Bose glass and superfuid phases of cavity polaritons

Abstract

We report the calculation of cavity exciton-polariton phase diagram which takes into account the presence of realistic structural disorder. Polaritons are modelled as weakly interacting two-dimensional bosons. We show that with increasing density polaritons first undergo a quasi-phase transition towards a Bose glass: the condensate is localized in at least one minimum of the disorder potential, depending on the value of the chemical potential of the polariton system. A further increase of the density leads to a percolation process of the polariton fluid which gives rise to a Kosterlitz-Thouless phase transition towards superfluidity. The spatial representation of the condensate wavefunction as well as the spectrum of elementary excitations are obtained from the solution of the Gross-Pitaevskii equation for all the phases.

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arXiv:cond-mat/0701713v1 [cond-mat.mtrl-sci] 29 Jan 2007
Bose glass and superfuid phases of cavity polaritons
G. Malpuech,1D. D. Solnyshkov,1H. Ouerdane,1M. M. Glazov,2and I. Shelykh3
1LASMEA, CNRS-Universit´ e Blaise Pascal, 24 Avenue des Landais, 63177 Aubi` ere Cedex France
2A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
3ICCMP, Universidade de Bras` ılia, 70919-970 Bras` ılia-DF, Brazil
(Dated: February 6, 2008)
We report the calculation of cavity exciton-polariton phase diagram which takes into account
the presence of realistic structural disorder. Polaritons are modelled as weakly interacting two-
dimensional bosons. We show that with increasing density polaritons first undergo a quasi-phase
transition toward a Bose glass: the condensate is localized in at least one minimum of the disor-
der potential, depending on the value of the chemical potential of the polariton system. A fur-
ther increase of the density leads to a percolation process of the polariton fluid which gives rise
to a Kosterlitz-Thouless phase transition towards superfluidity. The spatial representation of the
condensate wavefunction as well as the spectrum of elementary excitations are obtained from the
solution of the Gross-Pitaevskii equation for all the phases.
PACS numbers: 71.36.+c,71.35.Lk,03.75.Mn
Exciton-polaritons (polaritons) in microcavities are
composite 2-dimensional weakly interacting bosons [1, 2].
Despite their short radiative life time (∼ 10−12s) stim-
ulated scattering towards their ground state has been
demonstrated [3, 4], and quasi-thermal equilibrium was
recently reported [5]. In this quasi-equilibrium regime
cavity polaritons are expected to give rise to a Kosterlitz-
Thouless (KT) phase transition toward superfluidity [6].
The corresponding phase diagram has been established a
few years ago [7], and recently refined to fully take into
account the non-parabolic shape of the polariton disper-
sion [8]. Because of their light effective mass (typically
10−4times the free electron mass) polaritons show ex-
tremely small critical density and high critical tempera-
tures that can be larger than room temperature in some
cases. However, semiconductors were assumed to be ideal
in the approach used in Refs. [7] and [8] while experimen-
tal data clearly show strong localization of the condensate
because of structural imperfections. The phase observed
is in fact characteristic of a Bose glass [9] and no signature
of superfluidity has been reported thus far. In this Let-
ter we propose the derivation of a new polariton phase
diagram taking into account structural disorder whose
impact on the spatial shape of the wavefunction and the
dispersion of elementary excitations, is analyzed within
the framework of the Gross-Pitaevskii theory [10].
To give a qualitative picture of the model, we assume
that the polaritons are moving in a random potential
V (r) whose mean amplitude and root mean square fluc-
tuation are given by ?V (r)? = 0 and
respectively. The correlation length of this potential
???V (r)V (0)?dr/V2
system, there are here two types of polaritons states
[11]: the free propagating states and the localized states
with energy E < Ec, where Ec is the critical “delo-
calization” energy. The localization radius scales like
a(E) ∝ a0Vs
??V2(r)? = V0
As in any disordered
is R0 =
0.
0/(Ec− E)s, s being a critical index and
a0=
der of mean potential energy (i.e. 0 in our case), and
s ≈ 0.75 [11]. The quasi-classical density of states is
D0(E)∼= M/4π?2[1 + erf (E/V0)] [13].
exciton-polariton is a composite boson also containing
a photonic component which makes its trapping in the
state with the localization radius a = ac ? λ (where
λ is the wavelength of the incident light) not possible.
Thus the resulting effective density of states shows an
abrupt cut-off at E = E0which is self-consistently deter-
mined from a(E0) = ac: D(E) = D0(E) for E > E0and
D(E) = 0 otherwise.
Clearly, non-interacting bosons cannot undergo Bose
Einstein Condensation (BEC) as the number of parti-
cles which can be fitted to all the excited states of the
system (E > E0) is divergent. The situation is thus dif-
ferent from those for cold atoms trapped by power-low
potential in 2D, where the renormalization of the den-
sity of states makes “true” BEC possible [14]. Therefore,
even in the presence of disorder, BEC cannot take place
strictly speaking for cavity polaritons.
possible to define a quasi-phase transition which takes
place in finite systems [7]. Indeed, for the finite size R
system there is a finite number Ntrapof potential traps
for polaritons, thus, there is an energy spacing between
the single particle states. The typical energy distance
between the ground and excited states of the finite-size
system levels δ under the assumption of long-range po-
tential is approximately given by V0/Ntrapor ?2/2MR2
whichever is smaller. In this framework the critical den-
sity is given by the total number of polaritons which can
be accomodated in all the energy levels of the disorder
potential V (r) except the ground one [7]:
??2/mV0[12]. In two dimensions Ecis of the or-
However, an
However, it is
0
nc(T) =
?
i?=0
fB(Ei,E0,T), (1)
where fB(T,µ,T) is the Bose-Einstein distribution func-

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tion, µ is the chemical potential.
To evaluate the critical density nc(T,R), the discrete
sum is replaced by an integral in Eq. (1), and we find
nc(T) ≈ D(E0)kBT ln?1/(1 − eδ/kBT)?
is a smooth function. Above this density all additional
particles are accumulating in the ground state and the
concentration of condensed particles n0 satisfies n0 ≥
(n − nc), where n is the total density of polaritons. It is
not a real phase transition since the system has a discrete
energy spectrum and the value of the chemical potential
never becomes strictly equal to E0.
Interactions between particles start to become dom-
inant once the polaritons start to accumulate in the
ground state. The situation can be qualitatively de-
scribed as follows: particles start to fill the lowest energy
state which is therefore blue shifted because of interac-
tions (µ−E0> 0). Thus, for some occupation number of
the condensate the chemical potential reaches the energy
of another localized state, and this state starts in turn to
populate and to blue shift. The condensate, like a liquid,
fills several minima of the potential. It gives rise to the
spatial and reciprocal space pictures of Refs. [5, 6]. A
few localized states, covering about 20 % of the surface
of the emitting spot are all emitting light at the same
energy and are strongly populated. This characterizes a
Bose glass [9]. This situation occurs up to the achieve-
ment of the condition µ = Ec. This condition should be
accompanied by a percolation of the condensate which at
this stage should cover 50 % or more of the sample (in
the semiclassical representation). The delocalized con-
densate becomes at this stage a KT superfluid. More
precisely, the different sides of the finite-size system are
linked by the phase coherent path. Therefore we predict
two quasi-phase transitions driven by temperature and
particle density: first, with an increase of the polariton
density beyond nc(T) the system enters the Bose glass
phase, then with a further increase of the density the
polariton system becomes superfluid. The critical condi-
tion µ = Ecis valid only at low temperature where the
thermal depletion of the condensate is negligible.
The quantitative analysis can be carried out in the
framework of the Gross-Pitaevskii equation for the con-
densate wavefunction Ψ(r,t) which reads
assuming D(E)
i?∂
∂tΨ(r,t) =
?
−?2
2M△ + V (r) + g |Ψ(r,t)|2
?
Ψ(r,t),
(2)
where g is a constant characterizing the weak repul-
sive interaction between polaritons.
the Gross-Pitaevskii equation takes the form Ψ(r,t) =
Ψ0(r)exp(−iµt/?). Top panels (a), (b), and (c) of Fig. 1
show the real space distribution of the polaritons ob-
tained from the solution of the Gross-Pitaevskii equation.
The parameters are those of a realistic CdTe microcav-
ity at zero detuning. We have taken the polariton mass
The solution of
FIG. 1: (Color online) Spatial images (top panels) and quasi-
particle spectra (bottom panels) for a realistic disorder poten-
tial. The figures shown correspond to densities 0, 6×1010and
2×1012cm−2. The red lines are only guides to the eye, show-
ing parabolic, flat and linear-type dispersions. The colormap
of the panel 1.b is the same as 1.c.
m = 5 × 10−5m0, where m0 is the free electron mass,
and the interaction constant g = 3Eba2
Eb is the exciton binding energy (25 meV in CdTe),
aB = 34˚A is the exciton Bohr radius and Nqw = 16
is the number of wells embedded in the microcavity. We
have included a random disorder potential with V0= 0.5
meV and R0 = 3 µm. Figure 1.a corresponds to the
non-condensed situation. The spatial profile is given
by the statistical averaging over the all occupied states,
n(r) =?
ture is set to T = 19 K which corresponds to the effec-
tive polariton temperature measured in [5]. In this case
the total number of particles is small and thus nonlinear
terms in Gross-Pitaevskii equation can be neglected.
Once the quasi-condensate is formed, and for moderate
temperatures, one can neglect the thermal occupation of
the excited states and the spatial image of the polariton
distribution is given by the ground state wavefunction
which corresponds to solution of Eq. (2). We show the
resulting density below and above the percolation thresh-
old on Figs. 1.b and 1.c respectively. As expected, the
condensate is localized in a few minima of the random
potential as shown on Fig. 1.b. On Fig. 1.c the conden-
sate wave function still exhibits some spatial fluctuations
connected to disorder, but the condensate is nonetheless
well delocalized, covering the whole sample area.
To calculate the quasiparticle spectra shown in lower
panels 1.d, 1.e and 1.f of Fig. 1 we introduce a single-
particle Green’s function which takes the form
B/Nqw, where
jfB(Ej,T,µ(T))|Ψj(r)|2. Here the tempera-
Gω(r,r0) =
?
j
Ψj(r)Ψ†
?ω − Ej
j(r0)
, (3)

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where Ej and Ψj(r) are energies and eigenfunctions of
the elementary excitations [15], found numerically from
Eq. (2). The spectrum of elementary excitations is given
by the poles of the Green’s function in the (k, ω) repre-
sentation, and shown on the lower panels of Fig. 1. The
left panel 1.d shows typical parabolic dispersion broad-
ened by the disorder potential. The middle panel 1.e
shows parabolic dispersion with a flat part produced by
the localization of the condensate. The linear spectrum
on the right panel 1.f is the distinct feature of the su-
perfluid state of the system. Only the upper Bogoliubov
branch is shown. Figures 1.b and 1.e reproduce quite
well the experimental observations of Ref. [5] which are
characteristics of the formation of a Bose glass.
It is instructive to analyze both the variation of the
emission pattern and the quasi-particle spectrum in com-
parison with the behavior of the superfluid fraction of the
polariton system. The latter quantity can be calculated
using the twisted boundary conditions method [16]. Im-
posing such boundary conditions implies that the conden-
sate wavefunction acquires a phase between the bound-
aries, namely
Ψθ(r + Li) = eiθΨθ(r), (4)
where Li(i = x,y) are the vectors which form the rectan-
gle confining the polaritons and θ is the twisting param-
eter. The superfluid fraction of the condensate is given
by [16]
fs=ns
n= lim
θ→0
2ML2(µθ− µ0)
?2nθ2
, (5)
where µθis the chemical potential corresponding to the
boundary conditions Eq.(4) and µ0 is the chemical
potential corresponding to the periodic boundary con-
ditions (θ = 0). In the case of a clean system, V (r) = 0,
the plane wave is the solution of Eq. (2) and µθ− µ0=
n?2θ2/2ML2: the superfluid fraction is fs= 1. On the
contrary, for the strongly localized condensate the wave-
function is exponentially small at the system boundaries
and the change of the boundary condition (i.e. varia-
tion of θ) does not change the energy of the system, thus
fs∼ exp[−L/a(µ0)] and goes to 0 for the infinite system.
Due to the exponential tails of the localized wavefunc-
tions a small degree of superfluidity remains in the finite
size system. Equations (2) and (4) allow to study the de-
pletion of the superfluid fraction for arbitrary disorder.
The contribution of the disorder to the normal density of
polaritons can be represented as
nd
n= (1 − fs)n.(6)
Figure 2 shows the superfluid fraction calculated as
a function of the polariton density in the system for
?
????
?
?
FIG. 2: (Color online) Superfluid fraction as a function of the
density of particles, obtained from twisted boundary condi-
tions (black curve) and from the perturbative approach (red
curve).
T = 0 K. Due to the finiteness of the system considered
the superfluid fraction remains non-zero for any finite
density, but a very clear threshold behavior for densities
corresponding to the percolation threshold as observed
on Fig. 1, is also shown. For high values of the chemical
potential, where V2
0/µg ≪ 1, perturbation theory applies
and we obtain
nd
n=1
4
V2
0
µg,
(7)
for the normal density [17]. The resulting curve is shown
in red on Fig. 2. The twisted boundary conditions ap-
proach and Eq. (7) give coinciding results for high den-
sity of the polaritons.
In the rest of the paper, we concentrate on the estab-
lishment of the cavity polariton phase diagram. Similarly
to previous works [7], we roughly define a temperature
and density domain where the strong coupling is sup-
posed to hold. The limits are shown on Fig. 3 as thick
dotted lines [20]. The transition from normal to Bose
glass phase can be calculated from Eq. (1) and a realistic
realization of disorder. The lower solid line on Fig. 3
shows nc(T) for the same realization of disorder as for
Fig. 1. The free polariton dispersion is calculated using
the geometry of Ref. [5]. At T = 19 K we find nc =
2 × 108cm−2.
We now calculate the density for the transition be-
tween the Bose glass and the superfluid phase. In the
low temperature domain, this density is approximately
given by the percolation threshold µ = Ecand does not
depend significantly on temperature. This condition cor-
responds with good accuracy to the abrupt change of the
superfluid fraction fsshown in Fig. 2. However, at higher
temperature the thermal depletion of the condensate be-
comes the dominant effect. In that case the chemical

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potential of the condensate is much higher than the per-
colation energy Ecand the depletion induced by disorder
can be neglected compared to the thermal depletion of
the superfluid. The normal density then reads
n0
n(T) = −
2
(2π)2
?
E(k)∂fB[ǫ(k),µ = 0,T]
∂ǫ
dk, (8)
and the superfluid density in the system is given by
ns(T) = n − n0
n(T), (9)
which can be substituted into the Kosterlitz-Nelson for-
mula [18] to obtain a self-consistent equation for the tran-
sition temperature:
TKT=?2πns(TKT)
2M
. (10)
The joint solution of Eqs. (10) and Eq. (9) allows to
determine the superfluid phase transition temperature
TKT(n). The result of this procedure is shown on Fig. 3.
Below 120 K the critical density is given by the percola-
tion threshold and there is no temperature dependence.
Above 200 K the superfluid depletion is determined solely
by the thermal effects. In the intermediate regime the
crossover between the thermal and disorder contributions
takes place and our approximations are no longer justi-
fied. We also find that the superfluid transition takes
place very close to the weak to strong coupling thresh-
old and for densities 3 orders of magnitude larger than
the one of the Bose glass transition at 19 K. This sug-
gests that experimental observation of this phenomenon
remains a great challenge.
In conclusion, we have established the phase diagram
of cavity polaritons taking into account the effect of
structural imperfections. We predict that with increasing
density the polariton system first enters the Bose glass
phase before it becomes superfluid. The Bose glass pic-
ture is in good agreement with recent experimental data
[5]. The condensate wavefunctions as well as the spec-
tra of elementary excitations are obtained from the solu-
tions of the Gross-Pitaevskii equation including disorder.
Our work also shows that the presence of disorder has no
significant impact on the occurence of a bosonic phase
transition for polaritons. This explains why this phe-
nomenon has been observed in a rather disordered system
like CdTe. This also gives good hope for the observation
of such phase transition in even more disordered systems
like GaN [19]. However, since disorder strongly affects
the occurence of the superfluid phase transition, it could
bring renewed interest in cleaner systems like GaAs based
structures.
We thank K.V. Kavokin for enlightening discussions.
We acknowledge the support of the STREP ”STIM-
SCAT” 517769, and of the Chair of Excellence program
of ANR.
10
0
10
1
Temperature (K)
10
2
10
3
10
6
10
8
10
10
10
12
Polariton density (cm−2)
Bose Glass
Polariton diode
LED
VCSEL
Superfluid
FIG. 3: (Color online) Polariton phase diagram for a CdTe
microcavity containing 16 QWs. The horizontal and verti-
cal dashed lines show the limiting temperatures and densities
where the strong coupling holds. The lower solid line show the
critical density for the transition from normal to Bose glass
phase. The upper solid line shows the critical density for the
transition from the Bose glass to the superfluid phase. The
dashed part of the line shows the temperature range where
the validity of our approximations ceases.
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