Spin-orbit interaction and spin relaxation in a two-dimensional electron gas

0.24 MB·English

by  M. Studer

Spin-orbit interaction and spin relaxation in a two-dimensional electron gas
M. Studer,1,2 S. Scho¨n,3 K. Ensslin,2 and G. Salis1
1IBM Research, Zurich Research Laboratory, S¨aumerstrasse 4, 8803 Ru¨schlikon, Switzerland
2Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
3FIRST Center for Micro- and Nanosciences, ETH Zurich, 8093 Zurich, Switzerland
(Dated: January 6, 2009)
Usingtime-resolvedFaradayrotation,thedrift-inducedspin-orbitfieldofatwo-dimensionalelec-
trongasinanInGaAsquantumwellismeasured. Includingmeasurementsoftheelectronmobility,
9 the Dresselhaus and Rashba coefficients are determined as a function of temperature between 10
0
and80K.By comparingtherelativesize ofthesetermswith ameasured in-planeanisotropy of the
0
spin-dephasing rate, the D’yakonv-Perel’ contribution to spin dephasing is estimated. The mea-
2
sured dephasing rate is significantly larger than this, which can only partially be explained by an

n inhomogeneous g factor.
a
J
6 The possibility to manipulate spins in semiconduc- a) -V2/2 RS -V/2 b) [110]
1
tors is a requirement for future spin-based information

l] processing.1 Using the spin-orbit (SO) interaction2,3 is +V1/2 RS E
l a promising way to precisely control spin polarization B B
a RS I0 SIA BIA
s-h beleeccaturosdeeos.f4,i5tsMsaimnippluelaptrioinnciopflespbinasseudsionng ethxeteSrnOalingtaetre- y 200mm k f Bext Btot
me asuctcihonashabsublkeesnemshicoownnduinctvoarrsi6o,utswsoem-diicmoenndsuioctnoarlseylestcetmrosn, +xV2/2 RS Vxx q [110]
gases7 (2DEGs),andevenquantumdots containingonly
t. one single electron.8 On the other hand, the SO interac- FIG. 1: (a) The InGaAs/GaAs QW is shaped into a cross
a with 150-µm-wide arms contacted by Ohmic contacts. Ad-
m tion is a source for spin dephasing. In 2DEGs, the SO
ditional contacts on one arm of the cross allow a four-point
interaction induces a linear k-dependent splitting.9 This
- measurement of the voltage drop Vxx and the determination
d splitting gives rise to an effective magnetic field, leading of electron sheet density and mobility. (b) θ and φ are the
n todephasingofthepolarizedelectronspins.10 Thiseffect
angles of the magnetic and electric fields with respect to the
o isknownastheD’yakonv-Perel’(DP)mechanism,andits [110] axis. Btot is the vector sum of the external magnetic
c
control through manipulation of the SO interaction has field and thetwo SO effectivemagnetic field contributions.
[
been proposed11 as an alternative to the ballistic spin

2 transistor.4 A careful engineering of the SO interaction
v is therefore crucial for using it to manipulate the spin. suredspin-dephasingratesandtheirin-planeanisotropy,
8
wefindthatDPisnottheonlymechanismforspindecay
4
Ina2DEGatintermediatetemperatures,itisoftenas- in our samples at T between 10 and 80K.
8
sumedthatthespindecayisgovernedbytheDPmecha- The 2DEG we use in this work is located in an
2
. nism.12,13 Based on this assumption, information on the In0.1Ga0.9As/GaAs QW. Electrons are confined to a 20-
9 SO interaction in semiconductor quantum wells (QWs) nm-thick In Ga As layer that is n-doped (3×1016
0 was obtained from measurements of the spin-dephasing cm−3) to en0s.u1re 0a.9small electron scattering time such
8
rate.14,15,16 An independent measurementof the relative that we are in the dirty limit of the SO interaction9,
0
: size of the SO interactionin (110)-grownQWs using the where the frequency ωSO of spin precession about the
v
photogalvanic effect has been described in Ref. 17 and SO fields is small compared with the momentum scat-
Xi compared to the spin decay time. In this paper, we re- tering rate 1/τ (ω τ ≈ 10−5 for our samples). On
p SO p
r port on quantitative and independent measurements of both sides of this layer, there is a 10-nm-thick GaAs
a the SO interactionand the spin-dephasing rate in an In- spacer layer and a 10-nm-thick layer of n-doped GaAs.
GaAsQW,utilizingtime-resolvedFaradayrotation. Ina A 10nm cap of undoped GaAs completes the structure,
furtherdevelopmentofthemethoddescribedinRef.7,a grown by molecular beam epitaxy and forming a 2DEG
well-definedcurrentisappliedinthe2DEGusingOhmic 40nm below the surface. We use wet etching to pattern
contacts and a mesa structure (in Ref. 7, the electron a cross-shaped mesa as shown in Fig. 1(a) and create
drift was induced by an ac voltage applied to Schottky standard AuGe Ohmic contacts in the four ends of the
contacts in an unstructured 2DEG). The drifting elec- cross. Four additional contacts on one arm of the cross
trons see an effective SO magnetic field, in the following allow its use as a Hall bar to determine the resistivity
referred to as drift SO field. The sizes of its two con- and carrier density of the 2DEG. Two samples with the
tributions, the Rashba3 and the Dresselhaus2 fields, are same structure are glued into one chip carrier, whereby
◦
determinedasafunctionoftemperatureT fromthemea- one sample is rotatedby 90 to allow the SO interaction
sured influence of the in-plane electron drift velocity on tobemeasuredinonecool-down. At40K,thetwo-point
the spin precession. Comparing our results with mea- resistance of the crosses in x or y direction is 4.1kΩ.
2
We use additional resistors Rs = 4.7kΩ to compensate a) b)
for small variations in the contact resistance and apply
voltages V = V cos(φ) and V = V sin(φ) as shown in
1 A 2 A
Fig. 1(a). All angles are given with respect to the x axis
along[110],asseeninFig.1(b). Weobtaintheresistivity
of the 2DEG during optical experiments monitoring the
ratio of the voltage drop V and the current I through
xx 0
one armof the mesa [see Fig. 1(a)], and measure a value
of 770 Ω/sq at 40K. The voltages V and V create an
1 2
electric field E in the center of the cross in direction φ
and with an amplitude proportional to V . Because V
A xx
is monitored, the corresponding component of E can be
determined directly (see below). The electric field shifts
the Fermi circle by an amount of δk = m∗µE/~, where
∗
m is the effective electron mass, µ is the electron mo-
bility, and ~ is Planck’s constant divided by 2π. In the
dirty SO limit, the shift induces drift SO fields that can FIG. 2: (a) |Btot| in the center of the cross (symbols) as a
functionofthemeasuredelectricfield. Theslopeofthefitted
be expressed as9
line is 2.71×10−6Tm/V. (b) Dependence of the fitted slope
on the position x (crosses). Assuming a constant α−β of
2α δk 2β δk
B = y , B = y , (1) 4.9×10−14eVm,asimulation of theelectric field reproduces
SIA BIA
gµ (cid:18)−δk (cid:19) gµ (cid:18)δk (cid:19)
B x B x themeasured values (solid line).
with δk=(δk ,δk ), g is the electrong factor,µ is the
x y B
Bohr magneton, and α and β are the Rashba and Dres-
beam. The pump/probe intensity ratio is 10/1 and, un-
selhaus spin-orbit coefficients, respectively. The Rashba
lessstatedotherwise,thepumppoweris500µW,focused
field has its origin in the structure inversion asymme-
onto a spot 30µm in diameter. The circularly polar-
try (SIA) due to nonuniformdoping on both sides of the
ized pump pulse is tuned to the absorption edge of the
QW and the presence of the surface on one side of the
QW at 1.44eV and creates a spin polarization along the
QW. The Dresselhaus field is a consequence of the bulk
growth axis of the QW. With a pump power of 500µW
inversion asymmetry (BIA) of the zinc-blende structure.
and assuming an absorption of about 1%21 we obtain
Cubic Dresselhaus terms do not change the linearity of
B in δk, but introduce a correction of 1− 1k2/hk2i, a photo-excited carrier concentration on the order of a
BIA 4 F z few 1010cm−2, which is more than a magnitude smaller
wherek istheFermiwavenumberandhk2itheexpecta-
F z thantheequilibriumcarriersheetdensityintheQW(see
tion value of the squaredwave number along the growth
direction z. Taking a sheet density of 5.2× 1015 m−2 below). The Faraday rotation of the linear polarization
axisoftheprobepulsetransmittedisproportionaltothe
(see below) and approximating hk2i by (π/w)2, where
z spin polarization along the QW growth axis. Changing
w = 20nm is the QW width, we obtain k2 ≈ 1.32hk2i.
F z the delay ∆t between pump and probe reveals the spin
This gives a correction in β of about than 35%, which
dynamics of the system, and the Faraday rotation angle
will be neglected in the following. ∗
canbe describedbyAexp(−∆t/T )cos(Ω ∆t). Here,A
tioAnnθeaxstesrenenalimnaFgign.et1i(cbfi).elTdhBeexatngisleaspθplaierde9in0◦thfoerdsiaremc-- is the amplitude of the Faraday s2ignal anLd T2∗ the spin
ple1 and 180◦ for sample2. A transverse electron polar- dephasing time. A measurement of ΩL in a known mag-
neticfieldrevealsanelectrong factorof−0.29,assuming
ization precesses coherently about the vector sum18,19,20
that the g factor is negative.
of B and the drift SO fields defined in Eq. (1) with a
ext
Figure 2(a) shows B measured in the center of the
frequency givenby the modulus ofthis totalfield vector. tot
cross as a function of the electric field E = V /l be-
If B ≫ B ,B , the total field can be approxi- 0 xx
ext SIA BIA
tween the two contacts in the right arm. These con-
mated as7
tacts are separated by a distance of l=100µm. V is
2
set to ground. The temperature of the sample is 40K,
B (θ,φ)≈B +(B +B )cosθsinφ+ and B is oriented along the [110] direction, therefore
tot ext BIA SIA ext
(2) ◦
θ = φ = 90 . The data in Fig. 2(a) contain values from
(B −B )sinθcosφ.
BIA SIA
sweeps of V up and down. The up and down sweeps fit
1
Because of the different angular dependencies of the nicely to a straight line, showing that we can exclude a
Rashba and Dresselhaus SO fields, the two contribu- drift of ΩL over time, which might be caused by nuclear
tions can be distinguished. We use time-resolved Fara- polarization or drift of laser power or temperature.
day rotation to determine the Larmor frequency Ω = As a result of the sample geometry and the grounded
L
gµ B /~ of the spins precessing about B . For this, contactstotheleftandtherightofthecenter,theelectric
B tot tot
the output of a pulsed Ti:sapphire laser with a repeti- field in the center of the cross is reduced as comparedto
tion rate of 80MHz is split into a pump and a probe thevalueE measuredinthearm. Byscanningthelaser
0
3
a) 800V/m b) 880000V/m rier density n. We calculate the mobility µ using µ =
1700V/m 11600V/m 1/neρ, e being the electron charge. In the dark, the re-
2500VVV///mmm 2400VVV///mmm sistivityofthe2DEGisapproximately1000Ω/sqandde-
3300V/m 3200V/m
creasesto 770 Ω/sq under conditions of the optical mea-
surements. The Hall sheet densities are 5.8×1015 m−2
underilluminationand4.5×1015m−2inthedark. These
( (
( ( densitiesareconstantfrom10to 80K.FromShubnikov–
de Haas oscillations at T < 20K, we get a sheet density
of 5.2×1015 m−2 under optical illumination. A small
parallel conductivity of a doping layer could explain the
difference in the two numbers. Such a parallel conduc-
tivitydoesnotinfluencethe opticalexperimentsasthese
electrons do not contribute to the Faraday signal.
The mobilities extracted from the Hall measurement
( ( ( (
under illumination are shown in the upper inset of
Fig.4(a). Themobilitydoesnotchangesignificantlyover
FIG. 3: Btot as a function of the direction φ of the in-plane
thetemperaturerangemeasured. Usingtheresultsofthe
electric field for different amplitudes E and for two external
magnetic field directions (a) θ = 180◦ and (b) θ = 90◦. The transport measurements and assuming that m∗ =0.058,
solid lines are fits using Eq.(2). The data are obtained at wecancalculateδk anduseEq.(1)toobtainαandβ for
T =20K. allthetemperaturesmeasured. Theresultsaredisplayed
in Fig. 4(a). Error bars show a 2σ confidence interval.
The wafer usedfor this workis the same as the one used
beam along the x-axis and centered in the y-direction for sample3 in Ref. 7. We measure values for α and β
[see Fig. 1(a)], we obtain a cross section of the drift SO thatare by afactor of2-3smaller,whichwe attribute to
field that is related to the electric field distribution.22 a more precise determination of the electric field in this
The resulting slopes of the linear fits of B vs. E are work. In addition, different wafer processing and oxida-
tot 0
shown as a function of x in Fig. 2(b). We see a pro- tion of the wafer surface over time might influence the
nounceddipinthecenterofthecross,whichisexplained SO coefficients measured in this shallow 2DEG. Varia-
by the reduced electric field. The solid line in Fig. 2(b) tions in α and β for subsequent cool downs are within
represents the solution of a numeric simulation of the the error bar. Note that we extrapolate the mobility
electrostaticsusinga two-dimensionalpartialdifferential and the electric field in the center of the cross from a
equationsolver. Assumingthatαandβ areindependent transport measurement away from the center. We can-
of the position and using the measured mobility (see be- notexcludethatweunderestimatetheabsolutevaluesfor
low), the only fit parameter left is the difference α−β. α and β because of a reduced electron drift momentum
The measurement and the simulation are consistent and that might result from, e.g., screening by the optically
show that the electric field E in the middle of the cross excited charge carriers. The model of Vin˜a23 with the
is 0.71 times smaller than the measured value E . This results and the parameters used by Hu¨bner et al.24 pre-
0
correction is taken into account in the following when dicts the T dependence of the band parameters, from
indicating electric-field values. which we estimate the T dependence of α and β using
To disentangle α and β, we position the laser spot in k·ptheory.25 The calculatedT-inducedchangeinαand
thecenterofthecrossandrotatetheelectricfield. Wedid β between 10 and 80K is in the sub-percentage range
such experiments for different amplitudes of the electric and thus much smaller than our measurement error. In-
field up to 3.3kV/m and for two different configurations terestingly, in Ref. [15] a linear increase in α with T for
ofB . InFig.3(a),θ =180◦,andinFig.3(b),θ =90◦. higherT wasobservedona[110]QW,whichtheauthors
ext
Thedataareobtainedat20K.B oscillatesinφwithan could not explain with k·p theory.
tot
◦
amplitude thatis proportionalto α+β for θ =180 and We find no dependence of the SO coefficients on B ,
ext
◦
to α−β for θ = 90 . The difference in the amplitude in agreement with our assumption that the precession
for the two cases (note the different scales) shows that frequency is given by the modulus of the vector sum of
BSIA and BBIA are comparable in relative strength and Bext andthe driftSOfields giveninEq.(1). Figure4(b)
that the interplay of the two SO effects gives rise to an showsameasurementofα−β vs B . The insensitivity
ext
anisotropic spin splitting in k space. The solid lines are of the result on B excludes a significant admixture of
ext
a fit to the data using Eq. (2). Small deviations of the ak dependentandanisotropicg factor,aswasstipulated
data from theory in the φ direction could result from a in Ref. 26. To test the reliability of our method, we also
slight accidental off-center position of the laser spot. checked whether a lower pump power will influence the
TocalculatetheSOcoefficientsαandβ fromthemea- outcome of the measurement. This could occur from,
sured B and B , we need a value for the drift mo- e.g., a population of higher energy states with larger
SIA BIA
mentum of the electrons. This is obtained from a Hall pump power. We found, however, that α−β does not
measurementof the sheetresistivity ρ andthe sheet car- depend significantly on the pump power, as seen in
4
a) b) a)
(
m
(
*T2
1/
c)
Temperature ( (
,
b) c)
FIG. 4: (a) Measured µ, α and β vs. T. (b) Measured α−β
( (
shows no significant dependence on Bext. (c) Pump-power
dependenceof α−β. The data in (a) and (b) were obtained
with a pump power of 500µW, the data in (b) and (c) at *(T2 *(T2
T=40 K. 1/ 1/
Fig. 4(c).
When α and β are of similar magnitude, the spin life-
time is strongly anisotropicwith respect to the direction
( ( ( (
of B in the plane of the 2DEG.11,14,16,20,27,28 This
ext
anisotropy is a consequence of the DP mechanism be- ∗
FIG. 5: (a) Spin dephasing rate 1/T2 for two different in-
cause the spins precess about a SO field whose direction plane directions θ vs. T from 10 to 80 K. The average of
becomes independent of k for α ≈ β. From the mea- three measurements with 50µW pump power was used. (b)
sured anisotropy in T2∗ and the relative size of α and β, 1/T2∗ vs. Bext andlinearfitsforthreedifferentpumppowers.
we estimate the contributionfromthe DP mechanism to The data is measured in the θ = 90◦ configuration at 10K,
the spindephasing. InFig.5(a), the spin relaxationrate andat40Kforthe500µWcase. (c)Pump-powerdependence
1/T∗ is plotted as a function of T for the two orienta- of 1/T2∗ measured at 10K in the θ=90◦ configuration.
2
tions of B =0.99T. The T dependence is rather small,
ext
∗
and we can clearly see an anisotropy of 1/T , confirm-
2
ing the anisotropicspin splitting in our system. Because ference. Using the measured values for α and β, we get
our 2DEG is well in the dirty SO limit, we can use the C =6.6×1035m−2eV−2s−1. FromEqs.(3) and(4), this
motional narrowing limit of the DP mechanism,9 where yields relaxation rates for DP of about 0.6×109 s−1 for
the spin dephasing due to |k|-dependent SO fields is de- θ = 180◦, and of 1.0×109 s−1 for θ = 90◦. In Eq. (3),
creasedby spin-preservingscattering. This gives the fol- it is assumed that the SO splitting is linear in k. As
lowing expressions for the anisotropic spin decay rates: mentioned earlier, we are in a regime where kF2 ≈ hkz2i.
Taking into account the cubic Dresselhaus terms29 we
1 1 C
=C(α2+β2), = (α±β)2. (3) find only a small correction to the values for the spin
τz τx,y 2 relaxation rate obtained above. As the total 1/T2∗ lies
between 2.1×109 and 2.5×109 s−1, other spin-dephasing
Here, τ are the relaxation times of spins oriented
x,y,z mechanism must be present in our sample.
along x, y, or zk[001]. C is a constant that depends
To exclude optical recombination as a source of de-
on T, Fermi energy, scattering time, and the scattering
cayoftheFaradaysignal,wemeasuredthetime-resolved
mechanism.29 If we apply a largeexternal magnetic field
reflection,13 which exponentially decays with a decay
(Ω τ ≫1),wecanwritetheDPspin-dephasingrate
L x,y,z
time of less than 100ps (data not shown). Interpreting
as16
this time as the electron-hole recombination time pro-
1 1 1 sin2θ cos2θ vides evidence that the spin polarization, which is ob-
= ( + + ). (4)
τ (θ) 2 τ τ τ servable over a much longer time scale in the Faraday
DP z x y
signal,mustgetimprintedontotheequilibriumelectrons
For the difference, we get τ(190)− τ(1180) =Cαβ and read in the QW conductance band through recombination of
a value of about 0.4×109 s−1 in Fig. 5(a) for the dif- unpolarizedelectronsandholes.30 Itisthereforejustified
5
Relaxation mechanism DP ∆g Sum Measured of another spin-dephasing mechanism. A possible candi-
θ=90◦ (1/s)×109 0.6 0.7 1.3±0.2 2.1± 0.1 date is a random SO field originating from the Coulomb
θ=180◦ (1/s)×109 1.0 0.7 1.7±0.2 2.5± 0.1 potential of ionized dopants or from surface roughness
of the QW. It has been pointed out that such spatial
fluctuations might limit the spin lifetime for symmet-
aTsAeBstLimEaIt:eCdofnrtormibuthtieonmsetaosuthreedspaninisdoetrcoapyyraotfe11//TT2∗2∗,SaOt3c0oKn-, ric (110) QWs (Ref.34) or in the case where α = β.35
∗ The importance of this effect is probably smaller in our
stants α and β, the B-dependence of 1/T2, and the electron
samples where both α and β are finite but not equal
mobility. Measured values are obtained at a pump power of
50µW. in size. In a small-gap semiconductor, the Elliott-Yafet
(EY)mechanismcontributestothespindephasing.36 By
estimatingtheimportanceoftheEYmechanism12 inour
to interpret the decay time of the Faraday signal fitted sampleusingthemeasuredmobilityandtheknownband
in a window from 80 to 1000ps as the decay time T∗. parameters, we obtain a spin-relaxation rate on the or-
From the dependence of T∗ on B , information2 on der of 5×107s−1. This is negligibly small, but there are
2 ext
the mechanism of spin dephasing can be obtained. The indications37 that the EY spin-dephasing rate might be
DP spin dephasing rate does not depend on B in the larger than estimated with the equation derived for a
ext
motional narrowing regime and for Ω τ ≪ 1.31 In con- bulk semiconductor.
L p
trast to this, a B-dependence that is intrinsic to the DP
∗
The weak variationin 1/T with T shownin Fig. 5(a)
mechanismisobservedinhigh-mobilitysamples.28Inour 2
∗ canbeunderstoodasaconsequenceoflittletemperature
low-mobilitysamples,wefindalinearincreaseinT with
2 dependence of the individual contributions to 1/T∗. As
B , as shownin Fig. 5(b). Such a linear B-dependence 2
ext pointed out in Ref.29, the DP dephasing rate depends
isevidenceofaninhomogeneousdephasingduetoavari-
only weakly on T in the degenerate regime and in the
ation∆g oftheg factorinthe areaofthe 2DEGprobed,
intermediate temperature range, apart from its propor-
described by a dephasing rate 1/τ ,30,32
∆g tionalitytothe electronscatteringtime. As ourmobility
1 ∆gµ B is quite constant over the temperature range measured,
B ext
τ = 2~ . (5) we do not expect large variations. We observe no evi-
∆g dence ofa dependence ofthe g factorspreadonT. For a
We suspect that the g factor variation could be a conse- degenerateelectrondensityandaconstantmobility,also
quence of the in-well doping. That the sample is rather theT dependenceoftheEYmechanismshouldbesmall.
inhomogeneous is also seen in a photoluminescence ex- TheobservedweakT dependenceisthereforenotsurpris-
periment,inwhichweobserveabroadluminescencepeak ing, and has also been observed in other experiments.13
fromthe QW(not shown)witha full width at halfmax-
To conclude, we havemeasuredthe SO interactionco-
imum of about 20meV. We used different pump intensi-
efficients α and β as a function of T and find no signif-
ties and find a similar slope for the dashed linear fits in
icant T dependence. From α and β, the measured Hall
Fig. 5(b). The 500µW measurement was done at 40K, ∗
mobility and the anisotropy in 1/T , we estimate the
and the other two at 10K. From these data, we con- 2
contribution from DP spin dephasing, and find that DP
clude that ∆g is quite constant for different pump pow- ∗
alone cannot explain the measured 1/T . From a linear
ers. From the slopes in Fig. 5(a) and using Eq. (5), we ∗ 2
increase in 1/T with B , we identify an inhomoge-
obtain ∆g = 0.014. Unexpectedly in a doped sample 2 ext
neous broadening from a spread in the electron g fac-
with fast electron-hole recombination, the overall spin-
tor. These effects do not accountfor all of the measured
relaxation rate increases with increasing pump power;
spin-dephasing rate. We speculate that EY or an inho-
see Fig. 5(c). In a high-mobility sample, a decrease in
mogeneousSO field might induce anadditionalisotropic
thespinrelaxation-ratewithincreasinginitialspinpolar-
contribution. A more detailed study of the nature of
ization has been observed, which goes into the opposite
the elastic and inelastic electron-scattering mechanisms
direction.33 In an attempt to minimize this pump-power
involved might facilitate an exact attribution to the dif-
dependence,weusedalowpumppowerof50µWforthe
ferent decay mechanisms.
measurement in Fig. 5(a).
Table I summarizes the contributions to the We gratefully acknowledgehelpful discussions with T.
∗
anisotropic1/T forT =30K.Thecalculatedsumofthe IhnandI.ShorubalkoandthankB.Ku¨ngforevaporating
2
relaxation rate is by about 0.8×109s−1 lower than the contactmetals. ThisworkwassupportedbytheCTIand
measuredvalue. This discrepancy indicates the presence the SNSF.
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