Strongly correlated indirect excitons in quantum wells
1,2
P.Ludwig,
M.
and H.
2
Stolz
Institute of Theoretical Physics and Astrophysics, CAU-Kiel, Leibnizstrasse 15, D-24098, Germany
2 Institute of Physics, University of Rostock, Universitätsplatz 3, D-18051, Germany
Realization of a quantum Stark confinement
Abstract
●
A. V.
1
Bonitz,
Control parameter 1: tip to sample distance (ztip)
U
Wigner crystallization of electrons in semiconductor quantum dots has been recently
predicted [1,2]. This work has been extended to small clusters of indirect excitons in
double quantum wells (QW) [3,4]. Here we consider optically excited indirect excitons
in a single QW where spatial separation of electrons and holes is produced by an
electric field. The lateral confinement arises from the quantum-confined Stark effect
with typical trap size being of the order of several micrometers.
Trap properties and simulation results
tip
Field and quantum Stark confinement below the electrode
Ez, Er [kV/cm]
1
Exciton total energy [meV], L=30nm
R
QW
FWHM = 57µm
z
The (parabolic) lateral exciton confinement in the single QW is obtained
from the exciton total energy change (Stark shift) with respect to the
z-component of the field. The inhomogeneous field is produced by a tip
electrode.
2
=
12

=
[V ]
1

=
Quantum-confined Stark effect in QW
2
=

.6
12
QW
z[m]
=± x 2y 2
=± x y
2
2
2
With the substitutions r 1=  z− z tip 
2
2
2
r 2=  z  z tip  the potential
sample only
[V ]
.6
QW
q
2 1
4  0 1 2 r 1

for z≥0
for z≤0
continuity conditions at the boundary layer
z[m]
2
{

q
1  1−2 1

4  0  1 r 1  12 r 2
and
solves Poisson's equation and satisfies the Dirichlet
boundary condition r  ∞=0 as well as the
E 1  z=0= E 2   z=0, D 1 z  z =0= D 2 z  z=0.
Quantum well energy scheme
ε1=1, ε2=12.6, ztip=1µm, zQW=-1µm
An electric field induces:
CB
CB
AlGaAs
Ee
e-
Eg GaAs
Eh
VB
AlGaAs
electron energy
electron energy
Ez
hω
h+
L
●
AlGaAs
●
e
Ee
Eg GaAs
-
VB
hω
●
eff
Linear band shift:  E z =e⋅E z⋅z
Spatial separation of electrons and holes
(long radiative lifetime)
Lifetime of excitonic states increases
z
Ne
i=1
i =1
Nh
 =∑ H e ∑ H
 h −∑ ∑
H
AlGaAs
L
●
Ne
z
N e h
 e h= ∑
H
PIMC results for excitonic complexes in a homogeous field
i=1
●
●
●
[
i=1 j=1
e 02
●
4  0 r  r i −r j   z i − z j 
2
2
N e h
2
2
e
ℏ
F
0
− ∗ ∇ 2V QW

z
V
{E

r

,
z
}

∑
e  h
i
e h
z
i
i
2
2
2m e h
i j 4   0  r  
r i −r j   z i − z j 
]
The Ez-field is constant over the QW width: E z  ri , z i = E z  ri 
In the occupied trap region the radial field contribution is negligible
Spatial extension of the exciton is smaller than the mean particle distance (low and moderate density regime)
(1) Single exciton problem
(2) Effective X-X interaction
eff
i
●
●
●
i
Probability density of a free electron (red) and hole (green) in the growth direction of a QW of width
L = 3aB for different strengths of the electric field
The electric field induces a spatial charge separation
At low temperatures the attractive Coulomb-interaction leads to an excitonc bound state (grey circles)
of electrons and holes: excitons (X) and trions (X±)
ei
r e , z e  ,h r h , z h 
i
i
i
i
i
, see ref [7]
, see ref [7]
L=30nm
, see ref [7]
●
●
Exciton total energy (meV)
Binding energy (meV)
GaAs
Electric field [kV/cm]
●
biexciton, X2
Electric field [kV/cm]
General trend: electric field leads to reduction
of binding energy
Excitons: EB ≈ const for L ≈ aB = 10nm
Biexciton X2 becomes unstable already at fields
of 10-12kV/cm
exciton
●
●
●
eff
Stark shift
●
●
N=300
N=3000
U corr
k BT
crystal
fluid
gas
z eff
●
Classical coupling parameter as a function of number
of confined excitons
Depending on temperature excitons show solid-like,
liquid-like and gas-like behavior
Distance from trap center [µm]
●
●
Radial density distribution of excitons depending
on temperature
Inset: trap center density as a function of exciton
number
zeff corresponds to the electron hole binding energy
●
Determine an effective exciton confinement potential
i
Stark shift dependence on QW width
Quadratic and lineal Stark effect: exciton total energy
decreases with increase of the electric z-field
lateral
confinement of the excitons in the QW plane below a
point-like electrode (where is z-field is strongest)
Stark shift of 20meV for a field strength of 20 kV/cm
(L=30nm)
N=30
20µm
●
ext
F
 i =V QW

Hx R
V
e h
e h {E z  Ri }
Electric field [kV/cm]
Dependent parameters: exciton radiative
lifetime, X-X coupling, exciton density
GaAs
Number of excitons
2
eff
E B 
R=e 0 / 4  0 r z  
R≫ E xx  Rij 
Electric field [kV/cm]
Dipole moment and thus exciton-exciton
coupling are tuneable by changig tip voltage
Typical parameter values:
● Brückner parameter: r = a/a
s
B = 35 for Nx=2 and
rs = 11 for Nx = 3000
● Dipole parameter: a/zeff = 23 for N = 2 and a/zeff = 8
x
for Nx = 3000
U corr
k BT
 
R=e 0 z  
R
exciton
GaAs
ZnSSe
Electrode voltage Utip is proportional to the
QW field strength Ez,QW
/cm
=
exciton, X
LQW=30nm
=
Calculate average electron hole separation
i =⟨ z e − z h ⟩
zi  R
Electric field, Ez,QW [kV/cm]
Typical snap shots of thermodynamic Monte Carlo
simulations varying number of excitons Nx and
temperature T (trap frequency ω0 = 3.8 GHz)
Trap occupation range and exciton density increases
with number of confined particles
Decrease of temperature leads to a localization of
excitons and formation of a regular Wigner lattice
Separation of the z-problem by averaging over the QW thickness (PIMC simulation)
Reduction to an effective 2D-system of point dipoles (classical Monte Carlo)
●
Tuning the confinement allows
aribitrarily strong X-X coupling
Trap size and exciton density are tuneable
●
Ansatz
Ne
tip to sample distance [µm]
Exciton number and temperature dependence
Simulation idea
Eh h+
(Nx=30, T=50mK,
LQW=30nm)
Control parameter 2: applied electrode voltage
Electrostatical potential
tip and sample
dielectric
Changing tip to sample distance the potential geometry is controllable
Ez=20kV/cm
T=1K
2
ztip=2µm
o,QSC=65,7GHz
k BT
T=0.01mK
=± x y
0
sketch
Ez(r=0) = 20kV/cm
ztip=10µm
o,QSC=18,0GHz
=
U corr
Density [cm-2]
By changing the field strength and geometry, the excitation intensity (exciton number)
and temperature, the exciton-exciton correlations can be varied in broad ranges. Our
theoretical results allow us to predict the parameter range where interesting manyparticle states [5], including exciton crystallization should be observable in experiments currently set up at Rostock university [6].
2
substrate
ztip=50µm
o,QSC=3,87GHz
occupated
trap region
Electron hole separation [nm]
●
Using Path Integral Monte Carlo (PIMC) we compute, from first principles, the lateral
confinement of the excitons in the QW and the effective exciton-exciton interaction
potential in the presence of the electric field. These results are then used in classical
thermodynamic Monte Carlo simulations to investigate systems of several tens to
thousands of indirect excitions in GaAs-based QWs.
Coupling parameter Γ
●
exciton-exciton coupling
1
1
Filinov,
i
●
i
Effective interaction potential of two excitons in a
harmonic trap as a function of separation between
center of mass of excitons
For interparticle distance larger than 3.5aB classical
dipole-dipole approximation is valid
(3) N-particle Hamiltonian
Nx
[
Nx

H =∑ H ext
x  R i  ∑
i=1
i
i j
µ  Ri  µ R j 
3


4  0  r∣Ri − R j∣
]
●
●
The problem reduces to a 2D-system of N classical particles in
an effective quantum Stark confinement potential interacting
via dipole-dipole repulsion
The dipole moment
trap position

µ R=e

R depends on the exciton
0⋅z
eff
References
[1]
[2]
[3]
[4]
[5]
[6]
A. Filinov, M. Bonitz and Yu.E. Lozovik, Phys. Rev. Lett. 86, 3851 (2001)
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Yu.E. Lozovik, V.I. Yudson, JETP Lett. 22, 274 (1976); Yu.E. Lozovik and O.L. Berman, JETP 84, 1027 (1997)
for recent experiments cf. e.g.: L.V. Butov et al., Phys. Rev. Lett. 86, 5608 (2001); G. Chen, R. Rapaport, L.N. Pfeiffer,
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[7] A.V. Filinov, V. Golubnychiy, M. Bonitz, W. Ebeling, and J.W. Dufty, Phys. Rev. E 70, 046411 (2004)