# Two-dimensional electrostatic lattices for indirect excitons

**ABSTRACT** We report on a method for the realization of two-dimensional electrostatic lattices for excitons using patterned interdigitated electrodes. Lattice structure is set by the electrode pattern and depth of the lattice potential is controlled by applied voltages. We demonstrate square, hexagonal, and honeycomb lattices created by this method.

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Two-Dimensional Electrostatic Lattices for Indirect Excitons

M. Remeika, 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

Materials Department, University of California at Santa Barbara, Santa Barbara, California 93106-5050

(Dated: October 3, 2011)

We report on a method for the realization of two-dimensional electrostatic lattices for excitons using

patterned interdigitated electrodes. Lattice structure is set by the electrode pattern and depth of the

latticepotentialiscontrolledbyappliedvoltages. Wedemonstratesquare, hexagonal, andhoneycomb

lattices created by this method.

Studies of particles in periodic potentials are funda-

mental to condensed matter physics. While originally

experimental studies concerned electrons in crystal lat-

tices,avarietyofsystemswithparticlesinartificiallattice

potentials are actively investigated at present. Control-

ling the parameters of an artificial lattice provides a tool

for studying the properties of particles confined to the

latticeand,tosomeextent,foremulatingcondensedmat-

ter systems. Cold atoms in an optical lattice present a

prominent example of particles in artificial lattices. Phe-

nomena originally considered in context of condensed

matter systems, such as the Mott insulator – superfluid

transition, can be studied in the system of cold atoms in

optical lattices [1].

Excitons in artificial lattices present a condensed mat-

ter system of particles in periodic potentials [2–10]. In

particular, artificial periodic potentials, both static and

moving, can be created for indirect excitons [2, 3, 6, 10].

An indirect exciton in coupled quantum wells (CQW) is

a bound state of an electron and a hole in separate QWs

(Fig. 1a). Due to a dipole moment of indirect excitons

ed (d is close to the distance between the QW centers),

potential landscapes for excitons E(x, y) = edFz(x, y) ∝

V(x, y) can be created using a laterally modulated gate

voltage V(x, y) (Fz is the z-component of electric field

in the CQW layers) [2, 3, 6, 10–18]. Furthermore, due

to their long lifetimes, orders of magnitude longer than

that of regular excitons, indirect excitons can travel in

electrostatically created potentials over large distances

before recombination [6, 10, 11, 13, 15–18]. Also, due

to their long lifetimes, these bosonic particles can cool

to temperatures well below the temperature of quantum

degeneracy [19]. Therefore, the system of indirect exci-

tonsinelectrostaticlatticesgivesanopportunitytostudy

transport of cold bosons in periodic potentials.

Linear lattice potentials with energy modulation in

one dimension were created for indirect excitons by in-

terdigitated gates [2, 6, 10]. However, a number of phe-

nomena,includingtheMottinsulator–superfluidtransi-

tion, require in-plane energy modulation in both dimen-

sions[1]. Atwo-dimensional(2D)latticeforexcitonscan

be generated by a single electrode with a periodic array

of holes [3]. The lateral modulation of Fz, which deter-

mines the lattice depth, can be controlled by changing

the voltage applied to the electrode. However, within

this method, changing the lattice amplitude is accompa-

nied by changing the average electric field Favg

turn, lifetime and density of indirect excitons. An in-

dependent control Favg

z

and the lateral modulation of Fz

can be realized using multiple electrodes separated by

insulating layer(s) [20]. Superposition of the fields from

such electrodes can create the desired pattern of Fz(x, y).

However, within this method, the semiconductor struc-

ture is in series with the deposited insulator layer(s).

Therefore, considerable fraction of the applied voltage,

determined by the ratio between the conductance of the

insulator and semiconductor layers, can drop in the de-

posited insulator. Also, this fraction and, in turn, Fz(x, y)

may depend on the optical excitation.

Here, we present a method for creating 2D electro-

static lattices for indirect excitons that explores the op-

portunity to control exciton energy by electrode density

[21]. We demonstrate that 2D lattices for excitons can be

produced by patterned interdigitated gates. The lattice

constantandlatticestructurearedeterminedbytheelec-

trode pattern. Figures 1c, d, and e show the electrode

patterns for creating square, triangular, and honeycomb

lattices, respectively. The corresponding simulated exci-

tonpotentialprofilesareshowninFig.1f-k. Theaverage

field Favg

z

and spatial modulation of Fzcan be indepen-

dentlycontrolledbyvoltagesV0and∆V. Favg

indirect exciton regime and controls the exciton lifetime.

ModulationofFzformsthelatticepotential(Fig.1b). The

latticeamplitudecanbecontrolledinsituby∆V. Thein-

planeelectricfieldinthelatticeFxyissmallsothatitdoes

not cause the exciton dissociation: eFxyaB? Eex, aB∼ 20

nm and Eex ∼ 4 meV are the Bohr radius and bind-

ing energy for the indirect excitons, respectively [22, 23]

(Fig. 1b).

Advantages of this method include: (i) A variety of

2D lattice structures for excitons can be realized; (ii) The

depth of the lattice potential can be controlled in situ by

voltage; (iii) The average field can be controlled by volt-

age independently from lattice depth; (iv) Smooth 2D

z

and, in

z

realizesthe

arXiv:1109.6659v1 [cond-mat.mes-hall] 29 Sep 2011

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FIG. 1: (color online) (a) Energy diagram of the CQW. (b) Sim-

ulated electric field Fzand exciton energy edFzalong x (black)

and y (red) for square lattice. Lower plot shows lateral elec-

tric field Fr = (F2

schematics for square, triangular, and honeycomb lattices, re-

spectively. (f-k) Simulated exciton energy for these electrode

patterns. ∆V = 1V (see Supplementary information). (l-n)

SEM images of the electrode patterns.

x+ F2

y)1/2and eFraB. (c), (d), (e) Electrode

lattice potentials are realized by the electrode patterns;

(v) The lattice device can be fabricated using single layer

lithography with no deposited insulator layer.

We demonstrate experimental proof of principle for

creating2Dlatticesforexcitonsbythismethod. Asquare

latticepotential(Fig. 1i)wasusedforthedemonstration.

CQW structure was grown by MBE. n+-GaAs layer with

nSi = 1018cm−3serves as a homogeneous bottom elec-

trode. Semitransparent top patterned electrodes were

fabricated by evaporating 2nm Ti and 7nm Pt. CQW

with 8nm GaAs QWs separated by a 4nm Al0.33Ga0.67As

barrier were positioned 100 nm above the n+-GaAs layer

within an undoped 1µm thick Al0.33Ga0.67As layer. Exci-

tonswerephotogeneratedbyaTi:Sapphirelasertunedto

the energy of direct excitons in this sample (≈ 786 nm).

Exciton density was controlled by the laser excitation

power Pex. Photoluminescence (PL) images of the exci-

ton cloud were captured by a CCD with a filter selecting

FIG. 2: (color online) (a) PL intensity and (b) energy of excitons

inasquarelatticealongx(black). Thesamedatawithasmooth

curvesubtracted(red). (c,d)Similardataalong y. Dashedlines

are guides to the eye. Laser spot FWHM is 17µm along x and

14µm along y. Pex= 40µW, El= 4.2 meV. (e, f, g) Exciton PL

images in a square lattice for Pex= 11µW at El= 0, 2.1, and 4.2

meV respectively. (g) FWHM of exciton cloud emission along

x for El= 0 (black) and 4.2 meV (red) vs Pex.

photonwavelengthsλ = 800±5nmcoveringthespectral

range of the indirect excitons. The spectra were mea-

sured using a spectrometer with resolution 0.18meV.

Experiments were done at Tbath= 1.6K.

Figure 2a-d shows the emission profiles for excitons

in a square lattice along x or y. Each point in the x-

profiles was obtained by averaging over 5 lattice sites

along y and vice versa to reduce the noise in the data.

Another source of averaging is finite optical resolution,

see below (note that averaging reduced the amplitude

of the spatial modulations discussed below). The quan-

tity ?ω in Fig. 2b and d stands for the spectral average

?ω = M1/I, where I =

Iω?dω?is the total PL intensity

andM1=

Iω??ω?dω?isitsfirstspectralmoment. Asone

can see in Fig. 2, both I and ?ω are modulated with the

period matching the lattice constant revealing the exci-

ton confinement in the 2D lattice. The intensity maxima

match the energy minima demonstrating exciton collec-

tion in the lattice sites.

We also probed exciton transport in the lattice. Figure

2e-gshowsspatialimagesofexcitonPLatthreedifferent

lattice depths. As the lattice depth is turned up the ex-

citon cloud width becomes smaller and locations of the

lattice sites become apparent in the PL image. Figure 2h

shows the full width at half maximum (FWHM) of the

exciton cloud PL as a function of Pexfor lattice depths

El= 0and4.2meV.Atlowexcitondensities,theemission

?

?

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FIG. 3: (color online) Simulations for a square lattice. (a) Ex-

citon density. Emission (b) intensity and (c) energy with av-

eraging in y and accounting for spatial resolution as in the

experiment. El= 4.2 meV, γ = 7, T = 4 K, NA = 0.245.

spotisessentiallyequalinsizetothelaserexcitationspot

indicating that excitons are localized and do not travel

outside the laser excitation spot. At high exciton densi-

ties, the emission spot is larger than the laser excitation

spot indicating that excitons are delocalized and travel

outsidethelaserexcitationspot(Fig. 2h). Insimilarityto

the localization-delocalization transition studied in lin-

ear lattices [6], this behavior corresponds to the exciton

localization in the combined lattice potential and disor-

der potential at low densities and exciton delocalization

duetoscreeningofthepotentialbytherepulsivelyinter-

acting excitons (the amplitude of the disorder potential

inthesampleis∼ 0.6meV).Figure2hshowsthatahigher

exciton density and, in turn, higher interaction energy is

required for screening the potential with a higher lattice

amplitude, in agreement with this model.

In order to examine this agreement quantitatively, we

considered a mean-field model [6] where the local den-

sity n(r) of bright excitons is the solution of the equation

?

Here ν1= m/(2π?2) is the density of states per spin, γ is

the dimensionless interaction constant, ζ is the exciton

electrochemical potential, and T is the exciton tempera-

ture. Within this model, the first moment of the exciton

emission energy proves to be

ε(n) ≡ Tln1 − en/2ν1T?

= E(r) +

γ

ν1

n − ζ.

(1)

M1= (ζ − ε)n + 2ν1T2Li2

where Li2(z) is the dilogarithm function. From these two

quantitiesthelocalPLintensity andenergycanbecalcu-

lated (see above). For a more accurate comparison with

the experiment, we also included the effect of the finite

spatial resolution of our optical system. The choice of

the fitting parameters in the model δ ∼ 1µm for the de-

focussing parameter (see Supplementary information),

T = 3.6K, ζ = 5.0meV, and γ = 2.3 leads to a reason-

able agreement between the simulations (Fig. 3) and the

experiment (Fig. 2). However, this fitting should not

be overemphasized because of a number of the fitting

parameters and approximations made in the model.

?

eε/T?

,

(2)

In conclusion, we present a method for producing 2D

lattices for indirect excitons and experimental proof of

principle for this method. This work was supported by

the DOE Office of Basic Energy Sciences under award

DE-FG02-07ER46449. MF is supported by the UCOP.

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SUPPLEMENTARY INFORMATION

Electrostatic simulations

The electrostatic potential φ(r) in the system in the

absence of excitons was calculated numerically using

COMSOL Multiphysics 4.0 software package. The sys-

tem was modeled as a rectangular box 1µm thick in the

z direction and five or more lattice periods wide in the

x and y directions with the electrode pattern embedded

into the top surface of the box. The potential was cal-

culated by solving the Laplace equation in the volume

of the box. At the electrode surfaces the boundary con-

dition of constant potential was imposed, e.g., at the

ground plane (bottom surface) we have φ = 0. At all the

othersurfacesofthesimulationboxtheconditionofvan-

ishing electric displacement, D⊥= 0, was chosen. The

z-component of the electric field at 100nm from the bot-

tomplane(correspondingtothelocationofthequantum

wells) was used to calculate exciton energy E = edFz.

Optical resolution effects

The spatial resolution of the optical system is de-

scribed by its point spread function (PSF) P(r). We used

the following common model [1] for the PSF

P(r) =

?????

?

d2qΘ(Q − |q|)eiqr−iδ2q2/2

?????

2

.

(3)

This PSF has a finite width determined by the length

scale Q−1≡ λ/(2πNA) = 0.46µm set by the numerical

aperture NA of the system and by another length scale

δ ∼ 1µm that describes defocussing. The observable

intensity I(r) and its first spectral moment M1(r) were

calculated by taking the convolution of the PSF and the

“ideal” I and M1 derived from the mean-field theory

described in the main text.

[1] H.H.Hopkins, Proc.Roy.Soc.LondonSer.A231, 91(1955).

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