l. c. andreani, optical transitions, excitons, and polaritons in bulk and low-dimensional...

7/21/2019 L. C. Andreani, OPTICAL TRANSITIONS, EXCITONS, AND POLARITONS IN BULK AND LOW-DIMENSIONAL SEMICONDU

1/56

OPTICAL TRANSITIONS, EXCITONS, AND

POLARITONS

IN

BULK

AND

LOW-DIMENSIONAL SEMICONDUCTOR STRUCTURES

Lucio Claudio Andreani

Dipartimento

di Fisica A. Volta

Universita degli Studi di

Pavia

via A. Bassi, 6

1-27100 Pavia,

Italy

1 INTRODUCTION

Electrons in solids are

subject to the

crystal potential, as well as

to the mutual

electron

electron interaction.

The

resulting quantum-mechanical system represents a many-body

problem of great complexity.

In

weakly-correlated systems like

the

usual semiconduc

tors,

a good starting point is provided by the one-particle picture, in which the crystal

eigenstates are approximated by Slater

determinants

where

the

electrons occupy

the

one-particle eigenstates called

band

levels. This is only an

approximate picture,

since

the

electron-electron

interaction

yields corrections

to the

excited-state

spectrum

of

the

crystal. In particular,

two-particle excitations called excitons arise

at

energies below

the band gap,

and

excitonic corrections are found also

at

energies above the band gap.

The

electronic

states of

a crystal can

be

probed

using

an external

electromagnetic

field.

The study

of

the

optical properties gives very precise information on

the

elec

tronic structure of semiconductors. In analyzing

the

radiation-matter interaction,

it

is

useful to distinguish between instantaneous and retarded parts of

the

electromagnetic

field.

The unretarded

(c

- ;

00

part

describes

the

instantaneous Coulomb interaction

and

corresponds

to

the

longitudinal electromagnetic field, while

the retarded part

is

identified with

the

transverse electromagnetic field, i.e., with

the

physical photons. For

one-particle states, interaction with

the

transverse electromagnetic field gives rise to

interband

and

intraband

transitions. For two-particle (excitonic) states,

interaction

with

the

longitudinal

part

of

the

electromagnetic field corresponds

to

the

electron-hole

exchange interaction, while

interaction

with

the

transverse electromagnetic field gives

rise

to

polariton effects.

The

above picture applies to bulk semiconductors as well as to mesoscopic struc

tures

like

quantum

wells, wires

and

dots. However

both the

electronic states

and

onfined Electrons and Photons

Edited by

E.

Burstein and C. Weisbuch, Plenum Press, New York, 1995

57


2/56

the radiation-matter

interaction are modified by

the

reduced dimensionality. One of

the

most important modifications concerns

the

polaritonic effect. In infinite crystals,

the

conservation

of

crystal

momentum

implies

the

formation of quasi-stationary

states

called excitonic polaritons. In confined systems, the lack of crystal momentum con

servation

in

one

or

more directions implies that an exciton

interacts with

a

continuum

of photon states, thereby

providing a mechanism for intrinsic radiative decay of free

excitons.

In these lecture notes we give an outline of the theory of linear optical properties in

semiconductors, considering

both

bulk

and

confined systems. We consider one-particle

properties

(interband and

intraband transitions) as well as exciton

and

polariton effects.

One main point is

the

modification of exciton states

and

polari ton effects in going from

bulk

to

quantum-well systems, considering in particular

the

exciton radiative lifetime.

In

Sec. 2

we treat

optical properties

in

bulk semiconductors. After a review

of

the

classical theory

of

dielectric properties, including

the

Lorentz-oscillator model

and

Kramers-Kronig relations, we describe

the

calculation of

the

dielectric

constant

by a

semiclassical theory

of

the

radiation-matter

interaction. We

then

give an outline

of

the quantum theory

of excitons

and

polaritons. In Sec. 2.5 we summarize

the present

understanding of the radiative recombination of excitons in bulk semiconductors, which

is a basic problem but still not completely understood. In Sec. 3 we discuss optical

properties

in

confined systems, particularly in

the

quantum-well geometry. Following a

similar scheme, we first review

the

theory of one-particle transitions,

and

then consider

exciton

states and

polariton effects. In Sec. 3.4 we discuss

the

radiative lifetime of

excitons in going from bulk

to

confined systems. In Sec. 4 we summarize the

main

points

treated

in these lectures.

In the Appendix

we give rules for converting from the

Gaussian cgs)

to the

SI MKSA) system of units.

2 OPTICAL PROPERTIES IN BULK

SEMICONDUCTORS

2.1 Classical Theory

of

Dielectric

Properties

Classical electromagnetic

theory.

Maxwell equations in

matter

must be supple

mented by constitutive relations. We neglect magnetic effects,

and

consider only linear

response.

In

a homogeneous

and

isotropic medium, the Fourier components

of the

polarization

and

displacement fields are related to

the

electric field by Gaussian units)

P

D

XE

E + 47rP

=

tE,

1)

2)

where t(w,k) = 1 47rX w,k) is

the

frequency-

and

wavevector-dependent dielectric

function.

The

wavevector dependence of t is referred to as spatial dispersion

[1]. t

can often be neglected since

the

wavelength of radiation is much larger than

the lattice

spacing. In a crystal, which is invariant only under translations by lattice vectors, the

macroscopic dielectric function contains local-field effects

[2]. In

anisotropic

media, the

dielectric function becomes a 3 x 3 tensor, which reduces

to

a scalar only for cubic

crystals in

the

long-wavelength limit.

The

displacement

current

appearing

in

Maxwell equations is given

by

J _ ~ a

d - 47r t

In the

presence of free charges, an induced

current

appear according

to Ohm s

law,

J

ind

= erE

58

3)

4)


3/56

Both

currents

(3),(4) can be taken

into

account by introducing a complex dielectric

function,

5)

At finite frequencies, the conductivity u(w)

is

also complex. Thus it is a matter

of

convention

to attribute the

dispersive properties

of

a

medium

to

a real dielectric function

plus a conductivity, or

to

a complex dielectric function E = E1 E2

The index of refraction,

N = n K

is also complex. The absorption coefficient

is

obtained as

w

a =

-K .

c

When E2

E1,

Eq. (7)

can

be approximated by

6)

7)

8)

This is a good approximation for semiconductors, but

it

may

fail for metals (see Sec.

2.2).

Oscillator model. A simple classical picture for the dielectric function is provided

by the Drude-Lorentz

model

[3, 4J. The electrons

in

a crystals are represented by a

collection of

damped

harmonic oscillators, which respond

to

an applied electric field

according

to the

equation

[

2 J E

mo x

ijX

W j X = - e loc,

9)

where Eloc is

the

local field acting on the oscillator and mo is the free-electron mass.

The

induced

dipole moment for each oscillator is

Pj = eXj =

2 2 . ).

mo Wj - W - t i j W

10)

The macroscopic polarization

is

found by multiplying Pj by the number of oscillators

per unit volume, which we denote by Ii V, and summing over all resonances. f the

local field is identified with the applied field (i.e., if local-field corrections

are

neglected),

the resulting dielectric function is

47re

2

1 Ii

E

w

=

1

mo

V L

~ _ w

2 _ ii W

J J J

(11)

The

oscillator strength

Ii

is a dimensionless quantity, which represents

the

number of

classical oscillators

at

frequency

Wj

Since

the total

number of oscillators equals

the

number

N of electrons in the crystal,

the

oscillator strengths

must

satisfy

the sum

rule

L i =

N

12)

j

This

sum rule

can be

derived

by

comparing the high-frequency limit of Eq. (11) with

the

known limiting form of

the

dielectric function,

13)

59


4/56

where w;

=

47rNe

2

/(moV)

is

the plasma frequency.

Equation 11) yields

the

real and imaginary parts of the dielectric function,

the

index

of refraction, absorption coefficient, etc. Close

to

a resonance frequency, the real

part l W) shows anomalous dispersion, while

2 W)

yields resonant absorption

[3,4].

Free charges

contribute

a frequency-dependent conductivity u(w)

= ne

2

/(m*(( - iw)),

where

m

is the effective mass.

The

relation between 11) and

the

electronic structure

is given by a microscopic calculation

of the

parameters

Wj,

Ii

and i j In

a solid,

in

the energy region

of

interband transitions, the frequencies W j form a continuum. A

quantum-mechanical calculation of

the

dielectric function, with a suitable definition of

the oscillator strength, will

be

described in Sec. 2.2.

Dispersion

relations

nd sum rules. General properties of

the

optical constants

follow from the causality principle. The displacement field is related

to

the electric field

by

D t) = E t)+47r l O x r)E t-r)dr,

14)

where the response function

x r)

can be taken to vanish for

r

,

bD

h

Ql

I i

Ql

0

HH1

x

o

-30

LH1

-90

HH3

LH2

(0)

1 5 0 0 L L ~ 1 L ~ ~ 2 ~

k L

x

=

3.0

k or

HH1

LH1

HH2

b)

o

1

2

k L

Figure 7. Dispersion

of

the valence subbands in a 80 A wide GaAs-Gao.6Alo.4As quantum well a)

with no applied stress b) with an applied stress X = 3.0 kbar along the growth direction.

absorption probability for a single

quantum

well is a

pure number,

which can

be

defined

as

) _ energy absorbed/unit time surface

w w - incident

energy/unit time

surface

91)

The absorption probability can

be

calculated

by

time-dependent perturbation theory,

as in Sec. 2.2. The result is

92)

where kll is

the

in-plane wavevector. Note that

w w)

is dimensionless, while

the

ab

sorption coefficient 30)

has the

dimensions of em 1 .

It

is important

to

realize that

the

absorption coefficient 29) of a

plane

wave for

a sample containing a single

quantum

well is zero in

the thermodynamic

limit. This

implies

that

a plane wave propagating along the layer planes

is

not attenuated. There

is no contradiction with experiment, since the case of an infinite plane wave is never

realized

in

practice. In the real case of an incident light beam with diameter

C

the

absorption per

unit length

is of

the

order of w w)

/

C

(apart

from a factor of order

unity which depends on the electric field profile).

Thus

the attenuation of a light beam

propagating

in the layer planes can

be

increased

by

decreasing the

spot

diameter. A

finite absorption of order w w)/C is also obtained for light propagating in a waveguide

configuration,

and

in this case

the

relevant length C is the thickness of the waveguide

[75] This is

to

be

contrasted

with

the

case of light propagating along

the

growth

direction, where

the

dimensionless absorption probability 91) is

independent

of

the

spot size.

81


26/56

Multiple

quantum

wells MQWs) are usually employed in order to increase

the

optical efficiency. For a sample with

N

periods, the transmittance

T

is related to

the

dimensionless absorption probability 91) by

T

= 1 - wt =

exp N

.log 1 -

w))

exp -Nw ,

93)

and

the usual

formula for exponential decay is recovered. Macroscopically,

the

absorp

tion coefficient per unit length is obtained as

N Nw w

a = - - l og 1 - w

-

= ,

w

b

94)

where w b is

the MQW

period

and

=

N Lw L

b

)

is

the

thickness of

the

sample.

This derivation shows that the

proper

length

by

which the dimensionless absorption

probability

has to

be

divided

is

the period

w b not

the well

width

Lw).

A

MQW

sample can also be considered as a uniaxial crystal with a unit cell of size

w b along

the

growth direction. The macroscopic absorption coefficient can again

be

defined as in Eq. 29), and

can be

calculated by

the

procedure

of

Sec. 2.2.

When

the

electronic states are localized along the growth direction,

the

analog of Eq. 28) is

E E 2 E = S E E

r

2dkll

=

1 E

r

2dk

l

l

V .

cv

k V cv JBZ

(27r)2 w

+ b cv

JBZ (27r)2

J J

(95)

where E

j

denotes

the

sum over

the

identical QWs and S is the

area

of the sample.

Thus we see that formula 94) gives the absorption coefficient of a MQW sample for

any

direction of propagation.

The

physical meaning of Eq. 94) is that

the

attenuation

of a light beam in a MQW sample is proportional to the concentration of absorbing

regions,

and

therefore

must

depend

on

both

well and barrier widths.

A

limiting case

of

Eq. 94) is the case of a single quantum well, when

the

barrier width b - - t and the

macroscopic absorption coefficient vanishes.

A description

of

optical absorption in quantum well

structures

which is

based

on

the absorption probabil ity has two advantages: first, the absorption probability for

interband

transition

is essentially

independent

of the well

width

see below),

without

the artificial

1/

w dependence which

is

sometimes

reported

in the literature as a result

of dividing the

optical density by

the

well width

[76J.

Second,

the

absorption probability

integrated

over

the

excitonic peak is directly related

to the

oscillator

strength per

unit

area , which is

the

basic

quantity

character izing quasi-two-dimensional excitons see Sec.

3.2).

Interband

transitions. For transitions between valence and conduction subbands,

the

momentum matrix element can be evaluated by keeping only the lowest-order

term

in Eq. 89), and by neglecting

the

gradient of

the

slowly-varying envelope function.

The

result is

(96)

Selection rules come from

the

envelope-function as well as from

the

Bloch-function

part.

In the approximation of infinite barrier height,

the

envelope functions of valence

and

conduction subbands

are

identical, which leads

to the

selection rule

D.n

= 0

[77J.

This is

only approximately

true

for a well of finite barrier height, since particles with different

effective masses have different penetration in

the

barriers. In any case, transitions with

D.n = 1,3, . . .

are forbidden by parity. Selection rules for

the matrix

element between

Bloch functions can be found using

the

Wigner-Eckart theorem

[70J,

or, in a more

82


27/56

elementary

way, by observing that heavy holes

at

kll

=

0 have an angular

momentum

s =

3/2,

while light holes

and

electrons have s = 1/2. For light polarized in

the

layer planes, both heavy- and light-hole transitions are allowed. However, for light

polarized along

the

growth direction,

the

operator

f

. p does

not

change

the

angular

momentum and only light-hole transi tions are allowed. The ratios of oscillator strengths

for

interband

transitions at kll = 0 are ~ ~ l

:

~ ' J )

: frJ)

= 4 : 3 : 1.

Equation

(96) shows

that

the interband momentum matrix

element

between

Bloch

functions is essentially

the

same for bulk and QW systems. Therefore, the oscillator

strength

(34) is also

the

same. Differences in

the

optical absorption come from

the

joint

density of states. In an ideal two-dimensional system,

the

joint density of states has a

staircase form [77].

Absorption

of a single quantum well is low, being of

the

order of a percent. This can

be easily estimated from Eqs. (92)

and

(96), which yield (for the heavy-hole transition in

the

case of infinite barrier heights) w

1Ie

2

/

nne ) fl,fm:).

The

absorption probability

is

to

a first approximation independent of

the

well thickness, since

the

two-dimensional

density of states P D = p,/ In2 is a constant. This has been verified experimental ly [78].

Corrections

to

this result

can

come from leaking

of

the

wavefunction

in the

barriers,

which reduces

the

overlap integral,

and

from

the

slight variation of

the

Sommerfeld

factor

with

the well width (see below).

The quantitative

evaluation of

interband

absorption in QWs is

made

complicated

by valence band mixing. At kll 0, heavy and light hole states become mixed and

transitions between all pairs of conduction and valence subbands are possible. The

ones which

are

forbidden at kll

=

0 acquire a larger oscillator

strength

at those values

of kll for which

the

corresponding valence

subband

is strongly mixed

with

the sub

band of an allowed transition. The negative curvature of a subband also increases the

absorption,

because

it

gives a large joint density of states.

An

example

of

calculated

interband

absorption for transitions

to the

first conduction

subband is

shown in Fig. 8.

A phenomenological width

r

= 2 meV has been introduced.

The

peak at the energy

of

the LHI-CBl transition

comes from

the

negative curvature of

the LHI subband

(see

Fig. 7).

t must be stressed that

the interband

absorption shown in Fig. 8 is

not

a measurable

quantity,

since

the measured

absorption contains excitonic effects.

The

Sommerfeld

factor in the strict two-dimensional limit has been calculated by Shinada

and

Sugano

[79],

with the

result

S w)

_

- 1

+

e-

(97)

This factor is two

at the

subband edge, and decreases slowly with an energy scale given

by the

effective Rydberg. For realistic

quantum

well excitons,

the

enhancement

factor

is calculated to

be

between

1.3

and

1.4

at

the

absorption edge

[80].

Including

the

excitonic effect,

the

absorption probability in a GaAs-Gal_xAlxAs single

quantum

well

is

w 0.75 for

the

first HH transition, and w 1.0 for

the

lowest HH and LH

transitions together

(see Fig. 8 and Ref. [78]).

A schematic plot of the absorption in two-dimensional systems is shown in Fig.

9. As for three-dimensional excitons, the absorption probability corrected for excitonic

effects is continuous

at the

interband absorption edge [79]. Recently, analytical formulas

for

the absorption

lineshape in

quantum

wells including

bound and continuum

exciton

states

have been developed [81, 82], which are based on

the idea that the

excitonic

spectrum

has a dimensionality intermediate between

2D

and 3D.

83


28/56

1.5

LH1

HH2

HH3

LH2

t

'

1.0

e z

-

c

a

:.;:;

Q..

e x

a

0.5

f)

L l

0

80 130

18

E - Egop

meV)

Figure 8. Interband absorption probability in a 80 Awide GaAs-Gao.6Alo.4As quantum well,

without

excitonic effects.

n=l n=2 n=3

E

Figure 9 Schematic picture

of

the absorption probability in two-dimensional systems,

without

exci

tonic effects dashed line) and with excitonic effects solid line).

84


29/56

Intraband transitions. In bulk crystals,

intraband

transitions can only occur in

the

presence of some scattering mechanism (see Sec. 2.2). In quantum-confined struc

tures, intraband (or intersubband) transitions are made possible by

the

discretization

of the energy levels. Such transitions can only be observed in doped or photoexcited

samples [70, 83, 84].

The

dipole

matrix

element for

an intersubband transition

between

states of the

form (89) describes

what

is called

an

envelope-state transition.

The term

containing

the

derivative of

the

Bloch function now gives no contribution, since

the momentum

matrix element between Bloch functions of either the conduction or

the

valence band

vanishes.

The term with the

gradient of

the

envelope function is of

the

same

order

as

that coming from the correction

to

the lowest-order wavefunction on

the

r.h.s. of Eq.

(89). This second

term

in fact dominates,

and

has

the

effect of multiplying

the

matrix

element by

maim:. The

result is

(98)

For light polarized along

the

growth direction, transitions are allowed only between

conduction (or valence) states of opposite parities. For the transition between

the

two lowest conduction levels,

the matrix

element

in

the

case of infinite barriers

is

8/3) -in/Lw . The

oscillator

strength

(34) for

the

lowest intersubband

transition

is then evaluated as

fz 256

ma

0 96

ma

,

271r2

m; m;

(99)

i.e., it is nearly identical to that for interband transitions [85 86]. However

the joint

density of states

is quite different: for parabolic

subbands

with

the

same effective mass,

the transition

energy would be

the

same

at

all wavevectors, resulting in a 5-function

line where all the oscillator

strength

is concentrated. Line broadening comes from

conduction band nonparabolicity, as well as from many-body effects [87]. Assuming a

Lorentzian broadening,

the intersubband

absorption probability can be calculated from

(92)

and

(98), with the result

(100)

where

ns

is

the

areal density of electrons in

the

first conduction

subband. Intersubband

absorption is smaller

than

interband absorption by a factor v fi

2

n

s

/ m:,)

[70,86].

When

the

transition

matrix

element is evaluated in

the

gauge

eE

r,

the

so-called

giant electric dipole is found [85]. Of course

the

physical results are independent of

the

choice of

the

gauge,

and

in fact the same expression (99) for

the

oscillator strength is

found in

the

two gauges.

The

physical meaning of

the

giant electric dipole is that an

oscillator strength comparable to that of interband transitions is found in

the

infrared,

i.e., at a much lower energy as compared

to interband

transitions.

For light polarized in

the

planes, intersubband transitions become second-class (i.e.,

the matrix

element is linear in

the

wavevector).

In the

valence

band they

become

allowed due

to

valence

band

mixing. A theoretical analysis of

intersubband

transitions

in the valence band has been given in Ref. [88].

3.2 Excitons in Confined Systems

Thin-film and QW regimes: overview. In

this Section we discuss a few aspects

of the theory of excitons in

thin

layers [89, 90]. Two regimes can be distinguished,

85


30/56

according to the relative values of

the

well width L and

the

exciton radius a o In

the

limit

L

ai3, which we shall call

the thin-film regime, the

excitonic

Rydberg R*

is

much

larger than the quantization energy

2

/ m* L2), and the exciton is only weakly

perturbed

by confinement.

The

internal electron-hole wavefunction is

undistorted,

but

the

motion of the center of mass of

the

exciton

is

quantized [91, 92]. In

the

limit

L

v ai3 which we call

the quantum-well regime), the

excitonic Rydberg is smaller than

the

quantization

energy

of

the

subbands,

and

separate quantization

of

the

electron

and

hole subbands occurs.

The

distortion of

the

internal exciton wavefunction

due

to

a decrease of the average electron-hole separation leads to

an

increase

of

the binding

energy

and

of

the

oscillator strength per unit area as the well width is reduced [93,94].

Excitons in semiconductor heterostructures are usually described within

the

envelope

function scheme, with the electron and hole confinement potentials being added to the

effective-mass Hamiltonian. Due

to

translational invariance in

the

layer planes,

the

exci

ton is characterized by

an

envelope function F p,

Ze,

Zh), where p is the in-plane relative

coordinate. A suitable description of

the

exciton in

the

thin-film regime neglecting

the

valence-band degeneracy and in the assumption of infinite barriers)

is

provided by the

variational wavefunction of D Andrea and Del Sole [92],

F p,Ze,Zh)

=

N[cos KZ) - Fc z)cosh PZ)

+

Fo z)sinh PZ)]e-

r

/

a

,

101)

where r is

the

electron-hole relative coordinate,

Z.is the

center-of-mass coordinate along

the

growth direction, N

is

a normalization factor, and P,a are variational parameters.

The

functions

Fe z), Fo z)

are determined by

the

fulfillment of

the

no-escape boundary

conditions

F ze

= ~ = F Zh = ~ = o

The

requirement that

the

wavefunction

has a continuous derivative leads to

the

following quantization condition for the center

of-mass wavevector K

KL PL

t a n T

Ptanh

T

= o

102)

The quantity

1/

P can be

interpreted

as a dead-layer thickness

[40]

and

is of

the order

of the exciton radius. For P L

1

the quantization condition reduces to the center-of

mass quantization K = r / L - 2/P).

The

exciton levels are those of a particle of mass

M in a

box of

width Lcff

=

L

-

2/P For thin layers, the center-of-mass

quantization

is

only approximately true, and the full quant izat ion condition 102) must

be

used. The

transition-layer thickness 1/

P

decreases with

the

well width.

In

the

quantum-well regime,

the

exciton wavefunction is more suitably

represented

in a basis consisting of products of conduction

and

valence subbands,

F p, Ze, Zh) = L

Ajjle-UIPCj Ze)Vj{Zh).

j l

103)

For very narrow wells, only one pair of valence/conduction subbands needs to kept. In

the limit L 0 if the barriers are taken of infinite height, the binding energy tends

to

the value

appropriate

to the two-dimensional Coulomb problem,

R;D =

4R;D

[79]

With finite barrier height,

the

binding energy reaches a

maximum and

tends

to the

value appropriate to

the

barrier material as the well width tends to zero [95]. For wide

wells, several pairs of subbands must be kept in

the

expansion 103).

The

wavefunctions

101) and 103) have been shown to match for L v 3ai3 [96].

A quantitative determination of the exciton binding energy in quantum wells is

complicated by the interplay of several effects

[73].

In real

structures,

the finite height

of

the

barriers

and the

variation of

the

band parameters from one material

to

the

other

must

be taken into account. Due

to the

degeneracy of the valence band, the kinetic

86


31/56

20

18

:>

16

-;;::

14

\: '

5

12

10

8

HHI-CBI Is)

6 ~ ~ ~ ~ ~ L ~ L ~ L ~ L ~ ~

o

50

100

150 200

well widUl

A)

100

80

~ 6

5

il 40

lS

t

20

\t

H H I C B I

Is)

\

x = O . 2 S

\

b) \ o x=OAO

,

,

,

,

O ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ L ~ ~ ~

o 50 100

150

2

well width

A)

Figure

10. a) Binding energy of the ground-state HHI-CBl exciton and

b)


per unit area

for in-plane polarization

of the

ground-state

HHI-CBl and

LHI-CBl excitons in

GaAs/Al Gal_ As quantum wells. From Ref.

[73]

energy of the holes must be described by the Luttinger Hamiltonian: thus the effective

mass Hamiltonian must be taken to be

2

H =

Ec -iVe)5

+T

/ - iv\)

- I e

5

+ Ve Ze) +Vh Zh))5

,. 104)

Eb

re

-

rh

When the

off-diagonal

terms

of

the

Luttinger Hamiltonian are neglected, heavy-

and

light-hole excitons become uncoupled; however, coupling between

the

two series has an

important

effect

on the

binding energies

and

on

the

oscillator

strengths.

Other effects of

a comparable size are nonparabolicity of

the

bulk conduction band

and

the difference in

dielectric constants between well and barrier materials. All the above effects go in

the

direction of increasing the binding energy,

and

taken together result in very high binding

energies see Fig.

lOa), which in

GaAs/

AlAs

quantum

wells can

be

even higher

than

the

two-dimensional limit [73]. This prediction has recently been confirmed experimentally

[81]

The

combined effect

of

nonparabolicity

and the

dielectric mismatch results in a

decrease of

the

critical well width, below which

the

exciton binding energy tends

to the

value appropriate to

the

barrier material. However a quantitative calculation of this

effect is still lacking. A similar problem arises in short-period superlatt ices: when

the

superlattice

period decreses below a critical value,

the

exciton wavefunction spreads

out

and extends

over several layers

[97]

However

the maximum

value

of

the

binding energy

is likely

to

be strongly increased by nonparabolicity

and the

dielectric mismatch.

In

general,

the

increase of the binding energy due to

the

smaller dielectric constant of

the

barrier

often called

dielectric confinement

[98]) is of great interest as a way

to

increase

the stability

of

the

exciton against

thermal

dissociation.

Oscillator strength. The oscillator strength is defined, as usual , by Eq. 34).

For excitons in

thin

layers,

the

oscillator

strength

is proportional

to the

area S of

the

87


32/56

10

8


33/56

accurate calculation of

the

oscillator

strength

requires theories which take into account

valence band mixing and coupling between excitons belonging to different subbands

[100, 73]. For quantum wells

of medium width

the two-dimensional approximation

overestimates

the

oscillator

strength

by a factor between three

and

four. The oscillator

strengths

of

the ground-state excitons in GaAs-Gal_xAlxAs quantum wells are shown

in Fig. lOb, and

are

in good agreement with absorption

and

reflectivity measurements.

Excitons in QWs:

symmetry

properties

and selection

rules.

Due to valence

band mixing, heavy- and light-hole subbands in quantum wells are coupled for finite

values of

the

in-plane wavevector. Since exciton states are built

up

from band states

with wavevectors v

lias

valence

band

mixing can be expected

to playa

role on

the

selection rules for excitonic transit ions. Selection rules for excitons in

quantum

wells

have been derived in a model-independent way using

symmetry arguments

[101, 102].

Here we give just a few relevant examples.

The point group Td of the zincblende structure becomes D2d in quantum wells, due

to the

reduction of

symmetry.

The

lowest heavy-hole and conduction

subbandshave

r6

symmetry

at

kll=O, while

the

lowest light-hole

subband

has

r7

symmetry.

The

symme

try of exciton

states

is given by the product decomposition of 61). For

ground-state

excitons,

the

envelope function transforms according

to the

identity representation.

Thus the ground-state HH1-CB1 exciton transforms as r6 r6 = r

1

9 r

2

9 r

s

, while

the ground-state

LH1-CB1 exciton has multiplicity

r6

r7

= r3 E9 r

4

E9 rs The

representations r

1

r

2

r

3

r

4

are nondegenerate, while rs is twofold degenerate. Since

the

z-component of

the

dipole operator has

4

symmetry, while the x components

transform according

to

the

rs

representation, we see that for in-plane polarization both

heavy- and light-hole excitons are allowed, while for light polarized along

the

growth

direction only

the

light-hole exciton is allowed. t can also be shown that the oscillator

strength for

the

z-polarized LH exciton is four times that for in-plane polarization.

Although

the zincblende lattice is not invariant

under

space inversion, effects re

lated to the violation of inversion symmetry are small and difficult to observe in III-V

semiconductors. Therefore, parity selection rules hold for excitons in

quantum

wells to

a very good approximation. For example, excitons like LH1-CB2 or HH2-CB1 which

belong to conduction and valence sub bands of opposite parities are forbidden in the

s-like

ground

state. Such excitons can be observed only in p-like excited

states.

Parity

symmetry is broken in the presence of

an

electric field applied

or

built-in). We refer

to

[101, 102,

73]

for details.

Parity

selection rules for two-photon

transitions

are op

posite to those for one-photon transitions, thereby allowing very useful complementary

information to be obtained

by

two-photon spectroscopy [102].

The above selection rules maintain their validity also when valence band mixing

is taken into account.

In

fact, selection rules for ground-state excitonic transitions

are identical with those for

the

subbands

at

kll=O, since excitons

at

kcx=O transform

according to irreducible representations of the crystal point group.

The

role of valence

band mixing is to give a finite oscillator strength to some excitons not in s-states like

LH1-CB2

2p).

Also, valence

band

mixing changes

the

ratio of

the


of the in-plane polarized ground-state HH1-CB1 and LH1-CB1 excitons from

the

value

3:1 characterist ic of

the

subbands at kll=O) to

about

two [73]

Excitons in QWs:

exchange

interaction. The electron-hole exchange interac

tion splits the exciton states corresponding to different irreducible representations of

the

point group. t is proportional to the singlet component of a given exciton state

via the spin-orbit

factor g

The spin-orbit factors for

QW

excitons are: for the r

89


34/56

LHI-CBl

HHI-CBl

4 1)

r

/

/

/

/

5

(2)

/ ~

/ /

1

:;

__

_X_I6_ 4_)--1.L____ _ 1_)

5

(2)

It

x I6 4)

x.y

x.y

Figure

12. Schematic representation

of

the effect

of the

exchange interaction on

the

ground-state

excitons

in quantum

wells. Numbers

in

parentheses denote the degeneracy

of the

state.

representation = 1

and

=

1/3

for HH

and

LH excitons, respectively; for

the

r

4

representation = 4/3 for the light-hole exciton. All other states have = 0 and

no exchange contribution to the energy. The exchange splittings of the

ground-state

heavy-

and

light-hole excitons in QWs are illustrated in Fig. 12.

In

bulk semiconductors,

the

dipolar part of

the

exchange interaction

is

nonanalytical

for k

ex

-- 0 and gives rise to a splitting between longitudinal and transverse states with

the symmetry of

the

dipole see Sec. 2.3). For an isolated quantum well, we use

the

following terminology: taking

the

exciton wavevector k

ex

=

kxx

along

the

x-axis, we

denote

by

T mode

the

rs exciton with polarization vector E ii we call L-mode

the

other

rs

state

with polarization E

x

and we call

Z mode the

r

4

exciton with Ell z.

The

Z mode exists only for the light-hole exciton.

Thus

the longitudinal-transverse spli tting

is defined

to

be

the

exchange splitting between

Land

modes with rs symmetry.

The exchange interaction for QW excitons has been calculated in Ref.

[26]

with

the

Wannier-function formalism, and in Ref. [103] with the k-space methods.

The

two procedures yield essentially

the

same results.

In

Fig. 13 we show

the

long-range

part of the exchange interaction for the light-hole exciton as a function of the in-plane

exciton wavevector k

ex

I t can be seen that

the

longitudinal-transverse splitting vanishes

linearly in k

ex

for k

ex

--) O

This is an effect of reduced dimensionality,

and

it can be

understood

from

the

fact

that the

T

splitting

at

k

ex

=

0 is proportional

to

the

oscillator

strength per unit volume see Eq. 68)): for a single QW, the dimensionless oscillator

strength is proportional

to the

area

and

the oscillator strength per

unit

volume vanishes

[26].

The situation should be different in the case of multiple quantum wells, since

the

oscillator strength per unit volume is finite and a nonvanishing T splitting between

rs excitons

at

kex=O is expected to occur [104].

t

can be seen from Fig. 13 that

the

z-polarized r4) light-hole exciton at k

ex

= 0 is

shifted upwards with respect

to the

rs exciton by an energy of the order of a meV. Such

a splitting, which is expected from symmetry arguments, arises from the depolarization

field for a polarization perpendicular to

the

interfaces. The

ZT

splitting of quantum well

90


35/56

x=O.4

1.0

LH exciton

g

>

r

.5

L

0.0

T

0

10 15

kL

Figure 13.

Long-range exchange energies

of the L,

and

Z

modes for

the

ground-state

LHI-CBl

excitons in a

60

A

wide GaAs/Alo.4Gao.6As

quantum

well. From Ref.

[26].

excitons has been observed experimentally [105, 106].

Other

features

of the k-dependent

exchange interaction

are

much more difficult to observe, since they are overwhelmed by

spatial

dispersion.

t

has already been mentioned in Sec. 2.3 that taking into account

the nonanalytic

part of the exchange interaction corresponds to including the depolarization field in the

dielectric response, i.e, to solve Maxwell equations in the instantaneous limit

[1,

29].

Thus it

is no surprise that

the

dispersion of

the

exciton taking into account

the

dependent

exchange interaction coincides with

that

found from

the

solution of

the

electrostatic

equations [107].

3.3 Polaritons

in

Confined

Systems

Division into non

radiative

and

radiative

modes. Polariton effects are defined

to

be

the

effects coming from

the

interaction between excitons

and the retarded

trans

verse)

part

of

the

electromagnetic field.

There

is a

fundamental

difference between

polariton

effects in bulk

and

in confined systems see Fig. 14).

In

bulk crystals, due

to

the

conservation of crystal

momentum

an exciton with a given wavevector k

ex

interacts

with

only one

photon

with

the

same wavevector

and

polarization: there is no density

of states

for radiative decay, and

the

mutual

interaction

gives rise

to

the

stationary

po

lariton states. In quasi-two-dimensional systems, due to the breaking of translational

invariance along

the

growth direction, an exciton with in-plane wavevector kll interacts

with

photons with the same in-plane wavevector

but

with all possible values of k

z

Thus

there is a density of states for radiative decay, given by

~ V n 2 i w

p(kll w) == L,o(/iw - k ~ + kn = sCi: ~ B k o -

kll ,

kz 7r C V o

-

ll

108)

where

n

is

the

index

of refraction,

o

=

nw/

c is

the

wavevector of light

in

the

sample,

and

B a::)

is

the

Heavyside function

B a::)

=

1 for

a

>

0,

B a::)

=

0 for

a

.(z)p>.(z ),

>

110)


37/56

where A are

quantum numbers

of

the

excited

states

of

the

crystal . We consider only

the

resonant

terms, and

incorporate

the

contribution of all nonresonant

terms

in

the

background dielectric constant. For excitons in thin layers, we have

k

_ g.xl u

c

l

er

lu

v

} 2

X x W - . )

Ii Wk

- W - 2

111)

p.x z) = F.x p =

O,z,z), 112)

where F.x p, Ze Zh) is

the

exciton envelope function.

Thus the

semiclassical description

of polariton effects in thin layers consists of the following three steps [111, 92, 112]:

1.

Determine the

excitonic levels

liw.x and

wavefunctions F.x p,

Ze,

Zh)

by

solving

the

Schrodinger equation for

the

exciton;

2. Identify

the

resonant terms

and

calculate

the

nonlocal susceptibili ty 111);

3. Solve Maxwell

equations

with

the

constitut ive relation 109).

Before presenting a few applications, let us discuss some features of

the

nonlocal theory.

III The formalism can be applied to both the thin-film and quantum-well regimes for

the

exciton. For

QWs

only one resonant level needs

to be

considered, while for

thin

films one has

to sum

over

the

quantized levels of

the

center of mass.

III The

expression 110) has the important feature that each

term

in

the

sum is

separable in z, z .

This

allows Maxwell equations to be solved analytically in

many

cases [111].

III

In the

thin-film regime,

the

formalism is intrinsically ABC-free. Additional bound

ary

conditions are embodied in

the

no-escape boundary conditions for electrons

and

holes separately. Also,

the

formalism can account for

phenomena

related

to

the distortion of

the

excitonic wavefunction close to the surface, like a decrease

of

the

dead-layer

depth

for

thin

layers.

III The

formalism takes full account of polariton effects, including the radiative broad-

ening i.e.

both

real and imaginary parts of

the

self-energy are found).

In

expres

sion 111), , is a

nonradiative

broadening

term: the

radiative width arises auto

matically from

the

solution of Maxwell equations with retardation,

and

manifests

itself as

an

additional broadening

of

absorption or reflectivity peaks.

At

this point we should mention that an alternative procedure has been developed,

known as Stahl's coherent wave approach [41] in which

the

Schrodinger equation for

the

excitons is formulated in a density-matrix scheme

and

is solved in

the

presence of

the

electromagnetic field.

Stahl's

approach gives equivalent results for linear propert ies,

but

can also be applied

to

nonlinear phenomena and

to study

polariton effects at frequencies

above the band edge.

Polariton effects in thin

films:

polariton

interference

vs.

e quantization.

Interference of polaritons was observed

in

CdS

and

CdSe slabs [55],

in

CuCI slabs [113],

and, most

recently, in GaAs

thin

layers [114]. An example

is

given

in

Fig. 15.

The traditional explanation

makes use of

the

dielectric function with

spatial

dis

persion, which gives the two polariton branches

j

= 1,2 as the solutions of

c

2

k

_ [ WLTWT ]

- Eoo 2

w

2

Wk -

w

2

- i w

J

113)

93


38/56

I

r

I gl h

b

ll

I I I

t;

I

I

n

F

a

I

1514 5 6 1518

152 1522

photon n rgy

me-V)

Figure

15.

Reflectance

spectrum

of

a

MBE

grown

GaAs

thin

layer,

in

the

spectral

region

of the

exciton resonance. From Ref. [114].

The observed interference pattern can be explained as follows

[55J:

below the transverse

frequency, a single period is observed, which corresponds to

Fabry-Perot

interference

of

the lower polariton according to the condition

2klL

= 27rm

L

is the

width

of the

slab). The period doubling above WL is interpreted as interference between UP and LP,

described by the condition (k

1

-

k2 L

= 27rm. Moreover, a longer period appear which

corresponds to

interference

of the UP

with itself.

The

reflectivity

can be

calculated

imposing additional

boundary

conditions like Pekar s ABC

I i

Pi

=

0

at the

surfaces.

For GaAs, the interference pattern is more complicated, due to the presence of two

exciton-polariton branches (heavy and light) [114J.

A

related

phenomenon which occurs in very

thin

layers is quantization of the center

of

mass

of

the exciton, which can be observed in luminescence and reflectivity [115, 116,

117J. Assuming

an

excitonic wavefunction with CM confinement, selection rules for

optical transitions can

be

derived, which are

in

reasonable agreement with

experiment

[115J.

The two

phenomena

can

be treated in

a unified way using

the

nonlocal scheme

described above. Numerical calculations of absorption and reflectivity profiles [112, 92J

show

that

the

conditions of polariton interference are recovered for frequencies

not

too close

to the

resonance region.

When the

polariton

is

excitonlike,

the

interference

condition

2kL = 27rm

corresponds

to the

quantization condition

k = 7r

/

L)m

for

the

center-of-mass wavevector: this is the case for the LP for

k

ka and for

the UP

close

to the

longitudinal frequency. For very

thin

layers,

the

quantized wavevectors are

too

large

to

involve

the

UP. Also, as we mentioned in Sec. 3.2, quantization

of the

center

of mass is only approximately described by

the

condition

k = 7r

/ Lerr

m, and the

more

precise quantization condition (102)

must be

used.

In the

resonance region between

WT

and WL

the

positions of absorption and reflectivity peaks

cannot in

general be simply

explained

by

any of the two pictures; instead the full nonlocal theory must

be

used.

Polariton effects manifest themselves

both

as shifts with respect

to the quantized

CM

levels,

and

as additional broadenings of the peaks

due

to

the

radiative width.

The

nonlocal approach has been developed

so

far only for

the

case of

normal

inci

dence. Polariton dispersion in

thin

films has been studied

in

Ref.

[118J

using a macro

scopic

theory

with additional

boundary

conditions. The dispersion of

stationary

po-

94


39/56

lariton states

has

been

calculated for

both T-modes

s-polarization)

and L, Z

modes

p-polarization).

The

case of

p

polarization is more complicated, since

the

longitudinal

mode

is also excited

in

layers with L

A

and

therefore

the

ABC have two components.

We refer

to

[118,

36]

for details.

Polaritons

in

quantum

wells:

nonradiative

modes.

In

the

quantum-well limit,

the nonlocal formalism can be used

to

find the dispersion of nonradiative waveguide)

modes by taking only one resonant

term

in

the

nonlocal susceptibility [26, 107, 119]

the

full quantum-mechanical theory is developed in Ref. [120]).

The quantity

p,\ z) =

FQw O)c z)v z), where

FQw p)

is

the

in-plane electron-hole wavefunction. We take a

propagation vector

k

=

kxx

along

the

x

axis. We enclose

the quantum

well

in

a large

box

of width

d,

such that

the

response function is essentially zero for Izl

Iz l > d/2.

We

assume an electric field which decays exponentially far from

the

well:

114)

where

the

decay

constant

is given by

2

_

k

2

W

)1/2

a - x - Eoo .

c

115)

Here we give

an example of the

solution of Maxwell equations for

the T-mode

T

E,

or

s-polarization).

The

electric

and

magnetic fields are given by

E

B

Maxwell equations imply

Since for TE modes V . E

=

0

the

above equation gives, for

Izl d/2,

116)

117)

118)

Maxwell

boundary

conditions require

that Ell Ell

are continuous

at z = d/2:

from 116),

BEy/Bz is also continuous. Hence boundary conditions can be conveniently expressed

in terms of E

y

, yielding

119)

t can

be shown by

direct substitution that

a solution of Eq. 118) with

boundary

conditions 119) is

Ey(z) =

J

z p(z )e-alz-z l.

120)

Substituting in

118), we obtain

the

dispersion relation of

T-mode

polaritons in

the

form

121)

95


40/56

1623

L=60 A x=0 4

51622

ko the effects of retardation are negligible,

and

the

dispersion of

the

exciton with the long-range exchange interaction Fig. 13) is

recovered.

Luminescence from the L-polarized QW excitonic polariton has been reported in

Ref. [121), where coupling to the nonradiative modes

is

achieved by use of a grating.

96


41/56

Possible evidence for

QW

excitonic polaritons is given

by the

time-of-flight experiment

of Ref.

[122J,

in which an increase of the delay

time

of a light pulse

propagating

in the

layer planes

is

observed close to

the

exciton resonance. This could represent the analog

of

the

classical group-velocity experiments

[57J.

I t is of interest to investigate to which extent the quantum well can be described

within

a local scheme with

an

effective dielectric function,

WLT

f W)

= foo[1

.

J

a -

W

-

Z Y

126)

where WLT is a parameter which can be called effective LT splitt ing .

The

relation

between

the two approaches can be

studied

for the normal-incidence reflectivity. Within

the nonlocal scheme, the reflectivity of an isolated

quantum

well

surrounded

by infinite

barriers

is found to

be

[123J

r

2

R w)

=

a ,

w

-

wa)2

T

ra)2

127)

where

ra = ~ ~ x y

nmaC S

128)

will be

interpreted

in

the

next section

to

be

the

decay rate of the electric field. Within

the local scheme, the reflectivity of a layer of width d is calculated

to

be

R w) = 1 f W) -1 1- e

2iqd

) 12

1 Vf W))2

-

1 -

VE w))2e

2iqd

129)

where q = V

W w

/ c.

The

two approaches give identical results

under the

condition

qd

1,

which can be written as

[124J

d)2

EooWLT

\ 1 1 1.

A a

-

W

-

Z Y

130)

I f this condition is satisfied, the relation between

the

phenomenological

parameter

WLT

and

the

microscopic parameters is WLT = r

a

/ k

a

d . For a single

quantum

well, d

coincides with

the

well

width L

w

,

while for a multiple

quantum

well

d must

be identified

with

the MQW

period

Lw+Lb

[123, 124J. A sufficient condition for

130) to

hold is that

the

width Y

be much larger than

the

effective LT splitting

[123J.

This is usually verified

experimentally. The parameter WLT can

be

expressed as

the

bulk LT splitting, multiplied

by an enhancement factor related

to

excitonic confinement. However we

remark

that

for a single quantum well WLT is not

the

exchange splitting between longitudinal and

transverse

rs excitons, which, as

we

have seen, vanishes for k

ex

---t

O. For

MQWs

on the other hand WLT is

expected

to coincide with

the

finite exchange splitting at

k/l

=

O.

A thorough comparison between local

and

nonlocal schemes for what concerns

the

exciton-polariton dispersion

is

done in Ref.

[124J,

also for the case

of

MQWs where

it

is shown that the condition for

the

validity of

the

local description is given again by

Eq.

130).

The nonlocal scheme

can be

used to calculate the reflectivity and attenuated total

reflection of a quantum well. Such a calculation

is

done in Ref. [125J for both

sand

p

polarizations. Excitonic polaritons in

the

radiative region

appear

as peaks or dips in

the reflectivity, while nonradiative polaritons manifest themselves as dips in

the

ATR

spectrum

[125J.

In Ref. [126J the nonlocal formalism

is

applied to a calculation of the

reflectivity in quantum wells, quantum wire, and quantum dot structures.

97


42/56

0.2

0.1

o 3

L

Ie)

Figure 17. Radiative decay rate of the k =

a

exciton as a function of crystal thickness.

From

Ref.

[130]

3.4 Exciton Radiative Lifetime: from

Bulk

to Confined Systems

Crossover

2D--t3D:

overview. Excitons in

thin

layers with thickness

,\

have

a finite radiative lifetime, as was first shown in Ref. [108]. For Frenkel excitons,

the

decay

rate is

of

the order

of

r

\2/a

rmob where

a is the

lattice spacing

and rmol

is

the

decay

rate

of an isolated molecule.

The

decay

is

called

superradiant ,

because

the

decay

rate is proportional to the

number of molecules within a wavelength, which

contribute

in phase

to the

decay [127].

The

superradiant

decay of excitons

in

anthracene

films was observed

in

Ref.

[128].

For Wannier-Mott excitons,

the

decay

is

still often

called

superradiant, although

it

is not

meaningful anymore

to

speak

of

an enhancement

with respect

to the

isolated molecule. I prefer

the

picture of Fig. 14, according

to

which

the

intrinsic radiative decay of a free exciton in low-dimensional systems

is

due

to

coupling with a continuum of photon states.

For excitons in

thin

layers,

the

superradiant decay

must go

over

to the

stable

polariton behavior as

the

film thickness becomes

>.. The

behavior of

the

decay

rate

as a function of thickness

is

calculated in Refs.

[129, 130]

(see Fig.

17,

which is

calculated for Frenkel excitons). For

L

\,

the

decay

rate

increases as

r

)

L:

for

Wannier-Mott excitons, this corresponds

to the

regime where

the

center of

mass

of

the

exciton is quantized,

and the

oscillator

strength

per

unit

area increases linearly with

the

well thickness (see Fig. 11). For >.

the

decay

rate

has an oscillatory behavior and

decreases as

1/

due

to the

reduced overlap between exciton

and photon

wavefunctions.

In fact, for

\

the

physical picture is expected

to

coincide with

the

bulk polariton

behavior, for which the decay

rate

is an escape rate which goes as 1/ (see Eq. (85)).

This

crossover from superradiant decay

to

bulk polaritons is therefore well described

by

the

results of Refs.

[129, 130].

In Ref. [130]

it

is also shown that

the

bulk polariton

behavior is

obtained at

all wavevectors, when proper approximations

to

the

radiative

self-energy are

made.

The

QW

regime

as

\,

where

the

oscillator

strength (and

therefore

the

decay

rate)

increases as

the

well width is reduced,

is not

described in Refs.

[129, 130].

98


43/56

Radiative lifetime

of

QW excitons. Now we calculate

the

radiative decay

rate

of a free exciton in an isolated QW,

under

the assumption that

the

in-plane wavevector

kll

= kxx is conserved [131, 119, 123, 132]. We use the Golden Rule (20), with the

interaction

Hamiltonian being given by

(21). The

initial

state

consists of

the

exciton

state with polarization vector e

and

no photons present, while

the

final

state

is

the

crystal

ground

state

plus a

photon

with wavevector k

=

(qll, kz)

and

polarization

>..

Expressing

the

vector

potential in

second quantization according

to (81),

the

matrix

element is calculated

to

be

(131)

The

squared

matrix

element summed over

the

photon polarizations can be expressed

in terms of

the

exciton oscillator strength as

2

11 e

2

li,2

LI(il.C fW = 2 -L E l e . g A )1

2

A

n ma

V

A

(132)

Now

the

Golden Rule gives

(133)

Evaluating the

one-dimensional density of states as

in (108)

gives

the

decay

rate in

terms

of

the

oscillator

strength

per

unit area

as

r kx) = 211

ka

L E Ie. g A)1

2

0(k

a

- k

x

,

n

mackz

A

S

(134)

where

now

kz = Jk6 - ki.

We must now specify

the

exciton polarization vector e

and

the photon

polarization vectors g A). For a given in-plane wavevector

kll

= kxx,

the

two

orthogonal

photon

polarization vectors can be chosen as

(135)

For the T-exciton, I:A f,le. g A)1

2

= fxy, where fxy is the oscillator strength for in-plane

polarization.

2

For

the

L-exciton, we

obtain

I:A

Ele.

g A)1

2

=

fxy(kz

/k

a

)2,

while for

the

Z-exciton

I:A Ele .

g A)1

2

=

fz k

x

/k

a

2.

Thus we obtain the

radiative widths

of the

T

L, and Z excitons for kx ka as follows:

211 e

2

fxy

ko

n mac S k

z

211 e

2

fxu kz

---:;;: mac

o

211 e

2

fz k;

n

mac

S

kokz

136)

(the

factor of two was missing in Ref.

[123]). The

decay

rate

vanishes for kx

>

ko For

the

light-hole exciton

fxy =

4fz. Optically inactive

states

obviously have zero radiative

width.

2Note

that

for a given exciton polarization vector

e, the

oscillator

strength

can

be thought to be

summed over all possible polarizations, since only the vector i =

e

contributes. Thus the oscillator

strengths

fxy, fz are

exactly those calculated e.g. in Ref.

[73], and

reported in Fig. 10.

99


44/56

At kll

0, Land

T

modes have

the

same decay

rate 2ro

27re2/ nmoc)) fxlJ/S).

Taking an oscillator strength

fx,,/ S

= 50

.10-

5

A 2 which is

appropriate

for

the

HH1-

CB1 exciton in

GaAs/

AlxGal_xAs quantum wells of

about

100 Awidth [73], we obtain

firo 0.026 meV.

The

decay time of an exciton

state

is

TO 1/ 2r

o

) 12 ps [132].

This coincides with the result found from

the

reflectivity calculation

with the

nonlocal

susceptibility Sec. 3.3):

the

decay

time

of

the

electric field is

l / r

o, while

the

decay

time

of

the

intensity is twice as short. Note that

the

Golden Rule gives directly the decay

rate

of

the

exciton

state.

This lifetime is much longer than found in Ref. [131], where

the

index of refraction is not considered

and

the two-dimensional limit for the exciton

is assumed. Observation of such short lifetimes has been

reported

in Refs. [133, 134].

The

radiative

width

136) represents

the

imaginary part of the exciton self-energy

due

to interaction with

photons.

The

real

part of the

self-energy, calculated e.g.

in

Refs.

[131, 119, 132], is a small effect.

The largest polariton effect or quantum-well excitons

is the radiative lifetime.

Note that

the

decay

rate

diverges as

kx

- - t

o

for the

T

and

Z

modes.

This

divergence, which can be

traced

back

to

the

density

of

states 108), is

integrable and disappears when

the

thermal average is taken. A slight broadening of

the

wavevector due

to

inelastic scattering would wash

out

this divergence, which we

believe to have no physical consequences.

The

intrinsic

radiative

decay of free excitons is calculated assuming conservation

of

the

in-plane wavevector, thereby disregarding

the

effects of interface roughness and

acoustic phonon scattering. Wavevector conservation is likely

to be

a good

assumption

when the coherence

length

of

the

exciton is longer than the wavelength of light. Conse

quently,

the

short intrinsic lifetimes can only

be

observed in carefully selected samples

at low

temperature

[133, 134].

In

general, several effects can change this simple picture.

At

low

temperature,

excitons can

be

bound to

impurities

or

interface defects. Also,

interface roughness

acts

as a disordered

potential

for

the

exciton motion,

and

produces

a mobility edge within

the

inhomogeneously broadened exciton line [135]: below

the

mobility edge

the

exciton is localized by

the

disorder, while above

the

mobility edge

the

exciton is mobile

and

interface roughness acts as a dephasing mechanism. Finally,

scattering with acoustic phonons has to be taken into account at finite

temperature.

The

interplay

between all these effects constitutes a complicated problem, which is only

partly

understood

at time of writing. In the following I shall discuss a few models which

have been proposed [136].

Effect ofthermalization.

Thermalization is due

to

inelastic scattering

with

acous

tic phonons [53], which changes

the

exciton wavevector but does not change the

total

exciton population scattering with optical phonons is not expected to be relevant for

thermal

energies smaller than

36

meV, which is

the

LO phonon energy in GaAs).

The

key point is comparing the scattering rate with

the

radiative lifetime. The scattering

rate

by acoustic phonons is measured to be linear in with a coefficient I 5 pe V /

K

for a QW of 135 Awidth [53] Thus thermalization processes are faster than radiative

decay for

>

10

K. There

is also

other

evidence for this conclusion, coming from

a high-energy Boltzmann tail of

the

exciton lines [137].

Thus it

can be assumed as a

working

assumption

that

thermalization processes are faster

than

radiative recombina

tion, i.e., that excitons always have a thermal distribution while they decay radiatively.

However the possibility of a failure of this assumption must be kept in mind, particu

larly for thermalization between dipole-allowed

and

triplet exciton states see Fig. 12),

since spin-flip scattering is likely

to

be slower. Also, having a thermalized distribution

depends

on the

conditions of excitation resonant or nonresonant)

and

on

the

time of

observation [134].

100


45/56

In the

assumption

of a rapid thermalization,

the

decay

rate

of

the

luminescence is

given by the

thermal

average of

the

decay rate (136).

The

two characteristic energies

are the

thermal energy kBT, and

the

kinetic energy of excitons which decay radiatively:

the latter is at

most

El =

2

k5 (2M),

where

M is the

exciton mass. Using

M =

0.25

mo,

we find

1i2k6/(2M)

1.1 K. This means that, for

T

1

K,

only a small fraction of

excitons occupy

the

states

with

kx

l. c. andreani, optical transitions, excitons, and polaritons in bulk and low-dimensional...

Documents