Current-induced spin polarization in InGaAs and GaAs epilayers with varying doping densities
M. Luengo-Kovac, S. Huang, D. Del Gaudio, J. Occena, R. S. Goldman, R. Raimondi, V. Sih
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Current-induced spin polarization in InGaAs and GaAs epilayerswith varying doping densities
M. Luengo-Kovac, S. Huang, D. Del Gaudio, J.Occena, R. S. Goldman, R. Raimondi, and V. Sih Department of Physics, University of Michigan Department of Materials Science and Engineering, University of Michigan Dipartimento di Matematica e Fisica, Universit`a Roma Tre
Abstract
The current-induced spin polarization and momentum-dependent spin-orbit field were measuredin In x Ga − x As epilayers with varying indium concentrations and silicon doping densities. Sampleswith higher indium concentrations and carrier concentrations and lower mobilities were found tohave larger electrical spin generation efficiencies. Furthermore, current-induced spin polarizationwas detected in GaAs epilayers despite the absence of measurable spin-orbit fields, indicating thatthe extrinsic contributions to the spin polarization mechanism must be considered. Theoreticalcalculations based on a model that includes extrinsic contributions to the spin dephasing and thespin Hall effect, in addition to the intrinsic Rashba and Dresselhaus spin-orbit coupling, are foundto qualitatively agree with the experimental results. < et al. [16] fora 2DEG, which includes intrinsic and extrinsic contributions to the spin dephasing and thespin Hall effect, as well as the inverse spin galvanic effect.Five InGaAs and two GaAs samples were studied, each consisting of a 500 nm epi-layer grown by molecular beam epitaxy (MBE) on a semi-insulating (001) GaAs substrate.All samples were Si-doped at different concentrations. The samples were etched into across-shaped channel with arms along the [110] and [110] crystal axes. This allows for theapplication of an electric field along an arbitrary in-plane crystal axis [2].2able 1 shows a summary of sample parameters. The indium concentrations are deter-mined from X-ray rocking curves (XRC), which also show the epilayers to be pseudomorphicor nearly pseudomorphic with the substrate, i.e. the strain relaxation is minimal. The car-rier concentrations are determined from Hall and van der Pauw measurements performedon the cross-shaped channels. The mobility and SO coefficients α and β , defined below, aredetermined from spin-drag measurements [17], and the spin dephasing time T ∗ is determinedfrom time-resolved Faraday rotation (TRFR) measurements. All values are measured at 30K. Spin-orbit coupling in semiconductors manifests as an effective internal magnetic field.In zinc-blende semiconductors, this is described by the Hamiltonian [18] H SO = α ( k y σ x − k x σ y ) + β ( k y σ x + k x σ y ) (1)for x k [110] and y k [110], where α includes Rashba-like contributions from structural in-version asymmetry and uniaxial strain, and β includes linear Dresselhaus-like contributionsfrom bulk inversion asymmetry and biaxial strain [17]. As these two components of the SOfield have different crystal axis dependences, the anisotropy of the SO field is characterizedby the parameter r = α/β . In our InGaAs samples, the maximum SO field is along [110]and minimum along [110] crystal axes.The SO fields are measured by performing pump-probe spin drag measurements on thesamples [19]. The samples are mounted on the cold-finger of a continuous flow cryostat,and all measurements are performed at 30 K unless otherwise noted. A tunable-wavelengthpulsed Ti:Sapph laser is split into pump and probe pulses, and the relative time delay of thetwo pulses can be varied using a mechanical delay line. The pump pulse is circularly polarizedin order to induce a spin polarization in the sample according to the optical selection rules.The Faraday (Kerr) angle of the transmitted (reflected) linearly polarized probe is measuredwith a Wollaston prism and balanced photodiode bridge. The InGaAs (GaAs) samplesare measured in a transmission (reflection) geometry. Transmission measurements are notpossible in the GaAs samples as the wavelength used to probe the epilayer is absorbed bythe substrate. The pump and probe are modulated by a photoelastic modulator and opticalchopper respectively in order to allow for cascaded lock-in detection. An electromagnetallows for the application of an external magnetic field in the plane of the sample.When an electric field is applied across the sample, the electron spins precess about the3
40 -20 0 20 40 600.00.10.20.3 c) d)b) A m p li t ude ( a r b . un i t s ) Pump-probe separation ( m) 0 V 1 V 2 Va) 0 1 20123 B S O , ( m T ) Drift Velocity ( m/ns)B
SO, = v d D r i ft V e l o c i t y ( m / n s ) Voltage (V)-40 -20 0 20 400.00.51.0 N o r m . F a r ada y R o t a t i on Magnetic Field (mT)
FIG. 1. Spin drag measurements for the determination of the SO field for Sample C. (a) Amplitude A ( x ) vs. pump-probe spatial separation for 0 V (black), 1 V (red) and 2 V (green) and pump-probe time delay ∆ t = 13 ns. The location of the center gives the drift velocity. (b) Drift velocityvs. applied voltage. (c) Faraday rotation vs. magnetic field for the same in-plane voltages as (a)at the center of the spin packet. Fits to Eq. 2 give the SO field. (d) The perpendicular componentof the SO field at the center of the spin packet vs. drift velocity. The slope κ gives the strength ofthe SO field. vector sum of the external and SO fields. The Faraday/Kerr rotation θ F,K can be describedby the equation θ F,K ( ~B ext , x ) = X n A n ( x ) × cos h gµ B ~ (cid:12)(cid:12)(cid:12) ~B ext + ~B int (cid:12)(cid:12)(cid:12) (∆ t + nt rep ) i (2)where A n ( x ) is the amplitude due to successive pump pulses, g is the electron g-factor, µ B is the Bohr magneton, ~B ext is the external magnetic field, ~B int is the internal SO field, ∆ t isthe time delay between the pump and probe pulses, and t rep = 13 .
16 ns is the time betweenlaser pulses.Spin drag measurements are performed with the electric field applied parallel to theexternal magnetic field along either the [110] or [110] crystal axes and the time delay fixedto ∆ t = 13 ns. The drift velocity v d is determined from the pump-probe spatial separationat the position with maximum A ( x ) (Fig. 1a,b). Along these two crystal axes, with this4
80 -40 0 40 80-505 d)c) b) 1 V 2 V 3 V F a r ada y R o t a t i on (r ad ) Magnetic Field (mT)a) 0 2 40510 e l ( m - ) Drift Velocity ( m/ns)0 2 42345 ( n s ) Drift Velocity ( m/ns) 0 2 4024 = v d ( m - n s - ) Drift Velocity ( m/ns)
FIG. 2. (a) CISP measurements for 1V (black), 2V (red) and 3V (blue), showing an odd-Lorentzianlineshape for Sample A. The spin density ρ el (b) and lifetime τ (c) are used to calculate the spingeneration rate γ (c). The slope η of γ with respect to the drift velocity is used to characterize thestrength of the CISP.Sample x In n µ m ∗ ~ α m ∗ ~ β r T ∗ ρ el /θ el ([110],[110])(10 cm − ) (cm /Vs) (neV ns/ µ m) (neV ns/ µ m) (ns) ( µ m − /µ rad)A 0.026 20.8 ± ±
200 26 ± ± ± ± ± ±
300 39 ±
17 5.7 ±
17 6.9 ±
21 5.58 ± ± ±
200 28 ±
13 2.9 ±
13 9.8 ±
43 7.67 ± ± ±
300 -4.2 ±
16 28 ±
16 0.15 ± ± ± ±
500 13 ± ± ± ± ± ±
200 - - - 6.8 ± ± ±
100 - - - 3.87 ± α , β , and r could not be determined. Furthermore, as the absorptionof the GaAs epilayers cannot be measured, the conversion between Faraday angle and spin densitycannot be calculated. -5 -4 -3 -2 -1 x10 b) r = 0.61r = 0.15r = 9.8r = 1.0 r = 6.9 ( m - ) ( C I SP ) (mT/( m/ns)) (SO) [1-10] [110] A B C D E a) 0 1 2 310 -3 -2 -1 t h ( m - ) ( C I SP ) (mT/( m/ns)) (SO) FIG. 3. (a) η (CISP) vs. κ (SO splitting) for all five InGaAs samples. Squares indicate sampleswith higher indium concentration (2.4%-2.6%) and triangles indicate samples with 2.0% indium.Filled in and open symbols are for measurements along the [110] and [110] crystal axes respectively. r = α/β characterizes the anisotropy of the SO field. There was a negative differential relationshipobserved between the two parameters in all five samples. (b) Theoretical calculations for η basedon the model (Eq. 5) using the material parameters for the five InGaAs samples. The modelpredicts the observed negative differential relationship. configuration of parallel electric and magnetic fields, the SO field is purely perpendicularto the external magnetic field and manifests as a reduction of the amplitude of the centerpeak of the magnetic field scans (Fig. 1c). We measure the magnitude of the SO field as afunction of applied voltage.The SO field is found to be linear with drift velocity (Fig. 1d), where the slope κ is usedto characterize the strength of the SO field. Measurements of κ for voltages along the [110]and [110] crystal axes allow us to extract the SO parameters α and β (Table 1).CISP is measured with the Faraday rotation of the probe beam in the absence of opticalpumping (Fig. 2a). This is described by the equation [1] θ F = θ el ω L τ ( ω L τ ) + 1 (3)where θ el is the amplitude of the electrically induced Faraday rotation, ω L is the Larmorprecession frequency, and τ is the transverse spin lifetime. The electrical induced spindensity can be related to the electrically induced Faraday rotation with the equation (seeSupplemental Material) ρ el = θ el ρ op θ op (4)where ρ op and θ op are the optically induced spin density and Faraday rotation respectively.6he ratio ρ el /θ el for the InGaAs samples is shown in Table 1. The quantity of interest is thedensity of spins oriented per unit time, given by γ = ρ el /τ .The measurement shown in Fig. 2 is performed for various voltages applied parallel tothe external magnetic field. Fit values for ρ el and τ are shown in Fig. 2b,c as a function ofthe voltage, which is given in terms of the drift velocity. γ is found to be proportional to thedrift velocity (Fig. 2d), and the slope η is used to characterize the electrical spin generationefficiency. Measurements are repeated for voltages along the [110] and [110] crystal axes.Figure 3a shows the parameter η for CISP versus the parameter κ for the SO fields for theInGaAs samples. A theory of the inverse spin galvanic effect solely based on the inclusionof intrinsic SO contributions would predict that the CISP should be proportional to the SOfield. However, consistent with previous measurements [2], we found that the crystal axiswith the smallest SO splitting had the largest CISP and vice versa.Samples with higher carrier concentrations were found to have greater CISP (Fig. S2a).Assuming the same rate of spin polarization, this would result in a larger spin density givena larger carrier concentration. Furthermore, samples with lower mobility had greater CISP(Fig. S2b). Since the mobility is proportional to the momentum scattering time, this indi-cates that samples with less time between scattering events had greater spin polarizations,and suggests that an extrinsic polarization mechanism dominates.We also found that samples with higher indium concentration had higher electrical spingeneration efficiencies (see Fig. S1a). Higher indium concentration causes more strain inthe InGaAs epilayer due the 7% lattice mismatch between InAs and the GaAs substrate.The higher strain results in larger SO splitting in the epilayer. Thus, this suggests thatthe amount of SO splitting is related to the amount of CISP, albeit not in the direct waydescribed by the model with only Rashba and Dresselhaus SO contributions. There wasno clear correlation between the spin dephasing time and the magnitude of CISP (see Fig.S1b).In contrast to InGaAs grown on GaAs substrates, GaAs epilayers do not have straininduced spin-orbit fields. However, we also observed CISP in GaAs (see Fig. S2). As withthe InGaAs samples, we found that the CISP was greater along the [110] axis than the [110]axis. Furthermore, we found that the GaAs sample with higher carrier concentration hadmore CISP, consistent with the measurements in InGaAs.The SO fields in the GaAs samples were very small ( < et al. derived the Bloch equation for a 2DEG including both intrinsic and extrinsicSO contributions to the spin dephasing, the spin Hall effect, and the spin-generation torque[16]. The change in the total spin polarization over time is given as: ∂ ~S∂t = − (Γ DP + Γ EY ) (cid:18) ~S − N ~B ext (cid:19) − ( ~B ext + ~B SO ) × ~S + (Γ DP − Γ EY ) N ~B SO + θ extSH θ intSH Γ DP N ~B SO (5)where N is the density of states, and θ int(ext)SH is the spin Hall angle due to intrinsic (extrinsic)contributions [14, 20]. Γ DP and Γ EY are the dephasing rate tensors for the two dominantmechanisms: D’yakonov-Perel’ (DP) dephasing [21], an intrinsic effect that is due to pre-cession of the spins about momentum-dependent spin-orbit fields between scattering events,and Elliot-Yafet (EY) dephasing [22], an extrinsic effect that is due to spin flips at scatteringevents [23].The relative strength of the DP and EY dephasing mechanisms can be determined fromtemperature dependent measurements of the spin dephasing time and mobility (see Supple-mental Material). At 30 K, the temperature at which all CISP and SO field measurementswere performed, the extrinsic EY dephasing mechanism was found to be comparable to ordominant over the intrinsic DP dephasing mechanism for all samples.Using Eq. 5, we can solve for the theoretical steady-state spin density ρ el,th , and thereforethe theoretical spin generation rate per unit drift velocity η th . The values for η th calculatedusing the material parameters of the five InGaAs samples are shown in Fig. 3b as a functionof the SO splitting along the [110] and [110] crystal axes. For the given material parameters,the model predicts a negative differential relationship between the CISP and SO splitting.In general, the relationship between the CISP and SO splitting may be either positiveor negative depending on the values of the spin Hall angles, r , and q (see SupplementalMaterial). Although the predicted values are an order of magnitude larger than the measuredvalues, the relative magnitudes of the predicted η th are qualitatively consistent with theexperimental results.Figure S2c,d, shows η th as a function of carrier concentration and mobility respectively,8 -5 -4 -3 -2 -1 ( m - ) ( C I SP ) Carrier Concentration ( m -3 ) 3 4 5 6 710 -5 -4 -3 -2 -1 d)c) b) [1-10], Low In [110], Low In [1-10], High In [110], High In ( m - ) ( C I SP ) Mobility (10 cm /(Vs))a)0 1x10 t h ( m - ) ( C I SP ) Carrier Concentration ( m -3 ) 2 4 60.11 t h ( m - ) ( C I SP ) Mobility (10 cm /(Vs)) [1-10] [110] FIG. 4. Measured values of η (CISP) for the [110] and [110] crystal axes as a function of (a)carrier concentration and (b) mobility. Squares indicate samples with higher indium concentration(2.4%-2.6%) and triangles indicate samples with 2.0% indium. Red and black symbols are formeasurements along the [110] and [110] crystal axes respectively. Calculations for η as a functionof (c) carrier concentration and (d) mobility using material parameters for Sample D. using the material parameters for Sample D. The model predicts that the CISP is largest insamples with high carrier concentrations and low mobilities, consistent with the experimentalresults.We performed measurements of CISP and SO splitting along the [110] and [110] crystalaxes in seven In x Ga − x As samples with different Indium concentrations and doping densities.In all samples, we found a negative differential relationship between the magnitude of theCISP and SO splitting. Theoretical calculations based on the model proposed by Gorini etal. are found to qualitatively agree with the experimental results. This model was derivedfor a 2DEG, whereas measurements were performed on bulk epilayers. A model that includes3-dimensional effects may provide better quantitative agreement between the model and theexperiment.Work by M.L.-K. and V.S. was supported by the U.S. Department of Energy, Officeof Basic Energy Sciences, Division of Materials Sciences and Engineering under Award9e-sc0016206. R.R. acknowledges stimulating discussions with C. Gorini, A. Maleki, K.Shen, I. Tokatly, and G. Vignale. S.H., J.O., and R.S.G. were supported in part by theNational Science Foundation (Grant No. DMR 1410282); D.D.G was supported in part bythe National Science Foundation (Grant No. ECCS 1610362).
SUPPLEMENTARY INFORMATIONConverting Faraday Angle to a Spin Density
It is possible to convert the electrically induced Faraday rotation θ el to a spin density bycomparing the Faraday rotation due to optical polarization to the Faraday rotation due toelectrical polarization [1]. With optical injection, the number of spins polarized per laserpulse is n op = ρ op × πσ x σ y d = 12 α (cid:18) P pump f rep / π ~ cλ (cid:19) (S1)where ρ op is the density of optically polarized spins, σ x and σ y are the widths of the Gaussianprofile of the pump spot, d is the thickness of the epilayer, P pump is power of the pump spot, α is the absorption of the epilayer, and f rep and λ are the repetition rate and wavelength ofthe laser.The Faraday rotation due to optical injection is given by θ op = Ad Z Z " ρ op e − (cid:18) x σ x + y σ y (cid:19) dxdy = πAdρ op σ x σ y (S2)where the factor of two in the exponent accounts for the spatial profiles of the pump andprobe beams. In this way, we can get the conversion factor A = θ op / ( πdρ op σ x σ y ) betweenthe spin density and the Faraday rotation.The density of electrically polarized spins ρ el is related to the Faraday rotation by theequation θ el = Ad Z Z " ρ el × e − (cid:18) x σ x + y σ y (cid:19) dxdy = 2 πAdρ el σ x σ y (S3)The electrically induced spin polarization is spatially uniform, and so the only spatial de-pendence comes from the spatial profile of the probe beam. By including the results for theproportionality factor A into Eq. S3, we arrive at the result for the steady-state density ρ el : ρ el = θ el ρ op θ op (S4)10 .0 2.2 2.4 2.610 -5 -4 -3 -2 -1 a) b) ( m - ) ( C I SP ) x In (%) 5 10 15 20 [1-10], Low In [110], Low In [1-10], High In [110], High In Spin Lifetime (ns)
FIG. S1. η (CISP) for the [110] (red) and [110] (black) crystal axes as a function of (a) indiumconcentration and (b) spin dephasing time. Square and triangles indicated samples with higher(2.4-2.6%) and lower (2.0%) indium concentrations respectively. The ratio of the electrically generated spin density ρ el to the Faraday angle θ el for themeasurements along both crystal axes of each sample are given in Table 1. Current Induced Spin Polarization vs. In concentration and T ∗ Samples with higher indium concentration had higher electrical spin generation efficiencies(Fig. S1a). There does not seem to be a relationship between the spin generation efficiencyand the spin dephasing time (Fig. S1b).
Current Induced Spin Polarization in GaAs epilayers
Fig. S2a shows CISP for 2 V applied across the higher doped GaAs sample along boththe [110] and the [110] crystal axes. Fig. S2b shows CISP for 4 V applied across the lowerdoped GaAs sample along the [110] and [110] crystal axes. There was no measurable CISPalong the [110] axis in the lower doped GaAs sample. In the GaAs samples, the epilayer andsubstrate absorb at similar wavelengths, and so we cannot measure the absorption of onlythe epilayer in order to calculate the conversion factor A . Because of this, the CISP in theGaAs samples can only be reported in terms of µ rad and the magnitude of CISP in thesesamples cannot be directly compared to the InGaAs samples.11
80 -40 0 40 80-0.50.00.5 b) 10 GaAs 4 V [1-10] [110] K e rr R o t a t i on (r ad ) Magnetic Field (mT) a) 10 GaAs 2 V -80 -40 0 40 80
Magnetic Field (mT) x5
FIG. S2. CISP for (a) 2 V across the n ∼ and (b) 4 V across the n ∼ (b) GaAssamples. The data in (b) has been magnified by a factor of 5. Black squares and red circlesindicate measurements along the [110] and [110] crystal axes respectively. There was no CISPdetected in the n = 10 GaAs sample along the [110] crystal axis. For both samples, CISP wasgreater along the [110] crystal axis. Furthermore, the sample with higher carrier concentration hadgreater CISP.
Determining the Relative Strength of the Spin Dephasing Mechanisms
The total spin dephasing time is given by [24]1 τ s = 1 τ EY + 1 τ DP = C EY µ − T + C DP µT (S5)where T is the temperature, and C EY and C DP are coefficients denoting the relative strengthof the EY and DP dephasing mechanisms.We performed temperature dependent measurements of the spin dephasing time τ s andmobility µ , in order to extract the relative strength of the EY and DP dephasing mechanisms.We can fit for C EY and C DP in Eq. S5 using a two independent variable fit, with T and µ as the independent variables. The relative strength of the two dephasing mechanisms isthen defined by q ( T ) = τ − τ − = C EY C DP µ − T − (S6)The value of q ( T ) calculated from the fits of the dephasing time and Eq. S6 is shown inFig. S3a. The fits of the dephasing time versus mobility and temperature are shown in Fig.S3b,c for the sample with n = 1 . × cm − .12 EY / D P Temperature (K)a) 40 60 80 100
Temperature (K)5000 7500 100000.02.0E84.0E86.0E8 D epha s i ng R a t e ( s ) Mobility (cm /Vs) FIG. S3. (a) Ratio of the dephasing rates due to the EY and DP dephasing mechanisms for theIn . Ga . As sample with n = 1 . × cm − , calculated from results of a two-independentvariable fit of the dephasing rate as a function of the temperature and the temperature-dependentmobility. q > Theoretical Spin Generation Efficiency vs r and q The theoretical value for η th was also calculated as a function of r and q using mate-rial parameters for Sample D. Although the anisotropy in the spin generation efficiency isgreatest at r = 1, the total magnitude of the efficiency is lowest for that value (Fig. S4a).Although it is necessary to include the EY dephasing mechanism in order to get the negativedifferential relationship between CISP and the SO splitting, samples in which the DP de-phasing mechanism dominates (i.e. q <
1) have larger electrical spin generation efficiencies(Fig. S4b).
Crystal Axis with Maximum CISP
For the materials under study in this work, CISP was measured to be maximum alongthe [110] crystal axis. However, depending on the values of the spin Hall angles, r , and q ,the model predicts maximum spin polarizations along either the [110] or the [110] crystal13 t h ( m - ) r = [1-10] [110]a) 0 50 1000.010.11 t h ( m - ) q = EY / DP FIG. S4. Calculations for η as a function of (a) r = α/β and (b) q = Γ EY / Γ DP using materialparameters for Sample D. The red (black) curve denotes calculations for the [110] ([110]) crystalaxis. axis.We begin by defining the following dimensionless parameters (in units of ~ = c = 1): b = 2 eτ βEτ DP s H = θ extSH + θ intSH θ intSH , (S7)the dimensionless matricesˆ γ tot = r + q r r r + q
00 0 2(1 + r ) ˆ γ rel = s H (1 + r ) − q s H r s H r s H (1 + r ) − q
00 0 2 s H (1 + r ) , (S8)and the dimensionless vector ~b = ˆ E x + r ˆ E y − ( r ˆ E x + ˆ E y )0 , (S9)in the { [100], [010], [001] } basis.In the absence of an external magnetic field, the Bloch equation for the model proposedby Gorini et al. [16] can be written as ∂ t ~S = − ˆ γ tot ( ~S − b ˆ γ − ˆ γ rel ~b ) − b~b × ~S (S10)14he steady-state solution for the in-plane spin polarization has the compact form (cid:18) ˆ µ + b r ) ˆ ω (cid:19) ~S xy = b ˆ ν~b xy (S11)where ˆ µ = r + q r r r + q ˆ ω = b y − b x b y − b x b y b x ˆ ν = s H (1 + r ) − q s H r s H r s H (1 + r ) − q (S12)To first order in b , the steady-state in-plane spin polarization in thus given by ~S (1) xy = b ˆ µ − ˆ ν~b (S13)This can also be written in the form ~S (1) xy = ( a ˆ σ + a x ˆ σ x ) ~b (S14)where ˆ σ is the identity, ˆ σ x is the Pauli matrix, and a = b (1 + r + q ) − r (cid:2) s H (1 − r ) + q ( s H − r ) − q (cid:3) a x = 2 rq ( s H + 1) b (1 + r + q ) − r (S15)When ~E k [110], ~b k [110] and ~S xy = ( a − a x ) ~b , and when ~E k [110], ~b k [110] and ~S xy = ( a + a x ) ~b .If s H + 1 >
0, then a x >
0. This happens as long as θ extSH /θ intSH > −
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