Solid State Communications, Vol. 99, No. 6, pp. 433438, 1996 Copyright 0 1996 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098196 $12.00 + .OO
Pergamon
PI1 SUO38-1098(96)00264-S COLLECTIVE
EXCITATIONS
IN COUPLED QUANTUM WELLS
Ji-Xin Yu and Jian-Bai Xia
National Laboratory for Superlattices and Institute of Semiconductors, (Accepted
and Microstructures,
CAS, P. 0. Box 912, Beijing, 100083,China 12 April 1996 by Z. Gun)
The dielectric response of a modulated three-dimensional
electron system composed
of a
periodic array of quantum wells with weak coupling and strong coupling are studied, and the dispersions of the collective excitations and the single particle excitations as functions of wave vectors are given. It is found that for the nearly isolated multiple-quantum-we!! case with several subbands occupation,
there is a three-dimensional-like
qr=O (qz is the wave-vector
in the superlattice
subband collective
component
excitations
plasmon when
axis). There also exist intermode when q&.
in addition to one intra-subband
intra-subband
mode has a linear dispersion relation with q//(the wave-vector
perpendicular
to the superlattice
axis) when qo is small. The inter-subband
The
component modes cover
wider ranges in q//with increasing values of qz. The energies of inter-subband
collective
excitations
single-particle
excitation
anisotropy
in the 2D
spectra.
are close
by the corresponding
The collective
excitation
inter-subband
dispersions
show
obvious
quantum limit. The calculated results agree with the experiment.
The coupling between
quantum wells affects markedly both the collective
and the single particle
excitations
excitations
spectra. The system shows gradually a near-three-dimensional
electron gas
character with increasing coupling. Copyright 0 1996 Published by Elsevier Science Ltd Keywords:
A. semiconductors,
A. quantum wells, D. dielectric response,
D. electron-
electron interactions
Over the last two decades, the collective excitations low-dimensional intensively crossover
electron
by
many
research
attracted
much
theoretically,
electron attention
have
groups.
from a two-dimensional
a one-dimensional
been
studied
Although
electron gas(2DEG)
gas( 1DEG) both
coupling
behavior
experimentally
Such
behavior.’ semiconductor
electron gas(3DEG)
system
superlattices.
of the barrier,
of wells
on
the
collective
excitations.
The
previous works were mostly focused on the weak-coupling limit, ’ in which
the to
the
wave
function
overlap
between
electrons in different wells can be ignored. Pinczuk et ~1.‘~
has
and Olego et ~1.~ have determined
and
subband
few works have been done on the crossover
from a three-dimensional
the height
systems
of
collective
modulation
to a 2DEG
(MQW)
excitations
doped GaAs-A&As
in the
the inter- and intraof
electrons
in
multiple-quantum
2D limit by inelastic
light
the wells
scattering.
the
Sooryakumar
et aL5 also have observed
If we control the width and
inter-subband
collective excitations of electrons confined in
can
be
realized
in
we can study the effect
GaAs MQW superlattices.
of 433
Theoretically,
the dispersion
of
Tselis and Quinn6
Vol. 99, No. 6
COLLECTIVE EXCITATIONS IN COUPLED QUANTUM WELLS
434
have used self-consistent-field (SCF) prescription to study
approximation, the single-particle wave function and the
the collective excitations of both type-1 and type-II
eigen energy are given by
superlattices, they restricted their attention in the case of flat miniband limit, which the electron wave functions in adjacent layers do not overlap. Their theory can take into
(1)
account many-body effects, magnetic fields, and electronwhere S represents the extension of the 3DEG in the x-y phonon coupling in a simple way. Katayama and Ando presented a theory of resonant inelastic light scattering in modulation
doped
GaAs-AlGaAs
expressed the scattering
superlattices.
They
plane, and z = (i,,,k,)
is a 3D wave-vector, where kz is
confined to the Brillouin zone
cross section by dynamical
poIa~~bili~ timctions, which are calculated by taking into account the Coulomb interaction between carriers, the
(
-15 a
k, < % appropriate 1
to the periodic modulation. pl&,(z) is the wave~n~tion of electron in z-direction.
dynamical exchange-correlation .effect, and the interaction with the LO phonons based on the subband structures
In the second quanti~tion
representation, the total
calculated self-consistently. They get the spectra in the
Hamiltonian of the modulated interaction 3DEG can be
MQW mode1 and used the values at zero center (k, =0) of
written as8
superlattice Brillouin zone for the wave 5mctions and energies.
Up to now, few works have studied the strongcoupling case and the transition from weak-coupling to
(a’&
annihilates (creates) an
electron with the wave function Ia) and the energy Eh
wave
given in equation (I). @ is the 3D wave-vector and V(g)
function overlap is important. In this paper, we study the
is the well-known Fourier transform of the 3D Coulomb
dielectric response of a real 3DEG system with a periodic
potential. To study the collective excitations, we apply the
modulation in z-direction, whose results can be compared
standard SCF formalism’ to equation (2). Using the
with experimental
random-phase
strong-coupling
case, in which the electronic
where the operator a,
systems with electrons
occupying
several subbands. We consider the motion of an electron in
(RPA), we obtain
the
dielectric function
z-direction with real con~nement potential, which can make the system go over to different cases (especially,
approximation
E(q,W)=
l+~&+C,)~
aa,
G
na -n,.
E,. -E,
+A(w+iv)
different coupling cases between the neighbor wells). The weakly coupled system and strongly coupled systems are
(774 o+)
considered, and the dispersions of the collective excitations and singte-particle excitation (SPE) as functions of qz and q/l are obtained. The dielectric response %nction contains
where n,
(3)
is the Fermi occupancy a;d Cm = (O,O,z) a
(n = O,fl,*Z;~~)are reciprocal-lattice vectors. If we take
both the coupling effect between different excitation modes
account of the spin degeneracy, an additional factor “2”
and the local-field effect.
should be included in the second term on the right hand side of equation (3). The local-field effect has been
The
three-dimensional
electron
system that
we
included by summing over G,,, and if we ignore it, the
consider is periodically modulated in the z-direction with
dielectric unction
period a.
been used to study metals by Ehrenreich and Cohen.’ The
In the framework of the
effective-mass
will go over to the equation that has
Vol. 99, No. 6
435
COLLECTIVE EXCITATIONS IN COUPLED QUANTUM WELLS
matrix elements in the equation reflect the effect of the
single-quantum-well
or
uncoupled
MQW
structures
modulation. If there is no modulation on the 3DEG, the
(namely, the weak-coupling case). On the other hand,
system will degenerate to the isotropic 3DEG, and the
when al xz2 and V,,, is small, the wave unctions
wave t%nction described by equation (1) will be a simple
electrons in neighbor wells are partly overlapped, and
plane-wave, and the equation (3) will reduce to the
electrons can move more freely in the z-direction than
Lindhard equation. r”
those in the weak-coupling condition. This is the strong-
of
coupling case (the coupled superlattices). In this paper, we In general, collective excitations of the 3DEG are given by setting to zero the real part of the dielectric function.
will study the collective excitations of the system in the weak- and strong-coupling conditions.
Taking the spin degeneracy into account, we get the following equation
We first study the weak-coupling condition with parameters ar=275Pf u&l
54 ~7,=4OOfiV==2OOmeV,and
the two-dimensional electron density in each well h%=7.3x (4)
1O”cm”. Thus, we obtain the Fermi energy EF =I&270
The imaginary part of the s(g, W) gives the lifetime of the
meV and the spacing between the first and second subband
excitation
Er2=10.46meV, which is similar to the condition in Olego’s
If we restrict our dis~ssion to T=OK (as we will persist in the present paper), we can get the SPE using the equation (5) with Zm.s($,w)f 0. We can see that the formulas we use to discuss the collective excitations and SPE in the modulated 3DEG are very similar to those in the modulated 2DEG,* except that 4 is the two-dimensional wave-vector and V(q^) is the Fourier transform of the twodimensional Coulomb potential in the modulated 2DEG case.
We consider a modulation-doped GaAs-AlGaAs super-
0
0.2
0.4
lattice with the width of well and barrier are ar and ~2,
0.6
0.8
1
q&la)
respectively. Thus, the period of superlattice is a=ar+uz. The doped AlGaAs layer is in the middle of the AlGaAs
Fig. 1: Dispersion relations of collective excitations for
barrier with width ax, and we take V, to be the height of
three q8 at weak-coupling
the GaAs well. As we have mentioned above, the variation
uz=61 S& u3=400& V,,,=200meV, and N~7.3 x 10”cm~2
of the parameters (such as aI , 02 , V, ) will make the
with &==18,27OmeV and Er2=10.46meV.
system go over to different cases. On one hand, when ar is
represent the collective excitations and the shaded areas
much smaller than a2 (ur
indicate the SPE areas. The stars indicate the intra-subband
electrons are confined in the GaAs we& and they can not
collective excitations given by Olego’s experiment (Ref.4).
transfer to neighbor wells, so they will behave like those in
(Further information is given in the text).
case in which ar=275fi,
The
lines
COLLECTIVE
436
EXCITATIONS
IN COUPLED QUANTUM
WELLS
Vol. 99, No. 6
experiment.4 The electrons in this system will behave like electrons
in
subbands
uncoupled
occupied.
MQW
In Fig.],
structures
we show
with
two
the collective
excitations and SPE spectra for three qz values. The shaded areas indicate the ranges corresponds
of SPE. For example,
to the intra-subband
SPErz represents
30
SPEri
SPE in the first subband;
the inter-subband
SPE between
the first
and second subband, and so on. It is found that there is only one collective excitation dispersion curve when q*=O. This is a 3DEG-like plasmon whose energy is very close to those of 3D plasmon (which is 11.62meV) with effective electron
density N,o- =Nsla=8.20x10’6cm’3
wavelength
limit. The physical
origin
in the long-
of this mode
is 0.1
evident, which has been pointed out in the previous work.8
0.2 9,=*.42,
When qi;tO, the collective excitation spectra are obviously
0.3
0.5
0.4
q&tln)
different from that for qz =O. There are three dispersion Fig. 2: Dispersion relations of collective excitations
in
curves, the upper two are located above the inter-subband SPE regions, and the lower one appears above SPEir area. Although
we have already
included
strong-coupling
case
with
aa=75A
and
as=25A for
qz=0.42da. Others are the same as in Fig. 1.
the mode-coupling
effect’ between different collective excitation modes in the calculation, which can be seen clearly in equation (4), we can regard the lowest
one as mainly the intra-subband
In order to compare with the weak-coupling
collective excitations of the electrons in the first subband,
we consider
and the upper two as mainly the inter-subbands
wells. The coupling can be introduced
associated
with electrons
transferring
excitations
between
the first
condition,
the system with coupling between
or/and V,. Experimentally,
different
by decreasing
aa
it is easy to reduce the width of
subband and the excited subbands. The lowest one likes
barrier when we grow sample. So we consider the coupled
the intra-subband collective excitations of ZDEG. The stars
quantum wells with small 4. Fig.2 shows the collective
shown in Fig. I are the experiment data given by Olego ef
excitation
aI..4 The intra-subband
a2=75A and a3=25A. Other parameters
collective
excitation
curve has a
dispersions
(q,=O.42da) with
of the system
are the same as
linear dispersion relation with q//when qjj is small and q&O,
Fig.1. From the figure we can see that the SPE spectra
which is in good agreement
have an obvious change because the widths of subbands
energy
The
with
qz
are larger than those in weak-coupling
for a given q/l value, and the differences
of
modes are also conspicuously
of the intra-subband
increasing
with the experiment.4 mode
decreases
case. The collective
different. The energies
energies between various qz values tend to diminish at large
inter-subband
qn. The inter-subband
SPE ranges, and the collective excitation dispersions
collective
modes appear near the
SPE ranges and have energies ,almost independent value, which agrees larger
energies
and
with the experiments.’ cover
wider
ranges
of q//
They have in qll with
more obvious
modes are distinctly higher than the nearby
compared
with the weak-coupling
especially for the lower one. The intra-subband excitation
of
shows an interesting
are case,
collective
behavior with an obvious
increasing values of qz, and they all enter the SPE regions
non-zero
at last. It is obvious that the collective excitations
similar with the modulated 2DEG case.8 We can explain it
system with weak-coupling
of the
exhibit anisotropic characters.
energy in the long-wave
length limit, which is
as a result that the system will change to a near-3DEG
Vol. 99, No. 6
COLLECTIVE
EXCITATIONS
40
IN COUPLED QUANTUM spectra
WELLS with a2=25A
of the system
dispersions
437 and u3=lOA. The
of subbands are so large” that the SPE covers
almost all the energy region where the collective mode may 30
SPE23
emerge.
So we can not observe the 3D-like plasmon in
Fig.3(a) with q,=O,
and the collective excitation curves in
s
[email protected]) with q,=O.42&
E 520 & EF
[The crosses
enter the SPE regions quickly.
in Fig.3(a)
indicate
the energies
of the
plasmon in pure 3DEG]. Comparing Fig.3(a) and (b), we
5
observe that the SPE are quite different and it exhibits an
SF'62
10
obvious isotropic
character.
The intra-subband
collective
_, mode 0 0
0.1
0.2
above
SPEzz in Fig.3(b)
has a 3D-like-plasmon
behavior.
This is similar to the modulated
strongest
coupling case, except that there may still exist
ZDEG in the
0.3 inter-subband
excitation modes if we consider the higher
cir=O, q&tfa) energy levels in the superlattice 40
_ ._._,_.s
axis of the modulated
2DEG system.
1
In conclusion,
we have calculated the single-particle
energy spectra of a coupled quantum wells system in the effective-mass
s ii
approximation.
The effect of the electronic
wave function overlap between included.
520 $ EF
Using
the total
quantum representation,
neighbor wells has been
Hamiltonian
in the second-
and in the framework of the RPA
5
we obtained the dielectric unction IO
consistent-field
method,
including the local-field
The equation of the dispersion excitation 0
[equation
0.1
0.2
0.3
relation of the collective
(4)] and the equation
as timctions
of qz and q// for the systems
coupling and strong-coupling Fig. 3: Dispersion relations of collective excitations rather stronger
coupling condition
(a) q,=O,
(b) qz=0.42da.
in which a2=25A
at and
Others are the same as in
found
that
subbands qz=O.
for
the
occupation,
When q$O,
excitation,
Fig. I.
when the coupling
excitations
plasmon with non-zero coupling become
will behave
increases,
like a near-3DEG-
energy. It is confirmed
rather stronger.
and the
when the
Fig.3 is the dispersion
with weak-
weak-coupling
case
with
several
there is a 3D-like plasmon
there is only one intra-subband
when
collective
relation with q/l
which has a linear dispersion
decreases, and the differences gradually
We
between quantum wells, and
when qIf is small, When qz become
collective
of the SPE
calculated the dispersions of collective excitations and SPE
q,=O.42, q~l(nlal
system
effect.
spectra [equation (S)] for the system were obtained. 0
aa=iOA.
[equation (3)] with self-
large,
its energy
of energies between various
qz tend to diminish with increasing
q/j.
inter-subband
and they have larger
collective excitations,
There also exist
energies and cover wider ranges in q// with increasing qz. The energies
of inter-subband
collective
excitations
are
COLLECTIVE EXCITATIONS IN COUPLED QUAN~M
438
WELLS
Vol. 99, No. 6
close to the corresponding inter-subband SPE spectra.
subband collective mode in the strong coupling case with
These results are in agreement with that of experiments4,’
non-zero q2.
With the coupling existing between wells, both the collective excitations and the SPE spectra change greatly, and the inter-subband collective modes have higher energy and their dispersions are more obvious than those in the
Acknowledgment -
This work is supported by the
weak-coupling case. The system shows a 3D-like intra-
Chinese National Science Foundation.
References
1. For reviews see: S. Das Sarma, in Light &‘c~&r~q in Se~~conducfor Structures and superlattices, edited by D. J. Lockwood and J. F. Young (Plenum Press, New York, 1991), p. 499 and references therein; A. Pinczuk and G. Abstreiter, in Light Scattering in Solids V, edited by M. Cardona and G. ~ntherodt
(Springer-
Verlag, Berlin, 1989) p. 153 and references therein.
Wiegmann, Phys. Rev. B31,2578 (1985). 6. A. C. Tselis and J. J. Quinn, Phys. Rev. B29, 33 18 (1984). 7. S. Katayama and T. Ando, J. Phys. Sot. Jpn. 54, 1615 (1985). 8. Ji-Xin Yu and Jian-Bai Xia, Solid State Commun. (1996).
2. A. Pinczuk, J. M. Warlock, II. L. Stormer, R. Dingle, W. Wiegmann,
and A. C. Gossard, Solid State
Commun. 36,43 (1980). 3. A. Pinczuk, J. P. Valladares, C. W. Tu, A. C. Gossard, and J. H. English, Bull. Am. Phys. Sot. 32, 756 (1987). 4. D. Olego, A. Pinczuk, A. C. Gossard, and W. Wie~ann,
Phys. Rev. B26,7867 (1982).
5. R. Sooryakumar, A. Pinczuk, A. Gossard, and W.
9. II. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959). 10. 3. Lindhard, Kgl. Danske Videnskab. Selskab,
[email protected] Medd. 288 (1954). I I. The dispersions of subbands are large enough that the SPEa region has contained SPErr region, which is on the contrary to the weak-coupling case.