Spin Dynamis Driven by the Eletron-Hole
Exhange Interation in Quantum Wells
M. Z.Maialle
UniversidadeS~ao Franiso,13251-900 Itatiba-SP,Brazil
ReeivedonApril23,2001
We present a theoretial study of the spindynamis of site-loalized exitons in semiondutor
quantum wells driven by the eletron-hole exhange interation. Using a simple model for the
distributionofsitesweobservedspindephasingleadingtoastrongspinpolarization deay.
I Introdution
In general, ontrol overthe degreeof spin orientation
ofphotoreatedhargearriersinsemiondutor
stru-turesisahievedbypolarized-lightexitationofoptial
transitions in onformitywith theangular momentum
seletionrules. Insemiondutorquantumwells(QWs)
thespindynamisresultsfromtheindividualdynamis
oftheeletronspin(omponents1/2,inunitsof~)or
heavyholespin(angularmomentumprojetions3/2),
orfrom thespin dynamis of the bound eletron-hole
pair (exiton). Individual eletron and hole spinips
follow from band spin mixing in addition to
momen-tum sattering, and they havelittle eÆieny on old
arriersattheband edgesof narrowQWsdue to
van-ishing spin mixing and redued available phase spae
forsattering.
In intrinsi QWs the band-edge optial properties
aredominatedbytheexitonstateforwhihthe
orre-lationbetweentheeletronandholespinsissetbythe
eletron-hole (e-h)exhange Coulomb interation.
Al-thoughsmall,theexhangeontributiontothespin
dy-namisbeomesimportantintheaforementioned
situa-tionofstronglyonnedarriersatthebandedges. The
long-rangepartofthee-hexhange interation(LRX)
dependson theexiton enter-of-mass(.m.)
momen-tum, suh that sattering leadsto anexiton spin
re-laxationaordingtoamotionalnarrowingproess,for
whih shorter sattering time
gives longer exiton
spin relaxation time
s
(whih is valid for
s ).
However, suh mehanism annot be diretly applied
forsamplesthatexhibitlateralloalizationofexitonin
sitesreatedbyQWinterfaeimperfetions. Loalized
exitons do not satter as frequently asfree exitons,
suh that the motional narrowing proess is
inappro-priateif
lo
s
. Also,theloalizationoftheexiton
inlargesitesquantizesthe.m. motionandasaresult
theLRXouplingforexitonspinstatesj+1i !j 1i
vanishesinsymmetrisites.[1℄
itonspindynamisviathee-hexhangehavebeen
ad-dressed experimentallyand theoretially by Nikolaus
et al.[2 ℄ for (Zn,Cd)Se/ZnSe QWs. They have found
that the dominant mehanism in the spin deay was
not a true relaxation proess, but instead a
dephas-ingofthemarosopispinpolarizationresultingfrom
the LRXmatrix elements, whih are subjetedto the
same inhomogeneity of the sites. Further evidenes
of the ombined role of the exiton loalization and
e-h exhange interation are provided by experiments
in QWswith transversemagneti eld (Voigt
ongu-ration) where Larmorpreessions of the eletron spin
are observed inthe asethat theexhangeinteration
laksstrengthto orrelatetheeletronandholespins.
Otherwise, the preessingeletron spinwould haveto
drag the heavy-holespin that, in rstapproximation,
is pinned along the growth axis. In GaAs/AlAsGa
QWs[3℄ eletron spin preessionswere observed,
how-ever,in (Zn,Cd)Se/ZnSe QWs,[4℄ thepreessionswere
seenonlyat hightemperatures(100K).
Inthispaper,wepresentabriefaountofthe
theo-retialstudy[5℄ofthespindynamisinZnSe/(Zn,Cd)Se
QWs inluding the eets of exiton loalization,
e-h exhange interation and spin preessions about a
transversemagnetield.
II Spin dynamis by the e-h
ex-hange interation
TheexhangeinterationmatrixforQWsisalulated
in the eetive-mass approximation using an exiton
ground-statewavefuntionwrittenas
1s;K (r
e ;r
h )=(z
e )(z
h )'
1s ()
e iKR
p
A
; (1)
whereandRaretherelativeand.m. e-hoordinates
in-theQW groundstatesforeletronandheavyholeare
respetively(z)and(z). Spinmixingbetweenheavy
holesandlightholesisnegleted.
TheLRXHamiltonian[6℄anbewritten as[7℄
H LR
(K)=F(K) 2
6
6
4
0 0 0 0
0 K Ke
2i
0
0 Ke 2i
K 0
0 0 0 0
3
7
7
5 ; (2)
where the matrix elements are arranged for the
ex-iton spin basis in the order j +2i, j+1i, j 1i, j 2i.
Also, K=(K ;) in polar oordinates and F(K) gives
the quasi{two-dimensionalexhange strength.
Assum-ingnomixingwiththelight-holestates,theshort-range
partof theexhange interation isdiagonal in the
ex-iton spinbasisandan writtenas[7℄
H SR i;j = SR Æ i;1 Æ j;1 ; (3) with SR
setting the splitting between the dark j2i
andoptiallyativej1iexitons.
II.1 Loalized exitons
We treat only the loalization of exitons in sites
larger than the exiton Bohr radius. In this way, the
exiton.m.motionintheQWplaneisassumedtobe
bound into the minima of the site potential. We
de-sribesuhboundstateintheharmoniapproximation
b (K)= b x b y 2 ~ 2 1 4 exp 1 2 (b x K 2 x +b y K 2 y ) ; (4) where b=(b x ;b y
) speies the shape of the site, with
b x =~= p MV xx
given by the seond derivative V
xx of
thesitepotentialalong thesiteprinipal axisxat the
minimum(andsimilarlyfory). TheexitonmassMis
forthedispersionontheQWplane.
AspinHamiltonianfortheLRXanbeobtainedby
rst-orderperturbationtheoryfortheloalizedexiton
ifweneglettheKdependeneonF(K)andsubstitute
K in thematrixEq.(2)by
K b (d=o) = Z dKj b (K)j 2 K 2 x K 2 y q K 2 x +K 2 y ; (5)
withthe+and signsholdingforthediagonal(d)and
o-diagonal (o) terms, respetively. The short-range
exhangetermforloalizedstatesremainsasinEq.(3)
sine itdoesnot depend onKand beauseof the
ap-proximation used that the relative e-h motion is not
modiedby.m. loalization.
We model the distribution of sites as done by
Wilkinson et al. [8℄ using a Gaussian random
fun-tiontosimulatethein-planepotentialV(r)reatedby
the QW imperfetions. The distribution of potential
is haraterized by two parameters: the variane 2
of the site potentialand theorrelation length l suh
thathV(0)V(r)i = 2 V exp( r 2 =2l 2
),wherethebrakets
denoteeitherensembleaverageorspatialaverage. The
absorptionoptial density in this model isa Gaussian
funtion of variane 2
V
. The density probability of
minima N(V) of this potential gives a narrower
dis-tributionenteredatalowerenergywhenomparedto
theabsorptionGaussianurve. Thismodelwasusedto
interpret the Stokesshift and Nikolaus et al.[2℄ have
usedittoderivedthedensityofminimaN(V;V
xx ;V
yy ),
now also asa funtion of the seond derivatives, that
givesthedistributionofsiteswithdierentshapes. The
inhomogeneity in the values of the exhange matries
forthesitesisalulatedusingthedistributionof
min-imaN(V;V
xx ;V
yy ).
=bf II.2 Equations of motion { Coherent spin
dynamis
Theequationofmotionforthedensitymatrix(K)
(with elements
i;j
=jiihjj; where jii=j2i;j1i are
theexitonspinstates)isgivenby
d(K)
dt =
i
~
[ (K);H(K)℄+
t
inoh
+G; (6)
whereG isthegeneration rate, t inoh ontains
in-oherentontributionssuhasthereombinationrates
for the optially ative exitons with spins j1i and
spinrelaxationforeletronsandholeduetospin-orbit
interation. An applied magnetield B introduesa
ZeemanterminH
H B e = B g e 1 2 e
B and H B h = B g h 3 2 h z B z ; (7) where B
isBohrmagneton,g
e(h)
istheeletron(hole)
gfatorand arePaulimatriesusedfortheeletron
(s= 1
2
) and heavy hole (m= 3
2
) spins. Notie that
only the z omponent of B ats on the heavy holes
(theotheromponentswouldinvolvesmaller
ontribu-tionsdue tomixing withthelightholes).[9℄ The
equa-tionofmotionEq.(6),withtheontributionsEqs.(2),
(3)and(7),isasystemofrst-orderlineardierential
equationsthat ouplesallthe16matrixelements
i;j .
Inthemostgeneralases,nophysiallysimplesolution
emergesfromtheequationsofmotionandinthe
follow-ing they are solved numerially. The time dependent
solutions are then average out using the distribution
ofminima N(V;V
xx ;V
yy
) as weightto aount forthe
distributionofsites.
III Results and disussion
We have solved the problem of a free exiton in a
Zn
0:8 Cd
0:2
Se/ZnSeQWusingatrialwavefuntionwith
two variationalparameters.[10℄ Theexhange indued
available experimentaldata E
exh
=0:25meV [4℄and
E
exh
=0:5 meV.[11℄ Reasonable values were found
when using the bulk exhange splittings E
SR =0:2
meVandE
LR
=0:8meV.Inwhatfollowsweonsider
a5nm-QWforwhih
SR
=0:3meVinEq.(3)andF=
14 meV
A in Eq. (2). In addition, we have used[11℄
g
e
=1.1andlookedat exitonswithenergyV= 1.5
V
(luminesenetail)[2℄with
V
=6meVandsite
orrela-tionlengthl=200
A.TheexitingpulseisaLorentzian
ofwidth 1ps enteredin t=0swith irular
polariza-tion +
exitingexitonsj+1i.
InFig.1(a)weshowthetimeevolutionofthe
ele-tron and hole spin omponents along the z diretion
(growthaxis)forzeromagnetield. Wehavesetzero
deaying rates
t
inoh
=0 in Eq. (6), suh that the
deay observed in Fig. 1(a) for the spin omponents
are atually due to spindephasing. The physial
rea-son for suh dephasing is that the LRX, through the
o-diagonaltermsinEq.(2),induespreessionsofthe
holeandeletronspinsabouteahother. The
distribu-tionofsitesyieldsasimilardistributionontheexhange
strengthandonsequentlyonthepreessionfrequeny
resultingin dephasing. In Fig.1(b)atransverse
mag-netieldisappliedB
x
=2T.Besidesthesite-exhange
indueddephasing,nowtheeletronspingoesthrough
Larmorpreessionabouttheeld.
In Fig. 1() we have used dierent orrelation
lengths along the x and y diretions,that is, there is
stilladistributionofinhomogeneityofsitesasinthe
re-sultsofthepreviousgures,butnowelongatedsitesare
morelikely. Inthisase, theLRXassumesa
distribu-tionaboutnonzerovalues,whih setanitefrequeny
for the exhange indued e-h spin preession. This is
seenbythelongperiodosillationonthetimeevolution
oftheeletronandholespinomponents.
Inonlusion,wehavemodeled thespindephasing
indued by the e-h exhange interation for exitons
loalized in sites reatedby imperfetions on theQW
interfaes. Thespindephasingyieldedquiteshortspin
deaytimes30ps whenusingtypialparametersfor
(Zn,Cd)Se/ZnSe QWs, whih shows the relevane of
suhproessinthestudyofspindynamisinthese
sys-tems. InGaAs QWstheroleofspindephasingbythe
e-h exhange for loalized exitons is notso lear
be-auseinthesesystemsexitonsexhibitweaker
loaliza-tionandtheexhangestrengthisonsiderablysmaller
(E
exh
0:05 meV). Eletron-hole spin preession
thatleadstodephasingisthenlesslikelytohappen.
Aknowledgments: This work was supported by
0
20
40
60
80
-1.0
-0.5
0.0
0.5
1.0
σ
e
σ
h
S
pi
n C
om
pom
ent
s
(
a.
u.
)
T im e (ps)
-1.0
-0.5
0.0
0.5
1.0
(c) B =2 T,
l
x
=150Å ,
l
y
=250Å
(b) B =2 T
l
x
=
l
y
=200Å
-1.0
-0.5
0.0
0.5
1.0
(a) B =0 T,
l
x
=
l
y
=200Å
Figure 1. Time evolution of the spinomponents of
ele-tron e
z
; infull lines, and heavy hole h
z
; in dashedlines.
(a)Noappliedeld B=0T,orrelationlengthl =200
Afor
thexandydiretions. (b)SameasbeforebutforB=2T.
() Timeevolutions alulated for B=2 T,l
x =150
A and
ly=250
A,simulatingelongatedsites.
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