Exiton Trapping in a Periodially
Modulated Magneti Field
J.A.K. Freire 1
,V.N. Freire 1
, G.A. Farias 1
, and F.M. Peeters 2
1
Departamento deFsia,UniversidadeFederaldoCeara, Centrode Ci^eniasExatas,
CampusdoPii,CaixaPostal6030,60455-760 Fortaleza,Ceara, Brazil
2
Departement Natuurkunde, UniversiteitAntwerpen(UIA),
Universiteitsplein1,B-2610Antwerp,Belgium
Reeivedon23April,2000
Thebehaviorofexitonsinspatiallymodulatedmagnetieldsisdesribedtakingintoaountthe
exitonspinontribution.Theresultsshowthattheexitontrappinginperiodimagnetieldsis
possibleanddependentonthemodulationprole.
I Introdution
Inviewof reentdevelopmentsin nanotehnology
fab-riation,thestudyofhargedpartilesunderthe
inu-eneofnonhomogeneousmagnetieldshavebeenthe
subjetofagreatnumberoftheoretial[1℄and
exper-imental [2℄ investigations. Among these systems, the
magnetotransportthroughspatially periodimagneti
eldshasreentlyyieldedveryinterestingresults.
Sev-eral works havebeenperformedon thepossibilitiesof
exiton trapping by using nonhomogeneous stress [3℄
andspatiallyvarying eletrields[4℄.
Reently,irularmagnetieldsproleswereused
to indue exiton trapping[5℄. Inthese magneti
sys-tems, thelow-eld gradientregion reatesaminimum
in theeetiveonnement potentialfor theexitons.
Thus, they an be trapped in the magneti eld
in-homogeneity. To the best of our knowledge, the
on-nement of exitons in periodi magneti elds has
notbeen investigated. The purpose of this work is to
presentresultsonthebehaviorofexitonsin spatially
modulatedmagnetields.
II The modulated magneti eld
struture
Onedimensionalperiodimagnetiprolesanbe
re-ated, e.g., by thedeposition of ferromagnetilms or
stripesontopofasemiondutorheterostruture,with
ahomogeneousmagneti eldapplied perpendiularly
to the planeof thestruture [6℄. These lms are
pat-ternedinsuhawaythatthemagnetidomainsonsists
ofparallelstripeswithmagnetizationperpendiularor
paralleltothelm,hangingsignfromonestripetothe
next,givingriseto aperiodiallymodulatedmagneti
eld in the plane of the semiondutor struture (see
sketh inFig. 1(a)).
Assumingamagnetization perpendiularto thexy
plane, with boundariesx ( a=2;a=2),y ( 1;1),
andz ( h=2;h=2),whereaandhisthestripewidth
andthikness,respetively,theorrespondingequation
forthezomponentofthemagnetieldemergingfrom
onestripeanbewritten asfollows[1,6℄:
B
z
(x)=B
a +B
(x;z+h=2) B
(x;z h=2); (1)
whereB
(x;z)=(
0
M=2)[artan(
+
) artan( )℄,
=(x0:5a)=z,Misthestripemagnetization, zis
thedistaneoftheenterofthestripeto thequantum
well,andB
a
theuniformmagnetield. Theresulting
periodi magneti eldis asuperposition ofthose due
to the individual magnetized stripes, in suh a way
that:
B Total
z
(x)= 1
X
N= 1 ( 1)
N
B
z
(x Nl ): (2)
where N is the numberof stripes and l is a onstant
related to the lattie periodiity. The magneti eld
prole for N = 4is shown in Fig. 1 (b)(see
dashed-dottedline).
TheHamiltoniandesribingtheexitonmotionina
nonhomogeneousmagnetieldanbewrittenas[6℄:
H =H ?
(z
i
)+W(r;z
e ;z
h )+H
2D
(R;r)+H mz
(X); (3)
where H ?
(z
i
) = (~ 2
=2m
i )(
2
=z 2
i )+V
i (z
i
) is the
Hamiltonian desribing the eletron and heavy-hole
onnement in the quantum well; W(r;z
e ;z
h ) =
(e 2
=")f=r [r 2
+(z z ) 2
℄ 1=2
dierenebetweenthe2Dand3DCoulombinteration
andis treatedasaperturbation;
H 2D
(R;r)=H CM
(R)+H rel
(r;R;r
R
); (4)
with H CM
(R) = (~ 2 =2M)r 2 R and H rel
(r;R;r
R )= (~ 2 =2)r 2 r (e 2 ="r)+u 1 +u 2
,desribestheexiton
motionin thexyplanesubjetedto theperiodi
mag-netield B
z
(X),where randRaretheexiton
rela-tiveandenter-of-massmotionoordinates,r=r
e r
h
and R=( m e r e +m h r h
)=M,respetively,withthe
to-tal exitonmassM =(m e +m h );u 1 (u 2
)is relatedto
the rst (seond)order dependene of therelative
o-ordinateswith thenonhomogeneousmagnetield [6℄,
and=m
e m
h
=M istheexitonreduedmass.
Figure1. (a)Experimentalsetupshowingthestripe
param-eters: a(h)isthestripewidth(thikness),ditsdistaneto
themiddleofthequantumwell,Mthemagnetization,B
a is
theappliedmagnetield,andListhequantumwellwidth.
(b)Eetivepotentialandrespetivemagnetieldforthe
exiton groundstateasafuntionoftheX oordinate. In
this gure,N =4,a=0:5 m,h=0:2m,d=0:08 m,
Ba=0T,andtheironmagnetizationM=1740emu/m 3
.
Finally,thelast termin equation(3),i.e.,
H mz
(X) =
B g e;z S e;z 1 3 g h;z J h;z B z
(X);(5)
desribes the exiton spin interation with the
non-homogeneous magneti eld. In the above equation,
m
z
= 1;2 is the exiton quantum spin number
whih is related to the eletron (S
e;z
= 1=2) and
heavy-hole(J
h;z
=3=2)spinnumbers,andg
e;z (g
h;z )
istheeletron(heavy-hole) gfator.
Weusedtheadiabatiapproahbeausethemotion
spinmotionarefastasomparedtotheenter-of-mass
motion[6℄. Wealsoassumethat alldisplaementsare
deoupledfromeahother insuhawaythat onean
writethetotalexitonwavefuntionas:
m
z
(R;r;z
e ;z
h
)='(R)(r)F(z
e ;z h )L m z (R); (6)
with'(R) =exp(iQ
Y
Y) (X), where Q
Y
is thewave
vetoroftheenter-of-massmotionin theY diretion
(exitonfreediretion).
Theeigenvaluesoftheonnementinthequantum
well, the exiton relativeand spin motion an be
ob-tainedbyapplying the exitonHamiltonian (Eq. (3))
totheabovewavefuntion [6℄. Fortheexiton
enter-of-mass motion, we obtain the following Shr
odinger-likeequation: d dX ~ 2 2M eff (X) d dX V eff
(X)+E
(X)=0; (7)
with M eff (X)=M= 1 e 2 ~ 2 M 2 2 n r m r 4 B z (X) 2 1 ; (8) V eff
(X)= ~ 2 2M eff (X) Q 2 Y + e 2 2 2 n r mr 2 B z (X) 2 + e~ 2 B z (X)m r 1 2 B g ex;z B z (X): (9)
Theaboveequationinludes thedierent
eigenval-uesfor thefast motionthat ontributeasan eetive
potential and also results in an eetive mass. Here,
nr mr and nr mr
(in units of a B 4 and a B 2 , respetively)
areonstantsrelatedtotherelativeradial(n
r
)and
an-gular(m
r
)quantumnumbers[5℄, =(m e 2 +m h 2 )=M, and g ex = g e;z +g h;z
. The eetive potential as a
funtionoftheX oordinateisshownin Fig. 1(b).
III The exiton trapping energy
Inorderto estimate theexitononnement, wehave
denedtheexitontrappingenergyasthedierenein
energybetweenanexitonistateinthehomogeneous
appliedeldB
a
anditsorrespondingstateinthe
peri-odimagnetield.Asaonsequeneofthisdenition,
the Q
Y
term that is not dependent on the magneti
eld ((~ 2
=2M)Q 2
Y
) is not inluded in the alulus of
E
T
,whihgreatlydereasestheexitondependeneon
thewavevetorQ
Y
. Indeed,the inuene ofthe rst
terminEq. (9) ontheexitononnementissosmall
that it will not be further onsidered. In all our
nu-merialalulations, we haveonsidered a = 0:5 m,
h=0:2m,d=0:08m,andM=1740emu/m 3
[3℄.
Thetrappingenergyoftheexitongroundstateas
a funtion of the applied magneti eld B
a
is shown
in Fig. 2 for several number of stripes. The energies
N 6=1,but theyareverysimilar whenN 2forboth
spinorientation,i.e.,theyareindependentofthestripe
numbers. Theexeptionisthes= 1spinstatease,
whih shows two dierent behaviors when N is even
andwhenitisodd.
Figure2. Trappingenergyoftheexitongroundstateasa
funtionoftheappliedeldBaforthenumberofstripes(a)
N =1,(b)N =2,()N =11and (d)N =20, withspin
m
z
=0(solid),m
z
=+1(dashed),andm
z
= 1(dotted).
Notiethattheexitonisalwaystrappedinthe
pe-riodimagnetipotentialwhenN iseven,independent
of the spin orientation, whih is not true when N is
odd. In this situation, the exiton is not trapped for
s= 1andB
a >0.
Theangular momentum and spin interation with
themagnetieldgivethestrongerontributiontothe
onnementpotential(see Eq. 11). The trapping
en-ergyfortheexitonexitedstates2p ,2s,and2p +
,as
afuntionoftheappliedeldB
a
isshowninFig. 3.
No-tiethatthe2sstateisanevenfuntionoftheapplied
magneti eld, and that the 2p and 2p +
states are
symmetriin B
a
with respetto eah other. This an
beexplainedasfollows: the 2
0
term(seeEq. (9))gives
astrongerontributionasomparedto 1
0
,butevenin
this situation of zeroangular momentum, theexiton
This does not our beause the exiton g fator for
a 80
A well width is very small (g
ex
0:3). Also
duetothesmallgfator,theangularmomentumgives
a bigger ontribution than the spin, and the angular
termis symmetriin B
a
. The2
statesipB
z (X)in
signal,whih explainssuh behavior. Alsonotie that
the trapping energies of the exited states are about
1 order of magnitude larger than those of the ground
state(Fig. 2).
Figure3. Trappingenergyoftheexitonexitedstatesasa
funtionoftheappliedmagnetieldBaforrelative
quan-tumstates(a)2p ,(b)2s,and()2p +
,withspinnumbers
mz =0(solid),mz=+1(dashed)andmz= 1(dotted).
IV Final remarks
The exiton trapping energy in GaAs/Al
0:3 Ga
0:7 As
quantum wells in a periodially modulated magneti
eld whih we have obtained are not within the
a-tualenergy detetionlimitof photoluminesene(PL)
experiments. However, our work undoubtedly shows
that the trapping in periodi magneti strutures
ex-ists,andgivesfurtherinformationonitsharateristis
that ouldguidefutureexperiments. Wewould liketo
suggestthattoinreaseE
T
oneanonnetheexiton
in wide gap semiondutors strutures. In these
sys-tems,theexitonenergyanbehigherthan100times
that oftheorrespondingGaAs situation. Notiethat
an inreaseof 10times should maketheexiton
Aknowledgments
This work was partially supported by CNPq,
CAPES and FINEP. F. M. Peeters was supported by
FWO-VI,IMEC, andIUAP-IV.
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