Localization-Delocalization Transition of Indirect Excitons in Lateral Electrostatic Lattices

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by  M. Remeika

Localization-Delocalization Transition of IndirectExcitons in LateralElectrostatic Lattices
M. Remeika, J.C. Graves, A.T. Hammack, A.D. Meyertholen, M.M. Fogler, and L.V. Butov
Department of Physics, University of California at San Diego, La Jolla, CA 92093-0319
M. Hanson and A.C. Gossard
MaterialsDepartment, Universityof CaliforniaatSantaBarbara, SantaBarbara, California93106-5050
(Dated: January10,2009)
9 WestudytransportofindirectexcitonsinGaAs/AlGaAscoupledquantumwellsinlinearlattices
0 created by laterally modulated gate voltage. The localization-delocalization transition (LDT) for
0 transportacrossthelatticewasobservedwithreducinglatticeamplitudeorincreasingexcitondensity.
2 The exciton interaction energy at the transition is close to the lattice amplitude. These results are
consistent with the model, which attributes the LDT to the interaction-induced percolation of the
n
excitongasthroughthe externalpotential. Wealsodiscussapplicationsofthe latticepotentialsfor
a
J estimatingthestrengthofdisorderandexcitoninteraction.

0
1
Transportofparticlesinperiodicpotentialsisabasic

] problem,whichconcernsavarietyofsystemsextending
l
l from condensed matter systems with electrons in ionic
a
lattices to engineered systems such as photons in pho-
h
- toniccrystalsandcoldatomsinopticallattices. Thepar-
s
ticlelocalizationandLDTareperhapsthemostinterest-
e
m ingtransportphenomena. Particularcasesofthelatter—
themetal-insulatorandsuperfluid-insulatortransitions
.
t —havebeenextensivelystudiedforelectrons,photons,
a
m andcoldatomsinlattices[1,2,3,4,5,6].
Tunabilityofsystemparametershasbeenessentialin
-
d studiesofcoldatomsinopticallattices,allowingtoprobe
n
localization of atoms with increasing lattice amplitude
o
[3, 4, 5, 6]. Here, we study a condensed matter system
c
[ withtunableparameters:excitonsinelectrostaticlattices

created by a gate voltage. In this system parameters
1
of both the lattice, e.g., the lattice amplitude, and the
v
9 particles,e.g.,theexcitondensity,canbecontrolled.
4 Anindirectexcitonincoupledquantumwells(CQW)
3 is a bound state of an electron and a hole in sepa-
1
rate wells (Fig. 1a). Lifetimes of indirect excitons ex-
.
1 ceed that of regular excitons by orders of magnitude FIG.1: (a)EnergybanddiagramoftheCQW.(b,c)Schematic
0
and they can travel over large distances before recom- electrodepattern. TheappliedbasevoltageV realizesthein-
0
9 bination [7, 8, 9, 10, 11, 12]. Also, due to their long direct regime while the voltage modulation ∆V controls the
0
latticeamplitude. Calculatedlatticepotential forindirectex-
: lifetime, these bosonic particles can cool to tempera-
v citons for ∆V = 1V is shown in (c). PL images of indirect
tureswellbelow the quantum degeneracytemperature
i excitonsforlatticeamplitude(g-i)∆V=0and(d-f)∆V=1.2V
X T = 2π~2n/(mgk ) [13]. (In the studied CQW, exci-
dB B forexcitationpowersP = 0.2(d,g),3.7(e,h),and12(f,i)µW.
ar taonndsThav≈et3hKemfoarstshmed=en0s.2it2ymp0e,rspspinindne/ggen=er1a0c1y0cgm=−24),. T=1.6K,λex =633nm,andV0 =3Vforthedata.
dB
Furthermore, indirect excitons in CQW have a dipole
momented,wheredisclosetothedistancebetweenthe
QW centers. This allows imposing external potentials Notethatin-planeelectricfieldF presentnearelectrode
r
E(x,y)=edFz(x,y)∝V(x,y)forexcitonsusingalaterally edges can lead to exciton dissociation [14]. Therefore
modulated gate voltage V(x,y), which creates a trans- theCQWlayersinourstructurewerepositionedcloser
verseelectricfieldFz(x,y)[7,12,14,15,16,17,18,19,20]. tothehomogeneousbottomelectrode. Thisdesignsup-
ThelatticepotentialforindirectexcitonsE(x)wascre- pressesF makingthefield-induceddissociationnegligi-
r
ated by interdigitated gates. Base voltage V = 3V re- ble[17]. Anexampleofthecalculated[17](unscreened)
0
alized the indirect regime where indirect excitons are E(x) is shown in Fig. 1(c). Zero energy corresponds to
lowerinenergythandirectexcitons,whilevoltagemod- zerovoltage,the4meVenergyshiftduetobindingen-
ulation ∆V controlled the lattice amplitude (Fig. 1b,c). ergyoftheindirectexcitonisnotshown. Thepotential
2
modulationisnearlysinusoidalE(x)=E cos2 qx/2 . Its
0
amplitudeisE0 =3meVfor∆V =1Vandscale(cid:0)sline(cid:1)arly
with∆V. Thelatticeperiod2π/q = 2µmisdetermined
bytheelectrodedimensions.
CQW structure was grown by MBE. n+-GaAs layer
with n = 1018cm−3 serves as a homogeneous bottom
Si
electrode. Semitransparent top electrodes were fabri-
cated by magnetron sputtering a 90nm indium tin ox-
ide layer. CQW with 8nm GaAs QWs separated by
a 4nm Al Ga As barrier were positioned 100 nm
0.33 0.67
abovethen+-GaAslayerwithinanundoped1µmthick
Al Ga As layer. Excitons were photogenerated by
0.33 0.67
a 633 nm HeNe or 786 nm Ti:Sapphire laser focused
to a spot ∼ 10µm in diameter in the center of the
150×150µmlattice. Excitondensitywascontrolledby
theexcitationpower. Photoluminescence(PL)imagesof
the excitoncloud werecapturedbyaCCDwith afilter
800± 5nm covering the spectral range of the indirect FIG.2: (Coloronline)(a)Theemissionimageinenergy–xco-
excitons. The diffraction limited spatial resolution was ordinatesforthelatticewith∆V = 1.2V(E0 = 3.7meV). The
imagewasmeasuredatthecenter y = 0oftheexcitoncloud
1.5µm(N.A.=0.28). Thespectraweremeasuredusinga
and integratedover∆y = 1.5µm. Thecorresponding(b) en-
spectrometerwithresolution0.18meV.
ergyand(c)intensityprofiles(black,leftscale). Thesamepro-
Figure 1 shows images of the exciton cloud at zero fileswith subtractedsmoothbackgroundareusedtopresent
(∆V = E = 0) and finite (∆V = 1.2V, E = 3.7meV) themodulationsinenergyandintensity(red,rightmagnified
0 0
scale). The energy minima correspond to intensity maxima.
lattice amplitude for different excitation powers P. At
The PL linewidth does not exceed 2 meV that is characteris-
lowP(Fig.1d,g),thecloudprofileessentiallycoincides
ticofexcitonicemission. Thedottedlineshowstheprofileof
with the laser excitation spot. This indicates that exci-
thelaserexcitationspot. Theshadedareacontains twodeep
tons do not travel beyond the excitation spot, i.e., they intensity minima caused by a defectin the spectrometerslit.
are localized. On the contrary, at high P (Fig. 1e,f,h,i), T=1.6K,P=35µW,λ =633nm,andV =3Vforthedata.
ex 0
theexcitonsspreadbeyondtheexcitationspotindicating
thattheyaredelocalized. TheLDToccursbothwithand
without the lattice. While the cloud is practicallysym-
innerringeffectpersistsinthelattice(Figs. 1e,f,2c).
metricat∆V =0(Fig. 1h,i),atfinite∆Vitiscompressed
The fullwidth athalfmaximum of the exciton cloud
inthelatticedirection(Fig. 1e,f).
in the x-direction is plotted in Fig. 3a,b. Initially, it is
Thelatticepotentialalsocausesperiodicmodulations
practically independent of P but then starts to grow as
of PL characteristics. Figure 2 presents a PL image in
Pincreases. Wedefinetheexcitationpoweratthetran-
energy–x coordinates in the delocalized regime. Both
sition P as the point where the extrapolation of this
LDT
the integrated PL intensity I(x) and the average PL en-
growthtosmallPbecomesequaltothelow-Pconstant.
ergy~ω(x) show small modulation at the lattice period
At the LDT, the exciton cloud starts to spread beyond
superimposedonasmoothlyvaryingprofile(Fig. 2b,c).
theexcitationspotandthecloudextensionchangesfrom
To demonstrate the modulations more clearly, we sub- constanttoincreasingwithP. Figure3ashowsthatthe
tracted the smooth component and plot the remainder transition is smooth. This yields P ≈ 2µW for low
LDT
on a magnified scale in Fig. 2b,c. Minima in energy latticeamplitudesE .1meV. AthigherE theLDTfor
0 0
correspond to the maxima in intensity. We define the
the x-direction shifts to higher excitation powers with
amplitude of energy modulation as the difference be-
increasing lattice amplitude (Fig. 3a,c). The LDT was
tweenadjacentmaximaandminima δω = ω −ω .
max min also observed with reducing lattice amplitude, see the
Figure2showsthat~δωismuchsmallerthanthelattice datafor P = 5µW(Fig. 3b). Note thatexcitons remain
amplitudeE0 = 3.7meVthatisdiscussedbelow. Noin- localizedatlowerP = 0.9µWanddelocalizedathigher
tensity or energymodulation was observed at ∆V = 0. P=21µWforallE inFig. 3b.
0
This indicates that the modulations in question are not
The exciton transport along the y-direction is only
duetothepartiallightabsorptioninthetopelectrodes. weakly effected by the lattice (Fig. 1d-i). The LDT for
The PL intensity has a maximum along a ring in the this direction shifts slightly to lower excitation powers
regimeofdelocalizedexcitons(Figs. 1e,f,h,i,2c). Thisso- withincreasingE (Fig. 3c).
0
calledinnerringwaspreviouslyobservedinPLpatterns The smooth component of the ~ω(x) also exhibits an
ofindirectexcitonswithoutlattices[9]. Itwasexplained interesting behavior. It increases with increasing exci-
in terms of exciton transport and cooling [9, 11]. The ton density, both with increasing P or reducing |x|. Let
3
the middle curve corresponds to the percolation point.
It also explains why P increases as E goes up, see
LDT 0
Fig.3d.
Addingdisorderdoesnotmodifythispicturegreatly
aslongasE remainslargerthanthecharacteristicampli-
0
tudeofE . Otherwise, the percolationisdetermined
rand
by the random potential [21, 22], so that the depen-
denceofP and∆ω onE saturates. Thesaturation
LDT LDT 0
pointgivestheestimateof E . FromFig. 3d,wefind
rand
E ∼ 0.8meV. This number is comparable to the PL
rand
linewidthatlowdensities, suggestingthatthedisorder
isresponsibleforbothoftheseenergyscales.
Tofurtherdevelopthismodel,wemakethefollowing
simplifying assumptions: 1) E ≡ 0, while E(x) can
rand
beconsideredslowlyvarying,2)excitonsreachaquasi-
equilibriumstatewithlocalchemicalpotentialµ(x)and
temperatureT(x), which arealsoslowly varying, 3)ex-
FIG.3: The FWHMof the excitoncloudacrossthe lattice (a) citon interaction is local (dipolar tails, see below, are
vs. the excitationpower Pforlattice amplitudesE = 0, 0.6, neglected). Undertheseassumption,weobtain
0
1.2, 1.9, 2.8, 3.7meV (∆V = 0, 0.2, 0.4, 0.6, 0.9, 1.2V) and (b)
vs. E0 for P = 0.9, 5, 21µW. (c) The excitationpower at the ζ(x)≡µ(x)+E(x)≃const, (2)
transition from the localized to delocalized regime P as a
LDT ∞
feuxncicttoionncloofuEd0a.t(tdh)eTtrhaensinittieornacfrtioomntehneelrogcyaliinzethdetocednetleorcaolfiztehde n(x)≃ gν1dǫ , (3)
(rce,gdi)mpere~s∆eωntLDthTeasdaatafufnorcttihoeneoxfcEit0o.nFtriallnesdp(oorpteanc)rossqsu(aarleosngin) Z0 exph(cid:16)ǫ+ReΣ(ǫ,x)−µ(x)(cid:17)/kBT(x)i−1
thelattice. T=1.6K,λex =786nm,andV0 =3Vforthedata. where ζ is the electrochemical potential, ν1 = m/(2π~2)
is the density of states per spin species, and Σ(ǫ) is the
self-energy(intheuniformstateofthesamen).
∆ω denote the difference between the value of ω at To find the equation for the PL energy shift ~∆ω we
LDT
thelowestPandattheLDTatx=0. Figure3dpresents take advantage of the smallness of Q, the range of in-
thedependenceof~∆ω onE . Weseethatitisfinite
LDT 0
atE =0. AtlargeE weobservearemarkablerelation
0 0
~∆ω ≈E , (1)
LDT 0
whichiscrucialtoourinterpretationofthe mechanism
oftheLDT,seebelow.
Letusnowdiscussasimplemodelthatattributesthe
observedLDTtotheinteraction-inducedpercolationof
exciton gas through the total external potential E (r),
tot
which is the sum of the periodic lattice potential E(x)
and the random potential E (r) due to disorder. The
rand
latter is an intrinsic feature of solid state materials. It
formsmainlyduetoQWwidthandalloyfluctuationsin
thestructure.
The ideaisillustrated inFig. 4for the caseof no dis-
order, E (r) ≡ 0. If the local exciton density n(x) is
rand
low, it is concentrated in the minima of the potential
FIG.4: (a)Excitondensityfor(toptobottom)ζ = 5,3.7, and
E(x). The crests are nearly depleted. As a result, the
2.5meV. Thefirstofthesecorrespondstodataatx∼10µmin
exciton transport from one period of the lattice to the
Fig.2. k T=0.15meV,E =3.7meV,andγ=2.3forallcurves.
B 0
next through thermal activation or quantum tunneling (b)LatticepotentialE(x)andthePLenergyshift~∆ω(x)forthe
isexponentiallyslow. Astheaveragedensityincreases samesetofparameters. Inset: Modulationδω = ω −ω
max min
and reaches a certain threshold — “percolation point” of the PL energy as a function of the interaction strength γ.
—the crestsbecome populated, which permitsa faster Theexperimentalδωcorrespondstoγ ≈ 2.3. The valueof γ
excitontransport,i.e.,theobserveddelocalization. This predictedbythe“capacitor”formula[25,26,27,29]isindicated
bythearrow.
scenario naturally leads to Eq. (1), see Fig. 4b, where
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