Quantum Interference Controls the Electron Spin Dynamics in n-GaAs
V. V. Belykh, A. Yu. Kuntsevich, M. M. Glazov, K. V. Kavokin, D. R. Yakovlev, M. Bayer
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Quantum Interference Controls the Electron Spin Dynamics in n -GaAs V. V. Belykh,
1, 2, ∗ A. Yu. Kuntsevich,
2, 3
M. M. Glazov,
4, 5, † K. V. Kavokin,
D. R. Yakovlev,
1, 4 and M. Bayer
1, 4 Experimentelle Physik 2, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991 Moscow, Russia National Research University Higher School of Economics, Moscow, 101000, Russia Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia Spin Optics Laboratory, St. Petersburg State University, 199034 St. Petersburg, Russia
Manifestations of quantum interference effects in macroscopic objects are rare.
Weak localization is one of the few examples of such effects showing up in the electron transport through solidstate. Here we show that weak localization becomes prominent also in optical spectroscopy viadetection of the electron spin dynamics. In particular, we find that weak localization controlsthe free electron spin relaxation in semiconductors at low temperatures and weak magnetic fieldsby slowing it down by almost a factor of two in n -doped GaAs in the metallic phase. The weaklocalization effect on the spin relaxation is suppressed by moderate magnetic fields of about 1 T,which destroy the interference of electron trajectories, and by increasing the temperature. The weaklocalization suppression causes an anomalous decrease of the longitudinal electron spin relaxationtime T with magnetic field, in stark contrast with well-known magnetic field induced increase in T . This is consistent with transport measurements which show the same variation of resistivitywith magnetic field. Our discovery opens a vast playground to explore quantum magneto-transporteffects optically in the spin dynamics.doi:10.1103/PhysRevX.8.031021 I. INTRODUCTION
The design of future spintronic and opto-spintronic de-vices requires a detailed understanding of the correla-tion between the electron conductivity and spin relax-ation in prospective material systems, such as semicon-ductors. The electron spin relaxation in semiconductorsdepends strongly on whether electrons are itinerant orlocalized [1, 2]. Across the metal-to-insulator transition(MIT) the spin relaxation changes as dramatically asdoes the conductivity [2, 3]. Indeed, in the insulatingphase both conductivity and spin relaxation criticallydepend on the overlap of the wavefunctions of donor-bound electrons at low temperatures and on the numberof delocalized electrons at higher temperatures. In themetallic phase, in semiconductors without an inversioncenter with GaAs as prototype system, the spin relax-ation is governed by spin-orbit coupling (Dyakonov-Perelmechanism) [4] and, similarly to the conductivity, thespin relaxation becomes suppressed by electron scatter-ing events. The spin relaxation rate is closely relatedto the electron diffusion coefficient [5–7], so that chargetransport phenomena are generally expected to manifestalso in spin relaxation processes [8–11]. The situationbecomes particularly involved in the vicinity of the MITwhere quantum effects become important [12, 13].While the mechanisms of electron spin relaxation insemiconductors were largely clarified in theory back inthe 1970s [14], for a long time experiments could ac-cess the electron spin dynamics only via the Hanle ef- ∗ [email protected] † [email protected]ffe.ru fect near zero magnetic field. Since the 1990s more ad-vanced techniques have become available such as pump-probe methods analyzing the Kerr/Faraday rotation[15, 16] or polarization-resolved photoluminescence [17–20], and elaborated methods like resonant spin amplifi-cation [21, 22], spin noise spectroscopy [23–25] and spininertia reorientation [26]. Each of these tools has limita-tions related to the achievable time resolution, the ad-dressable time range or the applicable magnetic field.So far, access to the relation between the electron dif-fusion and spin relaxation in the vicinity of the MIT washindered by experimental limitations. Only recently thepump-probe technique was extended to facilitate directmeasurements of arbitrarily long spin dynamics with pi-cosecond time resolution across a wide range of magneticfields [27].On the other hand, the transport properties of semi-conductors that directly provide information about elec-tron diffusion, are rather easily accessible in experiment.In weak magnetic fields the low temperatures magnetore-sistance is negative due to the weak localization effect : themagnetic field destroys the phase coherence of interferingpaths and increases the electron diffusion coefficient [28–35]. The spin-orbit interaction has a pronounced im-pact on the low-field magnetoresistance leading to posi-tive magnetoresistance, i.e., antilocalization, if the spincoherence of electrons is lost faster than their phase [28].Although weak localization/antilocalization is expectedto emerge in the spin dynamics [13, 36], it has not beenidentified in experiments so far.In this paper we demonstrate that weak localizationsignificantly slows down the itinerant electron spin relax-ation in the Dyakonov-Perel’ mechanism. Using the ex-tended pump-probe Faraday rotation technique we studythe longitudinal electron spin relaxation time T as afunction of external magnetic field in n -doped metallicbulk GaAs. While the classical theory [37] predicts anincrease of T with increasing field mainly due to the cy-clotron motion of the free carriers, we observe an anoma-lous decrease of T in moderate fields B . T . Wedevelop a theoretical model of the weak localization ef-fect in the spin relaxation of bulk semiconductors andfind very good agreement between the calculations andexperimental data. Our results establish a strict rela-tion between the electron diffusion and spin relaxationin metallic systems in the vicinity of the MIT. Therebyall-optical access to weak localization is provided and atool to probe locally electron transport phenomena is de-veloped. II. EXPERIMENTAL DETAILS
The results are obtained on Si-doped GaAs sampleswith electron concentrations of n e = 5 . × cm − (2- µ m-thick layer grown by the molecular-beam epitaxy),3 . × cm − and 7 . × cm − (140 and 170- µ m-thick bulk wafers, respectively).For optical measurements the samples are placed inthe variable temperature insert of a split-coil magne-tocryostat ( T = 2 −
25 K). Magnetic fields up to 6 Tare applied parallel to the light propagation directionthat is parallel to the sample growth axis (Faraday ge-ometry). The extended pump-probe Kerr/Faraday ro-tation technique is used to study the electron spin dy-namics. It is a modification of the standard pump-probe Kerr/Faraday rotation technique, where circularly-polarized pump pulses generate carrier spin polarization,which is then probed by the Kerr (Faraday) rotationof linearly-polarized probe pulses after reflection (trans-mission) from (through) the sample. Implementation ofpulse picking for both pump and probe beams in com-bination with a mechanical delay line allows us to scanmicrosecond time ranges with picosecond time resolution.Details of the technique are given in Ref. [27].Here, a Ti:Sapphire laser emits a train of 2 ps pulseswith a repetition rate of 76 MHz (repetition period T R = 13 . T R , 160 T R or 320 T R in order to clearly exceed the char-acteristic time of spin polarization decay. The samplewith donor concentration n e of 5 . × cm − is stud-ied in reflection geometry (Kerr rotation) with the laserwavelength set to 819 nm, close to the donor-bound ex-citon resonance. The samples with n e = 3 . × cm − and 7 . × cm − are studied in transmission geome-try (Faraday rotation) with the laser wavelength set to829 nm.Magnetoresistance measurements were performed us- ing a standard 4-terminal technique with a lock-in am-plifier. The measurement current (36 Hz, 100 µ A) waschecked not to overheat the sample at the lowest temper-ature. Ohmic contacts (with an almost T -independentresistance of about 100 Ohm) were obtained by anneal-ing of indium drops on top of the preliminary scratchedwafer (10 minutes at 400 o C in vacuum). A PPMS-9 cryo-stat and Cryogenics CFMS-16 system were used to setthe temperature (2-40 K) and magnetic field (up to 6 T).The magnetic field perpendicular to the sample surfaceand current direction was swept from positive to nega-tive values with subsequent symmetrization of the datato compensate inevitable contact misalignment.
III. RESULTS AND DISCUSSION
Experiment . The circularly polarized pump laser pulsecreates spin polarization along the magnetic field (Fara-day geometry B k z k [001]) which can be detected bythe delayed probe laser pulse via Faraday rotation of itslinear polarization. Figure 1(a) shows the dynamics ofthe spin polarization for exemplary values of the exter-nal magnetic field B = 0, 2 and 6 T for the metallic sam-ple with electron concentration n e = 3 . × cm − ,which is somewhat above the MIT threshold, n MITe ≈ (1 − × cm − . The signal decays monoexponen-tially with the longitudinal spin relaxation time T . It isseen from Fig. 1(a) that as the magnetic field grows, T first decreases, reaches a minimum and then increases.The non-monotonic dependence of T ( B ) with a mini-mum at about 1 . T ( B ) de-pendence becomes less pronounced for the sample witheven higher carrier concentration, while for the sampleswith lower donor concentrations, below MIT, T mono-tonically increases with increasing B [the open circles inFig. 1(b)].To investigate the anomalous T ( B ) dependence fur-ther we perform measurements at different temperatureswith the results summarized in Fig. 2(a). Interestingly,the minimum in the T ( B ) dependence at increased B (or, alternatively, peak at B = 0) is observed only at lowtemperatures T .
14 K. Furthermore, with increasingtemperature the minimum becomes less pronounced dueto the decrease of the zero-field T value. The decreaseof T with magnetic field or temperature increase are un-expected in view of existing theories of free-electron spinrelaxation in semiconductors [4, 7, 14, 37]. This calls fora detailed modeling of the spin relaxation process whichis presented below. Model.
In GaAs-like semiconductors, being in themetallic phase, the spin relaxation is controlled by theDyakonov-Perel mechanism [4, 14]: the electron spin pre-cesses around the effective, spin-orbit coupling-inducedmagnetic field and the spin precession is randomized byscattering events. The spin dynamics is described in theframework of a kinetic equation for the spin distribution n e (cm -3 ): T ( s ) B (T)
T = 2 K
50 100 150 200 25010 -2 S p i n po l a r i za ti on ( a r b . un it s ) Time (ns)B (T):6n e =3.7 10 cm -3 T = 2 K (b)(a)
FIG. 1. Longitudinal spin relaxation. (a) Dynamics of theelectron spin polarization (measured as Faraday rotation sig-nal) at different magnetic fields for the n -GaAs sample with n e = 3 . × cm − . (b) Magnetic field dependence ofthe longitudinal relaxation time T for samples with differ-ent donor concentrations. The arrow indicates the minimumin the T ( B ) dependence. (a),(b) Temperature T = 2 K. function s k [4, 37–39] ∂ s k ∂t + Λ k { s k } + s k × Ω k = Q { s k } . (1)Each term in Eq. (1) has a transparent physical meaning.The operator Λ k = − ω c [ k × ∂/∂ k ] describes the electroncyclotron motion in the external magnetic field, where ω c = e B /mc is the cyclotron frequency, m is the elec-tron effective mass and e is the electron charge (the Zee-man splitting is neglected). The term s k × Ω k describesthe precession of the electron spin around the effectivemagnetic field arising due to the spin-orbit interactionin a system with bulk inversion asymmetry. The corre-sponding precession frequency Ω k is cubic on the electronwavevector k . In the last term in Eq. (1), Q { s k } is the -1 0 1 2 3 4 5 6406080100120140 T ( n s ) Magnetic field (T)
T (K):2 (a)n e = 3.7 10 cm -3 T / T Magnetic field (T)T = 2 Kn e (10 cm -3 ):7.13.7 coherentbackscattering (b) (c) FIG. 2. Effect of weak localization on longitudinal spin re-laxation time T . (a) Magnetic field dependence of T atdifferent temperatures. n e = 3 . × cm − . The dashedlines show fits to the experimental data with Eq. (5). Insetshows relative variation of T with magnetic field for metallicsamples with different electron concentrations. (b) Scheme ofconstructive interference of clockwise and counter-clockwiseelectrons paths, starting at the same impurity and relatedby the time reversal symmetry. The interference gives riseto the weak localization effect by increasing the backscatter-ing efficiency. (c) The interference between the same pathsas in Fig. 2(b) is destroyed by the magnetic field due to theextra phase acquired by the electron traveling clockwise andcounter-clockwise. collision integral, i.e., the operator describing the redis-tribution of electrons between different states in k -space.It takes into account the electron scattering and can begenerally presented as Q { s k } = X k ′ ( W kk ′ s k ′ − W k ′ k s k ) , (2)describing the balance between the processes where anelectron leaves the state with wavevector k ′ and is pro-moted to the state with wavevector k with the rate W kk ′ and vice versa, accordingly. For the elastic scat-tering by the central potential of ionized donors rele-vant for the studied system, W kk ′ = W k ′ k , and re-laxation of different angular harmonics Y lm ( ϑ k , ϕ k ) ofthe distribution function ( ϑ and ϕ are the angles of thewavevector) occurs independently [4]. Thus, for the spindistribution s k = δ s k Y lm ( ϑ k , ϕ k ) the collision integral Q { δ s k Y lm ( ϑ k , ϕ k ) } = − τ − l δ s k Y lm ( ϑ k , ϕ k ) and it is de-scribed by a set of relaxation times τ l ( l = 1 , , , . . . ):1 τ l = X k ′ W k ′ k [1 − P l (cos ϑ k ′ )] , (3)responsible for the relaxation of different angular har-monics of the distribution function; P l ( x ) is the corre-sponding Legendre polynomial. Note, that τ = τ p de-scribes the momentum relaxation of electrons. In a clas-sical approach, these relaxation times are independent ofthe magnetic field.The electron scattering slows down the spin relaxationdue to randomization of the spin precession around thespin-orbit magnetic field: between the scattering acts theelectron spin rotates by a small angle ∼ Ω k τ ( τ is thecharacteristic relaxation time), while the scattering pro-cesses changes the wavevector k and, correspondingly,the spin precession frequency Ω k reducing the cumulativespin rotation angle. It follows from the solution of Eq. (1)that the longitudinal spin relaxation time for degenerateelectrons in bulk GaAs at B = 0 takes the form [4] T (0) = 10532 α ~ E g E τ , (4)where E F is the electron Fermi energy, E g = 1 .
52 eVis the band gap energy, α ≈ .
063 is the dimensionlessDresselhaus constant for GaAs recalculated from datain Refs. [41, 42], and τ is the relaxation time of thirdangular harmonics of the electron distribution over mo-mentum given by Eq. (3).A similar suppression of the spin relaxation takes placedue to the cyclotron motion of the electron in externalmagnetic field accounted for by the operator Λ k { s k } inEq. (1). Indeed, the field induces a rotation of the elec-tron velocity and the wavevector k , thus, resulting in arotation of the effective magnetic field ∝ Ω k . In this way,the magnetic field acts as an extra scattering source andslows down the spin relaxation [37, 40]. The magneticfield dependence of T was calculated in Ref. [37]: T ( B ) T (0) = [1 + ( ω c τ ) ][1 + 9( ω c τ ) ]1 + 6( ω c τ ) ≈ ω c τ . (5)The last approximate equality in Eq. (5) holds for ω c τ ≪
1. Equation (5) clearly demonstrates an increase in thespin relaxation time T with growing magnetic field. Thisexpression with the temperature-independent τ ≈
40 fsdescribes the experimental data at B & T extrapolated to B = 0, we obtain after Eq. (4) almost the same τ as thevalue obtained above from the B -dependence.The classical theory of Dyakonov-Perel spin relaxationmechanism, expressed by Eqs. (4) and (5), as well as ad-ditional possible mechanisms of spin relaxation due tothe g -factor spread [43] cannot, however, explain the siz-able decrease of the spin relaxation time T in rather weak magnetic fields B . T .
14 K. Clearly, other effects, not accounted for bythe approach in Refs. [4, 7, 14, 37–39] must play an im-portant role in our experiment. In fact, in the deriva-tion of Eqs. (4) and (5) the electron dynamics is as-sumed to be classical, i.e., the inequality E F τ p / ~ ≫ E F τ p / ~ just slightly exceeds unity and quantum effectsstart to play a role. In particular, for an electron trav-eling through a disordered medium the interference be-tween classical trajectories, as schematically depicted inFig. 2(b) becomes important. For electron waves trav-eling clockwise and counter-clockwise through the sameconfiguration of impurities, the phases acquired on thesetwo paths, φ (cid:8) = φ (cid:9) = H k d l , are the same. As a result,the two paths shown by the solid and dashed lines inter-fere constructively, leading to coherent backscattering.In effect, the scattering efficiency by the impurities in-creases ( τ p decreases) and the electron propagation slowsdown. This is the weak localization effect signifying theonset of the MIT with decreasing electron density. Im-portantly, a magnetic field destroys the constructive in-terference owing to the extra phase proportional to thefield flux through the trajectory acquired by the diffus-ing electron, see Fig. 2(c). Indeed, for clockwise andcounter-clockwise propagation the field-induced phasesare opposite, hence, the magnetic field suppresses theweak localization [28, 33–35, 44].In order to account for the interference effect we fol-low the semiclassical approach where, as illustrated inFig. 2(b),(c), the quantum effects are accounted for byrenormalization of the cross-section for electron scatter-ing by the impurity [13, 36, 45]. The momentum relax-ation time τ p acquires a correction δτ p of the form δτ p τ p = − mDπ ~ n e X s,s ′ = ± / C ss ′ s ′ s ( r = 0) , (6)where D = v τ p / v F is the Fermi velocity. In Eq. (6), C s s s s ( r = 0)is the Cooperon matrix describing the electron interfer-ence along the closed loops, which is calculated via astandard diagram technique [28]. Similarly, the inter-ference effects modify the relaxation time of the thirdangular harmonics of the spin distribution function, τ ,which defines the spin relaxation time [Eq. (4)], as: δτ τ = − mDπ ~ n e X s,s ′ = ± / C ss ′ s ′ s ( r = 0)(2 δ ss ′ − . (7)The factor (2 δ ss ′ −
1) is due to the spin vortices in thecorresponding diagrams [13, 36].The Cooperon matrix, i.e., the sum of maximallycrossed diagrams, describes the spin-dependent probabil-ity P ret of an electron to return to the initial point afteran arbitrary number of collisions conserving its phase,that is the probability to pass through a loop in thereal space [Fig. 2(b)]. Qualitatively, the interference ofelectron waves propagating clockwise and counterclock-wise on the loops, Fig. 2(b), gives rise to the coherentbackscattering effect and modifies the rate of the scat-tering by an impurity W kk ′ . It gives rise to a sharp peakin W kk ′ at k ′ ≈ − k , i.e., for backscattering [45]. The in-terference induced contribution δW kk ′ = W kk ′ − W cl kk ′ ∝ P ret , where W cl kk ′ is the classical value found without in-terference effects, is proportional to the return probabil-ity. It gives rise to the corrections δτ l to the relaxationtimes τ l in Eq. (3). Both δW kk ′ and δτ l are determinedby the interference of the trajectories in Fig. 2(b). Themagnetic field destroys the interference and suppresses δτ l affecting the electron transport and spin dynamics.We introduce the phase relaxation time τ φ associ-ated with inelastic electron-electron or electron-phononscattering processes, and consider hereafter the diffu-sive regime where τ φ ≫ τ p , τ and the magnetic length l B = p ~ c/ ( | e | B ) exceeds by far the mean free path, l B ≫ v F τ p . Moreover, we impose the condition of ratherweak spin-orbit interaction, T (0) ≫ τ φ , meaning thatthe electron spin is conserved during passage through theclosed loops in which the interference takes place. As aresult, we have δT ( B ) T (0) = δρ ( B ) ρ (0) = − m π ~ n e τ p r | e | B ~ c F (cid:18) B φ B (cid:19) . (8)Here B φ = ~ c/ ( | e | l φ ), l φ = p Dτ φ is the phase relaxationlength, and the function F ( x ) is defined as [33–35] F ( x ) = ∞ X n =0 " √ n + 1 + x − √ n + x ) − p n + 1 / x . Note that for x ≪ F ( x ) ≈ .
605 and for x ≫ F ( x ) ≈ / (48 x / ). Discussion and comparison of electron spin dynamicsand transport
To independently experimentally confirm the presenceof weak localization and estimate its magnitude in theconsidered system, we have also measured the magnetore-sitance on the same samples [see Fig. 3(a)]. The low-fieldnegative magnetoresistance is clearly seen, in agreementwith previous works it arises from the weak localizationeffect [29–35, 46]. At high fields positive magnetoresis-tance is observed, presumably due to the field-inducedcompression of electron wave functions on donors andalso possibly due to the onset of Shubnikov-de Haas os-cillations. The observed behavior is qualitatively similarto that for T ( B ) [Fig. 2(a)] and, in particular, the scaleof magnetic field, destroying the weak localization, is thesame. Further, the negative magnetoresistance persistsin the same range of temperatures as the decrease of T with B .Furthermore, according to Eq. (8), the relative changeof T and ρ due to the weak localization should bethe same. Figure 3(b) shows these relative variationsof T (the spheres) and ρ (the solid lines), δT /T ≡ -1 0 1 2 3 4 5 60.20.30.40.5 n e =3.7 10 cm -3
40 30 221814105 ( c m ) Magnetic field (T)T (K): 2
T = 2 K / Magnetic field (T)n e (10 cm -3 ):7.13.7 -0.4 -0.2 0.0 0.2 0.4-0.10.00.1 T /T T / T , Magnetic field (T)T (K):10B T ( T ) Temperature (K)T (c)(b)(a)
FIG. 3. Evidence of weak localization in resistivity measure-ments. (a) Magnetic field dependence of the resistivity ρ atdifferent temperatures. Inset shows relative variation of ρ with magnetic field for metallic samples with different elec-tron concentrations. (b) Relative variation of T (the sym-bols) and ρ (the solid lines) with magnetic field at differenttemperatures. The red dashed lines show fits to both δT /T and δρ/ρ with Eq. (8). The curves are vertically shifted forclarity. (c) Curvatures of the magnetic field dependencies of δT /T (the spheres) and δρ/ρ (the squares), κ in Eq. (9), asa function of temperature. The red dashed line shows a T − / dependence. (a)–(c) n e = 3 . × cm − . T ( B ) /T (0) − δρ/ρ ≡ ρ ( B ) /ρ (0) −
1, with mag-netic field, respectively. Equation (8) is, strictly speak-ing, valid if the quantum corrections are small, i.e. for δT /T , δρ/ρ ≪ δT /T and δρ/ρ are in remark-able agreement in weak magnetic fields. The analysis ofthe asymptotic form of Eq. (8) shows that in weak fields B ≪ B φ δT ( B ) T (0) = δρ ( B ) ρ (0) ≈ − κB , (9)with the prefactor κ ≈ . e/mc ) q τ p τ φ . In the stud-ied temperature range τ p is constant, as found above,and τ φ = A/T , where A is a constant, in accordancewith Refs. [35, 47]. Thus, κ ∝ T − / . The values of cur-vature κ corresponding to T and ρ extracted from the fitare shown in 3(c). They are in very good agreement andfollow a T − / dependence as shown by the red dashedline.The dashed lines in Fig. 3(b) show fits to the ex-periment by Eq. (8) using a reasonable set of parame-ters, namely τ p = 55 fs (temperature independent) and τ φ = A/T with A = 19 ps · K. Such inverse temperaturedependence of the phase relaxation time was observed fora similar GaAs system [35] with n e = 2 . × cm − giving a similar value of A ≈
12 ps · K. In order to com-pare the value τ p = 55 fs with the previously obtained τ = 40 fs we calculated the ratio of τ p /τ by angu-lar integration of the cross-section of partial scatteringat the screened Coulomb potential of charged impurities(see Supplemental material [48]). For the parameters ofour sample the ratio τ p /τ = 1 . τ obtained by con-sidering classical Dyakonov-Perel relaxation is in goodagreement with the time τ p derived from the weak local-ization anomaly.We have also studied the magnetic field dependenciesof T and ρ for a sample with higher electron concentra-tion n e = 7 . × cm − . The corresponding results arepresented in the insets in Figs. 2(a) and 3(a) and morein detail in the Supplemental material [48]. One can seethat the effect of weak localization is reduced by a fac-tor of about two for n e = 7 . × cm − compared to the sample with n e = 3 . × cm − as expected fromEq. (8) which contains n e in the denominator. The times τ , τ p and τ φ are similar for both samples. In conclusion , we have demonstrated that the weak lo-calization of electrons has pronounced impact on theirspin dynamics. The longitudinal spin relaxation time T in n -doped GaAs, being in the metallic phase, demon-strates an anomalous decrease with increasing magneticfield at low temperatures. This decrease is due to thefield-induced destruction of phase coherence for electronsresulting in the suppression of the weak localization. Thisshows that physics studied in transport experiments cap-turing the entirety of physical phenomena between theelectrical contacts may be studied locally using focusedoptical probes of the spin dynamics. The potential of thisapproach will be very prominent also in two-dimensionalsystems where one can expect visualization of the weaklocalization induced non-exponential tails in spin polar-ization. IV. ACKNOWLEGMENTS
We are grateful to S. A. Crooker for providing thesamples, and to E. Evers and A. Greilich for valu-able advice and useful discussions. We acknowledgethe financial support of the Deutsche Forschungsgemein-schaft in the frame of the ICRC TRR 160 (project A1).MMG was partially supported by the RFBR ProjectNo. 15-52-12012, Russian Federation President grantMD-1555.2017.2, and the program of RAS “Nanostruc-tures”. KVK and MMG acknowledge support fromSaint-Petersburg State University via a research grant11.34.2.2012. Magnetotransport measurements were per-formed using research equipment of the LPI SharedFacility Center and supported by Ministry of Educa-tion and Science of the Russian Federation (Grant No.RFMEFI61717X0001). A. Yu. K. was supported by Ba-sic research program of HSE. [1] R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V.Lazarev, B. Ya. Meltser, M. N. Stepanova, B. P.Zakharchenya, D. Gammon, and D. S. Katzer,“Low-temperature spin relaxation in n-type GaAs,”Phys. Rev. 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