Magnetotransport in 2D electron systems with a Rashba spin-orbit interaction M. V. Cheremisin, A. S. Furman A.F.Ioffe Physical-Technical Institute, St.Petersburg, Russia (Dated: February 2, 2008) ThebeatingpatternofShubnikov-deHaasoscillations in2D electron systeminthepresenceofa 6 Rashbazero-fieldspinsplittingisreproduced. Itisshown,takingintoaccounttheZeemansplitting, 0 thattheexplicitformulaeforthenodepositionwelldescribestheexperimentaldata. Thespin-orbit 0 interaction strength obtained is found to be magnetic field independent in an agreement with the 2 basic assumptions of theRashba model. n a PACSnumbers: 73.20.At,71.43.Qt,72.25.Dc, 73.61.-r J 3 There has been growing interest in the zero-magnetic- ε±n =η(n± γ2n+β2),n≥1 ] field spin splitting[1, 2, 4] of the 2D electron gas l where η = ~ω /µ is thpe dimensionless magnetic field; l (2DEG), associated with the spin-orbit interaction c a ω = eB, the cyclotron frequency; β = 1(1 − χ), the h (SOI) caused by the structural inversion asymmetry in tecrm comnctainingthe Zeemanspin splitting;2χ= gm , the - heterostructures[5]. Application of a gate voltage [7, 8] 2m0 s spinsusceptibility;andn,anintegersimilartothatinthe e is known to be the most effective method to control conventionaldescriptionoftheLandaulevels(LL).Then, m the SOI strength. These 2D systems have been sug- gested for application in future spintronics devices, such according to Ref.[5] γ = δ = αkF, where δ is the . η µ√η t a as spin-based field-effect transistors[9], spin-interference dimensionless SOI strengthqparameter; ∆ = 2αkF, the m devices[10, 11], and nonmagnetic spin filters based on a zero-field spin-orbit splitting at the Fermi energy; and - resonant tunneling structure[12]. Usually, the beating- ~kF, the Fermi momentum. Usually, the typical Fermi d pattern analysis of Shubnikov-de Haas oscillations ( energy µ ∼ 80meV exceeds the SOI- induced splitting n SdHO)[3]andtheweakantilocalizationmethod[13]are ∆∼1meV(see[3]),and,thereforeδ ≪1. Itisnotewor- o c usedtodeterminetheSOIstrengthin2Dsystems. How- thy that the conventional spin-up(down) energy states [ ever, the former approach is known to lead to a certain associated with n-th LL number correspond to ε+ and n 2 controversyindeterminingthezero-fieldspinsplitting∆. ε−n+1 states respectively. In the absence of SOI, Eq.(2) v Namely,the spinsplitting deducedfromthe SdHO beat- reproduces well-known LL energy spectrum. 0 ing node position at finite fields [3] is different from that In contrast to the conventional formalism extensively 1 ∆ expected for B = 0. In the present paper, this dis- used to find the low-B magnetoresistivity, we use the al- 4 crepancyisattributedto the contributionofthe nonzero ternative approach [14, 15, 16] which allows to resolve 1 Zeeman spin splitting at finite fields. We support our magnetotransport problem in both the SdHO and Inte- 0 5 idea by a rigorous analysis of the SdHO beating pattern ger Quantum Hall Effect (IQHE) modes. Moreover,this 0 caused by SOI spin splitting. The beating node posi- method was successfully used in a recent paper [17] to / tions reported in [3] agree well with those predicted by reproducethe SdHObeating structureinthe presenceof t a the theory. Then, we demonstrate thatthe SOI strength the zero-field valley splitting (Si-MOSFET 2D system), m is independent of the magnetic field. and in both the crossed- and tilted- field configurations. - Let us consider a 2DEGin the x-y plane, subjected to FollowingtheargumentationputforwardinRef.[16],well d n a magnetic field. In the Landau gauge, the one-electron abovetheclassicallystrongmagneticfieldrangeωcτ ≫1, o Hamiltonian including the Rashba spin-orbit term [5] is whereτ isthe momentumrelaxationtime, 2DEGcanbe c given by assumeddissipationlessinstrongquantumlimitwhenthe v: (p+eA)2 α gµ cyclotron energy ~ωc exceeds both the thermal energy i H = + [σ(p+eA)]n+ B(σB) (1) kT and the energy related to LL-width ~/τ . Here, τ is X 2m ~ 2 q q the quantum relaxation time. Under the above assump- ar where p is the 2D momentum; m, the effective mass; tions σxx,ρxx ≃ 0. Nevertheless, routine dc measure- g, the Zeeman factor; µ , the Bohr magneton; and, n, B ments yield [16] the finite magnetoresistivity associated the unit vector in the z-direction. Then, σ is the Pauli with a combination of the Peltier and Seebeck thermo- spin matrix; B, the totalmagnetic field; and, α, the SOI electric effects. Within the scenario suggested [16], we strength. obtain the above magnetoresistivity in the form Ithasbeenshown[5]thatthesolutiontoEq.(1)hasan explicitforminthecaseofaperpendicularmagneticfield ρ=ρ α22D (3) yx B = B = B. The spectrum for dimensionless energy L z ε=E/µ (µ is the Fermi energy) is given by [5] where α is the 2DEG thermoelectric power; ρ 1 = 2D −yx ε =ηβ, (2) Nec/B, the Hall resistivity; N = − ∂Ω , the 2D den- 0 ∂µ T (cid:0) (cid:1) 2 sity, Ω = −kTΓ ln 1+exp µ−εn , the thermody- kT n namic potential; ΓP= (cid:0)eB, the z(cid:0)ero-wi(cid:1)d(cid:1)th LL density of hc states; L = π2kB2 , the Lorentz number; k , the Boltz- 3e2 B mann constant. In fact, the 2D thermoelectric power in 3 strong magnetic fields is a universal quantity [18], pro- portionaltotheentropyperelectron: α =− S ,where S = − ∂Ω is the entropy. Both S2D,N, anedN, there- eV )2 ∂T µ m fore, α2D(cid:0),ρ(cid:1)are universal functions of the dimensionless D ( sample A (x=0.65) temperature ξ = kµT and the magnetic field η = 2/ν, 1 sample B (x=0.60) where ν = N /Γ is the conventional filling factor, and sample C (x=0.53) 0 N = m µisthezero-fielddensityofthestronglydegen- 0 π~2 erate 2DEG in the absence of a SOI-induced splitting. 0 Using the Lifshitz-Kosevich formalism and, then, ne- 1 3 5 7 9 11 13 glecting finite LL-width( ~/τ → 0 ), we derive in Node index j q Appendix asymptotic formulae for Ω, and, hence, for N,S,ρyx,ρ, which are valid at low temperatures and FIG.1: Zero-field SOIsplitting at theFermienergy vsnode weak magnetic fields ξ,η ≪1: indexj,deducedfromtheexperimentaldata[3],withthehelp of thenodecondition specified byEq.(5). Dotted lines( from ∞ sin(2πk/η) top to bottom ) represent the mean values ∆0 for samples N =N ξF (1/ξ)+2πξN R(η), (4) 0 0 0 B,A,C respectively. sinh(r ) k k=1 X TABLEI: Transportdata(at4.2K)andtheZero-fieldspin- ∞ S =S −2π2ξk N Φ(r )cos(2πk/η)R(η), orbitsplittingatFermienergyforInxGa1−xAs/In0.52Al0.48As 0 B 0 k 2D system reported in Ref.[3] k=1 X whereS0 =kBN0(2ξF1(1/ξ)−F0(1/ξ))istheentropyat Sample(x) µ0×104,cm2/Vs n×1012cm−2 µ,meV ∆0,meV B =0;Fn(z), the Fermiintegral;andΦ(z)= 1−zzsicnoht(hz()z). A(0.65) 13.4 1.75 78 2.34 At B = 0 both the thermopower and 2D den·sity are B(0.60) 9.5 1.65 74 2.57 constants,i.e. α = π2ξ2kB,N =N ,hencethemagne- C(0.53) 6.8 1.46 65 1.63 2D 3 e 0 toresistivityisgivenbyzero-fieldasymptoteρ= h π2ξ2η. e2 6 AccordingtoEq.(4),foractualfirst-harmoniccase(k =1 system ( m = 0.049m , g ≃ 4 ) we find β = 0.45, and, ) the magnetoresistivity can be viewed as the zero-field 0 therefore Eq.(5) cannot be satisfied for j = 1. With the background, on which the rapid SdHO modulated by help of Eq.(5), we analyze the nodes, reported in [3] for long-period beatings( see Fig.2 ) are superimposed. It’s three different samples, and then plot the dependence of worthwhile to mention that at the beat nodes( i.e. when thezero-fieldSOIsplittingattheFermienergy∆against the form-factor at k = 1 vanishes ) the magnetoresistiv- thenodeindex(seeFig.1),startingfromj =3. Forthese ityisgivenbyzero-fieldasymptote. Thisisnot,however, samples ∆ is nearly constant within the actual range of thecaseoflowtemperaturesand(or)highmagneticfields the magnetic fields, therefore we obtain the respective when the high-order terms(k > 1) in Eq.(4) may de- meanvalues ∆ denotedin TableI. Note thatthe minor termine the amplitude of magnetoresistivity at the beat 0 deviationof∆withrespecttoitsmeanvalueinhigh-field nodes. Itturns outthatthe data reportedin[3]point to limit( low-index nodes ) can be associated with possible the above feature. magnetic field dependence of the g-factor. In contrast, We now analyze in detail the form-factor R(η)( see the non-parabolicity effects [19] seem to be irrelevant[6] Appendix)whichdeterminesthebeatingpatternofS,N for the actual low-field case B <1T. and, hence, ρ. For the actual first-harmonic case (i.e., WeemphasizethatthenodeconditionsimilartoEq.(5) k =1),thebeatingnodescanbeobservedwhenR(η)=0 was previously discussed in literature. Following the or analysis done in Ref.[6], the nodes occur when the spin- δ j orbit-split subbands are shifted one with respect an- β2+ = , (5) s η2 4 other by half a period at the Fermi energy. Namely, 1 ≃ ε+n = (ε−n+s +ε−n+s+1)/2, where s = 0,1,2... cor- where we neglect the small quadratic term δ2/4η2 ≪ responds to the node index as j = 1+2s. For actual δ/η2 evaluating Eq.(9). Then, j = 1,3.. is the beat- high LL-number case n ≫ 1 this condition reproduces ingnodeindex. Weemphasizethatthe firstnodecannot Eq.(5). be observed in experiments, performed, for example, in Let us discuss the conventional method [3] often used Ref. [3]. Indeed, for real 2D In Ga As/In Al As to extract the zero-field SOI splitting at the Fermi en- x 1 x 0.52 0.48 − 3 the IQHE mode. We argue that the noticeable increase in SdHO am- plitude was observed [3] at B ≃ 0.3T. This value satis- 0,010 fies the criterion of the classically strong magnetic field since ω τ = 4 while the corresponding cyclotron energy c ~ω =8.2Kcorrelateswiththat∼9.8KexpectedfromT- c dependent SdHO-damping factor, i.e. when 2π2ξ/η ∼1. W We conclude that the energy associated with LL width r, 0,005 ∼ ~/τ is less or at least equal to the thermal energy. q The above estimates point to validity of zero-width LL model in this particular case. Nevertheless, since both the temperature and finite LL width known to suppress the SdHO amplitude in a rather similar manner, we es- 0,000 0,00 0,01 0,02 teemreasonableto reproduceinFig.2the SdHO beating h pattern using somewhat higher temperature T = 1.6K than that T =0.5K reported in [3]. FIG. 2: SdHO beating pattern calculated with the help of Note that our approach provides a correct number of Eq.(4) at k =1 for sample A(x=0.65) [3]: N0 =1.75∗1012 oscillations between the adjacent nodes. For example, cm−2, m = 0.049m0, g = 4, ∆0 = 2.34meV, T = 1.6K. Ar- the number of oscillations confined between j = 3,5 rows show the beating nodes at j =3,5... Zero-field asymp- nodes (37) correlates with that (35) observed in [3]. A tote is represented by dotted line minor point is that our approach predicts a somewhat loweramplitude ofSdHO,comparedwiththatinthe ex- ergy. Accordingto phenomenologicalargumentsputfor- periment[3]. Forexample,forj =3node(B =0.873Tin ward by Das et al [3, 4], the nodes may occur when Ref.[3] ) we obtain ρ=0.0035Ohm. Actually, one would cos π∆tot = 0 or ∆ = ±j~ω , where the to- expectthesameorderofmagnitudeforSdHOamplitude ~ωc tot 2 c between the proximate nodes ( see j = 3,5 in Fig.2 ). tal (cid:16)spin sp(cid:17)litting at the Fermi energy between spin- down ε−n+1 and spin-up ε+n states yields ∆tot = ~ωc − Our estimation is, however, less than both the absolute (2β~ω )2+∆2. As expected, the total spin split- magnetoresistivity40OhmatB =0.873TandSdHOam- c plitude ∼5Ohm reported in Ref.[3]. ting ∆ coincides with the zero-field −∆ and the Zee- pman χt~oωt spin splitting in low( high ) magnetic field In conclusion, we demonstrated the relevance of the c approach[16] regarding the beating pattern of SdHO limit respectively. With the help of the dimensionless caused by Rashba spin-orbit interactions. Taking into units the node condition suggested by Das et al reads account the Zeeman splitting, the rigorous analysis of β2+ ηδ2 = 1±2j/2, hence, reproduces our result if one experimentaldata [3] suggests a B-independent strength qselects ”+” set at j ≥ 1. We argue that straightforward of the Rashba SOI. The above finding is consistent with procedure ( see Fig.1 ) used to extract ∆0 is, however, the general theoretical assumptions [5]. Our approach preferable compare to zero-field extrapolation method can be helpful for estimation of the SOI strength. suggested in Ref.[3, 4]. Indeed, for low-density samples The authors wish to thank Prof. N.Averkiev and Dr. and(or) under the temperature enhanced conditions the S.Tarasenko for helpful comments. This study was sup- SdHOamplitudeissuppressed,hence,thelow-fieldnodes ported by the Russian Foundation for Basic Research become hidden. In this case the zero-field extrapolation (grant 03-02-17588)and LSF ( Weizmann Institute). method [3] may lead to a subsequent errors. Let us now reproduce( see Fig.2 ) the SdHO APPENDIX beating pattern with the nodes occurred in a typi- cal sample( sample A(x = 0.65) in [3]) at B = 0.873;0.46;0.291;0.227;0.183;0.153T using Eq.(4), and Using the conventional Poissonformulae previously extracted value of zero-field SOI splitting ∆ = 2.34meV. It’s worthwhile to mention that our ∞ ∞ ∞ ∞ 0 ϕ(n)= ϕ(n)dn+2Re ϕ(n)e2πikndn, (6) results differ with respect to those, which can be ob- tainedwithintheconventionalformalisminthefollowing: Xm0 Za Xk=1Za (i) the low-field quantum interference, classical magne- where m −1<a<m , m the initial value of the sum- 0 0 0 toresistivityand3Dsubstrateparallelresistivity[3]back- mation,thethermodynamicpotentialcanberepresented groundareexcludedwithinourapproach;(ii)incontrast as the sum Ω=Ω +Ω of the zero-field and oscillating 0 to conventional SdHO analysis, our method determines parts as follows ∼ the absolute value of magnetoresistivity, and, moreover, can lead to a gradual transition [16] from the SdHO to Ω =−N µξ2F (1/ξ), (7) 0 0 1 4 Ω =−N0µηξRe ∞ ∞e2πiknln 1+e1−ξε±n dn, bAesaetxinpgesctgeodv,etrhneedfobrmy-tfhaectfoorrims-rfeadcutocerdairneasbuspeenrcimeopfoSseOdI. ∼ Xk=1Z0 (cid:18) (cid:19) to a field-independent constant R(η) = cos(2πkβ), and, therefore,thebeatingstructureisabsent. Usingthecon- whereF (z)istheFermiintegral. Forsimplicity,weomit n ventionalthermodynamicdefinition,wecaneasilyobtain the SOI-induced splitting in zero-field term Ω because 0 both the entropy and the density of 2D electrons, speci- δ ≪1. The specialinterestofthe presentpaper is inthe fied by Eq.(4). oscillating term Ω of thermodynamic potential, which canbe stronglyaff∼ectedbyspin-orbit-splitsubbands(±). After a simple integration by parts, the oscillating term yields [1] G. Lommer, F. Malcher, and U. Rssler, Phys. Rev. B, 32, 6965 (1985). Ω =N µRe ∞ iη ∞ e2πikn± dε (8) [2] JR.evL.uBo,3H8.,M10u1n4e2ka(1ta9,88F)..F. Fang, and P.J. Stiles, Phys. ∼ 0 kX=12πk Z0 1+eε−ξ1 [3] BH.onDga,sP,.KD..CB.haMttiallcehr,arSa.yaD,aJt.tSai,ngRh.,RanedifeMnb.eJragffeer,,PWhy.Ps.. Rev. B 39, 1411 (1989). Using Eq.(2), for a certain energy we calculate the ac- [4] B.Das,S.Datta,R.Reifenberger,Phys.Rev.B41,8278 tual high-order LL-like numbers, associated with both (1990). the spin-orbit-split subbands as [5] E.I.Rashba,Fiz.Tverd.Tela(Leningrad)2,1224(1960) [Sov. Phys. Solid State 2, 1109 (1960)]; Y.A. Bychkov ε γ2 γ2ε γ4 and E.I. Rashba, J. Phys. C 17, 6039 (1984). n±(ε)= + ± β2+ + . (9) [6] Y.A. Bychkov, V.I. Mel’nikov, and E.I. Rashba, Zh. η 2 s η 4 Eksp. Teor. Fiz. 98, 717 (1990), [Sov. Phys. JETP 71, 401 (1990)]. It should be noted that the integrandequation in Eq.(8) [7] J.Nitta,T.Akazaki,H.Takayanagi,andT.Enoki,Phys. is a rapidly oscillating function, which is, in addition, Rev. Lett.78, 1335 (1997). strongly damped when ε > 1. The major part of the [8] G.Engels,J.Lange,Th.SchpersandH.Lth,Phys.Rev. magnitude of the integral results from the energy range B 55, R1958 (1997). close to the Fermi energy, when ε∼1. Therefore, n (ε) [9] S. Datta and B. Das, Appl.Phys.Lett. 56, 665 (1990). ± [10] Tie-Zheng Qian and Zhao-Bin Su, Phys. Rev. Lett. 72, can be regarded as smooth functions of energy, and, hence, can be re-written as n± = n±1 + ∂∂nε± 1(ε −1), [11] J23.1N1it(t1a9,94F)..E. Meijer, and H. Takayanagi, Appl. Phys. where we use the designation n±1 = n±(1(cid:0)) . U(cid:1)nder the Lett. 75, 695 (1999). above assumption, we can change the lower limit of in- [12] T. Koga, J. Nitta, H. Takayanagi, and S. Datta, Phys. tegration to −∞ and then use the textbook expression Rev. Lett.88, 126601 (2002). [13] T.Koga,J.Nitta,T.Akazaki,andH.Takayanagi,Phys. ∞ 1e+ikeyydy = sin−hi(ππk) for the integral of the above type. Rev. Lett.89, 046801 (2002). [14] C.G.M. Kirby and M.J. Laubitz, Metrologia 9, 103 −FR∞inally, the thermodynamic potential yields (1973). [15] M.V. Cheremisin, Zh. Eksp. Teor. Fiz. 119, 409 (2001), Ω=Ω +N µ2π2ξ2 ∞ cos(πk(n+1 +n−1))R(η) (10) [Sov. Phys. JETP, 92, 357, 2001]. 0 0 r sinhr [16] M.V. Cheremisin, Physica E, 28, 393 (2005). k k kX=1 [17] M.V. Cheremisin, Physica E 27, 151 (2005). [18] S.M. Girvin and M. Jonson, J.Phys.C 15, L1147 (1982). where we assume that ∂n± ∼ 1 is valid for the ac- [19] Can-MingHu,J.Nitta,T.Akazakietal,Phys.Rev.B60, ∂ε 1 η tual case of high- order Landau levels n ≫ 1, and 7736 (1999). (cid:0) (cid:1) ± r =2π2ξk/η is a dimensionless parameter related to T- [20] M. Dobers, K. von Klitzing, G. Weimann, Phys. Rev. B dkamping of SdH amplitude. Then, R(η) = cos(πk(n+− 38, 5453 (1988). 1 [21] M. Dobers, J.P.Viereit, Y. Guldner et al, Phys. Rev. B n−1)) is the form-factor. The oscillatory part of the 40, 8075 (1989). thermodynamic potential consists of rapid oscillations cos(πk(n+1 +n−1)) ≃ cos(2πk/η), on which long-period