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.
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-
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
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 .
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 . Therefore, the system of indirect exci-
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-
tion, require in-plane energy modulation in both dimen-
be generated by a single electrode with a periodic array
of holes . 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
and the lateral modulation of Fz
can be realized using multiple electrodes separated by
insulating layer(s) . 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
. We demonstrate that 2D lattices for excitons can be
produced by patterned interdigitated gates. The lattice
trode pattern. Figures 1c, d, and e show the electrode
patterns for creating square, triangular, and honeycomb
lattices, respectively. The corresponding simulated exci-
and spatial modulation of Fzcan be indepen-
indirect exciton regime and controls the exciton lifetime.
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]
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
arXiv:1109.6659v1 [cond-mat.mes-hall] 29 Sep 2011
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.
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
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-
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
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
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
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
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.
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 , this behavior corresponds to the exciton
localization in the combined lattice potential and disor-
der potential at low densities and exciton delocalization
acting excitons (the amplitude of the disorder potential
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  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) +
n − ζ.
M1= (ζ − ε)n + 2ν1T2Li2
where Li2(z) is the dilogarithm function. From these two
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.
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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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
ishing electric displacement, D⊥= 0, was chosen. The
z-component of the electric field at 100nm from the bot-
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  for the PSF
d2qΘ(Q − |q|)eiqr−iδ2q2/2
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.
 H.H.Hopkins, Proc.Roy.Soc.LondonSer.A231, 91(1955).