Spin responses and effective Hamiltonian for the two dimensional electron gas at oxide interface {LaAlO} 3 /{SrTiO} 3
SSpin responses and effective Hamiltonian for the two dimensional electron gas atoxide interface LaAlO /SrTiO Jianhui Zhou, ∗ Wen-Yu Shan, † and Di Xiao Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
Strong Rashba spin-orbit coupling (SOC) of two-dimensional electron gas (2DEG) at the oxideinterface LaAlO / SrTiO underlies a variety of exotic physics, but its nature is still under debate.We derive an effective Hamiltonian for the 2DEG at the oxide interface LaAlO / SrTiO and find adifferent anisotropic Rashba SOC for the d xz and d yz orbitals. This anisotropic Rashba SOC leadsto anisotropic static spin susceptibilities and also distinctive behavior of the spin Hall conductivity.These unique spin responses may be used to determine the nature of the Rashba SOC experimentallyand shed light on the orbital origin of the 2DEG. PACS numbers: 73.20.–r, 71.70.Ej, 72.25.Mk
The discovery of a high mobility two-dimensional elec-tron gas (2DEG) at the interface between two band insu-lators LaAlO and SrTiO (LAO/STO) [1] has attractedincreasing attention [2]. However, the origin of the 2DEGis still under active debate. According to the intrinsic po-lar catastrophe mechanism, there should be a half elec-tron (per unit cell) transfer from the top surface layerof LAO to the LAO/STO interface. The resulting car-rier density at the interface is roughly 3 . × cm − ,which mainly comes from the three t g orbitals of Ti inSTO. Several transport experiments, however, estimatethat the carrier density is only 10% of that due to thepolar catastrophe mechanism [3–5]. In addition, it hasbeen proposed that electrons in the d xy orbitals, whichare confined in the xy plane, are more likely to become lo-calized at the interface due to the impurities or electron-phonon coupling, while those in the d xz and d yz orbitalsare itinerant and contribute to transport [6]. Within thisscenario, the localized and itinerant electrons would ac-count for the observed magnetic order [7] and supercon-ductivity [8–11], respectively. It is therefore important tounderstand the transport properties of the 2DEG. Othermechanisms, such as oxygen vacancies [12, 13] and polardistortion [14, 15], have also been proposed.Recent magnetotransport experiments have providedus insight into the 2DEG at the oxide interface. In par-ticular, a strong and field-tunable Rashba spin-orbit cou-pling (SOC) was observed [16, 17] and was modeled usingthe standard k -linear form [18], i.e., H R = λ R ( k × σ ) · ˆ z . (1)Based on this k -linear Rashba SOC, theoretical workshave predicted a variety of unusual effects, such as Fulde-Ferrell-Larkin-Ovchinikov-type superconductivity coex-isting with ferromagnetism [19], spiral magnetic orderand skyrmions [20–23], and the spin Hall effect [24]. How-ever, a very recent magneto-conductivity measurementhas suggested the possibility of a k -cubed Rashba SOCof the 2DEG at the oxide interface [25, 26]. Accordingly,some authors proposed a k -linear Rashba SOC for the d xy orbital [27, 28] and a k -cubed one for the d xz and d yz orbitals [28]. On the other hand, first-principles cal-culations combined with the envelope function methodhave found an anisotropic nonparabolic spin-split sub-band structure for the d xz and d yz orbitals [29], whichcould not be explained by the standard k -cubed RashbaSOC. Thus, a detailed investigation of the low energy ef-fective model and the nature of the Rashba SOC is highlydesirable.In this Rapid Communication, we present a detailedderivation of the effective Hamiltonian of the 2DEGat the oxide interface. We find a different anisotropicRashba SOC of the following form, H ani R ∝ (cid:0) k x − k y (cid:1) ( k × σ ) · ˆ z, (2)for the d xz and d yz orbitals, and a standard k -linearRashba SOC for the d xy orbital. The anisotropy of theRashba SOC naturally leads to anisotropic spin suscepti-bilities that have been observed experimentally [10, 11].We also show that this anisotropic Rashba SOC resultsin different behavior of the spin Hall conductivity (SHC)when compared to the standard k -linear and k -cubedRashba SOCs. These distinctive spin responses can beused for determining the nature of the Rashba SOC inexperiments and to shed light on the orbital origin of the2DEG at the LAO/STO interface.We begin by constructing the low-energy effectivemodel of the 2DEG at the LAO/STO interface aroundthe Γ point in the Brillouin zone. The 2DEG is formedfrom the d orbitals of the transition-metal Ti. Here wefocus on the three t g orbitals, namely, d xy , d xz , and d yz ,since the e g orbitals are pushed up about 2 eV higherthan the t g orbitals by the octahedral crystal field. Onthe xy plane, electrons in the d xy orbital can hop alongeither the x or y direction to the d xy orbitals on the neigh-boring Ti, while electrons in the d xz ( d yz ) orbital can hopto its neighbor only along the x ( y ) direction. Thus, thecorresponding hopping Hamiltonian can be expressed inthe following matrix form, a r X i v : . [ c ond - m a t . m e s - h a ll ] J un H = h ( k ) 0 00 − t cos k x
00 0 − t cos k y , (3)where h ( k ) = − ∆ E − t (cos k x + cos k y ), t = t pd / ∆ pd isthe effective hopping parameter between nearest neigh-boring Ti, ∆ pd is the splitting between the oxygen p andTi t g energy levels, and ∆ E is the difference in the on-site energies between the d xy orbital and the d yz / d xz or-bital. Note that since d xy is even, and d xz and d yz areodd under the operation z → − z , hopping between thesetwo sets of orbitals is prohibited in the presence of themirror symmetry.To model the effect of the SOC, we introducethe atomic SOC H ξ = ξ l · σ in the basis {| d xy ↑(cid:105) , | d xy ↓(cid:105) , | d xz ↑(cid:105) , | d xz ↓(cid:105) , | d yz ↑(cid:105) , | d yz ↓(cid:105)} , H ξ = ξ − i − i − i − i i i − i − i , (4)where ξ denotes the strength of the atomic SOC. σ refersto the spin degree of freedom, while l is the orbital an-gular momentum of the electron.Finally, there is a mirror symmetry breaking at theinterface due to the polar displacement of Sr and Tiatoms relative to the oxygen octahedra, which leads tothe Rashba SOC. Physically, the mirror symmetry break-ing can induce the hopping process from the d xz ( d yz )orbital to the d xy orbital via the p x ( p y ) orbital of oxy-gen. The corresponding Hamiltonian can be written as[27, 28] H γ = γ − i sin k y − i sin k x i sin k y i sin k x ⊗ σ , (5)where γ refers to the effective hopping amplitude betweenthe d xy orbital and the d xz and d yz orbitals. σ is the2 × H TB = H + H ξ + H γ . There are three pairs of degenerate bands at the Γ point,which are plotted in Fig. 1(a) using the parameters givenin Ref. 28. It can be seen that the energy contour of themiddle two bands has a strong anisotropy as shown inFig. 1(b), whereas the lowest two bands are isotropic asshown in Fig. 1(c). Note that the splitting of the twolowest-energy bands due to the Rashba SOC is unnotice-able for the given energy. To derive the effective Hamiltonian, we apply thequasidegenerate perturbation theory [30]. Up to lead-ing order in the SOC strength ξ , we obtain the effectiveHamiltonian for the top pair of bands, H top ( k ) = k m top − α top ( k × σ ) · ˆ z, (6)the middle pair of bands, H mid ( k ) = k m mid + α mid (cid:0) k x − k y (cid:1) ( k × σ ) · ˆ z, (7)and the bottom pair of bands, H bot ( k ) = k m bot − α bot ( k × σ ) · ˆ z, (8)where ( m top , m mid , m bot ) and ( α top , α mid , α bot ) are theeffective masses and Rashba SOC strengths for the top,middle, and bottom pairs of bands, respectively (all theirspecific expressions are given in the Supplemental Ma-terial [31]). The top pair of bands is a mixture of allthree t g orbitals. The bottom pair mainly comes fromthe d xy orbital. The middle pair is a hybridization be-tween the d xz orbital and the d yz orbital. It is also clearthat the bottom pair of bands has the k -linear RashbaSOC that was proposed by previous works [27, 28]. Thisconcentric isotropic Fermi contour of the d xy orbital hadalso been demonstrated at the surface of bare SrTiO [32]. In the middle pair of bands, the Rashba SOCbecomes anisotropic and has a k -cubed energy disper-sion [33]. Two recent angle-resolved photoemission ex-periments had already observed the anisotropic Fermicontour of the d xz orbital and the d yz orbital at a highcarrier density [34, 35]. Note that the effective Hamilto-nian of each pair of bands is constructed with respect toits own bottom edge.The anisotropic Rashba SOC for the d xz and d yz or-bitals in Eq. (7) is our main result. In the rest of thisRapid Communication, we will study its effects on thestatic spin susceptibility and the spin Hall conductiv-ity [36, 37]. For convenience, we redefine the corre-sponding effective mass m = m mid / (cid:126) and Rashba SOCstrength β = α mid / (cid:126) . The effective Hamiltonian for themiddle pair can be recast into H mid ( k ) = (cid:32) (cid:126) k m − iβ (cid:126) (cid:0) k x − k y (cid:1) k − iβ (cid:126) (cid:0) k x − k y (cid:1) k + (cid:126) k m (cid:33) , (9)with k ± = k x ± ik y . Some simple algebra leads to theeigenvalues of H mid ( k ), ε k s = (cid:126) k m + sβ (cid:126) k | cos 2 θ k | , (10)and the corresponding eigenvectors, φ k s = 1 L e i k · r η k s , (11) -2024 ΓΧΜ
E/t=-3E/t=-2(c)(b) E /t (a) Γ FIG. 1. (Color online) (a) Band structure of TB model de-scribing the oxide interface. Energy contours near the Γ pointfor energies (b)
E/t = − E/t = −
3. Parametersare adopted from Ref. 28: ∆ E /t = − . ξ/t = 0 . γ/t = 0 . where the spinor is given by η k s = (cid:0) − isς k e − iθ k , (cid:1) T / √ s = ± L is the area ofthe 2DEG with ς k = cos 2 θ k / | cos 2 θ k | = ± θ k =arctan( k y /k x ). Since our model is only valid around theΓ point, we would like to introduce a momentum cutoff k c = 1 / m (cid:126) β via the turning point of the energy disper-sion ε k = k (cid:126) / m − β (cid:126) k . The corresponding energyof this turning point is given by ε turn = 1 / m β .In general, the free spin susceptibilities can be writtenas χ ij ( q ) = − k B T µ B (cid:88) n, k Tr [ σ i G ( k , ω n ) σ j G ( k + q , ω n )] , (12)where σ i are the Pauli matrices with i = x, y, z , G ( k , ω n )is the Matsubara Green’s function of an electron withmomentum k and frequency ω n , and µ B is the Bohrmagneton. After carrying out the standard analytic con-tinuation and frequency summation (more details of thederivation can be found in the Appendix of Ref. 39), wecan find the static spin susceptibilities in the limit q → , χ zz = − µ B (cid:88) k f ( ξ + ( k )) − f ( ξ − ( k )) ξ + ( k ) − ξ − ( k ) , (13) χ xx = − µ B (cid:88) k ,λ ∂f ( ξ λ ( k )) ∂ξ λ ( k ) + χ zz , (14) χ yy = χ xx , (15)where ξ λ ( k ) = ε λ ( k ) − E F is the energy of the electronmeasured relative to the Fermi energy E F , and the su-perscript 0 indicates the spin susceptibility with q = . χ / µ Β E F / ε turn χ xx χ zz FIG. 2. (Color online) The spin susceptibility of 2DEG withthe anisotropic Rashba SOC as a function of the Fermi energy E F in units of ε turn (measured from the bottom of the middlepair of bands). We set the dimensionless effective mass of theelectron m = 1, the dimensionless Rashba SOC parameter β = 0 .
01, and the temperature T =5 K. All of the other components vanish due to the symme-try of Fermi surface. The out-of-plane component χ zz isthe so-called van Vleck susceptibility and originates fromthe virtual inter band transition. The in-plane compo-nent χ xx or χ yy contains both the intraband contribution(the Pauli susceptibility) and the interband contribution(the van Vleck susceptibility).Numerical calculations of the spin susceptibilities of2DEGs with the anisotropic Rashba SOC show two mainfeatures, as shown in Fig. 2. First, the spin suscepti-bilities are anisotropic, i.e., χ zz (cid:54) = χ xx . Secondly, thespin susceptibilities have a strong Fermi energy depen-dence. Note that the momentum cutoff k c is used in ournumerical calculations.Previously the anisotropic spin susceptibility was alsofound using the k -linear Rashba model [38]. However,the spin susceptibility is anisotropic when only the lowerRashba spin-split band is occupied. As soon as both spin-split bands are occupied, the spin susceptibility becomesisotropic [39]. As such, the anisotropy only shows up in asmall energy window. In contrast, the spin susceptibilityin our model is always anisotropic (up to the turningpoint when the model is no longer valid) [40]. Therefore,our result may provide an alternative explanation for theobserved magnetic anisotropy [10, 11].Let us now turn to calculate the SHC of the 2DEGwith the anisotropic Rashba SOC. The general spin con-ductivity tensor in the spin space is given as σ σ i αx = (cid:126) πL (cid:88) k Tr (cid:104) J σ i α ˜ K x (cid:105) , (16)where ˜ K x ≡ U K x U † is the vertex function in the spinspace and K x = ˜ G R J x ˜ G A is the vertex function in theeigenvectors space of H mid ( k ). ˜ G R and ˜ G A are the re-tarded and advanced Green’s function of 2DEG,˜ G A k s ( (cid:15) ) = 1 (cid:15) − ε k s − iη , ˜ G R k s ( (cid:15) ) = 1 (cid:15) − ε k s + iη , (17)where η is a positive infinitesimal. J α stands for thevelocity operator in the eigenvectors space and is givenby J α = U † j α U , where j α = ev α is the current operatorof electron in the spin space and v α = ∂H mid /∂ ( (cid:126) k α )refers to the velocity operator with α = x, y . The spincurrent operators are represented by J σ i α = (cid:126) { v α , σ i } , (18)where { A, B } ≡ AB + BA is an anti-commutator, andthe 2 × U = 1 √ (cid:18) − iς k e − iθ k iς k e − iθ k (cid:19) . (19)After taking the trace over the spin degree of freedom,we have the nonzero component of the intrinsic SHC as σ σ z yx = ebλ (cid:126) π (cid:90) k c k dkE F − (cid:126) k / m (cid:90) π ς k sin θ k × cos 2 θ k [ δ ( E F − ε k − ) − δ ( E F − ε k + )] dθ k , (20)which indicates that a spin Hall current along the y di-rection and polarized in the z direction may exist whenan external electric field is applied along the x direc-tion. The symbols b and λ are defined as b = (cid:126) /m and λ = β (cid:126) , respectively. In the weak anisotropy limit[ β (cid:28) (2 m (cid:126) k F ) − ], we keep the leading-order contribu-tion to the intrinsic SHC and find σ σ z yx = − e π , (21)which is identical to that of the 2DEG with k -linearRashba SOC [36] but is different from 2DEG with the k -cubed Rashba SOC [41, 42]. The vanishment of theother components of the spin conductivity tensor is dueto the symmetry of the Fermi surface.Now we consider the impact of disorder on the SHCup to the vertex correction. It is more convenient toimplement the calculation in the eigenvector space. Weconsider the randomly distributed, identical point defectsthat are spin independent, V ( r ) = σ V (cid:80) i δ ( r − R i ),and the matrix element can be expressed as V ss (cid:48) kk (cid:48) = V L (cid:88) i e − i ( k − k (cid:48) ) · R i (cid:16) ss (cid:48) ς k ς k (cid:48) e − i ( θ k (cid:48) − θ k ) (cid:17) , (22)where V is the strength of defect potential and R i is theposition of the defect. The self-energy in the first-order Born approximation can be written as (cid:10) (cid:104) k s | V G V (cid:12)(cid:12) k (cid:48) s (cid:48) (cid:11)(cid:11) AV = nV L δ kk (cid:48) (cid:88) k s g k s (1 + ss (cid:48) ς k ς k (cid:48) )= δ s s δ kk (cid:48) nV L (cid:88) k s g k s = δ kk (cid:48) δ ss (cid:48) Σ k s , (23)where n = N/L is the density of impurities per unitarea and (cid:104)(cid:104)· · · (cid:105)(cid:105) AV denotes the ensemble averaging overthe impurity distribution. We have introduced the rela-tion of disorder-free Green’s function, (cid:104) k s | G | k s (cid:105) = δ s s δ k k g k s . Thus the disordered Green’s functionturns out to be (cid:10) (cid:104) k s | G (cid:12)(cid:12) k (cid:48) s (cid:48) (cid:11)(cid:11) AV = 1 g − k s − Σ k s δ ss (cid:48) δ k (cid:48) k = ˜ G k s . (24)For the ladder diagram correction to the velocity opera-tor, we have the following iterative equation˜ v xs ,s ( k ) = v xs ,s ( k ) + (cid:88) k (cid:48) (cid:88) s ,s (cid:10)(cid:10) V s s kk (cid:48) V s s k (cid:48) k (cid:11)(cid:11) AV × ˜ G Rs (cid:0) k (cid:48) (cid:1) ˜ G As (cid:0) k (cid:48) (cid:1) ˜ v xs ,s (cid:0) k (cid:48) (cid:1) , (25)where ˜ v x is the corrected velocity operator, s , , , = ± k F (cid:28) β (cid:126) k F (cid:28) (cid:126) k F m , the equation can be approxi-mately solved by keeping the leading order of β , where k F is the Fermi wave vector and Σ k F is the self-energy.After lengthy but straightforward calculations, we canfind the corrected velocity operator (its derivation is pre-sented in the Supplemental Material [31]),˜ v x ( k ) = v x ( k ) + βmE F σ y . (26)Following the similar procedure in Eq. (16), we can cal-culate the SHC with the vertex correction in the weakanisotropy limit and get (cid:2) σ σ z yx (cid:3) V = − e π . (27)It can be seen that in the weak anisotropy limit, thevertex correction reduces the magnitude of SHC by afactor of 2. In fact, this unique feature of SHC under theinfluence of disorder originates from the special form ofRashba SOC. Our result is qualitatively consistent withthe fact that the term αk ( k × σ ) · ˆ z would result in anonzero SHC even with the vertex correction [43]. On theother hand, the vertex correction of disorder can causethe intrinsic SHC of 2DEG with standard k -linear RashbaSOC to vanish identically [44], but does not affect theone with k -cubed Rashba SOC [45]. Hence, the distinctbehaviors of SHC can be used to determine the nature ofthe Rashba SOC at LAO/STO interface.In summary, we have developed an effective Hamilto-nian of the 2DEG at the oxide interface LAO/STO andfound a different anisotropic Rashba SOC. We have foundthat the static spin susceptibilities are anisotropic anddependent on the Fermi energy. 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LaAlO / SrTiO ” In this supplementary material, we provide the detailed derivation of the effective Hamiltonians and the correctedvelocity operator.
THE DERIVATION OF THE EFFECTIVE HAMILTONIANS
In this section, we construct an effective Hamiltonian around Γ point. We first decompose the TB Hamiltonian H TB into two parts H TB ≈ H ( k = 0) + H ( k ) , (28)where H ( k = 0) and H ( k ) stand for the Hamiltonian at the exact Γ point and its deviation, respectively. In a newbasis {| d xy ↑(cid:105) , | d xz ↓(cid:105) , | d yz ↓(cid:105) , | d xy ↓(cid:105) , | d xz ↑(cid:105) , | d yz ↑(cid:105)} , H ( k = 0) and H ( k ) take the following form H ( k = 0) = − ∆ E − t − iξ ξ iξ − t iξ ξ − iξ − t − ∆ E − t − iξ − ξ iξ − t − iξ − ξ iξ − t (29)and H ( k ) = tk − iγk y − iγk x tk x iγk y tk y iγk x − iγk y − iγk x tk iγk y tk x iγk x tk y . (30)It can be seen that in H ( k ) we just keep the leading order term in k for each component with k = k x + k y . At theΓ point, we can get the two-fold degenerate eigenvalues E t = − ξ − t (31) E m = − ∆ E − t + ξ (cid:115)(cid:18) ∆ E + 2 t + ξ (cid:19) + 2 ξ (32) E b = − ∆ E − t + ξ − (cid:115)(cid:18) ∆ E + 2 t + ξ (cid:19) + 2 ξ (33)and their eigenstates ψ t = C ( θ | d xy , ↑(cid:105) + i | d xz , ↓(cid:105) + | d yz , ↓(cid:105) ) (34) ψ t = C ( − θ | d xy , ↓(cid:105) − i | d xz , ↑(cid:105) + | d yz , ↑(cid:105) ) (35) ψ m = 1 √ − i | d xz , ↓(cid:105) + | d yz , ↓(cid:105) ) (36) ψ m = 1 √ i | d xz , ↑(cid:105) + | d yz , ↑(cid:105) ) (37) ψ b = C ( θ | d xy , ↑(cid:105) + i | d xz , ↓(cid:105) + | d yz , ↓(cid:105) ) (38) ψ b = C ( − θ | d xy , ↓(cid:105) − i | d xz , ↑(cid:105) + | d yz , ↑(cid:105) ) , (39)where the parameters are given by θ = E t + 2 t − ξξ , θ = E b + 2 t − ξξ , C , = 1 (cid:113) θ , + 2 . (40)Here the subscripts t, m, b refer to top, middle and bottom bands, respectively. In the basis { ψ t , ψ t , ψ m , ψ m , ψ b , ψ b } , the Hamiltonian H TB can be expressed as H TB = t k it k − t (cid:0) k x − k y (cid:1) it k + t k it k − t k − it k − t (cid:0) k x − k y (cid:1) − it k + t k tk / t (cid:0) k x − k y (cid:1) it k + tk / − it k − t (cid:0) k x − k y (cid:1) ∗ t k it k − t k , where the parameters are t = C (cid:0) θ (cid:1) t , t = − C γθ , t = − C t/ √ t = − C γθ , t = − C t/ √ t = − C θ γ/ √ t = C C (1 + θ θ ) t , t = − γC C ( θ + θ ), t = − C θ γ/ √ t = C (cid:0) θ (cid:1) t .We follow the standard quasi-degenerate perturbation theory and seperate the Hamiltonian into the unperturbatedand perturbated part H TB = H np + H per . The corresponding matrix elements of perturbation Hamiltonian H per read (cid:104) ψ t | H per | ψ t (cid:105) = t (cid:0) k x − k y (cid:1) + t k E m − E t + t k + t k E b − E t (41) (cid:104) ψ t | H per | ψ t (cid:105) = 2 it t (cid:0) k x − k y (cid:1) k + + t k E m − E t + 2 it t k k − E b − E t (42) (cid:104) ψ m | H per | ψ m (cid:105) = t (cid:0) k x − k y (cid:1) + t k E t − E m + t (cid:0) k x − k y (cid:1) + t k E b − E m (43) (cid:104) ψ m | H per | ψ m (cid:105) = 2 i (cid:18) t t E t − E m + t t E b − E m (cid:19) (cid:0) k x − k y (cid:1) k + (44) (cid:104) ψ b | H per | ψ b (cid:105) = t (cid:0) k x − k y (cid:1) + t k E t − E b + t k + t k E m − E b (45) (cid:104) ψ b | H per | ψ b (cid:105) = 2 it t k k − E t − E b + 2 it t (cid:0) k x − k y (cid:1) k + E m − E b (46)and the others can be obtained by the relations (cid:104) ψ a | H per | ψ a (cid:105) = (cid:104) ψ a | H per | ψ a (cid:105) (47) (cid:104) ψ a | H per | ψ a (cid:105) (cid:63) = (cid:104) ψ a | H per | ψ a (cid:105) , (48)where a = t, m, b are the band indices. Up to the leading order of the Rashba SOC, we could get the effectiveHamiltonian for the top pair of bands H t ( k ) = (cid:32) k m t iα t k − − iα t k + k m t (cid:33) , (49)for the middle pair of bands H m ( k ) = (cid:32) k m m iα m (cid:0) k x − k y (cid:1) k + − iα m (cid:0) k x − k y (cid:1) k − k m m (cid:33) , (50)and for the bottom pair of bands H b ( k ) = (cid:32) k m b iα b k − − iα b k + k m b (cid:33) , (51)where the effective masses m a and Rashba SOC strength α a ( a = t, m, b ) are given by12 m t = t + t E m − E t + t E b − E t (52)12 m m = t t E t − E m + t E b − E m (53)12 m b = t + t E t − E b + t E m − E b (54) α t = t , α b = t (55) α m = 2 i (cid:18) t t E t − E m + t t E b − E m (cid:19) . (56) THE DERIVATION OF THE CORRECTED VELOCITY OPERATOR
In this section, we turn to give a detailed calculation of vertex correction. As shown in the main text, we have theiterative equation ˜ v xs ,s ( k ) = v xs ,s ( k ) + (cid:88) k (cid:48) (cid:88) s ,s (cid:10)(cid:10) V s s kk (cid:48) V s s k (cid:48) k (cid:11)(cid:11) AV × ˜ G Rs (cid:0) k (cid:48) (cid:1) ˜ G As (cid:0) k (cid:48) (cid:1) ˜ v xs ,s (cid:0) k (cid:48) (cid:1) , (57)where Green’s function ˜ G R/As = 1 / ( E F − (cid:15) k s ∓ i ImΣ k s ) with ImΣ k s ≡ (cid:126) / τ k s and s , , , = ±
1. Eigenvalue reads (cid:15) k s = (cid:126) k m + sβ (cid:126) k | cos 2 θ k | , (58)and wave function reads φ k s = e i k · r √ L (cid:18) − isζ k e − iθ k (cid:19) , ζ k = cos 2 θ k / | cos 2 θ k | . (59)Velocity matrix element in the eigenvector space reads (cid:18) ( v x k ) ++ ( v x k ) + − ( v x k ) − + ( v x k ) −− (cid:19) = (cid:32) (cid:126) km cos θ k + β (cid:126) k ζ k (cos 3 θ k + 5 cos θ k ) iβ (cid:126) k ζ k (sin 3 θ k − sin θ k ) − iβ (cid:126) k ζ k (sin 3 θ k − sin θ k ) (cid:126) km cos θ k − β (cid:126) k ζ k (cos 3 θ k + 5 cos θ k ) (cid:33) , (60)where ( v x k ) ss (cid:48) ≡ (cid:104) φ k s | v x | φ k s (cid:48) (cid:105) . Based on these, one can evaluate the disorder-averaged correlation function (cid:68)(cid:68) V ++ k , k (cid:48) V ++ k (cid:48) , k (cid:69)(cid:69) AV = (cid:68)(cid:68) V −− k , k (cid:48) V −− k (cid:48) , k (cid:69)(cid:69) AV = (cid:68)(cid:68) V −− k (cid:48) , k V ++ k , k (cid:48) (cid:69)(cid:69) AV = nV L (1 + ζ k ζ k (cid:48) cos( θ k − θ k (cid:48) )) , (61) (cid:68)(cid:68) V + − k , k (cid:48) V − + k (cid:48) , k (cid:69)(cid:69) AV = (cid:68)(cid:68) V − + k , k (cid:48) V + − k (cid:48) , k (cid:69)(cid:69) AV = (cid:68)(cid:68) V + − k (cid:48) , k V + − k , k (cid:48) (cid:69)(cid:69) AV = nV L (1 − ζ k ζ k (cid:48) cos( θ k − θ k (cid:48) )) , (62) (cid:68)(cid:68) V − + k (cid:48) , k V ++ k , k (cid:48) (cid:69)(cid:69) AV = − (cid:68)(cid:68) V ++ k (cid:48) , k V + − k , k (cid:48) (cid:69)(cid:69) AV = − (cid:68)(cid:68) V −− k (cid:48) , k V + − k , k (cid:48) (cid:69)(cid:69) AV = nV L iζ k ζ k (cid:48) sin( θ k − θ k (cid:48) ) . (63)Now we focus on the weak scattering and weak anisotropy limit, i.e., ImΣ k F (cid:28) β (cid:126) k F (cid:28) (cid:126) k F m . In this sense, we havethe approximated expression of Fermi wave vector and density of states k s,F ≈ √ mE F (cid:126) − s βm E F (cid:126) | cos 2 θ k | , (64) N s,F ≈ m π (cid:126) − s βm π (cid:126) (cid:112) mE F , (65)where k ± ,F and N ± ,F are the Fermi wave vector and density of states for two branches of bands. And the relaxationtime τ k s is given by1 τ k , + = 1 τ k , − = 2 π (cid:126) (cid:88) k (cid:48) (cid:68)(cid:68) V ++ k , k (cid:48) V ++ k (cid:48) , k (cid:69)(cid:69) AV δ ( E F − (cid:15) k (cid:48) , + ) + 2 π (cid:126) (cid:88) k (cid:48) (cid:68)(cid:68) V + − k , k (cid:48) V − + k (cid:48) , k (cid:69)(cid:69) AV δ ( E F − (cid:15) k (cid:48) , − )= nV m (cid:126) . (66)According to Eq. (60), it is natural to assume that the modified velocity takes the following form(˜ v x k ) ++ = ( βA ζ k + A k ) cos θ k + βk ζ k ( A cos 3 θ k + A cos θ k ) , (67)(˜ v x k ) + − = βB ζ k sin θ k + βk ζ k ( B sin 3 θ k + B sin θ k ) , (68)(˜ v x k ) − + = βC ζ k sin θ k + βk ζ k ( C sin 3 θ k + C sin θ k ) , (69)(˜ v x k ) −− = ( βD ζ k + D k ) cos θ k + βk ζ k ( D cos 3 θ k + D cos θ k ) . (70)Substitute these matrix elements into Eq. (57), and by using the relations (cid:90) kdk π ˜ G R k , + ˜ G A k , − = (cid:90) d(cid:15) k , − (cid:126) m − β (cid:126) k | cos 2 θ k | ) iδ ( E F − (cid:15) k , − ) E F − (cid:15) k , − − β (cid:126) k | cos 2 θ k |− (cid:90) d(cid:15) k , + (cid:126) m + 3 β (cid:126) k | cos 2 θ k | ) iδ ( E F − (cid:15) k , + ) E F − (cid:15) k , + + 2 β (cid:126) k | cos 2 θ k | = − m (cid:126) i β (cid:126) k F | cos 2 θ k | , (71) (cid:90) kdk π ˜ G R k , − ˜ G A k , + = m (cid:126) i β (cid:126) k F | cos 2 θ k | , (72)one finally obtains these coefficients of A i , B i , C i and D i , A = mE F , A = (cid:126) m , A = (cid:126) , A = 5 (cid:126) ,B = imE F , B = i (cid:126) , B = − i (cid:126) ,C = − imE F , C = − i (cid:126) , C = i (cid:126) ,D = − mE F , D = (cid:126) m , D = − (cid:126) , D = − (cid:126) . (73)This indicates a relation between the unmodified and modified velocities (in eigenvector space) (cid:18) (˜ v x k ) ++ (˜ v x k ) + − (˜ v x k ) − + (˜ v x k ) −− (cid:19) = (cid:18) ( v x k ) ++ ( v x k ) + − ( v x k ) − + ( v x k ) −− (cid:19) + βmE F ζ k (cid:18) cos θ k i sin θ k − i sin θ k − cos θ k (cid:19) . (74)0When transformed into the spin space, this means˜ v x k = v x k + βmE F σ y ,,