Analytically solvable model to the spin Hall effect with Rashba and Dresselhaus spin-orbit couplings
Rui Zhang, Yuan-Chuan Biao, Wen-Long You, Xiao-Guang Wang, Yu-Yu Zhang
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Analytically solvable model to the spin Hall effect with Rashba and Dresselhausspin-orbit couplings
Rui Zhang, Yuan-Chuan Biao, Wen-Long You, Xiao-Guang Wang, Yu-Yu Zhang, ∗ and Zi-Xiang Hu † Department of Physics, Chongqing University, Chongqing 401331, People’s Republic of China College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, People’s Republic of China Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China (Dated: January 7, 2021)When the Rashba and Dresslhaus spin-orbit coupling are both presented for a two-dimensionalelectron in a perpendicular magnetic field, a striking resemblance to anisotropic quantum Rabi modelin quantum optics is found. We perform a generalized Rashba coupling approximation to obtain asolvable Hamiltonian by keeping the nearest-mixing terms of Laudau states, which is reformulatedin the similar form to that with only Rashba coupling. Each Landau state becomes a new displaced-Fock state with a displacement shift instead of the original Harmonic oscillator Fock state, yieldingeigenstates in closed form. Analytical energies are consistent with numerical ones in a wide rangeof coupling strength even for a strong Zeeman splitting. In the presence of an electric field, thespin conductance and the charge conductance obtained analytically are in good agreements withthe numerical results. As the component of the Dresselhaus coupling increases, we find that thespin Hall conductance exhibits a pronounced resonant peak at a larger value of the inverse of themagnetic field. Meanwhile, the charge conductance exhibits a series of plateaus as well as a jumpat the resonant magnetic field. Our method provides an easy-to-implement analytical treatment totwo-dimensional electron gas systems with both types of spin-orbit couplings.
I. INTRODUCTION
Spin-orbit coupling (SOC) enables a wide variety offascinating phenomena, which brings out a new growingresearch field of spin-orbitronics, a branch of spintron-ics that focuses on the manipulation of the electron spindegree of freedom . A prominent example is the spinHall effect, which is the conversion of a spin unpolarizedcharge current into a net spin current without chargeflow . It has been discussed intensively , in whichan electric field induces a transverse spin current. In thepresence of a perpendicular magnetic field, the interplayof Zeeman coupling and various spin-orbit interactionshas stimulated a lot of discussions on resonance spin Hallconductance , which may has potential applicationsin spintronics.There are basically two types of SOCs in nature, i.e.,the Rashba term with structural inversion asymmetry and the Dresselhaus term due to the bulk inversion asym-metry or interface inversion asymmetry . Usually,both types of SOC coexist in a material, such as GaAs-AlGaAs quantum wells and heterostructures , butwhich one plays a major part depends on the propertiesof the material. It has been recognized that Rashba andDresselhaus SOC can interfere with each other, and leadsto a number of interesting phenomenon by tuning the ra-tio between them, such as anisotropic transport , spinsplitting , control of spin precession and light scat-tering . It has been applied experimentally by variousways, such as external electric field, changing tempera-ture, or inserting extra layer etc .In previous work, many efforts have been devotedto the spin Hall effect with the Rashba SOC in two-dimensional electron gas systems (2DEGs), which iscaused by the Laudau level crossing near the Fermi en- ergy. A zero-field spin splitting induced by the RashbaSOC competes with the Zeeman splitting in presence of amagnetic field, and then compromise a resonant spin Halleffect at certain magnetic field . In contrast, Dressel-haus SOC enhances the Zeeman splitting and results ina suppression to the resonance . There is an analyticalsolution for the system with only either Rashba or Dres-selhaus coupling . While considering both of themtogether, an analytical solution is currently not avail-able due to the absence of a full closed-form solution.The perturbation method has been adopted to give riseto an approximated results, which is valid a small ra-tio of the Zeeman energy to the cyclotron frequency .To pursue the intrinsic spin Hall effect induced by thecoexistence of the Dresselhaus and Rashba SOCs, it isdesirable to develop theories which can solve systems in-cluding both types of the SOCs.In this work we develop a generalized Rashba SOC ap-proximation (GRSOCA) to give an analytical solutionto the 2DEGs with both the Rashba and DresselhausSOCs in a magnetic field. Using a displacement trans-formation and an expansion of even and odd functionsof Laudau states up to the nearest-mixing terms, a re-formulated Hamiltonian of the same form as that withonly Rashba term is obtained, resulting in eigenstates inclosed form in the transformed displaced-Fock subspace.The novel displaced-Fock state for each Landau level in-volves the mixing of infinite Laudau level states inducedby two types of SOCs, exhibiting an improvement overoriginal Fock states. Energy levels are obtained explic-itly for arbitrary strengths of both types of SOCs, whichagree well with numerical results in a wide range of cou-pling strength even for a strong Zeeman energy. By com-paring to the system with only Rashba SOC, we find thatthe spin Hall conductance exhibits a pronounced reso-nant peak at a larger value of the inverse of the magneticfield, which arises from the contributions of the Dressel-haus SOC. Resonance originates from the energy degen-eracy near Fermi energy, where the eigenstates consistof the n th displaced-Fock state of spin up and n + 1thdisplaced-Fock state of spin down. A series of plateaus ofthe charge conductance are observed, and a jump occursat the resonant point, which fit well with the numericalones.The paper is outlined as follows. In Sec. II, we deriveexpressions for the quantized Hamiltonian of the 2DEGswith the Rashba and Dresselhaus SOC in a magneticfield. In Sec. III, we obtain the analytical solution ofthe effective Hamiltonian for arbitrary ratio between theRashba and Dresselhaus SOCs. In Sec. IV, we study thecharge and spin Hall conductances analytically with thefirst-order corrections when an electric field is applied.Finally, a brief summary is given in Sec. V. II. HAMILTONIAN
We consider a single electron in a two-dimensional sys-tem subjected to a perpendicular magnetic field ~B = − B ˆ e z = ∇ × ~A , which is confined in the x − y plane ofan area L x × L y . The Hamiltonian in the presence ofspin-orbit coupling is given by ( ~ = 1) H = 12 m ( ~p + ec ~A ) − gµ B Bσ z + H so , (1)where g is the Lande factor of the electron with the ef-fective mass m , µ B is the Bohr magneton, and σ k arethe Pauli matrices. The Laudau gauge is chosen as ~A = B~r × ˆ e z = ( yB, , H so = H D + H R with H R = α (Π x σ y − Π y σ x ) and H D = β (Π x σ x − Π y σ y ), where the canonical momentumis ~ Π = ~p + e ~A/c . The Rashba SOC H R originates fromthe structure inversion asymmetry of the semiconductormaterial and the coupling strength α can be tuned by anelectric field. While the coefficient β of the Dresselhausterm H D is determined by the geometry of the hetereo-structure that stems from the bulk-inversion asymmetryof the semiconductor material.Due to the gauge choice, the system is translationallyinvariant along the x direction, and p x = k is a goodquantum number. The orbit part of wave function isobtained as ψ ( x, y ) = exp( ikx ) ϕ ( y − y ), where ϕ ( y − y )is the harmonic oscillator wave function with the orbitcenter coordinate y = l b k and the magnetic length l b = p ~ c/eB . By introducing the ladder operator a = (Π x + i Π y ) l b / √ a = 1 √ l b [ y + c ( p x + ip y ) eB ] , (2) one obtains the Hamiltonian H = H RSOC + √ βl b ( a † σ − + aσ + ) , (3) H RSOC = ω ( a † a + 12 ) − ∆2 σ z + i √ αl b ( aσ − − a † σ + ) , (4)where ω = eB/mc is the cyclotron frequency and ∆ = gµ B B is the Zeeman splitting. When only the RashbaSOC term is present, i.e. β = 0, the Hamiltonian isreduced to H RSOC which can be written in a matrix formin the basis {| ↓ , n i , | ↑ , n + 1 i} H RSOC = (cid:18) ω ( n + 1 /
2) + ∆ / i p n + 1) α/l b − i p n + 1) α/l b ω ( n + 1 + 1 / − ∆ / (cid:19) . (5)This Hamiltonian, like a Rabi model with only thecounter-rotating wave term in the quantum optics, can besolved analytically in closed subspace, which is so-calledRashba SOC (RSOC) approximation. However, once in-cluding the additional Dresselhaus SOC term (rotatingwave term in Rabi model), the subspace related to n isnot closed, rendering the complication of the solution. Inthat case, each Landau level is coupled to infinite num-ber of other Landau levels, and thus the exact analyticsolution is not available. III. ANALYTICAL SOLUTION
Following previous section, the Hamiltonian ( 3) ofa two-dimensional electron with both the Rashba andDresselhaus couplings can map onto the anisotropic Rabimodel in quantum optics, which have been studied exten-sively by various approximate analytical solutions .The crucial is to establish a new set of basis states.To facilitate the study, we write the Hamiltonian as H = ω ( a † a + 12 ) − ∆2 σ z + g σ x ( a † e − iθ + ae iθ ) − g iσ y ( a † e iθ − ae − iθ ) , (6)with g = √ p β + α /l b , and e iθ = ( β + iα ) / p β + α . By performing a unitary transformation U = exp[ σ x ( a † γ − aγ ∗ )] with a dimensionless variationaldisplacement γ ( γ ∗ ), we obtain a transformed Hamilto-nian H = U HU † = H ′ + H ′ , H ′ = ωa † a + η + σ z { η cosh[2( a † γ − aγ ∗ )]+ g sinh[2( a † γ − aγ ∗ )]( a † e iθ − ae − iθ ) } , (7) H ′ = σ x [ a † ( g e − iθ − ωγ ) + a ( g e iθ − ωγ ∗ )]+ iσ y {− η sinh[2( a † γ − aγ ∗ )] − g cosh[2( a † γ − aγ ∗ )]( a † e iθ − ae − iθ ) } , (8)where η = ω/ − g ( γ ∗ e − iθ + γe iθ ) + ωγγ ∗ and η = − ∆ / − g ( γe − iθ − γ ∗ e iθ ) in Appendix A. The displace-ment shift γ ( γ ∗ ) is associated with the Rashba SOCand Dresselhaus SOC strengths, which captures the dis-placement of the harmonic oscillator states for essentialphysics.Since the even hyperbolic cosine function can be ex-panded as cosh[2( a † γ − aγ ∗ )] = 1 + [2( a † γ − aγ ∗ )] + [2( a † γ − aγ ∗ )] + · · · , it is approximated by keeping theterms which only contain the number operator ˆ n = a † a as cosh[2( a † γ − aγ ∗ )] = G ( a † a ) + O ( γ γ ∗ ) . (9)The coefficient G ( a † a ) can be expressed in the harmonicoscillator basis | n i as G n,n = h n | cosh[2( a † γ − aγ ∗ )] | n i = e − γγ ∗ L n (4 γγ ∗ ) , (10)with the Laguerre polynomials L m − nn ( x ) = P min { m,n } i =0 ( − n − i m ! x n − i ( m − i )!( n − i )! i ! . Here, the higherorder excitations such as a † , a , · · · are neglected in theapproximation. Similarly, we expand the odd functionsinh[2( a † γ − aγ ∗ )] by keeping the one-excitation termsassinh[2( a † γ − aγ ∗ )] = R ( a † a ) a † − aR ( a † a ) + O ( γ γ ∗ ) . (11)Since the terms R ( a † a ) a † and aR ( a † a ) are conjugated toeach other, which corresponds to create and eliminate asingle excitation of the oscillator, we define R n,n +1 = − √ n + 1 h n | sinh[2( a † γ − aγ ∗ )] | n + 1 i = 2 γ ∗ n + 1 e − γγ ∗ L n (4 γγ ∗ ) = R ∗ n +1 ,n . (12)Similarly, the other operators can be expanded by keep-ing leading terms as follows:sinh[2( a † γ − aγ ∗ )]( a † e iθ − ae − iθ ) = F ( a † a ) + O ( γ γ ∗ ) , (13)cosh[2( a † γ − aγ ∗ )]( a † e iθ − ae − iθ ) ≈ T ( a † a ) a † − aT ( a † a ) , (14)where the coefficients can be expressed in terms of theoscillator basis | n i F n,n = h n | sinh[2( a † γ − aγ ∗ )]( a † e iθ − ae − iθ ) | n i = − e iθ ( n + 1) R n,n +1 − ne − iθ R n,n − , (15) T n,n +1 = − h n | cosh[2( a † γ − aγ ∗ )]( a † e iθ − ae − iθ ) | n + 1 i√ n + 1= e − iθ G n,n − √ n + 2 √ n + 1 e iθ G n,n +2 , (16)with G n,n +2 = h n | cosh[2( a † γ − aγ ∗ )] | n + 2 i =(2 γ ∗ ) exp[ − γγ ∗ ] L n (4 γγ ∗ ) / p ( n + 1)( n + 2). Finally, we obtain the reformulated Hamiltonian ˜ H =˜ H + ˜ H D , consisting of˜ H = ωa † a + η + σ z e ∆ + e αa † σ + + e α ∗ aσ − , (17)˜ H D = e βa † σ − + e β ∗ aσ + , (18)where the Zeeman energy is renormalized as e ∆ = η G ( a † a ) + g F ( a † a ), the effective Rashba and Dressel-haus SOCs strength are derived as e α = { g e − iθ − ωγ − η R ( a † a ) − g T ( a † a ) } and e β = { g e − iθ − ωγ + η R ( a † a )+ g T ( a † a ) } .The form of the transformed Hamiltonian ˜ H by con-sidering contributions of the Rashba and DresselhausSOCs is identical with the original Hamiltonian (6)only with Rashba SOC terms. To obtain the solvableHamiltonian ˜ H , the transformed Dresselhaus terms ˜ H D are required to be vanished by choosing a proper dis-placement γ and γ ∗ . Within the oscillator basis | n i and the eigenstates | ± z i of σ z , the matrix element h n, + z | ˜ H D | n + 1 , − z i equals to be zero. It yields0 = g e iθ − ωγ ∗ + η R n,n +1 + g T n,n +1 . (19)Since the displacement γ ( γ ∗ ) is smaller compared withthe unit, it approximately leads to L n (4 γγ ∗ ) ≃ L n (4 γγ ∗ ) ≃ n + 1, and L n (4 γγ ∗ ) ≃ ( n + 1)( n + 2) / g e iθ − ωγ ∗ + γ ∗ ∆ + g e − iθ = 0, resulting in γ ≈ g β ( ω + ∆) p α + β . (20)We obtain the solvable Hamiltonian ˜ H by consider-ing both of the Rashba and Dresselhaus SOCs, whichretains the Rashba SOC term aσ − and a † σ + . It is so-called GRSOCA. Different from the RSOC approxima-tion, the effective Rashba SOC strength and Zeeman en-ergy are renormalized, which leads to richer physics in-duced by both types of the SOCs. The effective Hamil-tonian obtained by the variational method is expected tobe prior to the original Hamiltonian H RSOC (6) only withthe Rashba SOC terms. The simplicity of the method isbased on its analytical eigenstates and eigenvalues.One can easily diagonalize the effective Hamiltonian˜ H in the basis of | n, − z i and | n + 1 , + z i ˜ H = ωn + e ∆ − ,n √ n + 1 e α n,n +1 √ n + 1 e α ∗ n,n +1 ω ( n + 1) + e ∆ + ,n +1 ! , (21)where the Zeeman energy is transformed into e ∆ ± ,n = η ± f ( n ) with f ( n ) = η G n,n + g F n,n , and the effectiveSOC strength is renormalized as e α n,n +1 = ( g e iθ − ωγ ∗ ) − η R n,n +1 − g T n,n +1 . One obtains approximately f ( n ) ≈ η − g [ e iθ γ ∗ + n ( γe − iθ + γ ∗ e iθ )], and e α n,n +1 ≈ g ( e iθ − e − iθ ) − ωγ ∗ − η γ ∗ .Similar to the Hamiltonian H RSOC in Eq. (5) with onlythe Rashba SOC, the eigenvalues are obtained as E n, ± = ω ( n + 12 ) + 12 [ e ∆ + ,n +1 + e ∆ − ,n ] ± q [ e ∆ + ,n +1 − e ∆ − ,n + ω ] + 4( n + 1) | e α n,n +1 | . (22)And the corresponding eigenstates are expressed in theclosed form as | ϕ + ,n i = cos θ n | n + 1 i | + z i + sin θ n | n i |− z i , (23) | ϕ − ,n i = sin θ n | n + 1 i | + z i − cos θ n | n i |− z i , (24)where θ n = arccos( δ n / p δ n + 4( n + 1) | e α n,n +1 | ) with δ n = ω + e ∆ + ,n +1 − e ∆ − ,n .The ground state is | , + z i with the eigenvalue E = η + ( η − γ ∗ g e iθ ) e − γγ ∗ . (25)As a consequence, the corresponding wave functions ofthe original Hamiltonian H in Eq.(6) can be obtainedusing the unitary transformation as | Ψ ± ,n i = U † | ϕ ± ,n i , | Ψ + ,n i = 1 √ θ n |− γ, n + 1 i d + sin θ n |− γ, n i d ) | + i x +(cos θ n | γ, n + 1 i d − sin θ n | γ, n i d ) |−i x ] , (26)and | Ψ − ,n i = 1 √ θ n |− γ, n + 1 i d − cos θ n |− γ, n i d ) | + i x +(sin θ n | γ, n + 1 i d + cos θ n | γ, n i d ) |−i x ] , (27)where |±i x = ( | + i z ± |−i z ) / √ σ x .Each Laudau state becomes the displaced-Fock state | n i|∓ γ, n i d = e ∓ ( γa † − aγ ∗ ) | n i , (28)which is the displacement transformation of the Fockstate | n i . Especially it reduces to the coherent state |∓ γ, i d = e ∓ ( γa † − aγ ∗ ) | i , which can be expanded as asuperposition state of Fock states. Since the Dressel-haus and Rashba SOCs induce infinite n -th Landau-levelstates coupling, it is challenge to give eigenstates a closedform. Fortunately, the novel displaced-Fock states as anew set of basis states exhibit an improvement over orig-inal Fock states.Fig. 1 displays the first eight energy levels as a functionof the effective coupling strength η R /ω = √ α/ ( l b ω ) for R / E n / R / E n / our method numerics RSOC ==0.6(a) (b) FIG. 1: Energy levels E n /ω obtained analytically (red solidline) as a function of effective coupling strength η R /ω = √ α/ ( l b ω ) for different ration between the Dresselhaus andRashba SOCs strength (a) β/α = 0 . β/α = 1.The results obtained by the numerical exact diagonalizationmethod (black circles) and under the RSOC approximation(blue dashed line) are listed for comparison. The parametersare ∆ /ω = 0 . l b = 1 and ω = 1. various values of the Dresselhaus coupling strength β . Inthe absence of the spin-orbit coupling η R = 0, one ob-serves two separated n th Landau levels induced by theZeeman energy ∆ = 0 . ω , in which the lower level isthe spin-up state and the higher level corresponds to thespin-down electron state. As η R increases, the higherlevel of the n th Landau level state becomes lower dueto the hybridization of the n th and n + 1th displaced-Fock states induced by both types of SOCs. Comparingwith the Rashba SOC approximation, the energy crossingoccurs at a larger value of the coupling strength as a con-sequence of the Dresselhaus SOC. It demonstrates thatthe Dresselhaus SOC enhances Zeeman splitting, whilethe Rashba SOC interplay with the Zeeman splitting inopposite ways.For the ratio between the Dresselhaus and RashbaSOC strengths β/α = 0 .
6, our analytical approach isin good agreement with the numerical results in a widerange of coupling strength η R /ω < . η R /ω = 0 . β = α , in Fig. 1(b), the de-viation becomes more obvious. Because the DresselhausSOC play a more important effect as η R /ω increases, andthe Rashba SOC approximation fails. Therefore, our ap-proach, which takes into account the effects of the Dres-selhaus SOC terms, provides a more accurate analyticalexpression to the energy spectrum of the 2DEG system. IV. SPIN CURRENT WITH A ELECTRICFIELD
Since the competition of the SOC and the Zeeman en-ergy induces an energy crossing, the spin Hall resonanceis closely related to the level crossing. When an externalelectric field is applied, the SOC of the 2DEG inducesthe spin Hall effect, which is the transverse spin cur-rent response to the electric field. As the electric field E is applied along the y axis, the Hamiltonian becomes H = H + eEy with the original Hamiltonian H definedin Eq. (1). Using the replacement of y by y + eE/mω in the oscillator operator a , one obtains the quantizedHamiltonian H = H + H E , H E = − E [ kemω + eω ( ασ y + βσ x )] , (29)where H is given in Eq.(1), and the constant − e E / mω is dropped. Similar to the transformedHamiltonian ˜ H in Eq. (21), we perform the unitarytransformation U = exp[ σ x ( a † γ − aγ ∗ )] to H E , resultingin ˜ H E = U H E U † = − E kemω − E βeω σ x − E αeω { σ y G ( a † a )+ iσ z [ R ( a † a ) a † − aR ( a † a )] } . (30)The wave function for the Hamiltonian with the electricfield can be given to the first-order correction in the per-turbation in ˜ H E as (cid:12)(cid:12)(cid:12) ϕ (1) ± ,n E = | ϕ ± ,n i + X n = k,l h ϕ l,k | ˜ H E | ϕ ± ,n i E n, ± − E l,k | ϕ l,k i , ( l = ± ) , (31)where the eigenvalues E n, ± and eigenstates | ϕ ± ,n i aregiven in Eqs. (22)-(24).The charge current operator of a single electron is givenby j c = − eυ x , (32) υ x = 1 i [ x, H ] = p x m + ωy + ασ y + βσ x , (33)and the spin- z component current operator is j zs = ~ S z υ x + υ x S z )= 12 [ r ω m ( a † + a ) − eEmω ] σ z . (34)The average current density of the N e electron system isgiven by I c ( s ) = 1 L x L y X nl h j c ( s ) i nl f ( E nl ) , ( l = ± , (35)where f ( E nl ) is the Fermi distribution function, and N e = P nl f ( E nl ). The charge Hall conductance is G c ( s ) = I c ( s ) /E. (36) S z our method numerics S z E n E n E n,+ E n,- =0.5=0 Energy crossingEnergy crossing 10 12 14 16 18 209 11 13 15 17 19 (b)(d)(a)(c)
10 12 14 16 18 20 9 11 13 15 17 19 7 9 11 13 15 18 20Energy crossing53 5 7 9 11 13 15Energy crossing18 20
FIG. 2: Energy levels E n and average spin h S z i obtained an-alytically for an electron as a function of 1 /B for β = 0(a)(b)and β/α = 0 . /ω = 0 . α = 0 . ω . The re-sults of h σ z i obtained by the numerical exact diagonalization(black circles) are listed for comparison. Under the first-order perturbation, the correspondingspin/charge current can be expressed as h j c ( s ) i ± ,n = h j (0) c ( s ) i ± n + h j (1) c ( s ) i ± n , where h j (0) c ( s ) i ± n = h ϕ ± ,n | U j c ( s ) U † | ϕ ± ,n i , (37) h j (1) c ( s ) i ± n = X n = k,l h ϕ ± ,n | ˜ H E | ϕ l,k i h ϕ l,k | U j c ( s ) U † | ϕ ± ,n i E k,l − E n, ± + h.c. (38)Under the zeroth approximation, one obtains analyticalsolutions D j (0) c E ± n = e EhN φ L x L y , h j z (0) s i ± ,n = − eE mω h σ z i ± n , (39)where h σ z i ± n is given in the Appendix B. With the av-erage current density I zs , the spin Hall conductance canbe derived under the zeroth order correction by G z (0) s = − h S z i E eEmω = − h S z i G c e , (40) h S z i = X nl h σ z i nl f ( E n,l ) , ( l = ± ) . (41)And the Hall conductance is given as in the Appendix B G c = e N e / (2 πN φ ) , (42)which is only dependent on the filling factor N e /N φ with N φ = L x L y eB/ ( hc ) .Fig. 2 shows energy levels and the spin polarization h S z i under the zeroth approximation. It is observed thatthe energy E n, + with spin-up state firstly enters intothe Fermi energy region, then it gives rise to the en-ergy E n +1 , − with spin-down state in Fig. 2(a)(c). As theenergy gap between E n, + and E n +1 , − becomes smaller,it yields energy crossing at certain magnetic field B ,which is given by E n +1 , − = E n, + in Eq.(22). When themagnetic field exceeds the critical value B , the spin-down state with E n, − emerge firstly, and then the spin-up state with E n +1 , + enters into the Fermi energy region.The corresponding expected value of h S z i is calculated inFig. 2(b)(d). It reaches maxima at odd integers n , andminima at even integers n . A discontinuous jump oc-curs at B . Below the critical value B , the maximalvalue of h σ z i occurs at even integers n . The jump of thespin polarization ascribes to the energy crossing of twoeigenstates with almost opposite spins. Especially, whenonly Rashba SOC is considered ( β = 0), one obtains theconstraint condition for the energy crossing2 ω = q ( ω − ∆) + 4( n + 1) η R + q ( ω − ∆) + 4( n + 2) η R , (43)with the displacement γ = 0 (see the Appendix C). Itrecovers results with only the Rashba coupling . Bycomparing to the results with only Rashba SOC, the crit-ical value of 1 /B shifts to a larger value in Fig. 2(d). Itdemonstrates that the Dresselhaus SOC plays a vital rolein suppressing the energy crossing, which is different fromthe effects of the Rashba SOC.In presence of the electric field, the spin Hall conduc-tance of the spin- z component current is the most inter-esting. Fig. 3 shows the charge conductance G c in Eq.(42) and the spin Hall conductance G z (1) s obtained by thefirst-order corrections in Eq. (38). A series of plateausin the charge G c are visible, and a jump between twoplateaus is observed at the critical magnetic field, wherethe spin conductance G z (1) s becomes divergent with a res-onant peak. The resonance ascribes to the interferenceof two degenerate levels near the Fermi energy. The res-onance point coincides with the jump point of h S z i withthe energy crossing. By comparing to the behaviors withonly the Rashba SOC, the charge G c and spin Hall con-ductance G z (1) s exhibit a shift value of the resonant point,which is induced by the Dresselhaus SOC effects. For alarge Zeeman splitting energy ∆ /ω = 0 .
5, the resonantpoint shifts to a larger value of 1 /B in Fig. 3(b). Itdemonstrate that the SOC interactions and the Zeemansplitting play an opposite role in the energy-levels cross-ing. Fortunately, the charge and spin conductance ob-tained by first-order approximation agree well with thenumerical results, exhibiting the validity of our approach. G sz ( ) G c G sz ( ) G c our method G sz(1) our method G c RSOC G c numerics G c ,G sz(1) (a) / =0.1 =0.5(b) / =0.5 =0.5 FIG. 3: Charge conductance G c and spin Hall conductance G z (1) s obtained with first-order corrections as a function of1 /B for different Zeeman splitting energy (a) ∆ /ω = 0 . /ω = 0 .
5. The ratio between the Dresselhaus andRashba SOCs is β/α = 0 . α = 0 . ω . The results obtained by the numerical exact diag-onalization (black circles) and under the RSOC approxima-tion (blue dotted line) are listed for comparison. The externalelectronic field is E/ω = 0 . N/C . V. CONCLUSION
When both the Rashba and Dresselhaus spin-orbit cou-plings are considered, we find the single electron Hamil-tonian in two-dimensional system subjected to a per-pendicular magnetic field can map onto an anisotropicRabi model. We perform the generalized Rashba SOCapproximation using the displacement unitary transfor-mation, and keep the single Landau level (nearest neigh-bor Landau level mixing) matrix element for even (odd)coupling function, and a solvable Hamiltonian is ob-tained in a similar form as that with only the Rashbaterm. The strengths of the both types of SOCs and Zee-man splitting are absorbed in the displacement-shift vari-able. With comparing the numerical diagonalization, ourmethod provides accurate energy levels up to a large Zee-man splitting. As a consequence of the Dresselhaus andRashba SOCs, each Landau state becomes a displaced-Fock state, which has a displacement shift by comparingto the original Harmonic oscillator Fock state. With theanalytical solved eigenstates, the spin current displays ajump at a larger value of the inverse of the magnetic field,which demonstrates that the Dresselhaus SOC plays anopposite way in the energy splitting by comparing to theRashba SOC. Moreover, in the presence of an electricfield, the spin Hall conductance obtained by the first-order corrections diverges at the resonant point, and aseries of plateaus of the charge conductance are observed,which fit well with numerical results. In conclusion, ourmethod provides an easy-to-implement analytical solu-tion to the 2DEGs with considering all SOCs in which allthe coupling strengths, including Rashba, Dresselhaus,and Zeeman splitting, are described by the displacementshift. This solution could be potentially useful in thefuture studies of the quantum version of the spin Halleffects and the interacting fractional quantum Hall sys-tems.
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (Grants No.12075040, No.11875231,and No.11974064), and by the Chongqing Research Pro-gram of Basic Research and Frontier Technology (GrantsNo.cstc2020jcyj-msxmX0890).
Appendix A: Deviation of the Hamiltonian by thedisplacement transformation
We perform the unitrary transformation U =exp[ σ x ( a † γ − aγ ∗ )] to the Hamiltonian H in Eq. (6). Oneeasily obtains U aU † = a − γσ x and U a † U † = a † − γ ∗ σ x .The first and second terms of H in Eq. (6) can be trans-formed into U a † aU † = a † a − σ x ( a † γ + aγ ∗ ) + γγ ∗ , (A1)and U σ z U † = σ z { σ z [2( a † γ − aγ ∗ )] + ... }− iσ y { a † γ − aγ ∗ ) + 13! [2( a † γ − aγ ∗ )] + ... } = σ z cosh[2( a † γ − aγ ∗ )] − iσ y sinh[2( a † γ − aγ ∗ )] . (A2)Meanwhile, two SOCs terms of H are derived explicitlyas U σ x ( a † e − iθ + ae iθ ) U † = σ x ( a † e − iθ + ae iθ ) − ( γ ∗ e − iθ + γe iθ ) , (A3)and U iσ y ( a † e iθ − ae − iθ ) U † = iσ y B − σ z [2 AB − ( γe − iθ − γ ∗ e iθ )]+ 12! iσ y [4 A B − A ( γe − iθ − γ ∗ e iθ )] − σ z [8 A B − A ( γe − iθ − γ ∗ e iθ ) ]+ 14! iσ y [16 A B − A ( γe − iθ − γ ∗ e iθ )] + ... = iσ y [cosh(2 A ) B − ( γe − iθ − γ ∗ e iθ ) sinh(2 A )] − σ z [sinh(2 A ) B − ( γe − iθ − γ ∗ e iθ ) cosh(2 A )] where the operators are given by A = a † γ − aγ ∗ and B = a † e iθ − ae − iθ . Thus, the transformed Hamiltonianis given in terms of H ′ and H ′ in Eqs. (17) and (8).By expanding the even and odd functions cosh[2( a † γ − aγ ∗ )] and sinh[2( a † γ − aγ ∗ )], the corresponding coeffi-cients are derived as G n,n = h n | cosh[2( a † γ − aγ ∗ )] | n i = 12 h n | { exp[2( a † γ − aγ ∗ )] + exp[ − a † γ − aγ ∗ )] } | n i = e − γγ ∗ n X i =0 ( − n − i n !( n − i )!( n − i !) i ! (4 γγ ∗ ) n − i = e − γγ ∗ L n (4 γγ ∗ ) , (A4)and √ n + 1 R n +1 ,n = h n + 1 | sinh[2( a † γ − aγ ∗ )] | n i = 12 h n | { exp[2( a † γ − aγ ∗ )] − exp[ − a † γ − aγ ∗ )] } | n i = 2 γe − γγ ∗ p n !( n + 1)! n X i =0 ( − n − i n !( n + 1)!( n − i )!( n + 1 − i !) i ! (4 γγ ∗ ) n − i = 2 γ √ n + 1 e − γγ ∗ L n (4 γγ ∗ ) . (A5) Appendix B: Spin current under the zerothcorrections
In presence of the electric field, the spin current is de-rived as h j zs i ± n = 12 h ϕ ± ,n | U [ r ~ ω m ( a † + a ) − eEmω ] σ z U † | ϕ ± ,n i . (B1)Using the eigenstates | ϕ ± ,n i in Eqs. (23) and (24), oneobtains h ( a † + a ) σ z i + n = h ϕ + ,n | U ( a † + a ) σ z U † | ϕ + ,n i = h ϕ + ,n | [( a † + a ) − ( γ + γ ∗ ) σ x ] U σ z U † | ϕ + ,n i = h n | ( a † + a ) sinh[2( a † γ − aγ ∗ )] sin ∗ θ n θ n | n + 1 i− h n + 1 | ( a † + a ) sinh[2( a † γ − aγ ∗ )] cos ∗ θ n θ n | n i = 0 , (B2)and h ( a † + a ) σ z i − n = h ϕ − ,n | U ( a † + a ) σ z U † | ϕ − ,n i = 0 . (B3)So the spin current is simplified as h j zs i ± n = − eE mω h σ z i ± n . (B4)where the average value of h σ z i ± n are derived in the fol-lowing h σ z i + n = h ϕ + ,n | U σ z U † | ϕ + ,n i = cos θ n ∗ θ n G n +1 ,n +1 − sin θ n ∗ θ n G n,n −√ n + 1(sin θ n ∗ θ n R n +1 ,n + sin ∗ θ n θ n R n,n +1 ) , (B5)and h σ z i − n = h ϕ − ,n | U σ z U † | ϕ − ,n i = sin θ n ∗ θ n G n +1 ,n +1 − cos θ n ∗ θ n G n,n + √ n + 1(cos θ n ∗ θ n R n +1 ,n + cos ∗ θ n θ n R n,n +1 ) . (B6)Meanwhile, the charge current j c = − ev x can be ex-pressed in terms of the harmonic oscillator a ( a † ) as j c = − e [ r ~ ω m ( a † + a ) + ασ y + βσ x − eEmω ] . (B7)The average value of the charge current is given by h j c i ± n = h ϕ ± ,n | U j c U † | ϕ ± ,n i = h ϕ ± ,n | e Emω | ϕ ± ,n i = e E πN φ L x L y , (B8)where U j c U † = − e [ p ω m ( a † + a ) − p ω m ( γ + γ ∗ ) σ x + ασ y cosh(2 A ) + αiσ z sinh(2 A ) + βσ x − eE/ ( mω )]. Thenwe obtain the charge Hall conductance G c = I c /E as G c = e E πEN φ X nl f ( E nl ) = e N e πN φ . (B9) Appendix C: energy-crossing conditions
Since the n + 1-th Laudau level of spin down and the n -th Laudau level of spin-up becomes crossing near theFermi energy, resulting in the resonance peak of the spinHall conductance at the energy crossing point. It leadsto the energy crossing at certain magnetic field B , whichsatisfies E n +1 , − = E n, + . It yields2 ω + f ( n + 2) − f ( n + 1) + f ( n )= q [ f ( n + 1) + f ( n ) + ω ] + 4( n + 1) | e α n,n +1 | + q [ f ( n + 2) + f ( n + 1) + ω ] + 4( n + 2) | e α n +1 ,n +2 | . (C1)Especially, when the Dresshaul SOC is neglected by set-ting β = 0, the displacement variable reduces into γ = 0.One can simplify the parameters as R n,n +1 = R n +1 ,n =0, T n,n +1 = − iα/ p α + β , η = ω/ η = − ∆ / f ( n ) = − ∆ /
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