Retardation effects on intra- and intersubband plasmons in quasi-one-dimensional quantum-well wires

Superlattices

and Microstructures,

501

Vol. 12, No. 4, 7992

RETARDATION EFFECTS ON INTRA- AND INTERSUBBAND PLASMONS IN QUASI-ONE-DIMENSIONAL QUANTUM-WELL WIRES L.Wendler

fiir

Institut

Theoretische

Physik,

and V.G.Grigoryan

Technische

D/O-4200

Universitcit Merseburg,

Geusaer

StrasJe,

Germany

Merseburg, and R.Haupt

Inatitut

und Theoretische

Peatk&pertheorie

fin

The

spectrum

dimensional in detail

by the progress

of the last

fabricate

decade

layered

research

in crystal

growth

These novel systems which

(Q2D)

arise

behaviour

the challenging

topics of current reduced

QlD

starting

wires (QWW)

nanometer

lithographic

quantum-confined

systems

One of

nanostructures potential

intra-

of the

quasi-one-

are splitted

excitations.‘-”

into

These excita-

with far-infrared

(FIR)

spectrosco-

and Raman-scattering.”

PY ‘l-l3 The

aim of this

tardation

effects

and intersubbad

dots

are confined

by

a great deal of

excitation

of semi-

In a QWW

in two spatial

04 $08.00/O

excitations gas (QlDEG)

and intersubband

study

or by using novel

is the plasmon.

collective

tions are measured

curves of the

form.

electron

QlD

high-resolution

tions and hence, caused by size-quantization

0749-6036/92/080501+

and

dimensional

the

direc-

the single-

plasmons

at

z

=

0.

z -

y plane

Following

the one-electron

and the corresponding

equation,

where the electron

potential potential

V,,,(y).

along

This

a

in the

envelope subband effective is confined

lateral

effec-

is a sum of the bare initial

VO(y), the Hartree-potential

exchange-correlation

We

y-direction

EL are given by an one-dimensional

tive confinement

the reof intra-

QWW’s.

In the

well is assumed.

[L(Y)

in an effective potential

in parabolic

approximation

wave function

relation

by a model in which the electrons

quantum

effective-mass energy

is to investigate

in a zero thickness

z-direction

parabolic

paper

on the dispersion

the QWW

Schrijdinger

collective acts

particle

involves sys-

in the past few years.

the confinement

The dispersion in graphical

namely

have been prepared

has gained

tensor is treated

have unique

and QOD quantum

techniques

to pre-

The study of these low-dimensional

The most prominent conductor

interest

from a Q2D system employing

of a quasi-one-

The polarization

the quasi-two-

dimension&y,

and QOD systems

growth techniques. attention

from

of the carriers.

tems of further quantum-well

tech-

heterostructures

cise in atomic-scale. dimensional

physics

Maz- Wien-Platzl,

plasmon-polaritons

and presented

and novel

Jena,

1992)

approximation.

which made it possible

semiconductor

properties

intersubband

Universitcit

Germany

wire is investigated.

are calculated

physical

(QD).

4 August

in many fields of semiconductor

was initiated niques

(Received

the random-phase

excitations

A broad range of fundamental applications

Jena,

of the intra-and quantum-well

within

collective

Optik, Friedrich-Schiller-

D/O-6900

potential

V&y).

VH(Y) and the In general it is

0 1992 Academic

Press Limited

502

Superlattices

and Microstructures,

Vol. 72, No. 4, 7992

nesessary to distinguish two different cases for V,ff(y):

neglecting image effects. Further, ~45’ (q2, w) of Eq.(l)

(i) The first is realized in semiconductor nanostructures

is the matrix polarisation tensor of the QlDEG

with electrostatic

by15

ing V,,,(y)

confinement.

In this case the result-

is parabolic for small electron densities. (ii)

The second case is realized in semiconductor nanostruc-

Xf$(¶zP)=

tures with a bare initial ideal parabolic confinement potential Vo(y) = mR2y2/2.

enter the well and screen this bare initial parabolic po-

P$‘(Q~,w)

+ P$YQ~,u)

if and

P$‘(Q~,w)

- J$,,(P~,w)

if and else

=

tential changing the shape of the resulting potential Here we are concerned with case (i).

The current-response of the QlDEG persion

relation

ing retardation

of the collective effects,

excitations

(Y=/~=x, L # L’ o = P? L # L’ o # A

where

the intra- and intersubband

plasmon-polaritons.

Plasmon-polaritons

collective

formed of the collective

excitations

includ-

0

I

gives the dis-

(7)

L = L’

if and

P:;‘(W)

Electrons which arise from

donor impurities located away from the quantum well,

V,ff(y).

defined

are coupled

ReP,L,L’(qZ,w)

=

excita-

_-C 4n,(t;+k;J)+g 4rmqz {

*

Z

tions of the solid and photons. The dispersion relation reads”

(8)

detIGarSLL16L~Lz-~oD~~L1L’L(pZ,~)x~~’(Qz,W)l = 0

k-+k;

(1) ~L1L&L4(QZ, w ) are the matrix elements of the elecaS trodynamic Green’s tensor of the inhomogeneous wave

* In l--i> k-k:

’

RePipL’ ( qz , w ) =

equation (9)

* Ck’Lz(Y)D,B(~lI,~,IW)CBLSLl(Y’)

(2)

where Zll = (I, y, 0) is the position vector in the I - y plane (cr,p = r,y,z) CkL’(Y)

and

= &z

rlLL’(Y)

kh

with kk’ f

+&v

S,LL’(Y),

(3)

=

[!$ - ~(w

particle

TILLJ(Y) = SLYLY

g,LL’(y)=

Eqs.(2)-(10)

ami

[L(y)+$ - [L!(y)y.

[F + $(w - ftLL#)]

=

- &LO)]

and kr,

=

. Outside of the single-

continua, ReP&!‘(q,,u)

ImP,LL’(q,,w)

with

kk f

= R~P~~‘(Q~,w)

ImP~rL’(q,,w)

=

0 is valid.

= In

m is electron effective mass and cd is static

dielectric constant of the semiconductor containing the (4)

QlDEG,

ks = [2m(EF

- EL)/~~]‘/’

if EF > EL and

zero for EF 5 EL. CI;~LL,I = (EL - ILI)/~- is the subband The electrodynamic

Green’s tensor of the inhomoge-

separation frequency and EF is the Fermi energy.

neous wave equation is given by Due to the spatial symmetry of the effective confinement

potential

V,ff(y)

the system

of algebraic

equations (1) splits into the dispersion relation of symmetric intra- and intersubband plssmons and the dispersion relation of antisymmetric with

intersubband

plas

mons. Here we are mainly interested in the investiga-

(6)

tion of the retardation effects on the plasmons. Hence, to solve the complicated

algebraic dispersion relation

Superlattices we restrict further,

and Microstructures,

ourself to the diagonal

approximation

indistinguishable.

and,

intrasubbsnd

we assume that only the lowest subband is oc-

cupied.

In the diagonal

between

the different

503

Vol. 12, No. 4, 1992

line cr = (qq --E+J~/c~)~/~ = 0. Further, curve is always located the w - Q= plane. o the dispersion

For numerical

a GaAs

work we have chosen

QWW

with

effect

Ml = 2 meV:

6,

causes

to approach

the coupling

glected.

Gal-,Al,As

retardation

modes is ne-

approximation

plasmon-polariton

The

plasmon-polariton

the

the light

this dispersion

to the right of this light line in

For small wave vectors relation,

Eq.(l),

qz and small

reads

-

= 12.87

w = ~~lqzlkl

and m = 0.06624mc.

In(&) *

(11)

ln(aZn) - 1

with w, = [e21c~/(a2&o&,mZ~)]1/2 and ln = [h/(mn)]‘l”

GC / I

z-

0

2

1

Dispersion polariton

relation (heavy

is occupied

0

of the intrasubband =

3.7 * 105cm-‘).

thin solid lines are the boundaries the single-particle

plasmon-

solid lines) where one subband

(niDEG

intrasubband

6.8887

The

wi and ws of

continuum.

The

dashed line is the light line a! = 0.

_

Figure

1 shows the full RPA

lines in the w - Q= plane region tions

exist.

Notice

this

region

and, hence,

that ImPi;

and Rep::

the intrasubband

Landau-damped. intrasubbsnd

From Figure

plasmon-polariton

The corresponding

of the

Inside of

axe nonzero

of the intrasubband

Fig.2

Dispersion

relation

plasmon-polariton

O.RL’4;

0.8248

of the (1 - 0) intersubband (heavy

unretarded

the

(dotted

where

is the intrasubeffect

plasmon is very small and, hence, plasmon-polariton

0.8?46

0.824:

corresponding

the retardation.

curves of the intrasubband

0.8244

are

for qz = 0 at

excitation

0.8243

;

_.A

wave vec-

1 it is seen that starts

&I) 6.8884

excita-

Our result is that the retardation

on the intrasubband the dispersion

collective

solid

plasmon-polaritons

w = 0. This is also true if one neglects band plasmon.

small

this region is very narrow.

Imp::,

thin

intrasubband

for such

6.8885

curve of

The

are the boundaries

where the single-particle

tors as plotted

dispersion

plasmon-polariton.

6.8886

; 0 N z T

the intrasubband

I

1

3

qr( 103cm-1)

Fig.1

/

plasmon and are practically

line)

one

( ~~DEG = 3.7*105cm-I). the boundaries intersubband

line)

subband

and

the

plasmon

is occupied

The thin solidlines

are

ws and wh of the single-particle continuum.

light line a = 0. the corresponding splitting.

The dashed line is the

(b) shows the dispersion

tion of the intersubband resonance

solid

intersubband

plasmon-polariton

intersubband

relaand

plasmon near the

504

Superlattices

is the effective For c + wavelength QlD

width of the parabolic

co Eq.(ll) approach

intrasubband

quantum

well.

results in the well-known of the dispersion

and Microstructures,

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relation of the

plasmon.

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dispersion

plasmon-polariton

curve of the

is plotted.

The thin

solid lines in the w - q2 plane are the boundaries single-particle

intersubbaud

continuum.

of the

It is seen that

the dispersion curve is again always located to the right of the light line a = 0 in the w--q= plane. In this region the difference between

the plasmon-polariton

plasmon is very small but more pronounced case of the intrasubband intersubband plotted

plasmon

modes.

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R.Haupt

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