PHYSICAL REVIEW B 84, 245318 (2011)

Selective confinement of macroscopic long-lifetime exciton and trion populations
M. D. Fraser, H. H. Tan, and C. Jagadish
Department of Electronic Materials Engineering, Research School of Physics and Engineering, The Australian National University,
Canberra, ACT 0200, Australia(Received 14 May 2011; revised manuscript received 26 October 2011; published 27 December 2011)
Confinement of cold macroscopic particle ensembles opens avenues to study many-body coherent states ofmatter in a controlled environment. We create pure macroscopic cold long-lifetime trion and exciton gases andarbitrary mixtures spatially confined with fabricated electrostatic potentials on GaAs/AlGaAs coupled quantum
well structures. The population densities are readily controlled by electric field and laser power. In the exciton
phase, we observe a strong contraction of the exciton mode area at high densities and small traps, indicating theonset of exciton quantum degeneracy, linking and establishing an experimental system in which to study potential
macroscopic quantum phases of cold trion gases.
DOI: 10.1103/PhysRevB.84.245318

PACS number(s): 71.35.Pq, 71.35.Lk, 67.85.Pq

Excitons1 are the most suitable bosonic quasiparticle for
conducting atomic-like experiments in a solid-state environment, having properties analogous to a low-mass finitelifetime hydrogen atom. Charged excitons, known as trions,2
are also stable excited states of semiconductors amenable
to manipulation.3 Macroscopic quantum states of bosonicatoms are created as Bose-Einstein condensates (BECs),4
while charged fermionic electron gases exhibit phases, suchas a charged Fermi liquid,5 or a macroscopic coherence via
Cooper pairing in the superconductivity phase.6 There exists
a continuous transition between the BEC and the Cooper pairmediated Bardeen-Cooper-Schrieffer (BCS) phases, known
as the BEC-BCS crossover.7,8 We present a novel system
where interconversion between bosonic exciton and charged
fermionic trion populations is readily achieved and controlled
in an environment where an exciton BEC-like state is
expected,4,9 opening the new possibility of the creation ofmacroscopic quantum states of trions.
We consider the confinement and manipulation of cold,
long-lifetime populations of both exciton and trion gases inthe same structure, where the relative population of eachspecies can be controlled by an applied electrical field andoptical excitation. The long-lifetime stable exciton gas isachieved with two-dimensional spatially indirect excitons XI
in a coupled quantum well (CQW) structure,10 which has long
been considered a strong candidate for a BEC-like state,11–13
though challenging notably due to disorder and low transition
temperatures.14 The prospects for condensation are promising,
given recent reports of exciton quantum degeneracy,15–17
an important precursor to observing exciton condensationand of long coherence lengths.18 Although our experiments
are conducted in the steady state, the exciton lifetime isconsiderably greater than the trion formation time, offering
the unique possibility to study dynamics of trapped mixturesof Bose and charged Fermi particles. We further establish theexperimental possibility of quantum phases of a trapped trion
gas by demonstrating signatures of quantum degeneracy in the
exciton phase in such a trap.
We study two molecular-beam-epitaxy (MBE) grown
GaAs/Al0.3 Ga0.7 As CQW samples (Sn and Sp ) with wellwidths of tw = 8 nm (Sn ) and tw = 12 nm (Sp ) and tunnel
barrier thickness tb = 4 nm. The CQW of Sn is symmetrically embedded in a n+ -i-n+ structure (ne ∼ 1018 cm−3 )1098-0121/2011/84(24)/245318(5)with intrinsic thickness ti = 320 nm, while in Sp , the CQW
is asymmetrically situated 50 nm from the substrate in ap+ -i-n+ structure (nh ∼ 1019 cm−3 ) with intrinsic thickness
ti = 328 nm. The key difference is the use of n+ -i-n+ andp+ -i-n+ heterostructures which limit sample current andbackground carrier concentrations. As such, Sn is used for
exciton-trion mixture experiments and Sp for studying dense
exciton phases, although excitons and trions can be observedin both structures. Applying a perpendicular electric field F⊥
to this sample breaks the z symmetry of the wells, separating
the electrons into one well, the holes into the other.10,14,19
The resulting neutral indirect excitons XI0 have an interwell
recombination transition [Fig. 1(b)] in contrast to the intrawell0transition of direct excitons XD[Fig. 1(a)]. Due to the effective
band tilting of the quantum confined Stark shift, the XI0
luminescence is linearly red shifted in response to F⊥ , andenhanced significantly over that of a single QW, putting XI0 in
a high-field seeking state.
Spatial variations in this applied electric field F⊥ are used
to create potential minima and localize the excitons laterally.
The n+ /p+ cap layer is patterned into a circular shape tospatially modify F⊥ within the CQW layer. Electrostatic
trapping of XI0 has previously been considered;20–23 however
we additionally add a thick (∼250 nm) SiO2 layer to thesurface of the regions outside the trapping area, which due to
its low dielectric permittivity (in contrast with Alx Ga1−x As),
effectively screens the electric field from these regions,allowing electrical contacts to the trapping regions without theassociated trap losses. Further, indium-tin-oxide (ITO) is used
as a transparent surface electrical contact (with good Ohmic
contacts to n+ /p+ -GaAs, and transparency >95%), depositeduniformly to the entire sample. The design advantages include
low exciton losses from in-plane field ionization,24 and small
electric fields relative to the ionization field are able to generate
deep trapping potentials, with almost arbitrarily complex timeand space dependent profiles. The structure transparency also
allows for largely distortion-free imaging of the spatial excitonand trion luminescence profiles, an important requirement forstudying spatial properties of the exciton and trion gases.
Trions are normally observed only in tightly confinedquantum dots25 as microscopic populations, or as unconfinedgases in modulation-doped quantum wells26 or resonant tunneling diodes.27 In this trap structure, we observe the efficient

245318-1©2011 American Physical Society

M. D. FRASER, H. H. TAN, AND C. JAGADISHPHYSICAL REVIEW B 84, 245318 (2011)(a) F⊥ = 31.4

40.0

48.6

57.1

65.7

0
−4
PL Energy (meV)formation of a controlled macroscopic trion population, expected to have long radiative lifetimes and be thermalized close
to the lattice temperature. Charged species (trions and free
carriers) also feel the spatially localized electric field F (x,y)(within the CQW plane), which attracts one charge while
repelling the other depending on the gate voltage VG polarity.
The simultaneous trapping of excitons and free carriers (ifinjected in sufficient densities) enhances the formation rate andstability of exciton-trions. To create trions our structure relies
on charged carrier injection by photoexcitation (above theAl0.3 Ga0.7 As barrier)28 and background free carriers/leakage
current. The heavier mass of holes leads to more efficient
capture than electrons resulting in a spatial region within thelaser excitation profile containing a steady-state population of
excitons and heavy holes. We observe a strong asymmetry in
the prevalence of trion formation with VG polarity and thus
assign the consistently easier direction to heavy-hole trions
XI+ [Fig. 1(c)].
The typical cross-sectional electric field amplitude profile
F (x,z) is shown in Fig. 1(d), with dielectric, semiconductor,
and conducting layers indicated. The electric field is localized
within the semiconductor region below the trapping contact,
though the entire sample surface is at a fixed potential. Both XI0
and XI+ are observed [Figs. 1(f) and 1(g)], with steady-state
relative populations controlled through pumping power Pl ,electric field F⊥ , or trap size. The XI0 and XI+ populations
are studied using an imaging microphotoluminescence (μPL)
setup with samples immersed in a liquid He cryostat at T =1.4 K. Laser illumination (λ = 633 nm HeNe CW) and PL

74.3kV /cm

XD0

−8
−12
−16
−20

XI0

−24
−28

XI+

−32
−36 Δx = 50µm(b)Pl = 50µW

100µW

250µW

500µW

0

1000µW

2500µW

XD0−4
PL Energy (meV)

FIG. 1. (Color online) Schematic CQW band diagrams for the (a)0, (b) XI0 , and (c) XI+ under differing bias/excitation conditions. (d)XD
Simulated electric field cross-sectional (through y axis) profile showsF(x,z) is well confined to the trap region despite being uniformly
applied to the sample surface. Experimental measurements of (e)0and XI0 ) showing long-range diffusion ofuntrapped excitons (XD
0
XI , (f) selective trapping of XI0 within the linear regime, and (g)simultaneous trapping of XI0 and XI+ within the nonlinear regime.

collection are both via the same aspheric objective lens (f =4 mm, numerical aperture = 0.6), also immersed in the liquid
He, where under superfluid He conditions clear images result.
The PL image is projected onto the entrance slit of a 750 mm
imaging spectrometer, measuring the spectral distribution ofa one-dimensional (1D) spatial trap cross section yielding theeffective trap depth and exciton distribution.
At a sample position without trapping potentials, weobserve cold long-lifetime excitons with radially diffusive
motion exhibiting XI0 diffusion lengths far exceeding the laser
diameter [Fig. 1(e)]. At excitation powers below the Mott
transition, the XI experience a mean-field energy blue shift
with increasing density.29 The rapid out-diffusion of excitons
creates a radially varying density profile, evidenced by achevron-shaped E − x profile,24 indicating a larger density
and mean-field interaction energy at the laser excitation center.
The mean-field blue shift of the overall cloud is small becauseof out-diffusion and lack of population buildup, though thischanges dramatically in trapped measurements.
In patterned regions of the sample at VG > 0, the XI0 cloudis completely confined to the trap area [Fig. 1(f)], while0(the laser is
excitation outside the trap results in only XD
larger than the trap in this measurement). Under experimental
conditions of larger laser powers, smaller trap sizes, or larger
voltages, a XI+ population is observed (a 2–3 meV lower
energy peak to the XI0 population) within the trap [Fig. 1(g)].
The XI0 trapped cloud has a higher density at the trap center, and
the XI+ population is usually slightly more centrally localized,
given lower free carrier populations at the trap edges. No clear+
XD
populations (i.e., untrapped trions) are observed.
Like XI0 in untrapped samples, the trapped XI0 and XI+
clouds also Stark shifts to lower energy with increasingF⊥ . Figure 2(a) shows a sequence of measurements of a

−8
−12

XI0

−16
−20

XI+

−24
−28 Δx = 50µm

FIG. 2. (Color online) Experimental measurements of a spectrally
resolved 50 μm-diameter trap cross section showing trapped XI0 and
XI+ formation as a function of (a) increasing applied electric field F⊥
and (b) laser excitation power Pl .

245318-2

SELECTIVE CONFINEMENT OF MACROSCOPIC LONG- . . .

PHYSICAL REVIEW B 84, 245318 (2011)

(c)
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0

XD0

XI0
XI0
XI+
no trap
no trap

X
X

0

20

40

60

80

1558
PL Energy (meV )

XD+

X

(b)
1560

XD0 no trap

100

1556
1554
1552

X exp.

1546

X exp.

102

Electric field F⊥ (kV · cm )(d)9
3.0 ×102.5

ΔE modelΔE exp.

0

20

40

60

80

100

Electric field F⊥ (kV · cm−1)

X model

1548

−1

2.0

103

104

103

104

Laser power (µW )

h
X
X

1.5
1.0
0.5
0.0

rate equation model30 where, under CW pumping and staticbias, all populations are in a steady-state chemical equilibrium+
XI0 + h+ 
 XI . This model contains two variables for eachstate i, namely, the energy Ei and density ni , which are afunction of pumping intensity Pl and electric field F. The
direct exciton energy EXD0 is taken from experiment. A density
dependent blue shift is observed to control the XI0 energy of theform EXMF (nXI0 ) = 4π e2 dnXI0 / derived from dipole-dipole
mean-field interaction contributions,29 with dipole strengthd ≈ 20 nm. The density of trions at a particular F and Pl
resulting from a population of excitons nX0 and heavy holes
nhh is then given by
2π h̄2nhh (F,Pl ) nX0 (F,Pl )kB T
mX0 + mhh −E/kB T
e
,
(1)
×
mX0 mhhwhere mi and ni are the masses and densities, respectively,of species i ∈ {X0 ,hh}.30 The XI+ density thus depends
notably on the injection of excitons and holes and on the
XI0 and XI+ energy splitting (E = EX+ − EX0 ). nh (F,Pl )is difficult to estimate as it is sourced by contributions frombackground doping, optical injection, and lateral charge imbalance/electrical injection through n+ -i-n+ junction leakage.
However, assuming that for a given F and Pl , the total exciton
population (nX = nXI0 + nXI+ ) is an exponential function of F0
→XI0 transition and logarithmic in Pl , specifabove the XD
ically nX (F⊥ ) ∝ exp (C2 (F⊥ − FXD0 →XI0 )) + log10 (Pl ) (where
C2 is a fitting parameter), nX+ (F,Pl ) and EX+ are successfully
predicted for the parameter space investigated experimentally.
E [Fig. 3(c)] is a complex combination of the trion
X+binding energy (EB I ≈ −2.5 meV),31 exciton interactions,Stark shifts, and binding energy screening effects, estimated
by


X+
EX+ = EXD0 − C1a F⊥ − FXD0 →XI0 + EB I+ EMF nXI0 ,nXI+ + C1b F⊥ − FXI0 →XI+ . (2)nX+ (F,Pl ) =X model1550

Density (cm−2)

(a)
1580
1570
1560
1550
1540
1530
1520
1510
1500
1490

Exciton-Trion Splitting ΔE (meV )PL Energy (meV )50 μm-diameter trap as a function of F⊥ , where the transition
0
from XD
to XI0 (FXD0 →XI0 ≈ 25 kV/cm) and the emergence+of a XI (FXI0 →XI+  50 kV/cm) peak are observed. Unlike
in untrapped samples however, where the Stark shift isapproximately linear, the trapped samples display a roughly
piecewise nonlinear energy evolution. At fields just above the0
XD→XI0 transition, the XI0 Stark shift is roughly linear thoughshallower than in untrapped samples due to the confinementinduced density enhancement. However, the appearance of a
XI+ population corresponds to a simultaneous reduction in the
Stark-shift slope. At large voltages, ionization of both XI0 and
XI+ occurs with rapid red shifts and population loss.
Figure 2(b) shows the evolution of the trapped population at
a fixed voltage in a 50 μm trap as Pl is increased, exhibiting a
rapid and complete macroscopic trion population appearance.
At intermediate Pl the XI0 peak completely transforms toa XI+ peak, red shifted by E ≈ 2.5 meV. Whereas in theF⊥ -controlled transition, the trap depth slowly enhances theexciton and free-carrier population, the increase of Pl induces
a much more dramatic increase. In modulation-doped QW experiments, only a partial population transformation is usuallyobserved; however, in these exciton-trion traps, we observe
dynamical conversion of 100% of the XI0 population to XI+ .
The energy dependence of the XI0 and XI+ peaks as a function of F⊥ and Pl [Figs. 3(a) and 3(b)] can be used to ascertain
the approximate densities of the various particles to model
the mechanisms by which populations are controlled. Basedon the observation of comparable luminescence intensities,
particularly across the XI0 -XI+ transition [Fig. 2(b)], we assume
the XI+ and XI0 gases have similar radiative lifetimes andpopulation densities (given the complete population transfer
in some measurements). The system of trapped XI and a freecarrier mixture can then be understood by considering a simple

102

Laser power (µW )FIG. 3. (Color online) For a 50 μm-diameter trap, (a) and (b)show the energy evolution of the trapped XI0 and XI+ populations(at the trap center) with F⊥ and Pl following from Figs. 2(a) and2(b), respectively, and fitted with theoretical data from a chemicalrate equation model, (c) shows XI0 -XI+ splitting and (d) shows theabsolute XI0 and XI+ populations estimated from experiments and fit
with the numerical model.

The trion energy will shift with the electric field at adifferent rate to neutral excitons (C1a ) because of its charge(with rate C1b ), which occurs after a turn-on field FXI0 →XI+
corresponding to the point where the population of holes
becomes large enough (due to either trapping or pumping
buildup mechanisms) to facilitate a chemical equilibrium oftrions. Approximate Coulomb interactions between trions and√holes ECMF (nXI+ ) = (e2 /4π ) nXI+ are also included when thetrion population becomes significant. The experimental data
is closely fit with this theory, though the ionization region of
E is neglected.
This model accurately describes the energy evolution, trion
splitting, and the XI0 → XI+ transition where all data for asingle trap size are fit with a single set of parameters. Excitonand trion densities [Fig. 3(d)] estimated from this modeland from experimental mean-field shifts also agree, falling
comfortably below the ionization density nion ∼ 1011 cm−2 .
We find, in general, that smaller trapping diameters enhancethe observed effects, namely, faster trap-induced populationbuildup of both XI0 and XI+ with VG and Pl . Finer control over245318-3

M. D. FRASER, H. H. TAN, AND C. JAGADISHPHYSICAL REVIEW B 84, 245318 (2011)

FIG. 4. (Color online) (a) Evolution of the width of a spatiallytrapped XI0 profile (with respect to trap diameter) for trap diameters of
100 and 25 μm, which show contrasting population buildup behavior.
The dashed lines are fit to the weighted average. (b) Observation of
XI+ populations in a 25 μm trap at elevated voltages and laser power
Pl = 2.5 mW in sample Sp . The trapped XI0 linewidths are taken onlyin the exciton phase.

integrated) spatial profile full width at half maximum (FWHM)
evolution with pumping power is plotted in Fig. 4(a). The
many points at each laser power represent different voltages,
covering the effect of trap-depth induced population buildup.
With increased density, interaction energies broaden thespatial profile of a repulsively interacting trapped gas, andthe corresponding monotonic increase in spatial width isobserved in the 100 μm trap. The 25 μm trap is markedly
different, showing first expansion, followed by a contraction
and subsequent re-expansion of the exciton cloud diameter, thechanges being of order 15%, indicating that in small Stark traps
at a temperature of 1.2 K and densities below the ionization
limit, a quantum degenerate state of excitons is reachedThe lower current and free carrier population in sample Sp
allowed for the study of gases largely consisting of XI , andthe measurements in Fig. 4(a) were collected in this phase.
However, even in this sample, when both the laser power andthe applied voltage are large enough to generate sufficient
free carriers, conversion between XI0 and XI+ can also be
observed. Figure 4(b) shows the (voltage-induced) evolutionof a XI+ -peak from the XI0 -peak when excited at Pl = 2.5 mW
and at voltages elevated beyond that of the pure excitongas. The voltage-induced conversion between populations isgradual, as in sample Sn ; however to better study the natureof the transition between a degenerate XI0 gas and a XI+ gas, asample with simultaneously large achievable XI0 density and alarge window of XI+ occupation is advantageous. This might
be more readily achieved using a fine-tuned combination ofresonant and off-resonant excitations and/or light doping in asimilar structure to sample Sp .
Macroscopic quantum phases of trions have not beenreadily discussed in the literature. A charged Fermi liquid
might be expected (provided sufficient binding energy), thougha mechanism (to parallel Cooper pairing) for driving coherencein trion gases is not yet established, possibly due to thelack of an experimental system. Here we have demonstrated
the selective confinement of cold long-lifetime trions andexcitons in arbitrary mixtures, wherein the exciton phaseexhibits signatures of quantum degeneracy. Not only is a cold
pure trion gas observed, but the long lifetime is expected topermit dynamic exciton-trion conversion, thus opening the new
possibility to explore potential macroscopic quantum states ina trion gas, and novel bosonic-charged fermionic crossoverphysics in a controlled environment.32

the relative populations could be achieved with a two-color
excitation experiment,26 using both an above band laser (for
XI0 and XI+ ) and a XD -resonant laser (only XI0 ).
To establish an experimental link between the cold trapped
trion gas and a potential macroscopic quantum phase of
excitons, we present evidence for the onset of quantum
degeneracy in the trapped exciton phase. In the n+ -i-n+ sample
Sn , the trion phase is usually observed prior to sufficient
exciton density accumulation. However, in the p+ -i-n+ sample
Sp , with significantly lower free-carriers and current, the trap
occupation is almost entirely XI0 at similar voltages, thus being
more ideal for investigations of a cold pure exciton gas. An
abruptly narrowing spatial distribution16 is considered partial
evidence of quantum degeneracy onset, though not previously
observed in a controllable trapped gas. For (lithographicallydefined) trapping diameters of 100 and 25 μm, the (spectrally

The authors acknowledge the Australian Research Council
for financial support.

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