We measure simultaneously the in-plane electron g factor and spin-relaxation rate in a series of undoped inversion-asymmetric (001)-oriented GaAs/AlGaAs quantum wells by spin-quantum beat spectroscopy. In combination the two quantities reveal the absolute values of both the Rashba and the Dresselhaus coefficients and prove that the Rashba coefficient can be negligibly small despite huge conduction-band potential gradients which break the inversion symmetry. The negligible Rashba coefficient is a consequence of the "isomorphism" of conduction- and valence-band potentials in quantum systems where the asymmetry is solely produced by alloy variations. © 2011 American Physical Society

Eldridge, P.S.

Harley, R.T.

Henini, M.

Hübner, Jens

Oertel, S.

Oestreich, Michael

Institutionelles Repositorium der Leibniz Universität Hannover

RAPID COMMUNICATIONSPHYSICAL REVIEW B 83, 041301(R) (2011)Spin-orbit fields in asymmetric (001)-oriented GaAs/AlxGa1−xAs quantum wellsP. S. Eldridge,1,* J. Hu¨bner,1 S. Oertel,1 R. T. Harley,2 M. Henini,3 and M. Oestreich11Institute for Solid State Physics, Leibniz University of Hannover, Appelstrasse 2, D-30167 Hannover, Germany2School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom3School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 4RD, United Kingdom(Received 19 November 2010; published 13 January 2011)We measure simultaneously the in-plane electron g factor and spin-relaxation rate in a series of undopedinversion-asymmetric (001)-oriented GaAs/AlGaAs quantum wells by spin-quantum beat spectroscopy. Incombination the two quantities reveal the absolute values of both the Rashba and the Dresselhaus coefficientsand prove that the Rashba coefficient can be negligibly small despite huge conduction-band potential gradientswhich break the inversion symmetry. The negligible Rashba coefficient is a consequence of the “isomorphism”of conduction- and valence-band potentials in quantum systems where the asymmetry is solely produced by alloyvariations.DOI: 10.1103/PhysRevB.83.041301 PACS number(s): 71.70.EjSymmetry is a thread which runs through all of physics, andsymmetry reduction discloses basic physical principles. Weemploy crystallographically engineered symmetry reductionto study the intricate effects of spin-orbit interaction on theelectron spin in semiconductor nanostructures. Symmetryreduction is an especially powerful tool in semiconduc-tor physics because the variety of crystallographic direc-tions combined with band-gap engineering allows enormousfreedom.The interplay between structure, symmetry, and electronspin in semiconductors directly affects the spin-relaxation rates and the effective electron Lande´ factor g. Early studies of sand g focused on bulk zinc-blende material where both entitiesare isotropic.1 Subsequently, the reduction in symmetry fromTd to D2d symmetry in symmetrical (001)-oriented quantumwells (QWs) was shown to give rise to anisotropy between thein-plane (x,y) and the out-of-plane (z) directions.2,3 Furtherreduction in symmetry to C2v is achieved in (001) quantumwells by removing the mirror symmetry of the quantum wellpotential and allows an in-plane, twofold symmetric anisotropyof both s (Ref. 4) and g.5Fundamentally, s and g are both determined by spin-orbitinteraction but the basic mechanisms for their anisotropies arequite different. Theoretically the in-plane anisotropy of g isproportional to the asymmetry of the electron wave functionin the z direction with the proportionality constant given by theDresselhaus or bulk inversion asymmetry (BIA) spin-splittingcoefficient γ .5,6 In contrast, s is in many cases dominated bythe Dyakonov-Perel (DP) spin-relaxation mechanism and therelated in-plane anisotropy depends on the ratio (α/β) of theRashba structural inversion asymmetry (SIA) to the BIA spinsplitting.4 The SIA component is determined in a rather subtleway by the asymmetry of the structure along the z direction.7,8In this work, we determine the absolute value of both theRashba and Dresselhaus coefficients for a series of quantumwell structures by simultaneously measuring the in-planeanisotropy of s and g by spin quantum beat spectroscopy.9The specially designed undoped (001) quantum well samples,with reduced C2v symmetry but without external electric fields,illustrate clearly the different origins of the two anisotropiesas they possess a strong anisotropy of g and nearly negligibleanisotropy of s .Anisotropies of s and g have been measured previouslyin symmetrically grown quantum wells in an external electricfield,10,11 but the decisive simultaneous evaluation of Dressel-haus and Rashba components has not been carried out so far.Hanle measurements in undoped asymmetric quantum wellswithout an applied electric field have revealed a strong in-planeanisotropy of the Hanle depolarization curve,12 but suchmeasurements do not distinguish between the anisotropies ofs and g.13 Recently, Ganichev and co-workers introduceda seminal technique that uses the angular distribution ofthe spin-galvanic effect and therewith measured the ratio ofthe Rashba and Dresselhaus coefficients in doped quantumwells.14,15 Salis and co-workers developed a technique that inprinciple yields the absolute values of the coefficients in dopedstructures by optically monitoring the angular dependence ofthe electrons’ spin precession.16 However, screening effectscan produce an uncertainty in the measured values.17We first summarize the theoretical mechanisms for g and sanisotropy.4,5 For g a small magnetic field in the x direction Bxdeflects the rapid zero-point motion of an electron quantizedin the z direction and yields a change of momentum in they direction. This additional momentum δpy changes theeffective Rashba R and Dresselhaus D precession vectorswhich read for (001) quantum wells in zinc-blende crystalsR(p) = α/h¯2⎛⎝ py−px0⎞⎠ , D(p) = β/h¯2⎛⎝−pxpy0⎞⎠ , (1)where α and β are coefficients and px,y,z are the componentsof the electron momentum. Inspection of Eq. (1) shows thatR converts δpy into an additional magnetic field which isparallel to the external magnetic field Bx and thereby altersthe diagonal component of the g tensor (gxx = gyy). Bycontrast,D converts δpy to an additional magnetic field in they direction, i.e., perpendicular to Bx, and thereby generates anoff-diagonal component gxy . A rigorous theoretical treatmentyields5gxy = gyx = (2γ e/h¯3μB)(〈p2z〉 〈z〉 − 〈p2z z〉) , (2)where μB is the Bohr magneton and 〈 〉 represents anexpectation value for the electron wave function. The two041301-11098-0121/2011/83(4)/041301(4) ©2011 American Physical SocietyRAPID COMMUNICATIONSP. S. ELDRIDGE et al. PHYSICAL REVIEW B 83, 041301(R) (2011)terms in Eq. (2) cancel and gxy vanishes if the electron wavefunction is symmetric. The anisotropy of the g tensor is thusproportional to the Dresselhaus coefficient γ and determinedby asymmetry of the electron wave function, which may beinduced by asymmetry of the confining (conduction band)potential for the electrons. The effective g factor for magneticfield oriented at angle φ to the (100) axis in the quantum wellplane is given byg(φ) = −√g2xx + g2xy + 2gxxgxy sin(2φ). (3)For the spin relaxation, which is dominated by the DPmechanism, the rate in the quantum well plane xys (φ)is proportional to 〈2〉 where  = D +R. It will beanisotropic as a result of interference of the components andis given by4xys (φ) =C2[α2 + β2 + 2αβ sin(2φ)], (4)where C is a constant which depends on the in-plane electronmomentum relaxation time. Thus the spin-relaxation rateanisotropy gives the ratio α/β, where β = 〈pz〉2 γ /h¯2.Experimentally, we measure the electron-spin-relaxationrate along the growth direction (z) for a magnetic field appliedin the quantum well plane. The magnetic field causes rapidLarmor precession of the electron spins and the measuredrelaxation rate is given by the average of zs = C(α2 + β2)and xys (φ) (Ref. 11):s(φ) = 12[zs + xys (φ)] = D[1 +(αβ)2+ 2α3βsin(2φ)],(5)where D = 3Cβ2/4. Therefore measurement of bothanisotropies yields simultaneously the absolute values of αand β.The samples are four molecular beam epitaxy grown, (001)-oriented GaAs/AlGaAs multiple quantum wells with varyingasymmetry. Sample A comprises five repeats of a 12-nmAl0.4Ga0.6As barrier, an 8-nm GaAs quantum well followedby a 30-nm alloy layer where the aluminium concentration isvaried linearly from 0.04 to 0.4. Samples B–D are equivalentstructures but the one-sided potential gradient is in the quantumwell and has been grown as a digital alloy with conduction-band gradients equivalent to an electric field of 100 kV/cm(sample B), 50 kV/cm (sample C), and 25 kV/cm (sample D).Figure 1 shows the calculated n = 1 electron states for samplesA and B obtained by numerical solution of the Schro¨dingerequation. The calculated confinement energies for electrons insamples A–D are 34, 91, 61, and 37 meV, respectively.The samples are mounted on a rotation stage in a liquid-helium flow cryostat in a superconducting magnet with themagnetic field oriented in Voigt geometry. The rotationaxis corresponds to the growth axis of the sample and isparallel to the direction of excitation. Spin oriented electronsare optically created by circularly polarized ps pulses froma mode-locked Ti:sapphire laser with a repetition rate of80 MHz, a laser wavelength of 740 nm, and a pulse intensityyielding excitation density ≈2 × 1010 cm−2. After excitationthe carrier momentum distribution rapidly thermalizes byemission of phonons and scattering with other carriers and3002001000Potential (meV)6040200Position z (nm)0.120.080.040.00|ψ| 26040200Position z (nm)(a)0.20.10.0-0.1Degree of polarization12008004000Time (ps) B || [110] B 90º [110](b)(c)FIG. 1. (Color online) Conduction-band potential profile andnumerical calculated electron wave function for the n = 1 states for(a) sample A and (b) sample B. (c) The measured spin quantumbeats at 125 K for sample A for 3 T in-plane magnetic field clearlyshowing the different electron g factors for B‖[110] and [1¯10] andsimilar spin-relaxation times.the holes lose their spin orientation within the momentumrelaxation time due to strong valence-band mixing andk-dependent spin splitting. The photo-luminescence (PL) isspectrally and temporally resolved by a spectrometer and asynchroscan streak camera with two-dimensional readout witha resolution of 0.5 nm and 8 ps, respectively. The degreeof circular polarization of the PL, which is proportional tothe electron-spin polarization, is measured by a switchableliquid-crystal retarder and a polarizer.Figure 1(c) depicts the time evolution of the degree ofcircular polarization for sample A at 3 T and 125 K for anin-plane magnetic field B along [110] and [1¯10] directions.The observed oscillations are electron-spin quantum beats,the frequency being ωL = gμBh¯−1B, and so are a directmeasure of g for the particular magnetic-field direction.9Measurements of beats in 〈Sz〉 in this way do not yield thesign of g but a comparison with previous measurements onsymmetric QWs identifies that g is negative for samples A, C,and D and positive for sample B.10,18 The two clearly distinctoscillation frequencies in Fig. 1(c) directly demonstrate thein-plane g anisotropy whereas the nearly identical decayof the two polarization transients indicate that s is nearlyisotropic.Figure 2 shows in more detail the dependence of g and son the direction of the magnetic field in samples A and B.The black (red online) solid curves in Fig. 2 depict fits of theanisotropy of g using Eq. (3), which directly yield both gxx andgxy . The diagonal components of the g tensor gxx = gyy havebeen previously investigated in symmetrical quantum wellswhere the dependence on well width, i.e., confinement energyand barrier penetration, is well described by k · p theory.18,19The solid squares in Fig. 3(a) show gxx for all four samples,041301-2RAPID COMMUNICATIONSSPIN-ORBIT FIELDS IN ASYMMETRIC (001)- . . . PHYSICAL REVIEW B 83, 041301(R) (2011)-0.200-0.190-0.180-0.170Electron g factor450-45Angle to [110]5.04.54.03.53.0Spin relaxation rate (ns-1) g τs-140363228Electron g factor (10-3)360270180900Angle to [110]1412108642Spin relaxation rate (ns-1)(a)(b)FIG. 2. (Color online) Extracted spin-relaxation rate and electrong factor for different magnetic-field orientations from fits to the spinquantum beat measurements for (a) sample A at 125 K and (b) sampleB at 25 K.confirming a similar strong dependence of gxx on confinementenergy for asymmetric QWs. The open squares in Fig. 3(a)show gxy and these values yield, by Eq. (2), the dependenceof the Dresselhaus spin splitting constant γ on confinementenergy [solid dots in Fig. 3(b)]. The excellent agreementwith data from Ref. 20 illustrates clearly that gxy providesan accurate measure of γ in asymmetric (001) quantum wells.The remaining deviations of γ from the trend probably resultfrom differences between the actual and the nominal samplestructures which lead to uncertainties in the calculation of thewave-function asymmetry. Theoretical analysis of the decreaseof γ with confinement energy is beyond the scope of this paper.Nonetheless, such a trend is expected from k · p theory and hassimilar origin to the change of gxx with confinement energy inFig. 3(a).7,20Next, we study in detail the anisotropy of the spin-relaxationrate. The open circles in Figs. 2(a) and 2(b) depict s(φ) forsamples A and B, respectively, and the gray solid curves arefits according to Eq. (5). Additional temperature- and density-dependent measurements confirm that the DP spin-relaxationmechanism dominates s . The measurements clearly showthat there is almost no in-plane anisotropy of s and thereforeα is close to zero even though the potential gradients in bothsamples are large (>90 kV/cm).Figure 4 compares α in our samples (solid circles) withprevious experiments in external and internal (Hartree) electricfields (open circles).21,22 The comparison of the measurementsclearly show that the Rashba spin splitting in AlGaAsheterostructures is large even for a modest external (or internal)electric field, but almost negligibly small in the case ofasymmetries produced by alloy variation. Although allowed-0.25-0.20-0.15-0.10-0.050.00Electron g factor121086420|γ| (eV)806040200Confinement Energy (meV)(a)(b)FIG. 3. (Color online) (a)Variation in gxx (solid squares) and gxy(open squares) with confinement energy for samples B and C (5–25 K)and sample A (25 K) (gray diamonds). (b) Experimental values of theDresselhaus spin splitting constant against confinement energy (solidcircles); open circles correspond to data from Ref. 20.to be nonzero by the C2v symmetry of the samples, thevalues of α that are required to fit the present data are zerowithin experimental uncertainties; they show both positiveand negative values with no clear trend and the value of α/βis in all cases less than 0.1. The measurements push downby an order of magnitude the previous upper limit of Rashbaspin splitting observed in samples with asymmetry from alloyvariation.8,23 The small values of α are a direct consequenceof the isomorphous band edges, that is, the conduction- andvalence-band potentials are related by a constant factor. Thisis due to the fact that the expectation value of the effectiveelectric field always vanishes in the conduction band dueto Ehrenfest’s theorem,7 and in isomorphous structures, asillustrated in the right-hand panel of Fig. 4, it will also vanish inthe valence band, and it is the latter which determines the spinsplitting.In conclusion, we have determined simultaneously theabsolute values for the Dresselhaus and the Rashba spin-orbit interaction in undoped low-symmetry (001) quantumwells. All samples show a distinctive anisotropy of theelectron g factor but essentially isotropic spin-relaxation rates.This difference highlights the different origins of the twophenomena; the first is a measure of the conduction electronwave-function asymmetry and the latter is a measure of theexpectation value of the valence-band potential on conduction-band states. Although a one-sided gradient of the conductionand/or valence band leads in general to a finite Rashbaspin-orbit interaction, the experiment proves that isomorphismof the valence and conduction band in GaAs/AlGaAs quantumwells proscribe a sizable, gradient-induced Rashba spin-orbitsplitting.041301-3RAPID COMMUNICATIONSP. S. ELDRIDGE et al. PHYSICAL REVIEW B 83, 041301(R) (2011)Evzzzz6005004003002001000α(eVm10-15)100806040200Conduction Band Potential Gradient (kV/cm)EcEvEcnon-isomorphous isomorphousisomorphousnon-isomorphous(schematic) (schematic)FIG. 4. (Color online) Extracted values of the Rashba spin-orbit constant vs conduction-band potential gradient for samples A–D (solidcircles). The open circles show values for a built-in Hartree electric field (Ref. 21) of ∼15 kV/cm in an n-modulation doped structure and foran externally applied electric field of 60 kV/cm in an undoped (110)-oriented multiple quantum well sample (Ref. 22). Right and left panelsshow schematic potential profiles and electron probability density (middle) and effective electric field for conduction (top) and valence bands(bottom) (after Ref. 7). For an “isomorphous” structure (right panel) the expectation value of the valence-band electric field will vanish but notfor a “nonisomorphous” structure (left panel), giving zero (finite) SIA spin splitting for the former (latter) (Ref. 8).We thank K. Ko¨hler for providing us with the samplesand W. W. Ru¨hle for helpful discussions. 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Spin-orbit fields in asymmetric (001)-oriented GaAs/AlxGa 1-xAs quantum wells

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Institutionelles Repositorium der Leibniz Universität Hannover