In semiconductors with inversion asymmetry, spin-orbit coupling gives rise to
the well-known Dresselhaus and Rashba effects. If one considers quantum wells
with two or more conduction subbands, an additional, intersubband-induced
spin-orbit term appears whose strength is comparable to the Rashba coupling,
and which remains finite for symmetric structures. We show that the conduction
band spin splitting due to this intersubband spin-orbit coupling term is
negligible for typical III-V quantum wells

D'Amico, I.

Kyrychenko, F. V.

Ullrich, C. A.

English

arXiv

https://mospace.umsystem.edu/jspui/submit#nullIn semiconductors with inversion asymmetry, spin-orbit coupling gives rise to the well-known Dresselhaus and Rashba effects. If one considers quantum wells with two or more conduction subbands, an additional, intersubband-induced spin-orbit term appears whose strength is comparable to the Rashba coupling, and which remains finite for symmetric structures. We show that the conduction band spin splitting due to this intersubband spin-orbit coupling term is negligible for typical III-V quantum wells.This work was supported by DOE Grant No.
DE-FG02-05ER46213, the Nuffield Foundation Grant NAL/01070/G, and by the Research Fund 10024601 of the Department of Physics of the University of York

Ullrich, Carsten A.

D'Amico, Irene

University of Missouri: MOspace

arXiv:0710.2866v1  [cond-mat.mtrl-sci]  15 Oct 2007Intersubband spin-orbit coupling and spin splitting in symmetric quantum wellsF. V. Kyrychenko and C. A. UllrichDepartment of Physics and Astronomy, University of Missouri, Columbia, Missouri, 65211, USAI. D’AmicoDepartment of Physics, University of York, York YO10 5DD, United KingdomIn semiconductors with inversion asymmetry, spin-orbit coupling gives rise to the well-knownDresselhaus and Rashba effects. If one considers quantum wells with two or more conduction sub-bands, an additional, intersubband-induced spin-orbit term appears whose strength is comparableto the Rashba coupling, and which remains finite for symmetric structures. We show that the con-duction band spin splitting due to this intersubband spin-orbit coupling term is negligible for typicalIII-V quantum wells.PACS numbers: 73.50.-h, 73.40.-c, 73.20.Mf, 73.21.-bKeywords: spintronics, spin Coulomb drag, spin-orbit coupling, quantum wellsI. INTRODUCTIONResearch in nanoscience is crucial for its technologi-cal implications and for the fundamental exploration ofthe quantum properties of nanostructures such as quan-tum wells, wires and dots. Of particular interest is thestudy of spin dynamics, which hopes to revolutionize tra-ditional electronics using the spin properties of the carri-ers (spintronics) [1]. In this context, the theoretical pre-diction [2] and experimental confirmation [3] of the spin-Coulomb drag (SCD) effect was of great importance, asthis effect results in the natural decay of spin current andintrinsic dissipation in AC-spintronic circuits [4]. Dueto Coulomb interactions between spin-up and spin-downelectrons, the up and down components of the total linearmomentum are not separately conserved. This momen-tum exchange between the two populations representsan intrinsic source of friction for spin currents, known asspin-transresistivity [5].In [4] we demonstrated that the SCD produces an in-trinsic linewidth in spin-dependent optical excitations,which can be as big as a fraction of a meV for intersub-band spin plasmons in parabolic semiconductor quan-tum wells (QWs). This intrinsic linewidth would beideal to experimentally verify the behavior of the spin-transresistivity in the frequency domain.In our proposed experiment, we suggested to use sym-metric parabolic QWs to avoid an undesired splitting ofthe spin plasmons due to Rashba spin-orbit (SO) cou-pling. We based our discussion on earlier work [6], inwhich collective intersubband spin excitations in QWswere described in the presence of Dresselhaus and RashbaSO interaction terms [7, 8] for strictly two-dimensional(2D) systems [9]. In symmetric QWs, the Rashba termvanishes and only bulk inversion asymmetry (Dressel-haus) interaction is present.However, as shown recently by Bernardes et al. [10],the Rashba SO coupling gives finite contributions evenfor symmetric structures, if treated in higher order per-turbation theory. As a consequence, for QWs with morethan one subband, there appears an additional intersub-band SO interaction, whose magnitude can become com-parable to that of 2D Dresselhaus and Rashba interac-tions. This interaction gives rise to a nonzero spin-Hallconductivity and renormalizes the bulk mass by ∼ 5% inInSb double QWs [10]. This raises the question whetherthis effect must be accounted for when extracting theSCD from intersubband spin plasmon linewidths [4].In this paper we are going to show that while intersub-band SO interaction may manifest itself in some specialcases, as for example in the double well analyzed in Ref.[10], it has little to no effect on spin splitting and spinmixing in QWs once the 2D Dresselhaus and/or Rashbaterms are taken into account.In Sec. II we present the general formalism of calcu-lating conduction band states in quantum structures in-cluding both 2D and intersubband SO interaction. InSec. III we consider the specific case of symmetric single-well quantum structures, and in Sec. IV we present re-sults for a parabolic model QW. Sec. V gives a briefsummary.II. GENERAL FORMALISMWe consider conduction electrons in a QW describedby the HamiltonianHˆ = Hˆ0 + Hˆso, (1)where Hˆ0 is spin independent and Hˆso is the SO inter-action projected on the conduction band. For simplic-ity we will consider only spin off-diagonal (spin-mixing)terms in Hˆso. The eigenfunctions associated with Hˆ0alone can be obtained by solving a single-particle equa-tion of the Schro¨dinger-Poisson or Kohn-Sham type, re-sulting in spin-independent subband envelope functionsψi(z,k‖) and energy eigenvalues εi, where i is the sub-band index and z is the direction of quantum confine-ment.2Let us now consider the two lowest conduction sub-bands of the QW. In the basis of the first two subbandspinors |ψ1 ↑〉, |ψ1 ↓〉, |ψ2 ↑〉, |ψ2 ↓〉, the Schro¨dingerequation with the full Hamiltonian (1) has the formε1 α1 0 βα∗1 ε1 β′ 00 β′∗ ε2 α2β∗ 0 α∗2 ε2A = εA, (2)whereα1 = 〈ψ1 ↑ |Hˆso|ψ1 ↓〉α2 = 〈ψ2 ↑ |Hˆso|ψ2 ↓〉,β = 〈ψ1 ↑ |Hˆso|ψ2 ↓〉,β′ = 〈ψ1 ↓ |Hˆso|ψ2 ↑〉. (3)To remove the off-diagonal terms mixing the ↑, ↓ stateswithin the same subband, we apply the unitary transfor-mation B = U ·A withU =1√21 − α1|α1| 0 01 α1|α1| 0 00 0 1 − α2|α2|0 0 1 α2|α2| . (4)Equation (2) then transforms intoε1 − |α1| 0 −γ1 γ20 ε1 + |α1| −γ2 γ1−γ∗1 −γ∗2 ε2 − |α2| 0γ∗2 γ∗1 0 ε2 + |α2|B = εB,(5)where the off-diagonal matrix elementsγ1,2 =12[βα∗2|α2| ± β′ α1|α1|](6)connect the first and second subbands. We treat thesecontributions to the conduction band Hamiltonian per-turbatively to second order, and obtain the following so-lutions of Eq. (5):ε±1 = ε1 ± |α1|+|γ1|2ε1 ± |α1| − ε2 ∓ |α2| +|γ2|2ε1 ± |α1| − ε2 ± |α2| ,ε±2 = ε2 ± |α2|+|γ1|2ε2 ± |α2| − ε1 ∓ |α1| +|γ2|2ε2 ± |α2| − ε1 ± |α1|andB−1 =10−γ∗1ε1 − |α1| − ε2 + |α2|γ∗2ε1 − |α1| − ε2 − |α2|, (7)B+1 =01−γ∗2ε1 + |α1| − ε2 + |α2|γ∗1ε1 + |α1| − ε2 − |α2|, (8)B−2 =−γ1ε2 − |α2| − ε1 + |α1|−γ2ε2 − |α2| − ε1 − |α1|10, (9)B+2 =γ2ε2 + |α2| − ε1 + |α1|γ1ε2 + |α2| − ε1 − |α1|01 . (10)The eigenvectors B±i are normalized up to first order inthe off-diagonal perturbation.In the absence of intrasubband (2D) terms, α1 = α2 =0, the intersubband SO interaction gives rise to spin mix-ing without lifting the spin degeneracy (ε+i = ε−i ); it onlycauses a spin-independent shift of the subband energies.By contrast, if an intrasubband interaction is present (orif spin degeneracy is lifted by other means, e.g., by a mag-netic field), the spin splitting is affected. For the lowestsubband it is given by ε+1 − ε−1 = ∆ε1, where∆ε1 = 2|α1|+ 2|γ1|2 |α2| − |α1|(ε2 − ε1)2 − (|α2| − |α1|)2− 2|γ2|2 |α2|+ |α1|(ε2 − ε1)2 − (|α2|+ |α1|)2 . (11)To proceed further we need the explicit form of the SOHamiltonian Hˆso.III. RASHBA AND DRESSELHAUS SOINTERACTION IN SYMMETRIC QWSBy folding down the 14 × 14 k · p Hamiltonian for aQW grown in [001] direction in a zinc-blende crystal toa 2 × 2 conduction band problem [11], one obtains aneffective SO Hamiltonian in the conduction band:Hˆso ≈(0 hsoh∗so 0), (12)wherehso = R(z)k− − iλk+ ∂2∂z2− iλ4(k2− − k2+)k− , (13)withλ = 4√23PQP ′(1(E∆ − ε)(E′v − ε)− 1(Ev − ε)(E′∆ − ε))3andR(z) =√23P 2[∂∂z(1Ev − ε −1E∆ − ε)]+√23P ′2[∂∂z(1E′v − ε− 1E′∆ − ε)]. (14)Here, k± = 1√2 (kx ± iky), ε is the electron energy, Ev(z)and E∆(z) are the position-dependent Γ8 and Γ7 valenceband edges, and P = −i ~m〈S|pˆx|X〉 =√Ep~22m is the mo-mentum matrix element. Primed quantities correspondto the higher lying Γ8 − Γ7 conduction band and Q isthe momentum matrix element between the valence bandand the higher conduction band. Along with the Rashbaand linear Dresselhaus terms in Eq. (12) we keep the cu-bic Dresselhaus term as well. During the derivation weassumed that the variation of the band edges is smallcompared with the energy gaps in the material.In symmetric structures, due to parity conservationthe intrasubband SO interaction contains only the Dres-selhaus contribution,α1 = − λ4√2k3 sin(2ϕ) e−iϕ +D11√2k ei(ϕ+pi2), (15)α2 = − λ4√2k3 sin(2ϕ) e−iϕ +D22√2k ei(ϕ+pi2), (16)and the intersubband SO interaction (between the lowesttwo subbands) involves only the Rashba termβ = β′∗ =R12√2ke−iϕ, (17)where ϕ is the polar angle of the in-plane vector k‖ mea-sured from the [100] direction, and k = |k‖|. Further-more,Dii = −λ〈ψi(z)∣∣∣∣ ∂2∂z2∣∣∣∣ψi(z)〉(18)andR12 = 〈ψ1(z)|R(z)|ψ2(z)〉. (19)The quantity R12 corresponds to the coupling parameterη derived in Ref. [10] using an 8-band k · p model.For small k the linear term in Eqs. (15)-(16) dominatesand we can approximateα1|α1| ≈α2|α2| ≈ ei(ϕ+pi2). (20)Then,γ1 =1√2R12k cos(2ϕ+pi2)(21)γ2 = − i√2R12k sin(2ϕ+pi2), (22)and the ground state spin splitting follows from Eq. (11)as∆ε1 ≈ 2|α1|− R212D11√2(ε2 − ε1)2k3− R212D22√2(ε2 − ε1)2k3 cos(4ϕ).(23)The intersubband interaction results thus in an addi-tional spin splitting proportional to k3.Next, we expand the spin splitting that is induced bythe intrasubband SO interaction. Up to order k3 we ob-tain|α1| ≈ D11√2k +λ8√2k3 − λ8√2k3 cos(4ϕ), (24)which gives the final expression for the subband splitting:∆ε1 =√2D11k +(λ4− R212D11(ε2 − ε1)2)k3√2−(λ4+R212D22(ε2 − ε1)2)k3√2cos(4ϕ). (25)One finds that the intersubband SO interaction producesan additional spin splitting of the same symmetry as theintrasubband cubic Dresselhaus term. We will now es-timate the magnitude of this additional contribution forGaAs parabolic QWs.IV. SUBBAND SPIN SPLITTING INPARABOLIC WELLSLet us consider a parabolic QW with conduction bandconfining potentialV (z) =12Kz2, (26)resulting in the noninteracting energy spectrumεj =√~2Km∗(j − 12), j = 1, 2, . . . (27)The first and second subband envelope functions areψ1(z) =4√2ξpie−ξz2, (28)ψ2(z) =4√32ξ3pize−ξz2, (29)where ξ =√m∗K/4~2. Straightforward calculationsgive〈ψ1∣∣∣∣ ∂2∂z2∣∣∣∣ψ1〉= −ξ, (30)〈ψ2∣∣∣∣ ∂2∂z2∣∣∣∣ψ2〉= −3ξ, (31)〈ψ1 |z|ψ2〉 = − 12√ξ. (32)4For our parabolic well, the positional dependence of thevalence band edge (the valence band potential) isEv = −14Kz2,corresponding to a valence band offset VBO=0.33. ForGaAs parameters (Eg = 1.42 eV, ∆ = 0.34 eV, Ep =22 eV) Eq. (14) gives R(z) ≈ − (∂Ev∂z)7A˚2. Using Eqs.(18), (19) and (30)–(32) we then getR12 = − (7A˚2)K4√ξ, D11 = λξ, D22 = 3λξ,andR212D22(ε2 − ε1)2 = ∆ε~22m∗A˚2214764λ ∼ 10−6λ,for m∗ = 0.065m0 and ∆ε = ε2− ε1 = 40 meV. The con-tribution of the intersubband SO interaction to the spinsplitting of the lowest conduction subband is six ordersof magnitude weaker than that of the cubic Dresselhausintrasubband term and thus can be completely neglected.The spin mixing induced by the intersubband SO in-teraction can be estimated from Eq. (7):∣∣∣∣ γ2ε2 − ε1∣∣∣∣2≈ R212k22(∆ε)2=4932 ∆ε~22m∗A˚2 k2A˚2 ∼ 10−7,for k = 0.01 A˚−1. This is seven orders of magnitudeweaker than the spin mixing induced by intrasubbandSO interaction and also can be completely neglected.Similar results were obtained for GaAs symmetric rect-angular QWs.V. CONCLUSIONSIn this paper, we have considered the effects of SOcoupling on the conduction subband states in symmetricQWs. Our work was motivated by Ref. [10], which dis-cussed a SO coupling effect specific to QWs with morethan one subband and showed that it can affect the elec-tronic and spin transport properties in some systems.We found that although the magnitude of this in-tersubband SO interaction can be comparable to thatof the 2D Dresselhaus and Rashba terms, its effect onthe spin splitting and spin mixing of conduction bandstates is several orders of magnitude weaker since it con-nects states with different energies. This is due to thefact that the spin splitting and spin mixing of conduc-tion band states are renormalized by the intersubbandenergy difference.Therefore, if one considers system with non-degeneratesubbands, one can completely neglect the intersubbandSO interaction compared to the usual 2D Dresselhausand Rashba terms. These findings provide an a posteri-ori justification for the approach used to calculate sub-band splittings and spin plasmon dispersions carried outin Ref. [6]. This opens the way for a comprehensive the-ory of collective intersubband excitations in QWs in thepresence of SCD and SO coupling.AcknowledgmentsThis work was supported by DOE Grant No.DE-FG02-05ER46213, the Nuffield Foundation GrantNAL/01070/G, and by the Research Fund 10024601 ofthe Department of Physics of the University of York.[1] Semiconductor spintronics and quantum computation,edited by D. D. Awschalom, N. Samarth, and D. Loss(Springer, Berlin, 2002)[2] I. D’Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000).[3] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J.Stephens, and D. D. Awschalom, Nature (London) 437,1330 (2005).[4] I. D’Amico and C. A. Ullrich, Phys. Rev. B 74,121303(R) (2006); I. D’Amico and C. A. Ullrich, PhysicaStatus Solidi (b) 243, 2285 (2006); I. D’Amico and C. A.Ullrich, Journal of Magnetism and Magnetic Materials316, 484 (2007)[5] I. D’Amico I and G. Vignale, Europhys. Lett. 55, 566(2001); I. D’Amico and G. Vignale, Phys. Rev. B 65,085109 (2002); I. 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Intersubband spin-orbit coupling and spin splitting in symmetric quantum wells

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