Table Of ContentDecoherence-assisted initialization of a resident hole spin polarization in a p-doped
semiconductor quantum well
M. Kugler,1 K. Korzekwa,2 P. Machnikowski,2,∗ C. Gradl,1 S. Furthmeier,1 M. Griesbeck,1
M. Hirmer,1 D. Schuh,1 W. Wegscheider,3 T. Kuhn,4 C. Schu¨ller,1 and T. Korn1,†
1Institut fu¨r Experimentelle und Angewandte Physik,
Universita¨t Regensburg, D-93040 Regensburg, Germany
2Institute of Physics, Wroc law University of Technology, 50-370 Wroc law, Poland
3Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
2
4Institut fu¨r Festko¨rpertheorie, Westfa¨lische Wilhelms-Universita¨t, D-48149 Mu¨nster, Germany
1
0 (Dated: January 20, 2013)
2
Weinvestigatespindynamicsofresidentholesinap-modulation-dopedGaAs/Al0.3Ga0.7Assingle
n quantumwell. Time-resolvedFaradayandKerrrotation,aswellasresonantspinamplification,are
a utilizedinourstudy. Weobservethatnonresonantorhighpoweropticalpumpingleadstoaresident
J hole spin polarization with opposite sign with respect to the optically oriented carriers, while low
9 power resonant optical pumping only leads to a resident hole spin polarization if a sufficient in-
1 planemagneticfieldisapplied. Thecompetitionbetweentwodifferentprocessesofspinorientation
stronglymodifiestheshapeofresonantspinamplificationtraces. Calculationsofthespindynamics
] intheelectron–holesystemareingoodagreementwiththeexperimentalKerrrotationandresonant
l
l spinamplificationtracesandallowustodeterminetheholespinpolarizationwithinthesampleafter
a
opticalorientation,aswellastoextractquantitativeinformation aboutspindephasingprocessesat
h
various stages of theevolution.
-
s
e
m I. INTRODUCTION Here,wepresenttime-resolvedstudiesofthecombined
. electronandholespindynamicsinap-modulationdoped
t
a quantum well under different excitation conditions. We
m The promising field of semiconductor spintronics1 has utilize time-resolved Kerr/Faradayrotation21, as well as
- stimulated a large number of studies of spin dynamics the related resonant spin amplification technique22, and
d andspindephasingmechanismsinavastvarietyofsemi- time-resolved photoluminescence. We identify two pro-
n
conductors and their heterostructures in recent years. cessesinwhichspinpolarizationistransferredtotheres-
o
c Especially the dynamics of conduction-band electrons in identholesafteropticalorientation,quantitativelymodel
[ compoundsemiconductorswithoutinversioncenter,e.g., the dynamics, and determine the contributions of these
GaAs,havebeenstudiedbymanygroups(see,e.g.,Ref.2 processes depending on excitation conditions. By fitting
2
for a recent review). By contrast, hole spin dynamics ourtheoreticalmodeltothe experimentalresults,weare
v
8 in these systems have been investigated with less inten- able to find the degree of spin polarization after the op-
3 sity. This is, in part, due to the sub-picosecond hole tical excitation and to extract the hole spin dephasing
3 spindephasingtime(SDT)inbulkGaAs3,4,whicharises times, as well as the degree of coherence loss during ex-
1 from the strong spin-orbit coupling within the p-like va- citation at various conditions of optical pumping. We
5. lence bands. In p-doped quantum wells (QWs), an in- determine transverse spin dephasing times T2 of almost
0 crease of the hole SDT to a few picoseconds has been 100 ns under weak, resonant excitation. Additionally,
1 observed experimentally by several groups5,6 and repro- weshowthatfastdephasingoftheholespinstateduring
1 duced in microscopic calculations7. Significantly longer andjustafterhigh-powerorblue-detunedpumpingleads
v: holeSDT havebeenobservedforopticallyorientedholes to polarization of the hole spins at zero magnetic field,
i in n-doped QWs8,9, and more recently in p-doped QW whileatfinitefieldsthisprocesscompeteswiththepolar-
X
systems in which localizationof holes occurs atlow tem- ization mechanism due to trion spin precession10,20. As
ar peratures10–13,andalsoinquantumdots14. Forlocalized these two competing processes lead to opposite spin ori-
electrons in quantum dots, the contact hyperfine inter- entation our findings show that the spin orientation can
action leads to ensemble spin dephasing on the 10 ns be controlled by modifying the optical excitation condi-
scale15. However,due to their p-like wavefunctions, this tions.
dephasing process is suppressed for holes, and only the
weaker dipole-dipole interactionhas to be taken into ac-
count16. Therefore, localized holes may be more suit-
able than electrons for the realization of future quan- The paper is organizedas follows. First, in Sec. II, we
tum computingschemes. Additionally,the largeorienta- presentthesampleandtheideaoftheexperiment. Next,
tionalanisotropyof the hole g factor inGaAs-basedhet- in Sec. III, the theoretical model is introduced. Sec. IV
erostructures17 stronglyinfluences holespin dynamicsin contains the presentation and discussion of the experi-
tilted magnetic fields18 and may allow for spin manipu- mental results and their theoretical modeling. Finally,
lation schemes based on electrical g factor control19,20. Sec. V concludes the paper.
2
II. SAMPLE STRUCTURE AND crease sensitivity. The RSA technique is based on the
EXPERIMENTAL METHODS interference of spin polarizations created in a sample by
subsequent pump pulses. It requires that the spin de-
phasing time is comparable to the time delay between
Our samples are single-side p-modulation-doped
pump pulses. For certain magnetic fields applied in the
GaAs/Al Ga As QWs (QW width 4 nm), contain-
0.3 0.7
sampleplane,theopticallyorientedspinpolarizationpre-
ing a two-dimensional hole system (2DHS) with a hole
density p = 1.1 × 1011 cm−2 and mobility µ = 1.3 × cesses by an integer multiple of 2π in the time window
104 cm2/Vs (measured at 1.3 K) from a single wafer between subsequent pump pulses, so that constructive
interferenceoccurs. Thisleadstopronouncedmaximain
grown by molecular beam epitaxy. Some samples from
the Faraday or Kerr rotation angle measured for a fixed
this wafer are thinned for measurements in transmis-
time delay as a function of the applied magnetic field.
sion. For this, the samples are first glued onto a sap-
In our measurements, the time delay is chosen to probe
phire substrate with optically transparentglue, then the
thespinpolarizationremainingwithinthesample100ps
semiconductor substrate is removed by grinding and se-
before the arrival of a pump pulse.
lective wet etching. The samples contain a short-period
GaAs/AlGaAssuperlattice,whichservesasanetchstop, Time-resolved photoluminescence (TRPL) measure-
leaving only the MBE-grown layers. Earlier studies of ments are performed using a Hamamatsu streak cam-
spin dynamics performed on similar systems10,20 indi- era system synchronized to the pulsed Ti-Sapphire laser
cate that in structures of this kind the resident holes system. For these measurements, the laser is detuned
are weakly trapped, most likely on QW width fluctua- to create electron-hole pairs at an energy about 30 meV
tions. This is confirmed by a rapid increase of hole spin above the photoluminescence (PL) energy of the heavy-
dephasingaboveacertainthresholdtemperature,associ- hole exciton and trion lines. The PL from the sample is
atedwiththermalreleaseofthecarriersfromthesebind- collectedusinganachromatanddispersedinaspectrom-
ing centers and the onset of spin-orbit-relateddephasing eter before being detected by the streak camera.
characteristic of free carriers. For initial characterization of the samples, PL mea-
surements using continuous-wave excitation with a
The resonant spin amplification (RSA) measurements
are performed in an optical cryostat with 3He insert, 532nmlaserareperformed. Agratingspectrometerwith
aPeltier-cooledchargecoupleddevice(CCD) detectoris
allowing us to lower the sample temperatures below
usedtocollectthe PL.Figure1showsatypicalPLtrace
400mK andto applymagnetic fieldsofupto 11.5Tesla.
Here, the samples are cooled by cold 3He gas. For some measuredatasampletemperatureof1.2K.ThePLfrom
the QW is a near-symmetrical single peak with a spec-
ofthesemeasurements,thinnedsamplesareusedandthe
tralwidthofabout5meV.Nofinestructureofthispeak,
experiment is performed in transmission (Faraday rota-
correspondingto,e.g.,neutralorpositivelychargedexci-
tion) to limit the amount of absorbed laser power. The
tons,isobservedinPLmeasurements. Inthesamefigure,
time-resolved Kerr rotation (TRKR) measurements are
a typical spectrum of the Ti-Sapphire laser system can
performed in a Helium flow cryostat, in which the sam-
be seen at higher energy. In the following, by “resonant
ples are mounted on the cold finger of the cryostat in
excitation” we will understand the situation where the
vacuum. A pulsed Ti-Sapphire laser system generating
Ti-sapphire laser system is tuned to achieve maximum
pulses with a length of 600 fs and a spectral width of
Kerr signal. Since our sample is doped, the states corre-
3-4 meV is used for the optical measurements. The rep-
sponding to the maximum of the PL emission are occu-
etition rate of the laser system is 80 MHz, correspond-
pied by resident holes, therefore, the absorption at this
ingtoatime delayof12.5nsbetweensubsequentpulses.
energy is suppressed. Maximum Kerr signal is observed
Thelaserpulsesaresplitintoacircularly-polarizedpump
for an energetic position of the laser at the high-energy
beam and a linearly-polarized probe beam by a beam
flank of the QW PL emission.
splitter. A mechanicaldelayline is usedto createa vari-
able time delay between pump and probe. Both beams
are focused to a diameter of about 80 µm on the sample
using an achromat. III. THEORETICAL MODEL
In the TRKR and RSA experiments, the circularly-
polarized pump beam is generating electron-hole pairs In order to interpret the experimental results we pro-
in the QW, with spins aligned parallel or antiparallel to pose a minimal, generic model that is able to account
the beam direction, i.e., the QW normal, depending on for all the features of the spin dynamics observed in
the helicity of the light. In the TRKR measurements, the experiment without specific assumptions on the de-
thespinpolarizationcreatedperpendiculartothesample tailed mechanism of spin decoherence. In accordance
plane by the pump beam, is probed by the time-delayed with the previous experimental findings10,20, we assume
probe beam via the Kerr effect: the axis of linear polar- that the optical response can be described in terms of
ization of the probe beam is rotated by a small angle, independent hole-trion systems, trapped in QW fluctu-
which is proportional to the out-of-plane component of ations. The state of each such system is represented by
the spin polarization18,23. This small angle is detected thedensitymatrixρ,restrictedtothefourrelevantstates
using an optical bridge. A lock-in scheme is used to in- |↑i,|↓i,|T↑i,|T↓i, representing the two hole states and
3
where α=↑,↓ and the upper (lower)signis for α=↑(↓).
The factors e−u and e−w describe the effects of occupa-
tion relaxationand additionalpure dephasing in the ref-
erence frame associated with the system symmetry axis
Laser spectrum
s) (coincidingwith the axisofopticalorientation),whichis
unt QW PL the relevant one here in view of the fast character of the
o
y (c process, as compared to the Larmor precession.
sit Itshouldbenotedthatthecombinationofcoherentex-
n
nte citation and instantaneous dephasing does not necessar-
L I ily reflect the actual microscopic kinetics of the system.
P
In particular, for off-resonant excitation, hole spin flips
during relaxation to low-energy states are also possible.
Whilethisprocessisclearlybeyondourfour-levelmodel,
1645 1650 1655 1660 1665 1670 its essential effect is bringing the hole spin polarization
energy (meV)
towards equilibrium and dephasing of the hole spin co-
herence. Both these effects are included in our model
FIG. 1. (Color online) Photoluminescence (PL) trace of the
in terms of the instantaneous relaxation and dephasing
sample measured at 1.2 K and spectrum of the Ti-Sapphire
factors e−u and e−w.
lasersystemusedinthetime-resolvedexperiments. Thebro-
In the third stage, the system evolution (Larmor pre-
ken lines show Gaussian fitsto thequantumwell PL (dotted
cession, recombination and spin decoherence) is mod-
line) and the laser spectrum (dashed line).
eled in terms of the Markovian Master equation (in the
Schr¨odinger picture with respect to the spin dynamics
but in the rotating frame with respect to the interband
thetwotrionstateswithdifferentspinorientations(with
transition energy)
respecttothesystemsymmetryaxis,normaltothesam-
ple plane). We neglect the influence of the weak probe i
pulse on the spin dynamics and calculate the spin polar- ρ˙ =−~[H0,ρ]+Lh[ρ]+Lt[ρ]+Lr[ρ], (2)
ization at the arrival of the probe, which is known18,23
to be translated by the probe into the Kerr or Faraday with the initial condition ρ(0)=ρ2. Here
signal.
1 1
The experiment is modeled by a sequence of three H0 =− µBBgˆhσh− gtµBB·σt,
2 2
steps:
First, the pump pulse transforms the initial state ρ0 is the hole and trion spin Hamiltonian, where µB is the
into a new state ρ1, described up to the second order in Bohr magneton, gˆh is the hole Land´e tensor, gt is the
the pulse amplitude by Land´e factor of the trion (i.e., essentially, of the elec-
tron),whichweassumetobeisotropic,andσ ,σ arethe
h t
i ∞ vectors of Pauli matrices corresponding to the hole and
ρ =− dt[H(t),ρ ]
1 ~Z l 0 trion spin, respectively (the hole is treated as a pseudo-
−∞
spin-1/2 system), in the basis of hole spin states |↑i,|↓i
1 ∞ ∞
− dt dt′[H(t),[H(t′),ρ ]], (1) and trion spin states |T↑i,|T↓i. This Hamiltonian ac-
~2 Z Z l l 0
−∞ −∞ counts for the spin pressionwith the Larmor frequencies
ω = µ |gˆ B|/~ and ω = µ g B/~ for the hole and
h B h t B t
where we use the fact that the pulse is very short com-
trion, respectively.
pared to the spin evolution time scales and assume,
The hole dissipator L is obtained within the stan-
h
for simplicity, that the excitation is coherent. Here,
dard weak-coupling approach24 from the hole spin-
Hl =(1/2)f(t)|↑ihT↑|+h.c. is the carrier-lasercoupling environment Hamiltonian H = σ · R(h), where R(h)
Hamiltonian with a pulse envelope f(t) and σ+ circular he
are certain environment operators. We derive the evo-
polarization is assumed for the laser pulse.
lution equation for the hole spin in the Markov limit as
Second, we allow for a fast partial decoherence of the
in Ref. 18 but without the secular approximation which
hole spin which takes place on time scales much shorter
does not hold in the general case of possibly low or van-
thanthesubsequentspindynamicsandisthereforemod-
ishing magnetic fields. As a result, we get the dissipator
eled as instantaneous. This leads to a system state ρ
2 (in the Schr¨odinger picture) in the form
with
L [ρ]=−π R(h)(ω )(σσ ρ−σ ρσ )
h↓|ρ |↑i=h↓|ρ |↑ie−u/2−w h Xh lj j l j j l
2 1 lj
+R(h)(−ω )(ρσ σ −σ ρσ) ,
and lj l l j j l
i
hα|ρ2|αi= 1h↑|ρ1|↑i 1±e−u + 1h↓|ρ1|↓i 1∓e−u , wPahuerlieml,ajtr=ice±s,i0n,tωh0e=ref0e,reωn+ce=fr−amω−ea=ssωohc,iaatneddwσ±it,h0 tahree
2 2
(cid:0) (cid:1) (cid:0) (cid:1)
4
x direction (the orientation of the field), whereµ ,µ′ ,andµ arethetriondecoherenceratesde-
α α α0
fined as in Eqs.(4a) and(4b), but with the trion-related
−σ +iσ
σ =σ , σ =σ† = z y. spectral densities R(t)(ω) taken at the trion Larmor fre-
0 x + − 2 αα
quency ω , and N is the trion occupation.
t t
The spectral densities for the hole reservoirare
Theopticalresponse,thatis,therotationofthepolar-
izationplane of the reflectedortransmitted probe pulse,
1
Rl(jh)(ω)= 2π~2 Z dteiωthRl(h)(t)Rj(h)i, l,j =±,0, is proportional to18,23
where R(h) = R(h), R(h) = R(h)† = −R(h) −iR(h), and ∆Σ=Σ −Σ .
0 x + − z y t h
R(h)(t) denotes the operator in the interaction picture
l
with respect to the environment Hamiltonian. Consis- As the time-resolved Kerr response is investigated for
tently with the assumed C4v symmetry of the system, experimentalconditionsofrelativelyhighspindephasing
we set R(h)(ω) = 0 for α,β = x,y,z, α 6= β and and relaxation rates one can assume that the evolution
αβ
R (ω) = R (ω). The trion spin dissipator L is ob- aftereachlaserrepetitionisindependentandstartsfrom
yy xx t
tainedinthesamewaywithasetoftrion-relatedspectral thethermalequilibriumstate. Afterthepumppulse,the
densitiesR(t)(ω). Weassumethatthereservoirscoupled trion and hole spin polarizations are
αβ
to electron (trion) and hole spins are uncorrelated.
The last term in Eq. (2) is the standard spontaneous Σt =−Σh =Σ(0).
emissiongeneratorthataccountsfortheradiativerecom-
bination of the trion (see Ref. 18) with the rate γR. As a consequence of the initial dephasing, the hole spin
Eq. (2) can be rewritten in terms of the three compo- polarization is reduced to
nents of the hole spin polarization
X =h↑|ρ|↓i+h↓|ρ|↑i, Y =i(h↑|ρ|↓i−h↓|ρ|↑i), Σ(0) =−Σ(0)e−u (6)
h h h
Σ =h↑|ρ|↑i−h↓|ρ|↓i
h
(we assume no fast dephasing of the trion spin polariza-
(andanalogousforthetrion). Fortheholespinpolariza- tion). Then, by solving Eq. (2) one gets the Kerr signal
tion, the equations of motion are at B =0 in the form
X˙ =−(κ +κ )X +(κ′ +κ′)N , (3a)
h z x h x z h
Y˙ =ω Σ −(κ +κ )Y , (3b) ∆Σ(Kerr) =ae−γtt−be−γht, (7)
h h h x0 z h
Σ˙ =−ω Y −(κ +κ )Σ +γ Σ , (3c)
h h h x x0 h R t wherea=(1+η)Σ(0), b=Σ(0)+ηΣ(0),η =γ /[γ −γ ].
t h t R t h
where Nh is the hole population and Here γh = κx + κx0 is the hole spin decoherence rate
and γ = µ + µ + γ is the trion spin decoherence
t x x0 R
κα =2π Rα(hα)(ωh)+Rα(hα)(−ωh) , κα0 =4πRα(hα)(0), rate. Since we do not propose any specific microscopic
h i
(4a) mechanismforthespindecoherencethe ratesγt, γh,and
γ are treated as independent parameters of the model.
κ′ =2π R(h)(ω )−R(h)(−ω ) , (4b) R
α αα h αα h FortheRSAsignal,thespinpolarizationsurvivingbe-
h i
tween subsequent laser repetitions is essential. In or-
for α = x,z. In order to find an interpretation of the
der to find the resonantly amplified spin polarization
dephasingratesappearinginEqs.(3a)–(3c)wenotethat
just before the pump pulse, we find the mapping of the
at B = 0 one has ω = 0, hence κ = κ and the de-
h α α0 hole spin-related variables X ,Y ,Σ corresponding to
coherence time for the spin polarizationalong the struc- h h h
the three-step state transformation described above, as-
ture axis is T(0) =1/(2κ ), while the decoherence time
z x0 suming that trion occupations and interband coherences
for the in-plane components of the spin polarization is
decay completely in the repetition interval. Moreover,
Tx(0y) = 1/(κz +κx0). On the other hand, in sufficiently the RSA measurements are performed under conditions
strong fields (for ωh ≫ κα,κα0), the longitudinal (with of long spin dephasing time, hence we assume that the
respect to the field orientation) spin relaxation time is holespindephasingratesaresmallcomparedtothetrion
T1 =1/(κz+κx) and the transverse relaxation (dephas- recombinationrate. The RSAsignalisthenfoundasthe
ing) time is T2 =2/(κz+κx+2κx0). fixed point of this three-step transformation to the lead-
Theequationsofmotionforthetrionspinpolarization ing (second) order in the pulse area, that is, in the weak
are excitation limit. The resulting spin polarization just be-
X˙ =−(µ +µ )X +(µ′ +µ′)N −γ X , (5a) fore the arrival of the pump pulse is proportional to
t z x t x z t R t
Y˙ =ω Σ −(µ +µ )Y −γ Y , (5b)
t t t x0 z t R t P
Σ˙t =−ωtYt−(µx+µx0)Σt−γRΣt, (5c) ∆Σ(RSA) ∼fQ, (8a)
5
A. Kerr response at B=0
(a) (b)
∆E =7.2 meV atio-0.2 First, we investigate the TRKR measurements at zero
nal (arb. units) ∆E = 4.5 meV h/t polarization r---000...864(c) hole s pin dephasing mmasasaemIangnptnfstluee:htneiatccsetffimriaoeepnqlfdueuo.nrefacnHtttcuiehyorreneed,poeowuftfuemt1hnpp5eienrppKgfuuo,slmrsemterphieeepfdsroe,ewtxqwpecueoriret.fsnaoetcrriymoiendseedowtfuaamnvtieeanlaegsfinuaxgrneteddh-
sig 1 trion decoherence was tuned from near-resonant excitation to higher laser
Kerr ττ, (ns)th etinoenrsg.y,Fwighuirceh2r(eas)usltheodwisnthnroene-rTeRsoKnaRnttreaxcceistafotriornescoonnadnit-
resonant 0.1 excitationandtwodifferentvaluesoflaserenergydetun-
ing (symbols). While the trace for resonant excitation
0 500 1000 0 3 6
showsanear-monoexponentialdecay,thetwotracesmea-
t (ps) Laser detuning ∆E (meV)
sured using larger laser energies display a more complex
behavior,withaveryrapidinitialdecayofthesignaland
FIG. 2. (Color online) (a) TRKR traces measured at 15 K a zero crossing, followed by a slower decay of the nega-
with different laser excitation energies (symbols). The solid tive signal. Additionally, we note that the Kerr signal
lines represent fits to the data according to the theory (ex- amplitude decreases as the laser energy is detuned from
tended to t < 0 for better visibility). (b) Ratio of hole
the resonance, limiting the detuning range accessible in
and trion spin polarizations after rapid initial hole dephas-
the measurementstoabout7meV.This isdue bothto a
ing, Σ(0)/Σ(0) = −e−u, extracted from the measurements as
h t reducedabsorptionofthepumppulseandtothespectral
a function of laser detuning. (c) Hole spin dephasing time
dependence of the Kerr rotationof the degenerate probe
(solid stars) and trion spin decoherence time (open symbols)
pulse.
as a function of laser detuning.
We interpret the traces as follows: under resonant ex-
citation conditions in the absence of a magnetic field,
both the optically oriented electrons and the optically
where
oriented holes retain their spin orientation during the
photocarrier lifetime. Electrons recombine with holes
ω2
f =1−e−u− t , (8b) that have a matching spin orientation according to the
γ2 +ω2
R t selection rules, thereby removing the optically created
P =(iω˜+κ′)eiω˜tr/2−iω˜e−u/2−w−κtr/2−(ω˜ →−ω˜), spin polarizationfrom the sample during recombination.
(8c) Therefore, no hole spin polarizationis transferredto the
resident holes. By contrast, under non-resonant excita-
Q=e−uP + (iω˜−κ′)e−u/2−w+iω˜tr/2−iω˜eκtr/2
tionconditions,afractionofthe holespinpolarizationis
(cid:2)
−(ω˜ →−ω˜) . (8d) rapidly lost. This can result from hole spin flips during
(cid:3) energy relaxation of the optically created holes or from
Here, t is the laser repetition period, κ = κ + κ + the thermalization of the spin orientationof the resident
r x z
holes resulting from binding part of the oppositely ori-
2κ , κ′ = κ −κ , ω˜ = 2 ω2−κ′2/4, and (ω˜ → −ω˜)
x0 z x q h ented holes with the optically created excitons into tri-
representsadditionalterms,obtainedfromthepreceding ons. On the other hand, the electrons seem to retain
onesby changingthe signofω˜. Itis found thatthe RSA their spin orientation. Upon recombination, these spin-
response in the weak excitation limit does not directly polarized electrons remove holes with matching spin po-
depend on detuning. In order to simulate the response larization from the partly depolarized hole system, leav-
fromaninhomogeneousensembleofholespins,theresult ing an excess of holes oriented opposite to the optically
obtainedfromEqs.(8a)–(8d)wasaveragedaccordingtoa created hole spin orientation (we will refer to this op-
Gaussiandistributionofholeg-factorswiththestandard posite orientation as negative). We note that both the
deviation ∆g. spin-polarized electrons and holes created by interband
absorption of circularly polarized light will lead to the
same Kerr rotation of a test beam, so that a priori, the
observed Kerr rotation does not allow us to identify the
IV. RESULTS AND DISCUSSION
type of spin-polarizedcarriersdirectly. The originof the
Kerr signal can be determined by applying a magnetic
In this section we present the results of TRKR and field perpendicular to the spin polarization and observ-
RSAmeasurementsandinterpretthem,basedonthethe- ing the spin precession, using the different g-factors of
oreticalmodelpresentedintheprevioussection. Wefirst electrons and holes. Naturally, in the case of a doped
discusstheKerrmeasurementsatzeromagneticfieldand samplelikeour2DHS,investigatingtheKerrrotationaf-
then the RSA results. ter photocarrierrecombination,sothat onlythe resident
6
carriers remain, also gives unambiguous results.
Theexperimentaltracesarewell-reproducedbyEq.(7) (a) o 0.0(b)
(thefitsareshownassolidlinesinFig.2(a))inthewhole ati
n r-0.2
time range except for the first few picoseconds after ex- o
coictcautrios,nw,ihnicwhhisicnhotthmeoradpelieddininitaiatlimdeep-rheassoilnvgedofmtahnenheorleins malized) P=PMax polarizati-0.4
tahbelethtoeoerxyt.rFacrotmthtehrealteiaosot-fstqhuearheoslefitapnadraelmecettreorns,(wtreioanre) al (nor P=0.13 PMax h/t -0.6(c)
n
stohpceicnuirnpsio.tliaaFrliigdz.ae2tpi(hoban)ssi,snhΣgo,(hw0b)s/etfΣoh(tre0e)cp=ahlco−utloea−ctaeurdr(isreeeresurEeltcqso.ma(s6b)ia)nafautfnitoecnr- Kerr sig P=6x10-3 PMax ττ, (ns)th00..115 htroiolen sdpeicno dheeprehnacseing
tion of the laser energy detuning from resonance. We
0.05
see that, for resonantexcitation,the ratio is close to −1,
indicating a hole spin polarization almost equal to and 0 500 1000 0.01 0.1 1
t (ps) Excitation density P/P
oriented opposite to the electron spin polarization. As Max
the laser energy is increased, this ratio is reduced sig-
nificantly but does not reach zero, indicating that some FIG. 3. (Color online) (a) TRKR traces measured at 10 K
part of the optically oriented holes retain their spin ori- with fixed, near-resonant laser excitation energy and various
entation during energy relaxation. For all values of the pump powers (symbols). The solid lines represent fits to the
detuning, we also extract the trion (electron) spin de- data according to the theory (extended to t < 0 for better
visibility). (b)Ratioofholeandtrionspinpolarizationsafter
coherence time τ = 1/γ , and the hole spin dephasing
t t
rapidinitialholedephasingextractedfromthemeasurements
time, τ =1/γ ,fromthefitstoexperimentaldatausing
h h as a function of excitation density. (c) Hole spin dephasing
Eq.(7). Theirvalues,depictedinFig.2(c),remainnearly
time(solidstars)andtrionspindecoherencetime(opensym-
constantthroughoutthe investigateddetuningrange,in-
bols) as a function of excitation density.
dicating that the photocarrier and hole spin dynamics
are not strongly influenced by the initial energy relax-
ation of non-resonantly excited carriers. It is also clear
long-time hole and trion spin dephasing times decrease
thattheelectron(trion)spinlifetimeremainsclosetothe
as the pump power is increased. This decrease is rather
photocarrierrecombinationtime. Using TRPL, we mea-
weak(byaboutafactorof2overmorethantwoordersof
sure a photocarrier recombination time of about 175 ps
magnitude of the pulse power) and may be due to sam-
at a temperature of 15 K, using nonresonant excitation
ple heating by the pump beam. A rapid decrease of the
with larger detuning than during the TRKR measure-
hole spin dephasing time with temperature has been ob-
ments. As the recombination time increases with detun-
served previously by several groups6,11,26. The decreas-
ing25, this value only provides an upper bound for the
ing trion spin lifetime observed in the power-dependent
photocarrier lifetime under the excitation conditions in
experiments is not limited by faster photocarrier recom-
the TRKR measurements. We may therefore conclude
bination, as we observe in TRPL under nonresonant ex-
that the electron spin coherence time is mostly limited
citation conditions that the photocarrier recombination
by the carrier lifetime for weak, nonresonant excitation.
timeinoursampleincreases asthetemperatureisraised,
Next, we discuss power-dependent TRKR measure-
from 150 ps at 4 K to 400 ps at 40 K. Such an increase
ments. For this series, the pump power was increased,
is typically observed in the low temperature-regime for
relativetothevaluesusedinthepreviousseries,bymore intrinsic, as wellas p- or n-doped QWstructures20,27–29.
than two orders of magnitude. Figure 3(a) shows three
Therefore, the reduction of the trion spin lifetime must
TRKR traces for different excitation powers. The laser
be caused by spin-related decoherence processes. Most
excitation energy was chosen to be near-resonant and likely,this is due to anincreasedeffective k vectorwhich
keptfixedthroughouttheseries,thesampletemperature
leadstolargerspin-orbitfieldsandmorerapiddephasing
was 10 K. While for weak pumping, the TRKR traces via the Dyakonov-Perelmechanism30.
show almost no negative part, it becomes quite pro-
nouncedfor higher pump powersand the subsequent de-
cayofthe negativesignalbecomesmorerapid. Asinthe
B. Resonant spin amplification
previous measurement series, our theory closely fits the
experimentaltraces,except for the first few picoseconds,
inwhichtheinitialholespindephasingtakesplace. From We now turn to the resonant spin amplification mea-
the extractedratio ofhole andelectronspin polarization surements. All series of measurements were performed
we see that for weak, near-resonant pumping, the hole at a nominal sample temperature of 1.2 K in Voigt ge-
spin polarization is significantly larger than for stronger ometry. In this temperature range, we previously ob-
pumping(Figure3(b)), mostlikelyindicatingtheimpor- served hole spin dephasing times above 70 ns11, which
tanceofspinnon-conservingcarrier-carrierscatteringfor exceedthelaserrepetitionperiodandleadtowell-defined
rapid hole spin decoherence. We also observe that the RSAsignals. Figure4showsthetwoprincipallydifferent
7
vationscanbe understoodinthe followingway: Forzero
magnetic field, a fast partial re-equilibration of the hole
∆E= 5.4meV spin polarization leads to negative final polarization of
the residentholesdue to removalofthe optically aligned
s)
nit holes by spin-conserving electrons upon recombination,
u
nal (arb. ∆E= 1.8meV aiinzsattdihoisencuacsbrsseeeadntecidenoiSnfepct.hreiIscVewsAsai.yonAbsyattshuBebs=heoqlu0e,enstpthienpsusmpairpneppsuotallsateircs-
g
si constructively interferes, leading to the observed nega-
R
F resonant tivezero-fieldpeak. Atnon-zerofields,this processcom-
R
T peteswiththespinalignmentduetothetrionprecession
discussedabove,whicheffectivelyleadstorandomization
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 of the trion spin orientation at the moment of recombi-
Magnetic field (T) nationsothatrecombinationwithanoppositelyoriented
holebecomes possible. As a resultofthis process,anex-
cess population of optically oriented holes is left behind
FIG.4. (Coloronline)ThemeasuredRSAtracesforresonant
after recombination, which results in a positive spin ori-
andoff-resonantexcitationconditionsatlowexcitationpower,
entation. Obviously, this mechanism is only effective at
for three different values of pump pulse detuning.
non-zeromagneticfieldsanditsroleincreasesasthefield
grows. This increasing compensation between these two
processes is manifested by reduced negative RSA peak
shapes of the RSA traces we observe in experiment: For
amplitudesatgrowingfields,visibleintheuppertracein
resonantexcitationconditions(lowercurve),inwhichthe
Fig.4. Atacertainmagneticfield,theprecession-related
opticallyorientedholespinpolarizationisconserveddur-
hole spin initialization process becomes dominant, lead-
ingthephotocarrierlifetime,theRSAtraceshaveachar-
ing to a positive spin orientation of the resident holes.
acteristic, batwing-like shape, and the maximum at zero
Hence,weobservethepolaritychangeintheRSApeaks.
magnetic field is absent. This peculiar shape arises from
Forevenlargermagneticfields,reducedRSApeakheight
the process in which the spin polarization of optically
isagainobservedduetotheinhomogeneousensemblede-
orientedholesisturnedinto the residentholespinpolar-
phasing.
ization after recombination10,11: At zero magnetic field,
the optically oriented hole spin polarization is removed Thecompetitionofthe twoorientationprocesseslead-
byphotocarrierrecombination,asdescribedabove. With ing to opposite spin polarizations is reflected in our the-
anappliedin-plane magneticfield, however,the strongly oryby the factorf in Eq.(8a) which(apartfromthe in-
different g factors of electrons and holes lead to incom- homogeneous dephasing) determines the envelope of the
mensurateprecessionoftheopticallyorientedspinpolar- RSA response. The hole spin relaxation which tends to
izations, allowing the electrons to recombine with unpo- re-equilibratetheholespinpolarizationafteropticalcre-
larized,residentholes,sothatsomeholespinpolarization ation of extra holes with positive spin orientation (or,
remains in the sample after photocarrier recombination. equivalently, depletion of the negatively oriented holes
Thisholespinpolarizationisorientedinthesamewayas by binding them with the optically createdexcitons into
the optically created holes (which we will refer to as the trions) is described by the first two terms in Eq. (8b).
positive orientation). If the hole precession frequency is Here, according to Eq. (6), e−u is the degree of the fast
an integer multiple of the laser repetition rate this spin spin polarization decay during or just after the excita-
polarizationisresonantlymagnifiedtoyieldpronounced, tion. The trion precession, which leads to orientation in
equally spaced peaks in the RSA signal. Their magni- thepositivedirection,isaccountedforbythethirdterm.
tude grows as the magnetic field increases due to the It is clear from the form of Eq. (8b) that a sign change
increased efficiency of the aforementioned initialization of f, corresponding to a change of the “polarity” of the
process. The decay of the RSA peak height with further RSA response, is possible only for u > 0, that is, in the
increase of the magnetic field stems from hole ensemble presenceofinitialspinrelaxation. Moreover,theposition
dephasing due to the g-factor inhomogeneity25. ofthistransitionisshiftedtohigherfieldsasuincreases,
leading to a growing number of inverted RSA peaks in
For off-resonant excitation, when a substantial part
the low field region.
of the optically oriented hole spin polarization de-
phases before photocarrier recombination (as discussed The growing number of inverted peaks is clearly seen
inSec.IVAabove),theRSAsignalshapebecomesmore in Fig. 4: For increasing laser detuning, first a single
complex, as the middle and upper traces in Fig, 4 show. negative RSA peak at zero field develops, then addi-
Here, we observe RSA peaks also at zero magnetic field, tional peaks of the same orientation are seen, so that
and the amplitude of these peaks initially decreases at the magnetic field for which a crossoverbetween the ini-
low fields. For a certain field, we observe a change of tialization mechanisms occurs is increased. This allows
polarity of the RSA peaks, then the RSA peak ampli- us to conclude that the increasing detuning of the pump
tude first increases, then decreases again. These obser- pulse towards higher energies leads to increased initial
8
constant in the correspondingrange of frequencies. This
is true in particular for an Ohmic reservoir in the high
(a) ∆E= 6.3 meV T (ns)215000(b) tinemthpeerwathuorleerreagnigmeeo,fwmhiacghniesttihceficealdseshsteurediaesd~.ωEhx≪amkpBleTs
R signal (arb. units) ∆E=r e3s.6o nmaenVt w-3g (10)00055...46024((dc))w h/t --- 100...099050 h/t pol. ratio ofisotrefneanllldcyteeearspdseta(pirrnsrteqosdFudoaiupgfrco.eseiso5nsm(fitaastle))sl.otbohfTubettthahfeaeienlamsemtoudoerydiaeinesselulitodnrhfsegitdsvhraweRelsauSeuyexAlstpfoet(trhbrriaaltmuchteeeeasnlpritaenoaresleqi)RutpinaSvroneeAt---
TRF -3∆g (10)1222....8024(e) taicigtaraletceivhmeaelrynatcctaleolrlsoieswttiscousstohtfeothemxeetharaosucletretsmhpeiennvtasylurseetsesumoltf.sv.IanrTiFohuigiss.pg5h(oybos)d-,
0.0 0.5 1.0 1.5 1.6 0 2 4 6 8 we showthe value ofthe transversespindephasing time,
Magnetic field (T) Detuning (meV)
T = 2/κ. This intrinsic dephasing time increases for
2
decreasing detuning and saturates for low detunings at
about100ns. Theintensitiesofthefastdecoherenceand
FIG. 5. (Color online) (a) Experimental RSA traces (red
the ratio of hole and trion spin polarization are plotted
points) and best fits according to Eqs. (8a)–(8d) (blue lines)
inFig.5(c). Theratioofholeandtrionspinpolarization
for selected values of thedetuning. (b)–(e) Parameter values
reaches -1 as the resonance is approached. This clearly
extracted from the fitting: The dephasing time (b), the fast
demonstratesthattherapidinitialdephasinginducedby
decoherenceparameterandtheratioofholeandelectronspin
polarization (c), the hole g-factor (d), and the standard de- theoff-resonantexcitationatlowtemperaturesleadsonly
viation of the g-factor distribution in the ensemble (e). (the toasmalllossoftheopticalorientationwhichdisappears
lines are guideto the eye) completelyatresonance. Onthecontrary,the dephasing
factorwremainsfiniteevenattheresonance. Thisisdue
to the fact that spin dephasing is induced by an optical
excitation due to selective coupling of the light field to
(a) 13.3 P 100 (b) oneofthespinstates(accordingtotheselectionrules)31.
0 s) From our fitting we conclude that the g-factor tends to
n
units) T (250 isnacmreeahseolsdlsigthrtuleyfworitthhgersotwanindgarddetduenviinagtio(Fniogf. t5h(de)e)n.sTemhe-
arb. 4 P0 0 bleg-factordistribution(Fig.5(e)). Bothoftheseeffects
signal ( atio-0.8 (c) mpualysebaesetxhpeldaienteudnibnyg iasrinedcruecaesdeda,blseoardpitnigontoofatrheedupcutmiopn
RFR P0 pol. r-0.9 of the sample temperature, and a smaller spin-polarized
T h/t holeensemble. Ashiftoftheholeg factortolargerabso-
-1.0 lute values with temperature reductionhas already been
0.0 0.5 1.0 1.5 0 10 20 30 observedbySypereket. al.10 Asmallerensembleofspin-
Magnetic field (T) Excitation density (P/P0) polarized holes is more susceptible to g factor inhomo-
geneity which arises from local fluctuations of, e.g., the
QW width or the disorder potential.
FIG. 6. (Color online) (a) Experimental RSA traces (red The theoretical curves turn out to be sensitive to all
points) and best fits according to Eqs. (8a)–(8d) (blue lines) the model parameters except for κ′. The latter affects
forselectedvaluesoftheexcitationdensity. Parametervalues
the shape of the curves only at very low magnetic fields
extracted from thefittingas a function of excitation density:
since,accordingtoEqs.(8c)and(8d),itcanbeneglected
(b) Dephasing time T . (c) Ratio of hole and trion spin po-
2 when κ′ ≪ω which holds already in the vicinity of the
larization. h
first peak. The values of κ′ obtained for non-zero detun-
ing range from 0.012 to 0.03 ns−1, which corresponds to
spin relaxation. This is in fact expected, as off-resonant about50%ofthevalueofκ=2/T2(exceptforthelargest
excitationsupplies extra energyto the system leading to detuning, where κ′ is lower). This suggeststhat the spin
additionalrelaxationprocessesthatusuallytakeplaceon decoherence is dominated by the in-plane dephasing,de-
picosecond time scales and can lead to spin flips. scribed by the rate κz, as opposed to the relaxation of
theprojectiononthestructurenormal(describedbyκ ).
Our theory allows us to closely model the RSA signal x
This is expected for heavy holes,as the relaxationof the
shape using Eq. (8a) integrated over the inhomogeneous
axial component would involve a spin transfer of 3~ and
distribution ofg-factors. Inthe theoreticalmodeling un-
therefore should be suppressed.
derlyingthe fitting,weassumethatthe intrinsicdephas-
ing rates are constant in the relevant range of the mag- As can be seen in Fig. 6(a), similar effects in the RSA
netic field. This amounts to assuming that the spectral traces are observed for increasing pump power at res-
densities of the reservoir coupled to the hole spins are onant excitation conditions: first, a zero-field peak ap-
9
pears in the RSA traces, then additional peaks are ob- verted spin polarization at low magnetic fields. Nega-
served for higher pump powers. Again, our theory al- tivespinorientationwasearlierobservedinluminescence
lows us to precisely model the shape of the experimental from n-doped quantum dot systems under off-resonant
RSAtracesandtoextractthefastdephasingparameters excitation32–34. OurKerrrotationandRSAresultsshow
andtheholespindephasingtime. Here,weattributethe thatitispossiblenotonlytoopticallypolarizeholespins
growingfastdephasing,indicatedbythereductionofthe in a p-doped system but also to control the sign of this
ratioofholeandtrionspinpolarization(Fig.6(c)),tothe polarizationbychangingeither theexcitationconditions
considerably increased amount of energy pumped into or the magnetic field.
the system. This leads to anincreaseddensity ofvarious Wedevelopedatheoreticalmodelwhichquantitatively
excitations and, in consequence, to stronger spin non- describesthetime-resolvedKerrandRSAsignalsandal-
conserving scattering, in the time window before pho- lowsustoattributethespinpolarizationatzerofieldtoa
tocarrier recombination takes place, as observed in the decoherence-assistedprocessinwhichtheholespinpolar-
TRKR measurements. An increase of the pump power ization partly relaxes towards equilibrium within a very
also influences the spin dephasing time, T , of the resi- short time after a high-power or off-resonant excitation.
2
dentholespins,i.e.,thespindynamicsafterphotocarrier Theverygoodagreementobtainedbetweenthemeasure-
recombination,mostlikelydue tosampleheating,ascan mentdataandthe modelresultsallowsus to extractthe
be seen in the drastic reduction of T (Fig. 6(b)). This parameters relevant to the hole spin dynamics, includ-
2
interpretationissupportedbythefactthattheholegfac- ing the ratio of hole and electron spin polarizations af-
tor slightly decreases with increasing pump power (not ter optical orientation, the intrinsic (homogeneous) spin
shown), as expected for an increasing sample tempera- coherence time T , and the g-factor distribution in the
2
ture. ensemble.
Remarkably, rapid initial hole spin dephasing on the
few-picosecond timescale and long hole spin dephasing
times reaching a hundred nanoseconds coexist under off-
V. CONCLUSION
resonant excitation conditions in low magnetic fields at
low temperatures. Thus, our findings open the way to
We have performed a time-resolved study of hole spin optical spin orientation under conditions that assure a
dynamics in a p-modulation doped quantum well under long lifetime of the oriented hole spins.
different excitation conditions. In time-resolvedKerr ro- This work was supported in parts by the DFG (Ger-
tation measurementsat zero magnetic field, we observed many) under SPP 1285 and SFB 689 (M. Kugler,
the appearance of a hole spin polarization oriented op- M. Griesbeck, T. Korn, C. Schu¨ller), by the Founda-
posite to the optically oriented holes for non-resonant tionfor PolishScience underthe TEAMprogramme,co-
or high-intensity excitation conditions. In resonant spin financed by the European Regional Development Fund,
amplificationmeasurementsunder non-resonantorhigh- (K. Korzekwaand P. Machnikowski),and by a Research
intensityexcitationconditions,weobserveacompetition Group Linkage Project of the Alexander von Humboldt
between two different initialization processes for a res- Foundation (P. Machnikowski and T. Kuhn). The au-
ident hole spin polarization, which leads to a complex thors would like to thank M. Glazov for highly fruitful
shape of the RSA traces and to the appearance of in- discussion.
∗ [email protected] G. Bacquet, and V. Thierry-Mieg, Solid State Communi-
† [email protected] cations 93, 57 (1995).
1 J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. 9 X.Marie,T.Amand,P.LeJeune,M.Paillard,P.Renucci,
Zutic, Acta Physica Slovaca 57, 565 (2007). L. E. Golub, V. D. Dymnikov, and E. L. Ivchenko, Phys.
2 M.W.Wu.,J.H.Jiang,andM.Q.Weng,PhysicsReports Rev.B 60, 5811 (1999).
493, 61 (2010). 10 M. Syperek,D.R.Yakovlev,A.Greilich, J. Misiewicz, M.
3 D. J. Hilton and C. L. Tang, Phys. Rev. Lett. 89, 146601 Bayer, D. Reuter, and A. D. Wieck, Phys. Rev. Lett. 99,
(2002). 187401 (2007).
4 K. Shen and M. W. Wu,Phys. Rev.B 82, 115205 (2010). 11 T. Korn,M. Kugler, M. Griesbeck, R.Schulz,A.Wagner,
5 T. C. Damen, L. Vin˜a, J. E. Cunningham, J. Shah, and M. Hirmer, C. Gerl, D. Schuh, W. Wegscheider, and C.
L. J. Sham,Phys. Rev.Lett. 67, 3432 (1991). Schu¨ller, New J. Phys. 12, 043003 (2010).
6 P. Schneider, J. Kainz, S. D. Ganichev, S. N. Danilov, 12 T. Korn, Physics Reports 494, 415 (2010).
U. Ro¨ssler, W. Wegscheider, D. Weiss, W. Prettl, V. V. 13 M.Studer,M. Hirmer,D.Schuh,W.Wegscheider,K.En-
Bel’kov, M. M. Glazov, L. E. Golub, and D. Schuh, J. sslin,andG.Salis, http://arxiv.org/abs/1104.1092v1 (un-
Appl.Phys. 96, 420 (2004). published).
7 C. Lu¨, J. L. Cheng, and M. W. Wu, Phys. Rev. B 73, 14 D. Heiss, S. Schaeck,H. Huebl,M. Bichler, G. Abstreiter,
125314 (2006). J. J. Finley, D. V. Bulaev, and D. Loss, Phys.Rev. B 76,
8 B.Baylac,T.Amand,X.Marie,B.Dareys,M.Brousseau, 241306 (2007).
10
15 J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. 26 I.A.Yugova,A.A.Sokolova,D.R.Yakovlev,A.Greilich,
Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and D. Reuter, A. D. Wieck, and M. Bayer, Phys. Rev. Lett.
A. C. Gossard, Science 309, 2180 (2005). 102, 167402 (2009).
16 J.Fischer,W.A.Coish, D.V.Bulaev,andD.Loss,Phys. 27 J.Feldmann,G.Peter,E.O.G¨obel,P.Dawson,K.Moore,
Rev.B 78, 155329 (2008). C. Foxon, and R. J. Elliott, Phys. Rev. Lett. 59, 2337
17 R. Winkler, S. J. Papadakis, E. P. De Poortere, and M. (1987).
Shayegan, Phys.Rev.Lett. 85, 4574 (2000). 28 V. Ciulin, P. Kossacki, S. Haacke, J.-D. Gani`ere, B. De-
18 P. Machnikowski and T. Kuhn, Phys. Rev. B 81, 115306 veaud, A. Esser, M. Kutrowski, and T. Wojtowicz, Phys.
(2010). Rev.B 62, R16310 (2000).
19 T.AndlauerandP.Vogl,Phys.Rev.B79,045307 (2009). 29 P. Olbrich, J. Allerdings, V. V. Bel’kov, S. A. Tarasenko,
20 M.Kugler,T.Andlauer,T.Korn,A.Wagner,S.Fehringer, D.Schuh,W.Wegscheider,T.Korn,C.Schu¨ller,D.Weiss,
R.Schulz,M.Kubova´,C.Gerl,D.Schuh,W.Wegscheider, and S. D. Ganichev,Phys. Rev.B 79, 245329 (2009).
P.Vogl, and C. Schu¨ller, Phys.Rev.B80, 035325 (2009). 30 M. I. Dyakonov and V. I. Perel, Zh. Eksp. Teor. Fiz 60,
21 J. J. Baumberg, D. D. Awschalom, N. Samarth, H. Luo, 1954 (1971), [Sov. Phys. – JETP 33, 1053 (1971)].
and J. K. Furdyna,Phys. Rev.Lett. 72, 717 (1994). 31 A.Grodecka,P.Machnikowski,andJ.F¨orstner,Phys.Rev.
22 J.M.KikkawaandD.D.Awschalom,Phys.Rev.Lett.80, A 79, 042331 (2009).
4313 (1998). 32 A.S.Bracker,E.A.Stinaff,D.Gammon,M.E.Ware,J.G.
23 I. A. Yugova, M. M. Glazov, E. L. Ivchenko, and A. L. Tischler, A. Shabaev, A. L. Efros, D. Park, D. Gershoni,
Efros, Phys.Rev.B 80, 104436 (2009). V. L. Korenev, and I. A. Merkulov, Phys. Rev. Lett. 94,
24 H.-P. Breuer and F. Petruccione, The Theory of Open 047402 (2005).
Quantum Systems (Oxford University Press, Oxford, 33 S. Laurent, M. Senes, O. Krebs, V. K. Kalevich, B. Ur-
2002). baszek, X. Marie, T. Amand,and P. Voisin, Phys. Rev.B
25 D. R. Yakovlev and M. Bayer, in Spin Physics in Semi- 73, 235302 (2006).
conductors, edited by M. I. Dyakonov (Springer, Berlin, 34 A. Shabaev, E. A. Stinaff, A. S. Bracker, D. Gammon,
2008), Chap. Coherent Spin DynamicsofCarriers, p.135. A. L. Efros, V. L. Korenev, and I. Merkulov, Phys. Rev.
B 79, 035322 (2009).
