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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.

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