similar results can also be applied to iron pnictides such as LiFeAs. Due to the topologically nontrivial surfacestates, the FeSe 1−xTexmaterials would be an ideal system for realizing possible topological superconductors and Majoranafermions on the surface. ACKNOWLEDGMENTS We acknowledge discussions with J. P. Hu and exper- imental assistance from Y . M. Xu. This work was sup-ported by NSFC (11474340, 11274362, 11204359, and11234014), the 973 Program of China (2011CBA001000,2011CBA00108, and 2013CB921700), and the “StrategicPriority Research Program (B)” of the Chinese Academy ofSciences (XDB07000000 and XDB07020100). Z.W. and P.Z. contributed equally to this work. M Γ X R Z A ΓZ -6-4-2024Energy (eV)EF FIG. 6. (Color online) Electronic band structure of LiFeAs. The pzcomponent is highlighted by the size of the red circles.APPENDIX A: STRUCTURAL PARAMETERS OF Fe X AND BAND STRUCTURE OF LiFeAs The experimental parameters of Iron chalcogenides (FeX) are used in Table I. For LiFeAs, we used the experimental lat- tice parameters [ 30] and relaxed the free internal coordinates. The experimental structure (space group P4/nmm , No. 129) has the Li sites in 2c positions, which lie above and below thecenters of the Fe squares opposite the As. The calculated DFTband structure is given in Fig. 6.T h e/Gamma1 − 2band shows a large dispersion and a band reversion along the zdirection. APPENDIX B: SYMMETRY AND PARITY ANALYSIS Since the system has inversion symmetry, it is convenient to combine these orbitals into bonding and antibonding stateswith definite parity as |D ± α/angbracketright=1√ 2(|Feα/angbracketright±| Fe/prime α/angbracketright), |P± β/angbracketright=1√ 2(|Xβ/angbracketright∓|X/prime β/angbracketright). Using the orbitals with definite parity defined in the main text, we can label all the non-spin-orbital (NSO) bands with theirreducible representations (IRs), which are given in Table II. The/Gamma1 − 2band is composed of the antibonding states with both dxyandpzcharacters. At both the /Gamma1andZpoints, the IRs of the D4hgroup are labeled as /Gamma1± n, whereas along the /Gamma1-Z high-symmetry line, we label these IRs as /Lambda1nbecause of its C4vsymmetry. In our conventions, the ( x,y)a x e sa r er o t a t e d by 45◦as compared to the crystallographic axes, so that the dxyorbital in our definition is the one pointing from Fe to chalcogen atoms, as shown in Fig. 1(a). The IR labels along the /Gamma1-Zline are shown in Fig. 7(a). Around EF,t h eD+ xz/yz bands can mix with the P+ x/yorbitals with the same /Gamma1+ 5representations, although the P-Dhybridiza- tion is not strong due to the little orbital overlap in the realspace. For the /Gamma1 − 2representation, the D− xycharacter can mix TABLE II. Combined orbitals and related irreducible representations D/P (+/−)z2yz/xz xy x2−y2x/y z D4h + /Gamma1+ 1 /Gamma1+ 5 /Gamma1+ 4 /Gamma1+ 3 /Gamma1+ 5/Gamma1+ 1 − /Gamma1− 3 /Gamma1− 5 /Gamma1− 2 /Gamma1− 1 /Gamma1− 5/Gamma1− 2 C4v + /Lambda11 /Lambda15 /Lambda14 /Lambda13 /Lambda15/Lambda11 − /Lambda14 /Lambda15 /Lambda11 /Lambda12 /Lambda15/Lambda11 115119-5ZHIJUN W ANG et al. PHYSICAL REVIEW B 92, 115119 (2015) FIG. 7. Electronic band structure along /Gamma1-Z. (a) The IRs are labeled in the NSO bands. (b) The parities are labeled in the SOCbands. with the P− zband, and the dxyandpzorbitals have a strong hybridization along the zdirection. However, as shown in Table I, the distance dzbecomes larger with the Te substitution content xincreasing, thus leading to weaker hybridization. As a result, the /Gamma1− 2band, consisting of dxyandpzorbitals, sinks down to EFand has a large dispersion in FeSe 0.5Te0.5. Moreover, due to selection rules, the bonding state of D+ xy cannot mix with the pzband, including along the /Gamma1-Zline. As a consequence, the /Gamma1+ 4band characterized by the D+ xy representation exhibits a weak dispersion in the zdirection, which is not sensitive to the height dz. Beside, all the IRs in NSO bands and parities in SOC bands are presented inTable III.“ (+,−)” denotes that the quadruple bands consist of two Kramers pairs with opposite parity. The nontrivial Z 2 index is presented beside the vertical line, which denotes the hypothetical Fermi level in the main text.-0.4-0.2 0 0.2 0.4Energy (eV) -0.8-0.4 0 0.4 0.8 M Γ X R Z A Γ ZEnergy (eV) FIG. 8. (Color online) Upper panel: LDA +DMFT calculations. Lower panel: SOC band structure only taking the LDA +DMFT on- site modification into consideration. APPENDIX C: LDA +DMFT CONFIRMATION The LDA +DMFT method has proven to be a powerful technique to study the electronic structure of correlatedsystems. In this section, we apply this method to FeSe 0.5Te0.5 with a local Coulomb integral U=F0=4.0 eV and a Hund’s coupling J=0.7 eV , and confirm that it has a topological character of band inversion by computing the topologicalinvariants within the DMFT framework. The one-electon spectral function is defined as A k(ω)=−1 πIm/Sigma1(ω) [ω+μ−/epsilon1k−Re/Sigma1(ω)]2+Im/Sigma1(ω)2 in terms of the LDA band dispersion /epsilon1kand the self-energy /Sigma1(ω). This momentum-resolved spectra Ak(ω)i ss h o w ni n the upper panel of Fig. 8, where the overall renormalization of the bands and the bandwidth reduction are apparent. However,the relative positions of different bands almost do not change.The/Gamma1 − 2and/Gamma1+ 5band crossing near EFremains in the correlated system. In a previous work [ 31], only the Green’s function at zero frequency G−1 k(0)=μ−/epsilon1k−/Sigma1k(0) is needed to determine TABLE III. The D4hIRs and parities for each band. IRs (NSO) PD /Gamma1/Gamma1+ 1 /Gamma1− 2 /Gamma1− 5 /Gamma1+ 5 /Gamma1+ 3 /Gamma1+ 1 /Gamma1− 3 /Gamma1+ 5 /Gamma1+ 4 /Gamma1− 2 /Gamma1− 1 /Gamma1− 5 Z /Gamma1− 2 /Gamma1+ 1 /Gamma1− 5 /Gamma1+ 3 /Gamma1+ 5 /Gamma1+ 1 /Gamma1− 3 /Gamma1− 2 /Gamma1+ 4 /Gamma1+ 5 /Gamma1− 1 /Gamma1− 5 Parities (SO) 1 2 3 4 56789 1 0 1 1 1 2 1 3 1 4 1 5 1 6 /Gamma1 +−−−++++−+++−−−− Z −+−−++++−−+++−−− X (+− )( +− )( +− )( +− )( +− )( +− )( +− )( +− ) M (+− )( +− )( +− )( +− )( +− )( +− )( +− )( +− ) R (+− )( +− )( +− )( +− )( +− )( +− )( +− )( +− ) A (+− )( +− )( +− )( +− )( +− )( +− )( +− )( +− ) Z2 1 115119-6TOPOLOGICAL NATURE OF THE FeSe 0.5Te0.5. . . PHYSICAL REVIEW B 92, 115119 (2015) the topology of the quasiparticle states, since renormalization does not change the nontrivial topological nature of the bulksystem. Following Refs. [ 32], we compute the topological invariant of an interacting system with the Hamiltonian definedbyH t(k,ω)=H(k,ω)+/Sigma1k(0)−μ. 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PhysRevB.100.235144.pdf

PHYSICAL REVIEW B 100, 235144 (2019) Global phase diagram of the one-dimensional Sachdev-Ye-Kitaev model at finite N Xin Dai,1,2Shao-Kai Jian,1,3and Hong Yao1,4,* 1Institute for Advanced Study, Tsinghua University, Beijing 100084, China 2Department of Physics, Ohio State University, Columbus, Ohio 43210, USA 3Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA 4State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China (Received 17 April 2018; revised manuscript received 11 December 2019; published 30 December 2019) Many key features of the higher-dimensional Sachdev-Ye-Kitaev (SYK) model at finite N remain unknown. Here we study the SYK chain consisting of N(N/greaterorequalslant2) fermions per site with random interactions and hoppings between neighboring sites. In the limit of vanishing SYK interactions, from both supersymmetric field theoryanalysis and numerical calculations we find that the random hopping model exhibits Anderson localization atfinite N, irrespective of the parity of N. Moreover, the localization length scales linearly with N, implying no Anderson localization only atN=∞ . For finite SYK interaction J, from the exact diagonalization we show that there is a dynamic phase transition between many-body localization and thermal diffusion as Jexceeds a critical value J c. In addition, we find that the critical value Jcdecreases with the increase of N, qualitatively consistent with the analytical result of Jc/t∝1 N5/2logNderived from the weakly interacting limit. DOI: 10.1103/PhysRevB.100.235144 I. INTRODUCTION The seminal Sachdev-Ye-Kitaev (SYK) model [ 1,2] presents a zero-dimensional cluster consisting of NMajorana fermions with random all-to-all interactions. In the large- N limit it is exactly solvable, exhibiting maximal quantum chaos[2–4], emergent SL(2,R) symmetry, as well as a holographic dual to dilaton gravity theory in nearly AdS 2geometry [ 2,3]. Owing to its solvability and intriguing properties, it has stimu-lated enormous excitement [ 2,5–46]. In particular, the large- N limit of the SYK model, after being properly generalized tohigher dimensions [ 14,33–46], could provide an insightful and promising avenue to investigate the spectral and transportproperties of non-Fermi-liquid states. Nonetheless, featuresof the higher-dimensional SYK models with finite N remain largely unknown. As the case of finite Nis directly relevant to possible experimental realizations [ 47–50] of SYK models, it is desired to understand the characterizing properties of thehigher-dimensional SYK models at finite N. Here we consider a generic SYK chain model of Majorana fermions respecting time-reversal symmetry, which includesfour-fermion random interactions and random hoppings be-tween neighboring sites as shown in Fig. 1[see Eq. ( 1) below]. Note that the neighboring fermion hopping on a bipartitelattice respects the time-reversal symmetry defined as γ j,x→ (−)xγj,xwhere γj,xrepresents the Majorana fermion with flavor j=1,..., Non site x. Both the random hoppings and the random interactions are characterized by Gaussian randomvariables with zero mean; their variances are given by t 2/N and 3! J2/N3, respectively. We first consider the noninteracting limit, namely, J=0, for which the model in Eq. ( 1) reduces to a one-dimensional *yaohong@tsinghua.edu.cn(1D) random hopping model [ 51]. The presence of time- reversal symmetry renders the Majorana system in the BDIclass [ 52,53]. In particular, when the system size Lis odd, there will be Nzero-energy single-particle modes in the band center due to the particle-hole symmetry. From numericalcalculations, we find that the zero modes are localized forfinite N(both even and odd), which implies that all single- particle wave functions are Anderson localized. Moreover, ourresults show that the localization length scales linearly withthe fermion flavor N, i.e.,ξ∝N, indicating the absence of An- derson localization only atN=∞ . Inspired by the pioneering work of Refs. [ 54,55], we further derive the corresponding supersymmetric field theory and find that the low-energyphysics can be described by the supersymmetric nonlinear σ model with a vanishing topological θterm. From supersym- metric field-theory analysis, we obtain that the correspondingconductance decays exponentially with system size and thelocalization length scales linearly with N, consistent with the numerical calculations. For the case of finite interactions, by performing ex- act diagonalization (ED) we show that there is a dynamicphase transition from the many-body localized (MBL) phase[56–59] to the thermal diffusive metal phase as the interaction strength exceeds a critical value J c. When J<Jc, the tendency to the MBL phase can be understood perturbatively: a weakinteraction is irrelevant to the Anderson localized phase inthe noninteracting limit so the system remains many-bodylocalized; namely, sufficiently weak SYK interactions cannoteffectively thermalize the system which is Anderson localizedin the noninteracting limit. Numerically we find that thedynamic phase transition is characterized by the critical ex-ponent ν≈1.1±0.1, which is consistent with previous works on MBL transition using small system size ED. Moreover,as shown in Fig. 1(b), we find that the critical interaction strength J cneeded to thermalize the system decreases with the 2469-9950/2019/100(23)/235144(8) 235144-1 ©2019 American Physical SocietyXIN DAI, SHAO-KAI JIAN, AND HONG Y AO PHYSICAL REVIEW B 100, 235144 (2019) FIG. 1. (a) The schematic representation of the SYK chain model. (b) The global phase diagram of the SYK chain model at finite N. The system is in the many-body localized (MBL) phase when the SYK interaction Jis relatively weak, but exhibits thermalization and diffusion when Jexceeds a critical value Jc. Note that the critical value Jcdecreases with increasing N. increase of N, which is consistent with the analytical result of Jc/t∝1 N5/2logNderived from the weakly interacting limit [ 57]. II. MODEL We consider the SYK chain model of Majorana fermions: ˆH=/summationdisplay x,jkitjk,xγj,xγk,x+1+/summationdisplay x,ijk lJijk l,xγi,xγj,xγk,x+1γl,x+1 +/summationdisplay x,ijk lUijk l,xγi,xγj,xγk,xγl,x, (1) where γj,xrepresent Majorana fermions with flavor index j=1,..., Non site x=1,..., L.H e r e Uijk l,xlabel the usual on-site SYK interactions while tjk,xandJijk l,xrefer to random hopping and interaction between neighboring sites that areGaussian random variables with mean t 0=0 and variance /angbracketleftt2 jk,x/angbracketright=t2/Nand/angbracketleftJ2 ijk l,x/angbracketright=J2/N3, respectively. The hopping of Eq. ( 1) represents the random hopping model in the strong disorder limit ( t/t0=∞ ). We shall show below that it is qualitatively different from the weak disorder limit ( t/lessmucht0) in terms of the localization physics studied in the literature[51,60–62]. It is obvious that the model in Eq. ( 1) respects the time-reversal symmetry defined as γ j,x→(−1)jγj,x.T h e time-reversal invariance then forbids onsite quadratic termiγ i,xγj,xin the Hamiltonian. In the following, we shall focus on the case of vanish- ing onsite interactions, namely, Uijk l,x=0, while varying the nearest-neighbor SYK interaction strength Jwith respect to the hopping strength t. This is partly because the onsite SYK interactions cannot be defined for the case of N=2 Majorana fermions. In contrast, a finite nearest-neighbor SYKinteraction Jis allowed for all N/greaterorequalslant2, including N=2. As the case of N=2 is numerically more accessible, we can obtain more reliable results up to a reasonably large systemsize L. Nonetheless, we would like to emphasize that the general feature of the global phase diagram and the universalproperties of the MBL transitions do not depend on thespecific SYK interactions we consider. In other words, weexpect that characters of the phase diagram and the transitionsobtained for the nearest-neighbor SYK interactions also applyto the case of onsite SYK interactions. As an illustration,we calculate the many-body level statistics with solely onsiteSYK interactions for N=4, the result of which shown inFIG. 2. (a) For N=2,3,4, we compute both the scaling be- havior of disorder averaged IPR (a) and ground-state entanglement entropy (EE) (b) with system size L. The Fermi level is set to zero in computing the half-chain EE. (c) The representative linear fit of ξwith NforN=20,21,..., 76 after 600 disorder realizations with L=6001. For clarity, we only show the scaling behavior of one zero mode for each Nand the results for all other zero modes are similar. Fig. 5is qualitatively the same as that of the nearest-neighbor SYK interactions. Consequently, we study the phase diagramas a function of NandJ/tto include the case of N=2, while setting the onsite interaction to be zero. III. NONINTERACTING LIMIT In the noninteracting limit, Eq. ( 1) is equivalent to the random hopping model in the strong disorder limit. It wasshown previously that, when Nis odd, the zero modes of the random hopping model in the weak disorder limit areextended rather than Anderson localized [ 51,60–62]; it is not known if the system is Anderson localized or not in thecurrent strong disorder case, especially for odd- NMajorana fermions. Thus, we numerically calculate the inverse partici-pation ratio (IPR) [ 63] of the zero-mode wave function, which is defined by IPR = /summationtextL x=1(ψ∗ xψx)2 (/summationtextL x=1ψ∗xψx)2, where ψxlabels a zero- mode wave function and Ldenotes the lattice size. Towards the thermodynamic limit L→∞ , the scaling behaviors of the ensemble averaged IPR can tell if the wave function islocalized (IPR ∝const), extended ( ∝ 1 L), or critical ( ∝1 Lζwith 0<ζ< 1). As shown in Fig. 2(a), the IPR saturates to some nonzero constants with increasing LforN=2,3,4, signaling a very strong localization behavior. As a benchmark, we also study the scaling behaviors of the half-chain entanglement entropy (EE) of the ground-statewave function of the random hopping chain using Klich’smethod [ 64]. It was shown in Refs. [ 65,66] that in the non- interacting system inspecting entanglement properties of theground state alone can tell if the system is localized or not. As shown in Fig. 2(b), the ground-state EE saturates to a constant value as L→∞ forN=2,3,4, implying a localized state. The scaling behaviors of both the IPR and ground-state EE with respect to the system size Lyield consistent results and suggest that the single-particle wave functions are Andersonlocalized in the noninteracting limit, in contrast to the case 235144-2GLOBAL PHASE DIAGRAM OF THE ONE-DIMENSIONAL … PHYSICAL REVIEW B 100, 235144 (2019) with constant diagonal hopping [ 51]. To see if the Anderson localization persists to larger N, we compute the IPR of the zero modes up to N=76 with fixed system size. The corresponding localization lengths can be extracted from therelation IPR ∝1/ξ[63]. From the log-log plot shown in Fig. 2(c), we find that the localization length ξofN∈[20,76] can fit linearly with N, namely, ξ∝NforN/greatermuch1. It is quite remarkable that a single linear fit works for both even and oddN; no discernible sign of parity oscillations can be observed. Note that this linear scaling relation of localization lengthholds for all zero-mode wave functions. Although a similar relation was observed in the weak disorder limit ( t/lessmucht 0)[51,60,67], there is an important and qualitative distinction with the present strong disorder limit(t/t 0=∞ ). For the case of the weak disorder limit, Anderson localization occurs only for even Nwhile all zero-energy wave functions are extended for odd N. Consequently, it is natural to infer that the topological protection of the delocalizationin the wave function for odd Nin the weak disorder limit fails in the strong disorder limit. Indeed, as we shall showbelow, the topological θterm in the supersymmetric nonlinear σmodel vanishes in the strong order case for both even and odd N, consistent with the numerical results discussed above. IV . SUPERSYMMETRIC FIELD THEORY To furnish a firm understanding of numerical results, we develop a field theory using the supersymmetry approach[68–71], which is a powerful tool for analyzing noninter- acting disorder problems. For simplicity, we only sketchthe derivation and the details can be found in the Appen-dices. While the supersymmetry method was originally devel-oped to deal with complex fermions, concerning the single-particle physics the results of the supersymmetry theory ap-ply for both complex and Majorana fermions as we arguebelow. Suppose the single-particle Hamiltonian for Majoranafermions takes the form of H(γ)=/summationtextit jk,xγj,xγk,x+1. Imag- ine there exists an identical “ghost” copy H(γ/prime) of the original H(γ) such that they add up forming the complex fermionic Hamiltonian H(χ)=H(γ)+H(γ/prime)=/summationtext jk[itjk,xχ† j,xχk,x+1+ H.c.] where χj=(γ1 j+iγ2 j)/2 are complex fermion annihi- lation operators. The localization properties of the complexfermion model H (χ) are identical to those of the Majorana fermion model H(γ) as they share the same single-particle matrix itjk,x. The basic idea of the supersymmetry method is to promote the original anticommuting fermionic field χto the superfield ψby adding a commuting bosonic counterpart φ, i.e.,ψ= (φ,χ )T, such that the disorder average can be performed at the very beginning, due to the cancellation of determinantsfrom the Gaussian integrals of complex and Grassmann vari-ables. After the disorder average, the partition function can bewritten as Z=/integraldisplay D(¯ψ,ψ )e x p/bracketleftbigg i/summationdisplay n¯ψn,μzψn,μ−2t2 N/summationdisplay n∈A m∈Bstrgμμ ngνν m/bracketrightbigg , (2)where summation over repeated indices is assumed, zis the frequency, str represents the supertrace, and gμμ n≡ψn,μ⊗ ¯ψn,μis the superfield bilinear living on AandBsublattices, respectively (for details see the Appendices). To proceed,we introduce two auxiliary supermatrix fields Q ± nm≡QA,n± iQB,mto decouple the quartic term and then integrate out the superfield ψto obtain the action in terms of the superfield Q. The next step is to get the saddle-point solutionδS δQ±=0. Then we perform gradient expansions around the ground-state manifold to identify the low-energy degrees of freedom. Theresulting effective action at z=0i s S[T]=−˜ξ 8/integraldisplay drstr(∂T−1∂T), (3) where ˜ξ=Nis in units of the lattice constant a. One key feature of the effective action of Eq. ( 3)i st h e absence of the topological term ( N/2) str T−1∂Twhich, ac- cording to Refs. [ 51,67,72], would lead to the delocalized zero modes for odd N. In other words, the vanishing topo- logical term in Eq. ( 3) implies Anderson localization for both even and odd N. From the effective action in Eq. ( 3), it is conceptually straightforward to calculate the physicalobservables. For instance, the conductance at a given energyEis the functional average of the corresponding retarded and advanced Green functions g(E)≡/angbracketleftG(E +)G(E−)/angbracketright. However, the actual evaluation using the supersymmetric nonlinear σ model is technically complex and we just show the resulthere. Using the transfer-matrix method [ 51], we obtain the conductance gat zero energy for L/greatermuch˜ξ: g≈/radicalBigg ˜ξ πLexp/bracketleftbigg −L ˜ξ/bracketrightbigg , (4) which is consistent with the numerically observed Anderson localization behavior. Moreover, from Eq. ( 4), it is clear that the coupling constant ˜ξin the effective action of Eq. ( 3) can be identified as the localization length, which scales linearlywith NforN/greatermuch1. This linear- Nlocalization length for N/greatermuch1 is consistent with the result obtained from numerical calcula-tions. V . FINITE SYK INTERACTIONS After establishing Anderson localization in the nonin- teracting limit, we are ready to consider finite interactionstrength, i.e., J>0. To investigate how the interactions can thermalize the system, we employ ED to calculate the many-body level statistics of the interacting Hamiltonian of Ma-jorana fermions in Eq. ( 1). Assuming that {e n}denotes the many-body energy level in an ascending order, we calculate the dimensionless ratio /tildewiderndefined by /tildewidern=min( sn,sn−1) max( sn,sn−1), where sn=en+1−en[58,73]. For the uncorrelated energy lev- els obeying Poisson distribution, /angbracketleft/tildewider/angbracketright→ 2l n2−1≈0.386, while, for the Gaussian orthogonal ensemble (GOE) of a ran-dom matrix, /angbracketleft/tildewider/angbracketright→ 0.53. When J=0,/angbracketleft/tildewider/angbracketright≈0.386 for N=2 and 4, as shown in Figs. 3(a) and3(b), respectively, indicating Poisson distribution that is consistent with the Anderson local-ized state for finite N. Moreover, /angbracketleft/tildewider/angbracketrightincreases as Jincreases, indicating that the SYK interactions tend to thermalize thesystem. For both N=2 and 4, it is clear that /angbracketleft˜r/angbracketrightbetween 235144-3XIN DAI, SHAO-KAI JIAN, AND HONG Y AO PHYSICAL REVIEW B 100, 235144 (2019) FIG. 3. Disorder averaged level statistics for N=2 (a) and N=4 (b). By varying the interaction strength J/t, there are crossings between adjacent system sizes L. The positions of the crossings gradually drift to smaller J/tbut the trend of slowing down with increasing Lcan be seen clearly. The finite-size data collapse for N=2 (c) and N=4 (d) in the vicinity of the crossing points. Jc= 0.23t,ν=1.1f o r N=2, and Jc=0.045t,ν=0.99 for N=4. All the results are obtained by setting t1=0.5t,t2=1.5t. adjacent system sizes Lcrosses at a critical interaction strength Jc, indicating that there is a dynamic quantum phase transition from the MBL phase ( J<Jc) to the thermalized phase ( J>Jc). Due to the finite-size effect, the crossing points drift gradually towards smaller JcasLincreases, which is common in the ED studies of many-body localizations [ 58]. Nonetheless, the tendency of the drift becomes slower forlarger L. Essentially, this implies that J cis nonzero and the MBL phase should persist below Jcin the thermodynamic limit. To characterize the MBL transitions, we explore the crit- ical behaviors of the dynamic transition. Around the MBLtransition, /angbracketleft˜r/angbracketrightshould obey a universal scaling function, i.e., /angbracketleft˜r/angbracketright= f[(J−J c)L1/ν], where νis the correlation /localization length critical exponent. By collapsing the data, as shown inFig. 3(c), we obtain the critical exponent ν≈1.1±0.1f o r N=2( f o r N=4, the data collapse shown in Fig. 3(d) gives rise to ν≈0.99±0.2). In order to improve the quality of data collapse, we set the variance of the random hopping inEq. ( 1)t ob e t 2 1/Nand t2 2/N,t1/negationslash=t2, for odd and even bonds, respectively. The staggered variances significantly shortenthe localization length (which is still proportional to N,a s shown in the Appendices); accordingly, the finite-size effectdecreases for the accessible system size. Our ED calculationsshow that the critical exponent ν≈1.1, which is consistent with previous ED works on MBL transition using smallsystem size [ 74–76] and is still quite different from the results obtained by real-space renormalization studies using largesystem size [ 77–81].As explicitly shown for the N=2 and 4 SYK chain, the finite- Neffect renders the MBL phase when the interaction strength Jis smaller than a critical value J c.T h ev a l u eo f Jc ofN=4 is smaller than the one of N=2, indicating that Jc decreases as Nincreases. Due to the absence of Anderson localization for N=∞ , it is clear that Jc=0f o r N=∞ . As the discussion of the SYK models generally relies on alarge- Napproximation to control the quantum fluctuations, it is interesting to further explore how the critical strength J c scales with 1 /N. In the weakly interacting limit, the energy scale corresponding to the MBL transition is given by Tc∼ δξ λ|logλ|[56,57], where δξ=1 ρξis the average level spacing of single-particle states within a localization length in the noninteracting limit. ρis the average density of single-particle states per unit volume, and the dimensionless quantity λ= J N3/2δξcharacterizes the interaction strength with respect to the average single-particle level spacing. It is known from the noninteracting calculations that ξ∝Nand the average density of states per unit volume is found to be ρ∝t−1N (see the Appendices), thus δξ∝tN−2and Tc∝t2 J1 N5/2logN.I t directly leads to a rough estimate of the critical interaction strength Jc/t∝1 N5/2logNfor the dynamic transition of full many-body localization (namely, requiring Tc∼twhere tis the order of the bandwidth). By using the numerical datashown in Fig. 3, we estimate that the critical strength scales as J c∝N−ηwithη≈2.4, which is close to the scaling behavior ofη=5/2 derived from the weakly interacting limit (up to a logarithmic correction). Note that this scaling is consistentwith the requirement that J cvanishes at N=∞ . VI. DISCUSSION AND CONCLUDING REMARKS We have shown that, in the noninteracting limit, all the single-particle states in the SYK chain at finite N(N/greaterorequalslant2) are localized irrespective to the parity of N, due to the vanishing topological θterm. Here we conjecture that the same localiza- tion physics should apply to the other four symmetry classesin one dimension based on the notion of superuniversality [67,72,82], which refers to the fact that in one dimension all five symmetry classes, including classes D and DIII, sharesimilar low-energy properties. We further showed that thesystem enters an MBL phase for weak SYK interactions butundergoes a dynamic phase transition from the MBL phase toa thermalized phase when the interaction Jexceeds a critical value J cwith Jc/t∼1 N5/2logN. Finally, we mention some future directions related to finite N. For instance, it would be desired to characterize the thermal phase at finite Nin full details including its Lyapunov exponent, specific heat, and transportbehaviors. Due to the finite- Neffect, it is expected that its characters should be renormalized from its large- Nlimit. ACKNOWLEDGMENTS We thank E. Altmann, Y .-F. Gu, S. A. Kivelson, and X.-L. Qi for helpful discussions. This work is supported inpart by the National Natural Science Foundation of Chinaunder Grant No. 11825404 (X.D., S.-K.J., and H.Y .), theMinistry of Science and Technology of China under Grants 235144-4GLOBAL PHASE DIAGRAM OF THE ONE-DIMENSIONAL … PHYSICAL REVIEW B 100, 235144 (2019) No. 2016YFA0301001 and No. 2018YFA0305604 (H.Y .), the Strategic Priority Research Program of Chinese Academyof Sciences under Grant No. XDB28000000 (H.Y .), BeijingMunicipal Science and Technology Commission under GrantNo. Z181100004218001 (H.Y .), and Beijing Natural ScienceFoundation under Grant No. Z180010 (H.Y .). APPENDIX A: DERIV ATION OF SUPERSYMMETRIC FIELD THEORY 1. Disorder average The derivation of the supersymmetric field theory largely follows the approach developed in Refs. [ 51,54,67]. Thehopping matrix elements satisfy /angbracketleftbig tμν nm/angbracketrightbig =0, (A1) /angbracketleftbig tμν nmtν/primeμ/prime nm/angbracketrightbig =λ2 Nδμμ/primeδνν/primeδm,n+1. (A2) In order to carry out the disorder average, we promote the fermionic field φto the two-component superfield: ψ=/parenleftbigg ψb ψf/parenrightbigg (A3) where the subscripts band fdenote the bosonic and fermionic field variables, respectively. Then we can proceed by integrat-ing over t: /angbracketleftBigg exp⎛ ⎝i/summationdisplay n∈A,m∈B,μν¯ψn,μtμν nmψm,ν+H.c.⎞ ⎠/angbracketrightBigg =C/integraldisplay dtexp⎛ ⎝i/summationdisplay n∈A,m∈B,μν¯ψn,μtμν nmψm,ν+H.c.−N 2λ2Trt2⎞ ⎠ =C/integraldisplay dtexp⎛ ⎝−/summationdisplay n∈A,m∈B,μν/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalbigg N 21 λtμν nm−iλ/radicalbigg 2 N¯ψn,μψm,ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2⎞ ⎠=exp⎛ ⎝−/summationdisplay n∈A,m∈B,μν2λ2 N¯ψn,μψm,ν¯ψm,νψn,μ⎞ ⎠ =exp⎛ ⎝−/summationdisplay n∈A,m∈B,μν2λ2 Nψn,μ¯ψn,μψm,ν¯ψm,ν⎞ ⎠=exp⎛ ⎝−2λ2 N/summationdisplay n∈A,m∈B,μνstrgμμ ngνν m⎞ ⎠ (A4) where Cis a normalization constant and we have introduced the bilinear term gμμ n≡ψn,μ⊗¯ψn,μ. (A5) In the last two identities of Eq. ( A4) we have made use of the cyclic invariance property of the supertrace [ 71]. Then we arrive at the partition function Eq. ( 3) in the main text. 2. Hubbard-Stratonovich transformation Now we perform the Hubbard-Stratonovich transformation by introducing a pair of supermatrix fields Q± nm≡QA,n±iQB,m, where QA,n(QB,m)l i v eo nt h e A(B) sublattice, respectively: Z=/integraldisplay D(¯ψ,ψ )e x p⎛ ⎝i/summationdisplay n,μ¯ψn,μzψn,μ−2λ2 N/summationdisplay n∈A,m∈B,μνstrgμμ ngνν m⎞ ⎠ ×/integraldisplay DQ±exp⎡ ⎣−/summationdisplay n∈A,m∈B,μν/parenleftBigg 1 λ/radicalbigg 1 2NQ−−iλ/radicalbigg 2 Nψn,μ¯ψn,μ/parenrightBigg/parenleftBigg 1 λ/radicalbigg 1 2NQ+−iλ/radicalbigg 2 Nψm,ν¯ψm,ν/parenrightBigg⎤ ⎦ =/integraldisplay DQ±D(¯ψ,ψ )e x p⎛ ⎝i/summationdisplay n,μ¯ψn,μzψn,μ+i N/summationdisplay n∈A,μν¯ψn,μ(Q+ n,n−1+Q+ n,n+1)ψn,μ +i N/summationdisplay m∈B,μν¯ψm,ν(Q− m,m−1+Q− m,m+1)ψm,ν−N 2λ2/summationdisplay n∈A,m∈BQ+ nmQ− mn⎞ ⎠. (A6) The next step is to integrate out ψand we arrive at S[Q±]=N 2t2/summationdisplay nstr(Q+Q−)−N/summationdisplay n∈Astr ln( z+Q+ n,n+1+Q+ n,n−1)−N/summationdisplay m∈Bstr ln( z+Q− m,m+1+Q− m,m−1). (A7) 235144-5XIN DAI, SHAO-KAI JIAN, AND HONG Y AO PHYSICAL REVIEW B 100, 235144 (2019) 3. Nonlinear σmodel in the strongly disordered limit It is clear that, for z=0, the action in Eq. ( A7) is invariant under the transformation Q+→T1Q+T2andQ−→T−1 2Q−T−1 1, where T1,T2∈GL(1|1), and GL(1 |1) is the generalization of the original fermionic symmetry. The overall factor Nenables us to seek the saddle-point solution, which is exact in the large- Nlimit. By assuming a uniform ansatz Q±=1 2(Q± n,n+1+Q± n,n−1), from the saddle-point condition (δS δQ±=0) we obtain Q∓=2λ2 z+Q∓/equal1⇒ Q± sp=1 2/parenleftBig −z±/radicalbig z2+8λ2/parenrightBig . (A8) To identify the low-energy degrees of freedom for z=0, we can parametrize Q±by ( Q+,Q−)=(PT,T−1P)i nE q .( A7), where both T,P∈GL(1|1) and Tstand for massless fluctuation while Pis the massive fluctuation that is incompatible with the symmetry of the ground state. Let us ignore the massive fluctuations by setting P=1, and the action is of the form Sfl[T]=N/summationdisplay n∈Astr ln ( Tn,n+1+Tn,n−1)+N/summationdisplay m∈Bstr ln/parenleftbig T−1 m,m+1+T−1 m,m−1/parenrightbig . (A9) We then expand Tnmas Tnm=Tn+a 2∂n,mTn+a2 8∂2 n,mTn+··· (A10) where ais the lattice constant and ∂n,mdenote the directional derivative from site n→m. Taking Eq. ( A10) into Eq. ( A9), 1 NSfl[T]≈/summationdisplay n∈Astr ln/parenleftbigg 2Tn+a 2∂n,n+1Tn+a 2∂n,n−1Tn+a2 8∂2 n,n+1Tn+a2 8∂2 n,n−1Tn/parenrightbigg +/summationdisplay m∈Bstr ln/parenleftbigg 2T−1 m+a 2∂m,m+1T−1 m+a 2∂m,m−1T−1 m+a2 8∂2 m,m+1T−1 m+a2 8∂2 m,m−1T−1 m/parenrightbigg ≈/summationdisplay n∈Astr ln 2 Tn−/summationdisplay m∈Bstr ln 2 Tm+a2 16/summationdisplay n∈A/parenleftbig T−1 n∂2 n,n+1Tn+T−1 n∂2 n,n−1Tn/parenrightbig +a2 16/summationdisplay m∈B/parenleftbig Tm∂2 m,m+1T−1 m+Tm∂2 m,m−1T−1 m/parenrightbig ≈a2 16/summationdisplay n∈A/parenleftbig T−1 n∂2 n,n+1Tn+T−1 n∂2 n,n−1Tn/parenrightbig +a2 16/summationdisplay m∈B/parenleftbig Tm∂2 m,m+1T−1 m+Tm∂2 m,m−1T−1 m/parenrightbig , (A11) where we have made use of the fact that/summationtext m∈B∂n,mTn=0. By taking the continuum limit/summationtext n∈A→1 2a/integraltext ,E q .( A11) can be written as Sfl[T]=Na2 8/bracketleftBigg/summationdisplay n∈Astr/parenleftbig T−1 n∂2Tn/parenrightbig +/summationdisplay m∈Bstr/parenleftbig Tm∂2T−1 m/parenrightbig/bracketrightBigg /similarequalNa 16/integraldisplay str(T−1∂2T+T∂2T−1)=−Na 8/integraldisplay str(∂T−1∂T) (A12) where the integration by parts is used in the last equality. APPENDIX B: LEVEL STATISTICS AT LARGE J/tAND ONSITE U As shown in Fig. 4,a sJ/tincreases, the /angbracketleftr/angbracketrightvalue increases towards the GOE value 0.531, for both N=2 and 4. FIG. 4. The level statistics at large J/tforN=2 (a) and N=4( b ) .Now we instead consider the onsite SYK interaction U while setting J=0. As shown in Fig. 5(a),t h e/angbracketleftr/angbracketrightvalue asymptotes to Poisson and GOE values at small and large U/t limits, respectively. If we zoom in, as shown in Fig. 5(b),w e FIG. 5. The level statistics as a function of U/tforN=4a n d J=0. 235144-6GLOBAL PHASE DIAGRAM OF THE ONE-DIMENSIONAL … PHYSICAL REVIEW B 100, 235144 (2019) can see there is a crossing around U/t≈0.13, indicating a dynamical MBL phase transition. APPENDIX C: DENSITY OF STATES AND LOCALIZATION LENGTH IN THE NONINTERACTING LIMIT In the noninteracting limit, there are N×Lsingle-particle states in total. Therefore, the single-particle density of statesρper unit length can be found as ρ=NL L1 /Delta1E=N /Delta1E, (C1) where /Delta1Eis the total bandwidth. As shown in Fig. 6(a),/Delta1E/t saturates to constant as N→∞ with fixed L. Consequently, we conclude that ρ∝N/t. In addition, we also computed the localization length ξin the presence of dimerization. The data shown in Fig. 6(b) give rise to ξ≈0.22Nαwithα=1.04±0.04, while inFIG. 6. (a) /Delta1Eas a function of 1 /Nwith N∈[10,48]. (b) The log-log plot for the localization length ξas a function of Nwith N∈ [15,35]. The system size L=1001. 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PhysRevB.71.153306.pdf

Ballistic spin currents in mesoscopic metal/In(Ga)As/metal junctions Minchul Lee1,2and Mahn-Soo Choi1,* 1Department of Physics, Korea University, Seoul 136-701, Korea 2Department of Physics and Astronomy, University of Basel, CH-4056 Basel, Switzerland sReceived 14 July 2004; revised manuscript received 7 February 2005; published 18 April 2005 d We investigate ballistic spin transport through a two-dimensional mesoscopic metal/semiconductor/metal double junction in the presence of spin-orbit interactions. It is shown that finite transverse and/or longitudinalspin currents can flow in the presence of the Rashba and Dresselhaus terms. DOI: 10.1103/PhysRevB.71.153306 PACS number ssd: 73.63. 2b, 72.10. 2d, 71.70.Ej I. INTRODUCTION Since the advent of “spintronics” to utilize an electron’s spin, rather than its charge, for information processing andstorage, 1there has been growing interest in generating spin currents.2–14Though injecting spin-polarized carriers electri- cally still remains a challenge,2there have been proposed various all-semiconductor devices based on ferromagneticsemiconductors 3or spin-orbit sSOdinteractions.4In particu- lar, the latter enables us to manipulate the spin by controllingthe orbital motion of electric carriers, say, by applying anelectric field. Moreover, it has been suggested that the SOcoupling gives rise to dissipationless spin currents perpen-dicular to the external electric field, which is known as theintrinsic-spin Hall effect. 4–7 Theoretically, the existence of the spin Hall current has been highly controversial. Sinova et al.5predicted a finite- spin Hall current and universal-spin Hall conductivity in aclean, infinite two-dimensional electron system s2DES d. 6 Different groups have provided mutually contradicting argu- ments on the effect of impurity scattering in an infinite2DES. 7,8Recently, it was claimed that vanishing bulk-spin conductivity is an intrinsic property of clean 2DES.9The spin Hall effect in ssemi dfinite-size systems was also studied: It was argued that a finite-spin Hall current flows in the vi-cinity of the contacts, while the spin current vanishes in aninfinite system. 10Numerical studies have also reported finite- spin conductances in four-terminal samples.11Another im- portant issue has been raised regarding how the predictednonequilibrium-spin current is related to sor is distinguished from dthe background-spin current, which exists even in equilibrium. 12,13 A recent experiment14reports a finite-spin accumulation, possibly due to the spin current, in the very clean samples. Itimplies that the spin current, while it may vanish in the bulklimit, can be nonzero in finite or semifinite systems. In this paper we study ballistic spin transport through aclean, mesoscopic double-junction system consisting of a semiconductor stripe sandwiched by two normal-metalleads ssee Fig. 1 d. We use coherent scattering theory and show that in the presence of SO couplings, bothlongitudinal and transverse spin currents can flow in asemiconductor. II. MODELAND SCATTERING THEORY We consider a two-dimensional electron system of a semi- conductor sSdbetween two normal sNd-metal leads. Wechoose such a coordinate system that the xaxis syaxisdis perpendicular sparallel dto the N/S interfaces, and the zaxis is perpendicular to the two-dimensional s2Ddplane sFig. 1 d. The length swidth dof the semiconductor is LsWd; we will consider the limit W!‘. Within the effective-mass approximation,15the Hamiltonian reads as H=−"2 2=·1 msxd=+Vsx,yd+HRsxd+HDsxd.s1d The position-dependent effective mass msxdhas values ofmeandme*;emmein the normal metals and in the semi- conductor s−L/2,x,L/2d, respectively. The confinement potential has a potential barrier of height V0inside the semiconductor, Vsx,yd=V0fQsx+L/2d−Qsx−L/2dg+Vsyd, s2d where Qsxdis the Heaviside step function and Vsydaccounts for the finite width W. The potential barrier height V0is lower than the Fermi energy EFin the normal metals, so that EF*;EF−V0.0. The Rashba16and Dresselhaus17SO cou- pling terms are given by HR=a "ssxpy−sypxdandHD=b "ssypy−sxpxd,s3d respectively, inside the semiconductors, while they vanish in the normal-metal sides. In Eq. s3d,s=ssx,sy,szdare the Pauli matrices. The Rashba term HRarises when the confining potential of the quantum well lacks inversion symmetry, whilethe Dresselhaus term H Dis due to the bulk-inversion FIG. 1. A schematic of the system.PHYSICAL REVIEW B 71, 153306 s2005 d 1098-0121/2005/71 s15d/153306 s4d/$23.00 ©2005 The American Physical Society 153306-1asymmetry. In some semiconductor heterostructures se.g., InAs quantum wells dHRdominates,18and in others se.g., GaAs dHDis comparable to HR.19The coupling constants may range around a,0.1 eVÅ and b,0.09 eVÅ, respectively. Inside the semiconductor, the electrons feel a fictitious, in-plane magnetic field in the direction nˆk=xˆcoswk +yˆsinwk, where wk=arg fsbkx−akyd+isakx−bkydg. Accord- ingly, the eigenstates with spins parallel sm=+dand antipar- allel sm=−dtonˆkfor a given wave vector k=ksxˆcosf +yˆsinfdare written in the spinor form, Ckmsrd=eik·r ˛2Fme−iwk/2 e+iwk/2G. s4d The eigenenergies are Emskd=s"2/2me*dfk2−2mksosfdkg, whereksosfd;sme*/"2d˛a2+b2−2absin2 f. From the con- tinuity equation for the charge density, one can get the ex- pression for the charge-current density associated with agiven wave function Csrd, 20 jc=eRefC†srdvCsrdg, s5d wherevis the velocity operator defined by v=p me*−a "ssyxˆ−sxyˆd−b "ssxxˆ−syyˆd. s6d In the same manner, we define the spin-current density,13 jssnˆd=" 2C†srdvsnˆ·sd+snˆ·sdv 2Csrd, s7d according to the continuity equation, ]tQs+=·js=Ss, s8d for the spin density swith respect to the spin direction nˆd, Qssnˆd;" 2fC†srdsnˆ·sdCsrdg, s9d and the spin source, Sssnˆd=" 2ReFC†srdi "fH,nˆ·sgCsrdG. s10d The spin-source term arises in Eq. s8dbecause the spin-orbit couplings break the spin conservation. Before going further, it will be useful to understand the origin of the spin current in physical terms. As illustrated inFig. 2, for a,bÞ0 the Fermi contours, kFmsfd=mksosfd+˛kso2sfd+kF*2, s11d withkF*;˛2m*EF*/", are no longer isotropic,26and the group velocities vmskd=Ckm†vCkmof the eigenstates in Eq. s4dare not parallel to the wave vector k.20–22Nevertheless, Eq. s11d reveals an important symmetry property of the group veloci- ties, uv+skF+du=uv−skF−du. It means that the two eigenstates with opposite spin orientations make the same contributions to the charge transport along the kˆdirection sand opposite contri- butions along the perpendicular direction d.The spin transport withnˆ=nˆkˆis to the contrary: two eigenstates contribute theopposite ssame dspin currents along sperpendicular to dkˆ. This implies that the net-spin current is perpendicular to thecharge current. Particularly interesting are the cases of a=±b, where all the spin orientations ± nˆkfor the different wave vectors are parallel or antiparallel to each others wk=p/4dfsee Fig. 2 sbdg. It results from the conservation of ssx±syd/˛2, and the spin state becomes independent of the wave vector.22,23 Now we study charge and spin transport in N/S/N junc- tions. A coherent scattering theory at the N/S interfaces wasalready developed in the previous studies, 20,21considering the Rashba SO effect and appropriate boundary conditions. Itis straightforward to extend the scattering theory to incorpo-rate the Dresselhaus effect. We use the transfer-matrix for-malism to calculate the conductance through the semicon-ductor ssee Refs. 20 and 24 d. We consider the electrons incident from the left lead. The wave vector of the incident electron is at angle uwith the normal to the interface ssee Fig. 1 d. Contrary to the Rashba effect, the Dresselhaus effect is not invariant under rotations,causing anisotropic transport. 22Hence the relative orienta- tion, j, of the crystal symmetry axes and the interface sFig. 1dstrongly affects the spin current. Below we will calculate the charge conductance Gnscdsud;Inscdsud/Vsn=x,ydin the n direction for a definite incident angle uas well as the angle- averaged quantity Gnscd=e−p/2p/2duGnscdsud, whereVis the volt- age difference between two contacts and Inscdis the corre- sponding charge-current density in Eq. s5d. Also calculated are the analogously defined spin conductances Gnss,nˆdsudand Gnss,nˆd, polarized in the direction nˆ.27 The typical values for the parameters used below are EF=4.2 eV, em=0.063, b=0.1 eVÅ, L=200 nm, and W=1mm.aranges from −2 bto +2 b, andEF*ranges from 0 to 20 meV. We assume sufficiently low temperatures skBT!EF*d. FIG. 2. sColor online dRelative configurations of group velocities and spin quantization axes on Fermi contours for sad 2a=b=0.25 "vF*andsbda=b=0.25 "vF*with vF*="kF*/m*. Legend: thin solid-dashed curve: Fermi contour for m=±; thick solid-dashed tallsbluedarrows: group velocities for m=±; thick solid-dashed short sreddarrows: spin quantization axes for m=±. The configura- tion is symmetric under inversion.BRIEF REPORTS PHYSICAL REVIEW B 71, 153306 s2005 d 153306-2III. NORMAL INCIDENCE Owing to the symmetry uv+skF+du=uv−skF−du fsee the discus- sion below Eq. s10dgfor normal incidence su=0d, the charge current is purely longitudinal; i.e., Gyscdsu=0d=0. For a single transverse mode, we obtain the longitudinal charge conduc- tance, Gxscdsu=0d=e2 h32k2 us1+kd2−s1−kd2e2iDkLu2, s12d where Dk;˛kso2s−jd+kF*2, k;Dk/emkF, and kF;˛2meEF/". On the other hand, the spin current has only a transverse component and is polarized entirely in the xyplane; i.e., Gxss,nˆdsu=0d=0 for any nˆandGyss,zˆdsu=0d=0. Thenˆxˆ-polarized spin conductance Gyss,nˆxˆdsu=0dis given by Gyss,nˆxˆdsu=0d=e 4pL W32sme*2/"4dabcos2 j emkFksos−jd 3s1+k2d−s1−k2dsin2 DkL 2DkL us1+kd2−s1−kd2e2iDkLu2. s13d Gxscdsu=0dandGyss,nˆxˆdsu=0dare plotted in Fig. 3 as functions ofEF*anda/bfor different crystal orientations j. The peaks inGxscdsu=0dandGyss,nˆxˆdsu=0das a function of EF*come from the Fabry-Perot interference, which gives rise to resonances for DkL=npsn=0,1,2,... d. Unlike the slongitudinal d charge current, the spin current is very sensitive to a,b, andj, as seen from the factor abcos2 jin Eq. s13d. Note that Gyss,nˆxˆdhas no contribution from background-spin currents; within our scattering formalism, we count only the contribu-tions from electrons between the two Fermi levels of the twometal leads. 13 IV. ANGLE-AVERAGED CONDUCTANCES For true one-dimensional s1Ddleads skFW!1d, where only a single transverse mode is allowed, one has only to consider normal incidence su=0dor at a certain fixed u.25In the opposite limit skFW!‘d, where many transverse modes contribute to the transport, we should add up all the contri- butions from uin the range s−p/2,p/2d. It is tedious to find the scattering states for nonzero incidence angle uand more convenient to work numerically. Apparently, the main contribution to the longitudinal charge current comes from the normal incidence. Conse-quently, as shown in Fig. 4, the u-averaged longitudinal con- ductance Gxscdis rather similar to the normal incidence case Gxscdsu=0d. This is not the case for the spin transport. Figure 5 shows theudependence of the spin conductances polarized in the nˆxˆandzˆ, respectively. Again, the peaks correspond to the Fabry-Perot-type resonances. When summing up, the contri- butions to the nˆxˆ-polarized spin current from different angles are mostly canceled with each other, and hence the angle- averaged spin conductance Gyss,nˆxˆdbecomes small compared with the charge conductance Gxscd. On the other hand, the zˆ-polarized spin current is not subject to such cancellations, and remains relatively large sstill smaller than the longitudi- nal charge current despecially for j=0fsee Fig. 6 sadg. This is reminiscent of the intrinsic-spin Hall effect.6However, in our FIG. 3. sColor online dThe charge conductance Gxscdsu=0df sad andsbdgand the spin conductance Gyss,nˆxˆdsu=0df scdandsddgfor the normal incidence as functions of EF*fsadandscdganda/bfsbdand sddg.I nsadthree curves overlap almost completely. FIG. 4. sColor online dAngle-averaged charge conductance Gxscd as a function of sadEF*andsbda/bwith j=0. FIG. 5. Angle dependences of the spin conductances Gyss,nˆxˆdsud andGyss,zˆdsudforEF*=14 meV, a/b=0.5, and j=0. FIG. 6. sColor online dsadTransverse spin conductance Gyss,zˆdas a function of a/b.sbdSpatial dependence of the spin current den- sityIyss,zˆdwith a=0.1 eVÅ and b=0.BRIEF REPORTS PHYSICAL REVIEW B 71, 153306 s2005 d 153306-3caseGyss,zˆddepends on a,b,j, the potential barrier, and the channel length, showing no universal characteristics; for in- stance,Gyss,zˆdincreases almost monotonically with EF*. Our result is different from that of Mishchenko et al.10as well: The spin current is finite through the semiconductor regionand oscillates with position, even alternating its sign fFig. 6sbdg. This feature is due to contributions from the coherent standing waves. In the presence of impurity scattering thecoherent oscillation inside the sample should die away andthe spin current will be manifested only near the contacts. 10 Finally we remark that in the presence of both SO couplings the angle-averaged longitudinal spin conductance Gxss,nˆdis finite, even if much smaller than Gxscd. It reflects that spin is not conserved for an oblique incidence, because the spin-quantization directions are not consistent with the boundaryconditions.V. CONCLUSION Ballistic spin currents with different spin polarizations through mesoscopic metal/2DES/metal junctions have beeninvestigated in the presence of spin-orbit interactions. Usingthe coherent scattering theory we showed that longitudinaland/or transverse spin currents can flow through a clean2DES. The spin coherence can induce spin current and po-larization, with properties that are different from the ones inthe diffusive limit. ACKNOWLEDGMENTS M.L. thanks W. Belzig, C. Bruder, and J. Schliemann for helpful discussions. This work was supported by the SK-Fund, the SKORE-A, and the eSSC at Postech. M.-S.C. ac-knowledges the support from KIAS. *Electronic address: choims@korea.ac.kr 1G. A. Prinz, Science 282, 1660 s1998 d; S. A. Wolf, D. D. Aw- schalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L.Roukes, A. Y. 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Halperin, Phys. Rev. Lett.93, 226602 s2004 d. 11L. Sheng, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett. 94, 016602 s2005 d; B. K. Nikoli æ, L. P. Zârbo, and S. Souma, cond- mat/0408693 sunpublished d; E. M. Hankiewicz, L. W. Molen- kamp, T. Jungwirth and J. Sinova, cond-mat/0409334 sunpub- lished d. 12E. I. Rashba, Phys. Rev. B 70, 161201 sRds2004 d. 13E. I. Rashba, Phys. Rev. B 68, 241315 sRds2003 d. 14J. Wunderlich, B. Kästner, J. Sinova, and T. Jungwirth, cond-mat/ 0410295 sunpublished d.15G. Bastard, Wave Mechanics Applied to Semiconductor Hetero- structures sHalstead,NewYork,1988 d;M.G.Burt,Phys.Rev.B 50, 7518 s1994 d. 16Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 s1984 d. 17G. Dresselhaus, Phys. Rev. 100, 580 s1955 d. 18B. Das, S. Datta, and R. Reifenberger, Phys. Rev. B 41, 8278 s1990 d; G. L. Chen, J. Han, T. T. Huang, S. Datta, and D. B. Janes,ibid.47, R4084 s1993 d; J. Luo, H. Munekata, F. F. Fang, and P. J. Stiles, ibid.41, 7685 s1999 d. 19G. Lommer, F. Malcher, and U. Rossler, Phys. Rev. Lett. 60, 728 s1988 d; B. Jusserand, D. Richards, H. Peric, and B. Etienne, ibid.69, 848 s1992 d; B. Jusserand, D. Richards, G. Allan, C. Priester, and B. Etienne, Phys. Rev. B 51, R4707 s1995 d. 20T. Matsuyama, C.-M. Hu, D. Grundler, G. Meier, and U. Merkt, Phys. Rev. B 65, 155322 s2002 d. 21L. W. Molenkamp, G. Schmidt, and G. E. W. Bauer, Phys. Rev. B 64, 121202 sRds2001 d; G. Feve, W. D. Oliver, M. Aranzana, and Y. Yamamoto, ibid.66, 155328 s2002 d. 22J. Schliemann and D. Loss, Phys. Rev. B 68, 165311 s2003 d. 23J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 s2003 d. 24Gonzalo Usaj and C. A. Balseiro, cond-mat/0405065 sunpub- lished d. 25V. Marigliano Ramaglia, D. Bercioux, V. Cataudella, G. De Fil- ippis, and C. A. Perroni, cond-mat/0403534 sunpublished d. 26The Rashba and Dresselhaus terms in Eq. s3dare equivalent up to a unitary transformation. However, in the presence of both ef-fects, the anisotropy due to the Dresselhaus term manifests it-self. 27In 2D, the conductance and conductivity have the same dimen- sion. In particular, the spin conductance Gyss,zdis identical to the spin Hall conductivity ssHin, e.g., Refs. 5 and 10.BRIEF REPORTS PHYSICAL REVIEW B 71, 153306 s2005 d 153306-4

PhysRevB.77.195116.pdf

Ab initio simulation of photoemission spectroscopy in solids: Plane-wave pseudopotential approach with applications to normal-emission spectra of Cu(001) and Cu(111) Nataša Stoji ć,1,2Andrea Dal Corso,1,2Bo Zhou,3and Stefano Baroni1,2 1Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Beirut 2-4, I-34014 Trieste, Italy 2Theory @ Elettra group, INFM-CNR Democritos, I-34012 Trieste, Italy 3TASC National Laboratory, INFM-CNR, SS 14, km 163.5, I-34012 Trieste, Italy /H20849Received 26 March 2008; published 19 May 2008 /H20850 We develop a method for simulating photoemission spectra from bulk crystals in the ultraviolet energy range within a three-step model. Our method explicitly accounts for transmission and matrix-element effects, ascalculated from state-of-the-art plane-wave pseudopotential techniques within the density-functional theory.Transmission effects, in particular, are included by extending to the present problem a technique previouslyemployed with success to deal with ballistic conductance in metal nanowires. The spectra calculated for normalemission in Cu /H20849001/H20850and Cu /H20849111/H20850are in fair agreement with previous theoretical results and with experiments, including a recently determined experimental spectrum. The residual discrepancies between our results and thelatter are mainly due to the well-known deficiencies of the density-functional theory in accounting for corre-lation effects in quasiparticle spectra. A significant improvement is obtained by the LDA+ Umethod. Further improvements are obtained by including surface-optics corrections, as described by Snell’s law and Fresnel’sequations. DOI: 10.1103/PhysRevB.77.195116 PACS number /H20849s/H20850: 79.60.Bm, 71.15.Ap, 71.15.Mb I. INTRODUCTION Photoemission spectroscopy /H20849PES /H20850is one of the most ba- sic techniques for investigating the electronic properties ofsolids. 1,2In practice, however, it is difficult to directly extract information from the observed spectra and theoretical con-siderations are necessary for precise determination of the un-derlying transitions. The modeling of photoemission, as wellas the type and accuracy of the information that can begained from experiments, depends on the energy range of theincident light. In the energy range of ultraviolet photoemis-sion spectroscopy /H20849UPS /H20850/H20849between 5 and 100 eV /H20850, the spec- tra are dominated by wave-vector-conserving transitions /H20849di- rect transitions /H20850with transition matrix elements significantly differing for any pair of initial and final states. Hence, inUPS, final-state effects play a major role. In empirical ap-proaches, these final states are often modeled by free-electron bands, but in reality, they are influenced by the crys-tal potential, especially at low photon energies, andtherefore, their proper description requires detailed calcula-tions. Early photoemission calculations, ranging from one- electron approaches to many-body formulations, 3–7covered various aspects of the photoemission process. One-electronphotoemission calculations started with the so-called three-step model, 8which breaks the photoemission process into three independent steps: excitation of the photoelectron, itstransport through the crystal up to the surface, and its escapeinto the vacuum. Inclusion of quasiparticle lifetimes throughadjustable parameters, within the multiple-scattering Green’sfunction formalism, led to the development of the one-stepmodel. 9Its modern versions can model surfaces by a realistic barrier10and have the potential to replace the previously adopted muffin-tin potentials by space-filling potential cellsof arbitrary shape, 11also taking into account relativistic effects.12Ab initio methods based on the density-functional theory /H20849DFT /H20850are today considerably developed.13,14In particular, the plane-wave /H20849PW/H20850pseudopotential /H20849PP/H20850formulation is be- ing applied to a wide range of properties and systems. Rela-tivistic effects can be included in the PP both at the scalar relativistic or at the fully relativistic level, thus accountingfor spin-orbit coupling. 15This methodology, in principle, contains many of the ingredients necessary to predict a pho-toemission spectrum from first principles, from which infor-mation on various physical properties of the system can thenbe extracted. In the case of x-ray photoemission, for instance,the observed spectra are routinely compared to the density ofelectronic states. Considerable complications, however, arisein the ab initio simulation of UPS spectra, as well as in their interpretation, so that the application of state of-the-art DFTPW-PP techniques to this problem has hardly been attemptedso far. First and foremost comes the difficulty of accountingfor the nonperiodic nature of the electronic states involved inthe photoemission process. One of the few attempts to com-pute photoemission spectra by using PPs was made byStampfl et al. , 16who constructed the final states by a low- energy electron-diffraction /H20849LEED /H20850computational technique. Second, and no less important, is the well-known inability ofDFT to properly account for self-energy effects on the qua-siparticle states that are the main concern of PES. This fail-ure of DFT to accurately describe quasiparticle states is thefield of intense research, currently mainly addressed by usingtechniques from the many-body perturbation theory, such as,e.g., the GW approximation. 17–19 In this paper, the first problem is thoroughly addressed by calculating the transmission of electrons from the crystallinemedium into the vacuum by a technique that was previouslysuccessfully employed to deal with the ballistic conductanceof an open quantum system within the Büttiker–Landauerapproach. 20This technique, which was originally formulated with norm-conserving PPs, has been generalized to ultrasoftPHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 1098-0121/2008/77 /H2084919/H20850/195116 /H2084911/H20850 ©2008 The American Physical Society 195116-1/H20849US/H20850PPs by Smogunov et al. ,21and this generalization is used here to calculate the transmission into the vacuum ofthe crystalline Bloch states. In addition to transmission, acompletely ab initio approach to PES would require the cal- culation of dipole matrix elements and a proper account ofself-energy effects on the electron band structure, as well asof the effects of the change in the dielectric function uponcrossing the surface /H20849surface-optics effects /H20850. Dipole matrix elements are calculated completely ab initio by using a tech- nique first described by Baroni and Resta in 1986. 22The real part of the self-energy shifts to DFT bands is accounted forsemiempirically by using the LDA+ Umethod, while the imaginary part /H20849lifetime effects /H20850is simply added as an em- pirical parameter. Finally, surface-optics effects are ac-counted for by Snell’s and Fresnel’s equations. While theseequations could, in principle, be implemented by using adielectric function calculated ab initio , for simplicity, we choose to implement them by using experimental data. 23 As an application of our approach, we calculate the bulk contributions to the normal photoemission spectra fromCu/H20849001/H20850and Cu /H20849111/H20850. Copper is a prototypical system for UPS studies, for which many theoretical results, as well asaccurate experimental measurements, are available. We com-pare our calculations to previous theoretical studies, whichwere performed within the one-step and three-step models,and to experimental data. Not surprisingly, the main limitingfactor in our calculations appears to be the poor descriptionof the Cu electron bands by the local-density approximation/H20849LDA /H20850, while transmission effects are correctly accounted for, thus providing a viable way to select and to weighamong the many available final conduction states only thosethat couple to vacuum states. The paper is organized as follows: in Sec. II, we describe the theoretical method used to calculate photoemission spec-tra, while in Sec. III, we give some numerical details. In Sec.IV , we first discuss the contributions of different terms in ourexpression for the photoemission intensity on one specificexample; we then present our ab initio results for the Cu/H20849001/H20850surface, which were obtained at the DFT level with- out empirical adjustments, followed by the results forCu/H20849001/H20850and Cu /H20849111/H20850obtained from LDA+ Ubands and ac- counting for surface-optics effects. Section V contains ourconclusions. II. THEORY In a three-step model, the photoemission current is pro- portional to the product of the probability that an electron isexcited from an initial bulk state, /H9274i, to an intermediate bulk state,/H9274n, of energies EiandEnand wave vector k,/H20841Mni/H20849k/H20850/H208412 /H20849in this transition, the electron momentum is supposed to be conserved, in spite of surface effects that break translationalsymmetry /H20850, times the probability that the electron in the in- termediate state is transmitted into the vacuum, T/H20849E n,k/H20850, conserving the energy and the component of the momentumparallel to the surface, k /H20648. By summing the composite prob- abilities all over the possible initial and intermediate states,we obtain the current Ias a function of the photoelectron kinetic energy E kinand photon energy /H6036/H9275by using the fol- lowing standard expression:1,24,25I/H20849Ekin,/H6036/H9275,k/H20648/H20850/H11008/H20858 ni/H20885dk/H11036/H20841Mni/H20849k/H20850/H208412T/H20849En,k/H20850 /H11003/H9254/H20851En/H20849k/H20850−Ei/H20849k/H20850−/H6036/H9275/H20852/H9254/H20851Ekin−En/H20849k/H20850+/H9021/H20852, /H208491/H20850 where k/H11036is the component of kperpendicular to the surface and/H9021is the work function. The three-step model that we use is, of course, an approximation which, in particular, does notproperly account for coherence between the excitation pro-cess occurring in the bulk and the escape of the electronoccurring at the surface. This coherence may give rise tointerference effects which are, therefore, neglected in ourapproach. The transition matrix element in Eq. /H208491/H20850,M ni, is calculated from the interaction operator: Hint=e 2mc/H20849A·p+p·A/H20850, /H208492/H20850 where Ais the vector potential of the electromagnetic field, pis the momentum operator of the electron, eandmare the electron charge and mass, cis the speed of light, and nonlin- ear effects have been neglected. We will consider only theA·p-type interaction, while the /H11612Ainteraction, originating from the second term in Eq. /H208492/H20850and giving rise to surface emission /H20849the divergence of Ais significant only in a very narrow region around the surface /H20850, is neglected in accord with common practice in photoemission calculations. 9In this paper, we shall consider normal photoemission only, i.e., k/H20648 =0. Energy conservation is imposed by the two /H9254functions. In the dipole approximation, Acan be considered spatially constant /H20849the wavelength of the photon beam, which is 120–2500 Å in the UPS energy range, is very large com-pared to the atomic spacing in crystal /H20850, and therefore, the transition matrix elements are proportional to the dipole ma-trix element between the propagating initial and intermediatestates: M ni=e mcA/H20855/H9274n/H20841p/H20841/H9274i/H20856. /H208493/H20850 The vector potential Acarries information on the light polar- ization, which depends on the polar /H20849/H9258, defined with respect to the surface normal /H20850and azimuthal /H20849/H9278/H20850angles of incidence of the photon beam. In this paper, we consider linearly po-larized electromagnetic radiation, with the following conven-tion: for p-polarized light, Ais contained within the plane formed by the directions of the incident light and outgoingelectron, while for s-polarized light, Ais perpendicular to this plane. Thus, for s-polarized light, Ais parallel to the surface for normal photoemission. In general, when an electromagnetic plane wave impinges on a metal surface, the value of the vector potential transmit-ted inside the metal differs from the value in the vacuum dueto the departure from unity of the medium dielectric func-tion, /H9280/H20849/H9275/H20850.23The transmitted vector potential Atcan be cal- culated from the incident field Ai, as described by Fresnel’s equations, which were derived by using the Maxwell’stheory and Snell’s law. 26Actually, the difference between AtSTOJI Ćet al. PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-2andAican be very large, especially for small photon ener- gies /H20849/H6036/H9275/H1102120 eV /H20850and large /H9258, as extensively discussed in literature.24,27–31 In order to calculate the transmission factor, T/H20849En,k/H20850,w e take into account that the final state of the photoemissionprocess is a time-reversed LEED /H20849TRL /H20850state, which suffi- ciently far from the surface is free-electron-like in thevacuum /H20849outer region /H20850, while inside the crystal /H20849inner re- gion /H20850, it is a linear combination of the Bloch states available at the intermediate energy E n. The TRL state can thus be obtained by solving the one-electron Kohn–Sham /H20849KS/H20850 Schrödinger equation, which is subject to the appropriateboundary condition in the outer region. This task is accom-plished by matching the wave functions in the inner andouter regions, using a method proposed by Choi and Ihm, 20 which was originally devised to cope with ballistic conduc-tance and later generalized to account for US PPs. 21In the outer region, the TRL state is a plane wave whose wavevector has a component perpendicular to the surface equal to k /H11036=/H208812mE kin //H60362−k/H206482. For given values of the photoelectron kinetic energy Ekinand parallel momentum k/H20648in the inner region, the TRL state reads /H9278Ekin,k/H20648TRL/H20849r/H20850=/H20858 m/H9274n/H20849r,km/H11036/H20850t/H20849En,k/H20648,km/H11036/H20850, /H208494/H20850 where the sum is over all the Bloch states available at the intermediate energy, En=Ekin+/H9021. In the intermediate region, the TRL depends on the details of the self-consistent poten-tial at the surface, and this dependence determines the rela-tive amplitude of the wave functions in the outer and innerregions, hence the transmission coefficient. In practice,the solution of the KS Schrödinger equation by the methodof Choi and Ihm 20provides the coefficients of the expansion of the final photoemission state in Bloch waves. Thesecoefficients, which usually yield the total transmission, andhence the ballistic conductance, can be used to separatelycalculate the transmission probability into vacuum,T/H20849E n,k/H20850=/H20841t/H20849En,k/H20850/H208412, for each conduction band. We note that in this approach, the scattering state is normalized in such away that both the incident plane wave and the Bloch statescarry unit current. We now discuss the way in which the two delta functions appearing in Eq. /H208491/H20850can be treated in practice. The first delta function imposes energy conservation in the excitation stepof the photoemission process, while the second one relatesthe kinetic energy measured outside the crystal to the inter-mediate state energy, accounting for the work function. Thefirst delta function is usually represented as a Lorentzian: /H9254/H20851En/H20849k/H20850−Ei/H20849k/H20850−/H6036/H9275/H20852=/H9003h/2/H9266 /H20851En/H20849k/H20850−Ei/H20849k/H20850−/H6036/H9275/H208522+/H9003h2,/H208495/H20850 which corresponds to the spectral function of the hole left behind by the excitation process32and results in the broad- ening of the initial state. The width of the distribution, /H9003h, gives the inverse lifetime of the hole and is equal to theabsolute value of the imaginary part of the hole self-energy.As in the majority of other photoemission studies, we take /H9003 h as an adjustable parameter. The second delta function shouldbe replaced by the analyzer resolution function, most com- monly expressed in the form of a Gaussian: /H9254/H20851Ekin−En/H20849k/H20850+/H9021/H20852 =1 /H208812/H9266/H9003detexp/H20875−1 2/H20851Ekin−En/H20849k/H20850+/H9021/H208522 /H9003det2/H20876. /H208496/H20850 /H9003detis determined by the detector resolution and is related to the experimental energy broadening.32 We conclude by noticing that the bulk-emission model of photoemission neglects electron damping caused by thepresence of a surface. It is implicitly assumed in Eq. /H208491/H20850 that k /H11036conservation is perfect and the delta function for this conservation law is omitted. The k/H11036conservation, /H9254/H20849kn/H11036−ki/H11036−G/H11036/H20850, is usually represented by a Lorentzian, whose broadening parameter describes damping.32The con- sideration of only bulk emission is a good approximation ifthe damping is small, i.e., if its inverse, which is the electronescape length l e, is long enough, at least a few lattice spac- ings. The escape length can be estimated from the relation-ship 1 /l e=/H9003n/H9254k/H11036//H9254En,33where /H9003nstands for the inverse life- time of the intermediate state and /H9254k/H11036//H9254Enis the inverse group velocity of the intermediate state. The escape lengthdepends on the band structure through the /H9254k/H11036//H9254Enterm and on the photon energy through the empirical dependence of/H9003 non the intermediate-state energy. Empirically, one can use the relationship /H9003n=0.065 En.32 III. NUMERICAL DETAILS All the ingredients necessary to apply the theory outlined in Sec. II are calculated by using DFT within the PW-PPapproach, as implemented in the PWscf code of the Quantum ESPRESSO distribution.34In particular, the calculation of transmission coefficients has been performed from the outputof the PWcond component contained therein. For the exchange and correlation energy, we use the LDA with the Perdew–Zunger parametrization. 35The interaction of the valence electrons with the nuclei and core electrons is described by aVanderbilt US PP. 36The use of the PP method for simulating PES deserves some comments and requires care. ModernPPs are usually designed to very faithfully reproduce theelectronic structure /H20849orbital energies and one-electron wave functions outside the atomic core region /H20850of occupied states. Standard arguments based on the transferability concept en- sure that the quality of the electronic structure predicted byPPs is as good in the energy range immediately above theFermi energy /H20849E F/H20850, i.e., in an energy range that extends up to, say, 10–15 eV above EF. In the present case, however, par- ticular care has to be taken in describing the intermediatestate of the transitions because these lie at a higher energythan the transferability range of currently available PPs. Forthis reason, we have decided to generate a highly accurateUS PP, especially designed for the purposes of the presentwork. We used the 3 d 104s14p0atomic configuration of Cu, with the core radii of rs=2.1, rp=2.4, and rd=2.0 a.u. and two projectors in each of the s,p, and dchannels, one of which was chosen to the energy of a corresponding atomicstate at a somewhat higher energy than usually done. 37TheAB INITIO SIMULATION OF PHOTOEMISSION … PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-3PP energy bands thus obtained agree within 0.05–0.20 eV with those calculated from a highly accurate all-electronmethod, using the WIEN2K code,38up to 40 eV above EF. Ordinary Cu PPs, generated without high-energy projectors,tend to miss some unoccupied electron bands and have largerdeviations from the all-electron results with increasing en-ergy. To calculate T/H20849E n,k/H20850, we used the self-consistent potential of a 9-layer tetragonal slab along the /H20851001/H20852direction sepa- rated by a vacuum space equivalent to 11 layers. We chose avacuum space with length equal to two bulk layers as theunit cell in the left lead and two central bulklike layers of theslab as the periodic unit cell of the right lead. We used theexperimental lattice constant /H20849a=3.62 Å /H20850without relaxing the surface layers. However, our calculation allowing relax-ations along the perpendicular direction predicts −2.8% re-laxation for the first layer, 0.6% for the second, and 0.2% forthe third. We checked that transmission factors only negligi-bly change with surface relaxations in this case. Kinetic en-ergy cutoffs of 45 and 450 Ry have been used for the expan-sion of the wave functions and of the charge density,respectively. These cutoffs, which are unusually large for USPPs, are a consequence of the improved transferability thatwe required from our custom-tailored PP. An 18 /H1100318/H110031 Monkhorst–Pack mesh 39was used for the slab calculation. The band structure and the matrix elements of the dipoleoperator were calculated from a bulk calculation in which k /H11036 was sampled by 860 kpoints. The matrix elements of the dipole operator have been compared to the matrix elementsfrom the WIEN2K code, finding agreement of the order of 4% for the states of interest in this paper. For the evaluationof the second delta function from Eq. /H208496/H20850, we calculated the work function as the difference between the Fermi leveland the electrostatic potential in the vacuum and found/H9021=4.84 eV, which is in agreement with previous calculations 40and very close to the experimental values ranging from 4.59 to 4.83 eV.41For Cu /H20849111/H20850, our LDA cal- culations gave the value of 5.08 eV, which is in good agree-ment with a previous calculation /H208515.10 eV /H20849Ref. 42/H20850/H20852and experiment /H208514.9 and 4.94 /H20849Ref. 43/H20850/H20852. The broadening param- eters which appear in Eqs. /H208495/H20850and /H208496/H20850are chosen as /H9003 h =0.04 eV and /H9003det=0.07 eV. IV. RESULTS In this section, we first illustrate our method by discussing the various contributions to the spectrum, as calculated fromEq. /H208491/H20850, for Cu /H20849001/H20850at one specific photon frequency /H20849/H6036 /H9275 =23 eV /H20850and for one specific angle of incidence /H20849/H9258=65° /H20850of the incoming photon beam of ppolarization. We then present calculations for a more extensive set of frequencies and in-cidence angles. Finally, we try to correct the two mainsources of errors in our calculations by studying how thespectra are modified by the LDA+ Ubands and by surface optics. The corrected spectra are also presented in the case ofthe Cu /H20849111/H20850surface. A. Illustration of the method In Fig. 1, we show our calculated energy bands of bulk Cu. The bands are plotted in the /H9003-Zdirection, along /H20851001/H20852.These correspond to the bands along the /H9003-Xdirection in the fcc Brillouin zone /H20849BZ/H20850refolded in the tetragonal BZ. On top of the empty bands in Fig. 1, we add information regarding T/H20849En,k/H20850wherever it is greater than 0. This is possible because T/H20849En,k/H20850explicitly depends on k/H11036of the intermedi- ate states.44Thus, reading from Fig. 1at each intermediate energy, we can find if the propagating final statesexist /H20849T/H110220/H20850, and if so, for which values of the intermediate k /H11036. In agreement with empirical intermediate-state determinations,45we find that a free-electron-like band prop- erly modified by the ionic PP has the strongest coupling tothe vacuum state. All the bands with nonvanishing transmis-sion probability belong to the /H9004 1representation. This is a result of the selection rules for normal photoemission, whichimpose that the intermediate state be totally symmetric withrespect to all the C 4vsymmetry operations. By combining this result with the symmetry properties of the dipole matrixelement, we obtain the allowed transitions for normal photo-emission from the /H20849001/H20850surface of an fcc crystal: for the z polarization, only the /H9004 1→/H90041transition is allowed, while for thex/ypolarization, /H90045→/H90041transitions are allowed.1,46 Note, however, that some bands with /H90041symmetry might not be transmitted into vacuum, so symmetry alone would not besufficient to identify the intermediate states. In the same fig-ure, we also display nine direct transitions present in the/H6036 /H9275=23 eV case. Out of these transitions, only four satisfy selection rules and have dipole matrix elements and trans-mission factors that are both nonzero. Two of these transi-tions are of the /H9004 1→/H90041type and the remaining two are of the/H90045→/H90041type. Only one transition /H20849/H90041→/H90041/H20850has a large transmission factor and a large dipole matrix element, whilethe other of the same type has a small transmission factor.Both transitions of the /H9004 5→/H90041type have a small dipole01020Energy (eV )0.0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1.0 0.05 0.1 0.15 0.2 0.25 0.3 0.3 5 k|(2π/a)01020Energy (eV ) ∆1 ∆2 ∆2’ ∆5ωhT Bands 0 FIG. 1. /H20849Color online /H20850Band structure of bulk Cu along the k/H11036 direction for the Cu /H20849001/H20850surface. Among the unoccupied bands, we indicated those that have a transmission probability into vacuumlarger than 0. The different point sizes and colors indicate differenttransmission probabilities. The zero of the energy is at the Fermilevel. The direct transitions for /H6036 /H9275=23 eV are presented by arrows, while the four direct transitions which are simultaneously allowedand have nonzero transmission and dipole matrix element areshown with thick arrows.STOJI Ćet al. PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-4matrix element and one of them even has a small transmis- sion factor /H20849smaller than 0.2 /H20850. In Fig. 2, we illustrate the influence of the dipole matrix elements and of the transmission factors on the shape of thespectrum. In panel /H20849a/H20850, we show all the direct transitions, regardless of the symmetry, their dipole matrix elements, andthe transmission coefficients. In panel /H20849b/H20850, we calculate thephotoemission spectrum setting the transmission factor to 1 for all the bands. We find that the peaks with nonzero dipolematrix elements correspond to /H9004 2→/H90042and/H90041→/H90041/H20849/H90042 →/H90045,/H90045→/H90045,/H90045→/H90042,/H90042/H11032→/H90045, and/H90045→/H90041/H20850transitions forzpolarization /H20849x/ypolarization /H20850, although the transitions with/H90045and/H90042intermediate states are not allowed for the normal emission. By including the dipole matrix elements,we obtain a spectrum in which the intensity of the peaks maychange. The polarization and the direction of the incidentphoton beam now play an important role. For /H9258=65°, the z-polarized transitions are enhanced with respect to transi- tions due to x/y-polarized light. However, neglecting the probability of the intermediate states to be transmitted tovacuum, we still have many transitions into /H9004 2and/H90045inter- mediate states which have finite intensity. Also, the relativeintensities of peaks with /H9004 1intermediate states are incorrect. The introduction of the transmission factor not only selectsthe intermediate states with /H9004 1symmetry but also modulates the peak intensities. Thus, two peaks shown in panel /H20849c/H20850 originate from the /H90041initial state, while the shoulder of the main peak on the high-energy side and almost invisibleshoulder to the high-energy peak originate from the /H9004 5initial states. We note that, at variance with the rest of the paper, inFig. 2, we used smaller broadening parameters, /H9003 h=/H9003det =0.015 eV, in order to separate the different peaks. B.Ab initio results In the left panel of Fig. 3, we present our calculated spec- tra for five photon frequencies between 13 and 23 eV andtwo incident angles /H9258=18° and /H9258=65° for the ppolarization. FIG. 2. /H20849Color online /H20850Influence of dipole matrix elements and transmission factors on the spectrum for /H6036/H9275=23 eV and /H9258=65°. Panel /H20849a/H20850presents all the allowed transitions with an indication of the symmetry of the initial and intermediate states. Panel /H20849b/H20850shows the effect of the dipole matrix elements. Panel /H20849c/H20850shows the effects of the transmission factors. -4 -3 -2 -1 0010203001020304002040Intensity (arb. units)02040600255075100 -4 -3 -2 -1 002550750204060020406004080050100150 23 eV 21 eV 20 eV 15 eV 13 eV 13 eV15 eV20 eV21 eV23 eV Initial state energy (eV)θ=1 8οθ=6 5ο -4 -3 -2 -1 001201230246Intensity (arb. units)024602468 -4 -3 -2 -1 00120123024605101501020304050 23 eV 21 eV 20 eV 15 eV 13 eV 13 eV15 eV20 eV21 eV23 eV Initial state energy (eV)θ=1 8οθ=6 5ο FIG. 3. /H20849Color online /H20850Calculated photoemission intensity for Cu /H20849001/H20850/H20849left panels /H20850. The experimental spectra /H20849dashed line /H20850, together with a previous calculation /H20849darker full line /H20850/H20849Ref. 47/H20850are shown on the right. The spectra in both panels are given for various photon frequencies and for two incident angles, /H9258=18° /H20849on the left of each panel /H20850and/H9258=65° /H20849on the right /H20850. The sticks in the left panels indicate the positions of the main experimental peaks.AB INITIO SIMULATION OF PHOTOEMISSION … PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-5In the right panel, we show for comparison the experimental and previous theoretical normal-emission spectra for thesame frequencies and angles. 47Previous calculations were performed within the one-step model,9which were based on the nonrelativistic empirical muffin-tin potential and takinginto account the surface optics through the application ofSnell’s law and Fresnel’s equations. Our spectra in Fig. 3do not include surface-optics effects and are calculated withinthe bulk photoemission model. The first approximation isquite severe and its effects will be discussed below, while thesecond should be sufficiently justified for this surface. Actu-ally, for the photon energies considered here, /H9254k/H11036//H9254Enis smaller than 0.05 Å−1eV−1and the damping /H20849for an average photon energy of 18 eV /H20850can be estimated to be less than 0.06 Å−1. This rough estimate yields the electron escape length of about ten lattice spacings in the direction perpen-dicular to the surface. The choice of the incidence anglesignificantly influences the spectra: for /H9258=18°, light is mostly polarized in the xyplane /H20849the initial state belongs to the/H90045representation /H20850, while for /H9258=65°, most emission is from/H90041-like states /H20849zpolarization /H20850. Overall, our calculation reproduces the majority of the experimental peaks, albeitwith a shift of about 0.2–0.6 eV toward higher energies andsomewhat altered relative intensities. It is well known thatthe LDA, as well as the generalized gradient approximation/H20849GGA /H20850, fails to accurately describe quasiparticle energy bands as measured by PES, and therefore, the incorrect po-sition of the peaks has to be attributed to the error in thecalculated bands. 48The calculation within the one-step model47is performed with an empirical potential, and there- fore, the energy bands correspond to the experimental bandsrather well. The error in the positions of the energy bandscan result also in a reduction or an increase in the number ofpeaks. At /H9258=18° and /H6036/H9275=13 eV, the theoretical spectrum shows two peaks, one at −1.76 eV and one at −1.38 eV. Theformer is due to a transition from the /H9004 5band, while the latter originates from the /H90041band. Experimentally, only a single peak is present at about −2.30 eV. The spectrum for/H6036 /H9275=15 eV shows a single peak as the experiment although at a higher energy. The spectra at /H6036/H9275=20, 21, and 23 eV show only one main peak and a small peak, missing some ofthe shoulders present in the experimental spectra, a featurethat our result has in common with the previous calculation.The lower-energy features in the spectra /H6036 /H9275=20 and 21 eV originate from the /H90041bands and the higher-energy features originate from the /H90045bands. In the spectrum /H6036/H9275=23, both peaks are predominantly from the /H90045initial state, although there are significant contributions on the low-energy sidesoriginating from the /H9004 1initial states. These contributions are hard to discern due to a large broadening and closeness ofthe peaks /H20849/H110110.1 eV /H20850. For /H9258=65°, the agreement is somewhat worse. The spec- trum for /H6036/H9275=13 eV has a barely visible feature in place of the main peak of the experimental spectrum, while the /H90041 peak is significantly overestimated and at a too low energy. As we will show below, this will be corrected in part byconsidering the surface optics. That type of correction isquite large for small photon energies and large incidenceangles. 24Similarly to the case /H9258=18°, the spectrum for /H6036/H9275 =15 eV has only one peak, which actually contains two tran-sitions /H20849from the /H90041and/H90045initial states /H20850. Due to our impre- cise energy bands, both transitions are accidentally at thesame energy. The experimental spectrum for /H6036 /H9275=20 eV has two peaks of roughly equal intensity, with a broad shoulderon the high-energy side, while our spectrum reproduces justone main peak and a much lower-intensity peak at a higherenergy. Our spectra for /H6036 /H9275=21 and 23 eV are underestimat- ing or missing a high-energy feature, which is present in theexperimental spectra. We note that in contrast to the /H9258=18° case, both peaks in the spectrum for /H6036/H9275=23 eV originate from the /H90041initial states, and transitions from the /H90045give small contributions to the high-energy sides, as seen in Fig.2. Finally, we note that the peak intensities can be compared only to one spectrum, i.e., the intensities for different /H6036 /H9275 cannot be compared in our calculation due to the neglect of electron damping, which is energy dependent. Consequently,it is clear that the standard ab initio approach needs further corrections to reproduce the fine details of the experimentalspectra. C. Empirical corrections In this section, we try to empirically correct two of the main shortcomings of our approach by using quite simplemodels. It is known that inclusion of self-energy effects, atleast within the GW model, would be mandatory for a real-istic description of the band structure. 17Presently, however, this is beyond our capabilities mainly because it would re-quire a nontrivial extension of the ballistic conductance code.Hence, we choose a simpler approach, using LDA+ U. 49,50 LDA+ Ugoes beyond LDA by differently treating exchange and correlation for a chosen set of states, which in this caseis the copper 3 dorbitals. The selected orbitals are treated with an orbital dependent potential with associated effectiveon-site Coulomb interaction U eff, which is a function of Cou- lomb and exchange interactions UandJ,Ueff=U-J.51,52The LDA+ Umethod is most commonly known as a cure for the inability of traditional DFT implementations to predict theinsulating state of some strongly correlated materials. 49Al- though the theoretical foundation of LDA+ Uis somewhat questionable, its range of applicability is wider, and thismethod has indeed been successfully applied to metallic sys-tems where the effects of electron correlations areintermediate. 53–55LDA+ Uis also being successfully em- ployed as a predictive tool in the chemistry of transition-metal molecules. 56Furthermore, in the specific case of bulk copper, there is evidence that an account of self-interactioneffects in LDA through the LDA+SIC approach leads to animprovement in the calculated bands. 57However, the LDA+SIC approach neglects screening effects on the self-interactions, which are instead accounted for to different de-grees of accuracy in the GW and in the LDA+ Umethods. While GW addresses screening in a more rigorous way,LDA+ Ucan be considered as the static limit of a kind of /H20849admittedly, rather crude /H20850approximation to the GW method. 49 As Cu has an almost completely filled dshell, the main effect of the LDA+ Uis the shift in the electron bands, while the eigenfunctions are expected to remain quite close to theSTOJI Ćet al. PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-6LDA ones.49Actually, we checked that, for the values of U used here, the overlap between the LDA and the LDA+ U wave functions is of the order of 0.99. Thus, we kept thesame transmission and dipole matrix elements calculatedwith the LDA wave functions correcting only the band struc-ture. Table Ipresents some results, such as the positions of the dbands and bandwidths, which were evaluated by using dif- ferent methods including DFT with LDA, self-energy correc-tions within the GW approximation 17calculated on top of ab initio DFT results, and LDA+ Ufor three values of Ueff, which were all compared to the average over several experi-mental values. 45The positions of the dbands at the /H9003point vary greatly for different methods. The LDA calculationfinds the band too shallow by 0.6 eV, while GW reproducesthe experimental value quite well. LDA+ Usignificantly im- proves with respect to the LDA value and for U eff=2 eV gives almost the experimental value. A similar level of pre-cision can be seen for the positions of the dbands at the L andXpoints, with somewhat larger deviations of the LDA and the LDA+ Ufrom the experimental value at the Lpoint. Again, among the three values of U eff,a tt h e LandXpoints, the best agreement with experiment is obtained for Ueff =2 eV. The width of the dband at the /H9003point is, instead, quite faithfully reproduced both by the LDA and theLDA+ U. For other special points given in Table I, the widths remain almost constant for different U eff. We note that also the positions of s/pbands and Lgap improve with the LDA+ U. Overall, we conclude that the LDA+ Ucan correct the LDA bands in a significant manner and has effects com-parable to the full self-energy calculation. Also, on the basisof comparison with the experimental results, we find thatU eff=2 eV gives the best results and we choose to use this value in the rest of the paper. The second main problem in the calculation of the intensities of the photoemission peaks comes from the factthat the vector potential inside the solid is different fromthe vector potential in the vacuum. Consequently, one shouldcorrect the intensities using Snell’s law and Fresnel’sequations. 24,27–31An accurate account of this effect is quite difficult. First of all, one should use a dielectric functionconsistently calculated within the same ab initio scheme used for the calculation of the other quantities. Furthermore,the effect of the surface should be properly taken into ac-count in the evaluation of the dielectric function. However,as this would require substantial effort, we choose just toestimate the effect by using the experimental dielectric func-tion from Ref. 23. We report in Fig. 4the photoemission spectrum calculated with the LDA+ Ubands with U eff=2 eV and including theTABLE I. Comparison of different theoretical /H20851LDA, LDA+GW corrections /H20849Ref. 17/H20850,L D A + U,U1 =1.5 eV, U2=2 eV, and U3=2.5 eV /H20852band energies and bandwidths for copper at high-symmetry points compared to the average over several experiments /H20849as reported in Tables I and XIII in Ref. 45/H20850. All values are in eV . LDA GW LDA+ U1 LDA+ U2 LDA+ U3 Experiment Positions of /H90031,2 −2.18 −2.81 −2.60 −2.75 −2.92 −2.78 dbands X5 −1.44 −2.04 −1.85 −2.00 −2.17 −2.01 L3 −1.60 −2.24 −2.00 −2.17 −2.31 −2.25 /H90031,2-/H90032,5 0.83 0.60 0.83 0.83 0.83 0.81 Widths of X5-X3 2.94 2.49 3.01 3.04 3.05 2.79 dbands X5-X1 3.40 2.90 3.53 3.57 3.61 3.17 L3-L3 1.44 1.26 1.47 1.48 1.49 1.37 L3-L1 3.46 2.83 3.55 3.56 3.60 2.91 Positions of /H90031 −9.37 −9.24 −9.25 −9.22 −9.19 −8.60 s/pbands L2/H11032 −1.00 −0.57 −0.88 −0.85 −0.83 −0.85 Lgap L1−L2/H11032 4.04 4.76 4.67 4.88 5.10 4.95 -4 -3 -2 -1 0010203001020300204060Intensity (arb. units ) 0204060020406080 -4 -3 -2 -1 005101505101501020300102030020406080 23 eV 21 eV 20 eV 15 eV 13 eV 13 eV15 eV20 eV21 eV23 eV Initial state energy (eV)θ=1 8οθ=6 5ο FIG. 4. /H20849Color online /H20850Calculated photoemission intensity for various photon frequencies for two incident angles, /H9258=18° and /H9258 =65°, by using the LDA+ Uapproach, with Ueff=2 eV. The sticks indicate the positions of the main peaks in the experimental spectra/H20849and previous calculation spectra /H20850.AB INITIO SIMULATION OF PHOTOEMISSION … PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-7surface-optics corrections for the same parameters as in Fig. 3. We checked that the spectra do not show a strong dependence on the choice of Ueff, e.g., for Ueff=1.5 eV, the spectra are almost identical to the Ueff=2 eV case, with a small shift in energy. In comparison with the experimental spectrum from Fig. 3, we see that the energy positions are mostly corrected. For /H9258=18°, the spectrum for /H6036/H9275=13 eV is improved with re- spect to the LDA spectrum /H20849Fig. 3/H20850and also in terms of the distance and relative intensities of the two peaks. For /H6036/H9275 =15 eV, the two transitions which were at the same initial energy split, giving rise to a new peak. Thus, the lower-energy peak originates from the /H9004 5band and the higher- energy one originates from the /H90041band. In the one-step cal- culation, there is only a hint of a shoulder on the high-energyside. For /H6036 /H9275=20 and 21 eV, we lose the shoulders originat- ing from the /H90041initial state, which is present in the LDA spectra. This is because the bands are shifted in such a way that the transitions happen for k/H11036at which the two bands are very close in energy, less than 0.2 eV difference. The spec-trum for /H6036 /H9275=23 eV loses a higher-energy peak originating from the /H90041band, as this peak comes under the main peak from the /H90045band. For/H9258=65°, all the spectra have a significantly lower in- tensity with respect to the LDA as a direct consequence ofsurface-optics corrections. In the spectrum for /H6036 /H9275=13 eV, the positions of the peaks are improved and also the peakintensity ratio goes in the right direction as a result of theinclusion of /H9280/H20849/H9275/H20850. Nevertheless, this correction does not suf- fice and the intensities of the two peaks remain wrong bothwith respect to experiment and with respect to the one-stepmodel. By inspecting the Cu /H20849001/H20850band structure in Fig. 1, we see that for /H6036 /H9275=13 eV, the two initial bands actually have different dispersions at the k/H11036at which transitions are taking place /H208490.17 and 0.18 2 /H9266/a/H20850, whereas the dispersion of the intermediate band /H20849the lowest unoccupied band /H20850does not significantly change between the two k/H11036. The low-energy peak, which is underestimated, originates from the initialband of /H9004 5symmetry and has a smaller slope at the k/H11036of the direct transition. By allowing nondirect transitions due toelectron damping, this peak would get many more contribu-tions than the peak originating from a /H9004 1band with a strong dispersion and would, thus, improve agreement with the ex-perimental spectrum. A similar argument also holds for thespectrum for /H6036 /H9275=15 eV, which gets a new peak with respect to the LDA spectrum but whose intensity is overestimated.For/H6036 /H9275=20 and 21 eV, the high-energy peaks /H20849of/H90045origin /H20850 from the LDA spectra are smeared into shoulders due to theshift of bands. For the same reason, a high-energy shoulderfrom the LDA spectrum for /H6036 /H9275=23 eV is lost. Overall, we conclude that the LDA+ Ucorrection is affecting all the spectra, causing the shift of all peaks, which also results in adecrease or increase in the number of peaks. The surface-optics corrections improve agreement for low photon ener-gies and have stronger effects for /H9258=65°. In general, both corrections improve the agreement with experiment. Figure 5compares our ab initio and corrected spectra to a recently measured experimental spectrum on the Cu /H20849001/H20850 surface for ppolarization, /H6036/H9275=17 eV and /H9258=45°. The ex- perimental spectrum was measured on the Cu /H20849001/H20850singlecrystal surface at the APE beamline /H20849TASC, Italy /H20850at room temperature. It was integrated over an angular window of 1°around the normal emission. The energy resolution was esti-mated to be 25 meV. Both theoretical spectra have twopeaks, the low-energy one originates from the /H9004 1initial band and the high-energy one originates from the /H90045initial band. The ab initio spectrum has wrong energy positions /H20849/H110110.15 eV too high /H20850, the distance between the two peaks is too large, and the peak ratio is overestimated /H208491.78 instead of 1.40 /H20850. The corrected spectrum shows better agreement with experiment. The positions of the peaks are closer to the ex-periment /H20849/H110110.07 eV too deep /H20850, the distance between peaks is correct, while the peak ratio is somewhat underestimated/H208491.23 /H20850. However, we cannot reproduce the high-energy broad structure present in the experimental spectrum. The experi-mental energy resolution and inverse lifetime of the electron hole /H20849estimated to be /H9003 h=0.006 Ei2+0.01 eV /H110150.04 eV /H2085032 cannot account for the discrepancy between theory and ex- periment. Also, for this photon energy, broadening due tofinite electron escape length should not be pronounced. Wesee that not all the peaks can be explained by direct transi-tions only and assuming a /H9004 1intermediate state, as imposed by the selection rules for normal photoemission. We note thatin Eq. /H208491/H20850, we disregarded the delta function describing the k /H20648 conservation. By performing an analysis similar to the one in Fig. 2, we have found that there are two direct transitions, forbidden by selection rules, /H90045→/H90045and/H90042/H11032→/H90045, located at −2.41 and −2.38 eV in the corrected spectrum, respec- tively. They, also taking into account our somewhat impre-cise energy positioning, might correspond to the missingpeak. They have a very large dipole matrix elements and itseems likely that even a small admixing of these transitionsmight result in an observable structure in the photoemissionintensity. The finite-acceptance angle of the electron detectormeans that electrons are collected from a finite part of thesurface Brillouin zone /H20849broadening of k /H20648/H20850. This implies that in the normal-emission spectrum, it is possible to have smallcontributions from the dipole-selection forbidden transitions.These issues, however, are left for future investigations. As in the case of the /H20849001/H20850surface, also for the /H20849111/H20850 surface, we present our ab initio calculation of the transmis- sion factors, given on top of the empty initial bands, andplotted versus the wave vectors perpendicular to the surface-3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 Initial state energy (eV)Intensity (arb. units )Experiment Ab-initio spectrum Corrected spectrum FIG. 5. /H20849Color online /H20850Comparison of our Cu /H20849001/H20850ab initio /H20849light-colored full line /H20850and corrected /H20849dark full line /H20850spectra with the experimental spectrum /H20849dashed line /H20850for/H6036/H9275=17 eV and /H9258=45°.STOJI Ćet al. PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-8/H20849corresponding to the /H9003-Lline in the fcc cell /H20850, in Fig. 6. For this surface orientation, the unit cell has three atoms and thesymmetry corresponds to the C 3vpoint group. Selection rules allow only /H90111→/H90111transitions for the case of zpolarization and/H90113→/H90111transitions for the case of x/ypolarization. From Fig. 6, we conclude that for photon frequencies below 20 eV, transmission will be close to 1 for all dipole-allowedintermediate states. By using the same reasoning as for theCu/H20849001/H20850case, we find that for the average photon energy of 8 eV, k /H11036broadening is about 0.04 Å−1. Our rough estimate yields the electron escape length of about 15 lattice spacings,which ensures that the bulk model can also be applied in thiscase. In Fig. 7, we compare our calculated, experimental, 58andprevious theoretical24photoemission spectra from Cu /H20849111/H20850 for various photon energies indicated on the figure. Our cal-culated spectra include the /H9280/H20849/H9275/H20850and LDA+ Ucorrections. The angle of incidence was 60° and the experiment was per-formed with 90% p-polarized light. The previous calcula- tions were done within the three-step formalism, 25using an empirical band structure generated by the combined-interpolation-scheme approach. 59In those calculations, the authors decided to additionally suppress the Azcontributions to the spectra in order to get better agreement with the ex-periment. Our intensities are not adjusted beyond the correc-tions imposed by Snell’s law and Fresnel’s equations. Byusing the bulk-only model, we cannot reproduce the surfacepeak present in the experimental spectra at about −0.5 eV,which is also missing in the previous calculation. 24However, our spectra show overall similarity to the experimental spec-trum, especially considering the very general trends ofchanging peak positions and intensities, with increasing pho-ton energy. In comparison with the previous calculation, 24 two spectra seem to be shifted by 0.5 eV in photon energy,i.e., our spectrum for /H6036 /H9275=7.5 eV is very similar to the one in Ref. 24for/H6036/H9275=7.0 eV. Furthermore, the emergence of the first bulk peak, originating from the /H92611band at /H6036/H9275=6.5 eV /H20849reduced in intensity in our calculation because of an over- estimation of the work function /H20850, and its shift to the deeper energies for increasing photon energy by 0.5 eV, with alarger weight on the lower states, correspond to the experi-mental spectra. For /H6036 /H9275=7 eV, our spectrum has one /H92611peak, while the transition from the /H92613band at lower energies is not seen because of the overestimated work function. Due to thepoor experimental resolution, it is not easy to deduce if thereare two peaks or one in the experimental spectrum. The /H9261 3 peak is present in our spectrum for /H6036/H9275=7.5 eV, which re- produces well the experimental spectrum, with respect toboth the peak positions and intensity. The previous calcula-tion has a larger weight on the /H9261 3peak, which is in contrast to the experiment. The single peak in our spectra for /H6036/H9275k|0102030Energy (eV)0.0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1.0 FIG. 6. /H20849Color online /H20850Transmission intensity plotted on top of the unoccupied electron bands along the k/H11036direction for Cu /H20849111/H20850. The symbol size and color codings for the transmission intensity aregiven in the legend. The Fermi level is at zero energy. FIG. 7. /H20849Color online /H20850In panel /H20849a/H20850we present our calculated pho- toemission spectra from Cu /H20849111/H20850, while the experimental spectra/H20849Ref. 58/H20850are in panel /H20849b/H20850. Previ- ous theoretical spectra /H20849Ref. 24/H20850 are in panel /H20849c/H20850.AB INITIO SIMULATION OF PHOTOEMISSION … PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-9=8–9 eV shifts weight from the right to the left shoulder /H20849higher to lower energy /H20850and contains contributions from both the /H92611and/H92613initial states /H20849which cross at those ener- gies /H20850. In the experimental spectrum, the same trend is present, but it starts from a lower photon energy /H208497.5 eV /H20850 and the spectra for lower photon energies have two peaks.For/H6036 /H9275=9.0 eV, a small peak at a lower energy emerges /H20849/H11011−3.1 eV /H20850, which corresponds to the /H92613band and can also be found in all spectra for higher photon energies. It is alsopresent in the previous calculation and in the experimentalspectra, where it emerges at /H6036 /H9275=8.5 eV. Our spectra for /H6036/H9275=9.5–11.5 eV have correct peak positions but a wrong intensity ratio of peaks, significantly overestimating thelower peak, originating from the /H9261 1band, with respect to the higher-energy peak, arising from the /H92613band. The previous calculations resolved this disagreement by artificially sup-pressing the z-polarized contributions. A likely reason for this disagreement is the neglect of surface effects in thethree-step model. V. CONCLUSIONS In this paper, we have introduced a method for the ab initio low-energy photoemission calculations based on thepseudopotentials and plane waves, which has an advantage in its simplicity and unbiased basis set, with the possibility tosignificantly reduce the number of empirical parameters. Ourmethod based on the bulk-emission model results in a rea-sonable agreement with experiment in the photon energyrange up to /H1101125 eV. Empirical corrections, including the LDA+ Uand surface optics, give significant improvements. Nevertheless, in comparison with the one-step model, theintensity ratios of the photoemission peaks are still not fullyreproduced. This is due to the neglect of surface damping inour model, which, in principle, could be accounted for withinour approach. With respect to the experiment, some broadstructures are absent in our spectra, which we interpret tooriginate from the forbidden transitions in the normal photo-emission, caused by the detector’s finite-acceptance angleand the related broadening of k /H20648. Further work in this direc- tion, i.e., consideration of off-normal photoemission, is nec-essary to assess these effects. ACKNOWLEDGMENTS We acknowledge useful discussions with Ivana V obornik, Alexander Smogunov, Nadia Binggeli, and Paolo Umari. 1S. Hüfner, Photoelectron Spectroscopy-Principles and Applica- tions , Springer Series in Solid State Science V ol. 82 /H20849Springer- Verlag, Berlin, 1995 /H20850. 2B. Feuerbacher and R. F. Wilis, J. Phys. C 9, 169 /H208491976 /H20850. 3I. Adawi, Phys. Rev. 134, A788 /H208491964 /H20850. 4G. D. Mahan, Phys. Rev. B 2, 4334 /H208491970 /H20850. 5P. J. Feibelman and D. E. Eastman, Phys. Rev. B 10, 4932 /H208491974 /H20850. 6W. L. Schaich and N. W. Ashcroft, Phys. Rev. B 3, 2452 /H208491971 /H20850. 7C. Caroli, D. Lederer-Rozenblatt, B. Roulet, and D. Saint-James, Phys. Rev. B 8, 4552 /H208491973 /H20850. 8C. N. Berglund and W. E. Spicer, Phys. Rev. 136, A1030 /H208491964 /H20850. 9J. B. Pendry, Surf. Sci. 57, 679 /H208491976 /H20850. 10M. Graß, J. Braun, and G. Borstel, J. Phys.: Condens. Matter 5, 599 /H208491993 /H20850. 11M. Graß, J. Braun, and G. Borstel, Phys. Rev. B 47, 15487 /H208491993 /H20850. 12M. Graß, J. Braun, and G. Borstel, Phys. Rev. B 50, 14827 /H208491994 /H20850. 13R. M. Martin, Electronic Structure: Basic Theory and Practical Methods /H20849Cambridge University Press, Cambridge, 2004 /H20850. 14S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 /H208492001 /H20850. 15A. Dal Corso and A. M. Conte, Phys. Rev. B 71, 115106 /H208492005 /H20850. 16C. Stampfl, K. Kambe, J. D. Riley, and D. F. Lynch, J. Phys.: Condens. Matter 5, 8211 /H208491993 /H20850. 17A. Marini, G. Onida, and R. Del Sole, Phys. Rev. Lett. 88, 016403 /H208492001 /H20850. 18F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 /H208491998 /H20850. 19S. Botti, A. Schindlmayr, R. Del Sole, and L. Reining, Rep.Prog. Phys. 70, 357 /H208492007 /H20850. 20H. J. Choi and J. Ihm, Phys. Rev. B 59, 2267 /H208491999 /H20850. 21A. Smogunov, A. Dal Corso, and E. Tosatti, Phys. Rev. B 70, 045417 /H208492004 /H20850. 22S. Baroni and R. Resta, Phys. Rev. B 33, 7017 /H208491986 /H20850. 23J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Koch, Optical Properties of Metals II /H20849Physics Data, Fachinformationszentrum, Karlsruhe, 1981 /H20850. 24N. V . Smith, R. L. Benbow, and Z. Hurych, Phys. Rev. B 21, 4331 /H208491980 /H20850. 25W. D. Grobman, D. E. Eastman, and J. L. Freeouf, Phys. Rev. B 12, 4405 /H208491975 /H20850. 26M. Born and E. Wolf, Principles of Optics /H20849Pergamon, Oxford, 1970 /H20850. 27P. J. Feibelman, Surf. Sci. 46, 558 /H208491974 /H20850. 28M. A. B. Whitaker, J. Phys. C 11, L151 /H208491978 /H20850. 29A. Goldmann, A. Rodriguez, and R. Feder, Solid State Commun. 45, 449 /H208491983 /H20850. 30H. Wern and R. Courths, Surf. Sci. 152/153 , 196 /H208491984 /H20850. 31H. Wern and R. Courths, Surf. Sci. 162,2 9 /H208491985 /H20850. 32R. Matzdorf, Surf. Sci. Rep. 30, 153 /H208491998 /H20850. 33R. Courths, S. Hüfner, and H. Schulz, Z. Phys. B 35, 107 /H208491979 /H20850. 34S. Baroni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi /H20849URL: http://www.pwscf.org /H20850. 35J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 /H208491981 /H20850. 36D. Vanderbilt, Phys. Rev. B 41, R7892 /H208491990 /H20850. 37The projector energies were −0.39 and 4.20 eV for the schannel, −0.05 and 2.00 eV for the pchannel, and −0.49 and 2.80 eV for thedchannel. 38P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz,STOJI Ćet al. PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-10WIEN2K , An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz, TechnicalUniversitat Wien, Austria, 2001. 39H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 /H208491976 /H20850. 40F. Favot, A. Dal Corso, and A. Baldereschi, J. Chem. Phys. 114, 483 /H208492001 /H20850. 41P. O. Gartland, Phys. Norv. 6, 201 /H208491972 /H20850; G. A. Haas and R. E. Thomas, J. Appl. Phys. 48,8 6 /H208491977 /H20850; T. Fauster and W. Stein- mann in Electromagnetic waves: Recent Developments in Re- search , edited by P. 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Turnbull /H20849Academic, New York, 1962 /H20850, V ol. 13, p. 305.AB INITIO SIMULATION OF PHOTOEMISSION … PHYSICAL REVIEW B 77, 195116 /H208492008 /H20850 195116-11

PhysRevB.85.205429.pdf

PHYSICAL REVIEW B 85, 205429 (2012) Graphene nanoribbons subject to gentle bends P. Koskinen* NanoScience Center, Department of Physics, University of Jyv ¨askyl ¨a, 40014 Jyv ¨askyl ¨a, Finland (Received 24 February 2012; revised manuscript received 25 April 2012; published 15 May 2012) Since graphene nanoribbons are thin and flimsy, they need support. Support gives firm ground for applications, and adhesion holds ribbons flat, although not necessarily straight: Ribbons with a high aspect ratio are proneto bend. The effects of bending on ribbons’ electronic properties, however, are unknown. Therefore, this articleexamines the electromechanics of planar and gently bent graphene nanoribbons. Simulations with density-functional tight-binding and revised periodic boundary conditions show that gentle bends in armchair ribbonscan cause significant widening or narrowing of energy gaps. Moreover, in zigzag ribbons sizable energy gaps canbe opened due to axial symmetry breaking, even without magnetism. These results infer that, in the electronicmeasurements of supported ribbons, such bends must be heeded. DOI: 10.1103/PhysRevB.85.205429 PACS number(s): 68 .65.Pq, 61 .48.Gh, 62 .25.−g, 73.22.Pr I. INTRODUCTION Graphene nanoribbons (GNRs) are atomically thin and only nanometers wide, which makes them the flimsiest materialsin the world. Today such ribbons, acclaimed for promisingapplications, are fabricated in many ways, 1,2and investi- gated for heat conduction,3edge features,4–6and electronic characteristics,7among many other properties. However, since the ribbons are flimsy, they need stabilizing support—althougheven then ribbons can get folded, torn, rippled, and bent. 7–10 Also supports are different, as the interaction with graphene can be either physical or chemical. In physisorption thesupport interaction is weak, graphene’s electronic structureremains decoupled, and adhesion arises from the dispersivevan der Waals interactions alone. 11In chemisorption the support interaction is stronger, and the presence of chemi-cal bonds alters graphene’s electronic structure. 12Therefore adhesion, responsible for holding ribbons planar, ranges from/epsilon1=4m e V /atom to 70 meV /atom. 12,13However, fabrication processes, surface inhomogeneities, pinning, AFM tip manip-ulation, heat treatment, or mechanical strains can make ribbonssubject to gentle bends, as sketched in Fig. 1(a). Indeed, planar and gentle bending can be directly seen in scanning tunnelingmicroscopy experiments. 7–10Distortions like twisting, on the contrary, are less relevant on supports.14–16Only gentle bends are interesting, as sharp bends are structurally unstable(ribbons would desorb and fold instead). 8,10 The purpose of this work, therefore, is to answer the following simple question: What happens to GNRs’ electronicstructure upon planar bending? It turns out that simple geo-metrical arguments, together with nearest (and next-nearest)neighbor tight-binding reasoning, are sufficient for a thoroughunderstanding of the electromechanics of bent GNRs. Theseinsights should hence help interpreting imperfect experimentswith these distortion-prone ribbons. II. SIMULATING PHYSISORBED RIBBONS WITH PURE BENDING I modeled GNRs as free standing, without explicit presence of the support; it was there merely as a planar constraint. Theunderlying justification was to model physisorption wherethe support and GNR electronic structures are essentially decoupled. For chemisorption the results are not directly valid. The focus is on the bent sections of very long ribbons, with bending viewed as a local property. Apart from bending, planar ribbons can also stretch and shear; those deformations havebeen investigated by conventional methods. 17,18The deforma- tion mode certainly depends on the experimental conditions,and especially in short ribbons the strain patterns can becomecomplicated. 19,20However, ribbons yield easily upon lateral forcing, and adjust themselves readily to minimum-energygeometries. 21–24In bent geometries sliding is particularly easy as the ribbon and the support are mostly out of registry. Longribbons pinned at two distant locations, therefore, can removehigh-energy shearing and stretching by sliding, and favor purebending. 25I remark that the central results, as discussed below, will be valid also beyond pure bending. I modeled the electronic structure by density-functional tight-binding (DFTB) method,26,27and the bent geometry itself by revised periodic boundary conditions.28–30Atoms in the simulation cell were from GNR translational cell of lengthL, and the associated symmetry operation was a rotation of an angle αaround a given origin, as in Refs. 29and 31 [see Fig. 1(b)]. This means, therefore, that the simulated systems were effectively GNR hoops, containing hundreds ofthousands of atoms. Since the bends are gentle and the chargetransfer with physisorption usually small, it’s reasonable toassume that simulation describes the properties of bends inGNRs in a local sense. 5At any rate, regarding the bending, simulations were exact and the sole approximation was the DFTB method itself. Throughout this article I will use the dimensionless parameter /Theta1=W 2R(1) to quantify the amount of bending.32Then, to simulate GNR of width Wwith bending close to /Theta1/prime, I chose α=L/R/prime, with R/prime=W/(2/Theta1/prime) as the initial guess for the radius of curvature, and optimized the structure. The only fixed parameter was α,s o Rand/Theta1were outcomes of the optimization, although R≈R/prime and/Theta1≈/Theta1/prime. Here I remark that, because α’s are small (down to 10−3radians), the optimization was arduous and required 205429-1 1098-0121/2012/85(20)/205429(5) ©2012 American Physical SocietyP. KOSKINEN PHYSICAL REVIEW B 85, 205429 (2012) (a) (b) FIG. 1. (Color online) (a) Supported graphene nanoribbon sketched for gentle, planar bends. (b) The red (dark gray) atoms constitute the unit cell, simulated with revised periodic boundaryconditions; the symmetry operation is the rotation of an angle αaround a given origin (not shown). The ribbon’s width Wis defined by the outmost carbon atoms and Ris the mean radius of curvature. The bending parameter of Eq. (1),h e r e /Theta1=0.1, is also equal (approximately) to the compressive strain at the inner edge (ε in=−/Theta1) and to the tensile strain at the outer edge ( εout=/Theta1). maximum force criteria as small as fmax<10−5eV/˚A (look at Ref. 30to see why). I conducted such simulations for hydrogen-passivated arm- chair ribbons ( N-AGNRs) and zigzag ribbons ( N-ZGNRs), withN=5...40, with Wup to 84 ˚A, with 74 different GNRs in total (see Ref. 33for GNR notations). Each GNR was optimized for ten bendings between /Theta1=0 and/Theta1=0.1. The reason for /Theta1=0.1 as the upper limit for bending that I term “gentle” will be clarified later. Finally, the number of κ points was 50 ˚A/Lfor geometry optimization and 500 ˚A/L for electronic structure analysis. III. BENT RIBBONS GET STRETCHED Let us now turn our attention to the results. Prior to discussing electronic properties, however, let us first look atenergy and geometry. The energy in bent ribbons, as hinted bynanoshell elasticity, 34comes chiefly from axial in-plain strain. A quick estimate yields energy per unit area as E/A=1 6k/Theta12, where k=25 eV ˚A−2is graphene’s in-plane modulus.35This simple estimate is in fair agreement with the simulations, asshown in the inset of Fig. 2. Only the narrowest ribbons deviate from this estimate, for two reasons: First, the comparisonofWbetween atomic and continuum methods is inherently ambiguous; for small Wthis ambiguity is emphasized. Second, in narrow ribbons the value of kis affected by distinct elastic properties near the edges. While I could remedy thesedeficiencies by improving the model, my main interest is notin the minutiae of narrow ribbons, but in the wider ribbons andtheir universal trends. If we set the adhesion energy /epsilon1equal to the strain energy, we get /Theta1 /epsilon1=/radicalbig 6/epsilon1/kA c (2) as a rough estimate for the limit where the ribbon rather desorbs and straightens than remains adsorbed and bent on the support.The maximum adhesion /epsilon1=70 meV /atom yields /Theta1 /epsilon1=0.08, justifying the upper limit /Theta1/lessorsimilar0.1 for a “gently” bent ribbon, even though /Theta1/epsilon1really depends on the substrate. This is only an order-of-magnitude estimate, as fluctuations and finite-sizeFIG. 2. (Color online) The cross-ribbon averaged strain εavgas a function of bending parameter /Theta1for AGNRs (solid lines) and for ZGNRs (dashed lines); the bold dashed line is an analytical estimate. Inset: Elastic energy density as a function of bending; the bold dashedline is an analytical estimate, Eq. (4). In both plots the line width is proportional to W. Hence both in ε avgand in E/A the largest deviations are for the narrowest ribbons. effects can cause ribbons to desorb earlier. Direct experimental evidence8shows how GNRs on SiO 2bend up to /Theta1≈0.01— still nearly half the simple-minded limit of /Theta1/epsilon1≈0.025 given by/epsilon1≈6m e V /˚A2.36 Given the definition for /Theta1, the strain on the ribbon’s inner edge is εin≈−/Theta1and on the outer edge εout≈/Theta1. With strains around 10% the bond anharmonicities begin to emerge, andstretching becomes cheaper, compression more expensive.This implies that the neutral line moves away from the origin(Rincreases) and the ribbon stretches. We can take this effect into account by a strain-dependent in-plane modulus,k(ε)=k 0(1−γε). Then, by minimizing the total energy per unit length /integraldisplayR+W/2 R−W/21 2k0(1−γε)ε2dr (3) with respect to R, we obtain the cross-ribbon averaged strain as εavg=1 2γ/Theta12. (4) This analytical estimate, given the value γ=1.7 obtained from DFTB simulations of stretched GNRs, agrees well withsimulations, as shown in Fig. 2. The largest deviations occur again for the narrow ribbons, albeit with opposite tendenciesfor AGNRs and ZGNRs due to different edge morphologies. IV . ARMCHAIR RIBBONS ARE DOMINATED BY STRETCHING Equipped with these geometrical notions, let us now turn our attention to the electronic properties, starting with AGNRs.The inset in Fig. 3(a)shows the energy gaps for the known three families of N-AGNRs, defined by q=mod (N,3). 33,37The gaps scale as Eg≈βW−1, where β≈13 eV ˚Af o r q=0,1 (forq=2 the scaling is a bit different). Figure 3(a) shows how these gaps respond to bending: They widen or narrowwith the same q-dependent families. Deviations occur only for the narrowest ribbons. These trends can be understood by the following model. The energy gaps in stretched q=0,1 AGNRs depend on the 205429-2GRAPHENE NANORIBBONS SUBJECT TO GENTLE BENDS PHYSICAL REVIEW B 85, 205429 (2012) (a) (b) (c) FIG. 3. (Color online) Electronic structure of bent AGNRs. (a) Bending-induced gap changes Eg(/Theta1)−Eg(0) for the three q families of AGNRs. Line width is proportional to W; dashed lines are estimates for q=0,1. Inset: Gaps in straight AGNRs. (b) Wave functions for the frontier orbitals in straight 16-AGNR ( q=1): the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). (c) Density of states for16-AGNR with straight ( /Theta1=0.0), bent ( /Theta1=0.1), and stretched (ε= 1 2γ(0.1)2=0.85%) geometries. strain as /Delta1Estraight g≈(−1)qεδwithδ=12 eV (fit for q=2i s just more complex).17,38The origin for this strain dependence is illustrated in Fig. 3(b) for 16-AGNR with q=1: The highest occupied orbital is bonding and the lowest unoccupied orbitalis antibonding along the ribbon’s axis, and therefore stretchingtends to narrow the gap (for q=0 AGNRs the situation is the opposite and for q=2 intermediate). 39Next, if we pretend, in effect, that the bent AGNRs experience only the average axial strain (even if the strain is uneven), and thus juxtapose εwith εavgfrom Eq. (4), we get /Delta1Eg(/Theta1)≈1 2(−1)qγδ/Theta12. (5) Figure 3(a) plots these estimates for q=0 andq=1 AGNRs by the dashed lines. The fair agreement suggests that theelectronic structure of AGNRs subject to bending is domi- nated by the cross-ribbon averaged strain . Similar physics have been observed previously in bent carbon nanotubesand twisted GNRs. 15,31,39,40Hence the argument is easily generalized to combined bending and stretching, where theelectronic structure is modified by the average strain ε stretch+ 1 2γ/Theta12.15These trends, as given by four-valence DFTB, are repro- duced by a π-only tight-binding Hamiltonian H=−tn.n./summationdisplay i,jc† icj−t/primenext-n.n./summationdisplay i,jc† icj, (6) with the nearest-neighbor (n.n.) hopping parameter t(r)=2.6e V−5.8e V/˚A(r−1.42˚A), (7) and with the next-nearest neighbor hopping equal to zero ( t/prime= 0, not shown). Figure 3(c) shows further that the stretching analogy extends beyond energy gaps, as the entire density ofstates (DOS) is well described by the stretched geometry. Thiscarries the average strain analogy also for optical transitions, asshown earlier. 31Note that, if gauged through the relative gap change |/Delta1Eg/Eg|≈0.8˚A−1W/Theta12, the influence of bending becomes more important as Wincreases. V . ZIGZAG RIBBONS ARE DOMINATED BY BROKEN SYMMETRY Let us now leave AGNRs and turn our attention to the electronic properties of ZGNRs. First I have to remind that, for (a) (b) (c) FIG. 4. (Color online) Electronic structure of bent ZGNRs. (a) Band structure of 10-ZNGR with straight and bent geometries. For the straight ribbon, having a reflection symmetry, the dashed linesdenote symmetric and solid lines denote antisymmetric states under reflection. (For the bent ribbons no such distinction can be made.) (b) Wave functions of the frontier orbitals in straight 10-ZGNR:highest occupied molecular orbital (HOMO, symmetric) and lowest unoccupied molecular orbital (LUMO, antisymmetric). (c) Density of states in straight and bent 10-ZGNR. The dashed lines are froma next-nearest-neighbor tight-binding model [Eq. (6)with Eq. (8)]. Inset: Gaps for all ZGNRs plotted as a function of /Theta1−/Theta1 crit(Eg=0 when/Theta1</Theta1 crit). 205429-3P. KOSKINEN PHYSICAL REVIEW B 85, 205429 (2012) straight ZGNRs as such, the spin-parallel DFTB simulations are dubious, given the prediction for a spin-polarized groundstate. 33The magnetic structure should arise when the curious flat bands near the Fermi level41[left panel of Fig. 4(a)]—the famous edge states—spin polarize, lift degeneracies, and opena gap (not shown). 33This way spontaneous magnetization can stabilize the electronic structure. It has been shown that, unlike in AGNRs,38the electronic structure in ZGNRs is unaffected by stretching.17,42Therefore, after discovering above the average-strain argument withAGNRs, it’s natural to guess that bending would leave ZGNRs’electronic structure unaffected. The right panel of Fig. 4(a) shows, however, that bending can open an energy gap in ZGNRs . That is, electronic structure is stabilized by sheer bending, and the cause for magnetic spin polarization is lost.Note how the band structure changes only near the Fermi level,while other bands remain stable. The mechanism of the gap opening is related to broken reflection symmetry, as clarified by the following three-stepreasoning. First step: The bands are called “flat” because theyhave small dispersion. In the absence of next-nearest-neighborhopping ( t /prime=0), the Hamiltonian (6)gives edge states whose dispersion and energy are essentially zero, independent oft. This is illustrated in Fig. 4(b), where flat band electrons appear localized to next-nearest-neighbor sites, separated byvacancies. Therefore the tin Eq. (7), even if strain dependent, doesn’t affect the flat bands—splitting and dispersion hencerequire next-nearest-neighbor hopping t /prime. Second step: Fitting t/primeto strained GNRs by DFTB gives t/prime(r)=0.25 eV −0.6e V ˚A(r−2.46˚A). (8) The Hamiltonian (6), with hoppings (7)and(8), reproduces the electronic structure fairly well, as shown by the DOS for10-ZGNR in Fig. 4(c). [Pure stretching leaves DOS intact (not shown).] Third step: the flat band energies (also the banddispersion) are proportional to t /primeand hence proportional to edge strain via Eq. (8). Upon bending, the reflection symmetry breaks and states localize on either of the edges with straindifference ε out−εin=2/Theta1; opposite edges hence get unequal hoppings /Delta1t/prime=t/prime out−t/prime in∝/Theta1. Because energy splitting is proportional to /Delta1t/prime, it is also proportional to /Theta1. This is themechanism by which bending splits the flat bands with direct proportionality to /Theta1. As mentioned above, since t/primegives flat bands a small dispersion, splitting does not open the gap immediately. WhenWincreases, the span of the flat region in k zspace increases, and gap opening requires larger splitting. A fit to all ZGNRsyields a critical value for opening a gap as /Theta1 crit≈W/200 nm (orRcrit≈100 nm for all W), yielding the energy gap as EZGNR g≈4e V (/Theta1−/Theta1crit). (9) The gaps, displaying values up to 0 .4 eV , are plotted in the inset of Fig. 4(c). VI. CONCLUSIONS The physics in AGNRs and ZGNRs hence appear quite different: AGNRs are governed by average strain, whereasZGNRs are governed by broken reflection symmetry. Theeffects of broken symmetry on AGNRs or average strainon ZGNRs surely exist, but they are just less important. InZGNRs bending can have particular impact on transport, sincethe localization of edge states depends on the direction ofbending; if the ribbon has bends both to the left and to the right,the current-carrying electrons need to jump from one edge tothe other, suggesting width-dependent resistivity. 43Although it’s plausible that bent ZGNRs indeed acquire gaps and turnnonmagnetic, spin-polarized calculations would be opportune,even if the existence of magnetism has been disputed also forthe straight ribbons. 44 I obtained similar results also for unpassivated GNRs, observing similar phenomena. Thus it appears that these cleartrends arise from simple physics with plausible explanations,and it’s unlikely that, say, higher level electronic structuremethods should change the picture. I believe, therefore, thatthese general trends are helpful enough to serve as rules ofthumb to aid GNR device fabrication and analysis. ACKNOWLEDGMENTS I acknowledge Teemu Peltonen for discussions, the Academy of Finland for funding, and the Finnish IT Centerfor Science (CSC) for computer resources. *pekka.koskinen@iki.fi 1D. V . Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price, and J. M. Tour, Nature (London) 458, 872 (2009). 2J. Cai, P. Ruffieux, R. Jafaar, M. Bieri, T. Braun, S. Blankenburg,M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng et al. ,Nature (London) 466, 470 (2010). 3N. Wei, L. Xu, H.-Q. Wang, and J.-C. Zheng, Nanotechnology 22, 105705 (2011). 4X. Jia, M. Hofmann, V . Meunier, B. G. Sumpter, J. Campos-Delgado, J. M. Romo-Herrera, H. Son, Y .-P. Hsieh, A. Reina,J. Kong et al. ,Science 323, 1701 (2009). 5K. Suenaga and M. Koshino, Nature (London) 468, 1088 (2010). 6M. Acik and Y . J. Chabal, Jpn. J. Appl. Phys. 50, 070101 (2011).7X. Wang, Y . Ouyang, L. Jiao, H. Wang, L. Xie, J. Wu, J. Guo, and H. Dai, Nat. Nanotechnol. 6, 563 (2011). 8X. Li, X. Wang, L. Zhang, S. Lee, and H. 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A. Kinloch, I. Riaz, R. Jalil, and K. S. Novoselov, ACS Nano 5, 3079 (2011). 20Short means that the ribbon’s length is of the same order of magnitude as its width. 21F. Bonelli, N. Manini, E. Cadelano, and L. Colombo, Eur. Phys. J. B70, 449 (2009). 22Z. Lu and M. L. Dunn, J. Appl. Phys. 107, 044301 (2010). 23C. Lee, Q. Li, W. Kalb, X.-Z. Liu, H. Berger, R. W. Carpick, and J. Hone, Science 328, 76 (2010). 24S. Cahangirov, C. Ataca, M. Topsakal, H. Sahin, and S. Ciraci, P h y s .R e v .L e t t . 108, 126103 (2012). 25If the long ribbon is pinned to the substrate at points with a distance larger than the ribbon’s equilibrium length, the lowest-energy deformation will have uniaxial stretching; if the long ribbon ispinned to the substrate at points with a distance smaller than the ribbon’s equilibrium length, the lowest-energy deformation willhave pure bending. 26D. Porezag, T. Frauenheim, T. K ¨ohler, G. Seifert, and R. Kaschner, P h y s .R e v .B 51, 12947 (1995). 27P. Koskinen and V . M ¨akinen, Comput. Mater. Sci. 47, 237 (2009). 28P. Koskinen and O. O. Kit, P h y s .R e v .L e t t . 105, 106401 (2010).29T. Dumitrica and R. D. James, J. Mech. Phys. Solids 55, 2206 (2007). 30O. O. Kit, L. Pastewka, and P. Koskinen, P h y s .R e v .B 84, 155431 (2011). 31P. Koskinen, P h y s .R e v .B 82, 193409 (2010). 32S. Malola, H. H ¨akkinen, and P. Koskinen, Phys. Rev. B 78, 153409 (2008). 33Y .-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97, 216803 (2006). 34K. Kudin, G. Scuseria, and B. Yakobson, Phys. Rev. B 64, 235406 (2001). 35L. D. Landau and E. M. Lifshitz, Theory of elasticity ,3 r de d . (Pergamon, New York, 1986). 36R. H. Miwa, T. M. Schmidt, W. L. Scopel, and A. Fazzio, Appl. Phys. Lett. 99, 163108 (2011). 37V .B a r o n e ,O .H o d ,a n dG .E .S c u s e r i a , Nano Lett. 6, 2748 (2006). 38O. Hod and G. E. Scuseria, Nano Lett. 9, 2619 (2009). 39D. Gunlycke, J. Li, J. W. Mintmire, and C. T. White, Nano Lett. 10, 3638 (2010). 40D.-B. Zhang and T. 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PhysRevB.88.195416.pdf

PHYSICAL REVIEW B 88, 195416 (2013) Conductance across strain junctions in graphene nanoribbons D. A. Bahamon and Vitor M. Pereira* Graphene Research Centre and Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Received 12 August 2013; published 14 November 2013) To address the robustness of the transport gap induced by locally strained regions in graphene nanostructures, the effect of disorder and smoothness of the interface region is investigated within the Landauer-B ¨uttiker formalism. The electronic conductance across strained junctions and barriers in graphene nanoribbons iscalculated numerically, with and without various types of disorder, and comparing smooth and sharp strainjunctions. A smooth strain barrier in graphene is seen to be generically as efficient in suppressing transport at lowdensities as a sharp one, and the critical density (or energy) for the onset of transmission depends on the strainorientation with respect to the ribbon. In addition, hopping (or strain) inhomogeneity and work function mismatchat the interface region do not visibly degrade the transport gap. These results show that the strain-induced transportgap at a strain junction is robust to more realistic strain conditions. DOI: 10.1103/PhysRevB.88.195416 PACS number(s): 81 .05.ue, 73.63.−b, 77.65.Ly I. INTRODUCTION The ability to control the flow of charge through nanoscale devices, in particular being able to efficiently establish clearly defined on and off states, is one of the perennial driving forces and goals of research towards electronic devices of smaller dimensions or with new functionalities. The advent ofgraphene has brought an entire new realm of possibilities in this front. 1The intrinsic two dimensionality (2D) of graphene and a large number of other emerging strictly 2D crystals means that they can be readily integrated with the conventional 2D device fabrication paradigm in electronics.2But, more importantly, being a pure surface opens several new possibilities of significantly modifying their electronic-structure and transport properties, such as by tailoring the interaction with sub- strates, with the chemical environment, with electromagnetic radiation, or doping. In the case of graphene, in particular,having the strongest covalent bonding in nature leads to its record high tensional modulus of ∼1T P a ,w h i l e ,a tt h es a m e time, being able to withstand planar and elastic deformations as high as 15%–20%. 3,4Hence, unlike more conventional condensed matter systems whose intrinsic brittleness rarelyallows elastic deformations of even the order of 1%, 5large mechanical deformations in graphene are a reality6–8and a potential new means for external control of its electronic properties.9–11Its extraordinary mechanical robustness further allows the exploration of the third dimension, with the potentialfor manipulating this electronic membrane with strain and curvatures tailored for particular functionalities. 12–14 Apart from these practical and functional advantages of having a strong crystalline metallic membrane, the couplingof lattice deformations to electrons in graphene is alsoof great interest from a fundamental point of view. Thenature of the honeycomb lattice has the consequence that, inaddition to the conventional coupling through the so-calleddeformation potential, electrons in graphene feel the locallattice deformations through an additional pseudomagneticvector potential coupling. 15,16The direct consequence is that local nonuniform deformations translate directly intoan effective pseudomagnetic field. 17,18This fact, realized theoretically long ago, has been recently established with theexperimental observation of the reconstruction of the Dirac spectrum into Landau levels in regions of locally strainedgraphene. 6,7The magnitude of the effective pseudomagnetic fields extracted from these experiments can easily fall into the300–600 T range, vigorously attesting to the strong impact thatmechanical deformations can have in the electronic structureof graphene. Against the backdrop of these experiments and these num- bers, the prospect of strain-engineering graphene’s intrinsicelectronic response gains more traction, as reflected by theincreased interest in exploring and characterizing strainedgraphene electrically, optically, or magnetically. 19–23One of the simplest ideas whereby local strain can be used to modifythe transport characteristics of a graphene channel consists of alinear interface separating two regions of graphene in differentstates of uniaxial strain, thus establishing a one-dimensionalstrain junction or step. 9,24–26Similar ideas have been proposed for other semiconducting systems such as Si (Ref. 27) and, of course, films of heteroepitaxial Si have been instrumental intransistor mobility optimization. 28 In graphene, such a simple interface and its variations in the form of barriers or superlattices have the potential to generatea transport gap at low densities 24[which is to say that the dependence of the electronic conductivity σon the electron density neis modified such that σ(ne<n∗)=0], as well as to spatially confine some electronic modes to the barrierregion. 9,29,30If strain is uniform, albeit different, on both sides of the interface, a basic qualitative picture arises that allowsus to understand the underlying physics. It can be summarizedin that the associated pseudomagnetic potential vector will bedifferent in the two regions defined by the interface, with theconsequence that the Fermi circles for electrons on those tworegions will appear centered at different points in the absolute(undeformed) reciprocal space. If momentum conservationalong the interface direction is assumed, this displacementof the Fermi circle reduces the phase space available fortransmission across the interface, which is entirely suppressedwhen the density is so low that the displacement is larger thanthe Fermi momentum. 9 This paper aims at analyzing and quantifying the robustness of the main transport feature, the transport gap, when the 195416-1 1098-0121/2013/88(19)/195416(10) ©2013 American Physical SocietyD. A. BAHAMON AND VITOR M. PEREIRA PHYSICAL REVIEW B 88, 195416 (2013) idealized conditions under which these strain barriers have been previously studied are relaxed. In particular, we willbe interested in the fate of the transport gap when the straininterface becomes smooth, rather than the idealized steplikestrain step. In addition to this necessary generalization, weshall also look at the effect of disorder at the interface.Electronic disorder can arise as a result of inhomogeneousand random strain at, and within some distance from, theinterface, or as a result of impurities or enhanced interactionswith the substrate at the interface region, depending on thestrategy used to establish the strain step. In order to approachthese situations in both an unbiased and unified way, theconductance of graphene through such strain steps and barriersis calculated directly on the lattice, numerically, within theLandauer-B ¨uttiker formalism, where the energy-dependent transmission probability is calculated with the lattice Green’sfunction technique. Our calculations based on a graphene nanoribbon ribbon (GNR) geometry show that the strain-induced transport gapis robust under these more realistic strain barrier situations.More specifically, barrier smoothness and hopping (or strain)disorder have no impact in the presence or magnitude of thetransport gap. In fact, a “smooth” strain barrier is “better” thana “sharp” one in that it considerably improves the quality ofconductance quantization. This is quite the opposite of whathappens for an electrostatic step or barrier in graphene, wherea smooth interface region leads to qualitative changes in thetransmission properties, such as the considerable suppressionof the (Klein) tunneling amplitude except for precisely normalincidence. 31Local potential disorder is seen to have only a minor effect if it is short ranged, and long range is theonly model of disorder explored here that has a noticeabledetrimental effect. Despite our utilization of the conductance extracted for finite-width GNRs, some of our conclusions should transferdirectly and unchanged to the thermodynamic limit, notablythe robustness with respect to the barrier smoothness. At anyrate, the effort towards bottom-up synthesis of GNRs withatomically precise edges has already lead to various successfulstrategies, 32–34and hence the prospect of narrow GNRs free from edge roughness is quite real, therefore allowing thedirect testing of our calculations in that case. For example,the dependence of the Fano factor in strain magnitude couldbe used experimentally to quantify the strength of deformationin the sample. We emphasize from the outset that, here, we are not interested in the modulation of the intrinsic finite-size gapand the transition between metallic and insulating characterof perfect GNRs that have been amply discussed by otherauthors. 35–38 Even though direct signatures of strain in the elec- tronic structure of graphene have been detected by Ramanspectroscopy 39–41and local probe spectroscopy,6,7,42,43its effect on the electronic transport has not been experimentallyinvestigated yet. We trust that the current drive to utilizegraphene in flexible electronics, where mechanical defor-mation is inevitable to some extent, as well as our currentresults establishing the robustness of the transport gap, willmotivate direct studies of the transport characteristics understrain.(a) (b) y x θ L L W (c) FIG. 1. (Color online) Schematic representation of the unstrained-strained junction (a) and of the strain barrier (b). (c) Honeycomb lattice with the convention used for the nearest- neighbor vectors δ1=(√ 3a/2,−a/2), δ2=(0,a), and δ3= (−√ 3a/2,−a/2). II. MODEL AND METHODOLOGY The dynamics of electrons in graphene is modeled here as being described by a nearest-neighbor π-band tight-binding Hamiltonian for a hexagonal lattice H=/summationdisplay /angbracketlefti,j/angbracketrighttij(c† icj+c† jci), (1) where cirepresents the fermionic annihilation operator on site i,tijis the hopping amplitude between nearest-neighbor pz orbitals (in the unstrained lattice tij=t0≈−2.7e V ) .T h e leading effect of uniaxial strain to the π-band electrons arises through the modification of the carbon bond distances inequilibrium δ 0 i[cf. Fig. 1(c)], according to δi=(1+∇u)·δ0 i, where∇uis the plane displacement gradient tensor. Since we are interested in characterizing the effects on the quantumtransport across nonstrained-to-strained junctions, throughoutthis work we shall consider only the representative case ofuniaxial and planar deformations. In this case, the deformationgradient tensor can be exactly replaced by its symmetricdecomposition, the small strain tensor: ε=∇u/2+∇u /latticetop/2. In the coordinate system shown in Fig. 1, the strain tensor reads as10 ε=ε/parenleftbiggcos2θ−σsin2θ(1+σ) cosθsinθ (1+σ) cosθsinθsin2θ−σcos2θ/parenrightbigg ,(2) where εis the magnitude of the applied tension, θis the orientation with respect to the xdirection (see Fig. 1), and σ=0.165 is the Poisson ratio. The strain-induced changes in the three nearest-neighbor vectors δimodify the value of the corresponding hopping amplitudes. Their dependence on thedistance between neighboring p zorbitals is modeled to vary according to ti=t0e−3.37(δi/a−1),10witha=δ0=0.142 nm being the unstrained C-C bond length. Of special significance are the cases of strain applied along the lattice zigzag (e.g., θ=0) and armchair (e.g., θ=π/2) directions.10These two particular tension directions leave t1=t3in both cases, and the nearest-neighbor distances are 195416-2CONDUCTANCE ACROSS STRAIN JUNCTIONS IN ... PHYSICAL REVIEW B 88, 195416 (2013) modified explicitly as θ=0:|δ1|=|δ3|=/parenleftbigg 1+3ε 4−εσ 4/parenrightbigg a, (3a) |δ2|=(1−εσ)a (3b) and θ=π 2:|δ1|=|δ3|=/parenleftbigg 1+1ε 4−3εσ 4/parenrightbigg a, (3c) |δ2|=(1+ε)a. (3d) Effects of strain in the spectrum of GNRs have been considered elsewhere by various authors. But, these studiespertain mostly to situations where the entire nanostructure isunder strain, and not the cases of strain barriers or junctionsconsidered here. For example, it has been shown that straindoes not qualitatively affect the intrinsic band structure ofzigzag GNRs, 35–37whereas band-structure calculations for armchair GNRs (AGNRs) have shown a variable intrinsic gapwith a sawtooth shape as a function of the applied strain.However, in the latter case the value of the intrinsic gapstill scales with the inverse width of the ribbon, and hencebecomes rapidly insignificant as the transverse dimensionincreases. 38For calculational convenience, our analysis of the strain-induced transport gap generated by strain junctions willfocus only on AGNRs since these allow an analytical closedform for the propagating modes, which will be an importantfactor in the interpretation and analysis of the results. Withreference to Fig. 1, ribbons of different width are identified by the number of atoms along one of the vertical zigzag chainsW. Moreover, given that we are interested in following the evolution of the transport gap, the incoming (and outgoing)contact in the case of the junction (barrier) is always metallic.This means that we shall always choose W=3m+2, where mis a positive integer. The width of the nanoribbon is denoted byW, and is given by W=√ 3a(W−1)/2. In the case of the unstrained-strained junction, we define three regions in the infinite nanoribbon: an unstrained semi-infinite contact acting as a waveguide for the incident electrons,a strained semi-infinite contact acting as a waveguide forthe transmitted electrons, and a central region of length L acting as the scattering region (in this work we consider L equal to the width Wof the GNR). The transition from the unstrained to the strained region is not abrupt. Instead, strainis smoothly incremented over a length L s(Ls<L), which is a parameter that we can vary to assess the influence of the barriersmoothness on the conductance suppression at low densities.This is achieved with the following position-dependent strain: ε(y)=ε 0 e−(y−L/2)/Ls+1, (4) where y=0 and Lmark the end of the unstrained left and the beginning of the right-strained contact, respectively. Forthe study of the strain barrier, the infinite ribbon is likewisedivided in three regions: two semi-infinite unstrained contactsand a strained central region of length L=W.T h es t r a i n magnitude is also smoothed at the interfaces of the centralregion with the contacts as described above. The conductance of the graphene nanostructures is eval- uated within the Landauer-B ¨uttiker formalism, wherebyG(E)=G 0T(E), with G0=2e2 h.T(E) is the transmission function, and is obtained using the recursive lattice Green’sfunction technique. 44 III. CONDUCTANCE OF AN UNSTRAINED-STRAINED JUNCTION In order to understand how strain modifies the conductance of an AGNR and to compare it with the results obtained withinthe continuum approach based on the Dirac equation, 9,24,45 let us begin by describing the quantum transport features of the unstrained-strained junction. Figure 2(a) shows how the conductance of an AGNR varies in general with the Fermienergy. In view of the particle-hole symmetry of this problem, we show only the results for the conductance at positiveenergies. The value of the full transport gap is E g=2/Delta1ε, where /Delta1εis the transport gap observed in plots such as those in Fig. 2, where only positive energies are shown. Throughout this work, we shall refer to /Delta1εas the “transport gap,” being clear that it corresponds to half of Eg. The specific parameters for this system are W=15.2 nm (ribbon width), ε=0.03 (strain magnitude), θ=π/2 (strain direction), and the onset of strain is smoothed over different lengths Ls.I t is evident that, although the GNR is metallic in the absenceof any strain, a transport gap /Delta1 εdevelops, and its magnitude (/Delta10.03=0.0435t0) is insensitive to Ls. This means that, as far as the existence and magnitude of the transport gap at lowenergies is concerned, smoothing of the unstrained-strainedinterface has no effect (in the absence of disorder). The effectis therefore robust with respect to the degree of sharpness ofthe strain barrier, which is an important result since a realstrain interface will never be abrupt, and strain will alwaysdevelop incrementally. This is similar to what happens in aquantum point contact: when the width of the constrictionis reduced gradually (adiabatically), the intersubband mixingis reduced and the accuracy of the conductance quantizationmarkedly improves. 46One consequence of this is that, as seen E/|t0| E/|t0| G/G0 (a) (b) FIG. 2. (Color online) (a) Conductance of an unstrained-strained junction with W=15.2 nm, ε=0.03,θ=π/2, and different values of the smoothing length Ls. Notice that the plateaus become increasingly well defined with the introduction of a smoothing lengthL s/negationslash=0, but the fundamental suppression of conductance at low energies remains unaffected. (b) Conductance for the same system withLs=12.8 nm at different strain magnitudes. 195416-3D. A. BAHAMON AND VITOR M. PEREIRA PHYSICAL REVIEW B 88, 195416 (2013) in Fig. 2(a), the smoother the junction, the better defined the quantization plateaus become. In this sense, smootherjunctions are not only a necessity imposed by the actual elasticbehavior of the system and experimental conditions, but alsoa desirable situation as far as observation of conductancequantization is concerned. Figure 2(b) shows the typical behavior of the conductance of a smooth strain junction withincreasing strain magnitude: the transport gap increases withε(/Delta1 0.06=0.087t0), but the conductance quantization remains unaffected since plateaus are created at integer values of G0, with strain changing only the energies of the conductancesteps. It is crucial not to confuse the appearance of the transport gap with the finite subband spacing in a narrow nanoribbon.In other words, one could be tempted to think that, since thestrained region is still a nanorribon of the same width but withthe subbands slightly rearranged in energy, one would expecta small transport gap if the band arrangement in the strainedregion is such that it corresponds to an insulating nanoribbon.Even though this is true, it leads only to a small “intrinsic”gap scaling with ∼1/W, which is rapidly overshadowed by the strain-induced gap, whose scale is set by εand is largely insensitive to the width W. In order to understand the behavior of the conductance reported in Fig. 2, it is instructive to review the band structure of the system and consider the characteristics of the incoming,reflected, and transmitted modes. The band dispersion of anAGNR with θ=0o rπ/2 (i.e., t 1=t3) is given by E(qx,ky)= ±|φ(qx,ky)|, where φ=t2e−ikya+2t13eikya 2cos/parenleftbigg√ 3a 2qx/parenrightbigg (5) andt13=t1=t3. The transverse momentum is quantized as qx=2π√ 3a(W+1)p, where Wis the number of sites along a zigzag line in the transverse direction. The Wpossible values ofqxare identified by the mode index p=1,2,...,W.F o r a given energy and transverse momentum qxthere are two possible values of longitudinal momentum ky. All real values ofkyfor a given qxconstitute a one-dimensional subband and, hence, the mode index palso labels the subbands. If we use ito enumerate the conduction subbands in terms of increasing energy at ky=0, we can then refer to their respective mode numbers as pi.I nF i g . 3(a),w eh a v ea n example of this notation: the first band above E=0i st h e one corresponding to the mode p1, the second to the mode p2, and so on. In addition, since our unstrained ribbons are always metallic, we can use the fact that Wcan be cast as W=3m+2, with ma positive integer, and obtain the mode number of the lowest subband: p1=2m+2.47The usefulness of this labeling will now be made apparent with a particularexample. A. Transport gap and conductance quantization Consider Fig. 3where the lowest subbands of a particular AGNR are plotted, with and without strain. Strain naturallymodifies the electronic structure of the AGNR by (i) creating,in general, a small W-dependent intrinsic gap (0 .0033t 0in this particular case), and (ii) by reordering the relative positionand energy of the subbands corresponding to the same mode(a) (b) ky ky E/|t0| ε=0.03 ε=0 p1 p3 p5 p1 p3 p5 FIG. 3. (Color online) Band structure of an unstrained (a) and strained (b) AGNR with W=15.2n m( W=125,ε=0.03, and θ=π/2). The first, third, and fifth conduction bands of the unstrained nanoribbon are numbered and their new position in the strainednanoribbon is highlighted. index. In Fig. 3(a), we highlight the first, third, and fifth bands of the unstrained AGNR (whose mode indices arep 1=84,p3=83, and p5=82), and show in Fig. 3(b) how the bands with the same mode numbers appear at differentenergies. It can be seen that the fifth band becomes thelowest-energy band of the strained AGNR and the first band ofthe unstrained AGNR becomes the fourth. Figure 4illustrates this band rearrangement from a perhaps more instructiveperspective. To understand why tracing the mode indices in theunstrained and strained regions is significant, we should recallthat, as stated above, the mode index defines the transverse FIG. 4. (Color online) Illustration of the subbands of a GNR as “slices” of the 2D dispersion of graphene at equidistant values of thetransverse momentum (here W=23). In 2D graphene the effect of uniaxial strain is to displace the Dirac point by δKin reciprocal space. Since the momentum quantization is determined solely by the widthand chirality, the quantized momenta will be the same in the strained and unstrained regions. Consequently, subbands corresponding to the same transverse momentum (or mode number) will appear at differentenergies in the strained region, when compared to the unstrained one. In this example, the transport gap would be given by E p=15(ky=0), and its thermodynamic limit when W→∞ is noted as EG. 195416-4CONDUCTANCE ACROSS STRAIN JUNCTIONS IN ... PHYSICAL REVIEW B 88, 195416 (2013) momentum via qx=2π√ 3a(W+1)p. Since strain varies only along the longitudinal direction, the transverse momentum (or mode index) should be preserved across the interface.Additionally, since our geometry assumes an identical ribbonwidth in both strained and unstrained regions, the value of q x is identical on both. This means that an incoming mode pkwill only propagate across the interface if that mode is “open” in thestrained region, which is to say, if the corresponding band liesabove E F. This is precisely what happens, and what determines the transport gap. Inspecting the energy eigenvalues at the /Gamma1 point of the strained AGNR, it is found that the value of thetransport gap ( /Delta1 0.03=0.0435t0, cf. Fig. 2) does not tally with an eigenenergy of the strained AGNR, but, instead, it coincideswith an eigenenergy of the unstrained AGNR correspondingto the third band ( p 3=83). In other words, as EFis increased from zero, the incoming electrons belong to mode p1up to EF=0.0435t0; this mode can only propagate in the strained region for EF/greaterorsimilar0.08t0; hence, the conductance is zero. The lowest common band to the unstrained and strained regionsis the one associated with the mode p 3. Only when EFis increased past the minimum of this band can we have modeconservation, and this is what determines the transport gap. To delve more into this point, we can follow the evolution of the eigenenergies at k y=0 as a function of strain. The results are plotted in Fig. 5(a), where the evolution of the (b) (c) ∆/|t0| (a) W (nm) W (nm) ε E/|t0| p5 p3 p1 ≈vF W FIG. 5. (Color online) (a) Evolution in energy of the modes at the/Gamma1point as a function of εfor the AGNR of W=15.2n ma n d θ=π/2. The evolution of the first (blue), third (green), and fifth (red) modes has been highlighted as well as the intrinsic gap ≈¯hvF/W.T h e yellow shaded region indicates the evolution of /Delta1ε. (b) Transport gaps /Delta10.03and/Delta10.06as a function of ribbon width for ε=0.03 and 0 .06, respectively. The black (red) dot in the right vertical axis marks the value of the transport gap /Delta10.03≈0.04t0(/Delta10.06≈0.08t0) calculated using Dirac’s equation. (c) Transport gap /Delta1δtas a function of width forδt=0.1a n d0 .2, where t2=t0−δt. The values of δtwere chosen to approximate the values of t2forε=0.03 and 0 .06.energies for modes p1,p3, andp5is highlighted. The variation of the eigenenergies is linear with strain, which is expected inconnection with the illustration of Fig. 4: first, the displacement of the “enveloping Dirac cone” is linear in strain, 9,17,48and, second, the energy is linear in the momentum close to theDirac point. As a result of the variation of the individual subbands with energy, the intrinsic gap in the strained ribbon oscillates asstrain increases, with an amplitude ≈¯hv F/W; this is simply a consequence of the displacement of the Dirac cone inducedby strain, and the constancy of the quantized transversemomenta. 36,38,49–51The transport gap /Delta1ε, however, increases steadily with strain (except for a sawtooth modulation≈¯hv F/W). This can be appreciated with the aid of Fig. 5(a) that shows the minimum of each band as a function of strain.For example, following the vertical dashed line at ε=0.03, the first eigenenergy is 0 .0033t 0; this value corresponds to modep5=82. Since this mode is evanescent in the unstrained contact, there is no transmission. The second eigenenergy is0.0402t 0; this value corresponds to mode p3=83. For this energy, this mode is evanescent in the unstrained contact andconsequently there is no transmission either. When the Fermienergy reaches the first horizontal dashed line, that is, whenthe mode p 3becomes propagating in the unstrained contact and starts to transmit, the transmission gap /Delta10.03=0.0435t0is reached since the modes p3=83 are opened in both unstrained and strained contacts. Increasing the Fermi energy to the valueof the eigenenergy 0 .0474t 0corresponding to mode p7=81, there is no effect on the conductance because for this energythep 7mode is evanescent in the unstrained contact. Such a fine-grained analysis can be used to understand the positionand width of the conductance plateaus in Fig. 2. Consider, for example, the very narrow 2 G 0plateau for ε=0.03 in Fig. 2.I t appears when the Fermi energy reaches the fourth eigenvalue(0.0831t 0) of the strained contact, and both modes (in the unstrained and strained regions) with p1=84 are propagating and can transmit. Finally, when the Fermi energy comes tothe second horizontal dashed line 0 .0876t 0, the mode p5=82 of the unstrained contact finally is propagating and there istransmission in that mode. This also explains the origin ofthe shorter plateaus observed in the conductance of Fig. 2. Following the same procedure for ε=0.06, the transmission gap/Delta1 0.06=0.0876t0is extracted from Fig. 5(a), it can be seen that the transmission gap is created when the modes labeledwithp 5=82 are propagating in both contacts. Simply generalizing this reasoning and procedure, we can trace the dependence of the transport gap for any value ofstrain ( ε) and width ( W). Figure 5(b) shows such results for /Delta1 0.03and/Delta10.06as a function of the width of the nanoribbon. It can be seen that the transport gap has strong oscillationsfor narrow ( <100 nm) nanoribbons that rapidly die off with increasing W, on account of the reduction of spatial confinement in the transverse direction. For wide junctions, thetransport gap eventually converges at the asymptotic values of/Delta1 0.03=0.042t0and/Delta10.06=0.079t0, for the particular values of strain considered in the figure for illustration. These resultsare in complete agreement with those calculated using Dirac’sequation. 45 In addition, in order to make a direct comparison with the predictions for the transport gap predicted for the 2D graphene 195416-5D. A. BAHAMON AND VITOR M. PEREIRA PHYSICAL REVIEW B 88, 195416 (2013) system under uniaxial strain, we calculated the transport gap /Delta1δtof an unstrained-strained junction where only the hopping t2=t0−δtis modified in the strained contact. The values of δtwere chosen to be equivalent to the values of t2obtained withε=0.03 and 0 .06. In this ideal situation it is expected that/Delta1δt=δt/2( R e f . 9) and, indeed, observing Fig. 5(c)it can be seen that this is the asymptotic value for wide nanoribbons. B. Work function mismatch To efficiently inject carriers from one material to another, a good band alignment is required. The work function differencedetermines how the bands of different materials align whenthey are put in contact. It has been shown that the workfunction of graphene can be engineered through chemicaldoping 52or strain.53,54In particular, strain is known to increase or decrease the work function depending on whether thelattice is, respectively, strained or compressed. Consequently,different work functions in the strained and unstrained regionslead to band misalignment and this appears to require areformulation of the model and calculations presented in theprevious section. The band mismatch created by homogeneousstrain can be regarded as an effective scalar potential 53relating the Fermi energies in the unstrained and strained regions asE s F=Eus F−(φs−φus), where φus(φs) is the work function in the unstrained (strained) region. For AGNR it has been foundthat the work function increases linearly with uniaxial tensilestrain up to 12%, regardless the width of the nanoribbon, 54 and with a magnitude that can depend on the details ofedge passivation. Since we are interested in the generalconsequences of a work function mismatch to the transportgap and conductance quantization, without compromising thisgenerality we extracted φ us=4.2 eV and φs(ε=0.04)= 4.35 eV . These values correspond to the mismatch predicted in Ref. 54for hydrogen-passivated edges. With these parameters, the Fermi energy in the strained region Es Fcan be written in terms of its unstrained counterpart Eus Fand strain magnitude εas Es F=Eus F−3.75ε(eV). (6) According to this, the effect of the strain-induced work func- tion mismatch can be modeled by adding a strain-dependentonsite energy of −3.75εto the sites in the strained region. The addition of this effective scalar potential does not affectthe methods used in the previous section since transversemomentum conservation is still valid (the scalar potential is afunction of strain, and this varies only along the longitudinaldirection). We therefore used this approach to model andinvestigate the effects of work function mismatch. Since thestrain variation is smoothed according to Eq. (4), the effective local potential will vary smoothly as well. The effect of incorporating explicitly this work function mismatch in our conductance calculations is a downwardsdisplacement of the eigenenergies in the strained region which,consequently, modifies the quantization signatures in theconductance trace, as well as the transmission gap. Since theconductance trace G(E) is no longer particle-hole symmetric [Fig. 6(b)], the transport gap is now determined by E g= /Delta1+ ε+|/Delta1− ε|, which requires the explicit calculation of the transmission threshold at positive and negative energies /Delta1± ε.(b) (c) Eg/|t0| (a) W (nm) ε E/|t0| p5 p3 p1 G/G0 p7 p3 p1 E/|t0| p5 FIG. 6. (Color online) (a) Evolution in energy of the modes at the/Gamma1point as a function of εfor the AGNR of W=15.2n ma n d θ=π/2, including the work function mismatch as a scalar potential. The evolution of the positive and negative branches of the first (blue),third (green), and fifth (red) modes has been highlighted, as well as the negative branch of the seventh mode (orange). For positive energies, the yellow shaded region indicates the evolution of /Delta1 + ε,w h i l ef o r negative energies it indicates the evolution of /Delta1− ε. (b) Conductance of an unstrained-strained junction with W=15.2n ma n d θ=π/2, including the effect of the work function mismatch. (c) TransportgapE g=/Delta1+ ε+|/Delta1− ε|as a function of ribbon width for ε=0.03 and 0.06, respectively. The black and red dots on the right vertical axis mark the value of the transport gap calculated without the workfunction mismatch: 2 /Delta1 0.03=0.084t0andEg=2/Delta10.06=0.158t0 [see Fig. 5(b)]. For example, resorting to Fig. 6(a)that shows the evolution of the different modes with strain, we see that /Delta1+ 0.03=0.0435t0 and/Delta1− 0.03=−0.0777t0, leading to a transport gap Eg= 0.1212t0that is larger than the value 2 /Delta10.03=0.087t0obtained without work function mismatch. A larger transmission gapis also observed for ε=0.06 [E g=(0.0435+0.132)t0= 0.1755t0]. In this case, the asymmetry induced by the local potential has the consequence that the modes responsiblefor the gap have changed, and /Delta1 + 0.06=/Delta1+ 0.03is determined by mode p3=83 from the positive energy branch, while /Delta1− 0.06=−0.132t0is defined by mode p7=81 from the negative branch. These considerations are confirmed by adirect calculation of the conductance, which is shown inFig. 6(b) for these two particular values of strain [compare with Fig. 2(b)]. The width dependence of the gap is shown in Fig. 6(c). The asymptotic values for the full gap of E g=0.084t0and 0.16t0at, respectively, ε=0.03 and 0 .06, are seen to be equal to the full gaps obtained without workfunction mismatch [see Fig. 2(c)]. This is highlighted by the dots located on the right vertical axis, which mark the positionsE g=2/Delta10.03=0.084t0and 2/Delta10.06=0.158t0, where /Delta1εare the ones calculated in Fig. 2(c). The main message from here is that the work function mismatch displaces the center of the gap from E=0t o 195416-6CONDUCTANCE ACROSS STRAIN JUNCTIONS IN ... PHYSICAL REVIEW B 88, 195416 (2013) E=(/Delta1+ ε+/Delta1− ε)/2≈−3.75ε/2, but its magnitude remains unchanged relative to the case where work function mis-match is disregarded. This is a direct consequence of thelinear spectrum of graphene which causes the strain-inducedtransport gap in the 2D limit to be independent of a uniform,but different, potential energy in the two regions. Moreover,the analysis of the mode evolution with strain from Fig. 6(a) shows that the finite transport gap is still a consequence ofthe conservation of mode index (transverse momentum), andthat the gap is determined by the lowest-energy modes that aresimultaneously “open” in the strained and unstrained regions.Finally, since we are ultimately interested in the behavior ofthe transport gap as a function of strain, these facts allowus to concentrate only on strain junctions where the explicitwork function mismatch is ignored without losing generalityas far as the magnitude and strain dependence of the gap areconcerned. In the remainder of this paper, we therefore willnot consider explicitly the work function mismatch. C. Mode mixing When the strain is not in the transverse or longitudinal directions ( θ/negationslash=0,π/2), Eq. (5)is no longer valid to calcu- late the dispersion relation, consequently, the values of thequantized transverse momentum in the strained contact willdiffer form the values of the quantized transverse momentumin the unstrained contact. This will lead to mode mixing,even for a smooth junction or a junction without any kindof disorder. An electron incident upon the junction in a givenmode will be mixed with a number of modes of the strainedcontact with similar transverse momentum and symmetry ofthe wave function; 55,56as a consequence, the transport gap will be reduced or disappears since its existence was due tothe matching of the lowest-energy propagating modes. Modemixing will be small while the angle does not deviate toomuch from ideal situations ( θ=0 andπ/2), but this effect will reduce the value of the transport gap, as can be seen in Fig. 7 where the conductance of a strained junction of W=15.2n m withε=0.03 is plotted for different angles of the applied strain. For θ=85 ◦we obtained a transport gap /Delta10.03=0.04t0, and for θ=80◦we have /Delta10.03=0.027t0. When the band structure is calculated for the strained AGNR, using the lattervalues, it can be corroborated that, in fact, there exists atransport gap since there are lower-energy propagating modes E/|t00123 θ=90° θ=85° θ=80° θ=60° θ=45° θ=0° 0.02 0.04 0.06 0.08 0.1 0| G/G0 FIG. 7. (Color online) Conductance of an unstrained-strained junction ( W=15.2 nm) with ε=0.03 for different directions of uniaxial tension θ.that are not transmitting. For θ=60◦one sees an apparent transport gap /Delta10.03=0.002t0, but it is only apparent because when the band structure is calculated, this value correspondsto the lowest-energy propagating mode and, hence we areobserving the intrinsic gap, not a transport gap. When θ=45 ◦ there is no transport or intrinsic gap. Observing Fig. 7it is clear that any amount of mode mixing destroys the conductancequantization: some line shapes of the conductance at higherenergies can be traced back in the band structure of the strainedcontact. However, for low energies the effect is completely dueto the degree of mode mixing induced by angle of the appliedstrain. Another way to induce mode mixing is by adding disorder to the problem. To be specific, we consider the effectof adding disorder in a region of length L=Wbetween the perfect unstrained and strained contacts. Four disordermodels were analyzed: (i) edge disorder, (ii) hopping disorder,(iii) short-range bulk disorder, and (iv) long-range bulkdisorder. Edge disorder was implemented by removing sitesfrom the outermost row of carbon atoms with a probability p e. In the hopping disorder model, every hopping at the disorderedregion was modified according to ˜t ij=tij+/Delta1t, where /Delta1tis uniformly distributed in the interval [ −δt/2,δt/2]. The short- range bulk disorder was modeled via a random onsite energyu i=δu, where δuis a random number uniformly distributed over [−δU/2,δU/ 2]. Finally, long-range bulk disorder was generated by randomly distributing Nimpimpurities, each mod- eled by a Gaussian function of width ξ, and with an amplitude Unselected from the uniform distribution [ −δUg,δUg]. These impurities cause a modification of the onsite energy given by ui=Nimp/summationdisplay n=1Une−(r−Rn)2/2ξ2(7) and its strength is quantified with the dimensionless parameter K0, which in the dilute limit can be expressed as K0≈ 40.5nimp(δUg/t0)2(ξ/a 0)4, where nimpis the impurity density anda0is the lattice constant. Our strategy was to introduce a small amount of disorder to see how, for a given strainmagnitude ε, the transport gap /Delta1 εand the conductance were modified. The result is shown in Fig. 8, where we show the conductance averaged over 100 disorder realizations for eachmodel and for two values of strain ε=0.03 (left column) andε=0.06 (right column). Focusing on the transport gap, it can be seen that it is more strongly affected by edge andlong-range disorder, its suppression being largely insensitiveto the disorder strength in both situations. On the one hand,edge defects will induce strong backscattering, particularly inlow-energy modes of the unstrained contact 57–60which are just the modes reflected at the clean unstrained-strained junction.This backscattering leads to mixing with the propagatingmodes below /Delta1 εin the strained contact and, consequently, the transmission gap disappears. This effect is reduced in widerjunctions (not shown), as well as in unstrained AGNR. 57,58 Onsite long-range disorder, on the other hand, will lead electron-hole puddles that strongly impact the low-energystates. 61This again will introduce mode mixing between the low-energy modes of the unstrained contact with thelow-energy propagating modes below /Delta1 εin the strained contact. Turning our attention now to the other two types of 195416-7D. A. BAHAMON AND VITOR M. PEREIRA PHYSICAL REVIEW B 88, 195416 (2013)G/G0 E/|t0| (a) (b) (c) (d) (e) (g) (f) (h) E/|t0| G/G0 G/G0 G/G0 ε=0.03 ε=0.06 FIG. 8. (Color online) Average conductance of a disordered unstrained-strained junction ( W=15.2 nm, θ=π/2). The length of the disordered region is L=W=15.2 nm with ε=0.03 (left column) and ε=0.06 (right column). The disorder models shown are (a), (b) edge disorder with pe=0.1a n d0 .2, (c), (d) hopping disorder with δt=0.2a n d0 .4, (e), (f) onsite short-range disorder with δU=0.2a n d0 .4, (g), (h) onsite long-range disorder with K0=0.6, nimp=0.02,ξ=3a0andK0=5,nimp=0.04,ξ=3a0, disorder [hopping in Figs. 8(c) and8(d) and short range in Figs. 8(e) and8(f)], one sees that mode mixing is moderate since the transport gap is only reduced, and the conductancequantization is still preserved to a very good degree. 62,63 Comparing the left and right panels of Fig. 8, one can see that the same amount of disorder washes out more efficientlythe conductance features of the junction under the higheststrain. This can be explained as arising from the fact that thelarger the strain, the more high-energy modes of the unstrainedcontact become low-energy propagating modes of the strainedcontact, as shown in Fig. 5(a). Therefore, in the energy range under consideration here, these modes can only be conductingbecause of the mode mixing. IV . CONDUCTANCE OF A DISORDERED STRAINED BARRIER We now focus on the conductance of a disordered strain barrier. As defined earlier, this system consists of a strainedregion ( θ=π/2) of length L=Wbetween two perfect unstrained semi-infinite contacts [as illustrated in Fig. 1(b)]. In the absence of disorder and when the strain barrier is smooth,the resulting conductance is exactly the same to that observedin the unstrained-strained junction. This is expected because electron transmission can only occur when a given mode is opened in the three regions. Leftand right contacts are exactly the same, and so the same modewill be accessible at the same energy in both contacts. In thiscase, the reasoning of Fig. 5(a) can be repeated, resulting in the same quantum conductance. The situation is analogous to G/G0 E/|t0| E/|t0| G/G0 G/G0 (a) K0 L (nm) (b) (c) (d) FIG. 9. (Color online) Average conductance (a) and average Fano factor (b) of a disordered strained barrier ( W=L=15.2 nm, θ=π/2) for ε=0.03 and 0 .06 in the presence of onsite long- range disorder characterized by {K0=0.6,nimp=0.02,ξ=3a0} and{K0=5,nimp=0.04,ξ=3a0}. (c) Average conductance for E=0.01t0as a function of disorder strength of the same barrier studied in (a) and (b). The potential was smoothed over ξ=3a0for different values of the impurity density nimp. (d) Average conductance forE=0.01t0as a function of the length of the disordered region (W=15.2 nm,ξ=3a0, and different values of the impurity density nimphave been used, keeping K0=0.6 and 5). the quantum point contact example: the quantization of the conductance is given by the narrower constriction. Since we established above that onsite long-range disorder is the most efficient mode mixer, this is the only disorder modelthat we report on now. The resulting conductance averagedover 100 disorder realizations is presented in Fig. 9(a). There is no transport gap in the average conductance because of therelatively strong mode mixing in the strained disordered barrier(and there is no intrinsic gap in the contacts either). For lowlevels of disorder ( K 0=0.6), it is evident that mode mixing is lower since the line shape of average conductance bearssome resemblance to that of the unstrained-strained junction.It is even possible to observe the appearance of the smallplateaus that were discussed above. It can also be concludedthat larger strain transmits fewer modes (the conductance fora given energy is lower), and that the value of the conductanceis small ( G≈0.1G 0) in the energy range that corresponds to /Delta10.03and/Delta10.06of the unstrained-strained junction. When the disorder strength increases ( K0=5), there is no appreciable difference between the average conductance with ε=0.03 and 0.06, and, in fact, the overall features of the conductance traces are no different from the conductance of a disorderedunstrained barrier 57,64(but still, there is an increase of the average conductance, and the clear definition of the first twoplateaus with values G≈0.3G 0and 0.7G0). Shot noise is quantified by the dimensionless Fano factor F, in particular, it is a quantity that reveals information about the transport dynamics in the device. F=/summationtext pTp(1− Tp)//summationtext pTp, where Tpis the transmission probability of mode 195416-8CONDUCTANCE ACROSS STRAIN JUNCTIONS IN ... PHYSICAL REVIEW B 88, 195416 (2013) p.46,65In the eigenmode representation, the device can be seen as a parallel circuit of Windependent transmission modes, to access this representation in graphene (without using theanalytic expression for metallic contacts), 66–69it is necessarily a numerical method.48,70,71Figure 9(b)shows the average Fano factor, using the same values of the average conductance ofFig. 9(a). For low disorder ( K 0=0.6) and energies in the range of /Delta1ε, the Fano factor F≈0.95 indicates a tunnel barrier behavior: this is Tp/lessmuch1, as can be expected since there is no mode matching for that energy range. For the sameenergy range, it is seen that the Fano factor oscillates, a clearindication that mode mixing is the main mechanism for theconductance enhancement. For higher energies than /Delta1 ε,t h e Fano factor begins to decay because of the mode matchingbetween propagating modes in the contacts and in the strainedbarrier. For K 0=5, since there is no qualitative difference between the Fano factor for different values of strain, disorderenhances the mode mixing and the transmission is increased(Fano factor is smaller), especially in the energy range of thefirst two plateaus of the average conductance in Fig. 9(a),a s can be seen in the higher oscillations of Fano factor in thatenergy range. For higher energies, the oscillations are dampedand the Fano factor saturates. We followed the evolution ofFano factor for higher values of disorder up to K 0=10 (not shown) finding no significant changes in the line shape andvalues of the average Fano factor. Looking more closely at the effect of disorder on the conductance and its relation with strain, especially in theenergy range of /Delta1 ε, we fixed the Fermi energy at E=0.01t0 and increase the impurity density ( nimp) in the strained barrier. After averaging over 100 realizations [the average conductanceis plotted in Fig. 9(c)], it can be seen that the effect of strain is completely washed out by disorder, and there is no appreciable difference between the curves with ε=0.03 and 0 .06. With the increase of K 0the conductance is enhanced from G≈0.01G0 for low disorder to G≈0.3G0in the intermediate disorder regime. We switch now to examine the effect of length of thestrained barrier in the conductance; again we set the Fermienergy to E=0.01t 0and averaging over 500 disorder real- ization, the resulting conductance is plotted in Fig. 9(d).I ti s observed that the average conductance decreases exponentiallywith the length for all disorder strength and applied strain. ForK 0=5, it could be said that the localization length is roughly the same and that there are no appreciable differences in theline shape and values of the average conductance for differentvalues of strain. For K 0=0.6, the localization length is shorter for the higher strain value; this effect will be appreciated inlower conductance values for larger barriers, however, there is no formation of a transport gap. V . SUMMARY The electronic transport across strained junctions and barri- ers in a graphene nanoribbon has been studied in the frameworkof Landauer and B ¨uttiker, implemented using nonequilibrium Green’s functions. A clear strain-dependent transport gapappears for strain applied along the zigzag and armchairdirections of nondisordered strain junctions and barriers. Thetransport gap is a result of the perfect matching betweenthe propagating modes in both regions. A different angle ofapplied strain or disorder induces mode mixing, which tendsto degrade the gap as well as the conductance quantization,both signatures of the electronic transport across a strainedregion. For Fermi energies in the energy range of the transportgap, the presence of unmatched propagating modes in thestrained region means that they become active in the presenceof disorder, leading to a conductance plateau that is sustainedfor a broad range of disorder strength. The conductance in thisplateau decays exponentially, indicating that strain induceslocalization in the low-energy single-channel regime. 72 We have shown results for disordered square junctions and barriers with W=15.2 nm. However, our results can be easily extrapolated to junctions and barriers of different aspect ratios.For larger disordered regions, the number of scattering centersgrows, the conductance decays, and the quantization plateausare destroyed for energies higher than the transport gap ofthe clean junction. For energies below the transport gap, theconductance is enhanced leading to a broad plateau whoseconductance value can be smaller that G 0. 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PhysRevB.84.064517.pdf

PHYSICAL REVIEW B 84, 064517 (2011) Relaxation and frequency shifts induced by quasiparticles in superconducting qubits G. Catelani, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman Departments of Physics and Applied Physics, Yale University, New Haven, CT 06520, USA (Received 13 June 2011; revised manuscript received 29 July 2011; published 23 August 2011) As low-loss nonlinear elements, Josephson junctions are the building blocks of superconducting qubits. The interaction of the qubit degree of freedom with the quasiparticles tunneling through the junction represents anintrinsic relaxation mechanism. We develop a general theory for the qubit decay rate induced by quasiparticles,and we study its dependence on the magnetic flux used to tune the qubit properties in devices such as the phaseand flux qubits, the split transmon, and the fluxonium. Our estimates for the decay rate apply to both thermalequilibrium and nonequilibrium quasiparticles. We propose measuring the rate in a split transmon to obtaininformation on the possible nonequilibrium quasiparticle distribution. We also derive expressions for the shift inqubit frequency in the presence of quasiparticles. DOI: 10.1103/PhysRevB.84.064517 PACS number(s): 74 .50.+r, 85.25.Cp I. INTRODUCTION The operability of a quantum device as a qubit requires long coherence times in comparison to the gate operation time.1 Over the years, longer coherence times in superconductingqubits have been achieved by designing new systems inwhich the decoupling of the quantum oscillations of theorder parameter from other low-energy degrees of freedomis enhanced. For example, in a transmon qubit, 2the sensitivity to background charge noise is suppressed relative to thatof a Cooper-pair box. Irrespective of the particular design,in any superconducting device, the qubit degree of freedomcan exchange energy with quasiparticles. This intrinsic re-laxation mechanism is suppressed in thermal equilibrium attemperatures much lower than the critical temperature, dueto the exponential depletion of the quasiparticle population.However, both in superconducting qubits 3and resonators,4 nonequilibrium quasiparticles have been observed, which canlead to relaxation even at low temperatures. In this paper,we study the quasiparticle relaxation mechanism in qubitsbased on Josephson junctions, both for equilibrium andnonequilibrium quasiparticles. Quasiparticle relaxation in a Cooper-pair box was con- sidered in Ref. 5. In this system, the charging energy is large compared to the Josephson energy and quasiparticlepoisoning 6,7is the elementary process of relaxation: a quasi- particle entering the Cooper-pair box changes the parity (evenor odd) of the state, bringing the qubit out of the computationalspace consisting of two charge states of the same parity. Morerecently, the theory of 5was extended to estimate the effect of quasiparticles in a transmon.2In this case, the dominant energy scale is the Josephson energy, so that quantum fluctuationsof the phase are relatively small, while the uncertainty ofcharge in the qubit states is significant. As mentioned above,the advantage of the transmon is its low sensitivity to chargenoise. The possible role of nonequilibrium quasiparticles insuperconducting qubits was investigated in Ref. 3. While the properties of many superconducting qubits, e.g., the phaseand flux qubits, 8the split transmon, and the newly developed fluxonium,9can be tuned by an external magnetic flux, the effect of the latter on the quasiparticle relaxation rate has notbeen previously analyzed. Elucidating the role of flux is themain goal of this work. In particular, we show that studying theflux dependence of the relaxation rate can provide information on the presence of nonequilibrium quasiparticles. The paper is organized as follows: In the next section, we present results for the admittance of a Josephson junctionand the general approach to calculate the decay rate andenergy level shifts due to quasiparticles in a qubit with asingle Josephson junction. In Sec. III, we consider a weakly anharmonic qubit, such as phase qubit or transmon, andrelate its decay rate, quality factor, and frequency shift tothe admittance of the junction. The cases of a Cooper-pairbox (large charging energy) and of a flux qubit with largeJosephson energy are examined in Sec. IV. Some of the results presented in Secs. II–IVhave been reported previously 10in a brief format. In Sec. V, we describe the generalization to multijunction systems and study, as concrete examples, thetwo-junction split transmon and the many-junction fluxonium.We summarize the present work in Sec. VI. Throughout the paper, we use units ¯ h=k B=1 (except otherwise noted). II. GENERAL THEORY FOR A SINGLE-JUNCTION QUBIT We consider a Josephson junction closed by an inductive loop (see Fig. 1). The low-energy effective Hamiltonian of the system can be separated into three parts ˆH=ˆHϕ+ˆHqp+ˆHT. (1) The first term determines the dynamics of the phase degree of freedom in the absence of quasiparticles: ˆHϕ=4EC(ˆN−ng)2−EJcos ˆϕ+1 2EL(ˆϕ−2π/Phi1e//Phi1 0)2, (2) where ˆN=−id/dϕ is the number operator of Cooper pairs passed across the junction, ngis the dimensionless gate voltage, /Phi1eis the external flux threading the loop, /Phi10=h/2e is the flux quantum, and the parameters characterizing thequbit are the charging energy E C, the Josephson energy EJ, and the inductive energy EL. The second term in Eq. ( 1) is the sum of the BCS Hamiltonians for quasiparticles in the leads ˆHqp=/summationdisplay j=L,RˆHj qp, ˆHj qp=/summationdisplay n,σ/epsilon1j nˆαj† nσˆαj nσ, (3) 064517-1 1098-0121/2011/84(6)/064517(24) ©2011 American Physical SocietyCATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) FIG. 1. (a) Schematic representation of a qubit controlled by a magnetic flux [see Eq. ( 2)]. (b) Effective circuit diagram with three parallel elements, capacitor, Josephson junction, and inductor, characterized by their respective admittances. where ˆ αj nσ(ˆαj† nσ) are quasiparticle annihilation (creation) operators and σ=↑,↓accounts for spin. The quasiparticle energies are /epsilon1j n=/radicalBig (ξj n)2+(/Delta1j)2, with ξj nand/Delta1jbeing the single-particle energy level nin the normal state of lead jand the gap parameter in that lead, respectively. The occupationsof the quasiparticle states are described by the distributionfunctions f j/parenleftbig ξj n/parenrightbig =/angbracketleftbig/angbracketleftbig ˆαj† n↑ˆαj n↑/angbracketrightbig/angbracketrightbig qp=/angbracketleftbig/angbracketleftbig ˆαj† n↓ˆαj n↓/angbracketrightbig/angbracketrightbig qp,j=L,R (4) assumed to be independent of spin; double angular brack- ets/angbracketleft/angbracketleft.../angbracketright/angbracketrightqpdenote averaging over the quasiparticle states. Hereinafter, we assume for simplicity equal gaps in the leads/Delta1 L=/Delta1R≡/Delta1. The last term in Eq. ( 1) describes quasiparticle tunneling across the junction and couples the phase and quasiparticledegrees of freedom ˆH T=˜t/summationdisplay n,m,σ/parenleftbig eiˆϕ 2uL nuRm−e−iˆϕ 2vR mvL n/parenrightbig ˆαL† nσˆαR mσ+H.c. (5) The electron tunneling amplitude tin this equation determines the junction conductance gT=4πe2νLνR˜t2in the tunneling limit ˜t/lessmuch1, which we are considering. From now on, we assume identical densities of states per spin direction in the leadsνL=νR=ν0. The Bogoliubov amplitudes uj n,vj ncan be taken real, since Eq. ( 5) already accounts explicitly for the phases of the order parameters in the leads via the gauge-invariant phase difference 11in the exponentials. Accounting for the Josephson effect and quasiparticles dynamics byEqs. ( 2)–(5) is possible as long as the qubit energy ωand characteristic energy δEof quasiparticles (as determined by their distribution function and measured from /Delta1) are small compared to /Delta1(Ref. 5):ω,δE /lessmuch2/Delta1. In this low-energy limit, we may further approximate uj m/similarequalvj n/similarequal1/√ 2. Then, the operators e±iˆϕ/2in Eq. ( 5), which describe transfer of charge ±eacross the junction, combine to give ˆHT=˜t/summationdisplay n,m,σisinˆϕ 2ˆαL† nσˆαR mσ+H.c. (6) Starting from this low-energy tunneling Hamiltonian, in the next section we calculate the dissipative part of the junctionadmittance.A. Response to a classical time-dependent phase We consider here the “classical” dissipative response of a Josephson junction to a small ac bias to show that Eq. ( 6) correctly accounts for the known11junction losses in the low- energy regime. These “classical” losses are directly related tothe decay rate in the quantum regime, as we explicitly show inthe next section. We assume a time-dependent bias v(t)=vcos(ωt)o f frequency ω> 0 superimposed to a fixed phase difference ϕ 0. In other words, we take the phase to be a time-dependent number, which, by the Josephson equation dϕ/dt =2ev(t), has the form ϕ(t)=ϕ0+2ev ωsin(ωt). (7) Here, we focus on the linear in vresponse in the low-energy regime. Expressions for the current through the junction validbeyond linear response can be found, for example, in Ref. 11. Substituting Eq. ( 7) into Eq. ( 6), expanding for small v, and keeping the linear term, we find for the time-dependentperturbation δˆH(t) causing the dissipation δˆH(t)=ˆH ACsin(ωt), (8) ˆHAC=i˜tcosϕ0 2ev ω/summationdisplay n,m,σˆαL† nσˆαR mσ+H.c. The average dissipated power can be calculated using Fermi’s golden rule: it is given by the product of the transition ratetimes the energy change in a transition between quasiparticlestates caused by the perturbation. The energy change in atransition is ±ωby energy conservation, with the two signs corresponding to the events giving energy to or taking energyfrom the system. The average power Pis P=2π/summationdisplay {λ}qp/angbracketleft/angbracketleft|/angbracketleft{λ}qp|ˆHAC|{η}qp/angbracketright|2ω ×[δ(Eλ,qp−Eη,qp−ω)−δ(Eλ,qp−Eη,qp+ω)]/angbracketright/angbracketrightqp, (9) where Eη,qpandEλ,qpare the total energies of the quasiparti- cles in their respective initial {η}qpand final {λ}qpstates. We use Eq. ( 8) to evaluate the matrix element, average over initial quasiparticle states, and sum over final states to find P=1 2ReYJ(ω,ϕ 0)v2(10) with12 ReYJ(ω,ϕ 0)=1+cosϕ0 2ReYqp(ω). (11) Here, Re Yqpis the real part of the quasiparticle contribution to the junction admittance at zero phase difference: ReYqp(ω)=gT2/Delta1 ω/integraldisplay∞ 0dx1√x√x+ω//Delta1{fE[(1+x)/Delta1] −fE[(1+x+ω//Delta1 )/Delta1]}. (12) In deriving these formulas, we have approximated the standard BCS density of states functions as /epsilon1√ /epsilon12−/Delta12,/Delta1√ /epsilon12−/Delta12∼/radicalBigg /Delta1 2(/epsilon1−/Delta1)≡1√ 2x(13) 064517-2RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) and taken equal quasiparticle occupations in the two leads fL=fR≡f. The latter assumption considerably simplifies the calculations; moreover, it is physically justified in manydevices since the leads are often fabricated with the samematerial and deposition technique and, hence, have identicalproperties. We indicate with f Ethe energy mode of the distribution function fE(/epsilon1)=1 2[f(ξ)+f(−ξ)], (14) where /epsilon1=/radicalbig ξ2+/Delta12. Equation ( 11) for the real part of the admittance, valid at ω> 0, agrees with the linear response, low-energy limit of the nonlinear I-Vcharacteristic presented in Ref. 11. Extension to ω< 0 is found by noticing that Re Yqp is an even function of frequency. In thermal equilibrium and at low temperatures T/lessmuch/Delta1,t h e distribution function can be approximated as fE(/epsilon1)/similarequale−/epsilon1/T, (15) and Eq. ( 12) gives, at arbitrary ratio ω/T , ReYeq qp(ω)=gT2/Delta1 ωe−/Delta1/Teω/2TK0/parenleftbigg|ω| 2T/parenrightbigg [1−e−ω/T]. (16) Here, K0is the modified Bessel function of the second kind with asymptotes K0(x)/similarequal/braceleftbigge−x√π/2x, x /greatermuch1 ln 2/x−γE,x/lessmuch1(17) withγEthe Euler gamma. For a generic distribution function, we can relate Re Yqp to the density of quasiparticle nqpin the high-frequency regime ω/greatermuchδE, where δEindicates the characteristic energy of quasiparticle (measured from the gap) above which theoccupation of the quasiparticle states can be neglected; inthermal equilibrium, δE∼T. Under the assumption ω/greatermuchδE, we obtain from Eq. ( 12) ReY hf qp(ω)=1 2xqpgT/parenleftbigg2/Delta1 |ω|/parenrightbigg3/2 , (18) where xqp=nqp 2ν0/Delta1(19) is the quasiparticle density normalized to the Cooper-pair density and nqp=2√ 2ν0/Delta1/integraldisplay∞ 0dx√xfE[(1+x)/Delta1] (20) is the density written using the approximation in Eq. ( 13). Note that in thermal equilibrium at low temperatures [Eq. ( 15)], we have neq qp=2ν0√ 2π/Delta1Te−/Delta1/T. (21) Then, using Eq. ( 17), it is easy to check that for T/lessmuchωEq. ( 16) takes the form given in Eq. ( 18). The real and imaginary parts of the admittance satisfy the Kramers-Kr ¨onig relations. However, when taking the Kramers-Kr ¨onig transform of the real part, a purely inductive contribution to the imaginary part can be missed. Indeed, atlow energies, the complex junction admittance (obtained from the expressions in Ref. 11) can be written as YJ(ω,ϕ)=1−2xA qp iωLJcosϕ+Yqp(ω)1+cosϕ 2, (22) where xA qp=fE(/Delta1) (23) can be interpreted as the population of the Andreev bound states,13and the inverse of the Josephson inductance is 1 LJ=gTπ/Delta1 qp (24) [the subscript qp in /Delta1qpis used to indicate that in this expression it may be necessary to account for the effect ofquasiparticles on the gap (see Secs. II C andIII B )]. Unlike the Andreev states, free quasiparticles contribute to both dissipative and nondissipative parts of the total admittanceY Jvia the complex term Yqp. The real part of the quasiparticle admittance is defined in Eq. ( 12), while its imaginary part is given by the Kramers-Kr ¨onig transform of that expression: ImYqp(ω)=−gT2/Delta1 ωP π/integraldisplay∞ 0dx√x/integraldisplay∞ 0dy√y{fE[(1+x)/Delta1] −fE[(1+y)/Delta1]}/bracketleftbigg1 x−y+ω//Delta1−1 x−y/bracketrightbigg , (25) where Pdenotes the principal part and ω> 0. Using that ImYqpis an odd function of frequency, we can simplify the above expression to a form with a single rather than doubleintegral: ImY qp(ω)=gT2/Delta1 ω/bracketleftbigg/integraldisplay|ω|//Delta1 0dxfE[(1+x)/Delta1]√x√|ω|//Delta1−x−πxA qp/bracketrightbigg . (26) As discussed above for the real part, an analytic expression for ImYqpcan be obtained in thermal equilibrium, ImYeq qp(ω)=−gT2/Delta1 ωe−/Delta1/Tπ/bracketleftbigg 1−e−|ω|/2TI0/parenleftbigg|ω| 2T/parenrightbigg/bracketrightbigg . (27) Here,I0is the modified Bessel function of the first kind with asymptotes I0(x)/similarequal/braceleftbiggex√1/2πx, x /greatermuch1 1+x2/4,x /lessmuch1.(28) For arbitrary distribution functions satisfying the high- frequency condition ω/greatermuchδE, we find ImYhf qp(ω)=1 2gT2/Delta1 ω/bracketleftbigg xqp/radicalBigg 2/Delta1 |ω|−2πxA qp/bracketrightbigg . (29) Using Eq. ( 21) and the large- xlimit in Eq. ( 28), it is easy to show that, for T/lessmuchω,E q .( 27) reduces to the general expression in Eq. ( 29). In the high-frequency regime, real 064517-3CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) and imaginary parts of the quasiparticle admittance can be combined into the complex admittance Yhf qp(ω)=−2 iωLJ/bracketleftbiggxqp π/radicalbigg /Delta1 iω−xA qp/bracketrightbigg . (30) By substituting Eq. ( 30) into Eq. ( 22), we find that, in the total admittance YJ, the coefficient multiplying xA qpis proportional to (1−cosϕ) and vanishes for ϕ=0. This is in agreement with the absence of Andreev bound states when there is nophase difference across the junction. B. Transition rates The effects of the interaction between quasiparticles and qubit degrees of freedom Eq. ( 5) can be treated perturbatively in the tunneling amplitude ˜t. The interaction makes possible, for example, a transition between two qubit states (initial |i/angbracketright and final |f/angbracketright, differing in energy by amount ωif>0) by exciting a quasiparticle during a tunneling event. The ratefor the transition between qubit states can be calculated usingFermi’s golden rule /Gamma1 i→f=2π/summationdisplay {λ}qp/angbracketleft/angbracketleft|/angbracketleftf,{λ}qp|ˆHT|i,{η}qp/angbracketright|2 ×δ(Eλ,qp−Eη,qp−ωif)/angbracketright/angbracketrightqp. (31) We remind that in our notation Eη,qp(Eλ,qp) is the total energy of the quasiparticles in their initial (final) state {η}qp({λ}qp), and double angular brackets /angbracketleft /angbracketleft ···/angbracketright /angbracketright qpdenote averaging over the initial quasiparticle states, the occupation of which isdetermined by the distribution function. In the low-energy regime we are considering, the transition rate factorizes into terms accounting separately for qubitdynamic and quasiparticle kinetics /Gamma1 i→f=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftf|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Sqp(ωif). (32) Equation ( 32) is one of the main results of this work: it shows that the qubit properties affect the transition rate via the wavefunctions |i/angbracketrightand|f/angbracketrightentering the matrix element, while the quasiparticle kinetics is accounted for by the quasiparticlecurrent spectral density S qp: Sqp(ω)=16EJ π/integraldisplay∞ 0dx1√x√x+ω//Delta1(fE[(1+x)/Delta1] ×{1−fE[(1+x)/Delta1+ω]}), (33) where ω> 0 and we used the relation EJ=gT/Delta1/8gK (34) withgK=e2/2πthe conductance quantum. The expression forSqpatω< 0 is obtained by the replacements x→x− ω//Delta1 ,ω→−ωin the integrand in Eq. ( 33). The spectral density Sqpdepends on the detail of the distri- bution functions. In thermal equilibrium at low temperaturesT/lessmuch/Delta1,b yu s i n gE q .( 15), we find S eq qp(ω)=16EJ πe−/Delta1/Teω/2TK0/parenleftbigg|ω| 2T/parenrightbigg . (35)Note that the equality Seq qp(−ω) Seq qp(ω)=e−ω/T(36) implies that, in thermal equilibrium, the transition rates are related by detailed balance /Gamma1f→i /Gamma1i→f=e−ωif/T. (37) The similarity between Eq. ( 35)f o rSqpand Eq. ( 16)f o r ReYqpis not accidental. In thermal equilibrium, the following fluctuation-dissipation relation holds: Seq qp(ω)+Seq qp(−ω)=ω π1 gKReYeq qp(ω) coth/parenleftbiggω 2T/parenrightbigg . (38) Moreover, in the low-energy regime for an arbitrary distribu- tion function, the two quantities are also related by Sqp(ω)−Sqp(−ω)=ω π1 gKReYqp(ω). (39) In the high-frequency regime ω/greatermuchδE, we can simplify the above relation to Shf qp(ω)=ω π1 gKReYhf qp(ω). (40) For the transition rates, this corresponds to neglecting the downward transitions with ωif<0, in which a quasiparticle loses energy to the qubit, compared to the upward ones. Thisis a good approximation since the assumption ω/greatermuchδEmeans that there are no quasiparticles with energy high enough toexcite the qubit. Equation ( 40) can be checked by comparing Eq. ( 18)t o S hf qp(ω)=xqp8EJ π/radicalbigg 2/Delta1 ω(41) withEJgiven in Eq. ( 34) and the normalized quasiparticle density xqpin Eq. ( 19). C. Energy-level corrections In addition to causing transitions between qubit levels, the quasiparticles affect the energy Eiof each level iof the system. We can distinguish two quasiparticle mechanisms thatmodify the qubit spectrum and, hence, separate two terms inthe correction δE ito the energy δEi=δEi,EJ+δEi,qp. (42) First, in the presence of quasiparticles, the Josephson energy takes the form EJ,qp=gT 8gK/Delta1qp/parenleftbig 1−2xA qp/parenrightbig (43) withxA qpdefined in Eq. ( 23). As mentioned after Eq. ( 24), we use/Delta1qpto distinguish the self-consistent gap in the presence of quasiparticles from the gap /Delta1when there are no quasiparticles. At leading order in the quasiparticle density, we have /Delta1qp/similarequal/Delta1(1−xqp). (44) 064517-4RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) By treating these modifications to the Josephson energy as perturbations, the correction to the energy of level iis δEi,EJ=EJ/parenleftbig xqp+2xA qp/parenrightbig /angbracketlefti|cos ˆϕ|i/angbracketright. (45) Second, the virtual transitions between the qubit levels mediated by quasiparticle tunneling cause a correction thatcan be expressed in terms of the matrix elements of sin ˆ ϕ/2a s δE i,qp=/summationdisplay k/negationslash=i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Fqp(ωik), (46) where ωik=Ek−Ei. (47) The derivation of the above formulas and the definition of function Fqpin terms of the quasiparticle distribution function [Eq. ( A19)] are given in Appendix A. Here, we give the relation between Fqpand the imaginary part of the quasiparticle impedance, Fqp(ω)+Fqp(−ω)=−ω 2π1 gKImYqp(ω), (48) which we will use in the next section to obtain the quasiparticle-induced change in the qubit frequency. III. SINGLE JUNCTION: WEAKLY ANHARMONIC QUBIT As an application of the general approach described in the preceding section, we consider here a weakly anharmonicqubit, such as the transmon and phase qubits. We start withthe semiclassical limit, i.e., we assume that the potentialenergy terms in Eq. ( 2) dominate the kinetic energy term proportional to E C. This limit already reveals a nontrivial dependence of relaxation on flux. Note that, assuming EL/negationslash=0, we can eliminate ngin Eq. ( 2) by a gauge transformation.14 In the transmon, we have EL=0 and the spectrum depends onng, displaying both well-separated and nearly degenerate states (see Fig. 2). The results of this section can be applied to the single-junction transmon when consideringwell-separated states. The transition rate between these statesand the corresponding frequency shift is dependent on n g. However, since EC/lessmuchEJ, this dependence introduces only small corrections to /Gamma1n→n−1andδω; the corrections are exponential in −√8EJ/EC. By contrast, the leading term in the rate of transitions /Gamma1e↔obetween the even and odd states is exponentially small. The rate /Gamma1e↔oof parity switching is discussed in detail in Appendix C. The potential energy in Eq. ( 2) is extremized at phase ϕ0 satisfying EJsinϕ0+EL(ϕ0−2π/Phi1e//Phi1 0)=0. (49) ForEJ<E L, there is only one solution at the global minimum. For EJ>E L, however, there can be multiple minima; their number depends both on the ratio EJ/ELand the external flux /Phi1e. Here, we assume that the flux is such that distinct minima are not degenerate; in particular, this meansthat the flux is tuned away from odd-integer multiples of halfthe flux quantum. 15For the transmon with EL=0, we can takeωpoe1 10 0.0 0.2 0.4 0.6 0.8 1.0 ng FIG. 2. Schematic representation of the transmon low-energy spectrum as a function of the dimensionless gate voltage ng. Solid (dashed) lines denote even (odd) states (see also Sec. IV A ). The amplitudes of the oscillations of the energy levels are exponentiallysmall (Ref. 2) (see Appendix B); here, they are enhanced for clarity. Quasiparticle tunneling changes the parity of the qubit sate. The results of Sec. IIIare valid for transitions between states separated by energy of the order of the plasma frequency ω p[Eq. ( 56)] and give, for example, the rate /Gamma11→0. For the transition rates between nearly degenerate states of opposite parity, such as /Gamma1(1) o→e, see Appendix C. ϕ0=0 as a solution to Eq. ( 49). Next, we expand the potential energy around a minimum and find, at quadratic order, ˆH(2) ϕ=4ECˆn2+1 2(EL+EJcosϕ0)(ˆϕ−ϕ0)2.(50) Fluctuations of the phase around ϕ0are small under the assumption nEC ω10/lessmuch1, (51) where ndenotes the energy level and ω10=/radicalbig 8EC(EL+EJcosϕ0) (52) is the qubit frequency in the harmonic approximation. Note that anharmonicity and quality factor Qdetermine the operability of the system as a qubit.8The anharmonic correction to the transition frequencies can be calculated by considering theeffect on the spectrum of the next order in the expansion aroundϕ 0(cubic for the phase qubit, quartic for the transmon), which defines an anharmonic potential well of finite depth U. Then, the operability condition can be expressed as Q/n w/greatermuch1, where nwis the number of states in the potential well nw∼U/ω 10.16In a weakly anharmonic system, nwcan be large; however, if the quality factor is larger, the system can beused as a qubit despite the weak anharmonicity, as it is indeedthe case for the transmon. 2 The condition for small phase fluctuations in Eq. ( 51) enables us to calculate the matrix element of operator sin ˆ ϕ/2 by expanding around ϕ0up to the second order and using standard expressions for the matrix elements of the posi-tion operator between eigenstates |n/angbracketright,|m/angbracketrightof the harmonic 064517-5CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) oscillator [cf. Eq. ( 50)]. To first order in EC/ω10, we find (see also Appendix D) /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftm|sinˆϕ 2|n/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =δm,n/bracketleftbigg 1−2EC ω10/parenleftbigg n+1 2/parenrightbigg/bracketrightbigg1−cosϕ0 2 +EC ω10[nδm,n−1+(n+1)δm,n+1]1+cosϕ0 2.(53) Note that, in the first term on the right-hand side, the corrections due to the nonlinearity of sine (the second terminside the square brackets) are indeed small if condition ( 51) is satisfied. In addition, we have neglected here the anharmoniccorrections to the states used to calculate the matrix element;this is a good approximation for low-lying levels n/lessmuchn w.17 For the transmon ( ϕ0=0), the leading term in Eq. ( 53)i s of linear order in EC/ω10/lessmuch1; as we show in Appendix E, by including the first anharmonic correction to the states, thenext nonvanishing term in the square of the transmon matrixelement is cubic in E C/ω10, rather than quadratic as for the harmonic oscillator. Therefore, in the case of the transmon,keeping only the leading term is a better approximation thannaively expected. Equation ( 53) shows that at leading order we can restrict our attention to transitions involving only neighboring levels.Concentrating here on low-lying levels, using Eqs. ( 32), (39), and ( 53), we find the following relation between transition rate and impedance: /Gamma1 n→n−1−/Gamma1n−1→n=n CReYqp(ω10)1+cosϕ0 2,(54) where we also used EC=e2/2C. In the high-frequency regime, the upward transition rate can be neglected, /Gamma1n−1→n/similarequal 0, and the above expression simplifies to19[see also Eq. ( 40) and the text that follows it] /Gamma1n→n−1=n CReYhf qp(ω10)1+cosϕ0 2 =nω2 p ω10xqp 2π/radicalBigg 2/Delta1 ω10(1+cosϕ0). (55) In the last expression, we used Eq. ( 18) and introduced the plasma frequency ωp=/radicalbig 8ECEJ. (56) The above equation can also be obtained by substituting directly Eq. ( 40) into Eq. ( 32). Forn=1 andϕ0=0, Eq. ( 55) reduces to the transition rate presented in Ref. 3. The transition rate in Eq. ( 55) is proportional to the (possi- bly nonequilibrium) quasiparticle density xqpand depends on the external flux /Phi1eviaϕ0andω10[see Eqs. ( 49) and ( 52)]. The flux dependence is, in general, sensitive to the statesinvolved in the transition. This sensitivity can already beseen for transitions between harmonic-oscillator states: dueto the nonlinear interaction between phase and quasiparticles[see Eq. ( 6)], transitions between distant levels are possible. These transitions are suppressed by the smallness of phasefluctuations when E C/ω10/lessmuch1. For example, the rate for the 2→0 transition is /Gamma12→0=2ω10 π1 gKReYhf qp(2ω10)/parenleftbiggEC ω10/parenrightbigg21−cosϕ0 4. (57) Note that, in contrast to Eqs. ( 55) and ( 57) can not be written in terms of the real part of the total admittance of the junction:While in Eq. ( 55) the phase enters via the factor (1 +cosϕ 0)a s in Eq. ( 11), in Eq. ( 57), ReYqpis multiplied by (1 −cosϕ0). To obtain /Gamma12→0, we substituted into Eq. ( 32) the high-frequency relation ( 40), while the explicit form of the squared matrix element |/angbracketleft0|sin( ˆϕ/2)|2/angbracketright|2is found by setting n=2, and keeping the leading term in EC/ω10in the formula /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleft0|sinˆϕ 2|n/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =e−EC ω10/parenleftbiggEC ω10/parenrightbiggn1−(−1)ncosϕ0 2n!(58) derived in Appendix D. Equation ( 58) is valid for any ratio EC/ω10for transitions between eigenstates of the harmonic oscillator. When ϕ0=0, Eq. ( 58) gives vanishing matrix elements for even n: this is an example of the more general selection rule according to which only transitions betweenstates of different parity are allowed at ϕ 0=0. The rate for transitions between excited states and the ground state in the case of large phase fluctuations can beobtained using Eq. ( 58) when E L/lessmuchECandEJ/lessorsimilarωLC=√8ECEL. The latter condition enables us to neglect the Josephson energy term in Eq. ( 2). Then, using Eqs. ( 32) and (41) with ωif=nωLC, we find that the transition rate has a maximum for n=n0withn0≈EC/ωLC: /Gamma1n→0/similarequalω2 p ECxqp 2π/radicalBigg 2/Delta1 EC[1−(−1)ncos 2π/Phi1e//Phi1 0] ×1√2πn 0exp/bracketleftbigg −(n−n0)2 2n0/bracketrightbigg . (59) Here, we have approximated e−yyn/n!√n/similarequalexp[−(n− y)2/2y]/√ 2πy; the approximation is valid for y/greatermuch1 and |n−y|/lessorsimilar√2y. Equation ( 59) shows that when the charging energy is the dominant energy scale, dissipation is the strongestfor transitions between states whose energy difference ( nω LC) corresponds to the energy change ( EC) caused by the transfer of a single electron through the barrier, as in the “quasiparticlepoisoning” picture for the Cooper-pair box. 5We stress that, in the present case, charge is not quantized due to the finite valueof the inductive energy E L.14We will comment on the relation between Eq. ( 59) and the transition rate in the Cooper-pair box in Sec. IV A . A. Quality factor Returning now to the semiclassical regime of small EC, Eq. ( 55) with n=1 enables us to evaluate, in the high- frequency regime, the inverse Qfactor for the transition between the qubit states 1 Q10=/Gamma11→0 ω10=1 πgKReYhf qp(ω10)EC ω101+cosϕ0 2. (60) We stress that this formula is valid not only in thermal equilibrium, but also in the presence of nonequilibriumquasiparticles with characteristic energy δE/lessmuchω 10. We can 064517-6RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) generalize Eq. ( 60) to account for the possible coexistence of nonequilibrium and thermal quasiparticles. We take thedistribution function in the form f E(/epsilon1)=fne(/epsilon1)+feq(/epsilon1), (61) where fneis the nonequilibrium contribution, insensitive to temperature and satisfying the high-frequency conditionω 10/greatermuchδE, andfeqis the equilibrium distribution of Eq. ( 15). Noting that, within our assumption, the two terms in fE contribute separately to the transition rates and that for the thermal part we can not in general neglect the “upward”transitions, using Eqs. ( 32), (35), (41), and ( 53), we find 1 Q10=/Gamma11→0+/Gamma10→1 ω10=1+cosϕ0 2πω2 p ω2 10/bracketleftbigg xne/radicalBigg 2/Delta1 ω10 +4e−/Delta1/Tcosh/parenleftbiggω10 2T/parenrightbigg K0/parenleftbiggω10 2T/parenrightbigg/bracketrightbigg , (62) where xneis the normalized nonequilibrium quasiparticle density [cf. Eq. ( 19)]. Recently, good agreement between theory Eq. ( 62) and experiment has been shown for single-junction transmons(ϕ 0=0,ω10=ωp) in the temperature range 10–210 mK.20 However, while these measurements indicate that thermal quasiparticles are the main cause of relaxation above ∼ 150 mK, one can not conclude that nonequilibrium quasi-particles are present from the lower temperature data: byMatthiessen rule, any other relaxation mechanism that isindependent of (or weakly dependent on) temperature wouldhave the same limiting effect on Q 10as the first term in square brackets in Eq. ( 62). As we will discuss in more detail in Sec. VA, similar measurements on a flux-sensitive device should enable one to decide on the presence of nonequilibriumquasiparticles since Eq. ( 62) [and its analogous for the split transmon, Eq. ( 127)] describes the effect of flux on both equilibrium and nonequilibrium quasiparticle contributions toQ 10, and other sources of relaxation respond differently to the flux. B. Frequency shift A further test of the theory presented in Sec. IIis provided by the measurement of the qubit resonant frequency. In thesemiclassical regime of small E C, the qubit can be described by the effective circuit of Fig. 1(b), with the junction admittance YJof Eq. ( 22),YC=iωC , andYL=1/iωL [the inductance is related to the inductive energy by EL=(/Phi10/2π)2/L]. As discussed in Ref. 10, for parallel elements, the total admittance Yis the sum of their admittances Y=YJ+YC+YL, (63) and the resonant frequency ωris the zero of the total admittance Y(ωr)=0. In the absence of quasiparticles, we find ωr=ω10 withω10of Eq. ( 52). In the presence of quasiparticles, by considering their effect on the junction admittance at linear order in the quasiparticledensity x qpand Andreev level occupation xA qp, we obtain ωr=ω10+δω (64)with δω=i 2CYqp(ω10)1+cosϕ0 2−πgT/Delta1 Cω 10xA qpcosϕ0 −πgT/Delta1 2Cω 10xqpcosϕ0. (65) The last term in Eq. ( 65) originates from the gap suppression by quasiparticles [cf. Eq. ( 44)]. This term was neglected in Ref. 10 as it is subleading in the high-frequency regime consideredthere [see Eq. ( 73)]. The correction δωhas both real and imaginary parts. The imaginary part coincides 10with half the dissipation rate in Eq. ( 55)f o rt h e n=1→0 transition. Here, we show that the real part of δωrobtained in the effective circuit approach agrees with the quantum mechanical calculation. Within the harmonic approximation of Eq. ( 50), the energy difference ωibetween the neighboring levels Ei+1andEi, ωi≡Ei+1−Ei=ω10, (66) is of course independent of the level index i. The quasiparticle corrections to energy levels of Sec. II Ccause a correction δωi toωi: δωi=δEi+1−δEi. (67) As we show below, at leading order in EC/ω10, this correction is also independent of level index, i.e, it represents a renormal-ization of the system resonant frequency. As in Eq. ( 42), we separate the contributions due to change in the Josephson energy and due to quasiparticle tunneling δω i=δωi,EJ+δωi,qp. (68) For the first term on the right-hand side, we use Eq. ( 45) together with the matrix element of cos ˆ ϕat first order in EC/ω10[see Eq. ( D9)], /angbracketlefti|cos ˆϕ|i/angbracketright/similarequalcosϕ0/bracketleftbigg 1−4EC ω10/parenleftbigg i+1 2/parenrightbigg/bracketrightbigg , (69) to find δωi,EJ=−1 2ω2 p ω10cosϕ0/parenleftbig xqp+2xA qp/parenrightbig . (70) As discussed in Sec. II C, the term proportional to xqpis due to the gap suppression in the presence of quasiparticles Eq. ( 44), while xA qpaccounts for the occupation of the Andreev bound states. For the quasiparticle tunneling term, we substitute Eq. ( 53) into Eq. ( 46) to get δωi,qp=EC ω10[Fqp(ω10)+Fqp(−ω10)]1+cosϕ0 2. (71) Finally, using the relation ( 48) and adding the two terms, we arrive at δωi=−1 2CImYqp(ω10)1+cosϕ0 2 −1 2ω2 p ω10cosϕ0/parenleftbig xqp+2xA qp/parenrightbig . (72) This expression agrees with the real part of Eq. ( 65). We note that by extending the above consideration to include the nextorder in E C/ω10, anharmonic corrections to the spectrum can 064517-7CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) be calculated. They are dominated by the anharmonicity of the cosine potential in Eq. ( 2), with quasiparticles contributing negligible additional corrections. For the case of the transmon,the leading anharmonicity can be found in Ref. 2. In the high-frequency regime, using Eq. ( 29), the relative frequency shift is δω i ω10=1 2ω2 p ω2 10/bracketleftbigg xA qp(1−cosϕ0) −xqp/parenleftbigg1+cosϕ0 2π/radicalBigg 2/Delta1 ω10+cosϕ0/parenrightbigg/bracketrightbigg . (73) Note that, in the limit ω10/lessmuch/Delta1, we can neglect the cosine compared to the term multiplied by the square root inside theround brackets. However, this cosine term is the appropriatesubleading contribution since the terms neglected in derivingthe energy corrections presented in Sec. II Care suppressed by ω 10//Delta1with respect to the leading contribution. In recent experiments with single-junction transmons,20 relative shifts of order 10−5have been measured at temper- atures ∼200 mK, in agreement with Eq. ( 72). Together with the above-mentioned measurements of the transition rates inthe same devices, this is an additional, independent check ofthe validity of the present theory in the regime T/greaterorsimilar150 mK. While in the transmon ( ϕ 0=0) there are no Andreev bound states [indeed, in this case their contribution to the frequencyshift is absent; see Eq. ( 73)], in a phase qubit, both Andreev levels occupation x A qpand free quasiparticle density xqp affect the frequency. Assuming that the two quantities are proportional, xA qp∝xqp, the ratio between frequency shift Eq. ( 73) and transition rate Eq. ( 55) in the high-frequency regime is independent of the quasiparticle density. Theconstancy of this ratio has been recently verified by injectinga variable (but unknown) number of quasiparticles in a phasequbit. 21 IV . SINGLE JUNCTION: STRONG ANHARMONICITY Here, we consider the regime, complementary to that of the previous section, of qubits with large anharmonicities. Westudy first the single-junction Cooper-pair box (CPB); as forthe transmon, it is insensitive to flux, but in contrast to thetransmon, the CPB properties are strongly affected by the valueof the dimensionless gate voltage n g. Then, we analyze a flux qubit for which the external flux is tuned near half the fluxquantum /Phi1 e≈/Phi10/2. A. Cooper-pair box The CPB is described by Eq. ( 2) with EL=0 and EC/greatermuchEJ. In this limit, it is convenient to rewrite the Hamiltonian in the charge basis as5 ˆH=EC/summationdisplay q(q−2ng)2|q/angbracketright/angbracketleftq| −1 2EJ/summationdisplay q(|q/angbracketright/angbracketleftq+2|+|q+2/angbracketright/angbracketleftq|). (74) The eigenstates have definite parity (even or odd) and are given by linear combinations of even or odd charge states. The CPBoperating point is, without loss of generality, at ng=1/2. Near this operating point, the CPB is well described by the reducedHamiltonian H CPB=⎛ ⎜⎝EC(2ng)20 −EJ/2 0 EC(2ng−1)20 −EJ/20 EC(2ng−2)2⎞ ⎟⎠.(75) The reduced CPB Hamiltonian has a single odd eigenstate, the|q=1/angbracketrightcharge state, |o,0;ng/angbracketright=| 1/angbracketright, (76) withng-dependent eigenenergy E0(ng)=EC(2ng−1)2, (77) and two even eigenstates |e,±;ng/angbracketright, with energies E±(ng)=EC+E0(ng)±1 2ω10(ng). (78) The qubit frequency depends on the gate voltage as ω10(ng)=/radicalBig (4EC)2(2ng−1)2+E2 J. (79) Note that, at the operating point, we have ω10(1/2)=EJ and that the frequency rises quickly at a narrow distance from the optimal point, more than doubling for |ng−1/2|∼ EJ/EC/lessmuch1. In terms of the charge states, the two even eigenstates are |e,−;ng/angbracketright=cosθ|0/angbracketright+sinθ|2/angbracketright, (80) |e,+;ng/angbracketright=sinθ|0/angbracketright−cosθ|2/angbracketright, where cosθ=1√ 2/radicalBigg 1−4EC(2ng−1) ω10(ng). (81) The nonvanishing matrix elements of sin ˆ ϕ/2 can be readily obtained using the charge basis form of this operator: sinˆϕ 2=1 2i/summationdisplay q(|q+1/angbracketright/angbracketleftq|−|q/angbracketright/angbracketleftq+1|). (82) For the states in Eqs. ( 76) and ( 80), we find /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketlefto,0;n g|sinˆϕ 2|e,±;ng/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =1 4/bracketleftbigg 1±EJ ω10(ng)/bracketrightbigg . (83) We stress that the transitions are not between the qubit (i.e., even) states, but between the even and odd states;the corresponding transition frequencies are ω ±(ng)=EC± ω10(ng)/2[ s e eE q s .( 77) and ( 78)]. Therefore, the tunneling of a quasiparticle into the CPB changes the parity of the state,an effect known as “quasiparticle poisoning.” 6Substituting the matrix element ( 83) into Eq. ( 32) and using the high-frequency expression ( 41), we find /Gamma1e,+→o,0=/bracketleftbigg 1+EJ ω10(ng)/bracketrightbigg2EJ πxqp/radicalBigg 2/Delta1 ω+(ng)(84) 064517-8RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) for the transition between even excited and odd states. In thermal equilibrium with T/lessmuchω+(ng), using Eq. ( 21), we obtain /Gamma1e,+→o,0=/bracketleftbigg 1+EJ ω10(ng)/bracketrightbigg4EJ√π/radicalBigg T ω+(ng)e−/Delta1/T.(85) Within our approximations, this expression reproduces (after implementing the corrections described in Ref. 22and up to a numerical prefactor) the decay rate calculated in Ref. 5for the “open” qubit at the operating point ng=1/2. For the transition between even ground and odd states, the matrixelement in Eq. ( 83) vanishes at the operating point. This vanishing is a consequence of the low-energy approximationthat leads to Eq. ( 6): As the results of Refs. 5and 18show, the contributions that we neglect cause a finite transition rate,which is suppressed by a small factor of order E C/2/Delta1in comparison with the transition rate from even excited to oddstate. We note that, while in all the above expressions the distance |2n g−1|from the operating point can be large compared to the small parameter EJ/EC/lessmuch1, the description based on Eq. ( 75) is valid if other charge states can be neglected, which limits the range of validity to |2ng−1|<1/2 (with |2ng−1|−1/2/greatermuchEJ/EC). For example, at 2 ng−1/similarequal1/2, the charge states |0/angbracketrightand|3/angbracketrightare nearly degenerate and we can expect an enhanced transition rate /Gamma1e,+→o,3in comparison to the rate /Gamma1e,+→o,0, which we have considered above. Finally, let us comment on the relationship between the transition rate in the CPB and in the inductively shuntedJosephson junction with large charging energy [see theparagraph containing Eq. ( 59)]. As shown schematically in the right panel of Fig. 3and discussed in detail in Ref. 14, the spectra of the two systems are distinct, even in the limit ofsmall inductive energy E L:I nt h eC P B( EL=0), the energy levels form bands as ngvaries, while for any nonzero EL, the gate voltage ngcan be “gauged away” and the spectrum consists of discrete levels that become denser as ELdecreases. 0.3 0.4 0.5 0.6 0.70.00.20.40.60.81.01.21.4 ngEEC 0.00.20.40.60.81.01.21.4EEC FIG. 3. Left panel: spectrum of the reduced CPB Hamiltonian Eq. ( 75) around the operating point ng=1/2f o r EJ=0.1EC. Dashed line: energy of the odd state Eq. ( 77). Solid lines: energies of ground (bottom) and excited (top) even states Eq. ( 78). Right panel: in the presence of a small inductive energy EL, the CPB bands act as potentials in the quasimomentum space (see Ref. 14). Dense horizontal lines represent a few energy levels near the edges of the bands.Despite these differences, the ac responses of the two systems due to charge coupling agree in this limit.14Similarly, we now show agreement for the quasiparticle transition rates. We notethat, when taking the limit E L→0, the condition EJ/lessorsimilarωLC for the validity of Eq. ( 59) for the rate /Gamma1n→0requires that we also take EJ→0.23Moreover, since the final state considered in deriving the rate /Gamma1n→0is the lowest possible state, the corresponding final state in the CPB is either the even groundstate at n g=0 or the odd ground state at ng=1/2. Indeed, the width of the ground state [in quasimomentum space (see Fig. 3 and Ref. 14)i s∝(EL/EC)1/4, so that as EL→0, the state is localized at the bottom of the band. Note that, followingthe same procedure detailed above, it is straightforward toshow that the transition rate /Gamma1 o,+→e,0atng=0 coincides with/Gamma1e,+→o,0atng=1/2; hence, for our purposes, the two possibilities are equivalent. At finite EL, the total transition rate to the ground state is obtained by summing Eq. ( 59) over all initial levels n. Due to the Gaussian factor in the second line of Eq. ( 59), the number of levels that contribute to the total rate is approximately√n0∝(EC/EL)1/4, which grows as the inductive energy diminishes. However, the energy ofthe contributing levels tends to the charging energy, as can beseen by rewriting identically the argument in the exponentialof the Gaussian factor as −(E n−EC)2/2ECωLC, where En= nωLC; this agrees with frequency for the e,+→o,0 transition atng=1/2 in the CPB being approximately ECin the small EJlimit. Using Eq. ( 59), performing the sum over levels, and taking the limit EL→0, we find lim EL→0/summationdisplay n/Gamma1n→0=4EJ πxqp/radicalBigg 2/Delta1 EC, (86) which coincides with the leading term of Eq. ( 84) in the limit EJ→0 at the operating point ng=1/2. B. Flux qubit As a second example of a strongly anharmonic system, we consider here a flux qubit, i.e., in Eq. ( 2) we assume EJ>E Land take the external flux to be close to half the flux quantum /Phi1e≈/Phi10/2. Then, the potential has a double-well shape and the flux qubit ground state |−/angbracketright and excited state |+/angbracketrightare the lowest tunnel-split eigenstates in this potential8(see Fig. 4). The nonlinear nature of the sin ˆ ϕ/2 qubit-quasiparticle coupling in Eq. ( 6) has a striking effect on the transition rate /Gamma1+→− , which vanishes at /Phi1e=/Phi10/2 due to destructive interference: for flux biased at half the fluxquantum, the qubit states |−/angbracketright,|+/angbracketrightare, respectively, symmetric and antisymmetric around ϕ=π, while the potential in Eq. ( 2) and the function sin ϕ/2i nE q .( 32) are symmetric. Note that the latter symmetry and its consequences are absent in theenvironmental approach in which a linear phase-quasiparticlecoupling is assumed. Analytic evaluation of the matrix element determining the transition rate Eq. ( 32) at finite /Phi1 e−/Phi10/2 is possible when EC/lessmuchEJand the tunnel splitting ¯ /epsilon1is small compared to inductive and plasma energies ¯ /epsilon1/lessmuch2π2EL/lessmuchωp; an estimate for the splitting is given below in Eq. ( 100). With the above assumptions, we can use a tight-binding approach. Neglectingtunneling, the wave functions |m/angbracketrightare, as a first approximation, 064517-9CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) FIG. 4. (Color online) Potential energy (in units of EL) for a flux qubit biased at /Phi1e=/Phi10/2 with EJ/EL=10. The horizontal lines represent the two lowest energy levels, with energy difference ¯ /epsilon1given in Eq. ( 100). ground-state wave functions of the harmonic oscillator with frequency ωpand oscillator length /lscript=/radicalbig8EC/ωplocalized around the (flux-dependent) minima ϕmof the potential energy /angbracketleftϕ|m/angbracketright=/parenleftbigg1 π/lscript2/parenrightbigg1/4 e−(ϕ−ϕm)2/2/lscript2. (87) The minima are found by solving Eq. ( 49) approximately, using the condition EL/lessmuchEJ(which follows from the above assumptions) to get ϕm/similarequal2π/bracketleftbigg m−EL EJ(m−f)/bracketrightbigg ,f=/Phi1e /Phi10. (88) The energies of the localized states are (up to a constant term) Em=2π2¯EL(m−f)2, (89) where ¯EL=EL/parenleftbigg 1−1 β/parenrightbigg ,β=EJ EL(90) takes into account corrections small in 1 /β/lessmuch1. The above results are valid for |mEL/EJ|/lessmuch 1. Still neglecting tunneling, the matrix element of sin ˆ ϕ/2 between states localized in different wells vanishes, but the diagonal matrix element isfinite due to the shift of the minima away from 2 πm [see Eq. ( 88)]. Using the states in Eq. ( 87), we obtain /angbracketleftj|sinˆϕ 2|m/angbracketright/similarequal− (−1)mπEL EJ(m−f)δm,j. (91) To include the effect of tunneling, we allow for the possibility of transitions between neighboring wells with amplitude ¯ /epsilon1/2. As we are interested in the two lowest eigenstates for fnear 1/2, we consider only the m=0,1 wells and the effective Hamiltonian has the form ˆH=/parenleftbigg 2π2¯ELf2−¯/epsilon1/2 −¯/epsilon1/22 π2¯EL(1−f)2/parenrightbigg . (92) The eigenenergies are [cf. Eqs. ( 78)–(81)] E±(f)=π2 2¯EL[1+(2f−1)2]±1 2ω10(f) (93)with the flux-dependent qubit frequency ω10(f)=/radicalBig ¯/epsilon12+[(2π)2¯EL(f−1/2)]2, (94) while the eigenstates are |−/angbracketright = cosθ|0/angbracketright+sinθ|1/angbracketright, (95) |+/angbracketright = sinθ|0/angbracketright−cosθ|1/angbracketright, with cosθ=1√ 2/radicalBigg 1−(2π)2¯EL(f−1/2) ω10(f). (96) The tunnel splitting ¯ /epsilon1entering in the above formulas can be estimated by noting that, due to the assumption β/greatermuch1, the wells are nearly symmetric. Neglecting the asymmetry [i.e.,considering the potential in Eq. ( 2)a tf=1/2], the width and height of the tunnel barrier are approximately 2 π(1−1/β) and 2 ¯E J, respectively, with ¯EJ=EJ/bracketleftbigg 1−π2 41 β/parenleftbigg 1−1 β/parenrightbigg/bracketrightbigg . (97) To account for the height and width at EL/negationslash=0, we treat the two wells as cosine potentials with renormalized coefficients. Thatis, we consider each well to be described by the Hamiltoniangiven in Eq. ( B3) with the substitutions E J→¯EJandEC→ ¯EC, where ¯EC=EC1 (1−1/β)2. (98) Then, we can use the known asymptotic formula2,14,24for the splitting /epsilon10in the periodic cosine potential (i.e., for EL=0; see Appendix Bfor a derivation of this formula) /epsilon10=4/radicalbigg 2 πωp/parenleftbigg8EJ EC/parenrightbigg1/4 e−√8EJ/EC(99) to find ¯/epsilon1=2/radicalbigg 2 π/radicalbig 8¯EJ¯EC/parenleftbigg8¯EJ ¯EC/parenrightbigg1/4 e−√ 8¯EJ/¯EC. (100) Here, the numerical prefactor is smaller by factor of 2 in comparison with Eq. ( 99) to account for tunneling being between two wells rather than in a periodic potential.24 Turning now to the matrix element /angbracketleftj|sin ˆϕ/2|m/angbracketright,t h e diagonal elements j=m=0,1 are still approximately given by Eq. ( 91). Tunneling introduces finite but exponentially small off-diagonal elements, which, similarly to the splitting,can be calculated using the semiclassical approximation. Usingthe wave functions derived in Appendix B,w ea r r i v ea t [cf. Eq. ( C6)] /angbracketleft1|sinˆϕ 2|0/angbracketright/similarequalD/parenleftbigg¯EJ ¯EC/parenrightbigg1/3¯/epsilon1 2√ 2¯EJ(101) 064517-10RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) withD≈1.45 [see Eq. ( C7)]. We can now calculate the matrix element of sin ˆ ϕ/2 between qubit states |±/angbracketrightin Eq. ( 95)u s i n g Eqs. ( 91) and ( 101) to obtain /angbracketleft−|sinˆϕ 2|+/angbracketright =π(f−1/2)¯/epsilon1 ω10(f)/bracketleftbiggEL EJ+√ 2πD¯EL ¯EJ/parenleftbigg¯EJ ¯EC/parenrightbigg1/3/bracketrightbigg . (102) Here, the first term in square brackets is the combination of the two intrawell contributions Eq. ( 91), while the second one originates from the under-barrier tunneling Eq. ( 101). Comparing Eq. ( 102) to numerical calculations, we find that near half the flux quantum |f−1/2|/lessorsimilar¯/epsilon1/2π2EL,t h et w o approaches give the same dependence on flux and agree onthe order of magnitude of the matrix element, with Eq. ( 102) providing a smaller estimate than the numerics by a factor ofabout 2 /3. For |f−1/2|/greaterorsimilar¯/epsilon1/(2π) 2EL, the flux dependence in Eq. ( 102) via the factor ( f−1/2)/ω10(f) can be neglected and the right-hand side reduces to a flux-independent constant.However, this behavior is an artifact of our approximations: forthese larger deviations of flux, from half the flux quantum thematrix element acquires additional flux dependence, beyondthat given in Eq. ( 102), once the asymmetry of the potential is taken into account. Moreover, for very small flux |f|/lessorsimilar (¯/epsilon1/4√ 2π2EL)2, mixing of the state localized in well m=1 with that localized in well m=− 1 can not be neglected and the matrix element has a narrow peak around zero flux.Substituting Eq. ( 102) into Eq. ( 32), keeping the leading contribution, and using the relation ( 40), we find for the transition rate in the high-frequency regime 25 /Gamma1+→−=ω10 π1 gKReYhf qp(ω10)/parenleftbigg¯/epsilon1 4π¯EJ/parenrightbigg2/parenleftbigg 1−¯/epsilon12 ω2 10/parenrightbigg ×(√ 2πD)2/parenleftbigg¯EJ ¯EC/parenrightbigg2/3 (103) with Re Yhf qpof Eq. ( 18). T h er a t ei nE q .( 103) depends on reduced flux fvia the qubit frequency [see Eq. ( 94)]. In particular, for external flux equaling half the flux quantum, we have ω10(1/2)=¯/epsilon1and the transition rate vanishes, as discussed above. In the previoussection, we mentioned in the text after Eq. ( 85) that, for the Cooper-pair box, the vanishing of the rate at the operating pointis valid up to small corrections, being a consequence of thelow-energy approximation for the tunneling Hamiltonian inEq. ( 6). The same is true for the flux qubit; in the present case, the parameter suppressing these corrections is exponentiallysmall, being given by ¯ /epsilon1/2/Delta1. Note that, if keeping in Eq. ( 5) the contributions beyond the low-energy approximation, theoperators accounting for the qubit-quasiparticle interactioncan not be reduced to sin ˆ ϕ/2; therefore, for these additional contributions, the symmetry argument given at the beginningof this section for the vanishing of the transition rate at f=1/2 does not hold. FIG. 5. Left: schematic representation of the split transmon with two (possibly different) junctions. Right: in the fluxonium,a weaker junction ( j=0) is connected to a large junction array (j=1,..., M ). V . MULTIPLE-JUNCTION QUBITS: GENERAL THEORY AND APPLICATIONS In this section, we generalize the theory of Sec. IIto the case of systems containing multiple junctions. This generalizationwill enable us to consider the flux dependence of the transitionrates in the two-junction split transmon and in the many-junction fluxonium. These two qubits are particular examplesof the general case in which M+1 junctions separate M+1 superconducting islands forming a loop. We use the conventionthat junction j=0,..., M is between islands jandj+1 and identify island j=M+1 with island j=0 (see Fig. 5). When the loop-inductive energy is much larger than theJosephson energies of the junctions (i.e., the loop inductance issmall), the phases are subject to the flux quantization constraint M/summationdisplay j=0ϕj=2π/Phi1e//Phi1 0. (104) This constraint must be taken into account to derive the Hamil- tonian ˆH{φ}of the M-independent phase degrees of freedom φ,φk(k=1,..., M −1) starting from the Lagrangian26L{ϕ} for the M+1 constrained phases ϕj: L{ϕ}=M/summationdisplay j=0/bracketleftbigg1 2Cj/parenleftbigg/Phi10 2π˙ϕj/parenrightbigg2 +EJjcosϕj/bracketrightbigg , (105) where the overdot denotes derivative with respect to time, Cjis the capacitance of junction j, andEJjits Josephson energy. In Appendix F, we derive the Hamiltonian assuming Mof theM+1 junctions to be identical, which is relevant for both the split transmon ( M=1) and the fluxonium ( M/greatermuch1). Explicit expressions for the Hamiltonian in these two cases arepresented below. The total Hamiltonian ˆHof the system consists of three terms, as in Eq. ( 1): ˆH=ˆH {φ}+ˆHqp+ˆHT. (106) In addition to ˆH{φ}discussed above, the second contribution is the quasiparticle Hamiltonian ˆHqp=M/summationdisplay j=0ˆHj qp, ˆHj qp=/summationdisplay n,σ/epsilon1j nˆαj† nσˆαj nσ. (107) 064517-11CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) Here, the index jdenotes the superconducting island; other symbols have the same meaning as in Eq. ( 3) and we assume equal gaps in all islands /Delta1j≡/Delta1. The final contribution to ˆHis the tunnel Hamiltonian, given by the following sum [cf. Eq. ( 6)]: ˆHT=M/summationdisplay j=0˜tj/summationdisplay n,m,σisinˆϕj 2ˆαj† nσˆαj+1 mσ+H.c. (108) The transition rate between qubit states can again be calculated using Fermi’s golden rule as in Eq. ( 31). We assume that the quasiparticle distribution functions are the same in allislands and that tunneling across each junction is not correlatedwith tunneling in nearby junctions; this is a good assumptionif the mean level spacing in the finite-size superconductors issmall compared to the gap. Then, the total rate for the transitionbetween eigenstates of Hamiltonian ˆH {φ}is /Gamma1i→f=M/summationdisplay j=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftf {φ}|sinˆϕj 2|i{φ}/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 EJj˜Sqp(ωif), (109) where for convenience we have extracted the Josephson energy prefactor from the spectral density ˜Sqp=Sqp/EJ, with Sqp defined in Eq. ( 33). Similarly, the correction δEito the energy of state i{φ}is given by sums over junctions, which generalize Eqs. ( 45) and ( 46), δEi=δEi,EJ+δEi,qp, (110) δEi,EJ=M/summationdisplay j=0EJj/angbracketlefti{φ}|cos ˆϕj|i{φ}/angbracketright/parenleftbig xqp+2xA qp/parenrightbig , (111) δEi,qp=M/summationdisplay j=0EJj/summationdisplay k/negationslash=i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk {φ}|sinˆϕj 2|i{φ}/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ˜Fqp(ωik),(112) where ˜Fqp=Fqp/EJ. In the next sections, we use Eq. ( 109) to calculate the transition rates for the split transmon and thefluxonium and Eq. ( 110) to find the frequency shift in the split transmon. The flux-dependent transition rate between the twolowest even and odd states of a split Cooper-pair box has beenrecently considered in Ref. 27for gate voltage tuned at the operating point. A. Split transmon A split transmon consists of two junctions j=0,1i na superconducting loop (see Fig. 5). Therefore, there is only M=1 degree of freedom, which we denote with φ, governed by the Hamiltonian ˆHφ=4ECˆN2−EJ0cos( ˆφ−2πf)−EJ1cosˆφ (113) (see Appendix F). Here, ˆN=−id/dφ ,f=/Phi1e//Phi1 0, and the charging energy ECis related to the junctions’ capacitances by EC=e2 2(C0+C1). (114) Note that the Hamiltonian is periodic in fwith period 1, so we can assume |f|/lessorequalslant1/2 without loss of generality (i.e., we can measure the normalized flux from the nearest integer). Aftershifting φ→φ+πf, the sum of the two Josephson terms can be rewritten as EJ0cos( ˆφ−2πf)+EJ1cosˆφ→EJ(f) cos( ˆφ−ϑ), (115) where the effective Josephson energy EJis modulated by the external flux EJ(f)=(EJ0+EJ1) cos(πf)/radicalBig 1+d2tan2(πf) (116) with d=EJ0−EJ1 EJ0+EJ1(117) and tanϑ=dtan(πf). (118) After a further shift φ→φ+ϑ,w ea r r i v ea t ˆHφ=4ECˆN2−EJ(f) cos ˆφ, (119) which has the same form of the Hamiltonian for the single- junction transmon [i.e., Eq. ( 2) with EL=0], but with a flux-dependent Josephson energy Eq. ( 116). Therefore, the spectrum follows directly from that of the single-junctiontransmon (see Fig. 2) and consists of nearly degenerate and well-separated states. The energy difference between well- separated states is approximately given by the flux-dependent frequency [cf. Eq. ( 56)] ω p(f)=/radicalbig 8ECEJ(f). (120) Note that, for the system to be in the transmon regime, EJ(f)/greatermuchEC (121) at some flux, a necessary condition is EJ0+EJ1/greatermuchEC. (122) Then, we can distinguish two cases. First, in the nearly symmetric case of junctions with comparable Josephsonenergies |E J0−EJ1|/lessorsimilarEC, the condition ( 121) is satisfied not too close to half the flux quantum, |f|−1/2/greatermuchEC/π(EJ0+EJ1). (123) On the other hand, if the Josephson energies are sufficiently different |EJ0−EJ1|/greatermuchEC, then Eq. ( 121) is satisfied at any flux. The transition rate /Gamma11→0between the qubit states |0/angbracketright,|1/angbracketright can be calculated using Eq. ( 109) if we know the relation between ϕjandφ; the same relation is also needed to calculate the transition rate /Gamma1o→ebetween nearly degenerate states (see Appendix C 2 for details). According to Appendix F,f o rt h e variable φin Eq. ( 113), we have ϕ1=φandϕ0=2πf−φ. Accounting for the two changes of variables performed toarrive at Eq. ( 119), we obtain ˆϕ 0=πf−ϑ−ˆφ, (124) ˆϕ1=πf+ϑ+ˆφ. 064517-12RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) In the transmon regime ( 121), we proceed as in the derivation of Eq. ( 53) to find /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleft0|sinˆϕ j 2|1/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =EC ωp(f)1+cos(πf±ϑ) 2, (125) where the upper (lower) sign is to be used for j=1(j=0). Substituting this result into Eq. ( 109) and using Eq. ( 41) with ω=ωp(f), we find in the high-frequency regime [cf. Eq. ( 55)] /Gamma11→0=xqp 2π/radicalBigg 2/Delta1 ωp(f)ω2 p(f)+ω2 p(0) ωp(f). (126) For the transition quality factor, we consider, as in Sec. III A , the coexistence of equilibrium and nonequilibrium quasiparti-cles [see Eq. ( 61)] to find 1 Q10=1 2π/parenleftBigg 1+ω2 p(0) ω2p(f)/parenrightBigg/bracketleftbigg xne/radicalBigg 2/Delta1 ωp(f) +4e−/Delta1/Tcosh/parenleftbiggωp(f) 2T/parenrightbigg K0/parenleftbiggωp(f) 2T/parenrightbigg/bracketrightbigg .(127) In Fig. 6, we show with solid lines the quality factor as a function of temperature for four different values of flux fin a symmetric transmon ( d=0). As we discussed in Sec. III A , an extrinsic relaxation mechanism could be limiting the low-temperature quality factor. Characterizing this mechanism by aconstant quality factor Q extand assuming that only equilibrium quasiparticles are present, the transition quality factor has theform 1 Q10,tot=1 Qext+2 π/parenleftBigg 1+ω2 p(0) ω2p(f)/parenrightBigg ×e−/Delta1/Tcosh/parenleftbiggωp(f) 2T/parenrightbigg K0/parenleftbiggωp(f) 2T/parenrightbigg .(128) 0.0 0.2 0.4 0.6 0.8 1.0 1.20500 0001.0 1061.5 1062.0 1062.5 106 2Tωp0Q FIG. 6. (Color online) Quality factor as a function of 2 T/ω p(0) in a symmetric split transmon. Solid lines are obtained from Eq. ( 127) using a small nonequilibrium quasiparticle density xne=3.8×10−7 and a gap value such that /Delta1/ω p(0)=6.9 [these parameters are taken from experiments on single-junction transmon (Ref. 20)]. Flux increases from top to bottom; we show curves for f=0, 0.2, 0.3, and 0.4, respectively. We plot Eq. ( 128) with dashed lines for the same values of flux. The extrinsic quality factor is chosen so that solid and dashed lines match at f=0.The dashed lines in Fig. 6show Q10,totas a function of temperature for the same values of flux; the quality factorQ extis chosen so that the zero-flux curve coincides with the zero-flux curve described by Eq. ( 127). The change of quality factor with flux is markedly different in the two limitingcases (namely, presence of nonequilibrium quasiparticles andno extrinsic relaxation mechanism versus extrinsic relaxationwith no nonequilibrium quasiparticles) described by Eqs. ( 127) and ( 128). Therefore, the measurement of the temperature and flux dependencies of the quality factor should give indicationson the presence of nonequilibrium quasiparticles. For example,the low-temperature measurements reported in Ref. 28are compatible with a flux-independent quality factor; to explainthe data with Eq. ( 127) rather than Eq. ( 128), one would need to assume a quasiparticle density that decreases with increasingflux. Since magnetic fields are known to break pairs and thusincrease the quasiparticle density, for the transmons consideredin Ref. 28, it is unlikely that nonequilibrium quasiparticles are the source of the low-temperature qubit decay. The frequency shift for the split transmon is obtained, as in Sec. III B , by calculating the difference between correction to energies of nearby levels Eq. ( 67). The matrix elements appearing in Eqs. ( 111) and ( 112) are given by Eqs. ( 53) and (69) with ω 10=ωp(f),ϕ0=θ+πfforj=1, and ϕ0= θ−πf forj=0 [cf. Eqs. ( 124) and ( 125)]. Using those expressions, we find δEi+1,EJ−δEi,EJ=−1 2ωp(f)/parenleftbig xqp+2xA qp/parenrightbig (129) and δEi+1,qp−δEi,qp =1 16ω2 p(0)+ω2 p(f) ωp(f){˜Fqp[ωp(f)]+˜Fqp[−ωp(f)]}. (130) Then, using the relation ( 48) and Eq. ( 29), we arrive at the high-frequency result δω(f) ωp(f)=1 2/braceleftBigg xA qp/parenleftBigg ω2 p(0) ω2p(f)−1/parenrightBigg −xqp/bracketleftBigg 1 2π/parenleftBigg ω2 p(0) ω2p(f)+1/parenrightBigg/radicalBigg 2/Delta1 ωp(f)+1/bracketrightBigg/bracerightBigg . (131) At zero flux, this expression agrees with Eq. ( 73) applied to a single-junction transmon ( ω10=ωp,ϕ0=0). However, similarly to the flux qubit, at finite flux, the split transmonfrequency shift is sensitive to the occupation of the Andreevbound states (see the first term in curly brackets). B. Fluxonium In the fluxonium, an array of many identical junctions (M/greatermuch1) of Josephson energy EJ1/greatermuchEC1is connected to a weaker junction with EJ0<E J1. Then, the Hamiltonian ˆH{φ}for the M-independent degrees of freedom can be 064517-13CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) approximately separated into independent terms for the qubit phase φand the M−1 phases φk: ˆH{φ}=ˆHφ+M−1/summationdisplay k=1ˆHk, ˆHφ=4ECˆN2−EJ0cosˆφ+1 2EL(ˆφ−2πf)2,(132) ˆHk=4EC1ˆN2 k+1 2EJ1ˆφ2 k, where EL=EJ1 M,1 EC=1 EC0+1 MEC1(133) (see Appendix F). There, we also give Eq. ( F5) the relation between the M+1 (constrained) ϕjvariables and the M- independent φkvariables. Accounting for the changes of variables that bring the Hamiltonian in the form given above,we have schematically ϕ 0=φ, ϕ j=Lj({φk})+(2πf−φ)/M, (134) j=1,..., M. Here, Lj({φk}) denote linear combinations of the variables φk,k=1,..., M −1, whose specific form can be found in Appendix Fbut is not needed here, while we show explicitly the dependence of the constrained variables ϕjon the qubit phaseφ. As in the previous section, we take |f|<1/2 without loss of generality. As an example of the calculation of the transition rate for such a system, we assume that the plasma frequencyω p1=√8EC1EJ1of the array junctions is larger than the other relevant energy scales (namely, quasiparticle energy δE and qubit frequency ω10). Then, we can take the many-body state of the system |/Psi1{φ}/angbracketrightin the product form |/Psi1{φ}/angbracketright=|ψφ/angbracketrightM−1/productdisplay k=1|0k/angbracketright, (135) where |ψφ/angbracketrightis a low-energy eigenstate of ˆHφand|0k/angbracketrightis the ground-state wave function of the kth oscillator. The approximations used to derive ˆH{φ}in Eq. ( 132) imply that, in the formula ( 109) for the transition rate, we can linearize the sine for j=1,..., M . Therefore, for the transition rate between two states of the form ( 135), we obtain /Gamma1i→f=˜Sqp(ωif)/bracketleftbigg EJ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftf φ|sinˆφ 2|iφ/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +EL/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftf φ|ˆφ 2|iφ/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracketrightbigg . (136) In the weak tunneling limit ¯ /epsilon1/lessmuch2π2EL/lessmuchωp=√8ECEJ0 (with ¯/epsilon1the tunnel splitting of the qubit states at f=1/2), we can use directly the results of Sec. IV B : the flux-dependent qubit frequency ω10(f)i sg i v e nb yE q .( 94) and the first excited state|iφ/angbracketright=| + /angbracketright and ground state |fφ/angbracketright=| − /angbracketright are the linear combination of states localized in wells m=0 ,1i nE q .( 95).For the first term in square brackets in Eq. ( 136), the matrix element is given by Eq. ( 102). To evaluate the second term in the same regime, we note that for states |m/angbracketright,|j/angbracketright, that is, states localized in wells mandjas in Eq. ( 87), we have /angbracketleftj|ˆφ 2|m/angbracketright=π/bracketleftbigg m/parenleftbigg 1−EL EJ0/parenrightbigg +EL EJ0f/bracketrightbigg δj,m. (137) Therefore, for the states in Eq. ( 95), we find /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleft−|ˆφ 2|+/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =/parenleftbiggπ 2/parenrightbigg2/parenleftbigg 1−EL EJ0/parenrightbigg2¯/epsilon12 ω2 10(f). (138) In contrast to the matrix element of sin ˆ ϕ/2 considered in Sec. IV B , the contribution due to tunneling can be neglected in this case. Substituting this result and the leading term fromEq. ( 102) into the square brackets of Eq. ( 136), we get 29 π2 4¯/epsilon12 ω2 10(f)EL/bracketleftbigg (2π)2EL EJ0/parenleftbigg f−1 2/parenrightbigg2 (√ 2D)2/parenleftbiggEJ0 EC/parenrightbigg2/3 +/parenleftbigg 1−EL EJ0/parenrightbigg2/bracketrightbigg . In this expression, the first term in square brackets originates from the weak junction and the second one from the array.Note that, when considering flux near half the flux quantum,we can neglect the first term in comparison to the second andthe losses due to the array dominate over those due to the weakjunction. Keeping only the leading contribution in Eq. ( 139) and using Eq. ( 41)i nE q .( 136), we arrive at the expression for the rate in the high-frequency regime /Gamma1 +→−=xqp/radicalBigg 2/Delta1 ω10(f)2πELω2 10(1/2) ω2 10(f)(139) withω10(f) defined in Eq. ( 94). Note that since the frequency increases as the reduced flux fmoves away from 1 /2, the tran- sition rate is the largest at half the flux quantum. Interestingly,due to the factor E L, the rate scales as 1 /M[see Eq. ( 133)] and thus can be made small in an array with M/greatermuch1 junctions. The decrease of the rate as the number of junctions increase is due tothe suppression of the phase fluctuations in the array junctions[see Eq. ( 134)]. VI. SUMMARY In this paper, we have presented in detail a general approach to study the effects of quasiparticles on relaxationand frequency of superconducting qubits. The theory isapplicable to any qubit; the case of single-junction systemsis considered in Sec. IIand the generalization to multijunction systems is given in Sec. V. Our analysis is valid for both thermal equilibrium quasiparticles and arbitrary nonequilib-rium distributions, so long as the quasiparticle energy issmall compared to the qubit frequency; this condition, notnecessary in thermal equilibrium, ensures that quasiparti-cles primarily cause relaxation and not excitation of thequbit. For single-junction qubits, we have studied in Sec. IIIthe weakly anharmonic limit. For small phase fluctuations, bothquality factor (Sec. III A ) and frequency shift (Sec. III B ) 064517-14RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) are determined by transitions between neighboring qubit levels and can be related to real and imaginary parts ofthe “classical” junction admittance, respectively. The smallfluctuation case applies to phase and transmon qubits andour results in Eqs. ( 62) and ( 73) have been successfully tested in recent experiments 20,21with these qubits. For strong anharmonicity, we have presented in Sec. IVresults for the quasiparticle transition rate in the Cooper-pair box and the fluxqubit. We have considered two examples of multijunction qubits, the two-junction split transmon in Sec. VA and the many- junction fluxonium in Sec. VB. In particular, we argue that measuring the temperature and flux dependencies of the qualityfactor of a split transmon could help resolve the questionof whether nonequilibrium quasiparticles are present at lowtemperatures [see Eqs. ( 127) and ( 128) and Fig. 6]. ACKNOWLEDGMENTS We thank L. Frunzio, A. Kamal, and J. Koch for stim- ulating discussions and help with numerical calculations.This research was funded by DOE (Contract No. DE-FG02-08ER46482), by Yale University, and by the Office ofthe Director of National Intelligence (ODNI), IntelligenceAdvanced Research Projects Activity (IARPA), through theArmy Research Office (Contract No. W911NF-09-1-0369). APPENDIX A: CORRECTION TO ENERGY LEVELS To calculate the correction to the energy levels as presented in Sec. II C, we must account for both quasiparticle and pair tunneling. Note that, due to energy conservation, the latterdoes not affect the transition rate /Gamma1 i→fbetween states |i/angbracketrightand |f/angbracketrightso long as ωif<2/Delta1; for this reason, the pair tunneling Hamiltonian Hp Twas neglected in Eq. ( 1). More generally, the total Hamiltonian of the single-junction system is ˆHtot=ˆH0+ˆHT+ˆHp T+ˆHEJ (A1) with ˆH0=ˆHϕ+ˆHqp. (A2) The Hamiltonians ˆHϕ,ˆHqp, and ˆHTare defined in Eqs. ( 2), (3), and ( 5), respectively, and the pair-tunneling term is ˆHp T=˜t/summationdisplay n,m/bracketleftbig/parenleftbig eiˆϕ 2uL nvR m+e−iˆϕ 2uR mvL n/parenrightbig ˆαL† n↑ˆαR† m↓ +/parenleftbig e−iˆϕ 2vR muLn+eiˆϕ 2vL nuRm/parenrightbig ˆαR m↓ˆαL n↑/bracketrightbig +(L↔R). (A3) The last term in Eq. ( A1), ˆHEJ=EJcos ˆϕ, (A4) is necessary to avoid “double counting”: the Josephson energy originates from pair tunneling, so its inclusion inthe effective Hamiltonian ˆH ϕ[Eq. ( 2)] must be compen- sated for by subtracting the same term here. We will showbelow that this treatment is justified for small quasiparticledensity.In both the quasiparticle tunneling Hamiltonian ˆH T [Eq. ( 5)] and the pair-tunneling one in Eq. ( A3), using the definitions given after Eq. ( 3), the (real) Bogoliubov amplitudes are /parenleftbig uj n/parenrightbig2=1−/parenleftbig vj n/parenrightbig2=1 2/parenleftBigg 1+ξj n /epsilon1j n/parenrightBigg ,j=L,R. (A5) As in the main text, we assume equal gaps and distribution functions in the leads /Delta1L=/Delta1R≡/Delta1andfL=fR≡f. Moreover, we neglect the contributions of the charge modef Q(/epsilon1)=[f(ξ)−f(−ξ)]/2 since they are suppressed by the small factor δE//Delta1 /lessmuch1 compared to the leading contributions due to the energy mode fE[Eq. ( 14)]; for simplicity, in this appendix we drop the subscript E. We want to evaluate the correction δEito the energy of leveliof the qubit at second order in the tunneling amplitude ˜tfor small quasiparticle density. Thus, ˆH0in Eq. ( A2) is the unperturbed Hamiltonian, and we distinguish threecontributions to δE i, δEi=δE(1) i+δE(2) i+δE(3) i, (A6) caused, respectively by ˆHT,ˆHp T, and ˆHEJ. Noting that the latter is already of second order in ˜t, we treat it within first-order perturbation theory to write δE(3) i=EJ/angbracketlefti|cos ˆϕ|i/angbracketright. (A7) The quasiparticle tunneling correction δE(1) iis obtained by second-order perturbation theory δE(1) i=−/summationdisplay k,{λ}qp/angbracketleftBigg/angbracketleftBigg |/angbracketleftk,{λ}qp|ˆHT|i,{η}qp/angbracketright|2 Eλ,qp−Eη,qp−ωik/angbracketrightBigg/angbracketrightBigg qp,(A8) where ωik=Ek−Ei (A9) and the notation is the same as in Sec. II:{η}qpand{λ}qpdenote quasiparticle states, Eλ,qpandEη,qptheir energies, and /angbracketleft/angbracketleft.../angbracketright/angbracketrightqp averaging over {η}qp. Performing the averaging, after lengthy but straightforward algebra, we arrive at δE(1) i=4EJ π2/Delta1P/summationdisplay k/integraldisplay∞ /Delta1qpd/epsilon1L/integraldisplay∞ /Delta1qpd/epsilon1R ×/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 A+(/epsilon1L,/epsilon1R)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|cosˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ×A−(/epsilon1L,/epsilon1R)/bracketrightbigg/bracketleftbiggf(/epsilon1L)[1−f(/epsilon1R)] /epsilon1L−/epsilon1R−ωik −[1−f(/epsilon1L)]f(/epsilon1R) /epsilon1L−/epsilon1R+ωik/bracketrightbigg , (A10) where we introduced the functions A±(/epsilon1L,/epsilon1R)=/epsilon1L/radicalBig /epsilon12 L−/Delta12qp/epsilon1R/radicalBig /epsilon12 R−/Delta12qp ±/Delta1qp/radicalBig /epsilon12 L−/Delta12qp/Delta1qp/radicalBig /epsilon12 R−/Delta12qp(A11) describing combinations of BCS densities of states. Both these functions and the lower integration limit depend on 064517-15CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) the self-consistent gap /Delta1qp; however, since the integrand in Eq. ( A10) is at least linear in the distribution function f,w e can neglect the gap suppression by quasiparticles [see Eq. ( 44)] and approximate /Delta1qp/similarequal/Delta1. We note that the combinations of distribution functions in the last term of Eq. ( A10) restrict to low energies only one of the energy integrals, while the other integral is logarithmicallydivergent. To isolate this divergence, we add and subtract theterm obtained by setting ω ik=0 in the denominator; more precisely, we define P1 /epsilon1L−/epsilon1R=1 2lim ω→0+1 /epsilon1L−ω−/epsilon1R+1 /epsilon1L+ω−/epsilon1R(A12) and separate in δE(1) i=δE(1),f i+δE(1),d i a finite term δE(1),f i=8EJ π2/Delta1P/summationdisplay k/negationslash=i/integraldisplay∞ /Delta1d/epsilon1L/integraldisplay∞ /Delta1d/epsilon1R ×/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 A+(/epsilon1L,/epsilon1R)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|cosˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ×A−(/epsilon1L,/epsilon1R)/bracketrightbigg f(/epsilon1L)[1−f(/epsilon1R)]/bracketleftbigg1 /epsilon1L−/epsilon1R−ωik −1 /epsilon1L−/epsilon1R/bracketrightbigg , (A13) from a divergent one δE(1),d i=4EJ π2/Delta1P/integraldisplay∞ /Delta1d/epsilon1L/integraldisplay∞ /Delta1d/epsilon1Rf(/epsilon1L)−f(/epsilon1R) /epsilon1L−/epsilon1R ×/bracketleftbigg/epsilon1L/radicalBig /epsilon12 L−/Delta12/epsilon1R/radicalBig /epsilon12 R−/Delta12 −/angbracketlefti|cos ˆϕ|i/angbracketright/Delta1/radicalBig /epsilon12 L−/Delta12/Delta1/radicalBig /epsilon12 R−/Delta12/bracketrightbigg .(A14) To obtain this expression, we used the identities /summationdisplay k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|cosˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =1 (A15) and /summationdisplay k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 −/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|cosˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =− /angbracketlefti|cos ˆϕ|i/angbracketright.(A16) Equation ( A13)f o rδE(1),f i can be further simplified using the relations /epsilon1L/epsilon1R/bracketleftbigg1 /epsilon1L−/epsilon1R−ωik−1 /epsilon1L−/epsilon1R/bracketrightbigg =/epsilon12 L/bracketleftbigg1 /epsilon1L−/epsilon1R−ωik−1 /epsilon1L−/epsilon1R/bracketrightbigg −/epsilon1Lωik /epsilon1L−/epsilon1R−ωik /similarequal/Delta12/braceleftbigg/bracketleftbigg1 /epsilon1L−/epsilon1R−ωik−1 /epsilon1L−/epsilon1R/bracketrightbigg −ωik//Delta1 /epsilon1L−/epsilon1R−ωik/bracerightbigg , (A17)where the approximation is valid because the distribution function restricts the integral over /epsilon1Lto low energies above the gap. As discussed in Sec. II, the matrix elements of operators e±iˆϕ/2describe the transfer of a single charge. For this reason, for a low-lying level i, the main contribution in the sum over states kcomes either from levels with energy difference ωik∼EC/lessmuch/Delta1(when ECis large compared to EL, EJ), or from nearby levels (for small EC). In both cases, we haveωik/lessmuch/Delta1since, at large energy differences, the matrix elements quickly decrease; this is evident, for example, in theexpressions for the matrix elements in Sec. III. Then, according to Eq. ( A17), the term proportional to A −is suppressed by the small parameter ωik//Delta1in comparison to the leading term in A+, and we can approximate δE(1),f i as δE(1),f i/similarequal16EJ π2/Delta1P/summationdisplay k/negationslash=i/integraldisplay∞ /Delta1d/epsilon1L/integraldisplay∞ /Delta1d/epsilon1R/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ×/Delta1/radicalBig /epsilon12 L−/Delta12/Delta1/radicalBig /epsilon12 R−/Delta12f(/epsilon1L)[1−f(/epsilon1R)] ×/bracketleftbigg1 /epsilon1L−/epsilon1R−ωik−1 /epsilon1L−/epsilon1R/bracketrightbigg . (A18) Defining the function Fqpby Fqp(ω)=16EJ π2/Delta1P/integraldisplay∞ /Delta1d/epsilon1L/integraldisplay∞ /Delta1d/epsilon1R/Delta1/radicalBig /epsilon12 L−/Delta12/Delta1/radicalBig /epsilon12 R−/Delta12 ×f(/epsilon1L)[1−f(/epsilon1R)]/bracketleftbigg1 /epsilon1L−/epsilon1R−ω−1 /epsilon1L−/epsilon1R/bracketrightbigg , (A19) we arrive at the expression for the quasiparticle correction to the energy δEi,qpgiven in Eq. ( 46). The treatment of the pair-correction term δE(2) iin Eq. ( A6) is similar to the above one for δE(1) i. The pair correction is found by calculating the matrix element of ˆHp T[Eq. ( A3)] rather than ˆHTin Eq. ( A8); we find δE(2) i=4EJ π2/Delta1P/summationdisplay k/integraldisplay∞ /Delta1qpd/epsilon1L/integraldisplay∞ /Delta1qpd/epsilon1R/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|sinˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ×A−(/epsilon1L,/epsilon1R)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftk|cosˆϕ 2|i/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 A+(/epsilon1L,/epsilon1R)/bracketrightbigg ×/bracketleftbiggf(/epsilon1L)f(/epsilon1R) /epsilon1L+/epsilon1R−ωik−[1−f(/epsilon1L)][1−f(/epsilon1R)] /epsilon1L+/epsilon1R+ωik/bracketrightbigg . (A20) Note that, in this expression there is a term independent of the distribution function for which the approximation /Delta1qp/similarequal/Delta1is not applicable. Since /epsilon1L+/epsilon1R/greaterorequalslant2/Delta1qp, repeating the argument preceding Eq. ( A18), we can neglect ωikin the denominator 064517-16RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) and use identities ( A15) and ( A16) to obtain δE(2) i/similarequal4EJ π2/Delta1P/integraldisplay∞ /Delta1qpd/epsilon1L/integraldisplay∞ /Delta1qpd/epsilon1Rf(/epsilon1L)+f(/epsilon1R)−1 /epsilon1L+/epsilon1R ×⎡ ⎣/epsilon1L/radicalBig /epsilon12 L−/Delta12qp/epsilon1R/radicalBig /epsilon12 R−/Delta12qp +/angbracketlefti|cos ˆϕ|i/angbracketright/Delta1qp/radicalBig /epsilon12 L−/Delta12qp/Delta1qp/radicalBig /epsilon12 R−/Delta12qp⎤ ⎦.(A21) Both in this expression and in Eq. ( A14), the first term in square brackets does not depend on the level index i. Therefore, it leads to an unimportant common shift of all the levels, whichwe neglect. 30Keeping only the second term in each square bracket, we write δE(1),d i+δE(2) i≈δE/Delta1 i+δEA i, (A22) where, separating the terms independent of and proportional to the distribution function f,w eh a v e δE/Delta1 i=−4EJ π2/Delta1/angbracketlefti|cos ˆϕ|i/angbracketrightP/integraldisplay∞ /Delta1qpd/epsilon1L/integraldisplay∞ /Delta1qpd/epsilon1R ×/Delta1qp/radicalBig /epsilon12 L−/Delta12qp/Delta1qp/radicalBig /epsilon12 R−/Delta12qp1 /epsilon1L+/epsilon1R(A23) and δEA i=8EJ π2/Delta1/angbracketlefti|cos ˆϕ|i/angbracketrightP/integraldisplay∞ /Delta1d/epsilon1L/Delta1/radicalBig /epsilon12 L−/Delta12f(/epsilon1L) ×/integraldisplay∞ /Delta1d/epsilon1R/Delta1/radicalBig /epsilon12 R−/Delta12/bracketleftbigg1 /epsilon1L+/epsilon1R−1 /epsilon1L−/epsilon1R/bracketrightbigg .(A24) In both expressions, the integrations can be performed analyt- ically [using in Eq. ( A24) the definition ( A12)]. We obtain δE/Delta1 i=−EJ /Delta1/Delta1qp/angbracketlefti|cos ˆϕ|i/angbracketright (A25) and δEA i=2EJf(/Delta1)/angbracketlefti|cos ˆϕ|i/angbracketright. (A26) Finally, using Eqs. ( 23), (44), and ( A7), we arrive at δE/Delta1 i+δEA i+δE(3) i=EJ/parenleftbig xqp+2xA qp/parenrightbig /angbracketlefti|cos ˆϕ|i/angbracketright,(A27) which is the correction δEi,EJin Eq. ( 45). This result, together with Eqs. ( A18) and ( A19), concludes the derivation of the formulas presented in Sec. II C. APPENDIX B: GATE-DEPENDENT ENERGY SPLITTING IN THE TRANSMON The transmon low-energy spectrum is characterized by well-separated [by the plasma frequency ωp,E q .( 56)] and nearly degenerate levels, the energies of which, as shown inFig. 2, vary periodically with the gate voltage n g. Here, we derive the asymptotic expression (valid at large EJ/EC)f o r the energy splitting between the nearly degenerate levels. Weconsider first the two lowest-energy states and then generalize the result to higher energies. Using the notation of Sec. II, the transmon Hamiltonian is ˆHϕ=4EC/parenleftbigˆN−ng/parenrightbig2−EJ(1+cos ˆϕ). (B1) Its eigenstates can be written exactly in terms of Math- ieu functions.2However, since EJ/greatermuchEC, a tight-binding approach31can be used in which the two lowest (even and odd) eigenstates /Psi1eand/Psi1oare given by sums of localized wave functions /Psi1e(ϕ;ng)=eingϕ1√ L/summationdisplay jψ(ϕ−2πj)e−ing2πj, (B2) /Psi1o(ϕ;ng)=eingϕ1√ L/summationdisplay jψ(ϕ−2πj)e−ing2πje−iπj, where L/greatermuch1 is the number of sites, labeled with index j, and ψis the ground state of the Hamiltonian ˆH=4ECˆN2+V(ˆϕ)( B 3 ) with V(ϕ)=/braceleftbigg−EJ(1+cosϕ),|ϕ|<π 0, |ϕ|>π .(B4) This potential is such that/summationtext jV(ϕ−2πj)=−EJ(1+ cosϕ). Note that the even (odd) state is a linear combination of even (odd) charge eigenstates, as can be shown by consideringthe overlap of /Psi1 e(o)with the charge eigenstate einϕ/ 2for arbitrary integer n[in Eq. ( B2), what distinguishes the odd state from the even one is the last exponential in the expressionfor/Psi1 o, which changes the sign of the localized wave function at odd sites j]. The energy difference ωeobetween the two states is ωeo=/angbracketleft/Psi1o|ˆHϕ|/Psi1o/angbracketright−/angbracketleft/Psi1e|ˆHϕ|/Psi1e/angbracketright. (B5) Using Eq. ( B2), the contributions to ωeodue to products of wave functions ψlocalized at the same site cancel. The leading contribution to ωeooriginates from products of wave functions localized at nearby sites, ωeo=/epsilon10cos(2πng)( B 6 ) with /epsilon10=− 4/integraldisplay dϕψ (ϕ)ψ(ϕ−2π)V(ϕ). (B7) To estimate the above integral, the behavior of the wave function ψnearϕ=πis needed; in this region, a good approximation is given by the semiclassical wave function32 ψ(ϕ)/similarequal⎧ ⎨ ⎩C0 2√p(ϕ)exp/bracketleftbig −/integraltextϕ adφp (φ)/bracketrightbig , a<ϕ<π A0exp/bracketleftbig −/radicalBig EJ 2EC/parenleftbig 1−ωp 4EC/parenrightbig (ϕ−π)/bracketrightbig ,ϕ > π (B8) where C0andA0are constants, p(ϕ)=/radicalBigg EJ 4EC/radicalbigg 1−ωp 2EJ−cosϕ, (B9) 064517-17CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) andais the classical turning point defined by p(a)=0. The constant C0is determined by the normalization condition of the wave function, and A0then follows from continuity of the wave function. For states with large quantum number, thesemiclassical approximation can be used also in the classicallyaccessible region |ϕ|<a; the corresponding estimate for the normalization constant, which we indicate with C ∞,i s C∞=/radicalbigωp/4ECπ(see Ref. 32). Here, we are interested in the ground state (and more generally in low-lying states)for which C ∞is known to underestimate the normalization factor.24,33To evaluate C0, we note that for |ϕ|/lessmuchπ,t h e potential V(ϕ)i nE q .( B4) is well approximated by that of the harmonic oscillator; therefore, the semiclassical wavefunction ( B8) should match the normalized wave function of the harmonic oscillator given in Eq. ( 87) (with ϕ m=0) in the region a/lessmuchϕ/lessmuchπ. Indeed, in this region, we expand the cosine in Eq. ( B9) and rescale variables ( φ=˜φ/radicalbigωp/EJ) to find /integraldisplayϕ adφp (φ)/similarequal/integraldisplay˜ϕ 1d˜φ/radicalBig ˜φ2−1 =1 2[˜ϕ/radicalbig ˜ϕ2−1−ln( ˜ϕ+/radicalbig ˜ϕ2−1)] /similarequal1 2EJ ωpϕ2−1 4−1 2ln(2ϕ/radicalbig EJ/ωp). (B10) Using this expression, and p(ϕ)/similarequalϕ√EJ/8ECin the denom- inator, Eq. ( B8) becomes ψ(ϕ)/similarequalC0e1/4 √ 2/parenleftbigg8EC EJ/parenrightbigg1/8 e−ϕ2EJ/2ωp. (B11) This function matches Eq. ( 87) by setting C0=/radicalbiggωp 4EC(πe)−1/4=C∞/parenleftbiggπ e/parenrightbigg1/4 . (B12) The last form shows that the correct normalization factor is larger than the usual semiclassical estimate. Having found the normalization constant, we now consider the wave function in the region near ϕ=π. There, we can further simplify Eq. ( B8) as follows: We rewrite the integral in the exponential in the first line of Eq. ( B8)a s /integraldisplayϕ adφp (φ)=/integraldisplayπ adφp (φ)−/integraldisplayπ ϕdφp (φ). (B13) Then, the first integral on the right-hand side is /integraldisplayπ adφp (φ)=/radicalBigg 2EJ EC[E(k)−(1−k2)K(k)],(B14) where EandKdenote the complete elliptic integrals with modulus k, which has the value k2≡1−k/prime2=1−ωp 4EJ. (B15) Here, we are interested in the limit k→1 in which the complete elliptic integrals behave as E(k)/similarequal1+1 2k/prime2/parenleftbigg ln4 k/prime−1 2/parenrightbigg , (B16) K(k)/similarequalln4 k/prime.The last integral in Eq. ( B13) can be approximated as /integraldisplayπ ϕdφp (φ) /similarequal/radicalBigg EJ 2EC/bracketleftbigg/radicalbigg 1−ωp 4EJ(π−ϕ)−(π−ϕ)3 24/radicalbig1−ωp/4EJ/bracketrightbigg . (B17) Substituting Eqs. ( B13)–(B17) into Eq. ( B8), using p(π)/similarequal√EJ/2ECin the square root in the denominator of the first line, and requiring continuity of the wave function, we arriveat ψ(ϕ)=⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩A 0exp/braceleftBig −/radicalBig EJ 2EC/bracketleftBig/radicalBig 1−ωp 4EJ(ϕ−π) −(ϕ−π)3 24√ 1−ωp/4EJ/bracketrightBig/bracerightBig ,ϕ /lessorsimilarπ A0exp/bracketleftBig −/radicalBig EJ 2EC/radicalBig 1−ωp 4EJ(ϕ−π)/bracketrightBig ,ϕ > π (B18) with A0=1 (2π)1/4/parenleftbigg8EJ EC/parenrightbigg1/8 e−√2EJ/EC. (B19) The wave function near ϕ=−πcan be obtained by substi- tuting ϕ→−ϕin Eq. ( B18). We can now proceed with the calculation of the integral in Eq. ( B7). Using Eqs. ( B4) and (B18), expanding the potential for ϕ/lessorequalslantπ, and changing the integration variable ( ϕ→π−ϕ), we find /epsilon10/similarequal2EJA2 0/integraldisplay 0dϕϕ2exp/bracketleftBigg −/radicalBigg EJ 2ECϕ3 24/radicalbig1−ωp/4EJ/bracketrightBigg /similarequal8ωpA2 0=4/radicalbigg 2 πωp/parenleftbigg8EJ EC/parenrightbigg1/4 e−√8EJ/EC, (B20) where, going from the first to the second line, we neglect the subleading correction originating from the denominator in theargument of the exponential. The final expression for /epsilon1 0agrees with the known asymptotic formula,2,14,24thus validating our approach. The above result can be generalized to calculate the splitting between nearly degenerate even and odd states ofapproximate energy nω pabove the ground state by letting ωp→ωp(2n+1) in Eqs. ( B8) and ( B9) and those that follow [this replacement is appropriate so long as ωp(n+ 1/2)/lessmuch2EJ]. Matching the semiclassical wave function to the excited eigenstates of the harmonic oscillator, we find thatthe normalization coefficient depends on n: C n=/radicalbiggωp 4EC/parenleftbigg2 πe/parenrightbigg1/4/parenleftbiggn+1/2 e/parenrightbiggn/2/parenleftbigg√n+1/2 n!/parenrightbigg1/2 . (B21) Note that Cnapproaches C∞asngrows. Repeating the above calculation, we find the energy splitting /epsilon1n=/epsilon10(−1)n22n n!/parenleftbigg8EJ EC/parenrightbiggn/2 , (B22) also in agreement with the expression in the literature. 064517-18RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) APPENDIX C: RATE OF PARITY SWITCHING INDUCED BY QUASIPARTICLES IN THE TRANSMON The spectrum of the transmon, as described in Appendix B, comprises both well-separated and nearly degenerate levels ofopposite parity (see also Fig. 2). The leading contribution to the transition rate between states of different parity separatedin energy by (approximately) the plasma frequency is givenby Eq. ( 55) with ϕ 0=0 and is independent of ng. Here, we consider the quasiparticle-induced transitions betweenthe nearly degenerate states /Psi1 eand/Psi1o. We first consider a single-junction transmon to show explicitly that the ratedepends on n gand is exponentially small. Next, we study the experimentally relevant case of a split transmon; its rateis qualitatively different, not displaying such exponentialsmallness. 1. Single-junction transmon According to Eq. ( 32), the quasiparticle transition rate /Gamma1o→e between states /Psi1oand/Psi1ecan be written as /Gamma1o→e=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleft/Psi1 e|sinˆϕ 2|/Psi1o/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Sqp(ωeo). (C1) This rate depends on the gate voltage ngvia the states in the matrix element as well as via their energy difference ωeo[see Eq. ( B6)]. For the matrix element, we use Eq. ( B2) to find /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleft/Psi1e|sinˆϕ 2|/Psi1o/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/similarequal|sin(2πn g)|s, (C2) where s=2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay dϕψ (ϕ)ψ(ϕ−2π)s i nϕ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (C3) The matrix element in Eq. ( C2) vanishes at half-integer values ofng, as in the case of the Cooper-pair box [see Eq. ( 83)]. In fact, the vanishing holds at arbitrary ratio EJ/EC, as can be shown using the symmetry properties of Mathieu functions.For example, at n g=0, 1/2 the two lowest eigenstates of the transmon Hamiltonian Eq. ( B1) can be written in the charge basis as34 |/Psi1e/angbracketright=∞/summationdisplay m=0A(0) 2m[|2m/angbracketright+|− 2m/angbracketright], (C4) |/Psi1o/angbracketright=∞/summationdisplay m=0A(1) 2m+1[|2m+1/angbracketright+|− (2m+1)/angbracketright] and |/Psi1e/angbracketright=∞/summationdisplay m=0A(1) 2m+1[|2m+2/angbracketright+|− 2m/angbracketright], (C5) |/Psi1o/angbracketright=∞/summationdisplay m=0A(0) 2m[|2m+1/angbracketright+|− 2m+1/angbracketright], respectively, where the coefficients A(0) 2m,A(1) 2m+1depend on the ratio EJ/EC.35Using the charge-basis representation of sin ˆϕ/2i nE q .( 82), it is easy to check the vanishing of its matrix element between the above states for both values of ng.In the transmon limit EJ/EC/greatermuch1 under consideration, the product of wave functions localized at the same sitedoes not contribute to the matrix element in Eq. ( C2): the intrawell integral vanishes because ψ 2(ϕ) is a symmetric function ( ψbeing the ground state of a symmetric potential), which is multiplied by the antisymmetric function sin ϕ/2; the vanishing of the intrawell term has, thus, the same origin ofthe vanishing of the matrix element for a weakly anharmonicqubit at zero phase bias [see Eq. ( 53) with n=mandϕ 0=0]. To estimate the interwell contribution sin Eq. ( C3), we use Eq. ( B18) and that near ϕ=πwe have sin ϕ/2/similarequal1. After changing integration variable ( ϕ→π−ϕ), we arrive at s/similarequal4A2 0/integraldisplay 0dϕexp/bracketleftbigg −/radicalBigg EJ 2ECϕ3 24/radicalbig1−ωp/4EJ/bracketrightbigg /similarequalD/parenleftbiggEC EJ/parenrightbigg1/6/epsilon10 ωp, (C6) where (with /Gamma1denoting here the gamma function) D=21/63−2/3/Gamma1/parenleftbig1 3/parenrightbig ≈1.45. (C7) Due to the factor /epsilon10in Eq. ( C6), the transition rate in Eq. ( C1) is indeed exponentially small. Turning now to the factorS qpin Eq. ( C1), we note that its argument ωeois usually small due to its exponential suppression at large EJ/EC[see Eq. ( B20)]. Therefore, the “high-frequency” condition ωeo/greatermuch δE(with δEthe characteristic quasiparticle energy) is in general not satisfied and use of Eq. ( 41) expressing Sqpin terms of the quasiparticle density is not appropriate. In thermalequilibrium, one can use Eq. ( 35) for arbitrary ratio ω eo/T. Assuming /epsilon10/lessmuchT, using Eqs. ( 17) and ( 35), and the above results, we rewrite Eq. ( C1)a s /Gamma1o→e=16EJ πe−/Delta1/T/bracketleftbigg ln4T |/epsilon10cos(2πng)|−γE/bracketrightbigg ×/parenleftbiggEC EJ/parenrightbigg1/3/parenleftbigg D/epsilon10 ωp/parenrightbigg2 sin2(2πng). (C8) Generalization of this result to the transition rate /Gamma1(n) o→ebetween nearly degenerate states of higher energy is obtained bythe substitution /epsilon1 0→/epsilon1n. Except at the degeneracy points ng=1/4, 3/4 (where this expression diverges), we can estimate the rate in order of magnitude by assuming sin(2 πng), cos(2πng)≈1. For low-lying states, this estimate shows that the rate /Gamma1(n) o→eis small compared to the rate /Gamma11→0determining the relaxation time of the transmon [see Eq. ( 55)]. This smallness is due to the exponentially suppressed o→e matrix element Eq. ( C6) as a function of the ratio EJ/EC, in comparison with the weak power-law suppression of the1→0 matrix element as given by Eq. ( 53) with ϕ 0=0, m=1, and n=0. The relationship between the two rates is qualitatively different in the split transmon, as we discussnext. 2. Split transmon The above calculation of the even to odd transition rate in the single-junction transmon can be easily modified to yield therate for a split transmon. As discussed in Sec. VA, the effective 064517-19CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) Hamiltonian and therefore the form of the eigenstates are the same in the single and split transmon. The difference betweenthe two cases arises in the evaluation of the matrix elementspertaining to each junction [cf. Eq. ( 125)]. For the even to odd matrix element, we find /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleft/Psi1 e|sinˆϕj 2|/Psi1o/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 /similarequal1−cos(πf±ϑ) 2, (C9) where the upper (lower) sign applies to junction j=1(j= 0),fis defined in Eq. ( 88), and ϑin Eq. ( 118). In contrast with the single-junction transmon case considered above, herethe matrix element is dominated by the intrawell contributionhaving the same form of Eq. ( 53)a tn=m=0 and finite phase bias πf±ϑ. Substituting Eq. ( C9) into Eq. ( 109) and assuming thermal equilibrium quasiparticles [cf. Eq. ( 35)], we obtain /Gamma1 o→e=8(EJ0+EJ1) πe−/Delta1/Teωeo(f)/2TK0/parenleftbigg|ωeo(f)| 2T/parenrightbigg ×/parenleftBigg 1−ω2 p(f) ω2p(0)/parenrightBigg (C10) withωp(f)o fE q .( 120). The frequency ωeo(f) depends both on gate voltage, as in Eq. ( B6), and on flux via /epsilon10(f); the latter quantity is given by Eq. ( B20) with the substitutions ωp→ωp(f),EJ→EJ(f) [see Eq. ( 116)]. As above, the rate/Gamma1(n) o→eof transitions between nearly degenerate levels of higher energy is obtained upon the substitution ωeo(f)→ /epsilon1n(f)s i n ( 2 πng)i nE q .( C10). Note that the rate vanishes at integer multiples of the flux quantum; at those values offlux, exponentially small contributions to the matrix elementanalogous to those calculated above should be included. Atnoninteger values of reduced flux f,E q .( C10) should be compared with the transition rate between qubit states inducedby thermal quasiparticles /Gamma1 1→0=8(EJ0+EJ1) πe−/Delta1/Teωp(f)/2TK0/parenleftbigg|ωp(f)| 2T/parenrightbigg ×EC ωp(f)/parenleftBigg 1+ω2 p(f) ω2p(0)/parenrightBigg , (C11) obtained using Eq. ( 125). The ratio between these two quantities /Gamma1o→e /Gamma11→0=eωeo(f)/2TK0/parenleftBig |ωeo(f)| 2T/parenrightBig eωp(f)/2TK0/parenleftBig |ωp(f)| 2T/parenrightBigωp(f) ECω2 p(0)−ω2 p(f) ω2p(0)+ω2p(f) (C12) depends on temperature through the first factor on the right- hand side. Experimentally, measurements for the rate areperformed near n g=1/2, so that the relevant even and odd frequencies are ωeo(f)∼/epsilon10(f),/epsilon11(f); they are gener- ally two to three orders of magnitude smaller than ωp(f) (∼2π×4 GHz), while the latter is usually larger than twice the temperature ( T∼20–200 mK). Under these conditions, the first factor in Eq. ( C12) can be approximated, in order of magnitude, by 5 to 10. The last factor in Eq. ( C12) varies between 0 at f=0 and 1 at f=1/2; as flux is used to suppress the qubit frequency from its maximum value(/greaterorsimilar10 GHz), we can approximate the last factor by 1 /2. Finally, the central factor can be rewritten as√8EJ(f)/EC; since EJ(f)/ECusually is varied between 10 and 30, we arrive at the order-of-magnitude estimate /Gamma1o→e /Gamma11→0∼20–80 (C13) in the experimentally relevant ranges of parameters. This is an example of the more general statement that, except close tointeger values of f, the even and odd transition rate in a split transmon is faster than its decay rate. This result is qualitativelyin agreement with experimental bounds for the even and oddtransition rate in split transmons. 36,37 APPENDIX D: MATRIX ELEMENTS FOR THE HARMONIC OSCILLATOR In this Appendix, we present an analytic expression for the matrix elements of sin ˆ ϕ/2 between harmonic-oscillator states |n/angbracketrightand|m/angbracketright. Let us introduce the displacement operator ˆD(μ)=eμˆa†−μ∗ˆa, (D1) where ˆa(ˆa†) is the harmonic-oscillator annihilation (creation) operator. The matrix elements of ˆDare38 /angbracketleftm|ˆD(μ)|n/angbracketright=⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩e −|μ|2/2/radicalBig m! n!(−μ∗)n−mL(n−m) m (|μ|2), m/lessorequalslantn e−|μ|2/2/radicalBig n! m!(μ)m−nL(m−n) n (|μ|2), m/greaterorequalslantn (D2) where L(α) nare the generalized Laguerre polynomials. Since the position operator is ˆ ϕ=/lscript(ˆa+ˆa†)/√ 2, where /lscript=1/√mωis the oscillator length for an oscillator of mass mand frequency ω, we can write eiˆϕ/2=ˆD/parenleftbiggi/lscript 2√ 2/parenrightbigg . (D3) Note that for the harmonic oscillator described by Eq. ( 50), we have /lscript=2√ 2/radicalBigg EC ω10. (D4) To allow for fluctuations around a finite phase, we shift ˆϕ→ϕ0+ˆϕin the argument of sine and rewrite the resulting expression in terms of exponentials: sinϕ0+ˆϕ 2=1 2i(eiϕ0/2eiˆϕ/2−e−iϕ0/2e−iˆϕ/2). (D5) Then, using Eqs. ( D2) and ( D3), we find for m/lessorequalslantn /angbracketleftm|sinϕ0+ˆϕ 2|n/angbracketright =e−/lscript2/16/radicalbigg m! n!/parenleftbigg/lscript 2√ 2/parenrightbiggn−m ×L(n−m) m/parenleftbigg/lscript2 8/parenrightbigg sinϕ0+π(n−m) 2. (D6) 064517-20RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) The matrix element for m/greaterorequalslantnis obtained by exchanging n↔min the right-hand side. Equation ( 53) can be obtained from Eq. ( D6) by Taylor expansion for small /lscript, which for the Laguerre polynomials gives L(α) m(x)=(m+α)! m!α!−(m+α)! (m−1)!(α+1)!x+O(x2).(D7) Equation ( 58) follows from Eq. ( D6) with m=0u s i n g L(α) 0(x)=1. Using Eq. ( D2), we can also find the expectation value of the operator cos ˆ ϕ. After shifting the phase variable as done above, and since the expectation value of sine vanishes bysymmetry, we find /angbracketleftn|cos(ϕ 0+ˆϕ)|n/angbracketright=cosϕ0/angbracketleftn|cos ˆϕ|n/angbracketright. (D8) Writing the cosine in exponential form, using eiˆϕ= ˆD(i/lscript/√ 2), we arrive at /angbracketleftn|cos(ϕ0+ˆϕ)|n/angbracketright=cosϕ0e−/lscript2/4L(0) n/parenleftbigg/lscript2 2/parenrightbigg .(D9) APPENDIX E: MATRIX ELEMENTS FOR THE TRANSMON Here, we want to show that corrections to Eq. ( 53)f o rt h e transmon ( ϕ0=0) are of cubic order in EC/ωp, as claimed in the text following that equation. The transmon Hamiltonianis given by Eq. ( B1) and we neglect exponentially small corrections by setting n g=0( s e eR e f . 2and Appendices B andC). Numbering the eigenstates |ψn/angbracketrightstarting with n=0f o r the ground state, even- (odd-) numbered states are even (odd)functions of ϕ, due to the symmetry of the potential energy. Since sin ϕ/2 is an odd function, the matrix element between states of the same parity vanishes, /angbracketleftψ n±2j|sinˆϕ 2|ψn/angbracketright=0,j=0,1,2,.... (E1) Due to the smallness of the charging energy EC/lessmuchEJ,a sa first approximation, we can expand the Josephson energy inEq. ( B1) up to the fourth order in ϕ. In terms of creation and annihilation operators (cf. Appendix D, note that in the present case /lscript=2√ 2/radicalbigEC/ωp/lessmuch1), the approximate transmon Hamiltonian is ˆH=ˆH0+δˆH, (E2) ˆH0=ωp/parenleftbigˆa†ˆa+1 2/parenrightbig , (E3) δˆH=−EC 12(a+a†)4. (E4) To first order in EC/ωp, expressed in terms of harmonic- oscillator states, the transmon eigenstates are therefore |ψn/angbracketright=|n/angbracketright+|δψn/angbracketright,|δψn/angbracketright=−/summationdisplay j/negationslash=n|j/angbracketright/angbracketleftj|δˆH|n/angbracketright Ej−En, (E5) En=ωp/parenleftbigg n+1 2/parenrightbigg ,and, including the first anharmonic corrections to the eigen- states, the matrix elements are /angbracketleftψm|sinˆϕ 2|ψn/angbracketright/similarequal/angbracketleftm|sinˆϕ 2|n/angbracketright+/angbracketleftm|sinˆϕ 2|δψn/angbracketright +/angbracketleftδψm|sinˆϕ 2|n/angbracketright. (E6) Using Eq. ( D6), we find that the leading contribution to the first term on the right-hand side is /angbracketleftn±(2j+1)|sinˆϕ 2|n/angbracketright∝/parenleftbiggEC ωp/parenrightbiggj+1/2 ,j=0,1,2,.... (E7) Since we are interested in calculating the square of the matrix elements up to second order in EC/ωp, we can neglect transitions with j/greaterorequalslant1. For j=0, using Eq. ( D6)a tn e x tt o leading order, we find /angbracketleftn±1|sinˆϕ 2|n/angbracketright /similarequal/radicalBigg/parenleftbigg n+1 2±1 2/parenrightbiggEC ωp ×/bracketleftbigg 1−1 2/parenleftbigg n+1 2±1 2/parenrightbiggEC ωp+O/parenleftbiggEC ωp/parenrightbigg2/bracketrightbigg .(E8) Consider now the case m=n−1i nE q .( E6). Using Eqs. ( E4) and ( E5), and the leading term in Eq. ( E8), the central term in the right-hand side is approximately /angbracketleftn−1|sinˆϕ 2|δψn/angbracketright /similarequal− /angbracketleftn−1|sinϕ 2|n−2/angbracketright/angbracketleftn−2|δˆH|n/angbracketright En−2−En /similarequal−1 24/parenleftbiggEC ωp/parenrightbigg3/2√ n−1/angbracketleftn−2|(a+a†)4|n/angbracketright.(E9) To calculate the last factor, we note that (a+a†)2|n/angbracketright=/radicalbig n(n−1)|n−2/angbracketright+(2n+1)|n/angbracketright +/radicalbig (n+1)(n+2)|n+2/angbracketright. (E10) Shifting n→n−2 and taking the scalar product, we arrive at /angbracketleftn−2|(a+a†)4|n/angbracketright=4/radicalbig n(n−1)/parenleftbig n−1 2/parenrightbig , (E11) and substituting this expression into Eq. ( E9), we obtain /angbracketleftn−1|sinˆϕ 2|δψn/angbracketright=−1 6/parenleftbiggEC ωp/parenrightbigg3/2√n(n−1)/parenleftbigg n−1 2/parenrightbigg . (E12) Proceeding as above, we also find /angbracketleftδψn−1|sinˆϕ 2|n/angbracketright=1 6/parenleftbiggEC ωp/parenrightbigg3/2√n(n+1)/parenleftbigg n+1 2/parenrightbigg . (E13) 064517-21CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) Finally, substitution of Eqs. ( E8), (E12), and ( E13)i n t o Eq. ( E6)g i v e s /angbracketleftψn−1|sinˆϕ 2|ψn/angbracketright=/radicalBigg nEC ωp+O/parenleftbiggEC ωp/parenrightbigg5/2 . (E14) Repeating the above calculations for the case m=n+1 and using Eq. ( E7), we conclude that the square of the matrix element is /vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftψm|sinˆϕ 2|ψn/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 =EC ωp[nδm,n−1+(n+1)δm,n+1]+O/parenleftbiggEC ωp/parenrightbigg3 .(E15) APPENDIX F: MULTIJUNCTION HAMILTONIAN The aim of this Appendix is to derive the Hamiltonian for a multijunction system starting from the Lagrangian Eq. ( 105). We consider a loop of M+1 junctions and assume Mof them, denoted by index jwithj=1,..., M , to be identical, so that their capacitances and Josephson energies are, respectively,C j=C1andEJj=EJ1for 1/lessorequalslantj/lessorequalslantM. These Mjunctions will be referred to as the array junctions to distinguish themfrom the j=0 junction, whose capacitance C 0and Josephson energy EJ0can differ from those of the array junctions. While the system comprises M+1 junctions, there are only M-independent degrees of freedom due to the flux quantization constraint Eq. ( 104). Using that equation to eliminate the phase ϕ0, the Lagrangian is L{ϕ}=1 2C0 (2e)2⎛ ⎝M/summationdisplay j=1˙ϕj⎞ ⎠2 +1 2C1 (2e)2M/summationdisplay j=1˙ϕ2 j +EJ0cos⎛ ⎝M/summationdisplay j=1ϕj−2π/Phi1e//Phi1 0⎞ ⎠+EJ1M/summationdisplay j=1cosϕj. (F1) We introduce a new set {φ}ofMindependent variables φ=M/summationdisplay j=1ϕj, (F2) φk=ϕk−αM−1/summationdisplay l=1ϕl+ϕM√ M,k=1,..., M −1 (F3) where α=/parenleftbigg 1+1√ M/parenrightbigg1 M−1. (F4) The inverse transformation is given by ϕk=φk−αM−1/summationdisplay l=1φl+1 Mφ, k =1,..., M −1 (F5) ϕM=1√ MM−1/summationdisplay l=1φl+1 Mφ.In terms of the Mvariables φ,φk(k=1,..., M −1), the Lagrangian is L{φ}=1 8e2/parenleftbigg C0+C1 M/parenrightbigg ˙φ2+1 8e2C1M−1/summationdisplay k=1˙φ2 k−U({φ}) (F6) with potential energy U({φ})=−EJ0cos(φ−2π/Phi1e//Phi1 0) −EJ1M−1/summationdisplay k=1cos/parenleftBigg φk−αM−1/summationdisplay l=1φl+φ M/parenrightBigg −EJ1cos/parenleftBigg 1√ MM−1/summationdisplay l=1φl+φ M/parenrightBigg . (F7) Introducing the Mconjugate variables N=∂Lφ/∂φ andNk= ∂Lφ/∂φk(k=1,..., M −1), the Hamiltonian is H{φ}=N˙φ+M−1/summationdisplay k=1Nk˙φk−L{φ} =4ECN2+4EC1M−1/summationdisplay k=1N2 k+U({φ}), (F8) where EC=e2 2(C0+C1/M),E C1=e2 2C1. (F9) The Hamiltonian in Eq. ( F8) governs the dynamics of the M-independent degrees of freedom of the M+1 junction system with flux quantization and Midentical array junctions. For a two-junction system, we have M=1 and all the sums in Eqs. ( F7) and ( F8) are absent. Then, the Hamiltonian is that given in Eq. ( 113). 1. Fluxonium The fluxonium consists of M+1 junctions such that a “weak” junction j=0 with EJ0<E J1is connected to a large array of Mjunctions ( M/greatermuch1) with small phase fluctuations EC1/lessmuchEJ1. These conditions enable us to drastically simplify the last two terms of the potential energy U({φ})f o rt h e M- independent variables φ,φk(k=1,..., M −1) in Eq. ( F7). We consider small fluctuations of variables φkaround the configuration φk=0,k=1,..., M −1, which is an extremum of Ufor any value of φ[as can be checked by differentiating Uwith respect to φkand using Eq. ( F4)]. We further assume that typical values of φare small compared to 2πM (note that since Mis large, this weak restriction on φand its fluctuations still allows for phase slips through the weak junction). Then, we can expand the last two terms inEq. ( F7) to quadratic order in φ kandφ/M to find U({φ}) /similarequal−EJ0cos(φ−2π/Phi1e//Phi1 0)+1 2ELφ2+1 2EJ1M−1/summationdisplay k=1φ2 k (F10) 064517-22RELAXATION AND FREQUENCY SHIFTS INDUCED BY ... PHYSICAL REVIEW B 84, 064517 (2011) with EL=EJ1 M. (F11) Hence, in this approximation, the Hamiltonian ( F8)f o r theM+1 junction fluxonium separates into independent Hamiltonians for each of the Munconstrained variables φ,φk: H{φ}=Hφ+M−1/summationdisplay k=1Hk,Hφ=4ECN2−EJ0cos(φ−2π/Phi1e//Phi1 0)+1 2ELφ2, Hk=4EC1N2 k+1 2EJ1φ2 k. (F12) Up to a change of variable φ→2π/Phi1e//Phi1 0−φand redefinitions of symbols, Hφcoincides with Hϕof Eq. ( 2). The relations in Eq. ( 133) between the parameters of the M+1 junctions and the energies ECandELentering the effective qubit Hamiltonian Hφfollow from Eqs. ( F9) and ( F11), respectively. 1D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000). 2J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf,P h y s .R e v .A 76, 042319 (2007). 3J. M. Martinis, M. Ansmann, and J. Aumentado, Phys. Rev. Lett. 103, 097002 (2009). 4P. J. de Visser, J. J. A. Baselmans, P. Diener, S. J. C. Yates, A. Endo, and T. M. Klapwijk, P h y s .R e v .L e t t . 106, 167004 (2011). 5R. M. Lutchyn, L. I. Glazman, and A. I. Larkin, P h y s .R e v .B 72, 014517 (2005). 6K. A. Matveev, M. Gisself ¨alt, L. I. Glazman, M. Jonson, and R. I. Shekhter, P h y s .R e v .L e t t . 70, 2940 (1993). 7P. Joyez, P. Lafarge, A. Filipe, D. Esteve, and M. H. Devoret, Phys. Rev. Lett. 72, 2458 (1994). 8M. H. Devoret and J. M. Martinis, Quantum Inf. Process. 3, 163 (2004). 9V . E. Manucharyan, J. Koch, M. H. Devoret, and L. I. Glazman,Science 326, 113 (2009). 10G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, P h y s .R e v .L e t t . 106, 077002 (2011). 11A. Barone and G. Patern `o,Physics and Applications of the Josephson Effect (Wiley, New York, 1982). 12Note that unity and cos ϕin the denominator in Eq. ( 11)h a v e the same sign, in agreement with Ref. 11. To obtain the correct sign, it is important to calculate the matrix element of the time-dependent perturbation in Eq. ( 8), rather than the matrix element of the tunneling Hamiltonian itself, as was done in A. W. Overhauser,Phys. Rev. B 62, 3040 (2000). The latter procedure leads to the wrong sign for the cos ϕterm. 13C. W. J. Beenakker, in Transport Phenomena in Mesoscopic Systems , edited by H. Fukuyama and T. Ando (Springer, Berlin, 1992). 14J. Koch, V . E. Manucharyan, M. H. Devoret, and L. I. Glazman,P h y s .R e v .L e t t . 103, 217004 (2009). 15More precisely, we need to assume that low-lying states in a well are separated from those in nearby wells. This means that we neglecttunneling between wells [which is suppressed when E J/greatermuchEC,s e e Eq. ( 99)]. Tunneling is taken into account in Sec. IV B . 16Here and below, in writing the conditions for the validity of our approximations, we neglect factors that, for the flux-biased phasequbit, depend on ϕ 0. This dependence is through trigonometric functions with ϕ0as argument, and for typical flux biases, these functions would modify the applicability conditions by factorsof order unity. This is consistent with our assumption that the flux is sufficiently far from odd-integer multiples of half the fluxquantum. 17For both the transmon and the flux-biased phase qubit, this conditionconcides, up to numerical factors of order unity, with Eq. ( 51). 18R. M. Lutchyn, L. I. Glazman, and A. I. Larkin, Phys. Rev. B 74, 064515 (2006). 19In the case of the transmon, an estimate for the rate /Gamma11→0 was given in Eq. (4.10) of Ref. 2by generalizing the results obtained in Ref. 18 for the Cooper-pair box. Note that the presentmore rigorous treatment gives a parametrically larger rate (bya factor ∼√ /Delta1/T ) when applying Eq. ( 55) to a single-junction transmon. 20H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani,A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. Glazman,and R. J. Schoelkopf, e-print arXiv:1105.4652 . 21M. Lenander, H. Wang, R. C. Bialczak, E. Lucero, M. Mariantoni, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner,T. Yamamoto, Y . Yin, J. Zhao, A. N. Cleland, and J. M. Martinis,Phys. Rev. B 84, 024501 (2011). 22R. M. Lutchyn, L. I. Glazman, and A. I. Larkin, Phys. Rev. B 75, 229903(E) (2007). 23We note that in Ref.14 the limit EL→0 was considered at a fixed ratioEJ/EC. 24J. N. L. Connor, T. Uzer, R. A. Marcus, and A. D. Smith, J. Chem. Phys. 80, 5095 (1984). 25The factors in the second line in Eq. ( 103) were missed in Eq. ( 22)o fR e f . 10. Those factors increase the magnitude of the transition rate but do not affect its functional dependence onfluxf. 26In this section, we neglect for simplicity the dimensionless gate voltage ng. As discussed in Sec. III, this amounts to neglecting exponentially small corrections since, for both split transmon andfluxonium, we consider the regime of Josephson energy largecompared to charging energy. 27J. Lepp ¨akangas, M. Marthaler, and G. Sch ¨on, e-print arXiv:1104.2800 . 28A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret,S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 101, 080502 (2008). 29Here, we neglect for simplicity the renormalization of the parame-ters that appear in Eq. ( 102). 064517-23CATELANI, SCHOELKOPF, DEVORET, AND GLAZMAN PHYSICAL REVIEW B 84, 064517 (2011) 30In the terms we are neglecting, we can distinguish contributions proportional to the distribution function ffrom a contribution independent of f. The contribution proportional to foriginating fromδE(1),d i is logarithmically divergent, but this divergence is canceled by the similar contribution from δE(2) i. The contribution independent of finδE(2) iis linearly divergent, but independent of the gap /Delta1; therefore, this divergence can be eliminated by considering energy differences between the superconducting andnormal state. 31See, e.g., W. A. Harrison, Solid State Theory (Dover, New York, 1980). 32L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Butterworth- Heinemann, Oxford, 1981).33W. H. Furry, Phys. Rev. 71, 360 (1947). 34See, e.g., Sec. 28.4, in Digital Library of Mathematical Functions , Release date 2010-05-07, National Institute of Standards andTechnology [ http://dlmf.nist.gov/ ]. 35Note that the two lowest eigentstates of the reduced Cooper-pair box Hamiltonian ( 75)a tng=1/2[ s e eE q s .( 76)a n d( 80)] have the form of the m=0 contributions to the sums in Eq. ( C5). 36J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J .M .C h o w ,J .M .G a m b e t t a ,J .M a j e r ,L .F r u n z i o ,M .H .D e v o r e t ,S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. B 77, 180502(R) (2008). 37L. Sun et al. (unpublished). 38J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). 064517-24

PhysRevB.77.075315.pdf

Coherent defect-assisted multiphonon intraband carrier relaxation in semiconductor quantum dots A. N. Poddubny1and S. V. Goupalov1,2 1A.F . Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg 194021, Russia 2Department of Physics, Jackson State University, Jackson, Mississippi 39217, USA /H20849Received 18 May 2007; published 15 February 2008 /H20850 A defect-assisted mechanism of multiphonon intraband carrier relaxation in semiconductor quantum dots, where the carrier is found in a coherent superposition of the initial, final, and defect states, is proposed. It isshown that this mechanism is capable of explaining the observed trends in temperature dependences of theintraband relaxation rates for PbSe and CdSe colloidal nanocrystal quantum dots. DOI: 10.1103/PhysRevB.77.075315 PACS number /H20849s/H20850: 78.67.Hc, 71.38. /H11002k I. INTRODUCTION Recent experimental studies of PbSe colloidal nanocrys- tals /H20849NCs /H20850have revealed an unexpectedly fast intraband re- laxation in semiconductor quantum dots /H20849QDs /H20850with energy separation between adjacent electron /H20849or hole /H20850levels exceed- ing by far the optical phonon energies.1,2A pronounced tem- perature dependence of the observed relaxation rates2sug- gests that the underlying mechanism should involvemultiphonon transitions, while Auger-like relaxationprocesses 3are ruled out due to a nearly symmetric energy spectra in the conduction and valence bands of PbSe. The multiphonon transitions between otherwise orthogo- nal quantized electronic states in a QD may occur due to thenonadiabaticity of the electron-phonon system. 4–7A model study performed in conjunction with the experiments onPbSe NCs revealed that, in order for this mechanism to beresponsible for the experimentally observed temperature de-pendence of the relaxation rate, the electron-phonon cou-pling should be stronger than one would expect. 7However, the study of detailed models of electron-phonon interactionin quantum dots /H20849upon which the judgement about the cou- pling strength is based /H20850has always been a weak point in approaches to energy relaxation 8and Raman9processes. In the case of PbSe NCs, the situation is further obscured by thefact that the widely accepted model of their electronicstructure 10which would be natural to pick for an estimate of electron-phonon coupling strength fails to describe absorp-tion spectra of PbSe NCs 1and is therefore not quite reliable. Thus, instead of seeking to either confirm or rule out thismechanism, we will concentrate on examining other possi-bilities for the fast energy relaxation. As it has been proposed for epitaxially grown QDs, 11such possibilities can be provided by a localized state of an impu-rity or defect close to the QD surface. In the case of theexperiment of Ref. 1, such states can correspond to deep impurities in the silica host surrounding PbSe NCs occurringin an appreciable proximity to the NC surface. Otherwise,they can be due to surface states. An electron trapped to theimpurity is strongly localized /H20849as compared to the states of size quantization in the QD potential /H20850, which assures its strong coupling to local lattice vibrations. One can then con-sider a relaxation process involving the levels of size quan-tization in the QD potential as the initial /H208492/H20850and final /H208490/H20850states and the localized state at the impurity as an interme- diate state /H208491/H20850. The energy diagram for such a system is sketched in Fig. 1, where it is assumed that an electron trapped to the impurity strongly couples to a single localvibrational mode of frequency /H92750and normal coordinate q. Although carrier relaxation in QDs involving energy levels depicted in Fig. 1has been theoretically investigated,8,11,12all these studies considered the relaxation as a two step process. At the first stage of such a process, thesubsystem “electron plus local lattice vibrations” undergoesan energy conserving transition 2 →1 from the state, where the electron is in the excited state of size quantization in theQD potential /H208492/H20850and the lattice is at equilibrium, into the state, where the electron is bound to the impurity /H208491/H20850and the local lattice vibrations are excited. The lattice distortion lo-calized at the impurity site then spreads out through the en-tire heterogeneous system, and the lattice regains equilib-rium, so that the energy of the subsystem electron plus localvibrations is lower than it had been before the transition 2→1 took place. At the second stage, the subsystem electron plus local vibrations undergoes another energy conservingtransition /H208491→0/H20850to the state where the local vibrations are excited and the electron is found in the low-energy state of size quantization in the QD potential /H208490/H20850. This transition is again followed by the lattice relaxation. The two-step process described above implies instant lat- tice relaxation or negligible lifetime of the local vibrationalmode excitations. 6This is a legitimate assumption for epi- taxially grown QDs, where, despite the lattice constant mis- FIG. 1. Energy diagram representing adiabatic potentials for the quantum dot states /H208490 and 2 /H20850and the impurity level /H208491/H20850.PHYSICAL REVIEW B 77, 075315 /H208492008 /H20850 1098-0121/2008/77 /H208497/H20850/075315 /H208497/H20850 ©2008 The American Physical Society 075315-1matches between different constituents of a heterostructure, one has a crystalline system. The colloidal QDs and theirenvironment, on the other hand, represent a much more in-homogeneous system and one should expect local vibrationalexcitations to have longer lifetimes. Thus, for colloidal QDs,the lifetime of the local vibrational excitations is a relevanttime scale which can be longer than the time of the transition1→0 in the two-step process described above. However, in this case, the transition 2 →0 must be treated as a coherent quantum mechanical process. Such process represents a newmechanism of phonon-assisted carrier relaxation in QDs andits study is the scope of the present work. Similar processeshave been considered for the capture or generation ofelectron-hole pairs by impurities in bulk semiconductors. 13,14 As we will show, the lifetime of a state where several phonons are excited is inversely proportional to the numberof phonons. Thus, our mechanism is most efficient when theintermediate state has on average just a few phonons associ- ated with it, while the final state has a lot of phonons. Thefirst condition assures that the lifetime of the intermediatestate of the subsystem electron plus local vibrations is longenough for the proposed mechanism to be possible. The sec-ond condition allows one to neglect the time of the latticerelaxation, once the electron is in the state “0.” This meansthat the final state of the subsystem electron plus local vibra-tions has a large energy uncertainty and enables one to relaxthe requirement imposed on the frequency of local vibrationsby the energy conservation. 6,8,11,12In the meantime, the en- ergy of the intermediate state of the subsystem ‘‘electron pluslocal vibrations’’ should be close to that of the initial state inorder to ensure the large value of the transition matrix ele-ment. Another system where the proposed mechanism can play the key role is provided by CdSe colloidal nanocrystals. Inthe recent experiments on such quantum dots, the fast intra-band relaxation of electrons has been observed in the ab-sence of the holes. 15The electrons were injected into the quantum dots instead of being optically generated and thepump-probe measurements were subsequently carried out. 15 These experiments showed that intraband relaxation timesvary significantly with surface modification suggesting theinvolvement of a surface state such as the empty sstate of the surface Cd 2+atoms.15A faster relaxation rate was ob- served for lower temperatures,15though more thorough stud- ies of the temperature dependences of the relaxation rate inCdSe colloidal nanocrystals are desirable. It has been noticed 7that multiphonon relaxation processes are analogous to tunneling problems in electron transportutilizing the tunneling Hamiltonian. It turns out that the re-laxation mechanism proposed in this paper is analogous tothe problem of electron tunneling through a potential barrierplaced between two leads and containing localized electronicstates coupled to phonons and resonant with the energy ofthe tunneling electron. The latter problem was studied byGlazman and Shekhter. 16There is, however, a substantial difference between the two problems due to the discrete en-ergy spectrum for carriers confined in a QD. Indeed, in theproblem of electron tunneling, there is a continuum of bothinitial and final states in the leads. Thus, this problem has acharacteristic time related to the elastic tunneling of the elec-tron from the localized state to either of the leads and deter- mining the width of the resonance. The rate of the phonon-assisted tunneling can then be calculated using somegeneralization of the Fermi golden rule with the /H9254function accounting for the energy conservation being integrated outwhile summing over the continuum of final electronic states.In our problem, the initial and final electronic states are dis-crete. However, the spectrum of the entire system “electronplus phonons” is continuous and the scheme of calculation ofthe transition rate proposed in Ref. 16can still be applied, although some changes should be introduced. First, the widthof the resonance will be determined by the lifetime ofphonons in the intermediate state. Second, the phonon den-sity of states will enter the expression for the transition rate.It turns out that these changes can be more naturally incor-porated using a more traditional approach based on the cal-culation of overlap integrals of eigenfunctions belonging toshifted parabolic potentials /H20849see, e.g., Ref. 12/H20850. Another ad- vantage of this method is that it allows one to make somequalitative conclusions without having done actual calcula-tions. In this paper, the proposed mechanism of carrier relax- ation will be studied in detail for a model QD system, whilediscussions of its relevance to particular realizations of QDsare left for future studies. The rest of the paper is organizedas follows. In Sec. II, we formulate the problem in a formmost convenient for comparison with the work by Glazmanand Shekhter 16and with the case of nonadiabaticity-induced phonon-assisted transitions between intrinsic QD states.7The relaxation rate is expressed through a correlation functionwhich should be calculated taking into account phonon de-cay. In Sec. III, we devise an approximation allowing one toavoid evaluation of this correlation function. The lifetime ofthe intermediate state in the relaxation process is derived in aconsistent way. In Sec. IV, results of Sec. III are imple-mented into the calculation of the relaxation rate using directevaluation of the overlap integrals. Dependences of the re-laxation rate on temperature and electron-phonon couplingstrength are discussed in Sec. V. Our results are summarizedin Sec. VI. II. MODEL The Hamiltonian describing multiphonon nonradiative transitions between electronic levels of size quantization inthe QD potential with energies /H9255 2,/H92550/H20849/H92552/H11022/H9255 0/H20850and the level of electron localized at the impurity with the unperturbed energy /H92551can be represented as H=H0+V, where H0is the phonon Hamiltonian describing free local vibrations, H0=/H92750b†b and V=/H92550a†a+/H20851/H92551+/H9004/H20849b†+b/H20850/H20852c†c+/H92552d†d+/H20849B01a†c+ H.c. /H20850 +/H20849A12c†d+ H.c. /H20850. /H208491/H20850 Here, a,c,dare the electronic annihilation operators of the electronic states with the energies /H92550,/H92551,/H92552, respectively, and bis the phonon annihilation operator. We will first neglectA. N. PODDUBNY AND S. V. GOUPALOV PHYSICAL REVIEW B 77, 075315 /H208492008 /H20850 075315-2the phonon decay and introduce it later on. Parameter /H9004mea- sures the coupling strength of a carrier bound to the impurityto local vibrations. It determines the shift between theminima of the vibrational potentials /H20849corresponding to local vibrations and associated with different electronic states /H20850 whose dependence on the configuration coordinate issketched in Fig. 1. It is assumed that for an electron localized at the impurity, coupling with local vibrations is strong,while for the electron states of size quantization in the QDpotential, it is negligible. A similar assumption is valid for anelectron localized at a deep-level impurity, as opposed to freecarriers in bulk semiconductors. 5,6The matrix elements A12 and B01describe the electronic tunneling between the QD states and impurity level /H20849see Ref. 11for details /H20850. We write the electronic wave function as /H9023ˆ/H20849t/H20850=/H20851uˆ/H20849t/H20850c†+vˆ/H20849t/H20850d†+wˆ/H20849t/H20850a†/H20852/H208410/H20856, /H208492/H20850 where /H208410/H20856is the electronic vacuum and uˆ/H20849t/H20850,vˆ/H20849t/H20850,wˆ/H20849t/H20850are the operators acting on the phonon subsystem. The wave func- tion /H20851Eq. /H208492/H20850/H20852satisfies the Schrödinger equation in the inter- action representation with respect to the Hamiltonian H0, i/H11509/H9023ˆ/H20849t/H20850 /H11509t=Vˆ/H20849t/H20850/H9023ˆ/H20849t/H20850, /H208493/H20850 with initial conditions uˆ/H20849t=0/H20850=wˆ/H20849t=0/H20850=0 , vˆ/H20849t=0/H20850=1 . /H208494/H20850 Substituting Eqs. /H208491/H20850and /H208492/H20850into Eq. /H208493/H20850, we obtain the fol- lowing system of equations: i/H11509uˆ /H11509t=/H20851/H92551+Dˆ/H20849t/H20850/H20852uˆ/H20849t/H20850+B10wˆ/H20849t/H20850+A12vˆ/H20849t/H20850, /H208495/H20850 i/H11509wˆ /H11509t=/H92550wˆ/H20849t/H20850+B01uˆ/H20849t/H20850, /H208496/H20850 i/H11509vˆ /H11509t=/H92552vˆ/H20849t/H20850+A21uˆ/H20849t/H20850, /H208497/H20850 where Dˆ/H20849t/H20850=/H9004/H20849b†ei/H92750t+be−i/H92750t/H20850. We will follow Refs. 7and16and introduce the auxiliary operator uˆ0/H20849t/H20850as the solution of the following Cauchy prob- lem: i/H11509uˆ0 /H11509t=Dˆ/H20849t/H20850uˆ0,uˆ0/H20849t=0/H20850=1 . In an explicit form, it reads uˆ0= exp/H20877/H9004 /H92750/H20851b/H20849e−i/H92750t−1/H20850− H.c. /H20852 +/H20873/H9004 /H92750/H208742 /H20849i/H92750t−isin/H92750t/H20850/H20878. With the help of this operator, Eq. /H208495/H20850can be written asi/H11509 /H11509t/H20849uˆ0−1uˆ/H20850=/H92551uˆ0−1uˆ+uˆ0−1/H20851B10wˆ/H20849t/H20850+A12vˆ/H20849t/H20850/H20852. /H208498/H20850 Using Eqs. /H208496/H20850and /H208498/H20850, the operator wˆ/H20849t/H20850can be calculated in the lowest order in the tunneling matrix elements B01,A12 as wˆ/H20849t/H20850=−B01A12/H20885 0t dt0/H20885 0t0 dt1ei/H20849t0−t/H20850/H92550ei/H20849t1−t0/H20850/H92551e−i/H92552t1 /H11003uˆ0/H20849t0/H20850uˆ0−1/H20849t1/H20850. /H208499/H20850 The probability of the electron transition is given by W2→0=1 /H92702→0= lim t→/H11009/H20855wˆ†/H20849t/H20850wˆ/H20849t/H20850/H20856 t. /H2084910/H20850 The average here is taken over the initial states of the phonon subsystem at t=0. It is assumed that these states are equilib- rium states. We also have taken into account that, at t=0, there is no interaction-induced renormalization of the phononstates since the impurity electronic level is empty. Substitut-ing Eq. /H208499/H20850into Eq. /H2084910/H20850, we obtain W 2→0= lim t→/H11009Z t/H20885 0t dt0/H20885 0t0 dt1/H20885 0t dt0/H11032/H20885 0t0/H11032 dt1/H11032e−i/H20849t0/H11032−t0/H20850/H92550 /H11003ei/H20849t1−t0−t1/H11032+t0/H11032/H20850/H92551ei/H92552/H20849t1/H11032−t1/H20850U/H20849t0,t1,t0/H11032,t1/H11032/H20850, /H2084911/H20850 where we have introduced the correlation function U/H20849t0,t1,t0/H11032,t1/H11032/H20850=/H20855uˆ0/H20849t1/H11032/H20850uˆ0−1/H20849t0/H11032/H20850uˆ0/H20849t0/H20850uˆ0−1/H20849t1/H20850/H20856, /H2084912/H20850 and Z=/H20841B01/H208412/H20841A12/H208412. /H2084913/H20850 The correlation function /H20851Eq. /H2084912/H20850/H20852is analogous to the one which appeared in the problem studied by Glazman andShekhter. 16It was evaluated in Ref. 16neglecting phonon decay. In our case, phonon decay should be taken into ac-count as it determines the width of the resonance correspond-ing to the intermediate state in the relaxation problem understudy. However, the problem of evaluating the correlationfunction /H20851Eq. /H2084912/H20850/H20852taking the phonon decay into account is very difficult. Therefore, in the next section, we devise anapproximation which allows one to account for phonon de-cay without evaluating the correlation function /H20851Eq. /H2084912/H20850/H20852. III. WIDTH OF THE RESONANCE In order to understand the origin of the width of the reso- nance corresponding to the intermediate state in our relax-ation problem, we will consider an auxiliary problem whichwill help us to apply the concept of quasistationary states tothe states of the system “electron localized at the impurityplus phonons.” More precisely, let us set B 10=A12=0 and consider the electron localized at the impurity and coupled tophonons. One can introduce the retarded Green’s function ofthe electron as G R/H20849t/H20850=−i/H20855/H208550/H20841c˜/H20849t/H20850c˜†/H208490/H20850+c˜†/H208490/H20850c˜/H20849t/H20850/H208410/H20856/H20856/H9258/H20849t/H20850, /H2084914/H20850 where c˜/H20849t/H20850is the electron annihilation operator in the Heisen- berg representation and /H20855/H208550/H20841¯/H208410/H20856/H20856denotes a quantum me-COHERENT DEFECT-ASSISTED MULTIPHONON … PHYSICAL REVIEW B 77, 075315 /H208492008 /H20850 075315-3chanical average over the electronic vacuum and a statistical average over the phonon degrees of freedom described by the Hamiltonian H0. As the electron creation operator c˜†/H208490/H20850 enters the Green’s function at t=0, the latter reflects what happens to the electron when it is suddenly brought to theimpurity site from a remote location /H20849where it does not ex- perience any coupling or entangling to phonons /H20850att=0. Therefore, this auxiliary problem is essentially equivalent tothe initial relaxation problem. The only difference is that, asA 12=B01=0, the auxiliary problem does not describe dynam- ics of the electron tunneling to the impurity site. The Green’sfunction /H20851Eq. /H2084914/H20850/H20852can be written in terms of the operator uˆ 0,17 GR/H20849t/H20850=−ie−i/H92551t/H20855uˆ0/H20849t/H20850/H20856/H9258/H20849t/H20850. /H2084915/H20850 The spectral function defined as17 A/H20849/H9275/H20850=−2 I/H20885 −/H11009/H11009 dtG R/H20849t/H20850ei/H9275t/H2084916/H20850 consists of a series of /H9254-functional peaks separated by the phonon energy /H92750. Now, if there are processes leading to the phonon decay, then the expression for /H20855uˆ0/H20849t/H20850/H20856becomes18 /H20855uˆ0/H20849t/H20850/H20856= exp/H20875−/H90042/H20896n¯0+1 /H20851/H9253/H20849/H92750/H20850+i/H92750/H208522/H20853e−/H20851/H9253/H20849/H92750/H20850+i/H92750/H20852t−1/H20854 +n¯0 /H20851/H9253/H20849/H92750/H20850−i/H92750/H208522/H20853e−/H20851/H9253/H20849/H92750/H20850−i/H92750/H20852t−1/H20854 +t/H9253/H208490/H20850/H208492n¯0+1/H20850−i/H92750 /H20851/H9253/H208490/H20850/H208522+/H927502/H20897/H20876. /H2084917/H20850 Here, n¯0/H11013n¯/H20849/H92750/H20850stays for the Planckian factor and /H9253/H20849/H9275/H20850is the frequency-dependent phonon decay rate defined through the imaginary part of the phonon polarization operator.18At zero frequency, /H9253/H208490/H20850=0 /H20849which reflects the fact that the zero- phonon absorption linewidth is not affected by the phonon decay /H20850.18,19Thus, when Eq. /H2084917/H20850is substituted into Eqs. /H2084915/H20850 and /H2084916/H20850, each peak of the spectral function, except for the one corresponding to no phonons, becomes broadened. It isnatural to expect that it is this kind of broadening whichcauses the finite width of the resonance associated with theimpurity in our relaxation problem. Assuming /H9253/H20849/H92750/H20850/H11013/H9253/H11270/H92750, the shape of each peak of the spectral function can be described analytically. Equation /H2084917/H20850 yields /H20855uˆ0/H20849t/H20850/H20856= exp „S/H20853e−/H9253t/H20851/H208492n¯0+1/H20850cos/H92750t−isin/H92750t/H20852+i/H92750t −/H208492n¯0+1/H20850/H20854…, /H2084918/H20850 where the Huang-Rhys factor Sis given by S=/H20873/H9004 /H92750/H208742 . Substituting Eqs. /H2084918/H20850and /H2084915/H20850into Eq. /H2084916/H20850, we obtain A/H20849/H9275/H20850=/H20858 l=−/H11009/H11009 Al/H20849/H9275−/H9275l/H20850,/H9275l=/H92551−S/H92750+l/H92750,Al/H20849/H9275/H20850=2el/H9252/H92750/2−/H208492n¯0+1/H20850SR/H20885 0/H11009 dtIl/H208512S/H20881n¯0/H20849n¯0+1/H20850 /H11003exp /H20849−/H9253t/H20850/H20852ei/H9275t, /H2084919/H20850 where Il/H20849x/H20850is the modified Bessel function of order l. Equa- tion /H2084919/H20850contains the averaged number of phonons, n¯0.I ti s convenient to write out explicitly contributions of the initialstates with given numbers of phonons. To this end, we willuse I l/H20849x/H20850=/H20873x 2/H20874l /H20858 k=0/H11009/H20873x 2/H208742k1 k!/H20849k+l/H20850! and introduce z=exp /H20849−/H9252/H92750/H20850so that n¯0=z//H208491−z/H20850. Substitut- ing this into Eq. /H2084919/H20850, we obtain Al/H20849/H9275/H20850=2e−S/H20858 k=0/H11009/H9253/H208492k+l/H20850 /H92752+/H208492k+l/H208502/H92532zk /H208491−z/H208502k+l /H11003exp/H20873−2Sz 1−z/H20874S2k+l k!/H20849k+l/H20850!. /H2084920/H20850 This expression can be recast using the generating function for associated Laguerre polynomials,20 zk /H208491−z/H208502k+lexp/H20873−2Sz 1−z/H20874=/H208491−z/H20850zk/H20858 n=0/H11009 znLn2k+l/H208492S/H20850. As a result, Eq. /H2084920/H20850can be recast in the form Al/H20849/H9275/H20850=/H208491−e−/H9252/H92750/H20850/H20858 m=0/H11009 Alm/H20849/H9275/H20850e−m/H9252/H92750, /H2084921/H20850 where Alm/H20849/H9275/H20850=2e−S/H20858 k=0m/H208492k+l/H20850/H9253 /H92752+/H208492k+l/H208502/H92532S2k+l k!/H20849k+l/H20850!Lm−k2k+l/H208492S/H20850. /H2084922/H20850 Each term on the right-hand side /H20849rhs/H20850of Eq. /H2084921/H20850corre- sponds to a transition from the initial state with mphonons into the impurity state with m+lphonons and describes this transition’s contribution into the lth peak of the spectral func- tion Al/H20849/H9275/H20850. The latter is contributed by all processes where exactly lphonons are emitted. Each of the contributions Alm/H20849/H9275/H20850consists of /H20849m+1/H20850Lorentzians. The Lorentzians with small kon the rhs of Eq. /H2084922/H20850can have negative weights, while the weight of the Lorentzian with k=mis always posi- tive. The latter Lorentzian determines spectral wings ofA lm/H20849/H9275/H20850and it is natural to associate its width, /H9253/H208492m+l/H20850, with the linewidth of the transition between the states with mand m+lphonons. From the theory of spectral linewidths, it is well known21that the spectral width of a line corresponding to a transition between two quasistationary states is given bythe sum of their decay rates. As /H9253is the phonon decay rate and we have mphonons in the initial state and /H20849m+l/H20850 phonons in the impurity state, we finally arrive at the assign- ment of the width /H9253/H20849m+l/H20850to the state of the electron at the impurity entangled to /H20849m+l/H20850phonons. This assignment has aA. N. PODDUBNY AND S. V. GOUPALOV PHYSICAL REVIEW B 77, 075315 /H208492008 /H20850 075315-4transparent probabilistic interpretation: the state of the sys- tem ‘‘electron at the impurity plus phonons’’ decays as soonas any one of the phonons decays, and the probability of thisprocess is given by the probability of phonon decay times thenumber of phonons. IV. TRANSITION RATE We have seen that the lifetime of the intermediate state in the relaxation problem under study depends on the numberof phonons associated with it. In this case, it is more conve-nient to consider the relaxation process in the framework ofan approach based on the calculation of overlap integralsbetween eigenfunctions belonging to shifted parabolic poten-tials. These potentials for our system are schematically de-picted in Fig. 1. For the quantum dot states, the potentials are given by 6 V0/H20849q/H20850=M/H927502q2 2+/H92550, /H2084923/H20850 V2/H20849q/H20850=M/H927502q2 2+/H92552. /H2084924/H20850 Here,/H92750is the frequency of the local vibrational mode, qis the corresponding configurational coordinate, and Mis the mass of the impurity’s ion. The adiabatic potential for theintermediate state is given by 6 V1/H20849q/H20850=M/H927502/H20849q+q0/H208502 2+/H9255˜1, /H2084925/H20850 where q0=/H208812S//H20849M/H9275/H20850, /H9255˜1=/H92551−S/H92750. /H2084926/H20850 Let us introduce the eigenfunctions /H9274n/H208492/H20850/H20849q/H20850,/H9274n/H208491/H20850/H20849q/H20850, and /H9274n/H208490/H20850/H20849q/H20850in the adiabatic potentials V2/H20849q/H20850,V1/H20849q/H20850, and V0/H20849q/H20850, respectively. These are eigenfunctions of the harmonic oscil- lator satisfying the following relation: /H9274n/H208492/H20850/H20849q/H20850=/H9274n/H208491/H20850/H20849q−q0/H20850 =/H9274n/H208490/H20850/H20849q/H20850. Then, the transition rate W2→0is given by the Fermi golden rule W2→0=2/H9266Z /H92750/H20855/H20841M2→0/H208412/H20856, /H2084927/H20850 where /H20855/H20841M2→0/H208412/H20856=/H208491−e−/H9252/H92750/H20850/H20858 m=0/H11009 e−/H9252m/H92750/H20841Mm/H208412, /H2084928/H20850 Mm=/H20858 n=0/H11009/H20855/H9274m/H208492/H20850/H20841/H9274n/H208491/H20850/H20856/H20855/H9274n/H208491/H20850/H20841/H9274m+p/H208490/H20850/H20856 /H20849m−n/H20850/H92750+/H92552−/H92551+S/H92750+i/H9253n, /H2084929/H20850 andZis given by Eq. /H2084913/H20850. The energy /H9254function in Eq. /H2084927/H20850 has been integrated out yielding the phonon density of states /H92750−1, while the value of /H92750is adjusted to make the numberp=/H20849/H92552−/H92550/H20850//H92750integer.22We also took into account that, ac- cording to Sec. III, the width of the intermediate state is given by the number of phonons in that state times the pho-non decay rate. The overlap integrals entering Eq. /H2084929/H20850can be expressed in terms of the generalized Laguerre polynomials 23 /H20855/H9274m/H208490/H20850/H20841/H9274n/H208491/H20850/H20856=/H20881m! n!e−S/2S/H20849n−m/H20850/2Lmn−m/H20849S/H20850. This expression is valid for m/H33355n, while for m/H11022n, the iden- tity /H20855/H9274m/H208490/H20850/H20841/H9274n/H208491/H20850/H20856=/H20855/H9274n/H208490/H20850/H20841/H9274m/H208491/H20850/H20856/H20849−1/H20850n−mshould be used. The relaxation is most efficient when the energy of the intermediate state is different from that of the initial and finalstates in the amount not exceeding its decay rate. We willassume that this condition is always fulfilled and set /H9255 2−/H92551+S/H92750=l/H92750,l/H33528N. In this case, the term with n=l+mbecomes dominant in Eq. /H2084929/H20850and we obtain Mm/H11015−i /H9253/H20849l+m/H20850/H20855/H9274m/H208492/H20850/H20841/H9274l+m/H208491/H20850/H20856/H20855/H9274l+m/H208491/H20850/H20841/H9274m+p/H208492/H20850/H20856. /H2084930/H20850 V. RESULTS AND DISCUSSION At zero temperature, only the contribution with m=0 sur- vives in Eq. /H2084928/H20850, and the dependence of the relaxation rate on the strength of the electron-phonon coupling is governedby the dependence of /H20841M 0/H208412on the Huang-Rhys factor S. This dependence is shown in Fig. 2/H20849a/H20850/H20849solid curve /H20850. Both Sand/H92551 were changed in such a way that the energy difference /H92552 −/H92551+S/H92750/H11013l/H92750remained fixed. This is equivalent to a hori- zontal shift of the adiabatic curve 1 in Fig. 2/H20849b/H20850. As one can see from Fig. 2/H20849a/H20850, the dependence /H20841M0/H208412/H20849S/H20850has a maximum yielding the optimal value of the Huang-Rhys factor at whichFIG. 2. /H20849Color online /H20850/H20849a/H20850The dependence of the matrix ele- ments /H20841Mm/H208412on the Huang-Rhys factor Scalculated for different values of m=0,1,2,3 with l=1, p/H92750=200 meV, p=13, /H9253 =0.2 ps−1. The vertical dashed line indicates the maximum of the dependence /H20841M0/H208412/H20849S/H20850atS=5.7. /H20849b/H20850Adiabatic curves, Eqs. /H2084923/H20850–/H2084925/H20850, calculated for S=5.7. The horizontal lines indicate positions of the energy levels in the potential V2with m=0 and m=1.COHERENT DEFECT-ASSISTED MULTIPHONON … PHYSICAL REVIEW B 77, 075315 /H208492008 /H20850 075315-5the relaxation is the most efficient. This can be understood using quasiclassical language. According to the quasiclassi-cal theory, the overlap integral of eigenfunctions belongingto different adiabatic potentials is maximum at those energyvalues where the adiabatic curves intersect. 6The optimal value of the Huang-Rhys factor corresponds to the situationwhere both intersection points are close to the energy levelwith m=0, as shown in Fig. 2/H20849b/H20850. This figure also explains why the maximum of the curve /H20841M 1/H208412/H20849S/H20850, also shown in Fig. 2/H20849a/H20850, is close to the maximum of /H20841M0/H208412/H20849S/H20850. As temperature is increased, thermal activation leads to increasing importance of the terms on the rhs of Eq. /H2084928/H20850 with m/H110220. In the meantime, relative contribution of the term with m=0 to the transition rate decreases due to the factor of 1−exp /H20849−/H9252/H92750/H20850in Eq. /H2084928/H20850. Therefore, the character of the temperature dependence of the relaxation rate is determined by the ratio of /H20841M0/H20841and /H20841M1/H20841. One can see from Fig. 2/H20849a/H20850that, for l=1 and Huang-Rhys factor close to its optimal value, /H20841M1/H20841exceeds /H20841M0/H20841and the relaxation time decreases with temperature, as shown in Fig.3/H20849solid curve /H20850. For l/H110221, intersection points would be closer to the level with m=0 than to the level with m=1, resulting in/H20841M 0/H20841/H11022/H20841M1/H20841. This will lead to increasing temperature de- pendence of the relaxation time, as shown in Fig. 3. Therefore, the present mechanism can reproduce both in- creasing and decreasing temperature dependences of the re-laxation time. Those are two types of temperature depen-dences observed experimentally. 2,15Note that when the temperature sweeps from 0 to 300 K, the relaxation time inFig.3changes within one order of magnitude which is con- sistent with experimental observations. 2,15 VI. CONCLUSIONS We have proposed a new defect-assisted mechanism of multiphonon intraband carrier relaxation in semiconductorQDs, where the carrier is found in a coherent superpositionof the initial, final, and defect states, and the defect state isstrongly coupled to local lattice vibrations. The proposedmechanism is the most efficient when the intermediate andfinal states of the subsystem “electron plus local vibrations” have, respectively, small and large numbers of nonequilib-rium local phonons. Combination of different approaches hashelped us to devise a controllable approximation for the re-laxation rate. We have shown that the proposed mechanismis capable of reproducing both decreasing, as it was observedfor PbSe NCs, 2and increasing, as it was reported for CdSe QDs,15temperature dependences of the relaxation time. For reasonable values of parameters, the change in the relaxationtime with temperature rise from cryogenic to room tempera-tures was found within one order of magnitude, in agreementwith experimental observations. 2,15 ACKNOWLEDGMENTS We are indebted to I. N. Yassievich for useful discussions and constant encouragement. This work was funded by theRussian Foundation for Basic Research under Grant No. 07-02-00469-a. A.N.P. acknowledges support by the DynastyFoundation-ICFPM. 1J. M. Harbold, H. Du, T. D. Krauss, K.-S. Cho, C. B. Murray, and F. W. Wise, Phys. Rev. B 72, 195312 /H208492005 /H20850. 2R. D. Schaller, J. M. Pietryga, S. V. Goupalov, M. A. Petruska, S. A. Ivanov, and V. I. Klimov, Phys. Rev. Lett. 95, 196401 /H208492005 /H20850. 3A. L. Efros, V. A. Kharchenko, and M. Rosen, Solid State Com- mun. 93, 281 /H208491995 /H20850. 4K. Huang and A. Rhys, Proc. R. Soc. London, Ser. A 204, 406 /H208491950 /H20850; K. Huang, Sci. Sin. 24,2 7 /H208491981 /H20850. 5B. K. Ridley, Quantum Processes in Semiconductors /H20849Clarendon, Oxford, 1999 /H20850, Chap. 6. 6V. N. Abakumov, V. I. Perel, and I. N. Yassievich, Nonradiative Recombination in Semiconductors /H20849North-Holland, Amsterdam, 1991 /H20850.7S. V. Goupalov, Phys. Rev. B 72, 073301 /H208492005 /H20850. 8X.-Q. Li and Y. Arakawa, Phys. Rev. B 56, 10423 /H208491997 /H20850. 9T. D. Krauss and F. W. Wise, Phys. Rev. B 55, 9860 /H208491997 /H20850. 10I. Kang and F. V. Wise, J. Opt. Soc. Am. B 14, 1632 /H208491997 /H20850. 11P. C. Sercel, Phys. Rev. B 51, 14532 /H208491995 /H20850. 12D. F. Schroeter, D. J. Griffiths, and P. C. Sercel, Phys. Rev. B 54, 1486 /H208491996 /H20850. 13V. N. Abakumov, A. A. Pakhomov, and I. N. Yassievich, Pis’ma Zh. Eksp. Teor. Fiz. 53, 167 /H208491991 /H20850/H20851JETP Lett. 53, 176 /H208491991 /H20850/H20852. 14I. N. Yassievich, Semicond. Sci. Technol. 9, 1433 /H208491994 /H20850. 15P. Guyot-Sionnest, B. Wehrenberg, and D. Yu, J. Chem. Phys. 123, 074709 /H208492005 /H20850. 16L. I. Glazman and R. I. Shekhter, Zh. Eksp. Teor. Fiz. 94, 292 /H208491988 /H20850/H20851Sov. Phys. JETP 67, 163 /H208491988 /H20850/H20852.FIG. 3. /H20849Color online /H20850Temperature dependences of the relax- ation time calculated for different values of the impurity level po-sition /H20849l=1,2,3,5,7 /H20850with Z=/H208492.5/H20850 4meV4,S=5.7. Other param- eters are the same as in Fig. 2.A. N. PODDUBNY AND S. V. GOUPALOV PHYSICAL REVIEW B 77, 075315 /H208492008 /H20850 075315-617G. D. Mahan, Many-Particle Physics , 2nd ed. /H20849Plenum, New York, 1990 /H20850, Sec. 4.3. 18S. V. Goupalov, R. A. Suris, P. Lavallard, and D. S. Citrin, IEEE J. Sel. Top. Quantum Electron. 8, 1009 /H208492002 /H20850. 19It is convenient to think of A/H20849/H9275/H20850as of an optical absorption spec- trum of a hypothetical system with optical gap at the frequencyof the zero-phonon line. 20A. Messiah, Quantum Mechanics /H20849Dover, Mineola, NY, 1999 /H20850. 21V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics /H20849Pergamon, Oxford, 1982 /H20850, Sec. 62.22Remember that here we assume that the transition 2 →0 is fol- lowed by instant lattice relaxation, as state “0” has a large num-ber of nonequilibrium local phonons. Therefore, the final state ofthe subsystem ‘‘electron plus local vibrations’’ has a large en-ergy uncertainty. This allows one to consider a transition to a group of states of neighboring energies rather than consider a transition to a particular state . 23A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Special Functions , Integrals and Series Vol. 2 /H20849Gordon and Breach, New York, 1986 /H20850, Eq. 2.20.10.COHERENT DEFECT-ASSISTED MULTIPHONON … PHYSICAL REVIEW B 77, 075315 /H208492008 /H20850 075315-7

PhysRevB.88.064506.pdf

PHYSICAL REVIEW B 88, 064506 (2013) Density of states of disordered topological superconductor-semiconductor hybrid nanowires Jay D. Sau1and S. Das Sarma2 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA (Received 8 May 2013; published 12 August 2013) Using Bogoliubov-de Gennes equations we numerically calculate the disorder averaged density of states of disordered semiconductor nanowires driven into a putative topological p-wave superconducting phase by spin-orbit coupling, Zeeman spin splitting, and the s-wave superconducting proximity effect induced by a nearby superconductor. Comparing with the corresponding theoretical self-consistent Born approximation (SCBA)results treating disorder effects, we comment on the topological phase diagram of the system in the presence ofincreasing disorder. Although disorder strongly suppresses the zero-bias peak (ZBP) associated with the Majoranazero mode, we find some clear remnant of a ZBP even when the topological gap has essentially vanished in theSCBA theory because of disorder. We explicitly compare effects of disorder on the numerical density of states inthe topological and trivial phases. DOI: 10.1103/PhysRevB.88.064506 PACS number(s): 03 .67.Lx, 03 .65.Vf, 71 .10.Pm I. INTRODUCTION The theoretical prediction1–6that the combination of spin- orbit (SO) coupling, Zeeman spin splitting, and ordinarys-wave superconductivity could lead to an effective topological superconducting phase under appropriate (and experimentally achievable) conditions has led to an explosion of theoretical and experimental activities 6in semiconductor nanowires (InSb or InAs) in proximity to a superconductor (NbTi or Al) inthe presence of an external magnetic field. The experimentalfinding 7of a zero-bias peak (ZBP), in precise agreement with the theoretical predictions in the differential tunnelingconductance of an InSb nanowire (in contact with a NbTiNsuperconducting substrate) at a finite external magnetic field (B∼0.1−1 T), followed by independent corroborative observation 8–11of such a ZBP both in InSb and InAs nanowires in contact with superconducting Nb and Al by several groups,has created excitement in the condensed matter physicscommunity as well as in the broader scientific communityas perhaps the first direct evidence supporting the existence ofthe exotic, the elusive, and the emergent unpaired Majorana bound state in solids. Such excitement has invariably been fol- lowed by a wave of skepticism as one would expect in a healthyand active scientific discipline with questions ranging all theway from whether such a ZBP could arise from another (i.e.,non-Majorana) origin to whether all aspects of the observedexperimental phenomenology are consistent with the putativetheoretical predictions on the topological superconductivity underlying the existence of the Majorana mode. One particular issue, which is also the subject of the current work, attracting a great deal of theoretical attention 12–24is the role of disorder in the Majorana physics of superconductor-semiconductor hybrid structures. Disorder plays a key role in the Majorana physics because the underlying topologi- cal superconducting phase hosting the Majorana mode (atdefect sites) is essentially an effective spinless p-wave superconductor, 1–5which, unlike its s-wave counterpart, is not immune to nonmagnetic elastic disorder (i.e., spin-independent momentum scattering) as was already known10 years ago. 25Thus, even the simplest kind of disorder,namely, zero-range random nonmagnetic point elastic scat- terers in the wire, could strongly affect the topological super-conductivity in contrast to ordinary s-wave superconductivity, which is immune to nonmagnetic disorder, and the associatedMajorana bound states by suppressing the (topological) su-perconducting gap 13and/or creating Andreev bound states in the superconducting gap near zero energy14complicating the observation and the interpretation of the ZBP. We do, however,mention that elastic disorder or momentum scattering in thesuperconductor itself, no matter how strong (as long as it doesnot destroy the superconductor), does not affect the topologicalsuperconducting phase in the semiconductor. 21In addition, it has recently been emphasized16–18that elastic disorder by itself could create a ZBP (essentially, an antilocalization peakassociated with the disorder-induced quantum interference)in the nontopological phase in the presence of SO couplingand Zeeman splitting. Since the precise topological quantumcritical point (as a function of the applied magnetic field) sep-arating the topological and the trivial superconducting phaseis in general not known in the experiments, 7–12one cannot be absolutely sure that the observed ZBP is indeed a Majoranabound state (MBS) signature in the topological phase and nota trivial antilocalization peak in the nontopological phase. In the current work we consider disorder effects on MBS physics by directly calculating the density of states (DOS)of finite disordered nanowires in the presence of proximity-induced superconductivity taking into account SO couplingand spin splitting arising, respectively, from Rashba andZeeman effects in the wire. We use random uncorrelated pointscatterers with δ-function potential in the wire to represent the elastic disorder. The theory follows the standard prescription 26 of an exact diagonalization of the Bogoliubov-de Gennesequations in a minimal tight-binding model including super-conducting pairing, Rashba SO coupling, and Zeeman splittingin the Hamiltonian. The diagonalization of the discretizedtight-binding Hamiltonian leads to the exact eigenstates ofthe system, which then immediately give the DOS (theoreticaldetails are available in Ref. 26and are not repeated here.) Com- parison with theory in the presence of disorder necessitates 064506-1 1098-0121/2013/88(6)/064506(7) ©2013 American Physical SocietyJAY D. SAU AND S. DAS SARMA PHYSICAL REVIEW B 88, 064506 (2013) the ensemble averaging over many different impurity config- urations since each disorder configuration produces its ownunique result with random impurity-induced δ-function peaks in the superconducting gap. II. DOS IN THE DISORDERED TOPOLOGICAL PHASE In this section, we consider the effect of disorder on the semiconductor (SM)/superconductor(SC) system startingfrom the topological phase in the clean limit. In Fig. 1(with eight panels, each representing a different strength of disorderkeeping all other parameters fixed), we show our numericalDOS as a function of energy ( E) for different relevant parameter sets in the topological phase of the system. Weshow both the ensemble-averaged DOS using many-impurityconfigurations (but using the same disorder strength, i.e., thesame impurity density and potential strength, changing onlythe random locations of the impurities along the wire) and the λ = 2 μm λ = 18 μm λ = 1 μm λ = 0.2 μm λ = 0.5 μm λ = 0.03 μm λ = 0.07 μm λ = 5 μm FIG. 1. (Color online) Disorder-averaged DOS in the topological phase for the semiconductor nanowire in a magnetic field with Zeeman splitting VZ=5 K, proximity-induced pairing potential of amplitude /Delta10=3 K, Rashba SO-coupling strength αR=0.3e V−˚Aa n d μ=0. The crosses in the plots show the energy levels for a typical disorder realization for each of these mobililties. For this choice of parameters theclean quasiparticle gap /Delta1=1.3 K and coherence length ξ=0.3μm. The difference panels (a–h) correspond to different disorder strengths characterized by E s=¯h/2τ=0.01, 0.05, 0.1, 0.2, 0.46, 1.3, 3.3, and 7.3 K. The corresponding mean-free paths are λ=vFτin the panels and the self-consistent Born gaps ESCBA=1.2, 1.1, 0.9, 0.8, 0.13, 0, 0, and 0 K. Typical spectra. 064506-2DENSITY OF STATES OF DISORDERED TOPOLOGICAL ... PHYSICAL REVIEW B 88, 064506 (2013) typical DOS for a single impurity configuration in each case. Each panel corresponds to a specific disorder strength (i.e.,a fixed impurity density) and shows results for two differentlengths of the nanowire. Any given experiment can be expectedto represent a specific impurity configuration and therefore thesingle impurity realization is what is more directly relevantto experiments. It is clear from the plot of the typical DOSfor single impurity realizations that with increasing disorder,where the zero-energy peak in the disorder-averaged DOS isbroader, a given single impurity realization might not showa low-energy state even though the disorder-averaged DOSshows a peak [see Figs. 1(f) and 1(g)]. However, the zero- energy peak in the disorder-averaged density of states mightstill be directly relevant to experiments since a zero-energypeak in this case means that states near zero energy continueto persist when parameters such as gate voltage and Zeemanpotential are tuned. Since our calculations follow either Ref. 26for the exact numerical treatment or Ref. 13for the self-consistent Born approximation (SCBA) theory, we refer the reader to thosereferences for the technical details, which are actually prettystandard. 19–22 The superconductor (SC)-semiconductor (SM) hybrid nanowire structure is characterized by a large number ofindependent parameters, both for the actual experimentallaboratory systems and for our minimal theoretical model. Theminimal set of parameters necessary to describe the system arethe proximity-induced SC gap ( /Delta1 0) in the SM, the parent SC pairing potential ( /Delta1s), the Rashba SO coupling ( αR), the spin splitting ( VZ), the chemical potential ( μ), the SM effective mass ( m) which defines the SM tight-binding hopping param- eter, the nanowire length ( L), and disorder (which we take to be the uncorrelated random white noise potential associated withthe randomly located δfunction in real space spin-independent scattering centers). There are a few additional physicallyimportant parameters which are, however, not independentparameters of the theory: the coherence length in the nanowire(ξ), the SM mean-free path ( λ) due to disorder, and the number of occupied subbands (i.e., transverse quantized levels) in thenanowire which we take to be one throughout assuming thesystem to be in the one-dimensional limit. The single subbandapproximation is made for the convenience of keeping thenumber of parameters in the model to be tractable. In thelimit where the inter-subband energy separation is large 7–9 compared to the other relevant energy scales ( /Delta10,VZ, andμ for the topmost and therefore topological subband). Note thatin this regime, all bands other than the topmost band (i.e.,nearest to the Fermi level) are deep in the nontopologicalphase. While disorder can lead to scattering between differentsubbands, we expect the stronger SO coupling (larger k F)i n the lower bands to lead to a larger superconducting gap15and therefore not qualitatively alter the physics of the topmostnear topological band. In addition, there is an independentparameter defining the hopping amplitude across the SC/SMinterface which controls the proximity-induced SC pairinggap/Delta1 0in the SM in terms of the parent gap /Delta1sin the SC. Finally, the proximity gap ( /Delta1) in the SM in the presence of SO coupling and finite Zeeman splitting is reduced from /Delta10 in a known manner. All the details for modeling the disorder are given in Ref. 13where the SCBA was used in contrastto our exact numerical diagonalization in the current work following Ref. 26. One specific goal of our current work is to compare the analytic and simple SCBA theory13with the exact tight-binding numerical analysis to test the limits of validityand the applicability of the SCBA theory which, being analytic,can be used rather easily. We choose parameters approximately consistent with the InSb/Nb systems studied experimentally in Ref. 7. These are /Delta1 0=3 K and αR=0.3e V−˚A (corresponding to an effective SO-coupling strength m∗α2 R=2.5 K). Since our interest is in disorder effects, we focus on a range of magnetic fields witha spin splitting V Z∼5 K (we vary it in a few cases only to change the proximity to the topological quantum phasetransition point, separating the topological and the trivial SCphase). Given that the condition for the topological SC phaseto be realized in the SM is given by 2V2 Z>/Delta12 0+μ2, where /Delta10is the proximity-induced gap in the SM, the system should be in the MBS carrying topological phase for /Delta1< 5 K (since μ=0). Given that /Delta1</Delta1 0=3 K, with the reduction of /Delta1 below /Delta10arising from the existence of VZ/negationslash=0, our system is deep in the topological phase for the results shown in Fig. 1 sinceVZ(=5K)/greatermuch/Delta1(=1.3 K) and μ=0. Each panel in Fig. 1corresponds to a different disorder strength in the system, characterized by the correspondinglevel broadening E s=¯h/2τ(where τis the scattering time, τ=∞ in the absence of disorder) or equivalently the mean- free path λ=vFτ(where vFis the Fermi velocity), both calculated in the SCBA13for the given disorder in the wire. In each panel (and for each disorder) we show our DOSnumerical results for two distinct wire lengths: L=1.5μm andL=3μm. In each case, we show both the ensemble averaged DOS results using an averaging over many ( >1000) random impurity configurations (keeping E s,λ, etc., fixed) and the result for a typical single impurity configuration (thedistinct crosses or dots denoting δfunctions for the DOS at the value of energy). We emphasize that for an infinitely long wire(L/greatermuchξ≈0.5μm for our case) the DOS in the absence of disorder will vanish throughout the gap ( ±1.3 K in our case) with a δ-function peak at E=0 associated with the MBSs at the wire edges. To characterize the disorder strength for the results in Fig. 1 [with panels (a) to (h) with increasing disorder keeping allother parameters fixed], we use SCBA. 13The SCBA theory provides us with the SC gap in the topological phase for agiven disorder strength, allowing us to compare our direct (andexact) numerical calculation in the presence of disorder withthe SCBA theory. We show the calculated SCBA gap in eachcase in the figure captions for the sake of direct comparisonwith the exact results in the figures. (The SCBA theory isobviously an ensemble-averaged theory for the infinite systemand does not depend on L.) It is clear from the results of Fig. 1 that the analytical SCBA theory 13is in qualitative agreement with the exact ensemble-averaged numerical results for theDOS even for E s≈/Delta1, where the topological gap essentially vanishes [panel (f) in Fig. 1] both according to the SCBA (i.e.,ESCBA=0) and in our numerical results. While the disorder-averaged DOS calculated by exact diagonalizationdoes not stricly vanish inside the gap, the gap calculatedwith the SCBA can be identified with the peaks in the DOSat the edges of the gap. The closing of the gap within the 064506-3JAY D. SAU AND S. DAS SARMA PHYSICAL REVIEW B 88, 064506 (2013) SCBA coincides with the disappearance of the dips in the DOS around zero energy. The DOS peak at E=0 associated with the MBS is continuously suppressed with increasingdisorder, but quite amazingly there is a discernible DOSpeak at E=0e v e nf o r E s(=3.3K )/greatermuch/Delta1(=1.3 K), where ESCBA=0, and at best the topological superconductivity is gapless. The remarkable result, which is quite apparent in our Figs. 1(f)and1(g), is that the MBS peak of the DOS at E=0i s actually very robust to disorder and survives disorder strengthsubstantially larger than that (typically E s∼/Delta1) destroying the induced superconducting gap /Delta1. Thus, in Figs. 1(f) and1(g), although the SCBA theory and our exact numerical resultsboth show the system to be gapless with E s>/Delta1 , the DOS peak at E=0 associated with the MBS persists until Es/greatermuch/Delta1 as in Fig. 1(g), where Es=7K(/greatermuch/Delta1=1.3 K). It is not only that the MBS feature in the ensemble-averaged DOS survivesup to very strong disorder (e.g., the mean-free path λ=0.5, 0.2, and 0.07 μm, respectively, in Figs. 1(e)–1(g) which are smaller than the wire lengths of L=1.5 and 3 μm used in our numerical work), the typical DOS for single random impurityconfiguration also shows peaks at E=0 as can clearly be seen in Figs. 1(f)and1(g) [and as well as in Figs. 1(a)–1(d)], but not in Fig. 1(h) where the very large disorder strength (λ=0.03μm) suppresses both the ensemble-averaged MBS peak and the single configuration peak at E=0. The survival of the zero-energy DOS peak well above the point where theSC gap is completely suppressed by disorder is an importantnew result of our exact numerical work, directly establishingthe possible theoretical existence of a gapless topological SCphase. A. Griffiths effects T h eo r i g i no ft h e E=0 peak in the DOS in the strongly disordered case can be related to the Griffiths effect previouslyconsidered for the spinless p-wave superconductor. 25The Griffiths effect in the semiconductor nanowire at finite VZ arises from the disorder-induced variation of the chemical potential, which can lead to a transition from a topologicalphase to a nontopological phase and vice versa. The variationof the effective chemical potential can lead to domain wallsbetween topological and nontopological regions, each ofwhich would support a local zero-energy MBS. Since eachregion is of finite extent, the MBSs are a finite distance apartand split into conventional states with a nonzero energy. Infact, by carefully considering the distribution of the distancesbetween the MBSs, it has been shown 25that this splitting typically leads to a singular peak in the DOS at E=0, which diverges as E→0a sap o w e rl a wi n E. The topological phase is of course characterized by true zero-energy edgestates, whose energy is exponentially small in the length of thesystem L. Thus, in the long wire limit, the topological phase and the nontopological phase both have power-law-divergentDOS due to the Griffiths effects, but the topological phasehas a pair of zero modes exponentially close to zero energy. 12 This distinction between exponential versus power law in thelength seems to appear in the DOS plotted in Fig. 1.A ts m a l l disorder, there is a sharp peak which is very weakly split. Atlarger disorder [Figs. 1(e)–1(h)], which is approximately whenthe gap vanishes, i.e., E SCBA→0, theE∼0 peak in the DOS is broadened. For finite length systems, the sharp transition between the topological and nontopological phase at finite disorderbecomes a crossover. While the topological invariant canbe computed even for a disordered, but strictly infinitesystem, 12,23,24we have not done so in the present work because we restrict ourselves to systems of wire lengths comparable tothe experimental systems. For such finite wires the distinctionbetween the topological and nontopological phase is not sharp.In fact, the zero modes arising from the Griffiths effect arethemselves MBSs. The MBSs characterizing the topologicalphase are special only in the sense that they occur near theends of a finite system and are therefore separated from otherlow-energy MBSs by a distance of the order of the lengthof the wire. This can also occur from the Griffiths effectin some realizations of disorder, even in the nontopologicalparameter regime. Therefore, the Griffiths singularities seen inthe disordered Fig. 1indicate the presence of several low-lying MBSs in the spectrum of the finite wire. For very large disorderthe system becomes completely nontopological (and gapless)as in Fig. 1(h). There are three important messages following from our DOS results for the topological phase in the presence ofdisorder: (1) the zero-energy DOS peak associated with theMBS is strongly suppressed by disorder; (2) the SCBA is anexcellent quantitative approximation for calculating the SCgap in the topological phase including effects of disorder;and (3) most importantly, the zero-energy MBS peak is veryrobust against disorder and survives well after the SC gap in thetopological phase has been suppressed to zero and disappearsonly when the mean-free path λ/lessorsimilarξ/6o rE s/greaterorsimilar6/Delta1(for the systems are studied), whereas the topological SC gap vanishesforλ/lessorsimilarξ. Given the quantitative validity of the SCBA, we can calculate the phase diagram of the SC/SM structure in thepresence of disorder by using the analytical SCBA theory. 13 We show our SCBA-calculated quantum phase diagram ofthe system in Fig. 2.I nF i g . 2, we fix all parameters except ESVZ FIG. 2. (Color online) Quasiparticle gap /Delta1(in the color bar) versus Zeeman potential VZand scattering rate Esin the topological phase. The black region represents the gapless phase and is therefore not topologically robust. 064506-4DENSITY OF STATES OF DISORDERED TOPOLOGICAL ... PHYSICAL REVIEW B 88, 064506 (2013) for disorder ( Es) and spin splitting ( VZ) with the color representing the calculated SC gap ESCBA in the presence of disorder (by definition ESCBA=/Delta1forEs=0). The black region represents gapless superconducivity. We note, basedon our numerical results of Fig. 1, that a large part of the black region in the phase diagram allows for well-defined zero-energy DOS MBS peaks, although the systemis essentially a gapless topological superconductor in thisregime. Much of this SCBA gapless regime is dominatedby Griffiths physics except for very large disorder when thesystem eventually becomes nontopological (i.e., even the finitetopological segments disappear). Much of this SCBA gaplessregime is dominated by Griffiths physics except for very largedisorder when the system eventually becomes nontopological.Except for our current results showing the well-defined robustpersistence of the MBS peak even in the gapless topologicalregime (the black region in Fig. 2), one would have concluded that SCBA predicts a rather gloomy picture for the existence ofthe Majorana mode in the disordered SC/SM hybrid structuresince, without our current exact results, the conclusion wouldhave been that there cannot be any MBS in the black regionof the phase diagram. Of course the quantitative details of theSCBA phase diagram depend on the SO-coupling strength andthe topological phase with a large SC gap is easily achievedby large (small) SO coupling (disorder). Therefore, the absence of a topological superconducting gap within the SCBA in the dark region of Fig. 2does not rule out the presence of zero-energy MBSs. As discussedabove, the physics of this dark region in Fig. 2is complicated by the existence of disorder-induced subgap states—so thatMBSs can still appear as a result of Griffiths effects from asuperposition of many of these localized low-energy states. III. DOS IN THE DISORDERED NONTOPOLOGICAL PHASE Next we comment on the effect of disorder in the nontopo- logical SC regime, i.e., for VZ</radicalBig /Delta12 0+μ2for the SC/SM hybrid structure. In Figs. 3(a) and3(b), we show our calculated numerical DOS in the nontopological SC phase (for μ=0) forVZ=0 [Fig. 3(a)] and 1 K [Fig. 3(b)] in the presence of disorder. All parameters other than VZare exactly the same as in Fig. 1. We see that, at least for μ=0, the behavior of the DOS is similar to a classic s-wave SC (even for VZ=1K , which simply reduces the gap from /Delta10=3Kt o2K )w i t h essentially no discernible effect on the DOS. Even for disorderas large as E s>7K>2/Delta10, we do not see any structure developing in the SC DOS gap which remains completelyflat. For V Z=0 [Fig. 3(a)] this is a direct consequence of Anderson’s theorem27where the robustness of the gap arises from time-reversal symmetry. We see in Fig. 3(b) that this behavior persists for small Zeeman fields as long as VZ/lessorsimilar/Delta1. This behavior can be expected based on previous studies onthe bound states of single impurities in SO-coupled nanowiresin proximity to superconductors. 28There it was found that the low-energy subgap states appear for short-ranged impuritiesonly in the topological phase. However, this conclusion mightnot apply to longer-range disorder because, in principle, atany nonzero value of Zeeman potential puddles can lead tothe formation of a pair of low-energy MBSs. In our numerical(b)(a) FIG. 3. (Color online) Disorder-averaged DOS in the nontopolog- ical phase for the semiconductor nanowire in a magnetic field withZeeman splitting V Z=0 and 1 K [panels (a) and (b), respectively) atμ=0 for different disorder strengths characterized by scattering ratesEs=¯h/2τ. work here we restrict ourselves only to zero-range white noise disorder in the semiconductor. The situation, however, changes qualitatively when we consider the limit VZ/greaterorsimilar/Delta1. We show these results in Fig. 4 where VZ=1K ,μ=5 K, and VZ=μ=5Ka r es h o w n for several values of the disorder parameter Es(≡¯h/2τ). All other parameters are the same as in Figs. 1and 2. We note that the situation in Fig. 4describes the nontopological phase since VZ=μ</radicalBig μ2+/Delta12 0. In both panels, large disorder has a strong effect on the SC DOS, shrinking the SCgap considerably. However, in Fig. 4(b), where V Z/greatermuch/Delta10, eventually for Es=0.2 and 1.3 K (and VZ=μ=5K), the DOS seems to “flip” and the dip at E=0 becomes a peak at E=0 with a concomitant vanishing of any SC gap feature in the data. For even larger Es(Es=7Kin Fig. 4) the peak feature in the DOS is suppressed, but there is a clear E=0 DOS peak for intermediate (but still large with Es>/Delta1 0) disorder ( Es=0.2 and 1.3 K in Fig. 4) where the SC gap has disappeared, but a peak has developed in the DOS at zeroenergy. We believe that the zero-energy DOS peak in Fig. 4(b) forE s=0.2 and 1.3 K has the same origin as the physics recently discussed in several publications.16–18The hallmarks of this “trivial” DOS peak (arising from the competition amongspin splitting, SO coupling, and superconductivity) are that (1)it arises only for large Zeeman splitting in the nontopological 064506-5JAY D. SAU AND S. DAS SARMA PHYSICAL REVIEW B 88, 064506 (2013) (b)(a) FIG. 4. (Color online) Disorder-averaged DOS in the nontopolog- ical phase for the semiconductor nanowire in a magnetic field with Zeeman splitting VZ=1 and 5 K [panels (a) and (b), respectively] at μ=5 K for different disorder strengths characterized by scattering ratesEs=¯h/2τ. phase; (2) it occurs only after the SC quasiparticle gap has been completely suppressed by disorder; and (3) it exists onlyin an intermediate disorder range where the superconductingquasiparticle gap has been suppressed, vanishing for largerdisorder and the peak becoming a dip (i.e., the SC gap) forsmaller disorder. In order to specify where precisely in the phase diagram we are obtaining our exact numerical results in the presenceof disorder we finally show in Fig. 5our calculated SC gap as a function of V Z(for zero disorder) for the parameters chosen in our calculations. For μ=0 (i.e., Figs. 1–3), the topological quantum phase transition (TQPT) happens at VZ=/Delta10(=3K in our choice), and our Fig. 1belongs to the topological phase (VZ=5 K, to the right of the TQPT) whereas our Fig. 3 belongs to the nontopological ( VZ=0a n d1K )t ot h el e f t of the TQPT. Results for Fig. 4are obtained for VZ=μ (andVZ<μ), which can never manifest a TQPT since the VZ=/radicalBig μ2+/Delta12 0condition cannot be satisfied except for /Delta1=0 (i.e.,VZ→∞ )—but even the nontopological SC is strongly affected by disorder (for VZ/greatermuch/Delta1) here since the combination of SO coupling and Zeeman splitting makes the Andersontheorem moot. Thus the DOS peak in Fig. 4, associated with antilocalization, 16–19arises purely in the trivial phase with a completely suppressed quasiparticle gap which mightenable its experimental distinction from the MBS peak in thetopological SC phase.FIG. 5. (Color online) Quasiparticle gap ( /Delta1) versus Zeeman potential VZfor different values ( μ=0 and 5 K) for the chemical potential μ. The vanishing of the quasiparticle gap marks the topological quantum phase transition from the nontopological phaseat small V Zto the topological phase at large VZ. IV . DISCUSSION Finally we mention that the SO-coupled semiconductor system thus has two distinct TQPTs in the presence of thesuperconducting proximity effect and Zeeman splitting. Thefirst one is driven by the Zeeman field as originally predicted by Sau et al. 2with the TQPT defined by VZ=/radicalBig μ2+/Delta12 0 assuming a low-disorder situation. The second one is driven by increasing disorder ( Es) in the finite Zeeman splitting situation (i.e.,VZ>/radicalBig μ2+/Delta12 0) where the topological SC phase is destroyed by disorder (for Es/greaterorsimilar/Delta1), leading to a gapless nontopological phase dominated by the Griffiths physics asoriginally envisioned by Motrunich et al. 25Since the effective SC gap for VZ/negationslash=0 depends on VZand, in particular, /Delta1∝V−1 Z forVZ/greatermuch/Delta10(see Refs. 5and 13for details), we expect that there are two distinct magnetic-field-driven TQPTs in thesemiconductor nanowires—the first one is the TQPT predictedin Refs. 2–5taking the system from the trivial s-wave SC (induced by proximity effect) to the (effective p-wave) topo- logical Majorana carrying p-wave SC phase at low disorder (i.e.,/Delta1/greatermuchE s), and then the second one is at much higher Zeeman splitting (so that /Delta1/lessorsimilarEs) where disorder drives the system from a gapless topological SC to a nontopologicalGriffiths phase. It is unlikely that this second (purely disorderdriven) TQPT would be experimentally accessibly since thegapless nature of the SC phase [i.e., black region in Fig. 2 or the situation corresponding to Figs. 1(e)–1(h)] would make the finite temperature of the experimental system behave like avery high temperature (i.e., T/greatermuch/Delta1), making any experimental study of this disorder-driven TQPT difficult, if not impossible.Thus, the effective gapless nature of the system in Figs. 1(e)– 1(g) would make it very unlikely that the DOS peak (which is quite obvious in our theoretical results) could be studiedexperimentally. The best hope for the direct experimentalstudy of the MBS physics is therefore to have a large SC gap[as in Figs. 1(a)–1(d) or in the nonblack region of Fig. 2]i n the topological phase which necessitates having low effectivedisorder ( E s/lessmuch/Delta1), a condition guaranteed by having very clean semiconductor wires and or/very strong SO coupling in 064506-6DENSITY OF STATES OF DISORDERED TOPOLOGICAL ... PHYSICAL REVIEW B 88, 064506 (2013) the material. We add here that our theoretical results presented in this paper cannot be compared (or connected) with theexperiments in any way since we only calculate bulk DOS ofthe system, which cannot be directly probed experimentally. V . CONCLUSION In summary, we have studied the disorder-averaged DOS of a disordered SO-coupled nanowire in proximity to asuperconductor in both the topological and the nontopologicalparameter regime. We find that while some features in the DOSassociated with the superconducting gap in the topologicalphase appear to be in good quantitative agreement with theSCBA from previous work, 13the zero-energy peak persists to a much higher level of disorder than the previous SCBA worksuggests. Consistent with previous results, 13we find that the dips in the DOS associated with the quasiparticle gap disappearfor relatively modest ( Es/greaterorsimilar/Delta1) amounts of disorder. However, a zero-energy peak associated with MBSs generated by theGriffiths effect 23survives to much higher levels of disorder. The DOS peak associated with different levels of disorderarising from the Griffiths effect starting from the topologicalphase and the antilocalization peak 16–19starting from the non- topological phase appear to be different enough that one mightdistinguish them qualitatively. Of course, at this point one doesnot expect a sharp distinction between the peak arising fromthe Griffiths effect and that arising from the antilocalizationeffect because they are zero energy peaks in the DOS in thenontopological phase in the same symmetry class. i.e.. class D. ACKNOWLEDGMENT This work is supported by Microsoft Q, JQI-NSF-PFC, and the Harvard Quantum Optics Center. 1C. W. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. Lett. 01, 160401 (2008); M. Sato, Y . Takahashi, and S. Fujimoto, ibid. 103, 020401 (2009). 2Jay D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010). 3R. M. Lutchyn, Jay D. Sau, and S. Das Sarma, P h y s .R e v .L e t t . 105, 077001 (2010). 4Y . Oreg, G. Refael, and F. von Oppen, P h y s .R e v .L e t t . 105, 177002 (2010). 5J. D. Sau, S. Tewari, R. Lutchyn, T. Stanescu, and S. Das Sarma,P h y s .R e v .B 82, 214509 (2010). 6See, e.g., C. W. J. Beenakker, Annu. Rev. Con. Mat. Phys. 4, 113 (2013), for a review; J. Alicea, Rep. Prog. Phys. 75, 076501 (2012); M. Leijnse and K. Flensberg, Semicond. Sci. Technol. 27, 124003 (2012); T. Stanescu and S. Tewari, J. Phys.: Condens. Matter 25, 233201 (2013). 7V .M o u r i k ,K .Z u o ,S .M .F r o l o v ,S .R .P l i s s a r d ,E .P .A .M .B a k k e r s ,and L. P. Kouwenhoven, Science 336, 1003 (2012). 8M. T. Deng, C. L. Yu, G. Y . Huang, M. Larsson, P. Caroff, and H. Q. Xu,Nano Lett. 12, 6414 (2012). 9A. Das, Y . Ronen, Y . Most, Y . Oreg, M. Heiblum, and H. Shtrikman, Nat. Phys. 8, 887 (2012). 10A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K. Jung, and X. Li, P h y s .R e v .L e t t . 110, 126406 (2013). 11H. O. H. Churchill, V . Fatemi, K. Grove-Rasmussen, M. T. Deng, P .C a r o f f ,H .Q .X u ,a n dC .M .M a r c u s , P h y s .R e v .B 87, 241401(R) (2013). 12P. W. Brouwer, M. Duckheim, A. Romito, and F. von Oppen, Phys. Rev. Lett. 107, 196804 (2011); Phys. Rev. B 84, 144526 (2011).13J. D. Sau, S. Tewari, and S. Das Sarma, P h y s .R e v .B 85, 064512 (2012). 14A. M. Lobos, R. M. Lutchyn, and S. Das Sarma, P h y s .R e v .L e t t . 109, 146403 (2012). 15T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, P h y s .R e v .B 84, 144522 (2011). 16J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Phys. Rev. Lett. 109, 267002 (2012). 17D. Bagrets and A. Altland, Phys. Rev. Lett. 109, 227005 (2012). 18P. Neven, D. Bagrets, and A. Altland, New J. Phys. 15, 055019 (2013). 19D. I. Pikulin, J. P. Dahlhaus, M. Wimmer, H. Schomerus, and C. W.J. Beenakker, New J. Phys. 14, 125011 (2012). 20S. Takei, B. M. Fregoso, H-. Y . Hui, A. M. Lobos, and S. Das Sarma, Phys. Rev. Lett. 110, 186803 (2013). 21R. M. Lutchyn, T. D. Stanescu, and S. Das Sarma, P h y s .R e v .B 85, 140513(R) (2012). 22E. Prada, P. San-Jose, and R. Aguado, Phys. Rev. B 86, 180503(R) (2012); D. Chevallier, P. Simon, and C. Bena, arXiv:1301.7420 . 23I. C. Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B 83, 155429 (2011). 24I. Adagideli, M. Wimmer, and A. Teker, arXiv:1302.2612 . 25O. Motrunich, K. Damle, and D. A. Huse, P h y s .R e v .B 63, 224204 (2001). 26C. H. Lin, J. D. Sau, and S. Das Sarma, Phys. Rev. B 86, 224511 (2012). 27P. W. Anderson, J. Phys. Chem. Solids 11, 26 (1959). 28J. Sau and E. Demler, arXiv:1204.2537 . 064506-7

PhysRevB.74.224431.pdf

Synthesis and study of the heavy-fermion compound Yb 5Pt9 M. S. Kim,1M. C. Bennett,1D. A. Sokolov,1M. C. Aronson,1J. N. Millican,2Julia Y. Chan,2Q. Huang,3 Y. Chen,3,4and J. W. Lynn3 1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120, USA 2Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, USA 3NIST Center for Neutron Research, NIST, Gaithersburg, Maryland 20899, USA 4Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA /H20849Received 29 June 2006; revised manuscript received 28 September 2006; published 26 December 2006 /H20850 We report the synthesis of single crystals of a binary heavy-fermion system Yb 5Pt9. An unusual double phase transition is observed in specific heat CPmeasurements at 0.6 K and 0.65 K, signaling a magnetically ordered ground state. The complete magnetic field-temperature phase diagram of Yb 5Pt9is obtained from the magnetic field dependence of the electrical resistivity /H9267/H20849T/H20850and specific heat CP/H20849T/H20850, and consists of two phase lines terminating at finite temperature critical endpoints. Electrical resistivity and specific heat measurementsshow that the magnetically ordered state is a Fermi liquid with strong electronic correlations, absent in theparamagnetic state. At higher temperatures, strong magnetic anisotropy is observed, which we ascribe to crystalelectric field effects acting on a well localized Yb 3+moment, yielding a well separated doublet ground state, confirmed by inelastic neutron scattering measurements. Our measurements show that Yb 5Pt9is a heavy- fermion compound which is very near a quantum critical point. DOI: 10.1103/PhysRevB.74.224431 PACS number /H20849s/H20850: 75.30.Mb, 75.20.Hr, 71.27. /H11001a I. INTRODUCTION Heavy-fermion systems based on Ce and U intermetallic compounds have attracted much attention, because the com-petition between the intrasite Kondo screening and the inter-site exchange interaction induces many interesting phenom-ena, especially quantum critical points and their associatednon-Fermi-liquid behaviors. 1,2Recently, unconventional su- perconductivity was found in the ferromagnetic heavy fer-mion compounds UGe 2and URhGe.3,4These discoveries have renewed interest in finding heavy-fermion compoundswhich order ferromagnetically, particularly those with quan-tum critical points where superconductivity may compete orcoexist with types of magnetic order. Although theRuderman-Kittel-Kasuya-Yosida /H20849RKKY /H20850interaction which is responsible for magnetic order in heavy-fermion com-pounds enables both ferromagnetic and antiferromagnetic or-der, heavy-fermion compounds with ferromagnetic orderingare rarely reported. Thus, the discovery of ferromagneticheavy fermion compounds is important due to the richness ofthe physics, particularly near ferromagnetic quantum criticalpoints. 5 The electronic configuration of f13for Yb can be consid- ered the hole analogue to the f1electronic configuration of Ce, suggesting that the same phenomena can be realized in Yb-based compounds. However, much less experimental at-tention has focused on Yb-based compounds, because thesynthesis of Yb-intermetallic compounds is difficult due tothe high vapor pressure of Yb. Most of the Yb-based Kondolattice systems investigated so far orderantiferromagnetically, 6–9but several have been found to or- der at least partly ferromagnetically.10–12We report here the synthesis of a Yb-based intermetallic compound, Yb 5Pt9, and present the results of transport, magnetic, and thermal prop-erties.II. EXPERIMENTAL DETAILS Single-crystalline samples of Yb 5Pt9were grown using a Pb flux. The single crystals are hexagonal prisms with typicaldimensions of 0.5 /H110030.5/H110031 mm. 3Electron-probe mi- croanalysis was carried out on polished single crystals usinga Cameca SX100 microprobe system with elemental Yb andPt standards. The 5:9 atomic ratio of Yb:Pt is uniform withina systematic error of 2% over the whole crystal surface. In-deed, the electron backscattering image indicates homoge-neous composition without surface phases or inclusions.Several other Yb-Pt binary compounds have beenreported 14–20and their magnetic properties described,21–23 but Yb 5Pt9has not previously been reported. Single crystal x-ray diffraction measurements indicate that Yb 5Pt9crystal- lizes in an orthorhombic Cmmm /H20849No. 65 /H20850space group with lattice parameters of a=13.5550 /H208495/H20850,b=13.3720 /H208495/H20850, c=5.6540 /H208493/H20850Å, and V=1024.83 /H208498/H20850Å3, with Z=19. The structure was solved and refined using the SHELXL-97 program.24Corrections were made for absorption and extinc- tion, and the atomic positions were refined with anisotropicdisplacement parameters. Atomic coordinates and thermalparameters are listed in Table I, and a schematic representa- tion of the unit cell of Yb 5Pt9is shown in Fig. 1. Laue x-ray diffraction revealed that the orthorhombic caxis is along the prism axis of the crystal. The electrical resistivity was measured by the conven- tional four-probe method between 0.4 and 300 K in zerofield and in magnetic fields as large as 0.1 T. Magnetizationand magnetic susceptibility were measured using a quantumdesign magnetic property measurement system in magneticfields up to 7 T. Specific heat measurements were performedusing a quantum design physical property measurement sys-tem for temperatures between 0.4 and 70 K. Inelastic neu-tron scattering measurements were performed on a sampleprepared by powdering four grams of Yb 5Pt9single crystals.PHYSICAL REVIEW B 74, 224431 /H208492006 /H20850 1098-0121/2006/74 /H2084922/H20850/224431 /H208496/H20850 ©2006 The American Physical Society 224431-1These experiments were performed at the NIST Center for Neutron Research using the BT-7 double focusing triple-axisspectrometer with a fixed final energy of 14.7 meV, and atseveral fixed wave vectors. We used the double focusing PGmonochrometer, and focusing analyzer to get the maximumintensity. Neutron diffraction measurements were carried outon the identical powder sample using the BT-1 powder dif-fractometer at NIST, equipped with a Ge 311 vertically fo-cusing monochrometer, at a neutron wavelength of 2.079 Å. III. RESULTS AND DISCUSSION The temperature dependence of the electrical resistivity /H9267 is metallic, as shown in the inset of Fig. 2/H20849a/H20850. The room temperature value of /H9267is 35/H9262/H9024cm. With decreasing tem- perature, /H9267decreases approximately linearly to about 25 K,suggesting that electron-phonon scattering dominates the re- sistivity, at least for temperatures above the Debye tempera-ture, which we estimate is about 250 K. Increasing curvatureis observed in /H9267/H20849T/H20850below 25 K, suggesting that magnetic scattering mechanisms are relatively more important at low temperatures. As shown in Fig. 2/H20849a/H20850,/H9267is nearly independent of temperature in the range between 2 and 25 K, although aTABLE I. Crystallographic data for Yb 5Pt9. Atom Site xy z U eq/H20849Å2/H20850a Yb1 2b 1/2 0 0 0.0071 /H2084910/H20850 Yb2 2d 1/2 1/2 1/2 0.0060 /H208498/H20850 Yb3 8p 0.32260 /H2084914/H20850 0.63083 /H2084919/H20850 0 0.0062 /H208497/H20850 Yb4 8q 0.36889 /H2084915/H20850 0.82253 /H2084915/H20850 1/2 0.0043 /H208497/H20850 Pt1 2a 1/2 1/2 0 0.0060 /H208498/H20850 Pt2 2c 1/2 0 1/2 0.0071 /H208499/H20850 Pt3 8n 1/2 0.69230 /H2084916/H20850 0.7502 /H208492/H20850 0.0037 /H208497/H20850 Pt4 8o 0.30769 /H2084913/H20850 0 0.7499 /H208492/H20850 0.0033 /H208497/H20850 Pt5 8p 0.38884 /H2084912/H20850 0.83037 /H2084914/H20850 0 0.0041 /H208496/H20850 Pt6 8q 0.33043 /H2084911/H20850 0.61154 /H2084915/H20850 1/2 0.0046 /H208497/H20850 aUeqis defined as one-third of the trace of the orthogonalized Uijtensor. FIG. 1. Schematic representation of the unit cell of Yb 5Pt9crys- tallizing in an orthorhombic Cmmm /H20849No. 65 /H20850. There are 20 Yb atoms and 36 Pt atoms in the unit cell of Yb 5Pt9. FIG. 2. /H20849a/H20850The temperature dependence of the electrical resis- tivity/H9267/H20849T/H20850for Yb 5Pt9below 4 K. The arrow indicates the magnetic phase transition at 0.66 K. The inset shows the resistivity over anexpanded range of temperatures. /H20849b/H20850 /H9267as a function of T2in mag- netic fields of 0, 0.02, 0.05, 0.07, and 0.1 T. Solid line indicates fitsto /H9267/H20849T/H20850=/H92670+AT2.KIM et al. PHYSICAL REVIEW B 74, 224431 /H208492006 /H20850 224431-2rounded decrease is observed below 2 K, perhaps due to par- tial Kondo compensation. A sharp drop is found at 0.66 K,marked with an arrow as a magnetic transition temperature /H20849T C/H9267/H20850. Figure 2/H20849b/H20850shows /H9267plotted as a function of T2below 1 K in magnetic fields of 0, 0.02, 0.05, 0.07, and 0.1 T. As the magnetic field increases, the transition temperature TC/H9267 gradually decreases and a negative magnetoresistance is found above TC/H9267, indicating that the magnetic transition is sensitive even to the weakest magnetic fields. Such a sharpdrop and negative magnetoresistance is characteristic of theloss of spin disorder scattering found at a magnetic phase transition. In the low temperature ordered state T/H33355T C/H9267,w e find that /H9267/H20849T/H20850is well described by the Fermi liquid expres- sion /H9267/H20849T/H20850=/H92670+AT2, with the residual resistivity /H92670 =1.83 /H9262/H9024cm. The low value of /H92670attests to the high quality and crystallinity of our samples. The coefficient Ais 1.31/H9262/H9024cm/K2, comparable with that found in the heavy fermion superconductor UPt 3/H20849=1.6/H9262/H9024cm/K2/H20850.25This sug- gests that the quasiparticles in the magnetically ordered state are strongly interacting. Figure 3/H20849a/H20850shows the magnetic susceptibility /H9273/H20849T/H20850and the inverse magnetic susceptibility 1/ /H9273/H20849T/H20850of Yb 5Pt9with the measuring field of 0.1 T oriented along the caxis /H20849/H9273/H20648/H20850and perpendicular to the caxis /H20849/H9273/H11036/H20850./H9273/H11036is 7 times larger than /H9273/H20648 at 1.8 K, indicating that the magnetization easy axis is per- pendicular to the caxis. With decreasing temperature, 1/ /H9273/H20648decreases linearly down to 100 K and at lower temperatures deviates below the linear behavior. 1/ /H9273/H11036deviates above the linear extrapolation below 100 K. These strong departuresfrom Curie-Weiss behavior can result from crystal electricfield /H20849CEF /H20850effects, from the Kondo effect, and from precur- sor effects related to magnetic order. Considering the largemagnetic anisotropy, we believe that CEF effects are particu-larly strong, resulting from the low crystal symmetry of theorthorhombic structure. As shown in Fig. 3, the magnetic susceptibility /H9273/H20849T/H20850is well described by the Curie-Weiss ex- pression between 100 K and 300 K, for both field orienta- tions. The Curie temperatures found from these fits are −149and 27 K for /H9273/H20648and/H9273/H11036, respectively. The effective moments of 4.41 and 4.24 /H9262B, respectively, are close to 4.54 /H9262Bex- pected for free Yb3+, indicating that the magnetic moments are well localized and trivalent at high temperatures. Theinset of Fig. 3/H20849b/H20850shows the magnetization for fields up to 7 T applied along both crystal axes, for temperatures from1.8 K to 7 K. A nonlinear curvature is observed which be-comes increasingly pronounced at the lowest temperatures. At 1.8 K, the magnetic moment saturates above 2 T at thevalues of 2.0 and 2.5 /H9262Bfor fields parallel and perpendicular to the caxis, respectively. While this saturation can be char- acteristic of incipient ferromagnetic order, Fig. 3/H20849b/H20850shows instead that the magnetization collapses onto a universalcurve when the field is scaled by the absolute temperature.Further, the collapsed data are well described by a S=1/2 Brillouin function, which suggests that the curvature in themagnetization curves reflects normal paramagnetic behavior.We note that the saturation moments are much reduced fromthe fully degenerate value of 4.5 /H9262B, but are consistent with values found in other Yb-based compounds with the ortho-rhombic structure. 6,9,10,12,13This suggests that magnetic order in Yb 5Pt9arises from a doublet ground state caused by the orthorhombic CEF. In order to determine the ground state and energy level scheme for the Yb3+moments, we have measured the spe- cific heat CPof our sample for temperatures from 0 . 4Kt o7 0K /H20849Fig.4/H20850. The magnetic part of the specific heat was isolated from CPby estimating the phonon contribution Cph/H20849T/H20850.Cph/H20849T/H20850was calculated by fitting the high temperature CPby the Debye model. The Debye temperature, /H9008D =236 K is extracted from the linear extrapolation of the plot ofCP/TvsT2above 6 K, with the electronic specific heat coefficient /H9253h=15 mJ/Yb mol K2, as shown in the inset of Fig.4. We note that the Debye temperature obtained from the specific heat data is in good agreement with the value of/H11011250 K deduced from the linear temperature dependence of resistivity above 150 K /H110110.6/H9008 Din the inset of Fig. 2/H20849a/H20850. The magnetic part of the specific heat is presented on a logarith-mic temperature scale in Fig. 4, displaying a sharp lambda- like anomaly near the magnetic phase transition at 0.66 K,and a gradual decrease at higher temperatures. The promi-nent tail of the heat capacity between T Cand 2 K may be ascribed to the specific heat of an S=1/2 Kondo compen- sated moment, with a Kondo temperature TKof approxi- mately 1 K.26The inset of Fig. 5shows the electronic part of the specific heat in the ordered phase Cel=CP−Cph=/H9253lT, where the large value of /H9253l=355 mJ/Yb mol K2reveals that magnetic order in Yb 5Pt9results in the formation of a Fermi FIG. 3. /H20849a/H20850The temperature dependence of the magnetic suscep- tibility /H9273and inverse magnetic susceptibility 1/ /H9273in the magnetic field B=0.1 T parallel to the caxis /H20849/H9273/H20648/H20850and perpendicular to the c axis /H20849/H9273/H11036/H20850./H20849b/H20850A plot of the magnetization MvsB/T. The inset shows the magnetic field dependence of Mat low temperatures with fields as large as 7 T applied along both sample axes.SYNTHESIS AND STUDY OF THE HEAVY-FERMION … PHYSICAL REVIEW B 74, 224431 /H208492006 /H20850 224431-3liquid comprised of strongly interacting quasiparticles. Com- bining these results with those of the resistivity measure- ments, we find that A//H9253l2is 1.2 /H1100310−5/H9262/H9024cm K−2//H20849mJ mol−1K−2/H208502, very close to the value of 1.0 /H1100310−5/H9262/H9024cm K−2//H20849mJ mol−1K−2/H208502, known as the Kadowaki-Woods ratio, observed in many heavy-fermion systems.27 The temperature dependence of the entropy confirms the crystal field scenario which we proposed on the basis ofmagnetization measurements. The temperature dependenceof the entropy Swas determined by integrating C mag/T, and is overplotted in Fig. 4. Just above the magnetic phase tran- sition, S=0.65 Rln 2 and gradually increases to Rln 2 at 2 K. Sremains approximately constant until /H1101110 K, showing that magnetic order develops from a doublet ground state. Above10 K, Sbegins to increase again and finally attains the value Rln 4 at 70 K. This is consistent with the observation of the Schottky anomaly at 40 K, and the fit to this anomaly shownin Fig. 4finds that the splitting between the ground doublet and the first excited state, also a doublet, is /H1101170 K.Direct evidence for the splitting of the ground and excited doublets comes from inelastic neutron scattering measure-ments, shown in Fig. 6. A sharp peak is observed at 8.27 meV, whose intensity decreases with wave vector inapproximate agreement with the Yb 3+magnetic form factor while the energy is approximately independent of wave vec-tor, indicating that this is a crystal field excitation. The scat-tering at higher energies is found to increase in intensity withincreasing wave vector suggesting that this scattering is pre-dominantly lattice dynamical in origin. Data at a wave vectorof 2 Å −1were also collected at a series of temperatures from 5 K to 300 K. There is only a modest decrease in the inten-sity of this crystal field transition up to /H11011100 K, which in- dicates that this is the first crystal field excited state. At higher temperatures this peak decreases in intensity andbroadens substantially, and is essentially unobservable atroom temperature. The phonon scattering, which extendsinto the 8 meV range /H20849and below /H20850, increases steadily with increasing Tas expected for the thermal population factor for bosons. We note that the splitting between the ground stateand the first excited state deduced from heat capacity mea-surements /H2084970 K /H20850is in quite reasonable agreement with the direct measurement from inelastic neutron scattering mea- surements /H208498.3 meV, 96.4 K /H20850, validating our crystal field scheme. The low temperature specific heat C P, shown in Fig. 5, demonstrates that the magnetic phase transition in Yb 5Pt9is highly unconventional. The specific heat reaches a hugevalue of 12 J/Yb mol K, and consists of two large peaks at0.6 K and 0.65 K, followed by a shoulder at 0.66 K, where asharp drop is found in /H9267. However, we see from Fig. 5that/H9267 has no anomaly at either of the temperatures where the spe- cific heat displays maxima. The specific heat anomalies arenot ideally lambda-like, but no hysteresis was found in thespecific heat on cooling and heating, indicating that thephase transitions are at best weakly first order. We repeatedthese measurements on several other crystals from differentpreparation batches, and the results in Fig. 5were exactly reproduced in each case. Neutron diffraction experiments have been used to inves- tigate the nature of the magnetic order at the 0.6 K and FIG. 4. The logarithmic temperature dependence of the specific heat CPfrom which we have subtracted an estimated phonon con- tribution Cph, and the associated entropy S. The dashed line indi- cates the interpolation of /H20849CP−Cph/H20850/TvsT2below 0.4 K. The dot- ted line indicates a fit to a Schottky expression, yielding a peak atabout 40 K. The inset represents the plot of C P/TvsT2from which the electronic part of the specific heat, /H9253Tis extracted. FIG. 5. The temperature dependence of the specific heat CPand electrical resistivity /H9267below 0.8 K. FIG. 6. Inelastic neutron scattering spectra at 5 K for wave vec- tors of 2, 2.9, and 4 Å−1. The first excited crystal field level is observed at 8.27 /H208495/H20850meV. The scattering extending to higher ener- gies is dominated by the phonon density of states.KIM et al. PHYSICAL REVIEW B 74, 224431 /H208492006 /H20850 224431-40.65 K phase transitions in Yb 5Pt9. Figure 7compares the powder pattern obtained at 1.8 K, where the sample is para-magnetic, to the powder pattern at 0.3 K, where the sampleis magnetically ordered. We note that the 1.8 K powder pat-tern is in full agreement with the structure determined fromsingle crystal x-ray diffraction. While the details of the mag-netic structure will be presented elsewhere, 28the magnetic ordering wave vector was found to be wholly commensurate,adding to the intensities of the nuclear Bragg peaks withreduced temperature. Every peak at 0.3 K can be indexedwithin the reported orthorhombic structure. There are twopossible interpretations of this finding. One possibility is thatthe ground state is ferromagnetic, or alternatively, given thatthere are four inequivalent Yb atoms per unit cell, it is pos-sible that the order is antiferromagnetic with a wave vectorcommensurate with the high temperature reciprocal lattice.While a definitive answer to this question awaits a determi-nation of the magnetic structure, 28the field dependence of the heat capacity suggests that the magnetic ground state hasa ferromagnetic component. As expected for ferromagnetic states, Fig. 8shows that both peaks in the specific heat are strongly suppressed bymagnetic field, with the upper transition being more sensitiveto field than the lower transition. Above 0.05 T, the hightemperature peak evolves into a rounded shoulder, and thelow temperature peak is broadened and shifted to lower tem-peratures. A field of only 0.1 T is sufficient to totally sup-press both peaks, yielding an extremely broad but still fielddependent maximum. Above 0.1 T, both peaks merge into asingle broad peak, whose intensity decreases and shifts tohigher temperatures with increasing field. The magnetic fielddependences of the two specific heat transition temperaturesare plotted in Fig. 9. We have plotted the jumps /H9004Cin the specific heat at the two transitions in the inset of Fig. 9. The thermodynamic signatures of the magnetic phase transitions/H9004C highand/H9004Clowgo to zero at fields of 0.07 T for the upper transition, and 0.1 T for the lower temperature transition. Itis remarkable that the transition temperatures themselves re-main finite in these fields, suggesting that the phase linesreproduced in Fig. 9terminate in critical endpoints.IV. CONCLUSION We have reported here the synthesis of single crystals of Yb5Pt9, which we show is a heavy-fermion magnet. Para- magnetic behavior is found at high temperatures, involvingthe full Yb 3+moment. With decreasing temperature, the mo- ment degeneracy is lifted by the crystal electric field, result-ing in strong magnetic anisotropy and a doublet ground statewell separated from the excited doublet states. Magnetic or-der results in a low temperature Fermi liquid state, revealedby resistivity and specific heat measurements to have quasi-particle interactions as strong as those found in well-studiedheavy-fermion compounds like UPt 3. Neutron diffraction measurements find that the ordering wave vector of theground state is commensurate with the reciprocal lattice inthe paramagnetic state. Specific heat measurements find anunusual magnetic ordering transition which proceeds by two FIG. 7. A comparison of the neutron powder patterns for Yb 5Pt9 at 1.8 K /H20849a/H20850, 0.3 K /H20849b/H20850, and their difference /H20849c/H20850. All the peaks can be indexed in the crystal unit cell, indicating that the magnetic order ispossibly ferromagnetic. FIG. 8. The temperature dependence of the specific heat CPin magnetic fields as large as 1 T. /H9004Chighand/H9004Cloware the heat capacity steps at the higher and lower temperature phase transitions. FIG. 9. The phase diagram of Yb 5Pt9.TC/H9267/H20849/H17005/H20850is the transition temperature found in /H9267/H20849T,B/H20850in Fig. 2/H20849b/H20850, and Tmaxhigh/H20849/H17034/H20850andTmaxlow /H20849/L50098/H20850are the transition temperatures taken from the specific heat data /H20849Fig. 8/H20850. The inset shows the magnetic field dependence of the height of two peaks, /H9004Chighand/H9004Clowat the high and low transi- tion temperatures, respectively.SYNTHESIS AND STUDY OF THE HEAVY-FERMION … PHYSICAL REVIEW B 74, 224431 /H208492006 /H20850 224431-5steps, the first at 0.65 K and an additional one at 0.6 K. We find that very modest magnetic fields are sufficient to sup-press both magnetic ordering transitions to lower tempera-tures, while the specific heat anomaly at each transition dis-appears even more rapidly. The magnetic phase diagram thusconsists of two magnetic phase lines, which are at bestweakly first order, terminating at finite temperature and/orfinite field critical endpoints. We conclude that while the or-dering temperatures of Yb 5Pt9are quite low, this system ef- fectively avoids the generation of a quantum critical point,where the critical endpoints are suppressed to zero tempera-ture. This fact suggests that the typical signs of quantum criticality should be absent in this material, and indeed wefind evidence for only conventional Fermi liquid phenomenain the magnetically ordered phase.ACKNOWLEDGMENTS The authors are grateful to C. Henderson for assistance with the electron-probe microanalysis, which was performedat the University of Michigan Electron Microbeam AnalysisLaboratory /H20849EMAL /H20850. Work at the University of Michigan is supported by the National Science Foundation under GrantNo. NSF-DMR-0405961. Work at Louisiana State Universityis supported by Grant No. NSF-DMR-0237664 and by theAlfred P. Sloan Foundation. Identification of commercialequipment in the text is not intended to imply recommenda-tion or endorsement by the National Institute of Standardsand Technology. 1S. Doniach, in Valence Instabilities and Related Narraw Band Phenomena , edited by R. D. Parks /H20849Plenum, New York, 1977 /H20850, p. 169. 2B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 /H208491988 /H20850. 3S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian,P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheikin, D. Braith-waite, and J. Flouquet, Nature /H20849London /H20850406, 587 /H208492000 /H20850. 4D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J. P. Brison, E. Lhotel, and C. Paulsen, Nature /H20849London /H20850413, 613 /H208492001 /H20850. 5See, for instance D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod. Phys. 77, 579 /H208492005 /H20850; H. v. Lohneysen, A. Rosch, M. Vojta, and P. Wolfle, cond-mat/0606317 /H20849to be published /H20850. 6D. T. Adroja, B. D. Rainford, S. K. Malik, M. Gailloux, and K. A. Gschneidner, Phys. Rev. B 50, 248 /H208491994 /H20850. 7P. Gegenwart, J. Custers, C. Geibel, K. Neumaier, T. Tayama, K. Tenya, O. Trovarelli, and F. Steglich, Phys. Rev. Lett. 89, 056402 /H208492002 /H20850. 8E. Morosan, S. L. Bud’ko, P. C. Canfield, M. S. Torikachvili, and A. H. Lacerda, J. Magn. Magn. Mater. 277, 298 /H208492004 /H20850. 9M. A. Avila, M. Sera, and T. Takabatake, Phys. Rev. B 70, 100409 /H20849R/H20850/H208492004 /H20850. 10P. Bonville, P. Bellot, J. A. Hodges, P. Imbert, G. Jéhanno, G. Le Bras, J. Hammann, L. Leylekian, G. Chevrier, P. Thuéry, L.D’Onofrio, A. Hamzic, and A. Barthélémy, Physica B 182, 105 /H208491992 /H20850. 11K. Katoh, G. Terui, Y. Niide, S. Yoshii, K. Kindo, A. Oyamada, M. Shirikawa, and A. Ochiai, J. Alloys Compd. 360, 225 /H208492003 /H20850. 12Y. Muro, Y. Haizaki, M. S. Kim, K. Umeo, H. Tou, M. Sera, andT. Takabatake, Phys. Rev. B 69, 020401 /H20849R/H20850/H208492004 /H20850. 13O. Trovarelli, C. Geibel, R. Cardoso, S. Mederle, R. Borth, B. Buschinger, F. M. Grosche, Y. Grin, G. Sparn, and F. Steglich,Phys. Rev. B 61, 9467 /H208492000 /H20850. 14J. L. Moriarity, J. E. Humphreys, R. O. Gordon, and N. C. Baen- ziger, Acta Crystallogr. 21, 840 /H208491966 /H20850. 15W. Bronger, J. Less-Common Met. 12,6 3 /H208491967 /H20850. 16Q. Johnson, R. G. Bedford, and E. Catalano, J. Less-Common Met. 24, 335 /H208491971 /H20850. 17B. Erdmann and C. Keller, J. Solid State Chem. 7,4 0 /H208491973 /H20850. 18A. Iandelli and A. Palenzona, J. Less-Common Met. 43, 205 /H208491975 /H20850. 19J. LeRoy, J-M. Moreau, D. Paccard, and E. Parthe, Acta Crystal- logr., Sect. B: Struct. Crystallogr. Cryst. Chem. B34,9/H208491978 /H20850. 20A. Iandelli and A. Palenzona, J. Less-Common Met. 80,7 1 /H208491981 /H20850. 21F. Oster, B. Politt, E. Braunm, H. Schmidt, J. Langen, and N. Lossau, J. Magn. Magn. Mater. 63-64 , 629 /H208491987 /H20850. 22H. Schmidt, B. Politt, F. Oster, and E. Braun, J. Less-Common Met. 134, 187 /H208491987 /H20850. 23P. Bonville, P. Imbert, G. Jehanno, F. Oster, and B. Politt, J. Phys.: Condens. Matter 2, 4727 /H208491990 /H20850. 24G. M. Sheldrick, SHELXL-97, Program for Crystal Structure Re- finement, University of Göttingen, Germany, 1997. 25A. de Visser, J. J. M. Franse, and A. Menovsky, J. Magn. Magn. Mater. 43,4 3 /H208491984 /H20850. 26V. T. Rajan, J. H. Lowenstein, and N. Andrei, Phys. Rev. Lett. 49, 497 /H208491982 /H20850. 27K. Kadowaki and S. B. 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PhysRevB.80.014112.pdf

Generation and growth of sp3-bonded domains by visible photon irradiation of graphite Hiromasa Ohnishi1,*and Keiichiro Nasu1,2 1Solid State Theory Division, Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba 305-0801, Japan 2Graduate University for Advanced Study and CREST JST, 1-1 Oho, Tsukuba 305-0801, Japan /H20849Received 28 April 2009; revised manuscript received 18 June 2009; published 23 July 2009 /H20850 We theoretically study possible domain type collective structural changes with interlayer /H9268bonds and periodic buckling patterns, induced by visible photon irradiation onto graphite. The adiabatic potential-energysurface relevant to the generation and growth of this domain in the multidimensional local distortion coordi-nates of carbons is clarified by means of the semiempirical Brenner theory. The electronic states at variouslocal minima in this adiabatic potential surface are also calculated, in a good agreement with the observedscanning tunneling microscopy image. We will show that the first diaphite domain can be generated by a fewvisible photons, and grow up to become various nanosized domains by successive photoexcitations. DOI: 10.1103/PhysRevB.80.014112 PACS number /H20849s/H20850: 64.70.Nd, 73.20.At, 78.67. /H11002n I. INTRODUCTION There are a variety of materials, even though they are composed only of carbons, and the differences among themare due to the two types of possible bondings: sp 2andsp3. There has been interest in the macroscopically condensedphase of carbons at absolute zero temperature and atmo-spheric pressure, in which the most stable state is the graph-ite, characterized by the sp 2bonds, with the diamond, char- acterized by the sp3bonds, being 0.02 eV/atom higher than the graphite.1By the total-energy calculation with the local- density approximation /H20849LDA /H20850, the energy barrier between the graphite and the diamond is estimated to be about0.3–0.4 eV /atom. 2–4Thus, a macroscopic or tremendous energy is required to realize the direct graphite-diamond con-version. Hence the corresponding real synthesis is the appli-cation of a high temperature and a high pressure /H208493000 °C, 15 GPa /H20850, 5,6or strong x ray.7,8 In contrast to this situation, the recent experiment by Ka- nasaki et al.9may elucidate a new aspect for this graphite- diamond conversion. After the irradiation of visible photonsonto graphite, a nanoscale structural change is detected, andit may come from sp 3-like bonds with the contraction of the interlayer distance. Here we briefly recapitulate the essentialexperimental aspects: /H208491/H20850the exciting laser, with an energy of 1.57 eV, should be polarized perpendicular to the graphitelayers while the one polarized parallel to the graphite layergives no effect. It means that only the interlayer charge-transfer excitation can trigger this process. /H208492/H20850The exciting light should be a femtosecond pulse while the picosecondone gives almost no contribution. It means that only a tran-sient generation of an excited electronic wave packet in thesemimetallic continuum can efficiently trigger this process./H208493/H20850The process is quite nonlinear but less than the ten- photon process. /H208494/H20850The scanning tunneling microscopy /H20849STM /H20850measurement shows a nanoscale domain, in which one third of carbons sinks down and the residual two thirdsrise up from the layer in each six membered ring. Carbonsthat have sunk down are expected to form /H9268bonds between graphite layers, whereby the sp3-like structure is induced. /H208495/H20850 The resultant domain includes more than 1000 carbons and isstable for more than 10 days at room temperature. The electron-diffraction investigation has been performed by Ra-man et al. 11They revealed that the interlayer distance after the photoirradiation has contracted up to 1.9 Å from theoriginal 3.35 Å. From the above experimental facts, a possible mechanism of the present process is as follows: 10when the femtosecond pulse is irradiated onto graphite, an electron-hole pair, span-ning two layers, is generated. This electron-hole pair may beunstable, being easily dissipated into the semimetallic con-tinuum of the graphite as plus and minus carriers. Undoubt-edly, this is the most dominant relaxation channel of photo-generated carriers as long as the graphite is a goodconductor. However, by a small but finite probability, thiselectron hole is expected to be bounded with each other, justlike exciton, through the interlayer Coulomb attraction. Thisexciton-like electron-hole pair will be self-localized at a cer-tain point of the layer by contracting the interlayer distanceonly around it. Thus, a local interlayer /H9268bond is formed. This photoinduced phenomenon is quite in contrast to the aforementioned direct graphite-diamond conversion. Theunique structure is realized only through the iterative localdomain formation by a successive visible photon excitation, to which only a finite and microscopic order of energy isrequired. The unique domain structure, thus appeared, is notthe conventional diamond but an intermediate state betweenthe graphite and the diamond. Hence, we will refer it as“diaphite 12” hereafter for the convenience of explanation. The present phenomenon should be clarified by the con- cept of the photoinduced structural phase transition/H20849PSPT /H20850. 13–15Electrons, just after being excited by lights, are usually in the Franck-Condon state with no lattice motionfrom the starting ground state. However, they afterward in-duce a motion of surrounding lattice because of a suddenchange in their charge distribution. As a result, the wholesystem reaches a new state within the excited state as a co-operative phenomenon including these electrons and the lat-tice. The present process surely starts from the graphite, and the destination may be the diamond. In between, however,there may be various local minima in the adiabatic potential-energy surface spanned by the multidimensional lattice dis-PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 1098-0121/2009/80 /H208491/H20850/014112 /H208498/H20850 ©2009 The American Physical Society 014112-1tortion coordinates of carbons. Then, in this paper, we will be mainly concerned with the minimal domain; in the sense, itis expected to be the most nearest potential minimum fromthe starting graphite in this multidimensional potential sur-face. The growth process of this minimal domain will also beclarified. The whole process of this PSPT will be a complicated one. Then, in this paper, we will clarify the adiabatic path ingenerating the minimal /H20849or the first /H20850diaphite domain from the perfect graphite. We will also see several more higherenergy domains that can grow up from this first diaphite/H20849FD/H20850domain. We start with the two-layer AB-stacking graph- ite, and the local lattice distortion is introduced for variousdomain formations. Since the diaphite domain is constantlyimmersed in the vast semimetallic graphite, we expect thatall carbons are almost neutral throughout the deformationprocess. To theoretically calculate such a diaphite domain ina vastly surrounding graphite, the whole carbon clustershould be as large as possible. In our previous application oftheab initio LDA theory, 16,17the cluster size is limited to around 300 carbons per layer due to the restriction from thecomputational cost. In that case, it is very difficult to elimi-nate the cluster size effect. Hence, in this paper, the totalenergy of the system is estimated by means of the semi-empirical Brenner potential, 1with the cluster that consists of about 104carbons per layer. The electronic states related with the diaphite are also clarified, by means of thetight-binding approximation with the Slater-Kosterparametrization. 18 This paper is organized as follows: in Sec. II, methods of the total energy and electronic state calculations are ex-plained. In Sec. III, the adiabatic path in generating the first diaphite domain is clarified. In Sec. IV, the diaphite domain aggregation process is clarified. In Sec. V, few higher energy states are clarified in connection with the growth process ofthe first diaphite domain. Finally, in Sec. VI, we make a conclusion. II. METHOD Let us clarify the adiabatic potential surface related with domain type collective structural changes by means of thesemiempirical Brenner potential. 1The total binding energy by this method is given as Etot=1 2/H20858 i,j/H20849/HS11005i/H20850/H20853VR/H20849rij/H20850−B¯ijVA/H20849rij/H20850/H20854f/H20849rij/H20850, /H208491/H20850 where VRandVAare repulsive and attractive radial two-body pair potentials. While the bond order function B¯ijtakes into account the three- or four-body force as a function of thebond angle, and f/H20849r ij/H20850is the cutoff function of the force range with the bond length /H20849rij/H20850between ith and jth carbons. We have adopted the Brenner potential Iin the original method,1 which is empirically deduced to reproduce almost all exist- ing experimental and theoretical data of various carbon clus-ters. In fact, this parametrization well reproduces the bindingenergy of the graphite and the diamond. We have also esti-mated the adiabatic barrier between the graphite and the dia-mond under the uniform transformation condition. 2It is es- timated to be 0.35 eV/atom, which agrees well with the LDAresults. 2,4 To take into account the local domain formation, we have performed the calculation on a large carbon cluster with atwo-layer AB-stacking structure, consisting of about 10 4car- bons per layer. Although the system does not have the trans-lational symmetry due to the local lattice distortion, we haveapplied the two-dimensional periodic boundary condition inthe directions parallel to the layer. Thus, carbons at aroundthe cluster edge are fixed. We start from the complete graphite structure with the intralayer bond length 1.42 Å and the interlayer bond length3.35 Å. The local deformation of the lattice starting fromthis complete graphite is introduced by taking into accountthe following displacements: /H208491/H20850the disk type intrusion from the graphite surface, perpendicular to the graphite layers,with the following trial function for the first layer in Fig. 1as z= /H20902z¯−/H9004z /H20849l/H11349L0/H20850, z¯−/H9004ztanh /H20851/H9258/H20849l−L/H20850/H20852−1 tanh /H20851/H9258/H20849L0−L/H20850/H20852−1/H20849l/H11022L0/H20850,/H20903/H208492/H20850 where z¯is the zcoordinate of the original graphite, and l/H20849=/H20881x2+y2/H20850is the distance from the deformation center in the xyplane. Throughout the simulation, we restrict degrees of1st layer 2nd laye r FIG. 1. /H20849Color online /H20850A schematic picture of the trial distortion pattern. /H20849a/H20850The disk type intrusion from the graphite layer. /H20849b/H20850The buckling pattern inside the domain, inferred from the STM image.One third of carbons sink down with the amplitude /H9254and two thirds of carbons rise up with the amplitude /H9254/2 in each six membered ring, to keep its center of mass unchanged. Then the total intrusionfrom the graphite layer is given as /H20849/H9004z+ /H9254/H20850. The local shear dis- placement is introduced to the opposite direction in each layer. Itsvalue at the domain center is referred to /H9004x.HIROMASA OHNISHI AND KEIICHIRO NASU PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 014112-2the deformation to keep the total center of the mass of the cluster unchanged. Hence the deformation pattern in the sec-ond layer is just the inverse of the first layer against the xy plane at the center of the mass of the cluster. /H208492/H20850The ampli- tude /H9254/H20849/H113500/H20850of the buckling: two thirds of carbons rise up, and one third of carbons sink down in each six memberedring, keeping its center of mass unchanged. /H208493/H20850The local shear displacement /H9004x/H20849/H110220/H20850: each layer is shifted to the op- posite direction to improve the interlayer stacking sequence.A schematic picture of this deformation pattern is given inFig. 1. Here we define the domain as the region that the carbons sunk down more than /H9004z/2, and the number of car- bons in this domain is referred to as N d. All the above defor- mation parameters are variationally determined by systemati-cally changing values. Electronic states are estimated by means of the tight- binding approximation with the Slater-Kosterparametrization. 18Values of binding energies: /H9280sand/H9280p, and hopping integrals: Vss/H9268,Vsp/H9268,Vpp/H9268, and Vpp/H9266, are used, simi- lar to those in Ref. 17, which are fitted to reproduce the LDA result for the graphite and the diamond. Then, /H9280p−/H9280s =8.346, Vss/H9268=−4.143, Vsp/H9268=5.689, Vpp/H9268=7.758, and Vpp/H9266= −2.489 in eV unit. The hopping integrals are taken into ac-count up to nearest neighbors, and scaled by the Harrison’sr −2rule:19 V/H20849r/H20850=V/H20849r0/H20850/H20873r0 r/H208742 , /H208493/H20850 where r0is the reference bond length and fixed at 1.54 Å. III. GENERATION OF THE FIRST DIAPHITE DOMAIN The adiabatic energy generating the FD domain is clari- fied. The estimated adiabatic path to this diaphite domain isgiven in Fig. 2. The energy is referenced from that of the starting complete graphite throughout this paper. The local minimum, corresponding to the FD, has ap- peared with the energy 1.700 eV at L 0=1.878 Å, L =2.588 Å, /H9258=0.20, /H9004z=0.75 Å, /H9254=0.12 Å, and /H9004x =0.002 Å, after overtaking the barrier with an energy of1.925 eV. Hence the N dbecomes 48 atoms. The averaged interlayer distance becomes 1.85 Å, as shown in Fig. 2/H20849c/H20850, which is in a good agreement with the result of the electron-diffraction investigation: 1.9 Å. 11 The height of the barrier on the FD domain is a little larger than the energy of the irradiated visible light. Then itneeds only two or three light quanta to overtake this barrier.We can, thus, conclude that the energy barrier between thestarting graphite and the FD can be overtaken by few visiblephotons. Once the system reaches this FD domain, the struc-ture is well stabilized against the thermal fluctuation ataround room temperature. It should be noted that the realprocess of the generation of the diaphite domain is theFranck-Condon excitation and the lattice relaxation there-from, as schematically depicted in Fig. 2/H20849b/H20850. The local density of state /H20849LDOS /H20850of this FD is given in Fig. 2/H20849d/H20850, together with that of the graphite. This LDOS is calculated at the central carbon site, which forms the inter-layer /H9268bond. Peaks around /H110061 eV have newly appeared only in the case of the diaphite. The emergence of these peak structures is easily under- stood from the new bonding between distant graphitic layers.Hence, the system shows the pseudogap, which is character-istic to the insulator immersed in the semimetallic graphite. IV. DIAPHITE DOMAIN AGGREGATION Thus, the FD domains may be generated randomly in the graphite surface. They, however, are expected to aggregate ormerge afterward into more larger domains during the relax-ation. To clarify these situations, we have estimated the adia- batic energy against the change in the distance between twoFD domains. The result is given in Fig. 3. We can consider two aggregating directions due to the periodicity of the di-aphite buckling, and those are depicted in right-hand side ofFig.3. The /H20849a/H20850direction is parallel to the row of carbons that has sunk down from the original graphite surface. The dis-tance between centers of each FD is referred to as R. The calculation has been performed at only R’s, which are com- mensurate with the lattice structure. At larger Rthan 34.08 Å, the energy is just twice of the energy of the FD domain because of no mutual interference.While at R=4.26 Å, these two domains are completely ag- gregated, sharing their boundaries. These two situations areseparated by a barrier. The height of this barrier is only about0.07 eV, being sufficiently smaller than the original one forthe generation itself /H20851Fig.2/H20849b/H20850/H20852. Just after the generation of the FD domain, the whole system is still during the latticerelaxation, being not in the equilibrium completely, althoughit is around the new minimum, which is shown in Fig. 2/H20849b/H20850. Hence, during the relaxation, the whole system will overtakethe above energy barrier /H208490.07 eV /H20850. Consequently, neighbor- ing two FD domains can easily merge by using the residualenergy during the lattice relaxation. AtR=4.26 Å, the deformation parameters for the aggre- gated state along the /H20849a/H20850direction is slightly changed from those of the FD to /H9004z=0.75, /H9254=0.14, and /H9004x=0.006. Al- though this state has a higher energy than the segregated one,this state can also be realized through the aforementionedprocess. The aggregated state along the /H20849b/H20850direction has the same deformation parameters with the FD. Thus, the FD do-main can grow up to become more larger domains. V. HIGHER ENERGY STATES In the previous sections, we have clarified the generation of the FD domain, and its naive aggregation process. Theestimated FD domain can grow up to become larger do-mains, passing through various local minima of the adiabaticpotential surface by further additional visible photon excita-tions, and can finally reach the nanosized domain that isobserved in the experiment. 9 In this section, we will theoretically clarify such typical higher energy states by increasing the domain radius /H20849L0/H20850, shown in Fig. 1.GENERATION AND GROWTH OF sp3-BONDED … PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 014112-3The calculated higher energy states are summarized in Table I. We have shown four higher energy states, which have different domain radii. Here let us classify them intotwo groups for the convenience of the explanation. The firstgroup is represented with the symbol /H20849S/H20850, which has the large intrusion, /H9004z=0.75, and the small buckling amplitude, /H9254 /H110110.2. The other is represented with symbol /H20849L/H20850, which has the relatively small intrusion, /H9004z/H110110.5, and the large buck- ling amplitude, /H9254=0.40, compared with the group /H20849S/H20850. The adiabatic path to the /H20849S1/H20850structure from the FD is given in Fig. 4. The /H20849S1/H20850structure with the energy of 10.96 eV at L0=5.68 Å has appeared after overtaking the barrier at L0=4.97 Å with the energy of 11.41 eV along the L0direc-tion. Other optimized parameters are given in Table I. Other higher energy states that are given in Table Ihave also ob- tained as local minima of the adiabatic potential surface inthe multidimensional distortion coordinates although theiradiabatic paths are not given. The states shown in Table Ihave higher energy than the FD domain. As mentioned before, a tremendous energy willbe necessary for the instantaneous generation of such largedomains if it will be the conventional uniform phase transi-tion. For example, the energy barrier to the local minimumfor the /H20849S1/H20850structure along the /H20849/H9004z+ /H9254/H20850direction is higher than that along the L0direction in Fig. 4/H20849a/H20850. However, these states will be realized by iterative visible photon excitations/CID1z+δ(Å)/CID1x(Å) Total energy (eV) 0.20 Total energy (eV) /CID1z+δ(Å)Franck-Condon excitationLattice relaxation (0.81, 1.925) (0.87, 1.700) 2.144 1.431 1.4201.6100.180 0.1801.6101.850 LDOS (arb.units) Energy(eV)(b) (a) (c) (d) FIG. 2. /H20849Color online /H20850The adiabatic path to the FD. The energy is referenced from that of the starting complete graphite. /H20849a/H20850The adiabatic energy surface as a functional of the total intrusion /H20849/H9004z+/H9254/H20850and the local shear displacement /H9004x./H20849b/H20850The cross sectional view of the adiabatic energy surface at /H9004x=0.002, corresponding to the red line in /H20849a/H20850. Parenthesized values represent /H20849/H9004z+/H9254, total energy /H20850at crossing points of dashed lines, respectively. In the real process, the diaphite is realized through the Franck-Condon excitation and the latticerelaxation process therefrom. /H20849c/H20850The estimated diaphite structure. The values are given in angstrom unit /H20849Å/H20850./H20849d/H20850The LDOS of the graphite and the diaphite. The LDOS is calculated at the central carbon site, which forms the interlayer /H9268bond. The zero of the energy is the Fermi energy of each state.HIROMASA OHNISHI AND KEIICHIRO NASU PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 014112-4from the FD, only passing through the various local mini- mum of the adiabatic potential surface, as schematically de-picted in Fig. 4/H20849b/H20850. The states with larger domain radius take a larger local shear displacement, /H9004x, in both groups. One can easily infer that the shear displacement makes the interlayer stacking se-quence effective since it makes the sp 3structure more natu- ral. In the present case, however, the shear displacement isintroduced in the local region of the lattice, which leads to astress within a layer, and consequently the system loses thebinding energy. Hence, the effectiveness of the local sheardisplacement is determined by the competition between theincrease in the shear stress and the energy gain due to theimprovement of the interlayer stacking sequence. The scaleof the shear stress within a layer is almost saturated if it isintroduced with a sufficiently large radius /H20851/H11015/H20849L+10 /H20850Å/H20852. The energy gain by the improvement of the interlayer stack-ing sequence becomes larger and larger according to the in-crease in the domain radius L 0because of the increase in interlayer /H9268bonds. Then, states with large domain radius favor those with large /H9004x. In the case of the group /H20849S/H20850, the structure of each layer is closer to the graphite due to thesmall buckling. Then, the group /H20849S/H20850needs a large domain radius than the group /H20849L/H20850to overcome the shear stress. Estimated structures for the /H20849S2/H20850and the /H20849L2/H20850are given in Figs. 5/H20849a/H20850and5/H20849b/H20850, respectively. The averaged interlayer dis- tance for the /H20849S2/H20850becomes 1.85 Å, which is in a good agree-ment with the experiment 11structure. The averaged inter- layer distance for the /H20849L2/H20850becomes 2.23 Å. Although this value is larger than that of the experiment, the structure isstabilized by the large buckling amplitude. The estimatedsuperstructure of the /H20849L2/H20850type diaphite domain around the domain center is given in Fig. 5/H20849c/H20850as an example. One canR (a)(b) sinking down rising up (4.26, 3.310)(4.26, 3.424) (34.08, 3.400)(a)-direction (b)-directionTotal energy (eV) R(Å) FIG. 3. /H20849Color online /H20850The adiabatic energy against the change in the distance between two FD domains. The parenthesized valuesrepresent /H20849R, total energy /H20850at the crossing points of dashed lines. In the top of the figure, the situation is schematically depicted. Theapproaching directions are represented in the right-hand side of thefigure, as the /H20849a/H20850and /H20849b/H20850directions, respectively. The red and yellow circles represent carbons that sink down and rise up, respectively, toindicate the direction of the diaphite buckling. TABLE I. Estimated higher energy states. EminandEtoprepresent the energy at the local minimum and the barrier top, respectively. L0 /H20849Å/H20850L /H20849Å/H20850 /H9258/H9004z /H20849Å/H20850/H9254 /H20849Å/H20850/H9004x /H20849Å/H20850Etop /H20849eV/H20850Emin /H20849eV/H20850Nd /H20849atom /H20850 /H20849S1/H20850 5.68 6.39 0.22 0.75 0.14 0.01 11.41 10.96 192 /H20849S2/H20850 11.36 12.07 0.40 0.75 0.21 0.27 77.91 77.75 408 /H20849L1/H20850 4.97 5.68 0.30 0.45 0.40 0.02 14.43 13.90 84 /H20849L2/H20850 6.39 7.10 0.20 0.56 0.40 0.22 36.03 30.44 2160.95Total energy (eV ) L0(Å)⊿z+δ(Å) (1.878, 1.70)(5.68, 10.96)(4.97, 11.41) L0(Å)Iterative visible photon excitationsTotal energy (eV) (b)(a) FIG. 4. /H20849Color online /H20850The adiabatic path to the /H20849S1/H20850structure from the FD domain. /H20849a/H20850The adiabatic potential-energy surface as a functional of the L0and /H20849/H9004z+/H9254/H20850./H20849b/H20850The cross sectional view of the adiabatic energy surface, along the minimal ascending path, corre-sponding to the red line in /H20849a/H20850. The dashed curve schematically represents different valleys that belong to different lattice distortioncoordinates. The FD domain will grow up to become a larger do-main, only passing through the various local minima of the adia-batic potential surface by iterative visible photon excitations.GENERATION AND GROWTH OF sp 3-BONDED … PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 014112-5see that the interlayer bonds by the diaphite buckling are formed at the domain center. The LDOSs for above states are given in Fig. 6.I nt h e diaphite buckling, two types of carbon exist: /H20849a/H20850carbons forming the interlayer /H9268bonds by sinking down from the distorted layer, referenced as the /H9251carbon. /H20849b/H20850Carbons forming the surface by rising up from the distorted layer,referenced as the /H9252carbon. The LDOSs for the /H9251and/H9252 carbons20are individually plotted in Fig. 6.As given in Fig. 6/H20849a/H20850, the/H9251carbons in the group /H20849S/H20850show a straightforward opening of the pseudogap according to theincrease in the domain radius because the effect of the sur-rounding semimetallic graphite becomes weaker and weakerwith the increase in the domain radius. Although the /H9251car- bons in the group /H20849L/H20850also show a pseudogap and its opening, some peak structures have appeared around the Fermi en-ergy. These peak structures can also be seen in the scanningtunneling spectroscopy /H20849STS /H20850measurement, 9and its origin 1.850 1.4351.660 1.4201.8000.315 0.3151.570 1.420 1.7341.496 1.5420.600 0.6001.430 2.23 (c) (b) (a) FIG. 5. /H20849Color online /H20850The /H20849a/H20850and /H20849b/H20850represent the estimated structure for the /H20849S2/H20850and the /H20849L2/H20850, respectively. The values are given in angstrom unit /H20849Å/H20850. The superstructure around the domain center for the /H20849L2/H20850structure is given in /H20849c/H20850. The red and green carbons represent the/H9251and/H9252carbons, respectively. The explanation of the /H9251and/H9252carbons is given in the text. (a)α-carbon Energy (eV)LDOS (arb.units)(b)α-carbon (c)β-carbon (d)β-carbonLDOS (arb.units) LDOS (arb.units) LDOS (arb.units)Energy (eV) Energy(eV) Energy (eV) FIG. 6. /H20849Color online /H20850The LDOS for each higher state. The zero of the energy is the Fermi energy of each state. The LDOS of /H20849a/H20850the /H9251carbon for the group /H20849S/H20850,/H20849b/H20850the/H9251carbon for the group /H20849L/H20850,/H20849c/H20850the/H9252carbon for the group /H20849S/H20850, and /H20849d/H20850the/H9252carbon for the group /H20849L/H20850, with that of the graphite.HIROMASA OHNISHI AND KEIICHIRO NASU PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 014112-6will be argued in connection with the LDOS of the /H9252carbon after a while. The LDOSs for the /H9252carbons are given in Figs. 6/H20849c/H20850and 6/H20849d/H20850. In both groups, the LDOS around the Fermi energy is larger than that of the graphite. This behavior can be under-stood as follows: the origin of the different LDOSs betweenthe /H9251and/H9252carbons is due to the termination of the sp3 structure at the surface. Hence, even in the diaphite, surface /H9266bonds still exist. This /H9266bonds are weaker than that of the graphite and will become more weaker according to the sta-bilization of the sp 3structure. Then, the LDOS of the /H9252 carbons around the Fermi energy is larger than that of the graphite. This behavior of the /H9252carbons can be understood as the origin of the bright STM image of the diaphitedomain 9because the brightness of the STM image is propor- tional to the surface LDOS of the sample at the Fermilevel. 21 Finally, let us clarify about the LDOS of the /H9251carbons in the group /H20849L/H20850. As mentioned above, surface /H9266bonds between /H9252carbons still exist, and its nature is still itinerant just like the graphite. Hence, the /H9251and/H9252carbons are not simply distinguishable, and one can easily find an analogy in thepeak structures around the Fermi energy between Figs. 6/H20849b/H20850 and6/H20849d/H20850,/H20851and also between Figs. 6/H20849a/H20850and6/H20849c/H20850/H20852. Then, the peak structures around the Fermi energy are due to itinerantsurface /H9266electrons. It should be noted that the origin of the above peak structure is entirely different from that of the FD. The LDOS of the /H9251carbon of the /H20849L2/H20850diaphite domain is compared with the STS measurement in Fig. 7. The LDOS of the /H20849L2/H20850diaphite domain has shown peaks at around −0.65 and 0.4 eV although the experimental LDOS has peaks ataround −0.4 and 0.3 eV. Even though there is such a smalldifference, the estimated theoretical LDOS well reproducesthe characteristics of the experimental result. VI. CONCLUSION We have, thus, theoretically clarified the adiabatic path to the diaphite domain from the starting graphite by means ofthe semiempirical Brenner theory. The FD domain is nucle-ated by few visible photon excitations, and stable against thethermal fluctuation at around room temperature. This FD domain can grow up to become various larger domains by further additional visible photon excitations.Along this line, the states with various larger domain radiihave also been clarified, wherein the local shear displace-ment becomes gradually dominant. Electronic states of diaphite domains have also clarified by means of the tight-binding approximation. The LDOS ofthem has shown a site-dependent difference between the /H9251 and/H9252carbons due to the termination of the diaphite structure at the surface. The behavior of the /H9252carbon can be under-stood as the origin of the bright STM image of the diaphite domain. The LDOS of the /H9251carbon has shown the pseudogap, characteristic to the insulator immersed in thesemimetal. These two orders are closely related to eachother, and their origin is the itinerancy of /H9266electrons in the diaphite. The LDOS of the /H20849L2/H20850structure has well reproduced the experimental one obtained by the STS measurements. Hence,we expect that the diaphite domain, observed in theexperiment, 9have the /H20849L2/H20850type buckling structure. Thus, we have clarified the generation and growth of the diaphite domain by means of the semiempirical Brennertheory. The present method has led the smaller adiabatic bar-rier for the generation of the FD domain than our previousresult by the LDA calculation. 16,17In the LDA calculation, it is very difficult to eliminate the cluster size effect, as men-tioned before, while the present method can practically treatthe infinite size cluster. Although there are such differences,our main conclusion about the generation of the FD domainis not affected by them. ACKNOWLEDGMENTS The authors thank K. Tanimura, J. Kanasaki, and E. Inami for presenting their results prior to publication and valuablediscussions. This work is supported by the Ministry of Edu-cation, Culture, Sports, Science and Technology of Japan, thepeta-computing project, and Grant-in-Aid for Scientific Re-search /H20849S/H20850, Contract No. 19001002, 2007.LDOS (arb.units) (dI/dV)/(I/V)Energy (eV) Energy(eV)graphite graphitediaphite diaphite (L2) FIG. 7. /H20849Color online /H20850Comparison of the experimental /H20849upper figure /H20850and the theoretical /H20849lower figure /H20850LDOS. The experimental LDOS is obtained by the STS measurement /H20849Ref. 9/H20850. The theoreti- cal LDOS is the same as that for the /H9251-carbon of the /H20849L2/H20850diaphite.GENERATION AND GROWTH OF sp3-BONDED … PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 014112-7*ohni@post.kek.jp 1D. W. Brenner, Phys. Rev. B 42, 9458 /H208491990 /H20850. 2S. Fahy, S. G. Louie, and M. L. Cohen, Phys. Rev. B 34, 1191 /H208491986 /H20850. 3S. Fahy, S. G. Louie, and M. L. Cohen, Phys. Rev. B 35, 7623 /H208491987 /H20850. 4Y. Tateyama, T. Ogitsu, K. Kusakabe, and S. Tsuneyuki, Phys. Rev. B 54, 14994 /H208491996 /H20850. 5F. Bundy, J. Chem. Phys. 38, 631 /H208491963 /H20850. 6T. Irifune, A. Kurio, S. Sakamoto, T. Inoue, and H. Sumiya, Nature /H20849London /H20850421, 599 /H208492003 /H20850. 7F. Banhart, J. Appl. Phys. 81, 3440 /H208491997 /H20850. 8H. Nakayama and H. Katayama-Yoshida, J. Phys.: Condens. Matter 15, R1077 /H208492003 /H20850. 9J. Kanasaki, E. Inami, K. Tanimura, H. Ohnishi, and K. Nasu, Phys. Rev. Lett. 102, 087402 /H208492009 /H20850. 10L. Radosinski, K. Nasu, J. Kanazaki, K. Tanimura, A. Radosz, and T. Luty, Molecular Electronic and Related Materials- Control and Probe with Light , edited by T. Naito /H20849unpublished /H20850.11R. K. Raman, Y. Murooka, C. Y. Ruan, T. Yang, S. Berber, and D. Tománek, Phys. Rev. Lett. 101, 077401 /H208492008 /H20850. 12See Research highlight, Nature /H20849London /H20850458, 129 /H208492009 /H20850. 13K. Nasu, Photo-induced Phase Transitions /H20849World Scientific, Singapore, 2004 /H20850. 14K. Nasu, Rep. Prog. Phys. 67, 1607 /H208492004 /H20850. 15K. Yonemitsu and K. Nasu, Phys. Rep. 465,1/H208492008 /H20850. 16H. Ohnishi and K. Nasu, J. Phys.: Conf. Ser. 148, 012059 /H208492009 /H20850. 17H. Ohnishi and K. Nasu, Phys. Rev. B 79, 054111 /H208492009 /H20850. 18J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 /H208491954 /H20850. 19W. A. Harrison, Electronic Structure and The Properties of Sol- ids (The Physics of The Chemical Bond) /H20849Dover, New York, 1980 /H20850. 20Note that the present classification of the /H9251and/H9252carbons is different from the often used one, in that the /H9251and/H9252carbons are referred to as carbons having a neighbor in the adjacent layer ornot at the surface of the AB-stacking graphite, respectively. 21J. Tersoff and D. R. Hamann, Phys. Rev. B 31, 805 /H208491985 /H20850.HIROMASA OHNISHI AND KEIICHIRO NASU PHYSICAL REVIEW B 80, 014112 /H208492009 /H20850 014112-8

PhysRevB.95.224305.pdf

PHYSICAL REVIEW B 95, 224305 (2017) Nontrivial topology of cubic alkali bismuthides I. P. Rusinov,1,2I. Yu. Sklyadneva,3,4,5R. Heid,4K.-P. Bohnen,4E. K. Petrov,1,2Yu. M. Koroteev,5 P. M. Echenique,3,6,7and E. V . Chulkov2,3,6,7 1Tomsk State University, 634050 Tomsk, Russian Federation 2St. Petersburg State University, 199034 St. Petersburg, Russian Federation 3Donostia International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain 4Karlsruher Institut für Technologie, Institut für Festkörperphysik, D-76021 Karlsruhe, Germany 5Institute of Strength Physics and Materials Science, pr. Academicheskii 2/1, 634021 Tomsk, Russian Federation 6Departamento de Física de Materiales, Facultad de Ciencias Químicas, UPV/EHU, Apartado 1072, 20080 San Sebastián/Donostia, Basque Country, Spain 7Centro de Física de Materiales CFM - Materials Physics Center MPC, Centro Mixto CSIC-UPV/EHU, 20018 San Sebastián/Donostia, Spain (Received 27 January 2017; revised manuscript received 19 April 2017; published 28 June 2017) We report an ab initio study of the effect of pressure on vibrational and electronic properties of K 3Bi and Rb3Bi in the cubic Fm3mstructure. It is shown that the high-temperature cubic phase of K 3Bi and Rb 3Bi is dynamically unstable at T=0 but can be stabilized by pressure. The electronic spectra of alkali bismuthides are found to possess the bulk band touching at the Brillouin zone center and an inverted spin-orbit bulk bandstructure. Upon hydrostatic compression the compounds transform from the topologically nontrivial semimetal(K 3Bi)/metal (Rb 3Bi) into a trivial semiconductor (metal) with a conical Dirac-type dispersion of electronic bands at the point of the topological transition. In K 3Bi the dynamical stabilization occurs before the system undergoes the topological phase transition. DOI: 10.1103/PhysRevB.95.224305 I. INTRODUCTION The Bi-based binary compounds, A3Bi (A=Na, K, Rb), have recently attracted much attention due to their unusualelectronic properties. Using both first-principles calculationsand effective model analysis [ 1], as well as angle-resolved photoemission spectroscopy (ARPES) [ 2,3], it was shown that in the low-temperature crystal structure (a hexagonal phase)these semimetals possess 3D Dirac states protected by crystalsymmetry [ 4] with an inverted spin-orbit bulk band structure and nontrivial Fermi arcs on the surfaces [ 1,5]. Moreover, the compounds can be driven into various topologically distinctphases, such as topological insulator, topological metal (withnontrivial Fermi surfaces), or Weyl semimetal by breaking ofsymmetry [ 6]. On the other hand, it is well known that K 3Bi and Rb3Bi undergo a reversible transition from a hexagonal low-temperature phase to a cubic one, Fm3m,a t2 8 0◦C and 230◦C, respectively [ 7,8]. Na 3Bi, which crystallizes in a hexagonal structure regardless of temperature, becomes cubicunder hydrostatic compression [ 9,10]. As shown in Ref. [ 11] and in the present work, in the cubic structure these compoundsare also semimetals with an inverted band order between theconduction and valence bands at the center of the Brillouinzone (BZ). Such materials can be driven to topologicallydifferent phases by pressure-induced modifications of the bandstructure [ 12]. The prediction of new topological phases by using ab initio computational tools has become an effective way to search materials with new properties. The unique elec-tronic structure of the compounds also opens up opportunitiesto study topological phase transitions. The conical dispersionin such Dirac semimetals is realized at the critical point ofa topological phase transition and is achieved by tuning of asystem parameter (pressure), which makes the band structureeasy to engineer suitably [ 13,14].Inspired by recent theoretical studies of Na 3Bi and using first-principles band-structure calculations, we show that uponhydrostatic compression cubic K 3Bi and Rb 3Bi undergo a transition from a topological semimetal (K 3Bi)-metal (Rb 3Bi) to a conventional semiconductor (K 3Bi)-metal (Rb 3Bi). To examine the dynamical stability of K 3Bi and Rb 3Bi in the cubic phase and the effect of pressure on the lattice dynamics,we have calculated the phonon dispersion with and withouthydrostatic compression. II. CALCULATION DETAILS The electronic structure calculations were performed in the mixed-basis (MB) pseudopotential approach [ 15–17] with the exchange and correlation energy functional evaluatedwithin the generalized gradient approximation [ 18]. We used semilocal norm-conserving pseudopotentials [ 19], with p semicore states treated as valence states for alkali-metal atoms(3pfor K and 4 pfor Rb). The MB scheme employs a combination of local functions and plane waves. To expandvalence states, we used a plane wave basis set with a kineticenergy cutoff of 22 Ry ( ∼300 eV) augmented by local p- and d-type functions for alkali-metal atoms and s- andp-type local functions for Bi at each atomic site. The Fourier expansion ofthe crystal potential and charge density was found to be safelytruncated at 54 Ry. Integrations over the Brillouin zone (BZ)were performed by sampling an 8 ×8×8 mesh corresponding to 512 kpoints in the BZ and using a Gaussian broadening with a smearing parameter of 0.01 eV . To obtain phonon dispersions, the density-functional perturbation theory was used [ 20]. Its implementation in the MB scheme is described in Ref. [ 17]. We performed both scalar relativistic and including-spin-orbit-effect calcu-lations. Spin-orbit coupling (SOC) was incorporated into the 2469-9950/2017/95(22)/224305(7) 224305-1 ©2017 American Physical SocietyI. P. RUSINOV et al. PHYSICAL REVIEW B 95, 224305 (2017) pseudopotential scheme via the Kleinman’s formulation [ 21] and was treated fully self-consistently [ 22]. The dynamical matrices were computed on a (4 ×4×4) grid of wave vectors, and then the phonon frequencies along high-symmetrydirections of the BZ were obtained with the standard Fourierinterpolation scheme [ 23]. A. Wilson loop method To demonstrate the topological character of the compounds, we applied the Wilson loop (also known as Wannier chargecenter) method, proposed in Refs. [ 24,25]. The Wilson loop (ˆW) is defined as ˆW μ=Pexp/bracketleftbigg −i/integraldisplayπ −πdkˆAμ(k)/bracketrightbigg , (1) wherePis the path-ordering operator. ˆA(k) is the non-Abelian Berry connection, ˆAmn μ(k)=−i/angbracketleftumk|∂kμ|unk/angbracketright, (2) where m,n are band indexes, μis a space index, and unk are the Bloch eigenstates of the Hamiltonian. Practically, a tight-binding Wilson loop and discretized expression [ 26]a r e used, ˆWmn μ=/angbracketleftBigg um(k0)|k0+Gμ/productdisplay k=k0Pocc(k)|un(k0+Gμ)/angbracketrightBigg , (3) where Pocc(k)=/summationtextocc i=1|ui(k)/angbracketright/angbracketleftui(k)|,ui(k)=ui(k+Gμ)f o r eachμ, and Gμare basis vectors in the reciprocal space. The product is found on a finite mesh of closely adjacent kpoints which form a closed loop line. It should be noted that this line isperpendicular to the direction of kpoints where Wilson loops are calculated. The eigenspectrum of ˆW mn μdefines Wannier charge centers and makes it possible to graphically investigatethe topology of compounds. III. RESULTS AND DISCUSSION A. Structural parameters The high-temperature phase of K 3Bi and Rb 3Bi,Fm3m, is composed of four face-centered sublattices mutually shiftedby ( 1 4,1 4,1 4) along the body diagonal with two of them being symmetry equivalent. Atomic positions are shown in Table I. Also shown are the optimized lattice parameter obtained bytotal energy minimization together with available experimentaldata. In K 3Bi, the theoretical lattice parameter is very close to the experimental value, the deviation is ∼0.15%. For Rb 3Bi a TABLE I. Positions of atoms (O, octahedral hole; T, tetrahedral hole) and lattice parameters (in ˚A). The theoretical lattice constants, ath, were obtained with the account of SOC. The experimental parameters, aexpt, are taken from Refs. [ 7,27]. K(Rb) 3Bi Position athaexpt K(Rb)O(1/2,1/2,1/2) K 3Bi 8.818 8.805 K(Rb)T±(1/4,1/4,1/4) Rb 3Bi 9.218 8.98 Bi (0,0,0)larger underbinding of ∼2.6% is obtained. Such a deviation is rather typical for generalized gradient approximation–Perdew-Burke-Ernzerhof calculations. B. Dynamical stability The phonon spectra of K 3Bi and Rb 3Bi obtained at the equilibrium volume ( V0) show that the cubic phase is dynami- cally unstable at zero temperature. The phonon dispersions aregiven in Figs. 1(c) and1(d). The right panels show the density of phonon states. Since the unit cell contains four atoms, thereare 12 vibrational modes. The harmonic calculation yieldsimaginary phonon frequencies at the BZ boundary for twotransverse acoustic (TA) modes around the X point and forone TA mode near the K and U symmetry points. The modesare shown as negative ones in the figure. The correspondingatomic displacements are given schematically in Figs. 1(e) and1(f). The unstable modes are dominated by displacements of those K (Rb) atoms which occupy octahedral sites [K(Rb) O; see Table I]. To explore the effect of pressure, we have calculated the phonon spectra of K 3Bi and Rb 3B at different volumes. It is found that upon volume decrease, all phonon modes experi-ence an upward shift and at a volume of V∼0.92V 0(P∼ 1G P a )i nK 3Bi and at V∼0.82V0[P∼2.5G P a( ∼1G P a ) with respect to V0(Vexp)] in Rb 3Bi the TA modes in the phonon spectrum become stable. It means that at low temperaturesthe cubic Fm 3mphase of K 3Bi and Rb 3Bi is stabilized by pressure. Another alkali bismuthide, Na 3Bi, also becomes cubic under hydrostatic compression [ 9,10] (at 0.7–1.0 GPa) although the compound crystallizes in a hexagonal structureregardless of temperature. Unlike K 3Bi and Rb 3Bi, the cubic structure of Na 3Bi is still dynamically stable at ambient pressure, as shown by lattice dynamics calculations [ 10]. To show the structural transition under pressure, we have also calculated the enthalpy difference between the cubicand the hexagonal phases. The calculations were carried outwithin the projected augmented-wave approach realized in the V ASP code [ 28]. Lattice parameters and atomic positions were optimized for each pressure using the L-BFGS algorithm to solve unconstrained minimization problems [ 29]. The data for three possible hexagonal phases are presented in Fig. 2.L i k e Na3Bi, both compounds were first found to crystallize in the P63/mmc structure (Na 3As-type) [ 30]. For K 3Bi, two other hexagonal phases were then reported: P3c1 (anti-LaF 3-type, Ref. [ 31] and P63cm(Cu 3P-type; Ref. [ 32]). As compared to the P63/mmc phase, the unit cell of P3c1 and P63cm structures is increased from 8 atoms per unit cell to 24 as aresult of a distorted K(Rb)-Bi honeycomb lattice [ 33]. The calculated enthalpies show that (i) the P 3c1 and P63cm hexagonal structures are thermodynamically preferable whencompared to the earlier reported P6 3/mmc phase: /Delta1H (P= 0) forP3c1 andP63cmis about 0.01 meV /f.u. (in K 3Bi) and 0.04 meV /f.u. (in Rb 3Bi) smaller as compared to that for the P63cmstructure; (ii) the structural transition to the cubic phase indeed occurs above ∼1 GPa (with respect to the experimental volume). For Na 3Bi, the P3c1 phase was also experimentally confirmed by Raman measurements [ 34]. The observed num- ber of IR active modes ruled out the P63/mmc symmetry. 224305-2NONTRIVIAL TOPOLOGY OF CUBIC ALKALI BISMUTHIDES PHYSICAL REVIEW B 95, 224305 (2017) FIG. 1. (a) Atomic positions and (b) high-symmetry points in the bulk Brillouin zone. (c),(d) Phonon dispersions of K 3Bi (c) and Rb 3Bi (d) calculated with SOC are shown by black lines for the equilibrium theoretical volume V0a n db yg r a yl i n e sf o r V/V 0≈0.9( K 3Bi) and V/V 0≈0.8( R b 3Bi). The right panels show the density of phonon states at V0. The contributions coming from the motion of different atoms are given in corresponding colors. (e),(f) Atomic displacements for two TA modes at X (e) and one TA mode at K (d). Alkali-metal atoms areshown by solid (in the octahedral sites) and open (in the tetrahedral sites, they are out of the figure plane) circles. C. Electronic structure 1. Electronic structure at ambient pressure The calculated electronic spectra of K 3Bi and Rb 3Bi in the low-temperature P3c1 structure (a hexagonal phase) are shown in Figs. 3(a) and 3(d). The electronic bands were obtained with SOC included. The band dispersions are similarto those reported for the P6 3/mmc structure [ 6]. As follows from the figure, the hexagonal K 3Bi and Rb 3Bi are metallic with small electron pockets around the /Gamma1point, while Na 3Bi is a semimetal [ 1,6,33]. Like in the case of Na 3Bi, the conduction and valence bands are inverted at the bulk BZ center where theK/Rbsband is located at lower energy than the Bi p x,pyband. The band inversion at the /Gamma1point indicates a topologically nontrivial character of the electronic structure, which is also (a) (b) FIG. 2. Pressure-dependent enthalpy per formula unit (f.u.) of the hexagonal structures relative to that of the cubic one ( /Delta1H)f o rK 3Bi (a) and Rb 3Bi (b).manifested in the existence of topological surface states. The states with a Dirac-type crossing at the /Gamma1point are clearly seen on the (0001) surface [see Figs. 3(b) and 3(e)]i n s i d e the projected valence bands. However, another feature, a pairof Dirac nodes (band crossings) at the Fermi level on therotational k zaxis (the /Gamma1-A symmetry direction in momentum space), shows up only in K 3Bi. The crossing points are fourfold degenerate due to the time-reversal and inversion symmetries. The bulk Dirac nodes on the rotational axis away from the /Gamma1 point indicate that the Dirac states are symmetry-protected bythe threefold rotation crystal symmetry [ 3] which remains in theP 3c1 structure although the K/Bi sheets are buckled due to the shifting of alkali-metal atoms [ 1,6,33]. The projected bulk Dirac nodes and the nontrivial surface states (Fermi arcs)can be visible on the K 3Bi (01 10) surface [see Fig. 3(c)], where they are separated in the momentum space. So K 3Bi in the low-temperature hexagonal phase is a 3D topologicalDirac metal with two Dirac cones. Unlike K 3Bi, the electronic spectrum of Rb 3Bi shows only a band inversion in the BZ center. We could not find any indication of a Dirac-type bandcrossing on the rotational axis along the /Gamma1-A line [Fig. 3(f)]. It should be noted that the electronic properties of alkali bismuthides in the cubic structure have been studied beforewithin semirelativistic calculations [ 35]. Figure 4shows the electronic spectra of K 3Bi and Rb 3Bi obtained with the inclusion of spin-orbit coupling (a) and (b) and also displaysthe influence of SOC on the electronic bands (c) and (d) inthe vicinity of the BZ center. Without taking into account thespin-orbit coupling, the K 3Bi compound is a trivial semimetal with a small gap (see Table II)a tt h e /Gamma1point. The electronic 224305-3I. P. RUSINOV et al. PHYSICAL REVIEW B 95, 224305 (2017) FIG. 3. Electronic bands of (a) K 3Bi and (d) Rb 3Bi in the hexagonal ( P3c1) structure. Red circles (lines) indicate K(Rb) sstates. (b),(c),(e),(f) Projected spectral functions calculated at the equilibrium volume for the (0001) and (01 10) surfaces of K 3Bi (b),(c) and Rb 3Bi (e),(f). The projection is shown by a color scale reflecting the localization in the outermost surface layer (in units of states/eV). TSS means atopological surface state, DP means a Dirac point, and FAS is a Fermi arc state. structure of Rb 3Bi is characterized by a small electron pocket around /Gamma1and a hole pocket in the vicinity of the Xpoint. In both compounds the lowest conduction band at the /Gamma1point is composed of s-type electronic states and is located slightly above the three p-like bands, which are degenerate at the BZ center. Accounting of SOC modifies the dispersion of valencebands by splitting off the pbands at /Gamma1into two degenerate bands, /Gamma1 8, and a split-off state, /Gamma17so that the valence and conduction bands are now touching at the BZ center. SOCalso affects the s-type electronic states, band /Gamma1 6;t h e ya r e pushed below the Bi pbands, /Gamma18. As a result the energy gap εS(/Gamma16)−εP(/Gamma18) becomes negative and thereby the band order at the BZ center gets inverted, which indicates a topologicallynontrivial phase. Since the theoretical and experimental latticeparameters differ slightly, Table IIshows the gap values for both lattice constants. In addition, it is worth noting that noother band inversions are present for the rest of the high-symmetric points in the Brillouin zone. The band inversionin K 3Bi and Rb 3Bi is confirmed by using an improved approximation for the exchange energy, the hybrid exchange-correlation functional HSE06. It is based on a screenedCoulomb potential for the exchange interaction to screen thelong-range part of the Hartree-Fock (HF) exchange [ 36,37]. The HF term with the correct asymptotic behavior makesit possible to improve the accuracy of the band-structure prediction [ 38]. Na 3Bi in the strain-free cubic case is also a semimetal like K 3Bi because its conduction and valence bands touch only at the /Gamma1point. However, even in the case without SOC the Na −3sstates are energetically lower, by about 0.4 eV , than the Bi −6pstates at the /Gamma1point [ 11] and, what is more important, the sandpstates exhibit the opposite parities. When the SOC is taken into account, theinverted band ordering is further enhanced. The Na −3sstate (/Gamma1 6) is significantly reduced in energy, thereby increasing the energy difference with Bi pstates up to 0.85 eV . The same characteristic feature was reported for a ternary alkalibismuthide, KNa 2Bi [ 39]. Therefore, like KNa 2Bi, Na 3Bi in its strain-free cubic case is a topologically nontrivialzero-gap semimetal at P=0 GPa, whether or not the SOC is considered. To demonstrate the topological character of the considered compounds, we also applied the Wilson loop (Wannier chargecenter) method, proposed in Refs. [ 24,25]. Wannier charge centers (WCCs) were obtained from the occupied states onthe basis of the W ANNIER 90 code Hamiltonian [ 40]. The topological properties were investigated by checking twoplanes in the kspace: k z=0 andkz=π. The evolution of WCC 224305-4NONTRIVIAL TOPOLOGY OF CUBIC ALKALI BISMUTHIDES PHYSICAL REVIEW B 95, 224305 (2017) FIG. 4. Electronic band structures of (a) K 3Bi and (b) Rb 3Bi calculated with the account of SOC. (c),(d) Electronic bands of K 3Bi and Rb 3Bi obtained without (dashed lines) and with the inclusion of spin-orbit coupling (red solid lines) along the [110] ( /Sigma1) and [111] (/Lambda1) symmetry directions. (θ)f o rkx∈[0,π]a tfi x e d kz=0 (left) and kz=π(right) is shown in Fig. 5for K 3Bi. For each kx, overlap matrices have been obtained along string ky∈[−π,π]. Atkx=πn(n, integer) all the presented curves are doubly degenerate due totime-reversal symmetry in the considered systems. Owing tothe point symmetry, the results are similar for perpendicularplanes when k xandkyare fixed and the evolution of WCC is traced along the kyandkzdirections, respectively. From the figure it is clear that for the kz=0p l a n et h eW C C curve crosses the reference line once. However, WCC nevercrosses the reference line at k z=π. It should be noted that the reference line parallel to the kxaxis is purely arbitrary and can be moved to somewhere else, but the parity of crossingnumbers between the evolution lines and the reference linewill never change. This indicates the topological character ofK 3Bi and agrees with the band inversion at the /Gamma1point. The presented WCC curves also say that Z2is equal to (1;000) in the TABLE II. Calculated energy gaps at the /Gamma1point, εS(/Gamma16)− εP(/Gamma18) (in eV). The data are obtained for the theoretical equilibrium volume both with (SOC) and without (noSOC) spin-orbit coupling as well as for the experimental lattice parameter ( aexpt) with the account of SOC. K3Bi Rb 3Bi noSOC +0.057 +0.036 SOC −0.35 −0.40 SOC at aexpt−0.34 −0.16FIG. 5. The evolution of Wannier charge centers (WCCs) for K3Bi: (left) kz=0, (right) kz=π. A reference value is represented by the blue dashed line. topological phase of K 3Bi in spite of the semimetallic character of electronic structure. So the compound can be adiabaticallytransformed into an insulating phase with the same Z 2indexes by a small uniaxial extension, which was shown for KNa 2Bi [39]. 2. Changes induced by pressure To study the changes induced by hydrostatic compression, we calculated the electronic band dispersions of both com- pounds at different volumes. Figure 6shows the evolution of electronic bands under pressure for K 3Bi. At low pressures the compound is still a semimetal with the inverted band orderat the Brillouin zone center until the energy gap betweens-type and p-like bands, ε S(/Gamma16)−εP(/Gamma18), vanishes. Figure 7 shows the variation of the energy gap with volume ( V/V 0)f o r both compounds. For comparison, the data for a ternary alkalibismuthide, KNa 2Bi, which crystallizes in the cubic Fm3m structure [ 32,39] are also shown. Since the optimized and experimental lattice parameters for Rb 3Bi differ noticeably, the energy gap in Rb 3Bi as a function of V/V expis given in the inset. Right at the point of topological transition [ P≈1.8G P a for both compounds, Fig. 6(b)], two of the bands form a conical dispersion crossed by the third band [Fig. 6(b)]. Upon further increase of pressure an energy gap appears between the valenceand conduction bands and the topological semimetal turnsinto a conventional gapped semiconductor. Simultaneously,the band order around the /Gamma1point restores to normal when thes-type/Gamma1 6band is located above the /Gamma18bands composed ofp-type Bi states [Fig. 6(c)]. A similar transition occurs in KNa 2Bi, a topological semimetal, which upon hydrostatic compression become a trivial semiconductors [ 39]. Na 3Bi transforms to a regular insulator at ∼3.65 GPa [ 10]. Also, such semimetal-to-semiconductor topological transitions werereported in the crystals of Hg 1−xCdxTe by changing cadmium concentration [ 13]. So, in the cubic phase of the A3Bi (A=Na, K, Rb) compounds a 3D Dirac semimetal state is realized at the criticalpoint of the topological phase transition between a topologicalsemimetal and a normal band insulator by pressure-inducedmodifications of electronic structure (the so-called trivial Diracsemimetals [ 14]). Such a Dirac semimetal state features a 224305-5I. P. RUSINOV et al. PHYSICAL REVIEW B 95, 224305 (2017) FIG. 6. (a)–(c) Evolution of the K 3Bi electronic spectrum under hydrostatic lattice compression. Electronic bands near the /Gamma1point are shown for three different volumes: at V0(P=0, a topological semimetal), at the point of topological transition ( P≈1.8G P a ) ,a n d atP≈2.1 GPa (a trivial semimetal). Open circles show the bands formed by pstates, while the bands composed mainly of s-type states are depicted by solid (orange) circles. (d),(e) Projected spectral functions for the Bi and KOterminated (001) and KTterminated (100) surfaces of K 3Bi at the equilibrium volume, V0. The projection is shown by a color scale reflecting the localization in the outermost surface layer (in units of states /eV). single 3D Dirac cone at the /Gamma1point, where the bulk band inversion occurs. Unlike the pressure-induced case, the Diracstates in the hexagonal Na 3Bi and K 3Bi show a pair of bulk FIG. 7. Energy gap εS(/Gamma16)−εP(/Gamma18) as a function of V/V 0.T h e positive gaps correspond to a topologically trivial phase when the band inversion disappears. The inset shows the energy gap in Rb 3Bi as a function of V/V exp.Dirac nodes located on the rotation axis away from the /Gamma1 point and protected by the rotational symmetry (the topologicalDirac semimetal-metal) [ 6,14,33]. With respect to Rb 3B i ,i ti s a metal with an inverted band order in the bulk BZ center,but the Dirac band crossing on the rotational axis does notoccur. The topologically nontrivial character of semimetals is usually manifested in the existence of topological surfacestates. We calculated the surface electronic structure usingab initio -based tight-binding (TB) formalism implemented in the W ANNIER 90 package [ 40,41]. The TB model Hamiltonian was constructed by projecting the bulk energy bands obtainedin the first-principles calculation onto maximally localizedWannier functions (WFs). In our calculations, the spinor WF basis was chosen to be |p ↑ x/angbracketright,|p↑ y/angbracketright,|p↑ z/angbracketright,|p↓ x/angbracketright,|p↓ y/angbracketright,|p↓ z/angbracketright for Bi atoms and |s↑/angbracketright,|s↓/angbracketrightfor K(Rb) atoms. The projected surface states were then obtained from the surface Green’sfunction ( ˆG μν)[42–44].ˆGμνfor the semi-infinite systems is defined as /bracketleftbig (E+iδ)ˆIμη−ˆHsurf μη/bracketrightbigˆGην(E)=ˆIμν, (4) where Eis an energy parameter, δis a small smooth parameter (δ=0.5×10−3eV in our calculation), and ˆIμνis a unit matrix. μ,ν, andηenumerate principal layers which interact as nearest neighbors. The surface spectral function, A(E), was obtained from A(E)=−1 πImTr ˆG00(E). (5) The projected spectral functions calculated for the Bi and KOterminated (001) and KT(100) surfaces of K 3Bi (see Table Ifor notations) are shown in Figs. 6(d) and 6(e). The data are presented as a color intensity plot. The surface states with a Dirac-type crossing are clearlyseen inside the projected valence bands on both surfacespresented. In K 3Bi the pressure-induced topological transition to a trivial state occurs when the cubic phase is alreadydynamically stable. Thus, at T=0 K there is a pressure interval (see Fig. 7, the hatched area) where the compound is still a topological semimetal and is already dynamically stable(0.92/greaterorequalslantV/V 0/greaterorequalslant0.88, 0.95 GPa /lessorequalslantP/lessorequalslant1.8 GPa). Unlike K3Bi, the cubic phase of Rb 3Bi stabilizes at T=0 K only after the transition to a trivial semiconductor has occurred (atV/V 0≈0.82,P≈2.8G P a ) . IV . CONCLUSION We have performed an ab initio study of the effect of pressure on electronic and vibrational properties of alkalibismuthides, K 3Bi and Rb 3Bi, in the cubic Fm3mstructure. We find that at zero pressure the cubic compounds aretopologically nontrivial with the inverted band order at the /Gamma1 point and undergo a transition to a conventional semiconductor(metal) state upon hydrostatic compression. Simultaneously,the sequence of bands restores to normal. It is also shown that the cubic structure of K 3Bi and Rb 3Bi at low temperatures (i) is dynamically unstable and (ii) can be 224305-6NONTRIVIAL TOPOLOGY OF CUBIC ALKALI BISMUTHIDES PHYSICAL REVIEW B 95, 224305 (2017) stabilized by pressure. In K 3Bi, the stabilization occurs before the transition to a trivial semiconductor so that there is a pres-sure interval where the cubic phase is stable and the compoundis still topologically nontrivial. Unlike K 3Bi, the cubic phase of Rb3B stabilizes after the transition to a trivial semiconductor has occurred.ACKNOWLEDGMENTS This work has been supported by the Spanish Ministry of Science and Innovation (Grant No. FIS2016-75862-P), TomskState University Academic D.I. Mendeleev Fund Program(Research Grant No. 8.1.01.2017), and Saint Petersburg StateUniversity (Project No. 15.61.202.2015). [1] Z. Wang, Y . Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85,195320 (2012 ). [2] Z. K. Liu, B. Zhou, Y . Zhang, Z. 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PhysRevB.95.144306.pdf

PHYSICAL REVIEW B 95, 144306 (2017) Nonequilibrium-induced enhancement of dynamical quantum coherence and entanglement of spin arrays Zhedong Zhang,1,2Hongchen Fu,3and Jin Wang1,4,5,* 1Department of Physics and Astronomy, SUNY Stony Brook, Stony Brook, New York 11794, USA 2Department of Chemistry, University of California Irvine, Irvine, California 92697, USA 3School of Physics and Energy, Shenzhen University, Shenzhen 518060, China 4Department of Chemistry, SUNY Stony Brook, Stony Brook, New York 11794, USA 5State Key Laboratory of Electroanalytical Chemistry, Changchun Institute for Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, China (Received 25 September 2016; revised manuscript received 26 February 2017; published 12 April 2017) The random magnetic field produced by nuclear spins has long been viewed as the dominating source of decoherence in the quantum-dot based spins. Here we obtain in both exact and analytical manner the dynamicsof spin qubits coupled to nuclear spin environments via the hyperfine interaction, going beyond the weaksystem-bath interaction and Markovian approximation. We predict that the detailed-balance breaking producedby chemical potential gradient in nuclear baths leads to the rapid oscillations of populations, quantum coherenceand entanglement, which are absent in the conventional case (i.e., Overhauser noise). This is attributed to thenonequilibrium feature of the system as shown in the relation between the oscillation period and the chemicalpotential imbalance. Our results reveal the essentiality of nonequilibriumness with detailed-balance breaking forenhancing the dynamical coherence and entanglement of spin qubits. Moreover, our exact solution explicitlydemonstrates that the non-Markovian bath comprised by nuclear spins can preserve the collective quantum state,due to the recovery of coherence. Finally, we propose an experiment using ultracold trapped ions to observe thesenonequilibrium and memory effects. DOI: 10.1103/PhysRevB.95.144306 I. INTRODUCTION Recently, the quantum-dot-based spin qubits as a solid-state method were demonstrated to be successfulin quantum information science, such as the control ofelectron-spin qubits in GaAs quantum dots [ 1–4] and sensitive metrology [ 5]. This takes advantage of the controllability of the systems. Despite the controllability, the decoherenceowing to the influence of nuclei spins in the host materials stillremains challenging for maintaining a high fidelity duringthe quantum computing. Understanding the dynamics of thisprocess is of fundamental importance. In the materials the spins are subjected to the noise, due to the random magnetic field from the nuclear spins. This is theso-called Overhauser field under typical conditions, producingrandom magnitudes and directions. This practically results inthe decoherence in a typical timescale. So far, the dynamicsof spin arrays has been mostly explored under the influenceof Overhauser fields, through the considerable studies on howto mitigate the decoherence or alternatively electron-spin flip[6–8] and dynamical decoupling [ 9,10] to artificially eliminate the random noise. These studies often assume that the spinswill eventually relax to the equilibrium state. However, the nat-ural environments as the nuclei spins around GaAs or Si quan-tum dots always show the emergence of the inhomogenouscharge density, which gives the gradient of chemical potentials.Regardless of the polarization fluctuation, the inhomogeneityhere refers to the effect of magnitude randomness of the nucleispins, which is dictated by the charge density associated withnuclei spins. This generates an effective voltage, leading to *jin.wang.1@stonybrook.eduthe breakdown of detailed balance (time-reversal symmetry).Hence, the system will relax to a nonequilibrium steady statebreaking the time-reversal symmetry, rather than being ther-malized to the equilibrium. Inspired by these, it is necessary toconsider the issue from a new point of view of nonequilibriumquantum dynamics with detailed-balance breaking, althoughcertain properties can still be explored in the conventionalframework [ 11] under the equilibrium idea. Recently many in- teresting phenomenons, i.e., the improvement of stationary co-herence and current [ 12,13] in quantum dots, and quantum syn- chronization with robust phase locking [ 14–16], were found to be inherently nonequilibrium. These results showed that themore nonequilibriumness is, the higher coherence and entan-glement are. This clearly shows the inadequacy of the conven-tional ideas. Hence for the dynamical processes the new insightfrom the nonequilibriumness should be adopted for further pro-found understanding of the system relaxations. For instance,the propagation of the spin wave associated with the spin trans-fer in quantum dots would show certain remarkable behaviorthanks to the nonequilibriumness induced by inhomogeneityof charge density. This is what we will focus in this article. We uncover a novel phase, which shows the rapid oscillation of coherence, fidelity of the collective quantum states of spinqubits, and quantum entanglement, contrary to the pure decayunder the conventional equilibrium condition. This, in otherwords, reveals that the nonequilibriumness is essential forenhancing the dynamical coherence of the systems, besidessteady-state coherence [ 12,17,18]. This coherent-oscillation phase originates intrinsically from the nonequilibrium-inducednet current, which quantifies the degree of deviation fromequilibrium. Compared to the previous work, our exact andanalytical solution to the dynamics of spin qubits in the pres-ence of noisy environments goes beyond the weak system-bath 2469-9950/2017/95(14)/144306(9) 144306-1 ©2017 American Physical SocietyZHEDONG ZHANG, HONGCHEN FU, AND JIN W ANG PHYSICAL REVIEW B 95, 144306 (2017) coupling and Markovian approximations [ 13,19,20]. This demonstrates in a general scenario the revival of quantumcoherence arising from the non-Markovian effect, supportedby the pure numerical simulations [ 11]. We use ultracold trapped ions in the spirit of quantum simulation to proposea detailed experiment for observing this effect and to predictthe nonequilibrium-induced coherent oscillation can be seenin the system with as few as two ions, which is accessible inthe present experiments [ 21–23]. II. MODEL We consider the arrays of two quantum dots where each contains one electron spin (qubit) subject to its own randommagnetic field produced by nuclear spins via hyperfineinteractions. The inhomogenous charge density of nuclei spinsleads to the chemical potential imbalance in the environmentaround the quantum dots, regardless of the dimensional andgeometric details. Our purpose is to study the nonequilibriumeffect induced by this inhomogeneity of the nuclei environmentin the host materials (i.e., GaAs or Si). To this end, the disorderof nuclear spins in different dimensions as well as geometry[i.e., nuclear isotopes in three-dimensional (3D) volume] doesnot play a significant role so that it will be considered no longerhere. To capture the feature of inhomogenous charge density,we assume that the nuclei in the host materials are described byone-dimensional spin chains with different chemical potentialsand the nuclear spins are equally spaced by ain each chain. Thus, the original Hamiltonian takes the Heisenberg form of H=−Jσ 1·σ2−2/summationdisplay i=1N/summationdisplay n=1tSn,i·Sn+1,i +2/summationdisplay i=1N/summationdisplay n=1fσi·Sn,i, (1) where J, t are the spin-spin coupling strengths for electron and nuclear spins, respectively. In our model, nuclear andqubit spins are modeled as XY model including the transversecoupling only in order to generate the nuclear spin flips [ 24,25] and electronic spin-state transitions [ 25]. The longitudinal terms S z n,iSz n,i+1,σz 1σz 2are neglected here because of no transition induced. For the consideration of noise producedby nuclear spins, we only take into account of the magnituderandomness of nuclear spins and neglect the fluctuation of theirorientation. Thus, the longitudinal component of the randommagnetic field generated by nuclear spins dominates and thequbit-nuclear interaction is encoded as Ising type σ z iSz n,i.T h i s approximation can reasonably apply for the mesoscopic spinsystem formed by many nuclear spins in a semiconductorquantum dot, due to the fact that the nuclear spin flips areof the off-resonant frequency to the electronic ones especiallyat strong magnetic fields [ 25,27]. To this end, we will work under the following effective Hamiltonian: H eff=−J/parenleftbig σx 1σx 2+σy 1σy 2/parenrightbig +2/summationdisplay i=1N/summationdisplay n=1fσz iSz n,i −2/summationdisplay i=1N/summationdisplay n=1t/parenleftbig Sx n,iSx n+1,i+Sy n,iSy n+1,i/parenrightbig . (2)σ± 1=σ±⊗1,σ± 2=σz⊗σ±andS± n,i=Sz⊗Sz⊗···⊗ Sz⊗S±⊗1⊗···⊗ 1, based on the mapping between XXZ spin-1 2chain and 1D Fermi-Hubbard model [ 26], and σ±= 1 2(σx∓iσy),S±=1 2(Sx∓iSy) are the standard Pauli matri- ces. Therefore, σ± n,S± n,iobey the fermionic anticommutation relationship. III. TIME EVOLUTION OF THE DENSITY MATRIX OF SPIN QUBITS To solve the dynamics of the entire system, the follow- ing collective operators I± i=/summationtext2 j=1Oijσ± j, namely, I± 1= 1√ 2(−σ± 1+σ± 2),I± 2=1√ 2(σ± 1+σ± 2) and S± k,i=1√ NN/summationdisplay n=1S± n,ieinka,k=/parenleftbigg2m N−1/parenrightbiggπ a; m=1,2,..., N (3) in momentum space, will be first introduced to diagonalize the free Hamiltonian. The time evolution of entire system isgoverned by the operator U(t)=exp[− i ¯h/integraltextt 0Vint(τ)dτ] where Vint(t)=V(1) int(t)+V(2) int(t) in the interaction picture V(n) int(t)=f2/summationdisplay i,j=1/summationdisplay kO−1 niO−1 nje−i(ωi−ωj)t ×(I− iI+ j−I+ jI− i)⊗Sz k,n. (4) In order to obtain the dynamics of the system, one notices that the operators l=1 2(I− 1I+ 1−I+ 1I− 1+I− 2I+ 2− I+ 2I− 2),η 3=1 2(I− 1I+ 1−I+ 1I− 1−I− 2I+ 2+I+ 2I− 2),η 1= I− 1I+ 2+I− 2I+ 1,η 2=−i(I− 1I+ 2−I− 2I+ 1) satisfy [ l,ηj]=0, [ηi,ηj]=2i/epsilon1ijkηk, which gives rise to the Lie algebra su(2)⊕u(1). Thereby, the qubit-nuclear interaction in Eq. ( 4) does transform according to the irreducible representation D0⊕D0⊕D 1 2of the Lie group SU(2)×U(1). As inspired by the recent experiments [7,27] the spin qubits are initially engineered at the state |S/angbracketright=cosφ 2|1/angbracketright⊗| 0/angbracketright+eiθsinφ 2|0/angbracketright⊗| 1/angbracketright, which is the eigenstate of lwith the eigenvalue of 0. This further means the dynamical evolution of qubits according to the irreduciblerepresentation D 1 2of group SU(2)×U(1).ηjtake the form of 2×2 representation η1=/parenleftbigg 0−1 −10/parenrightbigg ,η 2=/parenleftbigg 0−i i 0/parenrightbigg ,η 3=/parenleftbigg −10 01/parenrightbigg . (5) Hence in terms of landηj, the representation of the time- evolution operator can be found U(t)=e−i ¯h/integraltextt 0V(1) int(τ)dτe−i ¯h/integraltextt 0V(2) int(τ)dτ =AN+iBN[σxRe(p)−σyIm(p)] (6) 144306-2NONEQUILIBRIUM-INDUCED ENHANCEMENT OF . . . PHYSICAL REVIEW B 95, 144306 (2017) by some algebra, where β≡ωt=4Jt/¯handCv≡ Sz v,1Sz v,2,Dv≡Sz v,1−Sz v,2. AN=1 2/bracketleftBiggN/productdisplay v=1(a+Cvb+i|p|Dv)+H.c./bracketrightBigg BN=1 2i|p|/bracketleftBiggN/productdisplay v=1(a+Cvb+i|p|Dv)−H.c./bracketrightBigg (7) and a≡cos2/parenleftbiggf 2Jsinβ 2/parenrightbigg ,b≡sin2/parenleftbiggf 2Jsinβ 2/parenrightbigg p≡1 2eiβ 2sin/parenleftbiggf Jsinβ 2/parenrightbigg . (8) In many circumstances, however, the dynamics of spin qubits is what people are mostly interested in. For this purposewe further trace out the degrees of freedoms of nuclear spinsin the entire density matrix ρ(t) to obtain the nonunitary time evolution of spin qubits. Suppose that initially the qubits areengineered at the state |/Psi1(0)/angbracketright=1 √ 2(|1/angbracketright⊗| 0/angbracketright+e−iθ|0/angbracketright⊗| 1/angbracketright)( 9 ) and the nuclear spins are at thermal equilibrium with different chemical potentials (Fermi energies). In other words, the initialcondition takes the product form ρ(0)=|/Psi1(0)/angbracketright/angbracketleft/Psi1(0)|⊗ρμ1 nucl⊗ρμ2 nucl ρμi nucl=Z−1 ie−¯β(H(i) nucl−μiNi); Hnucl=−t2/summationdisplay i=1N/summationdisplay n=1Sn,i·Sn+1,i;i=1,2, (10) where Ni=/summationtextN n=1S+ n,iS− n,iis the operator of total number of spin excitation in the nuclear spin chain. After a lengthy but straightforward calculation one can reach the 2 ×2 density matrix of spin qubits at moment t, by tracing out the nuclear spins as environments: ρs(t)=TrB[ρ(t)]=/parenleftbiggρ10,10(t)ρ10,01(t) ρ∗ 10,01(t)ρ01,01(t)/parenrightbigg , (11) where ρ10,10(ρ01,01) is the population on the state |1,0/angbracketright(|0,1/angbracketright) andρ10,01is the coherence between these two states. They can be written in the analytical form Reρ10,01(t)=/parenleftBiggN/productdisplay v=1Kv+Fe(μ1,μ2,t)/parenrightBigg cosθ 2+Fo(μ1,μ2,t)sinθcosβ 2 Imρ10,01(t)=/parenleftBigg cos2β 2N/productdisplay v=1Kv−sin2β 2+Fe(μ1,μ2,t) cos2β 2/parenrightBigg sinθ 2−Fo(μ1,μ2,t)cosθcosβ 2(12) ρ10,10(t)−ρ01,01(t)=/parenleftBigg 1+N/productdisplay v=1Kv+Fe(μ1,μ2,t)/parenrightBigg sinθ 2sinβ−2Fo(μ1,μ2,t) cosθsinβ 2, where two imbalance functions FeandFoare introduced to quantify the nonequilibrium contribution Kv=1−2/bracketleftbig fμ1 v/parenleftbig 1−fμ2 v/parenrightbig +fμ2 v/parenleftbig 1−fμ1 v/parenrightbig/bracketrightbig sin2/parenleftbiggf Jsinβ 2/parenrightbigg Fe(μ1,μ2,t)=/summationdisplay {me}(−4)c(me) 2sinc(me)/parenleftbigg2f Jsinβ 2/parenrightbigg/productdisplay v∈¯meKv/productdisplay q∈me/parenleftbig fμ1 q−fμ2 q/parenrightbig (13) Fo(μ1,μ2,t)=/summationdisplay {mo}(−4)c(mo)−1 2sinc(mo)/parenleftbigg2f Jsinβ 2/parenrightbigg/productdisplay v∈¯moKv/productdisplay q∈mo/parenleftbig fμ1 q−fμ2 q/parenrightbig andm=(m1,m2,..., m s),s=1,2,..., N is a nonempty subset of the set r=(1,2,..., N ); c(m) is the number of elements of subset m.me,mocorrespond to the m’s with even and odd numbers of elements, respectively. ¯m=r−m.fμin= [exp(¯hνn−μi kBT)+1]−1,¯hνn=ε+4tcos (2nπ/N ). So far, we have obtained the exact dynamics of spin qubits beyond theMarkovian and weak qubit-bath coupling approximations. Inthe forthcoming discussion, we will focus on the nonequilib-rium effects reflected by the imbalance functions F eandFo, which are governed by the chemical potential difference ofspin baths.IV . RECURRENCE FROM RELAXATION OF SPIN BATHS Based on our exact dynamics obtained above, we are able to explicitly explore the non-Markovian process beyondMarkovian approximation without memory. This recentlyattracted much attention in the study of dissipative quantumdynamics because the experimental measurements revealedthe non-Markovian noise from the nonexponential decay ofecho signal [ 27]. To measure the preservation of the quantum state, one can use the fidelity of the state defined as F(t)=√ /angbracketleft/Psi1(0)|ρs(t)|/Psi1(0)/angbracketrightif the system is initially in pure ensemble. 144306-3ZHEDONG ZHANG, HONGCHEN FU, AND JIN W ANG PHYSICAL REVIEW B 95, 144306 (2017) 0 1 2 3 4 5 6 70.20.40.60.81.0 ωtFidelity 0.00 0.02 0.04 0.06 0.08 0.10 0.120.20.40.60.81.0 ωt1.6 1.7 1.8 1.9 2.00.20.40.60.81.0 ωt(a) (b) (c) 0 1 2 3 4 5 6 70.40.20.00.20.4 ωtRe10ρ01 0.00 0.02 0.04 0.06 0.08 0.10 0.120.40.20.00.20.4 ωt1.6 1.7 1.8 1.9 2.00.40.20.00.20.4 ωt(d) (e) (f) 0 1 2 3 4 5 6 70.00.20.40.60.81.0 ωtConcurrence 0.82 0.84 0.86 0.88 0.90 0.92 0.940.00.20.40.60.81.0 ωt1.6 1.7 1.8 1.9 2.00.00.20.40.60.81.0 ωt(g)(h) (i) FIG. 1. Dynamics of (top) fidelity of quantum state |/Psi1π(0)/angbracketright=1√ 2(|1/angbracketright⊗| 0/angbracketright−| 0/angbracketright⊗| 1/angbracketright), (middle) the real part of coherence Re /angbracketleft1,0|ρs|0,1/angbracketright, and (bottom) concurrence of the spin qubits; ω≡4J/¯h. The red and black lines are for μ1=1.7,μ 2=0.5 (far-from-equilibrium), and μ1/similarequalμ2=0.5 (equilibrium), respectively. Other parameters are ε=1,t=0.2,kBT=0.1,N=100, and f=8J. Equivalently, the fidelity can be written as F(t)=/radicalbigg 1 2+cosθRe/angbracketleft1,0|ρs|0,1/angbracketright+sinθIm/angbracketleft1,0|ρs|0,1/angbracketright, (14) which shows a strong correlation to the quantum coherence. According to recent experiments [ 7], the spin qubits are initially prepared at a singlet state |/Psi1π(0)/angbracketright=1√ 2(|1/angbracketright⊗| 0/angbracketright− |0/angbracketright⊗| 1/angbracketright),(θ=π) in the regime of strong qubit-nuclear interactions, whose dynamical behaviors are illustrated inFig. 1. As is shown, the preservation of quantum state and revival of coherence are perfectly elucidated. Here we only show the behavior of the real part of coherence because it contributes to the preservation of state, based on Eq. ( 14). Physically the recovery of quantum coherence can be attributedto the non-Markovian effect [ 28,29], since the timescale of correlations in the environment is comparable to that of thesystem and the phase correlation of the system has a highchance to reconstruct. This implies the nonlocal correlation ofthe system in time domain and consequently the informationof initial state is memorized, which results in the recurrence of coherence. The coherence always shows a monotonic decay until reaching the stationary value. Furthermore, it isworth noticing that the system shows a perfect preservationof quantum state with fidelity of 100%. This feature canbe alternatively understood from the coherence dynamicsRe/angbracketleft1,0|ρ s|0,1/angbracketright, shown in Fig. 1(d), in which the magnitude of coherence periodically recovers to its initial value. The typeof state protection we discuss here is completely different from what has been achieved by dynamical decoupling and other control methods [ 9,10,30–32]. It is instead an intrinsic entanglement induced by slow relaxation of the baths leadingto the non-Markovian process. This means there is no needto artificially control over the spin-spin interactions to combatdecoherence. Moreover, no global decay occurs for the local peaks of either coherence or fidelity with initially being engineeredat the Bell state, as compared to that in the case |/Psi1 π 2(0)/angbracketright= 1√ 2(|1/angbracketright⊗| 0/angbracketright−i|0/angbracketright⊗| 1/angbracketright),(θ=π 2) as shown in the Supple- mentary Material (SM) [ 33] where such global decay does 144306-4NONEQUILIBRIUM-INDUCED ENHANCEMENT OF . . . PHYSICAL REVIEW B 95, 144306 (2017) exist. This is because of the conservation of the total angular momentum in the case θ=π:[Lz,H]=0(|/Psi1π(0)/angbracketrightis the eigenstate of total angular momentum Lz=σz 1+σz 2). To show the generality of the non-Markovian effect, we also perform the dynamics of fidelity and coherence as thespin qubits relax from another state |/Psi1 π 2(0)/angbracketright=1√ 2(|1/angbracketright⊗| 0/angbracketright− i|0/angbracketright⊗| 1/angbracketright),(θ=π 2), whose dynamics is illustrated in the SM. As is shown, the memory effect arising from the non-Markovian process always shows up, irrespective of whichinitial state is prepared. The generality of such non-Markovianeffect is further demonstrated by letting the qubits to relax fromother states, e.g., |/Psi1 π 4(0)/angbracketright=1√ 2(|1/angbracketright⊗| 0/angbracketright+e−iπ/4|0/angbracketright⊗| 1/angbracketright) as also shown in the SM. V. ENHANCEMENT OF DYNAMICAL COHERENCE FROM DETAILED-BALANCE BREAKING Now let us turn to the nonequilibrium effect with the detailed-balance breaking induced by the gradient of chemicalpotential from nuclear spins. From our analytical solution ofthe density matrix we know that the nonequilibrium contribu-tion can be quantified by the factor/producttext(f μ1q−fμ2q) in the imbal- ance functions FeandFo, which vanish under the time-reversal protection. The time evolution shown in Fig. 1and the figure in the SM illustrate that the nonequilibriumness produces therapid oscillations, through the comparison between the red andblack lines for far-from-equilibrium and equilibrium regimes,respectively. This reveals that the oscillations would not existunder equilibrium condition and the far-from-equilibriumregime is essential for observing the coherent oscillation ofspin qubits in the experiments. Having the rapid oscillationscompared to no oscillations gives the nonequilibrium biaswhich is directly related to the coherence enhancement asseen in Fig. 1. Later we will discuss the experimental implementation of such effect based on quantum simulation. To understand and explain the oscillation feature produced by the detailed-balance breaking, we need to introduce the spincurrent between the spin qubits, which provides a measureof the degree of deviation from equilibrium [ 12,34,35]. The current conservation gives d dtσ† 2σ2=ˆI1→2−ˆI2→bathwhere ˆI1→2denotes the spin current operator whose expectation I1→2=/angbracketleftˆI1→2/angbracketrightgives the spin current. In our model the current from the system to spin environment vanishes because of [σ† 2σ2,σz iSz n,i]=0;i=1,2. Thus, the spin current reads I1→2=d dt/angbracketleftσ† 2σ2/angbracketright=4J ¯hIm/angbracketleft1,0|ρs|0,1/angbracketright (15) based on the Heisenberg’s equation. This coincides with the form for curl quantum flux in our former work [ 12], in that microscopically the current strongly correlates to the curlquantum flux, vanishing under detailed balance at steady state.However, this current does not necessarily vanish during thenonequilibrium relaxation to the steady state. Even at thesteady state, the current is not necessarily zero due to the energyor information input from or output to the environments.As is shown by Eq. ( 15), the spin current governed by the nonequilibriumness generates a fast oscillation of coherence,furthermore the fidelity of quantum state. This can be under-stood as follows: The rapid oscillation of spin current inducedby chemical potential imbalance (nonequilibriumness) leads to the back and forth motion of spin waves between qubits, whichresults in the fast oscillations of coherence. For an analogy,this is in the similar spirit of limit cycle behavior in classicalstochastic processes [ 36,37], driven by the curl flux breaking the detailed balance at steady state, where a robust oscillationnetwork can be observed [ 38–40]. To further understand the nonequilibrium effect, we con- sider a certain limit where the nuclear spin environmentsevolve sufficiently slowly so that they can be well approxi- mated by quasistatic ensembles [ 28,29], namely dSz v,i dt/similarequal0o n the typical timescales of electron spin dynamics, around amicrosecond or less [ 41–43]. Then the entire system can be ap- proximately described by product state ρ(t)/similarequalρ s(t)⊗ρμ1nc⊗ ρμ2nc. By adopting the Heisenberg-Langevin theory [ 44] one can obtain the coherent oscillation: Im /angbracketleftσ+ 1σ− 2/angbracketright∝cos(/Omega1coht+φ) and Re /angbracketleftσ+ 1σ− 2/angbracketright∝1 /Omega1cohsin(/Omega1coht+φ) where the oscillation frequency reads /Omega1coh=4J ¯h/radicalbigg 1+f2V2 4J2,V=2N/summationdisplay q=1/parenleftbig fμ1 q−fμ2 q/parenrightbig .(16) Vserves as an effective voltage vanishing under detailed balance and it provides one type of quantification for thechemical potential imbalance. Equation ( 16) uncovers the analytical relation between the coherent oscillation andthe nonequilibriumness. It explicitly demonstrates that thedetailed-balance breaking is intrinsically responsible for therapid oscillation of coherence and subsequently the fidelityof quantum state, which confirms our argument above. Toverify the validity of our formula for /Omega1 coh, we perform a numerical calculation of the oscillation period and frequencywith respect to voltage V, illustrated in Fig. 2, which shows a perfect agreement with our analytical formula Eq. ( 16). It is worth pointing out that the case discussed above with static nuclear spin environment can be alternatively de-scribed by a back-of-the-envelope model with the HamiltonianH 1=−J(σx 1σx 2+σy 1σy 2)+σz 1(B+/Delta1B)+σz 2B. This is due to the static magnetic fields B+/Delta1B andBproduced by nuclear spins. The static magnetic field does not break the timereversal whereas our model studied before does, as indicatedby the damping oscillation. Some algebra gives rise to the Rabi frequency /Omega1 1=4J ¯h/radicalBig 1+/Delta1B2 4J2, which shows the increase of oscillation frequency by the magnetic field gradient /Delta1B. As/Delta1B∝/summationtext(/angbracketleftSz n,1/angbracketright−/angbracketleftSz n,2/angbracketright), the results in Eq. ( 16) can then be recovered. This coincidence further demonstrates the factthat the rapid oscillation of quantum coherence is attributed tothe nonequilibriumness produced by the inhomogenous chargedensity of the nuclei environment. VI. QUANTUM ENTANGLEMENT ENHANCED BY DETAILED BALANCE BREAKING The quantum nature usually is not only reflected by coherence, but is also revealed by quantum entanglement. Thelatter one takes the advantage over the former one becauseof its basis independence. In other words, the measure ofquantum coherence depends on the choice of basis, so thatit can be observed under some specific basis while it may 144306-5ZHEDONG ZHANG, HONGCHEN FU, AND JIN W ANG PHYSICAL REVIEW B 95, 144306 (2017) f=8J(a) 20 40 60 80 100 1200.020.030.040.050.060.07 VPeriod of oscillation 4J 20 40 60 80 100 1200100200300400 VOscillation frequency 4J f=0.4J(b) 20 40 60 80 100 1200.40.60.81.01.21.4 VPeriod of oscillation 4J 20 40 60 80 100 12005101520 VOscillation frquency 4J (c) f=8J 20 40 60 80 100 1200.020.030.040.050.060.07 VPeriod of oscillation of C 4J 20 40 60 80 100 1200100200300400 VOscillation frequency 4J(d) f=0.4J 20 40 60 80 100 1200.40.60.81.01.21.4 VPeriod of oscillation of C 4J 20 40 60 80 100 12005101520 VOscillation frquency 4J FIG. 2. (Large) Oscillation period and (small) oscillation frequency for (top) coherence and (bottom) concurrence vary as a function of effective voltage V, for (a), (c) strong ( f=8J) and (b), (d) weak ( f=0.4J) qubit-nuclear interactions; The triangle and circle markers are for the numerical calculations of oscillation period and frequency, respectively; For oscillation period, the purple triangle and red square markers correspond to the coherence and concurrence, respectively. All the smooth curves are obtained from the analytical result in Eq. ( 16). Other parameters are ε=1,t=0.2,kBT=0.1, and N=100. vanish by switching to other basis. Although the entanglement can be described by entanglement entropy in an elegant wayfor closed quantum systems, how to quantitatively measurethe entanglement for the open quantum systems is still achallenging issue. In spite of this, several quantifications havebeen proposed, i.e., negativity [ 45] and concurrence [ 46], each of which, however, has its own limitations. Here we willquantify the entanglement by exploring the concurrence ofthe spin qubits, taking advantage of the spin- 1 2feature. To obtain the concurrence that quantifies the quantum en- tanglement of the spin-qubit system, the spin-flipped operation ˜ρ=(σy⊗σy)ρ∗(σy⊗σy) to the system must be carried out at first and ˜ρ(t)=/parenleftBigg ρ01,01(t)ρ10,01(t) ρ∗ 10,01(t)ρ10,10(t)/parenrightBigg , (17) where ρ∗is the complex conjugate of ρ. Thus, the non- Hermitian matrix R=ρ˜ρcan be obtained R=/parenleftBigg (ρ10,10ρ01,01+|ρ10,01|2)2 ρ10,10ρ10,01 2ρ01,01ρ∗ 10,01 (ρ10,10ρ01,01+|ρ10,01|2)/parenrightBigg (18) whose eigenvalues are denoted as λi,i=1,2,3,4 in descending order. Obviously in our setup λ3= λ4=0 and subsequently the concurrence is C(ρ)= max(0 ,√λ1−√λ2−√λ3−√λ4), which gives rise to C(ρ)=√ρ10,10ρ01,01+|ρ10,01|−|√ρ10,10ρ01,01−|ρ10,01|| =2|ρ10,01|. (19)The positivity of the density matrix requires that the eigenval- uesp1,p2must be non-negative. Thus, p1p2=ρ10,10ρ01,01− |ρ10,01|2/greaterorequalslant0, which gives the result in the second line in Eq. ( 19). Equation ( 19) clearly shows that the entanglement will die when the coherence is destroyed. This, on the other hand, re-veals the intrinsic correlation between the quantum coherenceand entanglement. The concurrence can be properly employedto quantify the entanglement Edue to the fact that it is mono- tonically increasing with respect to Efor 0/lessorequalslantC/lessorequalslant1. Figures 1(g)–1(i) illustrate the dynamics of concurrence based on Eq. ( 19), which shows the similar rapid oscillation generated by nonequilibriumness as that occurs in coherence dynamics.This demonstrates that the entanglement between qubits can beenhanced by detailed-balance breaking. This further manifeststhe fact in a solid manner that the far-from-equilibrium regimecan not only promote the steady-state coherence [ 12,17], but can also improve the quantum nature in terms of entanglementin the dynamical processes. On the other hand, the non-Markovian effect is also displayed clearly by the recovery ofentanglement shown in Figs. 1(g)–1(i). To further explore the nonequilibrium contribution to the oscillation period, whichcan be measured through the experiments, we find that fromEq. ( 19) and the procedures for obtaining the oscillation fre- quency of coherence, the quantitative relation between voltageand the oscillation period of entanglement is of the form T concur=π¯h 2J/parenleftbigg 1+f2V2 4J2/parenrightbigg−1 2 , (20) which shares the same form as Eq. ( 16), owing to Eq. ( 19). Equation ( 20) predicts the rapid coherent oscillation of 144306-6NONEQUILIBRIUM-INDUCED ENHANCEMENT OF . . . PHYSICAL REVIEW B 95, 144306 (2017) FIG. 3. (a) Schematics of (left) our setup and (right) the stimulated two-photon Raman transition for realizing (red and blue) σx⊗σx+σy⊗ σyand (orange) σz⊗σzinteractions; (b) dynamics of coherence Im /angbracketleft1,0|ρs|0,1/angbracketrightwith the initial condition |/Psi1(0)/angbracketright=1√ 2(|1/angbracketright⊗| 0/angbracketright−| 0/angbracketright⊗| 1/angbracketright) and (c) the period (small for frequency) of coherent oscillation as a function of voltage V. The red and black lines in (a) are for μ1= 2.22 GHz ,μ 2=0.65 GHz (far-from-equilibrium) and μ1/similarequalμ2=0.65 GHz (equilibrium), respectively. The triangle and square markers are for the numerical calculations of oscillation period and frequency, respectively. The smooth curves are obtained from Eq. ( 16). Other parameters areε/similarequal1.31 GHz ,t/similarequal0.26 GHz ,T/similarequal1m K,J/similarequal0.22 GHz ,f/similarequal1.76 GHz, and N=16. entanglement in the far-from-equilibrium regime. The numerical calculations confirm our analytical formula foroscillation period, as illustrated by the red markers and linesin Figs. 2(c) and2(d). VII. EXPERIMENTAL IMPLEMENTATION FOR SIMULATING SPIN-QUBIT DYNAMICS In order to observe the non-Markovian and nonequilibrium effects investigated in this work, ultracold trapped ions seemsto be a good candidate to engineer our setup, since they wererecently used for the spin chain simulations [ 21–23,47]. The ions are usually confined by a linear radio-frequency (RF) trap with three-dimensional electrodes, whichsubsequently produces the internal levels of ions. Thespin- 1 2structure is realized by choosing two nearly degenerate sublevels of the ground state, split by Zeeman field. Onthe other hand, the ions reside on individual lattice sitesdue to the strong Coulomb repulsion, which still leads to the common motional modes of ions described by phonons. Such phonons allow long-range interactions to bemediated between spins associated with the ions. To engineerthe long-range spin-spin interactions, two detuning laserbeams with different frequencies ω 1,ω 2are required to perform stimulated two-photon Raman transition between thesublevels, through a third level with higher energy, as shown inFig. 3(right). Here the two particular processes are crucial: (i) the transition |↑/angbracketright|n /prime/angbracketright⇔| ↓ /angbracketright | n/angbracketright(n/prime/negationslash=n) with two laser beams detuned by the frequency difference between these two states;(ii) the transitions |↑/angbracketright|n /prime/angbracketright⇔| ↑ /angbracketright | n/angbracketright(n/prime/negationslash=n) and |↓/angbracketright|n/prime/angbracketright⇔|↓/angbracketright|n/angbracketright(n/prime/negationslash=n) with the two beams detuned by approximately the frequency of a motional mode. Here n/prime,ndenote the energy levels of ions. It can be directly shown that (i) generates thetransverse long-range interactions between spins, σ x⊗σxand σy⊗σydepending on the polarization of electric field, while (ii) generates the Ising type of interaction, σz⊗σz[48,49]. The array of nuclear spins can be implemented by an array of Ntrapped ions, showing a dipolar decay of spin-spin interaction Jij∼1 |i−j|3. In order to realize the XY couplings, the spin array can be manipulated to interact with detuning laser beams in terms of process (i) above. We now prepareother two ions trapped by linear RF trap, with the samesplitting between the sublevels of ground state as that in arraysof nuclear spins. The same technique can be employed asbefore to produce the transverse spin-spin interaction betweenthese two ions. To engineer the interaction between spin arraysand the two ions, the process (ii) can be used to generate theindividual coupling of ion 1 (2) to each ion in the spin array 1(2), by choosing the ˆzpolarization of the electric field in the laser beams, illustrated by Fig. 3(left). This indicates that 2 N pairs of detuning beams with ˆzpolarization are demanded in total, to simulate the qubit-nuclear interactions. We use 16 trapped ions to simulate each nuclear spin environment, by taking into account the conditions in re-cent experiments [ 22,23,47]. The two arrays of ions can be prepared initially with different Fermi energies and thetwo ions (qubits) are engineered at singlet state |/Psi1(0)/angbracketright= 1√ 2(|1/angbracketright⊗| 0/angbracketright−| 0/angbracketright⊗| 1/angbracketright). By choosing the parameters ε/similarequal1.31 GHz ,t/similarequal0.26 GHz ,T/similarequal1m K,J/similarequal0.22 GHz , 144306-7ZHEDONG ZHANG, HONGCHEN FU, AND JIN W ANG PHYSICAL REVIEW B 95, 144306 (2017) f/similarequal1.76 GHz ,μ 2/similarequal0.65 GHz, and N=16 based on the experimental feasibility [ 47,49], Fig. 3shows the dynamics of coherence and also the behavior of frequency of coherentoscillation with respect to effective voltage as introducedbefore. The non-Markovian effect is clearly shown by therevival of the coherence in Fig. 3(b). Moreover, it also shows the nonequilibrium effect (detailed-balance breaking),which generates the fast coherent oscillation, through thecomparison between the red and black curves. The dependenceof oscillation period (frequency) is further illustrated inFig. 3(c), which is measurable in the proposed experiment. VIII. DISCUSSION AND CONCLUSION In summary, we exactly and analytically solved the dynamics of spin qubits surrounded by the charge noiseproduced by nuclear spins. We found that the detailed-balancebreaking leads to a new phase of rapid coherent oscillationin spin dynamics, which was lacking in the conventionalensemble of nuclear spin bath preserving the detailed balance, i.e., Overhauser noise. The analytical relationship betweenoscillation frequency of coherence and entanglement andeffective voltage was further obtained. It suggests a quan-titative measure of the nonequilibrium effect, which wouldbe accessible in experiments. On the other hand, our resultshave the advantage over previous studies in purely numericalmanner for describing the non-Markovian process. Thus, wecan predict the recovery of coherence and the subsequentquantum-state preservation in a general scenario. These noveleffects we predict, especially the oscillations of coherence andentanglement arising from detailed-balance breaking, can beobserved in ultracold trapped ions we proposed in detail in thespirit of quantum simulation in the laboratory. ACKNOWLEDGMENTS We acknowledge the support from the Grant NSF-PHY- 76066. [1] D. Press, T. Ladd, B. Zhang, and Y . Yamamoto, Nature (London) 456,218(2008 ). [2] F. H. L. Koppens et al. ,Science 309,1346 (2005 ). [3] S. Foletti, H. Bluhm, D. Mahalu, V . Umansky, and A. Yacoby, Nature Phys. 5,903(2009 ). [4] C. Barthel, D. J. Reilly, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 103,160503 (2009 ). [5] P. Maletinsky et al. , Nanotechnol. 7, 320 (2012). [6] J. R. Petta et al. ,Science 309,2180 (2005 ). [7] H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V . Umansky, and A. Yacoby, Nat. Phys. 7,109(2011 ). [8] J. 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Das Sarma, P h y s .R e v .B 90,155306 (2014 ). [32] E. Barnes, X. Wang, and S. Das Sarma, Sci. Rep. 5,12685 (2015 ). [33] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.95.144306 for the different initial states that the system is engineered, with θ=π 2andθ=π 4. [34] H. Qian, J. Phys. Chem. B 110,15063 (2006 ). [35] M. Qian and Z. T. Hou, Reversible Markov Process (Hunan Scientific Publisher, Hunan, 1979). [36] G. Hu, T. Ditzinger, C. Z. Ning, and H. Haken, P h y s .R e v .L e t t . 71,807(1993 ). [37] W. J. Rappel and S. H. Strogatz, P h y s .R e v .E 50,3249 (1994 ). [38] D. Schultz, E. B. Jacob, J. N. Onuchic, and P. G. Wolynes, Proc. Natl. Acad. Sci. USA 104,17582 (2007 ). [39] J. Wang, L. Xu, and E. K. Wang, Proc. Natl. Acad. Sci. USA 105,12271 (2008 ). [40] C. H. Li, E. K. Wang, and J. Wang, J. Chem. Phys. 136,194108 (2012 ). [41] E. Barnes, L. Cywinski, and S. Das Sarma, Phys. Rev. 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PhysRevB.73.195113.pdf

Frequency and temperature dependence of the optical conductivity of granular metals: A path-integral approach V. Tripathi Theory of Condensed Matter Group, Cavendish Laboratory, Department of Physics, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom Y. L. Loh Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, Indiana 47907-2036, USA /H20849Received 23 January 2006; published 17 May 2006 /H20850 We study the finite-temperature optical conductivity /H9268/H20849/H9275,T/H20850of a granular metal using a simple model consisting of a array of spherical metallic grains. It is necessary to include quantum tunneling and Coulombblockade effects to obtain the correct temperature dependence of /H9268, and to consider polarization oscillations to obtain the correct frequency dependence. We have therefore generalized the Ambegaokar-Eckern-Schön /H20849AES /H20850 model for granular metals to obtain an effective field theory incorporating the polarization fluctuations of theindividual metallic grains. In the absence of intergrain tunneling, the classical optical conductivity is deter-mined by polarization oscillations of the electrons in the grains, /H9268/H20849/H9275/H20850=−/H20849ine2f/H9275/m/H20850//H20849/H92752−/H9275r2−i/H20841/H9275/H20841//H9270grain /H20850, where/H9275r=e/H20881/H208494/H9266/3m/H20850nis the resonance frequency, /H9270grain−1is the relaxation rate for electron motion within the grain, and fis the volume fraction occupied by the grains. At finite intergrain tunneling, we find that /H9268/H20849/H9275/H20850 =−/H20849ine2/H9275f/m/H20850//H20849/H92752−/H9275r2−i/H20841/H9275/H20841//H9270rel/H20850+/H9268AES/H20849/H9275,T/H20850, where/H9270rel−1is the total relaxation rate that includes the intra- grain relaxation rate /H9270grain−1as well as intergrain tunneling effects, and /H9268AES/H20849/H9275,T/H20850is the conductivity of the granular system from the AES model obtained by ignoring polarization modes. We calculate the temperatureand frequency dependence of the intergrain relaxation time, /H9003/H20849 /H9275,T/H20850=/H9270rel−1−/H9270grain−1, and find it is different from /H9268AES/H20849/H9275,T/H20850. For small values of dimensionless intergrain tunneling conductance, g/lessmuch1, the dc conductivity obeys an Arrhenius law, /H9268AES/H208490,T/H20850/H11011ge−Ec/T, whereas the polarization relaxation may even decrease algebra- ically,/H9003/H20849/H9275,T/H20850/H11011/H20849g/Ec2/H20850/H20851T2+/H20849/H9275/2/H9266/H208502/H20852, when/H9275,T/lessmuchEc. DOI: 10.1103/PhysRevB.73.195113 PACS number /H20849s/H20850: 78.67./H11002n, 72.80.Tm, 73.23.Hk I. INTRODUCTION An inhomogeneous mixture of metallic and insulating phases exhibits a transition between bulk metallic and bulkinsulating behavior. When the volume fraction of metal islarge, the composite material is a “dirty metal” containingisolated impurities; when the volume fraction of metal isvery small, it is a “dirty insulator.” Between these two ex-tremes, there is a third state consisting of large /H20849/H11011100 Å /H20850 metallic regions separated by insulating walls. Such systems are called granular metals. Granularity can arise automati-cally; for instance, electronic phase segregation has been di-rectly observed in the pseudogap phase of cupratesuperconductors 1and in two-dimensional electron gases in semiconductor heterostructures.2Granular metals can also be deliberately created by sputtering a metal onto an insulatingsubstrate, 3–5by lithographic deposition of quantum dots, or by self-assembly of metal nanoparticles coated with organicmolecules. 6Some of these methods allow the control of dis- order. Granular metals are very interesting as their transport properties—in particular, the dc conductivity—cannot be ex-plained by simple extrapolation from the neighboring metal-lic or insulating phases. Another probe of the metal-insulatortransition is the optical /H20849AC/H20850conductivity. In this paper we study the frequency and temperature dependence of the op-tical conductivity of granular metals. We begin by formingcomparative and contextual links with existing literature ontransport in dirty metals, dirty insulators, and granular metalsthemselves.A. Dirty metals A “dirty metal” consists of impurities embedded in a me- tallic host. The electronic states at the Fermi energy are de-localized throughout the solid, giving a finite conductivity atzero temperature. Thermal excitations are detrimental tocharge transport, so the dc conductivity has a “metallic” tem-perature dependence /H20849d /H9268/dT/H110210/H20850. At very low temperatures, electron-electron interactions and quantum coherence7–11can conspire to give “insulating” corrections to conductivity/H20851d/H20849/H9004 /H9268/H20850/dT/H110220/H20852, which are usually weak. The optical conductivity is well described by Drude theory, /H9268Drude /H20849/H9275/H20850=ne2 m/H9270Drude 1+i/H20841/H9275/H20841/H9270Drude, /H208491/H20850 where nis the conduction electron density and /H9270Drude is the relaxation time, which may be temperature dependent. Athigh frequencies the optical conductivity is dominated byelectronic inertia, /H9268Drude /H20849/H9275/H20850/H11015ne2/im/H9275. There are small co- herence corrections to the Drude result at low temperatures. B. Dirty insulators A “dirty insulator” or “dirty semiconductor” consists of impurities embedded in an insulating host. There is a finitedensity of states at the Fermi energy due to impurity states,but these states are all localized, so the dc conductivity iszero at T=0. Conduction occurs by thermally activated hop-PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 1098-0121/2006/73 /H2084919/H20850/195113 /H2084915/H20850 ©2006 The American Physical Society 195113-1ping between bound states, so the conductivity has an “insu- lating” temperature dependence /H20849d/H9268/dT/H110220/H20850; it obeys a variable-range-hopping law of the Mott12or Efros-Shklovskii13kind, depending on whether the long- range Coulomb interaction is screened. In this paper we willnot be studying the effects of long-range Coulomb interac-tion, and therefore, we discuss below only the Mott case. Forthe sake of completeness, a discussion of the Efros-Shklovskii case is provided in Appendix. A. Mott 14showed that the main contribution to optical con- ductivity comes from resonant absorption by pairs of states,one of which is occupied and the other empty. Mott’s argu-ment, which we recapitulate briefly, is valid when electroncorrelations due to long-range Coulomb interactions can bedisregarded. Let the two states in a pair have energies /H9280iand /H9280j. The resonance condition is satisfied when /H9275=/H9280j−/H9280i. The transition rate Pijin the presence of an electric field Ecos/H20849/H9275t/H20850is given by the Fermi Golden Rule, Pij=/H9266e2V/H20841xij/H208412E2/H9267/H20849/H9280j/H20850, /H208492/H20850 where/H9267/H20849/H9280/H20850is the density of /H20849impurity band /H20850states per unit volume and Vis the volume of the system. The conductivity /H9268/H20849/H9275/H20850is then found by multiplying by /H9275//H20849V/2/H20850E2, averaging over all occupied initial states with energies in the interval /H9280F−/H9275and/H9280F, and averaging over all unoccupied final states j. The result is14 Re/H20851/H9268/H20849/H9275/H20850/H20852=2/H9266e2V/H9275/H20885 /H9280F−/H9275/H9280F d/H9280/H20841xij/H20841av2/H9267/H20849/H9280/H20850/H9267/H20849/H9280+/H9275/H20850. /H208493/H20850 The best scenario for a hopping transition between the two localized states at low frequencies is that they are degenerateand the splitting of the levels to tunneling, /H9004 /H9280ij/H11011we−xij//H9264locis smaller than /H9275, or, in other words, the distance xijbetween the localized states should be large enough: xij/H11350r/H9275 =/H9264locln/H20849w//H9275/H20850. Here wis an energy scale of the order of the relaxation rate.15Localized states in a “shell” of thickness /H9264locaround r/H9275will also satisfy the condition for resonance. Using this in Eq. /H208493/H20850, we arrive at Mott’s optical conductivity for a disordered insulator, Re/H20851/H9268/H20849/H9275/H20850/H20852 /H11011 2/H9266e2nimp2/H20849/H9275//H9254/H208502/H20849r/H9275d−1/H9264loc/H20850r/H92752 /H110152/H9266e2nimp2/H20849/H9275//H9254/H208502/H9264locd+2lnd+1/H20849w//H9275/H20850, /H208494/H20850 where nimpis the number density of localized states, dis the dimensionality, and 1/ /H9254is the density of states at an impurity site. An important assumption in obtaining Eq. /H208494/H20850is that there is no inelastic scattering during the hopping process. At high frequencies, /H9275/greatermuchw, the electrons are not local- ized, and the optical conductivity reverts to the Drude ex-pression, Eq. /H208491/H20850, with n impas the conduction electron den- sity. At some intermediate frequency, the optical conductivityhas a maximum; however, this maximum is just due to acrossover between different behaviors, and is not associatedwith any special resonance. C. Granular metals A granular metal consists of metallic grains embedded in an insulating host. The electrons are localized within eachgrain due to the Coulomb blockade. Conduction occurs by intergrain tunneling of thermally excited charges, so the dcconductivity has an insulating temperature dependence/H20849d /H9268/dT/H110220/H20850. However, a granular metal differs from a dirty insulator, in that there is a large number of states N =/H20849aB/R/H20850−don each grain, so the mean level spacing /H9254 /H11011/H9280F/Nis very small. For temperatures /H20849or frequencies /H20850 higher than /H9254, these closely spaced levels may be treated as a continuum leading to incoherent or dissipative transportphenomena. 16–24Inelastic cotunneling, in particular, is the core of the variable-range-cotunneling mechanism of chargetransport in a disordered granular metal, 25,26and has an even greater effect on heat transport.27The low-energy particle- hole excitations within each grain also give rise to a metalliclinear-in- Tspecific heat. The standard model for studying dissipative transport in granular superconductors was obtained by Ambegaokar, Eck-ern, and Schön /H20849AES /H20850in 1982. 16This model has also been widely used to study normal granular metals.17–24It de- scribes the competition between incoherent intergrain tunnel-ing /H20849characterized by the dimensionless intergrain conduc- tance g/H20850that tends to delocalize charge, and Coulomb blockade /H20849characterized by the charging energy of the grain, E c/H20850that suppresses intergrain tunneling. These are quantum effects that are beyond the realm of classical electrodynamicsand circuit theory. The AES approach is valid at temperatureslarger than both the mean level spacing in a grain /H9254and the Thouless energy of intergrain diffusion.20,21In this regime, intergrain transport is incoherent and quantum interferenceeffects are unimportant. We now turn to optical conductivity. The study of optical properties of metal particles has a long history and occupiesa large body of literature. 28–33Effective-medium theories are perhaps the most common approaches.34The earliest of these is due to Maxwell Garnett, who in 1904 proposed usingfrequency-dependent dielectric functions in the expressionfor the effective dielectric constant of the granular metal thathad been obtained from electrostatics. 35Thus, if/H9255m/H20849/H9275/H20850and /H9255i/H20849/H9275/H20850are the bulk dielectric functions of the metallic and insulating phases, and /H9255eff/H20849/H9275/H20850is the effective dielectric con- stant of the composite, /H9255eff/H20849/H9275/H20850−/H9255i/H20849/H9275/H20850 /H9255eff/H20849/H9275/H20850+2/H9255i/H20849/H9275/H20850=f/H9255m/H20849/H9275/H20850−/H9255i/H20849/H9275/H20850 /H9255m/H20849/H9275/H20850+2/H9255i/H20849/H9275/H20850, /H208495/H20850 where fis the volume fraction of the metal. Alternatively, following Bruggeman,36one can treat the granular system as a fraction fof metal and 1− fof insulator immersed in an effective medium. The effective dielectric function is ob-tained by solving f/H9255 m/H20849/H9275/H20850−/H9255eff/H20849/H9275/H20850 /H9255m/H20849/H9275/H20850+2/H9255eff/H20849/H9275/H20850+/H208491−f/H20850/H9255i/H20849/H9275/H20850−/H9255eff/H20849/H9275/H20850 /H9255i/H20849/H9275/H20850+2/H9255eff/H20849/H9275/H20850=0 . /H208496/H20850 In 1908, Mie recognized the importance of polarization oscillations for the optical conductivity.37The classical opti- cal conductivity of a clean spherical metallic grain can beinferred from the equation of motion of the electrons. Sup-pose an external field E extei/H9275tacts on a spherical metallic particle and induces a polarization P. From classical electro-V. TRIPATHI AND Y. L. LOH PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-2dynamics, the field Eintinside the particle is Eint=Eext −/H208494/H9266/3/H20850P. Using the equation of motion of the electrons, −/H92752x/H9275=−eEint/m, together with the definition of the current density j/H9275=−ine/H9275x/H9275and its relation to the polarization, j/H9275 =i/H9275P, and the external electric field, j/H9275/H20849q=0/H20850=/H9268/H20849/H9275/H20850Eext,w e arrive at /H9268/H20849/H9275/H20850=−ine2f m/H9275 /H92752−/H9275r2, /H208497/H20850 /H9275r2=4/H9266 3ne2 m. /H208498/H20850 /H9275r, the frequency of resonant polarization oscillations, is smaller than the plasma frequency of the bulk metal, e/H208814/H9266n/m, by a factor of 1/ /H208813, and depends on the shape of the grain, but not on its size.49At very high frequencies, /H9275/greatermuch/H9275r, the optical conductivity approaches that of a free particle, because the inertia of the electrons prevents themfrom screening the external electric field. If the electrons inthe grain have a finite relaxation time, the equation of motion/H20849− /H92752+i/H20841/H9275/H20841//H9270rel/H20850x/H9275=−eEint/mgives /H9268/H20849/H9275/H20850=−ine2 m/H9275 /H92752−/H9275r2−i/H20841/H9275/H20841//H9270rel. /H208499/H20850 The Mie approach is entirely classical. In order to capture the temperature dependence of /H9268, which is determined by tunneling and charging effects, one has to use a quantumtreatment such as AES effective-field theory. In the originalAES model, the Coulomb interaction is approximated by acapacitance matrix; the electrostatic potential is uniform oneach grain /H20849although it may fluctuate in time /H20850. This amounts to assuming that the electrons are massless and can instanta-neously redistribute to suppress potential variations withinthe grain. Such a “monopole” approximation is adequate in-sofar as dc transport is concerned, because the bottleneck intransport is intergrain tunneling rather than electronic inertia.Optical properties, however, depend crucially upon the finitemass of the electrons and the possible polarization of indi-vidual grains /H20849see Fig. 1 /H20850. Indeed, a calculation of /H9268/H20849/H9275/H20850from the AES action alone misses the polarization resonance peak completely, and thus severely violates the sum rule. Purpose of this paper and results In this paper, we generalize the Ambegaokar-Eckern- Schön /H20849AES /H20850model for a regular array of spherical grains to include dipole /H20849polarization /H20850as well as monopole /H20849charge /H20850 degrees of freedom. Using this effective field theory, we areable to calculate the conductivity as a function of tempera-ture as well as of frequency /H208491/H20850Using a Kubo formula, we find that the optical conductivity of isolated grains is mainly due to intragraindipole oscillations, /H9268/H20849/H9275,T/H20850=−ine2f m/H9275 /H92752−/H9275r2−i/H20841/H9275/H20841//H9270grain, where/H9270grain−1is the relaxation rate for intragrain scattering and consists of, apart from the classical Drude relaxationin a bulk metal, additional finite-volume effects such as Landau damping.31,38,39 /H208492/H20850At finite intergrain tunneling, we find that there is a small additional “monopole” contribution /H9268AES/H20849/H9275,T/H20850due to intergrain charge oscillations, and that intergrain tunneling also imparts an extra width /H9003to the dipole resonance: /H9268/H20849/H9275/H20850/H11015/H9268AES/H20849/H9275,T/H20850−ine2f m/H9275 /H92752−/H9275r2−i/H20841/H9275/H20841//H9270rel, where/H9270rel−1=/H9270grain−1+/H9003/H20849/H9275,T/H20850. At a finite temperature, /H9268AES/H208490,T/H20850is finite and gives the dc conductivity of the granular array. /H9003/H20849/H9275,T/H20850depends on the intergrain dimen- sionless conductance, g, and the grain charging Ec.I ti s independent of /H9270grain−1, and has a different temperature de- pendence. /H208493/H20850The temperature and frequency dependence of the resonance width /H9003/H20849/H9275,T/H20850is different from /H9268AES/H20849/H9275,T/H20850, espe- cially when /H9275,Tare smaller than the effective charging en- ergy of the grains. At large /H9275,T, both/H9003and/H9268AESbecome independent of /H9275,Tand are proportional to the dimension- less intergrain tunneling conductance, g. Figure 2 illustrates the physical difference between the two. The qualitative dif-ference in the manner in which intergrain tunneling affects /H9268AESand/H9003cannot be explained by a simple effective me- dium approximation.40 /H208494/H20850The optical conductivity of a granular metal is physically different from a dirty insulator with Coulomb in-teraction even though both show similar features. For bothsystems, /H9268/H20849/H9275/H20850vanishes as a power law at low and high fre- quencies and has a maximum at intermediate frequencies. However, for a granular metal this maximum is due to reso-nant polarization oscillations, whereas for a dirty insulatorthe maximum is just due to a crossover and is not associatedwith any resonance. Our theory neglects the interactions between dipole de- grees of freedom on different grains. However, at high fre- FIG. 1. /H20849a/H20850Polarization in the standard AES model involves charge asymmetry between different grains. For weak intergrain tunneling, the energy /H9004Eassociated with the polarization is of the order of the charging energy of the grain, e2/R./H20849b/H20850Polarization due to uneven distribution of charge within a grain. Such polarizationexcitations cost less energy than case /H20849a/H20850but are not considered in the standard AES approach. The filled large circles denote grains,andeandhdenote electron excess and deficit, respectively.FREQUENCY AND TEMPERATURE DEPENDENCE OF THE ¼ PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-3quencies /H20849so that the metal dielectric function approaches unity /H20850or for small f, our approximation approaches the Max- well Garnett result, Eq. /H208495/H20850, when we use the standard rela- tion Re/H9268/H20849/H9275/H20850=−/H20849/H20841/H9275/H20841/4/H9266/H20850Im/H9255eff/H20849/H9275/H20850. In many experimental situations, such as two-dimensional granular arrays, it is pos- sible to screen out long-range Coulomb interaction withinthe sample by using a gate electrode, in which case an effec-tive medium treatment is not necessary. In any case, thedipole-dipole interactions can be modeled with a matrix ifnecessary, just as the monopole-monopole interactions areincluded in the original AES model as the capacitance ma-trix. For simplicity, we also ignore possible effects arisingfrom the nonuniformity of the shape of the metallic particles. D. Arrangement of the paper In Sec. II we introduce our model of granular metals: an array of spherical metallic grains with interacting electronsand a finite intergrain hopping. We then make a multipoleexpansion of the potential on the grain in terms of sphericalharmonics up to the l=1 /H20849dipole /H20850component. In Sec. III we develop an effective field theory in terms of the monopoleand dipole components of the potential fluctuations: this is ageneralization of the AES theory of transport in granularmetals. Optical conductivity is calculated in Sec. IV. First weconsider isolated grains and calculate the optical conductiv-ity using the Kubo formula approach as well as from thedielectric function. The calculation with the dielectric func-tion is much less tedious. The result agrees with classicalexpressions for the optical conductivity of isolated grains.Next we consider the case of finite intergrain tunneling andstudy the differences from the classical optical conductivity.An explicit expression for the broadening of the polarizationresonance, /H9003, is obtained. Its temperature and frequency de- pendence are found to be different from the conductivity ofthe granular metal obtained from the AES model. The paper concludes /H20849Sec. V /H20850with a discussion of the results and open problems for further study. II. MODEL We consider the following action for the granular metal array: S=e2 2/H20858 ij/H20885 /H9270xixj/H9267/H20849xi,/H9270/H20850/H9267/H20849xj,/H9270/H208501 /H20841xi−xj/H20841 +/H20858 i/H20885 /H9270xi/H9274/H9270xi†/H20851/H11509/H9270+/H9264/H20849−i/H11633xi/H20850/H20852/H9274/H9270xi+/H20858 /H20855ij/H20856/H20885 /H9270xixjtxi,xj/H9274/H9270xi†/H9274/H9270xj, /H2084910/H20850 where/H9267/H20849xi,/H9270/H20850=/H20849/H9274/H9270xi†/H9274/H9270xi/H9270−Q0i/H20850/H9008/H20849/H20841xi/H20841−R/H20850is the excess elec- tronic charge density at position xiin the ithgrain of radius R,/H9264/H20849−i/H11633i/H20850=/H9280/H20849−i/H11633xi/H20850−/H9262=pxi2/2m−/H9262,ais a lattice translation vector, and txi,xjis the intergrain hopping amplitude. Integrals over/H9270are understood to go from 0 to /H9252. We assume the intergrain hopping amplitude has a white-noise distribution, /H20855txi,xjtxk,xl/H20856=/H20841t/H208412/H9254/H20849xi−xl/H20850/H9254/H20849xj−xk/H20850, /H2084911/H20850 where angle brackets denote disorder averaging. Next we decouple the Coulomb interaction in Eq. /H2084910/H20850 through a Hubbard-Stratonovich field, Vi/H20849xi/H20850, that has the physical meaning of the electrostatic potential. The interac- tion part of the action is Sint=−1 2e2/H20858 ij/H20885 /H9270xixjC/H20849xi,xj/H20850Vi/H20849xi,/H9270/H20850Vj/H20849xj,/H9270/H20850 +/H20858 i/H20885 /H9270xi/H9274/H9270xi†Vi/H20849xi,/H9270/H20850/H9274/H9270xi, /H2084912/H20850 where /H20848xjC/H20849xi,xj/H208501 /H20841xj−xk/H20841=/H9254/H20849xi−xk/H20850subject to appropriate boundary conditions at the metallic grains; thus C/H20849xi,xj/H20850is proportional to the Laplace operator. For simplicity, we will consider grains sufficiently far apart so that the mutual interaction of electrons on differentgrains is small compared to the interaction of electronswithin individual grains. With this simplification, the inter-action part of the action becomes S int/H11015−1 8/H9266e2/H20858 i/H20885 /H9270xEi/H20849x,/H9270/H20850·Ei/H20849x,/H9270/H20850 +/H20858 i/H20885 /H9270xi/H9274/H9270xi†Vi/H20849xi,/H9270/H20850/H9274/H9270xi, /H2084913/H20850 Ei/H20849x,/H9270/H20850=−/H11633Vi/H20849x,/H9270/H20850is the electric field at xdue to charge on an isolated grain at i. The potential away from the boundary may be expanded in a basis of eigenfunctions of the Laplaceequation, V i/H20849x,/H9270/H20850=/H20858 lmAilm/H20849/H9270/H20850/H20873r R/H20874l Ylm/H20849/H9258,/H9278/H20850/H20849r/H11021R/H20850, FIG. 2. Physically different mechanisms involving intergrain tunneling processes that determine the dc conductivity /H9268/H20849that mea- sures the escape rate of an electron from the grain /H20850and the relax- ation rate of polarization oscillations /H9003./H20849a/H20850Coulomb blockade of intergrain tunneling, especially at small values of g, suppresses electron escape from the grains, leading to /H9268/H11011exp /H20849−Ec/T/H20850at low temperatures. /H20849b/H20850A polarization oscillation does not result in a net transfer of charge and therefore no strong Coulomb blockade at lowtemperatures. High temperatures and strong intergrain tunnelingboth wash out Coulomb blockade effects, whereby the two pro-cesses show the same temperature dependence. Drude theory pre-dicts the same temperature dependence for /H9268and/H9003.V. TRIPATHI AND Y. L. LOH PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-4Vi/H20849x,/H9270/H20850=/H20858 lmBilm/H20849/H9270/H20850/H20873R r/H20874l+1 Ylm/H20849/H9258,/H9278/H20850/H20849r/H11022R/H20850, where r=/H20841x/H20841. The continuity of Vi/H20849x,/H9270/H20850at the boundary re- quires that Ailm/H20849/H9270/H20850=Bilm/H20849/H9270/H20850. For the purposes of this paper, it is sufficient to retain just the monopole component /H20849average potential /H20850, Vi/H20849l=0 ;x,/H9270/H20850=/H20877Vi0/H20849/H9270/H20850, r/H11021R, Vi0/H20849/H9270/H20850/H20849R/r/H20850,r/H11022R,/H20878 and the dipole components /H20849electric field /H20850, Vi/H20849/H9251/H20850/H20849l=1 ;x,/H9270/H20850=/H20877Vi1/H20849/H9251/H20850/H20849/H9270/H20850/H20849x/H9251/R/H20850, r/H11021R, Vi1/H20849/H9251/H20850/H20849/H9270/H20850/H20849R2x/H9251/r3/H20850,r/H11022R./H20878 Using the definition Ei/H20849x,/H9270/H20850=−/H11633Vi/H20849x,/H9270/H20850, the interaction part of the action, Eq. /H2084913/H20850takes the form Sint/H11015−R 2e2/H20858 i/H20885 /H9270/H20873/H20851Vi0/H20849/H9270/H20850/H208522+/H20858 /H9251/H20851Vi1/H20849/H9251/H20850/H20849/H9270/H20850/H208522/H20874 +/H20858 i/H20885 /H9270xi/H9274/H9270xi†/H20873Vi0+/H20858 /H9251Vi1/H20849/H9251/H20850/H20849xi/H9251/R/H20850/H20874/H9274/H9270xi. /H2084914/H20850 Now make the gauge transformations /H9274/H9270xi→/H9274/H9270xie−i/H9272i0/H20849/H9270/H20850−i/H20858 /H9251/H9272i1/H20849/H9251/H20850/H20849/H9270/H20850/H20849xi/H9251/R/H20850, /H2084915/H20850 Vi0/H20849/H9270/H20850=i/H11509/H9270/H9272i0,Vi1/H20849/H9251/H20850/H20849/H9270/H20850=i/H11509/H9270/H9272i1/H20849/H9251/H20850. /H2084916/H20850 to eliminate Vi0/H20849/H9270/H20850and replace Vi1/H20849/H9270/H20850by a time-dependent vector potential, Sel=/H20858 i/H20885 /H9270xi/H9274/H9270xi†/H20875/H11509/H9270+/H9264/H20849−i/H11633xi/H20850+Vi0+/H20858 /H9251Vi1/H20849/H9251/H20850/H20849xi/H9251/R/H20850/H20876/H9274/H9270xi →/H20858 i/H20885 /H9270xi/H9274/H9270xi†/H20851/H11509/H9270+/H9264/H20849−i/H11633xi−/H9272i1/R/H20850/H20852/H9274/H9270xi. /H2084917/H20850 The gauge transformations also dress the tunneling element in Eq. /H2084910/H20850with monopole /H20849l=0/H20850and dipole /H20849l=1/H20850phase fluctuations, txi,xj→t˜xi,xj/H20849/H9270/H20850=txi,xjei/H20851/H9272i0/H20849/H9270/H20850−/H9272j0/H20849/H9270/H20850/H20852 /H11003exp/H20873/H20849i/R/H20850/H20858 /H9251/H20849/H9272i1/H20849/H9251/H20850/H20849/H9270/H20850xi/H9251−/H9272j1/H20849/H9251/H20850/H20849/H9270/H20850xj/H9251/H20850/H20874./H2084918/H20850 III. EFFECTIVE FIELD THEORY Integrating out the conduction electrons results in an ef- fective action for the l=0 and l=1 phase fluctuations: Seff/H20851/H9272/H20852=R 2e2/H20858 i/H20885 /H9270/H20851/H20849/H11509/H9270/H9272i0/H208502+/H20841/H11509/H9270/H9272i1/H208412/H20852 −t rl n /H20851G/H9272i1−1/H9254ij−/H20849t˜xi,xj/H9254j,i+a+i↔i+a/H20850/H20852,/H2084919/H20850 where−G/H9272i1−1=/H11509/H9270+1 2m/H20873pxi−/H9272i1 R/H208742 −/H9262 /H2084920/H20850 is the inverse of the electron Green’s function on grain iin the absence of intergrain tunneling, and t˜xi,xjis the dressed tunneling amplitude defined in Eq. /H2084918/H20850and the bare tunnel- ingtxi,xjhas a Gaussian distribution as in Eq. /H2084911/H20850. We study first the effective field theory for isolated grains and then consider the effect of finite intergrain tunneling. A. Isolated grains In the absence of tunneling, the “bare” effective action Seff/H208490/H20850/H20851/H9272/H20852is obtained by expanding the determinant in Eq. /H2084919/H20850 up to second order in /H9272i1/H20849/H9270/H20850, Seff/H208490/H20850/H20851/H9272/H20852=R 2e2/H20858 i/H20885 /H9270/H20851/H20849/H11509/H9270/H9272i0/H208502+/H20841/H11509/H9270/H9272i1/H208412/H20852 +1 2mR2/H20858 i/H20873/H20885 /H9270xiGi/H208490/H20850/H20849xi,xi;/H9270,/H9270/H20850/H9272i12/H20849/H9270/H20850/H20874 +/H208731 m/H20885 /H9270/H9270/H11032xixi/H11032/H9272i1/H20849/H9270/H20850·pxiGi/H208490/H20850/H20849xi,xi/H11032;/H9270,/H9270/H11032/H20850 /H11003/H9272i1/H20849/H9270/H11032/H20850·pxi/H11032Gi/H208490/H20850/H20849xi/H11032,xi;/H9270/H11032,/H9270/H20850/H20874, /H2084921/H20850 where Gi/H208490/H20850=−/H20851/H11509/H9270+/H208491/2m/H20850pxi2−/H9262/H20852−1is the bare electron Green’s function, Gi/H208490/H20850/H20849xi,xi/H11032;/H9270,/H9270/H11032/H20850=T/H20858 /H9261,n/H9274/H9261/H20849xi/H11032/H20850/H9274/H9261*/H20849xi/H20850 i/H9263n−/H9264i/H9261e−i/H9263n/H20849/H9270−/H9270/H11032/H20850 =/H20858 /H9261Gi/H9261/H208490/H20850/H20849/H9270,/H9270/H11032/H20850/H9274/H9261/H20849xi/H11032/H20850/H9274/H9261*/H20849xi/H20850. /H2084922/H20850 Note that /H20858 /H9261Gi/H9261/H208490/H20850/H20849/H9270,/H9270/H11032/H20850/H11015/H9263/H20849/H9280F/H20850T/H20858 n/H20885 −/H9270c−1/H9270c−1 d/H9264ie−i/H9263n/H20849/H9270−/H9270/H11032/H20850 i/H9263n−/H9264i =−2 i/H9263/H20849/H9280F/H20850T/H20858 ne−i/H9263n/H20849/H9270−/H9270/H11032/H20850cot−1/H9263n/H9270c,/H2084923/H20850 where/H9270c/H11011/H9280F−1is a short time cutoff. For /H20841/H9270−/H9270/H11032/H20841/greatermuch/H9270c, this simplifies to /H20858 /H9261Gi/H9261/H208490/H20850/H20849/H9270,/H9270/H11032/H20850/H11015/H9266T/H9263/H20849/H9280F/H20850 sin/H9266T/H20849/H9270−/H9270/H11032/H20850. /H2084924/H20850 We shall use this expression, unless stated otherwise. Equation /H2084921/H20850can be presented in a more recognizable form asFREQUENCY AND TEMPERATURE DEPENDENCE OF THE ¼ PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-5Seff/H208490/H20850/H20851/H9272/H20852=R 2e2/H20858 i/H20885 /H9270/H20851/H20849/H11509/H9270/H9272i0/H208502+/H20841/H11509/H9270/H9272i1/H208412/H20852 +4/H9266 3R 2e2/H20858 i/H9251/H9252/H20885 /H9270,/H9270/H11032Ki/H9251/H9252/H20849/H9270−/H9270/H11032/H20850/H9272i1/H20849/H9251/H20850/H20849/H9270/H20850/H9272i1/H20849/H9252/H20850/H20849/H9270/H11032/H20850. /H2084925/H20850 in terms of the bare electromagnetic response function of the ithgrain, Ki/H9251/H9252/H20849/H9270−/H9270/H11032/H20850=/H9254/H9251/H9252e2 Vm/H20885 xiGi/H208490/H20850/H20849xi,xi;/H9270,/H9270/H11032/H20850/H9254/H20849/H9270−/H9270/H11032/H20850 +e2 Vm2/H20885 xixi/H11032pxi/H9251Gi/H208490/H20850/H20849xi,xi/H11032;/H9270,/H9270/H11032/H20850pxi/H11032/H9252 /H11003Gi/H208490/H20850/H20849xi/H11032,xi;/H9270/H11032,/H9270/H20850, /H2084926/H20850 where V=/H208494/H9266/3/H20850R3is the volume of the grain. In a bulk metal, the two terms in the electromagnetic response func- tion in Eq. /H2084926/H20850would correspond to the diamagnetic and paramagnetic parts of the bulk conductivity /H9268/H20849/H9275/H20850=K/H20849/H9275/H20850/i/H9275. In a finite system, the situation is trickier. The frequency dependence of Kis Ki/H9251/H9252/H20849i/H9275m/H20850=/H9254/H9251/H9252ne2 m/H208731+2 mN/H20858 /H9261/H9261/H11032f/H20849/H9264i/H9261/H20850/H9264i,/H9261/H9261/H11032/H20841pi,/H9261/H9261/H11032/H9251/H208412 /H9275m2+/H9264i,/H9261/H9261/H110322/H20874, /H2084927/H20850 where/H9261,/H9261/H11032label the eigenvalues of the free electron Hamil- tonian of a grain, /H9264i,/H9261/H9261/H11032=/H9264i/H9261−/H9264i/H9261/H11032,f/H20849/H9264/H20850=/H20851e/H9252/H9264+1/H20852−1is the Fermi-Dirac distribution function, and pi,/H9261/H9261/H11032/H9251=/H20855/H9261/H20841pi/H9251/H20841/H9261/H11032/H20856.n =N/Vis the number density of electrons in a grain. If the temperature /H20849or frequency /H20850is much smaller than the level separation /H9254=/H9264/H9261/H9261/H11032avg/H11011/H9280F/N, we expand the right hand side of Eq. /H2084927/H20850in ascending powers of /H9275m: Ki/H9251/H9252/H20849i/H9275m/H20850/H11015/H9254/H9251/H9252ne2 m/H208731+2 mN/H20858 /H9261/H9261/H11032f/H20849/H9264i/H9261/H20850/H20841pi,/H9261/H9261/H11032/H9251/H208412 /H9264i,/H9261/H9261/H11032 −2 mN/H9275m2/H20858 /H9261/H9261/H11032f/H20849/H9264i/H9261/H20850/H20841pi,/H9261/H9261/H11032/H9251/H208412 /H9264i,/H9261/H9261/H110323/H20874+O/H20849/H9275m4/H20850./H2084928/H20850 The static part of Eq. /H2084928/H20850can be shown to vanish using the Reiche-Thomas-Kuhn sum rule,41,42 2 m/H20858 /H9261/H11032/H20841pi,/H9261/H9261/H11032/H9251/H208412 /H9264i,/H9261/H9261/H11032=−1 , /H2084929/H20850 along with the identity /H20858/H9261f/H20849/H9264i/H9261/H20850=N. Combining Eq. /H2084925/H20850and /H2084928/H20850, one finds that the surviving contribution in Eq. /H2084928/H20850 makes a finite size quantum correction to the RPA dielectric constant,28–33/H9255RPA=1−2 e2/m2R3/H20858/H9261/H9261/H11032/H20851f/H20849/H9264i/H9261/H20850/H20841pi,/H9261/H9261/H11032/H9251/H208412//H9264i,/H9261/H9261/H110323/H20852 /H110111−/H20849kFa0/H20850/H20849R/a0/H208502, where a0is a small length of the order of a lattice constant. As a result, even for metallic grains a few tens of lattice constants across, the static dielectric con-stant rapidly approaches bulk values /H20849where is it infinity /H20850,and the polarizability, /H9251=R3/H20849/H9255RPA−1/H20850//H20849/H9255RPA+2/H20850, approaches the classical value, /H9251classical =R3. The sum rule enables us to recast the electromagnetic re- sponse function as Ki/H9251/H9252/H20849i/H9275m/H20850=−/H9254/H9251/H92522e2 Vm2/H9275m2/H20858 /H9261/H9261/H11032f/H20849/H9264i/H9261/H20850/H20841pi,/H9261/H9261/H11032/H9251/H208412 /H9264i,/H9261/H9261/H11032/H20849/H9275m2+/H9264i,/H9261/H9261/H110322/H20850,/H2084930/H20850 which is a known result. For the rest of the paper, unless stated otherwise, we shall assume that the temperature /H20849or frequency /H20850is much larger than the level separation /H20849T//H9254/greatermuch1/H20850. Then, using Eq. /H2084929/H20850and /H2084930/H20850, we obtain Ki/H9251/H9252/H20849i/H9275m/H20850/H11015/H9254/H9251/H9252ne2 m,T//H9254/greatermuch1, /H2084931/H20850 that is, the value for the clean bulk metal; this is the diamag- netic response due to electron acceleration in an electricfield. Hence, Eq. /H2084925/H20850becomes S eff/H208490/H20850/H20851/H9272/H20852=R 2e2/H20858 i/H20885 /H9270/H20851/H20849/H11509/H9270/H9272i0/H208502+/H20841/H11509/H9270/H9272i1/H208412+/H9275r2/H20841/H9272i1/H208412/H20852,/H2084932/H20850 where/H9275ris the resonance frequency for a metallic sphere, /H9275r2=4/H9266 3ne2 m, /H2084933/H20850 we introduced in Eq. /H208498/H20850. In a collisionless bulk metal, the paramagnetic part of the electromagnetic response functiondefined in Eq. /H2084926/H20850vanishes. However, in a finite-size grain, if one can treat the quasiparticle excitations in the grain as acontinuum /H20849this is so if the temperature is not too low, T/greatermuch /H9254/H20850, the paramagnetic part is finite and gives rise to a finite relaxation of the oscillations through disintegration intoincoherent particle-hole excitations. 31,38The relaxation time has been shown in numerous works31,38,39to be of the order of the time of flight, R/vF. Physically, the relaxation is due to Landau damping of plasma oscillations at a finite wave vec-tor: the minimum wave vector in a grain of size Ris of the order of /H9266/R. Other inelastic processes such as phonon scat- tering will also contribute to relaxation. B. Finite intergrain tunneling We now obtain the effective field theory when intergrain tunneling is finite. At not too low temperatures,20,21 T/greatermuchmax /H20849/H20841t/H208412/H9254,/H9254/H20850, and for large enough43grains /H20849kFR/H208502/greatermuch1, it suffices to expand the electron determinant in Eq. /H2084919/H20850up toO/H20849t2/H20850, Sefftun/H20851/H9272/H20852=1 2/H20858 i,a/H20885 /H9270/H9270/H11032xixi/H11032xi+axi+a/H11032t˜xi/H11032,xi+a/H11032/H20849/H9270/H11032/H20850t˜xi+a,xi/H20849/H9270/H20850 /H11003G/H9272i1/H20849xi,xi/H11032;/H9270,/H9270/H11032/H20850G/H9272i+a,1/H20849xi+a/H11032,xi+a;/H9270/H11032,/H9270/H20850 =/H20841t/H208412 2/H20858 i,a/H20885 /H9270/H9270/H11032xixi+aG/H20849xi,xi;/H9270,/H9270/H11032/H20850G/H9272i+a,1/H20849xi+a,xi+a;/H9270/H11032,/H9270/H20850 /H11003ei/H20851/H9272ij,0/H20849/H9270/H11032/H20850−/H9272ij,0/H20849/H9270/H20850/H20852ei/H208491/R/H20850/H20851/H9272i1/H20849/H9270/H11032/H20850−/H9272i1/H20849/H9270/H20850/H20852·xi /H11003e−i/H208491/R/H20850/H20851/H9272j1/H20849/H9270/H11032/H20850−/H9272j1/H20849/H9270/H20850/H20852·xj, /H2084934/H20850V. TRIPATHI AND Y. L. LOH PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-6where/H9272ij,0/H20849/H9270/H20850=/H9272i0/H20849/H9270/H20850−/H9272j0/H20849/H9270/H20850. The /H9272i1dependence in Eq. /H2084934/H20850comes from the exponential as well as from the Green’s functions, G/H9272i1/H20849xi,xi;/H9270,/H9270/H11032/H20850, etc. We show in Appendix B that the contribution arising from the expansion of G/H9272i1/H20849xi,xi;/H9270,/H9270/H11032/H20850, etc., in powers of /H9272i1is insignificant com- pared to that coming from the exponential. Therefore we expand only the exponential in Eq. /H2084934/H20850up to second order in /H9272i1, etc. and ignore the /H9272i1dependence of the Green’s func- tions. Thus the tunneling part of the effective action is Sefftun/H20851/H9272/H20852/H11015/H20841t/H208412/H20858 i,a;/H9261/H9261/H11032/H20885 /H9270/H9270/H11032/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850Gi/H9261/H208490/H20850/H20849/H9270,/H9270/H11032/H20850Gi+a,/H9261/H11032/H208490/H20850/H20849/H9270/H11032,/H9270/H20850 −/H20841t/H208412 6R2/H20858 i,a;/H9261/H9261/H11032/H20885 /H9270/H9270/H11032/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850Gi/H9261/H208490/H20850/H20849/H9270,/H9270/H11032/H20850Gi+a,/H9261/H11032/H208490/H20850/H20849/H9270/H11032,/H9270/H20850 /H11003/H20853ri/H92612/H20851/H9272i1/H20849/H9270/H11032/H20850−/H9272i1/H20849/H9270/H20850/H208522+ri+a,/H9261/H110322/H20851/H9272i+a,1/H20849/H9270/H11032/H20850 −/H9272i+a,1/H20849/H9270/H20850/H208522/H20854/H20849 35/H20850 where /H9016i,i+a/H20849/H9270,/H9270/H11032/H20850= cos /H20851/H9272i,i+a,0/H20849/H9270/H20850−/H9272i,i+a,0/H20849/H9270/H11032/H20850/H20852 /H20849 36/H20850 andri/H92612=/H20855i/H9261/H20841rˆ2/H20841i/H9261/H20856are the matrix elements of rˆ2=/H20841xˆ/H208412with the eigenstates of grain i. For a spherical grain, the eigen- functions are spherical Bessel functions jn/H20849/H9260nlr/R/H20850Ylm/H20849/H9258,/H9278/H20850, where/H9260nlis the lth zero of jn/H20849x/H20850. Numerically evaluating the matrix elements, we find they range between 0.28 R2andR2. In particular, lim n→/H11009/H20849r2/H20850n1=R2, and lim l→/H11009/H20849r2/H20850nl=R2/3. Thus the matrix elements of r2do not vary strongly andare of the order of R2. Hence Eq. /H2084935/H20850may be written as Sefftun/H20851/H9272/H20852/H11015−/H9266gT2 2/H20858 i,a/H20885 /H9270/H9270/H11032/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850 sin2/H9266T/H20849/H9270−/H9270/H11032/H20850 +/H9266gT2b 2/H20858 i,a/H20885 /H9270/H9270/H11032/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850 sin2/H9266T/H20849/H9270−/H9270/H11032/H20850 /H11003/H20853/H20851/H9272i1/H20849/H9270/H11032/H20850−/H9272i1/H20849/H9270/H20850/H208522+/H20851/H9272i+a,1/H20849/H9270/H11032/H20850−/H9272i+a,1/H20849/H9270/H20850/H208522/H20854; /H2084937/H20850 here b/H110110.1 is a constant, gis the dimensionless intergrain tunneling conductance g=2/H9266/H20841t/H208412/H9263/H20849/H9280F/H208502, /H2084938/H20850 Eqs. /H2084932/H20850and /H2084937/H20850form the effective action, Seff/H20851/H9272/H20852=Seff/H208490/H20850 /H11003/H20851/H9272/H20852+Sefftun/H20851/H9272/H20852, which generalizes the AES action to include the physics of dipolar oscillations. This may be presented as Seff/H20851/H9272/H20852/H11015SAES/H20851/H92720/H20852+Spol/H20851/H92720/H20852, where Spol/H20851/H9272/H20852/H11015T 2/H20849e2/R/H20850/H20858 i,m/H20849/H9275m2+/H9275r2/H20850/H9272i1/H20849/H9275m/H20850·/H9272i1/H20849−/H9275m/H20850 +T 4/H20849e2/R/H20850/H20858 ia/H9003i,i+a/H20849i/H9275m/H20850/H20841/H9275m/H20841/H11003/H20851/H9272i1/H20849/H9275m/H20850 ·/H9272i1/H20849−/H9275m/H20850+/H9272i+a,1/H20849/H9275m/H20850·/H9272i+a,1/H20849−/H9275m/H20850/H20852,/H2084939/H20850 whereSAES/H20851/H92720/H20852=1 2/H20849e2/R/H20850/H20858 i/H20885 /H9270/H20849/H11509/H9270/H9272i0/H208502 −/H9266gT2 2/H20858 ia/H20885 /H9270/H9270/H11032/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850 sin2/H9266T/H20849/H9270−/H9270/H11032/H20850/H2084940/H20850 is the standard Ambegaokar-Eckern-Schön /H20849AES /H20850model for normal granular metals and /H9003i,i+a/H20849i/H9275m/H20850=e24/H9266gbT2 R/H20841/H9275m/H20841/H20885 /H9270/H208491−ei/H9275m/H9270/H20850/H9016i,i+a/H20849/H9270,0/H20850 sin2/H20849/H9266T/H9270/H20850./H2084941/H20850 The quantities /H9016and/H9003are functionals of /H92720. In principle, fluctuations of /H92720and of/H92721can influence each other since they both appear in Spol. In practice, it is sufficient to calcu- late the correlator /H20855/H9016/H20851/H92720/H20852/H20856forSAESalone, and to use this mean value in Spolto determine the fluctuations of /H92721.T o justify this, we show that fluctuations of /H92721have a negligible effect on the “kernel” for /H92720.I fi nE q . /H2084937/H20850we average over the fields /H9272i1using their bare propagator in the absence of tunneling /H20851see Eq. /H2084932/H20850/H20852, /H20855/H20849/H9272i1/H20849/H9270/H20850−/H9272i1/H208490/H20850/H208502/H20856=3/H20849e2/R/H20850T/H20858 n/H208491−e−i/H9275n/H9270/H20850 /H9275n2+/H9275r2 =3e2/R 2/H9275rcoth /H20849/H9275r/2T/H20850/H208491−e−/H9275r/H20841/H9270/H20841/H20850. /H2084942/H20850 Thus, at long times, /H9275r/H20841/H9270/H20841/greatermuch1, the correction to the tunnel- ing term of the AES model due to dipole modes is smallerthan the bare value by a factor of /H20849e 2/R/H20850//H9275r. In most com- mon cases of granular metals, this ratio is of the order of 10−2,a sEc/H11011102Kand/H9275r/H11011104K. At short times, /H9275r/H20841/H9270/H20841/lessmuch1, the correction is smaller than the bare value by a factor /H9275r/H20841/H9270/H20841/lessmuch1. Thus, under most common physical circum- stances, our approximation is valid. For finite tunneling, the propagator for the dipole modes is that of a damped harmonic oscillator, Dij/H9251/H9252/H20849i/H9275n/H20850=/H9254/H9251/H9252/H9254ij/H20849e2/R/H20850 /H9275n2+/H9275r2+/H9003/H20849/H9275n/H20850/H20841/H9275n/H20841, /H2084943/H20850 where the resonance linewidth is /H9003=/H20858 a/H9003i,i+a. /H2084944/H20850 IV. OPTICAL CONDUCTIVITY In this section we calculate the optical conductivity of isolated metallic grains and then generalize it to finite inter-grain tunneling. For isolated grains, we show that the opticalconductivity may be obtained in two ways: directly from theKubo formula and from the dielectric function. A. Isolated grains 1. Kubo formula approach We first calculate the optical conductivity for isolated grains using the Kubo formula approach. For this we intro-FREQUENCY AND TEMPERATURE DEPENDENCE OF THE ¼ PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-7duce an infinitesimal vector potential A/H9270xthat couples to the current j/H9270xand is related to the electric field through E/H9270x =i/H11509/H9270A/H9270x. The electronic kinetic energy becomes /H9280/H20849p˜/H9270xi/H20850→/H9280/H20851p˜xi−/H20849e/c/H20850A/H9270xi/H20852 =/H9280/H20849p˜xi/H20850−/H20849e/mc/H20850A/H9270xi·p˜xi+/H20849e2/2mc2/H20850A/H9270xi2,/H2084945/H20850 where pxi=−i/H11633xiandp˜/H9270xi=pxi−/H208491/R/H20850/H9272i1/H20849/H9270/H20850, and we have chosen the gauge /H11633·A=0. The optical conductivity tensor /H9268/H9251/H9252is the coefficient relating the q=0 component of the current, j/H9270/H9251/H20851A;q/H20852=−c Z0/H20885dxe−iq·x/H20885D/H20849fields /H20850/H9254S/H20851A/H20852 /H9254A/H9270x/H9251e−S/H20851A/H20852, /H2084946/H20850 to the q=0 component of the electric field, E/H9270x/H9252=i/H11509/H9270A/H9270x/H9252, /H9268/H9251/H9252/H20849/H9270,/H9270/H11032/H20850=Tc2/H20858 m/H9268/H9251/H9252/H20849i/H9275m/H20850e−i/H9275m/H20849/H9270/H11032−/H9270/H20850=/H20879/H9254j/H9270/H9251/H20851A;q=0/H20852 /H9254E/H9270/H11032/H9252/H20849q=0/H20850/H20879 A=0. /H2084947/H20850 Here Z0=Z/H20851A=0/H20852. Analytically continuing to real frequen- cies gives the well-known Kubo formula for the optical con- ductivity, /H20879/H9268/H9251/H9252/H20849/H9275,T/H20850=−ic2 /H9275Z0V/H20885dxdx/H11032/H20885 /H9270ei/H9275m/H9270/H11003/H20885D/H20849fields /H20850 /H11003/H20873/H92542S/H20851A/H20852 /H9254A/H9270x/H11032/H9252/H9254A0x/H9251−/H9254S/H20851A/H20852 /H9254A/H9270x/H11032/H9252/H9254S/H20851A/H20852 /H9254A0x/H9251/H20874/H20879 i/H9275m→/H9275,/H2084948/H20850 where Vis the volume of the system. The first term in Eq. /H2084948/H20850, as we shall see below, represents the inertial response of the electrons in the bulk metal. The second term, whichvanishes in the bulk, makes a finite contribution in the granu-lar metal. We denote these two contributions as /H9268/H9251/H9252/H20849/H9275,T/H20850=/H9268inertial/H9251/H9252/H20849/H9275,T/H20850+/H9268finite R/H9251/H9252/H20849/H9275,T/H20850. /H2084949/H20850 We can show that the “inertial” term is /H20879/H9268inertial/H9251/H9252/H20849/H9275,T/H20850=−ie2/H9254/H9251/H9252 /H9275mV/H20858 i/H20885dxi/H20855G/H9272i1/H20849xi,xi;/H9270,/H9270/H20850/H20856 −ie2 /H9275m2V/H20858 i/H20885dxidxi/H11032/H20885 /H9270ei/H9275m/H9270/H20855p˜/H9270xi/H11032/H9252p˜0xi/H9251 /H11003G/H9272i1/H20849xi/H11032,xi;/H9270,0/H20850G/H9272i1/H20849xi,xi/H11032;0,/H9270/H20850/H20856/H20879 i/H9275m→/H9275. /H2084950/H20850 This can be expressed in terms of the response function Ki/H9251/H9252 we defined in Eq. /H2084926/H20850. For simplicity we assume that all grains in the system are identical. Also, as in Appendix B, we approximate the Green’s functions G/H9272i1/H20849xi/H11032,xi;/H9270,0/H20850, etc. by their bare values. Then/H20879/H9268inertial/H9251/H9252/H20849/H9275,T/H20850=/H20873−if /H9275Ki/H9251/H9252/H20849i/H9275m/H20850−ie2 /H9275m2R2V/H20858 i/H9261/H20885 /H9270ei/H9275m/H9270 /H11003Gi/H9261/H208490/H20850/H20849/H9270,0/H20850Gi/H9261/H208490/H20850/H208490,/H9270/H20850/H20855/H9272i1/H9252/H20849/H9270/H20850/H9272i1/H9251/H208490/H20850/H20856/H20874/H20879 i/H9275m→/H9275, /H2084951/H20850 where fis the volume fraction occupied by the metallic spheres. The second term on the right hand side of Eq. /H2084951/H20850is smaller than the first by a factor of /H9263/H20849/H9280F/H20850/H20849e2/R/H20850//H20849N/H9275rmR2/H20850, where Nis the number of conduction electrons in a grain. Since 1/ /H20849mR2/H20850/H11011/H9254/H11011/H9280F/N, and/H9263/H20849/H9280F/H20850/H11011N//H9280F, the second term is smaller by a factor of about 1/ N. This is a small number since the number of conduction electrons in a grainin typical systems is of the order of 10 4. We have, dropping this term from Eq. /H2084951/H20850, /H20879/H9268inertial/H9251/H9252/H20849/H9275,T/H20850/H11015−if /H9275Ki/H9251/H9252/H20849i/H9275m/H20850/H20879 i/H9275m→/H9275=−ifne2 m/H9275/H9254/H9251/H9252, /H2084952/H20850 where we used Eq. /H2084931/H20850in the second line. This is indeed of the form of an inductive contribution. Now consider the finite size contribution to the conduc- tivity described in Eq. /H2084948/H20850and Eq. /H2084949/H20850, /H20879/H9268finite R/H9251/H9252/H20849/H9275,T/H20850=ie2 /H9275m2V/H20858 ij/H20885dxidxj/H20885 /H9270ei/H9275m/H9270 /H11003/H20855p˜/H9270xi/H9252p˜0xj/H9251G/H9272i1/H20849xi,xi;/H9270,/H9270/H20850 /H11003G/H9272j1/H20849xj,xj;0,0 /H20850/H20856/H20879 i/H9275m→/H9275. /H2084953/H20850 The diagonal matrix elements of the momenta pare identi- cally zero in a finite system, /H20855/H9261/H20841p/H20841/H9261/H20856=0; therefore we dis- card in Eq. /H2084953/H20850terms of the type /H20885dxipxi/H9252G/H9272i1/H20849xi,xi;/H9270,/H9270/H20850/H110130; this simplifies the finite size contribution to /H20879/H9268finite R/H9251/H9252/H20849/H9275,T/H20850=ie2 /H9275m2R2V/H20858 ij;/H9261/H9261/H11032/H20885 /H9270ei/H9275m/H9270/H11003Gi/H9261/H208490/H20850/H20849/H9270,/H9270/H20850Gj/H9261/H11032/H208490/H20850/H208490,0/H20850 /H11003/H20855/H9272i1/H9252/H20849/H9270/H20850/H9272j1/H9251/H208490/H20850/H20856/H20879 i/H9275m→/H9275. /H2084954/H20850 Here we have as usual approximated the Green’s functions G/H9272i1/H20849xi,xi;/H9270,/H9270/H20850by the bare values. Evaluating Eq. /H2084954/H20850gives the following finite size contribution:V. TRIPATHI AND Y. L. LOH PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-8/H20879/H9268finite R/H9251/H9252/H20849/H9275,T/H20850=ine2Nf /H9275m2R2/H20849e2/R/H20850 /H9275m2+/H9275r2/H20879 i/H9275m→/H9275 =ine2f /H9275m/H9275r2 −/H20849/H9275+i0+/H208502+/H9275r2. /H2084955/H20850 Here N=/H208494/H9266/3/H20850nR3is the total number of conduction elec- trons on a grain, nis the conduction electron density, and we used/H9275r2=/H208494/H9266/3/H20850ne2/m. Adding the inertial and finite size contributions from Eq. /H2084952/H20850and Eq. /H2084955/H20850, we arrive at the optical conductivity for isolated spherical grains, /H9268/H9251/H9252/H20849/H9275,T/H20850=−ine2f m/H9254/H9251/H9252/H9275 /H92752−/H9275r2−i/H20841/H9275/H208410+. /H2084956/H20850 Eq. /H2084956/H20850agrees with the expression for the optical conduc- tivity in Eq. /H208497/H20850that was obtained from a simple analysis of the equation of motion of the electrons in a clean grain. Fora finite intragrain relaxation time, /H9270grain, the optical conduc- tivity takes the form /H9268/H9251/H9252/H20849/H9275,T/H20850=−ine2f m/H9254/H9251/H9252/H9275 /H92752−/H9275r2−i/H20841/H9275/H20841//H9270grain. /H2084957/H20850 2. Conductivity from the dielectric function The optical conductivity of isolated grains that we ob- tained from a tedious Kubo approach could also be inferredfrom the dielectric function. In a Gaussian theory, the dielec-tric function /H9255for a single grain can be extracted from the effective action, S=−R 3 2e2/H20858 i,l,/H9251/H9252/H20885 /H9270,/H9270/H11032/H9255il/H9251/H9252/H20849/H9270−/H9270/H11032/H20850Eil/H9251/H20849/H9270/H20850Eil/H9252/H20849/H9270/H11032/H20850, /H2084958/H20850 where Eil/H9251/H20849/H9270/H20850are the multipole components of the electric field /H20851see Eq. /H2084913/H20850ff/H20852at the grains. The lowest possible angu- lar momentum component of an excitation on an isolatedgrain is l=1. That is, in the absence of intergrain tunneling, the simplest response to an electric field is a uniform polar- ization. Thus, we need to consider only E i1/H9251/H20849/H9270/H20850=Vi1/H9251/R =i/H11509/H9270/H9272i1/H9251/H20849/H9270/H20850/R. Furthermore, because of the high energy /H9275r /H11011e/H20881n/massociated with the dipole excitations, we can safely neglect in the effective action terms with higher pow-ers of /H9272i1/H20849/H9270/H20850. The following relation can be gathered from Eq. /H2084932/H20850, Eq. /H2084956/H20850, and Eq. /H2084958/H20850, /H20841/H9268/H9251/H9252/H20849/H9275,T/H20850/H20841g=0=−ine2f m/H9275/H9254/H9251/H9252 /H9255i1/H20849/H9275,T/H20850=−ine2f m/H9254/H9251/H9252/H9275 /H92752−/H9275r2−i/H9257/H20841/H9275/H20841. /H2084959/H20850 Such a cross relation has been discussed, for instance, by Hopfield44in 1965. Physically, the imaginary part of the di- electric function is associated with relaxation, so a strongerrelaxation implies weaker conduction. B. Finite intergrain tunneling The Kubo approach is the most reliable way to calculate the optical conductivity, but, as illustrated in Sec. IV A, it isvery tedious even for an isolated sphere. At finite intergrain tunneling, an even larger number of terms involving bothintragrain and intergrain currents would have to be calcu-lated. We also show that for Gaussian models, the dielectricfunction could be used to obtain the conductivity with sig-nificantly less effort. However, as the following discussionshows, the theory is Gaussian only in the two extreme casesof isolated grains, g=0, or strongly coupled grains, g/greatermuch1. So we resort to a combination of the Kubo and dielectric func-tion approach, using the Kubo approach for multipole modesthat cannot be considered in a Gaussian approximation, andretaining the dielectric function approach for modes that areeffectively Gaussian. At finite intergrain tunneling, an electric field can cause intergrain polarization /H20849opposite charges on adjacent grains /H20850 as well as intergrain polarization. We must therefore considerthe contribution of the monopole modes /H9272i0/H20849/H9270/H20850in the dielec- tric response function. Tunneling events are accompanied by fluctuations in electrostatic energy that can be large, of theorder of e 2/R, when intergrain tunneling is weak, g/lessmuch1. Therefore for weak but finite tunneling, we must considernon-Gaussian contributions for the monopole modes, V i0/H20849/H9270/H20850 =i/H11509/H9270/H9272i0/H20849/H9270/H20850. This is clear from the effective field theory at finite tunneling given by Eq. /H2084939/H20850and Eq. /H2084940/H20850. On the other hand, for strong intergrain tunneling, g/greatermuch1, monopole fluc- tuations are small because charges can easily flow to neutral-ize potential differences between the grains. In this case, weagain have an approximately Gaussian theory for the /H9272i0/H20849/H9270/H20850 modes. We write the total conductivity as a sum of the mono- pole and dipole contributions, /H9268/H9251/H9252/H20849/H9275,T/H20850/H11015/H92680/H9251/H9252/H20849/H9275,T/H20850+/H92681/H9251/H9252/H20849/H9275,T/H20850. /H2084960/H20850 The conductivity due to the monopole part has been obtained elsewhere21in the context of the AES model, /H20879/H92680/H9251/H9252/H20849/H9275,T/H20850=ia2−d /H9275/H20885 /H9270ei/H9024n/H9270KAES/H9251/H9252/H20849/H9270/H20850/H20879 /H9024n→−i/H9275, /H2084961/H20850 where ais the intergrain distance. It consists of diamagnetic and paramagnetic parts, KAES/H9251/H9252/H20849/H9270/H20850=KAES/H9251/H9252,dia/H20849/H9270/H20850+KAES/H9251/H9252,para/H20849/H9270/H20850, KAES/H9251/H9252,dia/H20849/H9270/H20850=/H9254/H9251/H9252e2/H9266gT2/H20885 /H9270/H11032/H20851/H9254/H20849/H9270/H20850−/H9254/H20849/H9270/H11032−/H9270/H20850/H20852 /H110031 sin2/H20849/H9266T/H9270/H11032/H20850/H20855cos/H20849/H9272i,i+e/H9251,0/H20849/H9270/H20850−/H9272i,i+e/H9251,0/H20849/H9270/H11032/H20850/H20850/H20856, /H2084962/H20850 KAES/H9251/H9252,para/H20849/H9270/H20850=−/H9254/H9251/H9252/H20858 i/H20855X0/H9251/H20849/H9270/H20850Xi/H9251/H208490/H20850/H20856, /H2084963/H20850 whereFREQUENCY AND TEMPERATURE DEPENDENCE OF THE ¼ PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-9Xi/H9253/H20849/H9270/H20850=e/H9266gT2/H20885 /H9270/H110321 sin2/H20851/H9266T/H20849/H9270−/H9270/H11032/H20850/H20852 /H11003/H20855sin/H20849/H9272i,i+e/H9253,0/H20849/H9270/H20850−/H9272i,i+e/H9253,0/H20849/H9270/H11032/H20850/H20850/H20856. /H2084964/H20850 In order to remind us of the AES origin of the l=0 compo- nent of the conductivity, we rename /H92680/H20849/H9275,T/H20850,/H9268AES/H20849/H9275,T/H20850. The contribution to the conductivity from the dipole part is written in terms of the l=1 component of the dielectric function /H20849see the previous discussion and in the previous section /H20850, /H92681/H9251/H9252/H20849/H9275,T/H20850=−ine2f m/H9275/H9254/H9251/H9252 /H9255i1/H20849/H9275,T/H20850=−ine2f m/H9254/H9251/H9252/H9275 /H92752−/H9275r2−i/H9003/H20849/H9275,T/H20850/H20841/H9275/H20841. /H2084965/H20850 Here/H9003/H20849/H9275,T/H20850is the intergrain relaxation rate defined in Eq. /H2084941/H20850./H9003involves the cosine correlator /H9016defined in Eq. /H2084936/H20850, and thus closely resembles the diamagnetic part of the AES conductivity, Eq. /H2084962/H20850. However, the full AES conductivity behaves very differently because of the paramagnetic contri- bution, Eq. /H2084963/H20850. In the presence of intragrain relaxation mechanisms such as impurity scattering or boundary scattering, we expect that /H9268/H9251/H9252/H20849/H9275,T/H20850=/H9254/H9251/H9252/H9268AES/H20849/H9275,T/H20850−ine2f m/H9254/H9251/H9252/H9275 /H92752−/H9275r2−i/H20841/H9275/H20841//H9270rel /H2084966/H20850 for consistency with Eq. /H208499/H20850and Eq. /H2084957/H20850, where the total relaxation rate /H9270rel−1is given by the Matthiessen rule, /H9270rel−1 =/H9270grain−1+/H9003. C. Some special cases The final expression for the optical conductivity contains the AES conductivity /H9268AES/H20849/H9275,T/H20850and the resonance width due to intergrain tunneling /H9003/H20849/H9275,T/H20850./H9268AES/H20849/H9275,T/H20850=/H92680/H20849/H9275,T/H20850has been explicitly defined in Eq. /H2084961/H20850through Eq. /H2084964/H20850, and /H9003/H20849/H9275,T/H20850has been defined in Eq. /H2084941/H20850and Eq. /H2084944/H20850. Below, we discuss a few special cases for a regular three dimensional array. Throughout we assume that the frequency lies in therange /H9275/T/greatermuch1,/H9275/H9270c/H11011/H9275//H9280F/lessmuch1, and the temperature much smaller than the charging energy, Ec/T/greatermuch1. (a)Consider first small intergrain tunneling conductance, g/lessmuch1. As the calculations are very complicated, we refer the reader to Appendices C and D for details. We show there thatat frequencies much larger than the charging energy, the con-ductivity tends to saturate, /H9268AES/H11015g/H20849e2/a/H20850. The same goes for the polarization resonance width, /H9003/H110154bzg /H20849e2/R/H20850, where zis the grain coordination number, and we used Eq. /H2084941/H20850and Eq. /H2084944/H20850. If the frequency is much smaller than the charging energy, the conductivity is dominated by thermal excitationof quasiparticles and obeys an Arrhenius law, /H9268AES /H110152ge2 ae−Ec/T,/H9275/lessmuchEc. /H2084967/H20850 In contrast, the resonance width does not obey an Arrhenius law:/H9003/H110154bzge2 R4/H9266 3Ec2/H20851T2+/H20849/H9275/2/H9266/H208502/H20852,/H9275/lessmuchEc. /H2084968/H20850 Suppose the charging energy is small compared to the reso- nance frequency, Ec/lessmuch/H9275r/lessmuch/H9280F. As significant changes in /H9268AESand/H9003occur on the scale of the Coulomb blockade energy Ec, the frequencies in the vicinity of the resonance are too large for Coulomb blockade physics to be significant. Inthis case,/H9003/H110154bzg 2/H20849e2/R/H20850is practically independent of fre- quency and temperature. Consider now the case where charging energy is large or comparable with respect to the resonance frequency, /H9275r /H11021Ec/lessmuch/H9280F. This can occur if the metal has a low enough conduction electron density, a large effective mass for theelectrons, and/or small grains. Increasing the volume fractionof the metal is another way in which the resonance frequencymay be reduced; we shall see in Sec. IV D that /H9275rrenormal- izes to/H9275r*=/H9275r/H208811−fasfis increased. This regime is very interesting because the resonance is in the low frequencyregime /H20849 /H9275/lessmuchEc/H20850for Coulomb-blockade physics. So near the resonance /H9275=/H9275r*, while/H9268AESstill obeys an Arrhenius law, Eq. /H2084967/H20850, the temperature dependence of /H9003can be qualita- tively different from /H9268AES. One expects here, following Eq. /H2084968/H20850,/H9003/H20849/H9275r*,T/H20850/H11008/H20851T2+/H20849/H9275r*/2/H9266/H208502/H20852/Ec2. (b)Finally, consider large intergrain conductance, g/greatermuch1. In this case both /H9268AES/H20849/H9275,T/H20850and/H9003/H20849/H9275,T/H20850evolve logarithmically21with temperature and frequency, /H9268AES/H20849/H9275,T/H20850/H11015g/H20849e2/a/H20850/H208751−1 /H9266gzln/H20873gEc max /H20849/H9275,T/H20850/H20874/H20876, /H9003/H20849/H9275,T/H20850/H110154bzge2 R/H208751−1 /H9266gzln/H20873gEc max /H20849/H9275,T/H20850/H20874/H20876, down to exponentially low temperatures and frequencies when perturbation theory is no longer valid. Below such lowtemperatures, the physics is similar to the g/lessmuch1 case dis- cussed previously. D. Comparison with Drude theory According to Drude theory, a bulk metal will have a frequency-dependent dielectric function, /H9255Drude /H20849/H9275/H20850=1−4/H9266ne2 m1 /H9275/H20849/H9275+i//H9270Drude /H20850, /H2084969/H20850 which when substituted in the Maxwell-Garnett formula, Eq. /H208495/H20850, yields the effective dielectric function for a homoge- neous system of metallic grains in vacuum, /H9255eff/H20849/H9275/H20850=/H92752−/H208494/H9266ne2/3m/H20850/H208491+2 f/H20850−i/H9275/3/H9270Drude /H92752−/H208494/H9266ne2/3m/H20850/H208491−f/H20850−i/H9275/3/H9270Drude./H2084970/H20850 From Eq. /H2084970/H20850one then infers the optical conductivity /H9268MG/H20849/H9275/H20850for the granular system in the Maxwell-Garnett ap- proximationV. TRIPATHI AND Y. L. LOH PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-10Re/H9268MG/H20849/H9275/H20850=fne2 m/H92752//H208493/H9270Drude /H20850 /H20851/H92752−/H9275r2/H208491−f/H20850/H208522+/H20849/H9275/3/H9270Drude /H208502. /H2084971/H20850 Equation /H2084971/H20850is not strictly correct because /H9270Drude does not include the Landau damping31,38,39that exists in the metallic grain but is absent in the bulk. /H20849See also the discussion in Sec. III. /H20850Besides, matching Eq. /H2084971/H20850to the correct dc con- ductivity requires that /H9270Drude /H11011/H9268AES/H20849T/H20850/H11011exp /H20849−Ec/T/H20850, whereas we have shown that the resonance width /H9003has a different temperature dependence from /H9268AES. Evidently, clas- sical arguments are unable to explain the full behavior of /H9268/H20849/H9275,T/H20850. Pending a proper theory of long-range interaction of di- poles in the granular metal, we nevertheless propose that theoptical conductivity of the granular metal in the Maxwell- Garnett approximation is given by Eq. /H2084971/H20850with /H9270rel−1=/H9270grain−1 +/H9003replacing /H208493/H9270Drude /H20850−1: Re/H9268/H20849/H9275,T/H20850=/H9268AES/H20849/H9275,T/H20850+fne2 m/H92752//H9270rel /H20851/H92752−/H9275r2/H208491−f/H20850/H208522+/H20849/H9275//H9270rel/H208502. /H2084972/H20850 The Maxwell-Garnett result, Eq. /H2084972/H20850, agrees with the dipole contribution in our stronger result for the optical conductiv-ity, Eq. /H2084966/H20850, that was derived for a dilute granular array, f/lessmuch1. Equation /H2084972/H20850shows that the resonance frequency un- dergoes an infrared shift, /H9275r*=/H9275r/H208811−f, as the volume frac- tion of the metal is increased. This dependence has beenpreviously obtained 45in the literature. One must take care not to extend Eq. /H2084972/H20850all the way to f=1 because the validity of our effective field theory is limited to the insulating phaseof the granular metal. V. CONCLUSIONS AND DISCUSSION We have developed an effective field theory of granular metals that is a generalization of the Ambegaokar-Eckern-Schön /H20849AES /H20850action to include polarization degrees of free- dom. This approach synthesizes the classical electrodynamictheories of Maxwell Garnett and Mie and the quantum me-chanical AES model for dissipative transport in order to cap-ture both finite-frequency and finite-temperature effects. It isvalid at temperatures larger than the mean level spacing /H9254in a grain. Using this effective field theory, we have calculated the frequency and temperature dependence of the optical con-ductivity of an array of spherical metallic grains. We haveshown that the temperature dependence of the polarizationresonance width /H9003differs qualitatively from that of the dc conductivity for frequencies and temperatures much smallerthan the charging energy of the grains. While the dc conduc-tivity obeys an Arrhenius law at low temperatures, /H9003de- creases only algebraically as a function of frequency andtemperature. We believe this prediction can be tested in ex- perimental situations where the condition /H9275r/H208811−f/H11021Eccan be satisfied. This can occur in systems where the conductionelectron density is low, the effective mass is large, and/or thegrains are small, and the volume fraction of the metal is large/H20849while still remaining in the insulating phase /H20850. This qualita- tive difference between the temperature dependences of thedc conductivity /H9268/H208490,T/H20850, and the collective mode damping /H9003/H208490,T/H20850, obeyed in certain granular metals is quite unlike the behavior seen46in pinned sliding density wave compounds where the temperature dependence of the collective modedamping is the same as the dc conductivity. Such a differ-ence could perhaps be used to distinguish between granular-ity arising from spontaneous electronic phase segregation instrongly correlated electron systems and density wave order. To keep our analysis simple, we have, in our field theo- retical treatment, ignored electrostatic interactions betweenmonopoles /H20849charges /H20850and dipoles /H20849polarizations /H20850on different grains. Strictly speaking, this is correct only in a dilutegranular array /H20849f/lessmuch1/H20850or at frequencies higher than the po- larization resonance. Renormalization of the resonance fre- quency due to the presence of neighboring grains, even in theabsence of tunneling, is one effect that is lost in this approxi-mation. Pending a general field theoretical treatment of long-range interaction of dipoles, we have used our result for theoptical response of a dilute array of grains as an input in aMaxwell-Garnett effective medium approximation to obtainthe optical conductivity at larger values of f. The shift that we obtain in the resonance frequency as a function of f agrees with earlier results in the literature. 45 Another aspect we have not considered is disorder, both in intergrain tunneling conductance and as a random back-ground potential due to quenched impurities in the insulating part. In the presence of a strong disorder, the dc conductivity obeys a soft activation law /H9268/H208490,T/H20850/H11011/H92680e−/H20881T0/Tinstead of an Arrhenius law; it should be interesting to consider the effect on optical conductivity. In principle it is possible to study theeffect of both kinds of disorder in our scheme. Finally, there are some fundamental limitations on AES- inspired treatments. Like the AES model, our dissipativetransport model is limited to the insulating side of a metal-insulator transition, and cannot describe the optical conduc-tivity through the transition; it also neglects quantum coher-ence effects, which are important at T/H11021 /H9254. The present level of rigor in our calculation is insufficient to study the various f−sum rules obeyed by the optical conductivity.45Our model is justified only for frequencies much smaller than the bandwidth, /H9275/lessmuch/H9270c−1/H11011/H9280F. At higher frequencies, or, in other words, for times shorter than thecutoff, /H9270/lessmuch/H9270c/H20851see the discussion following Eq. /H2084924/H20850/H20852, the dissipation kernel, T2/sin2/H20849/H9266T/H9270/H20850, that appears in the tunnel- ing terms in the effective field theory, Eq. /H2084937/H20850, is no longer valid. ACKNOWLEDGMENTS We are grateful to G. Blumberg, D. E. Khmelnitskii, Ši- mon Kos, and P. B. Littlewood for valuable discussions. V.T.thanks Trinity College, Cambridge for financial support.Y.L.L. thanks Purdue University for support. APPENDIX A: EFFECT OF LONG-RANGE COULOMB INTERACTION ON OPTICAL CONDUCTIVITY OF DIRTY INSULATORS Shklovskii and Efros generalized Mott’s treatment to in- clude the effect of long-range Coulomb interactions.15,47TheFREQUENCY AND TEMPERATURE DEPENDENCE OF THE ¼ PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-11difference is particularly significant when the particle-hole Coulomb energy at hopping distance r/H9275exceeds the optical frequency: e2//H9255r/H9275/greatermuch/H9275. Here/H9255is the dielectric constant of the medium. Physically, in the presence of Coulomb interac-tions, transitions to the final state /H9280jcan be made from an occupied level with energy in the range /H9280F−/H9275−e2//H9255r/H9275/H11021/H9280i /H11349/H9280F. Modifying the limits in Eq. /H208493/H20850accordingly,15 Re/H20851/H9268/H20849/H9275/H20850/H20852 /H11011 2/H9266e2nimp2/H9275 /H92542/H20873/H9275+e2 /H9255r/H9275/H20874/H20849r/H9275d−1/H9264loc/H20850r/H92752 /H110152/H9266e4nimp2/H9275 /H9255/H92542/H9264locd+1lnd/H20849w//H9275/H20850,e2//H9255r/H9275/greatermuch/H9275. /H20849A1/H20850 Equation /H20849A1/H20850assumes that the density of states is a con- stant; and this is correct as long as the energy /H20849/H9275+e2//H9255r/H9275/H20850is larger than the Coulomb gap, /H9004. At energies less than /H9004, the density of states at the chemical potential is not a constant,but instead has the form /H9267/H20849/H9280/H20850/H11011/H20841/H9280/H20841d−1/H20849/H9255/e2/H20850d. The Coulomb gap is the energy at which the density of states reaches the value in the absence of a Coulomb interaction; thus /H9004 /H11011/H20851e2dnimp//H20849/H9255/H9254/H20850/H208521//H20849d−1/H20850. Using this density of states, we get, for/H9004/H11022e2//H9255r/H9275/H11022/H9275, an optical conductivity, Re/H20851/H9268/H20849/H9275/H20850/H20852/H11008/H9275r/H92752−d/H11011/H9275/H9264loc2−dln2−d/H20849w//H9275/H20850. /H20849A2/H20850 At a finite temperature, /H9268/H20849/H9275=0/H20850is finite /H20849see previous /H20850.I ft h e temperature is low, /H20849T,/H9275/H20850/lessmuche2//H9255r/H9275, the frequency dependent conductivity in Eq. /H20849A1/H20850has an extra Boltzmann factor, Re/H20851/H9268/H20849/H9275,T/H20850/H20852 /H11011/H9268/H208490,T/H20850+2/H9266e2nimp2/H208491−e−/H9275/T/H20850 /H11003/H20849/H9275//H9255/H92542/H20850/H9264locd+1lnd/H20849w//H9275/H20850. /H20849A3/H20850 For high enough temperatures, T/greatermuch/H20849/H9275,e2//H9255r/H9275/H20850,or high enough frequencies, /H9275/greatermuch/H20849T,e2//H9255r/H9275/H20850, Mott’s result, Eq. /H208494/H20850is obtained.15 Our treatment so far has assumed that there is no inelastic scattering /H20849e.g., by phonons /H20850. At a finite temperature, phonons /H20849with characteristic frequency /H9275ph/H110111012Hz/H20850pro- vide an additional relaxation mechanism. Electrons maketransitions by emitting or absorbing phonons with energy ofthe order of /H9275rel/H11011/H9275phe−xij//H9264loc, so the main contribution to the optical conductivity from inelastic processes comes from fre-quencies of the order of /H9275rel. For such frequencies, we should use /H9275phinstead of win Eq. /H20849A3/H20850. APPENDIX B: EFFECTIVE ACTION CORRECTIONS FROM AN EXPANSION OF G/H9272i1IN POWERS OF /H9272i1 We explain how corrections to the effective tunneling ac- tion in Eq. /H2084934/H20850coming from the expansion of G/H9272i1/H20849xi,xi;/H9270,/H9270/H11032/H20850in powers of /H9272i1are small compared to the bare value when T/greatermuch/H9254. We expand Eq. /H2084920/H20850, G/H9272i1=/H208731−Gi/H208490/H20850 2mR2/H9272i12+Gi/H208490/H20850 mR/H9272i1·pxi/H20874−1 Gi/H208490/H20850, whereGi/H208490/H20850/H20849xi,xi/H11032;/H9270,/H9270/H11032/H20850=T/H20858 /H9261,n/H9274/H9261/H20849xi/H11032/H20850/H9274/H9261*/H20849xi/H20850 i/H9263n−/H9264i/H9261e−i/H9263n/H20849/H9270−/H9270/H11032/H20850 =/H20858 lGi/H9261/H208490/H20850/H20849/H9270,/H9270/H11032/H20850/H9274/H9261/H20849xi/H11032/H20850/H9274/H9261*/H20849xi/H20850, up to second order in /H9272i1. The resulting correction to the effective action of Eq. /H2084935/H20850is /H9254Sefftun/H20851/H9272i1/H20852=/H20841t/H208412 4mR2/H20858 ia;/H92611/H92612/H20885 /H9270/H9270/H11032/H20885 /H92701Gi/H92611/H208490/H20850/H20849/H9270,/H92701/H20850 /H11003Gi/H92611/H208490/H20850/H20849/H92701,/H9270/H11032/H20850Gi+a,/H92612/H208490/H20850/H20849/H9270/H11032,/H9270/H20850/H20855/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850/H20856/H9272i12/H20849/H92701/H20850 +/H20841t/H208412 2m2R2/H20858 ia;/H92611/H92612/H92613/H20885 /H9270/H9270/H11032/H20885 /H92701/H92702/H20841pi,/H92611/H92612/H9251/H208412Gi/H92611/H208490/H20850/H20849/H9270,/H92701/H20850 /H11003Gi/H92612/H208490/H20850/H20849/H92701,/H92702/H20850Gi/H92611/H20849/H92702,/H9270/H11032/H20850Gi+a,/H92613/H208490/H20850/H20849/H9270/H11032,/H9270/H20850 /H11003/H20855/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850/H20856/H9272i1/H20849/H92701/H20850·/H9272i1/H20849/H92702/H20850, /H20849B1/H20850 and/H9016i,i+a/H20849/H9270,/H9270/H11032/H20850is as defined in Eq. /H2084936/H20850. Next we simplify Eq. /H20849B1/H20850by integrating over /H9270,/H9270/H11032. Using the frequency rep- resentation and completing the integration over /H9270,/H9270/H11032,w e have /H9254Sefftun/H20851/H9272i1/H20852=/H20841t/H208412T2 4mR2/H20858 ia;/H92611/H92612;n,m/H20885 /H92701/H20855/H9016i,i+a/H20849/H9275m/H20850/H20856/H9272i12/H20849/H92701/H20850 /H110031 /H20849i/H9263n−/H9264i/H92611/H2085021 /H20851i/H20849/H9263n+/H9275m/H20850−/H9264i+a,/H92612/H20852 +/H20841t/H208412T2 2m2R2/H20858 ia;/H92611/H92612/H92613;n,m/H20885 /H92701/H92702Gi/H92612/H208490/H20850 /H11003/H20849/H92701,/H92702/H20850/H9272i1/H20849/H92701/H20850·/H9272i1/H20849/H92702/H20850/H11003e−i/H9263n/H20849/H92702−/H92701/H20850 /H20849i/H9263n−/H9264i/H92611/H208502 /H11003/H20841pi,/H92611/H92612/H9251/H208412/H20855/H9016i,i+a/H20849/H9275m/H20850/H20856 /H20851i/H20849/H9263n+/H9275m/H20850−/H9264i+a,/H92613/H20852. /H20849B2/H20850 It is convenient to perform the Matsubara sum over the fer- mionic frequencies. We have T/H20858 ne−i/H9263n/H20849/H92702−/H92701/H20850 /H20849i/H9263n−/H9264i/H9261/H2085021 /H20851i/H20849/H9263n+/H9275m/H20850−/H9264i+a,/H9261/H11032/H20852 =/H11509 /H11509/H9264i/H9261/H20873Gi/H9261/H208490/H20850/H20849/H92702,/H92701/H20850−Gi+a,/H9261/H11032/H208490/H20850/H20849/H92702,/H92701/H20850ei/H9275m/H20849/H92702−/H92701/H20850 i/H9275m+/H9264i/H9261−/H9264i+a,/H9261/H11032/H20874. /H20849B3/H20850 The first term in Eq. /H20849B2/H20850vanishes when we use Eq. /H20849B3/H20850 with/H92701=/H92702. HenceV. TRIPATHI AND Y. L. LOH PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-12/H9254Sefftun/H20851/H9272i1/H20852=/H20841t/H208412T 2m2R2/H20858 ia;/H92611/H92612/H92613;m/H20885 /H92701/H92702/H20841pi,/H92611/H92612/H9251/H208412 /H11003Gi/H92612/H208490/H20850/H20849/H92701,/H92702/H20850/H9272i1/H20849/H92701/H20850·/H9272i1/H20849/H92702/H20850/H20855/H9016i,i+a/H20849/H9275m/H20850/H20856 /H11003/H11509 /H11509/H9264i/H92611/H20873Gi/H92611/H208490/H20850/H20849/H92702,/H92701/H20850−Gi+a,/H92613/H208490/H20850/H20849/H92702,/H92701/H20850ei/H9275m/H20849/H92702−/H92701/H20850 i/H9275m+/H9264i/H92611−/H9264i+a,/H92613/H20874. /H20849B4/H20850 Equation /H20849B4/H20850contains two terms: one where the Green’s functions are on the same grain and the other where they areon different grains. The term with the Green’s functions onthe same grain can be simplified by summing over /H9264i+a,/H92613. The result of the summation in /H9263/H20849/H9280F/H20850sgn/H9275m. Since /H20855/H9016i,i+a/H20849/H9275m/H20850/H20856is an even function of /H9275m, summing over /H9275m makes the first term disappear. Thus, so far, /H9254Sefftun/H20851/H9272i1/H20852=−/H20841t/H208412T 2m2R2/H20858 ia;/H92611/H92612/H92613;m/H20885 /H92701/H92702Gi/H92612/H208490/H20850/H20849/H92701,/H92702/H20850 /H11003/H20841pi,/H92611/H92612/H9251/H208412/H9272i1/H20849/H92701/H20850·/H9272i1/H20849/H92702/H20850Gi+a,/H92613/H208490/H20850/H20849/H92702,/H92701/H20850 /H11003/H20855/H9016i,i+a/H20849/H9275m/H20850/H20856/H11509 /H11509/H9264i/H92611ei/H9275m/H20849/H92702−/H92701/H20850 i/H9275m+/H9264i/H92611−/H9264i+a,/H92613. /H20849B5/H20850 Now we integrate Eq. /H20849B5/H20850by parts with respect to the vari- able/H9264i/H92611using /H20858 /H92611↔/H9263/H20849/H9280F/H20850/H20848d/H9264i/H92611and the identity /H11509//H11509/H9264i/H92611 =2m/H20858/H9261/H208491/pi,/H9261/H92611/H20850/H20849/H11509//H11509pi,/H92611/H9261/H20850, /H9254Sefftun/H20851/H9272i1/H20852=/H20841t/H208412T/H9263/H20849/H9280F/H20850 mR2 /H20858 ia;/H92612/H92613;m/H20885d/H9264i/H92611/H20885 /H92701/H92702Gi/H92612/H208490/H20850/H20849/H92701,/H92702/H20850 /H11003/H9272i1/H20849/H92701/H20850·/H9272i1/H20849/H92702/H20850Gi+a,/H92613/H208490/H20850/H20849/H92702,/H92701/H20850 /H11003/H20855/H9016i,i+a/H20849/H9275m/H20850/H20856ei/H9275m/H20849/H92702−/H92701/H20850 i/H9275m+/H9264i/H92611−/H9264i+a,/H92613. /H20849B6/H20850 Now we complete the integration over /H9264i/H92611to get /H9254Sefftun/H20851/H9272i1/H20852=2/H9266/H20841t/H208412T/H9263/H20849/H9280F/H20850 mR2 /H20858 ia;/H92612/H92613;m/H20885 /H92701/H92702Gi/H92612/H208490/H20850/H20849/H92701,/H92702/H20850 /H11003/H9272i1/H20849/H92701/H20850·/H9272i1/H20849/H92702/H20850Gi+a,/H92613/H208490/H20850/H20849/H92702,/H92701/H20850 /H11003/H20855/H9016i,i+a/H20849/H9275m/H20850/H20856sgn/H20849/H9275m/H20850sin/H20851/H9275m/H20849/H92702−/H92701/H20850/H20852. /H20849B7/H20850 The right hand side of Eq. /H20849B7/H20850vanishes because the inte- grand is odd with respect to interchange of /H92701and/H92702. This proves that the correction to the effective action arising from /H9272i1fluctuations in G/H9272i1may be ignored. APPENDIX C: OPTICAL CONDUCTIVITY OF THE AES MODEL We calculate the paramagnetic and diamagnetic terms in the AES conductivity /H9268AESof the granular array. As a corol- lary, we also find that the resonance width /H9003is proportionalto the diamagnetic part of the AES conductivity. From Eq. /H2084961/H20850through Eq. /H2084964/H20850it follows that Re/H20851/H9268AES/H20849/H9275,T/H20850/H20852=1 /H9275Im/H20851KAESdia/H20849/H9275,T/H20850+KAESpara/H20849/H9275,T/H20850/H20852. /H20849C1/H20850 In the expression for KAESdia/H20849/H9275,T/H20850, we need to calculate the cosine correlator, /H9016/H20849/H9270/H20850=/H20855/H9016i,i+e/H9251/H20849/H9270/H20850/H20856. We discuss the case of weak intergrain tunneling g/lessmuch1 first. (a)For weak intergrain tunneling, /H9016/H20849/H9270/H20850may be evaluated perturbatively in increasing powers of g:/H9016/H20849/H9270/H20850=/H9016/H208490/H20850/H20849/H9270/H20850 +/H9016/H208491/H20850/H20849/H9270/H20850+¯, where the prefixes denote the power of g. The leading term /H9016/H208490/H20850/H20849/H9270/H20850can be shown to be21 /H9016/H208490/H20850/H20849/H9270/H20850=1 Z2/H20858 q1,q2=−/H11009/H11009 e−/H9252Ec/H20849q12+q22/H20850−/H208491−q1−q2/H208502Ec/H9270. /H20849C2/H20850 We similarly expand the diamagnetic response function KAESdia/H20849i/H9024n/H20850in powers of g,KAESdia=KAESdia,/H208491/H20850+KAESdia,/H208492/H20850+¯, where the prefixes in brackets denote the power of g. To obtain the leading order in gbehavior, we use Eq. /H20849C2/H20850in Eq. /H2084962/H20850and take the Fourier transform: KAESdia,/H208491/H20850/H20849i/H9024n/H20850=−g/H20849e2/a/H20850T Z2/H20858 q1q2/H20858 m/H20841/H9024m/H20841e−/H9252Ec/H20849q12+q22/H20850 /H11003/H20849e−2Ec/H9252/H208491−q1−q2/H20850−1/H20850/H20885d/H9024/H9254/H20851/H9024−2Ec/H208491−q1 −q2/H20850/H20852/H11003/H208751 i/H9024n−m−/H9024−1 i/H9024−m−/H9024/H20876. /H20849C3/H20850 Next, we perform the Matsubara sum over mfollowed by an analytical continuation i/H9024n→/H9275. The result is Im/H20851KAESdia,/H208491/H20850/H20849/H9275,T/H20850/H20852=g/H20849e2/a/H208501 Z2/H20858 q1q2e−/H9252Ec/H20849q12+q22/H20850/H11003/H20885d/H9024/H208491 −e−/H9252/H9024/H20850/H9024−/H9275 2/H20873coth/H9024−/H9275 2− coth/H9024 2/H20874 /H11003/H9254/H20851/H9024−2Ec/H208491−q1−q2/H20850/H20852. /H20849C4/H20850 The dc /H20849/H9275=0/H20850behavior in Eq. /H20849C4/H20850is dominated by single- charge excitations, /H20849q1,q2/H20850=/H208491,0 /H20850,/H208490,1 /H20850, whereas the ac be- havior at T=0 is dominated by “even” excitations /H20849q1,q2/H20850 =/H208490,0 /H20850,/H208491,1 /H20850: Im/H20851KAESdia,/H208491/H20850/H208490,T/H20850/H20852 /H11015 2/H9275g/H20849e2/a/H20850e−Ec/T, Im/H20849KAESdia,/H208491/H20850/H20849/H9275,0/H20850/H20850 /H11015/H9275g/H20849e2/a/H20850/H208491−2 Ec//H20841/H9275/H20841/H20850/H11003/H9008 /H20849/H20841/H9275/H20841−2Ec/H20850. /H20849C5/H20850 In the next order in g, the cosine correlator can be shown24to be /H9016/H208491/H20850/H20849/H9270/H20850=2/H9266gT2 Ec2sin2/H20849/H9266T/H9270/H20850,Ec/H9270/greatermuch1. /H20849C6/H20850 It follows that in the second order in g, the imaginary part of the spectral function KAESdia,/H208492/H20850isFREQUENCY AND TEMPERATURE DEPENDENCE OF THE ¼ PHYSICAL REVIEW B 73, 195113 /H208492006 /H20850 195113-13Im/H20851KAESdia,/H208492/H20850/H20849/H9275,T/H20850/H20852=/H92754/H9266g2T2e2 3aEc2/H20851T2+/H20849/H9275/2/H9266/H208502/H20852,/H9275/lessmuchEc. /H20849C7/H20850 The power law behavior of the second order /H20849ing/H20850diamag- netic response does not mean that the conductivity will fol-low a power law. This is because we also have a paramag-netic contribution, and one can show 24that the leading order paramagnetic response is second order in gand is equal and opposite toKAESdia,/H208492/H20850, KAESpara/H20849/H9270/H20850/H11015KAESpara ,/H208492/H20850/H20849/H9270/H20850=−KAESdia,/H208492/H20850/H20849/H9270/H20850. /H20849C8/H20850 Thus power law contributions cancel out in the conductivity and we are left with /H9268AES/H20849/H9275,T/H20850/H11015Im/H20851KAESdia,/H208491/H20850/H20849/H9275,T/H20850/H20852 /H9275, /H20849C9/H20850 where Im /H20851KAESdia,/H208491/H20850/H20849/H9275,T/H20850/H20852is given by Eq. /H20849C4/H20850. APPENDIX D: BEHAVIOR OF THE RESONANCE WIDTH Now we discuss the frequency and temperature depen- dence of the polarization resonance width /H9003. Note that /H9003/H20849/H9275,T/H20850is proportional to /H9268AESdia/H20849/H9275,T/H20850. In the absence of a canceling paramagnetic contribution, /H9003/H20849/H9275,T/H20850, unlike theconductivity, does show an algebraic behavior at low fre- quencies, /H9003/H20849/H9275,T/H20850/H110154bzg24/H9266e2 3REc2/H20851T2+/H20849/H9275/2/H9266/H208502/H20852,/H9275/lessmuchEc./H20849D1/H20850 Consider finally the case where the dimensionless inter- grain tunneling is large, g/greatermuch1. Except at very low tempera- tures /H20849explained next /H20850, both/H9268AESand/H9003are more or less determined by the diamagnetic contribution. 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PhysRevB.83.054515.pdf

PHYSICAL REVIEW B 83, 054515 (2011) Ginzburg-Landau theory of two-band superconductors: Absence of type-1.5 superconductivity V . G. Kogan and J. Schmalian Ames Laboratory and Department of Physics & Astronomy, Iowa State University, Ames, IA 50011 (Received 2 August 2010; published 23 February 2011) It is shown that within the Ginzburg-Landau (GL) approximation the order parameters /Delta11(r,T)a n d/Delta12(r,T) in two-band superconductors vary on the same length scale, the difference in zero- Tcoherence lengths ξ0ν∼ ¯hvFν//Delta1ν(0),ν=1,2 notwithstanding. This amounts to a single physical GL parameter κand the classic GL dichotomy: κ< 1/√ 2 for type I and κ> 1/√ 2 for type II. DOI: 10.1103/PhysRevB.83.054515 PACS number(s): 74 .20.De, 74 .25.Ha, 74 .25.Uv, 74 .70.Dd I. INTRODUCTION The physics of superconductors near their critical tem- perature, Tc, is based on the Ginzburg-Landau (GL) theory.1 The smallness of critical fluctuations in most superconductors justifies the usage of this mean-field theory except in a tinyregime close to the transition temperature. This includesmultiband superconductors with distinct sheets of the Fermisurface. A number of recent papers deal with two-bandmaterials with coefficients of the GL free energy (for thefield-free state), F=/summationdisplay ν=1,2/parenleftbig aν/Delta12 ν+bν/Delta14 ν/2/parenrightbig −2γ/Delta1 1/Delta12, introduced phenomenologically; see, for example, Ref. 2. Choosing these coefficients in various ways, one could arriveat a number of choice-dependent conclusions. 3,4However, there are important general constraints on the allowed values.For example, a straightforward analysis of the above freeenergy yields that the coefficients a νdo not have the familiar GL form α(T−Tc). Instead, they acquire a constant part, const+α(T−Tc), that is intimately related to the constant γ of the mixed Josephson-type term to ensure /Delta1ν∝√Tc−T nearTc. The coefficients of the GL free energy can furthermore be derived from microscopic theory; they are certain func-tions of the microscopic coupling constants responsible forsuperconductivity and temperature T. This was done several decades ago by Tilley 5and later by Zhitomirsky and Dao,6 who confirm, as expected, the const +α(T−Tc) behavior of theaνwithin a weak-coupling model. We show in this work that, independent of the microscopic origin of the GL coefficients, in the GL domain the ratio of theorder parameters is Tindependent, /Delta1 1(r,T)//Delta12(r,T)=const, (1) with the r-independent constant depending on interactions responsible for superconductivity. The one-dimensional (1D)version of Eq. ( 1) has been obtained while solving the GL problem of the interface energy between superconducting andnormal phases relevant for the distinction between type Iand type II two-band superconductors. 7For strong intraband scattering (the dirty limit), the result ( 1) has been obtained by Koshelev and Golubov provided the interband scattering couldbe disregarded. 8Here, we establish this result for anyproblem in the GL domain. We show that the equations for /Delta11(r,T) and/Delta12(r,T)a r e reduced to one independent GL equation. In other words, thereis a single complex order parameter describing the two-band superconductor in the GL domain and, as a consequence, asingle length scale ξfor spatial variation of both /Delta1 1(r,T) and /Delta12(r,T). Our results, along with the earlier critique9and a compre- hensive review by Brandt and Das,10question the validity of publications discussing properties of MgB 2within the GL framework where each band is attributed with its owncoherence length and sometimes even with its own penetrationdepth; see, for example, Ref. 11and references therein. We stress that our claim that the gap functions /Delta1 ν(r,T) change on the same length scale relates exclusively to thetemperature domain, however narrow it could be, where theGL theory is valid. Out of this domain and at low temperaturesin particular, different length scales ∼¯hv Fν//Delta1ν(0) may enter and result in properties substantially different from those in theGL region. Still, as long as the GL energy functional is used,the assumption of two coherence lengths cannot be justified. Below, we discuss the phenomenologic two-band GL theory and later confirm our conclusions within a weak-coupling microscopic scheme. II. TWO-BAND GL IN FIELD The two-band GL functional reads as follows: F=/integraldisplay dV/braceleftbigg/summationdisplay ν=1,2/parenleftbigg aν|/Delta1ν|2+bν 2|/Delta1ν|4+Kν|/Pi1/Delta1ν|2/parenrightbigg −γ(/Delta11/Delta1∗ 2+/Delta12/Delta1∗ 1)+B2 8π/bracerightbigg , (2) where/Pi1=∇+2πiA/φ0. Explicit expressions for the con- stantγalong with the coefficients a,b,K for a weak-coupling model will be given later. However, our results do not rely onthe validity of the weak-coupling theory and are more general. The GL equations are minimum conditions for the func- tional ( 2). One obtains, by varying Fwith respect to /Delta1 ∗ ν, a1/Delta11+b1/Delta11|/Delta11|2−γ/Delta1 2−K1/Pi12/Delta11=0, (3) a2/Delta12+b2/Delta12|/Delta12|2−γ/Delta1 1−K2/Pi12/Delta12=0. (4) We now recall that in the one-band GL equation, a/Delta1+b/Delta1|/Delta1|2−K/Pi12/Delta1=0, all terms are of the same order, (1 −T/T c)3/2=τ3/2(/Delta1∝ τ1/2,a∝τ, and/Pi12∝ξ−2∝τ). This is more subtle in the case of Eqs. ( 3) and ( 4) because γis a constant and aνcontain 054515-1 1098-0121/2011/83(5)/054515(5) ©2011 American Physical SocietyV . G. KOGAN AND J. SCHMALIAN PHYSICAL REVIEW B 83, 054515 (2011) constant parts. Keeping this in mind, we express /Delta12in terms of/Delta11from Eq. ( 3) and substitute the result in Eq. ( 4), keeping only terms up to order τ3/2: (a1a2−γ2)/Delta11+/parenleftbig b1a2+b2a3 1/γ2/parenrightbig /Delta11|/Delta11|2 −(a1K2+a2K1)/Pi12/Delta11=0. (5) Similarly, one obtains an equation for /Delta12: (a1a2−γ2)/Delta12+/parenleftbig b2a1+b1a3 2/γ2/parenrightbig /Delta12|/Delta12|2 −(a1K2+a2K1)/Pi12/Delta12=0. (6) In zero field, one has /Delta12 ν∝(a1a2−γ2), so that at Tc,a1a2− γ2=0, and therefore aνmust contain constant parts, aν=aνc−αντ, such that a1ca2c=γ2. Equations ( 5) and ( 6)f o r/Delta1νcan now be written as −ατ/Delta1 1+β1/Delta11|/Delta11|2−K/Pi12/Delta11=0, (7) −ατ/Delta1 2+β2/Delta12|/Delta12|2−K/Pi12/Delta12=0, (8) with α=α1a2c+α2a1c,K=a1cK2+a2cK1, (9) β1=b1a2c+b2a3 1c/γ2,β 2=b2a1c+b1a3 2c/γ2. We note that within the accuracy of the GL theory, up to O(τ3/2), these equations differ only in coefficients βof the nonlinear terms that determine the overall amplitude of thesolutions, whereas the rest of the coefficients are the same.The equations for /Delta1 1and/Delta12are coupled only via the vector potential. In particular, in zero field we have /Delta12 ν0=ατ/β ν, so that the ratio /Delta12 10(T) /Delta12 20(T)=β2 β1(10) comes out to be Tindependent in the GL domain. Furthermore, one easily checks that for any solution /Delta11(r,T)o fE q .( 7), Eq. ( 8) is satisfied by /Delta12(r,T)=/Delta11(r,T)/radicalbig β1/β2. (11) In particular, this implies that in equilibrium /Delta11(r,T) and /Delta12(r,T) must have either the same phases or phases differing byπ.12It is found in Ref. 6that for small γthe ratio /Delta12//Delta11 changes away from Tc; we note, however, that this deviation is beyond the GL accuracy. Reliable results beyond GL can beobtained only within microscopic approaches like the Gor’kovor Bogolyubov–de Gennes theories. Moreover, introducing the order parameters normalized on their zero-field values, /Delta1 1 /Delta110(T)=/Delta12 /Delta120(T)=/Psi1, (12) both Eqs. ( 7) and ( 8) are reduced to one: /Psi1(1−|/Psi1|2)=−K ατ/Pi12/Psi1. (13)Thus, the length scale of the space variation of both /Delta11and /Delta12, the coherence length, is given by ξ2=K/ατ. (14) III. MICROSCOPIC WEAK-COUPLING TWO-BAND MODEL NEAR Tc To establish a connection of GL equations with the two- band microscopic theory we turn to a weak-coupling modelfor clean and isotropic materials (not because these restrictionsare unavoidable, but rather due to the model simplicity). Perhaps the simplest formally weak-coupling approach is based on the Eilenberger quasiclassical formulation ofthe Gor’kov equations valid for general anisotropic orderparameters and Fermi surfaces. 13Eilenberger functions f,g for clean materials in zero field obey the system: 0=/Delta1g−¯hωf, (15) g2=1−f2, (16) /Delta1(k)=2πTN (0)ωD/summationdisplay ω>0/angbracketleftV(k,k/prime)f(k/prime,ω)/angbracketrightk/prime. (17) Here, kis the Fermi momentum; /Delta1is the order parameter that may depend on the position kat the Fermi surface. Further, N(0) is the total density of states (DOS) at the Fermi level per spin; the Matsubara frequencies are given by ¯ hω=πT(2n+ 1) with an integer n, and ωDis the Debye frequency; /angbracketleft.../angbracketright stands for averages over the Fermi surface. Consider a model material with the gap given by /Delta1(k)=/Delta11,2,k∈F1,2, (18) where F1,F2are two sheets of the Fermi surface. The gaps are assumed constant at each band. Denoting DOS on the twoparts as N 1,2, we have for a quantity Xconstant at each Fermi sheet, /angbracketleftX/angbracketright=(X1N1+X2N2)/N(0)=n1X1+n2X2,(19) where n1,2=N1,2/N(0); clearly, n1+n2=1. Equations ( 15) and ( 16) are easily solved: fν=/Delta1ν/βν,g ν=¯hω/β ν,β2 ν=/Delta12 ν+¯h2ω2,(20) where ν=1,2 is the band index. The self-consistency Eq. ( 17) takes the form /Delta1ν=/summationdisplay μ=1,2nμλνμ/Delta1μωD/summationdisplay ω2πT βμ, (21) where λνμ=N(0)Vνμare dimensionless effective interaction constants. The notation commonly used in literature, λ(lit) νμ= nμλνμ, includes DOS. We find our notation convenient since, being related to the coupling potential, our coupling matrix issymmetric: λ νμ=λμν. It is seen from the system ( 21) that/Delta11,2turns to zero at the same temperature Tcunless λ12=0 and Eqs. ( 21) decouple, the property that has been noted in earlier work.14–16AsT→ Tc,/Delta11,2→0, and β→¯hω. The sum over ωin Eq. ( 21)i s 054515-2GINZBURG-LANDAU THEORY OF TWO-BAND ... PHYSICAL REVIEW B 83, 054515 (2011) readily evaluated: S=ωD/summationdisplay ω2πT ¯hω/vextendsingle/vextendsingle/vextendsingle Tc=ln2¯hωD Tcπe−γ=ln2¯hωD 1.76Tc, (22) where γ=0.577 is the Euler constant. This relation can also be written as 1.76Tc=2¯hωDe−S. (23) The system ( 21)a tTcis linear and homogeneous: /Delta11=S(n1λ11/Delta11+n2λ12/Delta12), (24) /Delta12=S(n1λ12/Delta11+n2λ22/Delta12). The zero determinant gives Sand, therefore, Tc: S2n1n2η−S(n1λ11+n2λ22)+1=0, (25) η=λ11λ22−λ2 12. (26) The roots of this equation are S=n1λ11+n2λ22±/radicalBig (n1λ11−n2λ22)2+4n1n2λ2 12 2n1n2η. (27) Various possibilities that arise depending on values of λμν are discussed, for example, in Refs. 14–18. Introducing T- independent quantities, S1=λ22−n1ηS, S 2=λ11−n2ηS, (28) we write Eq. ( 25)a s S1S2=λ2 12, (29) the form useful for manipulations below. Ifλ12=0, Eq. ( 27) provides two roots: 1 /n1λ11and 1/n2λ22. The smallest one gives Tc, whereas the other corresponds to the temperature at which the second gap turnsto zero. We note that this situation is unlikely; it implies thatthe ever-present Coulomb repulsion is exactly compensated bythe effective interband attraction. Since the determinant of the system ( 24) is zero, the two equations are equivalent and give at T c/parenleftbigg/Delta12 /Delta11/parenrightbigg Tc=1−n1λ11S n2λ12S. (30) When the right-hand side is negative, the /Delta1’s are of opposite signs. Within the one-band BCS, the sign of /Delta1is a matter of convenience; for two bands, /Delta11and/Delta12may have equal or opposite signs.19 After simple algebra, Eq. ( 30) can be manipulated to /parenleftbigg/Delta12 /Delta11/parenrightbigg2 Tc=S1 S2. (31) We thus obtain, by comparing with Eq. ( 10), the ratio of phenomenological coefficients in terms of microscopiccouplings: β 1/β2=S1/S2. We have seen above that within the GL approximation this ratio remains the same at any Tin the GL domain not only for a uniform field-free state (or forγ→∞ as in Ref. 20) but for any situation with /Delta1’s depending on coordinates in the presence of magnetic fields.We note that the proportionality of /Delta1 1and/Delta12has also been shown to hold within microscopic weak-coupling theoryin the dirty limit by Koshelev and Golubov. 8It is also worth mentioning here that the proof of this proportionality in thepreceding section based on the GL approach is quite generaland holds for any scattering, gap anisotropies, etc. In the following we use the GL coefficients obtained in Refs. 5and6. In our notation they read a ν=N(0) η(Sν−nνητ),bν=N(0) W2nν,W2=8π2T2 c 7ζ(3), (32) γ=N(0) ηλ12,K ν=N(0)¯h2v2 ν 6W2nν, where the energy scale W∼πTcis introduced for brevity and vνare the Fermi velocities in two bands which for simplicity is assumed isotropic. We, in fact, confirmed Eqs. ( 32)o f Zhitomirsky and Dao employing different methods (except ourb νis by a factor of 2 larger than that of Ref. 6). It is worth noting that the microscopically derived aνare not proportional to τas in the standard one-band GL unless one of the parameters Sνis zero; given the condition ( 29) this may happen only if λ12=0. This feature of the two-band GL is sometimes overlooked.21,22 As stressed in Ref. 6,t h et e r m Kν|/Pi1/Delta1ν|2with order parameter gradients is the only possible in the GL energy,although the symmetry may allow for other combinations ofgradients. The coefficients entering the GL Eqs. ( 7) and ( 8)a r e α=N(0) 2C η,K=¯h2˜v2N(0)2 6W2η,s β ν=N(0)2DSν ηW2λ2 12,(33) where ˜v2=n1S2v2 1+n2S1v2 2 (34) has the dimension of a squared velocity and C=n1S2+n2S1,D=n1S2 2+n2S2 1 (35) are constants. Hence, we can express the length scale ( 14) of the space variation of both /Delta11and/Delta12in the GL domain in terms of microscopic parameters: ξ2=¯h2˜v2 2W2Cτ. (36) The upper critical field follows: Hc2=φ0/2πξ2. The one- band limit is obtained by setting n1=1,n2=0s ot h a t C=S2 and ˜v2=S2v2/3,which yields ξ2=7ζ(3)¯h2v2/48π2T2 cτas it should. Variation of the free energy Fwith respect to the vector potential Agives the current density. Following the standard procedure we obtain for the penetration depth of a weakmagnetic field, 1 λ2=32π3 φ2 0/summationdisplay ν=1,2/Delta12 ν0Kν=16πCN (0)e2˜v2 c2Dτ. (37) In the one-band limit this yields the correct result: λ−2= [16πe2N(0)v2/3c2]τ. 054515-3V . G. KOGAN AND J. SCHMALIAN PHYSICAL REVIEW B 83, 054515 (2011) A straightforward calculation yields the equilibrium zero- field free energy: F0=−N(0)W2C2 2Dτ2. (38) The thermodynamic field Hcfollows: H2 c/8π=−F0. One can show that the relative specific heat jump at Tcdiffers from the one-band value 12 /7ζ(3)=1.43 by a factor C2/D < 1.23 One can now form the dimensionless GL parameter, κ2=λ2 ξ2=c2W2D 8πN(0)e2¯h2˜v4, (39) and verify the standard relation Hc2/Hc√ 2=κ. Finally, the equilibrium energy is evaluated by substituting the solutions of the GL equations to the functional ( 2): F=H2 c 4π/integraldisplay dV/braceleftbigg b2−1 2|/Psi1|4/bracerightbigg , (40) where b=B/H c√ 2 is the dimensionless field. Thus, the theory of a two-band superconductor near Tcis mapped onto the standard one-order parameter GL scheme. In particular, this mapping means that the GL problem of the interface energy between normal and superconductingphases has the same solution as in the one-band case, that is, κ=1/√ 2 separates type I and type II superconductors. This has been demonstrated in Ref. 7by solving numerically the nonlinear system of GL Eqs. ( 3) and ( 4) without discarding the terms O(τ2) employed here. A. Remark on boundary conditions T h es o l u t i o n( 11) for the two gap functions of the GL Eqs. ( 7) and ( 8) holds indeed provided the boundary conditions for /Delta12are the same as for /Delta11multiplied by the factor√β1/β2. This is clearly the case for the 1D problem of the S-N interface energy discussed in Ref. 7. The same is true for the problem of the single-vortexstructure: both /Delta1’s are zero at the vortex center and approach /Delta1 ν,0with the correct ratio at infinity. However, for example, for proximity situations with a two-bandsuperconductor on one side of the contact with a normalmetal, the condition on the superconducting side far fromthe boundary is satisfied, whereas the question of boundaryconditions at the interface remains open. In this case, onecannot claim that both /Delta1(r)’s are proportional to each other. Nevertheless, as is seen from Eqs. ( 7) and ( 8), the length scale ξ=√ K/ατ is still the same for both order parameters. IV . DISCUSSION Two-band GL equations have been used in a number of publications where the coefficients in the GL energy functionala ν,bν,Kνandγwere varied and possible consequences were discussed. Moreover, different ξ’s and even λ’s were assigned to the two bands along with two different κ’s. This led to speculations that situations may exist whereone of the bands behaves as a type II superconductor withκ 1>1/√ 2, while the other may have κ2<1/√ 2 and behave near Tcas the type I; the superconductivity in such situations was called “type 1.5.” MgB 2has been suggestedas such an example; see, for example, Ref. 11and references therein. The present work argues that such situations do not exist. The point is that the GL equations are derived from themicroscopic theory within certain approximations that lead tothe free energy near T cbeing proportional to (1 −T/T c)2and the order parameter (or parameters) varying as (1 −T/T c)1/2. Formally, the nonlinear system of GL Eqs. ( 3) and ( 4)f o r two-band materials can be solved with whatever accuracy onechooses. However, physically there is no point in going foraccuracy higher than that of equations themselves; whateverresults obtained along these lines will be unreliable. To geta near- T cdescription more accurate than GL, one should go back to microscopic theory that generates many extra termsin the free energy expansion even for the one-band situation;see, for example, Ref. 24, so that the multiband generalization of such an approach is unlikely to produce a useful theory. Itis demonstrated on a one-dimensional problem of Ref. 7and is shown for a general case in this paper that within the GL accuracy , both order parameters of a two-band superconductor vary on the same length scale ξof Eq. ( 14) contrary to requirements of “1.5-type superconductivity.” We note that this conclusion holds for the “GL domain” defined as the temperature interval near T cwhere the GL expansion can be justified. We do not specify this domainexplicitly because its size may vary from one case to another;for example, it is argued in Ref. 8that for two dirty bands (with no interband scattering) of MgB 2, the domain of GL applicability shrinks practically to zero. However, whateverthis size is, within this domain the two order parameters varyon the same length scale. Therefore, attempts to employ the GLfunctionals, on the one hand, and to assume different lengthscales, on the other, cannot be justified. Moreover, we show that—within the GL accuracy—the two GL equations for the two-band case are reduced to asingle equation for the normalized order parameter; in otherwords, the two-band superconductor is described by a singlecomplex order parameter. This excludes possibilities of having“fractional vortices” with exotic properties such as thosediscussed in Refs. 25and26. Microscopically, our results were derived within a weak- coupling theory of clean superconductors. We believe, how-ever, that our conclusions go beyond that. For our results tohold it is crucial that due to the finite interband Josephsoncoupling γ, the coefficients a νin the GL energy remain finite atTc. Once this is guaranteed our qualitative conclusions remain unchanged, even if assumptions of the weak coupling,no scattering, and isotropy do not apply. Note added in proof . In the recent paper by Shanenko et al. , 27our conclusion on a single length scale ξin two-band superconductors near Tcis confirmed. Extra terms in the GL expansion discussed in this work are, by construction, smallcorrections and do not change our conclusion that the ideaof 1.5-type superconductivity is not warranted by the GLtheory. ACKNOWLEDGMENTS Discussions with R. Fernandes, J. Geyer, J. Clem, R. Prozorov, M. Das, M. Milosevic, M. Zhitomirsky, 054515-4GINZBURG-LANDAU THEORY OF TWO-BAND ... PHYSICAL REVIEW B 83, 054515 (2011) A. Gurevich, R. Mints, J. Berger, and L. Bulaevskii are grate- fully appreciated. The work was supported by the DOE-Officeof Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE- AC02-07CH11358. 1V . L. Ginzburg and L. D. Landau, Sov. Phys. JETP 20, 1064 (1950). 2I. N. Askerzade, A. Gencer, and N. G ¨ucl¨u,Supercond. Sci. Technol. 15, L17 (2002). 3E. Babaev and M. Speight, P h y s .R e v .B 72, 180502 (2005). 4E. Babaev and J. Carlstr ¨om, e-print arXiv:1007.1965 (to be published). 5D. R. Tilley, Proc. Phys. Soc. London 84, 573 (1964). 6M. E. Zhitomirsky and V .-H. Dao, Phys. Rev. B 69, 054508 (2004). 7Jani Geyer, Rafael M. Fernandes, V . G. Kogan, and J ¨org Schmalian, P h y s .R e v .B 82, 104521 (2010). 8A. E. Koshelev and A. A. Golubov, P h y s .R e v .L e t t . 92, 107008 (2004). 9A. Gurevich (private communication). 10E. H. Brandt and M. P. Das, e-print arXiv:1007.1107 (to be published). 11V . Moshchalkov et al. ,Phys. Rev. Lett. 102, 117001 (2009). 12This is not the case in the resistive state: A. Gurevich and V . M. Vinokur, P h y s .R e v .L e t t . 90, 047004 (2003). 13G. Eilenberger, Z. Phys. 214, 195 (1968). 14V . A. Moskalenko, Phys. Met. Metallogr. 8, 503 (1959) [Fiz. Met. Metalloved. 8, 503 (1959)].15H. Suhl, B. T. Matthias, and L. R. Walker, P h y s .R e v .L e t t . 3, 552 (1959). 16B. T. Geilikman, R. O. Zaitsev, and V . Z. Kresin, Sov. Phys. SolidState 9, 642 (1967). 17I. I. Mazin and J. Schmalian, Physica C: Superconductivity 469, 614 (1995). 18V . G. Kogan, C. Martin, and R. Prozorov, Phys. Rev. B 80, 014507 (2009). 19A. A. Golubov and I. I. Mazin, Physica C: Superconductivity 243, 153 (2009). 20R. Geurts, M. V . Milosevic, and F. M. Peeters, Phys. Rev. B 81, 214514 (2010). 21M. Karmakar and B. Dey, J. Phys. Condens. Matter 22, 205701 (2010). 22Shi-Zeng Lin and Xiao Hu, e-print arXiv:1007.1940 (to be published). 23V . A. Moskalenko, M. E. Palistrant, and V . M. Vakalyuk, Usp. Fiz. Nauk 161(8), 155 (1991). 24L. Neumann and L. Tewordt, Z. Phys. 191, 73 (1966). 25E. Babaev, Phys. Rev. Lett. 89, 067001 (2002). 26M. A. Silaev, e-print arXiv:1008.1194 (to be published). 27A. A. Shanenko et al. ,Phys. Rev. Lett. 106, 047005 (2011). 054515-5

PhysRevB.71.195321.pdf

Excitonic wave packet dynamics in semiconductor photonic-crystal structures B. Pasenow, M. Reichelt, T. Stroucken, T. Meier, and S. W. Koch Department of Physics and Material Sciences Center, Philipps University, Renthof 5, D-35032 Marburg, Germany sReceived 14 December 2004; revised manuscript received 1 March 2005; published 24 May 2005 d Significant aspects of the light–matter interaction can be strongly modified in suitably designed systems consisting of semiconductor nanostructures and dielectric photonic crystals. To analyze such effects, a micro-scopic theory is presented, which is capable of describing the optoelectronic properties of such hybrid systemsvia a self-consistent solution of the dynamics of the optical field and the photoexcitations of the material. Thetheory is applied to investigate the local excitonic resonances, which arise as a consequence of the modifiedCoulomb interaction in the vicinity of a structured dielectric medium. The excitation of a coherent superposi-tion of the spatially inhomogeneous optical transitions induces an intricate wave packet dynamics. In thepresence of dephasing and relaxation processes, the coherent oscillations are damped and the photoexcitedcarriers relax into spatially inhomogeneous quasi-equilibrium distributions. DOI: 10.1103/PhysRevB.71.195321 PACS number ssd: 71.35.Cc, 42.50.Md, 42.70.Qs, 71.35. 2y I. INTRODUCTION Photonic crystals, i.e., periodically structured dielectric materials, can be used to tailor the eigenmodes of the trans-verse electromagnetic field. 1–5Since the light–matter interac- tion in atoms, molecules, and solids is governed by thistransverse part of the field, the optical material properties canbe drastically changed with suitably designed photonic-crystal structures. Using a photonic band gap one can sup-press the spontaneous emission and thus increase the radia-tive lifetime of optical excitations by several orders ofmagnitude. 1,2,6–8By the interaction with localized defect modes in photonic crystals novel strong coupling effects canbe achieved. 9–12 In this context, semiconductor heterostructures are of par- ticular interest not only because they can be grown with al-most molecular precision but also because of their strongexcitonic resonances. 13,14The quantum efficiency of well- designed optoelectronic semiconductor devices approachesthe fundamental radiative lifetime limit and combinations ofsemiconductor nanostructures and photonic crystals allowfor the possibility to optimize characteristics of light-emitting diodes and lasers. 10,11,15–21In addition to this appli- cation potential, however, semiconductor photonic-crystalstructures are also of interest in the context of fundamentalphysics. For example, it has been demonstrated that the re-duced spontaneous emission due to photonic crystals resultsin strong modifications of the exciton statistics and Coulombmany-particle correlations inside a semiconductor material. 22 Beside the transverse field, also the longitudinal electro- magnetic field, i.e., the Coulomb potential between chargedparticles, can be modified significantly in suitably designedsemiconductor photonic-crystal structures. 23–26In such hy- brid systems, induced surface polarizations alter the opticalsemiconductor properties in comparison to a spatially homo-geneous excitation configuration, resulting in dielectric shiftsof the band gap and the energetic position of the excitonresonance. 27,28Furthermore, spatially inhomogeneous quasi- equilibrium carrier populations can be obtained, which maysubstantially influence the quasi-equilibrium gain spectra oflaser structures. 26The periodic modulation of the optical properties may even result in superradiant emission.29,30 In this paper, we present and apply a microscopic ap- proach, which is capable of describing the optical propertiesof spatially inhomogeneous semiconductor photonic-crystalstructures. In most of our previous treatment, 24–26where we have concentrated on longitudinal effects, the transverse op-tical field has been assumed to be homogeneous. In contrast,here, the coupled dynamics of the field and the material ex-citations are treated self-consistently by evaluating thecoupled Maxwell semiconductor Bloch equations sMS-BEs d by numerical integration. In each time step, Maxwells’equa-tions are solved using the finite-difference time-domainsFDTD dmethod 31with complex fields. The semiconductor Bloch equations sSBEd,13which are used here in a real-space basis in order to describe spatially inhomogeneous situations,are integrated using the standard leap-frog algorithm. 32With this combination it is possible to use the rotating wave ap-proximation for the semiconductor excitations and to restrictthe analysis to resonant excitations. This approach has suc-cessfully been applied in Ref. 33 to analyze optical absorp-tion spectra in the presence of quasi-equilibrium electron andhole populations. Here, it is extended and used to study be-sides the dynamics of the optical polarization also the spa-tiotemporal evolution of photoexcited populations. In particular, we analyze the local absorption spectra and discuss optical excitation of a coherent superposition of spa-tially inhomogeneous excitonic resonances. After the inci-dent laser pulse has decayed, the photoexcitations of thesemiconductor material oscillate coherently and display anintricate wave packet dynamics. The predicted effects shouldbe measurable by ultrafast nonlinear optical spectroscopywhere they show up as temporal modulations of thesignal. 34–36In the course of time, the coherent oscillations are damped due to dephasing and relaxation processes.Therefore, in the long time limit the photoexcited carriersapproach quasi-equilibrium distributions. Due to the modi-fied Coulomb interaction, these carrier distributions are inho-mogeneous in space. In Sec. II of this paper the microscopic approach and the self-consistent analysis of the light–matter interaction inPHYSICAL REVIEW B 71, 195321 s2005 d 1098-0121/2005/71 s19d/195321 s15d/$23.00 ©2005 The American Physical Society 195321-1semiconductor photonic-crystal structures is described. Nu- merical results on the excitonic absorption in spatially inho-mogeneous situations, the coherent dynamics of excitonicwave packets, and the decay of the coherent oscillations dueto dephasing and relaxation processes are presented and dis-cussed in Sec. III. The most important results are brieflysummarized in Sec. IV. II. SPATIALLY INHOMOGENEOUS MAXWELL- SEMICONDUCTOR BLOCH EQUATIONS In this section, we present our theoretical approach, which provides a self-consistent description of the coupled dynam-ics of the electromagnetic field and the optical material ex-citations in semiconductor photonic-crystal structures.Within a semiclassical treatment, the material system is de-scribed quantum mechanically whereas the dynamics of theelectromagnetic field is treated classically. As shown in Sec.II A, the transverse components of the field are determinedby Maxwell’s equations with a spatially varying dielectricfunction. These equations include the coupling to the opticalmaterial polarization whose time derivative appears as asource in the equation for the electric field. Besides the time-dependent field, also the static field, i.e., the Coulomb inter-action among charged particles, is modified in the vicinity ofa spatially structured dielectric environment.The generalizedCoulomb potential describing the interaction of the chargecarriers in a semiconductor near a photonic crystal can beobtained by solving an integral equation, 23–26as described in Sec. II B. The Hamiltonian, which governs the dynamics ofphotoexcited electrons and holes in the semiconductor, is in-troduced in Sec. II C.The equations of motion describing thedynamics of the optical material excitations are presented inSec. II D where also the inclusion of nonradiative dephasingand relaxation is discussed. A. Inhomogeneous Maxwell’s equations in photonic crystals Propagation of electromagnetic waves through a macro- scopic material is governed by the inhomogeneous Maxwellequations =·D= r, s1d =·B=0, s2d =ˆE+] ]tB=0, s3d =ˆH−] ]tD=j. s4d Here,EandHare the macroscopic electric and magnetic fields,DandBare the dielectric displacement and magnetic induction fields, and randjare the free charges and currents in the material, respectively. Maxwell’s equations constitutea set of 8 inhomogeneous differential equations for the 12field components, where the charges and currents act assource terms. For an interacting system, Maxwell’s equationsare solved together with the equations of motion for the par- ticle charges and currents, that are in turn driven by the op-tical fields EandH. In order to solve these equations, we need the constitutive relations D=DfEgandB=DfHgrelat- ing the macroscopic to the microscopic fields. In general, the constititive relations can be arbitrarily complicated, relatingthe components of the dielectric displacement and magneticinduction in a nonlocal, anisotropic, frequency dependent,and nonlinear manner to the components of the electric andmagnetic fields. As we are interested in optics in dielectricphotonic crystals, we shall assume nonmagnetic media in thefollowing, i.e., B= m0H. For the dielectric displacement, we make the ansatz D=e0esrdE, s5d where the photonic crystal structure is described via a peri- odically varying dielectric function esrd, which is assumed to be a local, scalar, and frequency independent. All other con- tributions are mediated by the resonant light–matter interac-tion and are determined by explicitly solving the microscopicequations of motion for the particle system. As is well known, the current and charge density couple to the vector and scalar potential, respectively, rather than tothe electromagnetic field components. These are introducedwith the aid of the homogeneous Maxwell equations as B ==ˆA,E=−A˙−= f. Inserting the potentials in the inho- mogeneous Maxwell equations, we obtain =ˆ=ˆA+esrd c2]2 ]t2A=−esrd c2=f˙+m0j, s6d =·esrdS1 c] ]tA+=fD=−r/e0. s7d From Eq. s7d, it is obvious that the scalar potential is not truly an independent variable, but is determined by the vec-tor potential, the dielectric function, and the charge density.In homogeneous media, the scalar potential can be expressedin terms of the charge density only with the aid of the Cou-lomb gauge =·A=0. Within a semiclassical treatment, the interactions induced by this part of the electric field aretreated fully quantum mechanically, whereas that part of theelectromagnetic field that is associated with the vector poten-tial is treated classically. In inhomogeneous media, the stan-dard Coulomb gauge leads to a nonvanishing scalar potentialeven in the absence of external charges. Therefore, it is ad-vantageous to introduce the generalized Coulomb gauge=· esrdA=0.37,38This gauge corresponds to a division into a transverse and longitudinal part of the dielectric displace- ment rather than the electric field, i.e., D=DT+DLwithDT =−esrdA˙andDL=−esrd=f. Within a semiclassical descrip- tion, the transverse part of the dielectric displacement and the magnetic field are treated classically and obey thecoupled wave equations =ˆD T e0esrd+] ]tB=0, s8dPASENOW et al. PHYSICAL REVIEW B 71, 195321 s2005 d 195321-2=ˆB−m0] ]tDT=m0jT, s9d wherejTis the transverse part of the current, which appears as a source term for the classical part of the electromagneticfield. In our numerical solutions, the spatiotemporal evolution of the electromagnetic field is obtained by solving the FDTDequations. 31These equations are self-consistently integrated together with the SBE, see Sec. II D, that determine the ma-terial excitations. B. The Coulomb interaction Within our semiclassical treatment, the field energy asso- ciated with the longitudinal part of the dielectric displace-ment is treated quantum mechanically and results in the Cou-lomb interaction among charged particles. Inserting thedefinition of D Land Eq. s1dinto the expression for the field energy, we obtain HC=1 2Ed3rDL·DL e0esrd =−1 2Ed3r=f·DL =1 2Ed3rf=·DL =1 2Ed3rfsrdrsrd. s10d The scalar potential is the solution of a generalized Pois- son equation −=·fesrd=fsr,tdg=rsr,td/e0, s11d with the charge density rsr,tdas inhomogeneity. Defining the generalized Coulomb potential VCsr,r8das the solution of the generalized Poisson equation with a d-function inho- mogeneity −=·fesrd=VCsr,r8dg=dsr−r8d/e0, s12d the scalar potential can be expressed as fsr,td=Ed3r8VCsr,r8drsr8,td, s13d yielding HC=1 2Ed3rEd3r8rsrdVCsr,r8drsr8ds14d for the Coulomb energy. The generalized Coulomb potential describes the interaction among charged particles in an inho-mogeneous dielectric environment. With the exception of a few analytically solvable geom- etries, like, e.g., two dielectric half spaces separated by aplane or a single sphere embedded into a material of differentdielectric constant, 39Eq.s12dhas to be solved numerically for general situations. For this purpose, it is advantageous tostart from the integral equation 23,26VCsr,r8d=−1 4pEd3r9=91 ur9−ru·Elsr9,r8d,s15d whereElsr,r8d=−=VCsr,r8dis the electric field at the posi- tionrdue to a unit charge at r8. If the dielectric function is piecewise constant, which is usually the case in photoniccrystals, one can partially evaluate the volume integral ap-pearing in Eq. s15dand use the boundary conditions for the electric field E land the dielectric displacement Dlat the interfaces ]Dij, which separate regions DiandDjof different dielectric functions. As shown in Ref. 23, one obtains VCsr,r8d=1 4pe01 esr8d1 ur−r8u −1 4pe0o ijS1 ei−1 ejDE ]Dijda91 ur9−runi9·Dlsr9,r8d =V0sr,r8d+dVsr,r8d, s16d whereni9denotes the unit vector normal to the surface at r9. According to Eq. s16d,VCis given by the sum of two con- tributions. V0has the usual 1/ ur−r8uspatial variation and is additionally statically screened with the local value of thedielectric function 1/ esr8d. The second term, dV, appears as a result of induced surface polarizations at the interfaces ]Dij, which separate the regions of different e, over which is integrated. The magnitude of dVmay be large, if the dielec- tric contrast is large and, in particular, if the charge is closeto an interface since in this case the interaction with theinduced surface polarizations is strong. In contrast to the situation in spatially homogeneous me- dia, the generalized Coulomb potential is a function not onlyof the relative coordinate r rel=r−r8but also of the center of mass coordinate rc.m.=sm1/Mdr+sm2/Mdr8withm1+m2 =M. The dependence of the generalized Coulomb potential onrc.m.obeys the same symmetry properties as the dielectric function esrd. In systems where the dielectric function is varying periodically in space, e.g., in photonic crystals, also VCexhibits this periodicity. The generalized Coulomb potential can be evaluated nu- merically using an integral equation for the dielectric dis-placement D lat the interfaces ]Dij. This equation can be obtained by applying nisrd·=to Eq. s16d, wherenisrdde- notes the unit vector normal to the surface at r. Defining the normal component of the dielectric displacement by Dnsr,r8d=nisrd·Dlsr,r8d, s17d one obtains25,26 Dnsr,r8d=nisrd·r−r8 ur−r8u3+ lim g!0+1 4pS1−e1 e2D 3E ]Dijda9nisrd·rg−r9 urg−r9uDnsr9,r8d,s18d withrg=r−gnisrd. To determine Dn, Eq. s18dcan be solved by matrix inversion on a grid in real-space using the Nystrom method.32Inserting this solution into Eq. s16dal-EXCITONIC WAVE PACKET DYNAMICS IN … PHYSICAL REVIEW B 71, 195321 s2005 d 195321-3lows one to determine the generalized Coulomb potential VC in all spatial regions of interest. C. Hamilton operator The Hamiltonian describing the optical properties of semiconductors and semiconductor nanostructures consistsof three terms 13 Hˆ=Hˆ0+HˆI+HˆC. s19d Here,Hˆ0contains the single-particle band structure, HˆIde- notes the interaction of the semiconductor with the classical part of the electromagnetic field, and HˆCdescribes the many- body Coulomb interaction among charged particles includingthe modifications due to the dielectric structuring discussedearlier. According to the minimal coupling approach, wehave Hˆ 0+HˆI=Ed3rc†srdF1 2m0sp−eAd2+VGsrd +VconfsrdGcsrd. s20d Here,m0is the electron mass, VGis the periodic lattice po- tential,Vconfis the confinement potential in a system of re- duced dimensionality, and c†,care the Heisenberg creation and annihilation operators for the electrons.The total particlecurrentjentering as source term into the dynamical equa- tions for the transverse field components is obtained from theinteraction Hamiltonian via j=−1 m0dHˆI dA=−e m0m0c†Sp−e cADc. s21d Using the Heisenberg equation of motion for the field opera- tors, it is easily verified that the current can be expressesalternatively as the time derivative of the polarization, i.e., j=1 m0] ]tP=1 m0] ]tc†erc=−1 m0d dAEd3rP˙A.s22d Since the total Hamiltonian can be changed by a total time derivative without altering the equations of motion, the dy-namics of the system can also be obtained from the dipoleinteraction Hamiltonian H I=−Ed3rET·P, s23d where we used the notation ET=−A˙=DT/e0esrd. Note that ETis not transverse but corresponds to the classical part of the electromagnetic field. This is not yet the dipole approxi-mation but is exact. The macroscopic polarization currents ]/]tdkPlhas to be computed from the Heisenberg equations of motion i"] ]tkPl=kfP,Hgl. s24d In order to solve these equations, it is convenient to expand the creation and annihilation operators in terms of eigenfunc-tions ofH0. In general, the eigenfunctions of H0in a crystal can be written as products of lattice periodic functionsu m,ksrdand the envelope funtions fm,ksrd =exp siki·ridwm,k’sr’dthat vary on a length scale much larger than the lattice constant, i.e., csrd =om,kum,ksrdfm,ksrdam,k. Here, mdenotes the band index, k all other relevant quantum numbers, riare the coordinates in the extended directions of the semiconductor nanostructure,r ’the coordinates perpendicular to the structure, and wm,kis the confinement function. For simplicity, we restrict theanalysis to a two-band situation considering only the lowestconfinement subbands, i.e., a single conduction and a singlevalence band, respectively. It is, however, straightforward toextend this approach to multiple bands by including summa-tions over the relevant bands in the following expressions. Applying the electron hole picture by c k=ac,kanddk=av,−k+, the single particle part of the Hamiltonian is given by H0=o kekeck+ck+ekhd−k+d−k, s25d where the sum over kis taken parallel to the heterostructure only and the subscript ihas been dropped for better readibil- ity. Here, eke=Egap+"2k2/2meandekh="2k2/2mh,me/hare the effective masses for the electrons/holes, respectively, andE gapis the gap including the confinement energy. To obtain a transparent description of the spatial inhomo- geneities in semiconductor nanostructures close to photoniccrystals, we perform a coarse graining on the length scale ofan elementary cell, yielding the real-space representation Hˆ 0=Eddr1Fcˆ1+SEG−"2=12 2meDcˆ1+dˆ 1+S−"2=12 2mhDdˆ1G, s26d where the operators cˆ1+sdˆ 1+d=Eddk s2pddexps−ik·r1dck+sdk+d andcˆ1sdˆ1dcreate and destroy electrons of mass meand charge −esholes of mass mhand charge + edatr1, respec- tively, and dis the effective dimensionality of the system. For the light–matter interaction, we employ the dipole approximation, which is obtained by a multipole expansionover an elementary cell where only the dipole moment istaken into account. A real space representation is again ob-tained by a subsequent coarse graining, yielding Hˆ I=−Eddr1E1std·msdˆ1cˆ1+cˆ1+dˆ 1+d, s27d with the operator for the interband polarization Psr1,td =msdˆ1cˆ1+cˆ1+dˆ 1+d. Here, E1std;Esr1,tdis the space- dependent classical electric field, mis the interband dipole matrix element, which is treated as a real space-independent material constant, and dˆ1cˆ1scˆ1+dˆ 1+ddescribes the local inter- band coherence, which corresponds to destroying screating d an electron–hole pair at r1.PASENOW et al. PHYSICAL REVIEW B 71, 195321 s2005 d 195321-4The Coulomb interaction is obtained by using rsr1d =−esc1+c1−d1+d1din Eq. s14dfor the charge density. It is worthwhile to note that the Coulomb energy contains the interaction between mutually different electrons and holes aswell as the self-interaction of the charge density with its ownpotential. While the self-interaction with the bulk part of theCoulomb potential is unphysical and can be removed by tak-ing the normally ordered product of the field operators, theself-interaction with induced surface polarizations at the in-terfaces of the photonic crystal is physically meaningful and must be included. 27,28Using the notation VCsr1,r2d;V12C =V120+dV12for the generalized Coulomb potential, the Cou- lomb Hamiltonian reads23,26 HˆC=e2 2Eddr1Eddr2V12Cscˆ1+cˆ2+cˆ2cˆ1+dˆ 1+dˆ 2+dˆ2dˆ1−2cˆ1+dˆ 2+dˆ2cˆ1d +e2 2Eddr1dV11scˆ1+cˆ1+dˆ 1+dˆ1d. s28d Here, the terms in the double integral represent the repulsive electron–electron and hole–hole interactions, as well as theattractive interaction between electrons and holes. The lastterm of Eq. s28ddescribes the self-interaction of the electron and hole with their respective image charges.The self-energyse 2/2ddV11acts as a spatially varying single-particle poten- tial for the electrons and holes. One could thus add the terms involving se2/2ddV11toH0and solve the resulting single- particle Hamiltonian by calculating Bloch-type electronic eigenfunctions using the spatial periodicity corresponding tothe period of the photonic crystal, see Sec. II D 2. Due to thevery different length scales involved, i.e., the long wave-length of light and the small lattice constant, numericalevaluations of this Bloch approach are rather demanding.Nevertheless, this approach can be used to analyze thedensity-dependent optical absorption in quasi-equilibriumsituations. 26,33 D. Equations of motion The dynamical properties of the semiconductor system are described by the Heisenberg equations for the relevantquantities describing the material excitations. 13The equation of motion for the expectation value of an arbitrary operator O=kOˆlis obtained from i"] ]tOstd=kfO,Hˆgl. s29d Whereas the commutators with Hˆ0andHˆIlead to a set of closed equations of motion on the single-particle level, i.e.,optical Bloch equations, the many-particle part of the Hamil- tonian,Hˆ C, introduces coupling to an infinite hierarchy of correlation functions.13,14,40To be able to analyze the optical properties of a spatially inhomogeneous systemwithin reasonable numerical limits, we restrict our presentanalysis to the level of the time-dependent Hartree–Fockapproximation. 13As an example, we discuss the dynamics of the off- diagonal interband coherence p12=kpˆ12l=kdˆ1cˆ2l. Considering the contribution of the Colulomb interaction to the time de- rivative of the interband coherence we obtain i"] ]tup12uC=kfpˆ12,HˆCgl =e2S1 2dV11+1 2dV22−V12CDp12 +e2Eddr3sV32C−V13Cdskc3+d1c2c3l+kd3+d1d3c2ld. s30d As a consequence of the many-body Coulomb interactions, we find on the right-hand side of Eq. s30da coupling to four-operator expectation values, which is the beginning ofthe usual infinite hierarchy problem of many-bodyphysics. 13,14,40Aclosed set of equations is obtained by using the time-dependent Hartree–Fock factorization, which ap-proximates the four-operator expectation values by productsof two-operator expectation values. 13,14,40This means that the four-operator terms appearing in Eq. s30dare approxi- mated by kc3+d1c2c3l<kc3+c3lkd1c2l−kc3+c2lkd1c3l=n33ep12−n32ep13, kd3+d1d3c2l<kd3+d1lkd3c2l−kd3+d3lkd1c2l=n31hp32−n33hp12, s31d where the electron sholedpopulations and intraband coher- encesn12e=kcˆ1+c2lsn12h=kdˆ 1+d2ldhave been introduced. Using such factorizations also in the equations of motion forneandnhand evaluating the remaining commutators with Hˆ0andHˆI, we obtain a closed set of coupled equations of motion determining the dynamics of the expectation valuesof all two-operator quantities. These equations are known asthe SBE in time-dependent Hartree–Fock approximation. 13 For our inhomogeneous system the explicit form of the SBEis i"] ]tp12=FEG−"2 2mh=12−"2 2me=22+e2 2dV11+e2 2dV22−V12C −e2Eddr3sV13C−V32Cdsn33e−n33hdGp12 +e2Eddr3sV13C−V32Cdsn32ep13−n31hp32d −m·sE1d12−E1n12e−E2n21hd+i"] ]tup12ucorr,s32dEXCITONIC WAVE PACKET DYNAMICS IN … PHYSICAL REVIEW B 71, 195321 s2005 d 195321-5i"] ]tn12e=F"2 2mes=12−=22d−e2 2dV11+e2 2dV22 −e2Eddr3sV13C−V32Cdsn33e−n33hdGn12e +e2Eddr3sV13C−V32Cdsn13en32e+p31*p32d +m·sE1p12−E2p21*d+i"] ]tun12eucorr, s33d i"] ]tn12h=F"2 2mhs=12−=22d−e2 2dV11+e2 2dV22 +e2Eddr3sV13C−V32Cdsn33e−n33hdGn12h +e2Eddr3sV13C−V32Cdsn13hn32h+p13*p23d +m·sE1p21−E2p12*d+i"] ]tun12hucorr. s34d In Eqs. s32d–s34d, the terms denoted by ucorrrepresent all many-body correlations that are beyond the time-dependentHartree–Fock limit. 13,14,40In the analysis presented here, these correlation terms are either neglected completely ortreated at a phenomenological level. The single-particle self-energies dVappear as potentials in the homogeneous parts of the equations of motion, Eqs.s32d–s34d. For the electron–hole interband coherence p 12, the homogeneous part of the equation of motion is furthermore influenced by the electron–hole Coulomb attraction − e2V12C, which gives rise to excitonic effects in the optical spectra.Additionally, integrals over the generalized Coulomb poten-tialV Cand products of p’s andn’s appear in Eqs. s32d–s34d, and all equations of motion contain sources representing thedriving by the electric field. Equations s32d–s34dtogether with FDTD equations for the electromagnetic field allow for a self-consistent descrip-tion of the dynamical evolution of the coupled light and ma-terial system, where the field is driven by the material polar-ization that is in turn driven by the electric field. This set ofequations may be solved for arbitrary field intensities. It con-tains many-body Coulomb effects and can be used to inves-tigate high-intensity effects like, e.g., excitonic Rabi-flopping, in a self-consistent fashion. Due to the self-consistency of the solution, radiative decay processes areincluded automatically, yielding the correct radiative decayrates for the polarization and the carrier populations evenwithin a semiclassical description. The only aspect a semi-classical approach cannot account for is spontaneous emis-sion, since the source term contains the expectation value ofthe field only and no fluctuations. 1. Low-intensity limit In order to eliminate density-dependent shifts of the single-particle energies and to prevent rapid dephasing andrelaxation due to carrier–carrier scattering, the numerical re-sults presented in the following are obtained using incident laser beams of weak intensities. Therefore, one can describethe light–matter coupling perturbatively and classify the ma-terial excitations according to their power in the optical field. Let us assume that the semiconductor is in its ground state before the optical excitation, i.e., the electron and hole popu-lations as well as the intraband and interband coherencesvanish initially. In this case, the linear optical properties of asemiconductor are determined by the equation of motion forthe linear electron-hole interband coherence i" ] ]tp12s1d=FEG−1 2mh=12−1 2me=22+e2 2sdV11+dV22d −e2V12CGp12s1d−m·E1d12, s35d where the superscript s1dindicates that the optical polariza- tion is calculated in first order in the light field. By diagonal-izing the homogeneous part on the right-hand side of Eq.s35d, one can obtain the energies of the excitonic resonances eXand the corresponding eigenfunctions CXsr1,r2d. For ex- citation with a homogeneous light field the oscillator strength of each excitonic state is proportional to m2ueddrCXsr,rdu2, i.e., to the absolute square of the electron–hole overlap, since the field generates electrons and holes at the same position inspace, see Eqs. s27dands35d. For an inhomogeneous excita- tion the spatial overlap of the polarization eigenfunctions andthe light field redistributes the absorption strengths of theexcitonic states. In second order in the light field, carrier populations and coherences are generated. This process is described by i" ] ]tn12es2d=F"2 2mes=12−=22d−e2 2dV11+e2 2dV22Gn12es2d +Eddr3e2sV13C−V32Cdsp31s1dd*p32s1d +m·sE1p12s1d−E2sp21s1dd*d, s36d i"] ]tn12hs2d=F"2 2mhs=12−=22d−e2 2dV11+e2 2dV22Gn12hs2d +Eddr3e2sV13C−V32Cdsp13s1dd*p23s1d +m·sE1p21s1d−E2sp12s1dd*d. s37d By diagonalizing the homogeneous part on the right-hand side of Eqs. s36dands37done can determine the electron and hole eigenstates. These can be populated according to Fermifunctions and inserted into Eq. s32dif one wants to study the density-dependent absorption in quasi-equilibriumsituations. 13,26,33 Note that due to the spatial integrals which appear as a consequence of the many-body Coulomb interaction it re-quires a lot more effort to numerically solve Eqs. s36dand s37dthan the equation for the linear polarization, Eq. s35d, which contains no spatial integrals. However, if the dynam-ics is coherent and one considers only terms up to secondPASENOW et al. PHYSICAL REVIEW B 71, 195321 s2005 d 195321-6order in the field, i.e., in the coherent xs2d-limit, it is actually not necessary to solve the combined set of Eqs. s35d–s37d. Using the equations of motion, one can easily verify that inthis limit the carrier populations and intraband coherencesare determined by the interband coherence via the sum rules n 12es2d=Eddr3p32s1dsp31s1dd*, s38d n12hs2d=Eddr3p23s1dsp13s1dd*, s39d i.e., they are given by spatial integrals over products of linear polarizations. 2. Dephasing and relaxation Due to the self-consistent solution of Maxwell’s equations together with the SBE, the radiative decay of the photoex-cited optical polarization and the populations is automati-cally included in our description. However, in real systemsthe material excitations often decay on shorter time scales.The coupling between the electron and the phonon systemcan be responsible for this, as well as the many-body corre-lation contributions, which include Coulomb scattering pro-cesses. Since a microscopic treatment of these processes inspatially inhomogeneous situations requires a very high nu-merical effort, we model them phenomenologically by insert-ing decay and relaxation times into the equations of motion. The nonradiative decay of the interband polarization, of- ten called dephasing, is assumed to be exponential and is described by adding − i"p 12s1d/T2*to the right-hand side of Eq. s35d. The total polarization decay time T2contains radiative and nonradiative contributions and is obtained as T2−1=Trad−1 +T2*−1. In the coherent limit, the dephasing of the carrier populations and intraband coherences is induced purely bythe finite lifetime of the photoexcited carriers. Thus, the car-rier populations and intraband coherences decay with thetime constant T 2/2. Since the radiative contributions are in- cluded automatically through the self-consistent solution ofthe MSBE only the nonradiative contributions −i"n 12es2d/sT2*/2dand −i"n12hs2d/sT2*/2dmust be added to the right- hand sides of Eqs. s36dands37d. In this case, the sum rules, Eqs. s38dands39d, remain valid and thus n12es2dandn12hs2d can be calculated using p12s1d. However, in reality the coherent limit is often not well suited to describe the dynamics of the electron–hole excita-tions in semiconductors. Typically, the populations and intra-band coherences do not vanish on a time scale similar to thedephasing of the optical polarization, but rather become in-coherent and approach quasi-equilibrium distributions in therespective bands. This thermalization process can be mod- eled by adding − i"sn 12es2d−n12e,eqd/T1and −i"sn12hs2d−n12h,eqd/T1, wheren12e,eqandn12h,eqdenote the populations and intraband coherences in quasi-thermal equilibrium, to the right-handsides of Eqs. s36dands37d. Since in this case the sum rules, Eqs. s38dands39d, are not valid any more, the dynamics of n 12es2dandn12hs2dhas to be determined by solving Eqs. s36dand s37dtogether with Eq. s35d. However, also in this situation itis possible to avoid the evaluation of the space integrals in the equations of motion. This can be achieved by decompos- ingn12es2dandn12hs2dinto a coherent and an incoherent part via n12es2d=n12es2d;coh+n12es2d;incoh, s40d n12hs2d=n12hs2d;coh+n12hs2d;incoh. s41d The coherent parts of nes2dandnhs2dare obtained using Eqs. s38dands39d, i.e., they decay with a time constant of T2/2. By inserting Eqs. s40dands41dinto Eqs. s36dands37d, re- spectively, and subtracting the coherent parts, one can deter-mine the following equations of motion for the incoherentterms: i" ] ]tn12es2d;incoh=F"2 2mes=12−=22d−e2 2dV11+e2 2dV22Gn12es2d;incoh −i"n12es2d;cohS1 T1−1 T2*/2D−i"n12es2d;incoh−n12e,eq T1, s42d i"] ]tn12hs2d;incoh=F"2 2mhs=12−=22d−e2 2dV11+e2 2dV22Gn12hs2d;incoh −i"n12hs2d;cohS1 T1−1 T2*/2D −i"n12hs2d;incoh−n12h,eq T1. s43d Equations s42dands43dshow that the incoherent populations and intraband coherences have two sources. If T1ÞT2*/2 they are generated directly from the coherent terms. Furthermore,the relaxation toward quasi-equilibrium distributions alwaysleads to incoherent terms, since n e,eqandnh,eqare determined by the total scoherent and incoherent dpopulations, see the following. In order to determine the quasi-equilibrium distributions, first the effective single-particle Hamiltonian defined by Hˆsingle-particle =Hˆband structure +HˆCoulomb self-energies =Eddr1Fcˆ1+SEG−"2=12 2meDcˆ1+dˆ 1+S−"2=12 2mhDdˆ1G +1 2Eddr1dV11scˆ1+cˆ1+dˆ 1+dˆ1ds 44d is diagonalized. Note that, besides the kinetic energies also the spatially varying self-energies, which act as inhomoge-neous potentials, are included in Eq. s44d. By diagonalizing this Hamiltonian separately for electrons and holes using thespatial periodicity induced by the photonic crystal, the dis- persions ekn, where nlabels the mini-bands, and the corre- sponding Bloch-type eigenfunctions Fknsrdare obtained. Since in the situations analyzed in the following the spatial period is relatively large, sufficiently many sabout 100 in the one-dimensional situation considered here dof the energeti- cally rather narrow mini-bands have to be included in thecalculation.EXCITONIC WAVE PACKET DYNAMICS IN … PHYSICAL REVIEW B 71, 195321 s2005 d 195321-7In this single-particle basis, the states are populated ac- cording to a quasi-equilibrium distribution, i.e., nknis given by the Fermi function F. The total density nat temperature T can be expressed as n=o k,nnkn=o k,nFsekn,T,md, s45d wherenkndenotes the population of state Fknwith energy ekn. Since the total density ndepends on the chemical potential, mneeds to be determined self-consistently. Having obtained nkn, we then transform back to real space since this is numeri- cally advantageous for performing the dynamic calculations.In the time-dependent solutions of the equations of motionthe carrier density is changing during the optical excitation.Therefore, the numerically calculated time-dependent den- sitynstdis used to determine the quasi-equilibrium n 12e,eqstd andn12e,eqstdreal-space populations and intraband coherences. III. NUMERICAL RESULTS A. Semiconductor photonic-crystal structure Numerical solutions of the MSBE for semiconductor photonic-crystal structures typically require a considerableamount of computer time and memory. On the one hand, ingeneral situations a three-dimensional space discretization isnecessary for FDTD solutions of Maxwell’s equations. Sincethe optical wavelength and the photonic structure have to beresolved with a suitable accuracy, such evaluations have tobe performed with a high number of grid points. On the otherhand, due to the generalized Coulomb interaction and thecoupling to spatially inhomogeneous light fields, the analysisof the material excitations has to be performed taking intoaccount both the relative and the center-of-mass coordinates.Thus the SBE have to be solved for a spatially inhomoge-neous situation where different length scales have to be re-solved since typically the exciton Bohr radius is about oneorder of magnitude smaller than the optical wavelength. In order to keep these numerical complexities within rea- sonable limits, we chose for the analysis presented in thispaper a model system consisting of a one-dimensional arrayof dielectric slabs s e=13dwhich extend in zdirection and are separated by air se=1d, see Fig. 1. The substrate below this dielectric structure is made of the same material as the di- electric slabs. Light propagating in this structure may createphotoexcitation in an array of parallel semiconductor quan-tum wires, which extend in ydirection perpendicular to the slabs and are separated from the photonic structure by thedistanceS.I nydirection the unit cell with length Lis re- peated periodically. In addition, periodic boundary condi-tions are also used in zdirection with period D, which is the distance between adjacent quantum wires. The parametersused in the numerical calculations are as follows: the lengthof the unit cell in ydirection is L=180 nm, the height of the slabs isH=700 nm, and for the width Wof the slabs differ- ent values are used. The parallel wires are separated by D =30 nm and the distance to the photonic crystal is S =2.6 nm. This value is assumed to be small in order to ob-tain a significant modification of the Coulomb interaction inthe quantum wires.Due to the light propagation through the dielectric struc- ture, the optical field is spatially inhomogeneous at the posi-tions of the quantum wires. Furthermore, since the wires areoriented perpendicular to the dielectric slabs, the generalizedCoulomb interaction varies periodically along the wires. Ourmodel system thus includes both a space-dependence of theelectromagnetic field and space-dependent modifications ofthe semiconductor properties. For later purposes, we introduce two reference systems. The limit W=Lis referred to as the homogeneous case, since the dielectric–air interface is far away sS+H=702.6 nm d from the semiconductor quantum wires and thus the modifi- cations of the Coulomb interaction are negligible. In the sec-ond limit W=0, which is denoted as the half-space case, the planarair–dielectricinterfaceisveryclose sS=2.6 nm dtothe semiconductor wire array. Therefore, the Coulomb interac- tion is significantly modified. Due to the planar interfaces,the generalized Coulomb potential is homogeneous with re-spect to the center-of-mass coordinate along the quantumwire for both limiting cases. In our calculations, we solve Eq. s18dfor a single dielec- tric slab, insert the solution in Eq. s16dto obtain the modified Coulomb interaction, and add the resulting dVfor the two nearest slabs neglecting the surface polarizations betweenthem. Numerical tests have justified this approximation if thedistance between the slabs is not too small. The edges be-tween the dielectric substrate and the slabs have beensmoothed as shown in Fig. 1. This avoids numerical prob-lems when solving Eq. s18dusing the Nystrom method and takes into account that realistic photonic crystals made byetching techniques have no sharp edges. FIG. 1. Schematical drawing of the considered semiconductor photonic-crystal structure. The photonic crystal is a periodic one-dimensional array of dielectric slabs s e=13dwhich are separated by airse=1d.The length Lof the unit cell in ydirection is 180 nm.The substrate below the photonic crystal is made of the same material asthe dielectric slabs. It surrounds the array of parallel quantum wireswhich lies S=2.6 nm underneath the photonic crystal. The distance between adjacent wires, which is also the length of the unit cell inzdirection, is D=30 nm. For the width Wof the slabs we use values of 0, 60, 90, 120, and 180 nm, respectively.The situations W=0 and W=L=180 nm are denoted by half-space and homogeneous case, respectively, as explained in the text. The height Hof the slabs is always 700 nm.PASENOW et al. PHYSICAL REVIEW B 71, 195321 s2005 d 195321-8Figure 2 shows the Coulomb modifications dVsr2,r1din the quantum wires for three fixed, representative positions r2=−90 nm sad,r2=−45 nm sbd, andr2=0nm scdas function of position r1. Positions r1underneath the 90-nm-wide di- electric slabs are indicated by the gray shading. The Cou-lomb modifications dVare small for positions r2directly underneath the slabs, like in case scd, since in this situation the distance to the dielectric–air interfaces of the photonicstructure is large. dVis bigger in the regions between the slabs, as can be seen in case sad, because in this situation the distance to the dielectric–air interfaces is small. Betweenthese extremal values, there is a quite sharp transition whichtakes place within a few nanometers directly under the edgesof the slabs, i.e., at ±45 nm in Fig. 2. The single-particlepotential dVsr1,r1d/2, dashed line in Fig. 2, shows this sharp transition and follows the periodicity of the photonic crystal. For short distances ur1−r2u, the magnitude and space depen- dence of the modified Coulomb interaction calculated forr 2=−90 nm, i.e., case sadin Fig. 2, differ only marginally from those which arise for the half-space case sW=0d. The curves sadandsbdshow a strong decrease of the Coulomb modifications with increasing distance between r1andr2. For particle positions r2underneath the slab, i.e., case scd,dV depends only weakly on distance for distances smaller thanthe half slab width, and decreases significantly if the distanceexceedsW/2. Figure 2 shows that the spatially periodically varying di- electric environment introduces a periodic single particle po-tential for the electron and holes, with minima underneath the dielectric slabs. This single particle potential influencesthe optical and electronical properties of the quantum wirevia electron and hole confinement effects and a periodicmodulation of the effective band gap. Furthermore, excitonicproperties are additionally altered as a result of the modifiedspace-dependent electron-hole attraction. All these effectsare analyzed in the following.B. Linear excitonic absorption In this section, the excitonic resonances in the linear ab- sorption spectra are obtained by solving the linear polariza-tion equation, Eq. s35d. We assume the incoming external light field to be a plane wave propagating in negative xdi- rection. The incident electric field is linearly polarized in y direction, i.e., in the direction of the quantum wires. The y andzcomponents of the electric and magnetic field, E yand Hz, respectively, have slowly varying Gaussian envelopes and oscillate in time with central frequencies close to theband gap frequency, i.e., E G/". The linear spectra, which are analyzed below, have been computed using the net energyflux through the boundaries of our FDTD simulation space DE˙=Edsn·S, s46d whereS=EˆHis the Poynting vector. The net flux contains all information about the absorbed or gained energy per unittime. In spectrally resolved experiments, the net flux is mea-sured over all times and analyzed in frequency space DE=EdtDE˙=EdvasvdI0svd, s47d where asvd=1 I0EdssE*sr,vdˆHsr,vd+c.c. d·n,s48d andI0svdis the intensity of the incoming light field. The absorption spectra shown in the following have been ob- tained by computing asvdfrom Eq. s48d. The semiconductor parameters used in the following are m=3.5eÅeyfor the dipole matrix element, mh/me=4 and me=0.066m0for the electron and hole masses, and EG =1.42 eV for the energy gap. Considering a dielectric con- stant of e=13, these parameters result in a three-dimensional exciton binding energy of EB=4.25 meV and a Bohr radius ofaB<13 nm. In most of the calculations, nonradiative ho- mogeneous broadening is modeled by introducing a decay rate of g="/T2*=1 meV in the equation of motion of the interband polarization. The Coulomb potential forthe one-dimensional wires has been regularized using V 0 =1/suru+a0d.41,42The regularization parameter is chosen as a0=0.16aB. Except for changes of the nonradiative decay timeT2*in Secs. III C and III D, these parameters are kept constant in the following. Additionally, we consider dielec-tric slabs with a thickness of W=90 nm, except in Fig. 3 where the influence of Won the excitonic absorption is in- vestigated. To obtain the results presented in the following, the FDTD calculations are performed on a grid with a spatialresolution of 5 nm, which requires a temporal resolution ofdt=dx/s2cd.8.3310 −18s.31The SBE have to be solved with a resolution smaller than the exciton Bohr radius aB. Therefore, we use here 1.3 nm, i.e., <aB/10. The self- consistent solution of the MSBE is done in the followingscheme: With the electric field at time t, the magnetic field at t+dt/2 and the polarization at t+dtare computed. The po- FIG. 2. Modification of the Coulomb potential dVsr2,r1d =dV21ssolid din the quantum wires for positions r2=−90 nm sad, r2=−45 nm sbd, andr2=0nm scdas function of position r1. The single-particle potential dVsr1,r1d/2=dV11/2 is also displayed sdashed d. The dielectric slabs of the photonic crystal are W =90 nm wide and H=700 nm high. Positions r1underneath the slabs are indicated by the gray areas. The energy unit is the three-dimensional exciton binding energy of GaAs E B=4.25 meV.EXCITONIC WAVE PACKET DYNAMICS IN … PHYSICAL REVIEW B 71, 195321 s2005 d 195321-9larizations at tandt+dtare used to determine its time de- rivative at t+dt/2 which together with the magnetic field allows us to evaluate the electric field at t+dt. Then these steps are repeated. For the case of a semiconductor quantum well placed close to a planar dielectric–air interface, image charge effectscause a shift of the single-particle energies, i.e., the band gapshifts to higher energies. Since the electron–hole attraction isincreased close to air, the exciton binding energy increases aswell. 23,28In semiconductor photonic-crystal structures both the band gap and excitonic binding energy become spacedependent. 24,25The band gap variation induces potential val- leys underneath the dielectric slabs which give rise to con-fined single-particle and exciton states, see insets in Fig. 3.These local potentials affect the linear absorption spectra andcause the double-peaked 1 s-exciton resonance visible in Figs. 3 sbd–3sdd. Note that the spectra are plotted on a loga- rithmic scale which emphasizes the continuum absorption.For comparison, also the homogeneous and the half-spacecases are shown in Figs. 3 sadand 3 sed, respectively. Figure 3 demonstrates that the spectral positions of the two excitonicpeaks are hardly affected by varying the width of the dielec-tric slabs. Whereas the lower exciton energy agrees with theposition of the exciton in the homogeneous case, the upperone corresponds to the half-space case. In contrast to theenergetic positions, the heights of the maxima are influencedsignificantly by the width of the dielectric slabs. With de- creasing width, the absorption of the lower homogeneous-like peak becomes smaller while that of the upper half-space-like peak increases. Further information on the relevant exciton resonances is shown in the insets of Fig. 3. Diagonalizing the linear polar-ization equation, Eq. s35d, for one unit cell with periodic boundary conditions yields the energetic positions of the ex-citonic states. Since the optical field at the quantum wire isspatially varying, the oscillator strengths of the differentresonances have been computed by fitting the self-consistently evaluated linear absorption spectra by a sum ofLorentzian curves o iAi/ssE−eX,id2+g2d, where g=1 meV is the decay constant used in the equation of motion for the interband polarization. The insets of Fig. 3 show that exceptfor the homogeneous and the half-space cases, which aredominated by a single excitonic peak, more than two exci-tonic resonances contribute to the absorption. However, forall considered widths of the dielectric slabs the two moststrongly absorbing resonances are at the energetic positionsof the homogeneous and the half-space excitons. The number of quantized exciton states is determined by the width of the dielectric slabs. For a 180-nm-wide slab, i.e.,in the homogeneous case, the Coulomb modifications essen-tially vanish and only the homogeneous 1 s-exciton reso- nance contributes, see Fig. 3 sad. For slab widths in between 180 and 0 nm potential valleys appear, see Fig. 2. Themaxima and minima of the space-dependent potential lie be-tween the values of the half-space and the homogeneouscase. The potential valleys are deepest for intermediate slabwidths, i.e, for W=90nm= L/2. Correspondingly, Figs. 3sad–3sednicely reflect the transition from the homogeneous to the half-space case which takes place with decreasing slabwidth. The logarithmic plots of the linear absorption show that the band gap appears at 0 E Band<4EBfor the homoge- neous and the half-space case, respectively, see Figs. 3 sad and 3 sed. Thus the exciton binding energy, i.e., the energetic distance between the lowest exciton resonance and the onsetof the continuum, increases from 4 E Bin the homogeneous case to <7.2EBin the half-space case. Qualitatively similar results have been obtained for quantum wells close to two-dimensional photonic crystals. 24,25In Figs. 3 sadand 3 sed, the 2s-exciton resonances are visible as small peaks at <−0.5EBand <3.1EB. For energies above <3.4EBthe continuum absorption is smooth also for the spatially inho-mogeneous cases, Figs. 3 sbd–3sdd. The inhomogeneity of the system causes the appearance of small absorption peaks inbetween <−0.5E Band <3.1EBwhich can be viewed as modified higher excitonic resonances. For a more detailed understanding of the excitonic reso- nances, we show in Fig. 4 also the polarization eigenfunc-tions, which belong to the excitonic states of Fig. 3 scd. These polarization eigenfunctions C Xsr1,r2dhave been obtained by diagonalization Eq. s35dand are real, since we use one unit cell with periodic boundary conditions. Shown in Fig. 4 sbdis the spatial variation of the eigenfunction for equal electronand hole positions, i.e., C Xsr1,r1d.The three lowest states are localized in the potential valley underneath the dielectric slabs, see Figs. 4 sbdand 4 scd. They look similar to usual FIG. 3. Excitonic linear absorption spectra for dielectric slabs of widthsW=180 nm sad,W=120 nm sbd,W=90nm scd,W=60nm sdd, andW=0nm sed, respectively, on a logarithmic scale. The height of the slabs is 700 nm in all cases. The insets show thedecomposition of the excitonic resonances into the contributingbound excitonic states and their oscillator strengths on a linearscale: sadcorresponds to the homogeneous case and sedto the half- space case. The calculations have been performed using a dampingof g=1 meV for the interband polarization and a quantum wire length of five unit cells with periodic boundary conditions.PASENOW et al. PHYSICAL REVIEW B 71, 195321 s2005 d 195321-10quantum mechanical eigenfunctions of a particle which is confined in a box-shaped potential and show an increasingnumber of nodes with increasing energy. Due to its higherenergy, the fourth state has strong contributions for positionsin between the dielectric slabs, which explains its half-spacelike character. The energetically higher fifth and sixth statescontribute negligibly to the excitonic absorption. When diagonalizing Eq. s35d, we obtain also polarization eigenfunctions which are antisymmetric, i.e., C Xsr1,r1d =−CXs−r1,−r1d. Because of our initial conditions for the light field these antisymmetric solutions of the polarization are not excited and do not contribute to the absorption. Thisis due to the fact that the incident light field is a homoge-neous plane wave which propagates in negative xdirection. The photonic crystal destroys the spatial homogeneity of thefield but maintains its mirror symmetry with respect to themiddle of one unit cell. Therefore, the overlap between thesymmetric light field and an antisymmetric wave functionvanishes. For a spatially structured incident light field orpropagation in a different direction, the symmetry of the sys-tem is broken and the antisymmetric wave function can beexcited. C. Coherent wave packet dynamics So far, we have focused on the linear optical properties of the system, e.g., the excitonic resonances, their oscillatorstrengths, resonance energies, and space-dependent eigen-functions. In this section we start to investigate the intricatecoherent wave packet dynamics of the electron density afterresonant excitation of the excitonic resonances. The absorp-tion spectra shown in the previous section have been com-puted using five unit cells for the quantum wire with periodic boundary conditions. This number of unit cells is required toobtain a converged continuum absorption. The exciton reso-nances are, however, already stable for just one unit cell.Therefore, it is justified to reduce the numerical requirementsfor the following investigations by considering one unit cellwith periodic boundary conditions. The densities are calculated for excitation with laser pulses of weak intensities up to second order s xs2ddin the light–matter interaction. In this section, we focus on the fully coherent dynamics and neglect nonradiative dephasing and relaxation processes, i.e., the limit T1,T2*!‘. The coherent electron density is obtained by solving Eq. s35dand using Eq.s38d. Figure 5 shows the spatiotemporal dynamics of the electron density n11es2d;cohafter excitation with a Gaussian pulse of 2 ps full width at half maximum sFWHM dduration of the pulse envelope and a central frequency which is tunedto the four energetically lowest excitonic resonances shownin Fig. 4. For excitation at the three lowest resonances, Figs.5sad–5scd, the electron density is basically concentrated at spatial positions underneath the dielectric slabs, i.e., between±45 nm. Since the spectral width of the incident electric fieldof 0.3E BsFWHM of field intensity dis comparable to the energetic spacing between the resonances, the density is notconstant as function of time. The pulses generate a coherentsuperposition of the exciton transitions which leads to wavepacket dynamics. However, comparing Figs. 5 sad–5scdwith the electron densities corresponding to the three lowest reso-nances, see Fig. 6, shows that the resonantly excited excitonsgive the strongest contributions to the density.When excitingat the fourth excitonic resonance, Fig. 5 sdd, the electron den- FIG. 4. sadLinear excitonic absorption spectrum for the param- eters considered in Fig. 3 scd. The lines indicate the spectral posi- tions and the oscillator strengths of the contributing excitonic reso-nances. sbdEigenfunctions of the interband polarization obtained by diagonalizing Eq. s35d. Shown is the spatial variation of the polar- ization eigenfunction for equal electron and hole positions, i.e.,C X,isr1,r1d. The dotted lines indicate the eigenenergies eX,iand cor- respond to the zero polarization axes. scdCorresponding single- particle potential induced by the spatially-varying dielectricenvironment. FIG. 5. Contour plots showing the spatiotemporal dynamics of the coherent electron density n11es2d;coh, see Eq. s38d, along one quan- tum wire unit cell. The system is excited by a Gaussian pulses of2 ps duration sFWHM d. The central frequencies of the pulses, vL, are tuned to the four lowest exciton states: sad"vL=EG−4.05EB, sbd"vL=EG−3.83EB,scd"vL=EG−3.46EB, and sdd"vL =EG−3.16EB, respectively. The dielectric slabs are W=90nm wide and H=700 nm high. The calculations have been performed assuming a fully coherent situation, i.e., nonradiative dephasing andthermalization have not been considered sT 1,T2*!‘d. White corre- sponds to the maximal density and black to zero density in eachplot.EXCITONIC WAVE PACKET DYNAMICS IN … PHYSICAL REVIEW B 71, 195321 s2005 d 195321-11sity is concentrated underneath the air regions of the photo- nic crystal. In this case the density dynamics correspondsessentially to a coherent superposition of the fourth and thirdexcitonic resonances, see Fig. 6. By using spectrally narrower, i.e., temporally longer laser pulses, it is possible to selectively excite single exciton reso-nances. In this case the coherently excited electron density isconstant as function of time and shows a spatial profile cor-responding to the curves shown in Fig. 6.As an example, we consider in Fig. 7 the excitation with a Gaussian pulseswhich are tuned to the third lowest exciton state and havedifferent temporal durations. The short 1 ps pulse, Fig. 7 sad, generates essentially a superposition of the four lowest exci-tonic resonances. Consequently, the wave packet dynamicsshows a complicated pattern and the density covers all spa-tial positions. For the 2 ps pulse, Fig. 7 sbd, the time evolu- tion is basically dominated by the third and fourth excitonicstates since their energetic separation is smaller than the en-ergy difference between the third and the second states. Theinfluence of the fourth state is reduced when the pulse dura-tion is increased to 2.5 ps. Finally, for a 4.5 ps pulse thecoherently excited density shows no dynamics but just afixed shape which is determined by the wave function of thethird exciton resonance, cf. Fig. 6. A similar analysis can be performed by monitoring the time dependence of the spatially integrated optical polariza-tion for different widths of the incident laser pulses. Themodulus of the polarization when exciting with a 1 ps pulse,Fig. 8 sad, shows modulations with a few frequencies. For the 2 ps pulse, Fig. 8 sbd, and more clearly for the 2.5 ps pulse, Fig. 8 scd, the modulations are dominated by a single fre- quency, i.e., in these cases only two excitonic transitionscontribute strongly to the polarization. For the long 4.5 pspulse, Fig. 8 sdd, only a single exciton resonance contributes and consequently the polarization shows no modulations asfunction of time. Note that the very slow decay of the polar-ization is due to the radiative decay which is included in ourself-consistent solutions of the MSBE. Such investigations have also been performed for the holes. The results are qualitatively similar to the ones ob-tained for the electrons. However, due to the bigger effectivemass of the holes they are more strongly confined in thepotential valley and therefore their density shows somewhatmore pronounced maxima and minima. FIG. 6. Coherent electron densities calculated using Eq. s38d and the polarization eigenfunctions shown in Fig. 4 sbd. On the left side the energetic positions of the resonances are shown. The ver-tically displaced lines indicate zero density. FIG. 7. Contour plots showing the spatiotemporal dynamics of the coherent electron density n11es2d;coh, see Eq. s38d, along one quan- tum wire unit cell.The system is excited by Gaussian pulses of 1 pssad,2p s sbd, 2.5 ps scd, and 4.5 ps sddduration sFWHM d, respec- tively. The central laser frequency has been tuned to the third exci-ton resonance " vL=EG−3.46EB. The dielectric slabs are W =90 nm wide and H=700 nm high. The calculations have been performed assuming a fully coherent situation, i.e., nonradiativedephasing and thermalization have not been consideredsT 1,T2*!‘d. White corresponds to the maximal density and black to zero density in each plot. FIG. 8. Temporal dynamics of the modulus of the spatially in- tegrated macroscopic optical polarization for resonant excitation ofthe third exciton, i.e., " vL=EG−3.46EB, using Gaussian laser pulses of durations of 1 ps sad,2p s sbd, 2.5 ps scd, and 4.5 ps sdd sFWHM d, respectively. The system parameter are the same as in Fig. 7.PASENOW et al. PHYSICAL REVIEW B 71, 195321 s2005 d 195321-12D. Wave packet dynamics with dephasing and relaxation Exciting semiconductor heterostructures with a short op- tical laser pulse produces a coherent optical polarization inthe material. With increasing time this polarization decaysdue to a variety of dephasing processes. Depending on therelevant physical mechanisms and the excitation conditions,typical dephasing times can vary between several picosec-onds or just a few femtoseconds. Radiative decay due to thefinite lifetime of the excited states always contributes to thedecay of the optical polarization. However, in semiconduc-tors the dephasing is typically dominated by the interactionwith phonons, by the many-body Coulomb interaction, orsometimes by disorder. 43Simultaneous with the dephasing of the polarization, the initially coherently excited carrier distri-butions change their nature and gradually become incoher-ent. Due to the interaction with phonons and Coulomb scat-tering among the electrons and holes, these incoherentpopulations approach thermal quasi-equilibrium distributionswith increasing time. For spatially homogeneous systems, it is possible to de- scribe dephasing and relaxation at a microscopic level. 13,43 For spatially inhomogeneous systems such an analysis ismuch more complicated. For example, the evaluation ofCoulomb scattering processes in the presence of disorder arecomputationally very demanding and can be performed onlyfor very small systems, see, e.g., Ref. 44. Therefore, we de-scribe these processes here on a phenomenological level. Asoutlined in Sec. II D 2, the nonradiative decay of the polar- ization is modeled by a dephasing time T 2*and the carrier populations approach a quasi-equilibrium Fermi–Dirac dis-tribution within the relaxation time T 1. Figure 9 sbdshows the electron and hole quasi-equilibrium densities, i.e., n11e;eqandn11h;eq, for a small average density of n0=0.001/aBsdotted line dat a temperature of 50 K. Since the thermal energy kBT<1EBis smaller than the depth of the single-particle potential dVsr1,r1d/2of <2EB, see Fig. 9 sad, the quasi-equilibrium distributions are strongly concentratedin the regions of low potential energy, i.e., underneath the dielectric slabs. Due to the higher effective mass of the holes,the spatial variation of the hole distribution is steeper thanthat of the electron distribution in the regions underneath thedielectric–air interfaces s±45 nm in Fig. 9 dwhere the single- particle potential changes rapidly. The computed spatiotemporal dynamics of the electron density including dephasing and relaxation is displayed inFig. 10. Compared to Fig. 5 we have used the same structuralparameters and excitation conditions and only included theincoherent processes by using relaxation and nonradiative dephasing times of T 1=T2*=6 ps. These values are reason- able considering that excitonic transitions are excited withpulses of weak intensities. Figure 10 shows that immediatelyafter the excitation the densities exhibit signatures of coher-ent wave packet dynamics, similar to Fig. 5. Due to relax-ation and dephasing, this wave packet dynamics is dampedwith increasing time. In the limit of long times, i.e., t @T 1,T2*, the electron population approaches a quasi- equilibrium Fermi–Dirac distribution. Therefore, regardlessof the excitation conditions which determine the position ofthe initially created density, the electrons eventually accumu-late in the regions of low potential energy, i.e., underneaththe dielectric slabs, cf. Fig. 9. This localization of the carrierdensities makes such structures interesting for possible appli-cations in laser structures.The fact that in the regions of highdensity, population inversion can be reached at a lower over-all density can lead to a reduction of the laser threshold. 26,33 FIG. 9. sadSingle-particle potential dV11/2 and sbdspace- dependent quasi-equilibrium electron ssolid dand hole sdashed dcar- rier densities, i.e., n11e/h;eq, for an average density of n0=0.001/aB sdotted dat a temperature of 50 K. The width of the dielectric slabs is 90 nm. FIG. 10. Contour plots showing the spatiotemporal dynamics of the coherent electron density n11es2d, see Eq. s40d, along one quantum wire unit cell. The system is excited by a Gaussian pulses of 2 psduration sFWHM d. The central frequencies of the pulses, vL, are tuned to the four lowest exciton states: sad"vL=EG−4.05EB,sbd "vL=EG−3.83EB,scd"vL=EG−3.46EB, and sdd"vL =EG−3.16EB, respectively. The dielectric slabs are W=90nm wide and H=700 nm high. Nonradiative dephasing and relaxation processes have been included using T1=T2*=6 ps at a temperature of 50 K. Except for the dephasing and relaxation, the material pa-rameters and the excitation conditions are the same as in Fig. 5.White corresponds to the maximal density and black to zero densityin each plot.EXCITONIC WAVE PACKET DYNAMICS IN … PHYSICAL REVIEW B 71, 195321 s2005 d 195321-13IV. SUMMARY It has been shown that surface polarizations at the inter- faces between different dielectric materials may significantlymodify the Coulomb interaction in their vicinity.As a result,both the band gap energy and the electron–hole attractionvary periodically in space near a photonic crystal. The modi-fied Coulomb interaction leads to characteristic signatures inthe excitonic absorption spectra of semiconductor photonic-crystal structures. In particular, the 1 s-exciton resonance splits into a certain number of resonances with spatially in-homogeneous eigenfunctions. Coherent excitation of the excitonic resonances leads to an intricate wave packet dynamics of the carrier distribu-tions. This dynamics depends very sensitively on the centralfrequency of the laser pulses and also on their spectral width,since these parameters determine the strength with which aparticular transition contributes to the coherent superposi-tion. By using long laser pulses which are tuned to a specificexcitonic resonance, carrier distributions which reflect theshape of a particular exciton wave function can be generated. Due to dephasing and relaxation processes, the coherent wave packet dynamics is visible only in a certain time win-dow after the optical excitation. With increasing time, the oscillations of the carrier densities are damped and in thelimit of long times the carriers approach spatially inhomoge-neous quasi-equilibrium distributions. This means that thecarriers accumulate at the regions of low potential energy,i.e., underneath the dielectric slabs for the structures consid-ered here. Further investigations of the optoelectronic prop-erties of semiconductor photonic-crystal structures in thepresence of spatially inhomogeneous carrier distributions areplanned. In particular, investigations of possible superradiantlight emission in structures with periodically varying spacedependent densities would be very interesting. ACKNOWLEDGMENTS This work is supported by the Deutsche Forschungsge- meinschaft sDFG dthrough the Schwerpunktprogramm “Pho- tonische Kristalle” and by the Center for Optodynamics,Philipps University, Marburg, Germany. T.M. thanks theDFG for support via a Heisenberg fellowship sME 1916/1- 1d. 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PhysRevB.32.7561.pdf

PHYSICAL REVIEW B VOLUME 32,NUMBER 11 1DECEMBER 1985 Near-surface defectprofiling withslowpositrons: Argon-sputtered Al(110) A.Vehanen, J.Makinen, P.Hautojarvi, H.Huomo, andJ.Lahtinen Laboratory ofPhysics, Helsinki University ofTechnology, SF-02150 Espoo,Finland R.M.Nieminen andS.Valkealahti Department ofPhysics, University ofJyvaskyla, SF-40100 Jyvaskyla, Finland (Received 30May1985) Wereportonslow-positron measurements ofatomicdefectdistribution nearasolidsurface. Defects are produced byargon-ion bombardment ofanAl(110) surface inultrahigh vacuum, Defectprofiles havea0 0 typicalwidthof15—25Aandcontain abroader tailextending to50-100A.Thedefectdensityattheouter- mostatomiclayerssaturates athighargonfluences toafewatomicpercent, depending onsputtering condi- tions.Defectproduction rateat)1keVArenergies istypically 1—5vacancy-interstitial pairsperincident ion.Molecular-dynamics simulations ofthecollision cascade predictsimilardefectdistributions. Low-energy ionbombardment ofsolidsurfaces isexten- sivelyusedforsamplepreparation andmodification insur- facescience andtechnology. Whiletheyieldanddistribu- tionofsputtered atomsandionshavebeenwidelystudied, andimplantation profilesoftheprojectile atomscanbemea- sured, littleisknownaboutproduction anddistribution of pointdefects andtheiragglomerates afterlow-energy parti- cleirradiation. Defects haveamajoreffectonnear- surface material properties, andtheirannealing afterirradia- tionisoftenprerequisite forsurfacescienceexperiments. Wehavedeveloped anexperimental method toobtain quantitative information onthedistribution ofdefectsnear asolidsurface. InthisRapidCommunication wedemon- strateittomeasure defectprofiles inaluminum after argon-ion bombardment undervarious conditions. The method isbasedondetecting theprobabilityJofathermal- izedpositron todiffusebacktothesurface, asafunctionof thepositron implantation energyE.Thepositron mobility is affected6 bylatticedefects, whichcantrapafreelydiffusing positron, thusreducingJ.TheshapeofJvsEcurvesis analyzed toyieldspatialdistribution ofthevacancy-type de- fects,whicharecapabletotrappositrons. Alsotheannihilation characteristics ofpositrons inside thesample areaffected byionbombardment, asdemon- strated byTriftshauser andKogelinHe-irradiated nickel sample. Theexperimental facilityisavariable-energy (E=0to30 keV)positron beams withabasepressure of3nPa. Monoenergetic positrons (2X10 sec',SE+4eV)strike thesolidtargetandrapidly thermalize withafairlywell- known implantation profileP(z,E),h6aving ashapeclose toaderivative ofaGaussian function. Asubsequent dif- fusivemotionofthepositron results in(i)positron annihi- lationfromthedelocalized (freelydiffusing) state,(ii)trap- pingandannihilation atadefectsiteinsidethesample, and (iii)diffusion backtothesurface withaprobability J.'o"A varietyofprocesses takeplaceuponthepositron returning tothesurface,e.g.,formation andemission ofanorthoposi- tronium (o-Ps)atom.'oSinceo-Pscanonlyannihilate via 3yemission (contrary toallotherpositron states,which disintegrate by2yemission6),Jcanbereadilyobtained by measuring the3y/2yratiowithaGedetector."" TheAl(110) sample'3 wasprepared inaconventional way,whichincluded aheating upto800Kand(atlater stages) sputtering withlow-energy Ar+.Itwasanalyzedwithlow-energy electron diffraction andretarding field Auger-electron spectroscopy measurements. Acharacteris- ticlow-energy electron-diffraction pattern wasfoundand Auger-electron spectroscopy showed &1%ofamonolayer Candtrace0contamination. Thepressure throughout the experiments was&20nPa.Ar+sputtering wasperformed with—10p,A/cmioncurrent intheenergyrange0.1-3 keVandincident angle8from0'(normal) to75'.After eachsputteringJ(E)wasmeasured. Fromthemeasured backdiffusion probabilityJ(E)we calculate thequantity'4 K(E)=1—Jdcf(E)/Jb„w, (E), 0.8 O U06Ar'/AI(110) 3keV SS~~8 1.6 0 &tln U~Cl ~~e0.8 0.4 0 00 0 positron energy E(keV) FIG.-1.Experimental positron trapping fractionsKasafunction ofthepositron implantation energyEinAl(110) singlecrystalafter argon-ion bombardment. TheAr+fluence was—10cmand theincident angle8=75'. Ionenergies areindicated inthefigure. Thesolidlinesarebestfitstothepositron diffusion-annihilation equation inthepresenceofaspatialdefectdistribution.where Jb„il,andJdefareresults fromsamples before (nodefects) andaftersputtering, respectively. Thus K(0~K~1)isameasure oftherelative fractionofposi- tronstrapped bylatticedefects. Figure 1showsexamples of KvsEcurvesafterbombarding withvariousAr+energies 327561 1985TheAmerican Physical Society 7562 A.VEHANEN etal. atanincident angle8=75'. Thecurveshavebeenmea- suredunderconditions (Ar+fluence &5&10tscmz), wherethetrapping fractionK(E)becomes independent of theAr+fluence. Obviously thiscorresponds toadynamic equilibrium, wheretheyieldofsputtered atomsandthenet production rateoftheirradiation-induced defectsareequal. Thetrapping fractionKinFig.1increases strongly atlowE andapproaches aconstant levelatE—1keV.Thisvalue ofKcorresponds tohigh-energy positrons, whichareim- planted farbeyond" thedefected overlayer. Consequently, theyhaveanoverall(independent ofE)probability todif- fusethrough thedamaged regiontothesurface. Thusthe valueofKatE&1keVisaroughmeasure ofthetotal numberofdefects,'6whiletheshapeofKvsEcontains in- formation ontheshapeofthedefectdistribution. Toobtaindetailed knowledge ondefectprofiles wehave fittedtheK(E)datatosolutions ofthequasistationary' diffusion-annihilation equation forthepositron density n(z,E):30 N V ~2Q 10 1.5 CO C3~10aAr'IAl(110) e~g=&5 aot)=75 D+V'zn(zE)—itsn(zE)—p,c(z)n(zE)+P(zE)=0. (2) Here,D+(=0.5cm'/sec) isthepositron diffusion coeffi- cient,Xs(=6nsec)annihilation rateofafreepositron, p, thespecific trapping rateintodefects,sandc(z)thedefect concentration profile. Wehavesolved's Eq.(2)byitera- tion:Theprofilec(z)isfirstguessed andn(z,E)iscalcu- latednumerically. SinceJ(E)isproportional totheposi- trondiffusion current emerging fromthesurface J(E)= I—D+&n(0,E)I, properly normalized valuesofK(E)canbeobtained from Eq.(1).Finally,c(z)isvarieduntilbestfittothedatais found.Theprofiles wereparametrized intermsofaGauss- ianfunction followed byanexponential tail.Thefitstothe experimental dataareshownassolidlinesinFig.1,and theyhavetypicalpvaluesrangingfrom0.9to1.2. ThefittedKvsEcurvesare'ratherindependent ofthe parameters inthefittingprocedure, e.g.,coefficients describing theimplantation profileP(z,E).Byvirtueofits definition [Eq.(1)),Kisalsoratherinsensitive totheun- certainties inmeasuringJ(E)fromthe3y/2yratio(o-Ps fraction)."Theresulting profilesc(z)typically contain a narrow (—15A)peaknearthesurface followed bya broader shoulder extending downto50-100A. Figure2showsthetotalnumberofdefects perunitarea andthemeandepthofdefectsforsamples sputtered at roomtemperature withahighAr+fluenceofvaryingener- gywithincident anglesof8=45' and75'.Wechosethe specific trapping rateinEq.(2)tobep,=5X10'4 sec whichisthevaluefoundformonovacancies inAl.Because vacancies inaluminum canmigrate freelyatroomtempera- ture,'thedefectsexpected totrappositrons aresmalltwo- andthree-dimensional vacancy clusters andvacancy-argon complexes. Sincep,isexpected tobeproportional tothe numberofvacancies inasmallcluster,"ourchoiceofp,is therefore ameasureofthetotalnumberofvacancies associ- atedwiththesputtering-damage cascades. Thetotalnumberofdefectsperunitareadepicted inFig. 2depends strongly onthesputtering energy andangle, whereas themeandepthofdefects,(z),hasamuchweak- erdependence onsputtering conditions. Theintegrated de- fectdensity inFig.2increases linearly withAr+energy, in0.5 C3Q 1 Ar'energy(kev) FIG.2.Experimental (circles) andsimulated (squares) charac- teristicsofdefectprofiles inargon-bombarded Al(110) asafunction oftheAr+energyattwodifferent incident angles.Theintegralof thedefectdistribution (defects perunitarea)andthemeandepthof theprofile(z)areshown. Theintegralsofthesimulated vacancy profiles arescaledbyaconstant factor. agreement withtheclassical modelofKinchin andPease.' Defectproduction isnegligible belowAr+-ion energies of around150eV.Valuesofthetotalnumberofdefects(per unitarea)typically correspond toaboutoneemptyatomic layer,whilethelocaldefectconcentration intheoutermost layersisoftheorderofafewatomicpercent. Thelatter valueischaracteristic forthedensityofvacancy-type defects inamorphous metals.' Togainmoreinsightintothegeneration ofatomicdefects during low-energy sputtering, wehavesimulated the sputtering damage withfull-scale molecular-dynamics com- putercalculations.'sThesubstrate wasa(110)filmof70 layersofAlatomsinteracting via(Morse) pairpotentials. Theparameters ofthepotential werechosen togivea correct (bulk)cohesion energy, compressibility andlattice constant. Periodic lateralboundary conditions wereimposed onthebasiccomputational unitcellcontaining 1400atoms. Thesputtering particle(Ar+)withachosen energyandin- cidentanglewasallowed tostrikethesurface. TheAr-Al potential wasoftheMoliere form. Thetrajectories ofthe projectile andsubstrate atomswerecalculated indiscrete timesteps(At=10tssec).Toallowforelectronic slowing down,friction termsoftheformdE/dz=—k;Eswerein- troduced intotheequations ofmotionoftheatoms,i.e., electronic slowing downproportional toionvelocities'5 was assumed. Valuesfork;wereadapted fromarecent calcu- lationbothforArandAlionsmoving inmetallic Al.The simulation continued untilalltheatomshadslowed down NEAR-SURFACE DEFECT PROFILING WITHSLOW... 7563 belowthedisplacement threshold ofabout10eV.Toac- countforshort-term recovery, arecombination radiusof4 Aforvacancy-interstitial pairswasassumed inanalyzing the damage distribution. Allothervacancies wereincluded in counting thedefectprofiles forvarious sputtering condi- tions.Theresulting integral andmeandepth(z)ofthecal- culated profiles areshownforsomecasesinFig.2. (squares). Theagreement withtheexperimental datais good,although thecalculated profiles tendtobeshallower atverylowAr+energies andhigh-incident angles. Figure3showsatypicalprofilec(z),determined bothex- perimentally andtheoretically, whichcorresponds toanAr+ energyof400eVandincident angleof8=25'.Thesimu- latedvacancy profilehasapronounced peakattheouter- mostlayerswithatailextending to100A.Theshapeof thesimulated profile isinsatisfactory agreement withthe experimentally deduced distribution. Thesimulated inter- stitialatomprofiles (notshown) havearelatively lower peaknearthesurface. Thecalculated sputtering yieldis0.8 atoms/Ar, whichisalsoingoodagreement withthemea- suredlow-energy yields.' Thesimulated defectproduction rate(400eV,25')is aboutsixvacancy-interstitial pairspereachAr+ion.The simulated vacancy profileresultsfromasingleAr+ion, whereas theexperiments aredoneatanequilibrium state4 (—10"Ar+/cm'), wherethedefectprofile isindependent oftheAr+fluence. Therefore theintegrals oftheprofiles inFigs.2and3cannotbedirectly compared totheexperi- ments. Consequently, wehaveusedaconstant scalingfac- torforthesimulated defectdensities inFig.2.Wehave alsoperformed measurements withuseoflowerAr+flu- ences.Thedefectproduction rateduring750eVargon sputtering at8=45'wasestimated tobearound three vacancy-type defects perincident ion.Comparison ofthese resultswiththecomputations allowsustogetaroughidea ofthelong-term recombination probability, whichobviously isresponsible forthesmallerareasoftheexperimentally de-U 60 O C0 C2U OO o0 I 020I I 4060 Depthz(A)80100+40~ C3C) 20~ V) O 0o C3O FIG.3.(a)Experimental and(b)simulated defectprofiles in Ar+-sputtered Al(110). Theexperimental Arfluences (—10 cm)correspond toaregimeofdynamic equilibrium, wherethe profiles arefluence independent. Thesimulations areperformed with100incidentAr+ions,Eachbarinthegraphincludes twolat- ticeplanes. duceddefectprofiles. Inconclusion, wehaveforthefirsttimeobtained quanti- tativeinformation onatomic-scale defectdistribution neara single-crystal surface. AnAl(110) surface sputtered with low-energy argonionscontains uptoafewatomicpercent vacantlatticesitesattheoutermost layers,andthedefect distribution extends downto50-100A.Themethod utiliz- ingtheuniqueproperties ofslowpositron beamscanbeex- tendedtoanalyze similarproblems associated, e.g.,within- terfacial disorder, overlayer structures, anddefects inmul- tilayercomponents. Suchworkiscurrently inprogress. Sputtering byParticle Bombardment I,editedbyR.Behrisch, Topics inApplied Physics, Vol,47(Springer-Verlag, NewYork,1981). A.Zomorrodian, S.Tougaard, andA.Ignatiev, Phys.Rev.B30, 3124(19S4). R.Miranda andJ.M.Rojo,Vacuum 34,1069(1984). 4L.K.Verheij,J.A.VandenBerg,andD.G.Armour, Surf.Sci. 122,216(1982). 5B.Poelsema, L.K.Verheij, andG.Comsa, Phys.Rev.Lett.53, 2500(1984). Positrons inSolids,editedbyP.Hautojarvi, Topics inCurrent Phy- sics,Vol.12(Springer-Verlag, NewYork,1979);Positron Solid StatePhysics, edited byW.Brandt andA.Dupasquier (North- Holland, Amsterdam, 1983). W.Triftshauser andG.Kogel,Phys.Rev.Lett.48,1741(1982). 8A.Vehanen, J.Lahtinen, H.Huomo, J.Makinen, andP. Hautoj'irvi, inPositron Annihilation, Proceedings oftheSeventh International Conference onPositron Annihilation, NewDelhi, 6-11January1985,editedbyP.C.Jain,M.R.Singru, andK.P. Gopinathan (WorldScientific, Singapore, 19S5). S.Valkealahti andR.M.Nieminen, Appl.Phys.A32,95(1983). A.P.Mills,Jr.,Phys.Rev.Lett.41,1828(1978). K.G.LynnandD.O.Welch,Phys.Rev.B22,99(19SO). P.J.Schulz,K.G.Lynn,andH.H.Jorch,inProceedings oftheIn- ternational 8'orkshop onSlowPositrons inSurface Science, Pajulahti, Finland, June1984,editedbyA.Vehanen, Laboratory ofPhysics ReportNo.135,1984(Helsinki University ofTechnology, Hel- sinki,Finland).'Kominco Co.,purity99.9999+%, mosaicspread&0.2',residual resistivity ratio&40.000. A.Vehanen, J.Makinen, P.Hautojarvi, andP.Huttunen, inRef. 8. Themeanpositron penetration depthzvariesinaluminum as z=145A(E/keV)'; seeRef.10. Morespecifically, thelevelofEatE&1keVisrelatedtothein- tegrated defectdensity weighted withthedistance measured from thesurface, viz.fc(z)zdz,seeRef.18. 0'R.M.Nieminen, J.Laakkonen, P.Hautojarvi, andA.Vehanen, Phys.Rev.B19,1397(1979). J.Makinen, A.Vehanen, P.Hautojarvi, H.Huomo,J.Lahtinen, R.M.Nieminen, andS.Valkealahti (unpublished),'R.W.Balluffi,J.Nucl.Mater.69470,240(1978). Smallinterstitial agglomerates arenotexpected toactaspositron traps,see,e.g.,Ref.6. R.M.Nieminen andJ.Laakkonen, Appl.Phys.20,181(1979). G.H.Kinchin andR.S.Pease,Rep.Prog.Phys.18,1(1955). 23A.Vehanen, K.G.Lynn,P.J.Schultz,E.Cartier, H.-J. Guntherodt, andD.M.Parkin, Phys.Rev.B29,2371(1984). C.Varelas, andJ.Biersack, Nucl.Instrum. Methods 79,213 (1970). E.FermiandE.Teller,Phys.Rev.72,399(1947). M.J.PuskaandR.M.Nieminen, Phys.Rev.B27,6121(1983). Inthesimulations, theirradiation damage iscalculated andcount- edaftereachincident Ar+ion,whereafter aperfect latticeis resumed.

PhysRevB.98.075420.pdf

PHYSICAL REVIEW B 98, 075420 (2018) Buildup and transient oscillations of Andreev quasiparticles R. Taranko and T. Doma´ nski* Institute of Physics, M. Curie Skłodowska University, 20-031 Lublin, Poland (Received 31 May 2017; revised manuscript received 25 May 2018; published 21 August 2018) We study transient effects in a setup where the quantum dot is abruptly sandwiched between the metallic and superconducting leads. Focusing on the proximity-induced electron pairing manifested by the in-gap boundstates, we determine characteristic timescales needed for these quasiparticles to develop. In particular, we deriveanalytic expressions for (i) charge occupancy of the quantum dot, (ii) amplitude of the induced electron pairing, and(iii) the transient currents under equilibrium and nonequilibrium conditions. We also investigate the correlationeffects within the Hartree-Fock-Bogolubov approximation, revealing a competition between the Coulombinteractions and electron pairing. DOI: 10.1103/PhysRevB.98.075420 I. INTRODUCTION Quantum impurity hybridized with any superconducting bulk material is influenced by the Cooper pairs which leakinto its region, developing the quasiparticle states in thesubgap spectrum |ω|/lessorequalslant/Delta1(where /Delta1is the energy gap of superconducting reservoir) [ 1,2]. These Andreev (or Yu-Shiba- Rusinov) states have been observed in numerous scanningtunneling microscopy studies, using impurities deposited onsuperconducting substrates [ 3,4] and in tunneling experiments via quantum dots (QDs) arranged in the Josephson [ 5], Andreev [6], and more complex (multiterminal) configurations [ 7,8]. Since measurements can be nowadays done with state-of-the-art precision probing the time-resolved properties, we addressthis issue here and determine some characteristic temporalscales of the in-gap quasiparticles. Any abrupt change of the model parameters ( quantum quench ) is usually followed by a time-dependent thermal- ization of the many-body system, where continuum statesplay a prominent role [ 9]. Dynamics of these processes has been recently explored in the solid state and nanoscopicphysics [ 10]. From a practical point of view, especially useful could be nanoscopic heterostructures with the corre-lated QD embedded between external (metallic, ferromag-netic, or superconducting) leads, which enables measure-ments of the transport properties under tunable nonequilibriumconditions [ 11]. Transport phenomena through QD coupled between the normal or superconducting leads have been so far exploredpredominantly in the static cases. Since new experimentalmethods allow us to study the QDs subjected to voltage pulsesor abrupt changes of the system parameters, it would be verydesirable to calculate the time-dependent currents and theirconductances. In particular, one can ask the question: How fast does the QD respond to an instantaneous perturbation ? For this purpose, analytical estimation of the transient oscillations andlong-time (asymptotic) behavior of the measurable quantities *doman@kft.umcs.lublin.plwould be very useful. Some early theoretical works haveinvestigated time-dependent transport via QD between thenormal and superconducting leads [ 12–15], however, analytic results are hardly available. As regards the QD coupled toboth normal leads, the transient current and charge occu-pancy have been determined for abrupt voltage pulses orafter an instantaneous switching of constituent parts of thesystem [ 16–32]. Time-resolved techniques could provide an insight into the many-body effects. For instance, the pump-and-probeexperiments [ 33] and the time-resolved angle resolved photo- emission spectroscopy [ 34] have determined the lifetime of the Bogoliubov quasiparticles in the high-temperature su-perconductors. Transient effects have been investigated innanoscopic systems, considering mainly the QDs hybridizedwith the conducting (metallic) leads. There has been studied thetimescale needed for the Kondo peak to develop at the Fermienergy [ 35], dynamical correlations in electronic transport via the QDs [ 36], or oscillatory behavior in the charge transport through the molecular junctions [ 37]. Dynamical phenomena of the QDs attached to supercon- ducting bulk reservoirs have been studied much less inten-sively. There have been analyzed: photon-assisted Andreevtunneling [ 38], response time on a steplike pulse [ 39], temporal dependence of the multiple Andreev reflections [ 40], time- dependent sequential tunneling [ 41], transient effects caused by an oscillating level [ 42], time-dependent bias [ 43], waiting time distributions in nonequilibrium transport [ 44–47], short- time counting statistics [ 48–50], metastable configurations of the Andreev bound states in a phase-biased Josephson junction[9,51], finite-frequency noise [ 52], superconducting proximity effect in interacting QD, and double-dot systems [ 53,54]. None of these studies, however, addressed the timescale typical fordevelopment of the subgap quasiparticle states in a setup,comprising the QD coupled to the normal lead ( N) on one side and to the isotropic ( s-wave) superconductor ( S)o n the other side. Our present study reveals that a continuouselectronic spectrum of the metallic lead enables a relaxationof the Andreev states, whereas the superconducting electrodeinduces the (damped) quantum oscillations with a period 2469-9950/2018/98(7)/075420(16) 075420-1 ©2018 American Physical SocietyR. TARANKO AND T. DOMA ´NSKI PHYSICAL REVIEW B 98, 075420 (2018) FIG. 1. Schematics of the setup, comprising the quantum dot (QD) coupled to the normal ( N) and superconducting ( S) electrodes. Sudden coupling to the continuum states triggers the relaxationprocesses, whereas the superconductor induces the in-gap bound states, giving rise to quantum oscillations. sensitive to the energies of the in-gap quasiparticles. In what follows, we evaluate the timescale at which such Andreevquasiparticle start to form, and another when they are finallyestablished. The paper is organized as follows. In Sec. II,w ei n t r o - duce the microscopic model and discuss the method for thetime-dependent phenomena. Section IIIpresents analytical results for the uncorrelated QD, such as (i) charge occupancy,(ii) complex order parameter, and (iii) charge current for theunbiased and biased heterojunction. In Sec. IV, we discuss the correlation effects and, finally, in Sec. Vwe summarize the main results. II. MICROSCOPIC MODEL For a description of the N-QD- Sheterostructure (see Fig. 1), we use the single impurity Anderson Hamiltonian ˆH=/summationdisplay σεσˆd† σˆdσ+Uˆn↑ˆn↓+/summationdisplay β(ˆHβ+ˆVβ−QD),(1) where βrefers to the normal ( N) and superconducting ( S) electrodes, respectively. As usual, ˆdσ(ˆd† σ) is the annihilation (creation) operator for the QD electron with spin σand energy εσ. Potential of the Coulomb repulsion between the opposite spin electrons is denoted by U. We treat the external metallic lead as free fermion gas ˆHN=/summationtext k,σεkˆc† kσˆckσ, and describe the isotropic superconductor by the BCS model ˆHS=/summationtext q,σεqˆc† qσˆcqσ−/summationtext q/Delta1(ˆc† q↑ˆc† −q↓+ˆc−q↓ˆcq↑), where εk(q)is the energy measured from the chemical potential μN(S), and /Delta1denotes the superconducting energy gap. Hybridization between the QD electrons and the metallic lead is givenbyˆV N−QD=/summationtext k,σ(Vkˆd† σˆckσ+H.c.) and ˆVS−QDcan be ex- pressed by interchanging k↔q. Since our study refers to the subgap quasiparti- cle states, we assume the constant couplings /Gamma1N(S)= 2π/summationtext k(q)|Vk(q)|2δ(ω−εk(q)). In the deep subgap regime |ω|/lessmuch/Delta1(the so-called superconducting atomic limit), the coupling /Gamma1S/2 can be regarded as a qualitative measure of the induced pairing potential, whereas /Gamma1Ncontrols the inverse lifetime of the in-gap quasiparticles. As we shall see, boththese couplings play important (though quite different) rolesin transient phenomena. We assume that all three constituents of the N-QD- Shet- erostructure are disconnected from each other until t/lessorequalslant0. Let us impose the external ( N, S) reservoirs to be suddenly coupledto the quantum dot V k(q)(t)=/braceleftbigg 0f o r t/lessorequalslant0, Vk(q)fort>0,(2) inducing the transient effects. Later on, we shall relax this assumption. Our problem resembles the Wiener-Hopf method[55] applied earlier in the studies of x-ray absorption and emission of metals [ 56]. In what follows, we explore the time-dependence of phys- ical observables ˆO, based on the Heisenberg equation of motion i¯h d dtˆO=[ˆO,ˆH]. In particular, we shall investigate expectation values of the QD occupancy /angbracketleftˆd† σ(t)ˆdσ(t)/angbracketright,t h e induced on-dot pairing /angbracketleftˆd↓(t)ˆd↑(t)/angbracketright, and the transient charge currents flowing between the QD and external electrodes (bothunder equilibrium and nonequilibrium conditions). Our strategy is based on the following three steps: First, we formulate the differential equations of motion for the annihila-tion ˆd σ(t) and creation ˆd† σ(t) operators of QD and similar ones for the mobile electrons ˆck(q)σ(t) and ˆc(†) k(q)σ(t), respectively. Next, we solve them using the Laplace transformations, e.g.,forˆd σ(t) we denote ˆdσ(s)=/integraldisplay∞ 0e−stˆdσ(t)dt≡L{ˆdσ(t)}(s). (3) For the uncorrelated QD, the analytical expressions for ˆdσ(s) and ˆd† σ(s) can be obtained (see Appendix A). Finally, using the corresponding inverse Laplace transforms, we compute thetime-dependent expectation values of the QD occupancy, theQD pair amplitude, and currents flowing between QD and bothleads. For example, QD occupancy n σ(t)≡/angbracketleftˆd† σ(t)ˆdσ(t)/angbracketrightis given by nσ(t)=/angbracketleftL−1{ˆd† σ(s)}(t)L−1{ˆdσ(s)}(t)/angbracketright, (4) where L−1{ˆdσ(s)}(t) stands for the inverse Laplace transform ofˆdσ(s). In our calculations, we make use of the wide-band-limit ap- proximation ( /Gamma1β=const) and set e=¯h=kB≡1, so that all energies, currents, and time are expressed in units of /Gamma1S,e/Gamma1S/¯h, and ¯h//Gamma1S, respectively. We also treat the chemical potential μS=0 as a convenient reference energy point and perform the calculations for zero temperature. For experimentally availablevalue/Gamma1 S∼200μeV [ 57–59], the typical times and current units would be ∼3.3p s e ca n d ∼48 nA, respectively. III. UNCORRELATED QD CASE We start by addressing the transient effects of the uncorre- lated quantum dot ( U=0), focusing on the superconducting atomic limit ( /Delta1=∞ ) for which analytical expressions can be obtained. More general considerations are presented inAppendix A. A. Time-dependent QD charge Let us inspect the time-dependent occupancy nσ(t)d r i v e n by an abrupt coupling of the QD to both external leads. Thisquantity, defined in Eq. ( 4), can be determined explicitly for arbitrary /Delta1(derivation is presented in Appendix A). Here we shall consider the formula Eq. ( A12) simplified for 075420-2BUILDUP AND TRANSIENT OSCILLATIONS OF ANDREEV … PHYSICAL REVIEW B 98, 075420 (2018) the superconducting atomic limit: n↑(t)=e−/Gamma1Nt/braceleftBigg n↑(0)+[1−n↑(0)−n↓(0)] sin2/parenleftBigg√ δ 2t/parenrightBigg /Gamma12 S δ/bracerightBigg +/Gamma1N 2π/integraldisplay∞ −∞dω f N(ω)L−1/braceleftbiggs+iε−σ+/Gamma1N/2 (s−s1)(s−s2)(s−iω)/bracerightbigg (t)L−1/braceleftbiggs−iε−σ+/Gamma1N/2 (s−s3)(s−s4)(s+iω)/bracerightbigg (t) +/parenleftbigg/Gamma1S 2/parenrightbigg2/Gamma1N 2π/integraldisplay∞ −∞dω[1−fN(ω)]L−1/braceleftbigg1 (s−s1)(s−s2)(s+iω)/bracerightbigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)(s−iω)/bracerightbigg (t),(5) where fN(ω) is the Fermi-Dirac distribution function of the normal lead and we defined auxiliary parameters s1,2=i 2[(ε↑−ε↓)+i/Gamma1N±√ δ], (6) s3,4=i 2[−(ε↑−ε↓)+i/Gamma1N±√ δ],s34, (7) δ=(ε↑+ε↓)2+/Gamma12 S. (8) The occupancy n↓(t) can be obtained from the same expression Eq. ( 5) upon replacing the set ( s1,s2,s3,s4)b y(s3,s4,s1,s2). Expressions given in the second and third lines of Eq. ( 5) could be presented in a more compact analytical form in the case εσ= 0[ s e eE q s .( A14)–(A16)]. In the general case, they are rather lengthy (even though accessible), therefore we skip them. Another simplification of Eq. ( 5) is possible upon neglecting the normal lead ( /Gamma1N=0). QD occupancy is then characterized by nonvanishing quantum oscillations: nσ(t)=nσ(0)+[1−nσ(0)−n−σ(0)] sin2/parenleftBigg√ δ 2t/parenrightBigg /Gamma12 S δ. (9) Forεσ=0, Eq. ( 9) reduces to nσ(t)=cos2/parenleftbigg/Gamma1S 2t/parenrightbigg nσ(0)+sin2/parenleftbigg/Gamma1S 2t/parenrightbigg [1−n−σ(0)], (10) implying the period of transient oscillations T=2π//Gamma1S, except of the initial conditions nσ(0)=1 and n−σ(0)=0 when the QD occupancy is preserved. The formula Eq. ( 10), obtained in the case /Gamma1N=0, re- sembles the Rabi oscillations of a typical two-level quantumsystem. Indeed, the proximitized QD is fully equivalent to suchscenario. To prove it, let us consider the effective Hamiltonian ˆH=/summationtext σεσˆnσ+/Gamma1S 2(ˆd† ↑ˆd† ↓+H.c.), assuming that at t=0t h e QD is empty n↑(0)=0=n↓(0). For arbitrary time t>0, we can calculate the probability P(t) of finding the QD in the doubly occupied configuration n↑(t)=1=n↓(t) within the standard treatment of a two-level system [ 60]. This probability is given by P(t)=/Gamma12 S (E1−E2)2+/Gamma12 Ssin2/parenleftbiggt 2/radicalBig (E1−E2)2+/Gamma12 S/parenrightbigg , (11) where E1=0 and E2=ε↑+ε↓are the energies of empty and doubly occupied configurations, respectively. This resultexactly reproduces our expression Eq. ( 10).For the QD suddenly coupled to both the normal and superconducting leads ( /Gamma1 N,S/negationslash=0), such oscillations become damped (see Fig. 2). This effect comes partly from the expo- nential factor exp( −/Gamma1Nt) appearing in front of the first term in Eq. (5) and partly from the second and third contributions. This can be illustrated by considering the case εσ=0,μN=0, for which Eq. ( 5) implies nσ(t)=e−/Gamma1Nt/braceleftbigg cos2/parenleftbigg/Gamma1S 2t/parenrightbigg nσ(0) +sin2/parenleftbigg/Gamma1S 2t/parenrightbigg [1−n−σ(0)]/bracerightbigg +1 2(1−e−/Gamma1Nt).(12) Under such circumstances, the QD occupancy approaches asymptotically a half-filling, lim t→∞nσ(t)=1 2. Figure 2dis- playsn↑(t) obtained in absence of external voltage for several values of /Gamma1N, assuming εσ=0 andnσ(0)=0 for both spins. The quantum oscillations occur with a period 2 π//Gamma1Sand their damping is governed by the envelope function e−/Gamma1Nt indicating that a continuous spectrum of the metallic lead is responsible for the relaxation processes. For a weak enoughcoupling /Gamma1 N, these oscillations could indirectly probe the dynamical transitions between the subgap bound states, asrecently emphasized by J. Gramich et al. [8]. Figure 3shows the QD occupancies obtained for several ini- tial conditions, assuming μ N=μS=0 andεσ=0. The case n↓(0)=0=n↑(0) allows quantum oscillations between two eigenstates of the proximitized QD, which are damped due to 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 60t [1/ΓS]n (t) nσ(t) t [1/ΓS]σ = σ = ΓN / ΓS = 0.1 0.25 0.5 1 0 0.5 1 0 20 40 FIG. 2. Time-dependent occupancy n↑(t)=n↓(t) obtained for εσ=0, assuming the initial occupancy n↑(0)=0=n↓(0) in absence of external voltage ( μN=μS=0). Different lines correspond to various ratios /Gamma1N//Gamma1S, indicated in the legend. Inset shows the QD occupancies nσ(t) for the finite Zeeman splitting ε↓−ε↑=/Gamma1S, assuming /Gamma1N//Gamma1S=0.1. 075420-3R. TARANKO AND T. DOMA ´NSKI PHYSICAL REVIEW B 98, 075420 (2018) 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 60t [1/ΓS]n (t) (1,0) (0,0) (0,1) FIG. 3. The time-dependent QD occupancy n↑(t) obtained in absence of external voltage for εσ=0,/Gamma1N=0.1/Gamma1S. Different curves refer to various initial occupancies ( n↑(0),n↓(0)) indicated in the legend. coupling to the normal lead (see Fig. 2). For the initial condition nσ(0)=1,n−σ(0)=0, the transient effects are completely different. The first term in Eq. ( 12)f o r(n↑(0),n↓(0))=(1,0)or (0,1) equals e−/Gamma1Ntor vanishes and together with the last term they yield1 2(1−e−/Gamma1Nt)—see the upper curve in Fig. 3 or1 2(1+e−/Gamma1Nt)—the lower curve, respectively. This stems from the fact that proximity-induced pairing affects only theempty and doubly occupied configurations and it is inefficientin the case considered here. In consequence, the quantumoscillations are absent and the QD occupancy exponentiallyevolves towards a half-filling. Let us also remark that for/Gamma1 S=0, Eq. ( 5) simplifies to the standard formula obtained by the nonequilibrium Green’s function method [ 61] [see Eq. ( A17)]. B. Development of the proximity effect Occupancy of the QD only indirectly tells us about emer- gence of the subgap bound states. To get some insight intothe superconducting proximity effect, we shall study here thetime evolution of the order parameter χ(t)=/angbracketleftˆd ↓(t)ˆd↑(t)/angbracketright.T h e general formula is explicitly given by Eq. ( A21). Expressing its first two terms (which depend on the initial QD occupancy)the pair correlation function can be written as χ(t)=/bracketleftBig (ε↑+ε↓)/parenleftBig 1−cos/parenleftBig√ δt/parenrightBig/parenrightBig +i√ δsin/parenleftBig√ δt/parenrightBig/bracketrightBig e−/Gamma1Nt/Gamma1S(n↑(0)+n↓(0)−1) 2δ−i/Gamma1N/Gamma1S 4π/Phi1∗ ↑, (13) where /Phi1σ=−/integraldisplay∞ −∞dεfN(ε)L−1/braceleftBigg s+iε−σ+/Gamma1N 2 (s−s1)(s−s2)(s−iε)/bracerightBigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)(s+iε)/bracerightbigg (t) +/integraldisplay∞ −∞dε[1−fN(ε)]L−1/braceleftbigg1 (s−s1)(s−s2)(s+iε)/bracerightbigg (t)L−1/braceleftBigg s+iεσ+/Gamma1N 2 (s−s3)(s−s4)(s−iε)/bracerightBigg (t). (14) In Appendix A, we show that for μN=0, the real part of/Phi1↑vanishes. Let us next analyze Eq. ( 13) for different initial conditions and values of the QD energy levels. Forn σ(0)=0,n−σ(0)=1 andμN=0 the function /angbracketleftˆd↓(t)ˆd↑(t)/angbracketright is real and nonoscillating in time and is equal to −/Gamma1N/Gamma1S 4πIm/Phi1↑, regardless of εσ. However, for μN/negationslash=0, also the imaginary part of /angbracketleftˆd↓(t)ˆd↑(t)/angbracketrightequals −/Gamma1N/Gamma1S 4πRe/Phi1↑and is nonoscillating function. For the initial conditions ( nσ(0),n−σ(0))=(0,0) or (1,1), the picture is completely different. Depending onthe value of ε ↑+ε↓, the real part of /angbracketleftˆd↓(t)ˆd↑(t)/angbracketrightoscillates forε↑+ε↓=0 or is a smooth function of time for ε↑+ ε↓/negationslash=0. Simultaneously, the imaginary part of the QD on-dot pairing oscillates irrespective of εσ. The oscillatory parts of /angbracketleftˆd↓(t)ˆd↑(t)/angbracketrightare dumped via e−/Gamma1Ntfactor, emphasizing the crucial role of continuum states of the normal electrode inrelaxation processes. In Fig. 4, we show the imaginary part of the on-dot pairing /angbracketleftˆd ↓(t)ˆd↑(t)/angbracketright, assuming the initial QD occupancy nσ(0)=0. Period Tof the damped quantum oscillations depends on the excitation energy between the subgap Andreev quasiparticles [8]v i aT=2π/√ (ε↓+ε↑)2+/Gamma12 S.F o r μN=0, these os- cillations are related to the transient current jSσ(t)fl o w i n g between the proximitized QD and the superconducting lead(see Sec. III C) in analogy to the Josephson junction comprising two superconducting pieces, differing in phase of the orderparameter. On the other hand, the real part (Fig. 5)e v o l v e s monotonously to its asymptotic value, except of one particularcase/Gamma1 N=0, when the real part of /angbracketleftˆd↓(t)ˆd↑(t)/angbracketrightvanishes. C. Transient currents for unbiased system So far, we have discussed the quantities which are important, but unfortunately they are not directly accessible experimen-tally. Let us now consider the measurable currents j Nσ(t) and jSσ(t), flowing from the QD to the external leads. Formally, the -0.4-0.2 0 0.2 0.4 0 10 20 30 40 60 t [1/ ΓS]- Im χ(t) ΓN / ΓS = 0.1 0.25 0.5 1 FIG. 4. The imaginary part of the induced on-dot pairing /angbracketleftˆd↓(t)ˆd↑(t)/angbracketrightobtained for the same parameters as in Fig. 2. 075420-4BUILDUP AND TRANSIENT OSCILLATIONS OF ANDREEV … PHYSICAL REVIEW B 98, 075420 (2018) 00.20.40.60.81 ΓN/ΓS 0 10 20 30 t [1/ΓS] 0 0.1 0.2 0.3 0.4 0.5−Re χ(t) FIG. 5. The time-dependent real part of /angbracketleftˆd↓(t)ˆd↑(t)/angbracketrightobtained for /Gamma1S=1 and the same parameters as in Fig. 2. transient current is defined by jβσ(t)=/angbracketleftdˆNβ(t) dt/angbracketright, where ˆNβ(t) counts the total number of electrons in electrode β=N,S . For instance, jNσ(t) simplifies to the standard formula [ 61] jNσ(t)=2I m/summationdisplay kVk/angbracketleftˆd† σ(t)ˆckσ(t)/angbracketright. (15)Assuming the energies of itinerant electrons to be static εkσ(t)=εkσ, one obtains ˆckσ(t)=ˆckσ(0)e−iεkσt−i/integraldisplayt 0dt/primeVke−iεkσ(t−t/prime)ˆdσ(t/prime),(16) and within the wide-band-limit approximation, it yields jNσ(t)=2Im/parenleftBigg/summationdisplay kVke−iεkt/angbracketleftˆd† σ(t)ˆckσ(0)/angbracketright/parenrightBigg −/Gamma1Nnσ(t). (17) Finally, inserting the time-dependent operator ˆd† σ(t)[ E q .( A8)] to Eq. ( 15), we obtain jNσ(t)=−/Gamma1Nnσ(t)+/Gamma1N πRe/parenleftbigg/integraldisplay∞ −∞dω fN(ω)e−iωt ×L−1/braceleftbiggs+iε−σ+/Gamma1N/2 (s−s1)(s−s2)(s−iω)/bracerightbigg (t)/parenrightbigg .(18) To compute the transient current of opposite spin electrons, jN−σ(t), one should replace the set of auxiliary parameters (s1,s2,s3,s4) by the following one ( s3,s4,s1,s2). In particular, forεσ=0 we get jNσ(t)=/Gamma1N π/integraldisplay∞ −∞dω fN(ω)/braceleftBigg e−/Gamma1Nt/21 2/summationdisplay p=±ωpsin(ωpt)−/Gamma1N 2cos(ωpt) /parenleftbig/Gamma1N 2/parenrightbig2+ω2p+/Gamma1N/bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+/parenleftbig/Gamma1S 2/parenrightbig2+ω2/bracketrightbig /bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+ω2 −/bracketrightbig/bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+ω2 +/bracketrightbig/bracerightBigg −/Gamma1Nnσ(t),(19) where ω±=/Gamma1S 2±ω. In absence of the superconducting lead, this formula is identical with the result obtained by means ofthe nonequilibrium Green’s function method. In Fig. 6, we present transient behavior of the current j N↑(t) induced by an abrupt coupling of the QD to both external leadsforμ N=μS=0 (i.e., without any bias). Similar to the time- dependent QD occupancy (Fig. 2), we observe the quantum oscillations of the period 2 π//Gamma1Sexponentially decaying with the envelope coefficient e−/Gamma1Nt. Large value of the current att=0+(equal to e/Gamma1N/2h) is artifact of the wide band limit approximation [ 19] in presence of abrupt switching, Eq. ( 2). In realistic experimental situations, this effect would -0.1-0.05 0 0.05 0.1 0 10 20 30 40 60 t [1/ΓS]jNσ(t) [ e ΓS / h ] jNσ(t) t [1/ΓS]ΓN / ΓS = 0.1 0.25 0.5 1 -0.1 0 0.1 0 10 20 30 FIG. 6. Transient current between the QD and the normal lead induced by a sudden coupling in absence of any bias. Results are obtained for the same parameters as in Fig. 2. The inset shows the transient current obtained for the sinusoidal coupling profiles Vk,q(t), assuming /Gamma1N//Gamma1S=0.1.not be observed. To check how a smooth (gradual) coupling process does affect our predictions, we have computed thetransient currents, assuming the sinusoidal switching profile V k,q(t)=Vk,q 2(sin (π/Gamma1Nt−π/2)+1) for 0 <t/lessorequalslant1//Gamma1Nand keeping constant value Vk,qfort>1//Gamma1N.W eh a v es o l v e d this problem numerically. Some representative results (for/Gamma1 N//Gamma1S=0.1) are displayed in the inset in Fig. 6. We noticed that for t>1//Gamma1Nall the time-dependent quantities are not particularly affected. The only difference (in comparison tothe abrupt coupling) is in the early time region 0 <t< 1//Gamma1 N. For instance, the transient current jN↑(t) smoothly evolves from zero to its asymptotic behavior with the same period ofquantum oscillations. In similar steps, we have also determined the transient current j Sσ(t)=2I m/summationtext kVq/angbracketleftˆd† σ(t)ˆcqσ(t)/angbracketright. Effective quasi- particles in superconductors are represented by a coherentsuperposition of the particle and hole degrees of freedom, sofor this reason the time-dependent operator ˆc qσ(t) consists of four contributions [see Eq. ( A10)]. Final expression for jSσ(t) becomes rather lengthy, therefore we present it in Appendix A4. However, in absence of external voltage the current Eq. ( A26) simplifies to jSσ(t)=/Gamma12 S 2√ δsin(√ δt)e−/Gamma1Nt/bracketleftBigg 1−/summationdisplay σ/primenσ/prime(0)/bracketrightBigg .(20) When the QD is initially empty/full the transient current jSσ=±/Gamma12 S 2√ δsin(√ δt)e−/Gamma1Ntreveals the damped oscillations. Contrary to this behavior, for the initial occupancies nσ(0)=0 andn−σ(0)=1 the current Eq. ( 20) vanishes. We assign this 075420-5R. TARANKO AND T. DOMA ´NSKI PHYSICAL REVIEW B 98, 075420 (2018) feature to inefficiency of the proximity effect whenever the QD is singly occupied, because electron pairing operates only bymixing the empty with the doubly occupied QD configurations.Initial conditions have thus important influence on transientphenomena. Furthermore, Eq. ( A21)f o r/angbracketleftˆd ↓(t)ˆd↑(t)/angbracketrightand Eq. ( A26) imply the exact relationship jSσ(t)=−/Gamma1SIm/angbracketleftˆd↓ˆd↑/angbracketright, which is popular in considerations of charge transport through Joseph-son junctions [ 62]. The transient current j Sσ(t) can hence be simply inferred from Fig. 4. At this point we emphasize that the charge conservation of our heterostructure is properly satisfied: d dtnσ(t)=jSσ(t)+jNσ(t)≡jdis,σ(t), (21) where jdis,σ(t) stands for the transient displacement current (see, e.g., Refs. [ 19,22]).D. Transient currents for biased system We have seen so far that time-dependent QD occupancy and transient currents provide indirect information aboutthe subgap quasiparticle energies and dynamical transitionsbetween them. In absence of any voltage ( μ N=μS=0), these transient currents finally vanish, with a rate dependenton the relaxation processes caused by the coupling /Gamma1 Nwith a continuum of metallic lead. From the practical point ofview, a much more convenient way for probing the timescalescharacteristic for the Andreev/Shiba quasiparticles could beprovided by transient properties of the biased system μ N/negationslash= μS. Following the steps discussed in previous Sec. III C, we shall study here the time-dependent differential conduc-tanceG σ(μ,t)≡d dμjNσ(t) as a function of external voltage μ≡μN(throughout this paper, the superconducting lead is assumed to be grounded μS=0). At zero temperature, Eq. ( 18) implies Gσ(μ,t)=/Gamma1NRe/bracketleftbigg e−iμtL−1/braceleftbiggs+iε−σ+/Gamma1N/2 (s−s1)(s−s2)(s−iμ)/bracerightbigg (t)/bracketrightbigg −/Gamma12 N 2L−1/braceleftbiggs+iε−σ+/Gamma1N/2 (s−s1)(s−s2)(s−iμ)/bracerightbigg (t)L−1 ×/braceleftbiggs−iε−σ+/Gamma1N/2 (s−s3)(s−s4)(s+iμ)/bracerightbigg (t)+/Gamma12 N/Gamma12 S 8L−1/braceleftbigg1 (s−s1)(s−s2)(s+iμ)/bracerightbigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)(s−iμ)/bracerightbigg (t), (22) where the conductance is expressed in units of2e2 h. Expression for G↓(μ,t) can be obtained by the replacement ( s1,s2,s3,s4)→ (s3,s4,s1,s2). Using the corresponding inverse Laplace transforms, we find (for εσ=0,G↑=G↓=G) G(μ,t)=/Gamma1N/braceleftBigg e−/Gamma1Nt/2 2/summationdisplay p=+,−μpsin(μpt)−/Gamma1N 2cos(μpt) /parenleftbig/Gamma1N 2/parenrightbig2+μ2p+/parenleftbig/Gamma1N 2/parenrightbig/bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+/parenleftbig/Gamma1S 2/parenrightbig2+μ2/bracketrightbig /bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+μ2 +/bracketrightbig/bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+μ2 −/bracketrightbig/bracerightBigg −/Gamma12 N 2F1(μ,t)+/Gamma12 N/Gamma12 S 8F2(μ,t), (23) where F1(μ,t) andF2(μ,t)a r eg i v e ni nE q s .( A15) and ( A16), andμ+/−=μ±/Gamma1S/2. In the steady limit, t→∞ and for εσ=0, keeping only terms that survive at late times, we obtain the expression identical with the result derived for the samesetup within the Büttiker-Landauer approach [ 63] G(μ,∞)=/Gamma1 2 N/Gamma12 S 4/bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+μ2 −/bracketrightbig/bracketleftbig/parenleftbig/Gamma1N 2/parenrightbig2+μ2 +/bracketrightbig. (24) For/Gamma1S/greatermuch/Gamma1N, the local extrema of this expression occur at μ=±/Gamma1S 2and they correspond to the energies of subgap bound states. For an arbitrary set of model parameters, such infor-mation is encoded in Eq. ( 22) which quantitatively specifies development of the in-gap states driven by the sudden switch-ing att=0. In Fig. 7, we present the differential conductance obtained numerically for /Gamma1 N//Gamma1S=0.1 and 0.7. Let us notice that differential conductance approaches its steady-limit shapeG ↑(μ,t=∞ ) characterized by two Lorentzian quasiparticle peaks centered at ∼±/Gamma1S 2. Their broadening /Gamma1Nis related to the inverse lifetime. More careful examination of G↑(μ,t) indicates that devel- opment of the subgap quasiparticles proceeds in three stepswith typical timescales τ 1,τ2andτf, as can be deduced from Figs. 7and 8.I nF i g . 8, we show how the position of the quasiparticle maxima develops in time for different/Gamma1N.A tt=τ1, there emerge two maxima from the single broad structure where τ1changes approximately from 5 (for /Gamma1N=0.2) up to 10 (for /Gamma1N=0.9) units of time. These maxima move rapidly (essentially during 1–2 units) from μ=0u pt o some value of μwhich depends on /Gamma1N. Next, the position of the quasiparticle peaks evolve continuously to their steady-limit position μ=±/radicalBig /Gamma12 S−/Gamma12 Nwithτ2approximately changing from 15 (for /Gamma1N=0.9) up to 30 (for /Gamma1N=0.2) units of time. Finally, the asymptotic quasiparticle feature is achieved withthe evolve function 1 −exp(−t/τ f) where τf=2//Gamma1N,s e e Eqs. ( 23), (A15), and ( A16), where the terms proportional to exp(−/Gamma1Nt/2) are responsible for such asymptotic behavior. We also clearly see that, near the quasiparticle peaks, thetotal differential conductance/summationtext σG(μ,t→∞ ) acquires its optimal value 4 e2/hknown from the previous studies (see, e.g., Ref. [ 2]). IV . CORRELATION EFFECTS Local repulsive interactions Uˆn↑ˆn↓compete with the proximity-induced electron pairing. This issue has been ad-dressed in the steady limit by numerous methods [ 2]. In particular, it has been shown [ 64] that effective pairing (man- ifested by the in-gap states) is predominantly sensitive tothe ratio U//Gamma1 Sand depends on the energy level εσ. Various 075420-6BUILDUP AND TRANSIENT OSCILLATIONS OF ANDREEV … PHYSICAL REVIEW B 98, 075420 (2018) ΓN=0.1 −1−0.500.51μ/ΓS 0 10 20 30 40 50 t [1/ΓS] 0 0.2 0.4 0.6 0.8 1G↑(μ,t) ΓN=0.7 −1−0.500.51μ/ΓS 0 10 20 30 40 50 t [1/ΓS] 0 0.2 0.4 0.6 0.8 1G↑(μ,t) FIG. 7. The time-dependent differential conductance G↑(μ,t)= G↓(μ,t) (in units of4e2 h) obtained for εσ=0,/Gamma1N=0.1 (top panel) and/Gamma1N=0.7 (bottom panel). experimental realizations of the correlated QD in N-QD- S geometry [ 57,58,65,66] indicated that the Coulomb potential Usafely exceeds (at least one order of magnitude) the su- perconducting energy gap /Delta1. Under such circumstances, the correlation effects show up in the subgap regime |ω|</Delta1 merely by a quantum phase transition (or crossover) from thespinless (BCS-type) state u|0/angbracketright+v|↑↓/angbracketright to the spinful (singly occupied) configuration |σ/angbracketright. This changeover occurs upon increasing the ratio U//Gamma1 Sand, above some critical value of the Coulomb potential Ucr,there can be observed the subgap Kondo effect (even in the superconducting atomic limit) [ 66,67]. We shall briefly analyze some correlation effects, focusing on thetransient effects. A. Competition between pairing and correlations The aforementioned quantum phase transition can be qualitatively captured already within the lowest order 0 10 20 30 40 50 0 0.1 0.2 0.3 0.5μ / ΓSt [ 1/ ΓS ] ΓN / ΓS = 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 FIG. 8. Positions of the quasiparticle maxima vs. time and μ//Gamma1S appearing in the differential conductance G↑(μ,t) for a number of ratios/Gamma1N//Gamma1S, as indicated. For negative values of μ//Gamma1Sthe results are symmetrical.(Hartree-Fock-Bogoliubov) decoupling scheme: ˆd† ↑ˆd↑ˆd† ↓ˆd↓/similarequaln↑(t)ˆd† ↓ˆd↓+n↓(t)ˆd† ↑ˆd↑−n↑(t)n↓(t) +χ∗(t)ˆd† ↑ˆd† ↓+χ(t)ˆd↓ˆd↑−|χ(t)|2.(25) Using this approximation Eq. ( 25) one can incorporate the Hartree-Fock terms into the renormalized energy level ˜ /epsilon1σ≡ /epsilon1σ+Un−σ(t), whereas the anomalous (pair source and drain) terms rescale the effective pairing potential ˜/Gamma1S/2≡/Gamma1S/2+ Uχ(t). This decoupling procedure Eq. ( 25) can give a crossing of the subgap quasiparticle energies at some critical ratioU//Gamma1 S, dependent also on εσ. In the Josephson junctions, such effect would cause a reversal of the dc tunneling current, the so-called 0 −πtransition [ 62,68]. In our N-QD- Sheterostructure, its influence is noticeable but rather less spectacular. Analytical determination of the dynamical observables (dis- cussed in Sec. III) is unfortunately not feasible in the present case, because the renormalized energy level ˜ ε σ(t) and effective pairing potential ˜/Gamma1S(t) are time-dependent in a nonexplicit way and the method used in the previous section is usefulonly for consideration of systems with constant QD energylevels and couplings with the leads. Therefore, in what follows,we consider the Coulomb repulsion in the system of theproximitized QD coupled only to the normal lead, applying theHartree-Fock-Bogoliubov approximation Eq. ( 25). We have computed numerically n σ(t),/angbracketleftˆd↓(t)ˆd↑(t)/angbracketrightandjNσ(t), solving the closed set of differential equations for time-dependent func-tionsn σ(t) and/angbracketleftˆd↓(t)ˆd↑(t)/angbracketright, respectively (see Appendix B). At intermediate steps, we had to compute additionally theexpectation values /angbracketleftˆd † σ(t)ckσ(0)/angbracketrightand/angbracketleftˆdσ(t)ˆck−σ(0)/angbracketright. All these quantities have been determined within the Runge-Kutta nu-merical algorithm. Figure 9displays influence of the Coulomb potential Uon the induced order parameter χ(t) for the unbiased system. The imaginary part, which is strictly related to the transient cur-rent, exhibits the damped quantum oscillations. Their periodand amplitude are substantially suppressed by the Coulombpotential. We assign this fact to a competition between theon-dot pairing and local Coulomb repulsion. The real part ofχ(t) is characterized by the same quantum oscillations. The asymptotic value of the complex order parameter χ(t→∞ ) with respect to the Coulomb potential Uis shown in Fig. 10 forε σ=0,/Gamma1N//Gamma1S=0.2. Such monotonously decreasing Reχ(∞) confirms a competing relationship between the on- dot pairing and the local repulsion. In Fig. 11, we show influence of the Coulomb potential U on the QD occupancy n↑(t). Besides the quantum oscillations, similar to the ones observed in the complex order parameter(Fig. 9), we notice a partial reduction of the QD charge upon increasing U. Apparently, this is caused by the Hartree term Un −σ(t) that lifts the renormalized QD level ˜ εσ(t). In the next subsection we briefly discuss the time-dependent subgapKondo effect. B. Subgap Kondo effect In this subsection, we briefly discuss another characteristic timescale, which could be related to the subgap Kondo effect.Since we cannot account for this effect within the equationof motion approach, we make some conjectures based on 075420-7R. TARANKO AND T. DOMA ´NSKI PHYSICAL REVIEW B 98, 075420 (2018) 2 4 6U/ΓS 0 5 10 15 20 t [1/ΓS] 0 0.25 0.5− Re χ(t) 0 2 4 6U/ΓS 0 5 10 15 20 t [1/ΓS]−0.2 0 0.2Im χ(t) FIG. 9. Influence of the Coulomb potential Uon the real (upper panel) and imaginary (bottom panel) parts of the induced pairingχ(t)=/angbracketleftˆd ↓ˆd↑/angbracketrightobtained for the unbiased system, using εσ=0,/Gamma1N= 0.2a n d/Gamma1S≡1. (i) systematic study of the steady case [ 66–68] combined with (ii) time-dependent analysis of the Kondo physics of normalQDs [ 35]. Self-consistent treatment of this phenomenon, which is beyond the scope of the present paper, would be very muchwelcome. When the Coulomb potential Uis sufficiently large in comparison to /Gamma1 S, the QD ground state evolves towards the spinful (doublet) configuration |σ/angbracketright. Under such conditions, the effective spin exchange between the correlated QD and mobileelectrons of the metallic lead activate the subgap Kondo effect.It has been analyzed by many groups, using various techniques[2]. In the present context, we shall make use of basic facts pointed out recently by R. Žitko et al. [66] and independently by one of us [ 67,68]. The exchange interaction −/summationtext k,pJk,pˆSd·ˆSkpbetween the QD spin ˆSdand spins ˆSkpof the mobile electrons in the normal lead can be determined by means of the generalizing canonical 0 0.1 0.2 0.3 0.4 0 2.5 5 7.5 10 U/ΓS− Re χ(∞) 0 0.1 0.2 0.3 0.4 0 2.5 5 7.5 10 U/ΓSRe χ(∞) Im χ(∞) FIG. 10. Asymptotic value of complex on-dot pairing χ(t→∞ ) suppressed by the Coulomb repulsion Uobtained for the same model parameters as in Fig. 9.2 4 6U/ΓS 0 5 10 15 20 t [1/ΓS] 0 0.2 0.4 0.6 0.8n↑(t) FIG. 11. The time-dependent occupancy of the correlated quan- tum dot for /epsilon1σ=0,/Gamma1N=0.2,/Gamma1S=1 in absence of external voltage. Schrieffer-Wolff transformation. Adopting it to the N-QD- S setup, it has been found that for the superconducting atomiclimit, the exchange coupling near the Fermi energy J kF,kFis equal to [ 67] JkF,kF=U|VkF|2 εσ(εσ+U)+(/Gamma1S/2)2. (26) For a spinful configuration, the Kondo temperature can be estimated, e.g., using the Bethe-Ansatz formula TK∝ exp{−1/[2ρ(εF)JkFkF]}, where ρ(εF) is the density of states of the normal lead at the Fermi level. We have compared suchresults with the unbiased NRG calculations and it has beenfound that the Kondo temperature is expressed by [ 67] T K=η√/Gamma1NU 2exp/bracketleftbigg πεσ(εσ+U)+(/Gamma1S/2)2 /Gamma1NU/bracketrightbigg ,(27) withη≈0.6. In particular, for the half-filled QD ( εσ= −U/2), the exchange coupling Eq. ( 26) simplifies to JkF,kF=J(N) kF,kFU2 U2−/Gamma12 S, (28) where J(N) kF,kFstands for the normal case ( /Gamma1S=0). Upon ap- proaching a transition from the spinful doublet to the BCS-like(spinless) ground state, the Kondo temperature is substantiallyenhanced [ 66,67] T K=T(N) Kexp/bracketleftBigg π /Gamma1NU/parenleftbigg/Gamma1S 2/parenrightbigg2/bracketrightBigg . (29) To get some insight into the transient phenomena related with the subgap Kondo regime, we make use of the finalconclusions inferred in Ref. [ 35] from the time-dependent noncrossing approximation study. The characteristic time τ K needed for the Abrikosov-Suhl peak to emerge at the Fermi level has been found to scale inversely with the Kondotemperature, i.e., τ K∼1/TK. This information adopted to ourN-QD- Ssetup implies the following relative ratio for the half-filled QD: τK=τ(N) Kexp/bracketleftBigg −π /Gamma1NU/parenleftbigg/Gamma1S 2/parenrightbigg2/bracketrightBigg , (30) where τ(N) Kstands for the normal state value ( /Gamma1S=0). We plot this scaling in Fig. 12. Let us remark that many-body screening, Eq. ( 26), of the QD spin can be practically realized only in the 075420-8BUILDUP AND TRANSIENT OSCILLATIONS OF ANDREEV … PHYSICAL REVIEW B 98, 075420 (2018) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8τK / τK(N) ΓS / U FIG. 12. Characteristic timescale τK∼1/TKof the subgap Kondo effect obtained for the half-filled QD using U=10/Gamma1Nwith respect to varying ratio /Gamma1S/U. In this case, the spinful ground state exists in the region /Gamma1S∈(0,U). doublet ground state (which for the half-filled QD occurs when /Gamma1S<U ). By increasing the ratio /Gamma1S/U, the Andreev bound states tend to their crossing and simultaneously the Abrikosov-Suhl peak Eq. ( 29) quickly broadens [ 66,67]. This explains why the characteristic timescale τ Kstrongly decreases with respect to/Gamma1S/U. V . SUMMARY We have investigated transient effects driven by a sudden coupling of the QD to the metallic and superconducting leads.Our study has revealed a gradual buildup of the subgap Andreevquasiparticle states, which is controlled by the coupling /Gamma1 Nto a continuous spectrum of the metallic lead. Depending on theinitial QD occupancy, we have also found the damped quantumoscillations of the charge occupancy n σ(t), the complex order parameter χ(t), and the transient currents jNσ(t),jSσ(t). A period of these oscillations would be sensitive to the Andreevquasiparticle energies, which can be indirectly controlled viaa coupling /Gamma1 Sto the superconducting reservoir. Analogous effects (relaxation and quantum oscillations) have been recently reported in Refs. [ 9,51] in studies of the metastable subgap states for the Josephson junction, consid-ering finite value of the superconducting gap. We estimatethat in realistic systems, where /Gamma1 S∼0.2 meV , the period of quantum oscillations would be a fraction of nanoseconds or inpicoseconds regime (hence should be empirically detectable).Buildup of the subgap Andreev quasiparticle states is expectedto be formed in N-QD- Sjunctions on a much longer timescale, corresponding to a microsecond regime. Our estimations seemto reliable, when comparing them with dynamical transi-tions between the subgap bound states of nanotubes [ 8] and parity switchings observed in the superconducting atomiccontacts [ 69]. We also addressed the correlation effects by means of the Hartree-Fock-Bogoliubov approximation, revealing thatthe repulsive Coulomb potential Usuppresses the proximity- induced electron pairing. We have explored some time-dependent signatures of this competition. In particular, wehave found that /Gamma1 Ncontrols the rate at which the stationary limit behavior is achieved, whereas the period of the dampedquantum oscillations is dependent on the Coulomb potentialdue to its influence of the Andreev quasiparticle energies [ 64].Finally, we have tried to evaluate the characteristic timescale τ Kneeded for the subgap Kondo effect to develop. Upon approaching the quantum phase transition from the (spinful)doublet side, we predict the strong reduction of this scale τ K, originating from a subtle interplay between the induced on-dotpairing and the Coulomb repulsion [ 66,67]. We hope that such variety of dynamical effects of the proximitized QDs could beverified experimentally. ACKNOWLEDGMENTS We acknowledge instructive discussions with V . Janiš and thank A. Baumgartner for useful remarks on observability ofthe transient effects in multiterminal heterostructures. We alsokindly thank T. Kwapi´ nski for technical assistance. This work is supported by the National Science Centre (NCN, Poland)through Grants No. DEC-2014/13/B/ST3/04451 (T.D.) andNo. UMO-2017/27/B/ST3/01911 (R.T.). APPENDIX A In this Appendix, we derive the Laplace transforms ˆdσ(s) andˆcqσ(s), which are needed for calculating the statistically averaged physical quantities discussed in this work. We presentexplicit formulas for the QD occupancy, the pair correlationfunction, and the transient currents flowing between QD andexternal reservoirs for the case /Delta1=∞ andU=0. 1. Laplace transforms To calculate expectation values of the quantities studied in this paper, we need the time-dependent operator ˆdσ(t). We can obtain it by computing the corresponding inverseLaplace transform L −1{ˆdσ(s)}(t). To determine ˆdσ(s), we use a closed set of the equations of motion for the operators: ˆdσ(t),ˆd† −σ(t),ˆckσ(t),ˆc† k−σ(t),ˆcqσ(t),ˆc† −q−σ(t),ˆc† q−σ(t), and ˆc−qσ(t). Laplace transforms of these differential equations for σ=↑ (assuming arbitrary energy gap /Delta1and neglecting the correlations) take the following form: (s+iε↑)ˆd↑(s)=−i/summationdisplay k/qVk/qˆck/q↑(s)+ˆd↑(0),(A1a) (s+iεk)ˆck↑(s)=−iVkˆd↑(s)+ˆck↑(0), (A1b) (s+iεq)ˆcq↑(s)=−iVqˆd↑(s)−i/Delta1ˆc† −q↓(s)+ˆcq↑(0), (A1c) (s−iεq)ˆc† −q↓(s)=iVqˆd† ↓(s)−i/Delta1ˆcq↑(s)+ˆc† −q↓(0), (A1d) (s−iε↓)ˆd† ↓(s)=i/summationdisplay k/qVk/qˆc† k/q↓(s)+ˆc† ↓(0), (A1e) (s−iεk)ˆc† k↓(s)=iVkˆd† ↓(s)+ˆc† k↓(0), (A1f) (s−iεq)ˆc† q↓(s)=iVqˆd† ↓(s)−i/Delta1ˆc−q↑(s)+ˆc† q↓(0), (A1g) (s+iεq)ˆc−q↑(s)=−iVqˆd↑(s)−i/Delta1ˆc† q↓(s)−ˆc−q↑(0). (A1h) 075420-9R. TARANKO AND T. DOMA ´NSKI PHYSICAL REVIEW B 98, 075420 (2018) Here we have assumed εqσ=εq,εq=ε−qand the sub- script k(q) corresponds to the normal (superconducting) electrode. To obtain ˆd↑(s), we have to calculate ˆc† −q↓(s)f r o m Eq. ( A1d) and insert it into Eq. ( A1c) for the operator ˆcq↑(s). In a next step, we use ˆcq↑(s) in the expression for ˆd↑(s)g i v e n in Eq. ( A1a) along with ˆck↑(s) obtained from Eq. ( A1b). Inconsequence, we get ˆd↑(s)M(+) ↑(s)=ˆA(s)−iK(s)ˆd† ↓(s). (A2) Next, we repeat this procedure, Eqs. ( A1e)–(A1h), obtaining ˆd† ↓(s)M(−) ↓(s)=ˆB(s)−iK(s)ˆd↑(s), (A3) where M(±) σ(s)=s±iεσ+/summationdisplay kV2 k s±iεk+/summationdisplay qV2 q(s∓iεq) s2+ε2q+/Delta12, (A4) K(s)=/summationdisplay qV2 q/Delta1 s2+ε2q+|/Delta1|2, (A5) ˆA(s)=−i/summationdisplay kVkˆck↑(0) s+iεk−/summationdisplay qVq s2+ε2q+/Delta12(/Delta1ˆc† −q↓(0)+i(s−iεq)ˆcq↑(0))+ˆd↑(0), (A6) ˆB(s)=i/summationdisplay kVkˆc† k↓(0) s−iεk+/summationdisplay qVq s2+ε2q+/Delta12(/Delta1ˆc−q↑(0)+i(s+iεq)ˆc† q↓(0))+ˆd† ↓(0). (A7) Equations ( A2) and ( A3) yield ˆd↑(s)=M(−) ↓(s)ˆA(s)−iK(s)ˆB(s) M(+) ↑(s)M(−) ↓(s)+K2(s). (A8) To determine ˆd↓(s), one should repeat the same procedure for a set of equations of motion for the operators: ˆd↓,ˆd† ↑,ˆck↓,ˆc† k↑,ˆcq↓,ˆc† −q↑,ˆc† q↑andˆc† −q↓(t), respectively. Effectively, we get ˆd↓(s)=M(−) ↑(s)ˆB†(s)+iK(s)ˆA†(s) M(+) ↓(s)M(−) ↑(s)+K2(s). (A9) In similar steps, we can compute ˆcqσ(s), which is needed in expression for the current flowing between the QD and the superconductor. From Eqs. ( A1a)–(A1h), we obtain ˆcqσ(s)=1 s2+ε2q+|/Delta1|2(−iVq(s−iεq)ˆdσ(s)+α/Delta1Vqˆd† −σ(s)−iα/Delta1ˆc† −q−σ(0)+(s−iεq)ˆcqσ(0)), (A10) where α=+(−)f o rσ=↑(↓). Laplace transforms of ˆd† σ(s) can be obtained taking the Hermitian conjugate of the operators ˆdσ(s)g i v e ni nE q s .( A8) and ( A9). Note that in the wide-band-limit approximation, the functions M(±) σ(s) andK(s) simplify in the superconducting atomic limit /Delta1=∞ tos±iεσ+/Gamma1N/2 and/Gamma1S/2, respectively. As an example, we present here the explicit form of the Laplace transform for ˆd↑(t): ˆd↑(s)=1 (s−s3)(s−s4)/braceleftBigg/parenleftbigg s−iε↓+/Gamma1N 2/parenrightbigg/bracketleftBigg ˆd↑(0)−i/summationdisplay kVkˆck↑(0) s+iεk−i/summationdisplay qVq(s−iεq)ˆcq↑(0) s2+ε2q+/Delta12−/summationdisplay qVq/Delta1ˆc† −q↓(0) s2+ε2q+/Delta12/bracketrightBigg −i/Gamma1S 2/bracketleftBigg ˆd† ↓(0)+i/summationdisplay kVkˆc† k↓(0) s−iεk+i/summationdisplay qVq(s+iεq)ˆc† q↓(0) s2+ε2q+/Delta12+/summationdisplay qVq/Delta1ˆc−q↑(0) s2+ε2q+/Delta12/bracketrightBigg/bracerightBigg . (A11) The creation operator ˆd† ↑(s) can be simply obtained from the Hermitian conjugate of Eq. ( A11) and using the replacement (s3,s4)↔(s1,s2). Note, that the operators ˆdσ(s) and ˆd† σ(s) depend on the superconducting energy gap /Delta1. In the main part of this paper, we have focused on the superconducting limit /Delta1→∞ , calculating the average values of /angbracketleftˆd† σ(t)ˆdσ(t)/angbracketright,/angbracketleftˆdσ(t)ˆd−σ(t)/angbracketright, and/angbracketleftˆd† σ(t)ˆck/primeσ(t)/angbracketright. 075420-10BUILDUP AND TRANSIENT OSCILLATIONS OF ANDREEV … PHYSICAL REVIEW B 98, 075420 (2018) 2. QD occupancy Let us determine the QD occupancy, nσ(t), expressed by the formula Eq. ( 4) computing expectation value of a product of the corresponding inverse Laplace transforms. Initially, at t=0, the QD is decoupled from both external reservoirs, therefore the only nonvanishing terms comprise the following averages /angbracketleftˆd† σ(0)ˆdσ(0)/angbracketright=nσ(0),/angbracketleftˆdσ(0)ˆd† σ(0)/angbracketright=1−nσ(0),/angbracketleftˆc† kσ(0)ˆck/primeσ(0)/angbracketright= δk,k/primefN(εk), and /angbracketleftˆckσ(0)ˆc† k/primeσ(0)/angbracketright=δk,k/prime[1−fN(εk)], where fN(εk) is the Fermi-Dirac distribution of the normal lead. Other terms, corresponding to itinerant electrons of the superconducting lead, can be neglected in the limit /Delta1=∞ , but they could be also included when considering the finite energy gap (we shall return to this issue later on). Using Eq. ( A8), we obtain the QD occupancy given by nσ(t)=L−1/braceleftBigg s+iε−σ+/Gamma1N 2 (s−s1)(s−s2)/bracerightBigg (t)L−1/braceleftBigg s−iε−σ+/Gamma1N 2 (s−s3)(s−s4)/bracerightBigg (t)/angbracketleftˆd† σ(0)ˆdσ(0)/angbracketright +/parenleftbigg/Gamma1S 2/parenrightbigg2 L−1/braceleftbigg1 (s−s1)(s−s2)/bracerightbigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)/bracerightbigg (t)/angbracketleftˆd−σ(0)ˆd† −σ(0)/angbracketright +/parenleftbigg/Gamma1S 2/parenrightbigg2/summationdisplay k,k/primeVkVk/primeL−1/braceleftbigg1 (s−s1)(s−s2)(s+iεk)/bracerightbigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)(s−iεk/prime)/bracerightbigg (t)/angbracketleftˆck−σ(0)ˆc† k/prime−σ(0)/angbracketright +/summationdisplay k,k/primeVkVk/primeL−1/braceleftBigg s+iε−σ+/Gamma1N 2 (s−s1)(s−s2)(s−iεk)/bracerightBigg (t)L−1/braceleftBigg s−iε−σ+/Gamma1N 2 (s−s3)(s−s4)(s+iεk/prime)/bracerightBigg (t)/angbracketleftˆc† kσ(0)ˆck/primeσ(0)/angbracketright. (A12) Forσ=↓, one should make the replacement ( s1,s2,s3,s4)→(s3,s4,s1,s2). Using the wide-band limit approximation, we can recast summations over momenta of the third and fourth terms in Eq. ( A12) by the integrals /Gamma12 S 4/Gamma1N 2π/integraldisplay∞ −∞dε[1−fN(ε)]L−1/braceleftbigg1 (s−s1)(s−s2)(s+iε)/bracerightbigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)(s−iε)/bracerightbigg (t) +/Gamma1N 2π/integraldisplay∞ −∞dεfN(ε)L−1/braceleftBigg s+iε−σ+/Gamma1N 2 (s−s1)(s−s2)(s−iε)/bracerightBigg (t)L−1/braceleftBigg s−iε−σ+/Gamma1N 2 (s−s3)(s−s4)(s+iε)/bracerightBigg (t). (A13) The final formula for nσ(t) is quite lengthy, therefore we present its simpler explicit form, corresponding to εσ=0: nσ(t)=nσ(0)e−/Gamma1Nt+[1−nσ(0)−n−σ(0)]e−/Gamma1Ntsin2/parenleftbigg/Gamma1S 2t/parenrightbigg +/Gamma1N 2π/integraldisplay∞ −∞dε f N(ε)F1(ε,t)+/Gamma1N 2π/Gamma12 S 4/integraldisplay∞ −∞dε[1−fN(ε)]F2(ε,t). (A14) Functions F1(ε,t) andF2(ε,t) are defined by F1(ε,t)=1 A(ε)/braceleftbigg/Gamma12 N 4+ε2+e−/Gamma1Nt 2/bracketleftbigg/parenleftbigg/Gamma12 N 4−/Gamma12 S 4+ε2/parenrightbigg cos(/Gamma1St)−/Gamma1N/Gamma1S 2sin(/Gamma1St)+/Gamma12 N 4+/Gamma12 S 4+ε2/bracketrightbigg −e−/Gamma1Nt/2/bracketleftbigg 2/parenleftbigg/Gamma12 N 4+ε2/parenrightbigg cos(εt) cos/parenleftbigg/Gamma1S 2t/parenrightbigg −/Gamma1N/Gamma1S 2cos(εt)s i n/parenleftbigg/Gamma1S 2t/parenrightbigg +/Gamma1Sεsin(εt)s i n/parenleftbigg/Gamma1S 2t/parenrightbigg/bracketrightbigg/bracerightbigg ,(A15) F2(ε,t)=1 /Gamma1SA(ε)/braceleftbigg e−/Gamma1Nt/bracketleftbigg−2 /Gamma1S/parenleftbigg/Gamma12 N 4−/Gamma12 S 4+ε2/parenrightbigg cos(/Gamma1St)+/Gamma1Nsin(/Gamma1St)+2 /Gamma1S/parenleftbigg/Gamma12 N 4+/Gamma12 S 4+ε2/parenrightbigg/bracketrightbigg +e−/Gamma1Nt/2[2(ε−cos(ε+t)−ε+cos(ε−t))−/Gamma1N(sin(ε+t)−sin(ε−t))]+/Gamma1S/bracerightbigg , (A16) andA(ε)=(/Gamma12 N 4+ε2 −)(/Gamma12 N 4+ε2 +), where ε+/−=ε±/Gamma1S 2. It should be noted that for /Gamma1S=0 the formula Eq. ( A12) coincides with the standard expression nσ(t)=nσ(0)e−/Gamma1Nt+/Gamma1N πe−/Gamma1Nt/2/integraldisplay∞ −∞dε f N(ε)cosh(/Gamma1Nt/2)−cos((ε−εσ)t) /Gamma12 N 4+(ε−εσ)2(A17) obtained by the nonequilibrium Green’s function (NEGF) technique [ 61]. We are not aware of any results available for /Gamma1S/negationslash=0, but it seems that our approach would be simpler in comparison to the NEGF method in which the QD occupancy is formally 075420-11R. TARANKO AND T. DOMA ´NSKI PHYSICAL REVIEW B 98, 075420 (2018) expressed via the lesser Green’s function nσ(t)=−iG<(t,t). In practice, it can be determined from the Keldysh equation G<=(1+Gr/Sigma1r)G< 0(1+/Sigma1aGa)+Gr/Sigma1<Ga. (A18) In particular, for nσ(0)=0E q .( A18) simplifies to G<=Gr/Sigma1<Ga(because G< 0=0[19,61]). In other cases, however, determination of the lesser Green’s function is much more demanding. We now return to the discussion of the terms appearing in formula Eq. ( 4), which contain the operators ˆcqσ. Let us analyze one of such terms, e.g., /angbracketleftBigg L−1/braceleftBigg s+iε↓+/Gamma1N 2 (s−s1)(s−s2)/summationdisplay qVq(s+iεq)ˆc† q↑(0) s2+ε2q+/Delta12/bracerightBigg (t)L−1⎧ ⎨ ⎩s−iε↓+/Gamma1N 2 (s−s3)(s−s4)/summationdisplay q/primeVq/prime(s−iεq/prime)ˆcq/prime↑(0) s2+ε2 q/prime+/Delta12⎫ ⎬ ⎭(t)/angbracketrightBigg , (A19) which can be reduced to the form /Gamma1S 2π/integraldisplay+∞ −∞dεfS(ε)L−1/braceleftBigg/parenleftbig s+iε↓+/Gamma1N 2/parenrightbig(s+iε) (s−s1)(s−s2)(s2+ε2+/Delta12)/bracerightBigg (t)L−1/braceleftBigg/parenleftbig s−iε↓+/Gamma1N 2/parenrightbig(s−iε) (s−s3)(s−s4)(s2+ε2+/Delta12)/bracerightBigg (t), (A20) where we made use of the equality /angbracketleftˆc† q↑(0)ˆcq/prime↑(0)/angbracketright=δqq/primefs(εq). We have checked (by numerically integrating the product of the corresponding inverse Laplace transforms) that this integral becomes smaller and smaller upon increasing /Delta1, and it finally diminishes to zero in the limit /Delta1=∞ . Similarly, we have checked that all other terms comprising the operators ˆcqσ(0) disappear for/Delta1=∞ as well. 3. QD pair correlation function Using the explicit formulas for ˆdσ(s), presented in Eqs. ( A8) and ( A9) and performing similar calculations as for the QD occupancy, we obtained the induced on-dot pairing given by /angbracketleftˆd↓(t)ˆd↑(t)/angbracketright=i/Gamma1S 2/bracketleftBigg n↑(0)L−1/braceleftbigg1 (s−s1)(s−s2)/bracerightbigg (t)L−1/braceleftBigg s−iε↓+/Gamma1N 2 (s−s3)(s−s4)/bracerightBigg (t) −(1−n↓(0))L−1/braceleftBigg s−iε↑+/Gamma1N 2 (s−s1)(s−s2)/bracerightBigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)/bracerightbigg (t) −/Gamma1N 2π/integraldisplay∞ −∞dω[1−fN(ω)]L−1/braceleftBigg s−iε↑+/Gamma1N 2 (s−s1)(s−s2)(s+iω)/bracerightBigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)(s−iω)/bracerightbigg (t) +/Gamma1N 2π/integraldisplay∞ −∞dω fN(ω)L−1/braceleftbigg1 (s−s1)(s−s2)(s−iω)/bracerightbigg (t)L−1/braceleftBigg s−iε↓+/Gamma1N 2 (s−s3)(s−s4)(s+iω)/bracerightBigg (t)/bracketrightBigg . (A21) 4. Transient current from superconducting lead In analogy to Eq. ( 15), we can define the transient current flowing from the superconductor to the quantum dot jSσ(t)=2Im/summationdisplay qVq/angbracketleftL−1{ˆd† σ(s)}(t)L−1{ˆcqσ(s)}(t)/angbracketright. (A22) Laplace transforms of ˆd† σ(s) and ˆcqσ(s) are given in Eqs. ( A8) and ( A10), so we can repeat the calculations similar to the ones discussed in preceding subsections. Let us consider the term proportional to n↑(0), which takes the form 2n↑(0)Im/braceleftBigg −i/summationdisplay qVqL−1/braceleftBigg s+iε↓+/Gamma1N 2 (s−s1)(s−s2)/bracerightBigg (t) ×/bracketleftBigg L−1/braceleftBigg Vq(s−iεq)/parenleftbig s−iε↓+/Gamma1N 2/parenrightbig (s−s3)(s−s4)/parenleftbig s2+ε2q+/Delta12/parenrightbig/bracerightBigg (t)+/Gamma1S 2L−1/braceleftBigg Vq/Delta1 (s−s3)(s−s4)/parenleftbig s2+ε2q+/Delta12/parenrightbig/bracerightBigg (t)/bracketrightBigg/bracerightBigg . (A23) For/Delta1=∞ , the first term in the bottom part of this equation vanishes and we can calculate the second term by interchanging summation over qwith the Laplace transformation: 2n↑(0)Im/bracketleftBigg −i/Gamma1S 2L−1/braceleftBigg s+iε↓+/Gamma1N 2 (s−s1)(s−s2)/bracerightBigg (t)L−1/braceleftBigg 1 (s−s3)(s−s4)/summationdisplay qV2 q/Delta1/parenleftbig s2+ε2q+/Delta12/parenrightbig/bracerightBigg (t)/bracketrightBigg . (A24) 075420-12BUILDUP AND TRANSIENT OSCILLATIONS OF ANDREEV … PHYSICAL REVIEW B 98, 075420 (2018) Since lim /Delta1=∞/summationtext qV2 q/Delta1 (s2+ε2q+/Delta12)=/Gamma1S 2the term proportional to n↑(0) simplifies to 2n↑(0)/Gamma12 S 4Im/bracketleftBigg −iL−1/braceleftBigg s+iε↓+/Gamma1N 2 (s−s1)(s−s2)/bracerightBigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)/bracerightbigg (t)/bracketrightBigg . (A25) In the same manner, we calculate the terms proportional to /angbracketleftˆd↑(0)ˆd† ↑(0)/angbracketright,/angbracketleftˆc† k↑(0)ˆck/prime↑(0)/angbracketright, and/angbracketleftˆck↑(0)ˆc† k/prime↑(0)/angbracketright, respectively. Any other terms containing expectation values of two superconducting lead electron operators vanish in the limit /Delta1=∞ . Finally, we get the transient current jSσ(t)=/Gamma12 S 2Re/bracketleftBigg −nσ(0)L−1/braceleftBigg s+iε−σ+/Gamma1N 2 (s−s1)(s−s2)/bracerightBigg (t)L−1/braceleftbigg1 (s−s3)(s−s4)/bracerightbigg (t) +(1−n−σ(0))L−1/braceleftbigg1 (s−s1)(s−s2)/bracerightbigg (t)L−1/braceleftBigg s+iεσ+/Gamma1N 2 (s−s3)(s−s4)/bracerightBigg (t)+/Gamma1N 2π/Phi1σ/bracketrightBigg , (A26) with/Phi1σdefined in Eq. ( 14). Forσ=↓, one should use the replacement ( s1,s2,s3,s4)→(s3,s4,s1,s2). In theequilibrium case (forμN=0), the formula Eq. ( A26) can be simplified, because Re /Phi1σ=0. To prove this property, let us focus on the case εσ=0, when we can express /Phi1σ=/summationtext4 j=1/integraltext∞ −∞dε(1−2fN(ε))Aj(ε) using the coefficients A1(ε)=−ie−/Gamma1Nt 2/Gamma1S/bracketleftBigg ei/Gamma1St /parenleftbig/Gamma1N 2−iε+/parenrightbig/parenleftbig/Gamma1N 2+iε−/parenrightbig−e−i/Gamma1St /parenleftbig/Gamma1N 2−iε−/parenrightbig/parenleftbig/Gamma1N 2+iε+/parenrightbig/bracketrightBigg , A2(ε)=ie−/Gamma1Ntε/parenleftBig /Gamma12 N 4+ε2 +/parenrightBig/parenleftBig /Gamma12 N 4+ε2 −/parenrightBig, A3(ε)=−/Gamma1N 2+iε/parenleftBig /Gamma12 N 4+ε2 −/parenrightBig/parenleftBig /Gamma12 N 4+ε2 +/parenrightBig, A4(ε)=−e−/Gamma1Nt/2 2⎡ ⎣e−iε+t /parenleftBig /Gamma12 N 4+ε2 +/parenrightBig/parenleftbig/Gamma1N 2−iε−/parenrightbig+e−iε−t /parenleftBig /Gamma12 N 4+ε2 −/parenrightBig/parenleftbig/Gamma1N 2−iε+/parenrightbig⎤ ⎦ −ie−/Gamma1Nt/2 /Gamma1S/parenleftbigg/Gamma1N 2+iε/parenrightbigg⎡ ⎣−eiε+t /parenleftBig /Gamma12 N 4+ε2 +/parenrightBig/parenleftbig/Gamma1N 2+iε−/parenrightbig+eiε−t /parenleftBig /Gamma12 N 4+ε2 −/parenrightBig/parenleftbig/Gamma1N 2+iε+/parenrightbig⎤ ⎦, (A27) andε+/−=ε±/Gamma1S 2. At zero temperature, this function can be given as /Phi1σ=/summationtext4 j=1/integraltext∞ 0dε(Aj(ε)−Aj(−ε)), using the following properties: A1(ε)=A1(−ε),A2(ε)=−A2(−ε),ReA3(ε)=ReA3(−ε) andA4(−ε)=A∗ 4(ε), which imply that Re /Phi1σ=0. The same conclusion is valid for εσ/negationslash=0 as well. APPENDIX B: MEAN FIELD APPROXIMATION In this brief Appendix, we consider the effective Hamiltonian of the proximitized QD coupled to the normal lead, treating electron correlations within the Hartree-Fock-Bogoliubov approximation: ˆH=/summationdisplay σ(εσ+Un−σ(t))ˆd† σˆdσ+/bracketleftbigg/parenleftbigg/Gamma1S 2+Uχ(t)/parenrightbigg ˆd† ↑ˆd† ↓+H.c./bracketrightbigg +/summationdisplay k,σ[Vkˆd† σˆckσ+H.c.]+/summationdisplay k,σεkσˆc† kσˆckσ. (B1) In general, all parameters εσ,/Gamma1S,Vk,εkσcan be time-dependent. In what follows, we outline the algorithm for numerical computation of the QD charge nσ(t) and the induced on-dot pairing χ(t)=/angbracketleftˆd↓(t)ˆd↑(t)/angbracketright. We have to solve numerically the following set of coupled equations of motion: dnσ(t) dt=2Im/parenleftBigg/summationdisplay kVkeiε/vectorkt/angbracketleftˆd† σ(t)ˆckσ(0)/angbracketright−¯/Delta1∗(t)χ(t)−/Gamma1Nnσ(t)/parenrightBigg , (B2) dχ(t) dt=i/summationdisplay kVke−iεkt(/angbracketleftˆd↑(t)ˆck↓(0)/angbracketright−/angbracketleft ˆd↓(t)ˆck↑(0)/angbracketright)−[i(¯ε↑(t)+¯ε↓(t))+/Gamma1N]χ(t)−i¯/Delta1(t)(1−n↓(t)−n↑(t)),(B3) 075420-13R. TARANKO AND T. DOMA ´NSKI PHYSICAL REVIEW B 98, 075420 (2018) where ¯ εσ(t)=εσ+Un−σ(t) and ¯/Delta1(t)=/Gamma1S 2+Uχ(t). In the wide-band-limit approximation and assuming εkσto be time- independent, the mixed functions appearing in Eqs. 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PhysRevB.95.245402.pdf

PHYSICAL REVIEW B 95, 245402 (2017) Origin of the core-level binding energy shifts in Au nanoclusters Alexey A. Tal,1,2,*Weine Olovsson,1and Igor A. Abrikosov1 1Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83 Linköping, Sweden 2Materials Modeling and Development Laboratory, National University of Science and Technology “MISIS”, 119049 Moscow, Russia (Received 15 March 2017; published 5 June 2017) We investigate the shifts of the core-level binding energies in small gold nanoclusters by using ab initio density-functional-theory calculations. The shift of the 4 fstates is calculated for magic-number nanoclusters in a wide range of sizes and morphologies. We find a nonmonotonous behavior of the core-level shift in nanoclustersdepending on the size. We demonstrate that there are three main contributions to the Au 4 fshifts, which depend sensitively on the interatomic distances, coordination, and quantum confinement. They are identified andexplained by the change of the on-site electrostatic potential. DOI: 10.1103/PhysRevB.95.245402 I. INTRODUCTION Nanoclusters with the size of nanometers demonstrate fascinating reactive, optical, electronic, and magnetic prop-erties, which are not observed in their bulk counterparts [ 1]. That behavior is determined by many factors such as highsurface-to-bulk ratio, electronic shell closing [ 2–4], geometric shell closing [ 2], and quantum confinement [ 5]. Spectroscopic measurements, especially such precise measurements as fromx-ray photoelectron spectroscopy (XPS), have been widelyused for the characterization of small nanoclusters [ 6]. The properties of nanoclusters make them extremely interesting forcatalysis applications, where reactions can be altered by smallchanges in structure or size. The high ratio of surface-to-bulkatoms in the clusters drastically increases their efficiency. Inthe recent paper by Ma et al. [7], the local coordination of gold atoms in small nanoclusters has been suggested as a parameterfor catalytic activity prediction. Moreover, in the work ofKaden et al. [8], it was shown that shifts of the binding energies of core electrons strongly correlate with the catalytic activity ofthe nanoclusters. Thus, an understanding of the origin of core-level binding energy shifts (CLS) in nanoclusters of differentsizes is important for their characterization, with a potentialfor designing nanoparticles with improved performance. Morphologies of small nanoclusters are very different from their corresponding bulk structures [ 1]. Thermodynamically favorable morphology of a nanocluster is determined bythe competition of surface energy and internal stress. Theanalysis of the thermodynamics of the gold nanoclustersperformed by Baletto et al. [9] showed that icosahedral and decahedral structures are the most favorable for smallnanoclusters. However, due to kinetics in the growth process,cubic and octahedral clusters are present as well [ 10]. In this work, we consider CLS in clusters with a magic numberof atoms with cubic, icosahedral, decahedral, and octahedralmorphologies. During the growth process, kinetics might notallow nanoclusters to transition into favorable morphology[11]. Moreover, a substrate where nanoclusters are collected affects the morphology and may induce a shift of the core statesdue to charge transfer [ 12]. All of these effects significantly *aleta@ifm.liu.secomplicate the analysis of nanoclusters. The use of theoretical modeling allows one to distinguish trends and illuminate onthe origin of positions of the core states in nanoclusters.Thus, we investigate core-level shifts of unsupported neutralnanoclusters with ideal structures, but it is worth emphasizingthat clusters of larger size should not be significantly affectedby the substrate. The great interest in gold nanoclusters inapplications for catalysis motivated our choice of the material[13–16]. Moreover, gold has the largest surface core-level shift of all noble metals [ 17], which makes it a good candidate for studying the behavior of the core levels and easier todistinguish trends. Several experiments have demonstrated that 4 flevels shift towards higher binding energies in Au nanoclusters with thedecrease of the nanocluster size [ 18–20]. Conventionally, these shifts are believed to be due to the final-state relaxation inducedby charging [ 19]. Besides the shift, a broadening of the 4 fpeak was observed in all experiments, explained by the effect of theelectrostatic potential. The negative surface core-level shiftsobserved in experiments are, to a large extent, an initial stateeffect, explained by the valence-band narrowing of the lesscoordinated surface atoms [ 17]. This causes a charge redistri- bution from 6 sto 5dand hence an increased charge density and screening of the core hole [ 21,22]. However, the structural effect on the CLS has not been fully understood. These aspectsmotivate the present work. In particular, we systematicallystudy the structural effects on the shifts of the core levelsand the relation between initial and final states. Moreover,we investigate the evolution of the core-state energies froman individual atom to bulk systems through atomic clusters.Although the charge-induced shift can be significant in thespectra, we focus our attention on structural effects. First-principles calculations of core-level shifts have proven to be a very accurate and useful tool for understandingof the XPS spectra and behavior of the core level in general[23,24]. Here we demonstrate that our calculation for the surface core-level shifts in Au(100) is 0.46 eV and in goodagreement with the experimental values of 0 .4±0.01 eV [ 17]. Based on this, we study how energy levels of the core statesdepend on the morphology and size of the nanoclusters andhow these levels change for different atoms in a nanocluster.Furthermore, the change of the core levels is analyzed in theatom-cluster-bulk sequence. 2469-9950/2017/95(24)/245402(6) 245402-1 ©2017 American Physical SocietyALEXEY A. TAL, WEINE OLOVSSON, AND IGOR A. ABRIKOSOV PHYSICAL REVIEW B 95, 245402 (2017) This paper is organized as follows. In Sec. II, the methods and details of the calculations are provided. The results arediscussed in Sec. III. Section III A describes the effects of the strain, coordination, and size on the shift of the core states.Section III B is dedicated to CLS in icosahedral, decahedral, and octahedral nanoclusters. Finally, Sec. IVpresents our conclusions. II. COMPUTATIONAL DETAILS We performed ab initio density-functional-theory (DFT) calculations of the core-level binding energy shifts [ 25] within the complete screening picture; in doing so, we include bothinitial-state (the shift of the on-site electrostatic potentialfor an atom in different environments) and final-state (core-hole screening by conduction electrons) effects. The electronwave functions were treated within the projector augmentedwave (PAW) method [ 26]. The plane-wave cutoff energy was 250 eV . The value of the cutoff was determined from conver-gence of core-level shifts in bulk and nanoclusters. All clusterstructures were optimized until the forces on relaxed atoms were less than 0 .02 eV ˚A −1. The integration of the Brillouin zone was performed in /Gamma1point for clusters larger than 50 atoms, and with denser k-point grid for small nanoclusters. The DFT calculations were performed with the generalized gradi-ent approximation (GGA) exchange-correlation functional inPerdew-Burke-Ernzerhof (PBE) [ 27] form as implemented in the Vienna ab initio simulation package, V ASP [28]. For the initial-state approximation CLS calculations, the Kohn-Sham equation is solved inside the PAW sphere for core electrons,after self-consistency with frozen core electrons has beenattained [ 29]. In calculations with the core hole, we assume that the hole at the ionized atom effectively acts as an extraproton. This assumption allows us to substitute the ionizedatom of atomic number Z with the next element in theperiodic table. This approximation is also called equivalentcore or (Z +1) approximation [ 23,25,30]. In calculations, CLS can be determined from the difference between ionizationenergies, which is the difference between total energy in theground state and total energy of the core-ionized state. Thus,ionization energy can be defined in the form of a generalizedthermodynamic chemical potential (GTCP) [ 30], μ=E ion−Egs 1/N, (1) where μis the GTCP and Nis the number of ionized atoms in the supercell ( N=1 in our calculations). The energies Egsand Eionare the total energies of the system in the ground state and ionized states, respectively. The shifts can be calculated from ECLS=/Delta1μi=μi−μRef i, (2) where iis the core level in the atom of the study and μRef icorresponds to a reference system. In this study, we have chosen pure bulk fcc Au as a reference system. It isimportant to notice that the shifts are calculated relative tothe Fermi level. The complete screening CLS approximation is known to be reliable and to reproduce experiments well[23,31]. According to the potential model [ 32], CLS can be approximated by the change of the on-site electrostatic2 4 6 810 12 Coordination−1.00−0.75−0.50−0.250.000.250.500.751.00Core-level shifts (eV) 92949698100102104 Lattice parameter distortion (%)Complete screening Initial state(a) (b) FIG. 1. (a) Au 4 fcore-level shifts of atoms with different coordination (number of nearest neighbors). Reduced coordinationsare obtained by cutting (124), (112), (100), and (111) surfaces. (b) Au 4fcore-level shifts of atoms in bulk structure with distorted lattice parameter. Distortion is denoted in percent of the perfect bulk latticeparameter. potential for an atom in different environments /Delta1V, Epot CLS=/Delta1V−/Delta1ER, (3) where /Delta1ERis core-hole relaxation energy or the screening of the core hole. The contribution of the first term into the bindingenergy shift is usually associated with initial-state CLS, whilethe second term is considered as a measure of the final-stateeffects. Because in our case the latter are small, as one seesfrom a comparison of our first-principles results obtainedwithin the complete screening and initial-state pictures(Figs. 1and2), we conclude that assuming the validity of the potential model of Gelius [ 32], the CLS is determined by the shift of the electrostatics potential to a very high extent. III. RESULTS AND DISCUSSION For reasons of clarity and comparison to other theoretical calculations as well as to experiment, we will forthwith discussthe 4fstates. The core states are very sensitive to the change of the local environment of an atom and, as will be shownbelow, the three main contributions to the shifts are number ofnearest neighbors (coordination), confinement, and the latticeparameter. We have found that the shifts correlate with thebehavior of the dband and can be explained by the change of the on-site electrostatic potential, as shown below. A. Contributions affecting shifts of the core states Coordination effects. Figure 1(a) shows how the position of the 4 fcore state depends on the coordination of the atom, where a coordination of 12 corresponds to an atom in thebulk and has zero CLS. These calculations were performedfor nanoclusters with fcc structure and with the bulk latticeparameter. Atoms from surfaces (124), (112), (100), and (111)were chosen as undercoordinated atoms. The largest shift 245402-2ORIGIN OF THE CORE-LEVEL BINDING ENERGY . . . PHYSICAL REVIEW B 95, 245402 (2017) −0.6−0.5−0.4−0.3−0.2−0.10.00.10.20.3 Complete screening Initial state 50 100 150 200 250 Number of atoms−7.5Core-level shifts (eV) Bulk Atom FIG. 2. Au 4 fcore-level shift as a function of the size of a nan- ocluster within initial-state and complete screening approximations.The structures are fcc unrelaxed clusters, where the position of the 4fstates is calculated for the central atom. was found for the most undercoordinated atoms. The shift decreases for atoms with larger coordination and saturates forcoordination of 10. The shifts with and without the final-stateeffect are very similar. This means that the effects of thecore-hole screening are not so significant. Strain effects. A ss h o w ni nF i g . 1(b),4fstates are very sensitive to the strain. Under uniform compression, the 4 k state shifts toward higher binding energies, and in the oppositedirection for a uniformly stretched lattice parameter. Thechange of the lattice parameter by 2% results in the shift of0.25 eV . Once again, the shifts calculated within initial andcomplete screening are very similar. Size effect. Another parameter that affects CLS in nan- oclusters is the size effect or confinement effect. In Fig. 2, the results for the calculated 4 fstate are shown. The shifts were calculated for the central atom in cubic with ideal(unrelaxed) fcc structure containing clusters from 13 to 256 FIG. 3. Shifts of 4 fstates in 108 cubic Au nanocluster with bulk lattice parameter on (100) facet (left) and its cross section (right).Colors of the atom correspond to the value of the shift.−6−5−4−3−2−1 0 1 E Ef(eV)d-band density of states (arb. units) 13 atoms32 atoms63 atoms108 atoms172 atoms256 atomsbulk FIG. 4. Density of states in the dband of the central atom in nanoclusters with different number of atoms, where red dots denote the center of the dband. atoms. Starting from the smallest nanocluster with 13 atoms, 4fstates are shifted towards smaller binding energies. Then, for larger clusters, this shift becomes smaller in magnitudeand approaches the position of the 4 fstate. It is important to −10 −8 −6 −4 −2 0 E Ef(eV)05101520d-band density of states (arb. units) CenterSecond layerThird layerSurface FIG. 5. Densities of states for the dband of four atoms, each from one layer (central/first layer, second layer, third layer, surface/lastlayer) in an Au icosahedral nanocluster with 147 atoms. 245402-3ALEXEY A. TAL, WEINE OLOVSSON, AND IGOR A. ABRIKOSOV PHYSICAL REVIEW B 95, 245402 (2017) FIG. 6. CLS in icosahedral nanoclusters: 55, 147, and 309 atoms. All atoms are colored corresponding to their CLS. Black bars denote calculated CLS. To facilitate comparisons with experimental data, aconvolution of the CLS with a 0.05 eV Gaussian is shown. emphasize that the shift nonmonotonically depends on the size and even changes the sign at around 100 atoms. Thus, we conclude that the three most significant effects contributing to the 4 fshifts in nanoclusters are the number of nearest neighbors or coordination, the distance to the nearestneighbors, and the size of the system or confinement effect.To discuss these effects in more detail for the relatively simplecase, we show the CLS in a 108 atom cubic nanocluster withbulk lattice parameter in Fig. 3. The internal atoms have zero shift as compared to their bulk position, while the states foratoms on the surface show big negative CLS. As mentionedearlier, these effects can be correlated with the behavior of the d band. That allows us to conclude that the CLS is determined bythe shift of the electrostatics potential to a very high extent. InFig.4, the change of the valence bandwidth is clearly observed. The red dots in Fig. 4denote the center of the dband. A decrease in the coordination leads to narrowing of the dband as shown in Fig. 5for 147 icosahedral cluster. As it was shown previously [ 21,30], narrowing of the dband accompanies the negative CLS for metals with more than a half or completelyfilled band. The confinement effect also appears together withthe band narrowing. B. Core-level shifts in nanoclusters In order to understand how the structure affects the core- level shifts in nanoclusters, we performed CLS calculations in FIG. 7. CLS in decahedral nanoclusters: 49, 146, and 318 atoms. All atoms are colored corresponding to their CLS. Black bars denote calculated CLS. To facilitate comparisons with experimental data, a convolution of the CLS with a 0.05 eV Gaussian is shown. nanoclusters of different size and morphology with a magic number of atoms. Magic numbers are the numbers of atomsin perfect cluster structures with all shells filled and all atomssitting in their ideal position [ 33]. Icosahedral nanocrystals. At small size, it is more energeti- cally favorable to minimize the surface energy of a nanoclusterby reducing the surface area. The icosahedral (Ih) structure hasthe smallest surface area among all cluster morphologies. Wehave analyzed icosahedral nanoclusters of three sizes: 55, 147,and 309 atoms. In Fig. 6(a), the icosahedral cluster consisting of 55 atoms is shown. Colors show the correspondence ofatom position to the CLS. One can see that undercoordinatedatoms of the surface have negative CLS of −0.5 eV , while the central atom has a positive CLS of 0.5 eV due to thestresses acting inside the cluster. The second layer of atomshas negative CLS of −0.1 eV . The next cluster size in Fig. 6(b) consists of 147 atoms or four complete layers. Similar trendsare observed. Undercoordinated atoms have negative CLS andatoms of the first and the second shell shifted toward higherbinding energies. In Fig. 6(c), the Ih cluster with 309 atoms or five layers is shown. The positive shift peak becomes broaderand its contribution increases compared to the surface shift’scontribution. The weight of the positive peak should increaseas a cube of radius, while the weight of the surface contributiongrows as a square. The CLS of the top layer approachessurface CLS of gold. This can be attributed to the fact that the 245402-4ORIGIN OF THE CORE-LEVEL BINDING ENERGY . . . PHYSICAL REVIEW B 95, 245402 (2017) FIG. 8. CLS in decahedral nanoclusters: 247, 318, and 389 atoms. All atoms are colored corresponding to their CLS. Black bars denotecalculated CLS. To facilitate comparisons with experimental data, a convolution of the CLS with a 0.05 eV Gaussian is shown. confinement effect at such size is negligibly small. In all three clusters, the largest shifts have been found on vertex atoms andcentral atoms. From the series of these three sizes, we concludethat the binding energy of core electrons in nanoclusters as afunction of the size is nonmonotonic and approaches the bulkvalue with increase of the size. Decahedral nanocrystals. In Fig. 7, the CLS for three decahedral (Dh) clusters of different size are shown.The sizes were chosen to be the closest magic number tothe size of the icosahedral nanoclusters, which is supposedto make the comparison of the nanoclusters with similarnumber of atoms easier. The decahedral structure is lesssymmetrical than the icosahedral one, which results in a moreeven distribution of the shifts. A 49-atom decahedral clusterhas many undercoordinated atoms, and thus the weight of thenegative CLS is much bigger. With the increase of the size upto 146 atoms, the amount of the internal atoms grows as wellas the weight of the positive shifts. The cluster with 318 atomshas two distinct peaks: one from the internal atoms and onefrom the surface. Similarly to the Ih cluster, surface atoms ofDh at these sizes approach (111) and (100) surface CLS ofgold, correspondingly. For all sizes, Dh clusters have broaderpeaks than Ih clusters, which is the result of the symmetryreduction. While icosahedral geometry can be described by one parameter, i.e., number of shells, a decahedral structure, as a FIG. 9. CLS in octahedral nanoclusters: 38, 116, and 201 atoms. All atoms are colored corresponding to their CLS. Black bars denote calculated CLS. To facilitate comparisons with experimental data, a convolution of the CLS with a 0.05 eV Gaussian is shown. less symmetrical structure, has more independent parameters. For example, decahedral can be considered as two identicalpyramids (top and bottom) with a free number of layersin between them. In order to understand how the structuraldifference of these configurations will affect the CLS, wecalculated the CLS of three Dh clusters with different numberof intermediate layers. In fact, the structure of the pyramidsis the same for all three structures, while the number of theintermediate layers is changed: 1, 3, and 5. In Fig. 8, one can see that the increase of the intermediate layers results ina broadening of the peaks and shift towards higher bindingenergies. In Dh structures, the largest negative CLS areobserved on (100) facets and not vertices as in Ih clusters. Octahedral nanocrystals. Truncated octahedron (TOh) is the least favorable structure at small sizes. The structure of TOhclusters is the most similar to the bulk fcc. Thus, TOh clustersshould have the least internal stress and the largest surfacearea. In Fig. 9, for all sizes, one can see two distinct peaks: surface CLS and internal CLS. At the largest size of 201 atoms,the surface peak splits into two: one from the /angbracketleft111/angbracketrightsurface with∼− 0.2 eV and another from /angbracketleft100/angbracketrightwith∼− 0.3e V . In the 201-atom TOh cluster, the lattice parameter is exactlythe same as in bulk, but the internal atoms have positive CLS,which is a clear manifestation of the confinement effect. All clusters have the largest negative CLS on vertices and edges due to the coordination effect, while atoms in the center 245402-5ALEXEY A. TAL, WEINE OLOVSSON, AND IGOR A. ABRIKOSOV PHYSICAL REVIEW B 95, 245402 (2017) of nanoclusters have large positive CLS due to strains. In most cases, small clusters (around 50 atoms) have the largest surfaceCLS. In Ih clusters, the largest CLS was found for vertex atoms,and in Dh and TOh clusters, the largest CLS is on (100) facets.All three considered morphologies have a similar signature inthe spectra: contributions from surface and bulk are separatedinto distinct peaks. Dh and TOh clusters have different typesof facets, which results in splitting of the surface peak forlarge clusters, while in Ih clusters, all surface atoms have verysimilar surface CLS. IV . CONCLUSION The evolution of the Au 4 fcore state from an individual atom to bulk through the cluster has been demonstrated.We have shown how the behavior of the core-level bindingenergy shift in nanoclusters is governed by three main effects:confinement, stress, and local coordination. In addition, theCLS in the complete screening picture was shown to be verysimilar to the initial-state one. The negative (positive) CLSin the initial-state model was explained by the shift of theelectrostatic potential. The difference between XPS features for icosahedral, dec- ahedral, and octahedral nanoclusters has been demonstrated.The largest CLS have been found on the edges and vertices ofsmall nanoclusters. Understanding of these trends combined with high-resolution XPS may allow one to distinguish themorphology of the nanoclusters from spectra. Moreover,judging by the magnitude and weights of the surface CLS,TOh with 116 atoms and Dh with 49 atoms could be expectedto have the largest absorption energy for other species and tobe the better catalysis than other clusters. ACKNOWLEDGMENTS The work was financially supported by the Knut and Alice Wallenberg Foundation through Grant No. 2012.0083 and theStrong Field Physics and New States of Matter 2014-2019(COTXS). Support by the Ministry of Education and Scienceof the Russian Federation (Grant No. 14.Y26.31.0005) isgratefully acknowledged. W.O. and I.A.A. acknowledge sup-port from the Swedish Government Strategic Research Areain Materials Science on Functional Materials at LinköpingUniversity (Faculty Grant SFO-Mat-LiU No. 2009 00971).The calculations were performed on resources provided by theSwedish National Infrastructure for Computing (SNIC) at theNational Supercomputer Centre (NSC) and the supercomputercluster provided by the Materials Modeling and DevelopmentLaboratory at NUST “MISIS”. [1] F. Baletto and R. Ferrando, Rev. Mod. Phys. 77,371(2005 ). [2] M. M. Kappes, R. W. Kunz, and E. Schumacher, Chem. Phys. Lett. 91,413(1982 ). [3] W. Ekardt, P h y s .R e v .L e t t . 52,1925 (1984 ). [4] W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y . Chou, and M. L. Cohen, Phys. Rev. Lett. 52,2141 (1984 ). [5] A. L. Efros, M. Rosen, M. Kuno, M. Nirmal, D. J. Norris, and M. Bawendi, Phys. Rev. B 54,4843 (1996 ). [6] R. Ferrando, J. Jellinek, and R. L. Johnston, Chem. Rev. 108, 845(2008 ). [7] X. Ma and H. 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PhysRevB.72.035124.pdf

Optical investigations on Y 2−xBixRu2O7: Electronic structure evolutions related to the metal-insulator transition J. S. Lee, S. J. Moon, and T. W. Noh School of Physics and Research Center for Oxide Electronics, Seoul National University, Seoul 151-747, Korea T. Takeda Division of Materials Science and Engineering, Graduate School of Engineering, Hokkaido University, Hokkaido 060-8628, Japan R. Kanno Department of Electronic Chemistry, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Yokohama 226-8502, Japan S. Yoshii *and M. Sato Department of Physics, Division of Material Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan /H20849Received 16 March 2005; published 18 July 2005 /H20850 Optical conductivity spectra of cubic pyrochlore Y 2−xBixRu2O7/H208490.0/H33355x/H333552.0/H20850compounds are investigated. As a metal-insulator transition /H20849MIT /H20850occurs around x=0.8, large spectral changes are observed. With increase ofx, the correlation-induced peak between the lower and the upper Hubbard bands seems to be suppressed, and a strong mid-infrared feature is observed. In addition, the p−dcharge transfer peak shifts to the lower energies. The spectral changes cannot be explained by electronic structural evolutions in the simple bandwidth-controlled MIT picture, but are consistent with those in the filling-controlled MIT picture. In addition, they arealso similar to the spectral changes of Y 2−xCaxRu2O7compounds, which is a typical filling-controlled system. This work suggests that, near the MIT, the Ru bands could be doped with the easily polarizable Bi cations. DOI: 10.1103/PhysRevB.72.035124 PACS number /H20849s/H20850: 71.30. /H11001h, 78.20. /H11002e, 78.40. /H11002q I. INTRODUCTION Pyrochlore ruthenium oxides A2Ru2O7−/H9254/H20851A=Bi, Tl, Pb, Y , and L/H20849=Pr-Lu /H20850/H20852are interesting materials, which show nu- merous electronic properties depending on the A-site ions. While Bi 2Ru2O7and Pb 2Ru2O6.5are good metals, Y 2Ru2O7 and L2Ru2O7−/H9254are insulators.1–3On the other hand, Tl2Ru2O7shows a temperature-dependent metal-insulator transition /H20849MIT /H20850around 125 K.4Therefore, it is clear that the electronic structures, especially near the Fermi level /H20849EF/H20850, should have some systematic evolutions depending on the A-site ions. Numerous investigations have been made to un- derstand how the electronic structures will evolve in thesepyrochlore ruthenates, 5–14however, there is no real consen- sus on this issue yet. One explanation is based on the fact that the different sizes of the A-site ions will control structural properties, such as RuuOuRu bond angle, which result in bandwidth changes of the Ru t2gbands. With strong electron-electron correlation effects,5–7,10–12these changes could result in a MIT, called the “bandwidth-controlled MIT.” In the earlydays, Cox et al. investigated Bi 2Ru2O7,Y 2Ru2O7, and Pb2Ru2O6.5by using photoemission spectroscopy and high- resolution electron-energy-loss spectroscopy and found thatthe density of states at E Fshould decrease in the sequence of Pb2Ru2O6.5,B i 2Ru2O7, and Y 2Ru2O7.5They concluded that Y2Ru2O7should be in the insulating state due to correlation- induced electron localization, namely the Mott insulator. Leeet al. calculated the bandwidth of the Ru t 2gbands using an extended Huckel tight binding method and confirmed thatthe MIT in the pyrochlore ruthenates should originate from the change of the relative size between the correlation energyand bandwidth, which was controlled by the A-site ion size. 7 The other explanation is based on the fact that Pb or Bi 6 p electrons should have very large wave functions, whichcould hybridize with the Ru t 2gwave functions. This hybrid- ization could result in large bandwidths of the Ru t2gbands and a net transfer of charge to the Ru t2gbands,8–11so the ruthenates could experience a MIT, called the “filling-controlled MIT.” Hsu et al. compared the x-ray photoemis- sion spectroscopy spectra of Pb 2Ru2O6.5and Bi 2Ru2O7with band-structure calculations and argued that the unoccupiedPb or Bi 6 pstates should be closely related to their metallic conductivity through mixing with the Ru 4 dstates via the ligand oxygen 2 pstates. 8Later, using the LDA calculation onA2Ru2O7−/H9254/H20849A=Bi, Tl, and Y /H20850, Ishii and Oguchi also ob- tained similar results that the Bi 6 pand the Tl 6 sstates could hybridize with the Ru t2gstates and should contribute to the density of states at EF.9Considering the fact that the formal charge valences of Pb and Bi are +3 just like those of theother pyrochlore ruthenates, the filling-controlled MIT israther surprising. Consequently, it is important to discrimi-nate whether the metallic state of the pyrochlore ruthenatesoriginates from the large bandwidth of the Ru 4 dbands or from the filling change. Optical spectroscopy has been used as a powerful tool to investigate electronic structures of highly correlated electronsystems. 15,16Several optical studies have been already done on some pyrochlore ruthenates and provided some clues tounderstanding of their electronic structure changes. 10,17–19PHYSICAL REVIEW B 72, 035124 /H208492005 /H20850 1098-0121/2005/72 /H208493/H20850/035124 /H208497/H20850/$23.00 ©2005 The American Physical Society 035124-1Earlier, we compared the optical spectra of Y 2Ru2O7, CaRuO 3, SrRuO 3, and Bi 2Ru2O7, and we already showed that Y 2Ru2O7should be a Mott insulator and that the metal- lic states of the perovskites could be understood in the band-width control picture. 10However, we stated that the metallic state of the pyrochlore Bi 2Ru2O7could not be easily under- stood. /H20849See comment 32 in Ref. 10. /H20850 In this paper, we report the optical conductivity spectra /H9268/H20849/H9275/H20850of Y 2−xBixRu2O7/H20849YBRO /H20850/H20849x=0.0, 0.5, 1.0, 1.5, and 2.0/H20850. YBRO is a good model system to investigate the MIT mechanism of the pyrochlore ruthenates, since it has bothinsulating and metallic end members. In addition, its solidsolution can be easily formed in all the region of x, and a MIT occurs around x=0.8. 2,3When the YBRO compound becomes metallic, the structural symmetry remains as a cu-bic; however, with increase of x, the RuuO bond length decreases and the Ru uOuRu bond angle increases. 2These xdependences are consistent with those in other pyrochlore ruthenates.13,14,20,21From the room temperature /H9268/H20849/H9275/H20850of YBRO, we observed that the variation of xcould result in systematic spectral changes, including large peak shifts andspectral weight redistributions in a wide energy range up to5 eV. Using these spectral changes, we will address the MITof YBRO from the bandwidth- and filling-controlled pic-tures. To clarify our argument further, we will also compare /H9268/H20849/H9275/H20850of YBRO with those of Y 2−xCaxRu2O7, which is a typi- cal filling-controlled system.22 II. EXPERIMENTAL Polycrystalline Y 2−xBixRu2O7samples were synthesized using the solid state reaction method.2Since the pyrochlore phase has a cubic structure, optical constants of the pyro-chlore ruthenates should be isotropic, so we can determinetheir optical constants from the reflectivity measurements ofpolycrystalline samples. Before reflectivity measurements,the sample surfaces were polished up to 0.3 /H9262m. We mea- sured reflectivity spectra from 5 meV to 30 eV using numer-ous spectrophotometers. 10After the optical measurements, thin gold films were evaporated on the samples and theirreflectivity spectra were measured again to correct the errorsdue to scattering from the sample surfaces. 23From the reflec- tivity spectra, we performed the Kramers-Kronig /H20849K-K /H20850 analysis to obtain /H9268/H20849/H9275/H20850. For this analysis, the reflectivity below 5 meV was extrapolated with a constant value for the insulating samples and the Hagen-Rubens relation for themetallic samples. Note that the dc conductivity values usedfor the Hagen-Rubens relation are larger by about two timesthe values obtained by the four-probe method, which origi-nates from the polycrystalline effects in the lattermeasurement. 24For a high-frequency region, the reflectivity value at 30 eV was used for reflectivities up to 40 eV, abovewhich /H9275−4dependence was assumed. In addition, we also independently measured /H9268/H20849/H9275/H20850between 0.7 and 4.0 eV using spectroscopic ellipsometry. The results of the K-K analysis agreed with the ellipsometry data, demonstrating the validityof our K-K analysis. 23 Sintered polycrystalline samples of Y 2−xCaxRu2O7were also synthesized by the solid reaction.22The sample densitieswere too low for the reflectivity measurements, so we de- cided to obtain their absorption spectra by measuring trans-mittance spectra of Y 2−xCaxRu2O7particles embedded in a KBr matrix, which is transparent up to around 4 eV. Mix-tures of Y 2−xCaxRu2O7and KBr powders were mixed to- gether thoroughly and pressed into pellets, whose thick-nesses were about 1 mm. The transmittance spectra weremeasured between 0.5 and 4.0 eV, and the absorption spectrawere evaluated by taking logarithms of the transmittancespectra and dividing them by the pellet thicknesses. 25 III. ASSIGNMENT OF SPECTRAL FEATURES OF Y2−xBixRu2O7BASED ON THE ELECTRONIC STRUCTURES OF END MEMBERS Figure 1 shows room temperature /H9268/H20849/H9275/H20850of Y 2−xBixRu2O7 /H20849x=0.0, 0.5, 1.0, 1.5, and 2.0 /H20850up to 10 eV. For x=0.0,/H9268/H20849/H9275/H20850 shows an insulating behavior with a small optical gap around 0.14 eV and the lowest interband transition around 1.6 eV.26 It also shows strong peaks around 3, 6, and 9 eV. As xin- creases, two intriguing spectral changes can be observed. /H208491/H20850 The spectral weight below around 1.0 eV increases, and astrong Drude-like peak appears for x=2.0. These behaviors are consistent with the xdependence of dc resistivity. 2,3/H208492/H20850 While the strength of the 6 eV peak increases, that of the9 eV peak decreases and almost disappears for x=1.5 and 2.0. FIG. 1. Doping-dependent /H9268/H20849/H9275/H20850of Y 2−xBixRu2O7at room tem- perature up to 10 eV. The indices of each peak appearing in /H20849a/H20850and /H20849e/H20850indicate the corresponding optical excitations, displayed in Fig. 2.LEE et al. PHYSICAL REVIEW B 72, 035124 /H208492005 /H20850 035124-2In order to understand spectral weight changes more eas- ily, let us adopt the schematic diagrams for the electronicstructures of the end members, i.e., Y 2Ru2O7and Bi 2Ru2O7, which were already presented in Ref. 10, and also displayedin Figs. 2 /H20849a/H20850and 2 /H20849d/H20850, respectively. For Y 2Ru2O7, the Ru ion has its formal valence of 4+ with four t2gelectrons. Since it is known as a Mott insulator,5,10the partially filled t2gstates split into the lower Hubbard band /H20849LHB /H20850and the upper Hub- bard band /H20849UHB /H20850,27as shown in the shaded area of Fig. 2/H20849a/H20850.28 The unoccupied egstates are located above the t2gstates by a crystal field splitting energy of about 3 eV.10The occu- pied O 2 pstates and the unoccupied Y 4 dstates are located below the LHB and above the egstates, respectively. On the other hand, Bi 2Ru2O7is close to a band metallic state,5 where the t2gstates form a single partially filled band at EF,as shown in the shaded area of Fig. 2 /H20849d/H20850. Instead of the unoccupied Y 4 dstates in Y 2Ru2O7,B i 2Ru2O7will have the unoccupied Bi 6 pstates, whose bandwidth might be much larger due to the extended nature of their correspondingwave functions. Other states, including the O 2 pstates and the Ru e gstates, should remain nearly the same. In each diagram, possible charge transfer excitations from the O 2 pstates are indicated as transitions A,B, and C, and possible transitions from the Ru t2gstates are indicated as transitions /H9251,/H9252, and/H9253. For Y 2Ru2O7, the lowest excitation around 1.6 eV can be assigned to the d−dtransition between the Hubbard bands, i.e., transition /H9251. And, the strong peaks around 3, 6, and 9 eV can be assigned to transitions A,B, andC, respectively. For Bi 2Ru2O7, the coherent peak below 1.5 eV should be attributed to the intraband transition of thepartially filled t 2gband, and the 3 eV peak can be assigned as transition A. The position of the 6 eV peak should corre- spond to the energies of transition Band an additional dipole-allowed transition, i.e., transition /H9253between the Ru t2g states and the Bi 6 pstates. The contributions from these two transitions can explain the large strength of this spectral fea-ture. The general trends of the high energy spectral changes, shown in Fig. 1, can be understood based on the electronicstructures of Y 2Ru2O7and Bi 2Ru2O7. Note that, as the Bi content increases, the 6 eV peak gains its spectral weight,and the 9 eV peak loses its spectral weight. These systematicxdependences confirm our peak assignments that the peaks around 6 and 9 eV should be related to the Bi ion states andthe Y ion states, respectively. Based on these peak assign-ments, we can argue that the spectral features in the lowenergy region below 5 eV should come from optical transi-tions related to the Ru t 2gand the O 2 pstates. IV . POSSIBLE MODELS FOR ELECTRONIC STRUCTURAL EVOLUTIONS OF Y 2−xBixRu2O7: BANDWIDTH- VS. FILLING-CONTROL Figures 2 /H20849b/H20850and 2 /H20849c/H20850show the possible electronic struc- tures near EFfor the intermediate compounds between Y2Ru2O7and Bi 2Ru2O7in the bandwidth- and the filling- controlled MIT pictures, respectively. The electronic struc-tures in the high energy regions are not displayed, since theyshould exhibit normal xdependences; that is, as xincreases, while the O 2 pstates and the Ru e gstates would not be much affected, the Y 4 dstates should be replaced by the Bi 6 p states. On the other hand, the details of the Ru t2gstates near EFshould change according to the mechanisms of the MIT. First, let us consider the bandwidth-controlled MIT, where the quasi-particle /H20849QP/H20850peak should appear near EFbetween LHB and UHB,27as displayed in Fig. 2 /H20849b/H20850. When the system becomes more metallic, the QP peak should increase with thereductions of the Hubbard bands. The corresponding /H9268/H20849/H9275/H20850in the low energy region should be composed of three spectral features: /H20849i/H20850a coherent peak centered at zero energy, /H20849ii/H20850an incoherent excitation between the Hubbard bands, and /H20849iii/H20850 strong p−dtransitions located at the higher energies. As shown in the inset of Fig. 3, /H9268/H20849/H9275/H20850of CaRuO 3,10which is known to be a correlated metal, exhibits such spectral fea- FIG. 2. Schematic diagrams of the electronic structures of Y2−xBixRu2O7/H20849a/H20850corresponds to that for x=0, i.e., Y 2Ru2O7. The occupied and unoccupied Ru t2gstates are located near EF, forming the lower Hubbard band /H20849LHB /H20850and the upper Hubbard band /H20849UHB /H20850, respectively. Outside these Ru t2gstates, there are the oc- cupied O 2 pstates and the unoccupied Ru egstates and the Y 4 d states. /H20849b/H20850sketches possible electronic structures of the Ru t2gstates in the intermediate region of x/H110111.0, when a metal-insulator transi- tion occurs through a change of the bandwidth. In the bandwidth-controlled system, a quasi-particle peak /H20849QP/H20850should appear be- tween the Hubbard bands. /H20849c/H20850sketches electronic structures, when a metal-insulator transition occurs due to the doping. In the filling-controlled system, a mid-gap state should appear. /H20849d/H20850corresponds to electronic structures for x=2.0, i.e., Bi 2Ru2O7. Compared to /H20849a/H20850, the Bi 6pstate replaces the Y 4 dstate, and the t2gstates form a single peak centered near EF.I n/H20849a/H20850and /H20849d/H20850, possible optical transitions are also indicated as A,B,C,/H9251,/H9252, and/H9253.OPTICAL INVESTIGATIONS ON Y 2−xBixRu2O7:… PHYSICAL REVIEW B 72, 035124 /H208492005 /H20850 035124-3tures quite well: namely, the Drude-like peak in the zero energy limit, the correlation-induced peak around 1.8 eV,and the p−dtransition peaks above 3 eV. As the system approaches the band metallic state, the coherent peak shoulddevelop, accompanied by the reduction of the incoherent ex-citation. Second, let us look into the filling-controlled MIT, where a mid-gap state should appear just below UHB /H20849above LHB /H20850 for the hole /H20849electron /H20850doping case, 27as displayed in Fig. 2/H20849c/H20850. Different from the QP peak in the bandwidth-controlled picture, this mid-gap state is not centered around EF, and it should provide an incoherent transport behavior in a moder-ate doping regime due to the disorder-induced carrierlocalization. 16In the hole-doping case, /H9268/H20849/H9275/H20850should have two additional excitations to the mid-gap state from LHB andfrom the O 2 pstates, each of which should appear at lower energies than transitions /H9251andA, respectively. As the carrier doping increases, the excitations to the mid-gap state shouldbecome stronger, accompanied by spectral weight reductionsof transitions /H9251andA. It should be noted that the lowest excitation comes from the transition between the LHB andthe mid-gap state, so that it should have a peak center at afinite energy. Considering large differences in the electronicstructures near E Fand the corresponding /H9268/H20849/H9275/H20850between the bandwidth- and filling-controlled MIT pictures, we will beable to address the possible MIT mechanism of YBRO byinvestigating their spectral changes at the low energy region.V . DISCUSSIONS ON THE METAL-INSULATOR TRANSITION MECHANISM OF Y 2−xBixRu2O7 A. Spectral changes of YBRO below 4.5 eV In order to investigate how the electronic structures of YBRO evolve near EFas the MIT occurs, we look into their /H9268/H20849/H9275/H20850in the low energy region. Figure 3 shows /H9268/H20849/H9275/H20850of YBRO up to 4.5 eV. As the MIT occurs, large spectral changes are observed for both the d−dand the p−dtransi- tions. For the d−dtransitions, while /H9268/H20849/H9275/H20850of the x=0.0 com- pound exhibits the correlation-induced peak around 1.6 eV, /H9268/H20849/H9275/H20850of the x=2.0 compound does not show the correlation- induced peak but exhibits a strong Drude-like peak extend- ing up to 1.5 eV. For the x=1.0 and 1.5 compounds, strong incoherent mid-infrared peaks can be observed. On the otherhand, the p−dtransition shows a systematic redshift with increase of x, as indicated by the solid triangles. Considering the fact that the spectral features in this energy region shouldbe mainly contributed by the Ru t 2gand the O 2 pstates, such systematic spectral changes with xindicate that there should be significant evolutions of these states near the MIT. B. Discussions based on the bandwidth-control picture Let us first discuss the evolution of /H9268/H20849/H9275/H20850of YBRO in the low energy region based on the bandwidth-control picture. Considering the x-dependent changes of the Ru uO bond length and the Ru uOuRu bond angle,2we can expect that the Ru t2gbandwidth should increase as xincreases. Actu- ally, the recent photoemission spectroscopy study on the Ru3dcore level by Kim et al. demonstrated that the electron correlation strength changes systematically with x, and it would play an important role in determining the electronicstructures of YBRO. 29In our optical data also, as discussed in Sec. III, the low energy spectral features of at least twoend members, i.e., the x=0.0 and 2.0 compounds, seem to exhibit typical behaviors of the Mott insulator and the bandmetal, respectively. However, when we look into the systematic changes of /H9268/H20849/H9275/H20850between end members, we can find a few intriguing features deviating from expected changes in the simple bandwidth-control picture. First, /H9268/H20849/H9275/H20850in the low energy re- gion of the intermediate compounds with x=1.0 or 1.5 are much different from that of a so-called correlated metal. As-suming that the bandwidth-controlled MIT should occur inYBRO, the x=1.0 and 1.5 compounds could be considered as correlated metals located between the Mott insulator andthe band metal. Then, according to the diagram in Fig. 2 /H20849b/H20850, their /H9268/H20849/H9275/H20850should exhibit a Drude-like peak centered at zero energy and a correlation-induced d−dtransition peak, which is observed for x=0.0 around 1.6 eV. Instead of such peak structures, they exhibit just a strong mid-infrared peakaround 1.0 eV for x=1.0 /H20849and 0.5 eV for x=1.5 /H20850. Second, the p−dtransition peak exhibits an unusual x-dependent evolution, which is different from typical be- haviors of the bandwidth-controlled system. We estimatedthe energy position /H9275p−dof the peak by fitting /H9268/H20849/H9275/H20850with a series of the Lorentz oscillators.10As shown in Fig. 4 /H20849a/H20850, /H9275p−dshows a gradual decrease with increase of x. Note that FIG. 3. Doping-dependent /H9268/H20849/H9275/H20850of Y 2−xBixRu2O7at room tem- perature up to 4.5 eV. The spectra for x=0.5, 1.0, 1.5, and 2.0 are shown with an upward shift by 600, 1200, 1500, and2600/H9024 −1cm−1, respectively. Inset shows /H9268/H20849/H9275/H20850of CaRuO 3at room temperature.LEE et al. PHYSICAL REVIEW B 72, 035124 /H208492005 /H20850 035124-4the Ru t2gbandwidth should be largely influenced by the p−dhybridization between the O 2 pand Ru t2gstates. If the p−dhybridization plays an important role, /H9275p−dshould be largely determined by the Ru uO bond length dRuuO:/H9275p−d should be inversely proportional to dRuuO.30–32Namely, the smaller dRuuOwould make the larger energy splitting be- tween the O 2 pstates and the Ru t2gstates, resulting in larger /H9275p−d. On the contrary to this expectation, /H9275p−ddecreases with decrease of dRuuO,2as shown in Fig. 4 /H20849b/H20850. This indi- cates that the observed dRuuOdependence of /H9275p−dcould not be simply related to the bandwidth-control effects.33 C. Discussions based on the filling-control picture In fact, the filling-controlled MIT picture can provide good explanations for most of the observed spectral changesof /H9268/H20849/H9275/H20850in the low energy region. As shown in Fig. 3, the d−dtransitions and the p−dtransitions for the intermediate compounds with x=0.5 and 1.0 seem to be broader than those of end members. And, as xincreases, both of them look like shifting to the lower energies. According to the sche-matic diagram in Fig. 2 /H20849c/H20850, which supposes the hole doping case for the Ru ions, the mid-gap state becomes strongerwith an increase of the hole doping for the Ru ions. Then theintermediate compounds should have two additional opticalexcitations, and each of them should be located just belowtransitions /H9251and A of the x=0.0 compound. This explains why the d−dand the p−dtransitions for the intermediate compounds become broader. If the finite widths of the peaksare taken into consideration, the apparent redshifts of the d −dandp−dtransitions could be attributed to the increased spectral weights of such additional excitations. Moreover, the d RuuOdependence of /H9275p−d, shown in Fig. 4/H20849b/H20850, is also consistent in the filling-controlled picture. As x increases, the amount of hole doping for the Ru ions in-creases, so d RuuOcould decrease.13,14In addition, with the increase of x, the mid-gap state becomes stronger, resulting in the apparent redshifts of /H9275p−d, as we addressed in theabove paragraph. Therefore, /H9275p−dwill decrease at smaller dRuuO. Although the filling-controlled MIT picture is consistent with the observed spectral changes of YBRO, it should benoted that the low frequency spectral distribution of YBROis too broad to clearly identify the appearance of the mid-gapstate. In recent photoemission studies, Park et al. observed an increase of the spectral weight near E F,12but the spectral features in the photoemission spectra were also too broad toclearly distinguish the appearance of the mid-gap state. Thebroad spectral features in the mid-infrared peak and the pho-toemission spectra might be related to the extended nature ofthe Bi 6 porbitals, 8–10which are hybridized with the Ru t2g bands. D. Comparison with the spectral changes of Y 2−xCaxRu2O7,a filling-controlled system To obtain further supports for the possible scenario of the YBRO MIT mechanism, we investigated absorption spectraof Y 2−xCaxRu2O7/H20849YCRO /H20850, which shows a MIT around x =0.5.22Since Y and Ca ions have different ionic states, as 3+ and 2+, respectively, the hole doping for the Ru 4 dstates should occur and increase with the increase of x. Therefore, the YCRO compounds will work as a model system with aclear filling-controlled MIT. Figure 5 shows the absorption spectra of the YCRO com- pounds between 0.5 and 4 eV. The spectra below 0.5 eVcould not be measured due to the multi-phonon absorption ofthe KBr pellet. For the Y 2Ru2O7sample, its absorption spec- trum is quite similar to its /H9268/H20849/H9275/H20850, displayed in the bottom of Fig. 3, demonstrating the validity of our approaches using transmittance spectra. It is quite interesting to find thatx-dependent changes of the absorption spectra for YCRO compounds are quite similar to those of /H9268/H20849/H9275/H20850for the YBRO compounds, shown in Fig. 3. Characteristic spectral features in YBRO, i.e., the redshifts of the d−dtransition peak and thep−dtransition peak, can also be observed in the absorp- tion spectra of YCRO. These similarities of optical spectra FIG. 4. /H20849a/H20850Doping-dependent changes of the p−dtransition energy /H9275p−dand /H20849b/H20850RuuO bond length dRuuOdependence of /H9275p−d. FIG. 5. Absorption spectra of Y 2−xCaxRu2O7at room tempera- ture. All of the spectra are shown in arbitrary units.OPTICAL INVESTIGATIONS ON Y 2−xBixRu2O7:… PHYSICAL REVIEW B 72, 035124 /H208492005 /H20850 035124-5between YBRO and YCRO strongly support our proposal that the MIT of the YBRO should originate from the dopingeffects by the Bi cations. E. Detailed mechanism of the doping effects Asxincreases, the Ru t2gbandwidth should become larger through the hybridization between the Bi 6 pstates and the Ru t2gstates.8–10,29However, our optical investigations clearly demonstrated that the hole doping will play a moreimportant role in determining the MIT in the YBRO com-pounds. How will the Bi substitution result in the hole dop-ing? According to the LDA calculations on the pyrochloreruthenates by Ishii and Oguchi, 9the electron orbitals at the Y site will not be mixed with the Ru t2gorbitals, but the unoc- cupied Bi 6 porbitals should be strongly hybridized with the Rut2gorbitals. These results imply that the substitution of the Y ion with the Bi ion could give rise to the hole-dopinginto the Ru t 2gbands. In particular, the extended wave func- tion of 6 pstates of the easily polarizable Bi cation makes the self-doping occur. According to the LDA calculations by Ishii and Oguchi,9 an antibonding band of the Tl 6 sand the O 2 porbitals in pyrochlore Tl 2Ru2O7is expected to cross EF, which could provide a self-doping to the Ru t2gstates. In our previous optical studies on Tl 2Ru2O7,17,18we reported an incoherent mid-infrared peak exhibiting unusual temperature depen-dences. Contrary to the YBRO case, the mid-infrared peakfeature could easily be distinguishable from other peaks, andits temperature dependences are similar to the doping depen-dences of the mid-infrared peak in an externally doped Mottinsulator. 34,35Therefore, we explained the temperature- dependent MIT of Tl 2Ru2O7in terms of a self-doping effect for the Ru ions by the easily polarizable Tl cation.17,18Note that similar self-doping has been observed for Tl- and Bi-based high-temperature superconductors. 36–39 It should be also noted that the geometrical frustration of the pyrochlore compounds might play important roles in de-termining their electronic structures. 40–42In the pyrochlore structures where the magnetic ions are located at vertices oftetrahedra, the antiferromagnetic interaction will experience a strong geometrical frustration. When the frustration is verystrong, heavy quasiparticles are formed around the Fermilevel by suppressing the short-range antiferromagnetic fluc-tuations. By decreasing the frustration, the heavy quasiparti-cle band splits, and the pseudogap begins to develop aroundthe Fermi level. We expect that such geometrical frustrationeffects could also influence the spectral evolutions of YBRO.Actually, a recent study on the magnetic and electric proper-ties of YBRO by Tachibana et al. demonstrated that there exists a strong coupling between charge and spin degrees offreedom, which could be a characteristic feature of a stronglycorrelated and geometrically frustrated system. 43However, the consideration of such geometrical frustration effects goesbeyond our current investigation. Further studies are stronglydesirable to understand how the geometrical frustrationshould modify the electronic structure changes in thebandwidth-control and the filling-control pictures. 44 VI. SUMMARY We reported the doping-dependent optical conductivity spectra of the cubic pyrochlore Y 2−xBixRu2O7/H208490.0/H33355x /H333552.0/H20850and investigated the mechanism of a metal-insulator transition of these alloy compounds. We reported that the spectral features below 5 eV exhibit large changes with avariation of x. We demonstrated that they should be under- stood in terms of the change in the filling state of the Ruions. Together with the previous optical studies onTl 2Ru2O7,17,18this work suggests that the metallic behaviors of some pyrochlore ruthenates, such as Bi 2Ru2O7and Tl2Ru2O7/H20849above 125 K /H20850, should be understood in terms of the self-doping for the Ru 4 dstates by the easily polarizable cations, i.e., Bi and Tl. ACKNOWLEDGMENTS We would like to thank S. Fujimoto for valuable discus- sions. 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A further investigation about this intriguing feature may provide us withan important clue to understanding the electrodynamics of themetallic pyrochlore compounds.OPTICAL INVESTIGATIONS ON Y 2−xBixRu2O7:… PHYSICAL REVIEW B 72, 035124 /H208492005 /H20850 035124-7

PhysRevB.86.184111.pdf

PHYSICAL REVIEW B 86, 184111 (2012) First-principles investigations of the atomic, electronic, and thermoelectric properties of equilibrium and strained Bi 2Se3and Bi 2Te3including van der Waals interactions Xin Luo, Michael B. Sullivan, and Su Ying Quek* Institute of High Performance Computing, Agency for Science, Technology and Research, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632 (Received 10 July 2012; revised manuscript received 7 November 2012; published 27 November 2012) Bi2Se3and Bi 2Te3are layered compounds of technological importance, being excellent thermoelectric materials as well as topological insulators. We report density functional theory calculations of the atomic,electronic, and thermoelectric properties of strained bulk and thin-film Bi 2Se3and Bi 2Te3, focusing on an appropriate description of van der Waals (vdW) interactions. The calculations show that the van der Waalsdensity functional (vdW-DF) with Cooper’s exchange (vdW-DF C09x) can reproduce closely the experimental interlayer distances in unstrained Bi 2Se3and Bi 2Te3. Interestingly, we predict atomic structures that are in much better agreement with the experimentally determined structure from Nakajima than that obtained from Wyckoff,especially for Bi 2Se3,where the difference in atomic structures qualitatively changes the electronic band structure. The band structure obtained using the Nakajima structure and the vdW-DFC09xoptimized structure are in much better agreement with previous reports of photoemission measurements, than that obtained using the Wyckoffstructure. Using vdW-DF C09xto fully optimize atomic structures of bulk and thin-film Bi 2Se3and Bi 2Te3under different in-plane and uniaxial strains, we predict that the electronic bandgap of both the bulk materials andthin films decreases with tensile in-plane strain and increases with compressive in-plane strain. We also predict,using the semiclassical Boltzmann approach, that the magnitude of the n-type Seebeck coefficient of Bi 2Te3 can be increased by the compressive in-plane strain while that of Bi 2Se3can be increased with tensile in-plane strain. Further, the in-plane power factor of n-doped Bi 2Se3can be increased with compressive uniaxial strain while that of n-doped Bi 2Te3can be increased by compressive in-plane strain. Strain engineering thus provides a direct method to control the electronic and thermoelectric properties in these thermoelectric topological insulatormaterials. DOI: 10.1103/PhysRevB.86.184111 PACS number(s): 62 .25.−g, 61.50.Lt, 71.15.Nc, 71 .20.Mq I. INTRODUCTION Bi2Se3and Bi 2Te3are members of the (Bi, Sb) 2(Te, Se) 3family of traditional thermoelectric materials—they can directly convert waste heat to electricity without any moving parts. These bulk thermoelectric materials were discoveredto have large Seebeck coefficients half a century ago andare now widely used in thermoelectric refrigeration. 1–3In recent years, there has been a surge of renewed interest in these thermoelectric materials—it was predicted and later, experimentally demonstrated,4–7that these materials consti- tute an exotic class of condensed matter, called topologicalinsulators. 8–10The topological insulators are distinguished by the existence of metallic spin-helical surface states, which are robust against the presence of nonmagnetic impurities ordisorder. 11,12These surface states have potential applications in spintronics,9,13quantum computation,8,10,14and thermo- electric energy conversion.15Importantly, these applications require a better fundamental understanding of the atomic and electronic structure of Bi 2Se3and Bi 2Te3when interfaced with other materials. It is well known that Bi 2Se3and Bi 2Te3belong to the tetradymite-type crystal with a rhombohedral structure (pointgroup R-3m). In the rhombohedral unit cell [Fig. 1(a)], there are three Se (Te) atoms that can be classified intotwo inequivalent types. We label these inequivalent atoms asSe 1(Te 1) (two of them in one unit cell) and Se 2(Te 2). The Bi atoms are equivalent. The Bi 2Se3and Bi 2Te3structure is also often alternatively described in the hexagonal representationwith a unit cell of 15 atoms, as shown in Fig. 1(b). Within thisrepresentation, it is clear that Bi 2Se3and Bi 2Te3have layered structures. Each Se 1(Te 1)-Bi-Se 2(Te 2)-Bi-Se 1(Te 1)f o r m sa so-called quintuple layer (QL), which is a slab with five atomiclayers. The QLs are stacked along the caxis with the weak van der Waals (vdW) interactions between neighboring QLs.The vdW interaction is relatively weak, but it can play a dom-inant role in interactions between atoms or layers separatedby empty space (so-called sparse matter). This interactionresults exclusively from long-range correlations, which areabsent from standard local and gradient-corrected densityfunctional theory (DFT) functionals. 16,17Much significant advancement has since been made that enables the treatmentof vdW interactions within DFT. In addition to the methodof dispersion-correction as an add-on to DFT, 18the recently developed van der Waals density functional (vdW-DF)19,20 incorporates the long-range dispersion effects as a perturbation to the local-density approximation (LDA) correlation term,and this method has been applied successfully in diversematerial systems. 17The choice of exchange functional is also important—the standard functional used within vdW-DF, revised Perdew–Burke–Ernzerhof (revPBE) 21typically gives vdW bond lengths that are a few percent too large.22 Most recently, Cooper developed an exchange functional thatreduces the short-range repulsion term in revPBE. 23 Although extensive electronic structure calculations have been performed for Bi 2Se3and Bi 2Te3,13,24–31most of them are calculated with experimental structures without full relaxationor in the slab calculations, with only the top four layers ofatoms in the top QL allowed to relax, fixing the inter-QLdistance. 13,32 184111-1 1098-0121/2012/86(18)/184111(15) ©2012 American Physical SocietyXIN LUO, MICHAEL B. SULLIV AN, AND SU YING QUEK PHYSICAL REVIEW B 86, 184111 (2012) FIG. 1. (Color online) Atomic structures of bulk Bi 2Te3and Bi2Se3. One QL contains five atoms in Se 1(Te 1)-Bi-Se 2(Te 2)-Bi- Se1(Te 1) series. (a) Rhombohedral unit cell. (b) Hexagonal unit cell (containing three QLs) Yet, inter-QL vdW interactions are essential for predicting atomic and electronic structures of Bi 2Se3and Bi 2Te3, when interfaced or intercalated with other materials. Indeed, muchof the current interest in these materials involves interfacingthem with other materials, and recent experiments indicatethat depositing Ag on Bi 2Se3results in Ag intercalation between QLs.11,33Furthermore, vdW interactions are required for accurate predictions of atomic structures of Bi 2Se3and Bi2Te3under strain, which can directly influence topological properties.28,34,35On the other hand, previous theoretical calculations found that pressure and uniaxial stress can greatlyinfluence the thermoelectric properties of Sb 2Te3,36and stress also plays an important role in the formation of defectsin these thermoelectric materials. 37,38Recent experiments and molecular simulations show that the lattice thermalconductivity of thermoelectric materials will be affected bydifferent strain conditions. 39,40How important is the vdW interaction in strain engineering, and how do they affect thethermoelectric properties of these materials? In this paper, we first explore the applicability of different exchange-correlation functionals, including those with vdWcorrections, on predicting atomic structures of Bi 2Se3and Bi2Te3. Next, using an appropriate vdW functional, we fully optimize the atomic structures of strained bulk and thin-filmBi 2Se3and Bi 2Te3. Based on these optimized reference structures, the effect of strain on atomic, electronic, and ther- moelectric properties are reported, and the importance of vdW interactions is elucidated by comparing the results with thoseobtained using structures optimized with the Perdew–Burke–Ernzerhof (PBE) functional (including spin-orbit interactions).We also explore the effects of vdW interactions and spin-orbitinteractions on the bulk moduli and phonon frequencies in theunstrained bulk systems.II. COMPUTATIONAL DETAILS Except for the thermoelectric transport properties (ad- dressed below), all our calculations are performed usingthe plane wave DFT code, Quantum-ESPRESSO (QE). 34 The norm-conserving pseudopotentials are generated usingthe Rappe-Rabe-Kaxiras-Joannopoulos (RRKJ) approach. Forstructural relaxation, the plane-wave kinetic energy cutoff isset to 56 Ry and the Brillouin zone is sampled with a 9 ×9×1 Monkhorst-Pack mesh—using a higher plane-wave cutoff of70 Ry and a 13 ×13×1k-point mesh changes the lattice constants and internal coordinates by less than 0.2%, withessentially no difference in resulting band structure. Phononfrequencies in the bulk are computed using a dense k-point mesh of 13 ×13×13. On the other hand, a plane-wave cutoff of 40 Ry is found to be sufficient (compared to 56Ry) for calculations of band structure and bulk moduli. Avacuum thickness of 16 ˚A is used in thin-film slab calculations (converged relative to a vacuum thickness of 20 ˚A). In the self-consistent calculation, the convergence threshold forenergy is set to 10 −9eV . All the internal atomic coordinates and lattice constants are relaxed, until the maximum componentof Hellmann-Feynman force acting on each ion is lessthan 0.003 eV /˚A. The spin-orbit coupling (SOC) effect, important for the heavy elements considered here, is treatedself-consistently in fully relativistic pseudopotentials for thevalence electrons. 41In order to investigate the importance of SOC and vdW interactions for different physical properties,we have performed a detailed investigation with differentexchange-correlation functionals. However, the main physicalinsights are obtained using atomic structures optimized usingthe vdW-DF functional 19,20,42with Cooper’s exchange23and electronic structure calculated using the PBE43functional with SOC. Many thermoelectric calculations44–46of similar materials are based on the WIEN2k package47with the semiclas- sical Boltzmann transport method in the relaxation-timeapproximation. 48Therefore, to compare our results with the literature, we use WIEN2k with BoltzTrap45for calculating thermoelectric transport properties of our previously relaxedstructures, with the same basis functions as in the reportedliterature, 44,45and using the PBE43functional with SOC49 as implemented in WIEN2k. The calculation of transport properties requires a very dense kgrid; here, a nonshifted mesh with 56 000 kpoints (4960 in the irreducible Brillouin zone) is used, which is found to be converged as compared toa denser sampling with 70 000 kpoints. Within BoltzTrap, the relaxation time τis assumed to be a constant with respect to the wave vector kand energy around the Fermi level, and the effect of doping is introduced by the rigid bandapproximation. Within the relaxation-time approximation, theSeebeck coefficient Scan be obtained directly from the electronic structure without any adjustable parameters. III. RESULTS AND DISCUSSIONS A. Structural properties of equilibrium Bi 2Se3and Bi 2Te3 Table Ishows the fully optimized lattice constants and internal coordinates obtained using different functionals. Thedifferent functionals give consistent internal coordinates, but 184111-2FIRST-PRINCIPLES INVESTIGATIONS OF THE ... PHYSICAL REVIEW B 86, 184111 (2012)TABLE I. Structural parameters for Bi 2Se3and Bi 2Te3. Within the rhombohedral representation, the unit cell is described by the lattice constant a0and angle α, the atom Se 2(Te 2) is set to be the origin (0, 0, 0), and the two Bi atoms occupy sites ±(μ,μ,μ ), while the two Se1(Te1) atoms are located at ±(υ,υ,υ ). The corresponding hexagonal cell edge is denoted bya0/primeandc0/prime, and the equilibrium inter-QL distance ( deqm) in hexagonal cell is also shown (see Fig. 1for notation). Exp. 1aExp. 2bLDA LDA +SOC PBE PBE +SOC revPBE PBE-D2 PBE-D2 +SOC vdW-DFC09x vdW-DFrevPBEx Bi2Se3 a0(˚A) 9.8405 9.841 9.616 9.552 10.889 10.617 12.458 9.938 9.8774 9.8771 11.18 α(o) 24.304 24.273 24.593 24.8507 22.123 22.792 19.493 23.905 24.152 24.105 21.895 μ(Bi) 0.4008 0.399 0.4009 0.4016 0.3946 0.3959 0.3868 0.3992 0.3998 0.4001 0.3946 υ(Se) 0.2117 0.206 0.2097 0.2081 0.2228 0.2199 0.2369 0.2127 0.2113 0.2115 0.2235 a0/prime(˚A) 4.143 4.138 4.096 4.111 4.178 4.195 4.218 4.116 4.133 4.125 4.246 c0/prime(˚A) 28.636 28.64 27.96 27.758 31.86 31.01 36.65 28.95 28.755 28.757 32.72 deqm(˚A) 2.579 2.253 2.406 2.298 3.574 3.302 5.145 2.668 2.570 2.580 3.722 2Q Ldeqma(˚A) 2.579 2.253 2.380 2.28 3.58 3.38 >4.48 2.68 2.68 2.48 3.58 Bi2Te3 a0(˚A) 10.476 10.473 10.175 10.124 10.829 10.628 12.399 10.664 10.565 10.368 11.138 α(o) 24.166 24.167 24.574 24.806 23.647 24.223 20.678 23.305 23.703 24.277 23.449 μ(Bi) 0.4000 0.400 0.4004 0.4012 0.3979 0.3996 0.3900 0.3985 0.3990 0.3999 0.3972 υ(Te) 0.2095 0.212 0.2083 0.2063 0.2146 0.2109 0.2294 0.2120 0.2099 0.2094 0.2168 a0/prime(˚A) 4.386 4.384 4.331 4.349 4.438 4.460 4.450 4.308 4.340 4.360 4.527 c0/prime(˚A) 30.497 30.487 29.589 29.423 31.565 30.934 36.390 31.109 30.792 30.174 32.48 deqm(˚A) 2.612 2.764 2.466 2.331 3.025 2.736 4.566 2.822 2.651 2.582 3.259 2Q Ldeqmc(˚A) 2.612 2.764 2.41 2.31 3.11 2.81 >4.31 2.71 2.71 2.51 3.41 aFrom Ref. 58(Nakajima). bFrom Ref. 59(Wyckoff). cdeqmfor 2 QL is obtained from Fig. 2. 184111-3XIN LUO, MICHAEL B. SULLIV AN, AND SU YING QUEK PHYSICAL REVIEW B 86, 184111 (2012) the lattice parameters, especially that related to the interlayer distance deqm(vertical distance between Se (or Te) atoms in adjacent QLs), vary significantly with different functionals.Focusing first on the LDA, PBE, and revPBE functionals,we find that LDA 50results are closest to experiment, with LDA underestimating deqmby 6.7 and 5.6% for Bi 2Se3and Bi2Te3, respectively, and PBE43overestimating deqmby 38.6 and 15.8% for Bi 2Se3and Bi 2Te3, respectively. The revPBE21 functional predicts an even larger deqmof>4.48 ˚A. Calcu- lations with the vdW-DF correlation functional together withrevPBE exchange lead to improved, smaller d eqmcompared to the revPBE functional; however, the predicted deqmis still overestimated by more than 20%. This error is much largerthan that reported for other vdW-bonded systems. 22In contrast, the semiempirical vdW correction based on Grimme’s scheme(PBE-D2) 18,51,52does help, predicting deqmwithin 3.4 and 8% of experiment for Bi 2Se3and Bi 2Te3, respectively. On the other hand, using the vdW-DF functional19,53with the most recently developed Cooper’s exchange (vdW-DFC09x),23we can obtain better agreement with experiment, with predicted deqmvalues of 2.58 ˚Af o rB i 2Se3and 2.582 ˚Af o rB i 2Te3, which are within 0.1 and 1.1% of experiment. We next consider the role of SOCin relaxing atomic structures. We find that SOC tends to reduced eqmin all cases; as a result, adding SOC increases the error for LDA but reduces the error for PBE and PBE-D2, thus givingd eqmvalues within 1.5% of experiment for PBE-D2 +SOC. As the vdW-DF functional has not been implemented with SOC,we did not check the effect of SOC on these results. However,since vdW-DF C09xcan already give excellent agreement with experiment, it is still unclear if SOC is truly important forstructural optimization. We note that the very reasonableresults of PBE-D2 and PBE-D2 +SOC are quite remarkable given that the correction is semiempirical. However, in thefollowing, we shall base most of our conclusions on structuresoptimized using the vdW-DF C09xfunctional. Since the interlayer vdW interactions are important, we focus on these interactions by computing, for simplicity, theinter-QL binding energy as a function of interlayer distance,for two QL Bi 2Se3and Bi 2Te3thin films (i.e., films with two QLs each containing five atomic layers), where the internalatoms within each QL are fixed to their experimental atomicpositions from the bulk (Fig. 2). Although the inter-QL distance for the two-QL film may be slightly different fromthe bulk, it has been shown that the vdW interactions aredominated by those between nearest-neighbor layers in otherlayered materials such as multilayer graphene and MoS 2.54,55 The close correspondence (with deviation of less than 5% for all functionals) between the equilibrium inter-QL distancescalculated for the bulk and for two QLs (Table I) further supports this assumption. From Fig. 2, we see that revPBE and revPBE +SOC give rise to repulsive interactions between the two QLs. Furthermore, the curvature of the energy versusdistance curve is quite different for different functionals.Although SOC does not change d eqmsignificantly, it does affect the interlayer force constants, which are given by thesecond derivative of the energy as a function of distance.We also note that the binding energies obtained using vdW-DF C09xor PBE-D2 are consistent with recent estimates obtained by rescaling results from the VV10 functional56 according to quantitative random-phase approximation (RPA) FIG. 2. (Color online) Inter-QL binding energy as a function of inter-QL separation distance for (a) Bi 2Se3and (b) Bi 2Te3two-QL thin films. The inter-QL energy is computed by taking the differencebetween the total energy of the two-QL film and twice that of a one QL film in the same unit cell. Here, the internal atoms within each QL are fixed to their experimental atomic positions from the bulk. Thegrey line shows the value of experimental distance in bulk, which is from Ref. 58. calculations.57These binding energies are larger for Bi 2Te3 than for Bi 2Se3, a finding consistent with the larger atomic size of Te compared to Se. Interestingly, we find that our first-principles-predicted structures using both PBE-D2 and vdW-DFC09xfunctionals are more consistent with the experimental structures reportedby Nakajima, 58rather than that from Wyckoff (Table I).59 Both experimental structures have been widely used in the theoretical literature, but importantly, the internal atomiccoordinates are different in these two structures (Table I), especially for Bi 2Se3, where the interlayer distance deqmis quite different (2.579 ˚A in Nakajima’s structure and 2.253 ˚A in Wyckoff’s structure). It is noted that the crystal structurefrom Wyckoff is obtained from an early electron diffractionstudy, 60while structures of Nakajima are obtained from x-ray diffraction powder analysis. As the interaction of electronswith matter are about 10 000 times stronger than that of xrays,multiple dynamical scattering will influence the intensityof electron diffraction patterns, thus making the structure 184111-4FIRST-PRINCIPLES INVESTIGATIONS OF THE ... PHYSICAL REVIEW B 86, 184111 (2012) FIG. 3. (Color online) Band structures of bulk (a) Bi 2Se3and (b) Bi 2Te3, computed using the GGA-PBE functional with SOC. The experimental ARPES data for Bi 2Se3(Ref. 61) is shown along the /Gamma1andZhigh-symmetry direction. The band structures of (c) Bi 2Se3and (d) Bi2Te3without SOC are also calculated with vdW-DFC09xand PBE functionals. determination from electron diffraction more difficult and less reliable than that from x-ray diffraction.58 B. Electronic properties of equilibrium Bi 2Se3and Bi 2Te3 Using PBE +SOC, we compute the band structures of the Nakajima and Wyckoff atomic structures and the vdW-DFC09x optimized structures. We find that the band structures for the vdW-DFC09xoptimized structures match very well to those of the Nakajima structures, as we expect from the abovediscussion. The band structures of the Wyckoff and Nakajimastructures are very similar in the case of Bi 2Te3[Fig. 3(b)] but are qualitatively different for Bi 2Se3[Fig. 3(a)]. Focusing now on these differences for Bi 2Se3, we note that the lowest conduction band along the /Gamma1-Z-Fhigh-symmetry direction is more rippled in the Wyckoff structure than in the Nakajima andvdW-DF C09xoptimized structures [Fig. 3(a).] Furthermore, the direct gap at Fis 0.40 eV smaller in the Wyckoff Bi 2Se3 structure than in the Nakajima structure, while the bandgap at /Gamma1is 0.20 eV smaller in the Nakajima Bi 2Se3structure than in the Wyckoff structure. (We note that the calculated bandstructures of Bi 2Se3based on the PBE-D2 +SOC relaxed structure and that of vdW-DFC09xoptimized structures are almost the same; not shown here.) Comparing the computedband structure with that measured in recent angle-resolvedphotoelectron spectroscopy (ARPES) experiments, 61we find that the band structure for the Nakajima structure is closer toexperiment than that for the Wyckoff structure, thus providingdirect experimental evidence favoring the coordinates from Nakajima and our optimization procedure. Specifically, thereare two key qualitative features indicating that the ARPESmeasurements agree better with the band structure for theNakajima structure. Focusing on the band structure along/Gamma1-Z, close to the Fermi level, we see (1) the highest-occupied valence band is flatter in the Nakajima structure, similar to thatfrom ARPES; (2) the energy separation between the highest-occupied valence band and the next-highest-occupied band isabout 0.75 eV for the Nakajima structure, in contrast to 0.3 eVfor the Wyckoff structure and similar to that from ARPES. Itis thus clear that agreement is much better for the Nakajimastructure than for the Wyckoff structure. Interestingly, theauthors of the ARPES paper 61had commented that the measured energy bands in the /Gamma1-Zdirection were significantly flatter than that predicted by DFT; our calculations stronglysuggest that the reason for this discrepancy was that theWyckoff structure was used for their calculations instead ofthe Nakajima structure. Since the main difference betweenthe Nakajima and Wyckoff Bi 2Se3structures is the larger interlayer separation in the former, these results underscorethe importance of a careful treatment of vdW interactions forprediction of band structures. We note that although DFT in many cases gives accurate qualitative predictions of band structure, the DFT Kohn-Sham values cannot quantitatively predict quasiparticle bandstructures. 62In this case, the calculated bandgaps of Bi 2Se3and Bi2Te3(using the vdW-DFC09xoptimized structures) are 0.30 184111-5XIN LUO, MICHAEL B. SULLIV AN, AND SU YING QUEK PHYSICAL REVIEW B 86, 184111 (2012) FIG. 4. (Color online) Thickness- dependent bandgaps at the /Gamma1point induced by the interaction between the surface states of thin film for (a) Bi 2Se3and (b) Bi 2Te3. Experimental results are reproduced fromRef. 66, PBE data are obtained from Ref. 65, and the GW calculation data are derived from Ref. 64. The slab structures in PBE and GW calculations are based on the Wyckoff structures. and 0.10 eV , slightly smaller than the experimentally measured values of 0.35 eV for Bi 2Se363and 0.17 eV for Bi 2Te36but consistent with the other DFT results.13,26 Although our calculations suggest that SOC is not crucial for predicting atomic structures, SOC has an important effecton the band structure. To illustrate this, we compare theband structures in Figs. 3(a) and 3(b) with those computed without SOC [Figs. 3(c) and 3(d)], using the vdW-DF C09x optimized structures. The band structures with and without SOC are quite different, consistent with large SOC-inducedband inversion at /Gamma1. Furthermore, we note that the band structures computed by the vdW-DF C09xfunctional and PBE are quite similar, suggesting that the vdW-DFC09xfunctional can produce band structures that are consistent with PBEand that vdW interactions have minimal effect on the bandstructure. This justifies our neglect of vdW effects in the bandstructure calculations. Recent interest in the topological metallic surface states in Bi 2Se3and Bi 2Te3slabs focus on the Dirac cone feature in the band structure at the /Gamma1point.13,30,64–66At very thin slab thickness, the interaction between the surface states atopposite interfaces opens the bandgap at /Gamma1, and this gap will gradually decay to zero as the slab thickness is increased.Previous PBE (with SOC) calculations on the unrelaxed slabsderived from Wyckoff structures found that the bandgap at /Gamma1 begins to close by three and four QLs for Bi 2Se3and Bi 2Te3, respectively.65To address the question of whether the atomic structure influences these predictions, we fully optimizedslab structures from one to five QLs using the vdW-DF C09x functionals; the fully optimized slab structures are similar to those directly derived from Nakajima structures. Based on theoptimized slab structures, we calculated the bandgap at /Gamma1with PBE+SOC, as shown in Fig. 4. Our results are close to that using the Wyckoff structures, except that the bandgap closesonly at four QLs instead of three QLs for Bi 2Se3. We note that compared to experiment and previous GW calculations,64the predicted gaps are too small. It is unclear if using the Nakajimainstead of the Wyckoff structures would affect the computedGW gaps; however, such calculations are beyond the scope ofthe current work.C. Bulk moduli and phonon frequencies of equilibrium Bi2Se3and Bi 2Te3 We further predict the bulk moduli and phonon frequencies of equilibrium Bi 2Se3and Bi 2Te3with vdW-DFC09xand other functionals and compare our predictions with availableexperimental literature. The bulk modulus Band its pressure derivative B /primecan be obtained by computing the changes in total energy of Bi 2Se3and Bi 2Te3with hydrostatic pressure and fitting the resulting energy-volume curves using theBirch-Murnaghan equation of state: E(η)=E 0+9BV 0 16[(η2−1)3B/prime+(η2−1)2(6−4η2)], (1) where η=(V0/V)1/3,V0is the equilibrium volume of the fully relaxed structure. The bulk moduli thus obtained arereported in Table II. In general, the values of Bpredicted for both Bi 2Se3and Bi 2Te3are similar ( ∼30–40 GPa). The experimental reports for Bdiffer, especially for Bi 2Se3, where an exceptionally high value of 53 GPa is obtained by Vilaplanaet al. 67However, the remaining experimental values are in reasonable agreement with our predicted values. Due tothe different experimental values, we are unable to draw aconclusion of the importance of vdW interactions or SOC indetermining the bulk modulus. More insights can be obtained by comparing computed zone center phonon frequencies with experiment. As canbe shown by group theory, the irreducible representationsof the zone center phonon modes in Bi 2Te3and Bi 2Se3are /Gamma1=2A1g+3A2u+2Eg+3Eu, among which 2 A1gand 2Eg are Raman (R) active; 3 A2uand 3Euare infrared (I) active, the three acoustic modes are composed by one A2uand two degenerate Eumodes. For the R-active modes, Egdescribes the shear mode of in-plane atomic vibrations, and A1gde- scribes the breathing mode of out-of-plane atomic vibrations.Except for results obtained with the vdW-DF functional, allphonon frequencies are computed within density functionalperturbation theory (DFPT) as introduced by Lazzeri andMauri 68in QE. Since the phonon frequencies cannot be 184111-6FIRST-PRINCIPLES INVESTIGATIONS OF THE ... PHYSICAL REVIEW B 86, 184111 (2012) TABLE II. The calculated bulk moduli B( G P a )o fB i 2Te3and Bi 2Se3. vdW-DFC09x PBE-D2 PBE-D2 +SOC LDA LDA +SOC PBE +SOC Exp. Bi2Se3 42.8 42.92 41.0 46.68 48.82 49.45 32.98a,5 3b Bi2Te3 40.2 35.61 32.5 42.46 46.56 42.55 39.47c, 32.5d aFrom Ref. 81. bFrom Ref. 67. cFrom Ref. 82. dFrom Ref. 83. computed with vdW-DF in QE, we adopt the force-constant approach69,70to compute the phonon frequencies with the vdW-DFC09xfunctional. In this method, we displace each atom in the primitive cell from its equilibrium position in the x, y, and zdirections by a distance of 0.015 ˚A and calculate the forces acting on each atom using the Hellmann-Feynmantheorem. Subsequently, the interatomic force-constant matrixis evaluated using a central finite-difference scheme. (We havechecked that within LDA, the phonon frequencies as calculatedwith the force-constant approach are essentially the same asthose obtained from the DFPT method.) First, comparing LDA and PBE results with and without SOC with experiment (Table III), we note that in all cases, SOC reduces the phonon frequencies. In general, inclusion of SOCthen leads to better agreement with experiment, as previouslyobserved by Cheng et al. 71We note that the importance of SOC in determining force constants is in contrast to our earlier ob-servation that SOC was not important for structural relaxation.The difference lies in the fact that force constants are related tothe rate of change of force with atomic displacements, whereasstructural relaxation is related only to the coordinates for theminima of the potential energy surface. This can be illustrated in the simplified picture in Fig. 2where the minima of the energy-distance curves are similar with and without SOC,but the curvatures of the curves are quite different. Next,although the PBE atomic structure is significantly differentfrom experiment compared to the PBE-D2 atomic structure, wefind that the PBE frequencies are quite similar to the PBE-D2frequencies, and in fact, the PBE-D2 frequencies are in manycases slightly farther from the experimental values. The PBE-D2 frequencies are quite similar to those obtained with thevdW-DF C09xfunctional. However, PBE-D2 +SOC frequen- cies give the best match with experiment, suggesting that PBE-D2+SOC can be used to make predictions on frequencies. D. Atomic and electronic properties of strained Bi2Se3and Bi 2Te3 Strain engineering is a mature technique for controlling the electronic properties of nanoscale semiconductors inindustry—mechanical strain can be imposed by microelec-tromechanical systems (MEMS) or by epitaxial growth of thin TABLE III. Calculated zone-center phonon frequencies for Bi 2Se3and Bi 2Te3bulks with and without SOC and the unit of frequency is in wave number per centimeter. The R- and I-active modes are denoted with R or I. LDA+SOC LDA PBE +SOCaPBEaPBE-D2 +SOC PBE-D2 vdW-DFC09x Exp. Bi2Se3Eg1(R) 42.80 44.272 38.893 42.128 38.83 43.92 43.048 37b A1g1(R) 75. 50 74.978 63.843 74.584 71.58 74.94 72.639 72.2c Eu1(I) 82.46 86.966 64.677 85.057 83.92 89.33 86.288 Eu2(I) 131.06 136.108 126.819 132.957 128.80 135.23 133.62 131.4c Eg2(R) 137.99 142.911 123.984 138.894 130.47 138.39 138.172 A2u1(I) 137.44 145.20 136.692 142.730 146.67 153.36 146.575 A2u2(I) 162.89 171.504 155.439 167.113 165.87 169.95 167.856 A1g2(R) 174.46 180.546 166.346 179.455 175.47 182.80 179.762 174c Bi2Te3Eg1(R) 42.04 43.34 35.457 36.358 38.51 41.84 40.737 34.356d A1g1(R) 62.64 65.6 53.869 53.903 59.99 61.54 62.184 62.042d Eu1(I) 64.28 69.4 48.399 63.142 66.77 76.18 68.945 Eu2(I) 94.92 99.65 91.228 97.399 95.46 102.68 100.393 Eg2(R) 104.67 112.72 95.931 104.404 102.75 111.19 109.670 101.735d A2u1(I) 96.8 104.03 95.064 102.569 104.30 111.89 103.956 A2u2(I) 120.5 129.47 118.613 128.220 126.81 137.21 128.644 A1g2(R) 131.91 140.38 127.219 137.2266 136.45 145.83 139.788 134.091d aPBE+SOC and PBE calculated data cited from Ref. 71. bReference 84. cReference 67. dReference 85. 184111-7XIN LUO, MICHAEL B. SULLIV AN, AND SU YING QUEK PHYSICAL REVIEW B 86, 184111 (2012) FIG. 5. (Color online) (a) and (b) Calculated bandgap as a function of in-plane strain applied to Bi 2Se3and Bi 2Te3(a) bulk materials and (b) two-QL thin films. (c) and (d) Bandgap at /Gamma1point as a function of in-plane strain for (c) bulk and (d) two-QL thin films. All band structures are calculated by PBE +SOC; the notation behind each material denotes which functional is used to relax the structure. films. To study the effect of strain on the atomic structures and electronic properties of Bi 2Se3and Bi 2Te3, we imposed different in-plane and uniaxial strains to the bulk material.The strain is defined by η=(α strain/α0−1)×100%, where αstrain is the lattice constant of the strained state, and α0is the optimized lattice constant of the unstrained bulk material. αis the in-plane lattice constant and out-of-plane lattice constantfor in-plane strain and uniaxial strain, respectively. In ourcalculation, a uniform in-plane strain from −3t o3 %i sa p p l i e d to Bi 2Se3and Bi 2Te3bulk and two-QL thin films. For the bulk material, the effect of uniaxial strain from −6t o6 % is also investigated. To predict the strained structures in thebulk, we start by constraining the in-plane (or out-of-plane)lattice constants to the strained state and then relaxing thelattice constant along the caxis (or uniformly in-plane), as well as the internal coordinates, with the vdW-DF C09xfunctional. Once we obtain the optimized atomic structures, we computethe electronic properties using PBE +SOC. With the in-plane strain changing from 3 to −3%, we find that the out-of-plane lattice constants of the relaxed bulkstructures increase approximately linearly, with a slope of0.342 ˚A (1.2% of out-of-plane lattice constant) and 0.353 ˚A (1.17% of out-of-plane lattice constant) per unit decrease inpercentage in-plane strain, for Bi 2Se3and Bi 2Te3, respectively. For uniaxial strain changing from 6 to −6%, a similar linear relationship was found, with the slope of 0.008 ˚A(0.19% of in-plane lattice constant) and 0.012 ˚A (0.27% of in-plane lattice constant) per unit decrease in percentageuniaxial strain for Bi 2Se3and Bi 2Te3, respectively. These results show that the out-of-plane lattice constant is stronglycoupled with the in-plane lattice constant when the structuresare optimized under different strain. The different percentage changes suggest strong anisotropy in the elastic properties of these layered materials. Although the overall band structuresof the strained systems are similar to those of the unstrainedones, the band structure near the Fermi level is influencedby the applied strains, resulting in significant changes in thebandgaps. Figure 5(a) shows the evolution of bandgap as a function of the in-plane strain for bulk Bi 2Se3and Bi 2Te3. Focusing on the vdW-DFC09xrelaxed structures, we note that the energy gap increased from 0.07 to 0.16 eV for bulkBi 2Te3, when the in-plane strain changed from 3 to −3%. For vdW-DFC09xoptimized bulk Bi 2Se3, the bandgap in general increases from extensive to compressive strain [from 0.25 eV(3.2% strain)] to 0.33 eV ( −1.8% strain). However, further increase in compressive strain to −2.8% reduces the bandgap (here, the kpoint for the valence-band maximum shifts from its original location near the Zsymmetry point to the shoulder of the M shape band near the /Gamma1high-symmetry point). To illustrate the importance of vdW interactions, we relaxed thestructures with both vdW-DF C09xand PBE +SOC functionals and then calculated the electronic structures with PBE +SOC. 184111-8FIRST-PRINCIPLES INVESTIGATIONS OF THE ... PHYSICAL REVIEW B 86, 184111 (2012) The bandgaps for the PBE +SOC optimized bulk Bi 2Te3 are similar to those for the vdW-DFC09xrelaxed structures. However, the bandgaps obtained for PBE +SOC optimized bulk Bi 2Se3are significantly different. These results are consistent with the fact that PBE +SOC overestimates deqm by 28% in bulk Bi 2Se3, but by only 4.7% in bulk Bi 2Te3. Moving now to the two-QL thin films, optimized using vdW-DFC09x, we note that the optimized in-plane lattice con- stants are smaller than the bulk—about 0.3 and 1.0% smallerfor Bi 2Te3and Bi 2Se3, respectively. However, the equilibrium inter-QL distances ( deqmis 2.589 ˚Af o rB i 2Se3and 2.595 ˚A for Bi 2Te3in the fully relaxed two-QL thin films) are similar to those obtained in Fig. 2. The bandgap for two-QL films is smaller than that for the bulk, because the bandgap in two-QLfilms is determined by interactions between metallic surfacestates on both sides of the film. As observed for the bulk,the bandgap for the two-QL films increases with compressivestrain and decreases with tensile strain, with the exceptionof Bi 2Te3thin films, where there is a very small decrease in bandgap at −3% compressive strain [Fig. 5(b)]. Further, the Bi2Se3and Bi 2Te3thin films become metallic when the tensile strain is larger than 3 and 1%, respectively. We note that inthis case, using the PBE +SOC relaxed structures results in bandgaps that are about twice as large or more, although thegeneral trend of how the bandgap evolves with strain are con-sistent with those for the vdW-DF C09xoptimized structures. It should be noted that in both the thin films and bulk material, the bandgap is indirect. The direct bandgap at the /Gamma1 point is an important issue for topological insulators, becauseof the Dirac cones at /Gamma1for the thin films. Figures 5(c) and5(d) show the bandgaps at /Gamma1for both the bulk and the two-QL thin film. We note that Bi 2Se3has a smaller direct bandgap at /Gamma1 than Bi 2Te3but has a larger indirect bandgap; this explains why Bi2Se3is more widely studied for its potential applications as a topological insulator, where it is important to distinguishmetallic surface-state carriers from intrinsic bulk carriers.Except for Bi 2Se3two-QL films, the direct bandgap at /Gamma1tends to decrease from extensive to compressive strain. Predictionsusing PBE +SOC relaxed structures result in similar trends for Bi 2Te3but not for Bi 2Se3. We note that Young et al. has investigated the evolution of the topological phase of bulk Bi 2Se3under mechanical strain,28using regression fits to obtain bandgap stress and stiffness tensors (the linear and quadratic coefficients relatingthe/Gamma1-point bandgap to strain). From these tensors, it was predicted that the topological phase transition will occur at6.4% uniaxial strain in the out-of-plane direction, relative tothe experimental structure. Here, we apply uniaxial strainson bulk Bi 2Se3and Bi 2Te3, in each case fully optimizing the internal coordinates and in-plane lattice constants usingthe vdW-DF C09xfunctional. In contrast to the case of in- plane strain, which affects the out-of-plane lattice constantsignificantly, out-of-plane uniaxial strain has much less effecton the in-plane lattice constant. In general, the gaps decreasewith increasing tensile uniaxial strain [Fig. 6(a)], and there is no clear quadratic or linear relation of the uniaxial strainat/Gamma1-point bandgap in Bi 2Se3. Our calculations also predict a topological phase transition (closing of /Gamma1-point bandgap) at 6% uniaxial strain for Bi 2Se3[Fig. 6(b)], which interestingly, is approximately consistent with the prediction by Young et al. FIG. 6. (Color online) (a) Direct bandgap at /Gamma1point as a function of uniaxial strain in bulk Bi 2Se3and Bi 2Te3. (b) The topological phase transition at 6% uniaxial stain for Bi 2Se3. Finally, we note that the /Gamma1-point bandgap is more sensitive to uniaxial strain than the indirect bandgap. To assess the practical feasibility of strain engineering, we compare the strain energy (the energy differences betweenthe strained states and their corresponding unstrained coun-terparts) with the vdW interaction energy. The strain energyis hundredths of electron volts per unit cell, and even at3% in-plane strain or 6% uniaxial strain, is still less than0.1 eV /unit cell. This is quite small compared to the vdW inter-QL interaction energy of about 0.20–0.25 eV /unit cell (Fig. 2), indicating that these layered compounds are likely to be able to withstand such strains without surface exfoliation,the prospects of using strain (e.g., via MEMS) to engineer theirbandgaps are promising. E. Thermoelectric properties As mentioned before, Bi 2Te3and Bi 2Se3are traditional thermoelectric materials that can generate electricity fromwaste heat once a temperature gradient exists. 46,72–74The thermoelectric performance is quantified by the figure of merit, 184111-9XIN LUO, MICHAEL B. SULLIV AN, AND SU YING QUEK PHYSICAL REVIEW B 86, 184111 (2012) FIG. 7. (Color online) (a) and (b) The Seebeck coefficient Sof vdW-DFC09xoptimized, bulk (a) Bi 2Te3and (b) Bi 2Se3as a function of dopant concentrations, with different in-plane strains. (c) and (d) For comparison, the results based on the PBE +SOC relaxed structure are shown in (c) and (d) for Bi 2Te3and Bi 2Se3, respectively. Please note that the scales are different for the different panels. ZT, where Tis the temperature and Zis defined by3 Z=S2σ (κe+kL), (2) andSis the Seebeck coefficient, σthe electronic conductivity, andκeandkLare the electronic and lattice thermal conduc- tivities, respectively. S2σis also known as the power factor. Superlattice materials based on Bi 2Te3have resulted in high ZTvalues >1.3A higher ZTcan be obtained by increasing the power factor and decreasing the thermal conductivity.75 In the following, we shall focus on analyzing the effect ofstrain on the power factor and the Seebeck coefficient, whichcan be computed within the linear response regime in thesemiclassical Boltzmann framework, 45as σ≡q2L0 (3) S=kB qL1 L0(4) with Lj=/integraldisplay∞ −∞−∂f0 ∂ED(E)v2τ/parenleftbiggE−μ kBT/parenrightbiggj dE, where qis the charge of carriers, f0is the Fermi distribution function of electrons, vis the Fermi velocity, τis the relaxation time,μis the chemical potential, and D(E) is the density ofstates. Using a constant relaxation-time approximation, the Seebeck coefficient can be completely determined from theband structure. We note that several scholars have done some pioneering thermoelectric studies on related materials, using PBE +SOC optimized structures or experimental structures. 36,44It was shown that the semiclassical Boltzmann transport methodwithin the relaxation-time approximation can predict thermo-electric properties of unstrained Bi 2Te3, in good agreement with experiment.44Furthermore, it was predicted that strain engineering can increase the power factor in Sb 2Te3.36,38 More recently, studies on Bi and Sb tellurides show that the lattice constants and volume expansion have an importantinfluence on the temperature behavior of Seebeck coefficient, 76 and strain can also play an important role in the anisotropyof electrical conductivity and Seebeck coefficient. 77Since most of these studies are carried out in WEIN2k using thesemiclassical Boltzmann transport method, 45for the sake of comparison, we use the same method (and the same,tested parameters from the literature 44) for calculation of thermoelectric properties.78 We study the effect of in-plane and out-of-plane strain on the in-plane Seebeck coefficient and power factor of bulkBi 2Se3and Bi 2Te3undern- andp-type doping, and at different temperatures, using the vdW-DFC09xoptimized structures. 184111-10FIRST-PRINCIPLES INVESTIGATIONS OF THE ... PHYSICAL REVIEW B 86, 184111 (2012) FIG. 8. (Color online) Seebeck coefficient Sof Bi 2Te3and Bi 2Se3under various strained states and both p-type doping and n-type doping are shown. The doping level is fixed to 1019cm−3. Please note that the scales are different for different panels. (The out-of-plane conductivity is very low, thus resulting in very small out-of-plane power factors.) We find that in-planecompressive strain can significantly improve the Seebeckcoefficient and power factor of Bi 2Te3(by as much as two times), while effects on Bi 2Se3are less significant. Overall, Bi2Te3has a power factor that is an order of magnitude larger than that of Bi 2Se3, even in the unstrained state; therefore, these results are significant for further enhancing the ZT of this excellent thermoelectric material. In contrast, we find thatuniaxial strain does not improve the power factor of Bi 2Te3. For Bi 2Se3, the power factor can be increased by compressive uniaxial strain in the case of ndoping and by 6% uniaxial tensile strain in the case of pdoping. Figure 7shows the calculated in-plane Seebeck coefficient Sat 300 K for both n-type and p-type doping as a function of carrier concentration. As the electron concentration increases,the Fermi level is shifted higher into the conduction band,resulting in less asymmetry between electrons and holes,therefore reducing the n-type Seebeck coefficient. Similar arguments can be made for p-doped systems. Focusing on the vdW-DF C09xrelaxed structures [Figs. 7(a) and7(b)], we see that the Seebeck coefficient of p-doped Bi 2Te3can be increased by 2% tensile strain for hole concentrations of10 19−1021cm−3and by 2% compressive strain for hole concentrations of 1017−1019cm−3. On the other hand, 2% compressive strain can increase the magnitude of Sinn-dopedBi2Te3for all dopant concentrations studied, while 2% tensile strain reduces the magnitude of S. This trend is consistent with the increase in bandgap from tensile to compressivestrain in Bi 2Te3, as well as to the steeper slope of conductivity versus energy [as can be inferred from Fig. 9(b)]. Forp-doped Bi2Se3, the overall Seebeck coefficient is significantly higher than Bi 2Te3due to the steeper variation of the density of states in the valence bands,46and the effect of strain on the improvement of Seebeck coefficient for Bi 2Se3is less obvious. However, 2% tensile strain can increase the mag-nitude of Sinn-doped Bi 2Se3for electron concentrations of 2×1018cm−3−1020cm−3. For comparison, we have also computed the thermoelectric properties of correspondingPBE+SOC optimized structures. The predicted thermoelec- tric properties can be qualitatively different, as shown in Figs. 7(c) and7(d), therefore underscoring the importance of vdW interactions in structural optimization and in predictingthe effects of strain on electronic and thermoelectric properties.In the following, we focus on vdW-DF C09xrelaxed structures. Figure 8depicts the temperature dependence of the Seebeck coefficient Sunder different in-plane strains, with the carrier concentration fixed to 1019cm−3. Figure 8(a) shows that compressive strain can increase the Seebeck coefficient ofp-doped Bi 2Te3when the temperature is higher than 350 K, and the peak value can be shifted from 300 to 400 K when a 3%compressive strain is applied. This result is very similar to the 184111-11XIN LUO, MICHAEL B. SULLIV AN, AND SU YING QUEK PHYSICAL REVIEW B 86, 184111 (2012) FIG. 9. (Color online) The conductivity, Seebeck coefficient and power factor of (a) Bi 2Se3and (b) Bi 2Te3under different in-plane strains. Negative-carrier concentrations denote electron doping. The power factor of Bi 2Se3is an order of magnitude smaller than that of Bi 2Te3, because of its much smaller conductivity. prediction of Ref. 76, in which the authors changed the lattice constants of Bi 2Te3to that of Sb 2Te3to model a compressive in-plane strain. Overall, the strain does not improve the p-type Seebeck coefficient of Bi 2Se3, except for a small enhancement at temperatures higher than 500 K [Fig. 8(b)]. As noted before, thep-type Seebeck coefficient, in general, is larger in Bi 2Se3 than in Bi 2Te3, reaching about 400 μV/K at 350 K. On the other hand, compressive strain increases the magnitude of the Seebeck coefficient in n-doped Bi 2Te3for all temperatures considered here, and the maximum of theSeebeck coefficient is shifted to higher temperature withlarger compressive strain [Fig. 8(c)]. Most notably, under 3% compressive strain, the magnitude of the Seebeck coefficient reaches a maximum of 305 μV/K at 350 K, roughly 45% higher than that of unstrained Bi 2Te3(210μV/K) at the same temperature. For n-doped Bi 2Se3, compressive strains instead reduce the magnitude of the Seebeck coefficient,but tensile strain increases this magnitude for temperaturesless than ∼500 K [Fig. 8(d)]. Furthermore, the maximum magnitude of the Seebeck coefficient is shifted to lowertemperatures at larger tensile strains. A maximum magnitudeof 256 μV/K is achieved in the Seebeck coefficient of Bi 2Se3under 1% tensile strain, 13% larger than the maximum magnitude in the unstrained system. It is interesting to notethat under 2% tensile strain, the maximum magnitude ofSeebeck coefficient of n-type Bi 2Se3(252μV/K) is at about 350 K, the same temperature where n-type Bi 2Te3has the largest Seebeck coefficient under 3% compressive strain.Since the in-plane lattice constant of Bi 2Se3is about 5%smaller than that of Bi 2Te3, it is possible that in a superlattice structure, the tensile strain in Bi 2Se3and compressive strain on Bi 2Te3result in a larger Seebeck coefficient. Together with the reduced lattice thermal conductivity in a superlat-tice structure, this may greatly improve the thermoelectricperformance. The electrical conductivity σcan be readily calculated within the constant relaxation-time approximation given anestimate for the relaxation time τ. In practice, we can estimate τby comparing our computed values of σ/τ and Sfor unstrained systems, with experimentally 74,79measured values of σandSat the same temperature (300 K), as described in Ref. 44. In this way, in-plane relaxation times of 2.2×10−14s and 2 .7×10−15s are derived for Bi 2Te3 and Bi 2Se3, respectively. In Ref. 44, it was shown that this derived relaxation time for Bi 2Te3gives good agreement with experiment for different doping concentrations; here, wefurther assume that the relaxation time is independent of strain,as is also assumed by other authors. 37 The resulting conductivities and power factors (at 300 K) are plotted in Fig. 9for different in-plane strains (positive- and negative-carrier concentrations denote the p-type and n-type doping, respectively). The conductivity of Bi 2Te3is 20 times larger than that of Bi 2Se3; this is consistent with the physical picture that Bi 2Te3has a smaller bandgap. Strain has a signifi- cantly larger effect on the conductivity of Bi 2Te3than Bi 2Se3. Tensile strain increases the conductivity while compressivestrain decreases the conductivity in Bi 2Te3, for both p-type andn-type doping, an observation that is consistent with the 184111-12FIRST-PRINCIPLES INVESTIGATIONS OF THE ... PHYSICAL REVIEW B 86, 184111 (2012) FIG. 10. (Color online) The conductivity, Seebeck coefficient and power factor of (a) Bi 2Se3and (b) Bi 2Te3under different uniaxial strains. Negative-carrier concentrations denote electron doping. increase in bandgap from tensile to compressive strain. The Seebeck coefficient is related with the conductivity and theelectron-hole asymmetry. When E−μ/greatermuchk BT,E q . (2)can be expressed in the Mott formulas,80 S=/parenleftbiggπ2k2 BT 3eσ/parenrightbiggdσ dE/vextendsingle/vextendsingle/vextendsingle/vextendsingle E=EF=π2k2 BT 3edlnσ dE/vextendsingle/vextendsingle/vextendsingle/vextendsingle E=EF.(5) Therefore, we can understand the Seebeck coefficient from the energy derivative of the log-scale conductivity.By comparing the conductivity and Seebeck coefficients inFigs. 9(a) and 9(b), we find that qualitatively, the carrier concentration can be used as an approximate proxy for theenergy scale—the Seebeck coefficient is larger when the slopeof the conductivity curve is steeper. This is also consistentwith the increased Seebeck coefficients with compressivestrain for Bi 2Te3. We further note that although the Seebeck coefficient of Bi 2Se3is in general larger than that of Bi 2Te3, under compressive strain, the n-type Seebeck coefficient of Bi2Te3can surpass that of Bi 2Se3. This large enhancement inn-type Seebeck coefficient of Bi 2Te3with compressive strain is in agreement with Ref. 77. Finally, we computed the power factor for Bi 2Se3and Bi 2Te3under different strain states. The power factor for Bi 2Te3is one order of magnitude larger than that for Bi 2Se3, due to the much larger conductivity in Bi 2Te3. Although the compressive strains can reduce the electron conductivity of n-type Bi 2Te3, it will also increase its Seebeck coefficient. The larger enhancement of the Seebeckcoefficient makes it possible to compensate the reductionof conductivity, resulting in an enhancement of the n-type power factor for Bi 2Te3, under compressive in-plane strain.This result is different from that in Ref. 77—the discrepancy may come from the different methods for imposing strain (inRef. 77, the strain is imposed by setting the in-plane and out-of-plane lattice constants of Bi 2Te3to that of Sb 2Te3, without relaxing the internal coordinates). Finally, we show in Fig. 10the thermoelectric properties of Bi 2Se3and Bi 2Te3under uniaxial strain. In contrast to the case for in-plane strain, tensile uniaxial strain decreasesthe conductivity, and compressive uniaxial strain increases theconductivity for Bi 2Te3, while tensile uniaxial strain increases Seebeck coefficient slightly, and compressive strain decreasesSeebeck coefficient. Unlike the case of in-plane strain, theseopposite effects on Seebeck coefficient and conductivity resultin no enhancement of the power factor for Bi 2Te3under compressive uniaxial strain. Uniaxial strain can be used toenhance the power factor of Bi 2Se3, however, compressive strain enhances the n-type power factor because of an increased magnitude of Seebeck coefficient (the power factor more thandoubles under compressive uniaxial strain of 4–6%, for dopingconcentrations between 10 19and 1020cm−3), while 2% com- pressive uniaxial strain can enhance the p-type power factor. IV . CONCLUSION In summary, we have performed a comprehensive inves- tigation of the effects of strain on the atomic, electronic,and thermoelectric properties of Bi 2Se3and Bi 2Te3, taking into account the vdW interlayer interactions using the vdW-DF C09xfunctional. Our optimized, unstrained structures are in much closer agreement with the experimental structurefrom Nakajima (determined by x-ray diffraction) than with 184111-13XIN LUO, MICHAEL B. SULLIV AN, AND SU YING QUEK PHYSICAL REVIEW B 86, 184111 (2012) that from Wyckoff (determined by electron diffraction).58,59 Importantly, the two experimental structures have qualitatively different band structures on Bi 2Se3—previously published photoemission results61on Bi 2Se3are in good qualitative agreement with the band structure of vdW-DFC09xoptimized structure and the Nakajima structure, but not the Wyckoffstructure, and the /Gamma1-point bandgap for the vdW-DF C09x optimized Bi 2Se3thin films closes at four QLs instead of three QLs, as previously reported using the Wyckoff structure.We predict that the bandgaps of these materials increasefrom tensile to compressive in-plane strain, suggesting thatcompressive strain may be used to increase the bulk bandgapof Bi 2Se3, thus making it easier to distinguish the metallic topological surface state from intrinsic bulk carriers. We alsoconfirm that a topological phase transition can occur in Bi 2Se3 at 6% uniaxial strain, as predicted by Young et al.28Strain can also be used to tune the thermoelectric properties of thesematerials: the n-type Seebeck coefficient of Bi 2Te3can be increased by compressive in-plane strain while that of Bi 2Se3 can be increased with tensile in-plane strain. The power factorofn-doped Bi 2Se3can be increased with compressive uniaxial strain while that of n-doped Bi 2Te3can be increased by compressive in-plane strain. Finally, we have compared theproperties of structures optimized using different functionalsand found that taking into account vdW interactions is crucialfor the predictions of electronic and thermoelectric propertiesof strained structures, while spin-orbit interactions are lessimportant for structure determination. 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PhysRevB.98.195439.pdf

PHYSICAL REVIEW B 98, 195439 (2018) General theory of the topological Hall effect in systems with chiral spin textures K. S. Denisov,1,2,*I. V . Rozhansky,1,2N. S. Averkiev,1and E. Lähderanta2 1Ioffe Institute, 194021 St. Petersburg, Russia 2Lappeenranta University of Technology, FI-53851 Lappeenranta, Finland (Received 20 July 2018; revised manuscript received 29 October 2018; published 27 November 2018) We present a consistent theory of the topological Hall effect (THE) in two-dimensional magnetic systems with a disordered array of chiral spin textures, such as magnetic skyrmions. We focus on the scattering regime whenthe mean-free path of itinerant electrons exceeds the spin texture size, and THE arises from the asymmetriccarrier scattering on individual chiral spin textures. We calculate the resistivity tensor on the basis of theBoltzmann kinetic equation taking into account the asymmetric scattering on skyrmions via the collision integral.Our theory describes both the adiabatic regime when THE arises from a spin Hall effect and the nonadiabaticscattering when THE is due to purely charge transverse currents. We analyze the dependence of THE resistivityon a chiral spin texture structure, as well as on material parameters. We discuss the crossover between spin andcharge regimes of THE driven by the increase of skyrmion size, the features of THE due to the variation of theFermi energy, and the exchange interaction strength; we comment on the sign and magnitude of THE. DOI: 10.1103/PhysRevB.98.195439 I. INTRODUCTION Among rich variety of transport phenomena in magnetic materials the special attention is now focused on the topologi-cal Hall effect (THE). THE is the appearance of an additionaltransverse voltage due to itinerant carrier exchange interactionwith chiral spin textures, such as magnetic skyrmions [ 1–4]. During the recent extensive experimental studies the detectionof THE signal has proved itself as an indicator that a samplemagnetization acquires a chiral structure. The observationof THE has been reported for various systems exhibitingdifferent chiral ordering of spins: skyrmion crystals [ 5–8], antiferromagnets (AFM) [ 9,10], spin glasses [ 11,12], and arrays of magnetic skyrmions [ 13–21]. Naturally, an appropriate microscopic theory of THE has to take into account the particular type of chiral spin or-dering. In the case of a regular noncollinear spin structurewith periodic or quasiperiodic spin arrangement [ 22–24], such as AFM lattices [ 9]o rs k y r m i o nc r y s t a l s[ 6,25,26], THE is usually described via the mean-field approach and using theadiabatic approximation [ 6,22,27]. The adiabatic theory is based on the geometric Berry phase interpreted as an effectivemagnetic field acting on an electron. Although the Berryphase description of THE has been widely applied for varioussystems [ 28–35], it is, however, invalid for a weak exchange coupling [ 36,37]. The latter case has been recently studied using different theoretical methods [ 38–42]. Another class of chiral spin systems studied experimentally is a disordered array of localized small chiral spin texturessuch as magnetic skyrmions [ 14–16,20,43]. The description of THE in terms of an effective mean magnetic field fails inthis case as there is no regular long-range chiral spin structurewhich can be described by a homogeneous effective magnetic *denisokonstantin@gmail.comfield. On the contrary, a carrier moves freely most of thetime with an occasional scattering on localized magnetizationvortices. The important feature of the individual scatteringregime is that the properties of THE strongly depend onwhether the carrier spin-flip processes are activated or notcorresponding to weak coupling regime and adiabatic regime,respectively [ 44]. The transverse electric response arises from the spin Hall effect [ 22,34] in the adiabatic regime and from the charge Hall effect in the weak coupling regime [ 38,41]. Thus, the complete theory of THE for the irregular dilute chi-ral systems requires an accurate treatment of carrier scatteringon a single chiral spin texture. In this paper we develop the theory of the topological Hall effect in disordered systems of chiral spin textures. Weconsider a two-dimensional (2D) metal with both electronspin subbands populated when THE can be generated eitherby charge or by spin transverse currents. Our approach isbased on the calculation of the exact scattering parameters onindividual spin textures presented in Ref. [ 44]. The paper is organized as follows: in Sec. IIthe kinetic theory of THE is described accounting for the carrier scattering on host impu-rities and noncollinear spin textures, in Sec. IIIthe properties of the exchange asymmetric scattering are discussed, Sec. IV covers the dependence of THE on material and spin textureparameters, we also describe the crossover between chargeand spin Hall regimes of THE driven by the suppression ofspin-flip scattering; in Sec. Vwe discuss the obtained results in the view of some real skyrmion systems. II. KINETIC THEORY Let us consider two-dimensional degenerate electron gas (2DEG) described by the Hamiltonian: H=p2 2m−α0S(r)·σ+/summationdisplay iu(r−ri), (1) 2469-9950/2018/98(19)/195439(12) 195439-1 ©2018 American Physical SocietyDENISOV , ROZHANSKY , A VERKIEV , AND LÄHDERANTA PHYSICAL REVIEW B 98, 195439 (2018) where the first term describes the electron free motion with an effective in-plane mass m, the second term represents the electron exchange interaction with a magnetic texturedescribed by a static spin field S(r), where α 0is an exchange coupling constant, σis the vector of Pauli matrices, and the last term describes scattering on host nonmagnetic impuritieslocated at r i. The topological Hall effect appears when S(r) has a noncollinear structure characterized by a nonzero spinchirality. We consider the case, when the spin field S(r) consists of two contributions: S(r)=S 0ez+/summationdisplay jδS(r−rj). (2) The first term is a background homogeneous field directed perpendicular to the 2DEG plane leading to the spin splitting/Delta1=α 0S0, we assume ferromagnetic exchange interaction (/Delta1>0). The energy dispersion for the two spin branches s=±1/2i sεs p=p2/2m−s/Delta1; the spin-dependent velocity is given by vs=√2(ε+s/Delta1)/m, where εis the electron energy. We further assume that the Fermi energy EFexceeds the spin splitting so that both spin subbands are populated(E F>/Delta1/2). The second contribution in Eq. ( 2) describes localized chiral spin textures δSof a few nanometer size located at rjand causing an additional elastic scattering of the carriers. While magnetic skyrmions are the typicalexample of such spin texture, our consideration covers a muchwider class of chiral spin textures, not necessarily having anonzero topological charge [ 44]. The feature of the chiral spin structures is that for a given incident electron flux there isa difference in scattering rates to the left and to the right,eventually leading to the Hall effect. We consider the classic transport regime ( k F/lscript/greatermuch1, where kF=√2mEF/¯his the Fermi wave vector, and /lscriptis the mean free path) on the basis of the Boltzmann kinetic equation: eE·∂fs(p) ∂p=St[fs(p)], St[fs(p)]=/summationdisplay p/prime,s/prime/parenleftbig Wss/prime pp/primefs/prime(p/prime)−Ws/primes p/primepfs(p)/parenrightbig ,(3) where fs(p) is the distribution function, pis 2D momentum, ands=±1/2 is the carrier spin projection on the axis normal to the motion plane, Eis an in-plane electric field, Wss/prime pp/primeis the elastic scattering rate from ( p/prime,s/prime)t o( p,s) state, and eis the electron charge. We solve Eq. ( 3) in linear approximation with respect to E. Expressing the scattering rate Wss/prime pp/primein the form of the Fermi’s golden rule we assume that it has two contributions: Wss/prime pp/prime=2π ¯h/parenleftbig ni|upp/prime|2δss/prime+nsk/vextendsingle/vextendsingleTss/prime pp/prime/vextendsingle/vextendsingle2/parenrightbig δ/parenleftbig εs p−εs/prime p/prime/parenrightbig ,(4) where the first term in parentheses describes the electron spin- independent scattering on nonmagnetic impurities, and thesecond term is driven by the scattering on chiral spin textures;interference effects between the two types of scatterers areneglected. Here n i,nskare the surface densities of impurities and localized magnetic textures, respectively, upp/primeis Fourier transform of the nonmagnetic impurity potential u(r) from Eq. ( 1),Tss/prime pp/primeis the exact Tmatrix of electron scatteringon the spin texture, and the delta function ensures energy conservation in the elastic scattering. Two contributions canbe distinguished in the square modulus of the Tmatrix: ν 2/vextendsingle/vextendsingleTss/prime pp/prime/vextendsingle/vextendsingle2=Gss/prime(θ)+Jss/prime(θ), Gss/prime(θ)=Gss/prime(−θ),Jss/prime(θ)=−Jss/prime(−θ). (5) Gss/prime(θ),Jss/prime(θ) are dimensionless symmetric and asymmetric scattering rates, respectively, θ=ϕ−ϕ/primeis the scattering angle, ϕ,ϕ/primeare the polar angles of p,p/prime, andν=m/2π¯h2 is the 2D density of states (per one spin). In the introduced notation we omit the dependence of Gss/prime,Jss/primeon the electron energy ε. It is the asymmetric part Jss/prime(θ) of an electron scattering on chiral spin textures that gives rise to the transversal current asthe scattering rates to the left and to the right become unequal.The scattering asymmetry acts as an effective magnetic field,which sign can be either the same for both spin projections ofan incident electron, hence leading to a charge Hall effect, oropposite for the opposite electron spin projections, leading tothe spin Hall effect. The properties of J ss/prime(θ) are discussed in Ref. [ 44] and summarized in Sec. III. The distribution function fs(p)=f0 s(ε)+gs(p) contains equilibrium f0 s(ε) and nonequilibrium gs(p) parts; the latter describes the appearance of the electric current in an externalelectric field E. Below the direction of Eis assumed along x axis, and the polar angle ϕof momentum pis counted from xaxis. In order to solve the kinetic equation ( 3) one should expand g s(p) and Wss/prime pp/primein a series of angular harmonics cosnϕ,sinnϕand perform the integration over the angle ϕ/primein the collision integral. The details of this calculation are givenin Appendix A. In this paper we focus on the linear regime in E,s og s contains only first angular harmonics: gs(p)=g+ scosϕ+g− ssinϕ. (6) The terms g+ s,g− sdetermine the longitudinal and transverse electric currents, respectively. Indeed, as Eis along xaxis, we get for the electric current j: /parenleftbigg jx jy/parenrightbigg =e/summationdisplay p,sgs(p)v/parenleftbigg cosϕ sinϕ/parenrightbigg =eν 2/summationdisplay s/integraldisplay vs(ε)dε/parenleftbigg g+ s(ε) g− s(ε)/parenrightbigg , (7) where the integration goes over the energy ε. The topological Hall effect related to the transverse electric current jyis thus determined by g− scoefficients. The explicit integration of the collision integral in Eq. ( 3) over p/primebrings us to the following system of equations on g± s(see details in Appendix A): eE⎛ ⎜⎜⎜⎝v ↑∂f0 ↑ ∂ε 0 v↓∂f0 ↓ ∂ε 0⎞ ⎟⎟⎟⎠=⎛ ⎜⎜⎜⎝−τ −1 ↑/Omega1↑↑ τ−1 ↑↓ /Omega1↑↓ −/Omega1↑↑−τ−1 ↑−/Omega1↑↓ τ−1 ↑↓ τ−1 ↓↑ /Omega1↓↑ −τ−1 ↓/Omega1↓↓ −/Omega1↓↑ τ−1 ↓↑ −/Omega1↓↓−τ−1 ↓⎞ ⎟⎟⎟⎠⎛ ⎜⎜⎜⎝g + ↑ g− ↑ g+ ↓ g− ↓⎞ ⎟⎟⎟⎠. (8) 195439-2GENERAL THEORY OF THE TOPOLOGICAL HALL EFFECT … PHYSICAL REVIEW B 98, 195439 (2018) Here we introduced the following parameters: τ−1 s=τ−1 0+ωs,τ−1 0=ni2π ¯hν/integraldisplay2π 0|upp/prime|2(1−cosθ)dθ 2π, ωs=nsk2π ¯h/integraldisplay2π 0[(1−cosθ)Gss(θ)+G¯ss]1 νdθ 2π, (9) τ−1 s¯s=nsk2π ¯h/integraldisplay2π 0Gs¯s(θ) cosθ1 νdθ 2π, /Omega1ss/prime=eBss/prime mc,B ss/prime=(nskφ0)/integraldisplay2π 0Jss/prime(θ)s i nθdθ, where τsis the total transport lifetime, τ0is the transport lifetime for the scattering on nonmagnetic impurities, and ω−1 s andτs¯sare the transport lifetimes for the scattering on chiral textures (here ¯sis the spin subband index opposite to s). The transverse Hall current due to the asymmetric scattering isdriven by /Omega1 ss/prime, which is analogous to the cyclotron frequency in the ordinary Hall effect; Bss/primeis the corresponding effective magnetic field, φ0=hc/|e|is the magnetic flux quantum, andcis the speed of light. Solving the system Eq. ( 8) and finding g± sallow us to calculate the resistivity tensor ρfor various transport scenarios as further discussed in Sec. IV. We would like to emphasize that both spin-conserving andspin-flip scattering channels contain asymmetric parts /Omega1 ss/prime and thus contribute to g− sandjy. III. ASYMMETRIC ELECTRON SCATTERING ON A CHIRAL SPIN TEXTURE In this section we consider the features of asymmetric electron scattering on a single chiral magnetic texture. Weexpress the scattering potential in the form V(r)=−α 0δS(r)·σ. (10) Outside the localized spin texture of a characteristic diameter athe magnetization is unperturbed so that δS(r>a / 2)→0 and the scattering potential vanishes. The topological Halleffect appears due to the asymmetry in the electron scatteringwhen the potential ( 10) can be characterized by a nonzero chirality. The details of the asymmetric scattering depend onthe particular distribution of spins in the texture and its size aswell as on the exchange interaction strength and the incidentelectron wavevector [ 44]. A. Chiral spin textures To describe a chiral spin texture in 2D the following parametrization is commonly used: δS(r)=⎛ ⎝δS/bardbl(r) cos (/kappa1φ+γ) δS/bardbl(r)s i n(/kappa1φ+γ) δSz(r)⎞ ⎠, (11) where r=(r, φ) is the polar radius vector, and r=0 corre- sponds to the center of the texture. The functions δS/bardbl,δSz/negationslash=0 depend on the distance from the center r. The vorticity /kappa1 describes the in-plane spin rotation with an initial phase γ. In what follows we consider that δSzis counted from the background magnetization S0, whose sign we denote as η= sgn(S0). In the last section we will also consider the case when S0andδSzare independent.η=+ 1 a/2 δSzSz(r)S0 −S0r skyrmion Q=1Λ1=π1−2r a Λ2=πsin2⎡ ⎣π 21+2r a⎤ ⎦chiral spin ring Q=0Λ3=π2r a1−2r a FIG. 1. The typical profiles Sz(r)=S0cos/Lambda1(r) of chiral spin textures. Note that for η=+1, theδSz(r)<0 is negative. Figure 1shows the profiles Sz(r)=S0cos/Lambda1(r)f o rt h r e e examples of chiral spin textures with η=+1 (we assume that δS2 /bardbl=S2 0−S2 z). Two of them describe a magnetic skyrmion (/Lambda11(r)=π(1−2r/a),/Lambda12(r)=πsin2[(π/2)(1+2r/a)]). The skyrmion has an opposite sign of spins in its centerwith respect to the background magnetization, which leadsto the appearance of a nonzero topological charge calledwinding number Q. The nonzero Qis particularly important for the thermal stability of skyrmions in ferromagnetic thinfilms [ 45–52]. The third magnetization profile Fig. 1[/Lambda1 3(r)= (2r/a)π(1−2r/a)] corresponds to a chiral spin ring with the orientation of spins in the center parallel to S0.C h i r a l rings have zero winding number, but they exhibit a similartopological Hall effect [ 44]. Such spin textures can appear in a material with spin-orbit interaction functionalized by mag-netic impurities, in a vicinity of a defect or impurity [ 53–56]. Let us notice that for the positive background spin orientation(η=+1)δS zis negative. Substituting ( 11)i n t o( 10) we get for the scattering potential: V(r)=−α0/parenleftbiggδSz(r) e−i/kappa1φ−iγδS/bardbl(r) ei/kappa1φ+iγδS/bardbl(r) −δSz(r)/parenrightbigg . (12) The potential V(r)i sa2 ×2 matrix, which depends on a polar angle φvia the off-diagonal components. When both functions Sz,S/bardblare nonzero, the angular dependence of the potential leads to the appearance of the asymmetric part in electron scattering rates Jss/prime(θ)=−Jss/prime(−θ), where θis the scattering angle. The sign of Jss/primedepends on /kappa1. The phase parameter γis an important characteristic of a skyrmion structure, i.e., γ=π/2,γ=0 correspond to Bloch and Néel skyrmions, respectively. However, γdoes not affect the scat- tering cross section and, therefore, appears to play no role inTHE. The role of ηis more complicated; we further explicitly specify the dependence of J ss/prime(θ,η)o nη. B. Asymmetric scattering features We consider the case when Fermi energy exceeds the background exchange splitting EF>/Delta1/2 so that both spin subbands are populated with electrons ( /Delta1=α0S0). The 195439-3DENISOV , ROZHANSKY , A VERKIEV , AND LÄHDERANTA PHYSICAL REVIEW B 98, 195439 (2018) symmetry upon the time inversion allows us to present the asymmetric scattering rates Jss/prime(θ,η) introduced in Eq. ( 5)i n the form (see the details in Appendix B) J↑↑(θ,η)=η/Gamma11(θ)+/Pi1(θ), J↓↓(θ,η)=η/Gamma11(θ)−/Pi1(θ), (13) J↑↓(θ,η)=J↓↑(θ,η)=η/Gamma12(θ), where/Gamma11,2(θ) and/Pi1(θ) have no dependence on the back- ground polarization η=sgn(S0). This representation is con- venient for treating the topological charge and spin Halleffects independently. Indeed, the terms η/Gamma1 1,2describe the asymmetric scattering in the same transverse direction de-termined by the texture orientation ηand independent of an initial carrier spin state. These terms, therefore, lead to thecharge Hall effect. On the contrary, the term /Pi1describes the scattering of spin up and spin down electron in the oppositetransverse directions independent of η. This process leads to spin Hall effect, it is absent for spin-flip channels. Both /Gamma1 1,2 and/Pi1change their sign upon /kappa1→−/kappa1. Which of the two contributions to the topological Hall effect (charge or spin) dominate strongly depends on whetherthe spin-flip processes are activated or not. Away from thethreshold E F/greatermuch/Delta1/2 the rate of the spin-flip scattering is controlled by the adiabatic parameter λa=(α0S0/¯h)τa, where τa=a/vFis an electron time of flight through the texture of diameter awith Fermi velocity vF=√2EF/m. In the case of λa/lessorequalslant1 the spin-flip processes are ef- fective, the asymmetric scattering arises from the interfer-ence between double spin-flip and single spin-conservingscattering events (so-called spin-chirality driven mecha-nism [ 36,37,39,41,42]). This process is sensitive to the spin chirality χdefined for any three spins δS 1,δS2,δS3forming the spin texture as χ=(δS1·[δS2×δS3]). The nonzero chi- rality of the spin texture in the weak coupling regime leads tothe charge Hall effect. The spin chirality based contribution isdescribed by /Gamma1 1,2.A tλa/lessorequalslant1 these terms dominate /Gamma11,2/greatermuch/Pi1, with spin-flip scattering prevailing /Gamma12=2/Gamma11. In the opposite case of large adiabatic parameter λa/greatermuch1 the spin-flip processes are suppressed in accordance with theadiabatic theorem. In this regime the scattering asymmetryis due to the Berry phase acquired by the wave functionof the electron moving through a noncollinear spin field inthe real space. The hallmark of this mechanism is that thesign of the effective magnetic field associated with the Berryphase appears to be opposite for spin up and spin downelectrons, thus leading to the spin Hall effect [ 22,30,32,33]. This adiabatic contribution to the Hall response is, therefore,described by /Pi1.A tλ a/greatermuch1 the spin Hall effect dominates /Pi1/greatermuch/Gamma11,2, and the charge Hall effect appears only due to nonzero carrier spin polarization Ps. The interplay between charge and spin topological Hall effects leads to a few nontrivial features discussed in thefollowing section. IV. TOPOLOGICAL HALL EFFECT In this section we discuss the topological contribution to the Hall resistivity ρT yxin the diffusive regime for different systems.A. Dilute array of chiral spin textures Let us consider a two-dimensional film containing spatially localized chiral spin textures such as magnetic skyrmions orchiral magnetic rings (see Fig. 1). We assume that all the textures have the same vorticity /kappa1, and the orientation η= sgn(S 0)=+1 is fixed, being determined by the background magnetization S0. We consider the dilute regime, when the scattering rate on spin textures is much smaller than that onnonmagnetic impurities ω sτ0/lessmuch1,/Omega1ss/primeτ0/lessmuch1, so the trans- port lifetime is given by τs=τ0. Solving the system ( 8)f o r g± sin the lowest order in ( /Omega1ss/primeτ0) we express the topological Hall resistivity ρT yxas a sum of two contributions (see details in Appendix C): ρT yx=ρc+ρa, ρc=1 nec(φ0nsk)/integraldisplay2π 0(/Gamma11+/Gamma12)sinθdθ, ρa=Ps1 nec(φ0nsk)/integraldisplay2π 0/Pi1sinθdθ. (14) The term ρcdescribes the charge transverse current (charge Hall effect) generated due to carrier asymmetric scatteringdue to spin-independent terms /Gamma1 1,2[Eq. ( 13)]. The term ρa describes the transverse spin current (spin Hall effect) driven by the spin-dependent contribution to the asymmetric scatter-ing/Pi1[Eq. ( 13)]. The spin current does not lead to a charge separation unless there is unequal number of spin up and spindown carriers in the system. Therefore, this contribution tothe Hall resistivity is proportional to the carrier spin polar-ization P s=(n↑−n↓)/(n↑+n↓)=/Delta1/2EF.I nE q .( 14)t h e notation n=n↑+n↓stands for the 2DEG sheet density. The relative importance of the two contributions ρaandρc in the appearance of the transverse charge current depends on the texture diameter aor the Fermi level EFas discussed in the following sections. 1. Crossover between charge and spin Hall effect Let us trace the dependence of ρT yx(14) on the spin texture diameter a. We assume that the Fermi energy EFsubstantially exceeds the exchange spin splitting so that both spin subbandsare populated and the spin polarization of the carriers isfar below 100%: P s=/Delta1/2EF/lessmuch1. The adiabatic parameter can be expressed as λa=Ps(ka), where k=/radicalbig 2EFm/¯h2. Figure 2shows the calculated dependence of charge ρc, adiabatic ρa, and total ρT yxHall resistivities on the skyrmion diameter afor the magnetic skyrmion with magnetization spa- tial profile /Lambda11(r) shown in Fig. 1. For the calculation results shown in Fig. 2the spin polarization was taken Ps=0.4, and the skyrmion surface density nsk=2×1011cm−2.T h e scattering rates Jss/primewere calculated using the phase function method [ 44]. As can be seen in Fig. 2,f o rλa/lessorequalslant1.8 the charge contribution ρcexceeds ρa, at that ρT yxis dominated by the purely charge current. For λa/greaterorequalslant4.5 the adiabatic term prevails ρa/greatermuchρcandρT yxappears due to the spin current converted into the charge current. In addition to the adiabatic parameter,the change of the texture size at the same time affects thewave parameter ka, which determines the properties of the scattering. As a result, the topological Hall resistivity ρ T yx 195439-4GENERAL THEORY OF THE TOPOLOGICAL HALL EFFECT … PHYSICAL REVIEW B 98, 195439 (2018) ρTyxρa ρc BT(kG)123456 0246810λa 2 4 6 8 10 12 140246810 kaρT yx(a.u.) FIG. 2. The dependence of ρT yxon magnetic skyrmion diameter kafor/Lambda11profile, and the crossover between charge and spin topo- logical Hall effects. The parameters Ps=0.4,nsk=2×1011cm−2. exhibits a nontrivial dependence on ain the intermediate region 4 .5/greaterorsimilarλa/greaterorsimilar1.8. As the spin-flip processes become suppressed, first the charge contribution ρcis decreased, and only later the adiabatic term ρastarts to increase. This effect results in the appearance of the local minimum for ρT yxin the crossover regime. The behavior of ρT yxin the crossover regime is highly sensitive to a particular magnetic texture profile. In Fig. 3 we present the dependence of ρT yxonafor three different spin texture spatial profiles shown in Fig. 1. As can be seen in Fig. 3the oscillating structure of ρT yxupon increasing kaexhibits a significant variation even for two very similar skyrmion configurations /Lambda11and/Lambda12. The strong dependence ofρT yxon/Lambda1(r) observed in the crossover regime is due to the significance of the interference in electron scattering as thewave parameter ka∼π[44]. The texture described by /Lambda1 2has larger spin gradients, so the adiabatic term activates at largerka, and the magnitude of ρ T yxfor/Lambda12in the crossover regime is smaller than that of /Lambda11. We would also like to stress out that the topological Hall effect exists as well due to scattering on chiral spin ringshaving zero winding number (orange curve in Fig. 3). The transverse conductivity ρ T yxdue to scattering on chiral spin rings possesses all the features described above including thethe existence of charge and spin Hall limiting regimes. Λ1Λ2 Λ3 BT(kG)23456 0246λa 468 1 0 1 2 1 40246 kaρT yx(a.u.) FIG. 3. The dependence of ρT yxon chiral texture diameter kafor different texture profiles in the region of crossover. The parameters Ps=0.4,nsk=2×1011cm−2.ρT yx/Δ3 ka=2ρT yx/Δ3 ka=3 ρT yx/Δ3(a.u.) 0.40.60.81ρT yx/Δ3(a.u.) 00 .20 .40 .60 .8−50510 Δ/2EF FIG. 4. The dependence of ρT yx//Delta13on the variation of /Delta1/2EFat ka=2 (blue curve) and ka=3 (red curve) for /Lambda11profile. 2. The magnitude of the topological Hall effect The magnitude of THE for the dilute systems can be ex- pressed in terms of the effective magnetic field BTintroduced as ρT yx=BT nec, BT=(φ0nsk)/integraldisplay2π 0(/Gamma11+/Gamma12+Ps/Pi1)sinθdθ. (15) The field BTshows the magnitude of the external magnetic field applied to the sample, at which the ordinary Hall effectcontribution to the transverse resistivity ρ O yxbecomes compa- rable with ρT yx. Usually in the THE estimates it is assumed that each skyrmion contributes via a magnetic flux quantum, so that inthe adiabatic regime |B T|≈Ps(φ0nsk)Q. However, our analy- sis shows that such an estimate does not take into account theimportant features of the scattering. According to Eq. ( 15), ρ T yxandBTlinearly depend on both the skyrmion surface density nskand the dimensionless scattering rates /Gamma11,2,/Pi1. Therefore, the magnitude of BTis renormalized differently depending on the scattering regime. For instance, in the weak coupling regime ( λa/lessorequalslant1)BT scales as /Delta13, as the perturbation theory couples /Gamma11,2with spin chirality [ 36,41] and, therefore, THE requires the third order in the exchange interaction. In Fig. 4the quantity ρT yx//Delta13is shown for two values of ka=2,3. As can be seen from the figure, the scaling /Delta13holds up to /Delta1/2EF≈0.2, the deviation from the scaling relation indicates that the perturbation theorybecomes invalid departing from the weak coupling regime. Let us note that although the asymmetrical scattering rates/Gamma1 1,2are small in the weak coupling regime [ /Gamma11,2are proportional to ( /Delta1/2EF)3(ka/2)8atλa<1], the magnitude ofBTcan be enhanced by increasing nsk. For example, for nsk=5×1012cm−2andλa=0.8(ka=2,Ps=0.4) one getsBT≈0.7T . The magnitude of BTin the intermediate and strong cou- pling regimes for nsk=2×1011cm−2can be seen in Figs. 2 and3and Figs. 6and7.A tnskφ0≈8 T the value of BTin the intermediate regime is of the order of several kG; while in thestrong coupling regime ( λ a/greatermuch1) it can go as high as several Tesla [ 57]. It is worth noticing that the conventional estimate |BT|≈Ps(φ0nsk)Qwidely used in the literature is applicable only in the adiabatic limit λa/greatermuch1 for a large skyrmion size ka/greatermuch1. In Fig. 5we show the dependence of ρT yxonkafor 195439-5DENISOV , ROZHANSKY , A VERKIEV , AND LÄHDERANTA PHYSICAL REVIEW B 98, 195439 (2018) Ps(φ0nsk) Λ1 Λ2 Λ3 BT6 8 10 12 14 01234λa 15 20 25 30 35010203040 kaρT yx(a.u.) (T) FIG. 5. The dependence of ρT yxon chiral texture diameter kafor different texture profiles in the adiabatic region. The parameters Ps= 0.4,nsk=2×1011cm−2. The saturation magnitude Ps(φ0nsk)≈ 3.3T . three texture profiles from Fig. 1atka/greatermuch1. Let us mention thatBTstarts to saturate at ka/greaterorequalslant30 for the textures with nonzero topological charge ( /Lambda11,2) approaching its maximum valuePs(φ0nsk)a tka/greaterorequalslant35. On the contrary, THE resistivity asymptotically falls to zero for /Lambda13spin texture having Q=0. Therefore, the topological charge is indeed important for THEin the quasiclassical limit ka/greatermuch1, while in all other cases the local chirality of the spin texture leads to emergence of THEindependently of Q. 3. The sign of the topological Hall effect In a real experiment when electron transport in a system with chiral spin textures is studied as a function of theexternal magnetic field B 0, it is often difficult to extract different contributions to the Hall effect. The total transverseresistivity ρ yxcontains three contributions ρyx=ρO yx+ρA yx+ ρT yx, where ρO yx,ρA yx, andρT yxare attributed to the ordinary, anomalous, and topological Hall effects, respectively. Herewe focus on the sign difference between ρ T yx=(BT/nec ) and ρO yx=(B0/nec ), thus we should compare the signs of BT andB0. We assume that the background magnetization S0is di- rected along the external magnetic field B0>0. In gen- eral, there is no any fixed relation between the signs of thetopological Hall resistivity ρ T yx, charge ρc, and adiabatic ρa contributions as can be seen in Fig. 2. In this figure ρachanges its sign with increase of ka. For some spin texture profiles the total topological resistivity ρT yxcan also change its sign in the crossover regime. This is the case for /Lambda13spin configuration as can be seen in Fig. 3. However, it is possible to specify the sign ofBTin the limiting regimes, i.e., away from the threshold EF/greatermuch/Delta1/2 and outside the adiabatic crossover λa≈1. Let us consider the weak coupling regime ( λa/lessorequalslant1), in which the charge current contribution to THE dominates ( ρc/greatermuchρa). In this regime, the effective magnetic field is proportional tothe chirality of the spin texture B T∝δS1·[δS2×δS3]. For /kappa1=+1,η=+1 the sign of the mixed vector product of any three spins δS1,δS2,δS3forming the skyrmion is negative andBT<0 due to δSz<0( s e eF i g . 1), thus the sign of BT appears to be opposite to B0. In the adiabatic regime ( ρa/greatermuch ρc) the electrons with positive spin projection (co-aligned with S0) retain the same type of scattering asymmetry as for small BT(kG)10 .50 .33 0 .25 0 .2 0510152025Ps 123450510152025 2EF/ΔρT yx(a.u.)βex=6 βex=4.5 βex=3 FIG. 6. The dependence of ρT yxon Fermi energy EFat different βex=/radicalbig m/Delta1/¯h2aparameter for /Lambda11profile, nsk=2×1011cm−2. λa. As these electrons constitute the majority at a positive spin polarization ( Ps>0), the effective magnetic field is also negative BT<0. We conclude that for /kappa1=+1,η=+1 configurations, the topological field BTusually has the opposite sign to the sign of the external field B0. For chiral spin configurations with negative vorticity /kappa1< 0 the fields BTandB0have the same sign. However, in the crossover regime λa∼1 and near the threshold EF≈/Delta1/2 there is no any fixed relation between B0andBTsigns. 4. Effect of the Fermi energy variation The dependence of ρT yxon the variation of the Fermi energy EFexhibits a number of distinctive features. At EF</Delta1/2 only one spin subband is occupied and spin polarization isP s=1. We start the analysis from the threshold EF/greaterorequalslant/Delta1/2, when the electrons start populating the second spin subband.In further consideration we keep /Delta1andaconstant changing only the Fermi energy E F. For this analysis it is convenient to combine /Delta1andainto a single parameter βex=λa/√Ps=/radicalbig m/Delta1/¯h2a, which is independent of EF. Figure 6shows the dependence of ρT yxonEFcalculated for the /Lambda11skyrmion configuration for three different values of βex. As can be seen from the figure, ρT yxdepends nonmonotonically on EF, with a maximum near the threshold and decreasing at a largerE F. The magnitude of ρT yxnear the threshold is controlled by skyrmion size a. As the spin-chirality driven mechanism relevant for a small skyrmion size does not work at EF</Delta1/2 (there is no spin-flip processes below the spin down subbandedge), the decrease of a(and hence β ex) suppresses ρT yxat EF=/Delta1/2. The suppression of ρT yxat a large EFoccurs from the one hand due to decrease of the spin polarization factor Ps inρa, and from the other hand due to decrease of the scattering cross section as the kinetic energy of the scattering electronexceeds the characteristic energy of the scattering potential(provided that λ a/lessorsimilar1) [44]. The variation of the Fermi energy affects the asymmetric part of the scattering cross section simultaneously throughkaandλ afactors and, therefore, gives rise to a number of interesting features in the transverse resistivity behavior.We demonstrate these peculiarities in Fig. 7, where the de- pendence of ρ T yx,ρc, and ρaonEFis plotted for βex=3 andβex=6. For βex=3 [Fig. 7(a)] the adiabatic term ρais 195439-6GENERAL THEORY OF THE TOPOLOGICAL HALL EFFECT … PHYSICAL REVIEW B 98, 195439 (2018) (a) (b)βex=3 ρaρc ρTyx BT(kG)32 .11 .71 .51 .31 .2 123456 2EF/Δ0246λa 0246ρT yx(a.u.) βex=6 ρaρcρTyx BT(kG)64 .94 .23 .83 .43 .2 11 .522 .533 .5 2EF/Δ0102030λa 0102030ρT yx(a.u.) FIG. 7. The dependence of ρT yx,ρc,ρaon Fermi energy EFat βex=3 (a) and βex=6( b )f o r /Lambda11profile, nsk=2×1011cm−2. negative when far from the threshold. This is due to the com- plex scattering pattern typical for the intermediate range ofthe adiabatic parameter values (1 /lessorequalslantλ a/lessorequalslant2). We have already encountered this effect considering the behavior of THE inthe crossover regime: ρ ais negative in Fig. 2for the same range of λaas in Fig. 7(a).F o rβex=6 [Fig. 7(b)]λais larger and the interference in the carrier scattering manifests itselfthrough the oscillation of ρ c,ρamagnitudes superimposed on the global suppression upon increasing of EF.T h es a m e oscillating peculiarities of transverse response can be seen inFig.2in the range 4 /lessorequalslantλ a/lessorequalslant5. Let us finally comment on the scattering rates behavior in the vicinity of the threshold EF≈/Delta1/2. Since the spin down and spin-flip scattering channels are absent for EF< /Delta1/2, we conclude that at EF≈/Delta1/2 the following relations are fulfilled: /Gamma12≈0, and /Gamma11(θ)≈/Pi1(θ). At that, only spin up scattering channel is activated with J↑↑≈2/Pi1(θ) (i.e., ρa≈ρc). It is worth mentioning that this relation holds in the vicinity of the threshold for any magnitude of the adiabaticparameter. B. Dense array of skyrmions In this section we apply the developed theory of THE to the case when the dominating scattering mechanism changesfrom scattering on nonmagnetic impurities to scattering onmagnetic textures. This transition affects the spin-dependentscattering time τ −1 s=τ−1 0+ωs, and correspondingly the be- havior of both the longitudinal and transverse resistivities.Let us consider the adiabatic regime of an electron scatteringassuming that the spin-flip scattering channels are completelyτ−1 0ρxx/angbracketleftτ/angbracketright−1∝nskMs Ps nsk/ni ρT yx/(φ0nsk) 10−210−110010110200.20.40.6ρxx(a.u.) (a.u.) FIG. 8. The dependence of ρxxandMson skyrmion surface density nsk. suppressed ( /Omega1↑↓=τ−1 ↑↓=0) and THE originates solely from the spin Hall effect ( ρa/greatermuchρc). At that the spin up and spin down channels are uncoupled and contribute independentlyto the conductivity. While /Omega1 ssτ0/lessmuch1 in the dilute regime ωsτ0/lessmuch1, the ratio /Omega1ssτs≈/Omega1ss/ωsremains small even in the dense skyrmion system ωsτ0/greatermuch1, as the symmetric scattering is more effective than the asymmetric one [ 57]. Thus, we can still solve the kinetic equation in the lowest order in /Omega1ssτsas described in the previous section. Keeping only the leadingterms with respect to /Omega1 ssτswe get for the resistivity tensor (see details in Appendix C): ρxx=m ne2/angbracketleftτ/angbracketright,ρT yx=Ms1 nec(φ0nsk)/integraldisplay2π 0/Pi1sinθdθ, /angbracketleftτ/angbracketright=1 2[(1+Ps)τ↑+(1−Ps)τ↓], Ms=1 2/bracketleftBigg (1+Ps)τ2 ↑ /angbracketleftτ/angbracketright2−(1−Ps)τ2 ↓ /angbracketleftτ/angbracketright2/bracketrightBigg . (16) HerePs=(n↑−n↓)/(n↑+n↓) is the spin polarization of the 2D free carriers. We have also introduced an averaged scatter-ing time /angbracketleftτ/angbracketright. The parameter M scontrols the conversion of the spin Hall to the charge Hall current. In Fig. 8we plot the dependence of ρxxand the spin/charge Hall factor Mson the skyrmion surface density nskvia the parameter ωsτcovering the transition between scattering on nonmagnetic impurities and skyrmions. The switching of the dominant scattering mechanism af- fects the spin dependent scattering time τs[Eq. ( 9)]. In the dilute regime of low skyrmion surface density ωsτ0/lessmuch1 con- sidered in the previous section, the total transport scatteringtimeτ sis independent of the carrier spin, being determined by scattering on host nonmagnetic impurities τs=τ0. Therefore ρxxdoes not depend on nsk. Increasing the skyrmion surface density turns the system into the dense skyrmionic regimeω sτ0/greatermuch1, when the total transport scattering time is deter- mined solely by the magnetic skyrmions and, hence, dependson the carrier spin state τ s=ω−1 s. When ωsexceeds τ−1 0,t h e longitudinal resistivity ρxx∝/angbracketleftτ/angbracketright−1increases linearly with nsk as shown in Fig. 8. According to Eq. ( 16) the topological Hall resistivity ρT yx is proportional to the skyrmion surface density nsk.T h e crossover in the dominating scattering mechanism affects ρT yx only via the Msfactor. In the dilute regime ( ωsτ0/lessmuch1) this parameter coincides with the carrier spin polarizationM s=Psas the scattering time on host impurities τ0is spin 195439-7DENISOV , ROZHANSKY , A VERKIEV , AND LÄHDERANTA PHYSICAL REVIEW B 98, 195439 (2018) ξ=+ 1 ξ=−1Sz(r)S0 −S0r γ=−π/2γ=+π/2 T FIG. 9. Two chiral spin textures with opposite orientation ξ= ±1a n d/kappa1=−1 connected by the time-inversion T. independent. However, in the dense regime ( ωsτ0/greatermuch1) the scattering time τsdepends on the carrier spin, this dependence creates an additional spin imbalance favoring the conversionof spin to charge currents. As a result, the M sfactor is renormalized accounting for τ↑/negationslash=τ↓. The general expressions for ρxxandρT yxin the adiabatic regime ( 16) are applicable for any spin-dependent scattering mechanisms, not necessarily due to skyrmions. We point outthat in the leading order with respect to /Omega1 ssτsthe effect of τsonρT yxcan be fully described by the replacement of the carriers spin polarization Psby an effective Msfactor, which accounts for τ↑/negationslash=τ↓. C. Paramagnetic chiral systems In the previous sections we considered THE in a 2D mag- netic layer with a background magnetization S0and local de- viations forming chiral magnetic textures. Unlike anomalousHall effect, THE does not necessarily require macroscopicspin polarization of the carriers in the sample. Therefore, THEis allowed in a system with no background magnetizationprovided it still has localized chiral spin textures. We will referto this situation as to a chiral paramagnetic case. In the absence of a preferred magnetization direction the chiral spin textures with opposite orientations can be createdin the same sample. We denote them by the orientation ofspins in the center ξ=sgn(S z|r→0)=±1. However, these two spin configurations are not independent, they must beconnected by the time-inversion symmetry. The example ofsuch a Kramers doublet of spin textures with /kappa1=−1i ss h o w n in Fig. 9. The presence of two spin textures with opposite orientation ξ=±1 in the same layer modifies the expression for the charge contribution ρ cto THE ( 14). Indeed, as S0≈0 the sign of the spin-chirality driven contributions /Gamma11,2to the carrier asymmetric scattering on the spin texture depends on itsorientation and the contributions to ρ cfrom textures with ξ= ±1 have opposite sign. We arrive at the modified expression forρcaccounting for both texture orientations ξ=±1: ρc=Pξ1 nec(φ0nsk)/integraldisplay (/Gamma11+/Gamma12)s i nθdθ, (17) Pξ=n+−n− n++n−, where n±are surface densities of ξ=±1s p i nt e x t u r e s , respectively, nsk=n++n−is the total surface density, and Pξis the polarization of the texture array in terms of theirorientations. Here we consider the dilute regime with τs=τ0. It follows from ( 17) that observation of THE in chiral param- agnetic systems is possible only when there is an imbalancein the texture orientations, i.e., P ξ/negationslash=0. Let us note that for the positive texture polarization Pξ>0, the sign of ρcis different to that of ρcfor magnetic skyrmions, or noncollinear rings in Fig. 1. Indeed, as we already men- tioned, δSz<0 for the magnetic skyrmions case leading to ρc>0. On the contrary, δSz>0 is positive for ξ=+1s h o w n in Fig. 9, so that ρc<0. V. DISCUSSION Let us summarize the hallmarks of the developed THE the- ory and consider some of the experimentally studied skyrmionsystems. First, the THE contribution to the resistivity ρ T yx= ρc+ρaconsists of two terms: the first one ρcdescribes the transverse charge current due to spin-independent asymmetricscattering, while the second one ρ aappears due to spin Hall effect converted to the charge transverse current via nonzero2DEG spin polarization. Domination of one of the two termsis controlled by the adiabatic parameter λ a, the crossover from charge Hall dominating to spin Hall dominating regime occursatλ a∼1 and is accompanied by a local minimum in the dependence of ρT yxon the skyrmion size. Let us estimate the characteristic values of λafor some real skyrmion systems. Skyrmions extensively studied byPanagopoulos’ group [ 16,17,47] in Ir/Fe/Co/Pt multilayers systems are in the range 40–80 nm in size. For an estimate wetake [ 58]a=50 nm, /Delta1=0.6e V ,E F=5e V ,t h ee f f e c t i v e in-plane mass m=m0and obtain λa≈30. Skyrmions studied by Fert’s group [ 2,3,15] in similar Co/Pt multilayers systems are somewhat larger so that the adia-batic parameter is also in a strong coupling range λ a≈60. This is also typical for other Co/Pt systems with the sizeof the skyrmions being around 100 nm [ 59]. An example of substantially different system is Ta/FeCoB/TaOx structurewith skyrmionic bubbles of ∼1μms i z e[ 60]. The electron transport in such systems is also in the adiabatic regime. It is worth discussing the role of the ratio P s=/Delta1/2EF in these systems. According to Fig. 6,ρT yxis significantly reduced when Ps/lessmuch1 as discussed in Sec. IV A 4 .F o rt h e parameters used in the estimation above Ps≈0.06, thus even at seemingly large skyrmion diameters ρT yxcan be rather small. Therefore, THE is expected to be more pronounced insystems with higher /Delta1/2E Fratio. The decrease of λadown to the order of unity leading to the nonadiabatic transport regime is expected for nanometer-sizechiral magnetic textures in metallic systems with typical fer-romagnets such as Co or Fe. In the recent studies of Weisen-danger’s group a few-nanometer size skyrmions were suc-cessfully stabilized [ 61]. For such nanoscale skyrmions the THE is rather sensitive to band structure parameters, i.e.taking a=5 nm and P s=0.5 one gets λa≈30 so the THE is size-independent [ 57], while for the smaller ratio of the exchange coupling to the Fermi energy Ps=0.06 one gets λa≈3 suggesting that the system is in the vicinity of the crossover from adiabatic to weak coupling regimes. Alternatively, the nonadiabatic scenario of THE can be achieved in the dilute magnetic semiconductors (DMS). The 195439-8GENERAL THEORY OF THE TOPOLOGICAL HALL EFFECT … PHYSICAL REVIEW B 98, 195439 (2018) existence of chiral spin textures in DMS with spin-orbit inter- action has been suggested both experimentally [ 18,19,62] and theoretically on the basis of chiral magnetic polaron [ 53,56] via the chiral paramagnet scenario discussed in Sec. IV C .T h e solid advantage of DMS is that both the Fermi energy andthe exchange interaction strength can be tuned, allowing us tocontrol the adiabatic parameter in a wide range covering theweak coupling and adiabatic regimes of THE. For example, taking a n-type Cd 1−xMnxTe-based quantum well (electron effective mass m≈0.1m0, the exchange inter- action constant x×220 meV) with a chiral spin texture radius equal to a typical Bohr radius of an impurity bound state3n m ,f o r x=0.08 and the electron sheet density n 1=5× 1011cm−2we get λa≈2.5. By decreasing the sheet density down to n2=1×1011cm−2or decreasing Mn fraction down tox=0.02 the adiabatic parameter can be adjusted to λa≈6 andλa≈0.8, respectively. Let us comment on material systems where the electron transport is affected by dense array of chiral spin textures.According to the results of Sec. IV B , the THE resistivity in the adiabatic regime ρ T yx∝Ms(φ0nsk) linearly depends on the textures surface density, while the coefficient Msis renormalized differently depending on whether the carrierscattering is dominated by host impurities or skyrmions. Theinterplay between the two scattering mechanisms, on thecontrary, manifests itself in ρ xx. One example is a ferromagnetic film in the vicinity of the phase transition, when thermally activated criticalmagnetic fluctuations lead to the peak in the longitudinalresistivity [ 28,63–65]. When spin-orbit interaction makes the critical fluctuations chiral a pronounced THE signal isexpected, as suggested in Refs. [ 28,29,65]. Let us mention that to describe the scaling properties of the resistivitiesρ yx(ρxx) in the vicinity of FM transition one should specify a specific model of the chiral fluctuations adequate for theconsidered material system. ACKNOWLEDGMENTS We thank M. Chsiev, V . Cross, A. Fert, A. Hallal, H. Jaffres, W. Legrand, D. Maccariello, C. Panagopoulos,M. Raju, and N. Reyren for very fruitful discussions and com-ments. The work has been carried out under the financial sup-port of Grants from the Russian Science Foundation (asym-metric scattering theory - Project No. 17-12-01182; numericalcalculations - Project No. 17-12-01265) and from RussianFoundation of Basic Research (Grant 18-02-00668). It wasalso supported by the Academy of Finland Grant No. 318500.K.S.D. and N.S.A. thank the Foundation for the Advancementof Theoretical Physics and Mathematics “BASIS.” APPENDIX A: INTEGRATION OF THE COLLISION INTEGRAL AND THE BOLTZMANN EQUATION In this Appendix we calculate the collision integral in Eq. ( 3) and derive the system of equations Eq. ( 8)f o rt h e function g± sfrom Eq. ( 6). We start from the collision integral St[gs(p)]=/summationdisplay p/prime,s/prime/parenleftbig Wss/prime pp/primegs/prime(p/prime)−Ws/primes p/primepgs(p)/parenrightbig ,(A1)where gs(p) is a nonequilibrium part of the full distribution function. The angular asymmetry of the transport responsiblefor THE arises from a complex dependence of g s(p)o n the polar angle of the momentum p=(p,ϕ), thus in what follows we denote this dependence explicitly as gs(ϕ). Let us write the scattering rates Wss/prime pp/primein the form Wss/prime pp/prime=1 νAss/prime(θ)δ/parenleftbig εs p−εs/prime p/prime/parenrightbig , Ass/prime(θ)=2π ¯hν/parenleftbig ni|upp/prime|2δss/prime+nsk/vextendsingle/vextendsingleTss/prime pp/prime/vextendsingle/vextendsingle2/parenrightbig , (A2) where θ=ϕ−ϕ/primeis the scattering angle, δis the delta func- tion, and ν=m/2π¯h2is the 2D density of states, we omit the dependence of Ass/primeon the electron energy ε. After integrating Eq. ( A1) over εwe arrive at St[gs(ϕ)]=/summationdisplay s/prime/integraldisplaydϕ/prime 2π(Ass/prime(θ)gs/prime(ϕ/prime)−As/primes(−θ)gs(ϕ)). (A3) The next step is to expand gs(ϕ) and Ass/prime(θ) in angular harmonics: gs(ϕ)=/summationdisplay n/greaterorequalslant1g+ s,ncosnϕ+g− s,nsinnϕ, Ass/prime(θ)=/lscript0,ss/prime+2/summationdisplay n/greaterorequalslant1/lscript+ n,ss/primecosnθ+/lscript− n,ss/primesinnθ, (A4) where /lscript+ n,ss/prime=2π ¯hν/integraldisplay2π 0dθ 2πcosnθ/parenleftbig ni|upp/prime|2δss/prime+nsk/vextendsingle/vextendsingleTss/prime pp/prime/vextendsingle/vextendsingle2/parenrightbig , /lscript0,ss/prime=2π ¯hν/integraldisplay2π 0dθ 2π/parenleftbig ni|upp/prime|2δss/prime+nsk/vextendsingle/vextendsingleTss/prime pp/prime/vextendsingle/vextendsingle2/parenrightbig , /lscript− n,ss/prime=2π ¯hνnsk/integraldisplay2π 0dθ 2πsinnθ/vextendsingle/vextendsingleTss/prime pp/prime/vextendsingle/vextendsingle2. The terms /lscript− n,ss/primemix odd ( −) and even ( +) angular parts of the nonequilibrium distribution function gs, hence they are responsible for the asymmetric scattering. In these terms wetake into account only the scattering on chiral spin textures(i.e., skew scattering due to host impurities leading to theanomalous Hall effect is neglected). It is especially convenientto write the asymmetric coefficients /lscript − n,ss/primeusing the dimen- sionless rates Jss/prime(θ) introduced in Eq. ( 5): /lscript− n,ss/prime=−e mc(nskφ0)/integraldisplay2π 0Jss/prime(θ)s i nnθdθ, (A5) where eis a negative electron charge, and φ0=hc/|e|is the flux quantum. After substituting the expansions Eq. ( A4) and integrating it over the angle ϕ/primewe arrive at the following expression for the collision integral: St[gs(ϕ)]=/summationdisplay nI+ s,ncosnϕ+I− s,nsinnϕ, I+ s,n=/summationdisplay s/prime[/lscript+ n,ss/primeg+ s/prime,n−/lscript− n,ss/primeg− s/prime,n−/lscript0,s/primesg+ s,n], I− s,n=/summationdisplay s/prime[/lscript+ n,ss/primeg− s/prime,n+/lscript− n,ss/primeg+ s/prime,n−/lscript0,s/primesg− s,n].(A6) 195439-9DENISOV , ROZHANSKY , A VERKIEV , AND LÄHDERANTA PHYSICAL REVIEW B 98, 195439 (2018) The number of nonzero angular harmonics g± n,sin the Boltz- mann equation Eq. ( 3) is determined by a particular transport scenario. For the considered in this paper linear response onthe electric field E, the field-driven part of Eq. ( 3) contains only the first angular harmonics n=1; therefore g ± s,n=0, I± s,n=0f o rn/greaterorequalslant2, and only the index n=1 is relevant, so we simplify the notation [see Eq. ( 6)]g± s≡g± 1,s.T h et e r m so f the collision integral for n=1a r eg i v e nb y I+ s,1=−τ−1 sg+ s+/Omega1ssg− s+τs¯sg+ ¯s+/Omega1s¯sg− ¯s, (A7) I− s,1=−τ−1 sg− s−/Omega1ssg+ s+τs¯sg− ¯s−/Omega1s¯sg+ ¯s, where we introduced τ−1 s=/lscript0,ss−/lscript+ 1,ss+/lscript0,¯ss,/Omega1ss/prime= −/lscript− 1,ss/prime, andτ−1 s¯s=/lscript+ 1,s¯s;¯sdenotes the spin state opposite to s. The detailed expressions for τs,τs¯s,/Omega1ss/primeparameters are given in Eq. ( 9). Assuming that Eis directed along the xaxis the Boltzmann equation for sspin subband is given by eEvs∂f0 s ∂εcosϕ=I+ s,1cosϕ+I− s,1sinϕ. (A8) Equating eEvs∂f0 s ∂ε=I+ s,1, andI− s,1=0 brings us to the system of algebraic equations Eq. ( 8)f o rg± 1,s. APPENDIX B: SYMMETRY OF SCATTERING RATES In this Appendix we derive the relations ( 13)f o rt h e dimensionless asymmetric scattering rates Jss/prime(θ,η). The background polarization η=sgn(S0)=±1 also determines the orientation of spins inside the core of a chiral spin texture.The starting point is the time-reversal invariance, which statesthat the scattering rate from ( p /prime,s/prime)→(p,s) with a scattering angle θ=ϕ−ϕ/primeshould be equal to that from ( −p,¯s)→ (−p/prime,¯s/prime) with the scattering angle −θand reversed polariza- tion of the spin texture S(r)→− S(r)(¯sdenotes the carrier spin state opposite to s). The spin texture reversal S(r)→ −S(r) implies η→−η,/kappa1→/kappa1, andγ→γ+π. Collecting these operations together we obtain Gss/prime(θ,η)+Jss/prime(θ,η)=G¯s/prime¯s(−θ,−η)+J¯s/prime¯s(−θ,−η). (B1) Taking into account that Gss/prime(θ,η)=Gss/prime(−θ,η) and Jss/prime(θ,η)=−Jss/prime(−θ,η) we get that the symmetric and asymmetric scattering rates satisfy Gss/prime(θ,η)=G¯s/prime¯s(θ,−η), Jss/prime(θ,η)=−J¯s/prime¯s(θ,−η). (B2) We further focus on Jss/prime. The relations ( B2) couple the two scattering channels with the opposite spin orientations. For thespin-conserving channels we have J ↑↑(θ,η)=−J↓↓(θ,−η). (B3) Let us introduce the symmetrized and antisymmetrized com- binations of J↑↑(θ,η),J↑↑(θ,−η) with respect to η: /Gamma1(θ,η)=1 2[J↑↑(θ,η)−J↑↑(θ,−η)], /Pi1(θ)=1 2[J↑↑(θ,η)+J↑↑(θ,−η)]. (B4) Since the background polarization can take only two values η=±1, the function /Pi1does not depend on η, while/Gamma1(−η)=−/Gamma1(η). The dependence of /Gamma1onηcan be specified explicitly as/Gamma1(η,θ)≡η/Gamma11(θ), where /Gamma11depends only on θand on the energy of the incident electron. Expressing the rates of the spin-conserving channels and using the symmetry ( B3) we arrive at the relations ( 13) J↑↑(θ,η)=η/Gamma11(θ)+/Pi1(θ), J↓↓(θ,η)=η/Gamma11(θ)−/Pi1(θ). (B5) As for the spin-flip scattering channels, there is an addi- tional symmetry J↑↓(θ,η)=J↓↑(θ,η) as the Hamiltonian is Hermitian, this symmetry leads to the absence of a spin Hallpart: J ↓↑(θ,η)=J↑↓(θ,η)=η/Gamma12(θ), (B6) where/Gamma12(θ) does not depend on η. APPENDIX C: DERIVATION OF THE RESISTIVITY TENSOR ρ In this Appendix we solve the system of Eq. ( 8) and calcu- late the longitudinal ( ρxx) and transverse ( ρT yx) resistivities for dilute and dense skyrmion systems [see Eqs. ( 14) and ( 16)]. 1. Dilute regime In the dilute regime the scattering is dominated by host nonmagnetic impurities: τs≈τ0/lessmuchω−1 s,/Omega1−1 ss/prime. In this case the longitudinal component g+ sis determined only by τ0: g+ s=τ0eEvs/parenleftbigg −∂f0 s ∂ε/parenrightbigg . (C1) The transverse part g− sin the lowest order in /Omega1ss/primeτ0is given by g− s=−τ0[/Omega1ssg+ s+/Omega1s¯sg+ ¯s]. (C2) The electric current jcalculated according to Eq. 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The product /Omega1ss/primeτs/lessmuch1 leading 195439-10GENERAL THEORY OF THE TOPOLOGICAL HALL EFFECT … PHYSICAL REVIEW B 98, 195439 (2018) to the Hall effect remains small as the asymmetric scattering rates are always smaller than the symmetric ones. Keepingthese conditions we get for g + sandg− s: g+ s=τseEvs/parenleftbigg −∂f0 s ∂ε/parenrightbigg ,g− s=−g+ s(/Omega1sτs). (C5) The electric current in Eq. ( 7) calculated with g± sfrom above is given by jx=σxxE, j y=σT yxE, σxx=σ↑+σ↓,σ s=nse2τs m, (C6) σT yx=− (σ↑/Omega1τ↑−σ↓/Omega1τ↓),where we take into account /Omega1≡/Omega1↑↑=−/Omega1↓↓for the adia- batic scattering regime. 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PhysRevB.73.205411.pdf

Filling factor and electronic structure of Dy 3N@C80filled single-wall carbon nanotubes studied by photoemission spectroscopy H. Shiozawa,1H. Rauf,1T. Pichler,1M. Knupfer,1M. Kalbac,1,2S. Yang,1L. Dunsch,1B. Büchner,1D. Batchelor,3and H. Kataura4 1IFW Dresden, P .O. Box 270116, D-01171 Dresden, Germany 2J. Heyrovský Institute of Physical Chemistry, ASCR, Dolejskova 3, 18223 Prague 8, Czech Republic 3Universität Würzburg, BESSY II, D-12489 Berlin, Germany 4Nanotechnology Research Institute, Advanced Industrial Science and Technology (AIST), Tsukuba 305-8562, Japan /H20849Received 17 November 2005; revised manuscript received 17 January 2006; published 16 May 2006 /H20850 We report on a detailed study of the filling factor and electronic structure of the trimetal nitride fullerene Dy3N@C80/H20849I/H20850filled single-wall carbon nanotubes /H20849SWCNTs /H20850, the so-called endohedral peapods /H20849Dy3N@C80@SWCNTs /H20850, using photoemission spectroscopy as a probe. The photoemission response close to the Fermi energy exhibits the manifestation of one-dimensional electronic structures, i.e., van Hove singulari-ties and a power-law /H20849Tomonaga-Luttinger liquid /H20850behavior. Within the experimental accuracy the spectral shapes are the same as those of the pristine SWCNTs reference sample. From a comparison between thevalence-band spectra of the peapods and the SWCNT measured with 400 eV photon energy, we determine thebulk filling of Dy 3N@C80@SWCNTs to be 74±10 %. In addition, from a detailed analysis of the resonant photoemission across the Dy 4 d-4fedge, we show that the effective Dy valency of Dy 3N@C80@SWCNTs is about 3.0 which is slightly larger than for Dy 3N@C80. DOI: 10.1103/PhysRevB.73.205411 PACS number /H20849s/H20850: 73.22. /H11002f, 79.60. /H11002i I. INTRODUCTION The endohedral peapods have a great potential to functionalize single-wall carbon nanotubes /H20849SWCNTs /H20850in a controlled manner because of a variety of the endo-hedral species in contrast to empty fullerenes. The first ex-amples for the formation of the endohedral peapods areGd@C 82@SWCNTs and La 2@C80@SWCNTs, which have been observed by high-resolution transmission electron mi-croscopy /H20849HRTEM /H20850. 1,2The latter reported a freezing of the free rotation of the La atoms at room temperature by theencapsulation of the fullerenes into SWCNT. 2The electric resistance of Gd @C82@SWCNTs exhibited a sharper in- crease with decreasing temperature down to 5 K comparedwith C 60@SWCNTs,3which was attributed to the additional electron scattering due to the local electrostatic potentialfrom the fullerenes. In addition, a band gap modulation ofthe SWCNT has been observed on Gd @C 82@SWCNT hav- ing a chirality of /H2084911,9/H20850by scanning tunneling microscopy.4 Raman spectroscopy of La 2@C80@SWCNTs observed an up-shift of the G-band mode from the pristine SWCNTs, which was attributed to the charge transfer from theSWCNTs to the endohedrals. 5This is the same tendency as a p-type semiconducting behavior at room temperature in Dy@C82@SWCNTs.6A study of La @C82,L a 2@C80, and Sc3N@C80by using the density functional theory /H20849DFT/H20850pre- dicted downshifts of the endohedral fullerene-derived mo-lecular orbitals which increase by increasing the tube-fullerene distance. 7This calculation also showed local radial deformations of the SWCNTs leading to shifts of the vanHove singularities /H20849vHs/H20850. A broadening of the S 11optical transition peak of the SWCNTs upon La @C82encapsulation can be understood as a consequence of the chirality-dependent energy shifts of vHs. 8Photoemission spectroscopy is a reliable tool to investi- gate the bulk electronic structures of such compositemolecules because its element and site selectivity make itpossible to extract the partial electronic structures of each of the constituents. In recent years photoemission studies con-tributed significantly to the understanding of the nature ofendohedrals as well as fullerene peapods /H20849e.g., C 60@SWCNTs /H20850.9–21 In the present study, the recently synthesized trimetal nitride fullerene,23Dy3N@C80/H20849isomer I /H20850, was success- fully encapsulated into the SWCNTs. The bulkelectronic structure of the endohedral peapods,Dy 3N@C80@SWCNTs, were investigated by photoemission spectroscopy. In this paper, we show that the one-dimensional electronic structure of the SWCNTs, i.e., vHsand a power-law /H20849Tomonaga-Luttinger liquid /H20850behavior, per- sists upon the formation of Dy 3N@C80@SWCNTs. In addi- tion, by taking advantage of the high atomic photoioniza-tion cross section of the Dy 4 fstates with 400 eV photon energy, we show that the effective valency of Dy is essen-tially trivalent. From the integrated intensity of the endohe-dral peapod spectrum relative to the SWCNT spectrumwe determine quantitatively the bulk filling factor ofDy 3N@C80@SWCNTs. For the given SWCNTs diameter distribution we observe a bulk filling factor of 74±10 %. Inaddition, a more quantitative analysis of the electronic con-figuration of Dy is performed by the resonant photoemissionspectroscopy across the Dy 4 d-4fedge. The Dy 4 fmultiplet profile extracted from the subtraction between on- and off-resonant spectra is well reproduced with two sets of multi-plets calculated for 4 f 9→4f8and 4 f10→4f9photoemission process, respectively. From the relative intensity of the twosets of multiplets, we estimate the bulk effective Dy valencyof Dy 3N@C80@SWCNTs to be 3.0, which is slightly largerPHYSICAL REVIEW B 73, 205411 /H208492006 /H20850 1098-0121/2006/73 /H2084920/H20850/205411 /H208496/H20850 ©2006 The American Physical Society 205411-1than the previously reported value of 2.8 for a Dy 3N@C80 film.19The valence change observed is attributed to the ad- ditional charge transfer between the endohedral fullerene andSWCNT. II. EXPERIMENTAL DETAIL The trimetal nitride fullerene Dy 3N@C80/H20849isomer I /H20850was produced using the modified Krätschmer-Huffman dc-arcdischarging method. 22High performance liquid chromatog- raphy and laser desorption time-of-flight mass spectroscopywere used to isolate and identify Dy 3N@C80fullerenes.23 The SWCNTs were synthesized by the laser ablation method and purified by the H 2O2treatment. A SWCNT film was prepared by drop coating from an acetone solution ofSWCNTs and cleaned by annealing at 450 °C in ultrahighvacuum /H20849UHV /H20850conditions as described previously. 20After the measurements on the SWCNT film, a Dy 3N@C80film with visible thickness was deposited onto the SWCNT filmby the sublimation at a furnace temperature of up to 600 °Cunder UHV conditions using an evaporator specially de-signed for the purpose of subliming tiny amounts of mol-ecules. The film was subsequently heated at 450 °C for morethan 24 h to encapsulate Dy 3N@C80into the SWCNTs: the temperature is slightly higher than the sublimation tempera-ture of Dy 3N@C80/H11011400 °C, but lower than 500 °C which closes the SWCNT ends. After the encapsulation procedure,an additional heat treatment at 600 °C was performed inorder to remove the excess Dy 3N@C80which did not enter the SWCNTs. A complete removal of the excess Dy 3N@C80 on the SWCNT film surface was confirmed by observing the disappearance of the Dy 3N@C80film visually. TEM obser- vations confirmed the SWCNTs to be filled. The synthesized endohedral peapod film was transferred into the analysis chamber equipped with a helium cryostatwithout breaking UHV conditions. The photoemission ex-periment was performed at beamline UE 52 PGM, Bessy II,using a hemispherical photoelectron energy analyzer SCI-ENTA SES 200. The absorption spectrum was obtained bymeasuring the drain current of the sample. The experimentalresolution and the Fermi energy /H20849E F/H20850were determined from the Fermi edge of a clean Au film. All spectra were recorded with an overall energy resolution better than 50 meV. Thebase pressure in the experimental setup was kept below3/H1100310 −10mbar. III. RESULTS AND DISCUSSION A. Filling factor We first determine the bulk filling factor of Dy3N@C80@SWCNTs by comparing the integrated inten- sity of the valence-band photoemission spectrum ofDy 3N@C80@SWCNTs with that of SWCNTs. For this pur- pose, we used the valence-band photoemission spectra ofDy 3N@C80@SWCNTs and the SWCNTs measured with 400 eV photon energy which are plotted in Fig. 1 togetherwith that of Dy 3N@C80. The spectra were normalized so that the ratio between the elastic and inelastic signals of the dif-ference spectrum Dy 3N@C80@SWCNTs−SWCNTs be-comes the same as that of the pristine Dy 3N@C80spectrum. While the SWCNT spectrum shows broad features, theDy 3N@C80@SWCNT spectrum exhibits two prominent peak structures at the binding energies around 7 and 10 eV,respectively. These peak structures are ascribed as the Dy 4 f multiplets, which will be discussed in detail in Sec. III C. Alarge contribution of the Dy 4 fstates in the valence-band photoemission spectrum is due to the high atomic photoion-ization cross section of the Dy 4 fstate much larger than those of the valence states of C and N with 400 eV photonenergy. The integrated intensity of the valence-band photoemis- sion from the Dy 3N@C80@SWCNTs with a bulk filling fac- tor of /H9263is expressed as IDy3N@C80@NT/H11008NNT/H9252NT/H9253NT+/H9263/H20858 M=Dy3,N,C80NM/H9252M/H9253M, /H208491/H20850 where NMis the number of the atoms of the constituents M=Dy 3,N ,C 80,S W C N Ti naD y 3N@C80@SWCNT with 100% filling, /H9252Mis the atomic photoionization cross section, /H9253Mis a factor to take into account the effect of a short pho- toelectron mean free path of 0.80±0.20 nm at 400 eV elec-tron energy. 21According to the previous photoemission work on C 60@SWCNTs,21when /H9253NT=1,/H9253C80is calculated as 0.75±0.10 for a C 80@SWCNT, with a SWCNT diameter of 1.42 nm, which is the mean SWCNT diameter of thepresent sample and it will be obtained in the next paragraph.Also, /H9253Dy3and/H9253Nshould be 0.75±0.10. From the atomic FIG. 1. Photoemission spectra of SWCNTs, Dy3N@C80@SWCNTs, and Dy 3N@C80collected with 400 eV photon energy. The difference spectrum betweenDy 3N@C80@SWCNTs and SWCNTs is also plotted.SHIOZAWA et al. PHYSICAL REVIEW B 73, 205411 /H208492006 /H20850 205411-2photoionization cross sections of 1.63 for Dy 4 f, 0.032 for N2s, 0.008 for N 2 p, 0.020 for C 2 s, and 0.002 for C 2 pat 400 eV photon energy, we have /H9252Dy3=1.63, /H9252N=0.040, and /H9252C60=/H9252NT=0.022. NMin the unit length of the C 80peas are NDy3=3, NN=1, NC80=80, and NNT/H11011185 for a Dy3N@C80@SWCNT with a SWCNT diameter of 1.42 nm and C 80–C80distance of 1.12 nm:24the latter is the sum of a C80/H20849Ih/H20850diameter of 0.79 nm /H20849Ref. 25 /H20850and the van der Waals distance in a graphite of 0.33 nm. By substituting the param- eters into Eq. /H208491/H20850and using the ratio between the inte- grated intensities of the Dy 3N@C80@SWCNT and SWCNT spectra after the subtraction of inelastic backgrounds,I Dy3N@C80@NT:INT=1.9:1, we obtain the bulk filling factor of Dy3N@C80@SWCNTs of /H9263=74±10%. Furthermore, by assuming a Gaussian distribution of the SWCNT diameter with a mean value of 1.42 nm and astandard deviation of 0.10 nm which will be obtained in thenext paragraph, and the condition that only the SWCNTwhose diameter is larger than a minimal value of d minis completely filled,21the estimated bulk filling factor /H9263 =74±10 % yields dmin=1.38±0.03 nm. This value is fully consistent with dmin/H110111.36 nm predicted theoretically for Sc3N@C80@SWCNTs.7 We point out that this method opens a very reliable method to determine the filling factor on a bulk scale and isa much simpler and more straightforward analysis than thedetermination of the bulk filling factors in C 60peapods using electron energy-loss spectroscopy26or x-ray diffraction.27 B. One-dimensional electronic structures As a second step we analyzed the electronic structure of Dy3N@C80@SWCNTs at low binding energies with special emphasis on the possible changes in the one-dimensionalelectronic properties, namely, vHs and Tomonaga-Luttingerliquid /H20849TLL/H20850behavior, in comparison with those of the SWCNTs. Figure 2 shows the valence-band photoemissionspectra of Dy 3N@C80@SWCNTs and the SWCNTs at bind- ing energies below 1.5 eV collected with 125 eV photon en-ergy. The sample temperature was kept at 17 K during mea-surements. Three peak structures are observed in bothspectra whose shapes are similar to each other and to thoseof the spectra measured in the previous studies on SWCNTsand C 60peapods.20,21,28,29According to the procedure per- formed by Ishii et al. ,28the photoemission spectra were re- produced by the tight-binding calculation taking into accountthe Gaussian distribution of the SWCNT diameter and a shiftofE F. As plotted in Fig. 2, the calculated spectrum repro- duces well the three peaks in both spectra with the sameparameters: the mean and the standard deviation of theSWCNT diameter distribution are 1.42 and 0.10 nm, respec-tively, and E Fis shifted by 0.18 eV towards unoccupied states. The first two peaks at 0.5 and 0.8 eV binging energiescorrespond to the first and second vHs peaks of the semicon-ducting tubes, while the third peak located at 1.1 eV bindingenergy is assigned as the first vHs peak of the metallic tubes.The estimated mean diameter of 1.42 nm of the SWCNTs issimilar to 1.44 nm of /H2084914,7/H20850SWCNT which has been re- ported to be the exothermic for encapsulation of La @C 82,La2@C80, and Sc 3N@C80by the DFT calculation.7Accord- ing to this calculation, the encapsulation of La @C82, La2@C80, and Sc 3N@C80in/H2084914,7/H20850SWCNT gives rise to a deformation only of the SWCNT with about 1% local radialexpansion leading to no appreciable shift /H20849/H110210.01 eV /H20850of the vHs peak positions. This is also confirmed by the present result where the vHs peak positions and widths exhibit nochange by the formation of Dy 3N@C80@SWCNTs. As a next point regarding the one-dimensional electronic structure, the TLL behavior of Dy 3N@C80@SWCNTs in comparison with that of the SWCNTs is investigated fromthe photoemission spectra near E Fwhich were plotted with a double logarithmic scale in the inset of Fig. 2. As seen in thefigure, both spectra have no Fermi edge, instead, they exhibitTLL behavior, namely, a power-law dependence /H9255 /H9251on the binding energy /H9255with a power-law factor /H9251=0.54±0.10 for the Dy 3N@C80@SWCNTs and 0.57±0.10 for the SWCNTs: the power-law factors were estimated by fitting the data inthe binding energy region from 0.05 eV up to 0.25 eV to thepower-law function. The estimated power-law factors are al-most identical. Within experimental accuracy, this is in goodagreement with the values reported in the previous works onSWCNTs and C 60@SWCNTs.20,28,29In summarizing, the present results regarding vHs and TLL behavior show thatthe encapsulation of Dy 3N@C80fullerenes does not affect FIG. 2. Photoemission spectra of Dy 3N@C80@SWCNTs and SWCNTs below 1.5 eV binding energy recorded with 125 eV pho-ton energy. The density of states /H20849DOS /H20850and photoemission spec- trum reproduced by the tight-binding calculation are also plotted.The inset shows log-log plots of the photoemission spectra near theFermi level.FILLING FACTOR AND ELECTRONIC STRUCTURE OF ¼ PHYSICAL REVIEW B 73, 205411 /H208492006 /H20850 205411-3the low-energy excitation nature of SWCNTs and their one- dimensional feature persists after the formation ofDy 3N@C80@SWCNTs. C. Effective valency of Dy We now turn to the valence-band electronic structure of the endohedral peapod with special emphasis on theresponse of the Dy ions which can be used to extract theeffective Dy valency. The subtraction between theDy 3N@C80@SWCNTs and SWCNTs photoemission spectra taken with 400 eV photon energy can be used to extract theDy 4 fstates. The obtained difference spectrum is, as a whole, similar to the pristine Dy 3N@C80spectrum taken with the same photon energy as seen in Fig. 1, and essen-tially exhibits the trivalent Dy multiplets. As reported inthe previous x-ray photoemission study of Dy 3N@C80using AlK/H9251radiation,19the prominent structures corresponding to the divalent Dy were observed around 4 eV binding energyin the pristine Dy 3N@C80spectrum. In contrast, the differ- ence spectrum has less structures in this energy region. Further details about the Dy valency are obtained from a resonance photoemission study in the Dy 4 d-4fexcita- tion region. The valence-band photoemission spectra ofDy 3N@C80@SWCNTs measured in the Dy 4 d-4fresonance region are plotted in Fig. 3. The photon energies were set tothe energies at which the prominent structures appear on theDy 4 d-4fabsorption spectrum plotted in the inset of Fig. 3. The absorption spectrum is similar to that of Dy metal inwhich Dy atoms are triply ionized. 30As observed in Fig. 3, the photoemission spectra have narrow structures over thewhole valence-band region which are the admixture betweenthe 2 sand 2 pstates of the carbon cages and the 4 fstates of the Dy ions. In the spectra taken with the photon energiesaround the giant peak in the Dy 4 d-4fabsorption spectrum, the three peak structures located at 6.7, 10.4, and 11.8 eVdominate. These peaks become most prominent in the spec-trum taken with the photon energy 161 eV which is exactlythe energy of the maximum of the giant peak in the Dy 4 d -4fabsorption spectrum. Therefore, these structures are as- signed as the resonant-enhanced Dy 4 fstates. With a 149 eV photon energy which is just below the onset of the first struc-ture in the Dy 4 d-4fabsorption spectrum, the Dy 4 fstruc- tures are undetectable due to the Dy 4 d-4foff-resonant con- dition, instead, the spectral features reflect predominantly thecarbon molecular structures. Comparison with the pristineSWCNT spectrum collected with 149 eV photon energy/H20849also plotted in Fig. 3 /H20850shows that the spectral shapes of Dy 3N@C80@SWCNTs are almost the same as those of the SWCNTs. The peak structure around 18 eV becomes promi-nent with increasing photon energy. Since the atomic photo-ionization cross section of the C 2 sstate relative to that of t h eC2 pstate increases with photon energy, this peak can be assigned as the C 2 sstate. The subtraction between the Dy 4 d-4fon-resonant spec- trum /H20849161 eV /H20850and the off-resonant spectrum /H20849149 eV /H20850is used to accurately extract the Dy 4 fmultiplets. It should be taken into account upon the subtraction that the photoemis-sion spectrum of SWCNTs also changes its shape slightlybetween the Dy 4 d-4fon- and off-resonances. Thus in order to extract the Dy 4 fmultiplets we used the four spectra, namely, the on- and off-resonant spectra of Dy 3N@C80@SWCNTs /H20849Ionp,Ioffp/H20850and those of SWCNTs /H20849Ions,Ioffs/H20850. The subtraction was done between Ionp−Ionsand Ioffp−Ioffs. The obtained Dy 4 fspectrum is plotted after the subtraction of the inelastic backgrounds in Fig. 4. The spec-trum has the four prominent peak structures at the bindingenergies 6.7, 10.4, 11.8, and 15.0 eV, which is similar to thephotoemission spectrum of Dy metal. 31This indicates that the effective Dy valency of Dy 3N@C80@SWCNTs is essen- tially trivalent. In addition to the trivalent Dy 4 fmultiplets, small lower binding-energy structures are observed around4 eV. This is in good agreement with the 4 fmultiplets of Dy 3N@C80reported recently,19and these structures can be safely identified as a contribution from the divalent Dy. According to the procedure performed in the previous paper,19the experimental profile was reproduced from the atomic calculation for 4 f9→4f8and 4 f10→4f9photoemis- sion processes.32The photoemission multiplets are convo- luted with V oigt function to take into account the experimen-tal resolution of 50 meV and the lifetime broadening. For thelatter, the different Lorentzian widths are used for two sets of FIG. 3. Valence-band photoemission spectra of Dy3N@C80@SWCNTs taken with the photon energies across the Dy 4 d-4fedge. The inset shows the Dy 4 d-4fabsorption spectrum of Dy 3N@C80@SWCNTs.SHIOZAWA et al. PHYSICAL REVIEW B 73, 205411 /H208492006 /H20850 205411-4multiplets. The lowest-energy term7F6of 4f9→4f8multi- plets was reduced to 30% to take into account its weak reso-nance enhancement. 33The result is shown in Fig. 4. The calculated photoemission profile reproduced well the experi-mental spectrum. The intensity ratio of 1:0.01 between thetwo sets of the photoemission multiplets, 4 f 9→4f8and 4f10→4f9, yields the effective Dy valency close to 3.0. This value is larger than 2.8 reported recently for pristineDy 3N@C80film,19which is the same tendency as observed with 400 eV photon energy. The Dy valence increase by theformation of the endohedral peapods can be attributed to theadditional charge transfer between the Dy 3N@C80and SWCNTs.34 IV . CONCLUSIONS We have investigated the bulk filling factor and the elec- tronic properties of the newly formed endohedral peapods,Dy 3N@C80@SWCNTs, using photoemission spectroscopy as a probe. The valence-band photoemission responses showthe electronic structure of pristine unfilled SWCNTs and thatof pristine endohedrals. A comparison of the photoemissionspectra taken with 400 eV photon energy betweenDy 3N@C80@SWCNTs and SWCNTs gives the filling factor of Dy 3N@C80@SWCNTs of 74±10 %. Close to EFwe ob- served the manifestation of one-dimensional electronic prop-erties, i.e., vHs and TLL behavior, which were within experi-mental accuracy identical to those of SWCNTs. The Dy 4 f multiplet profile extracted from the 4 d-4fresonant photo- emission spectroscopy of Dy 3N@C80@SWCNTs and SWCNTs was well reproduced with the atomic photoemis-sion multiplets, 4 f 9→4f8and 4 f10→4f9. From the relative intensity between two sets of the multiplets, we estimated theeffective valency of the Dy ions to be about 3.0, which islarger than 2.8 reported previously for pristine Dy 3N@C80.19 This valence change observed might be due to the additional charge transfer between the Dy 3N@C80and SWCNTs. 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Achiba, Appl.Phys. A 74, 349 /H208492002 /H20850. 28H. Ishii, H. Kataura, H. Shiozawa, H. Yoshioka, H. Otsubo, Y . Takayama, T. Miyahara, S. Suzuki, Y . Achiba, M. Nakatake, T.Narimura, M. Higashiguchi, K. Shimada, H. Namatame, and M.Taniguchi, Nature /H20849London /H20850426, 540 /H208492003 /H20850. 29H. Rauf, T. Pichler, M. Knupfer, J. Fink, and H. Kataura, Phys. Rev. Lett. 93, 096805 /H208492004 /H20850. 30S. Muto, S.-Y . Park, S. Imada, K. Yamaguchi, Y . Kagoshima, and T. Miyahara, J. Phys. Soc. Jpn. 63, 1179 /H208491994 /H20850. 31J. K. Lang, Y . Baer, and P. A. Cox, J. Phys. F: Met. Phys. 11, 121 /H208491981 /H20850. 32F. Gerken, J. Phys. F: Met. Phys. 13, 703 /H208491983 /H20850. 33M. Kai, A. Tanaka, and T. Jo, J. Electron Spectrosc. Relat. Phe- nom. 78,6 3/H208491996 /H20850. 34Unfortunately the effect of the charge transfer between the Dy3N@C 80and SWCNTs is not observed as an energy shift of the vHs peaks. This can be easily explained by the fact that thenumber of SWCNT carbon atoms is about 85 times larger thanthe number of the Dy ions in the peapods with the filling of74%. 0.2 electron doping per a Dy would correspond to a dopinglevel of the SWCNT of 0.0024 hole per a carbon atom, whichyields an energy shift below our detection limit expected fromresults on intercalated SWCNTs and C 60peapods.SHIOZAWA et al. PHYSICAL REVIEW B 73, 205411 /H208492006 /H20850 205411-6

PhysRevB.71.214513.pdf

Bare electron dispersion from experiment: Self-consistent self-energy analysis of photoemission data A. A. Kordyuk,1,2S. V. Borisenko,1A. Koitzsch,1J. Fink,1M. Knupfer,1and H. Berger3 1Institute for Solid State Research, IFW-Dresden, Helmholtzstrasse 20, D-01069 Dresden, Germany 2Institute of Metal Physics of National Academy of Sciences of Ukraine, 03142 Kyiv, Ukraine 3Institute of Physics of Complex Matter, EPFL, CH-1015 Lausanne, Switzerland sReceived 27 December 2004; published 9 June 2005 d Performing an in-depth analysis of the photoemission spectra along the nodal direction of the high- temperature superconductor Bi-2212 we developed a procedure to determine the underlying electronic struc-ture and established a precise relation of the measured quantities to the real and imaginary parts of theself-energy of electronic excitations. The self-consistency of the procedure with respect to the Kramers-Kronigtransformation allows us to draw conclusions on the applicability of the spectral function analysis and on theexistence of well-defined quasiparticles along the nodal direction even for the underdoped Bi-2212 in thepseudogap state. The analysis of the real part of the self-energy S 8svdfor an overdoped and underdoped Bi-2212 helps to distinguish the 70 meV “kink” from S8svdmaximum and conclude about doping dependence of the kink strength. DOI: 10.1103/PhysRevB.71.214513 PACS number ssd: 74.25.Jb, 71.15.Mb, 74.72.Hs, 79.60. 2i I. INTRODUCTION With modern angle-resolved photoemission spectros- copy1,2sARPES done gets a direct snapshot of the density of low-energy electronic excited states in the momentum-energy space of two-dimensional s2Ddsolids. 3–6All the in- teractions of the electrons which are responsible for the un-usual normal and superconducting properties of cuprates areencapsulated in such pictures, but are still hard to decipher.One way to take into account these interactions is to considerelectronic excitations as quasiparticles which, compared tothe noninteracting electrons, are characterized by an addi-tional complex self-energy. 7Extraction of the self-energy from experiment is thus of great importance to check thevalidity of the quasiparticle concept and understand the na-ture of interactions involved, but appears to be problematicsince the underlying band structure of the bare electrons isa prioriunknown. One can evaluate the interaction parameters taking the bare band dispersion from band structure calculations, 4how- ever, this unavoidably increases the uncertainty of any con-clusions on the strength and nature of the interactions in-volved. A direct determination of the bare band structurefrom experiment would be much more attractive in thissense. Previously, the bare band dispersion has been assignedto the high binding energy part of the experimentaldispersion. 8In Refs. 9 and 10 we have discussed that the bare Fermi velocity estimated from the nodalARPES spectrausing the Kramers-Kronig sKKdtransformation is in reason- able agreement with band structure calculations 11,12and with an analysis of the anisotropic plasmon dispersion,13although it has been pointed out that in order to quantify interactionparameters such as coupling strength 14or self-energy15a pre- cise and reliable approach of bare band determination isneeded. In this paper we introduce an approach to directly extract both the bare band dispersion and the self-energy functionsfrom ARPES spectra. We show that the approach is self- consistent within the highest experimental accuracy availabletoday.Applying the procedure to the spectra from the under-doped and overdoped Bi sPbd-2212 as well as for optimally doped Bi sLad-2201, we demonstrate the validity of the qua- siparticle concept in cuprates even in the pseudogap state. II. NODAL SPECTRAANALYSIS We start with a brief overview of the basics of the nodal spectra analysis within the self-energy approach. Measuringthe photoemission intensity as a function of the kinetic en-ergy and in-plane momentum of outgoing electrons IsE k,kd one obtains access to the spectral function of the one electron removal which is supposed to reflect the quasiparticle prop-erties of the remaining photohole: its effective mass and life-time. These properties can be expressed in terms of a quasi-particle self-energy S=S 8+iS9, an analytical function the real and imaginary parts of which are related by the KKtransformation ssee Sec. 1 of the Appendix d. Neglecting for the moment the effects of the energy and momentum resolu-tions as well as the influence of matrix elements, 6one can takeI~Asv,kd, where vis the energy of the remaining pho- tohole with respect to the Fermi level. In turn, the spectral function can be formulated in terms of the self-energy Asv,kd=−1 pS9svd fv−«skd−S8svdg2+S9svd2, s1d where «skdis the bare band dispersion. Within such a defi- nition, S9svd,0, and S8svd.0 for v,0. A. Linear dispersion In case there is no interaction, i.e. electronic excitations can live forever, the spectral function is a delta function withthe pole v−«skvd=0 and, e.g., for the nodal direction, can bePHYSICAL REVIEW B 71, 214513 s2005 d 1098-0121/2005/71 s21d/214513 s11d/$23.00 ©2005 The American Physical Society 214513-1represented by the solid line in Fig. 1. When interactions are present, the self-energy leads to a shifting and broadening ofthe noninteracting spectral function. The resulting picture isessentially that which is measured in ARPES sthe blurred region in Fig. 1 illustrates this d. If one neglects the momen- tum dependence of the self-energy, then, from Eq. s1d, the momentum distribution curves fMDC skd=Askd v=const ghave maxima at kmsvddetermined by v−«skmd−S8svd=0 for a given v. In other words, S8svd=v−«skmd, is that which is illustrated in Fig. 1 by the double-headed arrow. In the region where the bare dispersion can be considered as linear swith a slope vFdone can write S8svd=v−vFfkmsvd−kFg. s2d Assuming in addition weak kdependence of S9along a cut perpendicular to the Fermi surface ssee discussion in Sec. 5 of theAppendix d, the MDCs exhibit a Lorentzian line shape5 with the half width at half maximum Wand S9svd=−vFWsvd. s3d Thus, the determination of both the real and imaginary parts of the self-energy requires the knowledge of the baredispersion «skdsor, in the vicinity to E F, an “energy scale,” e.g., Fermi velocity vFd.9The KK transformation gives an additional equation which relates these functions: S8 =KK S9fe.g., Eq. sA1dg. This opens the way to extract all desired quantities from the experiment, but brings a new“problem of tails.” Under “tails” we mean the behavior ofS 9svdfor energies uvu.vm, where vmis a “confidence limit,” a maximal experimental binding energy to which both theWsvdandkmsvdfunctions can be confidently determined. Fortunately, as we show in Sec. 3 of the Appendix, the different but reasonable tails of S9svdalmost do not effect the low-energy behavior of S8svd. The influence of the high- energy region on the coupling strength sA3dcan be describedby mainly one parameter, the high-energy cutoff vc. This gives us the way to solve the whole problem, examining awider energy range of the ARPES data. B. Quadratic dispersion One more complication should be addressed here: in the wider energy range a deviation of the bare dispersion from aline should be taken into account. Along the nodal directionthe TB band in the occupied part can be well approximated by a simple parabola «skd= v0s1−k2/kF2d,9for which we still need one energy scale parameter: the bottom of the bare band v0or the bare Fermi velocity vF=−2v0/kF. Using this dis- persion in Eq. s1d, one can finally modify Eqs. s2dands3dto S8svd=vF 2kFfkm2svd−kF2g+v, s4d S9svd=−vF kFWsvd˛km2svd−W2svd. s5d C. Fitting procedure In short, the fitting machinery is based on Eqs. s4d,s5d, andsA1d. One can define three steps here. In the two first steps, the real part of the self-energy, for given v0,vc, andn swhich characterizes the tails, see below d, is calculated in two ways sidSdisp8by Eq. s4d,siidSKK8by Eq. s5dwith sub- sequent KK transform sA1d. Then, in step siiid, the param- eters v0,vc, andnfsee Eq. sA11dgare varied until Sdisp8svd and SKK8svdcoincide. In practice, we fit the difference Sdisp8−SKK8to a small contribution of experimental resolu- tion. The details of the procedure are given in Secs. 3 and 4of the Appendix. III. RESULTS We have applied the described procedure to the experi- mental data measured along the nodal direction for the fol-lowing samples: underdoped Bi sPbd-2212 sT c=77K d, over- doped sTc=75K dBisPbd-2212, and optimally doped Bi sLad- 2201 sTc=32K d, marked in the following as UD77, OD75, and OP32, respectively. The data for UD77 and OD75 were collected at 130 K, and for OP32 at 40 K. We have exploreda number of excitation energies in the range of 17–55 eVbut, as we show below, only at 27 eV, at which only theantibonding band is visible, 12the described procedure can be directly applied to the bilayer Bi samples. The experimentaldetails can be found elsewhere. 14,15 Figure 2 illustrates an example of the ARPES spectrum, photocurrent as a function of energy and momentum, takenfor UD77 Bi sPbd-2212 at 130 K along the nodal direction. On top of it, we plot the result of the fitting procedure, thebare dispersion. Another result of the procedure is the self-energy func- tions. They are shown in Fig. 3 for UD77 and in Fig. 4 forOD75 and OP32. We remind that the real part of the self- energy is represented by two functions S disp8andSKK8, ob- tained, as it is described above, from the experimental dis- FIG. 1. sColor online. dBare band dispersion ssolid line dand renormalized dispersion spoints don top of the spectral weight of interacting electrons. Though intended to be general, this sketchrepresents the nodal direction of an underdoped Bi-2212.KORDYUK et al. PHYSICAL REVIEW B 71, 214513 s2005 d 214513-2persion by Eq. s4dand from MDC widths with subsequent KK transform, respectively. The irreducible difference Sdisp8 −SKK8and the resolution function R8svd, to which the differ- ence is fitted, are also shown. Consequently, the interaction parameters which we give below should be referred to the Sdisp8functions. The imaginary part of the self-energy is presented by Swidth9svdfunction defined by Eq. s5d. In order to check the correctness of the KK numerics, we also plot the SKK9svd function which is obtained by back KK-transform sA2dof SKK8svd. The complete coincidence of Sdisp8andSKK8−R8functions in the whole accessible energy range substantiates that theself-energy constructed using Eqs. s4dand s5dis self- consistent within the experimental accuracy currently avail-able withARPES.This self-consistency shows, in addition tothe applicability of the self-energy approach to supercon-ducting cuprates, that the measured spectra belong to a singleband and are free of influence of any unaccounted additionalfeatures such as other bands, superstructures, or k-dependent backgrounds. It has been shown recently 12that although the electronic dispersion along the nodal direction in the bilayerBi-2212 is not degenerated, i.e., has a finite splitting about0.05 eV for the bare dispersion, the photoemission from thebonding band is highly suppressed at exactly 27 eV excita-tion energy. At other energies we do not expect that the de-scribed fitting procedure will work if applied directly. Figure4sbddemonstrates this showing the “best” fitting result that can be achieved for h n=38 eV. The difference between DS8svdandR8svdis apparent. At these “inconvenient” en- ergies the contributions of each band should be separated first, that complicates the analysis but can be done in prin-ciple by measuring several spectra at different h nor polar- ization se.g., see Ref. 16 d. In Table I we give the values of the experimental and calculated parameters for three investigated samples, forwhich the self-energy functions are shown in Figs. 3, 4 sad, and 4 scd. The Fermi momentum k Fand the renormalizedFermi velocity vRare determined experimentally; the energy of the bottom of the bare band v0, the bare Fermi velocity vF, and the coupling strength lare the results of the fitting procedure. Other fitting parameters, which characterize the high- energy tails of S9svd, are not so well defined as v0andlfor the reasons we discuss in Secs. 3 and 4 of theAppendix, but we can state that uvcu<uv0u/2. In case of the OD sample, the parameters vc=0.40±0.05 eV, n=4±0.5 are better deter- mined because of a higher confidence limit vm=0.45 eV at which one can see that S9svdstarts to saturate sSec. IV B d. IV. DISCUSSION The presented examples purpose to illustrate the applica- bility of the self-energy approach to Bi-cuprates. We believethat the described procedure gives a powerful technique topurify the ARPES data from artificial features and to build astrong experimental basis for understanding of the nature ofelectronic interactions in cuprates, but still a big work on thedata analysis should be performed. Nevertheless, some con-clusions can be made even on this stage. A. Well-defined quasiparticles The linear behavior of S8svdover a wide energy range uvu,uvkuindicates, using the criterion lim v!0S9svd/v=0, FIG. 3. sColor online. dReal and imaginary parts of the self- energy extracted from the experiment with the described procedure.A complete coincidence between the corresponding parts of theself-energy calculated from the two different experimental func-tions, the MDC dispersion and MDC width, demonstrates the fullself-consistency of the ARPES data treated within the self-energyapproach. FIG. 2. sColor online. dThe bare band dispersion along the nodal direction of an underdoped Bi sPbd-2212 ssolid parabola don top of its spectral weight at 130 K measured by ARPES. MDC sor renor- malized ddispersion shown by solid white sreddline.BARE ELECTRON DISPERSION FROM EXPERIMENT: … PHYSICAL REVIEW B 71, 214513 s2005 d 214513-3the existence of well-defined quasiparticles in the pseudogap state: for the underdoped Bi sPbd-2212 at 130 K the coher- ence factor Z=0.54±0.03. The offset of S9svdnot only comes from finite resolution but also finite temperature and scattering on impurities,17which are mostly energyindependent15and do not contribute to the slope of S8svd and, therefore, to the coherence factor. In Ref. 10 we have noticed that the scattering rate at room temperature looks more linear for underdoped samples than for overdoped ones that is in favor of the marginal Fermi FIG. 4. sColor online. dReal and imaginary parts of the self-energy extracted from the experiment with the described procedure.KORDYUK et al. PHYSICAL REVIEW B 71, 214513 s2005 d 214513-4liquid model sMFL d.18It is important to stress that S8svd, determined with better accuracy, exhibits a linear behavior below and above the kink energy vkssee Fig. 2 dwhich is now difficult to reconcile with the MFL model: as far as aslope in S 8svd, according to Eq. sA4d, is mainly determined by the coefficient at sv−vxd2term in the expansion of S9svd around vx, the straight sections on S8svdimply the regions where S9svdis precisely parabolic sexhibits constant curva- ture over some finite-energy regions d.B. High-energy cutoff It is interesting to note that even for the UD77 sample, for which the saturation of S9svdhas not been observed, it is not possible to reconcile the high-energy behavior of S9svdwith the saturation extreme sA9dforsA10dwithn=2g.This means that uS9svdureaches the maximum and starts to decrease at about vc, and, consequently, S8svdchanges the sign at ap- proximately the same frequency ssee Fig. 7 d. For OD75 and OP32 samples this conclusion is even more strict due tosmaller bandwidth. Fig. 5 shows the results for Bi sPbd-2212 OD75: sadS 8svdandS9svd;sbdkmsvdand«skdon top of the experimentally measured quasiparticle spectral weight. The fact that vcis not equal but roughly two times less than uv0uis consistent with presence of an essential electron- electron scattering channel, the doping independent Augerlike decay, 15which originates from the electron-electron Coulomb interaction and which mainly determines the life-time of quasiparticles at high frequencies. C. Doping dependence of the renormalization Another point arises as a consequence of the tight corre- lation between S8andS9. Recently we have shown15that two different channels can be distinguished in the scatteringrate: the doping independent Auger-like decay, mentionedabove, and the doping-dependent channel, which can benaturally associated with spin excitations. While such a de-composition of the scattering rate into two channels seems tobe becoming commonly accepted, 19there is still a contro- versy about the origin of the doping-dependent one. Thepresent analysis shows that regardless of the nature of thischannel, its doping and temperature dependence should ap-pear in the doping and temperature dependence of S 8and, consequently, of the renormalized dispersion, although it isclear that the variations in the latter should be marginal. It is really so, and, in Fig. 6, we plot together the real parts of the self-energy for UD77 and OD75 samples at130 K. Just from visual comparison of these data one canconclude that sidthe renormalization for UD77 is consider- ably higher than for OD75, siidthe energy of the maximum ofS 8svdfor the overdoped sample is lower than for the underdoped sample, it is about two times closer to the 70 meV “kink” energy, siiidthe kink feature is well defined in the underdoped case and becomes weaker with overdop-ing. Following this tendency one can expect that with over- doping the 70 meV kink vanishes while the renormalization FIG. 5. sColor online. dThe results of the fitting procedure for BisPbd-2212 OD75: sadreal sodd curve dand imaginary seven curves dparts of the self-energy; sbdthe experimental ssolid line d and bare sdashed line ddispersions on top of the experimentally measured quasiparticle spectral weight.TABLE I. Experimental and calculated parameters of the quasiparticle spectral function along the nodal direction in the normal state for three investigated samples: the Fermi momentum kFand the renormalized Fermi velocity vRare determined experimentally; the energy of the bottom of the bare band v0, the bare Fermi velocity vF, and the coupling strength lare the results of the described fitting procedure. Sample kFsÅ−1d vRseVÅ d v0seVd vFseVÅ d l BisPbd-2212 UD77 0.471 2.04±0.05 −0.90±0.04 3.82±0.17 0.87±0.12 BisPbd-2212 OD75 0.445 2.46±0.07 −0.86±0.03 3.87±0.14 0.57±0.10 BisLad-2201 OP32 0.47 2.04±0.10 −0.79±0.05 3.36±0.22 0.65±0.16BARE ELECTRON DISPERSION FROM EXPERIMENT: … PHYSICAL REVIEW B 71, 214513 s2005 d 214513-5maximum moves to lower frequencies faking a persistence of the kink in the whole doping range. Therefore, it is clearthat in order to clarify the origin of the kink feature a quan-titative measure of it is required. D. Phenomenology of the kink Keeping the visual definition of the kink as a sharp bend of the renormalized dispersion, we formalize it as a peak inthe second derivative of S 8svdand fitted it to a simple em- pirical function Slow8svd=−lv−Dl psv−vkd 3Sarctanvk d+arctanv−vk dD, s6d which gives a squared Lorentzian in a second derivative Ksvd=−d2Slow8svd dv2=2 pd3Dl fd2+sv−vkd2g2. s7d Fitting S8svdof the underdoped sample in uvu,170 meV energy range to this formula we have obtained an energy of the kink vk<−63 meV, a kink width fhalf width at quarter maximum of Ksvdgd<30 meV, and a strength of the kink eKdv=Dl<0.65. For the overdoped sample vk<−56 meV, Dl<0.45. We believe that a systematic study of this or similar quantitaties as a function of dopingand temperature will help to find the origin of the main elec-tronic interaction in superconducting cuprates. V. CONCLUSIONS We have demonstrated the full self-consistency of the data obtained using angle resolved photoemission and treatedwithin the self-energy approach.The extracted bare band dis-persion is in good agreement with the band structure calcu-lations and allows one to quantify the self-energy of the elec-tronic excitations in the real energy scale. The accuratelydetermined real and imaginary parts of the self-energy provethe existence of well defined quasiparticles along the nodaldirection even in the pseudogap state of Bi-2212. The demonstrated self-consistency of the procedure opens a way to validate the photoemission spectra: the KK sievecan be used to verify the spectra for the absence of the bandsplitting or artificial features. The preliminary analysis of thespectra certified in such a way shows that the overall renor-malization as well as kink in the nodal direction of Bi-basedcuprates is highly doping dependent, decreasing with over-doping. In the light of the present dilemma about the originof the main scattering boson in the cuprates, a systematicquantitative analysis of the KK verified spectra measured atdifferent temperature and doping level is indispensable. ACKNOWLEDGMENTS The project is part of the Forschergruppe FOR538 and is supported by the DFG under Grants Nos. KN393/4 and436UKR17/10/04 and by the Swiss National Science Foun-dation and its NCCR Network “Materials with Novel Elec-tronic Properties.” We are grateful to S. Ono and YoichiAndo for supplying the Bi-2201 sample andA. Chubukov, I.Eremin, D. Manske, and V. Zabolotnyy for discussions. APPENDIX 1. Kramers-Kronig transformation The quasiparticle self-energy S=S8+iS9in Eq. s1dis an analytical function the real and imaginary parts of which arerelated by the Kramers-Kronig sKKdtransformation 20 S8svd=1 pPVE −‘‘S9sxd x−vdx, sA1d S9svd=−1 pPVE −‘‘S9sxd x−vdx, sA2d where PV denotes the Cauchy principal value. It is instruc- tive to express some interaction parameters via both self-energy functions. The coupling strength FIG. 6. sColor online. dThe real parts of the self-energy for UD77 and OD75 samples at130 K: solid lines show the result of fitting thesereal parts to Eq. s6din a frequency range 0.17 eV , v,0 for UD77 and 0.12 eV ,v,0 for OD75; looking down arrows mark S8svd maxima; the dashed line denotes the 70 meV“kink” energy.KORDYUK et al. PHYSICAL REVIEW B 71, 214513 s2005 d 214513-6l=−SdS8 dvD v=0sA3d can be expressed in terms of S9differentiating the KK rela- tion sA1d: dS8svd dv=1 pPVE −‘‘S9sxd−S9svd sx−vd2dx. sA4d Here we use the fact that adding some constant to S9sxdin Eq. sA1ddoes not change the result. Then, for an even S9svd, Eq. sA4dleads to l=−2 pPVE 0‘S9svd−S9s0d v2dv. sA5d 2. Problem of tails In order to illustrate the problem, we rewrite Eq. sA5din an operator form l=−DS9and express the parameters of the bare dispersion and renormalization via the experimental val- ues of vR=sdkm/dvdv=0−1andDW: e.g., vF−1=vR−1−DW,o rl =1/Z−1, where Z=1− vRDW sA6d is the coherence factor s0,Z,1d. In case the MDC width Wsvddecays to zero or saturates on the scale covered by experiment, as it is expected for the scattering by phonons,21 theDWvalue can be easily defined, and all the mentioned parameters can be derived from experimental value of vRand Wsvdfunction. In cuprates, however, Wsvd, along the nodal direction, does not decrease or even saturate in the whole experimentally accessible energy region sup to vm=0.5 eV d. Equation sA5dcan give a certain feeling how the high- energy tails of the scattering rate S9svdforuvu.vminflu- ences l. For example, for a simple caseS9svd=−Hav2+Cforuvu,vc, 0 foruvu.vc,J sA7d where vc.0 is an energy cutoff and C;−S9s0d.0i sa n offset, Eq. sA5dgives l=2 pSavc−C vcD<2 pavc sA8d forC!vc. Using another ultimate model for S9tails, S9svd=−Hav2+Cforuvu,vc, avc2+Cforuvu.vc,J sA9d which approximates the saturation of scattering rate at high frequencies, one obtains l=4avc/p, twice of Eq. sA8d.I n the following sections we show how we solve this problem. 3. Calculation of SKK8 In order to perform a KK transform, high-energy tails should be attached to S9svdderived from Eq. s5d. Equations sA7dandsA9drepresent two extremes which can be enclosed in a simple analytical expression Smod9svd=−av2+C 1+Uv vcUn, sA10d as the ultimate cases with n!‘andn=2, respectively. For givennandvc, we construct S9svdfunction in a wide fre- quency range sup to uv0uor higher dassuming the particle- hole symmetry S9svd=HSwidth9suvudforuvu,vm, Smod9svdforuvu.vm,J sA11d where vmis a “confidence limit,” a maximal experimental binding energy to which both the Wsvdandkmsvdfunctions can be confidently determined, Swidth9svdis calculated from Eqs. s5dfor given v0, and Smod9svdis fitted to Swidth9svdin the confidence range in order to define aandC.Then, SKK8svdis obtained from S9svdby KK transform sA1d. FIG. 7. sColor online. dReal sthin blue dand imaginary sthick red dparts of the self-energy related by Kramers-Kronig sKKdtransform: S8=KK S9, for three models of S9tails.BARE ELECTRON DISPERSION FROM EXPERIMENT: … PHYSICAL REVIEW B 71, 214513 s2005 d 214513-7FIG. 8. sColor online. dIllustration of the fitting procedure: real parts of the self-energy Sdisp8svdsfilled squares dobtained by s4dand SKK8svdsopen circles dby Eqs. s5dandsA1d; the difference DS8svd=SKK8svd−Sdisp8svdssmall crosses dis fitted to R8svdscorresponding solid lined, the contribution of overall resolution determined by Eqs. sA12dandsA13d. In the first three panels v0=−0.9 but different n=3 ,4 ,a n d 6i nE q . sA10dare compensated by different vc=0.34, 0.45, and 0.52 eV, respectively. The last two panels, the “best” fitting results for slightly different v0’s.KORDYUK et al. PHYSICAL REVIEW B 71, 214513 s2005 d 214513-8Figure 7 shows the pairs of S9svdandS8svdfunctions obtained in such a way for the same v0but for three different models: Eqs. sA7d,sA9d, and sA10dwithn=4sdashed, dot- ted, and solid lines, respectively d. Since KK C=0, in order to simplify numerical calculation, the offset of S9svdcurves is set to S9sv0d=0. The experimental data are taken for UD77 sample. 4. Resolution function In step siiid, as we mentioned above, the difference DS8svd=SKK8svd−Sdisp8svdshould be fitted not to zero but to some small but detectable contribution of the overall resolu- tionR8svd. This difference can be easily understood by rea- soning that finite energy and angular resolutions mainly ef- fect the MDC’s width rather than its peak position and thatits contribution is frequency dependent. In order to illustratethis we can take into account the overall resolution, R,a s S width9svd=˛S9svd2+R2. Then one can consider its frequency-dependent contribution to the imaginary part of Ssvdas the difference between Swidth9svdand real S9svd: R9svd=˛R2+S9svd2−S9svd, sA12d and, due to additivity of the KK transform, SKK8=KK Swidth9 =KK S9+KKR9=Sdisp8+R8, construct v-dependent contribu- tion to SKK8as R8svd=KKR9svd. sA13d Although, in principle, the resolution effect R8svdcan be explicitely calculated from known energy and momentum resolutions, here we derive it empirically using Ras a pa- rameter. It is seen from Fig. 7 that different tails do not affectthe energy region u vu,0.25 eV, so, an irreducible difference in the slopes ssee Fig. 8 dD=dSKK8svd/dv−dSdisp8svd/dv .0 in the low-energy range uvu,0.07 eV swhile D=0 at higher energies 0.1 eV ,uvu,0.2 eV dis a measure of R8svd. In Fig. 8 we plot R8svdsetting the offset of S9svdto zero that gives the value of R=0.015 eV. For S9s0d,0 the pro-cedure gives larger Rvalues to accommodate the difference in slopes but this does not affect the fact that the irreducible difference between SKK8svdand Sdisp8svdis caused by the experimental resolution, and depends on frequency as is shown in Fig. 8: it vanishes at zero and high frequencieshaving a maximum around 0.1 eV. Thus, we can visualize the fitting procedure as fitting the difference DS 8svdtoR8svdfunction. The procedure has ap- peared to be robust with respect to the v0determination. Figure 8 illustrates this. First three panels show that for acorrect value of v0=−0.9 eV there is space for other param- eters to vary: different tails can be compensated by different vc’s, e.g., for n=3,4 ,a n d6i nE q . sA10d,vc=0.34, 0.45, and 0.52 eV, respectively. On the other hand, at slightly different v0’ssabout 10% lower and higher, see two right panels d, DS8svdcannot be fitted to R8svdin the whole frequency range. 5. Model assumptions Finally, we discuss two assumptions which have been made about the model self-energy: k-independence and particle-hole symmetry. It has been mentioned above that thesymmetric Lorentzian line shape of the MDC’s taken alongthe nodal direction was considered as an experimental evi-dence that the quasiparticle self-energy hardly depends onmomentum. 5Recently, however, it has been noticed that the necessary condition for the Lorentzian line shape is ]S9sk,vd/]k=0, but ]S8/]kcan be an v-independent constant.22This is especially interesting because the authors of Ref. 22 have shown that such a linear k-dependence of S8 can explain a nontrivial doping-dependent high-energy dis- persion observed for a variety of cuprates.23 As long as S8sk,vd=Sk8skd+Sv8svdand]S9/]k=0,k dependence of S8does not affect any result of the presented analysis except the bare dispersion. In this case, the real bare dispersion is just «realskd=«skd−Sk8skdorvFreal=vF −s]S8/]kdk=kF. Although our preliminary results, being in agreement with band structure calculations12and experimen- FIG. 9. sColor online. dPossible particle-hole asymmetry effect on S9svdsred/thick lines dand S8svdsblue/thin lines d: low-energy sdashed lines, vc=0.1 eV dand high-energy fsolid lines, by Eq. sA14dgcontributions shown on the top of the symmetric self-energy sshaded areas d.BARE ELECTRON DISPERSION FROM EXPERIMENT: … PHYSICAL REVIEW B 71, 214513 s2005 d 214513-9tal plasmon dispersion,13do not support strong kdependence ofS8, it will be interesting to examine its possible contribu- tion in a wide doping range and for different compounds. A possible particle-hole asymmetry is another complica- tion which can effect the results of the presented analysis. Ingeneral, one can expect an asymmetry of the self-energy dueto an asymmetric electron-boson interaction or as a simpleconsequence of asymmetric density of states. Without con-sidering the origin of the asymmetry, we examine its possibleinfluence based on the energy scale where it can appear. It iswell known that because of the possibility to performARPES at finite temperature one can get the informationabout quasiparticle spectral weight not only below thechemical potential but also from some region above. 24For T=300 K the MDC width can be measured up to 50 meV aboveEF, and, within the experimental uncertainty, it has appeared to be completely symmetric se.g., see Ref. 15 d.This means that if there is some asymmetry in the scattering rateat low-energy scale s,0.1 eV, a characteristic scale which can originate from an electron-boson interaction or from thevan Hove singularity in the occupied density of states of thehole-doped cuprates d, its magnitude is too small to be seen in theu vu,50 meV energy range and, consequently, hardly ef- fects the quasiparticle renormalization in the occupied regionsv,0d. Figure 9 illustrates this: the dashed curves, on top of the symmetric self-energy shown by shaded areas, represent a low-energy asymmetric contribution which is too big not tobe noticed in S 9svdsforuvu,50 meV dbut too small to in- fluence S8sv,0d. The solid curves in Fig. 9 present the case of high-energy asymmetry that can steam from the asymmetry of the bareband. 9We simulate it by an asymmetry part in the scattering rate Sa9svd=HSmod9sv,vc2d−Smod9sv,vcd,v.0, 0, v,0,J sA14d where Smod9is determined by Eq. sA10dwith vc=0.45 eV, vc2=0.66 eV, n=4,C=0. It is seen that although the influ- ence of Sa8svdon renormalization at −0.5 eV ,v,0e Vi s rather small scan be approximated at this stage by a linear contribution with a slope of about 20% of ldit can be, in principle, detected by more precise analysis, in which theinfluence of the energy and angular resolutions is taken intoaccount explicitly. 1A. Damascelli, Z.-X. Shen, and Z. Hussain, Rev. Mod. Phys. 75, 473s2003 d. 2J. C. Campuzano, M. R. Norman, and M. Randeria, in Physics of Conventional and Unconventional Superconductors , edited by K. H. Bennemann and J. B. Ketterson sSpringer-Verlag, Berlin, 2004 d, Vol. 2. 3T. Valla, A. V. Fedorov, P. D. Johnson, B. O. Wells, S. L. Hulbert, Q. Li, G. D. Gu, and N. Koshizuka, Science 285,2 1 1 0 s1999 d. 4P. V. Bogdanov, A. Lanzara, S. A. Kellar, X. J. Zhou, E. D. Lu, W. J. Zheng, G. Gu, J.-I. Shimoyama, K. Kishio, H. Ikeda, R.Yoshizaki, Z. Hussain, and Z. X. Shen, Phys. Rev. Lett. 85, 2581 s2000 d. 5A. Kaminski, M. Randeria, J. C. Campuzano, M. R. Norman, H. Fretwell, J. Mesot, T. Sato, T. Takahashi, and K. Kadowaki,Phys. Rev. Lett. 86, 1070 s2001 d. 6S. V. Borisenko, A. A. Kordyuk, S. Legner, C. Dürr, M. Knupfer, M. S. Golden, J. Fink, K. Nenkov, D. Eckert, G. Yang, S. Abell,H. Berger, L. Forró, B. Liang, A. Maljuk, C. T. Lin, and B.Keimer, Phys. Rev. B 64, 094513 s2001 d. 7A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskii, Quan- tum Field Theoretical Methods in Statistical Physics sPergamon, Oxford, 1965 d. 8P. D. Johnson, T. Valla, A. V. Fedorov, Z. Yusof, B. O. Wells, Q. Li, A. R. Moodenbaugh, G. D. Gu, N. Koshizuka, C. Kendziora,Sha Jian, and D. G. Hinks, Phys. Rev. Lett. 87, 177007 s2001 d. 9A. A. Kordyuk, S. V. Borisenko, M. Knupfer, and J. Fink, Phys. Rev. B67, 064504 s2003 d. 10A. Koitzsch, S. V. Borisenko, A. A. Kordyuk, T. K. Kim, M. Knupfer, J. Fink, H. Berger, and R. Follath, Phys. Rev. B 69, 140507 sRds2004 d. 11O. K. Andersen, A. I. Liechtenstein, O. Jepsen, and F. Paulsen, J. Phys. Chem. Solids 56, 1573 s1995 d.12A. A. Kordyuk, S. V. Borisenko, A. N. Yaresko, S.-L. Drechsler, H. Rosner, T. K. Kim, A. Koitzsch, K. A. Nenkov, M. Knupfer,J. Fink, R. Follath, H. Berger, B. Keimer, S. Ono, and YoichiAndo, Phys. Rev. B 70, 214525 s2004 d. 13N. Nücker, U. Eckern, J. Fink, and P. Müller, Phys. Rev. B 44, 7155 s1991 d; V. G. Grigoryan, G. Paasch, and S.-L. Drechsler, ibid.60, 1340 s1999 d. 14T. K. Kim, A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, M. Knupfer, H. Berger, and J. Fink, Phys. Rev. Lett. 91, 167002 s2003 d. 15A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, J. Fink, M. Kn- upfer, B. Büchner, H. Berger, G. Margaritondo, C. T. Lin, B.Keimer, S. Ono, and Yoichi Ando, Phys. Rev. Lett. 92, 257006 s2004 d. 16S. V. Borisenko, A. A. Kordyuk, S. Legner, T. K. Kim, M. Kn- upfer, C. M. Schneider, J. Fink, M. S. Golden, M. Sing, R.Claessen, A. Yaresko, H. Berger, C. Grazioli, and S. Turchini,Phys. Rev. B 69, 224509 s2004 d. 17For comparison of the offset value to transport data see T. Yoshida, X. J. Zhou, H. Yagi, D. H. Lu, K. Tanaka, A. Fujimori,Z. Hussain, Z.-X. Shen, T. Kakeshita, H. Eisaki, S. Uchida,Kouji Segawa, A. N. Lavrov, and Yoichi Ando, Physica B 351, 250s2004 d. 18C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996 s1989 d. 19X. J. Zhou, Junren Shi, T. Yoshida, T. Cuk, W. 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PhysRevB.84.125133.pdf

PHYSICAL REVIEW B 84, 125133 (2011) 13C NMR study on the charge-disproportionated conducting state in the quasi-two-dimensional organic conductor α-(BEDT-TTF) 2I3 Michihiro Hirata, Kyohei Ishikawa, Kazuya Miyagawa, and Kazushi Kanoda Department of Applied Physics, University of Tokyo, Bunkyo City, Tokyo, 113-8656, Japan Masafumi Tamura Department of Physics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan (Received 30 June 2011; published 28 September 2011) The conducting state of the quasi-two-dimensional organic conductor α-(BEDT-TTF) 2I3at ambient pressure is investigated with13C NMR measurements, which separate the local electronic states at three nonequivalent molecular sites (A, B, and C). The spin susceptibility and electron correlation effect are revealed in a locallyresolved manner. While there is no remarkable site dependence around room temperature, the local spinsusceptibility gradually disproportionates among the nonequivalent sites with decreasing temperature. Thedisproportionation ratio yields 5:4:6 for A:B:C molecules at 140 K. Distinct site and temperature dependences arealso observed in the Korringa ratio K i∝(1/T1T)iK−2 i(i=A, B, and C), which is a measure of the strength and the type of electron correlations. The values of Kipoint to sizable antiferromagnetic spin correlation. We argue the present results in terms of the theoretical prediction of the peculiar site-specific reciprocal-space ( k-space) anisotropy on the tilted Dirac cone, and discuss the k-dependent profiles of the spin susceptibility and electron correlation on the cone. DOI: 10.1103/PhysRevB.84.125133 PACS number(s): 76 .60.−k, 71.20.−b, 71.30.+h I. INTRODUCTION The organic charge-transfer salt α-(BEDT-TTF) 2I3(abbre- viated as α-I3hereafter) is a quasi-two-dimensional (Q2D) electron system with a 1 /4-filled hole band, which exhibits strong electron correlation effects [BEDT-TTF (ET hereafter) is the abbreviation of bisethylenedithio-tetrathiafulvalene;see Fig. 1(b)].α-I 3is comprised of alternately stacked 2D conducting layers of (ET)+1/2molecules and nonmagnetic insulating layers of triiodide anions (I 3)−1[Fig. 1(a)].1The unit cell contains four ET molecules, three of which are crys-tallographically nonequivalent [i.e., A, B, and C molecules; see Fig. 1(c)]. At room temperature, the band-structure calcu- lation predicts semimetallic Fermi surfaces. 2With decreasing temperature, resistivity shows weak temperature dependencedown to T CO∼135 K,3at which the system undergoes a first-order phase transition from a paramagnetic conductor to anonmagnetic insulator accompanied by an inversion symmetrybreaking. 3–13Below TCO, electrons are localized on ET lattice sites, forming the horizontal strip type of charge ordering (CO),5,10–14which is believed to be stabilized by the long-range Coulomb interactions.15,16 Under hydrostatic pressures, the phase transition tends to be suppressed.10,17,18Above PC∼1.5 GPa, the charge-ordered state completely disappears and the temperature-independent profile of resistivity extends down to several Kelvins. In this pressurized phase, recent band-structure calculations19–21 revealed that an unusual electronic structure with the lin- ear dispersion is realized around the Fermi level εF(i.e., massless Dirac cone), which is partly confirmed by recent magnetotransport experiments.22,23In contrast to the graphene monolayer, known as the typical massless Dirac-cone system with isotropic dispersion,24the Dirac cone in α-I3is strongly tilted;20the conical slant varies 10 times around the apex in the 2Dkspace.21The tilt stems from the low lattice symmetry inα-I3possessing only inversion symmetry (space group P-1). Because of this anisotropy of the band dispersion, a large imbalance in the local electronic density of states around εF is predicted among independent A, B, and C molecules.21 Recent13C nuclear magnetic resonance (13C NMR) exper- iments under hydrostatic pressures observed a large differ- ence in the local spin susceptibilities at these molecularsites. 25,26 The Dirac-cone picture has been discussed so far only at high hydrostatic pressures ( P>P C)i nα-I3. However, several experiments have recently suggested that it mightbe also applicable to the low-pressure conducting phase.Resistivity, 17,27Raman,10 13CN M R ,25,26x-ray,18Hall,27,28 and thermopower3,28measurements indicate that the ambient- and high-pressure conducting phases have many properties incommon. Moreover, the band-structure calculation 18suggests that the conical apex (Dirac point) locates at ∼8 meV below εFeven at ambient pressure,18,19and can be pushed up to εF by introducing a site-dependent potential imbalance between A, B, and C nonequivalent molecules.18,29,30Aside from this, the ambient-pressure conducting phase exhibits severalanomalous properties such as non-Drude optical conductivityjust above T CO,31and a large charge-density disproportionation among A, B, and C molecular sites as observed in x-ray,5 Raman,10and13CN M R( R e f . 11) experiments. On the whole, these findings probably indicate that the conducting state inα-I 3is closely associated with the Dirac cone and/or CO fluctuations. However, the local electronic properties are notyet fully understood quantitatively, in particular, the electroncorrelation effects. The purpose of this paper is to clarify the electronic nature in the ambient-pressure conducting state of α-I 3in microscopic and site-specific manners. Utilizing13CN M R experiments, we explored the local spin susceptibility andelectron correlation effects at each nonequivalent molecule 125133-1 1098-0121/2011/84(12)/125133(12) ©2011 American Physical SocietyHIRATA, ISHIKAW A, MIY AGAW A, KANODA, AND TAMURA PHYSICAL REVIEW B 84, 125133 (2011) (b) ba(c) (a) BI3 CA I3 C BAB BB CAC13 BEDT-TTF (ET) moleculeS S SS SS SS bc a FIG. 1. (Color online) (a) Side view of the crystal structure of α-I3. (b) Molecular structure of BEDT-TTF (ET). Arrows indicate the position of carbon atoms enriched by13C isotopes with 99% concentration. (c) Top view of the schematic crystal structure in the conducting layer. Rectangular bars represent nonequivalent ETmolecules ( =A, B, and C) in the unit cell. (i=A, B, and C) since NMR is capable of probing the local electronic states at these independent molecules separately. Asthe hyperfine coupling of a nuclear spin with its surroundingelectrons is highly anisotropic at the 13C site, one should know the field-angular-dependent characteristics of NMRspectra to make a reliable assignment of the NMR line anddetermine hyperfine tensors, which correspondingly providea quantitative basis for deducing the local spin susceptibilityand the electron correlation effects. Thus, we measured theangular dependences of the NMR Knight shift K iwith rotating the external field in the planes parallel and perpendicularto the conducting layers [i.e., the abplane in Fig. 1(c)]a t each temperature measured. The 13C hyperfine-shift tensors and the local spin susceptibility were determined from theangular-dependent profile of K iat each independent molecule. The nuclear spin-lattice relaxation rate (1 /T1)iwas measured under the in-plane fields in order to evaluate the KorringaratioK i∝(1/T1T)iK−2 i, which represents the strength and type of the electron correlations. The results are discussedin connection with the effect of CO fluctuations and Diraccone. II. METHODS To evaluate the local electronic properties in the ambient- pressure paramagnetic conducting state of α-I3, we performed 13C NMR measurements at external field Hof 6.00 T applied in the abandbcplanes in the temperature range from room temperature down to 140 K ( >T CO). We grew the single crystal of α-I3of the size 2 .5×0.5×0.1m m3by the conventional electrochemical oxidization method. Thecentral double-bonded carbon atoms in ET were selectivelyenriched with 13C isotopes (nuclear spin I=1/2,γn/2π= 10.705 MHz /T) at 99% concentration [see Fig. 1(b)], which means that the observed13C NMR spectra come from the central carbon sites. The13C NMR spectra were ob- tained by the fast Fourier transformation of the spin echosignals. As the origin of the NMR shift, we referred tothe resonance frequency of the 13C NMR signal of TMS [tetramethylsilane, (CH 3)4Si]. The spin-lattice relaxation rate 1/T1was obtained by the standard saturation and recovery method.III. RESULTS A. Angular dependence of the13C NMR line shapes and line assignment The assignment of the observed13C lines in the NMR spectra to A, B, and C nonequivalent molecular sites is thestarting point of the present work. This can be accomplishedby considering the crystal structure, the symmetry of A,B, and C molecules in the crystal, and the nuclear dipolarsplitting in the analysis of the angular dependence of 13CN M R spectra. In ET-based organic conductors, all crystallographically nonequivalent molecules contribute to distinct13C lines in the13C NMR line shapes. In the conducting state of α-I3,a t least three peaks are thus expected to be associated with thenonequivalent A, B, and C molecules [Fig. 1(c)]. However, each molecule has two neighboring 13C nuclei [Fig. 1(b)], which are coupled by the nuclear dipole interaction. Thiscoupling makes a single line split into a doublet or a quartetdepending on whether the molecule has inversion symmetryor not, respectively. 32,33The Pake doublet structure should be observed when the neighboring nuclei are equivalent,which is the case in molecules B and C. The Pake doubletis characterized by the nuclear dipolar splitting width (kHz) d given by d=3 2γ2 n¯h r3(1−3s i n2ζicos2ηi), (1) where γnis the gyromagnetic ratio of13C nucleus, 2 π¯his Planck’s constant, r=1.360 ˚A is the distance between two 13C nuclei,34and (ζi,ηi) are the angles formed between the direction of Hand the13C=13C vector of the i-site molecule ( i=A, B, and C), as illustrated in Fig. 3(b).T h e quartet structure is expected when the neighboring13C sites are nonequivalent as in molecules A. The quartet is composedof four lines ( ν=1−4), 33,35the lineshift δνof which is dependent on the nuclear dipolar splitting width dand the NMR shift difference between the two neighboring13C sites /Delta1δ; i.e.,δν=δ±d/3H±/radicalbig (d/3H)2+(/Delta1δ)2/2, where δis the averaged shift of the quartet and equals to the averageof the NMR shift at the neighboring 13C sites in ET without the nuclear dipole coupling. Thus, in the present case, eight 13C lines composed of two Pake doublets and one quartet are expected in the13C NMR spectra. In order to assign the NMR lines properly, we measured the magnetic-field angular dependences of the NMR spectra.Figure 2presents the typical data at 260 K with the external fieldHapplied within the abplane ( H/bardblab) and bcplane (H/bardblbc). The line shapes are strongly dependent on the direction of H.I nF i g . 2(a),t h e 13C lines can be divided into two groups, exhibiting an antiphase angular dependence as thedirection of H,ψis changed within the abplane [ ψ=0 ◦is set asH/bardbla; see the inset of Fig. 2(a)]. In Fig. 2(b), on the other hand, all the13C lines exhibit an in-phase variation when His rotated in the bcplane [ θ=0◦is set as H/bardblc; see the inset of Fig. 2(b)]. These phase relations can be adequately understood from the crystal structure;34i.e., the direction of13C-2pz orbitals within the unit cell. It is well known that the 2 pz electrons of13C atoms produce a dipole-type hyperfine field at the13C nuclear position, and give the largest contribution to 125133-213C NMR STUDY ON THE CHARGE- ... PHYSICAL REVIEW B 84, 125133 (2011) FIG. 2. (Color online) (a), (b) Angular dependences of the13C NMR line shapes under (a) H/bardblaband (b) H/bardblbcat 260 K. The definitions of angles ψandθare given in the insets. (Note that the origins of ψandθare set as H/bardblbandH/bardblc, correspondingly.) (c), (d) Nuclear dipolar splitting width dat A, B, and C nonequivalent molecules extracted from Figs. 2(a) and2(b), respectively. Calculated angular dependences are s h o w nt o g e t h e rb a s e do nR e f . 34(curves). In all figures, symbols represents the data relevant to molecules A (open circles), B (triangles), and C (crosses), respectively. the13C NMR shift.33,35As schematically shown in Fig. 1(c), molecules B and C, and molecules A form, respectively, a stackalong the aaxis. These stacks build a fishbonelike molecular arrangement in the abplane, and are approximately connected with mirror operation [( a,b,c)→(a,−b,c)]. The 2 p zorbitals, pointing perpendicular to the molecular plane [along the z axis in Fig. 3(a)], are parallel within the same stack, while they are mutually perpendicular between different stacks. Thisgeometry gives rise to an antiphase relation in the angulardependence of the NMR shift when the external field is rotatedin the abplane. On the other hand, all the 2 p zorbitals are roughly confined to the abplane, which is expected to cause an in-phase dependence of the NMR lines when the field isrotated within the bcplane. Referring to the considerations given above, we first assigned the 13C lines in Fig. 2(a). The four lines denoted by open circles are attributable to the quartet (moleculesA), and the remaining four lines to two pairs of the Pakedoublets (molecules B and C), as shown in Fig. 2(a).I ti seasy to distinguish molecules B and C when one focuses on the dihedral angles between molecules A and B ( ∼138 ◦), and A and C ( ∼131◦).34Since the former is larger than the latter, molecules B should exhibit larger phase difference frommolecules A than molecules C will in the angular dependence iiζ H ηxz y(a) y x z(b) S SS SS SSS FIG. 3. (Color online) (a) Molecular principal axes ( x,y,z)o f ET molecule. (b) Spherical polar coordinate ( ζi,ηi)o ft h ee x t e r n a l field vector Hati-site ET molecule ( i=A, B, and C) in the principal axes (x,y,z)( R e f . 34). 125133-3HIRATA, ISHIKAW A, MIY AGAW A, KANODA, AND TAMURA PHYSICAL REVIEW B 84, 125133 (2011) of the line shapes. We thus assigned the13C lines denoted by triangles and crosses to molecules B and C, respectively, asshown in Fig. 2(a). The results are consistent with the previous work. 11In Fig. 2(c), the observed angular dependences of the nuclear dipolar splitting width dfor A, B, and C molecules are presented with the corresponding symbols. The solid, dotted,and dashed lines are calculated angular dependences of dfor A, B, and C molecules, respectively, with r=1.360 ˚A and the molecular orientations obtained from Ref. 34. The data agree well with the calculations. Next, we focus on Fig. 2(b). By comparing the NMR spectra atθ=ψ=90 ◦in Figs. 2(a) and 2(b) (i.e.,H/bardblb), the13C lines under H/bardblbcwere assigned as shown in Fig. 2(b) [with the same symbols as in Fig. 2(a)]. Figure 2(d) presents the angular dependences of the dipolar splitting width dunderH/bardbl bc. Calculated angular dependences are presented together, which properly reproduce the experimental results. All of theseanalyses, thus, assure our assignments comprehensively. B. Angular dependence of the13CN M Rs h i f ta n d determination of the hyperfine-shift tensors The13C NMR shift is proportional to the hyperfine field at the13C site projected onto the direction of the external field H.T h e13C NMR shift at molecule i(=A, B, and C), δi,i s defined as the averaged shift of the lines at molecule i.I nF i g . 4, we present the representative angular dependences of the13C NMR shift δiat 260 K under H/bardblab[Fig. 4(a)] andH/bardblbc [Fig. 4(b)].δiis expressed as δi=[δxx i(Hx)2+δyy i(Hy)2+ δzz i(Hz)2]/|H|2, where x,y, andzare the ET principal axes [Fig. 3(a)],H=(Hx,Hy,Hz) is the external field in the ET principal axis [Fig. 3(b)], andδxx i,δyy i, andδzz iare the principal components of the13C-hyperfine-shift tensor. [Note that the shift tensor for the molecule A with two nonequivalent13C sites is defined as their average, although it can be determinedsite specifically. The reason why we do so is that the chargeand spin densities discussed below in terms of the shift ismolecular specific, and we need the shift value representativeof the molecular site. Moreover, this definition is beneficialfor the evaluation of electron correlation effects described inSec. III D ).] We fit the expression to the data in Figs. 4(a) and4(b) using the crystal-structure data given by Sawa et al. 34 The parameters optimized are listed in Table I. The fits in Fig. 4properly capture the sinusoidal angular dependences ofδiunder both field geometries. As seen in Table I,t h es h i f t tensors have different principal values among A, B, and C sites.This evidences differentiations in local spin susceptibility andcharge density among these nonequivalent sites in the unit cell. C. Temperature dependence of the13C NMR shift and the local spin densities Next, we proceed to separate the local charge density and spin density contributions in the observed shift data, andinvestigate the disproportionation of the local susceptibilitywithin the unit cell and its anomalous temperature dependence.Figures 5(a) and 5(b) show the temperature dependences of the 13C NMR spectra at ψ=110◦and 50◦(under H/bardblab), respectively. Large temperature dependences are seen in the FIG. 4. (Color online) Angular dependence of the13CN M R central shift at molecule i(=A, B, and C), δi, under (a) H/bardblab and (b) H/bardblbcat 260 K. Open circles, triangles, and crosses stand for the nonequivalent molecules A, B, and C, respectively. The solid, dotted, and dashed lines are fitted curves δi=[δxx i(Hx)2+ δyy i(Hy)2+δzz i(Hz)2]/|H|2(see text) with the optimized parameters ofδxx i,δyy i,a n dδzz ilisted in Table I. NMR shift δiwhen |δi|is large. On the other hand, δiis almost temperature independent when |δi|becomes small. The shift δiis the sum of the Knight shift and the chemical shift ( δi=Ki+σi), where the Knight shift Kiis related to the local spin susceptibility χithrough Ki=aiχi, and the chemical shift σioriginates from the orbital motion of electrons within ET molecules.36aiis the hyperfine-coupling constant between the13C nuclear and itinerant carrier spins. It is well known that the13C chemical shift in ET depends on the molecular valence, namely, the amount of hole on ET.37 TABLE I. The μ-axis component of the13C-hyperfine-shift tensor δμμ i(μ=x,y,z;s e eF i g . 3) at molecule i(=A, B, and C) deduced from the angular dependences of the NMR shift δi’s at 260 K shown in Figs. 4(a) and4(b). i (δi)xx(ppm) ( δi)yy(ppm) ( δi)zz(ppm) A 48 77 869 B 51 47 780C 59 130 929 125133-413C NMR STUDY ON THE CHARGE- ... PHYSICAL REVIEW B 84, 125133 (2011) FIG. 5. (Color online) (a), (b) Temperature dependences of the13C NMR spectra under Happlied within the abplane ( H/bardblab)a t (a)ψ=110◦and (b) ψ=50◦. [The angle ψis defined in the inset of Fig. 2(a).] (c), (d) Corresponding13C NMR Knight shifts at A, B, and C nonequivalent molecules. Because large charge-density disproportionations have been observed in α-I3among nonequivalent A, B, and C molecules above TCO,5,10,11,25,26,38σishould be determined for each molecule in this compound. Using the principal values of thechemical-shift tensors σ xx i,σyy i, andσzz i(in ppm) determined at 60 K,11and linearly interpolating the molecular valence ρi, determined by x-ray diffraction5to estimate ρiat 60 K, we get the following relations between the chemical-shift tensorsand valence ρ i:σxx i=130.8ρi+46.2,σyy i=30.4ρi+159.0, andσzz i=− 12.1ρi+61.9. In our case, the temperature dependence of σi(in ppm) is negligibly small, and the value ofσiis determined as σA∼165,σB∼120, and σC∼93 at ψ=110◦, andσA∼69,σB∼165, and σC∼166 at ψ=50◦, respectively. In general, these chemical-shift tensors allowus to calculate the chemical shift σ iunder arbitrary field orientation. Thus, to determine the angular dependence ofthe Knight shift in the following, we calculated the angulardependences of σ isite selectively and subtracted them from those of the shift δi. Figures 5(c) and5(d) depict the temperature dependences of the NMR Knight shift Ki(=δi−σi) under ψ=110◦and50◦(H/bardblab), respectively, with the use of the chemical shifts σidetermined above. The Knight shift Kiexhibits strong anisotropy within the abplane; for instance, KAis almost zero in Fig. 5(c), while it is large in Fig. 5(d). Since the total spin susceptibility χspin(∝2χA+χB+χC) is isotropic in α-I3,9the anisotropic behavior of Kishould stem from the anisotropy of the hyperfine interaction at the13C position, which is ex- pressed as Ki=[axx i(Hx)2+ayy i(Hy)2+azz i(Hz)2]χi/|H|2 withaxx i,ayy i, andazz ithe principal components of the13C hyperfine-coupling tensor [see Fig. 3(a)]. [This means that the small value of KAin Fig. 5(c) should be attributed to the vanishingly small hyperfine-coupling constant at ψ=110◦ geometry.] In ET-based materials, it is known that the profile of the highest occupied molecular orbital (HOMO) of ETdetermines the anisotropy of the coupling tensor aμμ i(μ=x, y, andz).33The temperature dependence of aμμ iis negligible since the spatial distribution of HOMO is most likely tobe temperature independent. Moreover, the spatial profileof HOMO is reasonably assumed to be the same at allmolecules in the unit cell because the observed differencesin the molecular structures are very small. 34Therefore, in 125133-5HIRATA, ISHIKAW A, MIY AGAW A, KANODA, AND TAMURA PHYSICAL REVIEW B 84, 125133 (2011) KiaveKnight Shift 0 Angle (degree)Kiamp FIG. 6. (Color online) Definitions of the average and amplitude of the angular dependence of the13C NMR Knight shift Kave iand Kamp i. the first approximation, we can assume that aμμ iis site and temperature independent. We also note that molecules A, B,and C are arranged in a nearly symmetrical manner withintheabplane, as we mentioned above [Fig. 1(c)]. Hence, ifHis rotated in the abplane, the average K ave iand the amplitude Kamp i of the angular dependences of the Knight shift (see Fig. 6) are both expected to reflect the magnitude of χi:Kave i=aaveχiandKamp i=aampχiwithaaveandaampthe averaged hyperfine-coupling constants. Figure 7shows the temperature dependences of Kave i andKamp i. For determining them, we calculated the angular 0100200300400500 120 150 180 210 240 270 300 Temperature (K)A B C (a) 13C NMR, 6.00 T in ab plane 0100200300400500 120 150 180 210 240 270 300 Temperature (K)A B C (b) 13C NMR, 6.00 T in ab plane FIG. 7. (Color online) (a) Average ( Kave i) and (b) amplitude (Kamp i) of the sinusoidal angular dependence of the13C NMR Knight shift at i-site ET molecule ( i=A, B, and C) under the external field Happlied parallel to the abplane ( H/bardblab).dependence of σi, using the empirical relations deduced above, and subtracted them from the angular dependencesof the shift δ iat all temperatures measured. The external fieldHwas rotated in the abplane because Kamp ibecomes largest in this field geometry and the line shapes are readilydiscerned owing to the antiphase angular dependence ofthe lines as mentioned above (in Sec. III A ). Around room temperature, there are little site dependences in the observedK ave iandKamp i. With temperature decreased, however, they exhibit site-specific temperature dependences, and then beginto decrease monotonically below T∼180 K at all sites down toT CO∼135 K. To deduce the local spin susceptibility χifrom Kave i (Kamp i), it is necessary to evaluate the hyperfine-coupling constant aave(aamp) defined above. This is achieved by comparing Kave i(Kamp i) with bulk spin susceptibility χspin at room temperature where the site dependence of Kave i (Kamp i) becomes negligible; that is, Kave i∝χspin(Kamp i∝ χspin). At room temperature, Kave i≈360 and Kamp i≈440 (in ppm) with little site dependences (Fig. 7) and χspin= 6.8×10−4emu/mol f.u. measured by Rothaemel et al.8In terms of them, the hyperfine-coupling constants are evaluatedasa ave≈5.9k O e /μBandaave≈7.2k O e /μB. To check whether these coupling constants are applicable under arbitraltemperatures, we compare the temperature dependences of/summationtext iKave i/4 (sharps) and/summationtext iKamp i/4 (squares) to that of the spin susceptibility χspin(crosses), as depicted in Fig. 8, where the summation is taken over all molecules in the unit cell (i.e.,molecules A, A, B, and C; see Fig. 1). All of these quantities are nearly scaled to each other (i.e.,/summationtext iKave i/4∝χspinand/summationtext iKamp i/4∝χspin), which assure our assumptions that aave andaampare site and temperature insensitive. Thus, it is allowed to determine the local susceptibility χiwith these coupling constants through Kave i=aaveχiandKamp i=aampχi at all temperatures measured. In Fig. 9(a), we show the temperature dependences of the hereby deduced local spin susceptibility χifrom Figs. 7(a) and 7(b). [Note that χiin Fig. 9(a) is an average among Kave i/aaveandKamp i/aampbecause these values are in good 0.60.70.80.91 120 150 180 210 240 270 300 Temperature (K)∑ iamp i4/K∑ iave i4/ Kχspin χA) ) FIG. 8. (Color online) Temperature dependences of the normal- ized susceptibility χspin(crosses), from Ref. 8,/summationtext iKave i/4( s h a r p s ) ,/summationtext iKamp i/4 (squares), and χA(circles), respectively. Summations are taken over all molecules in the unit cell. 125133-613C NMR STUDY ON THE CHARGE- ... PHYSICAL REVIEW B 84, 125133 (2011) 0.70.80.911.11.21.3 120 150 180 210 240 270 300 Temperature (K)A BC (b) 13C NMR, 6.00 T in ab plane0.01.02.03.04.0 120 150 180 210 240 270 300 Temperature (K)A B C (a) 13C NMR, 6.00 T in ab plane FIG. 9. (Color online) (a) Temperature dependences of the i-site local electron spin susceptibility χi(i=A, B, and C nonequivalent molecules). (b) Relative local spin density /angbracketleftsi/angbracketright=χi/[(2χA+χB+ χC)/4]. agreements with each other over the entire temperatures measured.] Susceptibility shows little site dependence aroundroom temperature. With decreasing temperature, however,large imbalance develops in χ iamong A, B, and C nonequiv- alent molecules below ∼270 K, which increases down to TCO.B e l o w ∼180 K, all χi’s decrease monotonically with decreasing temperature. The tendency χC>χ A>χ Bis con- sistent with previous works.11,25,26Just above TCO, its ratio reaches 5:4:6 for A:B:C molecules. This large imbalance cannot be expected by a simple single-band picture. Notice thatthe temperature dependence of χ Ahas the same profile as that ofχspin, namely, χA∝χspin, as seen in Fig. 8(circles). To highlight the temperature dependence of the spin-density disproportionation prominently, we depict the relative localspin density in Fig. 9(b), defined as /angbracketlefts i/angbracketright=χi/[(2χA+χB+ χC)/4], which reflects the relative contribution of the i-site HOMO to the conduction band around εF. Imbalance develops in the relative local spin densities at the B and C sites withdecreasing temperature. On the other hand, the A site showslittle temperature dependence ( /angbracketlefts A/angbracketright≈1), which is the direct consequence of χspin∝χA. At first glance, it seems to beplausible that the observed spin-density disproportionation can be qualitatively understood with the semimetallic band pictureas a redistribution process of B- and C-site HOMO contri-butions in the two bands. Indeed, band-structure calculationspredict two bands locating in the vicinity of the Fermi levelε F.18However, it is difficult to explain the strong decrease in χspin(Fig. 8) by the simple semimetallic framework; e.g., in the case of 2D Fermi surfaces, χspin[∝D(εF)] is expected to show little temperature dependence since the density of statesD(ε F) is not sensitive to the value of εF. Moreover, the trend of the spin-density disproportionation is opposite from whatis expected from the x-ray diffraction measurement 5[B site: (hole) rich, and C site: (hole) poor], which is also anomalousas a conventional semimetal. As we shall see in Sec. IV A , these features should be rather regarded as consequences ofthe anisotropic Dirac cone realized around ε F, which was originally predicted under hydrostatic pressures in α-I3.20 D. Temperature dependence of the13C nuclear spin-lattice relaxation rate and the electron correlation effects So far, the local electronic states in α-I3have been revealed in terms of the static susceptibilities. The nuclear spin-latticerelaxation rate 1 /T 1probes the fluctuations of electron spins. In this section, we show the site-dependent spin dynamicsuncovered by the site-selective measurements of 1 /T 1T, which allows one to see the correlation effects in site-specific orband-specific manners, as discussed in Sec. IV B . In the B and C molecules with inversion center, the two neighboring carbons are equivalent and give the identicalrelaxation rate. However, the two nonequivalent carbons in themolecule A without inversion center should exhibit differentrelaxation rates due to different hyperfine-coupling constantsas reported earlier. 39Nevertheless, the so-called T2process, which works to average the site-specific relaxation rates amongthese two carbons, tends to alter the two values toward someintermediate values in-between them. Then, the distinction ofthe two observed rates is not so informative, but their averageis a meaningful value specific to the A molecule. Thus, wedetermined the relaxation rate representative of the moleculeA from the relaxation curves of the whole spectra (i.e., thequartet), which were nearly single exponential. In Fig. 10(a) , we present the temperature dependence of the 13C nuclear spin-lattice relaxation rate 1 /T1divided by temperature Tatψ=110◦under H/bardblab. With decreasing temperature, 1 /T1Tdecreases monotonically at all sites from room temperature down to TCO. The temperature dependence is, however, different from site to site. The decreasing rates arelarge at A and B sites, while the C site exhibits only moderatetemperature dependence. In a conventional single-band metal,1/T 1Tis not expected to show site-specific temperature dependences because all sites probe the same electronicproperties in the conduction band. The distinct behaviors atA, B, and C sites indicate that the electronic structure in thissystem can not be interpreted within a simple single-bandmodel. Since the values of 1 /T 1TK2measure the degree of electron correlations in the conducting state, 1 /T1TK2was evaluated at each site. However, the Knight shift at A site is too small(K A=− 60∼− 70 ppm) at this field geometry in Fig. 5(c) 125133-7HIRATA, ISHIKAW A, MIY AGAW A, KANODA, AND TAMURA PHYSICAL REVIEW B 84, 125133 (2011) (ψ=110◦) to obtain reliable values of 1 /T1TK2, compared to B and C sites. We thus measured the spin-lattice relaxationrate 1/T 1on A site at ψ=50◦, where the Knight shift at A site is large and, hence, the relative error in 1 /T1TK2 becomes small. The results are shown in Fig. 10(b) . Then, we evaluated the values of 1 /T1TK2for the data series at ψ=50◦for the A site, and at ψ=110◦for the B and C sites, respectively. In the present case of anisotropic hyperfinecouplings, and in the presence of electron correlations, 1/T1TK2is expressed as the modified Korringa relation : (1/T1T)iK−2 i=(4πkB/¯h)(γn/γe)2βi(ζi,ηi)Ki(i=A, B, and C).32,33Here, γeis the gyromagnetic ratio of an electron, kB is the Boltzmann constant [which yields (4 πkB/¯h)(γn/γe)2= 2.397×105sec−1K−1],βi(ζi,ηi)i st h e i-site correction factor for the anisotropy of the hyperfine-coupling tensor as definedin (i=A, B, and C) βi(ζi,ηi)=/parenleftbig axx i/azz i/parenrightbig2(sin2ηi+cos2ζicos2ηi)+/parenleftbig ayy i/azz i/parenrightbig2(sin2ηi+cos2ζicos2ηi)+sin2ζi 2/bracketleftbig/parenleftbig axx i/azz i/parenrightbig sin2ζicos2ηi+/parenleftbig ayy i/azz i/parenrightbig sin2ζisin2ηi+cos2ζi/bracketrightbig2, (2) andKiis thei-site Korringa ratio , which reflects the type and strength of the electron correlations. The direction of H,(ζi,ηi) [see Fig. 3(b)] is determined at i=A, B, and C molecules for the present field geometries, as shown in Table II, based on the molecular orientations in Ref. 34. The principal components of the13C hyperfine-coupling tensor, axx i,ayy i, andazz i,a r e determined at each molecule i(listed in Table II) from the total shift tensors, given in Table I, and the estimated chemical-shift FIG. 10. (Color online)13C NMR nuclear spin-lattice relaxation rate divided by temperature 1 /T1Tat A, B, and C nonequivalent molecules under (a) ψ=110◦and (b) ψ=50◦(H/bardblab).tensors at 260 K (see Sec. III C ). The values of βi(ζi,ηi)a r e also presented in Table II. Figure 11represents the temperature dependence of the Korringa ratio Ki[∝(1/T1T)iK−2 i].Ki=1 means that there is no electron correlations. Figure 11shows relatively large Kifor all sites ( Ki∼4−11), indicating the presence of antiferromagnetic correlations. At room temperature, Ki shows little site dependence. Upon cooling, however, it begins to exhibit distinct behaviors for A, B, and C moleculesbelow ∼270 K. The Korringa ratios exhibit only small temperature dependences at A and C sites ( K A≈7 and KC≈4−6), while KBincreases below 200 K and reaches a value of 11 at 140 K. In the lowest temperature region, themagnitude relation of K iis given by KB>KA>KC.W e note that the well-studied dimer-type organic metal κ-(BEDT- TTF) 2Cu[N(CN) 2]Br,33,35,40which resides on the verge of the Mott transition, shows K∼5−10. On the other hand, in θ-(BEDT-TTF) 2I3,41known as a good 2D metal with a 1 /4- filled band, exhibits K≈2−3, pointing to weak correlations. In our case, Kiis intermediate or as large as in κ-type salt. This indicates the importance of electron correlations in α-I3, which is discussed in more details in Sec. IV B . IV . DISCUSSION As we mentioned in the preceding sections (Secs. III C and III D ), it is difficult to explain the observed anomalous decreases in spin susceptibility χspin(Fig. 7), temperature- dependent spin-density disproportionations (Fig. 9), and the large temperature and site dependences of spin fluctuationsK i[∝(1/T1T)iK−2 i] (Fig. 11) in terms of simple semimetallic or single band pictures. Then, how can we understand ourNMR results comprehensively? Intriguingly, there are two hints in the previous works. First, as we noted in Sec. I, resistivity shows weak temperature dependence in this systemboth at ambient and high pressures. 17,27Second, the NMR local spin susceptibilities at intermediate26and high25pressures exhibit very close behaviors to our results at ambient pressure.These results suggest that the ambient- and high-pressure con-ducting states have qualitatively similar features. Furthermore,the observed spin-density disproportionation (Fig. 9) reflects the presence of site-dependent potentials. The trend of thedisproportionation is the same range as that expected in the 125133-813C NMR STUDY ON THE CHARGE- ... PHYSICAL REVIEW B 84, 125133 (2011) TABLE II. Principal components of the hyperfine-coupling tensor aμμ i(μ=x,y,z) at molecule i, spherical polar angles ( ζi,ηi)[ d e fi n e di n Fig. 3(b)], and the correction factors for the anisotropy of the hyperfine-coupling tensor βi(ζi,ηi)[ s e eE q .( 2) in Sec. III D ]a ti=A(ψ=50◦), B(ψ=110◦), and C ( ψ=110◦) molecular sites. iaxx i(kOe/μB) ayy i(kOe/μB) azz i(kOe/μB) ζi(degree) ηi(degree) βi(ζi,ηi) A −1.11 −1.61 13 .35 21 .45 7 .20 .106 B −1.26 −2.25 11 .91 42 .66 7 .71 .139 C −0.74 −0.70 14 .32 31 .85 8 .80 .281 local site-potentials in Ref. 18, which is predicted to stabilize the Dirac cone.18,20,29,30All of these considerations thus imply that the Dirac cone dominates the electronic propertieseven at ambient pressure. In fact, as shall be discussedbelow, the observed features are properly captured by thetheoretical consequences of the anisotropic conical dispersionrealized around ε F.21We will focus on the local spin-density disproportionations in Sec. IV A , and the site-dependent spin fluctuation effects in Sec. IV B . A. Disproportionation of the local spin densities and the site-specific k-space anisotropy in the conduction band First, we consider the temperature and site dependences of the local spin densities. Let us assume that the wave function inthe conduction band is given by the Bloch sum of the HOMO atthei-site ET molecule ( i=A, B, and C) φ i[=φ(r−Rl−δi)] as follows: /Psi1k(r)=/summationdisplay Rl/summationdisplay i=A,B,Ceik·RlCi,kφ(r−Rl−δi), (3) where Rlstands for the position of one α-I3unit cell, and δiis the vector connecting the isite to the A site in the unit cell ( δA=0). The experimentally obtained i- site local spin susceptibility χiis proportional to the ther- mal average of φicontributing to the conduction band around εF,/angbracketleft/summationtext k|Ci,k|2/angbracketrightε=εF±kBT, which is explicitly given as21χi=−/integraltext∞ −∞dεD i(ε)f/prime(ε)=−/integraltext∞ −∞dε/summationtext k|Ci,k|2δ(ε− 02468101214 120 150 180 210 240 270 300 Temperature (K)A B C no corrlations (Korringa ratio = 1)13C NMR, 6.00 T in ab plane FIG. 11. (Color online)13C NMR Korringa ratio Ki[∝ (1/T1T)iK−2 i]a ti=A, B, and C nonequivalent molecules. Dashed line (Ki=1) corresponds to the free electrons case. Ki>1 implies the antiferromagnetic correlations, while Ki<1 stands for the ferromagnetic ones.ξk)f/prime(ε), with f(ε) the Fermi-Dirac distribution function, ξk the energy-momentum dispersion of the conduction band, and Di(ε) the local electronic density of states at molecule i.P r i m e stands for the derivative with respect to ε. There are two significant consequences of the band- structure calculations relevant to the present results;21one is the large tilting effect of the Dirac cone and the other isthe resulting strong angular dependences of |C i,k|2’s about the cone. The ratio of the steepest and gentlest slants (i.e.,anisotropy of the Fermi velocities) is estimated at about 10, 21 and the latter leads to a flat band dispersion giving the vanHove singularity around 10 meV above the Dirac points, 21 as schematically depicted in Fig. 12. Noticeably, |CB,k|2is largest around the steepest dispersion, denoted as S in Fig. 11, and shows a node around the gentle dispersion, denoted asG, whereas |C C,k|2has opposite characteristics. |CA,k|2is predicted to show no remarkable angular dependence. Thismeans that the local spin susceptibilities at B and C sitespreferentially probe the thermal excitations around the S andG portions of the cone, respectively, while the susceptibilityat the A site sees the average over the cone. The overalltemperature dependence of the spin susceptibility, shown inFig. 8, is basically explained in this context as follows: the conical dispersion has an energy-linear density of states,which gives rise to a linearly temperature-dependent spinsusceptibility. 21,42The decreasing total susceptibility with lowering temperature, observed below 200 K in Fig. 8, can be thought of as signifying this. The leveling off of kD -kD ΓE Dirac pointS GS G εF Dirac pointkΓ kD-kDQ ~ 2kD XY M FIG. 12. (Color online) Schematic representation of the k-space anisotropy of the tilted Dirac-cone spectrum predicted by the band- structure calculation given in Ref. 21. S and G stand for the steepest and gentlest portions of the dispersion, and kD(−kD)r e p r e s e n t s the position of the Dirac point. (Inset) First Brillouin zone of α-I3. Intervalley scattering vector with the wave vector Q∼2kD is represented by the bold arrow. 125133-9HIRATA, ISHIKAW A, MIY AGAW A, KANODA, AND TAMURA PHYSICAL REVIEW B 84, 125133 (2011) the susceptibility at higher temperatures, on the other hand, implies the breakdown of the cone picture at high energies.Actually, the calculated total spin susceptibility based on theband structure tends to saturate at high temperatures. 21 The spin-density disproportionation of χB<χ C, enhanced at low temperatures [see Figs. 9(a) and 9(b)], reasonably corresponds to the small and large local density of states atthe S and G portions on the cone. The peak formation ofχ Caround 190 K is attributable to the van Hove singularity located around G, where |CC,k|2shows the maximum.21On the contrary, |CB,k|2is vanishingly small at G, as mentioned above, which explains why χBcontinues to increase monotonously up to room temperature. Meanwhile, the A site shows temperaturedependences in-between B’s and C’s both in the local spinsusceptibility χ Aand the relative local spin density /angbracketleftsA/angbracketright[see Figs. 9(a) and9(b)].χAscales to the bulk spin susceptibility χspinover the whole temperature range ( χA∝χspin; see Fig. 7), which is reflected in the temperature-independent profile of /angbracketleftsA/angbracketrightshown in Fig. 9(b). These results indicate that the A site probes the whole conical dispersion in an averaged manner,and supports the theoretical prediction that |C A,k|2shows no remarkable angular dependence on the cone.21 At high temperatures of hundreds Kelvin, the electronic mean-free path in molecular conductors can reach the order of the unit-cell size due to the strong electron-phonon scatterings. The high-temperature equalization of the localspin susceptibilities [see Fig. 9(a)] might be partially due to this scattering effect, which will thermally average theanisotropy of the wave functions in the kspace. Several organic conductors exhibit bad metal natures such as a loss inDrude weight in optical conductivity, or broadening in angle- resolved photoemission spectroscopy (ARPES) spectra at high temperatures. 43,44Moreover, the correlation-induced effect such as charge ordering is disturbed at high temperatures ingeneral. In the present system, the charge disproportionation,if it is enhanced by electron correlations, is expected to bedepressed at high temperatures. This effect tends to makethe Dirac point pushed down below ε F, and may lead to a crossover from the Dirac system to a semimetal. Actually, the spin susceptibility and Korringa ratio are both temperature andsite independent around room temperature. It is probable thatthe picture of the massless Dirac electrons is broken by thesethermal effects. To illustrate how the hole density at the molecule i, i.e.,ρ i, relates to the above-discussed i-site spin susceptibility χi,w e made a comparison between ρiandχi. Here, the value of ρi is estimated from the intramolecular bond lengths determined by the x-ray diffraction study by Kakiuchi et al.5and the charge-sensitive modes in the vibrational spectroscopy byWojciechowski et al. 10They found that ρB>ρ C, which is supported by Hartree-Fock calculations based on the extendedHubbard model. 16This relation is, however, opposite to the spin-density profile χB<χ C. Because our material is a 3/4-filled electron-band (or a 1 /4-filled hole-band) system (see Sec. I), the charge (hole) density might correlate to the spin density, which seems irreconcilable with the present resultsat first glance. However, the charge-spin correlation holdswhen both charges and spins are spatially well localized.For itinerant electron systems, on the other hand, theyshould not necessarily match since the amount of the hole is given by ρ i∝/summationtext k|Ci,k|2[1−f(ξk)]≡/angbracketleft/summationtext k|Ci,k|2/angbracketrightε/greaterorequalslantεF [with/summationtext iρi=2(i: all molecules in unit cell)], while the spin density is expressed as χi∝/angbracketleft/summationtext k|Ci,k|2/angbracketrightε=εF±kBTas mentioned above. Obviously, χiis determined by |Ci,k|2 only in the vicinity of εF, while ρireflects the whole summation of |Ci,k|2over the conduction band above εF. Hence, we acquire the following relations from the experi- mental results: /angbracketleft/summationtext k|CB,k|2/angbracketrightε=εF±kBT</angbracketleft/summationtext k|CC,k|2/angbracketrightε=εF±kBT and/angbracketleft/summationtext k|CB,k|2/angbracketrightε/greaterorequalslantεF>/angbracketleft/summationtext k|CC,k|2/angbracketrightε/greaterorequalslantεF, which suggest that/summationtext k|CB,k|2</summationtext k|CC,k|2holds near εF, whereas/summationtext k|CB,k|2>/summationtext k|CC,k|2is valid well above εF. These con- trasting relations at low and high energies are consistentwith the picture of the conical band dispersion, which is characterized by the distinct anisotropies in |C B,k|2and |CC,k|2;21the portion G (in Fig. 12), which is contributed largely from the site C, is cut off by the Brillouin zone boundaryat low energies, while the portion S relevant to the site Bextends to higher energies and over a wide range of the k space covering the zone center (see Fig. 12). B. Korringa ratios and spin fluctuation effects Next, we turn our attentions to the site-specific spin fluctuation effects. As noted in Sec. III D , the Korringa ratio Ki∝(1/T1T)iK−2 iis a measure of the strength and type of spin fluctuations in the conduction band.45As represented in Fig. 11,Kiis strongly site and temperature dependent in our system, in contrast to a simple metal where Kishould be independent on sites and temperatures. This anomalousbehavior is understood based on the theoretical consequencethat each molecular orbital φ iat the site icontributes to the conical dispersion with a differing angular dependence in the k space, which is consistent with the discussion in the precedingsection. First, it should be reminded that |C A,k|2has a little angular dependence in the reciprocal space.21Hence, KAis expected to probe the fluctuations over the entire Dirac cone on averageand to serve as a benchmark for the strength of correlationsin the system. As we mentioned above, K Aexhibits a relatively large value ( KA≈7) in the whole temperature range (Fig. 11), pointing the presence of strong or intermediate antiferromagnetic spin fluctuations in the system. The spin-lattice relaxation rate at the site A, (1 /T1)A,i s given by45 (1/T1)A=2/parenleftbiggγn γe/parenrightbigg2kBT ¯h2(a⊥ A)2/summationdisplay Qχ/prime/prime(Q,ω) ω, (4) where a⊥ Ais the transverse component of the hyperfine- coupling tensor at the site A, ωis the NMR resonance frequency, and χ/prime/prime(Q,ω) is the imaginary part of the dynamical spin susceptibility. The wave vector Qand the frequency ω are characteristic for the spin fluctuations associated with theelectron-hole pair excitations at ε F. For the free electrons, χ/prime/prime(Q,ω) is expressed as χ/prime/prime 0(Q,ω)=πγ2 e¯h3ω[Dtot(εF)]/2,45 withDtot(εF) the total density of states at εF. In the presence of spin fluctuations, on the other hand, χ/prime/prime(Q,ω) are enhanced from the constant value χ/prime/prime 0(Q,ω) at the wave vector Q connecting the degenerate states around εF. The present 125133-1013C NMR STUDY ON THE CHARGE- ... PHYSICAL REVIEW B 84, 125133 (2011) system has two Dirac cones (at valley kDand−kD)i n the first Brillouin zone, which are mutually connected withtime-reversal symmetry. 20(kDstands for the position of the Dirac point inside the first Brillouin zone.) In this situation,two kinds of interactions are conceivable: (i) the intravalleyscattering with Q∼0, and (ii) the intervalley scattering with Q∼2k D. The former scattering gives ferromagnetic fluctuations, while the latter leads to antiferromagnetic ones.The result of K A≈7 in much excess of unity clearly shows that the intervalley scattering is the dominant sourceof the enhanced spin fluctuations in our system. This en-hanced electron correlation is naturally understood to helpstabilize the correlation-induced CO-insulating state below135 K. 18,29 In contrast to |CA,k|2,|CB,k|2and|CC,k|2have so distinct angular and energy dependences on the cone that KBandKC are expected to measure the fluctuation effects in preferential areas of the kspace, namely, the S and G portions in Fig. 12, correspondingly. Note that, roughly speaking, Ki probes the intensity of spin fluctuations per thermally excited quasiparticles instead of the intensity itself because (1 /T1T)i is divided by K2 i. At low temperatures, KBis much enhanced compared with KC; at 140 K, the value of KBreaches 11, which is twice of that of KC. The thermally excited states probed at the B site are well located around the Dirac point in the 2D kspace because the dispersion of the S portion is steep (see Fig. 12). In the G portion, on the other hand, the thermal excitation should be extended in a wide range of kstates because of the flat dispersion with the van Hove singularity located around150 K above ε F. The different k-space profile of thermal excitations probed at B and C sites should give a correspondingdifference in the intervalley scattering seen at B and C sites.The intervalley scattering probed at the B site is well restrictedto the vicinity of Q∼2k D, whereas that probed at the C site should be spread around Q∼2kD. The present results imply that the antiferromagnetic fluctuations are sharplyenhanced at a wave vector of Q∼2k D, and that the intervalley scattering is responsible for the electron correlations in thissystem. At elevated temperatures, K BandKCgradually approach each other, and all Ki’s become nearly the same around room temperature ( Ki≈6). This temperature dependence is understood in line with the diminishing spin-density dispro-portionation at higher temperatures, as discussed in Sec. IV A [see Fig. 9(b)]; the k-space averaging due to the intense phonon excitations makes the difference between the molecules A,B, and C indistinguishable and/or the possible depression ofcharge disproportionation pushes down the Dirac point below ε F, causing a semimetallic state. Theoretically, electron correlations stabilize the Dirac-cone dispersion in this system as we mentioned in Sec. I.18,20,29,30 Furthermore, the presence of the tilted Dirac cone can largely enhance the anisotropy of |CB,k|2and|CC,k|2in the kspace.21 The present site-dependent NMR characteristics well below room temperature are consistent with these predictions andgive an opportunity to look into the magnetic properties of thetilted Dirac cone in a k-dependent manner. V . CONCLUSION We performed13C NMR measurements on the charge- disproportionated conducting state in the layered organic conductor α-(BEDT-TTF) 2I3at ambient pressure. Reflecting the low crystal symmetry, we obtained separate NMR linesfor crystallographically nonequivalent three molecules in theunit cell ( i=A, B, and C). The temperature dependences of the resultant site-specific Knight shift K iare properly captured by the conical dispersion with the Dirac points closetoε F. The analyses of the site-selected nuclear relaxation rate (1/T1T)iandKipoint to the presence of strong or intermediate antiferromagnetic spin correlations. Exploiting the theoreti-cal prediction of the peculiar site-specific reciprocal-spaceanisotropy in the Dirac cone, the present results turn out toshow that the local spin susceptibility and electron correlationsare both strongly angular dependent on the cone. This outcomeis regarded as one of the outstanding aspects inherited fromthe tilted Dirac cone in α-(BEDT-TTF) 2I3. ACKNOWLEDGMENTS The authors thank R. Kondo and H. Sawa for informing us on the structural data and the local electronic density distribu-tion on triiodine anions before publication, and N. Tajima, Y .Suzumura, A. Kobayashi, A. Kawamoto, T. Takahashi, and H.Fukuyama for informing us on the detailed results of transportexperiments at ambient pressure, and valuable and helpfuldiscussions. This work was supported by MEXT Grant-in-Aids for Scientific Research on Innovative Area (NewFrontier of Materials Science Opened by Molecular Degreesof Freedom; Grants No. 20110002 and No. 21110519),JSPS Grant-in-Aids for Scientific Research (A) (GrantNo. 20110002) and (C) (Grant No. 20540346), and MEXTGlobal COE Program at University of Tokyo (Global Center of Excellence for the Physical Sciences Frontier; Grant No. G04). 1K. Bender, I. Henning, and D. Schweitzer, Mol. Cryst. Liq. 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PhysRevB.93.195137.pdf

PHYSICAL REVIEW B 93, 195137 (2016) Second-principles method for materials simulations including electron and lattice degrees of freedom Pablo Garc ´ıa-Fern ´andez,1Jacek C. Wojdeł,2Jorge ´I˜niguez,2,3and Javier Junquera1 1Departamento de Ciencias de la Tierra y F ´ısica de la Materia Condensada, Universidad de Cantabria, Cantabria Campus Internacional, Avenida de los Castros s/n, 39005 Santander, Spain 2Institut de Ci `encia de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra, Spain 3Materials Research and Technology Department, Luxembourg Institute of Science and Technology, Avenue des Hauts-Fourneaux 5, L-4362 Esch/Alzette, Luxembourg (Received 24 November 2015; revised manuscript received 8 April 2016; published 19 May 2016) We present a first-principles-based ( second-principles ) scheme that permits large-scale materials simulations including both atomic and electronic degrees of freedom on the same footing. The method is based on a predictivequantum-mechanical theory—e.g., density functional theory—and its accuracy can be systematically improvedat a very modest computational cost. Our approach is based on dividing the electron density of the systeminto a reference part—typically corresponding to the system’s neutral, geometry-dependent ground state—and adeformation part—defined as the difference between the actual and reference densities. We then take advantageof the fact that the bulk part of the system’s energy depends on the reference density alone; this part can beefficiently and accurately described by a force field, thus avoiding explicit consideration of the electrons. Then,the effects associated to the difference density can be treated perturbatively with good precision by working in asuitably chosen Wannier function basis. Further, the electronic model can be restricted to the bands of interest.All these features combined yield a very flexible and computationally very efficient scheme. Here we presentthe basic formulation of this approach, as well as a practical strategy to compute model parameters for realisticmaterials. We illustrate the accuracy and scope of the proposed method with two case studies, namely, the relativestability of various spin arrangements in NiO (featuring complex magnetic interactions in a strongly-correlatedoxide) and the formation of a two-dimensional electron gas at the interface between band insulators LaAlO 3and SrTiO 3(featuring subtle electron-lattice couplings and screening effects). We conclude by discussing ways to overcome the limitations of the present approach (most notably, the assumption of a fixed bonding topology), aswell as its many envisioned possibilities and future extensions. DOI: 10.1103/PhysRevB.93.195137 I. INTRODUCTION Over the past two decades first-principles methods, in par- ticular those based on efficient schemes like density functionaltheory (DFT) [ 1–5], have become an indispensable tool in applied and fundamental studies of molecules, nanostructures,and solids. Modern DFT implementations make it possibleto compute the energy and properties (vibrational, electronic,magnetic) of a compound from elementary information aboutits structure and composition. Hence, in DFT investigationsthe experimental input can usually be reduced to a minimum(the number of atoms of the different chemical species, anda first guess for the atomic positions and unit cell latticevectors). Further, the behavior of hypothetical materials canbe readily investigated, which turns the methods into theultimate predictive tool for application, e.g., in materialsdesign problems. However, interpreting or predicting the results of experi- ments requires, in many cases, to go beyond the time and lengthscales that the most efficient DFT methods can reach today.This becomes a very stringent limitation when, as it frequentlyhappens, the experiments are performed in conditions that areout of the comfort zone of DFT calculations, i.e., at ambienttemperature, under applied time-dependent external fields, outof equilibrium, under the presence of (charged) defects, etc. The development of efficient schemes to tackle such challenging situations, which are of critical importance inareas ranging from biophysics to condensed matter physicsand materials science, constitutes a very active research field. Especially promising are QM/MM multiscale approaches inwhich different parts of the system are treated at different levelsof description: The most computationally intensive methods[based on quantum mechanics (QM), as for example DFTitself] are applied to a region involving a relatively smallnumber of atoms and electrons, while a large embedding regionis treated in a less accurate molecular mechanics (MM) way(e.g., by using one of many available semiempirical schemes). Today’s multiscale implementations tend to rely on semiempirical methods—like tight-binding [ 6,7] and force- field [ 8,9] schemes—that were first introduced decades ago. In some cases, such schemes are designed to retain DFT-like accuracy and flexibility as much as possible. One rele-vant example are the self-consistent-charge density-functionaltight-binding (DFTB) techniques [ 10–12], and related ap- proaches [ 13–15], which retain the electronic description and permit an essentially complete treatment of the compounds. Another relevant example are the effective Hamiltonians developed to describe ferroelectric phase transitions and otherfunctional effects [ 16–18]; these are purely lattice models (i.e., without an explicit treatment of the electrons) basedon a physically-motivated coarse-grained representation ofthe material, and have been shown to be very useful evenin nontrivial situations involving chemical disorder [ 19] and magnetoelectric effects [ 20], among others. Such methods have demonstrated their ability to tackle many importantproblems (see, e.g., Refs. [ 15,21–23] for the DFTB approach), 2469-9950/2016/93(19)/195137(28) 195137-1 ©2016 American Physical SocietyPABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) and constitute very powerful tools. Nevertheless, they are limited when it comes to treating situations in which the key interactions involve minute energy differences (of theorder of meV’s per atom) and a great accuracy is needed,or where a complete atomistic description of the material isrequired. Another aspect in which many approximate approaches fail is in the simultaneous treatment, at a similar level ofaccuracy and completeness, of electronic and lattice degreesof freedom. Most methods in the literature are strongly biasedtowards either the electronic [ 24–26] or the lattice [ 8,9,16–18] properties. Further, the few schemes that attempt a realistic,simultaneous treatment of both types of variables usuallyinvolve very coarse-grained representations [ 27–29]. Here we introduce a new scheme to tackle the problem of simulating both atomic and electronic degrees of freedomon the same footing, with arbitrarily high accuracy, and at amodest computational cost. In essence, we will combine (i)an accurate model potential to describe the lattice-dynamicalproperties of the system with (ii) a tight-binding-like approachto describe the relevant electronic degrees of freedom andelectron-phonon interactions. Our scheme will be limited toproblems in which it is possible to identify an underlyinglattice or bonding topology that is not broken during thecourse of the simulation. As we will show below, such afixed-topology hypothesis permits drastic simplifications in the description of the system, yielding a computationally efficientscheme whose accuracy can be systematically improved tomatch that of a DFT calculation, if needed. Note that, whileour assumption of an underlying lattice may seem veryrestrictive at first, in fact it is not. There are myriads ofproblems of great current interest—ranging from electronicand thermal transport phenomena to functional (dielectric,ferroelectric, piezoelectric, magnetoelectric) effects and mostoptical properties—that are perfectly compatible with it.Further, this restriction can be greatly alleviated by combiningour potentials with DFT calculations in a multiscale scheme,a task for which our models are ideally suited. At its core, our new scheme relies on the usage of a force field to treat interatomic interactions, capable ofproviding a very accurate description of the lattice-dynamicalproperties of the material of interest. In particular, the schemerecently introduced by some of us in Ref. [ 30] constitutes an excellent choice for our purposes, as it takes advantage of theaforementioned fixed-topology condition to yield physicallytransparent models whose ability to match DFT results can besystematically improved. Then, a critical feature of our approach is to identify such a lattice-dynamical model with the description of the material inthe Born-Oppenheimer surface, i.e., with the DFT solutionof the neutral system in its electronic ground state. Sincethe force fields of Ref. [ 30] do not treat electrons explicitly, this identification implies that our models will not tackle thedescription of electronic bonding, as DFTB schemes do. Inother words, we will not be concerned with modeling theinteractions responsible for the cohesive energy of the material,or for the occurrence of a certain basic lattice topology andstructural features. Within our scheme, all such properties aresimply taken for granted, and constitute the starting point ofour models.Instead, our models focus on the description of electronic states that differ from the ground state. These are thetruly relevant configurations for the analysis of excitations,transport, competing magnetic orders, etc. By focusing onthem, and by adopting a description based on material- (andtopology-)specific electronic wave functions, we can afford avery accurate treatment of the electronic part while keepingthe models relatively simple and computationally light. As we will see, while it bears similarities with DFTB schemes, the present approach is ultimately more closelyrelated to Hubbard-like methods. Yet, at variance with theusual semiempirical Hubbard Hamiltonians, our models arefirmly based on a higher-level first-principles theory, treatingall lattice degrees of freedom, and the relevant electronic ones,with similarly high (full DFT at the asymptotic limit) accuracy.The term “second-principles,” used in the title of this paper, ismeant to emphasize such a solid first-principles foundation. II. OVERVIEW OF THE METHOD In this paper we introduce the formal framework of our new computational scheme to perform large-scale simulations. Theapproach is methodologically based on DFT and obtains allthe necessary information to simulate a material from thistechnique, so we have named it second-principles density- functional-theory (SP-DFT). Hence, the paper contains two main sections that describe (i) the development of the theoryour models are based on as a systematic approximation toDFT (Sec. III) and (ii) the approach to calculate the model parameters from DFT (Sec. IV). Given the generality of the approach and the many mag- nitudes that need to be calculated, we devote this section to(i) enumerate the most important foundations on which themethod is based, (ii) point to the corresponding sections ofthis paper where the reader can find a full description of thevarious approximations and the physical phenomena behindthem, (iii) present a comparison with previous methodologiesbased on similar ideas, and (iv) describe how to improve onthe approximations made in a systematic way to achieve fullDFT accuracy as the asymptotic limit. (1) As in DFTB approaches, the starting point of our method is the expansion of the DFT total energy (Sec. III B) with respect to charge density fluctuations around a referenceelectron density (Sec. III A ). Our first approximation involves limiting the expansion to second order. We choose ourexpansion so that the zeroth order term is the total DFTenergy for the reference electron density, which we representusing an accurate and efficient force field. The first andsecond order terms involve small corrections to this energyand require explicit electronic structure calculations. The keyto the success of this approach is that, contrary to otherelectronic structure calculation methods, the correction (andthe self-consistent equations, Sec. III H ) only depends on the difference density, and noton the total density. This magnitude involves few electrons and does not require knowledge of allthe electronic states of the system, and can thus be handledwith relatively modest computational resources. (2) The electronic wave functions are expanded in a minimal basis set made of localized Wannier functions (Sec. III C). The Hamiltonian matrix elements are represented in this basis set, 195137-2SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) and the resultant one- and two- electron integrals become the main parameters of the calculation (Sec. III C). While these parameters can, in principle, be computed from DFTsimulations, our approach here is to fitthem so that the model reproduces a training set of first-principles data. An extensionof the expressions to treat magnetic systems is developedin Sec. III D , where we see that Hubbard- and Stoner-like parameters appear in a natural way. (3) Although our basis set is localized in space, and therefore the relevant real space matrix elements are restrictedto only relatively close neighbors, their values can be affectedby the electrostatic interactions due to charges and electricdipoles located at distant regions of the material. Thus, we splitour one- and two- electron integrals into near-field and far-fieldcontributions. The latter are computed using the multipole expansion described in Sec. III E, while the former are obtained from DFT using the recipe provided in Sec. IV. (4) We introduce an explicit dependence of the electronic Hamiltonian on atomic positions by expanding the one-electron parameters as a function of distortions of a referencestructure. We thus account for electron-lattice couplings (Sec. III F). At this level a new approximation is introduced to neglect the dependence of the two-electron integrals with thegeometry. After finishing with the foundations of the method and gathering together all the previous ingredients, we find anexpression for the total energy of the system (Sec. III G ). Its minimization with respect to the coefficients of the wavefunctions in a Wannier basis gives rise to a set of Kohn-Sham-like equations that need to be solved self-consistently(Sec. III H ). After convergence, the forces on all the atoms and the stresses on the unit cell lattice vectors are computed(Sec. III I), allowing for structural-relaxation or molecular- dynamics simulations over very large length and time scales.To end the section on the method itself, we give in Sec. III J a brief account of some practicalities concerning an efficientimplementation of our approach. In Sec. IVwe propose a tentative approach for a systematic calculation of the model variables from first principles.Note that the development of a systematic—and automatic—strategy for the construction of models with predefinedaccuracy is a technically challenging task that remains forfuture work. In Sec. Vwe provide the details about the SCALE -UPcode package developed in the course of this work, stressing thecomputational efficiency of the simulations based on the newmethodology. This is followed by Sec. VIwhere we compare and highlight the novelty of our approach to two closely relatedtechniques, the effective-Hamiltonian and the DFTB methods. Then, we describe in Sec. VII a couple of nontrivial applications that were selected to highlight the flexibility ofthe models, the great physical insight that they provide, andtheir ability to account for complex properties with DFT-likeaccuracy. The two chosen systems present interactions ofvery different origin: (i) the magnetic Mott-Hubbard insulatorNiO and (ii) the two-dimensional electron gas (2DEG) thatappears at the interface between band insulators LaAlO 3and SrTiO 3. Note that, while accuracy will be highlighted, in these initial applications we have focused on testing the ability ofour scheme to tackle challenging situations from the physicsstandpoint, and to produce reasonable predictions beyond the information including in the training set. Finally we presentour conclusions and a brief panoramic overview of futureextensions of the method and possible fields of applicationin Sec. VIII. III. THEORY A. Basic definitions As customarily done in most first-principles schemes, we assume the Born-Oppenheimer approximation to separate thedynamics of nuclei and electrons. Hence, we consider thepositions of the nuclei as fixed parameters of the electronicHamiltonian. Our approach will give us access to the potentialenergy surface (PES), i.e., for each configuration of the nuclei,the total energy of the system will be computed. Our goal is to describe the electrons in the system, and the relevant electronic interactions, in the simplest possibleway. Hence, we will typically focus on valence and con-duction states, and will thus work with a lattice of ionic cores comprised by the nuclei and the corresponding core electrons, which will not be modeled explicitly. Here we useindistinctively the terms atoms, ions, and nuclei to refer to suchionic cores. Our method relies on the following key concepts: the reference atomic geometry (RAG henceforth) and the reference electronic density (RED in the following). As in the recent development of model potentials for lattice-dynamical stud-ies [30], the first step towards the construction of our model is the choice of a RAG, that is, one particular configurationof the nuclei that we will use as a reference to describe anyother configuration. In principle, no restrictions are imposedon the choice of RAG. However, it is usually convenient toemploy the ground state structure or, alternatively, a suitablychosen high-symmetry configuration. Note that these choicescorrespond to extrema of the PES, so that the correspondingforces on the atoms and stresses on the cell are zero. Further,the higher the symmetry of the RAG, the fewer the couplingterms needed to describe the system and, in turn, the numberof parameters to be determined from first principles. To describe the atomic configuration of the system we shall adopt a notation similar to that of Ref. [ 30]. In what follows, all the magnitudes related with the atomic structure will belabeled using Greek subindices. For the sake of simplicity, weshall assume a periodic three dimensional infinite crystal, withthe lattice cells denoted by uppercase letters ( /Lambda1,/Delta1,... ) and the atoms in the cell by lowercase letters ( λ,δ,... ). In this manner, the lattice vector of cell /Lambda1is/vectorR /Lambda1, and the reference position of atom λis/vectorτλ. In order to allow for a more compact notation, a cell/atom pair will sometimes be represented by a lowercase bold subindex, so that /vectorR/Lambda1λ↔λ. Any possible crystal configuration can be specified by expressing the atomic positions, /vectorrλ, as a distortion of the RAG, as /vectorrλ=(1+←→η)(/vectorR/Lambda1+/vectorτλ)+/vectoruλ, (1) where 1is the identity matrix,←→ηis the homogeneous strain tensor, and /vectoruλis the absolute displacement of atom λin cell /Lambda1 with respect to the strained reference structure. 195137-3PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) The second step is the definition of a RED, n0(/vectorr),for each possible atomic configuration . Our method relies on the fact that, in most cases, the self-consistent electron density, n(/vectorr), will be very close to the RED, so that changes in physicalproperties can be described by the small [with respect to n 0(/vectorr)] deformation density ,δn(/vectorr), defined as n(/vectorr)=n0(/vectorr)+δn(/vectorr). (2) Several remarks are in order about Eq. ( 2). First, with n(/vectorr) we denote the electron density that inte- grates to the number of electrons (i.e., it is positive). It istrivially related with the charge density (in atomic units it justrequires making it negative due to the sign of the electroniccharge). Second, this separation of the charge density into reference and deformation contributions is similar to what is commonlyfound in DFTB schemes, and even in first-principles meth-ods [ 31]. However, this parallelism may be misleading. Indeed, it is important to note that we make no assumption on the formof the RED. In most cases, e.g., nonmagnetic insulators, itwill be most sensible to identify the RED with the groundstate density of the neutral system. Nevertheless, as will beillustrated in Sec. VI B for Mott insulator NiO, other choices are also possible and very convenient in some situations. Third, our RED will typically be an actual solution of the electronic problem, as opposed to some approximate density,e.g., a sum of spherical atomic-like densities, possibly takenfrom the isolated-atom solution, as used in some DFTBschemes [ 15,32]. Fourth, the concepts of RAG and RED are completely independent: In our formalism, we define a REDfor every atomic structure accessible by the system, and notonly for the reference atomic geometry. Finally, let us remark,in advance to Sec. III J, that our method does not require an explicit calculation of n 0(/vectorr) (or any other function in real space, for that matter), a feature that allows us to reduce thecomputational cost significantly. In order to further clarify the concept of RED, let us discuss the application of our method to the relevant case of adoped semiconductor. As sketched in Fig. 1, our hypothetical semiconductor is made of two different types of atoms(represented by large green and small red balls, respectively)in a square planar geometry with a three-atom repeated cell.The RAG corresponds to the high-symmetry configuration inwhich the large atom is located at the center of the square,while the small atoms lie at the centers of the sides. In theneutral (undoped) case, a self-consistent DFT calculation ofthe RAG would yield an electronic configuration with all thevalence bands occupied and all the conduction bands empty, asillustrated in Fig. 1(e). The associated electron density would be our RED, n 0(/vectorr), represented by the green clouds in Fig. 1(b); the associated energy would be E(0), using the notation that will be introduced in Sec. III B. Now, if we dope the neutral system by adding or removing electrons, there will be a response of the electronic cloud,which will tend to screen the field caused by the extra charge.The doping electron (respectively, hole) will occupy the statesat the bottom of the conduction band (respectively, top ofthe valence band). The doping-induced charge redistributioncan be viewed as resulting from an admixture of occupiedand unoccupied states of the reference neutral configuration. FIG. 1. Schematic cartoon that represents the key physical concepts for the development of the second-principles models: the reference atomic structure and the reference and deformation electrondensities. Panels (a)–(c): the meaning of the balls (which represent the position of the atoms in a hypothetical semiconductor), and the green clouds (which represent charge densities) are explained in the maintext. Panels (d)–(f): the horizontal lines represent the one-electron energy levels obtained at the corresponding atomic structures and for the reference electronic configuration (neutral ground state). Full green circles represent full occupation of a given state by electrons. Half filled orange/green circles indicate partial occupation of aparticular level. The notations E (0),E(1),a n dE(2)for the energies are introduced in Sec. III B. The parameters γ,U, and,Iare defined in Secs. III C andIII D . Only the case of doping with electrons is sketched. Doping with holes would lead to an equivalent picture. The resulting state, described by the total charge density n(/vectorr), is sketched in Figs. 1(a)and1(d). The difference between the total electronic density and the RED is the deformation densityδn(/vectorr). Such a deformation density, which is the key quantity in our scheme, captures both the doping and the system’sresponse to it, as sketched in Figs. 1(c)and1(f). Finally, let us further stress the independence between RAG and RED. Note that all three quantities n(/vectorr),n 0(/vectorr), and δn(/vectorr) are in fact parametric functions of the atomic positions. This isillustrated in Fig. 2, which sketches a case in which one atom is displaced from the RAG. B. Approximate expression for the energy Let us consider an atomic geometry characterized by the homogeneous strain tensor←→ηand the individual atomic displacements {/vectoruλ}as described in Eq. ( 1). Our main objective is to find a functional form that permits an accurate approxi-mation of the DFT total energy at a low computational cost.The DFT energy can be written as E DFT=/summationdisplay j/vectorkoj/vectork/angbracketleftψj/vectork|ˆt+vext|ψj/vectork/angbracketright +1 2/integraldisplay/integraldisplayn(/vectorr)n(/vectorr/prime) |/vectorr−/vectorr/prime|d3rd3r/prime+Exc[n]+Enn.(3) 195137-4SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) FIG. 2. Schematic cartoon emphasizing that the division of the electron density into reference and deformation parts is carried out for any geometrical configuration of the system, as defined by the strain←→ηand atomic displacements {/vectoruλ}. The distortion of the reference atomic geometry is illustrated by the off-centering of one atom (indicated with a black arrow). The atomic distortion results in a modified n0(/vectorr) [panel (b)], as well as in additional changes depicted in panel (c), where the green and orange clouds denote positive and negative variations in the electronic density. Symbols have the same meaning as in Fig. 1. In this expression, the first term on the right-hand side includes the kinetic energy of a collection of noninteracting electrons ascomputed through the one-particle kinetic energy operator, ˆt; this first term also includes the action of an external potential,v ext, which gathers contributions from the nuclei (or ionic cores) and, possibly, other external fields. The second termis the Coulomb electrostatic energy, which in the contextof quantum mechanics of condensed matter systems is alsoreferred to as the Hartree term. The third term, E xc[n], is the so-called exchange and correlation functional, whichcontains the correlation contribution to the kinetic energy inthe interacting electron system, as well as any electron-electroninteraction effect beyond the classic Coulomb repulsion. Thelast term, E nn, is the nucleus-nucleus electrostatic energy. Note that Eq. ( 3) is written in atomic units, which are used throughout the paper. ( |e|=me=/planckover2pi1=aB=1, where |e|is the magnitude of the electron charge, meis the electronic mass, and aBis the Bohr radius). As already mentioned, we assume that the Born- Oppenheimer approximation applies, so that the positions ofthe nuclei can be considered as parameters of the Hamiltonian.We also assume periodic boundary conditions. (Finite systemscan be trivially considered by, e.g., adopting a supercellapproach [ 33].) Within periodic boundary conditions, the eigenfunctions of the one-particle Kohn-Sham equations, |ψ j/vectork/angbracketright, can be written as Bloch states characterized by the wave vector /vectorkand the band index j, with the occupation of a state given by oj/vectork.N o t e that Eq. ( 3) is valid for any geometric structure of the system, and we implicitly assume that the total energy ( EDFT), the one-particle eigenstates ( |ψj/vectork/angbracketright), and all derived magnitudes (such as the electron densities n,n0, andδn) depend on the structural parameters←→ηand{/vectoruλ}. The total energy of Eq. ( 3) is a functional of the density which, as described in Eq. ( 2), can be written as the sum of a reference part n0(/vectorr) and a deformation part δn(/vectorr). When we implement this decomposition, the linear Coulomb energyterm can be trivially dealt with. For the nonlinear exchangeand correlation functional, we follow Ref. ([ 12]) and expandE xc[n] around the RED as Exc[n]=Exc[n0]+/integraldisplayδExc δn(/vectorr)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0δn(/vectorr)d3r +1 2/integraldisplay/integraldisplayδ2Exc δn(/vectorr)δn(/vectorr/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0δn(/vectorr)δn(/vectorr/prime)d3rd3r/prime+···, (4) where we have introduced functional derivatives of Exc.I n principle, Eq. ( 4) is exact if we consider all the orders in the expansion. (Expansions like this one are frequentlyfound in the formulations of the adiabatic density functionalperturbation theory [ 34,35].) In practice, under the assumption of a small deformation density, the expansion can be cutat second order. As we shall show in Secs. III D andIII H , this approximation includes as a particular case the fullHartree-Fock-theory; hence, we expect it to be accurate enoughfor our current purposes. Within the previous approximation, we can write the total energy as a sum of terms coming from the contributions ofthe deformation density at zeroth (reference density), first, andsecond orders. Formally we write E DFT≈E=E(0)+E(1)+E(2), (5) where the individual terms have the following form [ 36]. For the zeroth-order term E(0), we get E(0)=/summationdisplay j/vectorko(0) j/vectork/angbracketleftbig ψ(0) j/vectork/vextendsingle/vextendsingleˆt+vext/vextendsingle/vextendsingleψ(0) j/vectork/angbracketrightbig +1 2/integraldisplay/integraldisplayn0(/vectorr)n0(/vectorr/prime) |/vectorr−/vectorr/prime|d3rd3r/prime+Exc[n0]+Enn.(6) The above equation corresponds, without approximation, to the exact DFT energy for the reference density n0. We can choose the RED so that E(0)is the dominant contribution to the total energy of the system, and here comes a key advantageof our approach: We can compute E (0)by employing a model potential that depends only on the atomic positions, wherethe electrons (assumed to remain on the Born-Oppenheimersurface) are integrated out. This represents a huge gain withrespect to other schemes that, like the typical DFTB methods,require an explicit and accurate treatment of the electronicinteractions yielding the RED as well as solving numerically forE (0)andn0for each atomic configuration considered in the simulation. The first-order term involves the one-electron excitations as captured by the deformation density, E(1)=/summationdisplay j/vectork/bracketleftbig oj/vectork/angbracketleftbig ψj/vectork/vextendsingle/vextendsingleˆh0/vextendsingle/vextendsingleψj/vectork/angbracketrightbig −o(0) j/vectork/angbracketleftbig ψ(0) j/vectork/vextendsingle/vextendsingleˆh0/vextendsingle/vextendsingleψ(0) j/vectork/angbracketrightbig/bracketrightbig . (7) Here, ˆh0is the Kohn-Sham [ 1] one-electron Hamiltonian defined for the RED, ˆh0=ˆt+vext−vH(n0;/vectorr)+vxc[n0;/vectorr], (8) where vH(n0;/vectorr) andvxc[n0;/vectorr] are, respectively, the reference Hartree, vH(n0;/vectorr)=−/integraldisplayn0(/vectorr/prime) |/vectorr−/vectorr/prime|d3r/prime, (9) 195137-5PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) and exchange-correlation, vxc[n0;/vectorr]=δExc[n] δn(/vectorr)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0, (10) potentials. It is important to note that Eq. ( 7) is different from the one usually employed in DFTB methods (see, for example,Refs. [ 10] and [ 12]): while typical DFTB schemes include a plain sum of one-electron energies, here we deal with thedifference between the value of this quantity for the actual system and for the reference one [see sketch in Fig. 1(f)]. Such a difference is a much smaller quantity, more amenable toaccurate calculations. Finally, the two-electron contribution from the deformation density, E (2), is given by E(2)=1 2/integraldisplay d3r/integraldisplay d3r/primeg(/vectorr,/vectorr/prime)δn(/vectorr)δn(/vectorr/prime), (11) where the screened electron-electron interaction operator, g(/vectorr,/vectorr/prime), is g(/vectorr,/vectorr/prime)=1 |/vectorr−/vectorr/prime|+δ2Exc δn(/vectorr)δn(/vectorr/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0. (12) Here, δ2Exc/δn(/vectorr)δn(/vectorr/prime) captures the effective screening of the two-electron interactions due to exchange and correlation.The latter magnitude is particularly important in chemistry, asit is related to the hardness of a material [ 37]. In summary, in this section we have shown how a particular splitting of the total density, into reference and a deformationparts, allows us to expand the DFT energy around n 0and as a function of δn. This expansion can be truncated at second order while keeping a high accuracy; nevertheless, this approachcan be systematically improved by including higher-orderterms in δn, in analogy to what is done, e.g., in the so-called DFTB3 method [ 21]. While the general idea is reminiscent of DFTB methods in the literature, [ 10–12] our scheme has two distinct advantages. On one hand, the zeroth-order termcan be conveniently parametrized by means of a lattice modelpotential, so that it can be evaluated in a fast and accurateway without explicit consideration of the electrons. On theother hand, the first-order term is much smaller, and can becalculated more accurately, than in usual DFTB schemes, as ittakes the form of a perturbative correction. C. Formulation in a Wannier basis 1. Choice of Wannier functions Our formulation requires the computation of the matrix elements of the Kohn-Sham one-electron Hamiltonian, asdefined for the RED, in terms of Bloch wave functions[Eq. ( 7)], as well as various integrals involving the deformation charge density [Eq. ( 11)]. To compute these terms, we will expand the Bloch waves on a basis of Wannier-like functions(WFs), |χ a/angbracketright, in the spirit of Ref. [ 38]. There are several reasons for our choice of a Wannier basis set over the atomic orbitalsmost commonly used in DFTB formulations [ 10–12,15]. First, the Wannier orbitals are naturally adapted to the specific material under investigation. In fact, they will betypically obtained from a full first-principles simulation ofthe band structure of the target material, which permits amore accurate parametrization of the system while retaining a minimal basis set. Second, the Wannier functions are spatially localized, and several localization schemes are available in the literature[38–44]. The localization will be exploited in our second- principles method to restrict the real-space matrix elements tothose involving relatively close neighbors, as will be explainedin Sec. IV. Third, the localized Wannier functions can be chosen to be orthogonal. Note that methods with nonorthogonal basisfunctions require the calculation of the overlap integrals thathave a nontrivial behavior as a function of the geometry of thesystem. Moreover, the one-particle Kohn-Sham equations inmatrix form become a generalized eigenvalue problem, whosesolution requires a computationally demanding inversion ofthe overlap matrix. The use of orthogonal Wannier functionsallows us to bypass these shortcomings. Fourth, the Wannier functions enable a very flexible description of the electronic band structure, as they can beconstructed to span the space corresponding to a specificset of bands [ 38,45]. Therefore, the electronic states can be efficiently split into: (i) an active set playing an important role in the properties under study; and (ii) a background set that will be integrated out from the explicit treatment. For instance,if the problem of interest involves the formation of low-energyelectron-hole excitons, our active set would be comprised bythe top-valence and bottom-conduction bands, and we woulduse the corresponding Wannier functions as a basis set. Typically, we will start from a set of Bloch-like Hamiltonian eigenstates |ψ (0) n/vectork/angbracketrightthat define a manifold of Jbands associated to the RED. Then, following, e.g., the recipe of Ref. [ 42], we have |χa/angbracketright≡|/vectorRAa/angbracketright=V (2π)3/integraldisplay BZd/vectorke−i/vectork·/vectorRAJ/summationdisplay m=1T(/vectork) ma/vextendsingle/vextendsingleψ(0) m/vectork/angbracketrightbig ,(13) where the Wannier function |/vectorRAa/angbracketrightis labeled by the cell Aat which it is centered (associated to the lattice vector /vectorRA) and by a discrete index a. Note that we use Latin subindices to label all physical quantities related with the electrons; to alleviate thenotation, we group in the bold symbol aboth the cell and the discrete index, so that a↔/vectorR Aa.I nE q .( 13),Vis the volume of the primitive unit cell, the integral is carried out over the wholeBrillouin zone (BZ), the index mruns over all the Jbands of the manifold, and the T (/vectork)matrices represent unitary transfor- mations among the JBloch orbitals at a given wave vector. Figure 3shows three paradigmatic examples: a nonmag- netic insulator [bulk SrTiO 3,F i g . 3(a)], a nonmagnetic metal [bulk Cu, Fig. 3(b)], and an antiferromagnetic insulator [bulk NiO, Fig. 3(c)]. In the first case, the valence bands are well separated in energy from other bands; further, theyhave a well-defined character strongly reminiscent of thecorresponding atomic orbitals. More precisely, three isolatedmanifolds corresponding to the occupied valence bands—withdominant O-2 s,S r - 4p, and O-2 pcharacter, respectively—are clearly visible; these bands are centered around 17 eV , 15 eV ,and 3 eV below the valence-band top, respectively. The Blocheigenstates for these bands can be directly used to computethe corresponding localized Wannier functions following the 195137-6SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) Γ X W L Γ K−20−15−10−50510Energy (eV) Γ X M R Γ M−20−15−10−50510Energy (eV) Γ X W L Γ K−10−50510Energy (eV) (a) (b) (c) O(2s)O(2p)Ti[3d(t2g)]Ti[3d(eg)] O(2s)Ni[3d(t2g)]+Ni[3d(eg)] +O(2p)Ni[3d(eg)] Cu(3d)+Cu(4s)Sr(4p) FIG. 3. Band structure, showing different band entanglements, in three archetypal cases: (a) insulating SrTiO 3, (b) metallic Cu, and (c) antiferromagnetic NiO. The groups of entangled bands are separated by energy gaps and colored differently. For SrTiO 3: bands with dominant O(2s), Sr(4 p), O(2p), Ti[3 d(t2g)], and Ti[3 d(eg)] characters are plotted, respectively in red, orange, blue, green, and magenta. For Cu: all the bands are entangled. For NiO: some weakly-dispersive bands at the top of the valence and bottom of the conduction regions (indicated by anarrow) share the Ni[3 d(e g)] character. On the leftmost and rightmost edges of the figure, the spatial shape of some of the Wannier functions for SrTiO 3and NiO are displayed, respectively. Isosurfaces corresponding to positive and negative values of the MLWFs are plotted with different colors. In these diagrams, golden, blue, red, and green spheres represent, respectively, Sr, O, Ti, and Ni ions. For NiO, the atoms in the cellused to simulate the antiferromagnetic ground state are shown. Dashed red lines mark the Fermi energy of the metal and the top of the valence bands of the insulators; in all cases such level is taken as the zero of energy. scheme of Ref. [ 42] or similar ones. In contrast, the bottom conduction bands of SrTiO 3have a dominant Ti- t2gcharacter, but overlap in energy with higher-lying (Ti- eg) conduction bands. The situation is even more complicated in the cases ofFigs. 3(b)and3(c), where the critical bands—i.e., those around the Fermi energy in the case of Cu, and those comprising theNi-3dmanifold in the case of NiO—are strongly entangled with other states. In such cases, we may need to use adisentanglement method—like, e.g., the one proposed inRef. [ 38]—to identify a minimal active manifold. Note that the inverse transformation from Wannier to Bloch functions reads /vextendsingle/vextendsingleψ (0) j/vectork/angbracketrightbig =/summationdisplay ac(0) ja/vectorkei/vectork·/vectorRA|χa/angbracketright, (14) where the connection between the c(0) ja/vectorkcoefficients and the transformation matrices in Eq. ( 13)i sg i v e ni nR e f .[ 36]. The Wannier functions corresponding to the RED [Fig. 1(b)]f o r ma complete basis of the Hilbert space. Hence, we can use them torepresent any perturbed electronic configuration of the system [e.g., the one sketched in Fig. 1(a)]a s |ψ j/vectork/angbracketright=/summationdisplay acja/vectorkei/vectork·/vectorRA|χa/angbracketright, (15) where the sum can be extended to as many bands as needed to accurately describe the phenomenon of interest. (As in theexample of Fig. 1, this might be the addition of an electron and the associated screening.) Finally, note that the Wannier function basis is implicitly dependent on the structural parameters←→ηand{/vectoru λ}, and it should be recomputed for every new RED corresponding tovarying atomic positions. Ultimately, our models will captureall such effects implicitly in the electron-lattice coupling terms,whose calculation is described in Sec. III F. Also, henceforth we will assume that each and every one of the WFs in our basis can be unambiguously associated witha particular atom at (around) which it is centered. Further, we will use the notation a∈λto refer to all the WFs associated to atom λin cell /Lambda1, an identification that will be necessary when discussing our treatment of electrostatic couplings inSec. III E. 2. Equations in a Wannier basis Using Eq. ( 15), we can write the electron density n(/vectorr)i n terms of the Wannier functions, n(/vectorr)=/summationdisplay abdabχa(/vectorr)χb(/vectorr). (16) We can assume we will work with real Wannier functions [ 45] and therefore drop the complex conjugates in our equations.In Eq. ( 16) we have introduced a reduced density matrix, d ab=/summationdisplay j/vectorkoj/vectorkc∗ ja/vectorkcjb/vectorkei/vectork(/vectorRB−/vectorRA), (17) which, following the nomenclature of Ref. [ 46], will be re- f e r r e dt oa st h e occupation matrix for the WFs. This occupation matrix has the usual properties, including periodicity when theWannier functions are displaced by the same lattice vector inreal space. Equation ( 16) can similarly be applied to the RED, n 0(/vectorr)=/summationdisplay abd(0) abχa(/vectorr)χb(/vectorr), (18) where the calculation of the occupation matrix is performed with the coefficients of the Bloch functions that define thereference electronic density, c(0) jα/vectork,a si nE q .( 14). In order to quantify the difference between the two densities defined in Eqs. ( 16) and ( 18), we introduce a deformation occupation matrix , Dab=dab−d(0) ab, (19) 195137-7PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) which will be the central magnitude in our calculations. Now the deformation density can be written as δn(/vectorr)=/summationdisplay abDabχa(/vectorr)χb(/vectorr). (20) Using these definitions, we can rewrite the E(1)andE(2)energy terms. Introducing Eq. ( 19) into Eqs. ( 7) and ( 11) we get, after some algebra [ 36] E(1)=/summationdisplay abDabγab, (21) and E(2)=1 2/summationdisplay ab/summationdisplay a/primeb/primeDabDa/primeb/primeUaba/primeb/prime, (22) respectively, where γabandUaba/primeb/primeare the primary parameters that define our electronic model. These parameters can beobtained, respectively, from the integrals of the one- and two-electron operators computed in DFT simulations, as γ ab=/angbracketleftχa|ˆh0|χb/angbracketright=/integraldisplay d3rχ a(/vectorr)h0(/vectorr)χb(/vectorr), (23) and Uaba/primeb/prime=/angbracketleftχaχa/prime|ˆg|χbχb/prime/angbracketright =/integraldisplay d3r/integraldisplay d3r/primeχa(/vectorr)χb(/vectorr)χa/prime(/vectorr/prime)χb/prime(/vectorr/prime)g(/vectorr,/vectorr/prime). (24) Alternatively, they can be fitted so that the model reproduces a training set of first-principles data. D. Magnetic systems The above expressions are valid for systems without spin polarization. The procedure to construct the energyfor magnetic cases is very similar, but there are subtletiespertaining to the choice of RED. In principle, one could use a RED corresponding to a particular realization of the spin order, e.g., the antiferro-magnetic ground state for a typical magnetic insulator, orthe ferromagnetic ground state for a typical magnetic metal.However, such a choice is likely to result in a less accuratedescription of other spin arrangements, which would hamperthe application of the model to investigate certain phenomena(e.g., a spin-ordering transition). Alternatively, one might adopt a nonmagnetic RED around which to construct the model. Such a RED might correspond toan actual computable state: For example, it could be obtained from a nonmagnetic DFT simulation in which a perfect pairingof spin-up and spin-down electrons is imposed. Further, as wewill see below for the case of NiO, in some cases it is possibleand convenient to consider a virtual RED whose character can be inspected a posteriori . This latter option follows the spirit of the usual approach to the construction of spin-phononeffective Hamiltonians [ 47], where the parameters defining the reference state cannot be computed directly from DFT, butare effectively fitted by requesting the model to reproduce the properties of specific spin arrangements.In the following we assume a nonmagnetic RED, and present an otherwise general formulation. The E (0)andE(1) terms thus describe the lattice and one-electron energetics corresponding to the nonmagnetic RED, and do not captureany effect related with the spin polarization. In contrast, thescreened electron-electron interaction operator [Eq. ( 12)] is spin dependent and equal to g(/vectorr,/vectorr /prime,s,s/prime)=1 |/vectorr−/vectorr/prime|+δ2Exc δn(/vectorr,s)δn(/vectorr/prime,s/prime)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0. (25) where sands/primeare spin indices that can take “up” or “down” values which we denote, respectively, by ↑and↓symbols. This distinction in the screened electron-electron operator leads usto introduce two kinds of Uparameters, Upar aba/primeb/prime=/integraltext d3r/integraltext d3r/primeχa(/vectorr)χb(/vectorr)χa/prime(/vectorr/prime)χb/prime(/vectorr/prime)g(/vectorr,/vectorr/prime,↑,↑) =/integraltext d3r/integraltext d3r/primeχa(/vectorr)χb(/vectorr)χa/prime(/vectorr/prime)χb/prime(/vectorr/prime)g(/vectorr,/vectorr/prime,↓,↓) (26) and Uanti aba/primeb/prime=/integraltext d3r/integraltext d3r/primeχa(/vectorr)χb(/vectorr)χa/prime(/vectorr/prime)χb/prime(/vectorr/prime)g(/vectorr,/vectorr/prime,↑,↓) =/integraltext d3r/integraltext d3r/primeχa(/vectorr)χb(/vectorr)χa/prime(/vectorr/prime)χb/prime(/vectorr/prime)g(/vectorr,/vectorr/prime,↓,↑), (27) which describe, respectively, the interactions between elec- trons with parallel ( Upar) and antiparallel ( Uanti) spins. As a consequence, E(2)in spin-polarized systems is E(2)=/summationdisplay s,s/prime1 2/integraldisplay d3r/integraldisplay d3r/primeg(r,/vectorr/prime,s,s/prime)δn(/vectorr,s)δn(/vectorr/prime,s/prime),(28) where δn(/vectorr,s)=/summationdisplay abDs abχa(/vectorr)χb(/vectorr), (29) andDs abis the deformation occupation matrix for the sspin channel, defined for the up and down spins as D↑ ab=d↑ ab−1 2d(0) ab(30) and D↓ ab=d↓ ab−1 2d(0) ab, (31) respectively. Replacing Eqs. ( 29)–(31) into Eq. ( 28), E(2)=1 2/braceleftBigg/summationdisplay ab/summationdisplay a/primeb/prime[D↑ abD↑ a/primeb/prime+D↓ abD↓ a/primeb/prime]Upar aba/primeb/prime +[D↑ abD↓ a/primeb/prime+D↓ abD↑ a/primeb/prime]Uanti aba/primeb/prime/bracerightBigg . (32) For physical clarity, and to establish the link of Eqs. ( 26) and ( 27) with Eq. ( 24), it is convenient to write UparandUanti in terms of Hubbard- ( U) and Stoner- ( I)-like parameters: Upar aba/primeb/prime=Uaba/primeb/prime−Iaba/primeb/prime (33) Uanti aba/primeb/prime=Uaba/primeb/prime+Iaba/primeb/prime, (34) 195137-8SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) so Uaba/primeb/prime=1 2/parenleftbig Upar aba/primeb/prime+Uanti aba/primeb/prime/parenrightbig , (35) Iaba/primeb/prime=1 2/parenleftbig Uanti aba/primeb/prime−Upar aba/primeb/prime/parenrightbig . (36) It is also convenient to introduce DU a/primeb/prime=D↑ a/primeb/prime+D↓ a/primeb/prime, (37) and DI a/primeb/prime=D↑ a/primeb/prime−D↓ a/primeb/prime, (38) so that Eq. ( 32) can be rewritten as: E(2)=1 2/summationdisplay ab/summationdisplay a/primeb/prime/braceleftbig DU abDU a/primeb/primeUaba/primeb/prime−DI abDI a/primeb/primeIaba/primeb/prime/bracerightbig .(39) Note that the value of Uin Eqs. ( 33) and ( 34) is consistent with the one in Eq. ( 24) if we consider a non-spin-polarized density (D↓ ab=D↑ ab). In addition, note that the newly introduced constant Iaba/primeb/primeonly plays a role in spin-polarized systems and is necessarily responsible for magnetism. Connection with other schemes The two-electron interaction constants— UandIdefined in Eqs. ( 33) and ( 34), respectively—are formally similar to the four-index integrals typically found in Hartree-Fock theory [ 4] and can be chosen to completely match this approach.However, one should note that the electron-electron interactionin our Hubbard-like and Stoner-like constants is not the bareone, but is screened by the exchange-correlation potentialassociated to the reference density, n 0[see Eq. ( 25)]. This fact brings our formulation closer to the so-called DFT +U[48,49] and GW [ 50,51] methods. Looking in more detail at our expressions for UandI, Uaba/primeb/prime=/integraldisplay d3r/integraldisplay d3r/primeχa(/vectorr)χb(/vectorr)χa/prime(/vectorr/prime)χb/prime(/vectorr/prime)gU(/vectorr,/vectorr/prime) (40) Iaba/primeb/prime=/integraldisplay d3r/integraldisplay d3r/primeχa(/vectorr)χb(/vectorr)χa/prime(/vectorr/prime)χb/prime(/vectorr/prime)gI(/vectorr,/vectorr/prime), (41) we find that they are very similar to those of Upar[Eq. ( 26)] andUanti[Eq. ( 27)], except that the operator involved in the double integral is, respectively, gU(/vectorr,/vectorr/prime)=1 |/vectorr−/vectorr/prime|+1 2/bracketleftbiggδ2Exc δn(/vectorr,↑)δn(/vectorr/prime,↑)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0 +δ2Exc δn(/vectorr,↑)δn(/vectorr/prime,↓)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0/bracketrightbigg , (42) and gI(/vectorr,/vectorr/prime)=1 2/bracketleftbiggδ2Exc δn(/vectorr,↑)δn(/vectorr/prime,↓)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0 −δ2Exc δn(/vectorr,↑)δn(/vectorr/prime,↑)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n0/bracketrightbigg . (43) Thus, we see that Ucontains the classical Hartree interactions, screened by exchange and correlation. Moreover, from Eq. ( 39)we see that U, as used here, is related with the deformation occupation matrix DU, that captures the total change of the electron density (i.e., the sum of the deformation occupationmatrix for both components of spins). Therefore, it is consis-tent with the usual definition U=d 2E/dn2, i.e., it quantifies the energy needed to add or remove electrons. On the other hand, Stoner’s [ 52–54]Ionly includes terms with quantum origin. In particular gIprovides the difference in interaction between electrons with parallel and antiparallelspins. E. Electrostatics 1. One-electron parameters The matrix element γab[Eq. ( 23)] gathers Coulomb inter- actions associated to the electrostatic potential created by bothelectrons and nuclei, which acts on the WFs χ aandχb.N o t e that these are the only long-ranged interactions in the system,since all other contributions (kinetic, exchange-correlation,external applied fields) can be considered local or semilocal. Inthe following we discuss the detailed form of this electrostaticpart of γ ab, which we denote γelec ab. Let us first consider the part of γelec abassociated to the electrostatic potential created by the electrons, γelec,e ab.W eh a v e γelec,e ab≡− /angbracketleftχa|vH(n0;/vectorr)|χb/angbracketright =/integraldisplay χa(/vectorr)/parenleftbigg/integraldisplayn0(/vectorr/prime) |/vectorr−/vectorr/prime|d3r/prime/parenrightbigg χb(/vectorr)d3r =/integraldisplay χa(/vectorr)/parenleftBigg/integraldisplay/summationtext co(0) c|χc(/vectorr/prime)|2 |/vectorr−/vectorr/prime|d3r/prime/parenrightBigg χb(/vectorr)d3r.(44) The expression of the reference electron density in terms of the occupation of Wanniers, o(0) c, and squares of Wannier functions in the reference state will be described in more detail inSec. III J. Following the criteria of Ref. [ 55], the one-electron matrix elements related with the Coulomb electron-electroninteraction can be split into two categories (see Fig. 4): (i) the near-field regime, where the two WFs ( aand b) significantly FIG. 4. Schematic representation of the near- and far-field inter- actions. The shape of the orbitals (represented here by two t2g-like WFs labeled aand b) is important in the determination of the short-range part of the γandUinteractions. In addition, the diagonal terms like γaaandUaabbalso include far-field effects due to charges and dipoles at distant regions of the material (see WF cin the figure). As regards the far-field interactions, the precise shape of the charge distributions generating the potential is not critical (illustrated by the diffuse orbital at point c), and can be approximated by a multipole expansion. 195137-9PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) overlap with the third WF ( c) that creates the electrostatic potential, and (ii) the far-field regime, where this overlap isnegligible. In the far-field regime, the electrostatic potential outside the region where a source charge χ cis located can be expressed as a multipole expansion (see Chapter 4 of Ref. [ 56]). More precisely, we can write the far-field (FF) potential created bythe charge distribution given by χ cas ve FF,c(n0;/vectorr)=−o(0) c/integraldisplay|χc(/vectorr/prime)|2 |/vectorr−/vectorr/prime|d3r/prime, (45) which applies to /vectorrpoints for which χc(/vectorr)≈0. Now, let λ label the atom—located at the RAG reference position /vectorτλ= /vectorR/Lambda1+/vectorτλ—around which χcis centered. It is convenient to shift the origin in the integral, /vectorr/prime/prime=/vectorr/prime−/vectorτλ, to write ve FF,c(n0;/vectorr)=−o(0) c/integraldisplay|χc(/vectorr/prime/prime+/vectorτλ)|2 |/vectorr−/vectorτλ−/vectorr/prime/prime|d3r/prime/prime. (46) Then, assuming that |/vectorr/prime/prime|/lessmuch| /vectorr−/vectorτλ|and using the superscript Tto indicate the transpose operation, as necessary to compute inner dot products, we get 1 |/vectorr−/vectorτλ−/vectorr/prime/prime|≈1 |/vectorr−/vectorτλ|+(/vectorr−/vectorτλ)T/vectorr/prime/prime |/vectorr−/vectorτλ|3+.... (47) Now, substituting Eq. ( 47) into Eq. ( 46) we obtain the multipole series ve FF,c(n0;/vectorr)≈qc |/vectorr−/vectorτλ|+(/vectorr−/vectorτλ)T/vectorpc |/vectorr−/vectorτλ|3+.... (48) The coefficient of the first term is the total charge (i.e., the monopole), and it is given by qc=−o(0) c/integraldisplay |χc(/vectorr/prime/prime+/vectorτλ)|2d3r/prime/prime=−o(0) c. (49) The coefficient of the second term is the electric dipole moment associated to χc, which amounts to /vectorpc=−o(0) c/integraldisplay |χc(/vectorr/prime/prime+/vectorτλ)|2/vectorr/prime/primed3r/prime/prime=−o(0) c(/vectorrc−/vectorτλ),(50) where /vectorrcrepresents the centroid of χc. Quadrupole and higher- order moments follow in the expansion, but here we assumethey can be neglected. Finally, the full FF potential created bythe electrons at point /vectorris simply given by v e FF(n0;/vectorr)=/summationdisplay c/primeve FF,c(n0;/vectorr), (51) where the prime indicates that we sum only over WF’s such thatχc(/vectorr)≈0. Let us now consider the part of γelec abassociated to the potential created by the nuclei, which we call γelec,n ab. In analogy with the electronic case, we write the FF electrostatic potentialcreated by the nuclei at point /vectorras v n FF(/vectorr)≈/summationdisplay λ/primeZλ |/vectorr−/vectorτλ|+/summationdisplay λ/prime(/vectorr−/vectorτλ)TZλ/vectoruλ |/vectorr−/vectorτλ|3+..., (52) where the primed sums run only over atoms λwhose associated WFs a∈λsatisfy χa(/vectorr)≈0.Then, adding all far-field contributions to γelec ab, and assign- ing each WF to its associated nucleus, we get vFF(n0;/vectorr)=/summationdisplay λ/primeqλ |/vectorr−/vectorτλ|+/summationdisplay λ/prime(/vectorr−/vectorτλ)T/vectorpλ |/vectorr−/vectorτλ|3+..., (53) where qλ=Zλ+/summationdisplay c∈λqc (54) is the charge of ionλ, while /vectorpλis the local dipole associated to that very ion. Note that we add together the contributionsfrom electrons and nuclei, which allows us to talk about ions in a strict sense [ 57]. We can further approximate this local dipole using the Born charge tensor←→Z ∗ λ, to obtain /vectorpλ=Zλ/vectoruλ+/summationdisplay c∈λ/vectorpc≈←→Z∗ λ/vectoruλ. (55) In order to get the final expression for the FF potential, we note that the electrostatic interactions described above donot take place in vacuum, but in the material at its referenceelectronic density. Thus, we need to take into account that theRED will react to screen such interactions, and that such ascreening can be modelled by the high-frequency dielectrictensor of the material at its RED. Thus, the far-field potentialat the center of WF χ ais vFF(n0;/vectorra)≈/summationdisplay λ/prime/bracketleftbig /vectoreT λa(←→/epsilon1∞)−1/vectoreλa/bracketrightbigqλ |/vectorτλ−/vectorra| +/summationdisplay λ/prime/bracketleftbig /vectorpT λ(←→/epsilon1∞)−1/vectoreλa/bracketrightbig |/vectorτλ−/vectorra|2, (56) where /vectoreλais a unitary vector parallel to /vectorτλ−/vectorra,←→/epsilon1∞is the high- frequency dielectic tensor, and the primed sums are restrictedin the usual way. We can now divide γ abin long-range (lr) and short-range (sr) contributions (see Fig. 4). Considering that χaandχbare strongly localized and orthogonal to each other, we define γlr ab as γlr ab=−vFF(n0;/vectorra)δab. (57) Then, we effectively define the short-range part of γabas γsr ab=γab−γlr ab. (58) Note that the short-range interactions defined in this way include electrostatic effects as well as others associated tochemical bonding, orbital hybridization, etc. These interac-tions do not have a simple analytic form; hence, in order toconstruct our models, they will generally be fitted to reproduceDFT results. It is important to note that the above derivation, and decomposition in long- and short-range parts, is exact and doesnot involve any approximation, except for: (i) the truncation ofthe multipole expansion and (ii) the analytic form introduced 195137-10SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) for the long-range electrostatic interactions, which strictly speaking only applies to homogeneous materials with a bandgap [ 58]. Finally, note that the γ abcouplings can be expected to be short in range, as they involve WFs χaandχbthat are strongly localized in space and decay exponentially as we move awayfrom their centers. Hence, the γmatrix can be expected to be sparse, which will result in more efficient calculations. Itis important to note that this short-ranged character of theγ abcouplings is expected despite the fact that the interactions contributing to γlr abare electrostatic and thus long ranged. 2. Two-electron integrals In a similar vein, we can split Uin short- and long-range contributions, so that Usr aba/primeb/prime=Uaba/primeb/prime−Ulr aba/primeb/prime, (59) where the long-range part will contain the classical FF interac- tion between electrons that can be approximated analytically,while the short-range part will contain all other interactionsincluding many-body effects. Similarly to the case above, we expect that (i) long-range two-electron integrals should be very small unless aoverlaps with band a /primeoverlaps with b/prime, (ii) we can truncate the multipolar expansion at the monopole level, and (iii) theelectrostatic interactions take place in a medium characterizedby the high-frequency dielectric tensor of the material at theRED. Under these conditions we choose U lr aba/primeb/primeto be Ulr aba/primeb/prime=/bracketleftbig /vectoreT aa/prime(←→/epsilon1∞)−1/vectoreaa/prime/bracketrightbig 1 |/vectorra/prime−/vectorra|δabδa/primeb/prime. (60) In order to avoid the divergence of this term, we assume that all one-body integrals ( a=a/prime) are fully included in the short- range part, Usr aba/primeb/prime. Assigning the Wannier functions to their closest nucleus, and summing over all the atoms in the lattice,we find that the total two-electron long-range energy adds to E (2),lr=1 2/summationdisplay λυ/bracketleftbig /vectoreT λυ(←→/epsilon1∞)−1/vectoreλυ/bracketrightbig/Delta1qλ/Delta1qυ |/vectorτυ−/vectorτλ|, (61) where /Delta1qλis the change in charge of the atom λwhen compared to the RED state [Eq. ( 54)]. Thus, the long-range part of Usimply updates the one-electron FF potentials due to the charge transfers between atoms. We would like to stress again that the separation in long- and short-range parts does not involve any approximation;indeed, effects usually considered important in many physicalphenomena, like, e.g., the anisotropy of the Wannier orbitalsat short distances [ 59], are included in U sr. F. Electron-lattice coupling The system’s geometry determines the reference density n0(/vectorr) as well as the corresponding Hamiltonian. In our scheme, such a dependence of the model parameters on the atomicconfiguration is captured by the electron-lattice couplingterms. Let us consider the lattice dependence of the one-electron integrals γ[Eq. ( 23)]. In Sec. III E 1 , these parameters were split in short- and long-range contributions. The explicit FIG. 5. Schematic representation of the effects of the expansion ofγsrin terms of atomic deformations. Panels (a) and (b) illustrate the electron-lattice coupling associated to diagonal, γsr aa, matrix elements that control the average energy of the corresponding bands. In (a) we sketch the forces on the atoms (represented by red spheres) asgenerated by electrons placed on a p xorpy-like WFs. In our method, these forces are captured by the tensor /vectorfin Eq. ( 62). In (b) we illustrate the change in the electronic structure as a consequence of theatomic displacement: If the atoms displace along xin the way shown in the top atomic chain of panel (a), then a variation in the position of thep xorbitals is induced, while the pylevel remains unaltered. Panels (c) and (d) illustrate the change in nondiagonal γsrmatrix elements between two neighboring orbitals when the intermediate atom moves, thus altering the bandwidth as illustrated in (d). The change in thebandwidth depends on the amount by which the atoms are displaced, represented in the cartoon by two different displacement vectors δ/vectorr 1 andδ/vectorr2. dependence of the long-range part with the distortion of the lattice is clearly seen in Eq. ( 55), where the electric dipole that enters in the multipole expansion of the far field potential[Eq. ( 56)] depends linearly with the atomic displacements, as computed with respect to the RAG. As regards the dependenceofγ sr abon the atomic configuration (see Fig. 5), we include it by expanding γsr ab=γRAG,sr ab+/summationdisplay λυ/bracketleftBigg −/vectorfT ab,λυδ/vectorrλυ +/summationdisplay λ/primeυ/primeδ/vectorrT λυ←→gab,λυλ/primeυ/primeδ/vectorrλ/primeυ/prime+···/bracketrightBigg , (62) where δ/vectorrλυ=←→η(/vectorRϒ−/vectorR/Lambda1+/vectorτυ−/vectorτλ)+/vectoruυ−/vectoruλ (63) quantifies the relative displacement of atoms λandυ.I n addition, /vectorfand←→gare the first- and second-rank tensors that characterize the electron-lattice coupling, closely relatedto the concept of vibronic constants [60]. We have checked that including quadratic constants is enough to describe typical changes in the value of γwith the geometry. For example, we have inspected the γparameters associated with the oxygen 2 p-like WFs of SrTiO 3, i.e., with the valence band of the material, and plotted in Fig. 6the three that are most sensitive to structural deformations: Theycorrespond to the diagonal elements of the σandπfunctions centered on the oxygen ions [see Fig. 3(a)] and a π−π 195137-11PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) FIG. 6. Variation of the γmatrix elements of bulk SrTiO 3that are most sensitive to the displacement of a Ti ion. Panels (a)–(c): the atoms at aT i O 2plane are represented by big blue (Ti) and small red (O) spheres. The displacement of the Ti4+is marked with an arrow. The Wannier-like functions related with the γmatrix elements under consideration are plotted: (a) a p-like orbital with lobules pointing perpendicularly to the Ti4+displacement ( π-bonding), (b) a p-like orbital with lobules pointing parallel to the Ti4+displacement ( σ-bonding), and (c) two p-like orbitals centered on different O atoms with lobules pointing perpendicular in one case and parallel in the other to the Ti4+displacement. In (a), and (b), the corresponding matrix element whose evolution is studied is diagonal (i.e., a=b), while in (c) the matrix element is off-diagonal (a/negationslash=b). Panels (d)–(f) show the variation of these matrix elements with respect to the Ti displacement. First-principles results are represented by blue crosses, while the model values [see Eq. ( 62)] are represented by solid blue lines. off-diagonal term. We find that, if we use a quadratic expansion to describe such a structural dependence, the errors are smallerthan 1% over a wide range of distortions, up to 0.3 ˚A. Hence, given the strong changes occurring in the hybridization offerroelectriclike materials like SrTiO 3, we consider that the approximation employed in Eq. ( 62) should be reasonable for most systems. Moreover, in the cases studied so far, we have found that the quadratic constants are typically much smaller than the linearones; further, among the quadratic constants, the diagonalones are clearly dominant. Hence, in Eq. ( 62) we restrict the expansion to two-atom terms, so that ←→g ab,λυλ/primeυ/prime≈←→gab,λυλ/primeυ/primeδλλ/primeδυυ/prime=←→gab,λυ. (64) The physical meaning of /vectorfab,λυis particularly obvious when a=b: it represents the force created by an electron occupying the WF χaover the surrounding atoms [see Figs. 5(a)and5(b)]. Such a parameter is key to quantify phenomena like the Jahn-Teller effect in solids [ 60] or polaron formation [ 61]. Off-diagonal terms in /vectorfdescribe the mixing of two WFs upon an atomic distortion and thus quantify changes incovalency [see Figs. 5(c)and5(d)]. They can be identified with the pseudo Jahn-Teller vibronic constants and are thus relevantto a wide variety of phenomena including ferroelectricity [ 60], spin-crossover [ 62], and spin-phonon coupling [ 63]. Finally, the geometrical dependence of the two-electron parameters, UandI, can be included in our model in a similar way. Nevertheless, since these terms are not explicitlydependent on the potential created by the ions, their valuecan be expected to be less sensitive to changes in the atomicconfiguration. Hence, in this paper, and in analogy to whatis customarily done in model Hamiltonian and DFT +U approaches [ 48], we will neglect such effects keeping in mind that these may be introduced, if necessary, in the future. G. Total energy Replacing the expressions for the one-electron [Eq. ( 21)], and two-electron [Eq. ( 39)] integrals into Eq. ( 5) for the total energy, we get E=E(0)+E(1)+E(2) =E(0)+/summationdisplay abDU abγab +1 2/summationdisplay ab/summationdisplay a/primeb/prime/parenleftbig DU abDU a/primeb/primeUaba/primeb/prime−DI abDI a/primeb/primeIaba/primeb/prime/parenrightbig ,(65) or, equivalently, in terms of the spin-up and spin-down densities, E=E(0)+/summationdisplay ab(D↑ ab+D↓ ab)γab +1 2/summationdisplay ab/summationdisplay a/primeb/prime{(D↑ ab+D↓ ab)(D↑ a/primeb/prime+D↓ a/primeb/prime)Uaba/primeb/prime −(D↑ ab−D↓ ab)(D↑ a/primeb/prime−D↓ a/primeb/prime)Iaba/primeb/prime}. (66) Now we introduce the decomposition of the γ[Eqs. ( 57) and ( 58)] and U[Eqs. ( 59) and ( 60)] parameters into long and short-range parts and gather together all the long-range 195137-12SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) terms to obtain E=E(0)+/summationdisplay abDU abγsr ab +1 2/summationdisplay ab/summationdisplay a/primeb/prime/parenleftbig DU abDU a/primeb/primeUsr aba/primeb/prime−DI abDI a/primeb/primeIaba/primeb/prime/parenrightbig +/summationdisplay aDU aa/parenleftBigg −vFF(/vectorra)+1 2/summationdisplay a/primeDU a/primea/primeUlr aaa/primea/prime/parenrightBigg .(67) Note that for the case of a non-spin-polarized system ( DI ab= 0) the expression for the total energy reduces to E=E(0)+/summationdisplay abDU abγsr ab +1 2/summationdisplay ab/summationdisplay a/primeb/primeDU abDU a/primeb/primeUsr aba/primeb/prime +/summationdisplay aDU aa/parenleftBigg −vFF(/vectorra)+1 2/summationdisplay a/primeDU a/primea/primeUlr aaa/primea/prime/parenrightBigg .(68) H. Self-consistent equations As it is clearly seen in Eq. ( 67), the total energy in our formalism depends on the deformation occupation matrixdefined in Eq. ( 19) and later generalized for the case of spin-polarized systems in Eqs. ( 30) and ( 31). This quantity is directly related with the deformation charge density, i.e.,with the difference between the total charge density and thereference electronic density. It can be computed from thecoefficients of the Bloch wave functions in the basis of Wannierfunctions, which are thus the only variational parameters of themethod. Solving for the ground state amounts to finding a point at which the energy is stationary upon variations in the electronicdensity, n(/vectorr). Following a textbook procedure [ 2–5], we obtain a set of self-consistent conditions analogous to the Kohn-Shamequations [ 1] /summationdisplay bhs ab,/vectorkcs jb/vectork=εs j/vectorkcs ja/vectork, (69) where, as defined above, εs j/vectork,i st h e jth band energy at wave vector /vectorkfor the spin channel s. The corresponding Hamiltonian matrix hs ab,/vectorkis hs ab,/vectork=/summationdisplay /vectorRB−/vectorRAei/vectork·(/vectorRB−/vectorRA)hs ab, (70) where hs abis the real-space Hamiltonian hs ab=γab+/summationdisplay a/primeb/prime/bracketleftbig/parenleftbig Ds a/primeb/prime+D−s a/primeb/prime/parenrightbig Uaba/primeb/prime −/parenleftbig Ds a/primeb/prime−D−s a/primeb/prime/parenrightbig Iaba/primeb/prime/bracketrightbig . (71) Note that this is a mean-field problem fully equivalent to that of the Hartree-Fock approach, and it must be solved self-consistently. The practical procedure for finding the solutionis straightforward: Given an initial guess for the deformationoccupation matrix ( D s ab), we compute the corresponding mean-field Hamiltonian ( hs ab); from the diagonalization of thismatrix we obtain a new deformation occupation matrix, and the procedure is iterated until reaching self-consistency. Notethat electrostatic effects are accounted for by computing thelong-range part of the γandUparameters; this is our scheme’s equivalent to solving Poisson’s equation, as customarily donein DFT and other approaches. Finally, note that in cases inwhich the system does not present any electron excitation, i.e.,whenever the full density is equal to the reference density andwe have D s ab=0, no self-consistent procedure is needed to obtain the solution. I. Forces and stresses Forces and stresses can be computed by direct derivation of the total energy [Eq. ( 65)] with respect to the atomic positions and cell strains. After some algebra, the result for the forces is /vectorFλ=−/vector∇λE=−/vector∇λE(0)−/summationdisplay abDU ab/vector∇λγab, (72) where λdenotes a specific atom in a certain cell; here we assume that electron-lattice couplings are restricted to the one-electron terms. The derivative of E (0)can be computed directly and exactly from the force-field on which our model is based. The deformation occupation matrix DU abdepends on the eigenvector coefficients and occupations. However, its deriva-tive with respect to the atomic displacement is not required,since the energy is stationary with respect to these coefficientsand occupations on the Born-Oppenheimer surface, and theHellman-Feynman theorem guarantees that their first-ordervariation will not modify the total energy, and therefore willnot affect the forces. Moreover, due to the orthogonality of thebasis set used, no orthogonality forces need to be included,as it is the case when using a basis of nonorthogonal atomicorbitals (see Appendix A of Ref. [ 31]). It is interesting to further inspect the similarity between the second term in our forces and the Hellmann-Feynmanresult [ 3], /vectorF λ=−/vector∇Enn−/summationdisplay j/vectorkoj/vectork/angbracketleftψj/vectork|/vector∇λˆh0|ψj/vectork/angbracketright, (73) as (via a Fourier transform) /vector∇λγabis analogous to /angbracketleftψj/vectork|/vector∇λˆh0|ψj/vectork/angbracketright, andDU abplays the role of the occupations oj/vectork. This connection should be considered with caution, though, asour forces have a dominant contribution from the RED state,which is also included in the Hellmann-Feynman expression.It is also interesting to note that, if we included the dependenceofU(andI) on the nuclear positions, we would have a Pulay term in Eq. ( 72)[64], reflecting the change of the WF basis set with the atomic displacements. Now we calculate the stress tensor in an analogous way. We adopt the standard definition [ 3] S αβ=−1 V∂/primeE ∂/primeηαβ, (74) where Vis the volume of the real-space cell and ∂/primedenotes derivative keeping the fractional coordinates of the atoms inthe system constant. We notice that there are only three termsin the energy that depend explicitly on the strain tensor←→η, namely, the RED energy E (0), the short-range one-electron 195137-13PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) term,γsr ab, and the electrostatic energy. Thus, we have Sαβ=−1 V/bracketleftbigg∂/primeE(0) ∂/primeηαβ+/summationdisplay abDU ab∂/primeγsr ab ∂/primeηαβ+∂/primeEelec ∂/primeηαβ/bracketrightbigg , (75) where Eeleccorresponds with the electrostatic energy as written in the fourth contribution to the total energy in Eq. ( 67). As in the case of the forces, the E(0)derivative is computed from the force field that describes the RED state. Similarly, thecalculation of the last, electrostatic term can be achieved viaEwald summation techniques (see, e.g., Ref. [ 65]). The only term that requires further manipulation is the derivative of γ sr ab, Eq. ( 62), with respect to the strain, which yields ∂/primeγsr ab ∂/primeηαβ=/summationdisplay λυ/bracketleftBigg −/vectorfT ab,λυ∂/prime(δ/vectorrλυ) ∂/primeηαβ +/summationdisplay λ/primeυ/prime∂/prime(δ/vectorrT λυ) ∂/primeηαβ(←→g)ab,λυλ/primeυ/primeδ/vectorrλ/primeυ/prime +/summationdisplay λ/primeυ/primeδ/vectorrT λυ(←→g)ab,λυλ/primeυ/prime∂/prime(δ/vectorrλ/primeυ/prime) ∂/primeηαβ/bracketrightBigg . (76) As in the case of the forces, the similarity between this result and the Hellmann-Feynman expression is apparent. To end this section, let us stress that only excited elec- trons/holes, which render Dab/negationslash=0, create forces and stresses not included in the underlying force-field described by E(0). In fact, in the typical case, the dominant contribution to bothforces and stresses will come from the derivative of E (0), with corrections that are linear in the difference occupation matrix. J. Practical considerations So far we have introduced a method for the simulation of materials at a large scale. We have presented the basic physicalingredients (reference atomic geometry, reference electronicdensity, deformation density, etc.), that allow us to approximatethe DFT total energy, forces and stresses. In this section we discuss some practicalities involved in the implementation of this method in a computer code to performactual calculations. Of course, different implementations arein principle possible; here we briefly describe some detailspertaining to our specific choices, which should be illustrativeof the technical issues that need to be tackled. 1. Definition of the RED The formulation above is written in terms of differences between the actual and reference states of the system in acompletely general way. However, from a practical point ofview, an appropriate choice of the RED, n 0(/vectorr), is a necessary first step towards an efficient implementation of our method. The most important ingredient to define n0is the reference occupation matrix that, following Eq. ( 17), amounts to d(0) ab=/summationdisplay j/vectorko(0) j/vectork/bracketleftbig c(0) ja/vectork/bracketrightbig∗c(0) jb/vectorkei/vectork(/vectorRB−/vectorRA), (77) where o(0) j/vectorkandc(0) ja/vectorkcharacterize the RED. While it would be possible to use the d(0) abresult computed from first principlesto perform second-principles simulations, in the following we shall simplify this expression in order to obtain a moreconvenient form. Note that the reference occupation matrix satisfies d(0) ab=0 for aand bbelonging to different band manifolds. (By definition, if aand bbelong to different bands, they cannot appear simultaneously in the expansion of a particular Bloch state [Eq. ( 14)], and the corresponding d(0) ab[Eq. ( 77)] will vanish.) It is thus possible to rewrite Eq. ( 77) and split the sum over states in two, one over manifolds Jand a second one over bands within a manifold. After having established this property, we impose that all the bands jthat belong to the same manifold Jhave the same occupation in the RED o(0) j/vectork=oJω/vectork=oJ Nk, (78) where ω/vectorkis the weight of each /vectork-point in the BZ. As we assume an homogeneous sampling in reciprocal space, ω/vectork= N−1 k, where Nkis the total number of points in our BZ mesh. Thus, for example, in a diamagnetic insulator [see Fig. 3(a)] (where the valence and conduction band always belong todifferent manifolds) we would choose the occupation for thereference states so that all the valence bands are fully occupiedwhile all conduction bands are completely empty. In this waythe reference electronic density for a diamagnetic insulator issimply the ground state density. For metals [Fig. 3(b)], and magnetic insulators [Fig. 3(c)], where a disentanglement procedure [ 38] has to be carried out to separate the desired bands from others with which theyare hybridized in a given energy window, the choice is notso simple. In such cases we distribute all the electrons of theentangled bands equally among the bands in the manifold. Forexample, in the case of metallic copper [Fig. 3(b)], which has an electronic configuration 3 d 104s1where the five 3 dfunctions cross with the 4 s-like band, we would distribute the eleven electrons over the six bands taking oJ=11/6. On the other hand, for NiO [Fig. 3(c)]s o m eN i ( 3 d) bands are occupied and entangled with the O(2 p) ones; at the same time, empty eg-like orbitals are part of the conduction band and entangled with other levels there. Here, we choose to disentangle thebands with strong Ni(3 d) and O(2 p) character from the other bands. Further, we assign the occupation by distributing thecorresponding electrons—eight 3 delectrons of Ni 2+and the six 2pelectrons of O2−—over the corresponding bands—five 3dbands and three 2 pbands—yielding oJ=14/8. Taking into account the spin polarization, the occupation per spinchannel is just o J=7/8. Using Eq. ( 78) to rewrite Eq. ( 77), and taking into account the relationship between the coefficients of RED Bloch statesand the unitary transformations between Bloch and Wannierrepresentations, we have [ 36] d (0) ab=oJδabδ/vectorRA/vectorRB=oJδab, (79) where we have used the properties of the unitary matrices. From this expression we see that oJis simply the occupation of the WF χain the reference state, o(0) a. Finally, inserting 195137-14SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) Eq. ( 79) into Eq. ( 18), we arrive to the conclusion that [ 36] n0(/vectorr)=/summationdisplay aoJ|χa(/vectorr)|2=/summationdisplay ao(0) a|χa(/vectorr)|2, (80) where we have used the fact that o(0) a=oJfor all WFs in the manifold. We would like to stress that this approach simply allows for a more efficient computational method since we can stillretrieve the full electron distribution in space by substitutingthe first-principles WFs into Eqs. ( 16) and ( 18). 2. Deformation electron density From the definition of the charge density in terms of the density matrix, Eq. ( 16), and the orthogonality of the Wannier basis functions, it trivially follows that the total number ofelectrons in the system is the trace of the density matrix, N=/integraldisplay n(/vectorr)d 3r=/summationdisplay adaa. (81) Therefore, the trace of the deformation matrix gives the number of extra electrons or holes that dope the system. From the very definition of the deformation matrix, we can deduce that if its diagonal element, Daa=daa−d(0) aa, (82) is negative (positive), that means that we are creating holes (inserting electrons) on that particular state, as illustrated inFig. 1. The fact that most of these electron/hole excitations take place around the Fermi energy has a very importantconsequence with regards to the efficiency of the method. Inorder to calculate D aa, and the total energy [see Eq. ( 67)], we do not need to obtain all the eigenvalues of the one-electron Hamiltonian, but just those around the Fermi energy.This opens up the possibility to use efficient diagonalizationtechniques that allow a fast calculation of a few relevant eigen-values, e.g., Lanczos. This approach allows us to speed up thecalculations in a very significant manner. (The diagonalizationof the full Hamiltonian matrix is one of the main computationalbottlenecks in electronic structure methods.) Along these lines,the possibility of obtaining linear scaling within our methodwill be discussed in a future publication. IV . PARAMETER CALCULATION The method presented above allows for the simulation of very large systems under operation conditions assuming thata few parameters describing one-electron and two-electroninteractions, as well as the electron-lattice couplings, areknown beforehand. For the sake of preserving predictingpower, it is important to compute those parameters from firstprinciples. All the electronic parameters of our models have well- defined expressions [see Eq. ( 23)f o rγ ab,E q .( 40)f o rUaba/primeb/prime, and Eq. ( 41)f o rIaba/primeb/prime], whose computation requires only the knowledge of the Wannier functions, the one-electronHamiltonian, and the operators involved in the double inte-grals, all of them defined for the RED. Since the chosen basisfunctions are localized in space, the required calculations couldbe performed on small supercells. Such a direct approach to obtain the model parameters is thus, in principle, feasible. Note that there has been significant work to calculate related integrals from first principles, as can be found, e.g.,in Refs. [ 42,48,49,51,66–73]. Yet, we feel that most of these approaches are too restrictive for the more general task thatwe pursue in this paper. For instance, the focus in the previousreferences is placed on strongly correlated electrons in a singlecenter, while we are also interested in multicenter integrals. A significant effort would thus be required to implement the calculation of the more complex interactions, including allthe potentially relevant ones, and developing tools to deriveminimal models that retain only the dominant parametersand capture the main physical effects. Note that, in a typicalsystem, the number of potentially relevant integrals will bevery large. In fact, the presence of four-index integrals likeU aba/primeb/primeandIaba/primeb/primeis the reason why Hartree-Fock schemes scale much worse than DFT methods with respect to thenumber of basis functions in the calculation [ ∼O(N 5)v s ∼O(N3), respectively]. Hence, at the present stage we have not attempted a direct first-principles calculation of theparameters, which is a challenge that remains for the future.Instead, we have devised a practical scheme to fitour models to relevant first-principles data. A. Parameter fitting Our procedure comprises several steps. 1. Training set First, we identify a training set (TS) of representative atomic and electronic configurations from which the rel-evant model parameters will be identified and computed.For example, the training set for a magnetic system shouldcontain simulations for several spin arrangements, so that themechanisms responsible for the magnetic couplings can becaptured. Additionally, if we want to study a system whosebands are very sensitive to the atomic structure, the trainingset should contain calculations for different geometries so thatthis effect is captured. Alternatively, if we want to describehow doping affects the physical properties of a material,then different DFT simulations on charged systems shouldbe carried out [ 74], etc. Let us note that it is typically possible to restrict the TS to atomic and electronic configurations compatible with smallsimulation boxes. This translates into (and is consistent with)the fact that, when expressed in a basis of localized WFs,the nonelectrostatic interactions in most materials are shortranged. We will use N TSto denote the total number of TS elements, noting that we will run a single-point first-principlescalculation for each of them. Further, N RAG is the number of TS configurations that correspond to the reference atomicgeometry. 2. Filtering the training set Leths ab(i) be the Hamiltonian of the ith TS configuration, in matrix form and as obtained from a first-principles (typically 195137-15PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) FIG. 7. Representation of the bands of SrTiO 3(top row) and NiO (middle and bottom rows, corresponding to the majority and minority spin channels, respectively) for various values of the Hamiltonian cutoff, δεh. Black lines represent the bands as obtained from first principles, while the results from the second-principles model are shown in green. On top of each plot we indicate the corresponding number of hs abmatrix elements per primitive cell that were included. DFT) calculation. We denote the whole collection of one- electron Hamiltonians in the training set by {hs ab(i)}. These Hamiltonians are expressed in a basis of localized WFs. Formally, they can be obtained by inverting Eq. ( 70) (see Sec. VI A of Ref. [ 45]), so that hs ab=(2π)3 V/integraldisplay BZd3k⎡ ⎣/summationdisplay j/bracketleftbig Ts(/vectork) ja/bracketrightbig⋆εs j/vectorkTs(/vectork) jb⎤ ⎦ei(/vectorRA−/vectorRB)/vectork, (83) where the Tmatrices are unitary transformations that convert the first-principles eigenstates into Bloch-like waves associ-ated to specific (localized) WFs. These transformations canbe obtained by employing codes like WANNIER 90 [66], which implements a particular localization scheme, i.e., a particularway to compute optimum Tmatrices [ 42,45]. Once the h s ab(i) Hamiltonians are known, we can identify the pairs of WFs with a large enough interaction and whichneed to be retained in the fitting procedure. In practice, weintroduce a cutoff energy δε hsuch that /vextendsingle/vextendsinglehs ab(i)/vextendsingle/vextendsingle>δ ε h,for at least one iin the TS , (84) defines the Hamiltonian matrix elements to be retained. (Diagonal elements, hs aa, are always considered independently of their value.) This condition allows us to identify the WFpairs ( a,b) to be included in the fitting procedure, regardless of the geometry or spin arrangement.In Fig. 7we compare the full first-principles bands for SrTiO 3and NiO with those obtained from models corre- sponding to different energy cutoffs. For all the δεhvalues considered, we also indicate the number of parameters in theresulting models. This allows us to estimate the size of themodel (and associated computational cost) needed to achievean acceptable description of the band structure. 3. Identifying most relevant model interactions Our models, even though we truncate them at second order of the expansion in Eq. ( 5), contain a daunting number of electron-electron interaction parameters. Constructing an ac-tual model usually involves further approximations regardingthe spatial range of the interactions, the maximum numberof different bodies (WFs) involved, etc. Hence, we need aprocedure to identify the simplest models that can reproducethe first-principles TS data with an accuracy that is sufficientfor our purposes. The scheme we have implemented is based on a very simple logic: We start from a certain complete model that may contain, in principle, all possible one-electron, two-electron,and electron-lattice parameters. We can then fit such a modelto reproduce the one-electron Hamiltonians {h s ab(i)}of our TS within a certain accuracy. Typically, by doing so, and bysystematically exploring different combinations of parametersin the model, we can identify the simplest interactions (i.e.,those that are shortest in range, involve fewest WFs, etc.) 195137-16SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) sufficient to achieve the desired level of accuracy; in other words, in this way we can identify noncritical couplings thatjust render the fitting problem underdetermined, do not im-prove the accuracy of the model, and can thus be disregarded.Naturally, this split between relevant and irrelevant couplingsis strongly dependent on the choice of the training set, whichshould be complete enough to capture the physical effects ofinterest. To better understand how the scheme works, consider the one-electron integrals γ abin the case of a nonmagnetic material like SrTiO 3. These parameters will be the only ones entering the description of the band structure of the RAG in the REDstate. Hence, we can fit them directly by requiring our modelto reproduce the Hamiltonian h abof this particular, reference state with a certain accuracy. More generally, the one-electron Hamiltonian correspond- ing to a TS configuration will reflect electronic excitationsdeparting from the RED state. More precisely, we can recallEq. ( 71) to write h s ab(i)=γab+/summationdisplay a/primeb/prime/bracketleftbig/parenleftbig Ds a/primeb/prime(i)+D−s a/primeb/prime(i)/parenrightbig Uaba/primeb/prime −/parenleftbig Ds a/primeb/prime(i)−D−s a/primeb/prime(i)/parenrightbig Iaba/primeb/prime/bracketrightbig , (85) where we restrict ourselves to TS configurations at the RAG, so that no electron-lattice term appears in this equation. It isconvenient to isolate the Ucontribution by defining h U ab(i)=h↑ ab(i)+h↓ ab(i) 2=γab+/summationdisplay a/primeb/primeDU a/primeb/prime(i)Uaba/primeb/prime,(86) and its average over all the RAG configurations in the training set, ¯hU ab=1 NRAG/summationdisplay ihU ab(i)=γab+/summationdisplay a/primeb/prime¯DU a/primeb/primeUaba/primeb/prime.(87) Analogously, the antisymmetrization of the Hamiltonian ma- trix elements with respect to the spin yields hI ab(i)=h↑ ab(i)−h↓ ab(i) 2=/summationdisplay a/primeb/primeDI a/primeb/prime(i)Iaba/primeb/prime. (88) We expect that the most important Uaba/primeb/primeandIaba/primeb/prime parameters will be, respectively, those involving WF pairs whose corresponding hU ab(i) and hI ab(i) are most strongly dependent on the TS state. Hence, we introduce the two-electron cutoff energy, δε ee, and retain only the ( a,b) pairs that satisfy, for at least one TS configuration, at least one ofthe following conditions: /vextendsingle/vextendsingleh U ab(i)−¯hU ab/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay a/primeb/prime/bracketleftbig DU a/primeb/prime(i)−¯DU a/primeb/prime/bracketrightbig Uaba/primeb/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δ ε ee,(89) or /vextendsingle/vextendsinglehI ab(i)/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay a/primeb/primeDI a/primeb/prime(i)Iaba/primeb/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δ ε ee. (90) Note that we gauge the hU abmatrix elements with respect to the ¯hU abaverage values so that the corresponding cutoff condition does not depend on the one-electron couplings γ.Once we have selected all the {(a,b)}pairs that fulfill such criteria, we can build the list of potentially relevantUandIconstants to be considered in the fit. Note that the number of free parameters is usually reduced by thefact that the U aba/primeb/primeandIaba/primeb/primeintegrals are invariant upon permutations of the ( a,b,a/prime,b/prime) indexes. In some cases, and in spite of the reduction of parameters due to symmetry, thelist of relevant interactions is excessively long and needs tobe further trimmed to successfully carry out the fitting. Insuch situations we introduce a third cutoff, δD, that operates over the difference occupation matrix to select the interactionsassociated to important changes of the electron density. Whendoing so we only accept Uconstants for which at least one pair of the associated indexes fulfills /vextendsingle/vextendsingleD U a/primeb/prime(i)−¯DU a/primeb/prime/vextendsingle/vextendsingle>δ D (91) and the corresponding expression for I /vextendsingle/vextendsingleDI a/primeb/prime(i)/vextendsingle/vextendsingle>δ D . (92) Let us also note that the most relevant γabparameters are trivially identified when we filter the TS one-electronHamiltonians as described above. 4. Fitting the RAG model Once our list of relevant γ,U, andIparameters is complete, we fit them to reproduce the {hs ab}matrix elements above the δεhenergy cutoff introduced previously. We have found it convenient to perform the fit in several steps, so that differenttypes of parameters are computed separately. More precisely,we first fit the Uparameters by requesting that our model reproduces the h U ab(i)−¯hU abmatrices [Eqs. ( 86) and ( 87)]. Analogously, we obtain the Iconstants by fitting to the hI ab(i) matrices [Eq. ( 88)]. Importantly, both of these fits are independent of the one-electron integrals, and have typicallyyielded well-posed, overdetermined systems of equations inthe cases we have so far considered. Finally, we obtain the γ parameters from the fitted U’s directly from Eq. ( 86). Direct comparison of the modeled bands with those obtained fromthe full first-principles {h s ab(i)}set provides an estimate of the goodness of the model (see the example in Sec. VI B and, particularly, Fig. 9). Note that, alternatively, one could try a direct fit of all the γ,U, andIparameters to the real-space Hamiltonians, using Eq. ( 71). However, we typically find that this strategy leads to nearly-singular problems in which very different solutionslead to comparably good results. In the general case, such adifficulty may be mitigated by extending the TS. However,here we adopted the simple and practical procedure describedabove, which permits a numerically stable method that yieldsaccurate and physically sound models. To end with this section we would like to stress that the γ ab constants obtained with this procedure contain both the short- and long-range contributions described above [Eq. ( 58)]. In order to isolate γsr ab, we simply subtract the corresponding electrostatic contribution [Eq. ( 57)] from the determined, full γabvalue. In order to calculate the electrostatic contribution [Eqs. ( 55)–(56)] we need first-principles results for the Born charge tensor,←→Z∗ λ, and the high-frequency dielectric tensor, 195137-17PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) ←→/epsilon1∞, that can routinely be obtained for systems where the RED is insulating [ 58]. 5. Relevant electron-lattice interactions As above, we assume that the deviations from the RAG only affect the one-electron integrals γ, and not the UandI parameters. We further assume that such a dependence on theatomic structure is given by the linear and quadratic electron- lattice constants /vectorfand←→gintroduced in Eq. ( 62). The selection of the most important electron-lattice cou- plings is performed by observing how much a particular matrixelement h s abchanges with a particular distortion of the lattice with respect to the RAG. To quantify this change, we need tocompare pairs of configurations iandi /primethat correspond to the same electronic state (e.g., to the same spin arrangement, to thesame amount of electron/hole doping, etc.) butdiffer in their atomic structure. More precisely, one of the configurationsmust correspond to the RAG state ( i), while the other one (i /prime) is characterized by a distortion given by {/vectoruλ(i/prime)}.F o r simplicity, here we will restrict to distortions involving onlyone displacement component of one atom, so that we only haveone specific u λα(i/prime)/negationslash=0, where αlabels the spatial direction. We then consider that a particular atom λparticipates in the electron lattice affecting the hs abelement if 1 |uλα|/vextendsingle/vextendsinglehs ab(i/prime)−hs ab(i)/vextendsingle/vextendsingle>δ f e−l, (93) where δfe−lis a new cutoff. Note that, for a large enough distortion |uλα|, this condition pertains to both the linear ( /vectorf) and quadratic (←→g) electron-lattice interactions. Yet, since we activate a single atomic displacement at a time, in the caseof←→gwe are only probing the diagonal elements. Restricting ourselves to the diagonal elements of←→gis justified by the observation that, in the systems we have so far studied, thoseare the only significant ones. At any rate, the scheme canbe trivially extended to check a possible contribution of off-diagonal terms. V . IMPLEMENTATION OF THE ALGORITHM: THE SCALE -UPCODE We have implemented this new method in the SCALE -UP (Second-principles Computational Approach for Lattice and Electrons) package, written in Fortran 90 and parallelized us-ing Message Passing Interface (MPI). Presently, this code canperform single-point calculations, geometry optimizations,and Born-Oppenheimer molecular dynamics using either fulldiagonalization or the Lanczos scheme mentioned above. The energy of the reference state, E (0), is obtained from model potentials like those introduced by Wojdeł et al. [30], which are interfaced with SCALE -UP. We have also developed an auxiliary toolbox ( MODELMAKER ) for the calculation of all the parameters defining E(1)andE(2), using as input DFT results for one-electron Hamiltonians in the format of WANNIER 90 [66]. As shown in Sec. VI, these implementations can be used to create models that match the accuracy of theDFT calculations at an enormously reduced computationalcost, opening the door to large-scale simulations (up to tensof thousands of atoms) of systems with a complex electronic structure, using modest computational resources. The input to the code is based on the flexible data format (fdf) library used in SIESTA [31] and contains several python- based tools to plot bands, density of states, geometries andother properties. VI. EXAMPLES OF APPLICATION In order to illustrate the method, we will discuss its application to two nontrivial systems with interactions of verydifferent origin. The first example consists in the calculationof the energy of a Mott-Hubbard insulator, NiO, for differentmagnetic phases. Our goal here is to show that the methodcan be used to deal accurately with complicated electronicstructures including phenomena like magnetism in transitionmetal oxides. In this example we will also show how themethod can tackle rather large systems (2000 atoms) thatare at the limit of what can be done with first-principlesmethods today, reducing the computational burden by ordersof magnitude. The second application involves the two-dimensional elec- tron gas (2DEG) that appears at the interface between bandinsulators LaAlO 3and SrTiO 3[75]. We will not discuss here the origin of the 2DEG, which has been treated in great detailin the bibliography [ 28,75–77]. Rather, we will check whether our approach can predict the redistribution of the conductionelectrons at the LaAlO 3/SrTiO 3interface, and the accompany- ing lattice distortion, as obtained from first principles. Thus,this example will showcase the treatment of electron-latticecouplings and electrostatics within our approach. A. Details of the first-principles simulations We construct our models following the recipes described in Sec. IV, and the first-principles data are obtained from small- scale calculations with the V ASP package [ 78–80]. The local density approximation (LDA) to density-functional theory isused to create the TS data for SrTiO 3. The calculations for NiO are also based on the LDA, but in this case an extra Hubbard- U term is included to account for the strong electron correla-tions [ 81], as will be discussed below. We employ the projector- augmented wave (PAW) scheme [ 80] to treat the atomic cores, solving explicitly for the following electrons: Ni’s 3 s,3p,3d, and 4s;O ’ s2 sand 2p;S r ’ s3 s,3p, and 4 s; and Ti’s 3 s,3p, 3d, and 4 s. The electronic wave functions are described with a plane-wave basis truncated at 300 eV for NiO and at 400 eVfor SrTiO 3. The integrals in reciprocal space are carried out using/Gamma1-centered 4 ×4×4k-point meshes in both cases. B. NiO magnetic couplings Transition metal oxides are very interesting as they present optical, magnetic, and structural properties that are, veryoften, tightly coupled with each other. This fact, togetherwith the large variety of functional properties that they candisplay, makes them a big focus of attention in both basicand applied materials science. From a theoretical point ofview their study is complicated, mostly because of the strongcorrelations associated to the electrons in the compact dshell (especially, those of first-row 3 dtransition metal ions) and 195137-18SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) the frequent presence of many competing magnetic phases. Naively, one may expect most of these oxides to be metallic dueto their open-shell nature while, in fact, many are insulators.This problem strongly affects computational methods; in fact,most common approaches, like DFT with local or semilocalexchange-correlation functionals, often fail to correctly repro-duce the magnetic or insulating properties of these compounds.Simulations at this level of theory yield too diffuse states withunderestimated interactions. The key to simulate successfullythese materials lies in the way electron-electron interactionsare handled. A panoply of methods have been developed todeal with this issue from first principles, ranging from theinclusion of a Hubbard-like term in the Hamiltonian in theso-called DFT +Umethods, to more sophisticated schemes with dynamically screened interactions, such as the GWapproximations [ 50,51]. For this first application of our scheme, we have chosen a simple transition metal oxide, NiO, a staple example of many new electronic structure simulation methods. Our goal is toshow how our second-principles scheme can handily be usedto compute the properties of strongly correlated materials,based on parameters obtained from first-principles LDA +U simulations. In particular we will study the band structure andmagnetism of this archetypical binary oxide. 1. Model parameters The geometric structure of NiO shows some subtle and not fully understood distortions associated to its mag-netism [ 82,83], but this issue is beyond the scope of the present paper. In order to keep the model as simple and illustrative aspossible, we neglect the lattice degrees of freedom in this case.As regards the spin order, neutron diffraction experiments [ 84] show that the ground state corresponds to the so-called AF2phase, where planes of spin-up and spin-down polarizednickels alternate along the /angbracketleft111/angbracketrightdirection of the conventional cell [see Fig. 8(c)]. Further experiments [ 85,86] evidence that the AF2 to paramagnetic transition is of second order andoccurs at a N `eel temperature T N=524 K. The simulations of NiO are carried out in the experimental rocksalt cell, with a lattice constant of 4.17 ˚A[87]. This cell is compatible with several spin arrangements ranging from fullyferromagnetic [FM; Fig. 8(a)] to various antiferromagnetic (AF) ones [Figs. 8(b) and8(c)]. Our TS includes the ground state spin arrangement (AF2) as well as the ferromagneticone, which we choose because it represents a relevant limitcase for spin-spin interactions. We use the LDA +Uapproach introduced by Dudarev et al. ,[81] with an effective Uvalue of 7 eV , applied only on the 3 dorbitals of Ni. These calculations indicate that the AF2 solution is more stable than the FM oneby 89 meV per formula unit (f.u.). Ligand-field theory predicts the magnetism in this lattice to be the result of the half-filled e gshell of the octahedrally coordinated Ni2+ion [see Fig. 8(d)]. Thus, we expect the levels around the Fermi energy to have this character. Aftercalculating the electronic structure from first principles withinthe LDA +U, we find that the top valence and bottom conduction bands are composed of several strongly entangledstates, as shown in Fig. 3(c). Thus, we project our WFs seeking to disentangle orbitals participating in the valence FIG. 8. Schematic cartoon of different magnetic configurations of bulk NiO in the conventional cell of its rocksalt structure: (a)ferromagnetic phase (FM), (b) antiferromagnetic phase with planes of spin-up (red arrows) and spin-down (blue arrows) polarized nickels alternating along the [001] direction (AF1), and (c) antiferromagneticphase with planes of spin-up and spin-down polarized nickels alternating along the [111] direction (AF2). (d) Scheme of the d levels associated to an isolated NiO 10− 6complex. band [Ni( t2g), Ni(eg), and O( p)] from others in the conduction band; to do this we use the tools provided within the WANNIER90 package. A graphical representation of the resulting orbitals[Fig. 3(c)] clearly shows that we are able to isolate bands with the expected chemical character: The isosurfaces of themaximally-localized WFs (MLWFs) at the right hand side ofFig.3(c)clearly resemble the shape of the O( p), Ni(d xy), and Ni(d3z2−r2) orbitals for the valence band, and of the Ni( d3z2−r2) orbital for the bottom of the conduction band. Given the strongentanglement of these bands, we use a reference occupationfor the construction of our model that is obtained by populatingequally all of them. As discussed in Sec. I I IJ1 , this amounts to assuming o J=7/8=0.875 electrons per band and spin channel. At this point we start with the analysis of the Hamiltonian as described in Sec. IV,u s i n gt h e {hs ab(i)}set obtained after the disentanglement procedure. First, we seek to choose areasonable value of δε hthat allows us to describe accurately the system’s bands without including an excessive numberofγ abterms. As can be seen in Fig. 7,δεh=0.05 eV is a reasonable choice; this involves the use of 71 γterms per f.u. In order to decide the values of δεeeandδDthat will determine the UandIparameters considered in the fit, we first examine the occupation difference of each of the WFs, Daa. Let us recall that the diagonal elements of the deformationoccupation matrices are defined as the difference between thereduced density matrix computed at the LDA +Ulevel for the corresponding configuration in the TS and the reference one. Inthe FM phase, the bands with e gcharacter for the majority spin channel are expected to be fully occupied ( daa=1), while they should be empty for the minority spin ( daa=0). Therefore, for WFs with egcharacter, we expect to have Daa=0.125 for the majority spins and Daa=−0.875 for the minority spins, for both Ni atoms. For the antiferromagnetic configuration, asthe spin in one of the Ni atoms is reversed, we expect the same 195137-19PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) TABLE I. The difference occupation of the Wannier functions (Daa) in the training set used for NiO. The results are presented for the ferromagnetic (FM) and AF2 antiferromagnetic states, and for the majority (Major) and minority (Minor) spin channels. State Spin Ni 1(eg)N i 1(t2g)O 1(p)N i 2(eg)N i 2(t2g)O 2(p) FM Major 0.125 0.125 0.125 0.125 0.125 0.125 FM Minor −0.756 0.124 0.047 −0.756 0.124 0.047 AF2 Major 0.122 0.125 0.083 −0.742 0.125 0.083 AF2 Minor −0.742 0.125 0.083 0.123 0.125 0.083 behavior for Daaas in the FM solution for one of the nickels, while the character of the majority and minority spins mustbe exchanged for the second Ni. In both cases (FM and AF2),thet 2gbands are fully occupied, so that the corresponding Daashould be 0.125. These tendencies are well reproduced in our TS configurations as can be seen in Table I, where the occupations are obtained using the {hs ab(i)}set obtained after applying δεh=0.05 eV filter. The differences with respect to the ideal ionic values are due to the chemical bonding betweenNi and O. If the results for the FM and AF2 configurations are compared, we can see that the only significant change pertainsto the occupation of the e g-like orbitals of the Ni ion whose spin is flipped: The majority (0.125) and minority ( −0.756) values of the difference occupation are basically exchanged aswe move from the FM to the AF2 calculation, as expectedfrom the localization of the magnetic moment over theseorbitals in Ni 2+ions (see Fig. 8). This is an indication that, in order to capture the magnetic interactions in this system,only electron-electron interactions involving e g-type WFs are necessary. Moreover, we can observe how, on one hand, theaverage orbital occupations vary very little (essentially by0.005, 0.001, and 0.003 electrons for the Ni( e g), Ni(t2g), and O(p)-like WF, respectively) between spin configurations. This fact translates into a similar value of DU[Eq. ( 37)]. As can be seen in Eq. ( 67), ifDUis the same for the different configurations of the TS, its contribution to the total energy isconstant, i.e., it does not play any role in the calculations ofthe relevant energy differences. On the other hand, the spin-up/spin-down differences of occupation are strongly changingbetween the FM and AF2 configurations. This indicates thatonly Stoner-type ( I) interactions are relevant to describe the relative stability of the magnetic phases in the training set.As can be seen in Table II, by playing with δε eeandδDit is straightforward to confirm that the eg−eginteractions drive magnetism in this system, while Ni( t2g)-like and O( p)-like energy levels have a secondary role. Nevertheless, includingthe latter interactions is necessary to accurately describe thebands. In Fig. 9we show the second-principles computed bands for two different set of parameters: (i) the first one, obtained aftera filtering the electron-electron interactions with a thresholdofδε ee=1.10 eV , was selected to include only couplings between Ni( eg)-like WFs. The results obtained with this set of parameters are labeled SP-LDAU-Ni( eg); (ii) the second one, obtained with a threshold of δεee=0.20 eV , corresponds to a case in which interactions between Ni( eg)-like and Ni( t2g)-likeTABLE II. Magnetic coupling constants of NiO obtained from various experiments and first-principles (upper part of the table) and second-principles calculations (lower part of the table). The latter have been modeled after the LDA +Ucalculations (highlighted) and, as can be seen, converge towards the results obtained with this method when reducing δεee. HSE stands for the hybrid exchange and correla- tion functional proposed by Heyd-Scuseria-Ernzerhof [ 90], PSIC for pseudo self-interaction-correction, ASIC for atomic self-interaction- correction, and GGA for generalized gradient approximation. Method J1(meV) J2(meV) neutron [ 88] 1.4 −19.0 neutron [ 91] −1.4 −17.3 HSE [ 89] 2.3 −21.0 PSIC [ 89] 3.3 −24.7 ASIC [ 89] 5.2 −45.0 GGA+U[92] 1.7 −19.1 LDA+U 2.6 −17.5 SP-LDAU-Ni( eg) −0.2 −19.1 SP-LDAU-Ni(3 d) −0.0 −19.1 SP-LDAU-Ni +O 3.3 −17.6 WFs at the same atom, as well as between Ni( eg)-like and nearest-neighboring O( p)-like WFs, are also included. The results are labeled SP-LDAU-Ni +O. As can be seen, the bands for the FM state are better reproduced in the secondcase, because of the correction of the diagonal spin-up/spin-down Ni( t 2g) and O( p) energies. These diagonal Hamiltonian matrix elements, which determine the center of mass of thecorresponding bands, vary with the Ni( e g) occupation as expressed in Eq. ( 71). Indeed, we can estimate the maximum error for the hs abterms as δhs ab=max/vextendsingle/vextendsinglehs ab(i)−hs ab/vextendsingle/vextendsingle, (94) where hs ab(i) is a matrix element directly obtained from the first-principles TS and hs abis computed from Eq. ( 71) for a given set of parameters. This maximum error reducesfrom 0.651 eV in the SP-LDAU-Ni( e g) case to 0.132 eV for SP-LDAU-Ni +O. We also considered a third, intermediate case with δεee=0.5 eV , where Ni( eg)-Ni(t2g) interactions are included but those with oxygen orbitals are neglected[SP-LDA-Ni(3 d) in Table II]. The associated maximum error is 0.29 eV for such a choice. It is interesting to note that our TS does not contain enough information to fit reliably all the Ni( e g)-Ni(eg) interactions compatible with our filters. More precisely, we find thatthere are only two relevant I ab,a/primeb/primeconstants: one related to the self-energy of the egstates ( Iaa,aa, where ais aeg-like basis function), and a second one quantifying the interactionbetween the two e gstates in the same atom (essentially, the exchange interaction known as Hund’s coupling). It isclear that our TS is not suitable to distinguish between suchinteractions. In both the FM and AF2 phases, the Ni 2+ions display a S=1 spin configuration; yet, the interplay between self-interaction and Hund coupling only appears when tryingto differentiate between the high- ( S=1) and low- ( S=0) spin intra-atomic states. 195137-20SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) −10−50510 −10−50510 −10−50510 −10−50510 −10−50510 −10−50510 −10−50510−10−50510(a) δεh=0.05 eV, δεee=1.10 eV δD=0.10 (b) δεh=0.05 eV, δεee=0.20 eV δD=0.10Ferromagnetic Antiferromagnetic Ferromagnetic AntiferromagneticEnergy (eV) Energy (eV)Energy (eV) Energy (eV) FIG. 9. Band structure of NiO as obtained for two different sets of cutoff parameters. Solid black lines represent first-principles bands obtained from LDA +Ucalculations, while solid green lines show the bands as obtained after filtering of hs abwith a cutoff of δεh=0.05 eV [see Sec. IV]. Dashed red lines represent the bands as obtained after a second filtering with (a) δεee=1.10 eV , δD=0.10 and (b) δεee=0.20 eV ,δD=0.10. We checked whether such an indeterminacy affects the energy difference between the FM and AF2 phases as obtainedfrom the model. To do so, we first add a Hubbard- Uconstant associated to the self-energy of the e gWFs, and make it equal to the corresponding Iconstant (i.e., we impose Iaaaa=Uaaaa). In this way the self-interaction of an electron placed in one ofthese orbitals is U−I=0, while the interaction between two electrons that only differ in their spin is U+I=2I=2U. Then, we vary this self-interaction parameter between 0 eV and6 eV , optimize the interaction between different e gorbitals to reproduce the bands, and calculate the FM-AF2 energy gap.We observe that the energy difference is quite insensitive to thevalue of I aaaa, varying by less than 5% in the explored range. Hence, we simply take Iaaaa=2 eV to fix the indeterminacy in the model.2. Results Magnetism in rocksalt structures is usually described using a Heisenberg Hamiltonian with coupling constants betweenfirst- ( J 1) and second- ( J2) nearest neighbors [ 86,88–92]. These constants can be obtained from the energy differencesbetween different spin arrangements by solving the equationsystem: E FM=Eref−6J1−3J2 EAF1=Eref+2J1−3J2 (95) EAF2=Eref+3J2, which involves the spin arrangements of Fig. 8.Erefstands for the energy of a reference phase. 195137-21PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) FIG. 10. Results for the DIdifference occupation matrix in real space. In panel (a) we display the 16-ion supercell employed in thecalculation along one of its main directions (left) and a general panoramic of the same cell (right). Panels (b)–(d) show the D I distribution in real space for the FM (b), AF1 (c), and AF2 (d) phases, respectively. Blue and red regions correspond to spin-up and spin-down magnetization, respectively. We employ our models to compute the energy of these phases, using a 2 ×2×2 supercell containing 16 atoms, as sketched in Fig. 10. After converging the calculations, we plot the spatial distribution of DI(using the spatial representation of the WFs in our basis) to check that the obtained solu-tions correctly correspond to the FM, AF1, and AF2 spinarrangements. These plots are shown in Fig. 10, where it can be seen that the electron distribution in the simulationsperfectly matches the spin orderings sketched in Fig. 8.W e can now check the numerical results for the phase energiesobtained with the three second-principles parametrizations[denoted, respectively, SP-LDAU-Ni( e g), SP-LDAU-Ni(3 d), and SP-LDAU-Ni +O in Table II] and compare them to our full DFT+Uresult and data in the literature (see, e.g., Ref. [ 89]). We find that the coupling constants computed from our models compare quite well with the first-principles results.Indeed, we find the J 2, running along the 180◦Ni-O-Ni bridge, to be much stronger than the J1coupling along the 90◦Ni-O-Ni path. It is worth noting that a parametrization as simple as thatof the SP-LDAU-Ni( e g) model captures this essential feature already. Then, when we include Icouplings between Ni( eg) and Ni( t2g) WFs, we obtain a very similar result, with a very small J1=−0.04 meV . Finally, when we include electron- electron couplings with the oxygen orbitals, we get a value forJ 1that is very close to the first-principles result. C. Electron gas at the LaAlO 3/SrTiO 3interface Now we tackle the well-studied electron gas appearing at the interface between LaAlO 3(LAO) and SrTiO 3(STO). The origin of the 2DEG has been intensively debated inthe literature [ 77,93]. Here we are going to consider an idealized defect-free interface in which the driving force for FIG. 11. Schematic representation of a polar (001) SrTiO 3/LaAlO 3interface. Atoms are represented by balls: O (red), Ti (blue), Sr (yellow), Al (black), and La (green). A free LaO-terminated surface of LaAlO 3is assumed. Numbers below each layer indicate formal ionic charge. The built-in polarization of SrTiO 3(null) and LaAlO 3is illustrated by black arrows. (b) Schematic representation of the energy bands in the case of partialcompensation of the polar discontinuity at the interface. /Delta1represents the LaAlO 3gap. (c) Set-up of the second-principles simulation for this interface. White and dark green squares represent, respectively,SrTiO 3and LaAlO 3cells. The metallic states in SrTiO 3, containing Neelectrons per cell, are represented here by a blue gradient. To be consistent with the electrostatic boundary conditions we usethe charge image method, represented here by a positive charge distribution (red gradient) on the LaAlO 3side. The meaning of NSTO, NLAO,a n dNfreeze is explained in the text. the 2DEG is the so-called polar catastrophe that was proposed originally [ 75,76], which arises from the charge discontinuity between LaAlO 3and SrTiO 3when the bilayer is grown along the (001) pseudocubic direction of the perovskite lattice [ 94]. In such a case, the occurrence of the metallic state stronglydepends on the electrostatic boundary conditions on each sideof the interface. [ 28] Let us look at them in some detail to establish the basic elements of the calculation. From simple electrostatic arguments we know that D LAO−DSTO=σfree, (96) where DLAO andDSTOare the normal components of the displacement field in LaAlO 3and SrTiO 3, respectively, and σfreeis the free charge density at the interface between the materials. Hence, depending on the particular values of DLAO andDSTO(which can be controlled in a simulation by varying the charges at the open surfaces of the layers [ 28]), a 2DEG appears at the interface according to Eq. ( 96). Figure 11(b) illustrates the case for a partial compensation ( DSTO=0, and DLAO<−0.5, both in units of electrons per surface unit cell), which correspond to the case of a partial transfer of chargefrom the LaAlO 3surface to the interface due to the crossing of the top of the valence band of LaAlO 3with the bottom of the conduction band of SrTiO 3. (The interface free carriers occupy states in the conduction band of SrTiO 3.) When there is no full compensation an electric field is present in the LaAlO 3layer. This setup is ideal to test our method, since the main physical effects are related to the negative doping of SrTiO 3, such a doping being controlled by the electrostatic boundaryconditions. Further, the properties of the 2DEG (e.g., spatial 195137-22SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) extension) depend essentially on the ability of SrTiO 3to screen these additional charges, which in turn involves theelectron-lattice couplings in our models. We simulate the LaAlO 3/SrTiO 3interface [Fig. 11(c) ]b y considering a slab of N=NLAO+NSTO5-atom perovskite unit cells, where the first NLAO cells are occupied by LaAlO 3and the following NSTOcells by SrTiO 3. Following Stengel [ 28], we do not consider the electronic details of the interface, as these were found to be of little relevanceto describe the main physical features of the 2DEG. In fact, asregards the construction of our model, we treat the entire slabas if it was made of SrTiO 3, but introducing the following modifications: (i) on the LaAlO 3side, the levels of the conduction band are shifted up (by appropriately modifying theγ aaself-energies) so that they do not interact with those on the SrTiO 3side, and (ii) to account for the large disparity between the LaAlO 3and SrTiO 3dielectric constants (the latter is around 25 times larger than the former at roomtemperature) we simply freeze the coordinates of the atoms onthe LaAlO 3layer at the RAG, to prevent atomic displacements from screening electric fields. As regards the electrostatic boundary conditions, we con- sider that DSTO=0 and DLAO=−Ne/S, i.e., we have Ne electrons per unit area Sdoping the slab. To impose such conditions, we first freeze into the centrosymmetric structurethe atomic positions of N freeze unit cells at the end of the SrTiO 3side of the slab [see Fig. 11(c) ]. Secondly, we use the image-charge method [ 56] [see Fig. 11(c) ] to introduce an electric field from the LaAlO 3side of the interface consistent withDLAO=−Ne/S. 1. Model parameters We now describe how we obtain the parameters for the SrTiO 3layer. As already mentioned, SrTiO 3is a nonmagnetic insulator and the RED corresponds to the ground state of theundoped system. This allows us to take the lattice potentialfor pure SrTiO 3described in Ref. [ 30]a st h e E(0)term of our model [see Eqs. ( 5) and ( 6)], using the LDA-relaxed cubic phase as RAG. [We slightly modified the force fieldof Ref. [ 30], by tuning the interaction between first-nearest- neighboring Ti and O pairs, to exactly reproduce a dielectricconstant of 500 for the cubic phase (see Fig. 12), as obtained from LDA calculations in Ref. [ 28]]. We then extended the model to include the electronic states associated with the bottom of the conduction band of SrTiO 3, which present a dominant Ti( t2g) character [see Fig. 3(a)]. We followed the recipe in Sec. IVto extract the γparameters describing these bands. Note that our focus in this application was to capture the electron-lattice effects that determine the properties of the2DEG, and we were not concerned with electron-electroncouplings beyond the LDA. Thus, we did not include Uor Iterms in our model, and used a TS that contains the RAG and distorted structures (with individual atoms displaced by0.05 ˚A, 0.10 ˚A, and 0.15 ˚A from their RAG positions), all of which where assumed to be in the RED state. We then found all γ,/vectorf, and←→gparameters [Eq. ( 62)] compatible with the choices δε h=0.05 eV and δfe−l= 1.0e V/˚A[95]. We observe that the electron-lattice constantsFIG. 12. Dielectric constant of SrTiO 3as calculated in LDA [ 28] (solid black line) and with our second-principles method (solid blue circles and line) as a function of the electric displacement field, i.e., as a function of the number of electrons per surface unit doping theinterface. associated to diagonal one-electron terms, γaa, are much more sensitive to the displacement of the ions than the off-diagonalones. The distortions that induce larger changes involve theTi-O bond, as expected according to the long literature oncovalency in ferroelectric oxides and related materials [ 60,96– 98]. Finally, we took the Born charges and high-frequency dielectric tensor used in our lattice model, E (0), to compute the electrostatic energy associated to the electronic degrees offreedom. 2. Results We now compare the results for the LaAlO 3/SrTiO 3inter- face obtained with the above-described model and the LDAresults of Ref. [ 28], where the calculations were performed forN LAO=5 and NSTO=12. Using these values, we carry out geometry optimizations with the constraints illustratedin Fig. 11(c) . The results of these calculations are shown in Figs. 13(a) and 13(b) , where we compare the electron densities from first-principles LDA and second-principles sim-ulations for N e=0.3 andNe=0.5, respectively. Moreover, in Figs. 13(c) and13(d) we show the obtained lattice distortions, in terms of the layer-by-layer rumpling, for the same cases. We can observe that the second-principles and LDA results match well for both atomic and electronic structure. Moreover,following the discussion in Ref. [ 28], we checked that our model captures correctly the influence that various physicalparameters (e.g., linear and nonlinear dielectric response ofthe lattice, etc.) have in the final result. As regards relativelysmall errors in the shape of the electronic density profiles, weattribute them to technical differences (pseudopotentials, etc.)in the LDA calculations of Ref. [ 28] and those performed to construct our models. D. Additional considerations To finish this section we would like to give some estimations of the computer time required to carry out the second-principles calculations for the systems discussed in the present 195137-23PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) 0.00.51.01.52.02.53.0 0.00.51.01.52.02.53.0 −0.10−0.050.000.050.10 −0.10−0.050.000.050.10Electron density (e/ Å3) Rumpling ( Å)(a) (b) (c) (d) Layer type Layer type FIG. 13. Results for the 2DEG at the LaAlO 3/SrTiO 3interface. The second-principles results are indicated with solid blue lines, while the LDA results are given by dashed red lines. Panels (a) and (b) show the electron density distribution for Ne=0.3, and Ne=0.5, respectively. Panels (c) and (d) show the rumpling of the lattice for the same two cases. work. We will focus first on NiO, as we carried out both the DFT and second-principles computations for the same unitcells and on the same computational platform, so a reliablecomparison of timings should be feasible. In Table IIIwe show the time necessary to perform a single-point calculationusing a single CPU, using the same reciprocal space samplingin the LDA +Uand second-principles simulations. The values for 4- and 16-ion cells clearly show the very large speed-up ofour second-principles simulations when compared to standardDFT even in small cells. Looking at Table IIIwe see that there are very small differences between the timing results betweenthe different parameterizations associated to different levelsof description of the electron-electron interactions in NiO. Fi-nally, in order to give an estimation of the scaling of the methodas it stands now (i.e., at an early stage of implementation), wecarried out a calculation of a 10 ×10×10 periodic supercell that contains 2000 atoms. This is approximately the size TABLE III. Simulation running times on a single CPU for the 4-ion 1 ×1×2, 16-ion 2 ×2×2, and 2000-ion 10 ×10×10 NiO supercells. The lower time obtained for the SP-LDAU-Ni(3 d) with respect to SP-LDAU-Ni( eg) is due to the smaller number of self- consistent steps required in the former simulation. Method 4 ion (s) 16 ion (s) 2000 ion (hours) LDA+U 65.0 3516.8 SP-LDAU-Ni( eg) 1.4 14.5 6.63 SP-LDAU-Ni(3 d) 1.4 14.1 6.59 SP-LDAU-Ni +O 1.5 15.8 6.97limit for single-point DFT calculations, and it would require significant computational resources and a highly parallelizedcode. However, this simulation at the SP-LDAU-Ni( e g)l e v e l took 6.6 hours of a single CPU, suggesting that calculationsincluding tens of thousands of atoms are within reach usingour models. Turning now to the simulation of the LaAlO 3/SrTiO 3 interface we note that each of the geometry optimizations involving a 85-atom supercell took about 13 minutes usinga single desktop CPU. VII. RELATION WITH OTHER METHODS As already highlighted in the Introduction, various methods have been proposed during the last few decades to bridgeover time and length scales while keeping DFT accuracy.They range from empirical potentials [ 99], to efficient linear- scaling DFT implementations permitting million atom simula-tions [ 100], or the use of the non-self-consistent (e.g., Harris) functionals [ 13], among others. A critical discussion of all of them is out of the scope of the present paper. Instead, wewill focus on the two well-established methods most closelyrelated to the present scheme, namely, the self-consistentcharge density functional tight-binding (SCC-DFTB), andthe effective Hamiltonians for lattice dynamical studies offerroelectrics and related materials. The former puts theemphasis on the description of the electronic structure, whilethe latter is a purely lattice model without an explicit treatmentof the electrons. Detailed connections of different aspects ofthe methodology have already been made at the corresponding 195137-24SECOND-PRINCIPLES METHOD FOR MATERIALS . . . PHYSICAL REVIEW B 93, 195137 (2016) sections where the main features of our model have been presented. Yet, once a complete view of our new schemehas been given, it seems appropriate to further compare thepresent work with these two previous proposals, recallingthe basic features of those methods, pointing to the specificsections where the equivalent approaches have been discussed(enclosed in parenthesis below), and emphasizing the newaspects and how we can go beyond the scope of the previousschemes. Our model is based on a simple and computationally- efficient, electron-free description of the lattice dynamicalor vibrational properties of the material of interest. Thefirst step on this direction was taken in the 90’s works ofVanderbilt, Rabe, and others, who introduced first-principlesmodel potentials (which they called effective Hamiltonians) todescribe the ferroelectric phase transitions of perovskite oxideslike BaTiO 3and PbTiO 3[16–18]. The effective-Hamiltonian approach involves a drastic simplification of the material,which is coarse-grained to retain only the lattice degreesof freedom associated with the ferroelectric properties. Thematerial is then described in term of a reduced number ofvariables (i.e., local polar modes and cell strains) whoseiterations are parametrized by means of a Taylor series ofthe energy around a suitably chosen reference structure.Such a scheme is physically-motivated, computationally veryefficient, and its precision can be improved, to some extent, ina well-defined way. The application of the original scheme toincreasingly complex oxides has shown its generality, the goodtransferability of the interatomic couplings among dissimilarchemical environments, and the reliability and predictivepower of the models. More recently, this scheme has beengeneralized by some of us [ 30] to retain all the atomic degrees of freedom, thus removing the coarse-grain approximation.These models can be trivially formulated for any material, theiraccuracy is systematically improvable, their interpretationis physically transparent, and they lend themselves to veryefficient schemes to compute the model parameters from firstprinciples. In the approach of Ref. [ 30], it is implicitly assumed that electrons follow the atoms adiabatically, so that, forany atomic configuration, the electrons are in their groundstate configuration. Hence, such models allow us to performBorn-Oppenheimer molecular-dynamics simulations, exactlyin the same way as most DFT codes do. Nevertheless, if wewanted to monitor the electronic properties during such asimulation, or consider situations in which the electrons are notin their ground state (e.g., excitations, additional carriers, etc.),we obviously need to extend the models. This is precisely thestep we have undertaken in this work, explicitly including theeffect of the electronic band structure using a tight-binding-likedescription. Among the most efficient implementations for large scale atomistic simulations, the SCC-DFTB has got a remarkablesuccess. Several approximations are shared with our newscheme. Among them, the second-order expansion of theDFT total energy with respect a charge density fluctuation(Sec. III B), the use of a localized minimal basis set to expand the electronic wave functions (Sec. III C), or the approximation (Sec. III C 2 ) and parametrization (Sec. IV) of interaction integrals. Within this framework, SCC-DFTB is comparablein speed with semiempirical methods, roughly 2–3 orders of magnitude faster than standard DFT [ 32]. Therefore, it has become a successful technique in the study of large organicand biological molecules. However, there are also some remarkable differences between SCC-DFTB and the present approach. In the SCC-DFTB method, the charge density fluctuations are quantifiedwith respect a reference electron density that is defined as thesuperposition of neutral atom densities; hence, the referencedensity is notrelated to the actual electronic structure of the specific material being simulated. Therefore, the fluctuationsinclude the difference in the charge density between the groundstate charge density and the reference one due to the chemicalbonding. As a consequence, all the valence band orbitalsmust be included in the calculation for an accurate enoughdescription of δρ. In contrast, we use a difference reference electron density in our case: We usually take the self-consistentsolution for the ground state for a representative atomicconfiguration of the material of interest. Our deformationcharge density thus represents a local deficiency/excess ofelectrons that occur in excited or perturbed electronic states. Inother words, our model focuses on the description of electronicexcitations and is not hampered by the need to account forchemical bonding. This opens the door to the treatment ofexcitons, polarons, transport properties, or the analysis ofcompeting magnetic orders, in a very accurate way. In contrast,our method does not allow us to compute cohesive energies andother basic quantities that are accessible within a SCC-DFTBscheme. The second difference is in the choice of the minimal basis set. The SCC-DFTB method employs atom-centered localizedorbitals [ 101], typically the product of a Slater orbital for the radial part times a spherical harmonic for the angular part. Atleast one radial function for each valence shell occupied in theisolated atom must be included. Moreover, the basis functionsare nonorthogonal, giving rise to an overlap matrix and to ageneralized eigenvalue problem. In contrast, in our scheme we can select the electronic bands that really play a role in the description of the properties underanalysis, and use a basis of Wannier functions coming fromunitary transformations of the corresponding manifold. Thisresults in a basis set that is perfectly adapted to the specificmaterial and property under investigation, and which mayreduce by orders of magnitude the number of basis functionsand dimension of the the associated Hamiltonian matrices.For example, in the case of SrTiO 3treated above in Sec. VI C , instead of requiring 30 atomic orbitals per formula unit [9atomic orbitals for Sr and Ti (1 s,3p, and 5 d) and 4 for the O (1 sand 3 p)], we can consider only the 3- t 2gorbitals of the conduction band of SrTiO 3. Since we use orthogonal Wannier functions, the mathematical problem of diagonalizingthe Hamiltonian is more amenable and does not require thecostly inversion of overlap matrices. Naturally, the reductionof the computational burden translates into an increase of thesize of the systems that can be simulated. The third difference comes from the evaluation of the matrix elements. In the SCC-DFTB method, only two-center Hamil-tonian and overlap matrix elements are treated and explicitlyevaluated, neglecting several three-center contributions to thecorresponding matrix elements. Moreover, (i) the diagonal 195137-25PABLO GARC ´IA-FERN ´ANDEZ et al. PHYSICAL REVIEW B 93, 195137 (2016) elements of the Hamiltonian are taken from the eigenvalues of the free atom [ 12], so the crystal field terms are not considered; and (ii) intra-atomic electron-electron interac-tions are averaged, without considering differences shell-by-shell [ 32]. In our case we explicitly retain many three- and four-center integrals in the parametrization process, and thediagonal matrix elements are sensitive to electrostatic effects,as explained in Sec. III E. This allows us to treat transition metal systems that traditionally have been challenging forSCC-DFTB approaches [ 102,103]. The last important difference is the treatment of the lattice and the interatomic interactions. In the SCC-DFTB method,the term that involves the ion-ion repulsion, together with theDFT double counting terms, are included in a short-rangerepulsive potential. It is approximated as a sum of two-bodypotentials, fitted from the difference of self-consistent DFT cal-culations and the corresponding tight-binding band-structureenergy for suitable reference systems at various interatomicdistances. In the new proposed scheme, we deal with thelattice dynamical part by using a model potential (forcefield) that is directly fitted to a training set of relevant DFTdata, achieving very good agreement with the first-principlesenergies (typically, accuracies below 1 meV per atom canbe achieved for the relevant part of the energy surface).Hence, the generality of the employed model potential, andthe flexibility in its definition and truncation, makes it possibleto have a very accurate description of the Born-Oppenheimersurface. VIII. CONCLUDING REMARKS In this paper we have presented a first-principles-based multiscale method, which we denominate second-principles , that makes it possible to compute the properties of materials atan atomic level, with an accuracy essentially equal to DFT, andat a very reduced computational cost. Our approach is basedon dividing the electron density of the system into a referencepart, usually corresponding to its neutral ground state at anygeometry, and a deformation part, defined as the differencebetween the actual and reference densities. We take advantageof the fact that the largest part of the system’s energy dependson the reference density and can be efficiently and accuratelydescribed by a force field with no explicit consideration of theelectrons. Then, the effects associated to the difference densitycan be treated perturbatively with good precision by workingin the Wannier function basis corresponding to the referencestate. Further, the electronic description can be restricted tothe bands of interest, which renders a computationally veryefficient scheme. Conceptually, the present approach constitutes a fresh look at the problem of how to describe lattice and electronic degreesof freedom simultaneously and effectively, introducing a con-venient partition of the energy that permits an accurate treat-ment of both types of variables and their mutual interactions.In our view, our method constitutes a significant step beyondthe usual techniques—ranging from molecular-mechanics totight-binding and quantum-mechanics/molecular-mechanicsschemes—towards a more unified model. As illustrated by the examples described here, the present approach allows us to obtain DFT-like accuracy in the analysisof subtle physical effects, like those determining the relative stability of the magnetic phases of NiO, or those involved inthe structural relaxations and screening processes associatedto the two-dimensional electron gas formed at the interface ofLaAlO 3and SrTiO 3. Note that these problems—which involve electron correlation effects, transition-metal ions, etc.—areusually hard to treat within DFTB schemes [ 103]. As currently formulated, our approach has only one essential limitation: It is restricted to systems in which it ispossible to (loosely) define an underlying bonding topologythat is to be preserved. Hence, while the method allows thesystem undergo significant structural modifications, e.g., likethose involved in typical ferroelectric or ferroelastic phasetransitions, it is not possible to study full-blown bond breakingdirectly with it. Nevertheless, this limitation can be overcomeby using our method in multiscale simulations that permit amore detailed treatment (e.g., with DFT) of the regions ofthe material in which the constant-topology condition is notsatisfied. It is also important to note that the constant-topology condition is perfectly compatible with the study of manystructurally nontrivial cases, such as nanostructured materials,surfaces, chemically-disordered solid solutions, coexistenceof different structural and electronic phases, etc. Hence, theapplication scope of our scheme is enormous. Let us also note that the present method can be extended to cover physical effects not mentioned here. For example, it ispossible to expand it to treat relativistic phenomena (as spin-orbit effects) or time-dependent nonequilibrium situations (asresulting from the interaction with light) in essentially the sameway as the initial DFT implementations were extended to doso (by implementing a noncolinear treatment of magnetism,time-dependent DFT, etc.). Further, since our method providesus with a Hamiltonian for the interacting system, one couldimagine solving the electronic problem in ways that go beyondthe mean-field approach adopted here, and thus better accountfor many body effects. The ability to simulate systems with thousands of atoms, treating both lattice and electrons accurately, may permit forthe first time predictive investigations of a variety of intrigu-ing phenomena—e.g., Mott transitions, coupled spin-latticedynamics, charged and conducting domain walls, polarontransport, etc.—in realistic conditions of temperature, appliedfields, etc. We thus believe that the present method has thepotential to significantly advance our understanding of some oftoday’s most interesting problems in condensed-matter physicsand material science. ACKNOWLEDGMENTS We thank M. Moreno and J. A. Aramburu for use- ful discussions. P.G.F. and J.J. acknowledge financial sup-port from the Spanish Ministry of Economy and Com-petitiveness through the MINECO Grant No. FIS2012-37549-C05-04. P.G.F. also acknowledges funding fromthe Ram ´on y Cajal FellowshipRYC-2013-12515. 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PhysRevB.97.165131.pdf

PHYSICAL REVIEW B 97, 165131 (2018) Multistage electronic nematic transitions in cuprate superconductors: A functional-renormalization-group analysis Masahisa Tsuchiizu,1,2Kouki Kawaguchi,2Youichi Yamakawa,2and Hiroshi Kontani2 1Department of Physics, Nara Women’s University, Nara 630-8506, Japan 2Department of Physics, Nagoya University, Nagoya 464-8602, Japan (Received 15 May 2017; revised manuscript received 17 December 2017; published 19 April 2018) Recently, complex rotational symmetry-breaking phenomena have been discovered experimentally in cuprate superconductors. To find the realized order parameters, we study various unconventional charge susceptibilities inan unbiased way by applying the functional-renormalization-group method to the d-pHubbard model. Without assuming the wave vector of the order parameter, we reveal that the most dominant instability is the uniform(q=0) charge modulation on the p xandpyorbitals, which possesses dsymmetry. This uniform nematic order triggers another nematic p-orbital density wave along the axial (Cu-Cu) direction at Qa≈(π/2,0). It is predicted that uniform nematic order is driven by the spin fluctuations in the pseudogap region, and another nematicdensity-wave order at q=Q ais triggered by the uniform order. The predicted multistage nematic transitions are caused by Aslamazov–Larkin-type fluctuation-exchange processes. DOI: 10.1103/PhysRevB.97.165131 I. INTRODUCTION In the normal state of high- Tccuprate superconductors, interesting unconventional order parameters emerge due tothe strong interference between the spin, charge, and orbitaldegrees of freedom. These phenomena should be directlyrelated to the fundamental electronic states in the pseudogapregion. The emergence of the charge-density-wave (CDW)states inside the pseudogap region has been confirmed by thex-ray and scanning tunneling microscopy measurements [ 1–6], as schematically shown in Fig. 1(a). The observed CDW pattern is shown in Fig. 1(b), in which the density modulations mainly occur on the oxygen p xandpyorbitals with antiphase (d-symmetry) form factor. The discovery of the CDW has promoted significant progress in the theoretical studies, suchas the spin-fluctuation-driven density-wave scenarios [ 7–14] and the superconducting-fluctuation scenarios [ 15–18]. The origin and nature of the pseudogap phase below T ∗ remain unsolved. For example, it is unclear whether the pseu- dogap is a distinct phase or a continuous crossover. The short-range spin fluctuations at T∼T ∗induce the large quasiparticle damping [ 19–21], which causes the pseudogap in the density of states. On the other hand, the phase transition around T∗has been reported by the resonant ultrasound spectroscopy [ 22], angle-resolved photoemission spectroscopy (ARPES) analy-sis [23], and magnetic torque measurement [ 24]. In particular, Ref. [ 24] discovered the C 4symmetry-breaking (nematic) transition, and its natural candidate is the uniform CDW withdsymmetry schematically shown in Fig. 1(c). A fundamental question, then, is what mechanism can account for such uncon-ventional multistage CDW transitions? No CDW instabilitiesare given by the mean-field-level approximations, such asthe random-phase approximation (RPA), unless large intersiteinteractions are introduced [ 13,25]. Therefore, higher-order many-body effects, called the vertex corrections (VCs), shouldbe essential for the CDW formation [ 7–13,26].In many spin-fluctuation-driven CDW scenarios, the CDW wave vector is given by the minor nesting vector Q aorQdin Fig. 1(d);Qais the “axial-wave vector” parallel to the nearest Cu-Cu direction, and Qdis the “diagonal-wave vector.” The experimental axial CDW is obtained if the Aslamazov–LarkinVCs (AL-VCs) are taken into account [ 13]. In addition, the uniform ( q=0) CDW instability has been studied intensively based on the Hubbard models [ 9,27–30]. In these studies, however, it was difficult to exclude the possibility that the CDWsusceptibility has the maximum at finite q. Theoretically, it is difficult to analyze the spin and charge susceptibilities with general wave vector qon equal footing by including the VCs in an unbiased way. For this purpose, inprinciple, the functional-renormalization-group (fRG) methodwould be the best theoretical method. The pioneering fRGstudies [ 9,28] were performed only in the weak-coupling region, so the obtained CDW instability is small and its q dependence is not clear. To overcome this problem, we haveto improve the numerical accuracy of the fRG method, andapply it to the two-dimensional Hubbard model in the strong-coupling region. In this paper, we study the orbital-dependent spin and charge susceptibilities for various symmetries on equal footing byanalyzing the higher-order VCs in an unbiased way usingthe improved fRG method. We find that the uniform CDWaccompanied by the p-orbital polarization ( n x/negationslash=ny), shown in Fig. 1(c), is driven by the antiferro spin fluctuations. In this uniform nematic CDW phase, another nematic CDWinstability emerges at the wave vector q=Q aas shown in Fig.1(b). The present study indicates that the uniform p-orbital polarization appears in the pseudogap region, and the axial q=QaCDW is induced at TCDW<T∗. These multistage CDW transitions in under-doped cuprates originate from thehigher-order AL-type VCs. In the present study, we use the functional RG + constrained RPA (RG +cRPA) method. The advantage of this 2469-9950/2018/97(16)/165131(9) 165131-1 ©2018 American Physical SocietyTSUCHIIZU, KAW AGUCHI, Y AMAKAW A, AND KONTANI PHYSICAL REVIEW B 97, 165131 (2018) (a) (b) (c) (d) (e) kxππ −π−π0d −4−202 (0,0) (π,0) (π,π) (0,0)[eV] Energy Wavevector (a) (b) (c) (d) (e) k x k k π π −π −π 0 d d d d d d −4 −2 0 2 (0,0 ) (π,0 ) (π,π ) (0,0) [eV] Energy Wav evectorTemperature hole concentration 0 AFAFCDWCDW SCT T* CDWTN TC 0.1 0.2 FIG. 1. (a) Schematic phase diagram of the high- Tccuprate super- conductors. T∗,TCDW,TN,a n dTcare the transition temperatures for the pseudogap state, CDW order, magnetic order, and superconductiv- ity, respectively. We study the 10%-doping case shown by the vertical broken line. (b) Schematic charge distribution in the d-symmetry pO-CDW state with the wave vector q=Qa≈(0.5π,0). (c) The uniform nematic pO-CDW state with nx/negationslash=ny. (d) The Fermi surface and (e) energy dispersion of the present d-pmodel. The lower-energy region ( |E|</Lambda1 0=0.5 eV) is divided into the Np=128 patches to perform the RG analysis. method had been explained in Refs. [ 31–34] and Appendix A in detail. II. MODEL AND THEORETICAL METHOD Here, we study a standard three-orbital d-pHubbard model [ 13,33,35] expressed as H=/summationtext k,σc† k,σˆh0(k)ck,σ+ U/summationtext jnd,j,↑nd,j,↓, where c† k,σ=(d† k,σ,p† x,k,σ,p† y,k,σ)i st h e creation operator for the electron on d,px, andpyorbitals, and ˆh0(k) is the kinetic term given as the 0MTO model in Refs. [ 35,36]. (The numerical results are unchanged if another realistic 1MTO model is used; see Appendix B.)U is the Hubbard-type on-site Coulomb interaction for the d orbital, and nd,j,σ=d† j,σdj,σat site j. Hereafter, we study the 10%-hole-doping case. The Fermi surface (FS) and theband structure are shown in Figs. 1(d) and1(e), respectively. By using the RG +cRPA theory in Ref. [ 33], we find that the spin susceptibility for delectrons, χ spin(q)=1 2/integraldisplay1/T 0dτ/angbracketleftSd(q,τ)Sd(−q,0)/angbracketright, (1) and the B1g-symmetry ( d-symmetry) charge susceptibility for pelectrons, χp-orb d (q)=1 2/integraldisplay1/T 0dτ/angbracketleftbig np-orb d(q,τ)np-orb d(−q,0)/angbracketrightbig , (2) are the most enhanced susceptibilities [ 33]. Here, Sd(q,τ)i s thedelectron-spin operator, and np-orb d(q)≡nx(q)−ny(q) [nx(y)(q)=/summationtext k,σp† x(y),k,σpx(y),k+q,σ]i st h e p-orbital charge- density-wave ( pO-CDW) operator with B1gsymmetry. Ifχp-orb d (q) diverges at q=Qa(q=0), the pO-CDW order shown in Fig. 1(b) [Fig. 1(c)], which is the CDW on porbitals, is realized. We verified that the charge susceptibilities withnon-B 1gsymmetries, such as the A1g-symmetry total charge susceptibility for n≡nd+nx+ny, remain small even in the strong-coupling region [ 33]. In the RG +cRPA method, we calculate the scattering processes involving higher-energy states |Ek,ν|>/Lambda1 0[νbeing the band index; see Fig. 1(e)] using the RPA with the energy- constraint and incorporate their contributions into the initialvertex functions of the RG equations [ 31–34]. Using the RPA, the higher-energy processes are calculated accurately bydropping the VCs, which are less important for higher-energyprocesses. The lower-energy scattering processes for |E k,ν|< /Lambda10are calculated by solving the RG equations, based on the Np-patch RG scheme [ 27,37]. Hereafter, we put Np=128 and/Lambda10=0.5e V .I nt h eR G +cRPA method, the numerical accuracy of the susceptibilities is greatly improved even in theweak-coupling region since the cRPA is used for the higher-energy processes, for which the N p-patch RG scheme is less accurate. We verified that the numerical results are essentiallyindependent of the choice of /Lambda1 0whenEF/greaterorsimilar/Lambda10/greatermuchT. By solving the RG equations, many-body vertices are grad- ually renormalized as reducing the energy scale /Lambda1l=/Lambda10e−l with increasing l(/greaterorequalslant0). In principle, the renormalization of the vertex saturates when /Lambda1lreaches ∼T[37,38]. Here, we intro- duce the lower-energy cutoff /Lambda1low(∼T) in the RG equations for the four-point vertex /Gamma1s(c) l, and stop the renormalization at /Lambda1l=/Lambda1low; see Appendix Aand Ref. [ 9]. [We do not introduce the lower-energy cutoff in the RG equations for χs,c(q).] In the previous study [ 33], we set a large cutoff /Lambda1low=πTto achieve stable numerical results. When /Lambda1low/greatermuchT, the uniform ( q=0) nematic susceptibility is especially underestimated comparedwith q/negationslash=0instabilities, as we discuss later. Since we have improved the numerical accuracy in solving the RG equations,we can use a smaller natural cutoff /Lambda1 low=T. For this reason, we can obtain the qdependence of the susceptibility accurately, including q≈0. We find that the numerical accuracy and stability are improved by employing the Wick-ordered scheme of the fRGformalism, in which the cutoff function /Theta1 /Lambda1 <(/epsilon1)=/Theta1(/Lambda1−|/epsilon1|) is used for the Green’s function [ 37]. In this scheme, in principle, the VCs due to the higher-energy processes areincluded more accurately compared with using another cutofffunction /Theta1 /Lambda1 >(/epsilon1)=/Theta1(|/epsilon1|−/Lambda1) based on the Kadanoff–Wilson scheme used in Ref. [ 33]. III. MULTISTAGE ELECTRONIC NEMATIC TRANSITIONS In Figs. 2(a) and2(b), we show the pO-CDW susceptibility χp-orb d (q) given by the RG +cRPA method for U=4.32 eV atT=0.1 eV . The large peaks obtained at q=0,Qa, and Qd originate from the VCs, since the RPA result is less singular, as seen in Fig. 2(b). As shown in Figs. 2(a)–2(c),t h em o s t dominant peak locates at q=0. This is consistent with the experimental uniform nematic transition at T∗(>T CDW)[24]. We also obtain the peak structures at q=QaandQd, consis- tently with our previous fRG study [ 33]. Figure 2(c) shows thatχp-orb d (0) monotonically increases with decreasing T, 165131-2MULTISTAGE ELECTRONIC NEMATIC TRANSITIONS IN … PHYSICAL REVIEW B 97, 165131 (2018) spinχ + + ··· + +· · ·0.1 0.11 0.120510 0.1 0.110.20.30.40.5 T[eV]q=Qq=Q q=0 Tχχχχ p−orb d(q) [eV][eV−1] q=Qa 00.20.4 (0,0) (π,π) (π,0) (π,π)RPARG+cRPA χχχχ p−orb d(q)Qd Qa 0[eV−1] Wavevector(0,0)(π,π) (π,0)(0,π) qq xy 00 ) ( ( 0 )(a) )c( )b( )e( )d(χχχχ p−orb d(q) 0.10.5 0.4 0.30.2 FIG. 2. (a), (b) The RG +cRPA result of the pO-CDW sus- ceptibility χp-orb d (q) obtained for U=4.32 eV at T=0.1e V .T h e RPA result is also shown for comparison in panel (b). The axial wave vector is Qa≈(0.37π,0) and the diagonal wave vector is Qd≈(0.40π,0.40π). Both QaandQdcorrespond to the wave vector connecting the hot spots shown in Fig. 1(b).( c )Tdependence of χp-orb d (q)f o rU=4.32 eV . (d) VCs due to MT processes. (e) VCs due to AL processes. consistently with the recent electronic nematic susceptibility measurement [ 39]. At low temperatures, χp-orb d (Qa) increases steeply and becomes larger than χp-orb d (Qd), shown in the inset of Fig. 2(c). Note that the temperature T=0.1e Vi s comparable to T∗∼300 K if the mass-enhancement factor m∗/m band∼3 is taken into account. The enhancement of χp-orb d (q) is caused by the spin fluctuations, due to the strong charge-spin interplay givenby the VCs. The moderate peak at Q dis caused by the Maki–Thompson (MT)-VCs, given by the series of the single-fluctuation-exchange processes shown in Fig. 2(d) [11,12]. However, the MT-VCs cannot account for the dominant peaksatq=0and Q a. Recently, it was found that the uniform nematic order in the Fe-based superconductors [ 26,40] and Sr3Ru2O7[31,41]i sd r i v e nb yt h eA L - V C ,g i v e nb yt h e series of the double-fluctuation-exchange processes shown inFig. 2(e). In fact, the first term in Fig. 2(e) is proportional to/summationtext kχspin(k+q)χspin(k), which takes a large value for q=0 whenχspin max/greatermuch1[26,42]. Later, we demonstrate that the AL-VC(a) )c( )b( f k kk xyq=0 0−10 q=QaΓdp−orb(q) Γspin (Qs) lq=0q=Qdq=Qs 1230 5 10 150.20.40.6 024051015 q=Qaq=0 χχχχspin maxχχχχ p−orb d(q) [eV−1][eV−1] q=Qdχχχχspin max [eV−1] [eV]URPA RPARG+cRPA FIG. 3. (a) RG +cRPA result of χp-orb d (q) at three peak positions as a function of χspin max(/Delta1Ep=0). The RPA results are also shown by lines. In the inset, the Udependence of χspin maxis shown. (b) Scaling flows of the effective four-point vertices for the pO-CDW with d symmetry, for U=4.32 eV at T=0.1e V . l(/greaterorequalslant0) is the scaling parameter. The scaling flows for spin channel are also shown where Qsis the nesting vector ≈(π,0.78π)o r( 0.78π,π ). (c) The optimized form factor fq=0(k) on the FS, which has dsymmetry. causes the uniform and axial CDW instabilities in the present d-pmodel. Next, we investigate the Udependencies of the susceptibil- ities. In the inset of Fig. 3(a), we show the Udependence of χspin max≡max q{χspin(q)}. Thanks to the numerical accuracy of the RG +cRPA method, χspin maxperfectly follows the RPA result for the wide weak-coupling region ( U< 4 eV). To clarify the close interplay between spin and orbital fluctuations, we plot the peak values of χp-orb d (q)a saf u n c t i o no f χspin maxin Fig. 3(a). In contrast to χspin max,χp-orb d (q) strongly deviates from the RPA result, indicating the significance of the VCs. With increasing U, the peak position of χp-orb d (q)s h i f t st o q=0atχspin max∼ 2.5, and χp-orb d (0) exceeds the spin susceptibility for χspin max/greaterorsimilar 10 eV−1. To understand the origin of the enhancement of χp-orb d (q), we analyze the scaling flow of the effective interaction for thepO-CDW introduced as /Gamma1p-orb d (q)≡/Gamma1c x;x(q)+/Gamma1c y;y(q)−/Gamma1c x;y(q)−/Gamma1c y;x(q), with /Gamma1c α;β(q)≡/summationdisplay k,k/prime/Gamma1c l(k+q,k;k/prime+q,k/prime) ×u∗ α(k+q)uα(k)uβ(k/prime+q)u∗ β(k/prime). 165131-3TSUCHIIZU, KAW AGUCHI, Y AMAKAW A, AND KONTANI PHYSICAL REVIEW B 97, 165131 (2018) Here/Gamma1c lis the charge-channel four-point vertex, which is a moderate function of the Fermi momenta in the parameterrange of the present numerical study. u α(k) is the matrix ele- ment connecting the porbitals ( α=x,y) and the conduction band [ 33] .T h es c a l i n gfl o wo f /Gamma1p-orb d (q) is shown in Fig. 3(b), with the scaling parameter l=ln(/Lambda10//Lambda1l). The negative effec- tive interaction drives the enhancement of the correspondinginstability. We also plot the effective interaction for the spinchannel, /Gamma1 spin(Qs). For the spin channel, /Gamma1spin(Qs)∼−Uat l=0, and it is renormalized like the RPA as /Gamma1spin l=/Gamma1spin 0/(1− c|/Gamma1spin 0|l)f o rl/lessorsimilarln(/Lambda10/T)=1.6, where cis the density of states. For the charge channel, although /Gamma1p-orb d (q)a tl=0i s quite small, it is strongly renormalized to be a large negativevalue. This result means that the CDW instability originatesfrom the VC going beyond the RPA. We also calculate the d-electron charge susceptibility with form factor f q(k), which is given as χd-orb(q)=1 2/integraldisplay1/T 0dτ/angbracketleftB(q,τ)B(−q,0)/angbracketright, (3) where B(q)=/summationtext k,σfq(k)d† k−q/2,σdk+q/2,σ. The numerically optimized fq(k)a tq=0is shown in Fig. 3(c), which has B1g symmetry. Its Fourier transformation gives the modulation of the effective hopping integrals, called the dx2−y2-wave bond order. Since the kdependence of f0(k)i nF i g . 3(c) is similar to that of |ux(k)|2−|uy(k)|2, the obtained χspin maxdependence ofχd-orb(0) with the optimized form factor is similar to that of χp-orb d (0) shown in Fig. 3(a). In Appendix C, we analyze the single- d-orbital Hubbard model, and find the strong enhance- ment of χd-orb(q) with the B1gform factor at q=0,Qa, and Qd, very similarly to the pO-CDW susceptibilities shown in Figs. 2and3. As shown in Fig. 2(c),χp-orb d (0) increases divergently at T∼0.1 eV , and the uniform p-orbital polarization with nx/negationslash= nydepicted in Fig. 1(c) appears below the transition tem- perature. To discuss the CDW instabilities inside the nematic phase, we perform the RG +cRPA analysis in the presence of the uniform pO-CDW order H/prime=−1 2/Delta1Ep[nx(0)−ny(0)]. In Fig. 4(a), we plot the peak values of χp-orb d (q) in the uniform pO-CDW state with /Delta1Ep=0.3 eV . Due to small /Delta1Ep>0, χp-orb d (q)a t q=Qx a(along the xaxis) strongly increases whereas that at q=Qy a(along the yaxis) decreases. Thus, thepO-CDW at q=Qx ais expected to emerge below TCDW, consistently with the phase diagram in Fig. 1(a). IV . ORIGIN OF NEMATIC ORDERS To confirm the mechanism of the nematic transition, we perform the diagrammatic calculation for the MT- and AL-VCs. These VCs can be obtained by solving the CDW equationintroduced in Ref. [ 43] in the study of Fe-based superconduc- tors. We analyze the linearized CDW equation introduced inAppendix Dand in Ref. [ 44]. By solving the CDW equation, both MT- and AL-VCs shown in Figs. 2(d) and2(e)with the op- timized form factors are systematically generated. Figure 4(b) shows the eigenvalue of the linearized equation, λ q,f o r/Delta1Ep= 0.1e V .H e r e , αSis the spin Stoner factor, and the horizontal axis is proportional to χspin max. The CDW susceptibility increases01 0 2 00.20.40.60.8 (0,0) (π,0)(0,π) ΔE=0.0 eV 0.3 eV(π,π) q=Qay χχχχspin maxχχχχ p−orb d(q) [eV−1][eV−1] q=Qdq=Qax ΔEp=0.3 eVq=0 0 100 200 300024 q=Qay 1/(1− αS)λq q=Qdq=Qax ΔEp=0.1 eVq=0)b( )a( FIG. 4. (a) The RG +cRPA result of χp-orb d (q)a tq=0,Qx,y a, andQdas a function of χspin maxfor/Delta1Ep=0.3 eV . The inset shows the FS. (b) The eigenvalues of the CDW susceptibility given by solving the linearized CDW equation in Appendix Dforγ=0.3e V . with the increase of λq.I nF i g . 4(b), we set the quasiparticle damping γ=0.3 eV . Note that λqdecreases with γ, whereas its overall qdependence is independent of γ,a ss h o w ni n Appendix D. The obtained results are qualitatively consistent with the RG +cRPA results in Fig. 4(a). In Appendix D,w e reveal that the CDW instabilities at q=0and q=Qaare given by the higher-order AL-type VCs. Finally, we explain why /Lambda1lowshould be set small. The RG equation for the q=0vertex, ¯/Gamma1c l(k;k/prime)≡/Gamma1c l(k,k;k/prime,k/prime), is given as d¯/Gamma1c(k;k/prime)/dl∝/Lambda1l˙f(/Lambda1l)/summationdisplay k/prime/primeδ(|Ek/prime/prime|−/Lambda1l)¯/Gamma1c l(k;k/prime/prime) ׯ/Gamma1c l(k/prime/prime;k/prime)+two other terms , where ˙f(/epsilon1) is the derivative of the Fermi distribution function. Since the factor |˙f(/Lambda1l)|is small for /Lambda1l/greaterorsimilar4T, the obtained ¯/Gamma1c l is strongly reduced if /Lambda1low/greatermuchT. In contrast, /Gamma1s,cforq/negationslash=0is not so sensitive to /Lambda1low. For this reason, the renormalization effect of /Gamma1c lis underestimated for q≈0if/Lambda1low/greatermuchT. Then, the obtained χp-orb d (0) is suppressed to be smaller than the peak values at q/negationslash=0if a large cutoff /Lambda1low/greatermuchTis used, similarly to the previous results for /Lambda1low=πT[33]. In the present study, largeχp-orb d (0) is correctly obtained thanks to the use of the small cutoff /Lambda1low=T. V . DOPING DEPENDENCE OF CHARGE-DENSITY-WA VE SUSCEPTIBILITIES In the above sections, we studied the spin and charge susceptibilities in the d-pHubbard model only for 10%-hole- doped case ( p=0.10). To understand the experimental phase diagram in Fig. 1(a), however, we have to study the doping dependence of susceptibilities. This issue is a very importantbut difficult goal for theorists. We find that the CDW is drivenby the strong spin fluctuations, which strongly develop whenthe hole-density papproaches zero experimentally. In the RG +cRPA theory, the CDW is driven by the strong spin fluctuations, and spin fluctuations develop asthe hole carrier papproaches zero experimentally. For this 165131-4MULTISTAGE ELECTRONIC NEMATIC TRANSITIONS IN … PHYSICAL REVIEW B 97, 165131 (2018) 0.05 0.1 0.15 0.201020 hole doping pχspin max χdp−orb(0)1/ p FIG. 5. Doping dependence of χp-orb d (0) obtained by the RG + cRPA theory ( Np=64) at T=0.1 eV . The obtained χp-orb d (0) linearly increases as papproaches unity, consistently with T∗in the experimental phase diagram in Fig. 1(a) in the main text. For p/lessorsimilar0.1, χp-orb d (0) exceeds χspin maxforT/lessorsimilar0.1e V . reason, as shown in Fig. 5, the uniform CDW susceptibility χp-orb d (0) linearly increases as pdecreases accompanied by the increment of χspin maxforp∼0. This result is consistent with the experimental pdependence of T∗in Fig. 1(a) in the main text. In Fig. 5, we modify Uslightly so that the experimental approximate relation χspin max∝1/pis satisfied, as performed in our previous study [ 13]. We put U=4.31, 4.25, and 4.06 eV forp=0.05, 0.10, and 0.20, respectively. In contrast to T∗,TCDW decreases near the half filling forp< 0.1, as depicted in Fig. 1(a). This behavior is also understood qualitatively based on the spin-fluctuation-drivenmechanism. In fact, the axial CDW wavelength Q ais given by the nesting vector between the neighboring hot spots, and |Qa| increases as papproaches zero. The CDW instability driven by the Aslamazov–Larkin vertex correction, which is qualitativelyproportional to/summationtext qχs(q)χs(q+Qa), is suppressed if |Qa| is very large, as we explained in Ref. [ 13]. Therefore, the difference in the doping-dependencies of T∗andTCDW is qualitatively understood. It is an important future issue toreproduce the experimental phase diagram in Fig. 1(a) more completely, which is one of the greatest goals in this field. VI. SUMMARY In summary, we studied various unconventional CDW instabilities in the d-pHubbard model by using the RG + cRPA method and predicted the multistage CDW transitionsin cuprate superconductors. Based on the proposed spin-fluctuation-driven CDW mechanism, the following under-standing has been reached: The short-range spin fluctuationsdrive the uniform nematic CDW around T ∗, and it triggers the axial q=QaCDW at TCDW successively. We also explained the doping dependence of T∗based on the RG +cRPA theory. These results naturally explain the phase diagram in Fig. 1(a), except for heavily under-doped region. Although the uniformCDW order cannot simply explain the pseudogap formation, the large quasiparticle damping [ 19–21] due to the short-range spin-fluctuations may induce the pseudogap for T/lessorsimilarT∗. ACKNOWLEDGMENTS We are grateful to Y . Matsuda, T. Hanaguri, T Shibauchi, Y . Kasahara, Y . Gallais, W. Metzner, T. Enss, L. Classen, andS. Onari for fruitful discussions. This work was supportedby Grant-in-Aid for Scientific Research from the Ministry ofEducation, Culture, Sports, Science, and Technology, Japan,and in part by Nara Women’s University Intramural Grant forProject Research. APPENDIX A: RENORMALIZATION GROUP EQUATIONS FOR THE FOUR-POINT VERTEX In the main text, we analyzed the d-pHubbard model by using the RG +cRPA method [ 32]. This method is the combination of the fRG theory and the cRPA. The RG +cRPA method enables us to perform reliable numerical study in theunbiased way. In this method, we introduce the original cutoffenergy /Lambda1 0in order to divide each band into the higher- and lower-energy regions: The higher-energy scattering processesare calculated by using the cRPA: The lower-energy scatteringprocesses are analyzed by solving the RG equations, in whichthe initial vertices in the differential equation are given by thecRPA. In the present model, the bare Coulomb interaction term on delectrons is given as H U=1 4/summationdisplay i/summationdisplay σσ/primeρρ/primeU0;σσ/primeρρ/primed† iσdiσ/primed† iρ/primediρ, (A1) U0;σσ/primeρρ/prime=1 2U0;s/vectorσσσ/prime·/vectorσρ/primeρ+1 2U0;cδσ,σ/primeδρ/prime,ρ,(A2) where U0;c=UandU0;s=−U. The antisymmetrized full four-point vertex /Gamma1(k+q,k;k/prime+ q,k/prime), which is the dressed vertex of the bare vertex ˆUin Eq. ( A2) in the microscopic Fermi-liquid theory, is depicted in Fig.6(a). Reflecting the SU(2) symmetry of the present model, /Gamma1is uniquely decomposed into the spin-channel and charge- d dΛ k2k1 k4k3 =k k’ k2k1 k4k3 +k k’ k3k1 k4k2 + k k’ k2k1 k3k4 ΓRG(k1,k2;k3,k4),( , ; , )kq k k q kσσ ρρΓ+ + = k,σ'k+q,σ k',ρ'k'+q,ρ Γ(a) (b)'''' FIG. 6. (a) Definition of the full four-point vertex /Gamma1σσ/primeρρ/prime(k+ q,k;k/prime+q,k/prime) in the microscopic Fermi-liquid theory. (b) The one-loop RG equation for the four-point vertex. The crossed lines represent the electron Green’s function with cutoff /Lambda1. The slashed lines represent the electron propagations having energy shell /Lambda1. 165131-5TSUCHIIZU, KAW AGUCHI, Y AMAKAW A, AND KONTANI PHYSICAL REVIEW B 97, 165131 (2018) channel four-point vertices by using the following relation: /Gamma1σσ/primeρρ/prime(k+q,k;k/prime+q,k/prime) =1 2/Gamma1s(k+q,k;k/prime+q,k/prime)/vectorσσσ/prime·/vectorσρ/primeρ +1 2/Gamma1c(k+q,k;k/prime+q,k/prime)δσ,σ/primeδρ/prime,ρ, (A3) where σ,σ/prime,ρ,ρ/primeare spin indices, and /vectorσis the Pauli matrix vector. We stress that /Gamma1c,sare fully antisymmetrized, so the requirement by the Pauli principle is satisfied. We note that/Gamma1 ↑↑↑↑=1 2/Gamma1c+1 2/Gamma1s,/Gamma1↑↑↓↓=1 2/Gamma1c−1 2/Gamma1s, and/Gamma1↑↓↑↓=/Gamma1s. In the RG formalism, the four-point vertex function is determined by solving the differential equations, called the RGequations. In the band-representation basis, the explicit formof the RG equations is given by [ 34] d d/Lambda1/Gamma1RG(k1,k2;k3,k4) =−T N/summationdisplay k,k/prime/bracketleftbiggd d/Lambda1G(k)G(k/prime)/bracketrightbigg ×/bracketleftbigg /Gamma1RG(k1,k2;k,k/prime)/Gamma1RG(k,k/prime;k3,k4) −/Gamma1RG(k1,k3;k,k/prime)/Gamma1RG(k,k/prime;k2,k4) −1 2/Gamma1RG(k1,k;k/prime,k4)/Gamma1RG(k,k 2;k3,k/prime)/bracketrightbigg , (A4) where G(k) is the Green’s function multiplied by the Heaviside step function /Theta1(/Lambda1−|Ek,ν|), and kis the compact notation of the momentum, band, and spin indices: k=(k,/epsilon1n,ν,σ ). The diagrammatic representation of the RG equations is shown inFig. 6(b). The first two contributions in the right-hand-side represent the particle-hole channels and the last contributionis the particle-particle channel. In a conventional fRG method, /Lambda1 0is set larger than the bandwidth Wband, and the initial value is given by the bare Coulomb interaction in Eq. ( A2). In the RG +cRPA method, we set /Lambda10<W band, and the initial value is given by the constrained RPA to include the higher-energy processeswithout over counting of diagrams [ 32]. In the main text, we introduced the lower-energy cutoff /Lambda1 low(∼T) in the RG equation for the four-point vertex: Eq. ( A4). For this purpose, we multiply the cutoff function [(/Lambda1low//Lambda1)ζ+1]−1to the right-hand side of Eq. ( A4). Here, ζis a parameter determining the width of this smooth cutoff, and we set ζ=10 in the main text. We do not introduce the lower-energy cutoff in the RG equation for the susceptibilities. APPENDIX B: RG +cRPA ANALYSIS OF THE d-pMODEL WITH DIFFERENT HOPPING PARAMETERS In Ref. [ 35], the authors derived the realistic d-pmodels for La-based cuprate by evaluating the hopping parameters onthe basis of the Nth-order muffin-tin orbitals ( NMTO). The model parameters for N=0 and N=1 are given in Table I. The band structure of the N=0 basis model (0MTO model) is very close to the local density approximation band structurenear the Fermi energy. For this reason, we have analyzedthe 0MTO model in the main text. On the other hand, theN=1 basis model (1MTO model) appropriately reproducesTABLE I. Hopping integrals for the N=0a n d N=1 models given in Ref. [ 35]. The units are eV . NMTO /epsilon1d−/epsilon1ptdd tpd t/prime pd tpp t/prime pp t/prime/prime ppt/prime/prime/prime pp N=0 0.43 −0.10 0.96 −0.10 0.15 −0.24 0.02 0 .11 N=1 0.95 0.15 1.48 0.08 0.91 0.03 0.15 0.03 the overall oxygen bonding band structure with deep bottom energy E/similarequal− 8 eV . To check the reliability of our RG +cRPA results, we analyze the d-pmodel with the 1MTO model parameters. Figure 7(a) shows the band structure of the 1MTO model. Here, we introduced the third-nearest d-dhopping t3rd dd= −0.1 eV to make the FS closer to Y-based cuprates. The FS of this model is shown in Fig. 7(b). Now, we analyze this model by using the RG +cRPA method. The parameters are the same as in the main text except for U. The number of patches is Np=128 and the initial cutoff is /Lambda10=0.5e V .T h e temperature is fixed at T=0.1e V . In Fig. 7(c), we show the obtained χp-orb d (q)f o rU=5.72 eV . The RPA results are also shown for comparison. It has thelargest peak at q=0and the second largest peak at q=Q a, respectively. The obtained qdependence of χp-orb d (q) is similar to Fig. 2(b). We also investigate the Udependencies of the spin and charge susceptibilities. As shown in the inset of Fig. 7(d), relatively large Uis required for the enhancement of χspin maxin the 1MTO model, since the density of states of the dorbital at the Fermi energy in the 1MTO model is smaller than that inthe 0MTO model [ 35]. In Fig. 7(d), we plot the peak values of χp-orb d (q) as functions of χspin max. The obtained results are quite similar to Fig. 3(a) in the main text. Thus the spin-fluctuation- driven CDW instabilities are universal phenomena in both the0MTO and 1MTO models. FIG. 7. (a) Energy dispersion and (b) FS of the d-pmodel with 1MTO model parameters. (c) RG +cRPA and RPA results for the pO-CDW susceptibility χp-orb d (q) withU=5.72 eV . (d) RG +cRPA result of χp-orb d (q) at three peak positions as a function of χspin max. 165131-6MULTISTAGE ELECTRONIC NEMATIC TRANSITIONS IN … PHYSICAL REVIEW B 97, 165131 (2018) In summary, we investigate the d-pmodel with 1MTO model parameters. We found that the results are very similar tothose for the 0MTO model given in the main text. Therefore, themechanism of the spin-fluctuation-driven CDW instabilitiesrevealed in the main text is universal, independently of thedetails of the model parameters. APPENDIX C: RG +cRPA ANALYSIS FOR SINGLE- d-ORBITAL HUBBARD MODEL In the main text, we studied the 0MTO d-pHubbard model based on the RG +cRPA theory and found that the pO-CDW susceptibilities develop strongly in the strong-spin-fluctuationregion. Similar results are obtained in the 1MTO model inwhich /epsilon1 d−/epsilon1pis 0.53 eV larger than that in the 0MTO model, as we show in Appendix B. In these d-pmodels, any Coulomb interactions on porbitals are not taken into account. Therefore, spin-fluctuation-driven CDW formation is also expected to berealized in the single- d-orbital Hubbard model with on-site Coulomb interaction. Here, we study the single- d-orbital Hubbard model with the first-, the second-, and the third-nearest hopping integralsast=− 0.50 eV , t /prime=0.083 eV , and t/prime/prime/prime=− 0.10 eV , respec- tively. The band structure and Fermi surface for n=0.90 are shown in Figs. 8(a) and 8(b), respectively. We calculate the d-electron charge susceptibility χd-orb(q) with the B1gform factor fq(k)=cos(kx)−cos(ky) introduced in Eq. ( 3)i nt h e main text. The results obtained are summarized in Fig. 8(c): Bothχd-orb(0) and χd-orb(Qa) are strongly enlarged in the strong spin-fluctuation region, very similarly to the pO-CDW susceptibility shown in Fig. 3(a) in the main text. −202 (0,0) (π,0) (π,π) (0,0) WavevectorEnergy [eV])π,π( )π,0( )0,π( )0,0(kx ky 05 1 0 1 50510 012051015 q=Qaq=0 χχχχspin maxχχχχd−orb(q) [eV−1][eV−1] q=Qdχχχχspin max [eV−1] [eV]URPARG+cRPA)b( )a( (c) FIG. 8. (a) Energy dispersion and (b) FS of the single- d-orbital Hubbard model. (c) RG +cRPA result of the d-electron charge sus- ceptibility with B1gform factor fq(k)=cos(kx)−cos(ky),χd-orb d(q), as a function of χspin max.Therefore, it was verified that our main numerical results in the main text are unchanged even in the single- d-orbital model once the B1gform factor is taken into account. We also analyzed the CDW equation for the single- d-orbital model, and obtained the strong CDW instability. The obtained formfactor /Delta1/Sigma1 0(k) hasB1gsymmetry. In real space, this is the bond order ( =modulation of hopping integrals) given by the Fourier transformation of the symmetry-breaking self-energy/Delta1/Sigma1 0(k). Thus, the robustness of the spin-fluctuation-driven CDW mechanism has been clearly confirmed. APPENDIX D: ANALYSIS OF LINEARIZED CHARGE-DENSITY-WA VE EQUATION In the main text, we analyzed the d-pHubbard model for cuprate superconductors in an unbiased manner by using theRG+cRPA method. We find that the nematic CDW with d form factor is the leading instability. The axial nematic CDWinstability at q=Q ais the second strongest, and its strength increases under the static uniform CDW order. This result leadsto the prediction that uniform nematic CDW occurs at thepseudogap temperature T ∗, and the axial CDW at wave vector q=Qais induced under T∗. In this section, we study the CDW formation mechanism in cuprate superconductors based on the diagrammatic methodto find which many-body processes cause the CDW order.Theoretically, the CDW order is given as the symmetrybreaking in the self-energy /Delta1/Sigma1(k). According to Refs. [ 26,44], the self-consistent CDW equation is /Delta1/Sigma1(k)=/parenleftbig 1−P A1g/parenrightbig T/summationdisplay qV(q)G(k+q), (D1) where PA1gis the A1g-symmetry projection operator, and G(k)=[G−1 0(k)−/Delta1/Sigma1(k)]−1is the d-electron Green’s function with the symmetry-breaking term /Delta1/Sigma1.V(q)= U2[3 2χs(q)+1 2χc(q)−χ0(q)]+U, where χs(c)(q)= χ0(q)/[1−(+)Uχ0(q)] and χ0(q)=−T/summationtext kG(k+q)G(k). To analyze the CDW state with arbitrary wave vector q,w e linearize Eq. ( D1) with respect to /Delta1/Sigma1: λq/Delta1/Sigma1 q(k)=T/summationdisplay k/primeK(q;k,k/prime)/Delta1/Sigma1 q(k/prime), (D2) where λqis the eigenvalue for the CDW for the wave vector q. The CDW with wave vector qappears when λq=1, and the eigenvector /Delta1/Sigma1 q(k) gives the CDW form factor. The kernel K(q,k,k/prime) is given in Fig. 9(a), and its analytic expression is [44] K(q;k,k/prime)=/parenleftbigg3 2Vs 0(k−k/prime)+1 2Vc 0(k−k/prime)/parenrightbigg ×G0(k/prime+q/2)G0(k/prime−q/2) −T/summationdisplay p/parenleftbigg3 2Vs 0(p+q/2)Vs 0(p−q/2) +1 2Vc 0(p+q/2)Vc 0(p−q/2)/parenrightbigg ×G0(k−p)[/Lambda1q(k/prime;p)+/Lambda1q(k/prime;−p)],(D3) 165131-7TSUCHIIZU, KAW AGUCHI, Y AMAKAW A, AND KONTANI PHYSICAL REVIEW B 97, 165131 (2018) (a) (b) (c)MT AL Hartree FIG. 9. (a) Schematic linearized CDW equation for general wave vector q. (b) Examples of the VCs generated by solving the linearized CDW equation. (c) Higher-order AL processes. where Vs 0(q)=U+U2χs 0(q),Vc 0(q)=−U+U2χc 0(q), and /Lambda1q(k;p)≡G0(k+q 2)G0(k−q 2)G0(k−p). The subscript 0 in Eq. ( D3) represents the functions with /Delta1/Sigma1=0. By solving the linearized CDW equation ( D2), many higher- order vertex corrections (VCs) are systematically generated.Some examples of the generated VCs are shown in Fig. 9(b).I f we drop the Hartree term and MT term in K(q;k,k /prime), we obtain the series of higher-order AL-VCs shown in Fig. 9(c).T h eA L terms drive the q=0CDW instability since its functional form ∝/summationtext kχs(k+q)χs(k)i sl a r g ef o r q≈0[43]. Figure 10(a) shows the obtained qdependence of λq forαS=0.995 at T=50 meV . Here, we introduced the quasiparticle damping γ=0.3e Vi n t o G0(k). Here, λqis the largest at q=0, and the second largest maximum is at q=Qa. αS≡Umax q{χ0 0(q)}is the spin Stoner factor. We also show the eigenvalue λAL q(and the second-largest eigenvalue λAL,2nd q ), which is obtained by dropping the Hartree and MT terms in thekernel. That is, λ AL qis given by the higher-order AL processes shown in Fig. 9(c).A t q=0and Qa,λAL qis almost equal to the true eigenvalue λq. In the present analysis, we dropped the /epsilon1ndependence of /Delta1/Sigma1 q(k) by performing the analytic continuation ( i/epsilon1n→/epsilon1) and putting /epsilon1=0. We also dropped the /epsilon1ndependence of the quasiparticle damping γ. Due to these simplifications, the obtained λqis overestimated. Therefore, we do not put the constraint λq<1 here. In Fig. 10(b) , we show the eigenvalue λMT q, which is obtained by dropping the Hartree and AL terms in the kernel.It is much smaller than λ qatq=0and Qa, whereas λq atq=Qdis comparable to the true eigenvalue. Therefore, the origin of the CDW instability at q=0and Qais the AL process, whereas that at q=Qdis mainly the MT process. Figure 10(c) shows the eigenvalues at q=0,Qa, and Qd as a function of αS. As the spin susceptibility increases ( αS/greaterorsimilar 0.98),λqis drastically enlarged by the VCs, and λq=0becomes the largest due to the AL processes. The form factor at q=0, /Delta1/Sigma1 0(k), has the d-wave symmetry, as shown in Fig. 10(d) . We stress that the eigenvalue λqis quickly suppressed by increasing γ, which is actually large in cuprates. Figures 10(e) FS p pa a a aa ad d d)b( )a( )d( )c( )f( )e( FIG. 10. (a), (b) qdependencies of λq,λAL q,λAL,2nd q ,a n dλMT qfor αS=0.995 and γ=0.3e V .( c ) λqatq=0,Qa,a n d Qdas function ofαSforγ=0.3 eV . (d) Form factor for q=0(dwave). (e), (f) λqas function of αSforγ=0.6 eV in the cases of /Delta1Ep=0a n d /Delta1Ep=0.1 eV , respectively. and10(f) show the CDW susceptibilities for the larger damping rateγ=0.6 eV in the cases of /Delta1Ep=0 and /Delta1Ep=0.1 eV , respectively. (Note that the damping rate is renormalizedto be ∼γ/5 in cuprates.) In Fig. 10(e) ,λ qreaches unity first at q=0with increasing αS. In the nematic state with /Delta1Ep=0.1 eV shown in Fig. 10(f) ,λqatq=Qx aexceeds λQd forαS>0.996. The corresponding eigenvalue is ∼1.4, which decreases with increasing γ. This result supports the main result of the present RG +cRPA study shown in Fig. 4(a)in the main text. In summary, we analyzed the linearized CDW equation based on the d-pHubbard model, by including both the MT and AL VCs into the kernel. When the spin fluctuationsare strong ( α S/greaterorsimilar0.98), the uniform nematic CDW has the strongest instability. The axial CDW instability is stronglymagnified under the uniform CDW order, as we explain themain text. The results obtained are consistent with the resultsby the RG +cRPA in the main text. Thus, it is concluded that the higher-order AL processes give the CDW orders at q=0 and Q a. 165131-8MULTISTAGE ELECTRONIC NEMATIC TRANSITIONS IN … PHYSICAL REVIEW B 97, 165131 (2018) [1] G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G.Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo,E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer,and L. 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PhysRevB.99.241404.pdf

PHYSICAL REVIEW B 99, 241404(R) (2019) Rapid Communications Monolayer VTe 2: Incommensurate Fermi surface nesting and suppression of charge density waves Katsuaki Sugawara,1,2,3Yuki Nakata,1Kazuki Fujii,1Kosuke Nakayama,1Seigo Souma,2,3 Takashi Takahashi,1,2,3and Takafumi Sato1,2,3 1Department of Physics, Tohoku University, Sendai 980-8578, Japan 2Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 3WPI Research Center, Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan (Received 11 March 2019; revised manuscript received 4 June 2019; published 18 June 2019) We investigated the electronic structure of monolayer VTe 2grown on bilayer graphene by angle-resolved photoemission spectroscopy (ARPES). We found that monolayer VTe 2takes the octahedral 1 Tstructure in contrast to the monoclinic one in the bulk, as evidenced by the good agreement in the Fermi surface topologybetween ARPES results and first-principles band calculations for octahedral monolayer 1 T-VTe 2.W eh a v e revealed that monolayer 1 T-VTe 2at low temperatures is characterized by a metallic state whereas the nesting condition is better than that of isostructural monolayer VSe 2which undergoes a charge density wave (CDW) transition to an insulator at low temperatures. The present result suggests an importance of the Fermi surfacetopology for characterizing the CDW properties of monolayer transition-metal dichalcogenides. DOI: 10.1103/PhysRevB.99.241404 Layered transition-metal dichalcogenides (TMDs) are a promising candidate for realizing the outstanding propertiesassociated with two-dimensionalization since bulk TMDs areknown to exhibit various physical properties such as mag-netism, the Mott-insulator phase, and charge density waves(CDWs), besides the wide range of transport properties (in-sulator, semiconductor, metal, and superconductor), most ofwhich are prone to the change in the dimensionality of ma-terials. When TMDs are thinned to a single monolayer [two-dimensional (2D) limit], they exhibit even more outstandingproperties distinct from the bulk, as represented by the room-temperature ferromagnetism in VSe 2in contrast to the non- magnetic nature of the bulk [ 1] and the change in the band-gap property from an indirect to direct transition in MoS 2[2]. The CDW is a most pronounced phenomenon widely seen in bothbulk and atomic-layer TMDs. In bulk TMDs, the interplaybetween the Fermi surface (FS) nesting and the energy-gapopening as well as its relationship to the strength of CDWproperties such as the CDW transition temperature ( T CDW) have been a target of intensive studies [ 3]. The role of dimensionality in the mechanism of CDW, in particular whether or not the CDW is more stable inthe 2D limit, is now becoming a target of fierce debates,being stimulated by a recent success in fabricating variousatomic-layer TMDs by exfoliation and epitaxial techniques.Recent studies on some TMDs such as TiSe 2,V S e 2, and NbSe 2[4–8] have shown a marked increase in TCDW upon reducing the thickness down to a few monolayers. In contrast,it has been reported that the CDW vanishes in monolayer TaS 2 and TaSe 2[9,10], suggesting an important role of the substrate and many-body effects. In 1 T-VSe 2, reducing the number of layers by exfoliating the bulk crystal leads at first to a gradualdecrease of T CDW, but at a critical film thickness of ∼10 nm, theTCDW exhibits a characteristic upturn and reaches 140 K in a few monolayers, much higher than bulk TCDW (110 K) [ 11]. Such an enhancement of TCDW in monolayer VSe 2wasalso revealed by angle-resolved photoemission spectroscopy (ARPES) wherein the role of FS nesting and electron-phononcoupling was intensively debated [ 4,6,7,12–14]. However, the nature of 2D CDW is still far from having reached aconsensus, as highlighted by a wide variety of periodic latticedistortions hitherto proposed for monolayer VSe 2(e.g., 4 ×4, 4×1,√ 7×√ 3, 4×√ 3) [6,7,12–14]. From a broader per- spective, it is still unclear to what extent the conventional FS-nesting picture can be applied to the 2D CDW materials andhow the FS topology and electronic interactions are relatedto the CDW properties such as the enhancement /suppression ofT CDW. To address these essential questions, a further study on the electronic structure with different monolayer TMDs ishighly required. In this Rapid Communication, we report a successful fab- rication of monolayer VTe 2on bilayer graphene /SiC and its ARPES investigation. While the monoclinic (1 T/prime/prime) phase is known to be stable in bulk VTe 2, our monolayer VTe 2film takes the octahedral 1 Tstructure [see Fig. 1(a)]. Importantly, this enables us to directly compare the electronic states andCDW properties with isostructural monolayer 1 T-VSe 2.O u r ARPES study on monolayer VTe 2signifies a large, nearly perfectly nested triangular FS centered at the Kpoint in the Brillouin zone (BZ). We also found that the V 3 dband apparently crosses the Fermi level ( EF) midway between the/Gamma1andKpoints, indicative of the metallic state at low temperatures, unlike monolayer VSe 2which shows a fully gapped insulating state below 140 K. We discuss the possibleorigins for such an intriguing difference in terms of thevariation in the FS-nesting condition and electron-phononcoupling. A high-quality monolayer VTe 2film was grown on bilayer graphene by the molecular-beam epitaxy (MBE) method.ARPES measurements were performed at the BL-28B beam-line in Photon Factory, KEK. First-principles band-structurecalculations were carried out using the QUANTUM ESPRESSO 2469-9950/2019/99(24)/241404(6) 241404-1 ©2019 American Physical SocietyKATSUAKI SUGAWARA et al. PHYSICAL REVIEW B 99, 241404(R) (2019) FIG. 1. (a) Crystal structure of monolayer 1 T-VTe 2.( b ) , (c) RHEED patterns of bilayer (BL) graphene and monolayer (ML) VTe 2on BL graphene, respectively. (d) AFM image of monolayer VTe 2. High (white), medium (blue), and low (dark) intensity regions are attributed to atoms /molecules adsorbed on VTe 2, clean mono- layer VTe 2, and atom molecules adsorbed on a graphene substrate, respectively. The reason why we did not attribute the dominant whiteand blue areas to graphene or 2 ML VTe 2is because we performed in situ ARPES measurements with the same sample and observed a dominant contribution from the monolayer energy bands to thetotal ARPES intensity. (e) Height profile along a cut shown by the magenta solid line in (d). (f) Photon-energy dependence of the EDC at the/Gamma1point in monolayer VTe 2. code with the generalized gradient approximation [ 15,16]. For details, see Sec. 1 of the Supplemental Material [ 17]. First, we present a characterization of monolayer VTe 2 film. Figure 1(b) shows the reflection high-energy electron diffraction (RHEED) pattern of pristine bilayer graphene ona SiC(0001) substrate. We clearly observe the 1 ×1 and 6√ 3×6√ 3R30◦streak patterns, which correspond to bilayer graphene and the underlying carbon-mesh layer on SiC, re-spectively [ 18]. After growing a VTe 2film by codepositing V and Te atoms onto the bilayer-graphene surface, the RHEEDintensity from the substrate disappears, and a sharp 1 ×1 streak pattern appears [Fig. 1(c)], similarly to the case of other monolayer TMD films grown on bilayer graphene [ 6,19,20], indicating the formation of VTe 2. The absence of additional streak patterns suggests no inclusion of a monoclinic phasewith a zigzag chain structure which is known to exist in bulkVTe 2below 482 K [ 21,22]. This situation is particularly suited FIG. 2. (a) Plot of ARPES intensity for monolayer VTe 2along the/Gamma1Mand/Gamma1Kcuts measured with hν=56 eV at T=40 K. (b) Band structure obtained from the first-principles band-structure calculations for monolayer 1 T-VTe 2with the input of an experimen- tal in-plane lattice constant (3.35 Å). Overall calculated bands werecontracted by 13% in the energy axis to find a reasonable matching with the experiment. (c), (d) Experimental band structure near E F, measured along the /Gamma1Mand/Gamma1Kcuts, respectively. Red and purple dashed curves are a guide for the eyes to trace the Te 5 pand V 3 d bands, respectively. for comparing the electronic states with other 1 Tmonolayer polymorphs. We come back to this point later in detail. Asseen in the ex situ atomic force microscopy (AFM) image in Fig. 1(d), large islands with a typical height of 0.8 nm, which correspond to that of a monolayer [Fig. 1(e)], are recognized on bilayer graphene. The energy distribution curve (EDC) atthe/Gamma1point in Fig. 1(f)signifies no detectable photon-energy (hν) variation in the energy position of the bands, supporting the 2D nature of the electronic states. We have estimated the in-plane lattice constant of mono- layer VTe 2asa∼3.35 Å at room temperature by compar- ing the relative position of the RHEED patterns betweengraphene and VTe 2. This value is in good agreement with that estimated from the absolute wave vector values at the M (1.08±0.03 Å−1) and K(1.24±0.03 Å−1) points relative to the/Gamma1point in the ARPES data (3 .35±0.09 Å) (including error bars due to the angular resolution and the angle-to- k conversion). Intriguingly, these values are much smaller thanthat of bulk octahedral 1 T-VTe 2(3.64 Å; obtained above 482 K where the 1 Tphase is stable). Taking into account that the coupling between graphene and the 1 T-VTe 2film could be sufficiently weak due to the van der Waals coupling nature,it is inferred that the value of 3.35 Å could be the most stablelattice parameter for free-standing monolayer VTe 2. Figures 2(a) and 2(b) display the plot of valence-band ARPES intensity for monolayer VTe 2measured at T=40 K along the /Gamma1Mand/Gamma1Kcuts with hν=56 eV , compared with the corresponding band dispersion obtained by thefirst-principles band-structure calculations for free-standing 241404-2MONOLAYER VTe 2: INCOMMENSURATE FERMI … PHYSICAL REVIEW B 99, 241404(R) (2019) monolayer 1 T-VTe 2with the input of an experimental lattice constant a. One can see in Fig. 2(a) several energy bands whose dispersion appears to be symmetric with respect to the/Gamma1point. A side-by-side comparison of Figs. 2(a) and2(b) also reveals a good agreement in the overall valence-band structurebetween the experiment and calculations, demonstrating thatthe fabricated monolayer VTe 2indeed takes the 1 Tstructure (see Sec. 2 of the Supplemental Material for details [ 17]). According to the calculations, energy bands lying at thebinding energy ( E B) of 1–6 eV , including the holelike bands which rapidly move toward EFon approaching the /Gamma1point, are attributed to the Te 5 porbitals, while the energy band within 0.5 eV of EFwith a relatively flat dispersion around the M point is assigned as the V 3 dband. To see more clearly the electronic states responsible for the physical properties, we show in Figs. 2(c)and2(d) the ARPES intensity near EFatT=40 K measured along the /Gamma1Mand /Gamma1Kcuts, respectively. A detailed spectral analysis by tracing the peak position of the EDCs suggests that the two topmostTe 5 pbands with different band velocities do not reach E F, but are topped at 60 meV below EFat the /Gamma1point. These bands are degenerate exactly at the /Gamma1point, consistent with the calculation in Fig. 2(b). We found that the shallow V 3 d band is also located at ∼60 meV below EFat the /Gamma1point and disperses toward higher EBon approaching the Mpoint [Fig. 2(c)], while it crosses EFmidway between the /Gamma1andK points, accompanied with a sudden drop in the spectral weight[Fig. 2(d)]. While the overall experimental band structure shows a good agreement with the calculated band structure forthe 1 Tphase, we found that some bands near E Fin the experi- ment are renormalized with respect to those in the calculations(for details, see Sec. 2 of the Supplemental Material [ 17]). It is noted here that we found no evidence for the energy splittingof bands associated with a possible exchange splitting due toferromagnetism, which is further corroborated by our x-raymagnetic circular dichroism measurement at 80 K showingno change in the V L 2,3-absorption edge across the magnetic- field reversal [ 23]. This suggests the absence of ferromagnetic order in monolayer VTe 2. At this stage, it is unclear why the ferromagnetism appears in monolayer VSe 2but not in monolayer VTe 2, though it is noted that the ferromagnetism in monolayer VSe 2itself is contradictory and is currently a target of fierce debate [ 1,6,7]. It is also unknown whether or not the ferromagnetic property is related to the CDW. To clarify the topology of FS, we have performed ARPES measurements in 2D kspace. Figures 3(a)–3(c) show the contour maps of ARPES intensity for different EBslices. At EB=EF[Fig. 3(a)], one can recognize a couple of fairly straight intensity patterns around the Mpoint running parallel to the /Gamma1Mdirection (red dashed line). This intensity pattern forms a large, almost perfectly triangular-shaped FS enclosingtheKpoint. Remarkably, this FS is well reproduced by the calculations for free-standing 1 T-VTe 2[Fig. 3(d)] with the input of an experimental lattice constant, confirming againthe 1 Tnature of our epitaxial film. The absence of any spurious intensity that could be associated with the bandfolding with (3 ×1) periodicity expected from the formation of a double zigzag-chain superstructure seen in bulk VTe 2[22] further corroborates the purely 1 Tnature of the film (see Sec. 4 of the Supplemental Material for details [ 17]). FIG. 3. (a)–(c) Plots of ARPES intensity at T=40 K as a function of 2D wave vectors, kxandky, at three representative energy slices at EB=EF, 0.2, and 0.4 eV , respectively. Energy contours were obtained by integrating the intensity within ±50 meV with respect to each EB. (d) Calculated FS obtained from the first-principles band-structure calculations for monolayer 1 T-VTe 2. (e) ARPES- derived band structure along three representative kcuts [cuts A–C in (a)] which cross the triangular FS. The systematic evolution of the V-shaped band dispersions from cut A to cut C indicates the holelike nature of triangular FS. Upon increasing EBto 0.2 eV [Fig. 3(b)], the experimental triangular pattern seen at EB=0 eV [Fig. 3(a)] transforms into an M-point-centered ellipsoid elongated along the /Gamma1M direction, which shrinks on further increasing EBto 0.4 eV [Fig. 3(c)]. This indicates that the triangular FS forms a hole pocket, consistent with the calculated band dispersion inFig. 2(b) in which the V 3 dband is located at ∼1 eV above E Fat the Kpoint. Figure 3(e) shows the ARPES-derived band structure along three representative kcuts (cuts A–C) which cross the triangular FS. On cut A, which touches the cornerof the triangular FS, one can see a couple of V-shaped bands in the vicinity of E F. These two V-shaped bands are gradually separated from each other on going from cut A to cuts B andC, indicating that the triangular FS is holelike. In Figs. 3(b) 241404-3KATSUAKI SUGAWARA et al. PHYSICAL REVIEW B 99, 241404(R) (2019) FIG. 4. (a), (b) Schematic FS together with a kcut and kpoints where high-resolution ARPES measurements were performed for monolayer VTe 2and VSe 2, respectively. (c), (d) Plots of ARPES intensity as a function of the wave vector and EBsymmetrized with respect toEF, measured at T=40 K along a cut crossing the corner of triangular pocket (cut A /B) for monolayer VTe 2and VSe 2, respectively. (e) EDCs near EFatT=40 K for monolayer VTe 2measured at various kFpoints in (a). (f) Same as (e) but symmetrized with respect to EF. Zero intensity for each spectrum is indicated by a dashed magenta line to highlight the absolute spectral weight. (g) Same as (f) but for monolayer VSe 2[6]. (h) Comparison of the FS topology between monolayer VTe 2and VSe 2. Red and blue circles correspond to the kFpoints for monolayer VTe 2and VSe 2, respectively. Orange and light blue arrows indicate possible nesting vectors qfor monolayer VTe 2and VSe 2, respectively. and3(c), one can also identify an intense circular spot at the /Gamma1point stemming from the Te 5 pbands. We emphasize again that although the proximity of Te 5 pbands to EFenhances the intensity at the /Gamma1point, these fully occupied bands do not participate in the FS. Therefore, the FS of monolayer VTe 2is solely dictated by the triangular hole pocket at the Kpoint, which greatly simplifies the discussion on the FS topologyand nesting, as detailed later. We have estimated the totalcarrier concentration to be 0 .98±0.08 electrons /unit cell, by evaluating the area of FS with respect to that of the whole BZ.This suggests that our monolayer film remains stoichiometricand no observable charge transfer from the substrate takesplace. Now that the FS topology is established, we shall address a key question regarding a possible energy gap opening associ-ated with the occurrence of CDW. We selected a kcut passing the corner of the triangular FS [blue line in Fig. 4(a)], and show the ARPES intensity at T=40 K plotted as a function of the wave vector and E Bsymmetrized with respect to EFin Fig. 4(c). One can clearly see a dispersive band reaching EF showing the brightest intensity at EF, indicating the absence of an energy gap. This is in sharp contrast to the result ofmonolayer VSe 2[Figs. 4(b) and4(d)] that signifies a marked suppression of intensity within ±0.1e Vo f EFatT=40 K due to the CDW-gap opening [ 6]. To see the low-energy spectral feature in more detail, we have performed high-resolution ARPES measurements along several cuts crossingthe FS, and show the EDCs at various k F(Fermi wave vector) points (points 1–7) covering the whole straight segment of thetriangular FS in the first BZ in Fig. 4(e). The correspond- ing symmetrized EDCs in Fig. 4(f) show a single peak at points 1 and 2 located around the corner of FS, while theEDCs at points 3–7 exhibit a weak dip structure at E Fthat is indicative of a large residual spectral weight at EF.W e attribute this spectral-weight suppression as the pseudogap,but not the CDW gap, since the spectral behavior resembles that of monolayer VSe 2at room temperature (well above TCDW∼140 K), which shows the coexistence of pseudogap and metallic Fermi-arc states [ 6]. Also, the pseudogap of VTe 2persists over a wide temperature range (10–300 K), similarly to VSe 2. It is noted that the pseudogap is unlikely to be due to some extrinsic effects such as the sample /surface quality and /or the experimental conditions (e.g., photoioniza- tion cross section and light polarization), but is an intrinsicfeature of monolayer VTe 2. The pseudogap may be explained in terms of the CDW fluctuations and /or the electron-phonon coupling associated with the CDW (see also Sec. 3 of theSupplemental Material [ 17]). The metallic state revealed in Figs. 4(c) and 4(e) in VTe 2is obviously different from the fully gapped insulating state below TCDW, as visible from the strong spectral-weight suppression around EFover the entire FS as seen in Fig. 4(g). We thus conclude that the CDW is suppressed in monolayer VTe 2. A key to understanding such contrasting behavior may lie in the difference in the FS topology between the twoV-dichalcogenide monolayers. Figure 4(h) directly compares the FS obtained from ARPES measurements of monolayerVTe 2and VSe 2[6]. One can immediately recognize that both monolayers show a similar triangular pocket at the K point whereas a circular hole pocket exists only in VSe 2. Extra hole carriers at the /Gamma1-centered pocket in VSe 2reside on the K-centered pocket in VTe 2, as seen from a larger triangular pocket in VTe 2. This is reasonable since the Se and Te atoms are isovalent and the total FS area should be thesame. The expansion of the triangular pocket would widenthe straight segment of the FS in VTe 2. Assuming that the nesting vector qis parallel to the /Gamma1Mdirection [ 6], this would lead to an enhancement of electronic susceptibility inVTe 2. Thus, one would naively expect that the CDW in VTe 2 is more stable than that in VSe 2according merely to the 241404-4MONOLAYER VTe 2: INCOMMENSURATE FERMI … PHYSICAL REVIEW B 99, 241404(R) (2019) FS-nesting picture. However, this is not the case since the CDW appears to be suppressed in VTe 2. Thus, one cannot sufficiently describe the CDW of monolayer VSe 2or VTe 2 simply in terms of the energy gain around EFin the elec- tronic system, which suggests the importance of consideringelectron-phonon coupling [ 24]. While a detailed discussion on the relevance of electron-phonon coupling requires sophisti-cated first-principles band-structure calculations, we point outhere a possibility that such an electron-phonon coupling couldbe linked to the electronic states via the commensurabilityof the nesting. We found that the nesting vector along the/Gamma1Mdirection in VSe 2is commensurate to the lattice (1 /4G, where Gis the reciprocal lattice vector) [ 6], while that in VTe 2is incommensurate (1 /4.6G). If such commensurability enhances the electron-phonon coupling at the correspondinglattice periodicity, it may stabilize the CDW. However, thisexplanation is still speculative, requiring further experimentaland theoretical studies to firmly pin down the CDW origin. Our band calculations of monolayer VTe 2show that a small (1 .5%) change in the in-plane lattice constant is suffi- cient to control the emergence /absence of a small hole pocket at/Gamma1and the concomitant change in the FS-nesting condition at the triangular pocket. Such a sensitivity of the FS topologyto the lattice parameters is essential due to the fact that thenarrow V 3 dband is located in the vicinity of E F. Therefore, we expect that small perturbations such as lattice strain andcarrier doping would easily trigger the change in the FStopology and the nesting vector, leading to the modulationof CDW properties. In this regard, the reported differencesin the FS topology around the /Gamma1point in VSe 2, which maylink to diverse periodic lattice distortions [ 4,6,7,13], may be interpreted in terms of a reflection of the high sensitivity ofthe CDW characteristics to the strain and carrier balance. Thepresent result suggests an importance of precisely controllingthe lattice strain and carrier concentration for manipulating theCDW. Such band engineering would be a main target of futurestudies. In conclusion, we have performed an ARPES study on monolayer 1 T-VTe 2grown on bilayer graphene by MBE. We found a large triangular FS at the Kpoint that satisfies a nearly perfect nesting condition, whereas the CDW is suppressed ashighlighted by the observation of E Fcrossing of bands at low temperature, in contrast to monolayer VSe 2that exhibits a well-defined CDW characterized by a fully gapped insulatingstate. The present result opens a pathway toward control-ling the physical properties of 2D TMDs through the bandengineering. We thank Y . Umemoto, K. Horiba, and H. Kumigashira for their assistance in the ARPES experiments. This workwas supported by the MEXT of Japan (Innovative Area“Topological Materials Science” JP15H05853), JST-PREST(No. JPMJPR18L7), JST-CREST (No. JPMJCR18T1), JSPSKAKENHI Grants (No. JP18K18986, No. JP18H01821, No.JP18H01160, No. JP17H04847, and No. JP17H01139), KEK-PF (Proposal No. 2018S2-001), and science research projectsfrom Murata Science Foundation, World Premier Interna-tional Research Center, Advanced Institute for Materials Re-search. Y .N. acknowledges support from GP-Spin at TohokuUniversity Grant-in-Aid for JSPS Fellows. [1] M. Bonilla, S. Kolekar, Y . Ma, H. C. Diaz, V . Kalappattil, R. Das, T. Eggers, H. R. Gutierrez, M.-H. Phan, and M. Batzill,Nat. Nanotechnol. 13,289(2018 ). [2] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105,136805 (2010 ). [3] K. Rossnagel, J. Phys.: Condens. Matter 23,213001 (2011 ). [4] P. Chen, Y .-H. Chan, X.-Y . Fang, Y . Zhang, M. Y . Chou, S.-K. Mo, Z. Hussain, A.-V . Fedorov, and T.-C. Ching, Nat. 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PhysRevB.84.115426.pdf

PHYSICAL REVIEW B 84, 115426 (2011) Spin orientation and sign of the Rashba splitting in Bi/Cu(111) Hendrik Bentmann,1Takuya Kuzumaki,2Gustav Bihlmayer,3Stefan Bl ¨ugel,3Eugene V . Chulkov,4,5 Friedrich Reinert,1,6and Kazuyuki Sakamoto2 1Experimentelle Physik VII and R ¨ontgen Research Center for Complex Material Systems (RCCM), Universit ¨at W ¨urzburg Am Hubland, D-97074 W ¨urzburg, Germany 2Graduate School of Advanced Integration Science, Chiba University, Chiba 263-8522, Japan 3Peter Gr ¨unberg Institut and Institute for Advanced Simulation, Forschungszentrum J ¨ulich and JARA, D-52425 J ¨ulich, Germany 4Donostia International Physics Center, 20018 San Sebasti ´an/Donostia, Basque Country, Spain 5Departamento de F ´ısica de Materiales and Centro Mixto CSIC-UPV/EHU, UPV/EHU, Apartado 1072, 20080 San Sebasti ´an/Donostia, Basque Country, Spain 6Karlsruher Institut f ¨ur Technologie (KIT), Gemeinschaftslabor f ¨ur Nanoanalytik,, D-76021 Karlsruhe, Germany (Received 5 July 2011; published 19 September 2011) Whereas the magnitude of the Rashba spin splitting at surfaces has been studied in detail, less is known about its sign which determines the precise spin orientation of the electronic states. We investigate the microscopicorigin of this sign by spin-resolved photoemission experiments and first-principles calculations on the stronglyspin-orbit coupled surface states in Bi/Cu(111). We conclude that the sign of the Rashba splitting is determinedby the particular charge asymmetry near the atomic cores. The precise spin orientation on heavy-element surfacesthus can comprise information on wave-function localizations and related aspects otherwise hardly accessible byother experimental methods. DOI: 10.1103/PhysRevB.84.115426 PACS number(s): 73 .20.At, 71 .70.Ej, 79.60.−i I. INTRODUCTION In a solid-state system that preserves both time-reversal and spatial inversion symmetry the electronic states are necessarilyspin degenerate. 1This degeneracy is broken by the spin- orbit interaction for electrons moving in two-dimensional(2D) surface-, interface- or quantum-well geometries whichgive rise to inversion-asymmetric confinement potentials. 2,3 This Rashba effect is an important mechanism in the field of spintronics as it can be utilized for the manipula-tion of spin-polarized currents in designated semiconductorheterojunctions. 4,5Particularly large Rashba splittings are observed in the electronic structure of clean or monolayercovered heavy-element surfaces. 6–12Related spin-splitting mechanisms are found for the surface states on topologicalinsulators. 13,14 The Rashba effect in a 2D electron gas leads to a splitting of the free-electron dispersion via a potential gradientperpendicular to the plane of confinement, E ±(k)=E0+¯h2k2 2m∗±|α||k|, (1) where m∗is the effective mass and the absolute value of the Rashba parameter αmeasures the size of the spin splitting [see Fig.1(a)]. The spin orientation of the states E±is then given by P±(k)=±α |α|(−ky,kx,0)/|k|. (2) In Eq. ( 2) we use the same sign convention as in Ref. 15. The spin polarization vector P±is oriented in plane and perpendicular to the wave vector. Furthermore, the branchesE ±have opposite spin orientations P+=− P−. The absolute directions of P±are determined by the sign of the Rashba parameter α.15,16Note that the sign of the effective mass defines whether the branch E+(and accordingly E−)i st h e outer or the inner branch in Fig. 1. As a result the spinorientation of the inner and of the outer branch depend on both the sign of the Rashba parameter and the sign of theeffective mass [see Fig. 1(b)]. Previous experimental and theoretical work gave increas- ingly detailed insights into parameters that determine themagnitude of the Rashba parameter in surface and thin-film systems. 6,11,17–29In the present paper we address a related but less explored issue, namely, the sign of the Rashba parameter at surfaces. At first glance one may not expect the sign of α to vary between different surfaces as the potential gradientbetween crystal and vacuum probed by the surface-state wavefunction should be similar for all systems. On the other hand,Rashba-split quantum-well states in Pb/Si(111) show a spin FIG. 1. (Color online) Dispersion in (a) and spin orientation in (b) for the Rashba model in a two-dimensional electron gas: The spin-orbit interaction splits the free-electron parabola (dashed line)into the branches E ±. The spin orientation, indicated by arrows, of the inner and outer branch depends on the sign of the Rashba parameter αand of the effective mass m∗. 115426-1 1098-0121/2011/84(11)/115426(6) ©2011 American Physical SocietyHENDRIK BENTMANN et al. PHYSICAL REVIEW B 84, 115426 (2011) orientation that is reversed when compared to the Shockley- type surface state on Au(111).30This has been attributed to competing effects at the two terminating interfaces of the Pbfilm. In this paper we investigate the sign of the Rashba splitting at surfaces and its microscopic origin using spin- and angle-resolved photoelectron spectroscopy (SARPES) for the surfacealloy Bi/Cu(111). The measured spin orientation in Bi/Cu(111)is in accordance with Eq. ( 2) and implies a negative Rashba parameter α. We confirm these findings by a first-principles calculation of the spin orientation. Employing a simple modelin combination with our calculations, we argue that the sign ofαis determined by local wave-function asymmetries in the vicinity of the Bi nuclei. Thus, more generally, our studyidentifies the sign of the Rashba parameter at surfaces asan important experimental observable. In particular, whencombined with calculations, it can provide information onwave-function localization, orbital character, and symmetriesof electronic states. We compare our results for Bi/Cu(111)with previous investigations on Au(111). 15 Noteworthy, the spin-split surface states of topological insu- lators show related spin orientations as depicted in Fig. 1(b).31 It is conceivable that similar mechanisms as described here for Rashba-split surface states also have an impact on the precisespin orientation of these topological states and hence on thesign of spin currents carried by them. The Bi/Cu(111) surface alloy forms a Rashba-split surface state with negative effective mass and mainly Bi sp zorbital character.27,28,32Similar to Au(111), this spzstate shows a dispersion perfectly matching the one prescribed by Eq. ( 1): two parabolic bands shifted in kand circular momentum distributions (see Ref. 28for further details). In addition, another surface state of pxpyorbital character has been found for Bi/Cu(111). In this paper we will focus on the Rashbasplitting and spin orientation of the sp zsurface state. II. EXPERIMENTAL AND THEORETICAL DETAILS Spin-resolved and spin-integrated experiments were carried out at room temperature (RT) using a newly designed SARPESsetup at Chiba University, Japan. The spectrometer consistsof a Scienta R4000 electron analyzer for energy and angularresolution as well as a Scienta Mott detector operated at25 keV for the spin analysis. The geometry of the setupallows for a parallel detection of two components of thespin orientation, in this case the component normal to thesurface and the in-plane component perpendicular to the wavevector (henceforth the Rashba direction). We employed amonochromated Xe discharge lamp (MB Scientific) as wellas a nonmonochromated He lamp as UV-light sources. Forall presented measurements we used excitation energies of8.44 eV (Xe I) or 21.22 eV (He I). We conducted the experiments at energy resolutions of ∼50 meV for the ARPES and 100–160 meV for the SARPES measurements. TheSAPRES spectra were taken with an acceptance angle of±3 ◦for Xe Iand±1.5◦for He I. The effective Sherman function for the SARPES experiments was 0.18. Careful in situpreparation of the single-crystalline Cu(111) substrate by repeated cycles of Ar-ion sputtering and annealing to ∼1000 K resulted in a clean and well-ordered surface as verified by ARPES intensity [a.u.] -1.2 -0.8 -0.4 0.0 Energy [eV]0°-20° 18°-10° 10°-0.20.00.2 Wave vector ky[1/Å] -0.6-0.4-0.20.00.2Energy [eV] -0.2 0.0 0.2 Wave vector kx[1/Å](a) (b)(c) 1 32KM FIG. 2. (Color online) Spin-integrated photoemission from Bi/Cu(111) obtained with Xe Iexcitation. (a) and (b) show the Fermi surface and the band structure along the ¯/Gamma1¯Kdirection. In (c) we display energy distribution curves around ¯/Gamma1obtained from the same data set as the map in (b). the photoemission linewidth of the L-gap surface state.33The surface alloy reconstruction Bi-Cu(111)(√ 3×√ 3)R30◦was obtained after evaporation of 1–2 monolayers (ML) of Bi andsubsequent postannealing at ∼500 K. We verified the qualtity of the surface by low-energy electron diffraction (LEED) andARPES. The base pressure for all experiments was lower than2×10 −10mbar. The calculation of the surface electronic structure is based on the full-potential linearized augmented plane-wave methodand density functional theory as implemented in the FLEUR code.34The surface was simulated in a slab geometry with a ten-layer Cu(111) film terminated on one end by theBi-Cu(111)(√ 3×√ 3)R30◦surface alloy reconstruction. For further specifications of the calculation, we refer the reader toRef. 28. III. SPIN ORIENTATION IN Bi/Cu(111) We first briefly discuss the spin-integrated electronic struc- ture of Bi/Cu(111) as obtained with Xe Iexcitation energy at RT (see Fig. 2). The band structure along ¯/Gamma1¯Kand the Fermi surface (FS) reproduce our earlier results in Ref. 28. The FS consists of one hexagonal and two circular contours.Three states labeled in Fig. 2(b) by ascending numbers cross the Fermi energy at ±0.1˚A −1,±0.17˚A−1, and±0.29˚A−1. We identify the features “1” and “2” with the inner and theouter branch of the sp zsurface state. Branch “3” corresponds to the pxpystate. Note that, despite temperature-induced spectral broadening, the three bands are still well resolvedat RT. This is particularly evident from Fig. 2(c), where we show energy distribution curves (EDCs) for emission angles 115426-2SPIN ORIENTATION AND SIGN OF THE RASHBA ... PHYSICAL REVIEW B 84, 115426 (2011) Spin-resolved photoemission intensity [a.u.] 1.2 0.8 0.4 0.0 Energy [eV]-3° -5° -8° -9° -10° -11° -11.5° -13.5° -15° -17° -18°(a) Xe I 0.8 0.4 0.0 Energy [eV]6° 8° 9° 10° 12°(b) 5° 6°Xe I He I 1.21.00.80.60.40.20.0-0.2-0.4 Energy [eV] -0.4 -0.2 0.0 0.2 0.4 Wave vector kx [1/Å] (c) (d)1.21.00.80.60.40.20.0-0.2-0.4 Energy [eV] -0.4 -0.2 0.0 0.2 0.4 Wave vector kx [1/Å] +y -y FIG. 3. (Color online) Spin-resolved electronic structure of Bi/Cu(111). In (a) and (b) we show spin-resolved energy distribution curves along ¯/Gamma1¯Kfor negative and positive emission angles. The respective excitation energies are indicated. The spectra in (a) were recorded with an energy resolution of 160 meV , and the ones in (b) with 160 meV (He I) and 100 meV (Xe I). In (c) we present a first-principles calculation of the spin-resolved surface electronic structure. (d) shows a high-resolution photoemission map of Bi/Cu(111) (taken from Ref. 28). Additional markers indicate the peak positions obtained from the spin-resolved datasets in (a) and (b). For all panels the spin-quantization axis is in planeand perpendicular to the wave vector k x(¯/Gamma1¯K). Red (light) and blue (dark) symbols correspond to a spin orientation parallel to yand−y, respectively. In (d) closed (open) symbols indicate data points obtained with Xe I (He I) excitation. θearound the surface normal in steps of 2◦(at the Fermi level θe=20◦corresponds to kx=0.35˚A−1for Xe I). The peak positions are indicated where red (upward pointing) and blue(downward pointing) markers refer to the anticipated oppositespin orientations of the sp zbranches imposed by the Rashba model. Black (rectangular) markers denote the pxpybranch. Note that while the inner spzbranch shows a fairly symmetric intensity behavior around ¯/Gamma1, the spectral weight of the outer spzstate is almost entirely suppressed for positive θe.T h i s is a result of strongly varying photoemission matrix elementswhich have been observed for other surface alloys in a similarfashion. 26 Having confirmed the results of previous studies on the spin-integrated electronic structure, we next present ourexperimental findings on the spin orientation in Bi/Cu(111). InFig.3we show spin-resolved EDCs N ↑(red upward triangles) andN↓(blue downward triangles) along ¯/Gamma1¯K(kx) for negative emission angles in Fig. 3(a)and for positive angles in Fig. 3(b). The spectra were recorded with Xe Iexcitation except for two spectra which were taken by He Iexcitation [see according to the indications in Figs. 3(a) and 3(b)]. Both panels refer to a spin quantization axis parallel to the yaxis (Rashba direction).Spin up ( ↑) and spin down ( ↓) correspond to a spin orientation pointing in the yand−ydirections, respectively. Note that we use a right-handed coordinate system with the zaxis pointing out of the surface plane toward the vacuum. Considering theEDCs in Fig. 3(a) for|θ e|/greaterorequalslant8◦we find peaks in N↑andN↓ which are separated in energy. Both of them disperse to higher binding energies for increasing |θe|. By comparison with the spin-integrated data in Fig. 2we can associate these peaks with the inner and the outer spzbranch. Hence, we find that these bands are spin polarized with opposite spin orientationsalong the Rashba direction. In the EDCs taken with Xe I forpositive θ ein Fig. 3(b) we observe a dispersive feature only forN↑which is ascribed to the inner spzbranch. We thus conclude that the inner spzstate reverses its spin orientation for opposite kdirections. The absence of a second peak is attributed to the aforementioned matrix element suppressionof the outer branch. In order to avoid the suppression ofthe outer sp zbranch, we collected additional data using He Iexcitation. Note that equivalent wave vectors correspond to approximately half the emission angle for He Icompared to Xe I. Indeed, the spectra in for He Iexcitation show an additional peak in N↓at lower binding energies which is attributed to 115426-3HENDRIK BENTMANN et al. PHYSICAL REVIEW B 84, 115426 (2011) the outer spzstate. This confirms that the spin orientation of the outer spzbranch is reversed for opposite kdirections as well. In our measurements of the out-of-plane component ofthe spin orientation (not shown) we did not find a significantspin polarization and it is hence estimated to be smallerthan∼5% . The experimental results on the spin-resolved electronic structure of Bi/Cu(111) are summarized in Fig. 3(d).W ep l o t a spin-integrated high-resolution ARPES map obtained withHe I(taken from Ref. 28) and, additionally, the peak positions obtained from the spin-resolved EDCs in Figs. 3(a) and 3(b). Again, red (upward) and blue (downward) symbols representa spin orientation in the yand−ydirections, respectively. Both datasets show a sound agreement especially for theinner state whereas the remaining discrepancies are attributedto the considerably reduced experimental resolution in thespin-resolved measurements. To conclude, we find a spinorientation of the sp zsurface state according to Eq. ( 2)a s prescribed by the Rashba model. The absolute directionsmatch those depicted in Fig. 1(b) for the two combinations (α> 0,m ∗>0) and ( α< 0,m∗<0). Thus, given the negative effective mass of the band dispersion, the experimentallydetermined Rashba parameter αfor Bi/Cu(111) is negative. To further corroborate the experimental findings, we con- sider our first-principles calculation of the spin-polarizedsurface band structure [see Fig. 3(c)]. Comparing Figs. 3(d) and3(c) we infer that the calculated surface band structure quantitatively reproduces the experimentally observed disper-sion. More importantly, for the purpose of the present study,also the calculated spin orientations of the individual branchesagree with our experimental results and yield a negative α. IV . SIGN OF THE RASHBA PARAMETER Based on our experimental and theoretical results, we will now discuss the origin of the sign of the Rashba parameter andthe corresponding spin orientation. Previous investigations onthe basis of first-principles calculations revealed a markedlylocal character of the Rashba effect at surfaces. 19The splitting size is determined within ∼0.2˚A around the nucleus where the atomic field gradients are largest. Reflecting the brokenstructural inversion symmetry, surface states can exhibita considerably asymmetric charge distribution around theatomic nuclei. In fact, such an asymmetry is a necessarycondition for a spin splitting to occur at all. Its precise form hasbeen shown to strongly influence the magnitude of α. 19,25,35 Hence, it is conceivable that also the sign of αis determined by details of the local wave-function asymmetry in the vicinityof the nuclei. To elaborate on this point, we adopt the simple expression for the Rashba parameter α=2/c 2/integraltext φ2(z)∂zVd3r, suggested in Ref. 25, assuming a free-electron behavior in the surface plane and a confined wave function φ(z) perpendicular to the plane. Note that for a strictly 2D case [ φ2(z)=δ(z−z0), with the Dirac-Delta function δ(z)] the above expression yields the same definition for αas the one used in Ref. 15, which then leads to the sign convention in Eq. ( 2). Close to the atomic cores the potential Vcan be approximated by the bare Coulomb term, and hence ∂zVis an antisymmetric function with respect to the nucleus at z=0. Thus, from the above model, we would expecta sign change of αdepending on whether an excess charge is localized on the vacuum ( z0>0) or on the substrate ( z0<0) facing side of the nucleus. Keeping this in mind, we nextconsider the calculated partial charge density of the sp zsurface state at ¯/Gamma1(see Fig. 4). Figure 4(a) shows a charge line profile along the zdirection and Fig. 4(b) displays a contour plot of the charge density in the [1 ¯10] plane. As anticipated, we find a clear asymmetry of the charge profile along the zdirection. More precisely, we notice an imbalance of the partial chargeclose to the Bi nucleus in favor of the substrate facing side(z 0<0). For this case we would expect a negative α, which is indeed what is found by our experiments and calculations.Hence, taken collectively, our results suggest that the negativesign of the Rashba parameter in Bi/Cu(111) is related to theparticular imbalance in the charge distribution around the Bicores. It is instructive to compare the present results for Bi/Cu(111) with the Rashba-split surface state of Au(111). A (b) (111) (112)-0.4 -0.2 0 0.2 0.4 z (A)00.20.40.60.8| Ψss |2(a) FIG. 4. (Color online) First-principles calculation of the partial charge density of the spzstate on Bi/Cu(111) at the ¯/Gamma1point. (a) One-dimensional charge density profile along the zdirection [(111) direction]. The Bi atom is located at z=0 and the vacuum side corresponds to positive zvalues. The charge was averaged in the xyplane within an interval of ±0.07 ˚A around the nu- cleus. (b) Two-dimensional cut through the charge density in the [1¯10] plane. One Bi atom in the center and two Cu atoms are indicated. 115426-4SPIN ORIENTATION AND SIGN OF THE RASHBA ... PHYSICAL REVIEW B 84, 115426 (2011) previous experimental and theoretical study on Au(111) found a positive Rashba parameter.15This result is nicely in line with our findings here because, contrary to Bi/Cu(111), the Au(111)surface-state wave function is localized predominantly on thevacuum facing side of the outermost Au layer and hence thesign of αis changed. 15,25Note, however, that for Au(111) with (α> 0,m∗>0) the actual spin orientations of the outer and of the inner branch are the same as for Bi/Cu(111) with(α< 0,m ∗<0) (compare Fig. 1). The particular wave-function localization for Bi/Cu(111) discussed above can be interpreted as a result of hybridizationof the adsorbate sp zorbitals with the underlying substrate states. This hybridization is directly inferred from Fig. 4(b), showing high partial charge at the Bi atom but also atthe Cu atoms. The sp zsurface state is thus involved in the adsorbate-substrate bonding and consequently stronglylocalized between the two. We expect a related behavior forthe isostructural surface alloys Pb/Ag(111) and Bi/Ag(111)which feature analogous Rashba-split surface states. 11,21,36 Our calculations show indeed that the Rashba parameter for these two systems is negative as for Bi/Cu(111), reflectinga similar wave-function localization predominantly on thesubstrate side of the adsorbate atoms. These considerationsexemplify that the sign of the Rashba splitting certainly bearsinformation on bonding properties and charge localization inheavy-element surface and thin-film systems. The findingsare easily generalized to a broader range of systems such as,for example, monolayer covered semiconductor surfaces withlarge spin splittings (see Refs. 12,24, and 37–39). Another interesting aspect where knowledge of the sign of the Rashba parameter may give additional insights concernsthe sensitivity of surface states toward adatom adsorbtion. Toillustrate this we compare previous results on the effect of Xeadsorption on the surface electronic structure of Au(111) andBi/Ag(111). For Au(111) a large change in binding energyof 150 meV after adsorption of a closed Xe layer has beenobserved and attributed to the direct overlap of surface-stateand adsorbate wave functions. 40On the other hand, the surface state of Bi/Ag(111) is only weakly influenced by aXe overlayer, suggesting a considerably smaller overlap.41 At least partly these observations can be traced back to the different wave-function localization of the two surface stateswhich is encoded in the sign of their Rashba parameter: Thesurface state on Au(111) features a higher partial charge on thevacuum side of the first layer than Bi/Ag(111), which resultsin a stronger interaction with adsorbates. V . SUMMARY We have shown that the sign of the Rashba splitting in surface and thin-film systems is determined by the precisecharge distribution asymmetry of a surface state close to theatomic nuclei. Thus, the sign of the Rashba parameter containsinformation on the real space localization of surface stateswhich can provide additional insights in adsorbate-substrateinteractions and related mechanisms. Specifically, we find anegative Rashba parameter for the surface alloy Bi/Cu(111).This result is explained by the particular wave-functionlocalization of this surface state which is involved in thebonding between the Bi adsorbate atoms and the Cu(111)substrate. It would be desirable to extend the present findingsto more complex systems, such as states with out-of-planeor other unconventional spin orientations beyond the Rashbamodel (see Refs. 32,36, and 37) as well as the surface states on topological insulators (see, e.g., Ref. 42). ACKNOWLEDGMENTS We gratefully acknowledge experimental support by Ryusei Tateishi and Yuta Yamamoto. G.B. would like to thankHugo Dil for helpful discussions. This work was supportedby the Grant-in-Aid for Scientific Research (A) 20244045,the G-COE programs (G-03), the Bundesministerium f ¨ur Bildung und Forschung (Grant Nos. 05K10WW1/2 and05KS1WMB/1), and the Deutsche Forschunsgsgemeinschaft(FOR 1162). H.B. acknowledges support by the Japan Societyfor the Promotion of Science. 1C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963). 2R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems , Springer Tracts in Modern Physics V ol. 191 (Springer, Berlin, 2003). 3Y . A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984). 4S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). 5H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M. Johnson, Science 325, 1515 (2009). 6S. LaShell, B. A. McDougall, and E. Jensen, P h y s .R e v .L e t t . 77, 3419 (1996). 7Y . M. Koroteev, G. Bihlmayer, J. E. Gayone, E. V . Chulkov,S. Bl ¨ugel, P. M. Echenique, and P. Hofmann, P h y s .R e v .L e t t . 93, 046403 (2004). 8D. 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PhysRevB.73.045121.pdf

Electronic structure and thermopower of Ni „Ti0.5Hf0.5…Sn and related half-Heusler phases L. Chaput,1J. Tobola,2P. Pécheur,1and H. Scherrer1 1Laboratoire de Physique des Matériaux, UMR 75560, ENSMN, Parc de Saurupt, 54042 Nancy, France 2Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Cracow, Poland /H20849Received 9 September 2005; revised manuscript received 13 December 2005; published 25 January 2006 /H20850 In this paper we investigate the electronic structure and the thermopower for Ni /H20849Ti0.5Hf0.5/H20850Sn and related half-Heusler compounds. Two different methods have been used to calculate the electronic structure, i.e., fullpotential linearized augmented plane wave method for ordered compounds and the Korringa-Kohn-Rostokermethod within the coherent potential approximation for disordered alloys. We show that these methods givevery close results if comparing the density of states obtained in both cases. Moreover, no peculiarities in theband structure have been revealed upon alloying the parent compounds and therefore the large value of thethermopower reported experimentally for Ni /H20849Ti 0.5Hf0.5/H20850Sn with respect to NiTiSn or NiHfSn, does not have an origin in the electronic structure behavior. The thermopower calculations performed for different half-Heuslercompounds rather suggest that the carrier concentration itself could be predominantly responsible for the largethermopower in Ni /H20849Ti 0.5Hf0.5/H20850Sn as well as in other half-Heusler phases. Therefore, the large negative values of the Seebeck coefficient are not limited to some specified half-Heusler semiconductors but seem to be therule for the n-type samples with moderately low carrier concentrations. DOI: 10.1103/PhysRevB.73.045121 PACS number /H20849s/H20850: 71.15.Mb, 72.10.Bg, 72.20.Pa I. INTRODUCTION Half-Heusler phases are well-known compounds studied for their wide variety of physical properties includingmagnetism, 1half-metallic ferromagnetism,2and shape memory effect.3Due to the fact that many 18-valence elec- tron half-Heusler systems are narrow band gap semiconduc-tors /H20849see e.g. 4,5/H20850, they were also examined for the generation of thermoelectricity.6,7In order to get the best efficiency of the thermoelectric material one should maximize the figureof merit ZT=S 2/H9268T//H9260, where S,/H9268, and /H9260represent ther- mopower, electrical, and thermal conductivity, respectively.Unfortunately, the half-Heusler systems were established tohave quite important thermal conductivity /H9260and not suffi- ciently large thermopower Sto compete with the “state of art” thermoelectric materials.8Interestingly, a very large ther- mopower /H20849about −300 /H9262V/K /H20850has been reported9,10in half- Heusler alloys with the general formula Ni /H20849Ti,Zr,Hf /H20850Sn, con- taining Hf and/or Zr atoms substituted on Ti-site. This largeSvalue is much greater than the thermopower values cur- rently measured for NiTiSn and NiHfSn parent compounds/H20849about −150 /H9262V/K, see, e.g., Ref. 9 /H20850. An understanding of these phenomena could be of great interest from the applica-tion point of view, since the thermopower appears squared inthe figure of merit. Note, that substitution with isoelectronicelements can also be beneficial for the thermal conductivitydue to the atomic mass defects /H20849without substantial modifi- cations of electronic properties /H20850. Hence, the goal of this paper was to search for the origin of the thermopower enhancement in the disordered half-Heusler materials. At first sight, such behavior could arise from some pecu- liarities in the electronic structure because of the quite dif-ferent nature of dorbital for Ti and Hf, which are known to play a decisive role in the binding of these compounds /H20849see Sec. II and Refs. 11 and 12 /H20850. However, as will be shown inSec. II, this is not the case in NiTi 0.5Hf0.5Sn. In Sec. III the calculated thermopower will be presented, as a function ofcarrier concentration, for NiTiSn, NiZrSn, NiHfSn,NiTi 0.5Hf0.5Sn, being in good agreement with available ex- perimental data. Then, this supports the suggestions that thelarge values of Seebeck coefficients might be the rule forhalf-Heusler systems with low carrier concentration and witha microstructural state with as few secondary phases as pos-sible. We believe that these indications should have someimportance for the optimization of the thermoelectric prop-erties in half-Heusler compounds. II. ELECTRONIC STRUCTURE FOR NiTi 0.5Hf0.5Sn A large thermopower has been reported for NiTiSn-based half-Heusler alloys, if the Ti atoms are partly replaced withHf or both with Hf and Zr atoms. We call these alloys withthe general formula Ni /H20849Ti,Zr,Hf /H20850Sn. The electronic structure of NiTi 0.5Hf0.5Sn has been inves- tigated in details and taken as a model case, since within allNi/H20849Ti,Zr,Hf /H20850Sn samples this composition was expected to ex- hibit the largest differences with respect to the well-knownNiTiSn and NiHfSn compounds. Since the thermopower hasbeen found much smaller /H20849about −150 /H9262V/K /H20850for the end- point compounds9some electronic structure peculiarities might give rise to a strong increase of thermopower. Oncethe chemical composition has been specified our model sys-tem is not fully defined. In fact, the analysis performed inRef. 9 does not reveal its crystal structure. We can thereforeconsider two extreme descriptions for this system: a fullyordered /H20849FO/H20850compound where exactly two of four Ti atoms per unit cell are substituted by Hf atoms, but also a fullydisordered /H20849FD/H20850alloy, where Hf atoms are distributed ran- domly on the Ti sites with two Hf atoms per unit cell only onaverage. The electronic structure of NiTi 0.5Hf0.5Sn has then been calculated in both cases.PHYSICAL REVIEW B 73, 045121 /H208492006 /H20850 1098-0121/2006/73 /H208494/H20850/045121 /H208497/H20850/$23.00 ©2006 The American Physical Society 045121-1A. Fully ordered (FO) compound A unit cell of the FO compound is shown in Fig. 1. While perfectly ordered half-Heusler compounds NiTiSn and Ni- HfSn have cubic symmetry with space group F4¯3m/H20849black cell in Fig. 1 /H20850, our FO model system is tetragonal and be- longs to the space group P4¯m2/H20849shown with a yellow cell in Fig. 1 /H20850. The electronic structure of this compound was calculated using the full potential linearized augmented plane wave/H20849FLAPW /H20850 WIEN 2k program.13The self-consistency cycle was achieved with 1000 kpoints in the Brillouin zone. The exchange-correlation potential was computed in the general-ized gradient approximation /H20849GGA /H20850approach using the Perdew-Burke-Erzenhof functional. 14The unit cell has been optimized employing the relation c=/H208812a. In our notation cis the lattice parameter of the cubic cell or equivalently the c axis for the tetragonal cell, and ais the lattice parameter in the basal plane of that cell. At equilibrium we foundc=11.452 a.u.; the value can be compared to those for NiTiSn and NiHfSn. The computed lattice parameters arethen reported in Table I together with those previouslycalculated 12and the experimental data. A good agreement was found, since for NiTiSn and Ni- HfSn our results are of order of 1% larger than the experi-mental values. The small differences with Ref. 12 might beattributed to a different choice of muffin-tin radius/H20849R mt=2.2 a.u. in all present calculations /H20850since the other pa- rameters were taken to be the same. We see also thatNiTi 0.5Hf0.5Sn almost obeys Vegard’s law indicating a con- tinuous change through the substitution of Ti by Hf. In Fig. 2 and Fig. 3 we show density of states /H20849DOS /H20850and energy bands E/H20849k/H20850for NiTiSn, NiHfSn, and FO NiTi 0.5Hf0.5Sn. A detailed explanation of the electronic struc- ture in NiTiSn can also be found in Refs. 11, 12, and 15. Atfirst sight, DOSs for all three compounds look quite similar.Thesstates of Sn are located about 0.6 Ry below E F. At the Fermi level we found a gap opened by hybridization betweendstates of Ni and Ti or Hf. The dstates below E Fcomes essentially from Ni dorbitals, whereas those above EFare mainly due to Ti dor Hf dorbitals. The only important difference between these compounds /H20849Fig. 2 /H20850comes from the location of d-like states with respect to EF. In NiTiSn and NiTi 0.5Hf0.5Sn Ti- dand Hf- dstates are located about 0.09 Ry above the gap, whereas Hf- dstates go up to 0.15 Ry in NiHfSn. This presumably results in a smaller energy gap forNiHfSn /H20849see Table I /H20850. Such behavior can be explained from the fact that orbitals are much more localized for Hf than forTi. To support our suggestion, the maps of electron densitieshave been plotted /H20849Fig. 4 /H20850along planes going through the same crystallographic position both in cubic /H20849along 101 plane /H20850and tetragonal /H20849along 112 plane /H20850structures. The hy- bridization is therefore stronger between Ni and Ti than be-tween Ni and Hf in both cases. Consequently, this leads tothe band gap shrinking in NiHfSn, since the states near theconduction band edge are still of Ni- dcharacter. Looking at the band structure for FO NiTi 0.5Hf0.5Sn, as illustrated in Fig. 3, it may be surprising that a direct gap at/H9003point appears, unlike the well-established indirect gap in the cubic NiTiSn and NiHfSn /H20849Fig. 3 /H20850. This comes simply from the folding of bands when going from cubic to tetrag-onal cell, since the /H9003Xvector of the cubic cell belongs to the reciprocal lattice of the tetragonal cell. B. Fully disordered (FD) compound In the next section the results of the thermopower calcu- lations will be presented for NiTiSn, NiHfSn, andTABLE I. Lattice parameter and band gap Egfor NiTiSn, Ni- HfSn and NiTi 0.5Hf0.5Sn. c/H20849a.u. /H20850 present calculationc/H20849a.u. /H20850 Ref. 12c/H20849a.u. /H20850 experimentsaEg/H20849eV/H20850 NiTiSn 11.271 11.261 11.187 0.45 NiHfSn 11.589 11.572 11.463 0.39NiTi 0.5Hf0.5Sn 11.452 - - 0.46 aRefs. 11 and 12. FIG. 1. /H20849Color online /H20850Cubic /H20849black /H20850and tetragonal /H20849yellow /H20850cell for the half-Heusler system. The dashed atoms belong to the planes/H20849101 /H20850and /H20849112 /H20850; see text. FIG. 2. /H20849Color online /H20850Density of states for NiTiSn, NiHfSn, and FO NiTi 0.5Hf0.5Sn.CHAPUT et al. PHYSICAL REVIEW B 73, 045121 /H208492006 /H20850 045121-2NiTi 0.5Hf0.5Sn using the electronic structure obtained for per- fectly ordered compounds. In order to verify if such elec-tronic structure approximation is satisfying for Ni /H20849Ti,Hf /H20850Sn alloys, we have also performed electronic structure calcula-tions assuming that the compounds are fully disordered. Thecharge self-consistent Korringa-Kohn-Rostoker /H20849KKR /H20850 method within the coherent potential approximation /H20849CPA /H20850has been then applied to compute density of states in the half-Heusler NiTi 1−xHfxSn with x=0.0, 0.25, 0.50, 0.75, and 1. The KKR-CPA calculations have been done in the sameway as for other half-Heusler phases 16/H20849more details about the KKR-CPA methodology can be found in Refs. 17 and18/H20850. The obtained DOS are quite close to those shown in Fig. 2 and therefore they have not been reproduced. However, in FIG. 3. Energy bands for NiTiSn, NiHfSn, and FO NiTi 0.5Hf0.5Sn.ELECTRONIC STRUCTURE AND THERMOPOWER OF … PHYSICAL REVIEW B 73, 045121 /H208492006 /H20850 045121-3Fig. 5 we zoom on the energy window near the conduction band that is relevant to the electron concentrations which aretaken into account in the thermopower calculations /H20849next section /H20850. We see from the KKR-CPA results that the DOS slope /H20849Fig. 5 /H20850increases more or less continuously when the concentration of Ti atoms increases from 0 to 1. This char-acterizes well a virtual crystal behavior. A similar tendency isobserved in the periodic FLAPW calculations /H20849Fig. 6 /H20850. So in our cases, similarities between FLAPW and KKR- CPA results would indicate that the electronic properties ofinvestigated materials are only weakly sensitive to the par- ticular choice of the crystal structure description. This al-lowed us selecting the most convenient structure to calculatethe transport coefficients. III. THERMOPOWER In the previous section we have seen that the larger ther- mopower observed in NiTi 0.5Hf0.5Sn does not originate from electronic structure anomalies with respect to the end-pointcompounds. This is also supported by direct calculations ofthe thermopower. Looking at experimental data of electron transport prop- erties reported for different half-Heusler semiconductingphases, we can notice a remarkable diversity of the measuredthermopower values /H20849even for nominally the same material, see Table II /H20850. As suggested by Uher et al. in Ref. 6, the measured thermopower significantly depends on the way thesamples have been obtained. For example, as the annealingtime increases, the amount of the secondary phases /H20849presum- FIG. 4. /H20849Color online /H20850Isodensity plot for NiTiSn, NiHfSn, and FO NiTi 0.5Hf0.5Sn. The unit is e/a.u.3 FIG. 5. KKR-CPA density of states calculated in FD NiTi 1−xHfxSn. FIG. 6. FLAPW density of states for NiTiSn, NiHfSn and FO NiTi 0.5Hf0.5Sn/H20849the same energy range of energy as in Fig. 5CHAPUT et al. PHYSICAL REVIEW B 73, 045121 /H208492006 /H20850 045121-4ably metallic /H20850decreases, leading to a larger thermopower. One may expect that the experimental conditions could alsobe the reason for the large difference between the ther-mopower of NiTiSn and NiTi 0.5Hf0.5Sn. But the question is if we should attribute these differences to an averaging of thethermopower between two different ordered phases, or to adrastic change in the scattering mechanism, or simply to thenumber of conducting electrons? In real systems, these phe-nomena are certainly connected, but it would be useful toknow which aspect is the most significant for future ther-mopower optimization in this family of compounds. To this end, the thermopower has been calculated for NiTiSn, NiTi 0.5Hf0.5Sn, and NiHfSn. The isoelectronic NiZrSn compound has also been considered, since a numberof experimental data exist in the literature. The experimentalresults are shown in Fig. 7 for the cases where both ther-mopower and electron carrier concentration data from Hallmeasurements /H20849see Table II /H20850are available. For other cases only the thermopower values have been reported. The thermopower was evaluated at room temperature us- ing the computational method recently applied to calculateelectron transport coefficients in CoSb 3based skutterudites.22 Since the theoretical background has been presented there,22 we only briefly comment how the thermopower is derived.First, the electronic structure obtained in Sec. II A was usedto calculate the transport function /H9268¯¯/H20849E/H20850=q2 V/H20858 kn/H9270knv/H6023knv/H6023kn/H9254/H20849E−Ekn/H20850, /H208491/H20850 which is the central quantity in the electron transport calcu- lations. In fact, this function contains all needed information for the investigated system as electron velocities v/H6023kn, relax- ation times /H9270kn, and energy levels Ekn. Next, the Onsager coefficients Lij, defining main transport coefficients as electrical conductivity /H9268, thermopower S, Hall concentration nH, and Lorenz factor L, are obtained from the following expression:Lij=/H20885dE/H20873E−/H9262 q/H20874i+j−2/H11509f0 /H11509/H9262/H9268¯¯/H20849E/H20850, /H208492/H20850 where the chemical potential /H9262derivation of the Fermi-Dirac function f0also appears. In particular, the thermopower is calculated from the well-known relation S=1 TL11−1L12 /H208493/H20850 =1 T/H20873/H20885d/H9255/H9268¯¯/H20849/H9255/H20850/H11509f0 /H11509/H9262/H20874−1/H20873/H20885d/H92551 q/H9268¯¯/H20849/H9255/H20850/H20849/H9255−/H9262/H20850/H11509f0 /H11509/H9262/H20874. /H208494/H20850 The most important steps of the applied procedure are sketched in Fig. 8 but more explanations are given in Ref.22.TABLE II. Experimental /H20849at room temperature /H20850thermopower and carrier concentration for investigated half-Heusler alloys. n His specified per cell of Ni 2A2Sn2, where A=Ti, Zr, or Hf. S/H20849/H9262V/K /H20850 nH Ref. NiTiSn −142 9 NiTiSn −270 0.000 83 19NiTiSn −318 20NiZrSn −167 0.021 6NiZrSn −171 0.0103 6NiZrSn −210 0.0024 6NiZrSn −176 9NiZrSn −520 4.7 10 −521 NiHfSn −124 9NiTi 0.5Hf0.5Sn −281 9 NiTi 0.5Hf0.5Sn −250 20 NiTi 0.5/H20849Zr0.5Hf0.5/H208500.5Sn −325 10 FIG. 7. /H20849Color online /H20850Calculated and experimental ther- mopower at room temperature. When not specified the relaxation isconsidered as constant. The dependence of the thermopower withthe electron concentration is obtained by varying the chemical po-tential in Eq. /H208491/H20850and using n=/H20848 /H9262d/H9255f0/H20849/H9255/H20850g/H20849/H9255/H20850. Note that in the upper panel the NiZrSn and NiTiSn /H20849/H9261=const /H20850curves respectively repre- sented as red dotted and black dashed-dotted line are very close.ELECTRONIC STRUCTURE AND THERMOPOWER OF … PHYSICAL REVIEW B 73, 045121 /H208492006 /H20850 045121-5Two particular cases have accounted for the present ther- mopower calculations: /H20849i/H20850The constant relaxation time approximation /H20849/H9270=const /H20850; in this case the thermopower Sbecomes indepen- dent on /H9270since the relaxation time cancels in Eq. /H208494/H20850. /H20849ii/H20850The constant mean free path approximation /H20849/H9261=/H9270kvk=const /H20850, which is in fact equivalent to the impurity scattering approach. The corresponding thermopower results obtained in the entitled half-Heusler systems are collected in Fig. 7. Thedark blue /H20849gray /H20850area presents the results obtained within /H9261=const, where the upper limit corresponds to NiHfSn and the lower one to NiTiSn. The light blue /H20849gray /H20850area illustrates the results gained within /H9270=const, where /H20849as in the previous case /H20850the upper limit corresponds to NiHfSn and the lower one to NiTiSn. Noteworthy, the results for NiZrSn andNiTiSn /H20849/H9261=const /H20850are nearly identical. In Fig. 7 the ther- mopower curve corresponding to NiTi 0.5Hf0.5Sn is also shown and is found slightly above the NiTiSn curve. In bothapproximations the curves computed for NiTi 0.5Hf0.5Sn and NiZrSn belong to the area bordered upward by the NiHfSncurve and downward by the curve for NiTiSn. In this casethe thermopower behaviors agree with expectations from thesimple considerations: Ti dstates are located closer E Fthan the Hf dstates, the slope of the density of states in the vicinity of EFis smaller in the case of Hf than for Ti /H20849see Fig. 5/H20850and therefore the thermopower follows more or less the DOS modifications. Moreover, the concentration dependentSeebeck coefficient variations are very close for NiZrSn andNiHfSn. An important observation /H20849in Fig. 7 /H20850is that the ex- perimental thermopower changes strongly with the carrierconcentration n. These variations are well reproduced by the calculations and much better than fitted with the free electroncurves. This shows that the carrier concentration nshould be regarded as an important factor when optimizing the ther-mopower of half-Heusler alloys, since the S/H20849n/H20850variation alone is sufficient to reproduce quite well the experimental value of S, when both thermopower and carrier concentration data are available. One can also conclude from Fig. 7 that the details of the scattering mechanism are less important thanthe carrier concentration. However, as expected, these curvessuggest that we start from a /H9261=const regime as the carrier concentration increases. These results can also be helpful to understand the uncon- ventionally large values reported for the thermopower ofNi/H20849Ti,Hf /H20850Sn and Ni /H20849Ti,Zr,Hf /H20850Sn alloys in Refs. 9 and 10. In fact, our calculations suggest that the electron carrier concen-tration could be at the origin of these large values /H20849see Table II/H20850as it is the case for NiTiSn compounds produced in Ref. 19. Even if there is no direct proof of this conclusion for allNi/H20849Ti,Zr,Hf /H20850Sn alloys, since the electron concentration has not been measured there, we do hope that the theoreticalresults will motivate further experimental investigations.Strictly speaking, the above-mentioned discussion can onlybe applied to Ni /H20849Ti,Hf /H20850Sn alloys since this case was consid- ered in the calculations. In Ni /H20849Ti,Zr,Hf /H20850Sn alloys, Zr atoms could give additional effects. However, we have shown thatNiTi xHf1−xSn follows a virtual crystal behavior, mainly due to the fact that Ti and Hf are isoelectronic. Since Zr is alsoisoelectronic to these atoms, a virtual crystal behavior is alsoexpected for Ni /H20849Ti,Zr,Hf /H20850Sn alloys. The discussion above should therefore also be valid but there is still no carrierconcentration measurement available in this case. IV. CONCLUSIONS We have shown using different electronic structure calcu- lations /H20849FLAPW and KKR-CPA /H20850, applied to extreme crystal- lographic approximations /H20849FO and FD /H20850, that there is no un- usual behavior of electronic structure in Ni /H20849Ti,Hf /H20850Sn alloys that could explain the marked enhancement ofthermopower 9,10with respect to NiTiSn and NiHfSn. On the other hand, the thermopower calculations within two different approximations for electron scattering/H20849 /H9270=const and /H9261=const /H20850have evidenced that the electron car- rier concentration ncan give itself such a large negative ther- mopower for the lowest carrier concentration. This might bean explanation for the large value of the thermopower inNi/H20849Ti,Hf /H20850Sn alloys as well as for NiTiSn compound reported in Ref. 19. However, this conclusion has still to be checked,since the Hall concentration has not been reported for thesealloys. Moreover, the theoretical results give also some in-sights into the thermopower measurements, 10which also in- spired this work. These authors10reported very high ZT value based on a large value of the thermopower. Followingthe above-mentioned discussion /H20849Sec. III /H20850the variations of the thermopower presented in Fig. 2 of Ref. 10 could beattributed to variations of the carrier concentration. FIG. 8. /H20849Color online /H20850The general scheme of the electron trans- port coefficients calculations. Note that only thermopower is ana-lyzed in the present work.CHAPUT et al. PHYSICAL REVIEW B 73, 045121 /H208492006 /H20850 045121-61J. Pierre, R. Skolozdra, J. Tobola, S. Kaprzyk, C. Hordequin, M. A. Kouacou, I. Karla, R. Currat, and E. Lelievre-Berna, J. AlloysCompd. 262 & 263 , 101 /H208491997 /H20850. 2R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 /H208491983 /H20850. 3T. J. Zhu, L. Lu, M. O. Lai, and J. Ding, Smart Mater. Struct. 14, S293 /H208492005 /H20850. 4F. G. Aliev, F. G. Brandt, V . Moshchalkov, R. V . Skolozdra, and A. I. Belogorov, Z. Phys. B: Condens. Matter 75, 167 /H208491989 /H20850. 5J. Tobola, J. Pierre, S. Kaprzyk, R. Skolozdra, and M. A. Koua- cou, J. Phys.: Condens. Matter 10, 1013 /H208491998 /H20850. 6C. Uher, J. Yang, S. Hu, D. T. Morelli, and G. P. Meisner, Phys. Rev. B 59, 8615 /H208491999 /H20850. 7S. J. Poon, Semicond. Semimetals 70,3 7 /H208492001 /H20850. 8G. D. Mahan, Good Thermoelectrics, Solid State Physics , edited by H. Ehrenreich and F. Saepen, V ol. 51 /H20849Academic Press, San Diego, 1998 /H20850. 9H. Hohl, A. P. Ramirez, C. Goldmann, G. Ernst, B. Wolfing, and E. Bucher, J. Phys.: Condens. Matter 11, 1697 /H208491999 /H20850. 10S. Sakurada and N. Shutoh, Appl. Phys. Lett. 86, 082105 /H208492005 /H20850. 11S. Öğüt and K. Rabe, Phys. Rev. B 51, 10443 /H208491995 /H20850. 12P. Larson, S. D. Mahanti, and M. G. Kanatzidis, Phys. Rev. B 62, 12754 /H208492000 /H20850.13P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2K : An Augmented Plane Wave /H11001Local Orbitals Program for Calculating Crystal Properties /H20849Karlheinz Schwarz, Techn. Universität Wien, Austria, 2001 /H20850, ISBN 3-9501031-1-2. 14J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 /H208491996 /H20850. 15J. Tobola, J. Pierre, S. Kaprzyk, R. Skolozdra, and M. A. Koua- cou, J. Magn. Magn. Mater. 159, 192 /H208491996 /H20850. 16J. Tobola, L. Jodin, P. Pecheur, H. Scherrer, G. Venturini, B. Malaman, and S. Kaprzyk, Phys. Rev. B 64, 155103 /H208492001 /H20850. 17S. Kaprzyk and A. Bansil, Phys. Rev. B 42, 7358 /H208491990 /H20850. 18A. Bansil, S. Kaprzyk, P. E. Mijnarends, and J. Tobola, Phys. Rev. B60, 13396 /H208491999 /H20850. 19B. A. Cook, J. L. Harringa, Z. S. Tan, and W. A. Jesser, 15th International Conference on Thermoelectrics /H20849IEEE, Pasadena, CA, 1996 /H20850. 20T. Katayama, S. W. Kim, Y . Kimura, and Y . Mishima, J. Electron. Mater. 32, 1160 /H208492003 /H20850. 21E. Arushanov, W. Kaefer, K. Fess, C. Kloc, K. Friemelt, and E. Bucher, Phys. Status Solidi A 177,5 1 1 /H208492000 /H20850. 22L. Chaput, P. Pécheur, J. Tobola, and H. Scherrer, Phys. Rev. B 72, 085126 /H208492005 /H20850.ELECTRONIC STRUCTURE AND THERMOPOWER OF … PHYSICAL REVIEW B 73, 045121 /H208492006 /H20850 045121-7

PhysRevB.82.094425.pdf

Magnetic-field control of the electric polarization in BiMnO 3 I. V. Solovyev1,*and Z. V. Pchelkina2 1National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan 2Institute of Metal Physics, Ural Division, Russian Academy of Sciences, 620041 Ekaterinburg GSP-170, Russia /H20849Received 11 July 2010; revised manuscript received 23 August 2010; published 15 September 2010 /H20850 We present the microscopic theory of improper multiferroicity in BiMnO 3, which can be summarized as follows: /H208491/H20850the ferroelectric polarization is driven by the hidden antiferromagnetic order in the otherwise centrosymmetric C2/cstructure; /H208492/H20850the relativistic spin-orbit interaction is responsible for the canted spin ferromagnetism. Our analysis is supported by numerical calculations of electronic polarization using the Berry-phase formalism, which was applied to the low-energy model of BiMnO 3derived from the first-principles calculations. We explicitly show how the electric polarization can be controlled by the magnetic field and arguethat BiMnO 3is a rare and potentially interesting material where ferroelectricity can indeed coexist and inter- play with the ferromagnetism. DOI: 10.1103/PhysRevB.82.094425 PACS number /H20849s/H20850: 75.85. /H11001t, 75.25.Dk, 75.47.Lx I. INTRODUCTION Today, the term “multiferroics” is typically understood in a broad sense, as the systems exhibiting spontaneous electricpolarization and any type of magnetic ordering. 1Such mate- rials have a great potential for practical applications in mag-netic memories, logic, and magnetoelectric sensors, andtherefore attracted enormous attention recently. Beside prac-tical motivation, there is a strong fundamental interest inunveiling the microscopic mechanism of coupling betweenelectric polarization and magnetic degrees of freedom. Nev-ertheless, the combination of ferroelectricity and ferro mag- netism, what the term “multiferroicity” was originally intro-duced for, is rare. Such a combination would, for example,provide an easy way for manipulating the electric polariza-tion Pby the external magnetic field, which is coupled lin- early to the net ferromagnetic moment, etc. The canonicalexample of the system, where spontaneous electric polariza- tion was believed to coexist with the ferromagnetic groundstate, is BiMnO 3. However, the origin of such coexistence is largely unknown. Originally, the ferroelectric activity inBiMnO 3was attributed to the highly distorted perovskite structure stabilized by the Bi 6 s“lone pairs.”2However, more resent experimental studies /H20849Ref. 3/H20850and first-principles calculations /H20849Ref. 4/H20850suggested that the atomic displacements alone result in the centrosymmetric C2/cstructure, which is incompatible with any ferroelectricity. In our previous papers/H20849Refs. 5and6/H20850, we put forward the idea that the ferroelectric activity in BiMnO 3could be improper and associated with some hidden antiferromagnetic order, which breaks the in-version symmetry. The purpose of this work is to provide thecomplete quantitative explanation for the appearance and be-havior of the ferroelectric polarization in BiMnO 3based on the Berry-phase formalism.7–9 II. METHOD The basic idea of our approach is to construct an effective Hubbard-type model,Hˆ=/H20858 ij/H20858 /H9251/H9252tij/H9251/H9252cˆi/H9251†cˆj/H9252+1 2/H20858 i/H20858 /H9251/H9252/H9253/H9254U/H9251/H9252/H9253/H9254cˆi/H9251†cˆi/H9253†cˆi/H9252cˆi/H9254 /H208491/H20850 for the Mn 3 dbands near the Fermi level and to include the effect of all other /H20849inactive /H20850states to the definition of the model parameters of the Hamiltonian Hˆ. Thus, the model is constructed in the basis of 40 Wannier functions in each unitcell /H20849including three t 2gand two egorbitals for each spin and for each of the four Mn sites /H20850, by starting from the electronic structure in the local-density approximation /H20849LDA /H20850. The Greek symbols in Eq. /H208491/H20850denote the combination of spin and orbital indices. All parameters of the model Hamiltonian /H208491/H20850 are defined rigorously, on the basis of the density-functionaltheory /H20849DFT /H20850. The details can be found in the review article /H20849Ref. 10/H20850and in our previous publications /H20849Refs. 5and6/H20850. Briefly, the one-electron part /H20849t ij/H9251/H9252/H20850is derived by using a gen- eralized downfolding method. One of the most important pa- rameters in tij/H9251/H9252is the large /H20849about 1.5 eV /H20850crystal-field split- ting between two eglevels, which is caused by the Jahn- Teller distortion and manifests itself in the orbital ordering.The screened Coulomb interactions /H20849U /H9251/H9252/H9253/H9254/H20850are derived by combining the constrained DFT technique with the random-phase approximation /H20849RPA /H20850, 10namely, the screening by outer electrons /H20849such as the 4 spelectrons of transition metals /H20850and the change in the spatial extension of the atomic wave func-tions upon the change in their occupation numbers can beeasily taken into account by solving the Kohn-Sham equa-tions within constrained DFT approach. On the other hand,the “self-screening” by the same type of electrons, whichcontribute to other bands due to the hybridization effects /H20849for example, the 3 delectrons in the oxygen band will strongly screen the Coulomb interactions in the 3 dband near the Fermi level /H20850, can be treated in the perturbative RPA scheme. The self-screening is a very important channel of screeningin solids, which substantially reduces the value of the effec-tive Coulomb repulsion U/H20849defined as the screened Slater integral F 0/H20850in the 3 dband of manganites.11In BiMnO 3,i ti s only about 2.3 eV,5that has important consequences on the behavior of interatomic magnetic interactions. The model /H208491/H20850is solved in the Hartree-Fock approximation,10PHYSICAL REVIEW B 82, 094425 /H208492010 /H20850 1098-0121/2010/82 /H208499/H20850/094425 /H208495/H20850 ©2010 The American Physical Society 094425-1/H20849tˆk+Vˆ/H20850/H20841Cnk/H20856=/H9255nk/H20841Cnk/H20856, where tˆkis the Fourier image of tˆij=/H20648tij/H9251/H9252/H20648and, if necessary, includes the relativistic spin-orbit interaction /H20849SOI /H20850,Vˆis the self-consistent Hartree-Fock method, and /H20841Cnk/H20856is the eigen- vector in the basis of Wannier functions /H20849where the spin in- dices are included in the definition of n/H20850. Once the orbital degeneracy is lifted by the strong lattice distortion, the Hartree-Fock theory provides a good approxi-mation for the ground-state properties. The effect of correla-tion interactions, which can be treated as a perturbation tothe Hartree-Fock solution, 10on the magnetic ground state of manganites is partially compensated by the magnetic polar-ization of the oxygen states: if the former tend to stabilizeantiferromagnetic structures, the latter favors the ferromag-netic alignment. 11Due to this compensation, the mean-field Hartree-Fock theory, formulated for the minimal 3 dmodel, appears to be rather successful for the analysis of the ground-state properties of manganites. III. MAGNETISM AND THE INVERSION SYMMETRY BREAKING First, let us explain the main idea of our previous work.5,6 What is the possible origin of multiferroic behavior of BiMnO 3and how can it be controlled by the magnetic field? /H208491/H20850The lattice distortion leads the orbital ordering, which is schematically shown in Fig. 1in two pseudocubic planes /H20849the orbital ordering in the y/H11032z/H11032plane is similar to the one in thez/H11032x/H11032plane /H20850. This orbital ordering predetermines the behavior of inter- atomic magnetic interactions, which obey some general prin-ciples, applicable for manganites with both monoclinic/H20849C2/c/H20850and orthorhombic /H20849Pbnm /H20850structure, 5,11namely, be- sides conventional nearest-neighbor interactions /H20849shown by hatched lines /H20850, one can expect some longer-range interac- tions between remote Mn atoms, which operate via interme-diate Mn sites. These sites are shown by arrows./H208492/H20850Why should the longer-range interactions exist? The answer is directly related to the fact that the on-site Coulombrepulsion Uis not particularly large. Therefore, besides con- ventional superexchange processes, there are other interac-tions, which formally appear in the higher orders of the 1 /U expansion and connect more remote sites. This mechanism israther similar to the superexchange interaction via interme-diate oxygen sites, except that the role of the oxygen stateshere is played by the unoccupied e gorbitals of the interme- diate Mn sites.11By mapping the Hartree-Fock total energies onto the Heisenberg model, one can obtain the followingparameters of interatomic magnetic interactions: 5,12JNN/H110115 and 6 meV /H20849where two slightly different values correspond to inequivalent bonds /H20850andJLR/H11011−3 meV. Thus, these interac- tions are at least comparable. Besides them, there are finite/H20849of the order −1 meV /H20850interactions in the bonds 1–2 and 4–4 across the inversion center, which define the final type of themagnetic ground state of BiMnO 3. /H208493/H20850Without spin-orbit coupling, the longer-range interac- tions tend to stabilize the antiferromagnetic ↑↓↓↑ structure /H20849where the arrows denote the directions of spins for the four Mn sites in the unit cell /H20850. This antiferromagnetic order de- stroys the inversion centers /H20849shown by “ /H11569” in Fig. 1/H20850and thus could be the cause of the ferroelectric activity. Since the↑↓↓↑ antiferromagnetic structure satisfies the symmetry op- eration Tˆ /H20002/H20853my/H20841R3/2/H20854/H20849where myis the mirror reflection y →−yassociated with the one half of the monoclinic transla- tion R3, and Tˆin the nonrelativistic case flips the directions of spins, which are not affected by my/H20850,Pis expected to lie in the zxplane.13There is an important difference between monoclinic BiMnO 3and orthorhombic systems, such as HoMnO 3.14In the latter case, the positions of the Mn sites coincide with the inversion centers. Therefore, in order tobreak the inversion symmetry by a magnetic order, the lattershould double /H20849triple, etc. /H20850the orthorhombic unit cell. In BiMnO 3, however, the inversion centers are located in inter- stitial positions and can be destroyed already by an antifer-romagnetic arrangement of spins within the same unit cell.Thus, we would like to emphasize again that the origin of theferroelectric polarization in BiMnO 3is essentially nonrela- tivistic. It is not related to a noncollinear spin texture either:the collinear antiferromagnetic arrangement of spins is suffi-cient to break the inversion symmetry and thus produce afinite electric polarization. /H208494/H20850Thus, the ferroelectric activity in BiMnO 3could be caused by the antiferromagnetic order. However, this conclu-sion seems to contradict to another experimental fact, ac-cording to which BiMnO 3is a good ferromagnet.3This con- tradiction can be reconciled by considering the relativisticspin-orbit interaction, which is responsible for the weak fer-romagnetism. Since the SOI-induced ferromagnetic magneti-zation is additionally stabilized by isotropic interactions J NN, the ferromagnetism is not so “weak,” and the magnetic struc-ture, obtained in the Hartree-Fock calculations for the low-energy model, is strongly noncollinear /H20849Fig.2/H20850. It belongs to the space group Cc, where the only nontrivial symmetry op- eration is /H20853m y/H20841R3/2/H20854and the magnetic moments in the rela- tivistic case are transformed by myas auxiliary vectors. Thus, the net ferromagnetic moment is aligned along the yaxis,3x2-r23y2-r23z2-r23z2-r23 3 334 4 4 41 1 1 12 2 2 2planex'y' JNN JNN JNNJNN 3x2-r23z2-r23y2-r23z2-r21 1 1 12 2 2 23 3 3 34 4 4 4planex'z' JLR JLR JLRJLRJNN JNN JNN JNN** FIG. 1. /H20849Color online /H20850Schematic view on the orbital ordering and corresponding interatomic magnetic interactions in thepseudocubic x /H11032y/H11032andz/H11032x/H11032planes. In the unit cell of BiMnO 3, there are four Mn sites /H20849indicated by numbers /H20850, which form two inequiva- lent subgroups: /H208491,2/H20850and /H208493,4/H20850. The nearest-neighbor ferromagnetic interactions JNNoperate in the hatched bonds. The atoms involved in the longer range antiferromagnetic interactions JLRare denoted by arrows. The inversion centers are marked by /H11569.I. V. SOLOVYEV AND Z. V. PCHELKINA PHYSICAL REVIEW B 82, 094425 /H208492010 /H20850 094425-2while the xand zcomponents form the antiferromagnetic structure. Other magnetic configurations have higher ener-gies. The details can be found in Ref. 6but here we would like to emphasize again that the role of the spin-orbit inter-action is to produce the ferromagnetic component of the spinmagnetization via the spin canting. It is notthe source of the ferroelectric polarization in BiMnO 3. By summarizing this part, the C2/csymmetry in BiMnO 3 is spontaneously broken by the hidden antiferromagnetic or- der. The true magnetic ground state of BiMnO 3is strongly noncollinear, where the ferromagnetic order along the yaxis coexists with the antiferromagnetic order, and related to itferroelectric polarization, along the xandzaxes. Our sce- nario not only explains the rare coexistence of ferroelectric-ity and ferromagnetism but also shows how the electric po-larization P/H20849and the symmetry of BiMnO 3/H20850can be controlled by the external magnetic field B=/H208490,By,0/H20850 coupled to the ferromagnetic magnetization. This idea wasformulated in Ref. 6. In the present work, we will further consolidate this picture by estimating the numerical values ofPand discussing details of its behavior in the external mag- netic field. IV . ELECTRONIC POLARIZATION Since the crystal structure of BiMnO 3has the inversion symmetry, there will be no ionic contribution to P, and the main mechanism, which will be considered below, is ofpurely electronic origin. In principle, the magnetoelastic in-teractions in the ↑↓↓↑ structure may cause the atomic dis-placements away from the centrosymmetric positions and give rise to the ionic term. Nevertheless, such calculationswould require the full structural optimization, which cannotbe easily incorporated in the model analysis. The first-principles calculations for HoMnO 3show that electronic and ionic terms are at least comparable.14Therefore, we expect that the electronic contribution alone could provide a goodsemiquantitative estimate for P. Moreover, the behavior of electronic polarization presents a fundamental interest as itallows one to explain how Pin improper multiferroics is induced solely by the magnetic symmetry breaking. The modern theory of electric polarization allows one to relate the change in Pto Berry’s phase of Bloch electrons. 7–9 It is particularly convenient to use the formulation proposed by Resta, where Berry’s phase is computed on the discretegrid of kpoints, generated by the N 1/H11003N2/H11003N3divisions of the reciprocal-lattice vectors /H20853Ga/H20854.9Then, the position of each point in the Brillouin zone is specified by the threeinteger indices /H208490/H11349s a/H11021Na/H20850, ks1,s2,s3=s1 N1G1+s2 N2G2+s3 N3G3, and three components of the electric polarization in the cur- vilinear coordinate frame, formed by G1,G2, and G3, can be found as9 /H9004Pa=−1 VNa N1N2N3/H20851/H9253a/H20849/H11009/H20850−/H9253a/H208490/H20850/H20852, /H208492/H20850 where Vis the unit-cell volume, /H92531=−/H20858 s2=0N2−1 /H20858 s3=0N3−1 Im ln/H20863 s1=0N1−1 detS/H20849ks1,s2,s3,ks1+1,s2,s3/H20850,/H208493/H20850 and similar expressions hold for /H92532and/H92533. Equation /H208492/H20850 implies that the only meaningful quantity in the bulk is thepolarization difference between two states that can be con-nected by an adiabatic switching process. 7–9 In the present case, S=/H20648/H20855Cnk/H20841Cn/H11032k/H11032/H20856/H20648is the overlap matrix, constructed from the Hartree-Fock eigenvectors /H20841Cnk/H20856in the occupied part of the spectrum, taken in two neighboring k points: k=ks1,s2,s3andk/H11032=ks1+1,s2,s3for/H92531, etc.15The polar- ization /H20851Eq. /H208492/H20850/H20852was first computed in the curvilinear coor-FIG. 3. /H20849Color online /H20850Magnetic-field dependence of the electric polarization, the angle /H9278between spin magnetic moments at the Mn sites 1 and 2, and three components of the vector of the magneticmoment at the site 1 in the Cartesian coordinate frame /H20849shown in the inset /H20850. FIG. 2. /H20849Color online /H20850Fragment of the crystal and magnetic structure corresponding to the lowest Hartree-Fock energy. The Biatoms are indicated by the big light gray /H20849yellow /H20850spheres, the Mn atoms are indicated by the medium gray /H20849red/H20850spheres, and the oxygen atoms are indicated by the small gray /H20849green /H20850spheres. The directions of spin magnetic moments are shown by arrows. Theinversion center is marked by the symbol /H11569. The left lower part of the figure explains the orientation of the Cartesian coordinate frame.The numerical values of the magnetic moments M=/H20849M x,My,Mz/H20850, measures in /H9262Bin the Cartesian coordinate frame, are M1,2=/H20849/H110070.08,1.45, /H110063.69 /H20850andM3,4=/H20849/H110060.97,2.02, /H110063.27 /H20850.6MAGNETIC-FIELD CONTROL OF THE ELECTRIC … PHYSICAL REVIEW B 82, 094425 /H208492010 /H20850 094425-3dinate frame and then transformed to the Cartesian frame shown in Fig. 2.13In all the calculations, we used the mesh of 72/H1100372/H1100336 points in the Brillouin zone. As will become clear below, one possible example of the adiabatic switching process, which can be used in the calcu-lations of P, is to restore the inversion symmetry by placing the system in the high magnetic field and then adiabaticallyswitching off the field. In practical calculations, however,one can typically use some particular choice of phase in/H20841C nk/H20856and enforce the equality /H9253a=0 for the centrosymmetric systems.15In this context, the discrete Eq. /H208493/H20850appears to be especially useful because it cancels out the contributions ofaccidental phases in /H20841C nk/H20856, which can emerge in the process of numerical diagonalization of Hartree-Fock equations indifferent kpoints. 9Moreover, it enforces the periodicity of the Hartree-Fock eigenvectors in the reciprocal space: /H20841Cnk/H20856 =/H20841Cnk+Ga/H20856, which corresponds to some particular choice of phase.7 First, let us discuss results without spin-orbit interaction. As pointed out in the previous section, the antiferromagneticalignment of spins at the sites 1 and 2 breaks the inversionsymmetry and yields finite electric polarization. However,the symmetry of the system also depends on the magneticconfiguration in the sublattice 3–4. The electric polarizationfor the ↑↓↓↑ structure lies in the zxplane /H20849P x =2.1/H9262C/cm2andPz=0.1/H9262C/cm2/H20850, in agreement with the symmetry arguments presented in Ref. 6. The ↑↓↓↑ struc- ture can be transformed to the ↑↓↑↓ one with the same energy by the symmetry operation /H20853Cy2/H20841R3/2/H20854/H20849where Cy2is the 180° rotation around the yaxis /H20850, which changes the di- rection of P:Px/H20849z/H20850→−Px/H20849z/H20850. On the other hand, the ↑↓↓↓ structure /H20849which has higher energy /H20850is transformed to itself by/H20853Cy2/H20841R3/2/H20854, and corresponding electric polarization is par- allel to the yaxis /H20849Py=4.8/H9262C/cm2/H20850. Other magnetic struc- tures, characterized by the ferromagnetic alignment of spinsat the sites 1 and 2 /H20849such as ↑↑↑↑ ,↑↑↑↓ , and↑↑↓↓ /H20850, pre- serve the inversion symmetry and result in zero net polariza-tion. Furthermore, without spin-orbit interaction one can easily evaluate separate contributions to Pof the states with differ- ent projections of spins /H20849↑and↓/H20850. For the ↑↓↓↑ structure, the vector of the electric polarization takes the following form: P ↑,↓=1 2/H20849Px,/H11006Py,Pz/H20850, where Py=5.7/H9262C/cm2, and the values of Pxand Pyare listed above. This result is very natural, because the distribution of the electron density foreach spin does not have any symmetry and, therefore, theelectric polarization P ↑,↓has all three components. On the other hand, the electron density with the spin ↑in the ↑↓↓↑ antiferromagnetic structure can be transformed to the onewith the spin ↓by the symmetry operation /H20853m y/H20841R3/2/H20854and, therefore, Py↑=−Py↓. Thus, in the total polarization P=P↑+P↓, the xandzcomponents with different spins will sum up, while the largest ycomponents will cancel each other. One can also evaluate the individual contributions to P coming from the t2gandegbands, which is separated by an energy gap.5This yields Pxt2g=−0.8 /H9262C/cm2,Pzt2g =−0.3 /H9262C/cm2,Pxeg=2.9/H9262C/cm2, and Pzeg=0.4/H9262C/cm2. Thus, the t2gband is polarized opposite to the egband, thatsubstantially reduces the value of P. Similar tendency was found in the first-principles calculations for orthorhombicmanganites. 16 The spin-orbit interaction results in the canting of spins away from the collinear ↑↓↓↑ antiferromagnetic state and toward the ferromagnetic alignment. It will reduce the value ofP. In the Hartree-Fock ground state /H20849see Fig. 2/H20850, the angle /H9278between spin magnetic moments at the sites 1 and 2 is reduced from 180° till 137°, and corresponding electric po-larization parallel to the xaxis is reduced from P x =2.1/H9262C/cm2till 1.6 /H9262C/cm2while the small component ofPparallel to the zaxis practically does not change /H20849Pz =0.1/H9262C/cm2/H20850. This effect can be further controlled by the magnetic field, which is applied along the yaxis and satu- rates the ferromagnetic magnetization. Since the absolutevalue of the local magnetic moment is nearly conserved, theincrease in the ferromagnetic component along the yaxis will be compensated by the decrease in two antiferromag-netic components along the xandzaxes. The corresponding ferroelectric polarization will also decrease. Results ofHartree-Fock calculations in the magnetic field are shown inFig.3. 17Sufficiently large magnetic field /H20849/H1101135 T /H20850will align the magnetic moments at the sites 1 and 2 ferromagnetically/H20849 /H9278=0/H20850and restore the C2/csymmetry.6The electric polar- ization follows the change in /H9278and completely disappears when/H9278=0. However, the decline of Pis much steeper, for example, PxandPzare reduced by factor two already in the moderate field By/H110115 T, corresponding to /H9278/H11011100°. More- over, Pzis always substantially smaller than Px. V . CONCLUDING REMARKS We have proposed the microscopic theory of improper multiferroicity in BiMnO 3, which is based on the inversion symmetry breaking by the hidden antiferromagnetic order.We have estimated the ferroelectric polarization and explic-itly shown how it can be controlled by the magnetic field.Our scenario still needs to be checked experimentally, andapparently one important question here is how to separate theintrinsic ferroelectricity in BiMnO 3from extrinsic effects, caused by the defects. For example, the values of the ferro-electric polarization obtained in the present work, althoughcomparable with those calculated for other improper ferro-electrics on the basis of manganites, 14are substantially larger than the experimental value 0.062 /H9262C/cm2/H20849at 87 K /H20850, which was reported so far for BiMnO 3.18Nevertheless, we believe that systematic study of manganites with the monoclinicC2/csymmetry and finding conditions, which would lead to the practical realization of scenario proposed in our work,presents a very important direction, because it gives a possi-bility for combining and intermanipulating the ferroelectric- ityandferromagnetism within one sample. ACKNOWLEDGMENTS This work is partly supported by Grant-in-Aid for Scien- tific Research /H20849C/H20850No. 20540337 from MEXT, Japan and Russian Federal Agency for Science and Innovations, GrantNo. 02.740.11.0217.I. V. SOLOVYEV AND Z. V. PCHELKINA PHYSICAL REVIEW B 82, 094425 /H208492010 /H20850 094425-4*solovyev.igor@nims.go.jp 1D. Khomskii, Physics 2,2 0 /H208492009 /H20850. 2R. Seshadri and N. A. Hill, Chem. Mater. 13, 2892 /H208492001 /H20850. 3A. A. Belik, S. Iikubo, T. Yokosawa, K. Kodama, M. Igawa, S. Shamoto, M. Azuma, M. Takano, K. Kimoto, Y. Matsui, and E.Takayama-Muromachi, J. Am. Chem. Soc. 129, 971 /H208492007 /H20850. 4P. Baettig, R. Seshadri, and N. A. Spaldin, J. Am. Chem. Soc. 129, 9854 /H208492007 /H20850. 5I. V. Solovyev and Z. V. Pchelkina, New J. Phys. 10, 073021 /H208492008 /H20850. 6I. V. Solovyev and Z. V. Pchelkina, Pis’ma Zh. Eksp. Teor. Fiz. 89, 701 /H208492009 /H20850/H20851JETP Lett. 89, 597 /H208492009 /H20850/H20852. 7D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48, 4442 /H208491993 /H20850. 8R. Resta, Rev. Mod. Phys. 66, 899 /H208491994 /H20850. 9R. Resta, J. Phys.: Condens. Matter 22, 123201 /H208492010 /H20850. 10I. V. Solovyev, J. Phys.: Condens. Matter 20, 293201 /H208492008 /H20850. 11I. Solovyev, J. Phys. Soc. Jpn. 78, 054710 /H208492009 /H20850. 12The interatomic magnetic interactions are defined as one half of the Hartree-Fock energy difference between antiferromagneticand ferromagnetic configurations in each bond. 13We use the following setting for the monoclinic translations: R1,2=1 2/H20849sin/H9252a,/H11007b,cos/H9252a/H20850andR3=/H208490,0, c/H20850. The positions of four Mn atoms in the unit cell are specified by the vectors: /H92701 =yMn/H20849R1−R2/H20850+1 4R3,/H92702=−/H92701,/H92703=1 2R1, and /H92704=1 2/H20849R2+R3/H20850. Theexperimental structure parameters were taken from Ref. 3. More detailed information about the settings, which were used for thecrystal structure of BiMnO 3, can be found in Refs. 5and6. 14S. Picozzi, K. Yamauchi, B. Sanyal, I. A. Sergienko, and E. Dagotto, Phys. Rev. Lett. 99, 227201 /H208492007 /H20850. 15Some caution should be taken when choosing the phases of the Bloch functions. In order to minimize the effect of the basis/H20849Wannier /H20850functions W R/H9270/H20849r/H20850, centered at the site /H9270of the unit cell R, the Bloch functions were defined as Wk/H9270/H20849r/H20850=1 /H20881N/H20858Reik·/H20849/H9270+R/H20850WR/H9270/H20849r/H20850. Then, the matrix elements /H20855WR/H9270/H20841r−R−/H9270/H20841WR/H9270/H20856ofPin the basis of /H20853WR/H9270/H20854will either vanish or cancel each other since the low-energy model and, therefore,/H20853W R/H9270/H20854themselves are defined by starting from the nonmagnetic LDA band structure /H20849Ref. 10/H20850, which preserves the inversion symmetry. Thus, the main contribution to Pin our model analy- sis arises from the evolution of /H20841Cnk/H20856. 16K. Yamauchi, F. Freimuth, S. Blügel, and S. Picozzi, Phys. Rev. B78, 014403 /H208492008 /H20850. 17The interaction term with the magnetic field is given by HˆB=−/H9262BB·/H208492sˆ+lˆ/H20850, where sˆandlˆare the operators of spin and orbital angular momentum, respectively. 18A. Moreira dos Santos, A. K. Cheetham, T. Atou, Y. Syono, Y. Yamaguchi, K. Ohoyama, H. Chiba, and C. N. R. Rao, SolidState Commun. 122,4 9 /H208492002 /H20850.MAGNETIC-FIELD CONTROL OF THE ELECTRIC … PHYSICAL REVIEW B 82, 094425 /H208492010 /H20850 094425-5

PhysRevB.80.115333.pdf

Magneto-orientation and quantum size effects in spin-polarized STM conductance in the presence of a subsurface magnetic cluster Ye. S. Avotina,1Yu. A. Kolesnichenko,1and J. M. van Ruitenbeek2 1B.I. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Avenue, 61103 Kharkov, Ukraine 2Kamerlingh Onnes Laboratorium, Universiteit Leiden, Postbus 9504, 2300 Leiden, The Netherlands /H20849Received 7 July 2009; revised manuscript received 25 August 2009; published 29 September 2009 /H20850 The influence of a single magnetic cluster in a nonmagnetic host metal on the spin current j/H20849s/H20850and the charge current jin the vicinity of a ferromagnetic scanning tunneling microscope /H20849STM /H20850tip is studied theoretically. Spin-flip processes due to electron interaction with the cluster are taken into account. We show that quantuminterference between the partial waves injected from the STM tip and those scattered by the cluster results inthe appearance of components perpendicular to the initial polarization of the spin current j /H20849s/H20850, which obtain a strongly inhomogeneous spatial distribution. This interference produces oscillations of the conductance as afunction of the distance between the contact and the cluster center. The oscillation amplitude depends on thecurrent polarization. We predict a strong magneto-orientational effect; the conductance oscillations may grow,shrink, or even vanish for rotation of the cluster magnetic moment /H9262effby an external magnetic field. These results can be used for the determination of the /H9262efffor magnetic clusters below a metal surface. DOI: 10.1103/PhysRevB.80.115333 PACS number /H20849s/H20850: 61.72.J /H11002, 73.40.Cg, 73.63.Rt, 74.50. /H11001r I. INTRODUCTION Subsurface defects, such as impurities and vacancies, re- sult in oscillations of the conductance as a function of theposition of a scanning tunneling microscope /H20849STM /H20850tip rela- tive to the defect position /H20849see, for example, Refs. 1–3/H20850. These Friedel-type oscillations originate from interferencebetween electron waves directly transmitted through the con-tact and waves that are scattered by the defect and reflectedback by the contact. The theory of STM conductance in thepresence of a single nonmagnetic pointlike defect below ametal surface in the vicinity of the contact has been devel-oped in Refs. 4and5. First the signature of Fermi-surface anisotropy in STM conductance in the presence of defectshad been analyzed theoretically in detail in Ref. 6.I nt h e paper 7the general results of Ref. 6was applied for the the- oretical investigation of the conductance of a tunnel pointcontact of noble metals in the presence of a single defectbelow surface. A pattern of the conductance oscillations,which can be observed by the method of scanning tunnelingmicroscopy, was obtained for different orientations of thesurface for the noble metals. 7Recently it had been confirmed experimentally that Fermi surfaces can be imaged in realspace with a low-temperature scanning tunneling microscopewhen subsurface point scatterers are present. 8The effect of Kondo scattering by a subsurface magnetic impurity on theconductance of a tunnel point contact has been analyzedtheoretically in Ref. 9. The applicability of STM can be extended for the study of magnetic objects below the surface of a conductor when amagnetic material is used for the STM tip such that the elec-tric current is spin polarized /H20849SP/H20850/H20849for review of SP-STM see Ref. 10/H20850. An important feature of SP-STM is that the spin- polarized current influences a magnetic object in a nonmag-netic matrix, producing so-called spin-transfer torque /H20849for re- view, see Ref. 11/H20850. For example, near a point contact, where the current density is large, the spin-transfer torque can bestrong enough to reorient the magnetization of ferromagnetic layers in magnetic multilayers. 12Such investigations are very important for the development of innovative high-densitydata-storage technologies. In this paper we consider theoretically the conductance of a tunnel point contact between magnetic and nonmagneticmetals in a SP-STM geometry. A magnetic cluster is embed-ded in the nonmagnetic metal in the vicinity of the contact.The changes in the spin-polarized current and the spin-transfer torque that influences the magnetic moment of thecluster are analyzed. We study the dependence of the ampli-tude of the conductance oscillations as a function of the STMtip position on the relative orientation between the spin po-larization of the tunneling electrons and the magnetic mo-ment of the cluster /H9262eff. II. MODEL OF THE CONTACT AND BASIC EQUATIONS Let us consider a tunnel contact between a semi-infinite half space z/H113500 of a nonmagnetic metal /H20849the sample /H20850and a sharp tip of a magnetic conductor /H20849Fig. 1/H20850. A voltage Vis applied between the tip and the sample. The electrical /H20849and spin /H20850current flows through a small region of the surface at z=0 near the tip apex that is closest to the sample. This system models the geometry of a SP-STM experiment. Thetip magnetization /H20849in real SP-STM the magnetization of the last atom 10/H20850, which we choose along the zdirection, defines the direction of the polarization of the tunnel current. Suchmagnetization can be obtained, for example, for a Fe/Gd-coated WSTM tip. 13In the nonmagnetic metal we place a spherical single-domain magnetic cluster having a magneticmoment /H9262eff/H20849Fig. 1/H20850. As first predicted by Frenkel and Dorfman14particles of a ferromagnetic material are expected to organize in a single magnetic domain below a criticalparticle size /H20849a typical value for this critical size for Co is about 35 nm /H20850. Depending on the size and the material, the magnetic moments of such particles can be /H9262effPHYSICAL REVIEW B 80, 115333 /H208492009 /H20850 1098-0121/2009/80 /H2084911/H20850/115333 /H208496/H20850 ©2009 The American Physical Society 115333-1/H11011102–105/H9262B, where /H9262Bis the Bohr magneton.15Below, we only consider electron scattering by the magnetic cluster,assuming the mean-free paths for all other processes/H20849electron-spin-diffusion length, electron-phonon mean-free path and others /H20850are much larger than the distance between the contact and the cluster center r 0. In order to describe the spin-polarized electron states of our system we use the approach proposed in the works ofSlonczewski and Berger: 16,21all calculations are performed by means of independent single-particle spinor wave func- tions/H9023ˆ/H20849rl;/H9268l/H20850of electrons with opposite spin directions, where rland/H9268lare the position vector and the spin direction, respectively, for each spin orientation l=1,2. W e use the representation /H92681,2=↑,↓in which the spin projection on the zaxis, sz=/H110061 2, is used. This approach corresponds to reduc- ing the many-particle problem of a partially polarized elec-tron system with nonzero average spin to a double-particleproblem for electrons in a pure /H20849completely polarized /H20850spin state. Neglect of electron-electron interactions enables us toseparate the double-particle Schrödinger equation into two independent equations for /H9023ˆ/H20849r l;/H9268l/H20850. In our case this separa- tion is valid if /H9262eff/H11271/H9262B, i.e., the electron-electron exchange interaction is negligible compared to electron exchange in-teraction with the cluster. Generally, the moment /H9262effof the cluster in a nonmagnetic metal takes an arbitrary direction.This direction /H20849the angle /H9251in Fig. 1/H20850can be held fixed by an external magnetic field H/H9251, the value of which is estimated asH/H9251/H11229T//H9262eff, where Tis the temperature /H20849see, for example, Ref. 17/H20850. For/H9262eff/H11229102/H9262BandT/H110111 K the field H/H9251is on the order of 0.1 T. We assume that H/H9251is much smaller than the magnetocrystalline anisotropy field of the STM tip, i.e., thedirection of the external magnetic field controls the directionof the cluster magnetic moment but its influence on the spin-polarization of the STM current is negligible. If the magneticmoment /H9262effof the cluster is “frozen” by the field H/H9251the problem becomes a stationary one. Also we take the distancebetween the contact and the cluster r 0to be much smaller than the radius rH=cpF/eH/H9251of the electron trajectory in theapplied external magnetic field H/H9251, and we do not take into account trajectory magnetic effects, which have been ana-lyzed in Ref. 18. The condition r 0/H11270rHtogether with inequal- ity/H9262BH/H9251//H9255/H112701/H20849/H9255is the electron energy /H20850allow neglecting the magnetic field in the single-electron Hamiltonian andconsidering H /H9251as an independent external parameter. The external magnetic field may result in a modulation of the tunnel current due to electron-spin precession.19For our problem such precession would become noticeable when thetransit time for the electron motion from the contact to thecluster t/H11229r 0/vFis larger than 1 //H9275L, where vFand/H9275Lare the Fermi velocity and Larmor frequency. The inequality men-tioned above, r 0/H11270rH, is equivalent to the condition /H9275Lt/H112701, so that the effect of electron-spin precession is negligible. Thus, under the assumptions outlined above the spinor wave functions /H9023ˆ/H20849rl;/H9268l/H20850satisfy the one-electron Schrödinger equation /H20873−/H60362 2m/H11569/H11612l2−/H9255/H20874Iˆ/H9023ˆ/H20849rl;/H9268l/H20850=−Uˆ/H20849rl/H20850/H9023ˆ/H20849rl;/H9268l/H20850, /H208491/H20850 where m/H11569is the effective mass of the electrons and Uˆ/H20849r/H20850is the interaction potential of the electrons with the cluster. The matrix Uˆconsists of two parts, describing the potential as well as the magnetic scattering, Uˆ/H20849r/H20850=/H20873gIˆ+1 2/H9262BJ/H9262eff/H9268ˆ/H20874D/H20849/H20841r−r0/H20841/H20850, /H208492/H20850 where gis the constant describing the nonmagnetic part of the interaction /H20849forg/H110220 the potential is repulsive /H20850,Jis the constant of exchange interaction, /H9262eff=/H9262eff/H20849sin/H9251,0,cos /H9251/H20850is the magnetic moment of the cluster, /H9268ˆ=/H20849/H9268ˆx,/H9268ˆy,/H9268ˆz/H20850with/H9268ˆ/H9262 the Pauli matrices and Iˆis the unit matrix. D/H20849/H20841r−r0/H20841/H20850is a spherically symmetric function localized within a region ofcharacteristic radius r Dcentered at the point r=r0, which satisfies the normalization condition /H20885dr/H11032D/H20849r/H11032/H20850=1 . /H208493/H20850 Equation /H208491/H20850must be supplemented with the common bound- ary conditions for continuity of the wave function and itsnormal derivative on the metal surface. We assume that the potential Uˆ/H20849r/H20850is small and use per- turbation theory. As a first step we should find the solutions /H9023ˆ /H208490/H20850/H20849rl;/H9268l/H20850of Eq. /H208491/H20850forUˆ/H20849r/H20850=0. Generally, this solution depends on the model chosen to represent the tunnel barrier.For any suitable model for the potential barrier the wave functions /H9023ˆ /H208490/H20850/H20849rl;/H9268l/H20850describe the spreading of the electron waves in the metal from the small contact region on thesurface. Here, we use the results of Refs. 9and20, in which the tunnel contact is modeled in the form of a circular orificeof radius a, with a large amplitude potential barrier U 0/H9254/H20849z/H20850. In order to describe the spin polarization of the STM cur- rent we introduce different magnitudes for the wave vectork /H9268for spin-up and spin-down electrons /H20849for the same energy /H9255/H20850before tunneling from the tip. This difference results in FIG. 1. Model of the contact in a SP-STM configuration with a subsurface magnetic cluster near the tunnel contact point. Spin-polarized electrons tunnel into the nonmagnetic metal in the smallarea below the STM tip. The trajectories of electrons that are scat-tered by the spherical magnetic cluster are shown schematically.AVOTINA, KOLESNICHENKO, AND VAN RUITENBEEK PHYSICAL REVIEW B 80, 115333 /H208492009 /H20850 115333-2different amplitudes t/H20849k/H9268/H20850of the electron waves injected into the nonmagnetic metal for different directions of the spin,9,20 t/H9268/H20849k/H9268/H20850/H11015/H60362k/H9268cos/H9277 im/H11569U0;/H20841t/H9268/H20841/H112701. /H208494/H20850 Here,/H9277is the angle between wave vector k/H9268and the normal to the surface z=0, pointing into the sample. The total effec- tive polarization Peffof the current depends on the difference between the probabilities of tunneling for different /H9268, Peff/H20849/H9255/H20850=/H20841t↑/H208412−/H20841t↓/H208412 /H20841t↑/H208412+/H20841t↓/H208412=k↑2−k↓2 k↑2+k↓2. /H208495/H20850 The functions /H9023ˆ/H208490/H20850/H20849rl;/H9268l/H20850for each spin direction have the same spatial dependence as those for a contact between non-magnetic metals. In following calculations we use the asymptotic expression for /H9023ˆ /H208490/H20850/H20849rl;/H9268l/H20850valid for ka/H112701/H20849Ref. 9/H20850 /H9023ˆ/H208490/H20850/H20849r;/H9268/H20850=t/H9268/H9023/H208490/H20850/H20849r/H20850/H9274ˆ/H9268, /H208496/H20850 where /H9023/H208490/H20850/H20849r/H20850=/H20849ka/H208502z 2ikr2eikr/H208731+1 ikr/H20874, /H208497/H20850 /H9274ˆ/H9268is the spinor /H9274ˆ↑=/H208731 0/H20874,/H9274ˆ↓=/H2087301/H20874, /H208498/H20850 andk=/H208812m/H11569/H9255//H6036is the magnitude of the electron wave vec- torkin the nonmagnetic metal. Note that the wave function /H9023/H208490/H20850/H20849r/H20850, Eq. /H208497/H20850, is zero in all points on the surface z=0, except the point r=0 /H20849at the contact /H20850where it diverges. This divergence is the result of taking the limit a→0 in the inte- gral expressions for /H9023/H208490/H20850/H20849r/H20850.5,20Yet, Eq. /H208497/H20850gives a finite value for the total charge current through the contact as ob-tained over a half sphere of radius rwith its center in the point r=0 for r→0. The solution to Eq. /H208491/H20850in linear approximation in the potential Uˆ, Eq. /H208492/H20850, can be written as /H9023ˆ/H20849r; /H9268/H20850 =t/H9268/H20853/H9023/H208490/H20850/H20849r/H20850/H9274ˆ/H9268+/H20851/H20849g˜/H11006J˜cos/H9251/H20850/H9274ˆ/H9268+J˜sin/H9251/H9274ˆ−/H9268/H20852/H9023/H208491/H20850/H20849r/H20850/H20854, /H208499/H20850 where the sign /H20849/H11006/H20850corresponds to /H9268=↑,↓. We have intro- duced the notation g˜=2m/H11569k /H60362g,J˜=m/H11569k /H9262B/H60362J/H9262eff /H2084910/H20850 for the dimensionless constants of interaction for potential and magnetic scattering, respectively. The wave functionscattered by the cluster is given by/H9023 /H208491/H20850/H20849r/H20850=−1 k/H20885dr/H11032G0+/H20849r,r/H11032/H20850D/H20849/H20841r/H11032−r0/H20841/H20850/H9023/H208490/H20850/H20849r/H11032/H20850,/H2084911/H20850 which undergoes specular reflection at the metal surface. Aiming for a solution for the wave function /H9023ˆ/H9268/H20849r/H20850in first approximation in the small parameter /H20841t/H9268/H20841/H112701 we substitute the electron Green’s function G0+/H20849r,r/H11032/H20850of the nonmagnetic half space in Eq. /H208499/H20850, G0+/H20849r,r/H11032/H20850=G00+/H20849/H20841r−r/H11032/H20841/H20850−G00+/H20849/H20841r−r˜/H11032/H20841/H20850, /H2084912/H20850 where r˜=/H20849/H9267,−z/H20850is the mirror image of the point rrelative to the metal surface and G00+is the Green’s function for free electrons, G00+/H20849R/H20850=−eikR 4/H9266R;R=/H20841r−r/H11032/H20841. /H2084913/H20850 The wave function, Eq. /H208499/H20850, enables calculation of the charge-current density jand spin-current density j/H20849/H9262/H20850, and the expectation value of the spin s. They are obtained as the sums of independent contributions of the two electron groups/H20849l=1,2 /H20850 j/H20849r/H20850=e/H6036 m/H11569/H20858 l=1,2Im/H20849/H9023ˆ/H11612/H9023ˆ+/H20850r1=r2=r, /H2084914/H20850 j/H20849/H9262/H20850/H20849r/H20850=i/H6036 2m/H11569/H20858 l=1,2/H20851/H20849/H11612/H9023ˆ+/H20850/H9268/H9262/H9023ˆ−/H9023ˆ+/H9268/H9262/H20849/H11612/H9023ˆ/H20850/H20852r1=r2=r, /H2084915/H20850 s/H20849r/H20850=/H20858 l=1,2/H20849/H9023ˆ+/H9268ˆ/H9023ˆ/H20850r1=r2=r. /H2084916/H20850 The Eqs. /H2084914/H20850–/H2084916/H20850with wave function, Eq. /H208499/H20850, describe the so-called tunneling contributions /H20849see, Ref. 16/H20850. They are proportional to the tunneling probability and define the mea-surable quantities which can be obtained after averaging overthe energies of the tunneling electrons and wave-vector di-rections /H20849see, Sec. III/H20850. In absence of the cluster s x=sy=0 and the local magneti- zation s0/H20849r/H20850due to itinerant spin-polarized electrons is ori- ented along the zaxis. The spin polarization in zeroth ap- proximation, sz0/H20849r/H20850, which is calculated from the wave function, Eq. /H208496/H20850, is a monotonic function of coordinates sz0/H20849r/H20850=/H20849/H20841t↑/H208412−/H20841t↓/H208412/H20850/H20873kza2 2r2/H208742/H208751+1 /H20849kr/H208502/H20876. /H2084917/H20850 The electron scattering by the spin-dependent potential, Eq. /H208492/H20850, changes the value and the direction of the vector s0/H20849r/H20850. Components sxandsyappear due to electron scattering by the cluster and they are subject to quantum interferencebetween transmitted and scattered waves. As a result of theinterference the spin components perpendicular to the zaxis are oscillatory functions of the coordinates while s z0acquires a small oscillatory component proportional to J˜2.MAGNETO-ORIENTATION AND QUANTUM SIZE EFFECTS … PHYSICAL REVIEW B 80, 115333 /H208492009 /H20850 115333-3/H20875sx/H20849r/H20850 sy/H20849r/H20850/H20876=/H20849/H20841t↑/H208412/H11006/H20841t↓/H208412/H20850J˜sin/H9251/H20841/H9023/H208490/H20850/H20849r/H20850/H20841/H20841/H9023/H208491/H20850/H20849r/H20850/H20841 /H11003/H20877cos sin/H20878/H20851/H92721/H20849r/H20850−/H92720/H20849r/H20850/H20852. /H2084918/H20850 Here /H20841/H9023/H20849i/H20850/H20849r/H20850/H20841and/H9272i/H20849r/H20850are the absolute values and the phases of the coordinate part of the wave functions, Eq. /H208499/H20850 /H20851see, Eqs. /H208497/H20850and /H2084911/H20850/H20852. Figure 2shows the spacial distribution of the xcompo- nent of the normalized spin density sx/H20849r/H20850/c0in the vicinity of the contact. The normalization constant c0=/H20849/H20841t↑/H208412 +/H20841t↓/H208412/H20850J˜/H20849ka/H208504sin/H9251/16/H9266. The figure demonstrates that the spin component sx/H20849r/H20850changes sign in the space of the normal metal. The sign of sxdepends on the difference of the phases /H9272i/H20849r/H20850. In Fig. 3we present a vector plot of the xcomponent of the spin-current density j/H20849x/H20850. An intricate image of the distribution in orientation of j/H20849x/H20850is visible. Note the lines at which the direction of the vector j/H20849x/H20850is inverted.III. CONDUCTANCE OF THE CONTACT, SPIN CURRENT, AND SPIN-TRANSFER TORQUE The total current through the contact can be evaluated by integration of the charge current density j/H20849r/H20850, Eq. /H2084914/H20850, over a half sphere centered at the point contact r=0 and covering the contact at z/H110220, and integration over all directions of the electron wave vector on the Fermi surface /H9255=/H9255F. In the Ohm’s law approximation and at zero temperature the totalcurrent through the contact Ican be written as I=eV /H9267/H20849/H9255F/H20850r2/H20885 /H9255=/H9255Fd/H9024k 4/H9266/H20885d/H9024/H9008 /H20849z/H20850/H9008/H20849kz/H20850jr/H20849r/H20850, /H2084919/H20850 where d/H9024andd/H9024kare elements of solid angle in real and momentum space, respectively, /H9267/H20849/H9255F/H20850is the electron density of states at the Fermi energy, /H9255F, for one spin direction, kzis zcomponent of the wave vector, jr/H20849r/H20850is the radial compo- nent of j/H20849r/H20850, Eq. /H2084914/H20850, and/H9008/H20849x/H20850is the Heaviside unit step function. After integration of Eq. /H2084919/H20850the conductance Gof the contact takes the form G=I V=G0/H208751+6 /H9266/H20849g˜+PeffJ˜cos/H9251/H20850W/H20849r0/H20850/H20876 /H9255=/H9255F,/H2084920/H20850 where G0is the conductance of the contact in absence of the cluster G0=/H20849kF↑2+kF↓2/H20850e2/H60363/H20849kFa/H208504 72/H9266/H20849m/H11569U0/H208502. /H2084921/H20850 Here, kF/H9268andkFare the absolute values of electron wave vectors at the Fermi level in magnetic and nonmagnetic met-als, respectively; P effis the effective spin polarization of the current injected through the contact /H20851see Eq. /H208495/H20850/H20852; the con- stants g˜andJ˜are given by Eq. /H2084910/H20850and W/H20849r0/H20850=/H20885dr/H11032D/H20849/H20841r/H11032−r0/H20841/H20850/H20873z/H11032 r/H11032/H208742 y1/H20849kr/H11032/H20850j1/H20849kr/H11032/H20850, /H2084922/H20850 where jl/H20849x/H20850and yl/H20849x/H20850are the spherical Bessel functions. Equation /H2084920/H20850coincides with the result for a tunnel point contact between nonmagnetic metals9when Peff=0 and krD /H112701. When the radius of action rDof the function D/H20849/H20841r −r0/H20841/H20850is much smaller than the distance between the contact and the cluster center r0,W/H20849r0/H20850is an oscillatory function of kr0forkrD/H113501, but the oscillation amplitude is reduced as a result of superposition of waves scattered by different pointsof the cluster. The integral W/H20849r 0/H20850, Eq. /H2084922/H20850, can be calculated asymptotically for r0/H11271rD,kr0/H112711, and krD/H114071. For a homo- geneous spherical potential D/H20849/H20841r/H20841/H20850=VD−1/H9008/H20849rD−r/H20850/H20849VDis the cluster volume /H20850the function W/H20849r0/H20850takes the form W/H20849r0/H20850/H112293 2/H20873z0 r0/H208742sin 2 kr0 /H208492kr0/H208502j1/H20849kd/H20850 kd, /H2084923/H20850 where d=2rDis the cluster diameter. The last factor in Eq. /H2084923/H20850describes the quantum size effect related with electron reflections by the cluster’s boundary. Such oscillations mayexist if the cluster boundary is sharp on the scale of theelectron wavelength. Figure 4shows the dependence of the amplitude of the conductance oscillations on the cluster di- FIG. 2. Grayscale plot of the spacial distribution of the xcom- ponent of the spin density, sx/H20849r/H20850/c0. The coordinate plane xzin real space crosses the contact and the cluster of the radius rD=k−1with its center in the point r0=/H208490,0,15 /H20850k−1. FIG. 3. Vector plot of the direction of the xcomponent of spin- current density, j/H20849x/H20850. The contour lines correspond to jz/H20849x/H20850=0. The plane xzcrosses the contact and the cluster; as in Fig. 2we have chosen rD=k−1andr0=/H208490,0,15 /H20850k−1.AVOTINA, KOLESNICHENKO, AND VAN RUITENBEEK PHYSICAL REVIEW B 80, 115333 /H208492009 /H20850 115333-4ameter. It demonstrates that a /H9266-phase shift may occur result- ing from interference of electron waves over a distance of thecluster diameter. In Eq. /H2084920/H20850the term proportional to P efftakes into account the difference in the probabilities of scattering of electronswith different /H9268by the localized magnetic moment /H9262eff.I t depends on the angle /H9251between the tip magnetization and /H9262effas cos /H9251. The same dependence was first predicted for a tunnel junction between ferromagnets for which the magne-tization vectors are misaligned by an angle /H9251,16and this was observed in SP-STM experiments.10 Similar to the electrical conductance, Eq. /H2084920/H20850, the total spin current for each spin component can be calculated I/H20849z/H20850=G0V e/H20875Peff+6 /H9266/H20849Peffg˜+J˜cos/H9251/H20850W/H20849r0/H20850/H20876 /H9255=/H9255F,/H2084924/H20850 I/H20849x/H20850=6G0V e/H9266sin/H9251/H20851J˜W/H20849r0/H20850/H20852/H9255=/H9255F. /H2084925/H20850 For our choice of the vector /H9262eff,I/H20849y/H20850=0. The value of the z component of the spin current I/H20849z/H20850, Eq. /H2084924/H20850, is determined to a large extent by the polarization Peff, Eq. /H208495/H20850, of the initial current. The oscillatory part of I/H20849z/H20850is modified by the addition of a term due to spin-flip processes on the cluster. The spin-current component perpendicular to initial direction of polar-ization, I /H20849x/H20850, has only a term that oscillates with r0and which disappears when the magnetic moment /H9262effis aligned with thezdirection. The spin-transfer torque Tacting on the magnetic mo- ment/H9262effis given by T=−J /H6036/H20885dr/H11032D/H20849/H20841r/H11032−r0/H20841/H20850/H9262eff/H11003/H20855s/H20849r/H11032/H20850/H20856, /H2084926/H20850 where/H20855s/H20849r/H20850/H20856=eV/H9267/H20849/H9255F/H20850/H20885 /H9255=/H9255Fd/H9024k 4/H9266s/H20849r/H20850, /H2084927/H20850 s/H20849r/H20850is defined by Eq. /H2084916/H20850. This torque is related with the spin-polarized electron tunnel current. In linear approxima- tion in J˜only the spin-density contribution sz0/H20849r/H20850, Eq. /H2084917/H20850, without interaction with the cluster, should be taken for thecalculation of the torque, Eq. /H2084926/H20850. In this approach T x=Tz =0. For large r0/H11271rDwe obtain Ty= sin/H92513G0V e/H9266/H20877PeffJ˜/H20873z0 kr02/H208742/H208751+1 /H20849kr0/H208502/H20876/H20878 /H9255=/H9255F./H2084928/H20850 The dependence of Ty, Eq. /H2084928/H20850, on the angle /H9251agrees with the dependence of the spin-transfer torque in tunnel junctionsbetween two ferromagnets with different directions ofmagnetization. 16,21 In this paper we do not aim to investigate the dynamics of the cluster magnetic moment. We only note that once thespin-polarized current-induced torque pulls the magnetic mo-ment away from alignment with H /H9251, the cluster moment will start precessing around the field axis. The Larmor frequencyis defined by the magnetic field due to combining the exter-nal field H /H9251and the effective field produced by the polarized current Heff/H11229−J/H20855s/H20849r0/H20850/H20856/gc/H9262B/H20849gcis the cluster “ gfactor” /H20850. The precession of the cluster magnetic moment gives rise toa time modulation of the SP-STM current as for clusters on asample surface. 22,23The study of nonstationary effects pro- vides a further means of obtaining information on the clusterand the spin polarization of the current inside the sample. IV. DISCUSSION In summary, we have studied theoretically the current and spin flows through a tunnel point contact between magneticand nonmagnetic metals when the tunnel current is spin po-larized in the geometry of SP-STM, Fig. 1. Electron-spin-flip processes due to a magnetic cluster situated in the nonmag-netic metal have been taken into account. These processesresult in the appearance of components of the spin-currentdensity j /H20849s/H20850/H20849r/H20850perpendicular to the source direction /H20849taken along the contact axis z/H20850, and a finite expectation value of the spin s/H20849r/H20850. We have analyzed the contribution of tunneling electrons to the spacial distribution of j/H20849x,y/H20850/H20849r/H20850andsx,y/H20849r/H20850.I t is found that these are nonmonotonic functions of the coor-dinates and undergo strong spacial oscillations originatingfrom quantum interference between partial waves transmit-ted through the contact and those scattered by the cluster/H20849see, Figs. 2and3/H20850. Specifically, between the contact and the cluster there are neighboring regions in which j /H20849x/H20850/H20849r/H20850flows in opposite directions /H20849Fig.3/H20850. The oscillatory correction, /H9004G, to the conductance G0of the ballistic tunnel point contact strongly depends on themagnetic scattering, Eq. /H2084920/H20850, /H9004G/H11011/H20849g ˜+PeffJ˜cos/H9251/H20850sin 2 kr0,kr0/H112711. /H2084929/H20850 The effective spin polarization Peffof the tunneling electrons and the dimensionless constants of potential scattering g˜and FIG. 4. Dependence of the oscillatory part of the conductance on the tip position on the metal surface for different cluster diam-eters. The /H92670coordinate is measured from the point /H92670=0 at which the contact is situated directly above the cluster; r0=/H208490,0,10 /H20850/kF; g˜=0.5; J˜=2.5; Peff=0.4; and /H9251=0.MAGNETO-ORIENTATION AND QUANTUM SIZE EFFECTS … PHYSICAL REVIEW B 80, 115333 /H208492009 /H20850 115333-5exchange scattering J˜are given by Eqs. /H208495/H20850and /H2084910/H20850. Gener- ally, for a single magnetic defect, which has a magnetic mo-ment on the order of one Bohr magneton, /H9262B, the spin part of electron scattering gives only a small contribution to theelectrical resistance. For a magnetic cluster with /H9262eff/H11271/H9262B, the exchange energy can be larger than the energy of spin- independent interaction /H20851J˜/H11022g˜, see Eq. /H2084910/H20850/H20852. An interesting phenomenon may be found when PeffJ˜/H11350g˜. A change in the direction of the vector /H9262eff/H20849the angle /H9251/H20850controlled by an external magnetic field is predicted to lead to a change in theamplitude of the oscillations, and for certain directions theamplitude may vanish, G osc=0. This large magneto- orientational effect provides a new mechanism for obtaininginformation on magnetic particles buried below a metal sur-face. Note that this effect is observable only for a spin- polarized current: if t ↑=t↓, in linear approximation in J˜the changes in the scattering amplitude for spin-up and spin-down electrons balance each other and the magneto-orientational effect is absent. As a consequence of spin flips due to the interaction of the electrons with the cluster the oscillatory part /H9004I /H20849z/H20850of the spin current in the original zdirection obtains a correction which depends on the exchange constant J˜and the orienta- tion of the magnetic moment, Eq. /H2084924/H20850 /H9004I/H20849z/H20850/H11011/H20849Peffg˜+J˜cos/H9251/H20850sin 2 kFr0,kFr0/H112711. /H2084930/H20850 A component of the spin current perpendicular to the zdi- rection, Ix/H20849s/H20850, is formed only by scattered waves and as theresult of quantum interference it becomes an oscillatory function of the distance r0, Eq. /H2084925/H20850 I/H20849x/H20850/H11011J˜sin/H9251sin 2 kFr0,kFr0/H112711. /H2084931/H20850 The spin currents, Eqs. /H2084930/H20850and /H2084931/H20850, appear even in the case of nonpolarized current through the contact and they are re-lated to the magnetic scattering by the cluster. The torque, which acts on the magnetic cluster due to spin polarization of electric current is a monotonic function of thedistance r 0, Eq. /H2084928/H20850, to linear approximation in the exchange constant Ty/H11011sin/H9251PeffJ˜/H20849z0/kFr02/H208502,kFr0/H112711, /H2084932/H20850 where z0is the depth of the cluster below metal surface. These results may be exploited in future experiments for detecting and investigating individual magnetic clusters bur-ied below the surface of a host metal. Specifically, a com-parison of the amplitude of the conductance oscillations fordifferent directions of the cluster magnetic moment allowsdetermination of the exchange energy J /H9262effand for a known value of the exchange integral Jto find /H9262eff. ACKNOWLEDGMENTS One of us /H20849Yu.K /H20850would like to acknowledge useful dis- cussions with B. I. Belevtsev, A. B. Beznosov, N. F. Kharch-enko, and D. I. Stepanenko. 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Vonsovsky, Magnetism /H20849John Wiley & Sons, New York, 1974 /H20850. 18Ye. S. Avotina, Yu. A. Kolesnichenko, A. F. Otte, and J. M. van Ruitenbeek, Phys. Rev. B 75, 125411 /H208492007 /H20850. 19F. J. Jedema, H. B. Heersche, A. T. Fllip, L. L. A. Baselmans, and B. J. van Weels, Nature /H20849London /H20850416, 713 /H208492002 /H20850. 20I. O. Kulik, Yu. N. Mitsai, and A. N. Omelyanchouk, Zh. Eksp. Teor. Fiz. 63, 1051 /H208491974 /H20850. 21L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 22H. C. Manoharan, Nature /H20849London /H20850416,2 4 /H208492002 /H20850. 23C. Durkan and M. E. Welland, Appl. Phys. Lett. 80, 458 /H208492002 /H20850.AVOTINA, KOLESNICHENKO, AND VAN RUITENBEEK PHYSICAL REVIEW B 80, 115333 /H208492009 /H20850 115333-6

PhysRevB.86.205324.pdf

PHYSICAL REVIEW B 86, 205324 (2012) Molecular nanoplasmonics: Self-consistent electrodynamics in current-carrying junctions Alexander J. White,1,*Maxim Sukharev,2,†and Michael Galperin1,‡ 1Department of Chemistry & Biochemistry, University of California at San Diego, La Jolla, California 92093, USA 2Department of Applied Sciences and Mathematics, Arizona State University, Mesa, Arizona 85212, USA (Received 2 October 2012; published 30 November 2012) We consider a biased molecular junction subjected to an external time-dependent electromagnetic field. We discuss local field formation due to both surface plasmon-polariton excitations in the contacts and the molecularresponse. Employing realistic parameters we demonstrate that such self-consistent treatment is crucial for theproper description of the junction transport characteristics. DOI: 10.1103/PhysRevB.86.205324 PACS number(s): 42 .50.Ct, 78.67.−n, 85.65.+h, 73.63.Kv I. INTRODUCTION Research in plasmonics is expanding its domains into several subfields due to significant advances in experimental techniques.1–6The unique optical properties of the surface plasmon-polariton (SPP) resonance, being the very foun-dation of plasmonics, find intriguing applications in opticsof nanomaterials, 7–9materials with effective negative index of refraction,10–12direct visualization,13,14photovoltaics,15–17 single-molecule manipulation,18–20and biotechnology.21–24 Theoretical modeling of the optical properties of metal nanos- tructures is conventionally based on numerical integrationof Maxwell’s equations, 25–29although simulations within time-dependent density functional theory appeared recently for small atomic clusters.30,31Moreover, current theoretical models are quickly advancing toward self-consistent simu-lations of hybrid materials: metal/semiconductor nanostruc- tures optically coupled to ensembles of quantum emitters. 32 This methodology, based on numerical integration of corre- sponding Maxwell-Bloch equations, brings new insights intonano-optics because it allows for the capture of collectiveeffects. The molecular optical response in close proximity to plasmonic materials is greatly enhanced by SPP modes leadingto the discovery of the single-molecule spectroscopy. 33–35Re- cently, experiments performed on current-carrying molecularjunctions have started to appear. 36–40Theoretical modeling of molecule-SPP systems utilizes the tools of quantum mechanicsfor the molecular part. In particular, studies of optical response of isolated molecules absorbed on metallic nanoparticles utilize Maxwell-Bloch (Maxwell-Schr ¨odinger) 32,41–44equa- tions or near-field–time-dependent density functional theoryformulations. 45,46 Realistic molecular devices are open quantum systems ex- changing energy and electrons with surrounding environment(baths). This is especially important in studies of molecules incurrent-carrying junctions interacting with external fields. 47 Usually in such studies the electromagnetic (EM) field isassumed to be an external driving force. 51–62Recently we utilized the nonequilibrium Green’s function technique tostudy the transport and optical response of a molecular junctionsubjected to an external EM field taking into account nearfields driven by SPP local modes, specific for a particularjunction geometry. 48,49Although the formulation allows us to describe the molecular junction with formation of the localfield by SPP excitations in the contacts taken into account explicitly, the molecular influence on formation of the localEM field was disregarded in these studies. Note that suchinfluence was shown to have measurable effects in plasmonicspectrum. 32,41,44,50 When a molecule located near a metal surface is driven by a strong EM field, one can expect to observe significantchanges in the total EM field due to radiation emitted by themolecule. Such radiation although quickly degrading with thedistance from molecular position can nevertheless noticeablyalter the local EM field. Since the latter is driving the molecule,transport characteristics of the junction may be significantlymodified. This calls for a self-consistent treatment, whereboth SPP excitations and molecular response participate information of the local EM field. Here we extend our previous considerations by taking into account complete electrodynamics and molecular junctionresponse in a self-consistent manner combining Maxwell’sequations with electron transport dynamics. The molecule istreated as a pointwise source in Ampere’s law. We demonstratethe importance of the molecular response in the formationof the local field for an open molecular system far fromequilibrium. The effect is shown to be important for properdescription of the junction transport characteristics. The paperis organized as follows. Section IIpresents a transport model of the molecular junction. Section IIIdescribes the methodology of computing the EM field taking into account molecularresponse. The results are presented in Sec. IV. Section V summarizes our work. II. MOLECULAR JUNCTION SUBJECTED TO EXTERNAL EM FIELD We consider a junction with a molecular bridge ( M) connecting between two contacts ( LandR). The bridge is formed by Dtwo-level systems with the levels representing ground ( g) and excited ( x) states of the molecule. Each of the two level systems is subjected to a classical local EM field/vectorE(t) (see Sec. IIIfor details of its calculation). Electron transfer is allowed along the chain of ground (excited) levelsof the bridge. The contacts are taken in the form of bow-tieantennas, and are assumed to be reservoirs of free electronseach in its own equilibrium with electrochemical potentialsμ LandμR, respectively (see Fig. 1). The Hamiltonian of the 205324-1 1098-0121/2012/86(20)/205324(8) ©2012 American Physical SocietyWHITE, SUKHAREV , AND GALPERIN PHYSICAL REVIEW B 86, 205324 (2012) y40 nm75 nm L R 1g1x DgDxz x FIG. 1. (Color online) Sketch of the junction. system reads (here and below e=¯h=1) ˆH(t)=ˆHM(t)+/summationdisplay K=L,R(ˆHK+ˆVK), (1) ˆHM(t)=/summationdisplay s=g,x/bracketleftBiggD/summationdisplay m=1εsˆd† msˆdms −D−1/summationdisplay m=1ts(ˆd† m+1sˆdms+H.c.)/bracketrightBigg −D/summationdisplay m=1(/vectorμmg,mxˆd† mgˆdmx+H.c.)/vectorEm(t), (2) ˆHK=/summationdisplay k∈Kεkˆc† kˆck, (3) ˆVK=/summationdisplay k∈K/summationdisplay s=g,x/parenleftbig Vk,mKsˆc† kˆdmKs+H.c/parenrightbig , (4) where ˆHM(t) and ˆHKare Hamiltonians of the molecular bridge ( M) and the contacts ( K=L,R ), and ˆVKis coupling between them. In Eqs. (2)–(4) ˆd† ms(ˆdms) and ˆc† k(ˆck)a r e creation (annihilation) operators for an electron on the levelsof the molecular bridge site mand state kof the contact, respectively. /vectorE m(t) is the local time-dependent field at bridge sitem, and/vectorμms,ms/prime=/angbracketleftms|ˆ/vectorμ|ms/prime/angbracketrightis the matrix element of the transition molecular (vector) dipole operator between states|ms/angbracketrightand|ms /prime/angbracketright. For simplicity below we assume that the transition dipole moment is the same for all bridge sites andhas only one nonzero component, μ mg,mx≡μgxfor any m.ts (s=g,x) andVk,mKsare matrix elements for electron transfer in the molecular bridge and between molecule and contacts,respectively, and m K=1(D)f o r K=L(R). Note that treating the external field classically allows us to account forarbitrary time dependence exactly (i.e., beyond perturbationtheory). 49 We follow the formulation of Ref. 49. Time-dependent current at interface K(LorR)i s63 IK(t)=− Im Tr/bracketleftbigg /Gamma1K/parenleftbigg G<(t,t)+/integraldisplayd/epsilon1 πfK(/epsilon1)Gr(t,/epsilon1)/parenrightbigg/bracketrightbigg , (5) where Tr[ ···] is a trace over the molecular subspace, fK(/epsilon1)≡ [e(/epsilon1−μK)/T+1]−1is the Fermi-Dirac distribution in contact K,/Gamma1Kis the molecular dissipation matrix due to coupling to contact K /Gamma1K m1s1,m2s2(/epsilon1)≡2π/summationdisplay k∈KVm1s1,kVk,m 2s2δ(/epsilon1−εk), (6) andG<(r)is a matrix in the molecular basis of the lesser (retarded) projection of the single-particle Green’s function,defined on the Keldysh contour as64 Gm1s1,m2s2(τ1,τ2)≡−i/angbracketleftTcˆdm1s1(τ1)ˆd† m2s2(τ2)/angbracketright. (7) HereTcis the contour ordering operator and τ1,2are the contour variables. In Eq. (5)Gr(t,/epsilon1) is the right Fourier transform of the retarded projection of Green’s function (7) Gr(t,/epsilon1)≡/integraldisplay dt/primeei/epsilon1(t−t/prime)Gr(t,t/prime). (8) Note that in Eq. (5)and below we assume the wide band limit65 in the metallic contacts. Green’s functions in Eq. (5)satisfy the following set of equations of motion:48,66 i∂ ∂tGr(t,/epsilon1)=I−/parenleftbig /epsilon1I−HM(t)+i 2/Gamma1/parenrightbig Gr(t,/epsilon1), (9) id dtG<(t,t)=[HM(t);G<(t,t)]−i 2{/Gamma1;G<(t,t)} +i/summationdisplay K=L,R/integraldisplayd/epsilon1 2πfK(/epsilon1)(/Gamma1KGa(/epsilon1,t) −Gr(t,/epsilon1)/Gamma1K), (10) where Iis the unity matrix, HM(t) is a representation of the op- erator (2)in the molecular basis, /Gamma1≡/summationtext K=L,R/Gamma1K,[...;...] and{...;...}are the commutator and anticommutator, and Ga(/epsilon1,t)≡[Gr(t,/epsilon1)]†. The first-order differential equations (9) and (10) are solved starting from the initial condition of the biased junction steady state in the absence of the optical pulse,E(t=0)=0 G r 0(/epsilon1)≡Gr(t=0,/epsilon1)=/bracketleftbigg /epsilon1I−HM(t=0)+i 2/Gamma1/bracketrightbigg−1 ,(11) G< 0≡G<(t=0,t=0) =i/summationdisplay K=L,R/integraldisplayd/epsilon1 2πGr 0(/epsilon1)/Gamma1KfK(/epsilon1)Ga 0(/epsilon1), (12) where Ga 0(/epsilon1)≡[Gr 0(/epsilon1)]†. Below we calculate the charge pumped through the junction by the optical pulse Q(t)≡/integraldisplayt 0dt/primeIL(t/prime)−IR(t/prime) 2−I0t, (13) where IL,R(t) are defined in Eq. (5), andI0is the steady-state current I0≡/integraldisplayd/epsilon1 2πTr/bracketleftbig /Gamma1LGr 0(/epsilon1)/Gamma1RGa 0(/epsilon1)/bracketrightbig (fL(/epsilon1)−fR(/epsilon1)).(14) III. SELF-CONSISTENT ELECTRODYNAMICS The time evolution of electric /vectorEand magnetic /vectorHfields is considered according to the set of Maxwell’s equations(written here in SI units) μ 0∂/vectorH(/vectorr,t) ∂t=−/vector∇×/vectorE(/vectorr,t), (15a) /epsilon10∂/vectorE(/vectorr,t) ∂t=/vector∇×/vectorH(/vectorr,t)−/vectorJ(/vectorr,t), (15b) where μ0and/epsilon10are the magnetic permeability and dielectric permittivity of the free space, respectively, and /vectorJ(t)i st h e 205324-2MOLECULAR NANOPLASMONICS: SELF-CONSISTENT ... PHYSICAL REVIEW B 86, 205324 (2012) electric current density. Note that magnetization is disregarded in Eqs. (15a) and(15b) , since we assume both molecule and contacts to be nonmagnetic. A molecule located at site m(≡/vectorrm) and driven by local electric field /vectorE(/vectorr,t), yields a time-dependent response, which enters Ampere’s law as a polarization current density /vectorJ(/vectorr,t)=∂/vectorPm(t) ∂tδ(/vectorr−/vectorrm), (16) where δis the Dirac delta function. The polarization depends on molecular characteristics through the molecular densitymatrix, which in turn is affected by the local field. In ourmodel, two-level systems of the molecular bridge (2)are assumed to occupy sites of the finite-difference–time-domain(FDTD) grid. Molecules contribute to the polarization at theirsite according to /vectorP m(t)=2I m [/vectorμmx,mgG< mg,mx (t,t)]. (17) The resulting system of coupled differential equations, Eqs. (15a) and(15b) , is solved simultaneously with EOMs for Green’s functions of the quantum system, Eqs. (9)and (10). Maxwell’s equations are discretized in time and space andpropagated using the FDTD approach. 67We employ three- dimensional FDTD calculations utilizing home-built parallel FORTRAN -MPI codes on a local multiprocessor cluster.68In spatial regions occupied by a plasmonic nanostructure (a bow-tie antenna in our case) we employ the auxiliary differentialequation method to account for materials dispersion. Thedielectric response of the metal is modeled using a standardDrude formulation with the set of parameters describingsilver. 48,49Green’s function EOMs are propagated with the fourth-order Runge-Kutta scheme. Within the described self-consistent model the local electric field/vectorEm(t)≡/vectorE(/vectorrm,t)i nE q . (2)driving a molecular junction is thus defined by both SPP excitations in the contacts andthe local molecular response. In the next section we showthat the molecular contribution changes the junction transportcharacteristics drastically, and in general cannot be ignored. IV . NUMERICAL RESULTS Here we present results of numerical simulations demon- strating the importance of a self-consistent treatment of thelocal EM field dynamics. Previous studies considered theinfluence of an isolated molecule on plasmon transfer, 41,42,45 molecular features in absorption,32,50,69,70and Raman29,31,46 spectra of molecules attached to nanoparticles. Below we discuss how molecular junctions and electron transport areinfluenced by a local EM field and vice versa in a self-consistent manner. Unless otherwise specified, parameters of the calculations areT=300 K, ε x=−εg=1e V ,tx=tg=0.05 eV , μgx= 32 D, /Gamma1L 1g,1g=/Gamma1R Dx,Dx =0.1 eV , and /Gamma1L 1x,1x=/Gamma1R Dg,Dg= 0.01 eV (other elements of the dissipation matrix are zero). The choice of parameters for the model was discussed in detailin our previous considerations. 48,49We note that according to the structure of the dissipation matrix, Eq. (6), and the characteristic highest occupied molecular orbital to lowestunoccupied molecular orbital (HOMO-LUMO) separation of∼2–3 eV , off-diagonal elements of the dissipation matrix(T 2dissipation) are much smaller than its diagonal elements (T1dissipation), and thus can be ignored. Note also that the T2type of dephasing is present in the model through the molecular coupling to the external field. Asymmetry inthe molecular coupling to the contacts represents a moleculewith a strong charge-transfer transition (see Refs 59and 61 for details). Such molecules are the primary candidates forconstruction of optically driven molecular charge pumps. TheFermi energy is taken at the origin, E F=0, and the bias is applied symmetrically, μL=−μR=Vsd/2. Following Ref. 48, the incoming incident field is taken in the form of a chirped pulse Einc(t)=Re/bracketleftbigg E0exp/parenleftbigg −(δ2−i¯μ2)t2 2−iω0t/parenrightbigg/bracketrightbigg , (18) where E0is the incident peak amplitude, ω0is the in- cident frequency, and δ2≡2τ2 0/[τ4 0+4/Phi1/prime/prime2(ω0)] and ¯ μ≡ −4/Phi1/prime/prime(ω0)/[τ4 0+4/Phi1/prime/prime2(ω0)] are parameters describing the incident chirped pulse ( τ0is the characteristic time related to the pulse duration). In the calculations below we useE 0=107V/m,ω0=2e V ,τ0=11 fs, and /Phi1/prime/prime(ω0)=3000 fs2. Figure 2shows instantaneous electric field strength distri- butions in a plane shifted by z=10 nm parallel to the xy plane. The distribution is calculated for a junction formedby bow-tie antennas with a single molecule ( D=1) placed in the center of the gap. Here ε x−εg=1.75 eV , /Gamma1L 1g,1g= /Gamma1R Dx,Dx =0.01 eV , /Gamma1L 1x,1x=/Gamma1R Dg,Dg=0.001 eV , and Vsd=0. Figure 2(a) presents simulations without molecular response. Figure 2(b) shows the results of a calculation where both SPP excitations in the contacts and molecular response are takeninto account. One can clearly see that even a single moleculedrastically changes the local electric field distribution. Sensitivity of the pulse temporal behavior to the molecular response is presented in Fig. 3(a). Here a local field affected by only SPP modes (dotted line) is compared to pulses calculatedwhen the molecular response is taken into account. The lattermay result in both enhancement (dashed line) or quenching(solid line) of the local field depending on the ratio of thepulse frequency ω 0to the molecular excitation energy εx−εg. In particular, quenching is observed for the laser frequencybeing below the threshold ( ω 0<εx−εg=2.25 eV), while frequency above the threshold ( ω0>εx−εg=1.75 eV) leads to enhancement of the field. To understand this behavior weperform a simple analysis treating coupling to the driving fieldas a perturbation, and neglecting the chirped character of thepulse. This leads to (see the Appendix) P 1(t)≈− E0cos(ω0t)|μgx|2/integraldisplayd/epsilon1 2π ×/parenleftbigg Im[G< 1g,1g(/epsilon1)]/epsilon1−(εx−ω0) [/epsilon1−(εx−ω0)]2+[/Gamma11x,1x/2]2 +Im[G< 1x,1x(/epsilon1)]/epsilon1−(εg+ω0) [/epsilon1−(εg+ω0)]2+[/Gamma11g,1g/2]2/parenrightbigg , (19) where G<is the lesser projection of Green’s function (7). Taking into account that in the absence of the chirp Einc(t)= E0cos(ω0t), the first term in the right side of Eq. (19) suggests that for a populated ground state, G< 1g,1g(/epsilon1)≈1, the molecular 205324-3WHITE, SUKHAREV , AND GALPERIN PHYSICAL REVIEW B 86, 205324 (2012) FIG. 2. (Color online) Map of the instantaneous electric field strength [ E2 x(/vectorr,t)+E2 y(/vectorr,t)+E2 z(/vectorr,t)]1/2at a distance of 10 nm from the molecule (the plane is parallel to xy) calculated (a) without and (b) with the molecular response. The distribution is shown for t= 77.8 and 81 .7 fs for (a) and (b), respectively. See text for parameters. polarization oscillates in phase with the field for ω0<εx− εg, and in antiphase for ω0>εx−εg. Thus according to Eqs. (15b) and(17) the molecular response quenches the field in the former case, and enhances it in the latter. Figure 3(b) illustrates this finding within the exact calculation showing themaximum of the total field for different molecular excitationenergies (circles) compared to the maximum of the EM fieldobtained without molecular response (triangles). Note that thecontribution of the second term in the right side of Eq. (19) is exactly the opposite that of the first term; however, since thecalculations presented in Fig. 3are performed at zero bias, the molecular excited state is initially empty, G < 1x,1x(t=0)≈0. While the local EM field cannot be measured directly, it is related to junction characteristics (in particular, itstransport properties) detectable in experiments. Figure 4(a) demonstrates the difference in the temporal buildup of thecharge pumped through the junction, when the molecule isconsidered to be driven by the field obtained within theself-consistent model vs the model with only SPP excitationstaken into account. The initial dip in the charge buildup(see dotted line) is related to a time delay of the moleculeinduced pulse for ε x−εg<ω 0[compare solid and dashed lines to the dotted line in Fig. 3(a)]. The delay is caused by the chirped nature of the incoming pulse, with initial pulsefrequency being lower than the molecular excitation energy,-404108E (V/m) 60 90 120 t( f s ) 2468108Emax(V/m) 1 1.5 2 2.5 x-g(eV)(a) (b) FIG. 3. (Color online) Local EM field at the molecular position. (a) Pulse calculated without (dotted line, black) and with ( εx−εg> ω0- solid line, red; εx−εg<ω 0- dashed line, blue) molecular response. (b) Maximum local field during the pulse vs molecular excitation energy calculated without (triangles, black) and with (circles, red) molecular response. See text for parameters. which results in suppression of the local field at the start of the pulse. Eventually, however, the incoming frequencybecomes higher than the molecular transition energy. Thecorresponding enhancement of the local field leads to anincrease in the charge pumped through the junction. Note thatforε x−εg>ω 0no delay is observed, and the local field is quenched throughout the pulse. Correspondingly the effective-ness of the charge pump is lower in this case [see solid linein Fig. 4(a)]. Figure 4(b) shows the total charge pumped through the junction during the pulse at different molecular excitationenergies. Clearly, the most effective EM field obtained withoutthe molecular response taken into account corresponds to theresonance situation, ω 0=εx−εg=2 eV . When molecular response is included in the model the situation is less straight-forward. Since local field enhancement is expected for lowmolecular excitation energies, ω 0>εx−εg[see Fig. 3(b)], the peak in the pumped charge distribution is shifted to theleft. Note that the lower height of the shifted peak is relatedto the fact that for a lower molecular gap, part of the opticalscattering channels is blocked due to partial population ofthe broadened excited and ground states of the molecule (seeRef. 49for detailed discussion). Note that the importance of the molecular response depends also on bias across the junction. Indeed, since high bias,V sd>εx−εg, may inject holes into the molecular ground state and electrons into the excited state, and since populating 205324-4MOLECULAR NANOPLASMONICS: SELF-CONSISTENT ... PHYSICAL REVIEW B 86, 205324 (2012) -0.9-0.6-0.30.00.3Q( e ) 60 90 120 t( f s ) 0.511.5Q( e ) 1 1.5 2 2.5 x-g(eV)(a) (b) FIG. 4. (Color online) Charge pumped through the junction. (a) Difference, /Delta1Q≡Q(sc)−Q(nosc), between results calculated with, Q(sc), and without, Q(nosc), molecular response vs time for εx−εg>ω 0(solid line, red) and εx−εg<ω 0(dotted line, blue). (b) Total charge pumped during the pulse vs molecular excitation energy calculated without (triangles, black) and with (circles, red)molecular response. See text for parameters. these states has opposite consequences for the local field enhancement [see Eq. (19) and the discussion following it], it is natural to expect that the molecular response is moreimportant at low biases, V sd<εx−εg. Figure 5illustrates this conclusion with results of our calculations within theself-consistent model. Here /Gamma1 L 1x,1x=/Gamma1R 1g,1g=0.05 eV . We observe that the difference in both the optically induced currentand the charge pumped through the junction (see inset) isalmost negligible at high biases. Similar reasoning indicatesthat the molecular response at strong incoming fields willbe less important also due to the population of the excitedmolecular state induced by external pulse. Asymmetry in the charge pumping relative to the sign of the chirp rate was discussed in our recent publication (see Fig. 4 in Ref. 48). One of the reasons for the asymmetry is related to the time spent by the local pulse in the region of frequenciesat and just below the resonance. This region provides the maincontribution to charge transfer (see discussion of Fig. 3in Ref. 48). Since time spent in this region by the positively chirped pulse is smaller than that by the pulse with equalnegative chirp rate (the positively chirped local pulse isshorter), one expects to observe an asymmetry as representedby the result of calculations using local EM field influenced-30310-6IL(A) 60 90 120 t( f s )-0.2-0.100.1 60 90 120 t( f s )Q( e ) FIG. 5. (Color online) Current at the left interface as a function of time. Shown are differences, /Delta1IL≡I(sc) L−I(nosc) L , between results calculated with, I(sc) L, and without, I(nosc) L , molecular response. The calculations are performed for Vsd=1.5 V (dashed line, blue) and 2 V (solid line, red). Inset shows corresponding difference in chargepumped through the junction. See text for parameters. only by SPP modes driving the junction (see curve with triangles in Fig. 6). However, as discussed above, it is this preresonance region where molecular response quenches localfield, thus diminishing (or even overturning) the asymmetryrelative to the chirp rate sign (see curve with circles in Fig. 6). Finally, we consider a three-site molecular bridge ( D=3) to model the spacial nonlocality of molecular polarization.Calculations are done for ε mx−εmg=2.25 eV , ω0=2e V , andVsd=0. Figures 7(a)–7(c) compare the pure plasmonic local field to the field calculated when the molecular responseis taken into account for the three sites of the bridge. Molecularpolarization decreases the local field amplitude on the first site[Fig. 7(a)], and enhances it on the rightmost site [Fig. 7(c)]. The field at the middle site [Fig. 7(b)] does not change. The effect can be understood following the discussion similar to -8-4010-2 02468 103 ’’(fs2)[Q( ’’)-Q(- ’’)]/Q avg FIG. 6. (Color online) Asymmetry in the charge transfer be- tween positively and negatively chirped incoming laser pulses,Q(/Phi1 /prime/prime)−Q(−/Phi1/prime/prime), normalized by their average, Qavg≡[Q(/Phi1/prime/prime)+ Q(−/Phi1/prime/prime)]/2. Shown are results calculated without (triangles, black) and with (circles, red) the molecular response. See text for parameters. 205324-5WHITE, SUKHAREV , AND GALPERIN PHYSICAL REVIEW B 86, 205324 (2012) -505108E(V/m) -505108E(V/m) -505108E( V / m) 60 90 t(fs)-0.100.1n -0.300.3108E(V/m) 00.511.5Q( e ) 01 0 0 2 0 0 t( f s )(a) (b) (c)(d) (e) (f) FIG. 7. (Color online) Effect of the self-consistent treatment on local field and level population in a three-site molecular bridge ( D= 3) as functions of time. Panels (a)–(c) show the local fields calculated without (dotted line, black) and with ( εx−εg>ω 0- solid line, red) molecular response for the three molecular sites. Panel (d) shows the difference in population of the ground, /Delta1n 1g≡n(sc) 1g−n(nosc) 1g(solid line, blue), and excited, /Delta1n 1x≡n(sc) 1x−n(nosc) 1x(dotted line, red), states for the first molecular site ( m=1). Panel (e) shows the scaled plot of the field on the central site ( m=2) for a longer period of time. The charge pumped through the three-site molecular bridge vs time is shown in panel (f). See text for parameters. that of Fig. 3. We find that for εx−εg>ω 0an increase in the population in the ground (a decrease in the excited) levelsof the molecular sites quenches the local field. Change inthe populations of the leftmost site [Fig. 7(a)] resulting from self-consistent treatment is shown in Fig. 7(d). We see that these changes are in agreement with the corresponding changein the local field. Similar considerations also hold for Figs. 7(b) and7(c) (the corresponding level populations are not shown). We note in passing that several parameters of the model definethe behavior of the bridge population during and after thepulse. In particular, strength and frequency of the external fielddefine the efficiency of charge transfer between ground andexcited states of the molecule: stronger coupling and resonantfrequency usually result in stronger population of the excitedstate (at equilibrium the excited state is empty in the absence ofthe pulse). Strength of the molecule-contact coupling definesthe lifetime of the excess population on the molecule. After theend of the pulse it takes ∼1//Gamma1for populations of molecular states to return to their steady-state values. The latter aredefined by the bias. The self-consistently calculated electric field on a site in the bridge shows a visible beat at large time scale [see Fig. 7(e)]. This behavior is related to the Rabi frequency due to theintersite coupling t s. Finally, Fig. 7(f) shows the charge transferred through the three-site junction as a function of time. The decrease in theeffectiveness of the pump is related to quenching of the local field on the first site of the bridge, where strong couplingto the left contact yields a quick resupply of the groundlevel population. Decreased efficiency in pumping the chargebetween ground and excited levels at this site is the reason forthe overall change in the effectiveness of the pump. V . CONCLUSION We consider a simple model of a molecular junction driven by external chirped laser pulses. The molecule is representedby a bridge of Dtwo-level systems. The contacts geometry is taken in the form of a bow-tie antenna. The FDTD techniqueis used to calculate the local field in the junction resultingfrom SPP excitations in the contacts. Simultaneously we solvetime-dependent nonequilibrium Green’s function equations ofmotion to take into account the molecular contribution to thelocal field formation. Note that many works on driven transport assume a pure incident field to be a driving force acting on the molecule.In our recent publications 48,49we considered the effects of local field formation due to SPP excitations in the contactson junction characteristics under external optical pumping.Here we make one more step by taking into account alsothe molecular response in the driving local field dynamics.Within a reasonable range of parameters we demonstrate thatthe latter is crucial for proper description of the junctiontransport. We compare our results with previously publishedpredictions, and show that the molecular contribution may leadto measurable differences (both quantitative and qualitative) inthe characteristics of junctions. This contribution is especiallyimportant at low biases and relatively weak external fieldsin the presence of a strong molecular transition dipole.In particular, we show that for laser frequencies shorter(higher) than the molecular excitation energy, the local SPPfield is usually quenched (enhanced) by molecular response.Extension of the approach to realistic ab initio calculations, taking into account a time-dependent bias, and formulatinga methodology for calculations in the language of molecularstates are the goals for future research. ACKNOWLEDGMENTS M.G. gratefully acknowledges support by the NSF (Grant No. CHE-1057930) and the BSF (Grant No. 2008282). APPENDIX: DERIVATION OF EQ. (19) To understand trends observed in the exact calculations based on Eqs. (9)and(10) and(15a) and(15b) , here we employ a simple consideration and derive an approximate expressionfor the molecular polarization, Eq. (17),g i v e ni nE q . (19).F o r simplicity we assume that only one projection of the moleculardipole is nonzero, and consider a single-molecule bridge ( D= 1). Then the molecular polarization is P 1(t)=− 2Im[μxgG< 1g,1x(t,t)]. (A1) Assuming the dissipation matrix of Eq. (6)is diagonal, the lesser projection of Green’s function in Eq. (A1) is given by 205324-6MOLECULAR NANOPLASMONICS: SELF-CONSISTENT ... PHYSICAL REVIEW B 86, 205324 (2012) the Keldysh equation of the form G< 1g,1x(t,t)=/summationdisplay s=g,x/integraldisplayt −∞dt1/integraldisplayt −∞dt2Gr 1g,1s(t,t1) ×/Sigma1< 1s,1s(t1−t2)Ga 1s,1x(t2,t), (A2) where /Sigma1< 1s,1s(t1−t2)=i/summationdisplay K=L,R/integraldisplayd/epsilon1 2πfK(/epsilon1)/Gamma1K 1s,1se−i/epsilon1(t1−t2) (A3) is the lesser self-energy due to coupling to the contacts. We start by neglecting a chirp of the incoming field Einc(t)=E0cos(ω0t), (A4) and treat the interaction between the molecule and incoming field Vss/prime(t)≡−δs/prime,¯sμs¯sEinc(t)( A 5 )within the first order of perturbation theory. Here ¯sindicates the state opposite to s, i.e., for s=g¯s=x. Within the approximations, the retarded Green’s function in Eq. (A2) can be expressed as (a similar expression can be written for theadvanced projection) G r 1s,1s/prime(t,t/prime)≈δs,s/primeG(0)r 1s,1s(t−t/prime) +/integraldisplay+∞ −∞dt/prime/primeG(0)r 1s,1s(t−t/prime/prime)Vss/prime(t/prime/prime) ×G(0)r 1s/prime,1s/prime(t/prime/prime−t/prime). (A6) Here G(0)ris the retarded projection of Green’s functions (7) in the absence of external field G(0)r 1s,1s(t−t/prime)=−iθ(t−t/prime)e−i(εs−i/Gamma11s,1s/2)(t−t/prime),(A7) andθ(...) is the Heaviside step function. Utilizing Eqs. (A4) –(A7) in Eqs. (A1) –(A3) leads to P1(t)≈− E0|μgx|2/integraldisplayd/epsilon1 2π/parenleftbigg Im/bracketleftbig G(0)< 1g,1g(/epsilon1)/bracketrightbig[/epsilon1−(εx−ω0)] cos( ω0t)−[/Gamma11x,1x/2] sin( ω0t) [/epsilon1−(εx−ω0)]2+[/Gamma11x,1x/2]2 +Im/bracketleftbig G(0)< 1x,1x(/epsilon1)/bracketrightbig[/epsilon1−(εg+ω0)] cos( ω0t)+[/Gamma11g,1g/2] sin( ω0t) [/epsilon1−(εg+ω0)]2+[/Gamma11g,1g/2]2/parenrightbigg , (A8) where we have used the Keldysh equation for the steady-state situation G(0)< 1s,1s(/epsilon1)=/summationtext K=L,RifK(/epsilon1)/Gamma1K 1s,1s [/epsilon1−εs]2+[/Gamma11s,1s/2]2. (A9)Assuming that detuning is much bigger than level broadenings, |ω0−(εx−εg)|/greatermuch/Gamma11s,1s(s=g,x), the term with sin( ω0t)i n Eq.(A8) can be ignored. Finally, dressing Green’s functions in Eq. (A8) , i.e. taking into account diagrams related to population redistribution in the molecule due to presence ofthe driving field, leads to Eq. 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PhysRevB.95.035422.pdf

PHYSICAL REVIEW B 95, 035422 (2017) Spin-orbit torque in two-dimensional antiferromagnetic topological insulators S. Ghosh*and A. Manchon† King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering Division (PSE), Thuwal 23955, Saudi Arabia (Received 5 September 2016; revised manuscript received 3 November 2016; published 23 January 2017) We investigate spin transport in two-dimensional ferromagnetic (FTI) and antiferromagnetic (AFTI) topological insulators. In the presence of an in-plane magnetization AFTI supports zero energy modes, which enablestopologically protected edge conduction at low energy. We address the nature of current-driven spin torque inthese structures and study the impact of spin-independent disorder. Interestingly, upon strong disorder the spintorque develops an antidamping component (i.e., even upon magnetization reversal) along the edges, which could enable current-driven manipulation of the antiferromagnetic order parameter. This antidamping torque decreaseswhen increasing the system size and when the system enters the trivial insulator regime. DOI: 10.1103/PhysRevB.95.035422 I. INTRODUCTION The successful manipulation of small magnetic elements using spin-polarized currents via spin transfer torque hasopened appealing perspectives for low power spin devices[1–3]. In the past ten years, it has been predicted [ 4–7] and observed [ 8–13] that noncentrosymmetric magnets with large spin-orbit coupling can also exhibit large spin torque,a phenomenon called spin-orbit torques (SOT). The physicsof SOT in homogeneous ferromagnets [ 14–20] and magnetic textures [ 21–26] has attracted a massive amount of attention since then. While these torques have been originally studiedin bulk noncentrosymmetric magnets [ 8,9] and ultrathin magnetic multilayers [ 10–13], their observation has been recently extended to magnetic bilayers involving topologicalinsulators [ 27–30]. A topological insulator (TI) is characterized by gapless edge/surface states in the absence of an external magneticfield [ 31]. The zero energy modes arise due to time reversal symmetry and are immune to nonmagnetic disorder [ 32,33]. This topological protection breaks down in the presenceof magnetization which destroys the zero energy modesand opens a gap [ 34]. This process is accompanied by the emergence of quantum anomalous Hall effect [ 35–37], as well as quantum magnetoelectric effect when the Fermilevel lies in the gap of the surface states [ 38,39]. Recently, three-dimensional TI have been used to achieve large SOTin an adjacent ferromagnet [ 27–30]. In spite of significant theoretical efforts to model the SOT exerted on homogeneousferromagnets [ 37,40–46] and magnetic textures [ 47–52], the exact nature of the torque observed experimentally remainsa matter of debate as it is not clear whether surface statesare still present and how bulk and surface transport contributeto the different components of the torque. Besides significantchallenges in terms of materials growth, the main difficultylies in the fact that magnetism itself breaks the topologicalprotection of surface states [ 53–58], which prevents us from taking full advantage of the gigantic spin-orbit coupling of *sumit.ghosh@kaust.edu.sa †aurelien.manchon@kaust.edu.sathe Dirac cones. Fortunately, ferromagnetism is not the onlyuseful magnetic order parameter that appears in nature. Recently, it has been realized that antiferromagnets can also be controlled by spin transfer torque [ 59–61], opening the emergent field of antiferromagnetic spintronics [ 62–64]. The nature of spin transfer torque has been investigatedtheoretically in antiferromagnetic spin-valves and tunneljunctions [ 65–72], as well as antiferromagnetic domain walls [73–79]. Most importantly for the present work, it has been recently predicted [ 80,81] and experimentally demonstrated [82] that SOT can also be used to control the direction of the antiferromagnetic order parameter. This naturally bringsTI as a possible testing ground due to their inherent strongspin-orbit coupling. Since antiferromagnetism only breakstime-reversal symmetry locally but not globally, it preservesthe topological nature of the surface gapless states [ 83,84]. Exploring the possibility of combining the topological natureof the surface or edge states in antiferromagnetic topologicalinsulators with the physics of SOT could therefore openappealing perspectives. In this paper, using scattering wave function formalism implemented on a tight-binding model, we explore the natureof spin transport and torque in two-dimensional ferromagnetic(FTI) and antiferromagnetic (AFTI) topological insulators. Wefind that AFTI is more robust against disorder than FTI, suchthat topological edge states are preserved even under weakdisorder. Most importantly, SOT possesses two components:a fieldlike torque ( odd under magnetization reversal) and an antidamping torque ( even under magnetization reversal). While the former is directly generated by the spin-momentumlocking at the edges, the latter arises upon scattering and isquite sensitive to disorder and size effects. II. METHOD We start from the Bernevig-Hughes-Zhang model [ 85]o n a square lattice. We use the basis (1 ↑,2↑,1↓,2↓)T, where 1,2 refer to two orbitals and ↑,↓refer to spin projections, and define the TI Hamiltonian by a 4 ×4 matrix, H(k)=/parenleftbigg h(k)0 0h∗(−k)/parenrightbigg , (1) 2469-9950/2017/95(3)/035422(7) 035422-1 ©2017 American Physical SocietyS. GHOSH AND A. MANCHON PHYSICAL REVIEW B 95, 035422 (2017) where h(k)i sg i v e nb y h(k)=[M+B(cos(kx)+cos(ky))]σz +Asin(kx)σx+Asin(ky)σy. (2) HereA,B,M are model parameters whose values depend on the real structure [ 31,85]. The topologically nontrivial phases appear for B>|M/2|, which is manifested as gapless edge states in quasi-one-dimensional systems. In case of a CdTe-HgTe quantum well this is achieved by tuning the width ofthe quantum well. For our calculations we choose Ato be the unit of energy and consider B=1.0A,M=−1.5Athat ensures the existence of topologically protected edge states fornonmagnetic TI. To map this bulk Hamiltonian ( 1) on a finite scattering region, we first extract the tight-binding parameters[83,86] by expressing H(k)=H 0+Hˆxeikx+Hˆyeiky+H† ˆxe−ikx+H† ˆye−iky,(3) with H0=⎛ ⎜⎝M 000 0−M 00 00 M 0 000 −M⎞ ⎟⎠, Hˆx=1 2⎛ ⎜⎝B −iA 00 −iA−B 00 00 Bi A 00 iA−B⎞ ⎟⎠, Hˆy=1 2⎛ ⎜⎝B−A 00 A−B 00 00 B−A 00 A−B⎞ ⎟⎠. We can use these hopping elements to construct a real space Hamiltonian for a finite system as H=/summationdisplay ic† iH0ci+/summationdisplay i/negationslash=jc† itijcj, (4) where tij=Hˆrij,ˆrij(=±ˆx,±ˆy) being the unit vector between nearest neighbor sites iandj, andc† i(ci) is the creation (destruction) operator for the state (1 ↑,2↑,1↓,2↓)Tat the ith site. The coupling between itinerant spins /vectorsand the local magnetization ( /vectormi), as well as the disorder potential Vr i,a r e introduced in the onsite energy as Himp m=/summationdisplay ic† i(H0+/vectormi·/vectors+Vr iI4)ci+/summationdisplay i/negationslash=jc† itijcj,(5) where Inis the nth rank identity matrix, and /vectors=(ˆsx,ˆsy,ˆsz), with ˆsx=/parenleftbigg 0 I2 I20/parenrightbigg ,ˆsy=/parenleftbigg 0−iI2 iI2 0/parenrightbigg ,ˆsz=/parenleftbigg I2 0 0−I2/parenrightbigg . (6) In the following, we consider five different configurations, defined by the spatial modulation of /vectormi: an ordinary, nonmag- netic TI as a reference (referred to as O), an FTI ( F), and A-,B-, and G-type AFTI configurations [( A), (B), and ( G)i n Fig.1]. The total system can be divided into three parts: (i) left lead, (ii) scattering region, and (iii) right lead. The leads areFIG. 1. Schematic of FTI and different types of AFTI. The green region shows one unit cell of the lead. Blue and red dots representpositive and negative /vectorm i. semi-infinite and can be characterized by one unit cell (green shaded region in Fig. 1) whereas the scattering region is defined by Eq. ( 5). Note that for A,G type AFTI we need to double the unit cell of the lead to maintain the translation symmetry.For this work, we consider a scattering region composed of40×20 sites arranged on a square lattice. To calculate the transport properties we adopt the wave function approach, asimplemented in the tight-binding software KWANT [ 87]. This approach is equivalent to the nonequilibrium Green’s functionformalism [ 88]. In this method one starts by defining the incoming modes at a particular energy in terms of eigenstatesof an infinite lead and subsequently obtain the wave functionwithin the scattering region by using the continuity relations.By applying this method throughout the scattering regionone can obtain the outgoing modes that can be exploited toconstruct the Smatrix of the system. The scattering wave function and the Smatrix are two basic outputs one can obtain from KWANT for any given system (see Sec. 2 of Ref.[87] for details). The conductance of the system is calculated from the Smatrix using Landauer-B ¨uttiker formalism. To calculate the nonequilibrium spin density at some given energyE F, we use a small bias voltage VBias=μL−μR, where eμL(R)=EF±eVBias/2 are the chemical potential of the left (right) lead. We use the scattering wave function calculated byKWANT and evaluate the expectation values of different spincomponents integrated over the bias window to get the totalnonequilibrium spin density as /vectorS neq i=/integraldisplayμL μR/angbracketleftψi(E)|/vectors|ψi(E)/angbracketrightdE, (7) where /vectorsis the onsite spin operator defined in Eq. ( 6), andψi(E) is the scattering wave function for the ith site at energy E. Once we get the nonequilibrium spin density we can calculate theonsite SOT as /vectorτ i=/vectormi×/vectorSneq i. (8) 035422-2SPIN-ORBIT TORQUE IN TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 035422 (2017) FIG. 2. Conductance ( G) of TI with different magnetic config- urations against random disorder for (a) out of plane ( /vectorm=0.2Aˆz) and (b) in plane ( /vectorm=0.2Aˆy) magnetic moments. The conductance is normalized to the conductance quantum G0=2e2/h. The boxed portion of (a) is enlarged in the inset. Finally, in order to introduce nonmagnetic disorder in the system we add to the Hamiltonian, Eq. ( 5), a random onsite energy Vr iuniformly distributed over the range [ −V0,V0]. This gets rid of any possible shift of energy spectrum that mightappear if one chose only positive amplitudes for the disorderpotential. The transport properties are then averaged over 1280random disorder configurations. III. ROBUSTNESS OF DIFFERENT MAGNETIC CONFIGURATIONS Let us first compute the impact of disorder on the conduc- tance in the various magnetic configurations. Figures 2(a)and 2(b) display the conductance behavior as a function of the disorder strength when the direction of the magnetic moments /vectormiis (a) out of plane and (b) in the plane. Here, the transport energy is taken as EF=0.25A. From Fig. 2(a) we see that when the magnetic moments lie out-of-plane FTI ( F) is comparatively more sensitive to disorder than AFTI ( A,B,G ) and nonmagnetic TI ( O), although the difference of robustness between the nonmagneticand magnetic TI is not very large. The initial quantizedconductance of FTI starts decreasing around V 0∼1.5Adue to the progressive quenching of the topological protection ofthe edge modes, while in contrast, AFTI and nonmagneticTI maintain their topological edge states up to V 0∼2A. The difference becomes quite significant when we set themagnetic order in the plane, see Fig. 2(b). Noticeably, F andBcases are very sensitive to disorder, while AandG cases are much more robust. This indicates that AFTIs withan in-plane staggered magnetic character along the transportdirection remain topological insulators even for weak disorder. For a better understanding of this effect, we calculate the density of states of the TIs in the absence of disorder, seeFig. 3(a). An in-plane magnetic order opens a gap for Fand B, while AandGcan preserve their gapless states similarly to the nonmagnetic TI ( O). Since the topological protection is stronger at lower energy, we calculate the robustness atE F=0.05A[Fig. 3(b)] and find the quantized conductance due to the edge states of A,G, andOcases survives longer compared to that evaluated at EF=0.25A.BandFtypes have a gap at that energy and hence show zero conductance.FIG. 3. (a) Total density of states (DOS) for TI, FTI and different AFTIs. (b) Conductance against random disorder at EF=0.05A (filled symbols) and at EF=0.25A(open symbols) for different configurations with an in-plane magnetic moment /vectorm=0.2Aˆy. From now on, we proceed with only FTI and G-type AFTI as they qualitatively behave similarly as B-type AFTI and A-type AFTI, respectively. IV . NONEQUILIBRIUM SPIN DENSITY AND SPIN-ORBIT TORQUE As mentioned in the introduction, spin transfer torque [59–61]a sw e l la sS O T[ 80,81] can be used to control the direction of the antiferromagnetic order parameter. The orderparameter can be controlled in two ways [ 62–64]: either using a time-dependent (ac) fieldlike torque (i.e., a torque that is odd under magnetization reversal), or using a time-independent(dc) antidamping torque (i.e., even under magnetization reversal). Our intention is to investigate the nature of SOTin the AFTI case, where both topologically protected edgetransport and antiferromagnetic order parameter coexist. First we calculate the total nonequilibrium spin density and the associated SOT in FTI [Figs. 4(a)and4(c)] and AFTI [Figs. 4(b)and4(d)], with an in-plane magnetic order ( /vectorm i∼ˆy) and in the absence of disorder. In these calculations, we set|m i|=0.2A,E F=0.25Aand (μL−μR)=0.02A. Figures 4(a) and 4(b) display the spatial distribution of the different components of the nonequilibrium spin densityin FTI and AFTI, respectively. The middle panels show thespatial profile of S y, the spin density component that is aligned along the magnetic order. This component is uniform in FTIand staggered in AFTI, as expected from the magnetic textureof these two systems. We observe finite S zon both edges, which is a characteristic feature of a TI (bottom panels) andmore interestingly finite and oscillatory S xon both edges (top panels). The oscillation is caused by the scattering atthe interfaces between the conductor and leads. Since S xand Syare not immune to scalar perturbation, the potential steps at the interfaces mix these components depending on the chiralityof each edge. Note that the amplitude of oscillation of S xin G-AFTI is two orders of magnitude smaller compared to that in FTI which denotes that the scattering in G-AFTI is weaker compared to that in FTI. From the symmetry we can easily recognize that Sz produces the so-called fieldlike torque [ /vectorτi F∼/vectormi×/vectorz] andSx gives rise to the antidamping torque [ /vectorτi D∼/vectormi×(/vectorz×/vectormi)]. Figures 4(c) and 4(d) represent the spatial profile of the 035422-3S. GHOSH AND A. MANCHON PHYSICAL REVIEW B 95, 035422 (2017) FIG. 4. Nonequilibrium spin densities for (a) FTI and (b) G- AFTI. We use a magnetization strength m=0.2Aalong ˆyand the spin densities are evaluated at EF=0.25Awith a bias voltage ( μL− μR)=0.02A. (c),(d) shows the corresponding fieldlike ( τF, magenta) and antidamping ( τD, green) SOT evaluated at the top edge for FTI andG-AFTI, respectively. fieldlike ( τF) and antidamping torques ( τD) at the top edges of the FTI and AFTI, respectively. To understand how thesetorques evolve in the presence of disorder, we further study therobustness of S xandSzin the presence of scalar disorder (see Fig.5). We define the (i) uniform spin density ( Sx,z u=/angbracketleftSx,z i/angbracketright) and (ii) staggered spin density ( Sx,z st=/angbracketleftsign (mi)Sx,z i/angbracketright) where the average is over the lattice sites. Since the spin density islocalized at the edges we calculate the robustness for the topedge only (Fig. 5). Similar results can also be obtained for the bottom edge. From Fig. 5we can see that, correspondingly with con- ductance, the nonequilibrium spin densities also fall downfaster in FTI compared to AFTI. Due to its periodic mod-ulation, S x uis initially zero for both FTI and AFTI. When FIG. 5. Variation of uniform ( Sx,z u) and staggered ( Sx,z st)s p i n densities as a function of disorder strength with an in-plane magnetic order|/vectormi|=0.2Afor (a) FTI and (b) G-AFTI. The green dot-dashed line shows the corresponding conductance. FIG. 6. Evolution of Sxat top layer with disorder strength for (a) FTI and (b) G-AFTI. The magnetic order and Fermi level is |/vectormi|= 0.2AandEF=0.25Arespectively. The spin density is averaged over 1280 configurations and calculated with a bias voltage 0 .02A. increasing the disorder, two different effects take place: (i) a progressive smearing of the edge wave function accompaniedby a reduction in S z u; (ii) an increase of disorder-induced spin-dependent scattering resulting in enhanced spin mixing.This disorder-induced spin mixing is at the origin of the S x st,u component observed in Figs. 5(a) and5(b). This mechanism has been originally established in metallic spin valves [ 89] and domain walls [ 90]. In a disordered ferromagnetic device submitted to a nonequilibrium spin density /vectorS0, spin dephasing and relaxation produce an additional corrective spin density of the form ∼/vectorm×/vectorS0. In the case of FTI, the magnetization is uniform so that a uniform Sx u∼Sz u/vectorm×/vectorzis produced [Figs. 5(a) and6(a)]. In the case of AFTI, the magnetization is staggered so that a staggered Sx st∼Sz u/vectormi×/vectorzis generated [Figs. 5(b) and6(b)]. In the latter, no uniform Sx uemerges. Notice that the buildup of Sx st,uupon disorder is a nonlinear process as disorder increases spin mixing and reduces Sz u, at the same time. Hence, one can identify three regimes ofdisorder. In the case of AFTI displayed in Fig. 5(b): (1) From V 0=0t oV0≈A,Sz u, remains (mostly) unaf- fected, while Sx st,uvanishes on average. (2) From V0≈AtoV0≈2A, topological protection breaks down progressively and Sz uis reduced upon disorder due to increased delocalization of the edge wave function. Duringthis process, disorder enhances spin mixing and therebyS x stincreases moderately. During this moderate increase the reduction of Sz uis compensated by the increase in spin mixing, thereby producing a finite Sx st. (3) For V0>2A,Sz uis further reduced and correspond- inglySx stdecreases too, as the disorder-driven spin mixing cannot compensate the reduction of Sz uanymore. Notice that although Sz u/greatermuchSx stin the weak disorder limit, both spin density components tend towards a similar valuefor large disorder ( V 0>2A),Sz u≈Sx st. In the case of FTI displayed in Fig. 5(a), the three regimes appear at different disorder strengths due to the weaker topological protection ofthe edges states. To illustrate the progressive buildup of S x st,uupon disorder, the spatial profile of the spin density at the edge of the 035422-4SPIN-ORBIT TORQUE IN TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 035422 (2017) FIG. 7. Staggered Sx stat/vectorm=0.2AˆyandV0=1.5Afor different lengths ( L) and widths ( W, given in legend). sample is reported in Fig. 6for (a) FTI and (b) AFTI. These calculations confirm that the overall increase in Sx stis a direct consequence of spin-dependent scattering upon disorder. Inthe absence of disorder, the component S xdisplays a smooth oscillation (green dots), as discussed previously. Following theprocess described above, in FTI with positive magnetization,the scattering creates mostly positive S xalong the top edge [Fig. 6(a)], while in AFTI due to the staggered magnetization Sxacquires a staggered nature [Fig. 6(b)]. It is worth mentioning that since the staggered Sx stemerges as a correction to Sz uupon scattering, its magnitude is not only sensitive to disorder but also to the dimension of the channel.As a matter of fact, Fig. 7shows that for a given amount of disorder, the magnitude of S x stdecreases with increasing the channel length Land (slightly) decreases when reducing the channel width W. We remind that the magnitude of Sxdepends on the amplitude of the edge wave function as well as on thestrength of the scattering potential, as mentioned above. Due tofinite size effect, the edge localization increases and graduallyreaches a saturation value as the width is increased. Thereforefor a given disorder strength, the edge states of a wider AFTIundergo smaller delocalization resulting in an enhanced S z uand thereby larger Sx st. Increasing the length Lfavors destructive interferences and therefore reduces Sx stprogressively. It is quite instructive to analyze our results in the light of the latest developments of spin torque studies on antiferromagnets[62–64]. As a matter of fact, it is well known that an external uniform magnetic field only cants antiferromagnetic momentsand is unable to switch the direction of the antiferromagneticorder parameter. Notwithstanding, time-dependent uniform magnetic fields (e.g., a magnetic pulse) can induce inertialantiferromagnetic dynamics [ 63,64]. In addition, it was re- cently proposed that a spin torque possessing an antidampingsymmetry [i.e., /vectorm i×(/vectorp×/vectormi), where /vectorp=/vectorzin our case] can manipulate the antiferromagnetic order parameter of acollinear antiferromagnet [ 60,80]. Applied to the AFTI studied in the present work, these considerations imply that a currentpulse can exert a torque on the antiferromagnetic orderparameter through the uniform spin density S z u, while a dc current can exert a torque via the staggered spin density Sx st.FIG. 8. The uniform and staggered spin density components for (a) G-AFTI at EF=0.05A, (b) G-AFTI at EF=0.25Aand (c) for TAF at EF=0.25A.F o rT A Fw eu s e B=1.0A, M =−2.2A. |/vectormi|=0.2Afor all three cases. Therefore, the staggered Sx stcomputed in Figs. 5(b) and6(b) can in principle be used to control the antiferromagnetic orderof an AFTI. Note that in the case of AFTI, we can choose E Fvery close to zero where the topological protection is stronger, withoutsignificantly affecting the magnitude of the current-driven spindensities [see Figs. 8(a) and8(b)]. By tuning the parameter Mone can weaken the topological protection and turn the AFTI into a trivial antiferromagnet (TAF). In this regime, thenonequilibrium spin density is more distributed within the bulkof the TAF and does not have any topological protection. As aresult and in spite of the strong spin-orbit coupling, S z uandSx st remain both very small [see Fig. 8(c)]. V . CONCLUSION In this work we present a detailed analysis of spin transport in two-dimensional FTI and AFTI. We show that topologicaltransport in AFTI is more robust compared to FTI in thepresence of both out-of-plane and in-plane magnetic order. Anin-plane magnetic order opens a gap in an FTI but preservesthe gapless states in an AFTI when the antiferromagneticorder is along the direction of transport, which allows anAFTI to operate at a much smaller energy. We also studythe robustness of the nonequilibrium spin density and SOTagainst scalar disorder and find that the in-plane spin densitiesget mixed up due to scattering. In the clean limit, this mixingis two orders of magnitude smaller in AFTI compared to FTI,which suggests that AFTI has stronger topological protectionagainst scalar disorder. The SOT possesses two components,a fieldlike torque arising from the spin-momentum locking atthe edges and an antidamping torque arising from scattering.This antidamping torque linearly decreases when increasingthe length of the sample due to destructive interferences. 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PhysRevB.92.224421.pdf

PHYSICAL REVIEW B 92, 224421 (2015) Effect of partial order on galvanomagnetic transport properties of ferromagnetic PdFe and PdCo alloys J. Kudrnovsk ´y and V . Drchal Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-182 21 Praha 8, Czech Republic I. Turek Institute of Physics of Materials, Academy of Sciences of the Czech Republic, ˇZiˇzkova 22, CZ-616 62 Brno, Czech Republic (Received 1 October 2015; published 15 December 2015) The effect of ordering on the galvanomagnetic properties such as the anomalous Hall conductivity (AHC) and the anisotropic magnetoresistance (AMR) ratio is studied using recently developed formalism which takesinto account both the Fermi surface and Fermi sea contributions to the conductivity tensor. As a case study wehave considered Pd-rich L1 2-Pd 70Fe30alloy and disordered fcc-Pd 1−xCoxalloy with a weak partial L10and L12orderings which can exist around stoichiometric compositions. We have found a nonnegligible effect of the Fermi sea term for the Pd-rich PdFe alloys while its effect for completely disordered fcc-PdCo alloy is small.The most important effect of inclusion of a weak partial order for fcc-PdCo alloys in theoretical calculations isimprovement of the agreement between theory and experiment for the resistivity but first of all for the AMRratio. An enhancement of the AMR ratio due to the layerlike L1 0ordering brings calculated and experimental values into a good agreement missing in the disordered alloy. The effect of the isotropic L12ordering on the AMR values is weak. Finally, the effect of the ordering on the resistivity and the AHC is much weaker, althoughobservable. DOI: 10.1103/PhysRevB.92.224421 PACS number(s): 72 .10.Fk,72.15.Gd,75.47.Np,75.50.Bb I. INTRODUCTION The influence of atomic ordering on the resistivity and other transport quantities represents an important problem inthe physics of metallic alloys [ 1]. The binary substitutional solid solutions of varying compositions on simple latticesaround specific concentrations exhibit ordered or partiallyordered superstructures depending on thermal treatment. Asan example let us mention the L1 0andL12superlattices which can occur on the parent fcc lattice for CuAu alloys with compo-sitions Cu 50Au50and Cu 75Au25/Cu 25Au75, respectively [ 2]. In ferromagnetic alloys with nonnegligible role of the spin-orbitinteraction, the atomic ordering modifies also the galvano-magnetic properties such as the anisotropic magnetoresistance(AMR) and the anomalous Hall conductivity (AHC) [ 3–8]. The AMR and AHC attract a considerable interest from the pointof view of technological applications. Their possible tuning oroptimization by varying order in the system has to be basedon a detailed understanding of underlying atomic processes.Alloys of late transition metals such as, e.g., of Pd-rich PdFeor PdCo alloys, are interesting both technologically as well ascase studies due to availability of corresponding experimentaldata which allows us to test the power of available theoreticaltools. The long-range ordered phases exist as stable phases in Pd-rich PdFe alloys with the L1 2structure around the composition Pd 75Fe25. Similarly, alloys with the L10structure exist around the equiconcentration composition, like, e.g., PtFealloys. The ideal L1 0andL12structures exist only around specific, stoichiometric alloy compositions mentioned abovewhile for nonstoichiometric alloys only partial order existswith a varying amount of disorder depending on annealing.Some other alloys, e.g., the PdCo alloys, form a continuoussolid solution with disordered fcc cubic structure. In this alloyno superlattice structures are found [ 9]. However, as discussedin Refs. [ 9–11], a detailed analysis indicates the presence of the metastable L1 2phase for high Pd concentrations (60% to 90%) and L10phase in a concentration window around the equiconcentration composition. This phase is accompaniedby two-phase regions at both sides of it [ 10]. Thus, some amount of order, a large one in Pd-rich PdFe and much smallerone in PdCo alloys, can be present in the system dependingon the sample preparation and can thus influence physicalproperties of those alloys. Particularly transport properties likethe residual resistivities AMR and AHC can be sensitive to theamount of order present. We have recently illustrated this fact on the study of galvanomagnetic properties of ordering L1 0-PtFe alloys [ 3]. In particular, the AHC requires a careful and consistent treatment. To this end we have employed the recently developed hybrid formalism [ 12] for the relativistic linear transport (the Kubo- Bastin approach) which includes, on equal footing, both thecontribution due to the states at the Fermi surface whichcontains most important scattering effects (the Fermi surfaceterm) and the term including all occupied states below theFermi level (the Fermi sea term). The latter one contributes essentially to the intrinsic part of the AHC (see below). In previous studies of disordered alloys the Fermi sea term,which exists also in ordered systems, has been neglected.This concerns also our early studies of the effect of orderon galvanomagnetic properties of Pd-rich PdFe alloys [ 4,5]. In Ref. [ 4] we have used a model in which the spin-orbit term was added to the scalar-relativistic Hamiltonian as aperturbation while the fully relativistic treatment of the Fermi surface term was employed in Ref. [ 5]. The present paper has two aims: (i) We wish to investigate the relevance of theFermi sea term on galvanomagnetic properties of well-orderednonstoichiometric Pd 70Fe30alloy with the L12structure; and (ii) we present a detailed study of galvanomagnetic propertiesof disordered fcc-PdCo. Such a system was studied recently 1098-0121/2015/92(22)/224421(8) 224421-1 ©2015 American Physical SocietyJ. KUDRNOVSK ´Y, V . DRCHAL, AND I. TUREK PHYSICAL REVIEW B 92, 224421 (2015) using the approach [ 13] which neglects the Fermi sea term. Authors of Ref. [ 13] obtained a good agreement between theory and experiment for the resistivity and the AHC while theAMR ratio was strongly underestimated for Co concentrationslarger than about 30%. We will show that inclusion of a weakpartial order, specifically the L1 0one, which can be present in the fcc-PdCo alloy in the concentration window around 50% of Co, at least as an admixture [ 10], can improve agreement between the theory and experiment, in particular as the AMRratio is concerned. As we shall discuss later, the reason isthe concentration inhomogeneity inherent in the layeredlikeL1 0lattice due to the partial long-range order which leads to structural anisotropy. A similar effect was also found recentlyin the partially ordered L1 0-PtFe alloy [ 3]. On the contrary, L12ordering has much smaller effect on the AMR ratio as the system remains an essentially isotropic one. II. FORMALISM The underlying electronic structure of the partially ordered alloys in the L12andL10lattices was studied in the frame- work of the multisublattice relativistic tight-binding linearmuffin-tin orbital (TB-LMTO) method in which the effect ofdisorder is treated using the coherent potential approximation(CPA) [ 14]. The CPA is a reliable approach as concerns concentration trends including the effect of partial ordering.The partial ordering on the L1 2andL10lattices can be characterized by the long-range order (LRO) parameter S.T h e full order ( S=1) exists only for the stoichiometric case, e.g., for the fcc-Pd 50Co50in the case of L10ordering. The value S< 1 for the nonstoichiometric case. We refer the reader to Ref. [ 2] for the definition of the partial long-range order parameter. As an example, we give the expression for Sfor the L10ordering in the Pd-rich Pd 1−xCoxalloys, i.e., S=(x−d)/[2x(1−x)], (1) where xdenotes the Co concentration ( x< 0.5) and dis an auxiliary parameter (0 /lessorequalslantd/lessorequalslantx) allowing us to introduce additional disorder. The case d=0 corresponds to the highest possible degree of order. In the present case it correspondsto a fully occupied Pd sublattice with off-stoichiometric Pdatoms distributed randomly on the Co lattice. The finite valueof the parameter dcharacterizes an amount of Co atoms on the originally fully occupied Pd lattice that were interchanged withthe same amount of Pd atoms on the Co sublattice. If d=x we obtain the random Pd-rich fcc-Pd 1−xCoxalloy. It should be noted that for the Co-rich alloy ( x> 0.5) the definition of the LRO parameter remains the same, but the role of xand 1−xis interchanged. The case of L12ordering is slightly more complicated [ 2]. The transport properties are studied using the recently de- veloped relativistic Kubo-Bastin approach [ 12] which allows us to treat both ordered and disordered systems on equalfooting. The main features of this approach formulated forthe zero temperature can be summarized as follows: (i) Theconductivity tensor consists of two terms, the first one whichcontains contribution only from the states at the Fermi energyis called the Fermi surface term. This term was previously usedin a number of applications to disordered systems includinga closely related approach based on the fully relativisticKorringa-Kohn-Rostocker (KKR) method [ 15]. This term also includes the most important elastic scattering effects due toimpurities. (ii) The second term depends on all occupied statesbelow the Fermi energy and it is thus called the Fermi seaterm. This term can be effectively evaluated by integration inthe complex energy plane over a contour which starts at theFermi energy and encircles all occupied valence states [ 12]. It is important to note that the Fermi sea term is regular inthe dilute limit of random alloy in contrast to the divergingFermi surface term, i.e., it has an intrinsiclike behavior.(iii) Consistently with the site representation of the TB-LMTOmethod we employ the hopping formulation of the velocityoperators in the transport formulation [ 16] which relies on the neglect of electron motion inside the atomic spheres. This isan excellent approximation for the linear response regime and,which is more important, it also leads to nonrandom effectivevelocity matrices as contrasted to random ones in the con-ventional formulation [ 15]. This fact significantly simplifies the evaluation of the disorder-induced vertex corrections [ 17]. (iv) A detailed analysis of properties of various contributionsto the AHC [ 12] allows us to identify the sum of the coherent part of the Fermi surface term (i.e., without vertex corrections)and the full Fermi sea term as an intrinsiclike part and theremaining vertex part of the surface term as an extrinsiclike partof the full AHC. As already mentioned, the main differenceis a strong concentration dependence of the extrinsic term in random alloy, in particular in the low-impurity limit, as contrasted to regular and concentration weakly dependingbehavior of the intrinsiclike term. (v) The effect of temperatureon transport properties is still one of the challenges in the field.A corresponding theory has to include the effect of phononsand magnons (spin disorder). In recent theoretical papers acombined effect of phonons and magnons is described interms of a multicomponent random alloy [ 18] which can be treated in the CPA or using the supercell approach [ 19,20]. The simplest version of such an approach which can beapplied to the spin disorder is the uncompensated disorderedlocal moment (DLM) approach, essentially a two-componentversion of a more general approach. The uncompensated DLMmodel allows us to make reasonable qualitative and sometimeseven quantitative conclusions. It should be noted that the DLMmethod does not give direct relation to the temperature whichis accessible in a more general approach [ 18]. In the past we have successfully applied the uncompensated DLM modelto study the effect of temperature on the galvanomagneticproperties of alloys with dominating spin disorder (see, e.g.,Refs. [ 3–5]). A related first-principles formulation of the AHE for disordered systems in which both the Fermi surface and Fermisea terms are included has appeared recently [ 6]. The main difference between this approach and the present one [ 12]i s the impurity potential. While in the present formulation weemploy a specific material-related spin-dependent impuritypotential, the spinless Gaussian disorder model treated in theframework of the second order perturbation theory with anempirical disorder parameter (containing the disorder strengthand the impurity concentration) is used in Ref. [ 6]. On the other hand, the full-potential formulation used in the latter approachcan be more accurate for ordered systems as compared to thepresent one employing spherical potentials. 224421-2EFFECT OF PARTIAL ORDER ON GALV ANOMAGNETIC . . . PHYSICAL REVIEW B 92, 224421 (2015) Once the components of the conductivity tensor σ(σμν, μ,ν=x,y,z ) are determined, the components of the resistivity tensor ρare found straightforwardly, ρ=σ−1. The total (isotropic) resistivity ρtotand the AMR ratio are determined asρtot=(2ρxx+ρzz)/3, and AMR =(ρzz−ρxx)/ρtot,i ft h e alloy magnetization is pointing along the zdirection. The AHC and anomalous Hall resistivity (AHR) are identified with σxy andρxy, respectively. We conclude this section with some details of numerical implementation. Calculations were done in the framework ofthe local density approximation using the V osko-Wilk-Nusairexchange correlation potential [ 21] and the spd-basis set. In selected cases, also the scalar-relativistic counterparts of thefully relativistic TB-LMTO-CPA codes were employed (seebelow). A small imaginary part Im z=10 −5Ry has been added to the Fermi energy for the evaluation of the Fermi surfaceterm (the corresponding longitudinal resistivity is smallerthan 0.1 μ/Omega1cm) while the Fermi sea term was evaluated using the contour integration (up to 40 energy points). Alarge number of kvectors of order 10 8in the Brillouin zone is required for the numerical convergency in transportcalculations. Unfortunately, there is no general rule for thenumber of points to be used and for each crystal or alloy theconvergence tests were done carefully. For further details werefer to our paper [ 12]. III. RESULTS A. Pd-rich L1 2-Pd 70Fe30alloy 1. Effect of ordering The resistivity and the AMR in ordering Pd 70Fe30alloy were studied in previous papers [ 4,5]. Here we wish to complete this study by evaluating the AHC and to investigatethe relevance of the Fermi sea term neglected in previousstudies [ 4,5]. We shall limit ourselves to samples with large values of the LRO parameter Sfor which a good agreement between experiment and theory was obtained for the resistivityand the AMR [ 5]. Results are summarized in Fig. 1in which the resistivities and the total Fermi surface and Fermi sea contributions tothe total AHC are plotted as a function of the parameter Sin the neighborhood of the maximal possible order. It should benoted that for the present nonstoichiometric alloy the maximalvalue of Sis equal to S max=0.833 and it corresponds to the case when Fe atoms fully occupy the native Fe lattice whileremaining ones are equally distributed on three equivalentPd sublattices of the L1 2lattice. Note that an opposite sign convention for σxyis used in Fig. 1in agreement with previous studies [ 4,5]. The following conclusions can be done: (i) The expected monotonic decrease (increase) of the resistivity (AHC) withthe increasing order is clearly seen. In particular, the resistivityforS=S max(dashed vertical line) is the smallest one for a given alloy composition while the AHC reaches its maximum.(ii) The Fermi sea contribution is weakly depending onthe alloy order and it has, contrary to the Fermi surfacecontribution, the same sign. Its effect on the AHC is thusrelatively stronger for smaller Sand relatively weaker for S close to the S max. On the other hand, the qualitative dependence 0 5 10 15 20 0.6 0.65 0.7 0.75 0.8 0.85-30-20-10 0 10 20 30 40 50 60 70ρtot (μΩ cm) σxy (kS/m) LRO parameter SPd70Fe30 alloy, Smax=0.833 FIG. 1. The residual resistivities (full boxes) and the total anomalous Hall conductivities σxy(full circles) of Pd 70Fe30are shown as a function of the long-range order parameter Son the L12lattice. The total anomalous Hall conductivities are resolved into the Fermi surface (empty circles) and Fermi sea (empty diamonds) contributions. The maximal possible value of Sfor the present composition ( Smax=0.833) is indicated by the dashed vertical line. of the AHC on Sremains unchanged, e.g., the critical value of Sfor which the sign of the AHC is changed is modified slightly. (iii) A large increase of the AHC close to Smaxis due to the dominating extrinsic contribution, i.e., the vertex part of the Fermi surface term. Such behavior of the AHC is characteristic for the low impurity limit (the local concentrations of the Featoms on Pd sublattices are only x Fe=0.067). No experimental data are available for the AHC values, but the present study can motivate new experiments for thisinteresting alloy system. 2. Effect of spin disorder In a previous study [ 5] we have demonstrated that a simple model of spin disorder, the uncompensated DLM model, isable to describe qualitatively and even quantitatively correctlythe temperature dependence of the resistivity and the AMR atroom temperatures. The effect of phonons cannot be generallyneglected and it leads to an essentially linear increase of thesample resistivity [ 18]. It is possible to include the effect of phonons approximately by performing calculations witha finite imaginary part added to the Fermi energy [ 22]. The spin-disorder contribution usually dominates for highertemperatures close to the Curie temperature [ 18,20]. Here we therefore concentrate on the effect of spin disorder only. While the effect of temperature is unambiguously related to the spin disorder in the system characterized by theratior=x[−]:x[+] of concentrations of oppositely oriented spinsx[−] andx[+], a detailed quantitative relation of this concentration and the temperature is beyond the scope of thismodel. Here we wish to extend this study also to the case of the 224421-3J. KUDRNOVSK ´Y, V . DRCHAL, AND I. TUREK PHYSICAL REVIEW B 92, 224421 (2015) TABLE I. The total resistivity ρtotand the total anomalous Hall conductivity (AHC tot)f o rP d 70Fe30alloy with 10% of Pd[Fe] antisites as a function of the amount of the spin disorder. The corresponding LRO parameter is S=0.714. See text as it concerns the meaning of the parameter r. Pd70Fe30with 10% of Pd[Fe] antisites r=x[−]:x[+] ρtot(μ/Omega1cm) AHC tot(kS/m) AHC surf(kS/m) AHC sea(kS/m) 0 (FM) 12.57 −23.63 −17.76 −6.07 1:79 23.02 −15.24 −9.52 −5.72 1:39 31.77 −13.73 −8.21 −5.22 1:19 46.15 −12.89 −7.76 −5.13 1:9 67.30 −12.08 −7.86 −4.22 1:1 (DLM) 115.35 0.0 0.0 0.0 AHC using the same model as in Ref. [ 5]. The total resistivity ρtotand the total AHC (AHC tot)o fP d 70Fe30alloy with 10% of Pd[Fe] antisites in the L12lattice is shown as a function of the amount of the spin disorder (the parameter r) in Table I.T h e corresponding LRO parameter is relatively large, S=0.714 (the maximal possible value is Smax=0.833). The amount of spin disorder increases from r=0 (the ferromagnetic state, FM, no spin disorder) to r=1 (the DLM state, full spin disorder which exists above the Curie temperature). In Table I we also show the total Fermi surface (AHC surf) and Fermi sea (AHC sea) contributions to the total AHC. It should be noted that the value r=1 : 19 roughly corresponds to the room temperature when compared to the experimental drop of valuesfor the resistivity and AMR at the room temperature [ 23]. The Fermi sea term depends weakly on the spin disorderas compared to the Fermi surface term. It should be notedthat the Fermi sea term is nonnegligible as it amounts toone third to one half of the dominating Fermi surface term.The decrease of the AHC with increasing spin disorder canbe clearly correlated with the reduction of the total magneticmoment with temperature as it is zero for the system which hasthe zero total moment, i.e., in the paramagnetic (DLM) state. B. fcc-Pd 1−xCoxalloys 1. Electronic structure The determination of the relativistic electronic structure of both the disordered and even partially ordered transition metalalloys is now a standard problem. We therefore present hereonly results which can be compared directly to available exper-imental results [ 24], namely, the concentration dependence of the total and atom-resolved magnetic moments for disorderedfcc-Pd 1−xCoxalloy. Theoretical moments together with avail- able experimental data [ 24] are shown in Fig. 2. The calculated total moments include orbital contribution while only thespin parts of atom-resolved moments are shown. The orbitalmoments for Co/Pd atoms over all studied concentration rangeare, respectively, 0 .03±0.005/0.105±0.015μ B. While Pd moments are nearly concentration independent, Co momentsdecrease almost linearly with the increasing Co content, butwith a small slope. This decrease of local Co moment has twoorigins: (i) a decrease of the alloy lattice constant, and (ii) adecrease of the impurity character of Co moment which actin accord, as contrasted to their mutual canceling in the caseof local Pd moments. The total moment increases with Co concentration because the Pd moment is essentially constant.We observe a very good overall agreement between calculatedand experimental total moments. We have also calculated local moments for L1 0andL12 phases at experimental lattice constants. Calculated values are shown in Fig. 2as diamonds at related Co concentrations, namely, at x=0.25 and x=0.5f o rt h e L12andL10 phases, respectively. They agree very well with disordered counterparts as it is known from other calculations of magneticmoments in disordered alloys employing the supercell ap-proach (see, e.g., Ref. [ 25]). The agreement with experimental values for local moments [ 26] is also good as concerns their sizes, but Co moments for the L1 0phase are slightly larger than those for the L12phase while the theory predicts an opposite trend. A possible explanation is the existence of the short-rangeorder in the L1 0phase [ 26] and a generally unknown amount of the sample order in the experiment which depends on thesample annealing. 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Magnetic moment ( μB) Co-concentrationstotal exp Co Pd FIG. 2. The total relativistic and atom-resolved magnetic mo- ments for the disordered fcc-Pd 1−xCoxalloy as a function of the Co-concentration x. The corresponding total moments (filled circles) are compared with the experiment (open circles). The calculated local moments for the ordered L10-PdCo and L12-Pd 3Co alloys (empty and filled diamonds) are shown at Co concentrations of 0.5 and 0.25,respectively. 224421-4EFFECT OF PARTIAL ORDER ON GALV ANOMAGNETIC . . . PHYSICAL REVIEW B 92, 224421 (2015) 2. Residual resistivity In this subsection we present results for residual resistivities of fcc-Pd 1−xCoxalloys and compare them with experimen- tal values. We have calculated concentration dependenceof residual resistivities both in the relativistic and scalar-relativistic model. This allows us not only to see the effectof spin-orbit coupling on the resistivity, but also to addressthe problem of spin-up ( ρ ↑) and spin-down ( ρ↓) resistivities estimated approximately in the experiment [ 27]. The measured resistivities and the AMR ratio were used in the framework ofthe the two-current model and theories based on its extendedversions. This has allowed the author of Ref. [ 27], using two different approaches, to estimate ρ ↑andρ↓. While results of these two approaches differed quantitatively, themain conclusion, namely, much larger ρ ↓as compared to ρ↑ was not changed. We have used the two-current conductivity (the scalar-relativistic limit) σ=σ↑+σ↓to estimate ρ↑ andρ↓theoretically by writing ρ↑=(σ↑)−1,ρ↓=(σ↓)−1. It should be noted that one-to-one correspondence betweenexperimental and theoretical estimates is not possible, butthe main features are reproduced by the present theory.Results are summarized in Fig. 3and we can make the following conclusions: (i) In agreement with experiment [ 27] we observe a much larger value of the ρ ↓resistivity as compared to the ρ↑one [Fig. 3(a)]. The calculated ρ↓exhibits the Nordheim-rule-like behavior while that estimated fromthe experiment is slightly asymmetric being centered aroundx=0.35. Calculated and experimental ρ ↑agree well in the sense that there is a maximum at x=0.15, although it is more pronounced in the theory. We observe an unusual localmaximum of the ρ ↑resistivity in the Co-concentration region around x=0.15. Such behavior is also observed in the total resistivity. According to the Supplement of Ref. [ 13] this can be traced down to the behavior of spectral densities alongtheX-Wline in the Brillouin zone in the close neighborhood of the Fermi energy. It should be noted that the calculatedconductivity of majority electrons in the framework of thetwo-current model is much larger than that of minority onesdemonstrating the dominating role of the majority electrons fortransport in fcc-PdCo alloys; and (ii) a reasonable agreementis obtained for the relativistic case although resistivitiescalculated for x> 0.3 are slightly larger than in the experiment [Fig. 3(b)]. There is also a good agreement with corresponding relativistic KKR-CPA calculations [ 13]. On the other hand, the scalar-relativistic calculations underestimate the resistivityindicating the relevance of the spin mixing due to the spin-orbitcoupling. The effect of relativistic corrections becomes smallerfor Co-rich samples (smaller spin-orbit coupling of the Coatoms as compared to the Pd ones). In Fig. 3(b) we have also included calculated relativistic resistivities assuming the partial L1 0long-range order for x=0.4, 0.5, and 0.6. Motivation for such study is an attempt to explain the strongly underestimated AMR values (seeSec. III B 3 for details). 3. Galvanomagnetic properties: The AMR and AHC for disordered samples The results of calculated AMR and AHC values for disordered fcc-Pd 1−xCoxas a function of the Co concentration 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Resistivity ( μΩ cm) Co-concentrationfr sr exp fr: L10-LRO(b) 0 5 10 15 20 Resistivity ( μΩ cm) total spin-up spin-down (x0.2)scalar-relativistic (a) FIG. 3. The residual resistivities of disordered fcc-Pd 1−xCoxal- loys as a function of the Co-concentration x: (a) The scalar-relativistic case in which the total resistivity (full box) is plotted along with spin-up (filled circles) and spin-down (empty circles) resistivities;and (b) the fully relativistic (fr, filled circles) and scalar-relativistic (sr, empty circles) cases are compared with the experiment [ 27] (expt., filled boxes). Also shown are relativistic resistivities for the partialL1 0(empty boxes) long-range order ( x=0.4, 0.5, and 0.6) for which an optimal agreement between experimental and theoretical AMR values was obtained (see Sec. III B 3 for more details). xare compared with the experiment in Fig. 4(AMR) and Figs. 5and6(AHC). The following conclusions can be made: (i) A reasonable agreement of the calculated and measuredAMR values is obtained only for samples with low Cocontent ( x/lessorequalslant0.3) while for higher Co content the AMR ratio is strongly underestimated similarly as in recent relativisticKKR-CPA calculations [ 13]. (ii) On the contrary, as shown in Fig. 5, a reasonable agreement between the calculated and experimental AHC is obtained. The main difference is the shiftof the calculated crossing point between positive and negativecalculated AHC values to a slightly higher Co concentration.Calculations also show that the Fermi sea term (filled squares,Fig.5) which is often neglected in theoretical calculations (e.g., also in related KKR study [ 13]) is negligible in the completely disordered sample. On the other hand, the L1 2-Pd 70Fe30alloy (Fig. 1) corresponds to a weakly disordered or well-ordered 224421-5J. KUDRNOVSK ´Y, V . DRCHAL, AND I. TUREK PHYSICAL REVIEW B 92, 224421 (2015) 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9AMR (%) Co-concentrationfcc Pd1-xCox alloy theory exp theory: L10-LRO FIG. 4. The calculated anisotropic magnetoresistance (AMR) for disordered fcc-Pd 1−xCoxas a function of the Co-concentration x (filled circles) is compared with the experiment (empty circles) [ 27]. The AMR values for alloys with the partial L10ordering around equiconcentration composition are shown as filled squares (see text for details). alloy with larger effect of the Fermi sea term. (iii) Another way of comparison of calculated and experimental AHC valuesis shown in Fig. 6. The total AHC σ xyfor disordered fcc- Pd1−xCoxis displayed here as a function of the corresponding total conductivity σtotfor the same Co concentrations as in Fig. 5. Corresponding experimental values were compiled from measured values of the total resistivity [ 28]. In this way we can compare, in a single plot, both the diagonal andoff-diagonal components of the conductivity tensor and it thus -2-1.5-1-0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9σxy (kS/cm) Co-concentrationfcc Pd1-xCox alloy σxy F. sea exp L10-LRO FIG. 5. The calculated anomalous Hall conductivity σxy(filled circles) for the disordered fcc-Pd 1−xCoxas a function of the Co concentration is compared with the experiment (expt., empty circles) [ 28]. Calculations also show that the Fermi sea term (F. sea, filled squares) is negligible in the present fully disordered sample. The AHC values for alloys with the partial L10ordering ( L10, empty squares) around equiconcentration composition are also shown (seetext for details).-2-1.5-1-0.5 0 0.5 1 1.5 50 60 70 80 90 100 110 120 130 140 150 160σxy (kS/cm) σtot (kS/cm)theory expfcc Pd1-xCox alloy FIG. 6. The anomalous Hall conductivity σxyfor the disordered fcc-Pd 1−xCoxalloy (filled circles) is calculated as a function of the total conductivity σtotfor Co concentrations same as in Fig. 5. Calculations are compared with corresponding experimental values (empty circles) compiled from data in Ref. [ 28]. There is a change of sign of σxyfor the Co concentration around x=0.3. represents a crucial test of the theory. A typical U-shape form of theσxyvs conductivity dependence found in the experiment is well reproduced by present calculations. There is a changeof sign of σ xyfor samples with the Co concentration around x=0.35. 4. Galvanomagnetic properties: The AMR and AHC for partially ordered samples Here we wish to address the problem of the underestimation of the AMR values for the completely disordered samplediscussed in a previous subsection. We suggest that the originof such discrepancy between the theory and experiment canbe traced down to the presence of the admixture of the L1 0 phase which exists in the alloy around the equiconcentration composition [ 10]. It should be noted that a systematic study of the multiphase alloy system is still a challenge for thetheory. We will simulate this effect by calculating the alloys forx=0.4, 0.5, and 0.6 with a partial LRO. No attempt was made to fit the experiment, but we rather wish to demonstrate thefact that the L1 0-partial ordering, which leads to the structural anisotropy, can be a possible explanation of the above-mentioned discrepancy between the theory and experiment.We will further support this explanation by presenting relatedcalculations for alloys (Co-rich ones) with L1 2ordering which do not exhibit the anisotropy inherent to the L10phase. Consequently, the effect of the L12ordering is much weaker than that to the L10phase. The effect is seen particularly clearly for the equiconcen- tration fcc-Pd 50Co50alloy. Calculated results are summarized in Table IIas a function of the LRO parameter S.T h e corresponding experimental values of the resistivity, AMRratio, and AHC totare, respectively, 11.76 μ/Omega1cm, 7.71%, and 0.5 kS /cm. It is seen that increasing order reduces the resistivity and strongly increases the AMR ratio. This increasecan be ascribed to increasing structural anisotropy of the L1 0 phase. A similar effect was found recently in partially ordered 224421-6EFFECT OF PARTIAL ORDER ON GALV ANOMAGNETIC . . . PHYSICAL REVIEW B 92, 224421 (2015) TABLE II. The total resistivity ρtot, the anisotropic magnetore- sistance (AMR), the total anomalous Hall conductivity (AHC tot), and its Fermi sea contribution AHC seafor the Pd 50Co50alloy with L10ordering for various values of the long-range order parameter S. The value S=0 corresponds to a completely random fcc- equiconcentration PdCo alloy. L10-Pd 50Co50with partial order Sρ tot(μ/Omega1cm) AMR (%) AHC tot(kS/cm) AHC sea(kS/cm) 0.0 14.93 2.83 0.419 0.064 0.1 14.86 3.37 0.414 0.0630.2 14.63 4.45 0.405 0.060 0.3 14.19 5.77 0.395 0.056 0.4 13.49 7.08 0.384 0.051 0.5 12.51 8.12 0.379 0.045 L10-PtFe alloys [ 3]. It should be noted that we have in mind the anisotropy due to the layeredlike structure of the the L10 phase and not the tetragonality due to the different atom sizes of Pd and Co atoms. Such tetragonality, however, should bevery small for studied cases with a small LRO parameter S when alloys are essentially cubic ( c/a≈1), but still with structural anisotropy which has a pronounced effect. We havethus neglected such tetragonality in present calculations. Thechoice S=0.4 brings calculated AMR and also resistivities into a good quantitative agreement with the experiment, inparticular the value of the AMR ratio is noticeably improved.The AHC value remains essentially the same and the Fermisea contribution to the AHC totis about 13% for S=0.4. Results for the resistivity, the AMR ratio, and the AHC are alsocompared with the experiment in Figs. 3,4, and 5, respectively. We have done similar calculations for alloys around the equiconcentration composition, namely, for Co concentrationsx=0.4 and 0.6. The results for the resistivity, the AMR ratio, and the AHC are also shown in Figs. 3,4, and 5, respectively, for corresponding Co concentrations. Specifically, for x= 0.4 we obtained an optimal agreement for S=0.2 while a somewhat larger value of S=0.5 was needed for a Co-rich case with x=0.6. But already for S=0.3 a significant improvement was obtained, i.e., the AMR ratio was 4.2% ascompared to 2.0% for the completely disordered case. In allcases results for the total AHC were modified only slightly andif at all, they were also improved. Finally, we note an increasingdifference between calculated (for completely random alloys)and experimental AMR values for x=0.4t ox=0.6. This is in agreement with increasing optimal values of Swhich are, respectively, S=0.2, 0.4, and 0.5 for x=0.4, 0.5, and 0.6. On the other hand, the L1 2ordering also reduces the resistivity and thus improves agreement with the experiment,but the AMR ratio is enhanced only weakly. The reason for this is a cubic structure even for ordered phases. For example,the AMR ratio for Co concentration x=0.7 is enhanced from about 1.5% to 2% only even for S=0.6. This value should be contrasted to the experimental value of about 6%.The resistivity is reduced from 10 to 7.4 μ/Omega1cm while the experimental value is 8.3 μ/Omega1cm. IV . CONCLUSIONS We have presented relativistic first-principles study of the magnetotransport properties in two related Pd-based alloys:(i) partially ordered L1 2-Pd 70Fe30and (ii) disordered as well as partially ordered fcc-based Pd 1−xCoxalloys over a broad range of Co concentrations. The main conclusions are:(i) The Fermi sea term neglected previously in the study of theAHC in ordering Pd-rich PdFe alloys is nonnegligible for wellordered phases although the qualitative conclusions are notchanged. The Fermi sea term is also nonnegligible when theeffect of temperature on the AHC is considered. (ii) Results fordisordered fcc-Pd 1−xCoxalloy (the resistivity, the AMR ratio, and the AHC) compare well with results of previous KKR-CPAcalculations [ 13] as expected. We have demonstrated, by comparing results for the resistivity calculated in the frame-work of the relativistic and scalar-relativistic models, the rel-evance of the spin-orbit coupling for a quantitative agreementwith experiment. Calculations also indicate the dominatingrole of majority electrons for transport in fcc-PdCo alloys.(iii) We have found a reasonable agreement between spin-resolved resistivities determined approximately in the exper-iment and theory, in particular, a much larger value of ρ ↓as compared to ρ↑. (iv) We have shown that the neglect of the Fermi sea term in a recent KKR study [ 13] represents only a small error. On the other hand, the relevance of the Fermisea term increases with the ordering in the alloy. (v) The mainresult of the study is suggestion of the partial L1 0ordering or the admixture of the L10phase detected in the experiment as a possible reason for large values of the AMR ratio inthe concentration window around the equiconcentration caseobserved experimentally. Physical origin is the structuralanisotropy of the layeredlike L1 0structure. On the other hand, an essentially isotropic or cubic like L12ordering improved agreement for the resistivity, but its effect on theAMR ratio is small. It should be noted that partial L1 0 ordering might be one of possible causes. In general, structural defects like short-range order or segregation due to the samplepreparation [ 13] are not included in the present study. ACKNOWLEDGMENT The authors acknowledge financial support by the Czech Science Foundation (Grant No. 15-13436S). [1] R. L. Rossiter, The Electrical Resistivity of Metals and Alloys (Cambridge University Press, Cambridge, 1987). [2] J. Banhart and G. Czycholl, Europhys. Lett. 58,264 (2002 ).[3] J. Kudrnovsk ´y, V . Drchal, and I. Turek, P h y s .R e v .B 89,224422 (2014 ). [4] J. Kudrnovsk ´y, V . Drchal, S. Khmelevskyi, and I. Turek, Phys. Rev. B 84,214436 (2011 ). 224421-7J. KUDRNOVSK ´Y, V . DRCHAL, AND I. TUREK PHYSICAL REVIEW B 92, 224421 (2015) [5] J. Kudrnovsk ´y, V . Drchal, and I. Turek, J. Supercond. Nov. 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Kudrnovsk ´y, V . Drchal, L. Szunyogh, and P. Weinberger, Phys. Rev. B 65,125101 (2002 ).[17] K. Carva, I. Turek, J. Kudrnovsk ´y, and O. Bengone, Phys. Rev. B73,144421 (2006 ). [18] H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min ´ar, and D. K ¨odderitzsch, P h y s .R e v .B 91,165132 (2015 ). [19] J. K. Glasbrenner, B. S. Pujari, and K. D. Belashchenko, Phys. Rev. B 89,174408 (2014 ). [20] Y . Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, M. van Schilfgaarde, and P. J. Kelly, Phys. Rev. B 91,220405(R) (2015 ). [21] S. H. V osko, L. Wilk, and M. Nusair, Can. J. Phys. 58,1200 (1980 ). [22] We have verified this by comparing results of this simple model for pure Fe and Gd with results of a more sophisticated approach,Ref. [ 18]. [23] Y . Hsu, S. Jen, and L. Berger, J. Appl. Phys. 50,1907 (1979 ). [24] R. M. Bozorth, P. A. Wolff, D. D. Davis, V . B. Compton, and J. H. Wernick, Phys. Rev. 122,1157 (1961 ). [25] K. Schwarz, P. Mohn, P. Blaha, and J. K ¨ubler, J. Phys. 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PhysRevB.80.081306.pdf

Half-metallic silicon nanowires: Multiple surface dangling bonds and nonmagnetic doping Zhuo Xu, Qing-Bo Yan, Qing-Rong Zheng, and Gang Su * College of Physical Sciences, Graduate University of Chinese Academy of Sciences, P .O. Box 4588, Beijing 100049, China /H20849Received 7 July 2009; published 14 August 2009 /H20850 By means of first-principles density functional theory calculations, we find that hydrogen-passivated ultra- thin silicon nanowires /H20849SiNWs /H20850along the /H20851100 /H20852direction with symmetrical multiple surface dangling bonds /H20849SDBs /H20850and boron doping can have a half-metallic ground state with 100% spin polarization, where the half-metallicity is shown to be quite robust against external electric fields. Under the circumstances withvarious SDBs, the H-passivated SiNWs can also be ferromagnetic or antiferromagnetic semiconductors. Thepresent study not only offers a possible route to engineer half-metallic SiNWs without containing magneticatoms but also sheds light on manipulating spin-dependent properties of nanowires through surface passivation. DOI: 10.1103/PhysRevB.80.081306 PACS number /H20849s/H20850: 73.21.Hb, 71.70.Ej, 73.22.Dj It is widely believed that nanowires can be the most promising candidates for the basic building blocks of futurenanoelectronics as they could be essentially useful in engi-neering diverse nanodevices such as field-effect transistors,logic gates, and DNA sensors. 1–4Thus, the nanowires with thin diameters5,6were actively studied both experimentally and theoretically in the past years. Recently, particular atten-tion was paid on the ultrathin silicon nanowires /H20849SiNWs /H20850 /H20849Refs. 7–12/H20850as they not only have unusual electronic prop- erties of scientific interest, but also could be potentially in-corporated into nowadays well-developed Si-based elec-tronic technology. The properties of ultrathin nanowires can be modified by adjusting diameters and orientations, conducting the surfacepassivation, or introducing dopants, etc. For SiNWs, amongother things, it is known that /H20849i/H20850due to surface reconstruc- tion, SiNWs along /H20851100 /H20852direction with diameters smaller than 1.7 nm prefer a square cross section with sharp cornerswhere the electrons are quite localized; 11–14/H20849ii/H20850the ultrathin SiNW that is saturated with H atoms on surface can show asemiconducting behavior, 8and if it is substitutionally doped with nonmagnetic elements such as B or P, the main bands ofimpurities distribute at the top of valence bands; 10/H20849iii/H20850 H-passivated ultrathin SiNWs along /H20851100 /H20852direction intersti- tially doped by a certain density of transition metals such asCo or Cr can have a half-metallic ground state; 15/H20849iv/H20850when the H-passivated nanowires or surfaces have an isolated or asingle row of surface dangling bonds /H20849SDBs /H20850, there is a band contributed by the SDBs crossing the Fermi level/H20849FL/H20850, 10,16–19and if the spin polarization is considered, the band splits into two discrete bands, respectively, below andabove the FL, 19leading to two band groups formed in the case of multiple SDBs.20 In the present Rapid Communication, by means of ab initio calculations we have found that, for an ultrathin H-passivated SiNW along /H20851100 /H20852direction with multiple SDBs at symmetrical positions and substitutionally doping Batoms at the center of cross section, a ferromagnetic /H20849FM /H20850 configuration is favored as the ground state, and such aSiNW is a half-metal with 100% spin polarization; namely, itis insulating for one spin direction while metallic for theopposite spin direction at the FL. We report a novel type ofhalf-metallic SiNWs with multiple SDBs and nonmagneticdoping in contrast to the previous study where the half-metallic SiNWs were obtained by doping transition metals. 15 This poses a possible route to make half-metallic SiNWs through manipulating the surface passivation and introducingnonmagnetic dopants. Our calculations were based on the density functional theory 21with generalized gradient approximation expressed by Perdew-Burke-Ernzerhof functional,22employing norm- conserving pseudopotentials23and linear combinations of atomic orbitals. The SIESTA code,24,25with double-zeta polar- ized basis sets, was used to perform spin-polarized calcula-tions. The energy cutoff was 180 Ry, and the total-energyconvergence criterion was 10 −5eV. All structures were fully relaxed by the maximum force tolerance of 0.015 eV /Å, and the lattice constant along the wire axis was optimized foreach structure. Among the SiNWs of 25 Si atoms per unit/H20851Fig.1/H20849a/H20850/H20852with different SDBs, the separation between two adjacent wires was kept as 15.27 /H20849/H110060.02 /H20850Å. The Brillouin zone along the axial direction was sampled with 15 kpoints. Figures 1/H20849a/H20850and1/H20849b/H20850depict the ultrathin SiNW along the /H20851100 /H20852direction passivated with H atoms without SDBs /H20849la- beled by SiNW-H12 /H20850, where there are 25 Si and 12 H atoms per unit cell. In SiNW-H12, the diameter is about 11 Å in-cluding H, and about 8 Å excluding H. The positions in the yaxis of the adjacent Si atoms at positions 3 and 4 /H20851Fig.1/H20849a/H20850/H20852 differ only by 0.009 Å. Note that the H saturated SiNW-H12is semiconducting. 8For the SiNW with single SDB per unit (b) SiNW -H12 (a) SiNW -H12 123 7 8 y x(c) SiNW -H11 z xzx1056 49 11 FIG. 1. /H20849Color online /H20850The /H20849a/H20850cross section and /H20849b/H20850side view of the optimized structures of H-saturated SiNW along the /H20851100 /H20852di- rection. The atomic positions marked by numbers in /H20849a/H20850are dis- cussed in the context. /H20849c/H20850The SiNW with one SDB per unit. Blue /H20849dark gray /H20850and orange /H20849light gray /H20850balls represent Si and H atoms, respectively.PHYSICAL REVIEW B 80, 081306 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 1098-0121/2009/80 /H208498/H20850/081306 /H208494/H20850 ©2009 The American Physical Society 081306-1/H20851denoted by SiNW-H11, Fig. 1/H20849c/H20850/H20852, there is one unpaired electron at the SDB in each unit, and the total magneticmoment /H9262is 1.00 /H9262Bper unit cell /H20849/H9262Bis Bohr magneton /H20850.I n SiNW-H11, the Si atom with SDB /H20851position 3 in Fig. 1/H20849a/H20850/H20852is stretched 0.047 Å outer than the Si at position 4 that is stillpassivated with H, and the energy for generating a singleSDB per unit is 3.52 eV. We compared different spin con-figurations by employing a supercell with two unit cells /H20849in- cluding two adjacent SDBs /H20850and disclosed that the FM con- figuration is favored as the ground state, with the energy of11.3 meV per unit cell lower than the antiferromagnetic /H20849AF /H20850 configuration, and 155.6 meV per unit lower than the non-magnetic /H20849NM /H20850configuration. It turns out that the SiNW-H11 is a FM semiconductor with an indirect gap of 0.59 eV. The electronic structure and density of states /H20849DOS /H20850of FM SiNW-H11 are presented in Fig. 2. It can be seen that there are two split bands just above and below the FL /H20851Fig. 2/H20849a/H20850/H20852. The upper band is of minority spins /H20849↓/H20850, while the lower is of majority spins /H20849↑/H20850, which correspond to the half- filled band crossing the FL if the spin polarization is ignored,similar to the case at C /H20849001 /H20850surface. 19For the bands away from the FL, there is merely a small displacement betweenthe spin-up and spin-down subbands, which is close to theSiNW-H12. The constituents of the two split bands areclearly manifested in the projected DOS /H20849PDOS /H20850: The Si atom with a SDB contributes 44.9% to the total DOS/H20849TDOS /H20850, which dominates overwhelmingly among all atoms, and the 8 Si atoms in the corner including the SDB contrib-ute 82.9% to the TDOS, as shown in Fig. 2/H20849b/H20850. This indicates that the two split bands are formed primarily by the unpairedelectron of SDB and the electrons closely around the SDB. Now let us look at the SiNW with multiple SDBs. With- out loss of generality, the case of four SDBs per unit celllocated symmetrically on each edge of the cross section willbe calculated, as shown in Figs. 3/H20849a/H20850and3/H20849b/H20850, labeled by SiNW-H8. In comparison to the SiNW-H11, the FM configu-ration of SiNW-H8 /H20851SiNW-H8FM, Fig. 3/H20849a/H20850/H20852has more split bands, which are separated into two groups above and belowthe FL, and each group includes four bands of the same spin species /H20849either spin up or down /H20850, where there is one band degenerate with another in each group. The magnetic mo-ment is 4.00 /H9262Bper unit cell. However, the SiNW-H8FM is merely a metastable state. The ground state has an AF con-figuration /H20851SiNW-H8AF, Fig. 3/H20849b/H20850/H20852, where the SDBs on ad- jacent edges have opposite spin alignments, with an energylower than SiNW-H8FM by 26.9 meV per unit cell. Theaverage energy for generating each SDB in this case is 3.55eV. The spin-up and spin-down bands of the SiNW-H8AFcoincide, and /H9262is zero, as revealed in Fig. 3/H20849b/H20850. Therefore, SiNW-H8 is an AF semiconductor with an indirect gap of0.47 eV. In the band structure of SiNW-H8FM /H20851Fig.3/H20849a/H20850/H20852, the indirect gap between the bands of majority and minorityspins is 0.34 eV, which is so small that it could be possible togenerate a perfect half-metallic behavior in the SiNW-H8 solong as the two conditions are satisfied: /H20849i/H20850the FL is slightly shifted upward or downward to intersect the split bands ofthe same spin species; /H20849ii/H20850The FM configuration of SDBs should be the ground state. To meet with the conditions, itmay be convenient to conduct nonmetallic dopings /H20849such as Bo rP /H20850. By substitutionally doping one B atom at the center of the FIG. 2. /H20849Color online /H20850/H20849a/H20850The band structure and /H20849b/H20850the total and projected DOS of ferromagnetic SiNW-H11. The Si with SDBis marked in pink /H20849gray /H20850. The eight Si atoms in the corner including the SDB are enclosed in the inset of /H20849b/H20850and marked blue /H20849dark gray /H20850. The red /H20849gray /H20850solid and green /H20849light gray /H20850dashed curves in band structures represent majority /H20849↑/H20850and minority /H20849↓/H20850spins, re- spectively, throughout the context. The Fermi energy is set to zero. FIG. 3. /H20849Color online /H20850The band structures of /H20849a/H20850SiNW-H8FM, /H20849b/H20850SiNW-H8AF, and /H20849c/H20850SiNW-H8BFM. /H20849d/H20850The DOS of SiNW- H8BFM. The structure and spin configuration of SDBs for /H20849e/H20850 SiNW-H8BFM and /H20849f/H20850SiNW-H8BAF. The dark blue /H20849dark gray /H20850, orange /H20849light gray, small /H20850, and wine /H20849solid /H20850balls represent Si, H, and B atoms, respectively; the pink /H20849gray /H20850and green /H20849light gray /H20850 atoms denote Si atoms with SDB for majority and minority spins,respectively.XUet al. PHYSICAL REVIEW B 80, 081306 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 081306-2cross section /H20851position 1 in Fig. 1/H20849a/H20850/H20852of the SiNW-H8FM /H20849labeled by SiNW-H8BFM /H20850, we found that the SiNW- H8BFM is the ground state, and the FL moves indeed down-ward intersecting the majority-spin bands, giving rise to ahalf-metallic behavior, as shown in Fig. 3/H20849c/H20850. This is also manifested in Fig. 3/H20849d/H20850, where the DOS of majority spin G ↑/H20849EF/H20850at the FL /H20849EF/H20850is 8.28/eV, while the DOS of minority spin G↓/H20849EF/H20850is zero, leading to the spin polarization P =/H20851G↑/H20849EF/H20850−G↓/H20849EF/H20850/H20852//H20851G↑/H20849EF/H20850+G↓/H20849EF/H20850/H20852=100 %. The indirect band gap of minority spin is 0.86 eV. The total magneticmoment of SiNW-H8BFM is 3.00 /H9262Bper unit cell, while the local moment of each Si atom with SDB is 0.51 /H20849/H110060.01 /H20850/H9262B, that is 0.11 /H9262Blower than SiNW-H8AF. As there are three spin-up bands at different kpoints in the Brillouin zone crossing the FL, it is hard to open a gap due to Peierlsinstability. 12Thus, the half-metallicity in the SiNW-H8BFM is robust. We also examined possible magnetic configura-tions /H20851FM, AF, and ferrimagnetic /H20849FI/H20850/H20852in a supercell with either one or two unit cells, respectively, together with NM/H20849SiNW-H8BNM, spin-unpolarized /H20850calculations and ob- served that apart from the FM ground state /H20851SiNW-H8BFM, Fig.3/H20849e/H20850/H20852, the lowest metastable state is an AF metal /H20851SiNW- H8BAF, Fig. 3/H20849f/H20850/H20852. The total energies of SiNW-H8BNM and SiNW-H8BAF are 309.2 and 21.5 meV per unit cell, respec-tively, higher than that of SiNW-H8BFM. Based on the en-ergy difference between the SiNW-H8BFM and SiNW-H8BAF, we can estimate that at the temperature lower than 250 K, the half-metallic property in the SiNW-H8B can be retained. Doping P at the center in a FM configuration of SiNW-H8 /H20849labeled by SiNW-H8PFM /H20850is similar to the SiNW-H8BFM except that the FL moves upward crossing the spin-downbands, resulting in the half-metallicity for minority spins.However, the ground state for this case is a FI metal, wherethe total magnetic moment /H9262=0.06/H9262Bper unit cell, and the total energy is slightly lower than SiNW-H8PFM by 3.2 meVper unit cell. Compared with the AF ground state of SiNW-H8, doping B at the center will decrease both the charge ofeach SDB and the diameter of the SiNW, leading to a FMground state, while doping P brings about inverse changes,making a FI ground state. Besides, it is unraveled that inspite of doping B, P or not, if there are two SDBs in onecorner /H20851at positions 3 and 8 in Fig. 1/H20849a/H20850/H20852or in one edge /H20849at positions 3 and 7 /H20850, the FM configuration appears not to be the ground state. For the SiNW-H8BFM, Fig. 4/H20849a/H20850indicates that the spin density /H20849 /H9267↑−/H9267↓/H20850distributes identically in each of the four corners with SDB, where /H9267↑/H20849/H9267↓/H20850is the number density of electrons with spin up /H20849down /H20850. The dopant B at the center has a homogeneous effect on the symmetrical SDBs, leading tothe half-metallicity. If doping B takes place at the edge /H20851po- sition 2 in Fig. 1/H20849a/H20850/H20852instead of the center with the FM con- figuration /H20851SiNW-H8B /H208492/H20850FM, Figs. 4/H20849b/H20850and4/H20849c/H20850/H20852, there are three spin-up bands just below the FL, and four spin-downbands and one spin-up band just above the FL. The four Siatoms with SDBs contribute to the PDOS distinctly, amongwhich the one /H20851red Si /H20849I/H20850in Fig. 4/H20849c/H20850/H20852adjacent to the B atom plays an overwhelming role. It is also confirmed by the spin- FIG. 4. /H20849Color online /H20850The contour plot of spin-density distribu- tion /H20849in unit of 10−3e/Bohr3/H20850of/H20849a/H20850SiNW-H8BFM and /H20849b/H20850SiNW- H8B /H208492/H20850FM at the cross section through the central atom. The gray, white, and dark blue /H20849dark gray /H20850balls stand for the positions of Si, H, and B atoms. /H20849c/H20850The band structure and DOS of SiNW- H8B /H208492/H20850FM. Four Si atoms with SDBs and one B atom are marked pink /H20849gray dash, I /H20850, dark yellow /H20849gray short dot, II /H20850, purple /H20849dark gray dot, III /H20850, blue /H20849light gray dash dot, IV /H20850, and wine /H20849solid /H20850, respectively. FIG. 5. /H20849Color online /H20850/H20849a/H20850The SiNW-H8BFM under an external electric field E/H6023. The DOS under /H20849b/H20850Ez=5.0 V /Å along the zdirec- tion; and /H20849c/H20850Ex=0.18, /H20849d/H208500.19, /H20849e/H208501.0, and /H20849f/H208501.8 V /Å along the xdirection.HALF-METALLIC SILICON NANOWIRES: MULTIPLE … PHYSICAL REVIEW B 80, 081306 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 081306-3density distribution in Fig. 4/H20849b/H20850, where the spin density of the 1/4 cross section around Si /H20849I/H20850is nearly zero, totally different from those in other three corners. The local moment of Si /H20849I/H20850, /H20849II/H20850,/H20849III/H20850, and /H20849IV/H20850is 0.09, 0.58, 0.63 and 0.61 /H9262B, respec- tively. In general, doping B at any position other than thecenter will give a semiconductor. The total energy per unitcell of doping B atoms at position 4, 2, 3, 5, 6, and 7 /H20851Fig. 1/H20849a/H20850/H20852is 0.36, 0.52, 0.70, 0.46, 0.58, and 0.73 eV, respec- tively, higher than doping at position 1 /H20849SiNW-H8BFM /H20850, suggesting the center is the most preferred doping positionfor B. For the FM configuration, one B dopant per unit will re- duce one electron below the FL, which, depending on therelative positions among dopants and SDBs, could changethe band structure in two ways. One is that the dopant affectsmainly one position of SDBs, moving one spin-up band toabove the FL, as SiNW-H8B /H208492/H20850FM. The other is that the dopant affects equally the multiple positions of SDBs, push-ing more spin-up bands crossing the FL, as the half-metallicSiNW-H8BFM. Consequently, in order to produce the half-metallicity, the SDBs should be arranged at symmetrical po-sitions and, the dopants must be located symmetrically to theSDBs. In addition, several cases with larger radius of 61 Siatoms and four symmetrical SDBs per unit cell are alsochecked. The results show that substitutionally doping eitherone B atom at the center or five around the center per unitcell can still generate the FM ground state and lead to thehalf-metallicity with 100% spin polarization. To examine whether the half-metallic property of the SiNW-H8BFM /H20851Fig. 5/H20849a/H20850/H20852is stable under external electric fields, we made calculations in presence of longitudinal andtransverse electric fields, respectively. By applying an elec- tric field along the nanowire direction /H20849along zaxis /H20850,w e found that the half-metallic behavior of the SiNW-H8BFMstill retains under a field as large as 5.0 V /Å/H20851Fig. 5/H20849b/H20850/H20852. When an electric field is applied perpendicular to the nano-wire axis /H20849along the xdirection /H20850, we observed that the half- metallic state is kept when the field is less than 0.18 V /Å /H20851Fig.5/H20849c/H20850/H20852; when the transverse field reaches 0.19 V /Å/H20851Fig. 5/H20849d/H20850/H20852or larger, the spin polarization decreases dramatically, and the half-metallic property is destroyed, which becomes aFM metal. When the electric field increases further, sayabout 1.0 V /Å, the half-metallic property comes back again /H20851Fig.5/H20849e/H20850/H20852. When the field exceeds 1.8 V /Å/H20851Fig.5/H20849f/H20850/H20852 ,i t becomes a NM metal. In summary, a type of half-metallic SiNWs along the /H20851100 /H20852direction by introducing multiple SDBs and boron doping is predicted by means of the spin-dependent ab initio calculations. The obtained results show that the half-metallicity in such SiNWs is quite robust against externalelectric fields. Under the circumstances with different SDBs,the H-passivated SiNWs can also be FM or AF semiconduc-tors. The present findings might be applicable in nanospin-tronics and nanomagnetism. The authors are grateful to X. Chen, S. S. Gong, X. L. Sheng, Z. C. Wang, and L. Z. Zhang for helpful discussions.The calculations are performed on the supercomputerNOVASCALE 6800 in Supercomputing Center of CAS. Thiswork is supported in part by the NSFC /H20849Grant No. 10625419 /H20850, the MOST of China /H20849Grant No. 2006CB601102 /H20850, and CAS. *Corresponding author; gsu@gucas.ac.cn 1J. Appenzeller, J. Knoch, M. T. Björk, H. Riel, H. Schmid, and W. Riess, IEEE Trans. Electron Devices 55, 2827 /H208492008 /H20850. 2D. Wang, B. A. Sheriff, and J. R. Heath, Nano Lett. 6, 1096 /H208492006 /H20850. 3Yu Huang, Xiangfeng Duan, Yi Cui, Lincoln J. Lauhon, Kyoung-Ha Kim, and Charles M. Lieber, Science 294, 1313 /H208492001 /H20850. 4Guo-Jun Zhang et al. , Nano Lett. 8, 1066 /H208492008 /H20850. 5A. M. Morales and C. M. Lieber, Science 279, 208 /H208491998 /H20850. 6D. D. D. Ma, C. S. Lee, F. C. K. Au, S. Y. Tong, and S. T. Lee, Science 299, 1874 /H208492003 /H20850. 7J. A. Yan, L. Yang, and M. Y. Chou, Phys. Rev. B 76, 115319 /H208492007 /H20850. 8E. Durgun, N. Akman, C. Ataca, and S. Ciraci, Phys. Rev. B 76, 245323 /H208492007 /H20850. 9A. K. Singh, V. Kumar, R. Note, and Y. Kawazoe, Nano Lett. 6, 920 /H208492006 /H20850. 10C. R. Leao, A. Fazzio, and A. J. R. da Silva, Nano Lett. 8, 1866 /H208492008 /H20850. 11R. Rurali and N. Lorente, Phys. Rev. Lett. 94, 026805 /H208492005 /H20850. 12R. Rurali, Phys. Rev. B 71, 205405 /H208492005 /H20850. 13S. Ismail-Beigi and T. Arias, Phys. Rev. B 57, 11923 /H208491998 /H20850. 14J. X. Cao, X. G. Gong, J. X. Zhong, and R. Q. Wu, Phys. Rev. Lett. 97, 136105 /H208492006 /H20850.15E. Durgun, D. Cakir, N. Akman, and S. Ciraci, Phys. Rev. Lett. 99, 256806 /H208492007 /H20850; E. Durgun, N. Akman, and S. Ciraci, Phys. Rev. B 78, 195116 /H208492008 /H20850. 16M.-V. Fernández-Serra, Ch. Adessi, and X. Blase, Nano Lett. 6, 2674 /H208492006 /H20850. 17H. Peelaers, B. Partoens, and F. M. Peeters, Nano Lett. 6, 2781 /H208492006 /H20850. 18R. Kagimura, R. W. Nunes, and H. Chacham, Phys. Rev. Lett. 98, 026801 /H208492007 /H20850. 19J.-H. Cho and J.-H. Choi, Phys. Rev. B 77, 075404 /H208492008 /H20850. 20S. Okada, K. Shiraishi, and A. Oshiyama, Phys. Rev. Lett. 90, 026803 /H208492003 /H20850. 21W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 /H208491965 /H20850. 22J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 /H208491996 /H20850. 23N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 /H208491991 /H20850. 24P. Ordejón, E. Artacho, and J. M. Soler, Phys. Rev. B 53, R10441 /H208491996 /H20850. 25Main properties have been confirmed in supercells of two and four units along the zaxis by SIESTA . We also used Quantum ESPESSO /H20849P. Giannozzi et al. , www.quantum-espresso.org /H20850with plane-wave basis sets and ultrasoft pseudopotentials to verify therelative energies of different spin configurations of SiNW-H8Bin a unit cell, which gives similar results with SIESTA .XUet al. PHYSICAL REVIEW B 80, 081306 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 081306-4

PhysRevB.101.075207.pdf

PHYSICAL REVIEW B 101, 075207 (2020) Modeling ultrafast out-of-equilibrium carrier dynamics and relaxation processes upon irradiation of hexagonal silicon carbide with femtosecond laser pulses G. D. Tsibidis ,1,*L. Mouchliadis,1M. Pedio ,2and E. Stratakis1,3 1Institute of Electronic Structure and Laser (IESL), Foundation for Research and Technology (FORTH), N. Plastira 100, Vassilika Vouton, 70013, Heraklion, Crete, Greece 2Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche (CNR-IOM), Trieste, Italy 3Department of Physics, University of Crete, 71003 Heraklion, Greece (Received 31 October 2019; revised manuscript received 5 January 2020; accepted 10 February 2020; published 24 February 2020) We present a theoretical investigation of the yet unexplored dynamics of the produced excited carriers upon irradiation of hexagonal silicon carbide (6H-SiC) with femtosecond laser pulses. To describe the ultrafastbehavior of laser-induced out-of-equilibrium carriers, a real-time simulation based on density-functional theorymethodology is used to compute both the hot-carrier dynamics and transient change of the optical properties.A two-temperature model (TTM) is also employed to derive the relaxation processes (i.e., thermal equilibrationbetween carrier and lattice through carrier-phonon coupling) for laser pulses of wavelength 401 nm, duration50 fs at normal incidence irradiation which indicate that surface damage on the material occurs for fluence∼1.88 J cm −2. This approach of linking real-time calculations, transient optical properties, and TTM modeling, has strong implications for understanding both the ultrafast dynamics and processes of energy relaxation betweencarrier and phonon subsystems and providing a precise investigation of the impact of hot-carrier population insurface damage mechanisms in solids. DOI: 10.1103/PhysRevB.101.075207 I. INTRODUCTION Over the past decades, the advances of ultrashort pulsed- laser technology have emerged as a powerful tool for manytechnological applications, in particular in industry andmedicine [ 1–14]. To this end, understanding of laser-driven physical phenomena such as electron excitation, scatteringprocesses, relaxation mechanisms, phase transitions, and ab-lation is important to elucidate many fundamental propertiesof solids that can lead to enhanced control of the laser energyfor numerous potential applications. One of the most challenging issues that influences laser- driven phenomena is the response of excited carriers scat-tering processes in the femtosecond time window. A betterdescription of those mechanisms is crucial for a detailedknowledge of laser-induced ultrafast processes. On the otherhand, the investigation of the ultrafast electron dynamicswithin the electron gas in a laser-heated material is a realchallenge. In principle, the extremely small electron-electroncollision time ( ∼10 fs), associated with the generation of highly hot and nonthermalized (i.e., out-of-equilibrium) elec-tron distribution during excitation, complicate direct observa-tion [ 15]. Nevertheless, advances in laser technology have al- lowed generation of out-of-equilibrium electron distributionswhile they have enabled observation of their relaxation inreal time, through predominantly the response of the materialoptical parameters [ 16–18]. *tsibidis@iesl.forth.grTo model laser-matter interaction and describe material’s response, a common approach that has been widely used is thetraditional two-temperature model (TTM) which, however,ignores the formation of nonthermal electron populations [19]. One major problem of the classical TTM is that it considers that these extremely hot excited carriers thermalizeinstantaneously which is not valid [ 20]. While this assumption yields precise quantitative results for the electron dynamicsthat agree with pump-probe and reflectivity experiments forpulse durations longer than 100 fs [ 15,21], inconsistencies have been observed at shorter pulses for which a strong pres-ence of out-of-equilibrium electron is expected [ 16,20,22]. To overcome the limitations originating from the overes- timation of the electron energy, various revised models havebeen proposed based on: (i) Boltzmann’s transport equations[23], (ii) three-temperature models [ 18,20], and (iii) two- temperature models with the introduction of two source terms[22,24]. The above approaches described successfully both the ul- trafast dynamics and thermal response of the irradiated mate-rial in many physical systems [ 17,18,20,23,25]. Nevertheless, although those methodologies appeared to illustrate efficientlythe role of the nonthermal electrons in the subsequent re-laxation processes (i.e., that achieve thermal equilibrationbetween carrier and lattice through carrier-phonon coupling),some of the above models were applied only for metals (i.e., consideration of an infinitesimal nonthermal, steplikechange of the electronic distribution due to the irradiationand promotion of electrons to the unoccupied states abovethe Fermi energy) [ 16,17,20,22,24–28]. One very intriguing challenge is whether similar models can be developed for 2469-9950/2020/101(7)/075207(12) 075207-1 ©2020 American Physical SocietyTSIBIDIS, MOUCHLIADIS, PEDIO, AND STRATAKIS PHYSICAL REVIEW B 101, 075207 (2020) other materials (i.e., semiconductors or dielectrics) where ex- citation and relaxation processes include more complex mech-anisms such as multiphoton/tunneling and impact ionizationas well as carrier recombination. It is evident that a revisionto existing models is required to account for the behavior ofout-of-equilibrium carriers in the conduction band and theirinteraction with thermalized carriers and lattice when theprocesses are considered. However, validity of a simplistic ex-tension of the aforementioned models is rather questionable.By contrast, due to the complexity of the physical mechanismsthat are involved, an approach based on quantum-mechanicalprinciples is regarded as a more precise technique to describethe underlying ionization processes and ultrafast dynamics.To address this need, simulations based on density-functionaltheory (DFT) have been applied in various systems [ 29,30] and the impact of out-of-equilibrium electrons in the subse-quent relaxation processes has been successfully evaluated.Nevertheless, one still unexplored process in these approachesis that they do not consider potential temporal variation ofthe optical parameters (and therefore energy absorption) ofthe irradiated material induced by the presence of hot carriers,which becomes significant at extremely short pulses. One very promising wide-band-gap material is SiC and its polymorphs due to its impact on numerous technologicalapplications. More specifically, the advantages of SiC devicesare opening up for advanced applications in the most impor-tant fields of electronics while its properties allow the perfor-mance of existing semiconductor technology to be extended[31,32]. Although the properties of this material have been widely explored, response upon extreme heating is an area thathas yet to be investigated. To address the above challenges and apply the method- ology to explore physical processes after irradiation of SiCwith single femtosecond laser pulses, a two-tiered approach isfollowed to describe two regimes: (i) a real-time simulationis presented to compute the ultrafast dynamics of the out- of-equilibrium excited carriers as well as the induced optical parameters for hexagonal Silicon-Carbide (6H-SiC) (Sec. II); (ii) a revised TTM for semiconductors is employed to providea description of the temporal evolution of the temperatures ofthe carriers/lattice population and recombination process forthe produced thermalized population of excited carriers and the energy relaxation between the carriers and the lattice sys-tems via carrier-phonon coupling. A detailed analysis of theresults of the theoretical model yields is presented in Sec. III for various values of the laser fluence while an estimationof the surface damage threshold is calculated. Concludingremarks follow in Sec. IV. II. THEORETICAL MODEL A. Structure of 6H-SiC Silicon carbide is a unique material as it occurs in some 250 polymorphs. A particular kind of polymorphism whichis called polytypism occurs in certain close-packed struc-tures: two dimensions of the basic repeating unit cell remainconstant for each crystal structure while the third dimensionis a variable of a common unit perpendicular to the planeswith the closest packing. Polytypes consist of layers withspecific stacking sequence where the atoms of each layer can FIG. 1. Structure of 6H-SiC: Silicon atoms are represented by large spheres (in red) correspond while carbon atoms are represented by small spheres (in blue) The cell parameters are a=b=3.095 Å, c=15.18 Å [ 35]. be arranged in three configurations in order to maximize the density [ 33]. The fundamental structural unit is a covalently bonded tetrahedron of four carbon (C) atoms with a singlesilicon (Si) atom at the center. On the other hand, each C atomis surrounded by four Si atoms. Among the various polytypesof SiC, the hexagonal 6H configuration [here, the Ramsdellclassification scheme is used where the number indicates thenumber of layers in the unit cell and the letter indicates theBravais lattice (H stands for hexagonal)] is one of the mostwidely studied [ 33,34] and it will be the focus of this work. In Fig. 1, the unit cell of 6H-SiC is shown which has a complex structure with 12 atoms (Fig. 1). B. First-principles calculations Polytypism has a strong influence on the material physical and chemical properties. In particular, the optical propertiesof SiC and their relation to the polytypic character havebeen extensively investigated [ 36–39]. These studies include measurements of the dielectric function, the refractive index,as well as the determination of the frequency-dependent di-electric function, optical absorption, and reflectivity spectraand are connected with the band structure of the material.A precise evaluation, however, of the optical properties forsystems in nonequibrium states due to excitation conditionsrequire also consideration of correlation effects (i.e., exci-tonic effects due to electron-hole Coulomb interaction) orplasmons. A consistent estimation of the role of excitoniceffects can be derived from the solution of the Bethe-Salpeterequation for the electron-hole Green’s function, within themany-body perturbation theory (MBPT) framework [ 36]. In this work, YA M B O was used to address the above is- sues [ 40].YA M B O is a consistent ab initio code for cal- culating quasiparticle energies [ 41] and optical parameters of electronic systems within the framework of MBPT. Al-though alternative approaches have been used, including ab initio molecular-dynamics techniques [ 42], the YA M B O code has proven to be an efficient, well-established algorithm todescribe excitation and dynamics following irradiation ofmaterials with intense sources (see Ref. [ 43] and references 075207-2MODELING ULTRAFAST OUT-OF-EQUILIBRIUM … PHYSICAL REVIEW B 101, 075207 (2020) therein). Using the YA M B O code, the equilibrium properties were computed starting from a self-consistent calculationof the Kohn-Sham eigenvalues and eigenstates in the DFTframework within the local-density approximation. DFT cal-culations were performed with the QUANTUM ESPRESSO code [44] using the Perdew-Burke-Ernzerhof functional [ 45] and norm–conserving pseudopotentials. Compared to other poly-types of SiC, the analysis of 6H-SiC in terms of band-to-bandtransitions is more demanding and complex due to the largenumber of bands being folded into the small Brillouin zone.A shifted 8 ×8×2k-point sampling for the ground state was used, while a kinetic energy cutoff of 100 Ry was considered.The quasiparticle corrections to the fundamental band gaphave been calculated from the standard GW approximation(Gstands for the one-body Green’s function and Wfor the dynamically screened Coulomb interaction) with the Godby-Needs plasmon-pole model and applied as a rigid shift toall the bands. Calculations of the quasiparticle energies andoptical susceptibilities have been performed using the YA M B O code [ 40] and a total of 100 bands for Green’s function expan- sion was used. The energy positions of the top of the valenceband and the bottom of the conduction band are calculatedalong with the direct and indirect band gaps through the YA M B O code. According to the calculations, the Fermi level is estimated to be at 10.33 eV , while the energy band gaps arecomputed to be equal to 2.03 and 3.16 eV for the indirect anddirect band gaps, respectively. Following the evaluation of the ground-state properties ( QUANTUM ESPRESSO is used to perform this step and de- termine the dielectric function as a function of the photonenergy), the evolution of the electronic system under intenselaser irradiation requires performance of real-time (RT) sim-ulations. YA M B O is, subsequently, employed to compute the out-of-equilibrium carrier distribution within the pulse dura-tion assuming the laser-pulse characteristics (energy, shape,duration, polarization). This step is implemented in YA M B O with a recently introduced feature that allows monitoring thereal time carrier dynamics within the nonequilibrium MBPTframework. By numerically integrating the time-dependentequation of motion for the density matrix expressed in thespace of the single-particle wave-functions, the time evolutionof the nonequilibrium carrier distribution is computed. In turn,the carrier occupations are subsequently used to monitor thetime evolution of optical parameters. An essential step of theRT simulation is the removal of all symmetries since theexternal laser field breaks the symmetry, which eventuallyleads to the development of polarization effects (for a detaileddescription, see Ref. [ 43].). Hence, polarization effects are expected to be closely related both to the photon energy andthe laser fluence and it will be reflected on the excitation levelof the carriers (i.e., values of the DFT-based calculated carrierdensities as shown in Sec. III). In the current work, single-shot laser pulses are used impinging normally to the sample. To simulate the tempo-ral profile of the pulse, it is important to set precisely thepropagation variables such as the time interval, the durationof the simulation, the integrator, and the pulse intensity. Inregard to the laser pulse shape and polarization, a linearlypolarized Gaussian pulse has been chosen that is centeredat the fundamental absorption peak in order to generate asignificant amount of carriers. The number of carriers is expected to increase as long as the pulse intensity is nonzero. Simulation results for the optical parameters of the irradi- ated material are illustrated in Fig. 2for various photon ener- gies that correspond to laser wavelengths in the range [73 nm,12μm] at 300 K. The calculation of the optical parameters such as the refractive index n, extinction coefficient k, and reflectivity Rof the material can be derived from the following expressions based on the results for the dielectric constant ε (Fig. 2): n=/radicalBigg |ε|2+Re(ε) 2 k=/radicalBigg |ε|2−Re(ε) 2(1) R=(n−1)2+k2 (n+1)2+k2, while the absorption coefficient is given by α=4πk/λL where λLstands for the laser wavelength. Results show the frequency dependence of the real and imaginary parts ofthe dielectric function along the transverse and longitudinaldirections, respectively [Figs. 2(a) and 2(b)]. A comparison with theoretical and measured values for the longitudinalcomponent of the dielectric function ε[Figs. 2(a) and 2(b)] reported in previous works shows a remarkable agreement thatillustrates the validity of the approach [ 36,38,46]. It is noted that while there is a discrepancy for εvalues between the two polarization directions, there is no difference in εfor the trans- verse and longitudinal components for the photon energy usedin this work (3.09 eV). An interesting aspect is the “metallic”behavior [i.e., Re( ε)<0] that is exhibited in both transverse and longitudinal spectra by the irradiated material at laserwavelengths λ L<179 nm (photon energies larger than 6.9 eV). It is also noted that the energy absorption and its spatialattenuation during the pulse is treated via kandα[Eq. ( 1)]. C. Energy and particle balance equations 1. Carrier excitation and carrier-phonon relaxation processes assuming instantaneous carrier thermalization To describe the carrier excitation and relaxation processes for semiconductors, the relaxation-time approximation toBoltzmann’s transport equation has been widely employed[19,47–53] to determine the spatial and temporal dependence (t) of the carrier density number, carrier energy, and lattice energy. The carrier system is assumed to be nondegenerate(i.e., Maxell-Boltzmann distributed) as the adoption of a morerigorous approach is not expected to lead to substantial dif-ferences in the evaluation of the main observable effects (i.e.,damage thresholds [ 50]). To describe the carrier dynamics and associated thermal effects, the following excitation, energy,and particle balance equations are used to derive the evolutionof the carrier density number N c, carrier temperature Tc, and lattice temperature TL[48,50,51]. More specifically, the balance equation for the lattice subsystem yields that the lat- tice energy density rate∂UL ∂tshould be equal to /vector∇·(KL/vector∇TL)+ g(Tc−TL) in which the first term describes energy transport 075207-3TSIBIDIS, MOUCHLIADIS, PEDIO, AND STRATAKIS PHYSICAL REVIEW B 101, 075207 (2020) 0 5 10 15−10−505101520 Photon Energy [eV]Re(ε) Longitudinal Tranverse (a) 0 5 10 15−50510152025 Photon Energy [eV]Im(ε) Longitudinal Tranverse (b) 0 5 10 1500.20.40.60.81 Photon Energy [eV]Reflectivity Longitudinal Tranverse(c) FIG. 2. Simulated results for the real and imaginary part of dielectric function (a), (b) and (c) reflectivity at various photon energies assuming longitudinal and transverse dielectric constantcomponents. in the lattice system ( KLis the lattice heat conductivity) and the second term describes the energy exchange of the latticewith the carrier system ( gstands for carrier-phonon couplingcoefficient). Given that C L=∂UL ∂TL, the following equation is derived: CL∂TL ∂t=/vector∇·(KL/vector∇TL)+Cc τc(Tc−TL). (2) By contrast, the balance equation for the carrier system is more complicated as the carrier-energy density is dependentnot only on T cbut also on Ncand on the energy band gap Eg [48,50,51]. Thus, the carrier-energy density rate∂Uc ∂tis given by the following expression: ∂Uc ∂t=Cc∂Tc ∂t+∂Nc ∂t∂Uc ∂Nc+∂Eg ∂t∂Uc ∂Eg, (3) where Ccis the carrier heat capacity ( Cc=3NckB,kBstands for the Boltzmann constant [ 48,50,51]). On the other hand, in a nondegenerate system, Ucis given by Uc=Nc(Eg+3kBTc) (i.e., equal to the product of the carrier number densityand the sum of the band-gap energy per unit volume andthe kinetic energy of the electrons and holes; the latter isequal to 2 ×3/2k BTcfor a nondegenerate carrier system to accommodate both the electron and hole densities [ 47,48,50]) while the balance equation for the carrier system is providedby the expression C c∂Tc ∂t=− g(Tc−TL)+LE1(Ephoton,I)( t h e first term in the second part describes the energy exchange be-tween the carriers and the lattice system while LE 1(Ephoton,I) is dependent on the intensity of the laser Iand the photon energy Ephoton which is related to the energy provided to the carrier system by the laser source [ 47]). The above discussion leads to the following equation for theTcrate ( g=Cc τc, where τcis the carrier-phonon energy relaxation time [ 47,48,51,52]): Cc∂Tc ∂t=−Cc τc(Tc−TL)−∂Nc ∂t(Eg+3kBTc) −Nc∂Eg ∂t+LE1(Ephoton,I). (4) Finally, the particle balance equation is related to the rate of the carrier density following excitation of the material andit is given by the expression ∂N c ∂t=−γNc3+LE2(Ephoton,I), (5) where the first term in the second part is related to Auger recombination ( γstands for the Auger recombination coef- ficient that leads to a gradual reduction of carrier densitywhile LE 2(Ephoton,I) includes various excitation mechanisms such as interband and intraband absorption processes, impactionization, etc. [ 47,48,50,51]). It is noted that no carrier current or heat-current density is considered (in previous stud-ies, simulations manifested that neglecting heat dissipationand particle transport are not expected to produce significantchanges to the material response [ 47,48,50]). The above set of coupled nonlinear equations [Eqs. ( 2), (4), and ( 5)] constitutes the main theoretical framework that is used to describe thecarrier density and thermal evolution ( N c,Tc, and TL) towards relaxation for a semiconducting material [ 47]. Despite the underlying complexity of the physical processes, the aboverate equations [Eqs. ( 2), (4), and ( 5)] have successfully de- scribed ultrafast phenomena and relaxation processes in awide range of materials such as metals, semiconductors, and 075207-4MODELING ULTRAFAST OUT-OF-EQUILIBRIUM … PHYSICAL REVIEW B 101, 075207 (2020) FIG. 3. Processes following irradiation with ultrashort pulses: Regime 1: energy absorption, production of hot (out-of-equilibrium)carriers, carrier thermalization, regime 2: Carrier cooling through lattice-carrier interaction and equilibration process. dielectrics [ 13,27,47,48,50–65]. An assumption that is usually made in modeling carrier excitation and relaxation processesis that carriers are considered to thermalize instantaneously (i.e., a delta function equilibration is assumed) which inprinciple is true for long pulses. Nevertheless , as noted in the Introduction, the use of these equations is questionableif an out-of-equilibrium carrier population is formed whichoccurs for very short pulses ( <100 fs). A detailed account of the energy of out-of-equilibrium carriers cannot be describedby the balance equations presented in the previous section asa thermalization of the out-of-equilibrium carriers to a hotFermi distribution is required. Thermalization of the carriersystem is achieved mainly through carrier-carrier scatteringprocesses that allows formation of a Fermi distribution for thecarriers with a well-defined electron temperature which meansthat before thermalization is completed, carrier temperature isnotdefined and therefore Eqs. ( 2)–(4)cannot be used. 2. Carrier-phonon relaxation processes assuming formation of out-of-equilibrium carrier population Therefore, in contrast to the traditional methodology of us- ing Eqs. ( 2), (4), and ( 5) to describe energy absorption, carrier excitation, carrier internal energy and densities, an alternativemethodology is used through, firstly, the employment of DFTapproaches presented in Sec. II B. On the other hand, a rapid carrier thermalization is assumed to have been completed at the end of the pulse. Therefore, todescribe the physical processes following irradiation 6H-SiCwith very short femtosecond pulses, two regimes are inves-tigated (Fig. 3): (i) Regime 1: for t<6τ p(τpstands for the laser pulse duration) where DFT calculations are performed todetermine laser energy absorption and ultrafast dynamics, (ii)Regime 2: for t>6τ pwhen laser is considered to have been switched off. In the latter case, as carrier thermalization is as-sumed to have been completed, a modified version of Eqs. ( 2), (4), and ( 5) should be used to derive the evolution of the carrier densities and temperatures of the carriers and lattice.Notably, in contrast to the traditional TTM that has a sourceterm which gives rise to carrier excitation, an appropriatemodification is required to show departure from the standardapproach. More specifically, in the revised TTM model whichis used in this work, it is assumed that LE 1(Ephoton,I)= 0f o r t>6τp(as pulse has been switched off) while thecarrier-density value attained at the end of the pulse is derived through DFT calculations as explained in Sec. II; on the other hand, Ncevolution after the laser pulse has been switched off takes into account feedback from those calculations (seethe next section). Similarly, LE 2(Ephoton,I)=0f o r t>6τp. Thus, the evolution of the carrier density for t>6τpis calcu- lated through the expression Nc=−/integraldisplayt t=6τpγNc3dt+PE(t=6τp)( 6 ) after integration of Eq. ( 5), where PE(t=6τp) is the carrier density at t=6τpwhich is computed through DFT calcula- tions and real-time simulations (see Sec. III).PEis attributed to polarization effects contributions as emphasized in Sec. II. Notably, this term is not included in the traditional TTM. On the other hand, equilibration of the thermalized carrier system with the material is performed through carrier-phononcoupling [i.e., Cc τc(Tc−TL)]; however, as the pulse duration is too short, it is assumed that its influence is not very significantbefore the laser pulse ends. Hence, the set of coupled set of nonlinear equations used to describe the carrier and thermalevolution of the system at t>6τ pis the following: Cc∂Tc ∂t=−Cc τc(Tc−TL)−∂Nc ∂t(Eg+3kBTc)−Nc∂Eg ∂TL∂TL ∂t CL∂TL ∂t=/vector∇·(KL/vector∇TL)+Cc τc(Tc−TL) ∂Nc ∂t=−γNc3. (7) The parameter values that are used in this work for 6H- SiC are the following: for CL, the lattice heat capacity, a temperature-dependent expression is derived through fit-ting of data in Ref. [ 66],γ=7×10 −31cm6/s[67],τc∼ 300−500 fs [ 48,50,53]), and Egthat corresponds to the TL- dependent energy band gap of 6H-SiC is taken to be equaltoE g=3.01−6.5×10−4×(TL)2/(TL+1200) eV [ 68,69] [this is the reason why∂Eg ∂tin Eq. ( 4) turns into∂Eg ∂TL∂TL ∂tin the first equation in Eq. ( 7)]. The aforementioned expression is used to provide the evolution of the band gap for t>6τp where a lattice temperature gradient occurs that could lead to Egshrinkage. In principle, the temperature effect on the en- ergy bands of the semiconductor and hence the band gap of thematerial is a cumulative effect of thermal lattice expansion andelectron-phonon interaction. On the other hand, as noted in theprevious section, at smaller timepoints t<6τ pand during the laser-based excitation time, Egis calculated from the YA M B O code and it corresponds to the difference between the top and bottom energy positions of the valence and conduction bands,respectively. It is noted that the direct energy gap (equal to3.16 eV as computed in Sec. II)i su s e df o r E gint<6τpas the indirect band gap corresponds to a phonon-assisted interbandtransition that is less likely to occur given the insignificantphonon system energy during the pulse. It is also emphasized,though, that in this work, single -shot simulated experiments were performed in which disorder or lattice deformation dueto laser irradiation were not considered. It is evident thatin multiple-shot conditions a more precise description ofthe response of the material should be obtained by taking 075207-5TSIBIDIS, MOUCHLIADIS, PEDIO, AND STRATAKIS PHYSICAL REVIEW B 101, 075207 (2020) into account the role of defects or disorder induced by laser heating in the calculations of the energy gap. On the otherhand, for t>6τ p, although remarkable changes to the thermal response of the system are not expected, a rigorous approachthrough the aforementioned temperature-dependent expres-sion [ E g=3.01−6.5×10−4×(TL)2/(TL+1200) eV] is used in this work. It is noted that thermally generated ef-fects (i.e., thermal expansion, strain propagation, plastic de-formation, defect formation, etc.) can also be developed asa result of the irradiation with intense femtosecond pulses([24,57,65,70]); however, such an investigation is beyond the scope of this work. Finally, K L=611/(TL−115) W cm−1K−1[66]. In pre- vious works, an anisotropic heat conductivity was reportedfor various polytypes of SiC including 6H-SiC in which itwas shown that the cross-plane thermal conductivity K (z) L (perpendicular to the hexagonal planes) of 6H-SiC is 30% lower than its in-plane thermal conductivity K(z) L(parallel to the hexagonal planes) [ 71,72]. Experimental observations indicate that the anisotropy in the thermal conductivity 6H-SiC is expected due to the hexagonal Bravais lattice structurewhich suggests that in general this difference should notbe ignored in a rigorous investigation. Nevertheless, in thisstudy, it is assumed that for the computation of the damagethresholds, results are not expected to be remarkably sensitiveto the 3D character of the heat diffusion. Certainly, a moreprecise exploration of the impact of the anisotropy on thermaleffects could provide a more detailed account of the roleof directional heat diffusivity; however, this investigation isbeyond the scope of the present study. Therefore, for the sakeof simplicity, a bulk material is considered while the laser spotradius is taken to be substantially larger than the thicknessof the material and, thereby, a 1D solution is consideredto sufficiently determine the carrier dynamics and thermalresponse of the system. III. RESULTS AND DISCUSSION A quantitative description of carrier excitation, relaxation processes, and thermal response of both the carrier and latticesystems is provided through the use of the aforementionedDFT+TTM combined model. To highlight the contribution of the nonthermal carriers to the transient dynamics of thesystem, femtosecond pulsed-laser beams of duration, signifi-cantly smaller than the carrier-phonon energy relaxation time,are assumed ( τ p=50 fs). The photon energy of the laser beam is ¯hω=3.09 eV which corresponds to laser-beam wavelength λL=401 nm and it is similar to the size of the material’s computed energy band gap ( ∼=3.16 eV). The initial conditions areTe(t=0)=TL(t=0)=300 K, and Ne=1012cm−3at t=0. The (peak) fluence is equal to E p=√πτpI0/(2√ ln2), where I0stands for the peak intensity. The optical parameters evolution (real and imaginary part of the dielectric function, and reflectivity R) are illustrated in Fig. 4for six various fluence (peak) values, 0.45, 0.6, 0.75, 0.9, 1.05, and 1 .88 J cm−2. For all fluence values, DFT calculations showed a decreasing reflectivity reaching aminimum at t=6τ pbefore a relaxation to the initial re- flectivity value [ ε(λL=401 nm) ∼=8.9+0.15ishown also in Fig. 2]. This behavior resembles that demonstrated by lower0 100 200 300 4004567891011 Time [fs]Re(ε) Ep=0.45 J/cm2 Ep=0.6 J/cm2 Ep=0.75 J/cm2 Ep=0.9 J/cm2 Ep=1.05 J/cm2 Ep=1.88 J/cm2 Intensity [Arb. Units](a) 0 100 200 300 40000.511.522.533.5 Time [fs]Im(ε) Ep=0.45 J/cm2 Ep=0.6 J/cm2 Ep=0.75 J/cm2 Ep=0.9 J/cm2 Ep=1.05 J/cm2 Ep=1.88 J/cm2 Intensity [Arb. Units](b) 0 100 200 300 4000.060.080.10.120.140.160.18 Time [fs]Reflectivity Ep=0.45 J/cm2 Ep=0.6 J/cm2 Ep=0.75 J/cm2 Ep=0.9 J/cm2 Ep=1.05 J/cm2 Ep=1.88 J/cm2 Intensity [Arb. Units](c) FIG. 4. Evolution of (a) real part, (b) imaginary part of dielectric constant, (c) reflectivity ( λL=401 nm, τp=50 fs). Black solid line shows the laser intensity profile. band-gap semiconductors upon irradiation with laser pulses of duration that is comparable with τc, and fluences that are not high enough to induce “metallization” (Re( ε)<0 [52,53]) of the irradiated material. Interestingly, both for the 075207-6MODELING ULTRAFAST OUT-OF-EQUILIBRIUM … PHYSICAL REVIEW B 101, 075207 (2020) fluences used in this work as well as for even larger values which correspond to intensities where the material appearsto undergo a phase transformation or even ablation, Re (ε) never becomes negative [Fig. 4(a)]. On the other hand, a noticeable variation of the imaginary part of the dielectricfunction Im (ε)is predicted [Fig. 4(b)] that is also related to the free-electron absorption coefficient and significant responseof the excited electron system. Furthermore, transient reflec-tivity calculations [Fig. 4(c)] illustrate a substantially large drop during the pulse duration that further increases the laserenergy absorption. By contrast, larger laser energies allowincrease of excitation at larger depths [Fig. 4(b)]. To quantify the carrier population in the simulations, it is noted that the volume of the unit cell equals 838.1109 atomicunits that corresponds to 1 .233 68 ×10 −22cm−3. The carrier- density evolution illustrated in Fig. 4(a) that results from DFT calculations indicates an initial increase of the carrierpopulation that reaches a peak value where laser intensity ishigher before a sharp decrease occurs. The temporal decreaseof reflectivity (Fig. 4) and, thereby, increase of the absorbed laser energy is projected on the increase of excited carrierdensity as higher excitation conditions are induced [Fig. 5(a)]. It is noted that the initial decrease (from a peak value) ofcarrier density that is shown in Fig. 5(a) is due to some kind of polarization effects. These effects are usually smallat resonances but they become more important outside theresonance regimes. Similar behavior has been reported in previous works in which the decrease of carrier density is attributed to recom-bination effects [ 30]. Variation of the carrier-density values (Fig. 5) with fluence demonstrates also that the laser energy affects the induced polarization effects. As noted in Sec. II, similarly, there is also a dependence of the polarization ef-fects on photon energy as excitation levels at different laserwavelengths are also expected to change. Notably, DFT calculations demonstrate that after the end of the pulse, the carrier-density evolution remains constant un-like an anticipated decrease predicted in other semiconductorsin different irradiation conditions [ 47,50,52,53]. The absence of a further decreasing behavior is due to the fact that (Augeror radiative) recombination processes are not included in theDFT model. Certainly, the incorporation of such processesin the quantum-mechanical approach would allow a moreprecise description of carrier transient evolution. Recombi-nation and other scattering processes could be introduced byselecting appropriate approximations for the self-energy andintroducing a dynamical character to the self-energy. Theseadditions, though, would make the approach more demanding,which is beyond the scope of the present study [ 73]. On the other hand, the model presented in this work is aimed to combine the DFT-based calculations and TTMresults by linking the description in the two different regimeswhere out-of-equilibrium (regime 1) and thermalized (regime 2) carriers are present. Therefore, to allow an efficient descrip-tion of carrier dynamics, some physically consistent method-ology is required to link the two regimes. To correlate thecarrier temperature of a thermalized population with theirdensity, it is assumed that at the end of the pulse, the carriershave reached their maximum thermal energy and maximumcarrier temperature T max cand that after that moment, they stop0 100 200 300 400024681012 Time [fs]Nc [1021 cm−3] Ep=0.45 J/cm2 Ep=0.6 J/cm2 Ep=0.75 J/cm2 Ep=0.9 J/cm2 Ep=1.05 J/cm2 Ep=1.88 J/cm2 Intensity [Arb. Units](a) 0 100 200 300 40002468101214 Time [fs]Nc [1021 cm−3] DFT TTM Intensity [Arb. Units] 0 5 10012345 Time [ps]Nc [1021 cm−3] (b) FIG. 5. Evolution of carrier density (a) through DFT calcula- tions, (b) results for Ep=1.88 J/cm2(λL=401 nm, τp=50 fs). Inset illustrates the carrier evolution at larger timepoints throughEq. ( 5). Black solid line shows the laser intensity profile. to receive energy from the laser source (while Tcstarts to drop due to carrier-phonon scattering processes) [ 47,48,52]. Furthermore, it is assumed that at the end of the pulse, carriershave thermalized and a Fermi-Dirac distribution with a well-defined temperature has been reestablished [ 15]. Given the anticipated insignificant variation of the lattice temperature within the pulse duration due to the small τ pand the large heat capacity of the lattice system for semiconduc-tors compared to C c,TLis approximately equal to T0 L=300 K at the end of pulse. It is noted that in other materials suchas metals with smaller heat capacity, hot electron-phononscattering processes lead to a rather significant increase ofthe lattice temperature within the pulse duration [ 24,26]. By contrast, similar notable increase of T Lis not expected for 6H-SiC. However, a more thorough investigation that providesa more conclusive estimation of the lattice temperature isbeyond the scope of the present work. To solve Eq. ( 7), an explicit forward time centered space finite-difference scheme is used [ 53]. While Eq. ( 7) can be also employed to provide a 3D solution assuming the spatialcharacteristics of the beam profile, for the sake of simplicity, 075207-7TSIBIDIS, MOUCHLIADIS, PEDIO, AND STRATAKIS PHYSICAL REVIEW B 101, 075207 (2020) and for the objectives of the present study, equations are solved in 1D, along a line between z=0μm and z=5μm. It is assumed that on the boundaries, von Neumann boundaryconditions are satisfied and heat losses at the front and backsurfaces of the material are negligible. A common approachfollowed to solve similar problems is through the employmentof a staggered grid finite-difference method which is found tobe effective in suppressing numerical oscillations. Tempera-tures ( T candTL) and carrier densities ( Nc) are computed at the center of each element while time derivatives of the displace-ments and first-order spatial derivative terms are evaluated atlocations midway between consecutive grid points [ 51]. Starting from the values of N ccalculated from the DFT approach [Fig. 5(a)], a correction to the carrier-density evo- lution is required to account for Auger recombination. Morespecifically, the third equation of Eq. ( 7) is used to produce the rate of the carrier density for t>6τ pwhile the initial carrier density to derive Tmax ccorresponds to the value of Ncat t=6τpfor which DFT calculations predict a carrier density that stops to decrease further. This value is taken to be thecontribution of polarization effect at t=6τ p[i.e., PE(t=τp) in Eq. ( 6)]. Considering the above assumptions, the maximum carrier temperature Tmax cis calculated by the first equation of Eq. ( 7) assuming∂Tc ∂t=0. Then, Eqs. ( 6) and ( 7) lead to the following expression for Tmax c: Tmax c∼=T0 L/parenleftbigCc τc+NcCc CLτc∂Eg ∂TL/parenrightbig −γNc3Eg Cc τc+NcCc CLτc∂Eg ∂TL−3kBγNc3. (8) Equations ( 7) and ( 8) allow the calculation of the evolution of the carrier densities (including the correction due to Augerrecombination), as well as the temporal dependence of thecarrier and lattice. Results and correction to the carrier-densityevolution profile are shown in Fig. 5(b) for 1.88 J cm −2(sim- ilar behavior is predicted for other fluences) while the insetdepicts the transient dynamics of N cat larger timepoints. In Fig. 5(b),t h e blue dashed line is derived from DFT calculations [see also Fig. 4(a)] while the reddashed-dotted line results from the use of Eqs. ( 7) and ( 8). Notably, the significant decrease of Ncresulting from the contribution of recombination processes is manifestly illustrated in Fig. 5(b) which indicates the Auger recombination role should not beignored. The significance of Auger recombination in both thecarrier dynamics [ 74] and surface modification processes has been also revealed in previous reports [ 75]. On the other hand, the thermal response of the carrier and lattice system on the surface of the material for laser fluenceequal to 1 .88 J cm −2is summarized in Fig. 6. It is evident that a maximum carrier temperature occurs at t=6τpthat is subsequently followed by a decrease due to carrier-lattice heattransfer and relaxation of the system. Relaxation processesand exchange of energy between the carrier and lattice sub-systems yield a similar behavior to what occurs in other ma-terials [ 47,50,52,53]. Furthermore, the simulated maximum T Lvalues allow an estimation of the damage threshold of the material ( ∼1.88 J cm−2). It is noted that, in this work, damage threshold is associated with the fluence value at which thesurface lattice temperature exceeds the melting point of thematerial [ 24,51,57,76](T melting=3100 K for 6H-SiC [ 77]).0 1 2 3 4 5 6012345 Time [ps]Temperature [104 K] Tc TL FIG. 6. Evolution of electron and lattice temperature ( λL= 401 nm, τp=50 fs, Ep=1.88 J/cm2). Certainly, the aforementioned methodology and predic- tions that are used to provide an estimate for the damagethresholds require validation of the model with experimentalresults. To the best of our knowledge, there are not similarreports with experimental observations for the pulse durationand laser wavelengths considered in this work. Nevertheless,experimental measurements for damage thresholds illustratedin Fig. 7at various laser wavelengths and pulse durations indicate that the theoretical value for the critical fluenceused for the simulations conditions in this work representsa reasonable prediction: Experimental measurements at var-ious wavelengths (Fig. 7) show a dispersion of the damage threshold estimations while the simulated value appears to bewithin the range of the measured values [ 78–82]. Certainly, other effects should also be taken into account to providea conclusive picture such as reflectivity changes at differentwavelengths [Fig. 2(c)] and role of multiphoton absorption. FIG. 7. Threshold fluences in various conditions: experimental measurements and prediction of RT +TTM model ( NAcorresponds to the numerical aperture of the lens used in the experiment [ 82]). 075207-8MODELING ULTRAFAST OUT-OF-EQUILIBRIUM … PHYSICAL REVIEW B 101, 075207 (2020) On the other hand, there is a number of reports about laser-induced periodic surface structures (LIPSS) which areformed on 6H-SiC crystal irradiated by femtosecond laserpulses at various wavelengths [ 83]. Experimental results for irradiation with multishot laser pulses at 400 nm indicatea measured fluence threshold for LIPSS formation whichis approximately equal to 0 .49 J/cm 2[83] while the model yields a fluence threshold approximately equal to 1 .88 J/cm2 for single-shot simulations. Similarly, bulk ablation of 6H-SiC at 785 nm takes place at a fluence of 1 .4J/cm2[84] while nanoripples have lower damage threshold than bulk singlecrystals which has also been observed in other studies [ 83]. A possible reason can be attributed to the fact that, in princi-ple, an experimentally observed formation of LIPSS requiresirradiation with multiple number of pulses ( NP>10 shots [85]). By contrast, it is known that in transparent materials and semiconductors [ 86,87], the damage threshold for surface modification at increasing NPdrops significantly (more than 1/4 of the value for NP=1) compared to the measured value for single-shot experiments due to the presence of defects andincubation. This is expected to provide a satisfactory agree-ment between the predicted single laser shot-based result withthe measured value (i.e., deduction of a predicted multishotdamage threshold around 0 .4J/cm 2which appears to agree with the experimental value). A challenging issue is whether the approach followed in this work can be used to describe LIPSS formation mecha-nisms upon 6H-SiC irradiation with very short femtosecondpulses ( τ p<100 fs). One physical process that is directly linked with the formation of periodic structures on solidsis the interference of laser pulses with surface plasmonwaves (SP) that are excited as a result of laser irradiation[49,51,88]. On the other hand, according to well-established theories, excitation of SP requires carrier densities that lead toRe(ε)<−1. However, according to the simulation results in Fig. 4(a), despite the large decrease of Re (ε)for 1.88 J cm −2, this parameter does not drop to sufficiently low values thatcan induce SP excitation despite the extremely high carrierdensities which are produced ( ∼8×10 21cm−3). This can be attributed, firstly, to the need to revise the dispersion relationthat is required for SP excitation [ 51,53,89]; more specifically, the Drude-model-based dielectric function expression differsif nonthermal contributions are included that indicates thatappropriate corrections have to be included. Therefore, thecarrier-densities evaluation for which Re( ε)<−1 is expected not to be the correct condition to determine the onset of SPexcitation. Secondly, given the significance of the incubationeffects, the precise role of defects in multipulse experiments(that lead to SP excitation and LIPSS formation [ 51,62,90]) and the variation caused to an effective dielectric constantshould be also taken into account. These are some issuesthat need to be elaborated on to determine the contributionof hot electrons in incubation-related processes and surfacemodification mechanisms. Certainly, a more accurate conclusion will be drawn if more appropriately developed experimental (for exam-ple, time-resolved experimental) protocols are also intro-duced to evaluate the damage thresholds at the onset of thephase transition; similarly, pump-probe experiments could beused to validate the reflectivity changes. Furthermore, theFIG. 8. Spatial distribution of carrier density at t=300 fs ( λL= 401 nm, τp=50 fs, Ep=1.88 J/cm2). aforementioned potential impact of anisotropy-related ef- fects on damage thresholds should be further explored.Anisotropies in visible pump-probe experiments have beenpreviously reported by pumping at 800 nm [ 91]. Although results in Fig. 6are shown for irradiation with laser pulses of a single wavelength (at 401 nm), the method-ology can be generalized at other frequencies for which alaser wavelength-dependent carrier is expected. More specif-ically, other values of the photon energies are expected toinfluence the energy absorption, optical parameters (Fig. 2), and maximum excited carrier densities that, in turn, leadto a variation of T max c. Similar results have been predicted at larger wavelengths, longer pulse durations, and variousfluence values in a wide range of materials [ 13,27,47,48,50– 65]. Nevertheless, extension of the investigation of the thermal response of the material following irradiation with very shortpulses of different photon energy is beyond the scope of thepresent study. It is also noted that results illustrated in this work aimed to underline the response of the system in laser conditions thatlead to surface damage and therefore, special emphasis wasgiven to N c,Tc, and TLvalues at z=0. As expected, the atten- uation of the laser beam energy inside the material is expectedto lead to a gradual decrease of the carrier density. Results inFig.8illustrate the spatial distribution of the maximum carrier density at t=6τ pforEp=1.88 J/cm2andλL=401 nm up to 600 nm below the surface of the material. It is evident that atthis wavelength the absorption coefficient reaches values up to15×10 4cm−1[Figs. 4(a) and4(b)]. Therefore, irradiation of 6H-SiC with intense femtosecond pulses leads to large valuesfor the absorption coefficient that is characteristic to materialsthat show a metallic behavior [ 92]. One aspect that is of paramount importance is whether the above methodology can also be used to cover a widerrange of potential photon energies extending to 100 eV (i.e..wavelength ∼12 nm). The latter corresponds to a spectral region in which free-electron lasers (FEL) can be used toenable unique ultrafast scientific research [ 93]. At the same time, the unique output characteristics of x-ray FEL present 075207-9TSIBIDIS, MOUCHLIADIS, PEDIO, AND STRATAKIS PHYSICAL REVIEW B 101, 075207 (2020) severe requirements on the optics used to guide and shape the x-ray pulses, and the detectors used to characterize them[93,94]. The limitation, though, that is raised in regard to the employment of the presented DFT-based methodology isthat the pseudopotential which is used for the calculations inthis work assumes that core electrons are not excited. Thisassumption at substantially larger photon energies rather leadsto an underestimation of the excitation levels which might bealso reflected in the response of the material. Therefore, theinvestigation of the optical parameter values of the irradiatedmaterial at higher energies requires a revised and more preciseexpression of the pseudopotential that is beyond the scope ofthe current work. Certainly, several parameters including a more rigorous description of the thermalization process of the carriers,influence of scattering processes, microscopic analysis ofnonequilibrium phase-transition mechanisms through the useof hybrid molecular-dynamics-TTM models [ 29,95,96] and a complete parametric investigation of the ultrafast dynamicsand relaxation processes at a large range of photon energiesand pulse durations should be considered towards providinga complete picture of the ultrafast processes. Nevertheless,the aforementioned framework is designed to provide a sat-isfactory methodology to link processes at two very smalltimescales (some hundreds of fs). IV . CONCLUSIONS A theoretical framework was presented that describes both the ultrafast dynamics and thermal response following irradia-tion of 6H-SiC with ultrashort pulsed lasers of duration that istoo short to assume an instantaneous thermalization of excitedcarriers. The dynamics of produced out-of-equilibrium carrierpopulation and thermalization process is described through a quantum-mechanical approach and real-time simulations.Equilibration of the thermalized carrier system with the latticethrough carrier-phonon scattering processes are presented viaa revision of the classical TTM that allows the prediction ofthe decrease of the carrier density which is not appropriatelyaccounted for in real-time simulations. Results predict thetemporal variation of the optical parameters and allow anestimation of the surface damage threshold. The theoreticalframework is expected to enable a systematic analysis ofthe impact of the yet-unexplored hot carriers on surface (oreven structural effects) on semiconductors through a com-bined DFT +TTM methodology. Predictions resulting from the above theoretical approach demonstrate that elucidatingultrafast phenomena in the interaction of matter with veryshort pulses ( <100 fs) can potentially set the basis for the development of tools for nonlinear optics and photonics fora large range of applications. ACKNOWLEDGMENTS The authors acknowledge financial support from Nanoscience Foundries and Fine Analysis (NFFA)–EuropeH2020-INFRAIA-2014-2015 (under Grant AgreementNo. 654360), HELLAS-CH Project No. MIS 5002735,implemented under the “Action for Strengthening Researchand Innovation Infrastructures,” funded by the OperationalProgramme “Competitiveness, Entrepreneurship andInnovation” and cofinanced by Greece and the EU (EuropeanRegional Development Fund), and COST Action TUMIEE(supported by COST-European Cooperation in Science andTechnology) .We would also like to acknowledge fruitful discussions with Davide Sangalli and Andrea Marini. [1] A. Y . V orobyev and C. 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PhysRevB.83.064410.pdf

PHYSICAL REVIEW B 83, 064410 (2011) Hubbard III approach with hopping interaction and intersite kinetic correlations Grzegorz G ´orski and Jerzy Mizia* Institute of Physics, University of Rzesz ´ow, ul. Rejtana 16A, 35-959 Rzesz ´ow, Poland (Received 21 April 2010; revised manuscript received 14 December 2010; published 15 February 2011) We analyze the Hubbard model with added hopping interaction within the full Hubbard III approximation. In the Green’s-function decoupling process, the intersite kinetic correlation functions are included. This is anextension of our previous paper [G. G ´orski and J. Mizia, Phys. Rev. B 79, 064414 (2009) ] in which the basic Hubbard model with the intersite kinetic correlations was analyzed in the framework of the coherent potentialapproximation (CPA). In the CPA method, the up-spin electrons propagated in the lattice of frozen down-spinelectrons. The full Hubbard III solution used now takes into account the itinerancy of down-spin electrons. Thecombined effect of the hopping interaction and intersite kinetic correlation leaves the position of spin bandsunaffected, but it deforms the density of states (DOS) of electrons, changing in this way the average electronenergy. It is the main driving force behind the ferromagnetism as opposed to the rigid shift of the entire band,which takes place in the conventional Stoner magnetism. In the numerical calculations, we have used the bandswith symmetrical DOS (semielliptic or bcc-like DOS) and also with asymmetrical DOS resembling the fcc DOS.The spontaneous ferromagnetic transition was obtained under the combined action of the hopping interactionand the intersite correlation in the systems that contain even a moderately strong peak in the DOS, such as thebcc- and fcc-like DOS. DOI: 10.1103/PhysRevB.83.064410 PACS number(s): 75 .10.Lp, 71 .10.Fd I. INTRODUCTION Itinerant-electron ferromagnetism is one of the basic problems in solid-state physics. The central model describingthis phenomenon is the single-band Hubbard model. 1,2Exact solution of this model is possible only for one-dimensional(1D) systems 3,4and for infinite-dimensional systems.5For that reason, different approximate methods exist. Introducing theHartree-Fock approximation, we obtain the well-known Stonercriterion for ferromagnetism, U Cρ(εF)=1, where UCis the critical value of on-site repulsion and ρ(εF) is the density of states (DOS) on the Fermi level εF. Hartree-Fock approxima- tion (called also the mean-field approximation) overestimatesferromagnetic ordering and the Curie temperature. It allows forthe ferromagnetic transition at relatively small Uand in the broad range of concentrations, but it neglects the correlationeffects that can change the shape of the DOS, even splitting itat higher U, and can change the width of the band. Obtaining a ferromagnetic solution in a higher approxi- mation is very elusive. Reliable solutions exist only in somespecific cases. One of the earliest results was obtained byNagaoka, 6who invented the model of the fully polarized ferromagnetic state (Nagaoka state) in the presence of a singlehole in a half-filled band at U=∞ . For such a model, he obtained saturate ferromagnetism for sc, bcc, fcc, and hcplattices. Looking at the magnetic problem from a differentperspective, Lieb 7obtained the ferromagnetic ground state for asymmetric bipartite lattices with finite Coulomb interactionand different numbers of sites in each sublattice. Mielke 8and Tasaki9reached the ferromagnetic ordering for the lattices with flat bands. M ¨uller-Hartmann10suggested that in the 1D model with next-nearest-neighbor hopping included, one canhave the ferromagnetic ground state in the system with doubleminima at the limit of low particle density. The dynamical mean-field theory (DMFT) 11has allowed for subsequent progress in solving the Hubbard model.Using the finite-temperature quantum Monte Carlo (QMC)technique within the DMFT equations, Ulmke 12obtained ferromagnetism for the fcc-type d=∞ lattice even at inter- mediate Coulomb interaction. Numerical calculations basedon QMC 13and the variational QMC method14have arrived at spontaneous ferromagnetism for the infinite- UHubbard model. To describe correctly the magnetic state, V ollhardt and co-workers15postulated (i) including the intersite interactions, (ii) including the correlation effects, and (iii) consideringhighly asymmetrical DOS with the peak away from the centerof the band. In addition to the on-site repulsion U=(ii|1/r|ii)i n real materials, the intersite interactions are also important, including the nearest-neighbor repulsion V=(ij|1/r|ij), the nearest-neighbor exchange interaction J=(ij|1/r|ji), the pair hopping interaction J /prime=(ii|1/r|jj), and the correlated hopping interaction /Delta1t=(ii|1/r|ij),which is also called the bond-charge interaction. Although in general they are smallerthanU, as postulated by Hirsch and co-workers, 16–19they play a key role in creating ferromagnetic ordering. Among these interactions, the correlated hopping interaction /Delta1t (Refs. 16and20–33) may be particularly important for creating ferromagnetism. The analysis carried out in the mean-fieldapproximation has shown that the interaction /Delta1tdecreases critical on-site repulsion U Cfor some carrier concentrations even to zero.16,30 One of the most accepted and frequently used approx- imations describing the correlation effects in the Hubbardmodel is the Hubbard III approximation. 2This approximation at high enough Usplits the spin band into two bands: the lower band centered around the atomic level T0and the upper band centered around the level T0+U. The width of these bands depends on electron concentrations with different spins.Unfortunately, this approximation did not produce the ferro-magnetic ground state; see Refs. 34and35. The scattering cor- rection of the Hubbard III approximation, which is equivalent 064410-1 1098-0121/2011/83(6)/064410(13) ©2011 American Physical SocietyGRZEGORZ G ´ORSKI AND JERZY MIZIA PHYSICAL REVIEW B 83, 064410 (2011) with the coherent potential approximation (CPA),36assumes that the σelectrons move in a frozen sea of −σelectrons, the possibility of dynamic correlations between those twogroups being ignored. The intersite correlations have been also ignored; the focus instead was on the on-site correlations alone. As a result, the self-energy was obtained as independent ofmomentum kand of spin. Therefore, to correct the Hubbard III approximation for the possibility of band ferromagnetism requires introducing the intersite correlations. In our previous paper (see Ref. 37), we described in great detail the Hubbard III approximation with the intersite kinetic correlation functionsincluded but without the intersite interactions. The Hubbard IIIapproximation has been simplified to the CPA type of solution. The intersite kinetic correlation functions /angbracketleftc + i−σcj−σ/angbracketrightand /angbracketleftˆniσc+ i−σcj−σ/angbracketrightwere originally ignored in the Hubbard III approach and in most of the subsequent papers by other authorsdevoted to this model. The self-energy in this approximationhas the spin-dependent, k-independent band-shift term and the k-dependent term. Nolting and co-workers 38–40have developed a similar model called the modified alloy analogy method approxima-tion (MAA). It is a combination of the CPA (or alloy analogy) 36 method and the spectral density approach (SDA).41–43Their approximation has led to spontaneous magnetization only forsome carrier concentrations and a highly asymmetric fcc typeof DOS. In our paper, 37we showed that the MAA method can be obtained as a simplified version of our approach, theapproach that was based on including the intersite correlationsdirectly into the Hubbard III or CPA scheme. Our conclusionsfor ferromagnetism were more restrictive than in the MAAmethod, since in addition to the band-shift term consideredin the MAA method, we also included the bandwidth changeterm that was neglected in that method. In this paper, we present an approach that includes the intersite hopping interaction /Delta1tand the intersite correlations arising in the decoupling process. The reason for introducingthe intersite hopping interaction in the full Hubbard III schemeis that this interaction has already created a bandwidth changeand a band shift (see Refs. 16and17) in the Hartree-Fock (HF) approximation, both of which enhance ferromagnetism.In Ref. 37, we reduced both the scattering and the resonance broadening effect to the CPA-like approach, in which the+σelectron moves in a frozen sea of −σelectrons. In the current improved approach, the +σelectron moves in a sea of −σelectrons defrozen by the resonance broadening effect [see Hubbard III, Eqs. (56)–(59)]. Therefore, even thesolution of the simple Hubbard model (with only repulsion U) will be improved compared to our previous paper. 37The interaction /Delta1twill be treated also in this full Hubbard III approach. We will show that in this model, there is aspontaneous magnetization at a broad interval of parameters.The analysis will be carried out in the Green’s-function for- malism using the equation-of-motion approach described byZubarev. 44,45 The paper is organized as follows. In Sec. II, the general Green’s-function chain equations for the Hubbard modelwith the intersite hopping interaction are calculated. Thescattering correction and the resonance broadening correctionare solved together. As a result, the self-consistent set ofequations for the self-energy and DOS are obtained. Thisshows how the hopping interaction combined with the intersitekinetic correlation deforms the DOS and produces the spin-dependent change of the average energy. In Sec. III,w e derive conditions for the spontaneous transition to ferromag-netism. Discussion of the numerical results for ferromag-netic ordering is presented in Sec. IV. Finally, Sec. Vis devoted to the conclusions and to summarizing the obtainedresults. II. THE MODEL We analyze the basic Hubbard model with added hopping interaction and the exchange field, H=−/summationdisplay ijσtijc+ iσcjσ+T0/summationdisplay iσˆniσ+U 2/summationdisplay iσˆniσˆni−σ +/summationdisplay ijσ/Delta1tij(ˆni−σ+ˆnj−σ)c+ iσcjσ−/summationdisplay iσ(μ+Finnσ)ˆniσ, (1) where the operator c+ iσ(ciσ) is creating (annihilating) an electron with spin σ=↑,↓on the ith lattice site, ˆniσ=c+ iσciσ is the electron number operator for electrons with spin σon theith lattice site, Uis the on-site Coulomb interaction, /Delta1tij is the hopping interaction, Finis the on-site atomic Stoner field (exchange field) in the HF approximation, and μis the chemical potential. The Stoner exchange field is introducedas the test for the ferromagnetic transition, which will takeplace when the value of F incalculated numerically drops to zero. In the many-body considerations presented later, thetermμ+F innσwill be absent, since it will be moved into the Fermi-Dirac statistics. Quantity tijis the hopping integral between the ith and jth lattice site and T0is the Bloch band center of gravity.2 To analyze Hamiltonian (1), we use the equa- tion of motion for the Green’s functions in Zubarevnotation, 44 ε/angbracketleft/angbracketleftA;B/angbracketright/angbracketrightε=/angbracketleft[A,B]+/angbracketright+/angbracketleft /angbracketleft [A,H ]−;B/angbracketright/angbracketrightε, (2) where AandBare the fermion operators. Using Hamiltonian ( 1)i nE q .( 2), we can find the following equation for the Green’s function /angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε: ε/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε=δij+T0/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε−/summationdisplay ltil/angbracketleft/angbracketleftclσ;c+ jσ/angbracketright/angbracketrightε+U/angbracketleft/angbracketleftˆni−σciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay l/Delta1til/angbracketleft/angbracketleft(ˆni−σ+ˆnl−σ)clσ;c+ jσ/angbracketright/angbracketrightε +/summationdisplay l/Delta1til/angbracketleft/angbracketleft(c+ i−σcl−σ+c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε. (3) 064410-2HUBBARD III APPROACH WITH HOPPING ... PHYSICAL REVIEW B 83, 064410 (2011) The equations of motion for functions /angbracketleft/angbracketleftˆni−σciσ;c+ jσ/angbracketright/angbracketrightεand/angbracketleft/angbracketleftˆnl−σciσ;c+ jσ/angbracketright/angbracketrightεhave the form ε/angbracketleft/angbracketleftˆni−σciσ;c+ jσ/angbracketright/angbracketrightε=n−σδij+T0/angbracketleft/angbracketleftˆni−σciσ;c+ jσ/angbracketright/angbracketrightε+U/angbracketleft/angbracketleftˆni−σciσ;c+ jσ/angbracketright/angbracketrightε−/summationdisplay ltil/angbracketleft/angbracketleftˆni−σclσ;c+ jσ/angbracketright/angbracketrightε −/summationdisplay ltil/angbracketleft/angbracketleft(c+ i−σcl−σ−c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay l/Delta1til/angbracketleft/angbracketleftˆni−σ(ˆni−σ+ˆnl−σ)clσ;c+ jσ/angbracketright/angbracketrightε +/summationdisplay l/Delta1til/angbracketleft/angbracketleftˆni−σ(c+ i−σcl−σ+c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay l/Delta1til/angbracketleft/angbracketleft(ˆniσ+ˆnlσ)(c+ i−σcl−σ−c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε. (4) and ε/angbracketleft/angbracketleftˆnl−σciσ;c+ jσ/angbracketright/angbracketrightε =n−σδij+T0/angbracketleft/angbracketleftˆnl−σciσ;c+ jσ/angbracketright/angbracketrightε+U/angbracketleft/angbracketleftˆnl−σˆni−σciσ;c+ jσ/angbracketright/angbracketrightε−/summationdisplay mtim/angbracketleft/angbracketleftˆnl−σcmσ;c+ jσ/angbracketright/angbracketrightε −/summationdisplay mtlm/angbracketleft/angbracketleft(c+ l−σcm−σ−c+ m−σcl−σ)ciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay m/Delta1tim/angbracketleft/angbracketleftˆnl−σ(ˆni−σ+ˆnm−σ)cmσ;c+ jσ/angbracketright/angbracketrightε +/summationdisplay m/Delta1tim/angbracketleft/angbracketleftˆnl−σ(c+ i−σcm−σ+c+ m−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay m/Delta1tlm/angbracketleft/angbracketleft(ˆnlσ+ˆnmσ)(c+ l−σcm−σ−c+ m−σcl−σ)ciσ;c+ jσ/angbracketright/angbracketrightε.(5) The notation of ˆnα iσ,nα σ, andεαfollows the original Hubbard paper,2and that for the hopping interaction is the following: /Delta1t+ il=/Delta1tiland/Delta1t− il=0. Following Hubbard,2we can express the single-electron Green’s function as /angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε=/summationdisplay α=±/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε, (6) where the higher-order Green’s function in the energy representation /angbracketleft/angbracketleftˆnα i−σciσ;c+ jσ/angbracketright/angbracketrightε(α=±) fulfills the equation ε/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε=nα −σ/parenleftBigg δij−/summationdisplay ltil/angbracketleft/angbracketleftclσ;c+ jσ/angbracketright/angbracketrightε/parenrightBigg +εα/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε−/summationdisplay ltil/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig clσ;c+ jσ/angbracketrightbig/angbracketrightbig ε −ξα/summationdisplay ltil/angbracketleft/angbracketleft(c+ i−σcl−σ−c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay l/summationdisplay β=±/parenleftbig /Delta1tα il+/Delta1tβ il/parenrightbig /angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbigˆnβ l−σclσ;c+ jσ/angbracketrightbig/angbracketrightbig ε +nα −σ/summationdisplay l/summationdisplay β=±/parenleftbig /Delta1tα il+/Delta1tβ il/parenrightbig/angbracketleftbig /angbracketleftbigˆnβ l−σclσ;c+ jσ/angbracketrightbig/angbracketrightbig ε+/summationdisplay l/Delta1til/angbracketleftbig/angbracketleftbigˆnα i−σ(c+ i−σcl−σ+c+ l−σci−σ)ciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε +ξα/summationdisplay l/Delta1til/angbracketleft/angbracketleft(ˆniσ+ˆnlσ)(c+ i−σcl−σ−c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε. (7) In a further analysis of Eq. ( 7), we follow the Hubbard III approach2and reduce Green’s functions of the higher order appearing in the third, fourth, fifth, seventh, and eighthterms to Green’s functions of the type /angbracketleft/angbracketleftˆn α i−σciσ;c+ jσ/angbracketright/angbracketrightεand /angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε. The third and fifth terms are approximated in this way in Appendix A, and the fourth and eighth terms are approximated in Appendix B. In the course of performing these approximations, we keep the intersite averages of thetype/angbracketleftc + i−σcj−σ/angbracketrightand/angbracketleftˆniσc+ i−σcj−σ/angbracketright. This is the main difference between this paper and Hubbard’s approach. For the seventhterm, responsible for the hopping interaction, we use the sametype of approximation, namely /summationdisplay l/Delta1til/angbracketleftbig/angbracketleftbigˆnα i−σ(c+ i−σcl−σ+c+ l−σci−σ)ciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε ≈/summationdisplay l/Delta1til/angbracketleftc+ i−σcl−σ+c+ l−σci−σ/angbracketright/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε =S/Delta1t σ/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε. (8)The parameter S/Delta1t σis the spin-dependent band shift created by the hopping interaction, S/Delta1t σ=/summationdisplay l/Delta1til/angbracketleftc+ i−σcl−σ+c+ l−σci−σ/angbracketright=2γ+I−σ,(9) where I−σis the intersite correlation parameter and γ±is the hopping interaction parameter defined as I−σ=1 N/summationdisplay iltil/angbracketleftc+ i−σcl−σ/angbracketright,γ+≡γ=/Delta1tij tij,γ−=0. (10) As a result, we obtain the equation /parenleftbig ε−S/Delta1t σ/parenrightbig/angbracketleftbig /angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε =nα −σ/parenleftBigg δij−/summationdisplay ltil/angbracketleft/angbracketleftclσ;c+ jσ/angbracketright/angbracketrightε/parenrightBigg +εα/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε 064410-3GRZEGORZ G ´ORSKI AND JERZY MIZIA PHYSICAL REVIEW B 83, 064410 (2011) −/summationdisplay ltil/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig clσ;c+ jσ/angbracketrightbig/angbracketrightbig ε −ξα/summationdisplay ltil/angbracketleft/angbracketleft(c+ i−σcl−σ−c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε +/summationdisplay l/summationdisplay β=±/parenleftbig /Delta1tα il+/Delta1tβ il/parenrightbig /angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbigˆnβ l−σclσ;c+ jσ/angbracketrightbig/angbracketrightbig ε +nα −σ/summationdisplay l/summationdisplay β=±/parenleftbig /Delta1tα il+/Delta1tβ il/parenrightbig/angbracketleftbig/angbracketleftbigˆnβ l−σclσ;c+ jσ/angbracketrightbig/angbracketrightbig ε+ξα/summationdisplay l/Delta1til ×/angbracketleft /angbracketleft(ˆniσ+ˆnlσ)(c+ i−σcl−σ−c+ l−σci−σ)ciσ;c+ jσ/angbracketright/angbracketrightε.(11) In further investigations, we will use the notation ε/prime= ε−S/Delta1t σ. Analysis of this equation will proceed along the lines of the Hubbard III approximation2with intersite correlation included in the way developed in our previous paper.37 In Eq. ( 11), there are terms coming from the commutator [ciσ,H]−. These are the second, third, fifth, and the sixth terms in Eq. ( 11). They lead to what is known as the “scattering correction.” The fourth and last terms, which come from thecommutator [ ˆn iσ,H]−, give the “resonance broadening” effect. The difference with our previous paper37is the presence of additional interaction /Delta1til. This interaction enriches the scattering correction and the resonance broadening correctionterms.The scattering correction term is expressed by the Green’s functions /angbracketleft/angbracketleft(ˆn α i−σ−nα −σ)clσ;c+ jσ/angbracketright/angbracketrightεand/angbracketleft/angbracketleft(ˆnα i−σ− nα −σ)ˆnβ l−σclσ;c+ jσ/angbracketright/angbracketrightε, which are given by Eqs. ( A3) and ( A6)i n Appendix A. As mentioned earlier, the fourth and last (seventh) terms in Eq. ( 11), which come from the commutator [ˆniσ,H]−, give the “resonance broadening” effect. The functions /angbracketleft/angbracketleftc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε,/angbracketleft/angbracketleftˆnlσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε, and /angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε, appearing in those terms, are found in Appendix Bas Eqs. ( B3), (B4), and ( B7). Now we insert the functions appearing in the scatter- ing correction and resonance broadening correction fromAppendixes A and B into Eq. ( 11). Further analysis does not include any additional approximations. Equation ( 11)i s solved directly in the momentum space in Appendix C.F r o m Eqs. ( C7) and ( C8) of Appendix C, we obtain the final relation for the Green’s function G σ k(ε), Gσ k(ε)=1 ε−/Sigma1σ tot,k(ε)−(εk−T0), (12) where the self-energy /Sigma1σ tot,k(ε)i st h es u mo ft h e k-independent term/Sigma1σ 0(ε) and the k-dependent term /Sigma1σ 1,k(ε), /Sigma1σ tot,k(ε)=/Sigma1σ 0(ε)+/Sigma1σ 1,k(ε), (13) which are given by /Sigma1σ 0(ε)=n+ −σε++n− −σε−+S/Delta1t σ+/bracketleftbig n− −σn+ −σ(ε+−ε−)+SB σ(ε/prime)/bracketrightbig [ε+−ε−−/Omega1+ σ(ε/prime)+/Omega1− σ(ε/prime)] ε−S/Delta1tσ−/Omega1totσ(ε/prime)−{n+ −σ[ε−−/Omega1−σ(ε/prime)]+n− −σ[ε+−/Omega1+σ(ε/prime)]}(14) and /Sigma1σ 1,k(ε)≡γ(εk−T0)/braceleftbiggSB σ(ε/prime)−n+ −σn− −σ[2(ε+−ε−)−/Omega1+ σ(ε/prime)+/Omega1− σ(ε/prime)−γ(εk−T0)] ε−S/Delta1tσ−/Omega1totσ(ε/prime)−{n+ −σ[ε−−/Omega1−σ(ε/prime)]+n− −σ[ε+−/Omega1+σ(ε/prime)]}−2n+ −σ/bracerightbigg . (15) The k-dependent term /Sigma1σ 1,k(ε) is proportional to the hopping interaction and vanishes at /Delta1t→0. III. FERROMAGNETIC SOLUTION To analyze the possibility of a ferromagnetic transition, we will use two coupled equations for electron number andmagnetization, n=n ↑+n↓,m=n↑−n↓, (16) where n±σare given by nσ=1 N/summationdisplay k/integraldisplay∞ −∞Sσ k(ε)fσ(ε)dε, (17) where Sσ k(ε) is the spectral density and fσ(ε) is the Fermi function with the exchange field Finnσcoming from the lastterm in the Hamiltonian ( 1), Sσ k(ε)=−1 πImGσ k(ε), (18) fσ(ε)=1 1+exp[(ε−μ−Finnσ)/kBT]. The spectral density function depends on the intersite correlation functions Iσand/angbracketleftˆnlσc+ l−σci−σ/angbracketright. Parameter Iσ defined by Eq. ( 10) after transforming to momentum space can be written as Iσ=−1 N/summationdisplay k(εk−T0)/integraldisplay∞ −∞Sσ k(ε)fσ(ε)dε. (19) The average /angbracketleftˆnlσc+ l−σci−σ/angbracketrightis calculated using the com- mutator [ H,c+ l−σ]−(as in Ref. 42), which in the case of our Hamiltonian (1) leads to the expression /angbracketleftˆnlσc+ l−σci−σ/angbracketright=1 U/braceleftBigg /angbracketleft[H,c+ l−σ]−ci−σ/angbracketright−T0/angbracketleftc+ l−σci−σ/angbracketright+/summationdisplay mtlm/angbracketleftc+ m−σci−σ/angbracketright −/summationdisplay m/Delta1tlm[/angbracketleftˆnlσc+ m−σci−σ/angbracketright+/angbracketleft ˆnmσc+ m−σci−σ/angbracketright+/angbracketleftc+ lσcmσc+ l−σci−σ/angbracketright+/angbracketleftc+ mσclσc+ l−σci−σ/angbracketright]/bracerightBigg . (20) 064410-4HUBBARD III APPROACH WITH HOPPING ... PHYSICAL REVIEW B 83, 064410 (2011) The four operators’ averages appearing in the preceding term of the hopping interaction are new with respect to Ref. 42.T h e importance of this interaction was pointed out in the HF approximation by Refs. 16and17. The averages present in the /Delta1t interaction term are of a higher level with respect to the averages preceding them in Eq. ( 20); therefore, we could approximate them as follows: /angbracketleftˆnlσc+ m−σci−σ/angbracketright≈nσ/angbracketleftc+ m−σci−σ/angbracketright,/angbracketleftˆnmσc+ m−σci−σ/angbracketright≈nσ/angbracketleftc+ m−σci−σ/angbracketright, (21) /angbracketleftc+ lσcmσc+ l−σci−σ/angbracketright≈/angbracketleftc+ lσcmσ/angbracketright/angbracketleftc+ l−σci−σ/angbracketright,/angbracketleftc+ mσclσc+ l−σci−σ/angbracketright≈/angbracketleftc+ mσclσ/angbracketright/angbracketleftc+ l−σci−σ/angbracketright. Using Eqs. ( 21)i nE q .( 20), we can write that 1 N/summationdisplay li(−tli)/angbracketleftˆnlσc+ l−σci−σ/angbracketright=1 U/braceleftBigg 1 N/summationdisplay li(−tli)/angbracketleft[H,c+ l−σ]−ci−σ/angbracketright+T0 N/summationdisplay litli/angbracketleftc+ l−σci−σ/angbracketright−1 N/summationdisplay ilmtli(tlm−/Delta1tlm2nσ)/angbracketleftc+ m−σci−σ/angbracketright +1 N/summationdisplay ilmtli/Delta1tlm(/angbracketleftc+ lσcmσ/angbracketright+/angbracketleftc+ mσclσ/angbracketright)/angbracketleftc+ l−σci−σ/angbracketright/bracerightBigg . (22) From the Green’s-function equation of motion ( 2) and the Zubarev relation,44 /angbracketleft[H,c+ l−σ]−ci−σ/angbracketright =−1 π/integraldisplay∞ −∞Im/angbracketleft/angbracketleftci−σ;[H,c+ l−σ]−/angbracketright/angbracketrightεf−σ(ε)dε, (23) we obtain 1 N/summationdisplay il(−tli)/angbracketleft[H,c+ l−σ]−ci−σ/angbracketright =1 N/summationdisplay k(εk−T0)/integraldisplay∞ −∞εS−σ k(ε)f−σ(ε)dε. (24) The second term in Eq. ( 22) has the following form: T0 N/summationdisplay litli/angbracketleftc+ l−σci−σ/angbracketright=T0I−σ. (25) The third term in Eq. ( 22) can be written as 1 N/summationdisplay ilmtli(tlm−/Delta1tlm2nσ)/angbracketleftc+ m−σci−σ/angbracketright =(1−2γnσ)1 N/summationdisplay k(εk−T0)2/integraldisplay∞ −∞S−σ k(ε)f−σ(ε)dε, (26) and the fourth term is given by 1 N/summationdisplay ilmtli/Delta1tlm(/angbracketleftc+ lσcmσ/angbracketright+/angbracketleftc+ mσclσ/angbracketright)/angbracketleftc+ l−σci−σ/angbracketright=− 2γIσI−σ. (27) Using the preceding results in Eq. ( C5), we obtain SB σ(ε/prime)=/braceleftBigg 2−γ U1 N/summationdisplay k(εk−T0)/integraldisplay∞ −∞[ε−(1−2γnσ) ×(εk−T0)]S−σ k(ε)f−σ(ε)dε +/bracketleftbigg2−γ U(T0−2γIσ)+1/bracketrightbigg I−σ/bracerightbigg Fσ H,0(ε/prime)C(ε/prime). (28)To calculate numerically the value of expressions ( 17), (19), and ( 28), we will use the relation 1 N/summationdisplay kfk(ε)=/integraldisplay∞ −∞ρ0(ε0)f(ε,ε0)dε0, (29) where ρ0(ε) is the initial DOS. For the initial DOS, we will assume the formula ρ0(ε)=1+/radicalBig 1−a2 1 πD√ D2−ε2 D+a1ε, (30) with the asymmetry parameter a1varying continuously from a1=0 corresponding to a symmetric semielliptic band (or Bethe lattice) to a1≈1 corresponding to the fcc lattice15 [see Fig. 1(a)]. Another DOS that will be used has the form ρ0(ε)=C(a2) D√ D2−ε2 D+a2(|ε|−D), (31) C(a2)=a2/2 1+(a2−1)π 2a2+√2a2−1 a2log1−a2 √2a2−1+a2, which for a2=0 is again the semielliptic DOS, and for a2→1 has a strong singularity at the center ( ε=0) resembling the bcc DOS [see Fig. 1(b)]. The preceding two types of DOS are used in this paper because at a1,a2→1 they represent 3 dtransition magnetic elements, which have fcc or bcc crystal structures. Theyboth have a strong peak within the DOS, and the fcc-type DOS has, in addition, a strong asymmetry-helpingferromagnetism. 15 IV . NUMERICAL RESULTS We apply the formalism developed here to analyze the magnetic ordering in the electron bands with symmetrical DOS, that is, the semielliptic DOS given by Eq. ( 30) with a1=0, or the bcc-like DOS given by Eq. ( 31) with a2/negationslash=0, and also in the electron bands with the asymmetric fcc-likeDOS where the maximum density is shifted toward their edge[Eq. ( 30) with a 1/negationslash=0]. Our main test of the ferromagnetic transition is a condition for the value of the critical on-siteexchange field F cr into drop to zero. In general, it is enough 064410-5GRZEGORZ G ´ORSKI AND JERZY MIZIA PHYSICAL REVIEW B 83, 064410 (2011) 0123 ]D/1[a =0.971 a= 0 . 91 a= 0 . 51 a= 01 -1 0 1 [D]-0.5 0.50123 ]D/1[a =0.972 a= 0 . 92 a= 0 . 52 a= 02(a) (b) FIG. 1. Model densities of states given by Eq. ( 30) (a) and Eq. ( 31) (b) shown for different values of parameters a1anda2. for ferromagnetism to have Fcr in<F in, where Finis the constant set by the material, but in this paper we imposethe rather rigorous condition for F into be zero. The critical Fcr inis calculated from equations for electron concentration and magnetization [Eqs. (16) and (17)] in the limit ofm→0. In Fig. 2, we present the dependence of F cr inon electron concentration for the symmetrical semielliptic DOS ( a1=0) in the strong correlation case U=15D. We compare the cases of the Hubbard III scattering effect (CPA) and the Hubbard IIIfull approximation (HIIIF) (with the resonance broadeningeffect included). Both cases are calculated with and without theintersite correlation. The effect of intersite kinetic correlationis reduced to zero when the lower Hubbard band is closed,n=1. In general, the curves calculated by means of the scattering effect lie lower than those for the full approxi-mation, which involves also the resonance broadening effectallowing for the −σelectrons to move through the lattice. Apparently, the more self-consistency we add to the solution,the farther away we are from spontaneous ferromagnetism.The curves with added intersite correlation (overlooked inthe original Hubbard solution) favor ferromagnetism, but donot create spontaneous magnetization without changing theDOS or adding the hopping interaction (see later in thepaper). In Fig. 3, we present the dependence of F cr inon electron concentration for the same case of the symmetrical semiellipticDOS ( a 1=0) and the strong correlation case U=15Das in0 0.25 0.5 0.75 1 n00.511.52 ]D[ Fnirc]D[ Fnirc FIG. 2. Critical value of the on-site exchange interaction as a function of electron concentration calculated in the Hubbard III scattering effect approximation (CPA) with the intersite correlation(thin solid line) and without it (thin dashed line). The same curves calculated in the Hubbard III full approximation with the intersite correlation (thick solid line) and without it (thick dashed line). Thesemielliptic DOS is used. The Coulomb repulsion U=15D, and the bandwidth D=1 eV. Fig. 2. We added the hopping interaction treated in the full Hubbard III approximation. The exchange field Fcr inrequired for ferromagnetism is strongly reduced under the influence ofhopping interaction as compared to the result obtained from thesame full Hubbard III approximation applied only to the on-siteUinteraction. The reduction of the exchange field F cr inby the hopping interaction is not sufficient to drive the transition tothe ferromagnetic state in this case of the semielliptic DOS,since this DOS is relatively flat and does not have strongpeaks. It has been shown within different approaches that the mod- erately strong peak in DOS enables ferromagnetism. 15,38–40 Therefore, in Fig. 4we show the dependence Fcr in(n) for such a DOS described by Eq. ( 31). This symmetrical DOS has a peak in the center that grows with increasing parametera 2[see Fig. 1(b)] and it resembles the bcc DOS. Figure 4 0 0.25 0.5 0.75 1 n00.511.52 ]D[ Fnirc]D[ Fnirc FIG. 3. Critical value of the on-site exchange interaction as a function of electron concentration calculated for the semielliptic DOS at different values of the hopping parameter γ. The Coulomb repulsion U=15Dand the bandwidth D=1 eV. The original Hubbard III solution (without the intersite correlation and without the hopping interaction) is shown as the dot-dashed line. 064410-6HUBBARD III APPROACH WITH HOPPING ... PHYSICAL REVIEW B 83, 064410 (2011) 0 0.25 0.5 0.75 1 n00.511.52 ]D[ Fnirc]D[ Fnirc -0.5 FIG. 4. Critical value of the on-site exchange interaction as a function of electron concentration calculated for the symmetricalDOS given by Eq. ( 31) with a 2=0.9, at different values of the hopping parameter γ. The Coulomb repulsion U=15Dand the bandwidth D=1 eV. The original Hubbard III solution (without the intersite correlation and without the hopping interaction) is shown for a2=0.9 as the dot-dashed line. shows that at a2=0.9 and rather large γ=0.4,we obtain spontaneous ferromagnetism. Minima of Fcr incorrespond to electron concentrations at which the Fermi energy is localizedclose to the peak in the DOS. V ollhardt and co-workers 15postulated that the afore- mentioned peak in the DOS is particularly supportive forferromagnetism when it is located away from the center ofthe band. Therefore, we performed numerical calculationsfor the asymmetric DOS of Eq. ( 30) resembling the fcc DOS, with a 1=0.7 and 0 .9[ s e eF i g . 1(a)]. In Fig. 5,w e present the dependence Fcr in(n) for this DOS. For the DOS witha1=0.7,we obtain spontaneous ferromagnetism at 0 0.25 0.5 0.75 1 n00.511.52 ]D[ Fnirc -0.5a = 0.71 a = 0.71 a = 0.71 a = 0.91 FIG. 5. Critical value of the on-site exchange interaction as a function of electron concentration calculated for the asymmetric DOS g i v e nb yE q .( 30) with a1=0.7o r0 .9, and for different values of the hopping parameter γ. The Coulomb repulsion U=15Dand the bandwidth D=1 eV. The original Hubbard III solution (without the intersite correlation and without the hopping interaction) is shown for a1=0.7as the double dot-dashed line.-1 0 1 14 15 16 13 [D]n= 0.55 m= 0.178 00.20.40.6 ]D/1[00.20.40.6 ]D/1[n= 0.55 m= 0 00.20.40.6 ]D/1[n= 0.55 m= 0(a) (b) (c) FIG. 6. Quasiparticle density of states as a function of energy for different values of the hopping parameter γ. In the calculations, we have used the asymmetric DOS with a1=0.7. Other parameters are U=15DandD=1 eV. Part (c) shows the magnetic solution. rather high values of the hopping parameter γ∼0.4 and electron concentrations around n≈0.6. The ferromagnetism exists in a much broader range of electron concentrations inthe case of larger asymmetry ( a 1=0.9). The symmetrical DOS with the strong peak did not bring the ferromag-netism in such a broad range of electron concentrations (seeFig.4). In Fig. 6, we show the quasiparticle DOS as a function of energy for different values of the hopping parameter γ.T h e results show that after taking into account both the scatteringand the resonance broadening corrections, the bandwidths ofthe lower and upper bands are equal to each other at any U. AtU/greatermuchD,we obtain two quasiparticle bands, with widths only slightly reduced with respect to the initial bandwidth 064410-7GRZEGORZ G ´ORSKI AND JERZY MIZIA PHYSICAL REVIEW B 83, 064410 (2011) of 2D. This is in contradiction to the well-known result of the scattering correction or the CPA, where the bandwidth ofthe lower σband was 2 D(1−n −σ)1/2and that of the upper band was 2 Dn1/2 −σ. The capacities of the lower and upper σ band are still 1 −n−σandn−σ,as in the CPA theory. In the case of ferromagnetic ordering, the effective bandwidthis the same for spins ↑and↓[Fig. 6(a)]. It is also distinct to the case of scattering correction, where for m/negationslash=0,bands with different spins had unequal widths given by the previousexpressions. In our model, the width of the spin bands does not depend on the strength of the hopping interaction. This interactionchanges the shape of the upper band. The shape of thelower band does not depend on the value of the hoppinginteraction, but rather on the carrier concentration. Thisbehavior is characteristic of the full Hubbard III approxima-tion. In this approximation, the resonance broadening effectcauses the dependence of /Sigma1 σself-energy on the itinerancy of −σelectrons through the self-energy /Sigma1−σ. Such a result goes against the intuitive but crude HF approximation,33in which it is clear that the intersite hopping interaction bringsboth the bandwidth change and the spin-dependent bandshift. In our approach, the most essential factor for obtaining ferromagnetism is the shift of the average energy of the spinbands while their position remains unchanged. This shift isdescribed by the factors S B σ(ε/prime) (depending on U) and S/Delta1t σ 0 0.25 0.5 0.75 1 n-0.2-0.100.10.2 ]D[ T1]D[ T114.85 14.8 14.75 14.6514.7]D[ T2]D[ T2 (a)(b) FIG. 7. Spin band center of gravity for different values of the hopping parameter γ. (a) The lower Hubbard band, (b) the upper Hubbard band. In calculations, the asymmetric DOS with a1=0.7 was used. Other parameters are U=15DandD=1 eV. The center for the original Hubbard III solution (without the intersite correlation)is shown as the dot-dashed line. In the magnetic state ( γ=0.4), the center of gravity is different for different spin directions (dotted and dashed lines).0 0.25 0.5 0.75 1 nm 00.4 0.20.6 m== n FIG. 8. Magnetization mas a function of band occupation n calculated for different values of the hopping parameter and for the asymmetric DOS [Eq. ( 30)] with a1=0.7. Other parameters are U=15DandD=1 eV. The dashed line is the line of the saturated ferromagnetism. (depending on /Delta1t) created by intersite correlation in the presence of the interactions mentioned earlier. In Fig. 7,w e show the center of gravity of the bands for different spins forthe lower and upper bands. They are defined as T iσ=/integraldisplayεmax,i εmin,iερi σ(ε)dε, (32) where i=1,2 stands for the lower and upper bands, respec- tively. Quantity ρi σ(ε) is the density of states in the ith band andεmin,i(εmax,i) denotes the lower and upper boundary of theith band. In the lower band, both factors SB σ(ε/prime) and S/Delta1t σshift the center of gravity toward higher energies. In the upper band, theeffective shift of the mean energy is the result of competitionbetween S B σ(ε/prime) and S/Delta1t σ, and it is toward lower energy. At the transition to ferromagnetism ( m/negationslash=0), the boundaries of the spin bands remain unchanged. The shift of the average m=n m 00.4 0.20.6 0 0.25 0.5 0.75 1 n FIG. 9. Magnetization mas a function of band occupation n calculated for different values of the hopping parameter and for theasymmetric DOS [Eq. ( 30)] with a 1=0.9. Other parameters are U=15DandD=1 eV. The dashed line is the line of the saturated ferromagnetism. 064410-8HUBBARD III APPROACH WITH HOPPING ... PHYSICAL REVIEW B 83, 064410 (2011) energy is caused by deformation of the density of states, unlike in the case of the conventional Stoner magnetism. Withoutthis average energy shift, caused by the intersite correlationscombined with the hopping interaction [see Eq. ( 9)], the ferromagnetic transition in this full Hubbard III approximationwould be impossible. In Fig. 8, we show the magnetic moment mas a function of band filling for the asymmetric DOS with a 1=0.7 and different values of the hopping parameter. It can be seen thatthe value of the magnetic moment is relatively low with respectto the saturation moment, m sat=n. For concentrations higher than concentration corresponding to mmax[n>n (mmax)], we have two solutions for mat one concentration. This is the evi- dence of a first-order phase transition. For n<n (mmax), there is only one solution and the transition is of second order. Thesame relation m(n) is shown in Fig. 9for the more asymmetric DOS with a 1=0.9. The spontaneous magnetization exists in a broader interval of electron occupations and for smaller valuesof the hopping interactions. V . CONCLUSIONS In this work, we have extended the classic Hubbard III approximation to include the intersite correlations and thehopping interaction. We have also changed the approximationfrom the CPA-like solution to the full Hubbard III scheme,taking into account the resonance broadening effect andallowing for the itinerancy of −σelectrons. These changes have led to two noteworthy effects: (i) Spin-dependent average band energy shift described by the parameters S B σ(ε/prime) and S/Delta1t σ. This band shift is really a distortion of band shape, which leads to the change in the bandcenter of gravity. It is expressed by the parameters S B σ(ε/prime) and S/Delta1t σ, and is different for both spin bands in the ferromagnetic case. At the same time, despite the shift in the center of gravity,the bands’ boundaries remain unchanged. (ii) The wave-vector dependence of the self-energy /Sigma1 σ 1,k(ε) created by the hopping interaction. This dependence causesshape distortion of both Hubbard bands. The DOS shapedistortion for the lower band is smaller than that for the upperband. The hopping interaction, which changes the shape of theHubbard bands, influences their widths only to a small degree. In effect, the bandwidths in the full Hubbard III approxi- mation remain almost constant. This result is in sharp contrastto the result of the scattering correction or the equivalent CPAapproximation, where in the case of strong correlation, theon-site Coulomb interaction Ualone changes the widths of the lower and upper Hubbard band to 2 D(1−n −σ)1/2and 2Dn1/2 −σ,respectively. We applied our model to the analysis of magnetic ordering in the system of interacting electrons. The numerical analysishas shown that including the intersite correlations and thehopping interaction reduces greatly the exchange field nec-essary for ferromagnetic ordering. In the systems with flatDOS such as the semielliptic DOS, the exchange field is notreduced to zero. In effect, the system does not undergo aspontaneous transition to ferromagnetism. The systems thatcontain even a moderately strong peak in the DOS may havethis transition at some electron concentrations and at relativelyhigh values of the hopping interaction. The main drivingforce for the magnetic transition is the shift in the centers of gravity of majority and minority spin bands producedby their distortion and described by the parameters S B σ(ε/prime) andS/Delta1t σ. At the same time, the positions and widths of spin bands remain roughly unchanged. The wave-vector-dependentself-energy /Sigma1 σ 1,k(ε) also does not change band boundaries. It causes band deformation different from the parametersS B σ(ε/prime) and S/Delta1t σ. It raises the maximum of DOS, narrows it (especially the upper band), and at the same time leaves theband boundaries roughly unchanged. Such changes to the DOShelp the ferromagnetic ordering, but the /Sigma1 σ 1,k(ε) influence is not a key factor for ferromagnetic alignment. We have shownin our previous paper 37that/Sigma1σ 1,k(ε), without the band shift caused by SB σ(ε/prime), does not cause the ferromagnetism for the fcc-type DOS. Herrmann and Nolting,46using the SDA approach in 3D, reached a similar conclusion. They showedthat for the simple-cubic lattice, the full self-energy with thek-dependent part favors ferromagnetism more strongly than the local self-energy, but for the bcc and fcc lattices, thecontribution of the k-dependent part of the self-energy to ferromagnetism is very small. Therefore, they recognized thatwith the increasing number of nearest neighbors, the impor-tance of the k-dependent part of the self-energy decreases rapidly. Momentum-dependent self-energy was also obtained in the cluster dynamical mean-field theory (CDMFT) 47and the dynamical cluster approximation (DCA).48,49The DCA and CDMFT are two generalizations of the DMFT methodto finite clusters that take into account short-range spatialcorrelations by adding the k-dependent term to the self-energy. These two methods were applied to the antiferromagneticordering and the d-wave pairing superconductivity (see, e.g., Ref. 50and references therein) since this superconductivity shows a strong momentum dependence of the self-energy asinferred from photoemission experiments. 51In the case of ferromagnetic ordering, the role of the k-dependent part of the self-energy within the DMFT method is still not clear.Using the LDA +DMFT model, Chioncel et al. 52pointed out that the k-dependent part of the self-energy may not be the decisive factor for ferromagnetic alignment. This conclusionis in agreement with our present calculations, in which wehave obtained the k-dependent part of the self-energy directly from the full Hubbard III solution, which included the intersitekinetic correlations. In our previous paper, 37we overestimated the ferromagnetic effect by using the set of CPA-like equations, although theintersite correlations were added in the Green’s-functiondecoupling process. In this paper, we have used in analyticaland numerical calculations the full Hubbard III solution, whichincludes the scattering and the resonance broadening effect.This is an improved approach, in which the +σelectron moves in a sea of −σelectrons defrozen by the resonance broadening effect [see Hubbard III, Eqs. (56)–(59)]. That sea of−σelectrons was frozen in the CPA-like approach equivalent to the Hubbard III scattering correction [see his Eqs. (37)–(40)]used in our previous paper. 37 This improved approach has weakened the ferromagnetic effect. The effect was brought back by the presence of thehopping interaction. The use of the hopping interaction /Delta1t for ferromagnetism has a potential of removing a magnetic 064410-9GRZEGORZ G ´ORSKI AND JERZY MIZIA PHYSICAL REVIEW B 83, 064410 (2011) paradox that has persisted for a long time. It was shown within the mean-field approximation (Ref. 32) that the magnetic moment adjusted to the experimental value at low temperatureby fitting the intersite interaction, /Delta1t, will decrease with the temperature much faster than the moment adjusted by fittingthe Stoner field. Therefore, the Curie temperature in the modelwith interaction /Delta1twill be lower in comparison to the Stoner type of estimates and closer to the experimental value. 32This decrease of TC, together with the decrease by another intersite effect, namely the spin waves, will bring this temperature intoagreement with the experimental value. The metal-insulator transition could be analyzed along the lines of similar models, 53,54but one should use the relatively small Ucomparable with half-bandwidth D. The importance of adding the hopping interaction for the metal-insulator phasetransition was already stressed by Schiller, 55who obtained the hopping interaction and the k-dependent single-particle self-energy in the simple Hubbard-like two-band model andused it for the metal-insulator transition.In summary, this approach, which includes intersite cor- relations, enables ferromagnetism after taking into consid-eration the hopping interaction. This model, in which wehave the k-dependent self-energy, may be used to analyze the other interesting phenomena existing in the stronglycorrelated electron systems (e.g., the metal-insulator transitionin transition-metal compounds, superconductivity in high-temperature superconductors, the half-metallic ferromagnets,and heavy fermion substances). APPENDIX A: SCATTERING EFFECT The scattering effect is expressed by the functions /angbracketleft/angbracketleft(ˆnα i−σ− nα −σ)clσ;c+ jσ/angbracketright/angbracketrightεand/angbracketleft/angbracketleft(ˆnα i−σ−nα −σ)ˆnβ l−σclσ;c+ jσ/angbracketright/angbracketrightε. To derive an equation of motion for these functions, we neglectthe Green’s-function terms coming from the commutator[ˆn iσ,H]−that are responsible for the “resonance broad- ening” correction in the higher-order equations, and weobtain (ε/prime−εβ)/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbigˆnβ l−σclσ;c+ jσ/angbracketrightbig/angbracketrightbig ε=/parenleftbig/angbracketleftbigˆnα i−σˆnβ l−σ/angbracketrightbig −nα −σnβ −σ/parenrightbig δlj−/summationdisplay mtlm/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbigˆnβ m−σcmσ;c+ jσ/angbracketrightbig/angbracketrightbig ε +/summationdisplay m/Delta1tlm/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbigˆnβ l−σ(ˆnl−σ+ˆnm−σ)cmσ;c+ jσ/angbracketrightbig/angbracketrightbig ε. (A1) Approximating the last two functions in Eq. ( A1) and summing over β=±,we arrive at /angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig clσ;c+ jσ/angbracketrightbig/angbracketrightbig ε =−1 Fσ H,0(ε/prime)/summationdisplay mteff lm,σ/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig cmσ;c+ jσ/angbracketrightbig/angbracketrightbig ε,(A2) where teff lm,σ=tlm−2n−σ/Delta1tlmis the effective hopping inte- gral. Equation ( A2) is of the same type as Eq. ( 25)i nR e f . 2. Therefore, its solution is assumed to be the following: /angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig clσ;c+ jσ/angbracketrightbig/angbracketrightbig ε =−/summationdisplay mWσ lm,i(ε/prime)teff mi,σ/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig ciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε(A3) where Wσ lm,i(ε)=gσ lm(ε)−gσ li(ε)gσ im(ε) gσ ii(ε), (A4)gσ lm(ε)=1 N/summationdisplay kexp[ik·(Rl−Rm)] Fσ H,0(ε)−/parenleftbig εeff k,σ−T0/parenrightbig, (A5) andεeff k,σis the effective dispersion relation. Using Eqs. ( A1), (A2), and ( A3), we can write the expression /angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbigˆnβ l−σclσ;c+ jσ/angbracketrightbig/angbracketrightbig ε =−nβ −σFσ H,0(ε/prime) ε/prime−εβ/summationdisplay mWσ lm,i(ε/prime)teff mi,σ/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig ciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε. (A6) APPENDIX B: RESONANCE BROADENING EFFECT To find the functions /angbracketleft/angbracketleftˆnα lσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightεappearing in the resonance broadening effect, we derive the equation ofmotion ε/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε=δij/angbracketleftbigˆnα lσc± l−σc∓ i−σ/angbracketrightbig −ξαδlj/angbracketleftc+ lσc± l−σc∓ i−σciσ/angbracketright+T0/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε −/summationdisplay mtim/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σcmσ;c+ jσ/angbracketrightbig/angbracketrightbig ε∓/summationdisplay mtim/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ m−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε±/summationdisplay mtlm/angbracketleftbig/angbracketleftbigˆnα lσc± m−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε −ξα/summationdisplay mtlm/angbracketleft/angbracketleft(c+ lσcmσ−c+ mσclσ)c± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε+U/angbracketleftbig/angbracketleftbigˆnα lσˆn± lσc± l−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε +/summationdisplay m/Delta1tim/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ(c+ i−σcm−σ+c+ m−σci−σ);c+ jσ/angbracketrightbig/angbracketrightbig ε 064410-10HUBBARD III APPROACH WITH HOPPING ... PHYSICAL REVIEW B 83, 064410 (2011) ±/summationdisplay m/Delta1tim/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ(c+ iσcmσ+c+ mσciσ);c+ jσ/angbracketrightbig/angbracketrightbig ε ∓/summationdisplay m/Delta1tlm/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ(c+ lσcmσ+c+ mσclσ);c+ jσ/angbracketrightbig/angbracketrightbig ε+/summationdisplay m/Delta1tim/angbracketleftbig/angbracketleftbigˆnα lσ(nm−σ+ni−σ)c± l−σc∓ i−σcmσ;c+ jσ/angbracketrightbig/angbracketrightbig ε ±/summationdisplay m/Delta1tim/angbracketleftbig/angbracketleftbigˆnα lσ(nmσ+niσ)c± l−σc∓ m−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε∓/summationdisplay m/Delta1tlm/angbracketleftbig/angbracketleftbigˆnα lσ(nmσ+nlσ)c± m−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε +ξα/summationdisplay m/Delta1tlm/angbracketleft/angbracketleft(nm−σ+nl−σ)(c+ lσcmσ−c+ mσclσ)c± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε. (B1) To truncate the infinite set of equations, we assume the following approximations in the higher-order Green’s functions: /angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σcmσ;c+ jσ/angbracketrightbig/angbracketrightbig ε≈/angbracketleftbigˆnα lσc± l−σc∓ i−σ/angbracketrightbig /angbracketleft/angbracketleftcmσ;c+ jσ/angbracketright/angbracketrightε,/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ m−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε≈δlmnα σn±−σ/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε, /angbracketleftbig/angbracketleftbigˆnα lσc± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε≈nα σ/angbracketleft/angbracketleftc± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε,/angbracketleft/angbracketleft(c+ lσcmσ−c+ mσclσ)c± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε≈0, /angbracketleftbig/angbracketleftbigˆnα lσ(nm−σ+ni−σ)c± l−σc∓ i−σcmσ;c+ jσ/angbracketrightbig/angbracketrightbig ε≈2n−σ/angbracketleftbigˆnα lσc± l−σc∓ i−σ/angbracketrightbig /angbracketleft/angbracketleftcmσ;c+ jσ/angbracketright/angbracketrightε, /angbracketleftbig/angbracketleftbigˆnα lσ(nmσ+niσ)c± l−σc∓ m−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε≈2nσδlmnα σn±−σ/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε, (B2) /angbracketleftbig/angbracketleftbigˆnα lσ(nmσ+nlσ)c± m−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε≈2nσnα σ/angbracketleft/angbracketleftc± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε, /angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ(c+ i−σcm−σ+c+ m−σci−σ);c+ jσ/angbracketrightbig/angbracketrightbig ε≈/angbracketleftc+ i−σcm−σ+c+ m−σci−σ/angbracketright/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε, /angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ(c+ iσcmσ+c+ mσciσ);c+ jσ/angbracketrightbig/angbracketrightbig ε≈/angbracketleftc+ iσcmσ+c+ mσciσ/angbracketright/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε, /angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ(c+ lσcmσ+c+ mσclσ);c+ jσ/angbracketrightbig/angbracketrightbig ε≈/angbracketleftc+ lσcmσ+c+ mσclσ/angbracketright/angbracketleftbig/angbracketleftbigˆnα lσc± l−σc∓ i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε, /angbracketleft/angbracketleft(nm−σ+nl−σ)(c+ lσcmσ−c+ mσclσ)c± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε≈0. All the approximations without the intersite averages, /angbracketleftc+ i−σcj−σ/angbracketrightand/angbracketleftˆniσc+ i−σcj−σ/angbracketright, follow the line of Hubbard.2Terms with the intersite averages are the additional terms taking into account the intersite kinetic correlation. In effect, for the function /angbracketleft/angbracketleftc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε,we can find the expression /angbracketleft/angbracketleftc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε=−/summationdisplay m/negationslash=iW−σ lm,i(ε−±ε±∓ε/prime)teff im,−σ/angbracketleft/angbracketleft(ˆn± i−σ−n± −σ)ciσ;c+ jσ/angbracketright/angbracketrightε +/summationdisplay α=±/angbracketleftbigˆnα lσc± l−σc∓ i−σ/angbracketrightbig ε/prime−(ε±±ε−∓εα)Fσ H,0(ε)/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε. (B3) For the function /angbracketleft/angbracketleftˆnlσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε,we obtain the following result: /angbracketleft/angbracketleftˆnlσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε=/angbracketleftˆnlσc± l−σc∓ i−σ/angbracketright ε/prime−(ε±±ε−∓ε+)Fσ H,0(ε)/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε−nσFσ H,0(ε−±ε±∓ε±2z/Delta1tI −σ) ε/prime−(ε±±ε−∓ε+) ×/summationdisplay m/negationslash=iW−σ lm,i(ε−±ε±∓ε/prime)teff im,−σ/angbracketleft/angbracketleft(ˆn± i−σ−n± −σ)ciσ;c+ jσ/angbracketright/angbracketrightε. (B4) We derive now the equation of motion for the second type of functions: /angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε, appearing in the resonance broadening effect in Eq. ( 11), ε/angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε =T0/angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε−/summationdisplay mtim/angbracketleft/angbracketleftˆniσc± l−σc∓ i−σcmσ;c+ jσ/angbracketright/angbracketrightε∓/summationdisplay mtim/angbracketleft/angbracketleftˆniσc± l−σc∓ m−σciσ;c+ jσ/angbracketright/angbracketrightε ±/summationdisplay mtlm/angbracketleft/angbracketleftˆniσc± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε−/summationdisplay mtim/angbracketleft/angbracketleft(c+ iσcmσ−c+ mσciσ)c± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε +U/angbracketleft/angbracketleftˆniσˆn± lσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay m/Delta1tim/angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ(c+ i−σcm−σ+c+ m−σci−σ);c+ jσ/angbracketright/angbracketrightε ±/summationdisplay m/Delta1tim/angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ(c+ iσcmσ+c+ mσciσ);c+ jσ/angbracketright/angbracketrightε∓/summationdisplay m/Delta1tlm/angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ(c+ lσcmσ+c+ mσclσ);c+ jσ/angbracketright/angbracketrightε +/summationdisplay m/Delta1tim/angbracketleft/angbracketleftˆniσ(nm−σ+ni−σ)c± l−σc∓ i−σcmσ;c+ jσ/angbracketright/angbracketrightε±/summationdisplay m/Delta1tim/angbracketleft/angbracketleftˆniσ(nmσ+niσ)c± l−σc∓ m−σciσ;c+ jσ/angbracketright/angbracketrightε ∓/summationdisplay m/Delta1tlm/angbracketleft/angbracketleftˆniσ(nmσ+nlσ)c± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε+/summationdisplay m/Delta1tim/angbracketleft/angbracketleft(nm−σ+ni−σ)(c+ iσcmσ−c+ mσciσ)c± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε.(B5) 064410-11GRZEGORZ G ´ORSKI AND JERZY MIZIA PHYSICAL REVIEW B 83, 064410 (2011) Keeping in line with the previous approximations made in Eq. (B2), we assume in Eq. ( B5) the following approximations: /angbracketleft/angbracketleftˆniσc± l−σc∓ i−σcmσ;c+ jσ/angbracketright/angbracketrightε≈0, /angbracketleft/angbracketleftˆniσc± l−σc∓ m−σciσ;c+ jσ/angbracketright/angbracketrightε≈δlmnσn± −σ/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε, /angbracketleft/angbracketleftˆniσc± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε≈nσ/angbracketleft/angbracketleftc± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε, /angbracketleft/angbracketleft(c+ iσcmσ−c+ mσciσ)c± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε≈0, /angbracketleft/angbracketleftˆniσ(nm−σ+ni−σ)c± l−σc∓ i−σcmσ;c+ jσ/angbracketright/angbracketrightε≈0,(B6) /angbracketleft/angbracketleftˆniσ(nmσ+niσ)c± l−σc∓ m−σciσ;c+ jσ/angbracketright/angbracketrightε ≈2nσδlmnσn± −σ/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε, /angbracketleft/angbracketleftˆniσ(nmσ+nlσ)c± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε ≈2nσnσ/angbracketleft/angbracketleftc± m−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε,which leads to the following relation: /angbracketleft/angbracketleftˆniσc± l−σc∓ i−σciσ;c+ jσ/angbracketright/angbracketrightε =−nσFσ H,0(ε−±ε±∓ε/prime) ε/prime−(T0+Un∓σ)/summationdisplay m/negationslash=iW−σ lm,i(ε−±ε±∓ε/prime)teff im,−σ ×/angbracketleft /angbracketleft(ˆn± i−σ−n± −σ)ciσ;c+ jσ/angbracketright/angbracketrightε. (B7) APPENDIX C: SCATTERING AND RESONANCE BROADENING CORRECTIONS Now we insert functions appearing in the scattering correc- tion of Appendix A[Eqs. ( A3) and ( A6)] and the resonance broadening correction of Appendix B[Eqs. ( B3), (B4), and (B7)] into Eq. ( 11), obtaining (ε/prime−εα)/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε=nα −σ/parenleftbigg δij−/summationdisplay ltil/angbracketleft/angbracketleftclσ;c+ jσ/angbracketright/angbracketrightε/parenrightbigg +nα −σ/summationdisplay β=±(γα+γβ)/summationdisplay ltil/angbracketleftbig/angbracketleftbigˆnβ l−σclσ;c+ jσ/angbracketrightbig/angbracketrightbig ε +/bracketleftbig 1−Xα σ(ε/prime)/bracketrightbig λσ(ε/prime)/angbracketleftbig/angbracketleftbig/parenleftbigˆnα i−σ−nα −σ/parenrightbig ciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε−ξα/bracketleftbig 1−XB 1,−σ(ε/prime)/bracketrightbig λ−σ(ε/prime)/angbracketleft/angbracketleft(ˆn− i−σ−n− −σ)ciσ;c+ jσ/angbracketright/angbracketrightε −ξα/bracketleftbig 1−XB 2,−σ(ε/prime)/bracketrightbig λ−σ(ε++ε−−ε/prime)/angbracketleft/angbracketleft(ˆn+ i−σ−n+ −σ)ciσ;c+ jσ/angbracketright/angbracketrightε+ξαSB σ(ε/prime)/angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε, (C1) where λσ(ε/prime)=/summationdisplay lmtilWσ lm,i(ε/prime)teff mi,σ,Xα σ(ε/prime) =/summationdisplay β=±(γα+γβ)nβ −σFσ H,0(ε/prime) ε/prime−εβ, (C2) XB 1,−σ(ε/prime)=γnσFσ H,0(ε/prime)/bracketleftbigg1 ε/prime−(T0+Un+σ)+1 ε/prime−ε+/bracketrightbigg , (C3) XB 2,−σ(ε/prime) =γnσFσ H,0(ε++ε−−ε/prime)/bracketleftbigg1 ε/prime−(T0+Un−σ)+1 ε/prime−ε−/bracketrightbigg , (C4)and SB σ(ε/prime)=1 N/summationdisplay il(−til)[(2−γ)/angbracketleftˆnlσc+ l−σci−σ/angbracketright −/angbracketleftc+ l−σci−σ/angbracketright]Fσ H,0(ε/prime)C(ε/prime). (C5) To solve this equation, we will use the following Fourier transforms: /angbracketleft/angbracketleftciσ;c+ jσ/angbracketright/angbracketrightε=1 N/summationdisplay kGσ k(ε)e x p [ik·(Ri−Rj)], (C6)/angbracketleftbig/angbracketleftbigˆnα i−σciσ;c+ jσ/angbracketrightbig/angbracketrightbig ε=1 N/summationdisplay k/Gamma1α k,σ(ε)e x p [ik·(Ri−Rj)], where the functions /Gamma1α k,σ(ε) fulfill the relation /Gamma1− k,σ(ε)+/Gamma1+ k,σ(ε)=Gσ k(ε). (C7) Taking into account the preceding relations, we can write Eq. ( C1) in a final form as /bracketleftbiggε/prime−/Omega1tot σ(ε/prime)−ε/prime ++2n+ −σγ(εk−T0)n+ −σγ(εk−T0) n− −σγ(εk−T0) ε/prime−/Omega1tot σ(ε/prime)−ε/prime −/bracketrightbigg/bracketleftBigg /Gamma1+ k,σ(ε) /Gamma1− k,σ(ε)/bracketrightBigg =/bracketleftbiggn+ −σ n− −σ/bracketrightbigg/braceleftbig 1+(εk−T0)Gσ k(ε)−/Omega1tot σ(ε/prime)Gσ k(ε)/bracerightbig −/bracketleftbiggn+ −σ/Omega1+ σ(ε/prime) n− −σ/Omega1− σ(ε/prime)/bracketrightbigg Gσ k(ε)+/bracketleftbigg+1 −1/bracketrightbigg SB σ(ε/prime)Gσ k(ε); (C8) where ε/prime α=εα−/Omega1α σ(ε/prime),/Omega1α σ(ε/prime)=Xα σ(ε/prime)λσ(ε/prime), (C9) /Omega1tot σ(ε/prime)=λσ(ε/prime)+/bracketleftbig 1−XB 1,−σ(ε/prime)/bracketrightbig λ−σ(ε/prime)−/bracketleftbig 1−XB 2,−σ(ε/prime)/bracketrightbig λ−σ(ε++ε−−ε/prime). (C10) In the functions XB 1,−σ(ε/prime),XB 2,−σ(ε/prime),X+ σ(ε/prime), and X− σ(ε/prime) we replace their argument by ε/prime=ε±−S/Delta1t σ. 064410-12HUBBARD III APPROACH WITH HOPPING ... 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PhysRevB.96.165411.pdf

PHYSICAL REVIEW B 96, 165411 (2017) Interelectron interactions and the RKKY potential between H adatoms in graphene Pavel Buividovich,1,*Dominik Smith,2,†Maksim Ulybyshev,1,‡and Lorenz von Smekal2,§ 1Institut für Theoretische Physik, Universität Regensburg, 93053 Regensburg, Germany 2Institut für Theoretische Physik, Justus-Liebig-Universität Gießen, 35392 Gießen, Germany (Received 29 March 2017; revised manuscript received 19 September 2017; published 6 October 2017) We use first-principles quantum Monte Carlo simulations to study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between hydrogen adatoms attached to a graphene sheet. We find that the pairwise RKKYinteractions at distances of a few lattice spacings are strongly affected by interelectron interactions, in particular,the potential barrier between widely separated adatoms and the dimer configuration becomes wider and thus harderto penetrate. We also point out that antiferrromagnetic and charge density wave orderings have very differenteffects on the RKKY interaction. Finally, we analyze the stability of several regular adatom superlattices withrespect to small displacements of a single adatom, distinguishing the cases of adatoms which populate eitherboth or only one sublattice of the graphene lattice. DOI: 10.1103/PhysRevB.96.165411 I. INTRODUCTION Functionalization of graphene with hydrogen adatoms or other admolecules which produce resonant scattering centersis currently a subject of intense research. First of all, itprovides a way to create a band gap in graphene [ 1–3] with a possibility to tune it and even to return the material to the initialsemimetallic state [ 4]. Also the magnetic moments induced around hydrogen adatoms due to interelectron interactions[3,5,6] play an important role in spin-relaxation processes [ 7] and can be used to tune the magnetic properties of graphene. The spatial distribution of adatoms plays a crucial role in the properties of hydrogenated graphene. For instance, magneticmoments of adatoms placed at different sublattices are coupledantiferromagnetically, while adatoms placed sufficiently closeto each other on the same sublattice induce ferromagneticordering [ 3]. The stability (or instability) of these adatom configurations determines the magnetic properties of thefunctionalized material and might also explain the arrangementof hydrogen adatoms in regular superlattices observed inrecent experiments [ 8,9]. Especially important is the case of functionalized graphene on top of boron nitride [ 8], where hydrogen adatoms tend to occupy only one sublattice at specialplaces in a moiré structure, forming islands of graphane. The smallness of the pairwise elastic interaction of hydro- gen adatoms in graphene, which does not exceed ∼10 meV for distances larger than the interatomic spacing [ 10] and decays as r −3at large distances [ 11], suggests that the Ruderman-Kittel-Kasuya-Yosida (RKKY) contribution fromconduction electrons might dominate the interadatom inter-actions in graphene. The RKKY potential between pairs ofadatoms was studied in a number of papers starting from theseminal article [ 12]. Typically, some analytic approximations to the fermionic Green’s function in the presence of resonantscatterers are used [ 13–16] in order to calculate the forces acting between adatoms. Also a noninteracting tight-binding *pavel.buividovich@physik.uni-regensburg.de †dominik.smith@theo.physik.uni-giessen.de ‡maksim.ulybyshev@physik.uni-regensburg.de §lorenz.smekal@theo.physik.uni-giessen.demodel [ 17], and density functional theory (DFT) [ 18,19]w e r e used in subsequent calculations. However, the influence ofelectron-electron interactions on the RKKY potential was notstudied so far despite the fact that they are quite noticeable ingraphene. Even the DFT approach is known to underestimatethe effect of interelectron interactions. For instance, it stronglyunderestimates the gap size in hydrogenated graphene [ 2], which is strongly enhanced by interactions even at moderateconcentrations of adatoms, as suggested by the recent QMCstudy in Ref. [ 3]. The importance of the effects of interelectron interaction was also discussed in [ 20]. In this paper we report on a first-principles quantum Monte Carlo (QMC) study of the RKKY interaction betweenhydrogen adatoms in graphene, consistently taking into ac-count interelectron interactions. We first consider pairwiseinteractions and demonstrate that for small distances betweenadatoms interaction effects dominate over the effects of finitetemperature and finite hybridization terms. Then we considerthe stability of adatom superlattices with respect to smallshifts of a single adatom, finding the conditions for a dynamic stability of various superlattices with adatoms occupyingonly one or equally both sublattices. It will be shown thatthe stability conditions are substantially different from thosepreviously defined in Ref. [ 1]. The effect of antiferromagnetic (AFM) and charge density wave (CDW) ordering is alsodiscussed, revealing an important feature of CDW order: thepossibility to stabilize the superlattice configurations in whichonly one sublattice is occupied by hydrogen adatoms. II. NUMERICAL SETUP We describe electrons in the conduction band of graphene using the standard tight-binding Hamiltonian with nearest-neighbor hoppings on the hexagonal lattice and electrostaticinterelectron interactions: ˆH=/summationdisplay /angbracketleftx,y/angbracketright,σ−txy(ˆa† y,σˆax,σ+H.c.)+1 2/summationdisplay x,yVxyˆqxˆqy,(1) where/summationtext /angbracketleftx,y/angbracketrightand/summationtext x,ydenote summations over all pairs /angbracketleftx,y/angbracketrightof nearest-neighbor sites and over all sites x,yof the graphene honeycomb lattice respectively. ˆa† x,σ,ˆax,σare 2469-9950/2017/96(16)/165411(7) 165411-1 ©2017 American Physical SocietyBUIVIDOVICH, SMITH, ULYBYSHEV , AND VON SMEKAL PHYSICAL REVIEW B 96, 165411 (2017) the creation/annihilation operators for electrons with spin σ=↑,↓in carbon πorbitals, txyare hopping amplitudes, ˆqx=−1+/summationtext σˆa† x,σˆax,σis the charge operator at site x, and Vxyis the interelectron interaction potential. Some of the results presented in this work concern the noninteracting limit(V xy=0), in case of which the model ( 1), with or without adatoms, can be solved exactly. Periodic spatial boundary conditions are imposed as in Refs. [ 21–23]: (x1+L1,x2)→(x1,x2), (2) (x1,x2+L2)→(x1+L2/2,x2), (3) where L1andL2define the spatial size of the lattice. In this work we use two models of hydrogen adatoms on the graphene sheet. The first is the simple vacancy modeldescribing hydrogen adatoms as missing lattice sites in thetight-binding Hamiltonian ( 1), so that hopping amplitudes t xy are equal to zero for all neighbors yof the lattice sites xto which adatoms are attached. Away from adatoms, all hoppingamplitudes are t xy=t=2.7 eV. Furthermore, we assume that each adatom has zero charge. The second model is the full hybridization model [ 24], in which hybridization terms ˆHhyb.=γ/summationdisplay x∈H,σ(ˆa† x,σˆcx,σ+H.c.)+Ed/summationdisplay x∈H,σˆc† x,σˆcx,σ (4) are added to the tight-binding Hamiltonian ( 1), where γ=2.0t is the hybridization parameter, Ed=−0.06tis the electron energy for the adatom,/summationtext x∈Hdenotes summation over all lattice sites with hydrogen adatoms, and ˆc† x,σare the creation operators for electrons on adatoms. As the hybridization modelsuffers from a fermion-sign problem (discussed in more detailbelow), which prevents the application of QMC, we use thefull hybridization model only in the noninteracting limit. It isused, among other things, to verify the validity of the vacancymodel and in cases where the effect of interactions can bemodeled by an explicit antiferromagnetic or charge densitywave (CDW) mass term. In order to treat interelectron interactions, we use the Suzuki-Trotter decomposition followed by the Hubbard-Stratonovich transformation and represent the partition func-tionZ=exp(−ˆH/T ) at temperature Tas a path integral over Hubbard-Stratonovich fields φ x,τin Euclidean time τ∈ [0,T−1] which is discretized; see Refs. [ 22,23] (to maintain exact particle-hole and sublattice symmetries one can use anexponential transfer matrix for the fermions between adjacenttime slices [ 25]): Z=/integraldisplay Dφ x,τe−S[φx,τ]det(Me[φx,τ])det(Mh[φx,τ]),(5) where Me=∂τ−hxy−iφx,τδxyandMh=∂τ−hxy+ iφx,τδxyare the fermionic operators for electrons and holes respectively. The matrix of the single-particle tight-bindingHamiltonian h xyis identical for electrons and holes unless we introduce hybridization ( 4) or an additional mass term modeling a charge density wave. If the matrix hxyis the same, fermionic determinants for electrons and holes are complexconjugate: det(Me[φx,τ])det(Mh[φx,τ])=|det(M[φx,τ])|2, (6) and the weight for the Hubbard fields in ( 5) is real and positive, which is a necessary requirement for a stochastic samplingofφ x,τ. That this is not true in the case of hybridization is the principle reason why we can use only the vacancy modelin QMC calculations. Fortunately, as is demonstrated in thenext section, this describes hydrogen adatoms with reasonableprecision. We sample the fields φ x,τwith the (manifestly positive) weight proportional to the integrand in ( 5) using the hybrid Monte Carlo algorithm. For the interelectron interaction po-tential V xywe use the potentials calculated with the constrained RPA method [ 26] for suspended graphene (see [ 3,23]f o r details). Within the interacting tight-binding model the RKKY interaction is nothing but the fermionic Casimir potential. Fora pair of adatoms we calculate it as the free energy F xyof the electrons on the graphene lattice with adatoms at sites xandy [13,27]. In the absence of interelectron interactions we simply compute the corresponding single-particle energy levels /epsilon1xy with adatoms numerically and obtain Fxyup to an irrelevant constant F0from Fxy=−T/summationdisplay /epsilon1xyln(1+e−/epsilon1xy/T)+F0. (7) The free energy cannot be calculated directly in hybrid Monte Carlo simulations. To overcome this, we calculate thedifferences /Delta1F=F x+l,y−Fx,ybetween free energies for adatom positions which differ by a shift along one carbon-carbon lattice bond l. We represent this difference as an integral /Delta1F=−T/integraldisplay 1 0dα∂ αlnZα, (8) Zα=/integraldisplay Dφx,τe−S[φx,τ]|det(Mα[φx,τ])|2, (9) where Mαlinearly interpolates between fermionic operators with adatoms at positions xandy(atα=0) and x+landy (atα=1). Differentiating the path integral ( 9)f o rZαbyα, we express /Delta1Fas /Delta1F=−2T/integraldisplay1 0dα/angbracketleftbig ReTr/parenleftbig M−1 α∂αMα/parenrightbig/angbracketrightbig , (10) where the expectation value is calculated with the same path- integral weight as in ( 9). The matrix ∂αMαis very sparse, allowing for an efficient calculation of Tr( M−1 α∂αMα). The integral over αis calculated using the six-point quadrature rule with six values of α∈[0,1], including α=0 andα=1. The above can be extended with no additional complicationsto cases with more than two adatoms. III. PAIRWISE INTERACTIONS To study interadatom interactions in QMC simulations we use the simple vacancy model, since QMC has a fermion-sign problem for the hybridization model ( 4). For hydrogen adatoms the hybridization parameter γ 2/greatermuchEdtis sufficiently large, so that the corresponding sp3state of the carbon atom 165411-2INTERELECTRON INTERACTIONS AND THE RKKY . . . PHYSICAL REVIEW B 96, 165411 (2017) FIG. 1. Interaction of two adatoms calculated within the free tight-binding model on a lattice with 72 ×72 cells: (a) profile along zigzag direction (zoomed version in the inset); (b) two-dimensional (2D) profile of RKKY potential for hybridization model ( 4) without interelectron interactions at room temperature T=310 K. is effectively unavailable for pzelectrons [ 28] and the simple vacancy model is a good approximation to ( 4). In Fig. 1(a)we demonstrate that without interelectron interactions the RKKYpotentials are very similar for the hybridization model ( 4) and the vacancy model. In all cases, the pairwise interaction haswell-known features: alternating signs for different sublattices[13] and the order-of-magnitude enhancement at some dis- tances [clearly seen in Fig. 1(b)], at which the two adatoms induce midgap states with zero energy [ 17,18]. With reasonable computational resources, QMC simula- tions are limited to rather high temperatures in physical units(T=0.09 eV =1040 K here) which are still relatively small, however, compared to the typical energy scales of the interact-ing tight-binding model in Eq. ( 1). In Fig. 1we demonstrate that at least in the absence of interelectron interactions suchtemperatures indeed do not affect the qualitative featuresof the pairwise RKKY interactions. This suggests that wemay safely use the vacancy model for adatoms in QMC atT=0.09 eV =t/30. We use lattices with 24 ×24 cells, for which finite-volume effects are smaller than statistical errors. FIG. 2. Pairwise RKKY interaction of hydrogen adatoms in the interacting tight-binding model ( 1) compared with the noninteracting case. Zoomed version in the inset; adatoms were modeled asvacancies. FIG. 3. Free energy change of the superlattice system upon displacement of a single adatom (zoom-in and overview; simplevacancy model was used in both interacting and noninteracting case); (c) superlattice structure with the zigzag profile used in (a) and (b) indicated by the red arrow (from simulations of a 24 ×24 lattice at T=0.09 eV). In Fig. 2we illustrate the effect of interelectron interac- tions on the RKKY potential along the zigzag direction bycomparison with the free electrons. In the free case we use thesame vacancy model and exact numerical computation of thefree energy according to Eq. ( 7). The potential is particularly strongly modified at distances of three and four C-C bonds,while at distances larger then eight to nine C-C bonds thechange of potential is too small to detect it with QMC. Themain physical effect of the electron-electron interactions isthat the local minimum at a distance of three bonds disappearsand the potential barrier between widely separated adatomsand the global minimum corresponding to a dimer configura-tion becomes harder to penetrate. IV . SUPERLATTICES A. Interaction effects We now consider superlattices of regularly distributed hydrogen adatoms as examples of functionalization with afinite adatom concentration. To address superlattice stability,we consider the variation of free energy accompanying theshift of a single adatom from its regular position. First wecompare interacting and noninteracting profile of the freeenergy accompanying the shift of one adatom along the zigzagdirection. These profiles are shown on Figs. 3(a)and3(b) both for interacting and noninteracting tight-binding models. Wechose the system with 5.56% coverage of hydrogen adatomspopulating only one sublattice, as illustrated on Fig. 3(c). The vacancy model is used in both the interacting andnoninteracting case (to avoid a sign problem for the formerand make a direct comparison meaningful). First we note that the overall scale of the RKKY interaction for superlattices is enhanced in comparison with pairwiseinteractions, so that the RKKY potential for a single adatom(with all other adatoms fixed) becomes comparable with the 165411-3BUIVIDOVICH, SMITH, ULYBYSHEV , AND VON SMEKAL PHYSICAL REVIEW B 96, 165411 (2017) FIG. 4. Change of energy levels of Tamm states in presence of of AFM and CDW mass terms: We illustrate the case in which one adatom moves from one sublattice to the other. The dashed line corresponds to the Fermi level. diffusion barriers /Delta1U∼0.3...1.0 eV for hydrogen adatoms [29]. Surprisingly, for superlattices interelectron interactions do not change the RKKY potential qualitatively, despite inducinga very large gap /Delta1/epsilon1∼1 eV in the midgap energy band [ 3]. To understand this observation, we recall that interelectroninteractions induce global antiferromagnetic (AFM) orderingfor graphene with adatoms [ 3], with the effective mass term ˆM AFM=±m/summationtext x(ˆa† x↑ˆax↑−ˆa† x↓ˆax↓) which has alternat- ing signs on different sublattices. We can estimate the change in free energy upon the shift of a single adatom to the neighboringlattice site (and thus to another sublattice) assuming that (1)this change is determined mostly by Tamm states localizednear this adatom and (2) the energy of this Tamm state canbe estimated as the mass term at nearest-neighbor sites of theadatom (as the wave functions of the Tamm states are mostlylocalized on these sites). Since the AFM mass has differentsigns for different spin components, the states with differentspins simply exchange their energies, and the overall sumof energies in ( 7) does not change in such a “mean field” approximation (see Fig. 4for illustration). The same argument applied to charge density wave (CDW) ordering leads to a completely different conclusion. Sincein this case the mass term has the same sign for both spincomponents, the energies of two spin components no longercompensate each other, and the change of free energy uponadatom shift can be estimated as 2 m(see Fig. 4). Unfortunately, verification of this scenario in full QMC simulations is verydifficult, since a CDW mass term causes a sign problem: thefermionic determinant in ( 5) is no more positive definite [ 25]. But at least we can illustrate the effects of both CDW and AFMmass terms on the RKKY potential for free electrons. Theresults are shown in Figs. 3(a) and3(b).W eu s e m=0.5e V , which approximately corresponds to the AFM mass inducedby interelectron interactions for this concentration of defects[3]. We indeed see that while the non-interacting result with AFM mass m=0.5 eV almost coincides with the QMC result, the CDW mass term completely changes the RKKY potentialand the locations of its minima. We note, however, that the energy gap size of 0 .5e V is an overestimate for the real graphene on boron nitridesubstrate. To demonstrate this, we have performed MonteCarlo simulations on 6 ×6 lattices with the bare CDW mass 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -π 0 πDensity Arg(det Mel.Mh.) FIG. 5. The distribution of the phase factor in the path integral ( 5) for the calculation with CDW mass term mCDW=40 meV on the 6 × 6 lattice at temperature T=0.09 eV. Calculations were performed for superlattice of vacancies shown in Fig. 3(c). termmCDW=0.04 eV which is close to that in real graphene on boron nitride [ 30]. At nonzero mCDWthe fermionic determi- nants for particles and holes are no longer complex conjugateto each other, thus the relation ( 6) is no longer valid and the path-integral weight in the partition function ( 5) acquires some complex phase. We treat this complex phase using the brute- force reweighting. Namely, we sample the configurations of Hubbard-Stratonovich fields with the probability proportionalto the absolute value of the path-integral weight, and includethe complex phase into the expectation values of physicalobservables like ( 10). Due to the fact that changing AFM mass term to the CDW mass term of the same value changesonly the complex phase of the fermionic determinant butnot the absolute value, we were still able to represent theabsolute value of the product of two determinants in ( 5) as the square of the absolute value of a single (electron orhole) determinant, and apply the same hybrid Monte Carloalgorithm as for the AFM mass term. We used the same setupas for the superlattice shown in the plots on Figs. 3(a)–3(c), but reduced the overall lattice size to 6 ×6. Already at this small lattice size the complex phase exhibits strong oscillationswhich require a lot of statistics for reweighting. To illustratethese difficulties, in Fig. 5we demonstrate that the distribution of the complex phase in the path integrals ( 5) is nearly flat, thus phase cancellations between different configurations arevery important and require a very large number of MonteCarlo samples to resolve with good statistical accuracy. Thesedifficulties in reweighting limit the simulations to rather smallvalues of the CDW mass term m CDW and to small lattice sizes. Nevertheless, even on small lattices we can obtain a rough estimate of the influence of CDW mass term inducedby boron nitride on RKKY interaction potential. Using thereweighting technique we computed the change of the energyaccompanying the shift of one adatom from its regularsuperlattice position to the nearest-neighbor site. This shiftcorresponds to position 1 in the plots in Figs. 3(a) and3(b). The change of the energy is /Delta1F=−0.61±0.16 eV in the case of CDW mass while for zero mass the same calculationyields /Delta1F=−0.506±0.018 eV . We conclude that such a small bare CDW mass term is not enough to change the sign of 165411-4INTERELECTRON INTERACTIONS AND THE RKKY . . . PHYSICAL REVIEW B 96, 165411 (2017) the/Delta1F. More generally, this means that real graphene is rather far from CDW phase transition so that the renormalization ofthe corresponding mass term due to interelectron interactionsis not very significant. On the other hand, the appearance ofa large CDW mass term could still be expected if the ratiobetween on-site and nearest-neighbor electrostatic interactionpotentials could be tuned to favor the CDW ordering [ 25]. B. Dynamic stability of superlattices In order to address the dynamic stability of superlattice configurations with only one or both sublattices populatedby adatoms, we use the hybridization model ( 4) without interelectron interactions. This seems fairly well justifiedbecause these interactions do not qualitatively change theRKKY potential and their quantitative effect can be mimickedby an explicit AFM mass quite well [see Fig. 3(b)]. Previously this subject was studied in the papers [ 1,14–16]. However, the free energy of the system with defects was calculated in[14–16] using some approximation for the fermionic propaga- tor (stationary phase approximation) and for the free energyitself. For instance, only pairwise interactions were taken intoaccount in the Monte Carlo study of the dynamic stability ofvarious spatial configurations of defects in [ 16]. Moreover, in all these papers, the calculation of the full free energy was usedto study which spatial configuration of adatoms is energetically favorable. Randomly generated adatom configuration with equally/unequally populated sublattices were used in thosestudies. But these calculations do not in general imply thestability of superlattices with respect to adatom displacements.The reason is that the real global energy minimum is acollection of dimers due to very large pairwise RKKYinteraction at nearest-neighbor position (see Figs. 1and2), so there is no guarantee that a given spatial configuration ofadatoms will not change into a collection of dimers after a setof energetically favorable shifts of adatoms’ positions. For thisreason we study the change of the energy of superlattice after ashift of one adatom, which automatically changes the relativeoccupation of sublattices. Results of our calculations are presented in Fig. 6.W e observe that the superlattices with adatoms on a singlesublattice at half filling are dynamically unstable in all casesconsidered here due to the fact that a change of position of anadatom to the opposite sublattice is energetically favorablein any case. In contrast, superlattices of adatoms whichequally populate both sublattices are stable for low adatomconcentration [see the structure in Fig. 6(b)]. For higher adatom concentrations, all superlattices become unstable. Thisinstability is likely to lead to the formation of dimers with alarge binding energy. Since a finite chemical potential can change the sublattice preferences of the pairwise RKKY interaction [ 17–19], the stability region of superlattices with only one sublatticeoccupied by adatoms might start from some finite chemicalpotential rather than at half filling. In Fig. 7we show the minimal change of the free energy among the three possiblenearest-neighbor shifts as a function of the chemical potentialμfor the same six adatom configurations as in Fig. 6. One can see that the single-sublattice superlattice with the lowest con-centration of 3% adatoms (labeled “a”) does indeed become0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4(c) FIG. 6. Change of the free energy of superlattice systems upon the displacement of a single adatom. The fixed positions of other adatoms in the superlattices are marked with black dots. All plotscorrespond to half filling. On the left: superlattices with only one sublattice populated by adatoms. On the right: superlattices of the the same densities of adatoms, but with equally populated sublattices.Calculations were performed for free electrons in the tight-binding model with hybridization term ( 4). stable above μ≈0.25 eV. At about the same value of μthe corresponding superlattice with equal population of adatomson both sublattices on the other hand becomes unstable. Thestructures with equally populated sublattices are stable mainlynear half filling for low concentrations of adatoms ( /lessorequalslant3%) or in FIG. 7. The minimal change of the free energy accompanying the shift of a single adatom to the nearest-neighbor position as afunction of chemical potential. The alphabetic labels correspond to superlattices in Fig. 6. Negative values correspond to unstable superlattices. Calculations were performed for free electrons intight-binding model with hybridization term ( 4). 165411-5BUIVIDOVICH, SMITH, ULYBYSHEV , AND VON SMEKAL PHYSICAL REVIEW B 96, 165411 (2017) FIG. 8. Pairwise RKKY interaction depending of the chemical potential. Free fermions with hybridization model ( 4) was used in calculations. Lattice size is 72 ×72, results are shown for zero temperature. some region of finite dopings for larger adatom concentrations. The larger the concentration, the smaller is the stability region,and vice versa for the single-sublattice configurations: thedenser these adatom configurations get, the larger a chemicalpotential is needed to stabilize them. A similar change can be observed in pairwise RKKY interactions, where positions of barriers and local minimainterchange after some value of chemical potential. Thisphenomenon is illustrated in Fig. 8for the same model of free electrons with hybridization term ( 4). The only difference that the critical chemical potentials seems to be much smaller(0.2 eV vs 0.4 eV) for superlattices. V. CONCLUSIONS We have calculated the RKKY interaction potential be- tween hydrogen adatoms on a graphene sheet, taking intoaccount effects of electron-electron interactions in fully non-perturbative first-principles QMC simulations. In particular,we have studied both pairwise potentials and free-energy differences with stability analyses for various configurationsof finite adatom densities. For the pairwise RKKY potentialwe found that the interelectron interactions tend to increasethe potential barrier between widely separated adatom anddimer configurations which implies some suppression of dimerformation in the process of random deposition of adatoms ana graphene sheet. For finite adatom concentrations, we have demonstrated that charge-density formation (CDW) and antiferromagneticorder (AFM) in the ground state, whether induced by substratesleading to staggered on-site potentials or dynamically by theinterelectron interactions, have very different effects. Whilean AFM mass term does not qualitatively change the RKKY-type interaction, the effect of a CDW mass term can be muchmore significant and even influence the sublattice ordering ofadatoms. Our stability analyses of different hydrogen superlattices show that single-sublattice configurations of adatoms areunstable at half filling but can be stabilized by an appropriateamount of doping with chemical potentials |μ|>μ c.T h e critical value μcthereby increases with increasing adatom con- centrations. Superlattices with equally populated sublatticesare stable near half filling for low concentrations of adatoms.More densely populated superlattices are likely to be unstabletowards dimer formation. As further plans we would like also to mention the variety of rich RKKY-related physics in bilayer graphene [ 31]. Taking into account that interaction effects might be also importantin bilayer graphene [ 32], similar calculations for it are in our plans for future work. ACKNOWLEDGMENTS We thank M. I. Katsnelson for useful and motivating discussions. This work was supported by the DeutscheForschungsgemeinschaft (DFG), Grants No. BU 2626/2-1and No. SM 70/3-1. 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PhysRevB.90.195109.pdf

PHYSICAL REVIEW B 90, 195109 (2014) Block Lanczos density-matrix renormalization group method for general Anderson impurity models: Application to magnetic impurity problems in graphene Tomonori Shirakawa1,2and Seiji Yunoki1,2,3 1Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan 2Computational Materials Science Research Team, RIKEN Advanced Institute for Computational Science (AICS), Kobe, Hyogo 650-0047, Japan 3Computational Quantum Matter Research Team, RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Received 12 September 2014; revised manuscript received 25 September 2014; published 5 November 2014) We introduce a block Lanczos (BL) recursive technique to construct quasi-one-dimensional models, suitable for density-matrix renormalization group (DMRG) calculations, from single- as well as multiple-impurity Andersonmodels in any spatial dimensions. This new scheme, named BL-DMRG method, allows us to calculate notonly local but also spatially dependent static and dynamical quantities of the ground state for general Andersonimpurity models without losing elaborate geometrical information of the lattice. We show that the BL-DMRGmethod can be easily extended to treat a multiorbital Anderson impurity model where not only inter- andintraorbital Coulomb interactions but also Hund’s coupling and pair hopping interactions are included. We alsoshow that the symmetry adapted BL bases can be utilized, when it is appropriate, to reduce the computationalcost. As a demonstration, we apply the BL-DMRG method to three different models for graphene with astructural defect and with a single hydrogen or fluorine absorbed, where a single Anderson impurity is coupled toconduction electrons in the honeycomb lattice. These models include (i) a single adatom on the honeycomb lattice,(ii) a substitutional impurity in the honeycomb lattice, and (iii) an effective model for a single carbon vacancyin graphene. Our analysis of the local dynamical magnetic susceptibility and the local density of states at theimpurity site reveals that, for the particle-hole symmetric case at half-filling of electron density, the ground stateof model (i) behaves as an isolated magnetic impurity with no Kondo screening, while the ground state of theother two models forms a spin-singlet state where the impurity moment is screened by the conduction electrons.We also calculate the real-space dependence of the spin-spin correlation functions between the impurity site andthe conduction sites for these three models. Our results clearly show that, reflecting the presence or absence ofunscreened magnetic moment at the impurity site, the spin-spin correlation functions decay as ∝r −3, differently from the noninteracting limit ( ∝r−2), for model (i) and as ∝r−4, exactly the same as the noninteracting limit, for models (ii) and (iii) in the asymptotic r,w h e r e ris the distance between the impurity site and the conduction site. Finally, based on our results, we shed light on recent experiments on graphene where the formation of localmagnetic moments as well as the Kondo-like behavior have been observed. DOI: 10.1103/PhysRevB.90.195109 PACS number(s): 71 .15.−m,73.22.Pr,75.20.Hr I. INTRODUCTION Recently, magnetic properties of graphene monolayers have attracted much attention [ 1–4]. Because of the characteristic electronic band structure with Dirac-like linear dispersionsnear the Fermi level ( E F) and the resulting “V-shape” elec- tronic density of states, ρ(ω)∼|ω|,a tEF[2,4,5], a unique electronic and magnetic behavior is expected in graphene.Among many, very recent experiments have revealed that hydrogen or fluorine adatoms as well as vacancies in graphene can induce magnetic moments with spin 1 /2 per adatom or vacancy [ 6,7], although pristine graphene is diamagnetic. Moreover, some experiments have observed the Kondo-like signature in the temperature dependence of the resistivity whenthe vacancies are introduced in graphene [ 8], even though the other experiments have found otherwise [ 6]. The early theoretical studies have considered a single mag- netic impurity coupled to the graphene conduction electrons and found that the magnetic impurity is completely isolated, i.e., unscreened by the conduction electrons, when the modelpreserves the particle-hole symmetry, while the magnetic moment can be screened when the model is strongly particle- hole asymmetric [ 9–15]. These theoretical results appear to contradict the experimental observation reported in Ref. [ 8],where the Kondo temperature is found to be symmetric with respect to the applied gate voltage, which changes the chemical potential of the conduction electrons. Thereforethe experiments indicate that the graphene with vacancies is close to the particle-hole symmetric point. However, the early theoretical studies predict no Kondo screening in this limit[9–15]. Motivated by these experiments [ 6–8], several models have been recently proposed to explain the origin of magneticmoment and the possible Kondo-like effect in graphene withstructural defects or adatoms [ 15–21]. One of the possible explanations of the emergent magnetic moment in graphenewith a structural defect is due to the partially filled danglingbonds of sp 2orbital on carbon atoms surrounding the vacancy [19–25]. It has been also pointed out that the scattering of defects drastically changes the electronic structures of πband and produces the logarithmic divergence at EFin the local density of states at the vicinity of defects [ 21,26,27]. Therefore a nonperturbative real-space theoretical approach that canincorporate the elaborate lattice geometry is highly desirable tounderstand the magnetic properties of graphene with structuraldefects or adatoms. The interests in the real-space aspects of magnetic im- purities is not only to study geometrically different lattice 1098-0121/2014/90(19)/195109(21) 195109-1 ©2014 American Physical SocietyTOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) structures of various systems but also to directly capture the real-space nature of the ground state, e.g., the spatialdistribution of “Kondo cloud” in the Kondo singlet state[28,29]. Indeed, as compared to the thermodynamics and the transport properties, the real-space nature of Kondo problemhas been much less studied both experimentally and theoreti-cally. However, very recently, scanning tunneling spectroscopyexperiments have successfully observed the long-range Kondosignature for single magnetic atoms of Fe and Co in a Cu(110)surface and found that the Kondo cloud seems rather spatiallyextended away from the magnetic atoms [ 30]. The recent experimental progress further encourages us to study themagnetic impurity problems and the Kondo physics in realspace. Theoretically, on the other hand, it is still difficult to study the real-space properties simply because the analyticalapproaches available are rather limited and also because eventhe most powerful and well accepted numerical method formagnetic impurity problems, i.e., numerical renormalizationgroup (NRG) method [ 31–33], can not treat the real-space dependence directly, where the high-energy scales are in-tegrated out by using logarithmic discretization of energy.Quantum Monte Carlo (QMC) methods can calculate spatiallydependent quantities [ 34–36]. However, the QMC calculations often suffer the negative sign problem at low tempera-tures and can not be applied to general Anderson impurity models. In the last decade, the density-matrix renormalization group (DMRG) method [ 37–40] has been successfully used to investigate limited properties of single- and multiple-impurityAnderson models. For example, the DMRG method has beenapplied to the single- and two-impurity Anderson/Kondomodels in one dimension to study correlation effects inthe conduction sites [ 41–43] and to evaluate the Kondo screening length [ 44–46]. Moreover, the DMRG method has been employed to address single- and multiple-impurityAnderson models in more than one dimension [ 47–49] and also applied as an impurity solver for the dynamical mean-fieldtheory (DMFT) [ 50,51]. However, these approaches encounter difficulties in calculating the spatially dependent quantitiessuch as spin-spin correlation functions. Therefore it is highlydesired to develop new numerical methods, which can computedirectly various physical quantities in real space in any spatialdimensions. To overcome the difficulties, here we introduce a block Lanczos (BL) recursive technique, which constructs, withoutlosing any geometrical information of the lattice, quasi-one-dimensional (Q1D) models, suitable for DMRG calculations,from single- as well as multiple-impurity Anderson models inany spatial dimensions. This new approach, named BL-DMRGmethod, enables us to calculate various physical quantitiesdirectly in real space, including both static and dynamicalquantities, with high accuracy. Thus the BL-DMRG methodis in sharp contrast to the NRG method since the NRGmethod has a severe limitation in calculating the spatiallydependent quantities because the logarithmic discretizationin energy space has to be introduced to construct the Wilsonchain [ 33]. The BL-DMRG method is also superior to the QMC methods because the BL-DMRG method can be easilyextended to a more involved impurity model such as amultiorbital single-impurity Anderson model where inter- and intraorbital Coulomb interactions as well as Hund’scoupling and pair hopping interactions are included. There-fore the BL-DMRG method has potential as a promisingimpurity solver of DMFT for multiorbital Hubbard models[52]. To demonstrate the BL-DMRG method, we apply this method to three different models for graphene with a structuraldefect and with a single absorbed atom, where a singleAnderson impurity is coupled to the conduction electrons inthe honeycomb lattice. These models include (i) an Andersonimpurity absorbed on the honeycomb lattice (model I),(ii) a substitutional Anderson impurity in the honeycomblattice (model II), and (iii) an effective model for a singlecarbon vacancy in graphene (model III). Our results of the localmagnetic susceptibility and the local density of states at theimpurity site reveal that, for the particle-hole symmetric caseat half-filling of electron density, the ground state of modelI behaves as an isolated magnetic impurity with no Kondoscreening, while the ground state of models II and III formsa spin singlet state where the impurity moment is screenedby the conduction electrons. To understand the real-space spindistribution of the conduction electrons around the impurity,we subsequently calculate the spin-spin correlation functionsbetween the impurity site and the conduction sites and finda qualitatively different asymptotic behavior when compared with the noninteracting limit, which results from the different screening characteristics: the spin-spin correlations decay as∝r −3, different from the noninteracting limit ( ∝r−2), for model I and as ∝r−4, exactly the same as the noninteracting limit, for models II and III. We also discuss the relevanceof our results to the recent experiments on graphene withstructural defects and with hydrogen or fluorine adatoms wherethe formation of local magnetic moments and the Kondo-likebehavior have been observed [ 6–8]. The rest of this paper is organized as follows. First, we introduce the BL-DMRG method for general Andersonimpurity models and describe the details in Sec. II.T h eB L recursive technique is employed to construct Q1D models fromgeneral Anderson impurity models in any spatial dimensionsand for any lattice geometry without losing the structuralinformation in Sec. II A. To optimize the DMRG calculations for Q1D models constructed by the BL recursive technique,symmetrization schemes of the BL bases are described inSec. II B. The numerical technique to calculate spatially dependent quantities in real space away from the impuritysite is explained in Sec. II C. The extension of the BL-DMRG method and the symmetry adapted BL bases for a multiorbitalsingle-impurity Anderson model are provided in Sec. II D. The BL-DMRG method is then demonstrated in Sec. IIIfor single-impurity Anderson models. The three different single-impurity Anderson models for graphene witha structural defect and with a single adatom are introducedin Sec. III A . After briefly explaining the numerical details of the calculations for these models in Sec. III B, the nature of the ground state is examined by calculating the local magneticsusceptibility at the impurity site in Sec. III C 1 and the local electronic density of states at the impurity site in Sec. III C 2 . The spin-spin correlation functions between the impuritysite and the conduction sites are evaluated in Sec. III C 3 . 195109-2BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) The relevance of our results to the recent experiments on adatoms or vacancies in graphene is discussed in Sec. IV.T h e possible further extension of the BL-DMRG method is alsobriefly discussed. The detailed derivation of the hybridizationfunction for general Anderson impurity models is described inAppendix. II. BL-DMRG METHOD In this section, we introduce the BL-DMRG method for general Anderson impurity models in any spatial dimensionsand for any lattice geometry. To this end, first we describein Sec. II A the BL recursive technique which enables us to transform exactly a general Anderson impurity model to aQ1D model without losing any geometrical information ofthe lattice. Once a Q1D model is constructed, we can usethe DMRG method to calculate both static and dynamicalquantities with extremely high accuracy. We then describe in Sec. II Btwo schemes to reduction the computational cost for DMRG calculations. One is to utilizethe lattice symmetry of the models to construct the symmetryadapted BL bases, which is similar to the one introduced inNRG calculations for multiimpurity problems [ 53–55]. The other is to use spin degrees of freedom to reduce the dimensionsof the local Hilbert space, which can be applied to moregeneral cases even if the models do not possess appropriatelyhigh lattice symmetry. The BL-DMRG procedure to calculatespatially dependent quantities such as spin-spin correlationfunctions is also explained in Sec. II C. Finally, the extension to a multiorbital single-impurity Anderson model is brieflydiscussed in Sec. II D. It should be emphasized that, although the BL-DMRG method shares some similarity with the NRG method [ 55], the BL-DMRG method can be readily extend to more generalmodels, one example discussed in Sec. II D, and has significant advantages in calculating spatially dependent quantities andalso in the computational cost by using the symmetry adaptedBL bases. We should also note that, very recently, the directapplication of a standard Lanczos technique to single-impurityAnderson and Kondo models [ 56] as well as its extension to a two-impurity Kondo model [ 57] have been proposed for DMRG calculations, which is somewhat similar to theBL-DMRG method introduced in this paper. However, weemphasize that the use of BL recursive technique in theBL-DMRG method significantly enlarges the applicability ofDMRG calculations not only to more general multiorbitalsingle- or multiple-impurity Anderson models but also to thecalculations of spatially dependent quantities. Moreover, asdescribed in Appendix, the BL bases representation of thehybridization function for general Anderson impurity modelsfurther enlarges the usefulness of the BL recursive techniquefor other numerical methods such as the QMC methods andthe NRG method. A. Q1D map of a general Anderson impurity model: a BL recursive technique We consider a general Anderson impurity model described by the following Hamiltonian: HAIM=Hc+Hd+HV+HU, (1)where Hc=/summationdisplay n,n/prime/summationdisplay σ/epsilon1c n,n/primec† n,σcn/prime,σ, (2) Hd=/summationdisplay m,m/prime/summationdisplay σ/epsilon1d m,m/primed† m,σdm/prime,σ, (3) HV=/summationdisplay m,n/summationdisplay σ(Vm,nd† m,σcn,σ+H.c.), (4) and HU=/summationdisplay m1,...,m 4/summationdisplay σ1,...,σ 4Uσ1σ2;σ3σ4 m1m2;m3m4d† m1,σ1d† m2,σ2 ×dm3,σ3dm4,σ4. (5) Here, c† n,σ(cn,σ) is the creation (annihilation) operator of an electron at site (or orbital) n(=1,2,..., N ) with spin σ (=↑,↓) in the conduction sites (or bands) and d† m,σ(dm,σ)i st h e creation (annihilation) operator of an electron at impurity sitei(=1,2,..., N i) and orbital α(=1,2,..., N d), denoted by m=(i,α)(=1,2,..., M , where M=NiNd) for simplicity, with spin σ. The individual terms, Hc,Hd,HV, and HU, describe the one-body part of the conduction sites (or bands),the one-body part of the impurity sites, the hybridizationbetween the impurity sites and the conduction sites (or bands),and the two-body Coulomb interaction part of the impuritysites, respectively. Notice that this Hamiltonian includes a widerange of Anderson impurity models, ranging from the simplestsingle-orbital single-impurity Anderson model ( N i=Nd=1) to a more complex multiorbital multiple-impurity Andersonmodel ( N i,Nd>1). Notice also that neither the spatial dimensions nor the lattice geometry is assumed for HAIM. We shall now show that the general Anderson impurity model HAIMgiven in Eq. ( 1) can be mapped onto a Q1D ladderlike model, for which the DMRG method is applied withhigh accuracy. This Q1D mapping can be achieved exactlywithout losing any geometrical information of the lattice byusing the BL recursive technique, which is a straightforwardextension of the basic Lanczos recursive procedure [ 58]. To simplify the formulation, let us first introduce the vectorrepresentation of fermion operators: c † σ=(d† 1,σ,d† 2,σ,..., d† M,σ,c† 1,σ,c† 2,σ,..., c† N,σ). (6) Then, the one-body part of the Hamiltonian, H0=Hc+Hd+ HV,i nE q .( 1) can be represented as H0=/summationdisplay σc† σˆH0cσ (7) with ˆH0=/parenleftbiggˆHdˆV ˆV† ˆHc/parenrightbigg , (8) where ˆHd,ˆHc, and ˆVareM×M,N×N, and M×N matrices with matrix elements ( ˆHd)m,m/prime=/epsilon1d m,m/prime,(ˆHc)n,n/prime= /epsilon1c n,n/prime, and ( ˆV)m,n=Vm,n, respectively. Next, let us construct the following matrix ˆP1composed of Mdifferent vectors em: ˆP1=(e1,e2,..., eM), (9) 195109-3TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) where emis a (N+M)-dimensional column unit vector with its element ( em)n=δm,nand thus ˆP1is a (N+M)×M matrix. Using ˆP1as the initial BL bases, the Krylov subspace of ˆH0is spanned with the BL bases ˆPl+1(l=1,2,...) generated through the three-term recurrences, ˆPl+1ˆT† l=ˆH0ˆPl−ˆPlˆEl−ˆPl−1ˆTl−1, (10) where ˆEl=ˆP† lˆH0ˆPl,ˆP0=0, and ˆT0=0. The left-hand side of Eq. ( 10) is obtained with a QR factorization of the ( N+ M)×Mmatrix in the right-hand side of Eq. ( 10). Thus ˆPl+1 is a column orthogonal ( N+M)×Mmatrix and ˆTlis a lower triangular M×Mmatrix with ( ˆTl)m,m/prime=0f o rm<m/prime, i.e., ˆTl=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝T (l) 11 00 ··· 0 T(l) 21T(l) 22 0··· 0 T(l) 31T(l) 32T(l) 33··· 0 ............... T (l) M1T(l) M2T(l) M3···T(l) MM⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠. (11) After repeating this procedure, the BL bases are constructed and they are gathered in the ( N+M)×(N+M)m a t r i x ˆP: ˆP=(ˆP 1,ˆP2,ˆP3,...), (12) which satisfies ˆP†ˆP=ˆPˆP†=ˆIwhere ˆIis the unit matrix. It should be noted here that, in practical calculations whereNis large, we usually terminate the BL iteration after the Lth iteration, for which ˆPis a rectangular ( N+M)×(LM) matrix satisfying only ˆP †ˆP=ˆIbut not ˆPˆP†=ˆI, in general. With this unitary matrix ˆP, the one-body part H0of the Hamiltonian can now be block-tridiagonalized, H0=/summationdisplay σc† σˆPˆP†ˆH0ˆPˆP†cσ=/summationdisplay σa† σˆHBL 0aσ (13) with ˆHBL 0=⎛ ⎜⎜⎜⎜⎜⎜⎝ˆE 1ˆT1 00 ··· ˆT† 1ˆE2ˆT2 0··· 0 ˆT† 2ˆE3ˆT3··· 00 ˆT† 3ˆE4... ...............⎞ ⎟⎟⎟⎟⎟⎟⎠(14) anda σ=ˆP†cσ[59]. Note that ˆTlis a lower triangular M×M matrix and thus ˆHBL 0has the bandwidth of 2 M+1. Hereafter, we will use the following convention for the indices of a† σ: a† σ=(a† 1,1,σ,a† 1,2,σ,..., a† 1,M,σ,a† 2,1,σ,..., a† l,m,σ,···),(15) and thus (aσ)lm=N+M/summationdisplay n=1(ˆP† l)m,n(cσ)n, (16) (cσ)n=N/M+1/summationdisplay l=1M/summationdisplay m=1(ˆPl)n,m(aσ)lm, (17) where ( aσ)lm=al,m,σ and cσis given in Eq. ( 6)[59]. It is important to notice that, because of the specific choice of theinitial BL bases ˆP1in Eq. ( 9), the new fermionic operators representing the impurity sites remain unchanged, i.e., (d† 1,σ,d† 2,σ,..., d† M,σ)=(a† 1,1,σ,a† 1,2,σ,..., a† 1,M,σ).(18) Therefore the two-body part HUof the Hamiltonian is exactly in the same form for the new fermionic operator aσ. This is the crucial point for the exact mapping of any Anderson impuritymodel onto a Q1D model. It is now apparent that, using the BL recursive technique introduced above, a general Anderson impurity model H AIM in any spatial dimensions and for any lattice geometry can be mapped exactly onto a Q1D model, i.e., a semi-infinite M-leg ladder model, described by the following Hamiltonian: HQ1D AIM=/summationdisplay σHQ1D 0,σ+HU, (19) where HQ1D 0,σ=L/summationdisplay l=1M/summationdisplay m,m/prime=1(ˆEl)m,m/primea† l,m,σal,m/prime,σ +L−1/summationdisplay l=1M/summationdisplay m,m/prime=1((ˆTl)m,m/primea† l,m,σal+1,m/prime,σ+H.c.) (20) andHUis the same two-body Coulomb interaction term given in Eq. ( 5). Notice that the index lin Eq. ( 20) corresponds to the one in the BL iteration in Eq. ( 10), which is terminated at the Lth iteration [ 60]. The schematic representation of the Q1D mapping for an Anderson impurity model with Ni=3 andNd=1 (thus M=3) is shown in Fig. 1. It should be noticed that the resulting Q1D ladder model HQ1D AIMin the BL bases with Lsites along the leg direction there- fore represents the original system HAIMwith approximately at least πL2and 4πL3/3 conduction sites (or orbitals) in two and three spatial dimensions, respectively (see, e.g., Figs. 4, 5, and 9). This implies that, as long as the impurity properties are concerned, the BL-DMRG method can treat quite largesystems with reasonable computational cost for a wide varietyof Anderson impurity models. B. Symmetrization of BL bases In this section, we shall describe how the symmetry of Hamiltonian can be used to further simplify the Q1Dmodel constructed by the BL recursive technique. This isbest explained by taking a specific model. Therefore, as anexample, we now consider a two-impurity Wolff model onthe honeycomb lattice [ 61], where two conduction sites on the honeycomb lattice are replaced by two impurity sites, asschematically shown in Fig. 2(a). The Hamiltonian of the Wolff model is H WM=HWM c+HWM V+HWM U, (21) 195109-4BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) l=6,E6^ ... ... ... m=2m=1 l=1, l=2, l=3, ...(a) (b) m=3E1^ T1^T2^T3^E2^E3^ l=4,T4^E4^ l=5,T5^E5^ FIG. 1. (Color online) Schematic representation of the Q1D mapping for a single-orbital three-impurity Anderson model ( Ni=3, Nd=1, and M=3) in three dimensions. Using the BL recursive technique, the Anderson impurity model HAIMis transformed exactly to a semi-infinite three-leg ladder model HQ1D AIMwithout loosing any geometrical information of the lattice. The conduction sites (or orbitals) are indicated by a blue cube in (a). The Coulomb interaction termHUis active only at the impurity sites (denoted by red circles with arrows) both in (a) and (b). Blue circles without arrows in (b) represent the “ripple” sites (i.e., BL bases) generated by the BL recursive procedure. The indices landmin (b), representing the site position of the resulting three-leg ladder model along the leg and rung directions, respectively, correspond to the ones in the new fermionic operator ( aσ)lm=al,m,σ in Eq. ( 15) and used to describe HQ1D AIMin Eq. ( 19). The “on-site potential” and “nearest-neighbor hopping” matrices, ˆEland ˆTl, respectively, in Eq. ( 20) are also indicated in (b). where HWM c=−t/summationdisplay /angbracketleftr,r/prime/angbracketright/summationdisplay σ(c† r,σcr/prime,σ+H.c.), (22) HWM V=V/summationdisplay /angbracketleftr,r/prime/angbracketright/prime/summationdisplay σ(c† r,σcr/prime,σ+H.c.), (23) and HWM U=/summationdisplay r∈Imp.Ur(nr,↑−1/2)(nr,↓−1/2). (24) Here,c† r,σ(cr,σ) is the electron creation (annihilation) operator at site ron the honeycomb lattice with spin σ(=↑,↓) and nr,σ=c† r,σcr,σ.T h es u m /angbracketleftr,r/prime/angbracketrightinHWM cruns over all pairs of nearest-neighbor sites except for the ones connecting tothe impurity sites, whereas the sum /angbracketleftr,r /prime/angbracketright/primeinHWM Vruns over all pairs of nearest-neighbor sites only involving the impuritysites.H WM Urepresents the on-site Coulomb interaction at the impurity sites with site dependent interaction Ur, and the sum inHWM Uincludes only the impurity sites. The two-impurity Wolff model described by HWMis a special case of the general Anderson impurity model HAIM in Eq. ( 1) with Ni=2, Nd=1, and M=2. Applying the BL recursive technique described in Sec. II A, we can readily show that the two-impurity Wolff model HWM m=2m=1 l=1, 2, 3, 4, 5, ...(a) (b) ... FIG. 2. (Color online) (a) Schematic representation of a two- impurity Wolff model on the honeycomb lattice described by HWMin Eq. ( 21) and (b) the semi-infinite Q1D ladder model obtained by the BL recursive technique. Solid red spheres with green arrows indicatethe impurity sites. White spheres next to the impurity sites in (a) represent the first ripple states generated by the second ( l=2) BL iteration in Eq. ( 10). The hybridization (with its strength V) between the impurity sites and the conduction sites are indicated by bold blue lines in (a). The indices landmin (b), representing the site position of the resulting Q1D ladder model along the leg and rung directions,respectively, correspond to the ones in the new fermionic operator (a σ)lm=al,m,σ used in Eq. ( 25). is mapped exactly to the Q1D ladder model described by the following Hamiltonian: HQ1D WM=2/summationdisplay m=1Urm(n1,m,↑−1/2)(n1,m,↓−1/2) +L/summationdisplay l=12/summationdisplay m,m/prime=1/summationdisplay σ/epsilon1l mm/primea† l,m,σal,m/prime,σ +L−1/summationdisplay l=12/summationdisplay m,m/prime=1/summationdisplay σtl mm/prime(a† l,m,σal+1,m/prime,σ+H.c.),(25) where rmrepresents the position of mth impurity site, /epsilon1l mm/prime=(ˆEl)m,m/prime,tl mm/prime=(ˆTl)m,m/prime, andn1,m,σ=a† 1,m,σa1,m,σ= c† rm,σcrm,σ. A schematic representation of this Q1D ladder model is shown in Fig. 2(b). In the presence of the reflection or C 2rotation point group symmetry at the center of two impurity sites, the Q1D model HQ1D WMin Eq. ( 25) can be further simplified by introducing symmetric and antisymmetric bases, γ1,1,σ=(a1,1,σ+a1,2,σ)/√ 2=(cr1,σ+cr2,σ)/√ 2, (26) γ1,2,σ=(a1,1,σ−a1,2,σ)/√ 2=(cr1,σ−cr2,σ)/√ 2, as the initial BL bases for the BL iteration. It is then readily shown that the two-impurity Wolff model HWMin Eq. ( 21) 195109-5TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) ... ... ... ...m=2m=1 l=1,l=2,l=3,l=4,... m=2 m=1 l=1,l=2,l=3,l=4,... l=4,l=3,l=2,l=1, ...,(a) (b) FIG. 3. (Color online) Schematic representation of a pure one- dimensional model mapped from the two-impurity Wolff modelon the honeycomb lattice shown in Fig. 2(a). Using the symmetry adapted BL bases, (a) the hopping terms between the legs are completely eliminated except for the impurity sites and thus (b) theladder model is further simplified to a pure one-dimensional model. Red solid circles with arrow indicate the impurity sites and blue circles without arrows represent the symmetry adapted BL bases generatedby the BL iteration. The indices landmrepresent the site position of the resulting one-dimensional model and correspond to the ones in the symmetry adapted fermionic operator γ l,m,σ used in Eq. ( 27). can be mapped onto the following Q1D model: ˜HQ1D WM=2/summationdisplay m=1Urm 4[(γ† 1,1,↑+(−1)m+1γ† 1,2,↑) ×(γ1,1,↑+(−1)m+1γ1,2,↑)−1] ×[(γ† 1,1,↓+(−1)m+1γ† 1,2,↓) ×(γ1,1,↓+(−1)m+1γ1,2,↓)−1] +L/summationdisplay l=12/summationdisplay m=1/summationdisplay σ˜/epsilon1l mγ† l,m,σγl,m,σ +L−1/summationdisplay l=12/summationdisplay m=1/summationdisplay σ˜tl m(γ† l,m,σγl+1,m,σ+H.c.).(27) Here,γl,m,σ is thelth BL bases generated by the BL recursive technique with the initial BL bases γ1,m,σgiven in Eq. ( 26). It should be noticed that, in contrast to the previous Q1D ladder model HQ1D WMin Eq. ( 25), the resulting Q1D model ˜HQ1D WMis now completely decoupled [see Fig. 3(a)], owing to the symmetry adapted BL bases, except for the “initial” sites ( l=1), i.e., the interacting impurity sites. This form is particularly useful forthe DMRG calculations because this Q1D model is regarded asa pure one-dimensional chain model, as schematically shown in Fig. 3(b). We should also note that, because of this choice of the initial BL bases in Eq. ( 26), the two-body Coulomb inter- action terms in ˜H Q1D WMcontain intersite interactions between the impurity sites, although in the original representation of theWolff model H WMthe two-body interaction terms are local. However, this slight complexity does not cause any difficulty in applying the DMRG method to ˜HQ1D WM. Three remarks are in order. First, exactly the same pure one- dimensional Hamiltonian ˜HQ1D WMgiven in Eq. ( 27), including the two-body interaction part, can be constructed by usingthe standard Lanczos tridiagonalization procedure applied separately to each symmetric and antisymmetric basis givenin Eq. ( 26) as the initial Lanczos basis. Indeed, a similar idea using the standard Lanczos tridiagonalization procedure hasbeen proposed for two-impurity models in the context of NRG[53,54]. Second, for the two-impurity Wolff model on the honeycomb lattice defined in Eqs. ( 21)–(24), the symmetric and antisymmetric BL bases in Eq. ( 26) can always decouple the Q1D ladder model H Q1D WMto a pure one-dimensional model ˜HQ1D WM, regardless of the location of two impurity sites. Third, this simplification is made possible solely because of thesymmetry of the one-body part of the original HamiltonianH WM. The similar simplification of the Q1D model using the symmetry adapted BL bases can be applied to more involvedmodels, an example being discussed below in Sec. II D. Let us now discuss the physical meaning of the BL bases generated by the BL recursive technique for the two-impurity Wolff model H WMon the honeycomb lattice. Since a† l,m,σ=/summationtext rc† r,σ(ˆPl)r,m[see Eq. ( 16)], the mth BL bases generated after the lth BL iteration is represented by ( ˆPl)r,m, where ris a two-dimensional vector on the honeycomb lattice. Figure 4 shows rdependence of ( ˆPl)r,mform=1 and 2 obtained with the initial BL bases ˆP1given in Eq. ( 9). In the standard Lanczos tridiagonalization procedure with the initial Lanczos basis sim-ilar to ˆP 1, e.g., e1in Eq. ( 9), the Lanczos basis generated after thelth Lanczos iteration forms a s-wave “ripple” around the impurity site for any land the size of the ripple increases with l[32,55,56]. Similarly, as shown in Figs. 4(a)–4(j),e v e r yB L iteration generates two orthogonal bases for m=1 and 2, and each basis is like a propagating ripple centered at each impuritysite. However, these BL bases generated are no longer s-wave- like once they overlap. This is simply because these two basesmust be orthogonal and thus they can not be s-wave-like once these two ripples overlap each other [see Figs. 4(k)–4(p)]. It is also interesting to see the ripples for γ † l,m,σ generated af- ter the lth BL iterations using the symmetric and antisymmetric initial BL bases given in Eq. ( 26). In general, the off-diagonal terms in ˆElas well as ˆTlare due to the interference between the two ripples for m=1 and 2 once the two ripples overlap [see Figs. 4(k)–4(p)]. However, because the BL iteration respects the symmetry of Hamiltonian HWM 0=HWM c+HWM V, the BL bases generated still preserve the symmetric andantisymmetric characteristics even for l>1 if the symmetric and antisymmetric initial BL bases are used. This can beclearly seen in Fig. 5. Both before [Figs. 5(a)–5(j)] and after [Figs. 5(k)–5(p)] the two ripples overlap, they are clearly symmetric and antisymmetric with respect to C 2rotation (or reflection) at the center of two impurity sites. Therefore theoff-diagonal elements in ˆE landˆTlare zero when the symmetry adapted BL bases are appropriately used. Indeed, al,m,σ andγl,m,σ relate to each other and the relation depends on the relative position of two impurity sites on thehoneycomb lattice. When the two impurity sites are located ondifferent sublattices of the honeycomb lattice, the parameters/epsilon1 l mm/primeandtl mm/primein Eq. ( 25) satisfy /epsilon1l 11=/epsilon1l 22,/epsilon1l 12=/epsilon1l 21, (28) tl 11=tl 22,tl 12=tl 21=0. 195109-6BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) 15 20 25 30rx −max |(P ) |r,ml rˆ 0 max |(P ) |r,ml rˆ 510152025 ry510152025 ry 15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx(a) (l,m)=(1,1) (c) (l,m)=(2,1) (e) (l,m)=(3,1) (g) (l,m)=(4,1) (i) (l,m)=(5,1) (k) (l,m)=(6,1) (m) ( l,m)=(7,1) (o) (l,m)=(8,1) (b) (l,m)=(1,2) (d) (l,m)=(2,2) (f) (l,m)=(3,2) (h) (l,m)=(4,2) (j) (l,m)=(5,2) (l) (l,m)=(6,2) (n) (l,m)=(7,2) (p) (l,m)=(8,2) FIG. 4. (Color online) Intensity plots of ( ˆPl)r,matr=(rx,ry), i.e., the real-space distribution of the mth BL bases generated after the lth BL iteration, for the two-impurity Wolff model on the honeycomb lattice HWMin Eq. ( 21). The BL bases for different landm(indicated in the figures) are generated starting with the initial BL bases ˆP1given in Eq. ( 9), thus corresponding to a† l,m,σ used in HQ1D WM[Eq. ( 25)]. The two impurity sites are located at r1=(12√ 3,12) and r2=(15.5√ 3,15.5), indicated by open black circles. Therefore, in this case, γl,m,σ for any l(>1) is related to al,m,σ via the following simple relations: γl,1,σ=(al,1,σ+al,2,σ)/√ 2, (29) γl,2,σ=(al,1,σ−al,2,σ)/√ 2. On the other hands, when the two impurity sites are located on the same sublattices, the parameters in Eq. ( 25) satisfy /epsilon1l 11=/epsilon1l 12=/epsilon1l 21=/epsilon1l 22=0. (30) Therefore γl,m,σ are determined so as to diagonalize 2 ×2 matrix tl mm/primewith respect to mandm/primein Eq. ( 25). Finally, we note briefly another scheme which can be used to reduce the computational cost in DMRG calculations. Thiscan be applied when the one-body part of the Hamiltonian is separated for up and down electrons, as in H AIM[Eq. ( 1)] [62]. In this case, the one-body part of the Q1D model obtained bythe BL recursive technique is also separated for up and downelectrons [see HQ1D AIMin Eq. ( 19)]. Therefore the Q1D model is described by two decoupled semi-infinite Q1D Hamiltonians,one for up electron sites and the other for down electronsites, which connect to each other via two-body part of the Hamiltonian at the impurity sites, as schematically shown in Fig. 6(a). By stretching the up electron part of the Q1D Hamiltonian to the left, we can finally obtain the infinite Q1Dmodel, as shown in Fig. 6(b). The total Hilbert space in DMRG calculations is propor- tional to s 2 Dm2D, where mDis the number of density-matrix eigenstates kept in DMRG calculations and sDis the number of local states for added sites, i.e., the number of local statesat each rung in the Q1D model. Therefore, in the case oftwo-impurity Wolff model H WM,sDis reduced form 16 to 4 by using this reduction scheme for spin degrees of freedom.Although we can no longer use the fact that the total S zis a good quantum number to reduce the dimension of the Hilbertspace, we find that this reduction scheme is still useful when 15 20 25 30510152025 rxry −max |(P ) |r,ml rˆ 0 max |(P ) |r,ml rˆ 510152025 ry 15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx15 20 25 30rx(a) (l,m)=(1,1) (c) (l,m)=(2,1) (e) (l,m)=(3,1) (g) (l,m)=(4,1) (i) (l,m)=(5,1) (k) (l,m)=(6,1) (m) ( l,m)=(7,1) (o) (l,m)=(8,1) (b) (l,m)=(1,2) (d) (l,m)=(2,2) (f) (l,m)=(3,2) (h) (l,m)=(4,2) (j) (l,m)=(5,2) (l) (l,m)=(6,2) (n) (l,m)=(7,2) (p) (l,m)=(8,2) FIG. 5. (Color online) Intensity plots of ( ˆPl)r,matr=(rx,ry), i.e., the real-space distribution of the mth BL bases generated after the lth BL iteration, for the two-impurity Wolff model on the honeycomb lattice HWMin Eq. ( 21). The BL bases for different landm(indicated in the figures) are generated starting with the symmetric and antisymmetric initial BL bases, γ† l,1,σandγ† l,2,σ,g i v e ni nE q .( 26), thus corresponding to γ† l,m,σ used in ˜HQ1D WM[Eq. ( 27)]. The two impurity sites are located at r1=(12√ 3,12) and r2=(15.5√ 3,15.5), indicated by open black circles. 195109-7TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) ... ... s =16D ... ...down sites ( σ= ) ... ... up sites ( σ= ) s =4D(a) (b)(m,σ)=(1, ) m=1 m=2l=1l=2l=3l=4... l=1l=2 l=2l=1 ... ...(m,σ)=(1, ) (m,σ)=(2, ) (m,σ)=(2, ) ... ... FIG. 6. (Color online) Schematic representation of a reduction scheme to save the computational cost in DMRG calculations by using spin degrees of freedom for a single-orbital two-impurity Anderson model HAIMwithNi=2,Nd=1, and M=2. (a) The semi-infinite ladder model obtained by the BL recursive technique. Here the up and down electron sites (indicated by cyan and orange circles, respectively) are explicitly represented. The impurity sitesare denoted by red spheres at the left edge. The red shaded plaquette at the left edge indicates where the two-body interaction part H Uis active at the impurity sites. The local degrees of freedom at each rungin this representation is s D=16. (b) Using the fact that the one-body part of the Q1D model is decoupled for up and down electron sites (except for the impurity sites), the up electron part in (a) can be simplystretched to the left to form an infinite ladder model, which contains less local degrees of freedom at each rung, i.e, s D=4. The indices landm, representing the site position of the ladder model along the leg and rung directions, respectively, correspond to the ones in the new fermionic operator ( aσ)lm=al,m,σ used in Eq. ( 20). there is no point group symmetry available to construct the symmetry adapted BL bases. We will use this reduction schemein Sec. III C 3 for systems where the symmetry adapted BL bases are not easily constructed. C. Calculations for spatially dependent quantities The BL-DMRG method allows us to calculate spatially de- pendent quantities in real space, such as correlation functionsbetween any sites and local density of states at any conductionsites. For example, to calculate correlation functions betweenthe impurity site r impand the conduction site r, we can simply take the impurity site(s) and the conduction site of interest asthe initial BL bases. The resulting Q1D model constructed bythe BL recursive technique contains explicitly the impurity siter impas well as the conduction site r, for which the correlation functions are readily evaluated using the DMRG method. Thisscheme is explained schematically for a single-impurity Wolffmodel in Fig. 7. Although a similar idea has been applied in the NRG method [ 55], the BL-DMRG method has several advantages over the NRG method in calculating spatially dependentquantities: (i) the BL-DMRG method can treat any conductionHamiltonians in real space, (ii) the reduction scheme to savethe computational cost is available for the BL-DMRG methodby using the symmetry adapted BL bases if the one-body partof the Hamiltonian has an appropriate symmetry, and (iii) the m=2m=1 l=1, 2, 3, 4, 5, ...(a) (b)rimp r ... FIG. 7. (Color online) Schematic representation of the Q1D mapping for a single-impurity Wolff model on the honeycomb latticeto calculate correlation functions between the impurity site (denoted by red sphere with green arrow) at r impand a conduction site (denoted by blue sphere) at r. White spheres in (b) indicate the ripple states (i.e., BL bases) generated by the BL recursive technique with taking the impurity site rimpand the conduction site ras the initial BL bases. The indices landmin (b) correspond to the site position of the resulting semi-infinite ladder model along the leg and rung directions, respectively. The same Q1D mapping is used to calculate, e.g., local density of states at the conduction site r. reduction scheme using spin degrees of freedom can also be applied in the BL-DMRG method if the one-body part of theHamiltonian is separated for up and down electrons. Instead of constructing a different Q1D model for each conduction site of interest, as shown in Fig. 7(b),i ti si n principle possible to calculate physical quantities involvingthe conduction sites by using Eq. ( 17) directly for a single Q1D model constructed with the initial BL bases containingonly the impurity sites. However, this approach suffers severalproblems. First of all, it is not necessarily true that thewell-defined nonsingular ( N+M)×(N+M) unitary matrix ˆPin Eq. ( 12) is always obtained by the BL iterations in Eq. ( 10). This is simply because the BL bases generated by the BL recursive technique belong to a certain irreduciblerepresentation determined by the initial BL bases. The basesbelonging to other representations are not generated becausethese bases are decoupled to the impurity sites. Second, theBL iterations are very often terminated with a finite numberLof iterations, specially when we consider the conduction sites in the thermodynamics limit N→∞ . In this case, ˆPis a rectangular ( N+M)×(LM) matrix and thus the inverse ofˆPcan not be defined to describe the operator c † σfor the conduction sites using the BL bases operator a† σ[59]. In spite of all these difficulties, if we obtained the well-defined unitarymatrix ˆP, we would then represent the physical quantities 195109-8BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) using the BL bases, e.g., c† n,σcn/prime,σ/prime=/summationdisplay l,l/prime/summationdisplay m,m/prime(ˆP† l)m,n(ˆPl/prime)n/prime,m/primea† l,m,σal/prime,m/prime,σ/prime.(31) However, we would still have to carry out these matrix multiplications for all a† l,m,σal/prime,m/prime,σ, separately, which is com- putationally very demanding. On the other hand, any operatorinvolving the conduction sites can be incorporated exactly inthe Q1D model generated by the BL recursive technique if theconduction sites are included explicitly in the initial BL bases(see Fig. 7). D. Multiorbital systems It is rather straightforward to extend the BL-DMRG method for a multiple-impurity Anderson model to a multiorbitalsingle-impurity Anderson model. For completeness and forpossible future applications, we shall here briefly describethe formulation of the BL-DMRG method for a multiorbitalsingle-impurity Anderson model and discuss the symmetry ofthe BL bases. As an example, we shall consider a five d-orbital single- impurity Anderson model. The Hamiltonian is given by Eq. ( 1) withN i=1 and Nd=5. Assuming that the impurity site is in a tetragonal environment with D4hpoint group symmetry, the five-fold degenerate dorbitals are reducible and contain the following irreducible representations: a1g(d3z2−r2orbital), b1g(dx2−y2orbital), b2g(dxyorbital), and ( eg:1,eg:2)[ (dyz,dzx) orbitals]. For the two-body part of the Hamiltonian, we canconsider, e.g., the most complete interactions, H d U=U/summationdisplay mnm,↑nm,↓+U/prime/summationdisplay m<m/prime/summationdisplay σnm,σnm/prime,¯σ +(U/prime−J)/summationdisplay m<m/primenm,σnm/prime,σ −J/summationdisplay m/negationslash=m/primec† m,↑cm,↓c† m/prime,↓cm/prime,↑ +J/prime/summationdisplay m/negationslash=m/primec† m,↑c† m,↓cm/prime,↓cm/prime,↑, (32) where U,U/prime,J, and J/primeare intraorbital Coulomb interac- tion, interorbital Coulomb interaction, Hund’s coupling, andpair-hopping, respectively. Here, m=(a 1g,b1g,b2g,eg:1,eg:2), nm,σ=d† m,σdm,σand ¯σindicates the opposite spin of σ. Applying the BL recursive technique, the five d-orbital single- impurity Anderson model is mapped onto a semi-infinitefive-leg ladder model, as shown in Fig. 8. Let us now discuss the symmetries of the BL bases, i.e., ripple states, generated by the BL recursive technique.For simplicity, we further assume that the conduction bandscoupled to the impurity site are formed by sorbitals on the square lattice and the impurity site is embedded in one ofthe sites forming the square lattice. Then, the five d-orbital single-impurity Anderson model is describe by the followingHamiltonian: H d=Hs c+Hsd V+Hd U, (33) ... ... ... ... ... l=1,l=2,l=3,l=4,...1gm=a m=b1g 2gm=b g:2m=eg:1m=e(a) (b) ... ... ... ... ... l=1,l=2,l=3,...1gm=a m=b1g2gm=b g:2m=eg:1m=e (c) l=1, l=2, l=3,..., FIG. 8. (Color online) Schematic representation of the Q1D mapping for a five d-orbital single-impurity Anderson model. The BL recursive technique transforms the five d-orbital single-impurity Anderson model onto a semi-infinite five-leg ladder model. A bluecube represents the conduction sites in (a) and blue circles indicate the BL bases in (b) and (c). Red circles with arrows denote the impurity site with orbital m(=a 1g,b1g,b2g,eg:1,a n deg:2). The yellow shaded regions in (b) and (c) indicate where the two-body Coulomb interaction Hd Uis active at the impurity sites. The indices landmin (b) and (c), representing the site position of the resulting Q1D modelalong the leg and rung directions, respectively, correspond to the ones in the new fermionic operator ( a σ)lm=al,m,σ in Eq. ( 15) and used to describe the Q1D model in Eq. ( 19). where Hs c=−t/summationdisplay /angbracketleftr,r/prime/angbracketright/prime/summationdisplay σ(c† r,σcr/prime,σ+H.c.) (34) and Hsd V=V1/summationdisplay e=±ex,±ey/summationdisplay σ/parenleftbig c† rimp+e,σda1g,σ+H.c./parenrightbig +V2/summationdisplay e=±ex/summationdisplay σ/parenleftbig c† rimp+e,σdb1g,σ+H.c./parenrightbig −V2/summationdisplay e=±ey/summationdisplay σ/parenleftbig c† rimp+e,σdb1g,σ+H.c./parenrightbig . (35) Here, the sum in Hs cruns over all nearest-neighbor sites, r andr/prime, excluding the impurity site rimp.Hsd Vrepresents the hybridization between the impurity site and the surroundingnearest-neighbor conduction sites. e xandeyare the lattice unit vectors along x- and y-directions on the square lattice, respectively. The symmetry of the dorbitals is reflected with the sign of the hybridization parameters in Hsd Vand also causes 195109-9TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) rx−10 0 10−10010 ry rx−10 0 10(a) (b) −max |(P ) |r,ml rˆ max |(P ) |r,ml rˆ 0 FIG. 9. (Color online) Intensity plots of ( ˆPl)r,matr=(rx,ry), i.e., the real-space distribution of the mth BL bases generated after thelth BL iteration, for the d-orbital single-impurity Anderson model on the square lattice Hdin Eq. ( 33). The BL bases for (a) m=a1gand (b) m=b1gare generated with taking the dorbitals as the initial BL bases. Here, the results for l=15 are shown. The impurity site is located at r=(0,0), indicated by open black circle. zero hybridization between dxy,dyz, anddzxorbitals and s orbital. Moreover, here we simply ignore the one-body term at the impurity site. Applying the BL recursive technique with taking the d orbitals as the initial BL bases, we can generate the BL baseswhich belong to the same irreducible representation with theinitial BL bases. This can be best seen in rdependence of (ˆP l)r,m[Eq. ( 16)], i.e., the mth BL bases generated after the lth BL iteration. A typical example is shown in Fig. 9. Although in this case only d3z2−r2anddx2−y2orbitals hybridize with the conduction sorbitals, Fig. 9clearly demonstrates that the each symmetry of the initial BL bases is preserved even after theBL iterations are executed. Because of the different irreduciblerepresentations, there is no matrix element in the resulting Q1Dmodel between the BL bases a l,m,σ with different m’s forl>1 [see Fig. 8(b)]. Even when the conduction bands formed by porbitals are considered, the same conclusion is reached as long as thesymmetry is respected correctly in the Hamiltonian. In thiscase, one can show that the five dorbitals are all coupled to the conduction sites with finite hybridization, and the BL basesgenerated by the BL iterations preserve the same irreduciblerepresentation of the five dorbitals when they are used for the initial BL bases. The resulting Q1D model is a semi-infinitefive-leg ladder model, where different legs belong to differentirreducible representations and thus the legs are decoupled toeach other except for the impurity site, as shown in Fig. 8(b). More generally, in many cases, a multiorbital single-impurityAnderson model possesses a specific point group symmetry,and therefore the corresponding Q1D model is decoupledaccording to the irreducible representation of the bases [ 63]. Let us finally discuss the reduction scheme for the mul- tiorbital systems to save the computational cost. First, it istrivial to apply the reduction scheme using the spin degrees offreedom (see Fig. 6). Second, as shown above, a Q1D model mapped from a multiorbital single-impurity Anderson model isa semi-infinite ladder model with decoupled chains, except for the impurity site, because there is no matrix element betweenthe bases with different irreducible representations. This canbe used to reduce the computational cost by, e.g., putting twoof the five semi-infinite legs on the left and the other three onthe right, ending up with an infinite ladder model with lessnumber of legs, as shown in Fig. 8(c). Although this scheme introduces an imbalance of the Hilbert space between the leftand right sides of the system when the impurity contains anodd number of orbitals, we still find this scheme to be veryeffective to save the computational cost. The BL-DMRG method introduced here can be readily extended to any multiorbital multiple-impurity Andersonmodels. Therefore we expect that the BL-DMRG methodis efficiently applied as an impurity solver of DMFT formultiorbital Hubbard models [ 52] and for realistic electronic structure calculations of correlated materials [ 64]. With straightforward extension, the BL-DMRG method is appliedalso to Kondo impurity models where localized spins arecoupled to conduction sites. III. APPLICATION: MAGNETIC IMPURITY PROBLEMS IN GRAPHENE In this section, using the BL-DMRG method introduced in Sec.II, we shall study three different single-impurity Anderson models for graphene with a single structural defect and with asingle adatom. We first introduce the models in Sec. III A and explain briefly the numerical details in Sec. III B, followed by the numerical results for the local magnetic susceptibility inSec. III C 1 , the local electronic density of states in Sec. III C 2 , and the spin-spin correlation functions between the impuritysite and the conduction sites in Sec. III C 3 . A. Models We study three different single-impurity Anderson models in this section. The Hamiltonians H/Gamma1of these three models (/Gamma1=I,II, and III) are given as H/Gamma1=Ht+HV+HU, (36) where Ht=−t/summationdisplay /angbracketleftr,r/prime/angbracketright/summationdisplay σ(c† r,σcr/prime,σ+H.c.), (37) HV=V/summationdisplay r∈S/summationdisplay σ/parenleftbig c† r,σcrimp,σ+H.c./parenrightbig , (38) and HU=U/parenleftbig nrimp,↑−1/2/parenrightbig/parenleftbig nrimp,↓−1/2/parenrightbig . (39) Here,c† r,σ(cr,σ) is the electron creation (annihilation) operator at site rand spin σ(=↑,↓).Htdescribes the conduction sites with the nearest-neighbor hopping tand thus the sum for/angbracketleftr,r/prime/angbracketrightruns over all nearest-neighbor pairs of conduction sites at randr/primeon the honeycomb lattice. HVdescribes the hybridization between the impurity site at rimpand the conduction site at rwhere the sum over r∈Sis taken for the conduction sites connected to the impurity site throughV.H Udescribes the impurity site with the on-site interaction 195109-10BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) (a) (b) (c) (a) (b)-4 -2 0 2 4ω/t00.10.20.30.40.5ρ0(ω) -4 -2 0 2 4ω/t00.10.20.30.40.5ρ0(ω) -4 -2 0 2 4ω/t00.10.20.30.40.5ρ0(ω)(d) (e) (f)model I model II model III FIG. 10. (Color online) (a)–(c) Schematic representation of (a) a single adatom on the honeycomb lattice (model I), (b) a substitutional impurity in the honeycomb lattice (model II), and (c) an effective model for a single structural defect (vacancy) in graphene (model III).A red sphere with a green arrow indicates the impurity site. The conduction sites connected to the impurity site through the hybridization V(denoted by bold blue lines) are represented by cyan spheres and other conduction sites are indicated by black dots. The hopping tis finite only between the nearest-neighbor conduction sites, indicated by thin black lines. (d)–(f) Local density of statesρ 0(ω) per spin for Htprojected onto the second Lanczos basis with l=2i nE q .( 10), i.e., the conduction sites connected to the impurity site through V, as indicated by cyan spheres in (a)–(c). Three models are indicated in figures (d)–(f). Uandnrimp,σ=c† rimp,σcrimp,σ. The models described by H/Gamma1 correspond to a special case of the general Anderson impurity model HAIMwithNi=Nd=M=1i nE q .( 1). These three models are different in the location of the impurity site and the way how the impurity site hybridizeswith the conduction sites. The first model, model I, is fora single impurity absorbed (i.e., a single adatom) on thehoneycomb lattice, as depicted in Fig. 10(a) . The impurity site is located on top of one of the conduction sites in thehoneycomb lattice and hybridizes with only this conductionsite. The second model, model II, represents a substitutionalimpurity in the honeycomb lattice, i.e., a single-impurity Wolffmodel, as depicted in Fig. 10(b) . One of the conduction sites in the honeycomb lattice is replaced by the impurity site, whichhybridizes with the three nearest neighboring conduction sites.The third model, model III, represents an effective modelfor a single structural defect in graphene [see Fig. 10(c) ]. In this model, the impurity site is composed of a localized sp 2 dangling orbital, which hybridizes with the two neighboring sites, as indicated in Fig. 10(c) . Model III is obtained from model II with deleting one of the hybridizing bonds betweenthe impurity site and the conduction sites in Model II. Model III deserves more explanation. In the presence of a single structural defect (i.e., vacancy) in graphene, three(a) (b) (c) (d) (e)xy xy xy zyE FIG. 11. (Color online) Schematic representation of local or- bitals of carbon atoms (black dots) around the vacancy (red dashed circles) and a local molecular orbital energy diagram for modelIII [ 19]. (a) sp 2dangling orbitals (yellow leaves) of the three carbon atoms surrounding the vacancy without structural distortion. (b) Same as (a) but with structural distortion reported by first-principles band structure calculations [ 22–25]. Two of the three carbon atoms surrounding the vacancy are closer to each other. (c) The resulting local molecular orbital energy diagram for (b).Without distortion and hybridization, the three dangling orbitals are degenerate. The lowest and highest levels correspond to the bonding and antibonding states, respectively, composed mostly of thedangling orbitals of the two carbon atoms closer to each other. The second lowest level corresponds to the nonbonding state composed mostly of the remaining dangling orbital. Since there are three electrons (arrows) in these dangling orbitals, the second lowest level is half-filled. (d) The half-filled dangling orbital (yellow leaf) andp zorbitals (green circles) of the other two neighboring carbon atoms surrounding the vacancy. Without additional distortion, the half-filled dangling orbital does not hybridize with other orbitals. (e) Same as(d) but the view from the in-plane axis of graphene. The green leaves indicate the p zorbitals. According to first-principles band structure calculations, the additional out-of-plane distortion takes place in thepresence of vacancy [ 22–25], which induces nonzero hybridization between the half-filled dangling orbital and the p zorbitals of the other two neighboring carbon atoms. dangling orbitals appear around the defect, which are formed bysp2orbitals of three carbon atoms surrounding the defect, each carbon atom contributing a single sp2orbital, and are pointing towards the defect [see Fig. 11(a) ]. Without additional structural distortion and hybridization, these three danglingorbitals are degenerate. However, according to first-principlesband-structure calculations [ 22–25], because of the additional structural distortion around the defect, these three fold de-generate dangling orbitals are split into three nondegeneratelevels, as shown in Figs. 11(b) and11(c) . As a result, two of the three unpaired electrons in the sp 2dangling orbitals occupy the lowest nondegenerate level and the remaining electronoccupies the second lowest level, forming the localized statelocated mostly at one of the nearest neighboring carbon atomsaround the defect [ 22–25]. Therefore we can ignore the paired 195109-11TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) electrons occupying the lowest level and consider only the half-filled second lowest level as an impurity site. Note thatthe second lowest level is mainly composed of the sp 2dangling orbital, which points towards the defect and thus hybridizesmostly with the p zorbitals of the other two neighboring carbon atoms surrounding the defect, due to the additionalout-of-plane distortion, as depicted in Figs. 11(d) and11(e) , but not with the p zorbital at the same carbon atom because it is symmetrically forbidden [ 19]. Therefore, in model III, the pz orbital is also present at the same site where the impurity exists, although there is no direct hybridization between these twoorbitals [see Fig. 10(c) ]. As mentioned above, the difference between model II and model III is the number of conductionsites which hybridize with the impurity site. As explained in details in Appendix, the impurity properties of Anderson impurity models are determined solely bythe hybridization function /Delta1(ω)[10,33]. The hybridization function for models I–III is expressed as /Delta1(ω)=π|T 1|2ρ0(ω), (40) where T1is the matrix element of Ht+HVbetween the first Lanczos basis, i.e., the impurity site, and the second Lanczosbasis (see Sec. II Aand Appendix). As described in Appendix, we can readily show that T 1=Vfor model I, T2=√ 3Vfor model II, and T1=√ 2Vfor model III. ρ0(ω)i nE q .( 40)i s the local density of states per spin for Htprojected onto the second Lanczos basis and is evaluated as ρ0(ω)=1 NN/summationdisplay k=1/parenleftBigg 1√NS/summationdisplay r∈Su(k) r,c/parenrightBigg2 δ(ω−/epsilon1c,k), (41) where u(k) r,cis the kth eigenstate of Htat site rwith its eigenvalue /epsilon1c,k. The sum over r∈Sin Eq. ( 41) is taken for the conduction sites connected to the impurity site through V,a s indicated by cyan spheres in Figs. 10(a) –10(c) , andNSis the number of these sites. Since the hybridization function /Delta1(ω)i s proportional to the local density of state ρ0(ω), we can capture the fundamental difference among the three models simply bycomparing ρ 0(ω). As shown in Fig. 10(d) ,ρ0(ω) for model I is exactly the same as the local density of states for the pure honeycomblattice model. Therefore model I is equivalent to the so-calledpseudogap Kondo problem [ 10,12–15]. The pseudogap Kondo problem has been studied both analytically and numericallybased on the low-energy calculations [ 9–15]. The previous studies have found that the ground state is always in the localmagnetic moment phase and hence no Kondo screening occursas long as the system is particle-hole symmetric. As will beshown below, our numerical calculations also find that theKondo screening is absent for model I when the particle-holesymmetry is preserved at half-filling. In the case of models II and III, ρ 0(ω) has a singularity at the Fermi level ( ω=0), as shown in Figs. 10(e) and10(f) . The appearance of the zero-energy singularity is understoodas follows. Recall first that ρ 0(ω) is the local density of states forHtprojected onto the conduction sites next to the impurity site connected through Vin the honeycomb lattice. Therefore, assuming that these conduction sites belong to Bsublattice, the number NAof the conduction sites on Asublattice is smaller by one than the number NBof the conduction sitesonBsublattice, i.e., NA=NB−1, where the total number Nof the conduction sites is NA+NB. Consequently, a single zero-energy state is induced when there is no hopping betweenthe same sublattices because the rank of N×Nmatrix for H t isN−1. The zero-energy state is localized mostly around the impurity site and the amplitude of the wave function ofthis state is finite only on Bsublattice. This zero-energy state causes logarithmically diverging behavior in ρ 0(ω)a tω=0 [26,27]. We thus expect that the impurity properties for models II and III would be similar but different qualitatively from theones for model I. B. Numerical details As already indicated in Eq. ( 39), in this paper, we consider only the particle-hole symmetric case at half-filling. Thereforethe local electron density is always one, including at theimpurity site, irrespectively of UandVvalues. To avoid unnecessary finite size effect [ 44], we always terminate the BL iteration at an even number Lof iterations when the Q1D model is constructed. Therefore the resultingQ1D model has the even number Lof sites along the leg direction. For the calculations of physical quantities dependingonly on the impurity site, the resulting Q1D model is apure one-dimensional chain and we consider Lup to 200 with keeping m D∼12Ldensity-matrix eigenstates in the DMRG calculations. For the calculations of physical quantitiesinvolving the conduction site, i.e., the spin-spin correlationfunctions between the impurity site and the conduction sites,the resulting Q1D model is a two-leg ladder model and weconsider Lup to 240 with keeping m D∼16Ldensity-matrix eigenstates. The discarded weights are typically of the order10 −8and the error of the ground state energy is ∼10−4t.W e should emphasize that the resulting Q1D model with hundredsofLsites along the leg direction corresponds to the original system H /Gamma1with tens of thousands of conduction sites Nin two spatial dimensions. For example, the Q1D model with L=240 represents the original model H/Gamma1with at least N∼180 000. To calculate the dynamical quantities, we employ the correction vector method [ 65,66]. Although the dynamical quantities can be evaluated with other methods, e.g., byexpanding spectral functions into a continued fraction [ 67,68], using Chebyshev polynomials [ 69], or Fourier transforming the corresponding real-time dynamics [ 70], the correction vector method is most promising for our purpose because itis a direct calculation of the dynamical quantity by includingthe Hilbert space for the excited states and thus there is noadditional error caused, e.g., by the numerical integration orby terminating the finite number of polynomials. C. Results 1. Local magnetic susceptibility at the impurity site Let us first examine the magnetic properties. For this purpose, here we calculate the local magnetic susceptibilityχ i(ω) at the impurity site defined as χi(ω)=−1 πIm/angbracketleftψ0|Sz rimp(ω+iη+H/Gamma1−E0)−1Sz rimp|ψ0/angbracketright, (42) 195109-12BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) 00.020.040.060.080.1χi(ω)tU/t=0 U/t=2 U/t=4 00.020.040.060.080.1χi(ω)tU/t=0 U/t=2 U/t=4 01234 56 7 8 ω/t00.020.040.060.080.1χi(ω)tU/t=0 U/t=2 U/t=40 0.02 0.04 1/L00.10.2χi(0)t 0 0.02 0.04 1/L00.0050.01χi(0)t 0 0.02 0.04 1/L00.010.02χi(0)t(a) (b) (c) FIG. 12. (Color online) Local magnetic susceptibility χi(ω)a t the impurity site calculated for models (a) I, (b) II, and (c) III. The parameters used are L=100,V=t,a n dη=20t/L forU/t=0 (circles), 2 (squares), and 4 (triangles). For comparison, χ0 i(ω)f o r the noninteracting limit calculated using Eq. ( 43) is also shown in red (black) dashed lines with L=100 (1000) and η=20t/L. (Insets) L dependence of χi(0) with keeping η=20/L. For comparison, χ0 i(ω) calculated using Eq. ( 43) is also plotted by red dashed lines. where Sz rimp=(nrimp,↑−nrimp,↓)/2i st h e z-component of spin operator at the impurity site rimp,|ψ0/angbracketrightis the ground state of H/Gamma1with its energy E0, andη(>0) is a broadening factor (a real number). In the noninteracting limit with U=0,χi(ω) can be obtained directly using the kth eigenstate u(k)with its eigen- valueεkof the one-body part of H/Gamma1described either by the conduction site bases cσas in ˆH0in Eq. ( 8) or by the Lanczos bases aσas in ˆHBL 0in Eq. ( 14), i.e., χ0 i(ω)=η 2π/summationdisplay k∈(εk<μ)/summationdisplay k/prime∈(εk/prime>μ)|u(k) rimpu(k/prime) rimp|2 (ω−εk/prime+εk)2+η2, (43) where u(k) ris the site rcomponent of u(k)andμis the chemical potential. Since models I, II, and III are all particle-holesymmetric at half-filling, the chemical potential is μ=0. It is very intriguing to find that χ 0 i(ω) can be calculated more accurately, in a sense that it is closer to the one inthe thermodynamic limit, by using ˆH BL 0than ˆH0, as long as the same matrix sizes of ˆHBL 0and ˆH0are taken. This is simply because more important degrees of freedom aroundthe impurity site are extracted in ˆH BL 0already for relatively smallL. The results of χ0 i(ω) for the three different models in the noninteracting limit are shown in Fig. 12. It is clearly observed in Fig. 12thatχ0 i(ω) diverges in the limit of ω→0f o r model I, while it converges to zero for models II and III.024 6 81 0 1 2 1 4 16 U/t0.60.650.70.750.80.85Sr model I model II model IIIimp FIG. 13. (Color online) Local spin ¯Srimpat the impurity site for V=t,L=200, and various values of U. For comparison, ¯Srimpin the strong coupling limit ( U→∞ ) is indicated by dashed line. The different behavior of χ0 i(ω) in the limit of ω→0 can be easily understood by recalling that χ0 i(ω) is proportional to the convolution of the local density of states, i.e., χ0 i(ω)∝/integraldisplay dωρ0 i(ω/prime−ω)ρ0 i(ω/prime)/Theta1(ω−ω/prime)/Theta1(ω/prime),(44) where /Theta1(ω) is the Heaviside step function and ρ0 i(ω)i st h e local density of state at the impurity site [see Eq. ( 50)]. The diverging behavior of χ0 i(0) for model I is due to the presence of the zero-energy state, which causes the zero-energy peak atthe Fermi level in the local density of state at the impurity site[see also in Fig. 14(a) ]. In contrast, the local density of states at the impurity site for models II and III has the pseudogapstructure at the Fermi level, i.e., ρ 0 i(ω)∝|ω|, and hence χ0 i(ω)∝ω3. The diverging behavior of the local density of states at ω=0 in the noninteracting limit for model I is due to the fact that the numbers of sites (including the impurity site)onAandBsublattices, N AandNB, respectively, are different for model I, but the same for models II and III, the similardiscussion being given in the last part of Sec. III A forρ 0(ω). The results of χi(ω) calculated using the dynamical DMRG method for the three models are shown in Fig. 12.F i r s t ,i ti s noticed in Fig. 12that the dynamical DMRG calculations well reproduce χ0 i(ω) obtained using Eq. ( 43) with the same Land ηfor the noninteracting limit. In the case of finite interaction U, we find that χi(ω) for model I diverges in the limit of ω→0, which indicates the presence of free magnetic moment at the impurity site. Although the diverging behavior of χi(0) for finite Useems similar to the one found in χ0 i(0) for the noninteracting limit, we find in Fig. 13that the local spin ¯Srimp at the impurity site, ¯Srimp=/radicalBig /angbracketleftψ0|Srimp·Srimp|ψ0/angbracketright, (45) is sizably large for finite Uas compared to the one for the noninteracting limit. Here, the spin operator Srat site ris 195109-13TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) 00.10.20.30.4ρi(ω)tU/t=0 U/t=2 U/t=4 00.10.20.30.4ρi(ω)tU/t=0 U/t=2 U/t=4 01234 56 7 8 ω/t00.10.20.30.4ρi(ω)tU/t=0 U/t=2 U/t=40 0.02 0.04 1/L00.20.4ρi(0)t 0 0.02 0.04 1/L00.020.040.06ρi(0)t 0 0.02 0.04 1/L00.040.080.12ρi(0)t(a) (b) (c) FIG. 14. (Color online) Local density of states ρi(ω) at the im- purity site calculated using the dynamical DMRG method for models(a) I, (b) II, and (c) III. The parameters used are L=100,V=t, andη=20t/LforU/t=0 (circles), 2 (squares), and 4 (triangles). For comparison, ρ 0 i(ω) for the noninteracting limit calculated using Eq. ( 50) is also shown in red (black) dashed lines with L=100 (1000) and η=20t/L. (Insets) Ldependence of ρi(0) with keeping η=20t/L. For comparison, ρ0 i(ω) calculated using Eq. ( 50) is also plotted by red dashed lines. defined as (Sr)ν=1 2/summationdisplay σ1,σ2c† r,σ1ˆσν σ1,σ2cr,σ2 (46) and ˆσν(ν=x,y,z )i st h e νcomponent of Pauli matrices. In addition, as will be discussed later in Fig. 14, the local density of states at the impurity site is zero at the Fermilevel for finite U, qualitatively different form the case for noninteracting limit. Therefore we conclude that in the groundstate of model I the local magnetic moment is not screened butrather isolated, and thus no Kondo screening occurs. This is ingood accordance with the previous studies for the pseudogapKondo problem [ 9–15]. On the other hand, as shown in Fig. 12,χ i(ω) for models II and III monotonically decreases with decreasing ωfor small ωand it becomes zero in the limit of ω→0, which indicates the absence of free magnetic moment at the impurity site.Since lim ω→0χ0 i(ω)→0 already in the noninteracting limit for models II and III, the absence of free magnetic moment foras m a l l Uregion is related to the formation of bonding orbital composed of the impurity site and the surrounding conductionsites. However, as shown in Fig. 13, the local spin ¯S rimpat impurity site indeed increases with increasing Usmoothly to the strong coupling limit (i.e., U→∞ ), where a single electron is completely localized at the impurity site and onlythe spin degree of freedom is left. Therefore these results implythat there is the crossover from a small Uregion to a large U region where the screening mechanisms are different: for asmallUregion, the absence of free magnetic moment is due to the formation of bonding orbital, whereas for a large Uregion the local magnetic moment is screened by the surroundingconduction electrons, i.e., the formation of a Kondo singletstate [ 32]. Other noticeable effects of Uonχ i(ω) are summarized as follows. First, the line shape of χi(ω) changes systematically with increasing U: the overall weight moves downward to a lower energy region with increasing U. This is associated with the decrease of the effective exchange interaction betweenthe impurity site and the conduction site with increasingUin the strong coupling limit. Second, the total spectral weight increases with U. Notice that the total spectral weight is related to the local spin ¯S rimpat the impurity site, i.e.,/integraltext∞ 0χi(ω)dω=¯S2 rimp/3. The larger Uincreases the tendency of single occupancy at the impurity site with less charge fluctuations, which in turn increases the local magneticmoment, as seen in Fig. 13. 2. Local density of states at the impurity site The local density of states ρ(r,ω) at site ris defined as ρ(r,ω)=/braceleftBigg −1 πImGe(r,ω+iη)f o r ω> 0 −1 πImGh(r,ω+iη)f o r ω< 0, (47) where Ge(r,z) andGh(r,z)a r e Ge(r,z)=/angbracketleftψ0|cr,σ(z−H/Gamma1+E0)−1c† r,σ|ψ0/angbracketright (48) and Gh(r,z)=/angbracketleftψ0|c† r,σ(z+H/Gamma1−E0)−1cr,σ|ψ0/angbracketright, (49) respectively. The local density of states ρi(ω) at the impurity siterimpis thus ρi(ω)=ρ(rimp,ω). Figure 14shows the results of ρi(ω) for the three models calculated using the dynamical DMRG method. Since thesemodels are particle-hole symmetric at half-filling and thespectra are symmetric at ω=0, we show ρ i(ω) only for ω/greaterorequalslant0 in Fig. 14. For comparison, we also calculate the local density of states ρ0 i(ω) at the impurity site for the noninteracting limit by numerically diagonalizing the one-body part of theHamiltonian H /Gamma1described by the Lanczos bases aσas in ˆHBL 0 in Eq. ( 14), i.e., ρ0 i(ω)=η π/summationdisplay k∈(εk>μ)|u(k) rimp|2 (ω−εk)2+η2(50) forω/greaterorequalslant0. As shown in Fig. 14, the dynamical DMRG calculations well reproduce ρ0 i(ω) obtained using Eq. ( 50) with the same Landη. Let us first focus on ρi(ω) for model I. As shown in Fig. 14(a) , the spectral weight is redistributed drastically with increasing U. The diverging behavior of ρi(ω)i nt h e limit of ω→0f o r U=0 is strongly suppressed and the low-energy spectral weight is transferred to a higher energyregion with increasing U. As shown in the inset of Fig. 14(a) , we find that ρ i(0) for finite Uapproaches to zero in the limit of L→∞ . This implies that for model I a small Uregion is qualitatively different from the noninteracting 195109-14BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) limit but rather smoothly connected to the strong coupling limit where the charge fluctuations are completely suppressedand only the spin degree of freedom is left at the impuritysite. It is also observed in Fig. 14(a) that the lowest peak in ρ i(ω) atω/t∼0.3( 0.5) for U/t=2 (4) becomes broader and the peak position shifts slightly to higher energy as Uincreases. This is indeed consistent with the previous study of the samemodel using the QMC method [ 71]. In addition, we find that with increasing Uthe spectral weight in a much higher energy region is enhanced and gradually forms a peak structure, e.g,atω/t∼3f o rU/t=4. We shall next examine ρ i(ω) for models II and III. As shown in Figs. 14(b) and14(c) , we find that (i) ρi(ω)f o rt h e low-energy region of ω/t/lessorsimilar0.5 is almost insensitive to the values of Uand (ii) ρi(0) clearly becomes 0 in the limit of L→∞ , thus exhibiting a pseudogap structure similar to the one for the noninteracting limit. The pseudogap structure inρ i(ω) is also found even for much larger U(not shown). The fact that ρi(ω) for the low-energy region is insensitive to Uis in good qualitative agreement with the previous study by theperturbation theory for the conventional Anderson impuritymodel, in which ρ i(ω)a tω∼0 for finite Uremains the same as the one for U=0[72,73]. We also find in Figs. 14(b) and14(c) that the spectral weight in the high-energy region of ω/t > 3 increases with U, which is transferred from the low-energy region below ∼3t.T h i s spectral weight redistribution with increasing Uis also very similar to the one in the conventional Anderson impurity model[72,73], where the spectral weight in the low-energy region is suppressed and the high-energy peaks, corresponding tothe lower and upper Hubbard peaks at ω∼±U/2, gradually emerge with increasing U, although the excitation energy of the high-energy peak found in Figs. 14(b) and 14(c) is significantly different from U/2. Therefore Udependence of the spectral weight for models II and III can be qualitativelyexplained by the conventional Anderson impurity picture,except for the absence of Kondo resonance peak, which issimply due to the pseudogap structure in ρ 0 i(ω)a tω∼0f o r the noninteracting limit. 3. Spin-spin correlation functions between the impurity site and the conduction sites Finally, we shall calculate the spin-spin correlation func- tionsSi(r) between the magnetic impurity site at rimpand the conduction site at r, Si(r)=/angbracketleftψ0|Srimp·Sr|ψ0/angbracketright. (51) As described in Sec. II B, the ladder model is constructed separately for each conduction site rto which the spin-spin correlation function Si(r) is evaluated. We should first note that one can easily construct the symmetric and antisymmetric BL bases for model II becausethe symmetry of the lattice structure remains the sameas the one for the honeycomb lattice. However, it is notstraightforward to construct the symmetric and antisymmetricBL bases for models I and III. In the case of model I,we can in principle perform the BL iterations taking as theinitial BL bases two conduction sites, i.e., the conduction−10010 −10 0 10 −10 0 10(a) (b) (c) −10010 max | S (r)|i1/4r max | S (r)|i1/4r −|S (r)|i1/4sign[ S (r)]i rxry ryrx FIG. 15. (Color online) Intensity plot of the spin-spin correlation functions Si(r) between the impurity site and the conduction sites for models (a) I, (b) II, and (c) III. The impurity site is located at rimp= (rx,ry)=(0,0) for (a) and (b), and rimp=(0,−0.5) for (c), indicated by black circles. The parameters used are U/t=4a n d V/t=1. The system size Lis chosen to satisfy L=lpath+100+mod(lpath,2) where lpathis the minimum path length to reach the conduction site from the impurity site in the honeycomb lattice. Notice that the color intensity used here is for |Si(r)|1/4sign[Si(r)], instead of Si(r) itself, for clarity. site of interest and the conduction site connected to the impurity site via V[the conduction site denoted by cyan sphere in Fig. 10(a) ]. After the BL iterations are completed, the impurity site can be added to the resulting ladder model.With this slightly modified implementation, we can readilyconstruct the symmetry adapted BL bases and the resultingladder model is essentially decoupled for the symmetric andantisymmetric BL bases, as explained in Sec. II B. However, this implementation naturally introduces an odd number ofsites, which is problematic in the DMRG calculations. In thecase of model III, it is generally difficult to construct thesymmetry adapted BL bases. Therefore we use the reductionscheme for spin degrees of freedom described in Sec. II B(see also Fig. 6) for models I and III. Figure 15shows the spatial distribution of the spin-spin correlation functions S i(r) for the three models. Because of the bipartite nature of the honeycomb lattice and the particle-holesymmetry at half-filling, we can clearly see in Fig. 15the alternating dependence of the sign of S i(r) for all models: the spin-spin correlation functions Si(r) at the conduction site belonging to the same (different) sublattice of the impurity siteis positive (negative). We can also notice in Fig. 15that model I exhibits relatively strong ferromagnetic correlations, whileantiferromagnetic correlations are dominant for models II andIII. The different behavior among the models is attributed tothe fact that the ground state of model I is characterized with 195109-15TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) the appearance of unscreened local magnetic moment but the ground states of models II and III are instead both spin singlet,as discussed above in Sec III C 1 and Sec. III C 2 . To discuss more details of the spin structures around the impurity site, the log-log scale plots of the spin-spin correlationfunctions S i(r) for the three models are shown in Fig. 16.W e should notice first that the correlation functions can be verysmall for large |r|,a ss m a l la s ∼10 −9–10−8at the maximum distance studied in Fig. 16. However, we can still distinguish clearly the significant difference in the asymptotic behavior ofS i(r) for these models. We find in Fig. 16that the spin-spin correlation functions Si(r) between the impurity site and the conduction sites decay as Si(r)∝/braceleftbigg 1/|r|3for model I 1/|r|4for models II and III(52) in the asymptotic |r|. These calculations thus demonstrate the capability of the BL-DMRG method to study spatiallydependent quantities with extremely high accuracy. To better understand these results, we also calculate the spin-spin correlation function S 0 i(r) for the noninteracting limit, which is given as S0 i(r)=3 2/summationdisplay k∈(εk<μ)/summationdisplay k/prime∈(εk/prime>μ)/parenleftbig u(k) rimp/parenrightbig∗u(k/prime) rimp/parenleftbig u(k/prime) r/parenrightbig∗u(k) r.(53) In the noninteracting limit, the spin-spin correlation functions between any two sites on the same sublattice are exactlyzero, whereas they are negative between any two sites on thedifferent sublattices. As shown in Fig. 16, we find that in the noninteracting limit the spin-spin correlation functions decayas S 0 i(r)∝/braceleftbigg 1/|r|2for model I 1/|r|4for models II and III(54) in the asymptotic |r|. Therefore the interaction Udrastically changes the exponent of the spin-spin correlation functionsfor model I. The asymptotic behavior of S i(r) for model I with finite Uis rather the same as the one for Ruderman- Kittel-Kasuya-Yosida (RKKY) interaction [ 74–76] between two magnetic impurities coupled through the Dirac electronson the honeycomb lattice at half-filling, which has been indeedfound to be as ∝|r| −3[77–83]. In sharp contrast, the exponent remains the same for models II and III with and without U.T h e different effect of Uon the asymptotic behavior of Si(r)f o r the three models is understood because the magnetic momentat the impurity site is not screened but rather isolated in theground state of model I while the impurity moment is screenedby the conduction electrons to form the spin-singlet groundstate for models II and III, as discussed in Sec. III C 1 and Sec. III C 2 . It is also noticed in Fig. 16that the absolute value of the spin-spin correlation functions are suppressed with increasingUfor model I, but they are enhanced for models II and III with positive (negative) values between the same (opposite)sublattices. These different behaviors are also understoodby considering the different nature of the ground states ofthese models. The former results are due to the increase ofunscreened local magnetic moment at the impurity site with10-810-610-410-2100|Si(r)| U/t=2.0 U/t=4.0 U/t=6.0 U/t=8.0 10-810-610-410-2100|Si(r)| U/t=2.0 U/t=4.0 U/t=6.0 U/t=8.0 1 1 0 100 |r|10-810-610-410-2100|Si(r)| U/t=2.0 U/t=4.0 U/t=6.0 U/t=8.0(a) (b) (c)|r|−3|r|−3|r|−2 |r|−3|r|−4 |r|−4 FIG. 16. (Color online) Log-log scale plots of the spin-spin cor- relation functions Si(r) between the impurity site and the conduction sites for models (a) I, (b) II, and (c) III. The impurity site is located atrimp=(x,y)=(0,0) and the conduction sites rare chosen along (0,1) direction (see Fig. 15). The parameters used are V/t=1a n d different values of U/t indicated in the figures. The system size L is chosen to satisfy L=lpath+100+mod(lpath,2) where lpathis the minimum path length to reach the conduction site from the impuritysite in the honeycomb lattice. The spin-spin correlation functions S 0 i(r) for the noninteracting limit calculated using Eq. ( 53)a r es h o w n by red open circles. For comparison, |r|−αwith different exponent α is also plotted by black dashed lines. increasing U( s e ea l s oF i g . 13). The latter results are because the ground states for models II and III are both spin singlet, 195109-16BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) where the increased ferromagnetic correlations have to be compensated by enhancing the antiferromagnetic correlations. IV . SUMMARY AND DISCUSSION We have introduced the BL-DMRG method for single- as well as multiple-impurity Anderson models in any spatialdimensions. The BL recursive technique is employed to map,without losing any geometrical information of the lattice, ageneral Anderson impurity model onto a Q1D model, to whichthe DMRG method can be applied with high accuracy. One ofthe key ideas in the BL-DMRG method is to include, as theinitial BL bases, the Anderson impurity sites where the two-body interactions are finite. With this choice of the initial BLbases, the two-body interactions remain local in the resultingQ1D model. We have also introduced two reduction schemesto save the computational cost for the DMRG calculations.One is to construct the symmetry adapted BL bases when theHamiltonian possesses a certain point group symmetry suchas rotation and reflection. The other is to use spin degreesof freedom when the one-body part of the Hamiltonian isseparated for up and down electrons. We have also discussedbriefly the extension of the BL-DMRG method and thesymmetry adapted BL bases for a multiorbital single-impurityAnderson model. Furthermore, we have demonstrated how theBL-DMRG method is applied to calculate spatially dependent quantities such as spin-spin correlation functions and local density of states at the conduction sites. We should emphasize that the resulting Q1D model in the BL bases with Lsites along the leg direction represents the original Anderson impurity model in real space withapproximately at least πL 2and 4πL3/3 conduction sites in two and three spatial dimensions, respectively. Therefore,as long as the impurity properties are concerned, the BL-DMRG method can treat quite large systems for a wideclass of Anderson impurity models, which are currently outof reach with the direct application of the QMC methodsand the Lanczos exact diagonalization method. The spa-tially dependent quantities are rather difficult to calculatedwith the NRG method. Therefore the BL-DMRG methodhas a great advantage on this aspect as well over theNRG method. As an application of the BL-DMRG method, we have stud- ied the ground state properties of single-impurity Andersonmodels for graphene with an adatom and with a structuraldefect (vacancy). For this purpose, we have considered threedifferent models: (i) a single impurity absorbed on thehoneycomb lattice (model I), (ii) a substitutional impurityin the honeycomb lattice (model II), and (iii) an effectivemodel for graphene with a single vacancy of carbon atomwhere the impurity site represents one of the sp 2dangling orbitals at the carbon atoms surrounding the vacancy (modelIII). We have focused only on the particle-hole symmetriccase at half-filling and thus the electron density is alwaysone, including at the impurity site. Our numerical results forthe local magnetic susceptibility, the local spin, and the localdensity of states at the impurity site clearly show that themagnetic moment at the impurity site is not screened but ratherisolated, and thus no Kondo screening occurs in the groundstate of model I, while the impurity moment is screened by theconduction electrons to form the spin-singlet ground state in models II and III. Moreover, we have applied the BL-DMRG method to cal- culate, with extremely high accuracy, the spin-spin correlationfunctions S i(r) between the impurity site and the conduction sites for the three models. We have found the qualitativedifference in the spatial distribution of the spin structures of theconduction electrons around the impurity site. The spin-spincorrelation functions S i(r) decay asymptotically as ∝|r|−3for model I, the same asymptotic behavior as the one for the RKKYinteraction between two magnetic impurities coupled to theDirac conduction electrons, but qualitatively district from theone for the noninteracting limit ( ∝|r| −2). On the other hand, the spin-spin correlation functions Si(r) decay asymptotically as∝|r|−4for models II and III, which are exactly the same as the ones for the noninteracting limit. This difference canbe understood because the magnetic moment in the groundstate of model I is isolated but the spin singlet is formed in theground state of models II and III. It is now interesting to discuss these results based on Lieb’s theorem [ 84]. According to Lieb’s theorem for bipartite lattice systems with no hopping between the same sublattices (exceptfor the on-site potential), the total spin S totof the ground state at half-filling is Stot=|NA−NB|/2, where NA(NB)i st h e number of sites belonging to Asublattice ( Bsublattice) [ 84]. Regardless of the rigorous condition for Lieb’s theorem [ 85], the theorem can be applied to the three models studied here because all models are bipartite and at half-filling. Since modelI has different number of sites (including the impurity site) onAandBsublattices, |N A−NB|=1, the theorem predicts the total spin of the ground state is 1 /2, which can be regarded as the isolated impurity spin. On the other hand, in modelsII and III, N A=NBand thus the theorem predicts that the ground state of these models is spin singlet, which is also inaccordance with our numerical results. We shall now discuss our results in comparison with the recent experiments on graphene. The experiments on graphenewith hydrogen or fluorine adatoms as well as with structuraldefects (vacancies) have revealed that these systems carrymagnetic moments with spin 1 /2 per adatom or vacancy and that these magnetic moments behave paramagneticallyeven at lowest temperatures [ 6,7]. Therefore these experiments strongly indicate that no Kondo screening occurs. On the otherhand, different experiments on graphene with vacancies haveobserved the Kondo-like signature in the temperature depen-dence of the resistivity [ 8]. Although we have focused only on single impurity models with the particle-hole symmetry athalf-filling, our results should be relevant to these experimentsas long as the number of adatoms or vacancies are dilute. Wehave found that the magnetic moment is unscreened but ratherisolated in the ground state of model I, which therefore canexplain, at least qualitatively, the spin 1 /2 free moment per adatom observed experimentally on graphene with hydrogenor fluorine adatoms [ 6,7]. On the other hand, we have found that the ground state of model III, a model for graphene witha single structural defect, is spin singlet and no free magneticmoment is found. Therefore our results for model III are not inaccordance with the experimental observation reported in Ref.[6] but seem to be consistent qualitatively with experiments in Ref. [ 8]. 195109-17TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) There are two comments regarding our results for model III and the experiments on graphene with vacancies reportedin Ref. [ 8]. First, it is reasonable that the impurity moment is screened to form the spin-singlet ground state in modelIII. The reason is as follows. The number of electrons andthe number of sites are both even in model III and thereforethe ground state is closed shell in the noninteracting limit.Assuming the adiabatic evolution of the ground state withinteraction U, the ground state must be total spin S tot=0 unless the correlation induces a level cross between the groundstate and a low-lying excited state. In the experiments, thenumber of electrons removed by introducing vacancies mustbe even (i.e., six electrons removed per vacancy), and thusthe system easily forms a close shell state with the totalspinS tot=0 or possibly nonzero integer spin, but not with Stot=1/2. Second, although our results for model III seem to be consistent with the experimental observation in Ref [ 8], there is the following fundamental discrepancy. By controllingthe number of electrons through gate voltage, it is foundexperimentally that the highest Kondo temperature appearsaway from half-filling [ 8]. This observation seems contradict to our calculations because the diverging hybridization function/Delta1(ω)a tω=0 should induce the most tightly screened state and thus the highest Kondo temperature at half-filling, but notaway from half-filling, for model III. The discrepancy betweenour results and the experiments as well as the disagreement between the two experiments suggest that the understanding of physics of graphene with vacancies and the correspondingmagnetic properties would be beyond the simple model studiedhere and deserve further investigation both theoretically andexperimentally. Finally, we shall briefly comment on further possible extensions of the BL-DMRG method. The method is quitegeneral and can be applied to general Anderson impuritymodels in any spatial dimensions. One major advantage ofthis method is its flexibility for the form of the conductionHamiltonian. In this paper, we have studied Anderson impuritymodels in the real-space representation. However, the BL-DMRG method can be applied, without any difficulties, toAnderson impurity models in the energy-space representation(see Appendix). The BL-DMRG method in the energy-spacerepresentation allows us, in principle, to do the calculations inthe thermodynamic limit once the hybridization function /Delta1(ω) is evaluated accurately (see Appendix). The implementationof these extensions is straightforward and we believe thatthe BL-DMRG method in the energy-space representationshould be valuable, e.g., for application as an impurity solverof DMFT for realistic electronic structure calculations ofcorrelated materials [ 64]. Research along this line is now in progress [ 86]. ACKNOWLEDGMENTS The authors are grateful to H. Watanabe, E. Minamitani, and W. Ku for valuable discussion. The computation has been doneusing the RIKEN Cluster of Clusters (RICC). This work hasbeen supported by Grant-in-Aid for Scientific Research fromMEXT Japan under Grant Nos. 24740269 and 26800171, andin part by RIKEN iTHES Project and Molecular Systems.APPENDIX: THE HYBRIDIZATION FUNCTION OF A GENERAL ANDERSON IMPURITY MODEL As mentioned in Sec. III A , the difference among different Anderson impurity models appears only through the hybridiza-tion function as long as the Anderson impurity terms arethe same. Therefore the hybridization function determines thephysics of Anderson impurity models. In this Appendix, weshall derive the hybridization function for a general Andersonimpurity model described by the Hamiltonian H AIMin Eq. ( 1), and show that indeed the model difference appears throughthe hybridization function. The hybridization function is alsorequired to apply the BL-DMRG method to Anderson impuritymodels in the energy-space representation. To this end, we shall use the path integral formulation for a general Anderson impurity model H AIM. The partition function ZforHAIMis given as Z=/integraldisplay Dd∗DdDc∗Dcexp[−S(d∗,d,c∗,c)], (A1) where S(d∗,d,c∗,c)=S0(d∗,d,c∗,c)+SU(d∗,d), (A2) and S0(d∗,d,c∗,c)=−Triωn/summationdisplay σ(d† σ(iωn),c† σ(iωn)) ×/parenleftbigg iωn−ˆHd −ˆV −ˆV†iωn−ˆHc/parenrightbigg/parenleftbigg dσ(iωn) cσ(iωn)/parenrightbigg . (A3) Here,S0(d∗,d,c∗,c)[SU(d∗,d)] is the one-body part (the two- body part) of the total action S(d∗,d,c∗,c), and d† σ(iωn)=(d∗ 1,σ(iωn),d∗ 2,σ(iωn),..., d∗ M,σ(iωn)) (A4) and c† σ(iωn)=(c∗ 1,σ(iωn),c∗ 2,σ(iωn),..., c∗ N,σ(iωn)) (A5) are Grassmann variables, corresponding to d† m,σandc† n,σ, respectively, at Matsubara frequency iωn.T riωnindicates the sum over the Matsubara frequencies. The matrices ˆHd,ˆHc, and ˆVare defined in Eq. ( 8). Carrying out the Gaussian integrals over variable c∗andcin Eq. ( A1), we obtain an effective action Seff(d∗,d) for variables d∗andd, i.e., Seff(d∗,d)=S0(d∗,d)+SU(d∗,d), (A6) where S0(d∗,d)=−Triωn/summationdisplay σd† σ(iωn) ×(iωn−ˆHd−ˆ/Gamma1(iωn))dσ(iωn)( A 7 ) andˆ/Gamma1(z) is the hybridization function for a complex frequency zdefined as ˆ/Gamma1(z)=ˆV(z−ˆHc)−1ˆV†. (A8) 195109-18BLOCK LANCZOS DENSITY-MATRIX RENORMALIZATION . . . PHYSICAL REVIEW B 90, 195109 (2014) To derive the above formula, we have used the following identity on Grassmann variables: /integraldisplayN/productdisplay i=1dx∗ idxiexp[−x†ˆAx+x†ˆB†y+y†ˆBx] =det(ˆA)e x p [ y†ˆBˆA−1ˆB†y], (A9) where ˆAis a regular N×Nmatrix, ˆBis aM×Nmatrix, and x†=(x∗ 1,x∗ 2,..., x∗ N), (A10) y†=(y∗ 1,y∗ 2,..., y∗ M), (A11) are the vector representations for Grassmann variables x∗ iand y∗ i, respectively. It is now obvious from Eqs. ( A6) and ( A7) that the effective action Seff(d∗,d) for the impurity sites depends on the conduction sites only through the hybridization function ˆ/Gamma1(z). Therefore all properties at the impurity sites are determinedsolely by ˆ/Gamma1(z) when the Anderson impurity term H dis the same. In other words, as long as the impurity propertiesare concerned, any models with the same H dandHU are equivalent if HcandHVgenerates the same ˆ/Gamma1(z). Therefore we can even consider Eqs. ( A6) and ( A7)a sa n effective Anderson impurity model in the complex-frequencyrepresentation which describes exactly the same physics of theoriginal model H AIMin the real-space representation. Next, to derive the relation between the hybridization function ˆ/Gamma1(z) for a complex frequency zand the noninteracting Green’s function, and also the recurrence relation for thenoninteracting Green’s function, we will use the followingbasic matrix algebra. Assuming that matrix ˆXis a regular square matrix, ˆX=/parenleftbiggˆX 11ˆX12 ˆX21ˆX22/parenrightbigg , (A12) with ˆX11being a r×rmatrix, the first r×relements ˆY11of the inverse matrix of ˆXis the inverse of the Schur complement ofˆX22, i.e., ˆY11=/parenleftbigˆX11−ˆX12ˆX−1 22ˆX21/parenrightbig−1, (A13) where /parenleftbiggˆX11ˆX12 ˆX21ˆX22/parenrightbigg/parenleftbiggˆY11ˆY12 ˆY21ˆY22/parenrightbigg =ˆ1, (A14) and we assume that ˆX22is a regular matrix. We shall now derive the formula for the hybridization function ˆ/Gamma1BL(z) in the BL bases a† σandaσ[Eq. ( 15)], which block-tridiagonalize ˆH0in the form of ˆHBL 0,a ss h o w ni n Eq. ( 14). First, notice that since the noninteracting Green’s function for a complex frequency zis defined as ˆG(z)=/parenleftbiggˆGdd(z)ˆGdc(z) ˆGcd(z) ˆGcc(z)/parenrightbigg =/parenleftbigg z−ˆHd−ˆV −ˆV†z−ˆHc/parenrightbigg−1 (A15) in the original conduction site bases c† σand cσ[Eq. (6)], the impurity-site components of the Green’s func- tion, ˆGdd(z), is related to the hybridization function ˆ/Gamma1(z)through ˆ/Gamma1(z)=z−ˆHd−ˆG−1 dd(z). (A16) Next, we introduce the following matrix ˆG(l) BL(z)i nt h eB L bases a† σandaσ, defined as a part of the block matrices in ˆHBL 0: ˆG(l) BL(z)=⎛ ⎜⎜⎝ˆG(l) 11(z)ˆG(l) 12(z)··· ˆG(l) 21(z)ˆG(l) 22(z)··· .........⎞ ⎟⎟⎠ =⎛ ⎜⎜⎜⎝z−ˆE L−l−ˆTL−l··· 0 −ˆT† L−lz−ˆEL−l+1··· 0 ............ 00 ···z−ˆEL⎞ ⎟⎟⎟⎠−1 , (A17) where l=0,1,2,...,L −1 and ˆG(0) BL(z)=ˆG(0) 11(z)=(z− ˆEL)−1. The noninteracting Green’s function is then expressed simply as ˆG(L−1) BL (z) in the BL bases obtained after the Lth BL iteration. Therefore the hybridization function ˆ/Gamma1BL(z)i nt h e BL bases is given as ˆ/Gamma1BL(z)=z−ˆE1−/bracketleftbigˆG(L−1) 11 (z)/bracketrightbig−1. (A18) It is important to notice here that because of the block- tridiagonal form of the matrix ˆG(l) BL(z)i nE q .( A17), the following recurrence relation is satisfied: ˆG(l) 11(z)=/parenleftbig z−ˆEL−l−ˆTL−lˆG(l−1) 11(z)ˆT† L−l/parenrightbig−1.(A19) This can be readily shown by using Eq. ( A13). Finally, using Eqs. ( A18) and ( A19), we obtain the following form for the hybridization function ˆ/Gamma1BL(z) for a complex frequency zin the BL bases: ˆ/Gamma1BL(z)=ˆT1ˆG(L−2) 11 (z)ˆT† 1 =ˆT1[z−ˆE2−ˆT2(z−ˆE3−··· )−1ˆT† 2]−1ˆT† 1.(A20) Clearly, this is a matrix extension of the continued fraction formula [ 87] and a similar formula has been used in the recur- sive Green’s function technique [ 88–90]. The recursive form forˆ/Gamma1BL(z)i nE q .( A20) allows us to evaluate the hybridization function very accurately as compared to the simple fulldiagonalization method since the recursive method can treatmuch larger matrix sizes. Now, recall that ˆG (L−1) BL (z)=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝z−ˆE 1−ˆT10··· 0 −−−−−−−−−−− − −ˆT† 1 0ˆG(L−2) BL (z)−1 ... 0⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠−1 , (A21) 195109-19TOMONORI SHIRAKAWA AND SEIJI YUNOKI PHYSICAL REVIEW B 90, 195109 (2014) and therefore the matrix representation of the local density of states ˆ ρ0(ω)f o rH0at the second BL bases (i.e., at the sites next to the impurity sites in the Q1D model HQ1D AIM) with ˆT1=0 [see also Fig. 1(b)]i s ˆρ0(ω)=−1 πlim δ→0+ˆG(L−1) 22 (z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=ω+iδ,ˆT1=0 =−1 πlim δ→0+ˆG(L−2) 11 (z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=ω+iδ =−1 πlim δ→0+(z−ˆE2−ˆT2(z−··· )−1ˆT† 2)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=ω+iδ. (A22) Here, 0+is positive infinitesimal and we have used Eq. ( A19) in the third equality. Hence we finally obtain the hybridizationfunction ˆ/Delta1(ω) for a real frequency ωas ˆ/Delta1(ω)=−Imˆ/Gamma1BL(ω+i0+) =πˆT1ˆρ0(ω)ˆT† 1, (A23) where we have used Eq. ( A20). 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