Physica 138A (1986) 557-572
North-Holland, Amsterdam
NONEQUILIBRIUM
GREEN’S FUNCTIONS AND KINETIC
EQUATIONS FOR HIGHLY EXCITED SEMICONDUCTORS
II. APPLICATION
TRANSPORT
TO THE STUDY OF NONLINEAR OPTICAL AND
PROPERTIES OF THE MANY-EXCITON
SYSTEM
K. HENNEBERGER,
Sektion
MathematiklPhysik
der Piidagogischen
G. MANZKE,
Hochschule
GDR
V. MAY
“ Liselotte
Herrmann”,
2600 Giistrow,
R. ZIMMERMANN
Zentralinstitut
fCr Elektronenphysik
Hausvogteiplatz
der Akademie der Wissenschaften
5-7, 1086 Berlin, GDR
Received 17 December
der DDR,
1985
Nonequilibrium Green’s function technique is applied to the many-exciton system under the
action of an externally driven light field. Starting with Dyson’s equation for the nonequilibrium
exciton propagator the shift and damping of exciton-levels due to exciton-exciton interaction are
calculated in a local approximation with respect to the density of excitons. As a consequence, the
Boltzmann equation of excitons contains many-exciton contributions in the diffusion- and driftterm as well as in the collision integral.
The corresponding diffusion equation is derived yielding a diffusion coefficient decreasing with
increasing density and a density dependent source term due to the action of the light field.
Numerical calculations are carried out for A,_,-excitons
of a CdS platelet considering a
two-beam as well as a one-beam experiment.
For the two-beam case we present the density-profile resulting from nonlinear diffusion and the
corresponding reflection spectrum around the A,=,-level. For the one-beam case solving the
diffusion equation and Maxwell’s equations simultaneously an optical bistability just below the
Mott-transition is predicted.
1. Introduction
In highly excited direct semiconductors the intermediate density region,
where individual excitons are present but already strongly interacting, is still a
matter of actual interest. Due to material specifics and experimental conditions
there is an interference between different complications in this case: (1) many
particle interaction including, e.g., Mott transition; (2) resonant, i.e. far from
equilibrium excitation of excitons; (3) spatially inhomogeneous; and (4) non0378-4371/86/$03.50 0 Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
558
K. HENNEBERGER
et al.
stationary behaviour. Even as a consequence of this interference
several
interesting phenomena, as e.g. optical bi- or multistability and domain formation and migration can be expected and has been discussed in a more or less
phenomenological way ’ ‘“).
In this paper we are going to describe the optical and transport behaviour of
a dense gas of excitons starting with a consequent microscopic approach. In
section 2 the nonequilibrium Green’s function technique3’4.5’6) is applied to the
electron-hole
pair propagator yielding a Dyson equation governing the spectral and kinetic behaviour of the exciton gas’). The self-energy diagrams are
considered up to second order in the Born collision approximation7’x) and the
Dyson equation is approximated up to first order derivatives concerning the
local variables
r = (r, + r,.)/2
and
t = (t, + t,,)/2 4,6) .
Instead of completely solving the kinetics a local thermodynamic equilibrium is
assumed and with this in section 3 the local spectral properties of excitons, i.e.
density dependent energy and damping, are calculated within a quasiparticle
approximation up to first order in the exciton density’.‘).
In section 4 the electromagnetic
field is explicitely taken into account
semiclassically”).
An externally driven light field obeying Maxwell’s equations
is introduced leading to a corresponding source term in the exciton kinetics.
Light propagation as well as exciton generation are governed by a nonequilibrium, i.e. local density dependent dielectric function E(W, n). In it as well as in
the drift and field terms of the Boltzmann-like transport equation all quantities
involved, as e.g. velocity and force fields, occur renormalized due to the
density dependence of exciton parameters. Correspondingly the well-known
diffusion approximation
leads to a density dependent diffusion coefficient
D(n). Due to Van der Waals attraction between excitons D decreases with
increasing density and becomes even negative at densities well below the Mott
condition.
In section 5 numerical results for CdS concerning unusual diffusion behaviour as well as nonlinear optical properties are given”). Beside the limiting
cases of hindered (D = 0), low density (D = Do) and effective (D = 00) diffusion the cases of “near resonant” and “far from resonant” excitation are
shown, whereby an optical multistability is indicated in the former case.
2. Nonequilibrium
Green’s function technique
Following7.‘) the nonequilibrium
two-time electron-hole
pair propagator
(1)
GREEN’S
FUNCTIONS
FOR
EXCITED
SEMICONDUCTORS
II
559
is introduced, where e’ (h’) denote electron (hole) creation operators in the
Heisenberg representation,
1, (lh) comprise electron (hole) momentum k,
(-k,,) and spin S, (-s,,), (Y,p equal to + (-) denote positive (negative) branch
of the double time contour3*5X6), and (. . *) means expectation value of
operators time-ordered
along %. By inspection of the equation of motion
hierarchy a Dyson equation has been established in ref. 7 for the exciton
representation of the electron-hole
pair propagator
c j-d2 (Gz-‘(1,2)
Y
- &,(l,
2))G,,(2,1’)
= &(l,
1’).
In it G represents the full exciton propagator, G,’ the inverse of the single
exciton propagator and on the rhs F contains beside a &contribution
also
Pauli-blocking factors resulting from real occupation of electron (hole) states
(phase-space occupation factor). 1 is the shortening for the quantum number LYE
of internal exciton state, wave vector k, and time rl.
The self-energy 2 will be considered within the second order Born collision
approximation with respect to the exciton-exciton interaction. It is diagramatitally shown in the electron-hole
pair representation in fig. 1. Note, that the
diagrams contain in each order a direct as well as an exchange contribution
with respect to electron-hole
pairs. Furthermore,
the r-vertex comprises a
direct interaction between all electrons and holes involved minus the same but
with electrons exchanged. By this electron, hole and electron-hole
pair
exchange is included in a consistent way.
In order to separate the matrix Dyson equation (2) into a spectral and a
kinetic part we write it by components:
Fig. 1. First and second order self-energy
diagrams of the electron-hole
pair propagator
K. The
r-vertex
comprises the direct interaction
between two electron-hole
pairs as well as the interaction
combined
by electron exchange.
(r* follows by shifting the electron exchange from the left to the
right of the y-boxes.)
The Coulomb-interaction
of the electron and the hole is contained
in the
y-vertex.
560
K. HENNEBERGER
d2 (G,‘(l,
d2 W,‘(L
et al
2) - s(ret’adv)(l, 2))GCretiadv)(2, I’) = ~(1, 1’) ,
1’) - Z”(1,
211-Z;‘,“;,GZ(2,
,
2)GCad”(2, I’)} =
0,
where the propagators G = G _ + , G < = - G + _ and retarded (advanced) functions G cret) = G,, + G+_ (Gcad”) = G
- G _+) have been introduced.
As has been demonstrated explicitly+& refs. 6, 7, 9, 10 we introduce in (3)
and (4) the variable set (kwrt) instead of (1, 1’) and assume all characteristic
functions to be slowly varying with respect to the local variables (rt) and
diagonal with respect to cz (characterizing the internal state of excitons). Then
the solution of (3) exact up to first order in the derivatives (Vr , d ldt) is
G(ret’adv)(A,
W) = F(A)
/(fiw
-
H(ret’adv)(
A, w) + i&)
(A = ok, rt)
(5)
(for the proof see ref. 6), which determines locally the spectral properties
supposed the self-energy to be given. H is an effective exciton hamiltonian
defined by the single exciton energy E,, and the self-energy
zPet’adv)(
A, w) =
E,,(ak)
+ 2
(ret’adv)(
A, w)
.
(54
With the same procedure applied to (4) we arrive at an equation for G”( A, 0)
(see (3.2,6) of ref. 6) representing a kinetic equation for excitons supposed the
selfenergies Z “(A, w) to be given. We renounce giving this equation explicitly,
because it will not be used in this general shape. Instead we introduce in the
next section the quasiparticle approximation for the spectral part and the local
equilibrium for the kinetic part. The corresponding approximations of (4) will
be given in section 4.
3. Density dependent exciton parameters
In the following we consider the spectral problem (5) within a quasiparticle
approximation,
i.e. we expand the self-energy at the poles of the retarded
propagator G”““( A, o),
E,(A) - iT( A) 12 =
H’““(
A,
W)(~o=E,-i~/Z
,
(6)
defining locally the renormalized energy E, and damping r. In the self-energy
all diagrams of fig. 1 are to be taken into account.
GREEN’S FUNCTIONS FOR EXCITED
SEMICONDUCTORS
561
II
By inspection of the corresponding analytical expressions, which are given
e.g. by eqs. (29) of ref. 7, one realizes the spectral problem (6) to depend on
the solution GZ(A, w) of the kinetic equation. Therefore we consider the
kinetic problem in the following way. First we introduce a frequency dependent distribution function cp(h, w) by the ansatz
G’(A, w> = @A,
o>(l + cp(A, w)),
G’(A, w) = G(A, o)cp(A,
w) .
(7)
Here the spectral function G = G’ - G’ = G”“” - GCad’) has been introduced, which is in the quasiparticle approximation given by
G( A, o) = -iF( A)T( A)l((no
- E,(A))* + r2( A) /4) .
(8)
Let the damping in the spirit of a quasiparticle picture be small enough and,
hence, the spectral function very well localized. Then upon integration over
frequency G’( A, 0) reduces to the Wigner-distribution of excitons
NJ A) =
-1 z
G<(A,w>
=~(A)cp(A,41nm=4
(9)
and, at the same time, the kinetic equation for G’( A, w) reduces to the one
for the Wigner-distribution
= ;
s:‘(A, dlfiw=& + N,(A)) -
$S’(A, w)j,,=,XNX(A).
(10)
Note the occurrence of the renormalized energy E, in the definitions of group
velocity V,E,ln, force field -V,E, and in the collision terms of (10).
Especially even when an external scalar field is absent an internal force may
be created by the gradient of the exchange and correlation contributions of
self-energy to the quasiparticle energy E,. Such contributions have been
introduced intuitively, e.g. in ref. 12.
For the time being, we renounce solving e.g. (10) with respect to the
k-dependence of N,(A). Instead we suppose a local thermodynamic equilibrium for excitons produced by an efficient scattering either with each other or
with phonons. Furthermore
we assume a unique global temperature
T*
characterizing this local equilibrium, i.e. we neglect thermodiffusion at all. In
this case the Wigner distribution (9) reduces further to a Bose distribution.
In all cases considered here we use a Boltzmann distribution for excitons
over the 1s ground state only,
562
K. HENNEBERGER
et al
(11)
In consequence of this assumption we can use the results of refs. 7, 8, 9 for the
self-energy, where the homogeneous exciton density there is to be understood
as the local one in our connection. The local exciton density n(rt) has still to be
determined, of course by the kinetic equation (10) summed up over ak. This
will be done in the next section for the more general case, where the exciting
light source is explicitly taken into account.
4. Explicit consideration
of light source in kinetics
Till now we neglected simply the coupling of electrons and holes to the
transverse electromagnetic field. It is well known, however, that this coupling
is essential in several respects even in highly excited direct semiconductors.
Generally it affects the spectral properties as well as the kinetic behaviour. In
subsequent papers13’14 a systematic approach to related problems as, e.g., to
the propagation of intense laser light in the interband frequency region’5”6 and
to the kinetics of polaritons and light scattering”“’
will be given. For the
investigation to be made here we take into account the transverse electromagnetic field semiclassically, i.e. by adding an external perturbation
2 E(k, t)P:(k,
&x,(t) = - ;
k
(12)
t)
to the Hamiltonian,
where P, is the interband or excitonic polarization
operator and E is the electric field strength obeying Maxwells equation
($+c’k’)E(k,t)=-4
T $
(Pdk,
t) + (P,(k,
In (13) P, = x,E represents a background
introduced.
Now the whole formalism developed here
inclusion of (12) in the Hamiltonian and the
the simplest case, i.e. if we neglect the effect
ation at all, the result is very easily understood
polarization holds from
(P,(k 4) =
c 1dt’ ,C(kt, k’t’)E(k’t’)
k’
,
t)))
polarization
(13)
phenomenologically
has to be supplemented by the
investigation of (P,) in (13). In
of E on the spectral renormaliz(for the details see ref. 10). The
(14)
GREEN’S FUNCTIONS FOR EXCITED SEMICONDUCTORS II
563
where
+
Gcad’)(a- k’t’, a’ - kt))
(15)
( p, - transition dipole moment of excitons) looks, indeed, like a linear response susceptibility; it depends however via the self-energies in e.g. (5) on the
on density-dependent
exciton
kinetic stage, i.e. in our approximation
parameters.
The kinetic equation (4) has to be supplemented by replacing
-W,
2)+P(l,
2) + Zrad(l, 2) )
(16)
where the radiative self-energy
2rad(1,
2, = ; j- d3 F(1,3)$(3)8*(2)
(17)
with
is to be added. &ad in (16) describes the action of light as a source in exciton
kinetics. If one introduces (16) into (4), the kinetic equation (10) is transformed into
= P(A) -
(y +1)
&(A)
+ Rcinto)(A) .
(19)
ret
In the rhs of e.g. (19) P(A) represents a generation rate for excitons formulated with Srradand 6. The second term gives the net rate of scattering out of
the state c& (here an additional recombination contribution is phenomenologitally considered), whereas the third term is the spontaneous scattering into the
state czk.
In the following we refer mainly to the locally defined density of excitons,
(20)
564
K. HENNEBERGER
for which a balance equation is obtained
$ n(rt)
+ V&t)
et al.
by summing up (19) over ak,
= p(rt) - n(d) h,,, .
(21)
In (21) the collision contributions of (10) due to exciton-exciton
interaction
have cancelled in consequence of the k-integration. VJ results from integrating
the drift and field terms of (lo), where the excitonic current density is given by
(22)
The generation
rate for excitons reads explicitly as
(23)
In (23) 6 is to be understood as acting via its arguments k and o as a
differential operator on the Fourier-transformed
version of (18) (8 contains
additionally the factor F).
5. Approximative treatment of basic equations
If a macroscopic homogeneous
and stationary situation is realized, the
kinetic equation (19) simplifies essentially (vanishing of the Ihs) and can be
better solved directly without regarding (21). This has been done for the cases
of a macrooccupation (resonantly excited) and a completely relaxed Boltzmann
distribution of excitons in ref. 10. Now we consider the still stationary but
spatially inhomogeneous case. Then Maxwell’s equation (13) and the kinetics
(19) have to be solved simultaneously. Concerning the former one spatial
dispersion effects in (14) will be neglected and, hence, (13) becomes for a
monochromatic,
linear polarized
wave of frequency
w: E = d?(r) x
exp(-iot)
+ c.c.,
A&r) +
$ E(O, n(r))&)
=0 .
E = 1 + 47rx follows from (15) in the stationary, local approximation,
using (5) in the vicinity of the 1S excitonic ground state,
e(k
= 00, r) =
F=l+:n.
E,
(
l-
or explicit
AEd
fiw - HcTet)(a = 1Sk = 00, r) > ’
(25)
GREEN’S FUNCTIONS FOR EXCITED
SEMICONDUCTORS
II
565
In (25) the background dielectric constant E,, the longitudinal transverse
splitting AE,, and the locally density dependent effective exciton hamiltonian
have been introduced. In the density region of interest the consideration of
Pauli-blocking (factor F, a,-Bohr radius of excitons) is not necessary (see ref.
10).
Concerning the exciton kinetics, the rhs of e.g. (19) is considered in a
relaxation time approximation, whereas on the Ihs N, is approximated by its
local equilibrium shape (11). Thus, the deviation SN, = N, - Nr’ follows
according to
6N
f V,E,V, - V,E, ; V, Np’ = ---L
7relax ’
(26)
(27)
Now the results of refs. 8, 9 for density dependent
used up to first order in their density dependence,
E,(lSR, r) = E,,(lSk)
r(lSk,
exciton parameters
i.e.
- an(r) ,
r) = r, + j+r(r) .
(28)
(29)
If then (28) and (29) are used in (26) and (17) and the corresponding
inserted into (22), one finds a generalized diffusion approximation
where the density dependent
will be
SN, is
diffusion coefficient is given through
D(n) = L&(1 - anlk,T*)/(l
+ m/r,>
(31)
with
D, = hk,T* /2M&
(32)
the low density diffusivity. It is interesting to note, that D decreases wth
increasing density and can even become negative if the density becomes
sufficiently high. This tendency is produced mainly by the red shift (28) of the
exciton energy and reflects the fact, that the corresponding Van der Waals
attraction acts against diffusion in order to contract the excitonic gas into a
liquid (dielectric) state.
K. HENNEBERGER
566
Having e.g. (30) for the current,
equation (19) the balance equation
approximate still the generation rate
the interband dielectric function and
et al.
we can solve instead of the Boltzmann
(21) for the local density. In (21) we
(23) consistently with expression (25) for
obtain
P(r) = Im E(O, n(r))I@r)12/7rtc ,
with again w the frequency
6. Numerical
(33)
of the incident monochromatic
light wave.
results for CdS and discussion
We present numerical results for A,,,-excitons
of a CdS-platelet of thickness
d. Assuming the spatial inhomogeneity in x-direction only, we have as basic
equations
d%(X)
+ 02E(0, n(x)) 2J%)
dX2
C
;
(
dn(x)
dx
w+))
)
+
=0
)
(34)
1%912
Im E(O, n(x)) 7
44
o
- 7
= ,
ret
(35)
which have to be solved simultaneously with the well-known boundary conditions. Previous results to this problem have been published in ref. 19, to which
we refer for further details.
In contrast to ref. 19, where density dependencies in (34) and (35) have
been considered only in the damping r of the dielectric function E, the real part
of self-energy according to (28) and the diffusivity according to (31) will be
taken into account too. The parameter set used in numerics is given in table I.
Note that our density linearized theory predicts for that set of parameters the
Mott-transition at about n = 4 X 1017 cmp3 ‘).
First we discuss a two beam experiment, where the platelet is non-resonantly
excited by e.g. interband absorption and probed near the exciton resonance.
TABLE I
2
=0.2
T*=30K
II
E,
=
AE,,
8
= 1.9 meV
E,,= 2.5528 eV
r, = 0.2 meV
if = 3.63 X 10-l’ cm3 meV
7 = 4.06 x 10-l’ cm3 meV
7_ = 0.5 x w9 s
L=e==0.5p.m
GREEN’S
FUNCTIONS
FOR EXCITED
SEMICONDUCTORS
II
567
Then the generation rate in the diffusion equation (35) does not follow from
solving (34) but corresponds essentially to the absorption profile of the
interband excitation, which has to be used as an input in (35). In fig. 2 density
profiles for a generation rate of excitons p(x) = Zo~pe-“~” (1 laP = 0.1 km) are
shown in dependence on different excitation intensities (in photons per cm2
and sec.) and compared with those of conventional diffusion (D = Do).
Because D is decreasing with n, the diffusion profile becomes more pronounced near the x = 0 plane at higher Z,, and, correspondingly n(0) and n(d)
deviate increasingly from their conventional values (inset). The reflectivity
spectrum, fig. 3, exhibits essential differences from the conventional one (a) as
\
\
I
9”
16
i..
0
4
.5
k-(m)-
Fig. 2. Diffusion profile in the case of nonresonant pumping for different pump intensities Z0 (in
10zl cmW2 s-l). Absorption
length of the pump beam 0.1 pm. Inset shows the density at the
forward (x = 0) and backward (x = d) surface of the platelet in dependence of I,,. - - - - D = D,;
D = D(n).
K. HENNEBERGER
568
et al.
Fig. 3. Pumped reflectivity for different diffusion profiles (l/a, = 0,l pm; I, = 2.5 X lo*’ cm-‘s-l)
(a) D = D,; (b) D = D(n); (c) constant density of 1.25 x 1Ol6cmm3 (the arrow indicates the shifted
exciton position).
well as from the homogeneous one (c). This results from averaging over local
density dependent resonance energies of the exciton. Similar behaviour as that
of curve (b) cannot be obtained by an effective damping constant because of
the presence of interference structures. For similar transmission spectra we
refer to ref. 22.
In the remaining we consider a one-beam experiment with frequency of
incident light 3 meV below the 1s exciton resonance. If diffusion is neglected at
all (D = 0) we arrive at a local connection between n(x) and the light intensity
Z(X) = c@(X)[*/AJ.
From (35) follows in this case locally
n=ZwIm.z(~,n)lc.
(36)
This dependence is shown in fig. 4 and compared with the case of very effective
diffusion (D-+w), where the exciton distribution becomes homogeneous, i.e.
n(x) = const. and IZcan be obtained from the incident intensity I,, simply by the
conservation law
-
n
7ret
=
2 (1 -
R(n) - T(n))
(37)
derived in ref. 19 (R (T) - reflectivity (transmission) of the platelet). Corresponding curves for d = 0.1 pm and d = 1 pm are shown also in fig. 4. It is seen
that diffusion tends to destroy the bistable behaviour obtained for D = 0.
The numerical solution of the complete system (34), (35) has been obtained
in nearly the same way as it was done in ref. 19. In fig. 5 the profiles of density
GREEN’S FUNCTIONS FOR EXCITED SEMICONDUCTORS
II
569
29
22
23
20
Fig. 4. Exciton density in dependence of Z0for nearly resonant pumping (60 = E,, - 3 me? The
curve labeled with “local” shows the local IO-dependence of 12(D = 0). The two other curvesstand
for the case of very effective diffusion (D = m) and different platelet thicknesses d.
and current of excitons for a fixed incident intensity Z,,are given and compared
with those for conventional diffusion (D = DAO).Here the modulation of j
corresponds to the interference structure of /E(x))’ and, hence, compensates
structures in the shape of n. The difference between both cases, i.e. the
influence of unconventional diffusion, is evident.
Fig. 5. Density and current profile within the platelet for resonant pumping (fiw = E,, - 3 meV,
z0 = 5 x 102’ cm-’ s-l); - - - D = D,; D = D(n).
K. HENNEBERGER
570
et al.
Fig. 6. Transmitted intensity I, versus incoming intensity IO for fiw = E,, - 3 meV. - - - D = m
(constant n); D = D(n) (cross-line indicates vanishing of D(n)).
Fig. 6 shows the transmitted versus incident intensity and represents the
main result of this section. The characteristics
resulting from a fictive
homogeneous exciton density (dashed line) exhibits already bistable behaviour.
This is amplified in the real characteristics (full line) where additional loops
appear due to interference structures. Unfortunately,
the curves in fig. 6 are
reasonable only up to a definite intensity I,, (see mark in fig. 6) where the
diffusion coefficient becomes zero and negative according to (31). After that
intensity the numerics zero and negative according to (31). After that
intensity the numerics failed to produce a stable solution. We expect nonstationary behaviour of density e.g. movement of domains as has been
discussed, e.g., in ref. 20 for another physical situation. In order to continue
the curve further the calculations were done with a fictious small but positive
diffusion coefficient of D,l 100.
7. Concluding
remarks
In this paper a straightforward way from the microscopic quantum mechanical equations to observable nonlinear effects in optical and transport properties
of the many exciton system has been presented. Thereby it was the main
purpose to demonstrate the ability and power of the nonequilibrium Green’s
function technique in principle rather than to obtain rigorous results. Thus a lot
GREEN’S
FUNCTIONS
FOR
EXCITED
SEMICONDUCTORS
II
571
of approximations were to be used in order to get the formalism tractable. One
can divide them into two groups. The first group is related to the general
theoretical framework and should be accepted here as standard without further
inspection. Such standard approximations are, e.g., the derivation of Dyson’s
equation, Boltzmann and other transport equations (linear in the differential
operator L3) 6), evaluation of spectral functions, self-energies, etc.
The second group of approximations concerns more directly the system
under consideration
and has to be viewed more critically, because they
influence the numerical results remarkably. First the evaluation of the selfenergy linear in density fails certainly at excitation intensities high enough.
Also the numerical evaluation of the corresponding graphs in refs. 7, 8 is not
yet conclusive enough to believe the red shift predicted by (28) to be correct.
The neglection of biexciton formation at all might be also questionable. Due to
the induced absorption process excitation energy can be accumulated in the
biexciton subsystem leading to important modifications of exciton kinetics.
The inclusion of thermodiffusion, domain formation and kink motion would
be also desirable for an improved understanding of the present results.
The nonlinear effects discussed in the present paper rely mainly on the
predicted exciton red shift. As mentioned above its numerical value is open to
further refinement, which is especially needed because of missing evidence for
a definite exciton shift in measured absorption spectra. Optical nonlinearities at
the A,=, exciton in CdS have been found recently’l) at excitations in the
MW/cm’ range which is three orders of magnitude above the intensities
relevant in our calculations. Here the findings have been explained successfully
in terms of a transition into an electron-hole
plasma. Excitonic nonlinearities
as investigated in the present paper are believed to play a major role in
experiments with multiple quantum well structures (MQWS) 23).
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