Influence of magnetic quantum confined Stark effect on the spin lifetime of indirect excitons

0.85 MB·

by  P. Andreakou

Influence of magnetic quantum confined Stark effect on the spin lifetime of indirect
excitons.
P. Andreakou,1 A. V. Mikhailov,1,2 S. Cronenberger,1 D. Scalbert,1 A. Nalitov,3 A. V. Kavokin,2,3,4
M. Nawrocki,5 L. V. Butov,6 K. L. Campman,7 A. C. Gossard,7 and M. Vladimirova1
1Laboratoire Charles Coulomb, UMR 5221 CNRS/ Universit´e Montpellier 2, F-34095, Montpellier, France
2Spin Optics Laboratory, St-Petersburg State University, 1, Ulianovskaya, St-Peterbsurg, 198504, Russia
3School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom
4Russian Quantum Center, 100, Novaya, Skolkovo, Moscow Region, 143025, Russia
5Institute of Experimental Physics, University of Warsaw, Hoz˙a 69, 00-681 Warsaw, Poland
6Department of Physics, University of California at San Diego, La Jolla, CA 92093-0319, USA
6
1 7Materials Department, University of California at Santa Barbara, Santa Barbara, California 93106-5050, USA
0
We report on the unusual and counter-intuitive behaviour of spin lifetime of excitons in coupled
2
semiconductor quantum wells (CQWs) in the presence of in-plane magnetic field. Instead of con-

n ventionalaccelerationofspinrelaxationduetotheLarmorprecessionofelectronandholespinswe
a observe a strong increase of the spin relaxation time at low magnetic fields followed by saturation
J and decrease at higher fields. We argue that this non-monotonic spin relaxation dynamics is a fin-

4 gerprint of the magnetic quantum confined Stark effect. In the presence of electric field along the
CQW growth axis, an applied magnetic field efficiently suppresses the exciton spin coherence, due

] to inhomogeneous broadening of the g-factor distribution.
s
a
g
- INTRODUCTION
t (a)
  Absorp5on
 (log
 color
 scale)

n
ua The effect of magnetic field on Wannier-Mott excitons V)
  DX
q is studied since late 1950s. Theoretical works by Elliott meV

t. and Loudon [1], Hasegawa and Hovard [2], Gor’kov and y
 (me 10

 1

 0.1

 0.01

a Dzyaloshinskii [3] describe the diamagnetic energy shift g5

m and fine structure of bulk excitons. Lerner and Lozovik ner2 IX
E Applied
 voltage,
 0
 -­‐
 1V

- expanded these studies to two-dimensional systems in-
d
cluding quantum wells (QWs) [4]. Later, a detailed the-
n
o ory has been developed for magneto-excitons in biased (b)
  (c)

c coupledquantumwells(CQWs),wherebothspatiallydi-
[ rect(DX)andindirect(IX)exctionstatesmayberealised

1 [5–7]. It has been shown that due to the joint action of
v normal-to-QW-planeelectricandin-planemagneticfields
7 the exciton dispersion can be strongly affected. Besides
2
this, magnetic fields strongly affect the exciton oscillator
4
strength, that is ability to absorb or emit light, as has
0
0 been pointed out in the seminal paper of Thomas and (d)
  (e)

. Hopfield [8]. Due to the opposite orientations of Lorentz
1
forcesactinguponelectronandhole,theexcitonacquires
0
6 astationarydipolemomentinthepresenceofamagnetic
1 field. This constitutes the magnetic Stark effect studied
: in various semiconductor systems [8, 9]. The conven-
v
i tional Stark effect is enhanced in CQWs compared to
X
single QWs [10]. In this paper we address the magnetic
FIG. 1. (color online) (a) Colormap of the excitonic absorp-
r Stark effect in CQWs.
a tion in a CQW structure studied in this work, calculated as
In CQWs, excitonic states are mixtures of traditional a function of applied electric bias. Sketch of the CQW band
intrawell, or DX states and interwell, or IX states, con- structureinthepresence(c),(e),orabsence(b),(d)ofelectric
sisting of an electron and a hole confined in different biasalongz-axis. Redandblueparabolasin(d-e)aredisper-
sions of two direct and two indirect states corresponding to
QWs, Fig. 1 (b, c). This mixing can be controlled by
optical transitions indicated in (b-c).
the external bias. The corresponding exciton energies
and oscillator strengths can be accurately calculated by
solving Schr¨odinger equations for different values of the
gate voltage [11]. An example of such calculation for a ormap of the excitonic absorption in a CQW structure
typical CQW sample is given in Fig. 1 (a), where a col- is shown in the energy/gate voltage plane. The absorp-
2
tion is inversely proportional to the exciton lifetime. In timeupto5T,followedbyadecreaseathigherfieldsand
the absence of applied bias, DX-like state is the ground a non-monotonic behaviour of the spin relaxation time.
state of the system, IX is several meV above it, and its We interpret this unusual behaviour as a consequence of
oscillatorstrengthisonly10timeslower. Bycontrast,at the magnetic Stark effect, which in CQWs converts DX
strong gate voltage, the IX state is about 20 meV below toIX,havingmuchslowerspinrelaxationrate,whilethe
the DX and has an oscillator strength 100 times lower distribution of g-factors plays much weaker role at zero
than the DX. At intermediate gate voltages, IX and DX bias.
states anticross. It is convenient to describe the system
intermsofdirectandindirectstates,interactingviaelec-
tron tunneling. SAMPLE AND EXPERIMENTAL SETUP
The spin lifetime of a pure DX state is short, ≈50 ps.
Indeed, an important source of DX spin relaxation is the Our sample consists of two 8 nm wide GaAs quantum
fluctuatingeffectivemagneticfield,whichoriginatesfrom wellsseparatedbya4nmAl Ga Asbarrierandsur-
0.33 0.67
themomentumdependentcomponentoftheexchangein- rounded by 200 nm Al Ga As layers. The voltage
0.33 0.67
teraction [12]. The fluctuations, due to the scattering of V applied between the conducting n-GaAs layers drops
g
theexcitoncenterofmassmomentum,areresponsiblefor in the insulating layer between them [16]. The sample is
the exciton spin relaxation, in the same manner as any placed in the helium bath magneto-optical cryostat.
othermotionalnarrowingspin-flipprocessesare,withthe We perform photoinduced Kerr rotation and reflectiv-
characteristic dependence of the spin-relaxation time on ityexperimentsat2K.Two-colormeasurementsarereal-
the inverse momentum scattering time. For pure IX the ized by spectral filtering of pump and probe pulses. The
exchangeinteractionisnegligibleduetothelowelectron- pulsedurationis1ps,thespectralwidthis1.5meV.The
hole overlap, so that the fluctuating wave-vector depen- Ti-Sapphire laser repetition rate is reduced to 20 MHz
dent exchange field does not affect spin relaxation. At in order to avoid exciton accumulation between pulses
lowexcitondensities, IXspinlifetimecanbeevenlonger at high gate voltage and high magnetic field. Typical
than the spin relaxation time of a two-dimensional elec- powers are 120 and 70 µW for pump and probe, respec-
tron gas (2DEG) systematically present in biased QWs tively,focusedona100µmdiameterspot[17]. Magnetic
[13]. This difference is essentially due to stronger locali- fields are applied in the plane of the structure (Voigt
sation of IXs in the QW disorder potential, as compared geometry). Spin-polarized DXs are optically excited in
to 2DEG or DX. The variation of the spin lifetime of ex- the CQW by a circularly polarized pump pulse, tuned in
citons with the gate voltage can be understood as being the vicinity of DX resonance. The resulting dynamics of
due to the mixing between the purely DX state charac- the spin polarization (exciton density) is monitored via
terized by a fast relaxation rate, and the purely IX state Kerr rotation (reflectivity) of the delayed linearly polar-
having a slow spin relaxation rate [13]. ized probe pulse.
Application of a magnetic field in the plane of the The probe energy is also tuned around the DX res-
CQWs provides a rich playground where the combina- onance, and is chosen independently of the pump en-
tion of magnetic and traditional Stark effect, disorder, ergy, in order to optimise the signal. We have shown in
interactions and mobility governs the spin dynamics in our previous work, that the spin dynamics of DXs, IXs,
the system. Indeed, in-plane magnetic field shifts the and residual 2DEG in coupled quantum wells can be ef-
dispersionofIXstatesink-space,asillustratedschemat- ficiently addressed in this configuration [13].
ically in Figure 1 (d, e) [5, 6]. In this work we study the
implication of this phenomenon for the spin dynamics
in CQWs, using the time-resolved Kerr rotation spec- RESULTS
troscopy. In the presence of an in-plane magnetic field
this technique allows us to determine the transverse spin Figure2presentsthemainresultofthiswork. Itshows
lifetime, which is limited by the exciton recombination the Kerr rotation signal measured at the pump energy
time,thespincoherencetime,andtheeventualpurespin E =1.571eVandtheprobeenergyE =1.569eVfor
pp pr
dephasing due to an inhomogeneous broadened distribu- different values of magnetic field ranging from zero to 10
tion of the g-factors. We identify two regimes of spin T. No gate voltage is applied, pump and probe energies
coherence, controlled by thestrength ofthe applied elec- were chosen to optimise the zero-field signal, and kept
tric field. At strong bias, zero-field spin lifetime reaches fixed for the set of measurements shown in Fig. 2 (a).
10 ns. The applied magnetic field leads to exciton spin One can see that the monotonous bi-exponential decay
dephasing, due to strong inhomogeneous broadening of at B = 0 is replaced by a much longer living oscillatory
the g-factor distribution in biased CQWs. A similar be- behaviour once the magnetic field increases up to about
haviour is observed in the 2DEG in the hoping regime 5T. The further increase of the magnetic field is accom-
[14, 15]. At zero bias, application of the in-plane mag- panied by the decrease of the decay time, back to the
neticfieldresultsinastrongincreaseofexcitonspinlife- zero-field value. All the curves measured in the presence
3
Fig. 2 (c), and persists whatever the pump power and
energy is.
WeattributetheoscillatorybehaviouroftheKerrrota-
tion signal to the precession of the electron spin, rather
than to the exciton spin precession. Indeed, in GaAs-
based QWs such precession has already been observed
[20]. It was shown, that when the hole spin relaxation
time τ is shorter than (cid:126)/∆ , the spin of an electron
h 0
boundintoanexcitonprecessesatthesamefrequencyas
the free electron spin [21]. Here ∆ stands for the short-
0
rangepartoftheexchangeinteraction. Inour8nmQWs,
∆ ≈ 70 µeV [22], and, at least for delocalised excitons
0
hole spin relaxation is fast τ < 10 ps. Therefore, we
h
conclude that the observed spin dynamics should be at-
tributed to the precession of the spin of electrons bound
to holes within excitons. We have checked, that in the
regime where zero-field precession of exciton spin is ob-
served, applicationofthemagneticfieldofonly0.15Tis
sufficient to overcome the hole exchange field acting on
the electron spin, and recover electron spin precession.
FIG. 2. (a) Waterfall plot of Kerr rotation measured at
Therefore, in what follows we consider that Kerr rota-
zero electric bias as a function of the pump-probe delay at
different magnetic field intensities, E = 1.571 eV, E = tion oscillations observed at zero bias are related to the
pp pr
1.569 eV. (b) Same measurements at B =0 for two different spin precession of electrons bound within excitons.
pump energies. (c) Magnetic field dependence of the slowest There are two very surprising findings shown in Fig.
relaxationtimemeasuredatzeroelectricbiasfortwodifferent
2. First of all, except for indirect excitons, typical relax-
excitation energies. Lines are fit to two models, based on
ation times observed for excitons in GaAs QWs do not
theLiouvilleequationwithLindbladterm(solidline)andon
exceed 100 ps [23], while in the present experiment we
themicroscopicanalysisoftheSchro¨dingerequation(dashed
line). deal with much longer times. Moreover, the spin life-
time is not expected to increase when magnetic field in-
creases. A constant or decreasing with in-plane mag-
of the magnetic field are well described by a fast (50 ps) netic field spin lifetime is typically observed in semicon-
exponential decay, followed by a slower decaying cosine ductor QWs [14, 24]. This is due to dephasing gov-
function. erned by the width of the electron g-factor distribution
√
We have shown in our previous work that at low exci- τ (∆g,B)= 2(cid:126)/(∆gµ B),anditisinverselypropor-
inh B
tationenergyandpower,onecanreachtheregimewhere tionaltotheappliedmagneticfield,inastrikingcontrast
exciton spin precesses even in the absence of the applied with our experimental observation.
magneticfield[13]. Thisprecessionisduetoasmallsplit- BeforegoingintotheanalysisoftheV =0results, let
g
ting between two perpendicularly polarized linear exci- usnowconsiderthespincoherenceinthepresenceofthe
tonstatesδ thatisgenerallypresentinQWstructures strong gate voltage V = 0.8 V, as shown in Fig. 3. In
xy g
[18, 19]. For exciton spin this splitting acts as an effec- this regime, IX is the lowest energy exciton state of the
tive in-plane magnetic field. Therefore, relaxation of the system and in the Kerr rotation signal at B = 0 we ob-
spin polarization is accompanied by its rotation around serve three exponentially decaying components (squares
this effective field. Such precession is very sensitive to inFig. 3(a))[13]. Theycanbeattributedtothespinre-
the excitation energy and is only observed at low pump laxationoftheDX,2DEG,whichformsinbiasedCQWs
energy and power, as shown in Fig. 2 (b). At low pump [25–27], and IXs. The latter has much longer spin relax-
energy, when excitons are essentially localised, we ob- ation time, up to 10 ns. At B = 1 T we still observe
serve a decay accompanied by an oscillation of the Kerr threecomponents(circlesinFig. 3(a)). Theexperimen-
rotation signal, while at high energy pumping simple bi- tal data are fitted to a linear superposition of one ex-
exponential decay is observed. The set of measurements ponential and two damped cosine functions. The fastest
shown in Fig. 2 (a) corresponds to the high energy exci- exponential decay, associated with the DX spin, is not
tation, where there is no spin precession at B = 0. The affected by magnetic field, as it is much faster than the
corresponding decay times of the precessing component precessionperiod. Thetwoothercomponentsexhibitthe
extracted from the fitting procedure are shown in Fig. oscillatory behaviour with different precession frequen-
2 (c). The non-monotonous behaviour as a function of cies and decay times. This is illustrated in 3 (c), were
the applied magnetic field can be clearly observed. It is FourierspectraofKerrrotationmeasuredatB =1Tare
robustwithrespecttotheexcitationenergy, asshownin shown. While at zero bias only one peak appears in the
4
2DEGandahighervalue,∆g =0.016,formorelocalised
IXs.
Let us summarise the dependence of the spin lifetime
inasystemofCQWsonboththein-planemagneticfield
andtheelectricfieldalongthegrowthaxis(definedbythe
applied gate voltage). Fig. 4 (a) shows the longest spin
lifetime extracted from the fitting procedure described
above. The values are given for different in-plane mag-
netic fields up to 5 T, for each field the gate voltage
dependenceisshown. Tworegimescanbedistinguished.
TheyareindicatedbydifferentbackgroundcolorsinFig.
4. Atlowvoltage,intheregimewhereIXenergyishigher
than that of DX (direct regime), spin relaxation time
increases with the magnetic field increase. The corre-
spondingg-factorslightlyincreaseswithbiasbutremains
aboveg =0.1, Fig. 4(b). Usingphoto-inducedreflectiv-
ity technique described in Ref. [13], we could not detect
any measurable modification of the exciton lifetime with
magnetic field in this regime, it remains slightly below
10nsupto10T.Athighvoltage,i. e. above≈0.3V,IX
FIG. 3. (a) Kerr rotation measured at Vg = 0.8 V as a becomesthelowestenergyexcitonstate(indirectregime)
function of the pump-probe delay at B = 0 and B = 1 T,
[31]. In this regime the g-factor decreases substantially
E = 1.568 eV, E = 1.569 eV. (b) Three characteristic
pp pr and spin lifetime time also changes its behaviour. It de-
decay times extracted from Kerr rotation measurements at
creaseswhenmagneticfieldincreases. Itwasshown,that
V = 0.8 V and at different in-plane magnetic fields. These
g
decay times are ascribed to DX, IX, and 2DEG spin relax- inthisregime,magneticfieldalsoleadstostrongincrease
ation. Solid line are fit the spin dephasing model, assum- oftheIXlifetime,duetotheshiftoftheIXdispersionin
ing g-factor distribution ∆g = 0.016 for IX and ∆g = 0.006 k-space[6]. Thisfeatureisalsoreproducedinourphoto-
for 2DEG. (c) Fourier spectra of Kerr rotation measured at induced reflectivity experiments. At V =0.8 V and 7 T
g
B = 1 T. Zero bias spectrum is compared to V = 0.8 V.
g exciton lifetime approaches the time delay between the
(d) Two precession frequencies extracted from Kerr rotation
laser pulses (48 ns), leading to the accumulation of ex-
measurements at V = 0.8 V and at different in-plane mag-
g
citons in the structure. The 1/B behaviour of the spin
netic fields. These frequencies are ascribed to IX, and 2DEG
spin rprecession lifetimeintheindirectregimecanbeunderstoodinterms
of the inhomogeneous broadening of the long-living and
strongly localized IX as shown in Fig. 3 (b). Strong
Fourier spectrum at V =0.8 V two peaks can be clearly gate voltages pushes electron and hole towards QW in-
g
distinguished, the lowest frequency corresponding to the terfaces, which contributes to the increasing role of the
slowestdecay. Theprecessionfrequencyoftheslowcom- disorder potential. Additional localisation also leads to
ponent is related to the IX spin (more precisely to the the decreasing g-factor [29]. By contrast, the increase of
precessionofthespinofelectronboundintoIX),andthe thespinlifetimeinthepresenceofthein-planemagnetic
fastcomponentisassociatedwiththebareelectronspin. field observed in the direct regime does not have analogs
The magnetic field dependence of the two precession fre- in other electronic or excitonic systems. We will show in
quencies is shown in Fig. 3 (d) [28]. One can see that it thetheoreticalpartofthepaper,thatthiseffectisdueto
corresponds to different g-factors, which may arise from the magnetic field induced mixing of DX and IX states,
the different mass and density and therefore different lo- characteristic of CQWs.
calization of IXs and electrons [13]. Indeed, the degree
oflocalizationisacrucialparameter,thatcontrolstheg-
factor values in GaAs/AlAs-based heterostructures [29]. THEORY
Note, that in undoped CQWs identical to the one stud-
ied here, previous studies have found the same value of This section presents the theoretical model of exciton
the g-factor g = 0.12 [30]. The decay times obtained at spin relaxation in CQWs in the absence of electric bias
V = 0.8 V are shown in Fig. 3 (b). While the shortest but in the presence of in-plane magnetic fields. Using
g
(DX) spin lifetime remains constant, both 2DEG and IX two approaches: semi-phenomenological Lindblad equa-
spin lifetimes decrease with the increase of the magnetic tion approach and the microscopic approach based on
field. Solid lines show the fit to 1/B behaviour, consis- Shr¨odinger equation, we will show, that magnetic field
tent with the inhomogeneous broadening expected from dependence of the exciton spin lifetime can by explained
thedistributionofg-factors,whichgives∆g =0.006fora by the mixing of DX and IX states. Let us consider
5
dispersion is shifted in the reciprocal space in perpen-
diculartothemagneticfielddirection(alongthex-axis),
oppositeforthetwopossibleorientationsofIXdipolemo-
ment [5, 6]. The IX and DX parabolic dispersions, char-
acterized by the effective mass m given by the sum of
X
electron and heavy hole in-plane effective masses, inter-
sectatsomemomentumdependingontheDX-IXenergy
splitting and the shift of the IX dispersion in magnetic
field k(B) = ±eBd/(cid:126), Fig. 1 (b). Here e is the electron
charge, d is the distance between QWs centres.
Resonant optical pumping generates DXs at the bot-
tom of the dispersion. However, excitons are accelerated
out of the excitation spot by repulsive exciton-exciton
interactions. For DX the characteristic kinetic energy
acquired by excitons is given by the interaction energy
6R a2n [37], where R and a are the exciton Rydberg
y B y B
energy and Bohr radius, respectively, and n is the 2D
exciton density, created by the optical pulse. In realis-
FIG.4. Gatevoltagedependenceof(a)theslowestdecaytime
tic conditions this blueshift is of the same order as the
measuredinKerrrotationscansatdifferentgatevoltagesand
energy difference between the DX and IX ground states
in-plane magnetic fields (b) Gate voltage dependence of g-
(∼1meV).Thisenergyisthereforesufficienttoreachthe
factor, associated with this slowest component. Two regimes
canbeidentified. AtsufficientlysmallV ,suchthatDXstate range of reciprocal space close to the intersection point
g
remains below IX (direct regime) , decay time increases with ofIXandDXdispersions. Excitonsinthisrangebecome
magnetic field. Under gate voltage such that IX becomes coupled to the long-living, spin conserving indirect state
the lowest exciton state in the system (indirect regime), the and define the longest spin relaxation times observed in
dependence is inverse, decay time drops dramatically in the
the experiment.
presence of the magnetic field.
To model the experiment we derive the full exciton
Hamiltonian, accounting for the electron Zeeman split-
ting, magnetic Stark effect. To construct the Hamilto-
CQWs with a pair DX states at energy E and and
DX nian we choose a basis of four exciton states, schemati-
a pair of IX states at E , Fig. 1 (b, d). At zero bias
IX callyshowninFig. 1(b): apairofspatiallydirectstates
E <E due to reduced binding energy of the latter.
DX IX and a pair of indirect ones. Taking into account the ex-
TheDXstateischaracterizedbyastrongelectron-hole citon spin structure, consisting of four spin states with
exchange interaction, while it is vanishingly small for IX total momenta projections on the growth axis ±2 and
state. The short range component of this interaction ∆0 ±1, we arrive to the 16×16 Hamiltonian matrix form:
splitsthebright(±1)andthedark(±2)excitonmomen-
 
tum states and prevents elastic bright-dark convertion HIX,1 0 J 0
processes. Spin-flip processes allowed by the selection H= 0 HIX,2 0 J  (1)
rules therefore imply simultaneous rotation of both elec-  J 0 HDX,1 0 
tron and hole spins. The long-range part of exchange in- 0 J 0 HDX,2
teraction∆iswavevectordependent. Itisresponsiblefor
Here the spin structure of each excitonic state is given
the Maialle-Andrada e Silva-Sham mechanism of exciton
by the diagonal 4x4 blocks H . The coupling
spin relaxation, which dominates in single QWs [12, 32]. IX(DX),1(2)
between them is given by a single block J = JI , de-
The corresponding spin relaxation time is much shorter 4x4
scribing spin conserving electron tunneling through the
than that of IX-bound electron spin relaxation, because
potential barrier between the two QWs. We neglect the
exchange interaction is vanishing within IX [19, 33–36].
analogous term describing the hole tunneling due to its
We will assume in the following that the spin relaxation
heavyeffectivemassintheCQWsgrowthdirection. This
rate IXs is governed by its interaction with DXs only.
allows to decouple the two DX-IX pairs and, without a
An in-plane (parallel to the y-axis) magnetic field af-
lossofgenerality, reducetheproblemtothe8×8Hamil-
fects the IX-DX coupling via magnetic Stark effect [8],
tonian:
whichmaybeinterpretedviaintroductionofaneffective
electric field acting on propagating carriers. This field is (cid:18)H J (cid:19)
H= IX . (2)
out of CQWs plane and shifts the IX states in full anal- J H
DX
ogy with the real electric field. The energy scheme and
band structure of unbiased CQWs under in-plane mag- Motion of dipolar excitons in CQWs is mainly due to
netic field is shown in Fig. 1 (b, d). The IX energy repulsion-induceddriftratherthanpurediffusion[33,38].
6
Thus,itcanbecharacterisedbyaquicklyfluctuatingmo- dau gauge vector potential A = Bde z, corresponding
x
mentum q on top of a slowly varying one K, resulting in to the external magnetic field Be . Here d is the mean
y
the total wavevector k=K+q, so that q (cid:28) K. Exci- electron-holeseparationwithintheIX,whichmaybeap-
ton spin relaxation is induced by the fluctuating part of proximated by the distance between the CQWs centers.
the long range electron-hole exhange field ∆(q)∝q. We Note that for IXs we neglect both long and short range
explore two different approaches that allow accounting parts of the electron-hole exchange.
for this effect. In the first one we neglect long-range ex-
changeinteractionintheDXHamiltonian,butintroduce
the fluctuating field via the phenomenological Lindblad Lindblad equation analysis
superoperator in the Liouville equation for the density
matrix. In the second one we directly introduce the fluc- In this subsection we use the semi-phenomenological
tuating field in the coherent Hamiltonian part and solve approach based on quantum Liouville equation with the
the Shroedinger equation on the timescales shorter than Lindblad superopeoperator in the right-hand part, also
the characteristic momentum scattering time. called Lindblad equation. We first assume the linear in
In the most general way the DX block in the basis of k dependence of the long range exhange ∆(k) [12]. It al-
spin states (−2,−1,+1,+2) reads: lowsustoisolatetheslowlyvaryingpartoftheexchange
field, corresponding to the wave vector K from the fast
 
−∆ ∆ 0 0 fluctuating part, linear in q : ∆(k) = ∆(K)+∆(q). We
0 B
HDX =EDX(k)I4x4+∆0B ∆(0k)∗ ∆(0k) ∆0B . (3) rceomunotvefotrhfleulcattuteartifnrgomfietldhevicaohthereeLnitnHdbalmadiltsounpiearnopanerdataocr-
0 0 ∆ −∆ term. TheLindbladequationfortheexcitondensityma-
B 0
trix ρ reads:
Here the DX energy E (k) = (cid:126)2k2/2m +E , ∆ is
DX X DX 0
dρ i
the short range part of the electron-hole exchange inter-
=− [H,ρ]+L(ρ). (5)
action, ∆(k) is its long range part, and ∆ = gµ B/2. dt (cid:126)
B B
istheelectronZeemansplitting. Notethatweneglectthe
Here the Lindblad superoperator L accounts for the DX
hole spin splitting due to the small heavy hole g-factor.
spin relaxation and decay. We neglect relaxation and
WealsoneglectthemagneticStarkeffectforDXsasitis
decay processes corresponding to the IX. The Lindblad
linear in electron-hole separation distance. On the other
termL(ρ)alsomayberepresentedinablockmatrixform:
hand, we keep it in the IX Hamiltonian block:
(cid:18) (cid:19)
0 −ρ γ /2
 0 ∆ 0 0  L(ρ)= DX−IX DX . (6)
B −ρ γ /2 L (ρ )
IX−DX DX DX DX−DX
HIX =EIX(k)I4x4+∆0B 00 00 ∆0B, (4) Here ρDX−IX and ρIX−DX are the top-right and bottom-
0 0 ∆B 0 left4x4blocksofthetotaldensitymatrix,whichdecayat
twice slower rate than the DX density γ /2. The L
DX DX
wheretheIXenergyE (k)= (cid:126)2 (cid:0)k− eBde (cid:1)2+E . block, which describes the fast relaxation of the direct
IX 2mX (cid:126) x IX
The correction to the IX energy appears due to the Lan- part, reads:
 
0 0 0 0
(cid:16) (cid:17)
0 ρ+1,+1 −ρ−1,−1 γ −ρ−1,+1 γ 0
 DX−DX DX−DX ex DX−DX ex 
LDX(ρDX−DX)=−ρDX−DXγDX+0 −ρ+1,−1 γ (cid:16)ρ−1,−1 −ρ−1,−1 (cid:17)γ 0, (7)
 DX−DX ex DX−DX DX−DX ex 
0 0 0 0
where γ = (2τ )−1 is the rate of spin relaxation due hole spin.
ex ex
to the Maialle-Andrada e Silva-Sham mechanism. We Thesolutionofequation(5)yieldsthedynamicsofthe
assume that the hole spin relaxation is by far the fastest Kerr rotation angle, measured in the experiment. The
processinthesystemandtakeasinitialconditionforthe effect itself is produced by spin-dependent exciton res-
Lindbladevolutionequationthedensitymatrixwithonly onance shifts, stemming from exciton-exciton exchange
two non-zero elements ρ+1,+1 =ρ−2,−2 =n/2, which interactions. Both direct and indirect components con-
DX−DX DX−DX
describestheexcitonicsystemofdensityn,pumpedwith tribute to the value of Kerr rotation angle. The value
a σ+ polarized optical pulse, with a fully relaxed heavy of the Kerr rotation angle δθ is a sum of two contri-
7
butions, linear in bright DX and bright IX spin polariza- Microscopic analysis
tionsρ+1,+1 −ρ−1,−1 andρ+1,+1−ρ−1,−1 respectively
DX−DX DX−DX IX−IX IX−IX
[39]. Furthermore, the coefficients before the two polar- Assuming a generic spin relaxation mechanism stem-
izations, given by the Coulomb carrier exchange, weakly ming from the spin precession in a stochastically fluctu-
dependontheelectron-holeseparationdistanceandmay ating field, the spin relaxation rate γ =1/τ scales with
s s
be assumed equal for IX and DX contributions, allowing both characteristic value of the fluctuating field Ω and
us to write: the fluctuation time scale, given by the momentum re-
laxationrateτ viaDyakonov-Perelformulaγ =Ω(B)2τ
s
δθ ∼ρ+1,+1 −ρ−1,−1 +ρ+1,+1 −ρ−1,−1. (8) [12,40]. Inprinciple,bothparametersdependonthede-
DX−DX DX−DX IX−IX IX−IX
gree of DX-IX coupling: Ω scales with the electron-hole
overlap, while τ in drift-diffusion regime depends on the
Kerr rotation angle decay, as well as the relative spin
value of the exciton dipole moment. Here we focus on
polarisation, may be extracted from the solution of Eqs.
the variation of Ω with magnetic field B, which allows
(5, 8) as functions of the magnetic field B. We fit the
us to explain the experimental measurements. Instead
experimental data assuming the exciton effective mass
of the Lindblad equation we solve the Shr¨odinger equa-
m = 0.22m [7] and the interwell distance d = 12nm.
X 0 tion,takingthefullHamiltonian(2)withthefield,stem-
TheelectronLandefactorg =0.1isobtainedfromexper-
ming from the long-range electron-hole exchange ∆(k).
iments,andthebright-darksplittingis∆ =70µeV[22].
0 Thisapproachisvalidonthetimescalesshorterthanthe
Inthesimulationweneglectedthecoherentelectron-hole
momentum relaxation time, as the effective field is mo-
exchange field ∆(K) compared with other fields in the
mentum dependent and its absolute value is in the first
system for both DX and IX. The decay time obtained
approximation linear in the momentum value [41].
with the fit parameters ∆E = E −E = 1.5 meV,
DX IX Taking a wavefunction, describing a DX with the fully
J = 0.15 meV, K = 90 µm−1, 1/γ = 2 ns, 1/γ =
DX ex relaxed hole spin and the electron spin s = −1/2, as
e
400 ps, is shown in Fig. 2 (c) by solid line. It has a
initial condition, one may trace the corresponding Kerr
maximum in the vicinity of B = 5T, which corresponds
rotation angle δθ(t). Treating it as bright exciton sub-
to the maximum DX-IX mixing. One can see, however,
space pseudospin projection, we numerically extract the
that the relation between the decay times at the peak
characteristic frequencies of its rotation Ω(B) from the
and at B = 0 is limited by 2. Indeed, in the ideal case
Fourier transform of δθ(t). Assuming Ωτ (cid:28) 1, we esti-
of negligible DX-IX coupling at B =0 where the exciton mate the exciton spin decay rate as γ = Ω(B)2τ. The
s
spin relaxation is given by that of DXs, the longest pos-
parameters of the numerical calculation are the same as
sibledecaytimeisachievedwhilsttheDXandIXmodes
for the Lindblad equation.
are resonant. Excitons in this case are half-indirect and
Thisapproachyieldsupto4timesincreaseofthemea-
thus lose the spin polarization at the twice reduced rate,
sured spin relaxation time τ =1/γ at the resonance of
s s
according to the solution of the Lindblad equation. In
DXandIXenergies, asthestochasticrotationfrequency
the realistic case, where the coupling J (cid:54)= 0, the exci-
can be twice lower for a coupled DX-IX state in compar-
ton spin dynamics is affected by the IX admixture even
ison with a pure DX state. The fit of the experimental
at B = 0, therefore this approach yields 2 as the upper
datausingthismicroscopicapproach(dashedlineinFig.
limit for the relation τ (B = 5)/τ (B = 0), whereas the
s s 2 (c)) is therefore more accurate in comparison with the
measured value is close to 3.
semi-phenomenological approach based on the Lindblad
The main reason for the discrepancy between this re- equation (solid line), even though it does not take into
sult and the experimental data lies in the phenomeno- account the dependence of the transport time τ on the
logical nature of γ , introduced as the relaxation rate magnetic field.
ex
of the direct exciton part. Note that this parameter dif-
fers from the exciton spin depolarization time in a single
QW of the same width as those composing CQWs. The CONCLUSIONS
classical Dyakonov-Perel picture gives an insight to this
difference. As long as the characteristic exciton trans- WehavestudiedspindynamicsofexcitonsinCQWsin
port time τ is longer than the electron tunneling time the presence of crossed magnetic and electric fields using
(J/(cid:126))−1, the exciton loses its spin as a whole, rotating time-resolved Kerr rotation spectroscopy. Two qualita-
in a stochastic effective magnetic field between scatter- tively different regimes of spin decoherence are identi-
ing events, rather than losing it via its DX and IX com- fied, depending on the strength of the electric field, ap-
ponents independently. Our further microscopic analy- plied along the growth axis. In the presence of the gate
sis of the spin relaxation gives similar qualitative result voltage, such that IX becomes the lowest energy exciton
as the semi-phenomenological Lindblad equation-based state of the system, the inhomogeneous spin coherence
model,butallowstoimprovethequantitativeagreement time is found to be inversely proportional to the mag-
between the theory and the experiment. nitude of the in-plane magnetic field. This behaviour is
8
understood in terms of the inhomogeneous distribution and E. A. Muljarov, Phys. Rev. B 85, 045207 (2012).
of g-factors, typical for QW structures. Inevitably, such [12] M.Z.Maialle,E.A.deAndradaeSilva, andL.J.Sham,
distribution leads to the broadening of the spin preces- Phys. Rev. B 47, 15776 (1993).
[13] P. Andreakou, S. Cronenberger, D. Scalbert, A. Nali-
sion frequency distribution between excitons, and thus
tov, N. A. Gippius, A. V. Kavokin, M. Nawrocki, J. R.
lineardependenceofthespindephasingrateonthemag-
Leonard, L. V. Butov, K. L. Campman, A. C. Gossard,
netic field. This inhomogeneity seems to be stronger for
and M. Vladimirova, Phys. Rev. B 91, 125437 (2015).
excitons than for electrons, probably due to stronger lo- [14] E.A.Zhukov,D.R.Yakovlev,M.Bayer,G.Karczewski,
calisation of excitons, which are heavier particles. Com- T. Wojtowicz, and J. Kossut, phys. stat. sol. (b) 243,
pletelydifferentmechanismsdominatethespincoherence 878 (2006).
in symmetric CQWs, when zero, or small electric field is [15] E.A.Zhukov,D.R.Yakovlev,M.Bayer,M.M.Glazov,
E. L. Ivchenko, G. Karczewski, T. Wojtowicz, and
applied,sothatIXenergyishigherthanthatofDXstate.
J. Kossut, Phys. Rev. B 76, 205310 (2007).
Inthisregime,wehavefoundmanifestationsofthequan-
[16] L. V. Butov, A. Imamoglu, A. V. Mintsev, K. L. Camp-
tumconfinedmagneticStarkeffectintheexcitonspinre-
man, andA.C.Gossard,Phys.Rev.B59,1625(1999).
laxation time dependence on the in-plane magnetic field [17] We have checked that reducing probe to pump power
in CQWs. The strongly non-monotonous behaviour of ratio does not change the signal dynamics.
the spin lifetime that may seem counter-intuitive finds [18] D. N. Krizhanovskii, D. Sanvitto, I. A. Shelykh, M. M.
its explanation in the magnetic field induced mixing of Glazov, G. Malpuech, D. D. Solnyshkov, A. Kavokin,
S. Ceccarelli, M. S. Skolnick, and J. S. Roberts, Phys.
DXsandIXsduetotheshiftoftheIXsdispersioncurve.
Rev. B 73, 073303 (2006).
This is the signature of the magnetic Stark effect. The
[19] A. A. High, A. T. Hammack, J. R. Leonard, S. Yang,
magnetic Stark effect appears to be a convenient tool
L. V. Butov, T. Ostatnicky´, M. Vladimirova, A. V. Ka-
of exciton spin engineering, that may complement tra- vokin, T. C. H. Liew, K. L. Campman, and A. C. Gos-
ditional quantum confined Stark effect in the structures sard, Phys. Rev. Lett. 110, 246403 (2013).
where inhomogeneity is important. [20] T.Amand,D.Robart,X.Marie,M.Brousseau,P.LeJe-
une, and J. Barrau, Phys. Rev. B 55, 9880 (1997).
Acknowledgments. We are grateful to K. V. Kavokin,
[21] M.Dyakonov,X.Marie,T.Amand,P.LeJeune,D.Ro-
A. P. Dmitriev, N. A. Gippius and M. I. Dyakonov
bart, M. Brousseau, and J. Barrau, Phys. Rev. B 56,
for valuable discussions and acknowledge the support of 10412 (1997).
EU ITN INDEX PITN-GA-2011-289968. LVB was sup- [22] E. Blackwood, M. J. Snelling, R. T. Harley, S. R. An-
ported by DOE Award DE-FG02-07ER46449. AK ac- drews, and C. T. B. Foxon, Phys. Rev. B 50, 14246
knowledgesthesupportfromtheRussianMinistryofEd- (1994).
[23] M. I. Dyakonov, ed., Spin Physics in Semiconductors,
ucation and Science, Contract No 11.G34.31.0067. MN
Springer series in solid-state sciences (2008).
acknowledges the support from the Polish National Sci-
[24] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett.
enceCenterunderdecisionDEC-2013/09/B/ST3/02603.
80, 4313 (1998).
[25] L.V.Butov,L.S.Levitov,A.V.Mintsev,B.D.Simons,
A. C. Gossard, and D. S. Chemla, Phys. Rev. Lett. 92,
117404 (2004).
[26] R.Rapaport,G.Chen,D.Snoke,S.H.Simon,L.Pfeiffer,
[1] R. J. Elliott and R. Loudon, Journal of Physics and K. West, Y. Liu, and S. Denev, Phys. Rev. Lett. 92,
Chemistry of Solids 15, 196 (1960). 117405 (2004).
[2] H. Hasegawa and R. E. Howard, Journal of Physics and [27] L.Butov,A.Imamoglu,K.Campman, andA.Gossard,
Chemistry of Solids 21, 179 (1961). JournalofExperimentalandTheoreticalPhysics92,260
[3] L. P. Gor’kov and I. E. Dzyaloshinskii, J. Exp. Theor. (2001).
Phys. 26, 449 (1968). [28] It was difficult to mesure the decay times at B > 7 T,
[4] I.V.LernerandY.E.Lozovik,SovietPhysicsJETP53, due to the smaller intensity of the signal.
763 (1981). [29] Z. Chen, S. G. Carter, R. Bratschitsch, P. Dawson, and
[5] A.A.GorbatsevichandI.V.Tokatly,SemiconductorSci- S. T. Cundiff, Nat Phys 3, 265 (2007).
ence and Technology 13, 288 (1998). [30] M. Poggio, G. M. Steeves, R. C. Myers, N. P. Stern,
[6] L.V.Butov,A.V.Mintsev,Y.E.Lozovik,K.L.Camp- A.C.Gossard, andD.D.Awschalom,Phys.Rev.B70,
man, andA.C.Gossard,Phys.Rev.B62,1548(2000). 121305 (2004).
[7] Y. E. Lozovik, I. V. Ovchinnikov, S. Y. Volkov, L. V. [31] L. V. Butov, A. A. Shashkin, V. T. Dolgopolov, K. L.
Butov, and D. S. Chemla, Phys. Rev. B 65, 235304 Campman, and A. C. Gossard, Phys. Rev. B 60, 8753
(2002). (1999).
[8] D. G. Thomas and J. J. Hopfield, Phys. Rev. 124, 657 [32] A. Vinattieri, J. Shah, T. C. Damen, D. S. Kim, L. N.
(1961). Pfeiffer, M. Z. Maialle, and L. J. Sham, Phys. Rev. B
[9] M. Lafrentz, D. Brunne, B. Kaminski, V. V. Pavlov, 50, 10868 (1994).
A. V. Rodina, R. V. Pisarev, D. R. Yakovlev, A. Bakin, [33] J. R. Leonard, Y. Y. Kuznetsova, S. Yang, L. V. Butov,
and M. Bayer, Phys. Rev. Lett. 110, 116402 (2013). T. Ostatnicky´, A. Kavokin, and A. C. Gossard, Nano
[10] Y. J. Chen, E. S. Koteles, B. S. Elman, and C. A. Lett 9, 4204 (2009).
Armiento, Phys. Rev. B 36, 4562 (1987). [34] M.Beian,M.Alloing,E.Cambril,C.G.Carbonell,J.Os-
[11] K. Sivalertporn, L. Mouchliadis, A. L. Ivanov, R. Philp, mond, A. Lemaˆıtre, and F. Dubin, EPL 110, 27001
9
(2015). [38] R. Rapaport, G. Chen, and S. H. Simon, Phys. Rev. B
[35] K. Kowalik-Seidl, X. P. Vo¨gele, B. N. Rimpfl, S. Manus, 73, 033319 (2006).
J. P. Kotthaus, D. Schuh, W. Wegscheider, and A. W. [39] A. V. Nalitov, M. Vladimirova, A. V. Kavokin, L. V.
Holleitner, Applied Physics Letters (2010). Butov, and N. A. Gippius, Phys. Rev. B 89, 155309
[36] A.Violante,R.Hey, andP.V.Santos,Phys.Rev.B91, (2014).
125302 (2015). [40] M.I.DyakonovandV.I.Perel,SovietPhysicsJETP33,
[37] C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, 1053 (1971).
and P. Schwendimann, Phys. Rev. B 58, 7926 (1998). [41] G.E.PikusandG.L.Bir,J.Exp.Theor.Phys.60,195
(1971).