J.

Phys.

I

France

(1992)

2

487-499

1992,

APRIL

487

PAGE

Classific3tion

Physics

Abstracts

73.00

72.40

78.55

optical

Oscillator strengths of the semiconductor superlattice

d'optique

Laboratoire

Vauquelin,

75231

(Received

h

appliqu£

ainsi

entrdment

with

envelope

Bloch

and +p recorded

parallel

field

formation

transitions

-p

low

at

of

temperature

qui

des

excitons

optical axis

%tre

peuvent Bloch.

+

p

semiconducteur

calcu16es

h

indirects,

d'excitons dons

l'dchelle

enregistrds

h

p

et

photocourant

l'aide

61ectrique

champ

Le

formation

la

transitions

transitions

be

can

in

de basse

field

and

ladder.

by

using the

a

of

between

oblique

the

photocurrent

the

model

present

interaction

electron-hole

asymmetry

Features

the

superlattice perturbative model

semiconductor

a

calculated

strength

induce

for

superr6seau

un

de

de

electric

both

accounted

be

can

rue

simple.

trks

Wannier-Stark

the

induit des

spectres

manikre the

dans

enveloppes

d'oscillateur

of

The

indirect

10

6December1991)

croissance,

de

fonctions

growth applied

the

to

functions.

of

direction

dlectron-trou,

strengths

oscillator

The

electric

Industrielles,

Chimie

et

accepted

optiques

transitions la

caract6ristiques pr6vues d'une

%tre

1991,

November

forces

les

entre

Physique

de

France

des

l'interaction

que

peuvent

Abstract.

inducing

21

Certaines

temp6rature

an

revised

asym£trie

une

Wannier-Stark.

under

05,

d'oscillateur

forces

Supdrieure

Ecole

Cedex

champ £lectrique parallble h de perturbation des avec

un

modme

d'un

Physique,

Pads

May 1991,

7

R4sum4.-Les soumis

in

spectra

simple

very

a

manner.

Introduction.

1. Due

to

the

applied towards

lack

of

coupling

resonant

along the growth the evenly spaced

transition

between

periods

axis

»

is

labelled

state away is respectively greater or smaller energy in the electron and a hole localized same

[2, 3]

Photoluminescence are

generally

than

lifetimes

The

of the

proportional

carriers. to

the

p the

well

It is

widely

overlap

of

the

known

value to

p

or

(Fig. I), the

-p

counterparts.

of p

and

on

the

show

The value

of

that

the

hole

matrix

element

envelope

for

functions.

that

of

ratio

the

applied

the

optical absorption in the radiative dependent the on and

is

that

its

to

an

0).

experiments

[2,4]

maximum

whose

sign indicating peak corresponding the

the

[5]

electron

wavefunction

a

of energy (peak labelled

their

transition

F.

+

than

photoconduction

and

weaker

intensities depends on the photocurrent is proportional photoluminescence strongly spectra are

p field

with

electron

an

hole

the

a

superlattice

ladder

and

adjacent quantum Wells when an electric field is (SL), the miniband spectrum converges iii. In a type I SL, the peak corresponding to the

between

of

Stark

«

hole

a

from

transitions

is

an

in a field

Tronc

P.

p

under

transitions electric

an

+p

to +p electric

while

structure

and

the

the

non-radiative

optical

transition

JOURNAL

488

PHYSIQUE

DE

N°

I

4

/

/

/

/

/ '

-2

'

-1

,

,

i

'j

,

,,

Sketch

I.

electric

al.

conduction

the

valence

and

potential

band

profiles

in

type I SL

a

under

an

field.

Dignam et

of

>,

II,

, '

/

/

O+1+2

,

~

Fig.

,'

/

',

al.

et

[4] using

their

[6]

fitted

have

Wannier

calculations

the

functions

of

asymmetry

an

photocurrent

of

measurements

and

variational

along

wavefunction

the

but

methods

the

Agullo-Rueda beginning of

by

made

imposing growth

the

at

direction

z

in

every

well.

2.

The

model.

Before developing proposed by Bleuse

type I SL excitons

is

After

equal

are

not

model

our

it

[I],

et

al.

if

the

readily

can

that

the

electron-hole

model,

Bleuse's

interaction

the

eigenfunction

SL

~

z cn,q

C~ ~

is

the

function

Bessel

(x

Jm

In same

hole

=

of

in

+

p

induces

is

=

two

different

wells

are

carrier

a

=

of

case

and

formation

the

functions

the

transitions

p

centered

z cn,

on

of

in

a

indirect

well

is

(1)

9~n(Z)

well

than

the

centered

miniband

~

at

(z ) the envelope the

origin.

Vfhen

A, C~,~ is given by

(A/2 eFd~

J~

:

q

quantum

larger

q-th

the

SL, d being the period and

isolated

:

(2)

~

integer

theory, the approximation is made approximation is made if two functions

located

special

the

of the

which

nd)

of the

(- i )m Jm (x )

this

of

(Z

9~

all the N wells where n is expanded over eigenfunction of the ground state of an Nefd the total potential drop in the SL

J~(x)

in

strengths

considered.

nq

where

shown,

be

oscillator

index ~

and

with

m,

(

:

~

Jm (x )2

that the ~~(z) functions corresponding respectively

considered.

(3)

=

orthogonal.

are

to

an

electron

The and

a

N° 4

SYMMETRY

A

overlap

The

of

the

STARK

OF

electron

nl)

Trill

SUPERLATTICES

IN

wavefunctions

hole

and

EFFECT

centered

at

489

origin

the

is

then

:

z Cl, Cl, 0(~S ~l)

(4)

0

~

m

runs

m

the

over

of the

wells

By translating corresponding to

S.

envelope

hole

the the

does

not

by

function

±

d

depend along

on

m

the

z

and is

axis,

equal to gets the overlap

taken one

to

be

transition

I

±

(~$ ~(

SL

=SZCS,oC$,1

(mini)

(5)

~

(nlln~i) =SZCS,oC$,-i m

It

readily

can

shown

be

(Fig. 2)

that C

i

~,

limited

gets

extension

of

since

SL

the

the

=

wavefunctions

C

+

(,

i.

localized

are

Neglecting

the

by the

electric

effect

of the

field,

one

:

(trim))

(nil n~ j) In

(- 1)~

~

the

same

it

manner

be

can

shown

that

(all n~ p)

(6)

=

:

(- IT

=

(all n))

(7)

,'

J

I I

i

i j i

/

,-, I

f i

,

/

I

I

',)

electron hole + d

hole

,

d ,

Fig.

Sketches

2.-

wavefunctions

In

carriers could to

be

conclusion, have also

of

the

be

considered

in

electron

the

by

translated

same

only

d

along

wavefunction the

z

centered

at

the

origin

and

of

the

heavy-hole

axis.

model, the + p and transitions between free p optical strengths when a type I SL is considered. This result, which optical absorption coefficient calculated in reference [II, has approximation ~~(z) functions actually because the not are

Bleuse's

the

deduced

±

oscillator from as

the an

strictly orthogonal. Nevertheless, it can be noticed that the carriers localization of the induced by the electric field, which is an important feature of the Bleuse's theory, has been unambiguously in all reported verified experiments [2-4, 7]. Let us general We shall perturbative model operating theory. tum to now our more use a functions introduced by Bastard [8]. The with the superlattice envelope of the carriers

JOURNAL

490

k

unperturbed superlattice being determined by Let

consider

us

the

SL

a

type I SL

p~

well

the

ll~

with

is the

The

mirror

allowed

the

formation

built

parallel

from

excitons,

of

0~

the

on

the

well

and

if

hole

a

and

located

considers

one

notices

one

and

electron

same

layers

the

to

of

values

conditions.

centered

system

is

$r~(z),

functions

Bloch

induces electron

image of the

mirror

well.

which an

N° 4

I

perturbative potential Q (z ). Moreover,

added

an

from

built

system

(-p)~

the

envelope functions are of cyclic boundery use

interaction

electron-hole

PHYSIQUE

DE

centered hole

a

the

at

in

that

a

the

on

centered

on

of

the

center

potential by the electron (hole) on the hole The interaction induced (electron) changed into f~P~l~~ when one goes from the first system to the second. the validity of using a perturbative model to take into An important step is now to prove electric efz and the excitons formation. the applied potential Q(z) account electric field F localizes the carriers in the SL. In III-V SLS We have that the applied seen localized effective If N is the of wells due to its large number the heavy hole is strongly mass. is is limited because potential which the electron wavefunction spread, N the drop over over well.

fP~l~~

is

=

spread

whole

the

A~

(if

of the

wavefunction

electron

probability

the

not

for

electron

an

Nefd from

N is odd

number

of

close

and N =

would

minibandwidth

zero)

be

:

(8)

A~

of N

can

be

checked

of

the

electron

value

of

the

first

electron

spread

the

and

the

wells

N

over

x

=

functions

being

I

displays been

has

Nefd/A~,

used

which

figure

in

shown

are

of the

;

functions

Bleuse's table

which

of

values

the

on

[3]

A~/2 efd

corresponding

of of the moduli these squares The voltage drop along N wells,

5.

=

with

J~(x)

the

over

=

value the

to

functions,

these

for

The

I.

to

3

N

a

expansion

the

truncate

approximation

good

wells

This

arguments.

symmetry

provide

which

exceed

cannot

tunnel

to

of A~, justifies heights of the SL

order

are

3 the

the to

all

for use

barriers generally widely smaller than the Moreover Rydbergs of the indirect for the electrons and the excitons small are electron when compared to A~ [9] showing that the perturbation arising from the hole induced by the electric field. interaction is widely weaker than the perturbation The electron wavefunction and hence of the electron-hole spread of the system being limited to N wells we have to use, to calculate the perturbed eigenenergies and the expansions eigenfunctions, the corresponding integration interval of the perturbed (of extension over z Nd~, the allowed values of k being those of a periodic wells. Indeed, after the array of N of

perturbative

a

electric

only k

N

to

Table the

field N

I.

J~(x)

model

has

or

(including Values, and

switched

been

wells

since A~ is holes. the

the

0

versus

value)

drastically and

N, ofx used

corresponding

N

electron

the

on,

This

more.

values

wavefunction reduces

drastically

to

truncate

the

increases

the

is the number

the

energy

expansion ofthe

of Nefd/A~.

x

=

A~/2 efd

Nefd/A~

3

1.4

1.07

5

2.4

1.04

7

3

1-1?

9

4

1.12

Ii

5.5

13

6

1.08

same

of

whether

allowed difference

Bleuse's

the

SL

values between

fiznctions

has

of the

over

N° 4

SYMMETRY

A

EFFECT

STARK

OF

491

SUPERLATTICES

IN

(a)

>

~

«

cc

i~

z

(b)

>

~

j

~

~

z

z

Fig.

3.

Square

N

3, b)

with

=

the

The

is

p)~

(n

pd It is

+

pd

z

axis,

also

at

holes. when

the

wavefunction

electron

clear

it

least,

and

indexes

of

the

Bleuse's

after

a) with

[II

model

ladder). fP~l~~ potential is

being drop along

to

A~,

the

constant,

values

(note that, in

of k

(Fig. 4)). envelope function

model,

Bleuse's

the

curve

of

the

the

of

centered

when

splits

field

electric

carrier

a

carrier

same

energies

the

on

the

on

the

under

the

centered

(Stark

m

minibands

cosine

a

envelope function Obviously translation.

pd

that

is

the

in a necessary the subspace

Higher compared

electron

E(k) being

from the

allowed

consecutive

two

translations

voltage

makes

over,

by

well

the

The

A~,

field

deduced

and

along

of

=

electric

well

modulus

M

5.

corresponding to unperturbed energy

states

the

n~

of

N

an

invariant

N wells being of perturbative model, corresponding to the

will

not

for

even

under

translation

md

a

of

the

exciton

integer.

the

functions envelope opposite parities and very hole

considered

be

the

order

of

first

miniband

because

holes.

Moreover

is

zero

for

weak

for

their

the

indexes

electron

electric

field

both

for

of

to

functions same

and

in

energy

the

overlap

from

parities

miniband

perturbation

electrons

the

difference

contributions

envelope

the

first

the

the

diagonalize

to

is

the

large of

the

minibands

with

different

from

but

[5]. Let

us

electron-hole

consider

a

SL

interaction.

with

an

Before

applied electric switching on

field the

but electric

without

field,

taking the

into

bottom

account

of

the

the

first

JOURNAL

492

PHYSIQUE

DE

N°

I

4

'

nld

-4nlsd

2nlsd

2nlsd

o

Rid

4n/5d

k

Fig.

Allowed

4.

values

corresponding

and

of k

of the

values

electron

energy

after

5

N

at

Bleuse's

=

model.

miniband

electron

k

wavevector

E

(k ) with k

# 0 is

perturbation

the

and

equal

to

hole

eigenenergies

both

degenerate

twice

heavy

the

of

top

zero;

since

E

(k )

E =

first

are

not

(- k

for

correspond

miniband

degenerate. SL

the

On

has

the

to

The

symmetry.

even

a

contrary

eigenenergies of the carriers. The allowed values of k are very few and the energy difference unperturbed eigenstates is between localization (see above) ; nevertheless it is not possible to in every large due to the assume that the lowest perturbed eigenenergy corresponds to the perturbed state originating case k=0 unperturbed with holes the heavy-hole from the (specially for the since one minibandwidth is very small when compared to the potential drop, which is of the order of the therefore miniband). We shall consider in i) the situation the previous first electron where and situation assumption is valid in 2) the where perturbed eigenenergy the lowest corresponds to a state originating from an unperturbed one with a ko different wavevector from

arising

least

at

zero

from

for

the

electrons

recombination

radiative

field

electric

the

the

or

place

takes

modifies

the

In

both

holes.

between

situations

electron

the

and

shall

we

the

hole

that

assume

with

the

is

weak

the

lowest

energies.

1)

of

effect

The

compared

to

It is

[X((z))

A

perturbed ~

will

envelope

are

+

Xi ~~

is

normalization

envelope function. For example, the overlap of the 0~ well is :

formation)

introduced

functions

of

in

both

which

the

hole

xi (z

functions

when

situations.

and

electron

the

envelope

Bloch

functions

X ))

finally

be

$r)~o(z)) perturbed by

cl X(

and

the

the

are =

These

(~ )~

that

assumed

recombine

field

(exciton

interaction

electron-hole

the

electric

the

which

$r(

=

o

radiatively (z )

and

potential Q(z). (A)~~[c(X(+x[) and respectively written are as $r((( (z ), expanded Bloch functions the envelope over coefficients and c(l~~ is the weight of x(l~~ in the perturbed

the

the

electric

envelope

functions

of the

electron

and

the

hole

centered

on

~~

~~~~~~~~

)~~-~~~~~~~j~~~~~llxl)1

~~~

=

with

:

lAol~

=

(~ol~ =

(Clxl+ xllclxl + xl) (c(x(+ x)lc(x(+ x)) (xllxl) (xllxl) =1. =

(Cl(~

=

=

(c((~

+ +

(xllxl) (x)lx))

(lo)

N° 4

A

The

element

matrix

OF

SYMMETRY

of

~l'$I" with

potential

electric

the

u(l~~(z)

where

of

is

periodic being

the

barriers

the

of

part

the

with

even

seen

E(k)

that

equal

is

~((i"

k)

z,

~

q~,

=

q~,~

_~

purely

is

coefficient

The

qjj>

the

E(k)]~

[E(0)

order, c~ is

first

iE (o)

(k )i-

E

z

i

~,,

At k is

The

is

easily ~k',

~

shown

be

that

:

k"

(13)

o =

real.

xi(z)

$r~,o(z)

over

is

of

function

a

the

elements.

matrix

At

c~

k"

q~jh)

ar~d

imaginary and q~, ~ the expansion of

of

function.

:

(12) can

~- k',

=

q_~,

envelope

Bloch

gets

one

it

and

a~d

*

qjl()~

therefore

to

Ui~~~(- z)

E(-

to

(~l'(I' )

~

~~ ~~

(hole)

electron

respect =

have

:

=

uif~(z) We

493

exp(ikz) u(l~~(z)

$r(l~~(z)

potential

is

SUPERLATTICES

IN

(#i(~~~(Z)(Q~~~~(Z)( $i(~~~(Z))

~

:

EFFECT

STARK

the

first

changed

order

into

equal

is

c_~

k.

c~ has c~

a~

=

+

iE (o)

to

order, one

b~ of

has

second

(k') i-

The

i

order

q~,

is

term

q~,,

~,

parity

second

and

a_~

can

a~

=

b_

b~ a~ is of first Moreover

second

E

-c~.

with

b~

The

~.

:

(14)

o

o

defined

no

qo

order

therefore

term

written

be

is

even as

when

:

k # 0

~

=

order.

:

pj(h)j ~/j,(h)) ($~(( $~") (Ao(~

~k

~

§~,

=

~kk'

[c([~+ z (a(+b((~

=

(15)

~'°

[~o(~= [c([~+ z (a)+b)[ ~ k*0

The

growth

overlap direction

for

the on

±

the

transition

p

$r)= o(z)

(x~(x~) =

by operating the perturbed by the Q(z)

calculated

is

function

(A?~ ~+~)~~ [(c()* c(fro+

x

la(

=

+

b(

*

[exp (

±

the

(16)

*P

Z

II

along

(x((x))

*P

(x

pd translation potential :

±

ikpd ) (at

+

b) ) ]

tr~

~~ k#0

with:

(A±P j2

ej2~ ~

~°

~e ~

~

bej2~ h~j2

~

bhj2- j~

~

(17)

~,~

ll±p j2

~0hj2~ ~ k+0

~k~~-k.

~h k

k

°

(~

JOURNAL

494

The

of

numerator

overlap

the

N

(C~)

#

p

+

*

is

then

PHYSIQUE

DE

N° 4

I

:

I

C~ ~0 + 2

kpd [(a()

CDS

a'

*

(b()

+

*

b'l

~k

(18)

k»°

z

2

±

I sin

kpd (b()

at

*

(al)

+

b)I

*

trk

k>0

One

(x~(X~)

from

goes

(x~(x~)

to

by changing

P

the

In

formulae,

above for

k

lE~(o)

the

term

E~(k )j~

jE~(o)

it

and

kpd

sin

equal

to

ar/Nd.

2

that

has

ii-pi

leads

This

rank

the

+p

cos

kpd

to

the

of the

which

will

(N

i

values

of

proportional

to

called

asymptotic

«

criterion

criterion

are

its

k has

strongest

the

lowest

asymmetry

those

which

finite

allowed

between such

are

value

that

which

is

:

(19)

NM

=

for

»

provides

relation

This

with

transitions

-p where

«

i~-pi =

appearing

reasons

integer

an

for

value

when p

exciton

the

side

each

on

formation of

NM

odd).

is If

x

be

considered.

i~pi

transitions

and

p

is

allowed

smallest

contribution

a

and

=

the

of

therefore

and

I

m

k

that

sees

predicted strengths

be

can

oscillator

a~.

the

from

comes

E~(k) j~

iApi

the

contribution

main with

the

readily

One

into

a,

-P

formation

the

(z )

$r~

over

, o

of

excitons

is

(z )

becomes

d~

taken

now

~k e~ is

a

function

of

matrix

the

elements

~

of

into

+

Ck

the

coefficient

the

account,

of the

expansion

~k electron

hole

interaction

:

g~ejh)_ j~ejh)(~~j ~rpejh)~~~j ~ejh)~~~j k' ',k" k" The

symmetry

of

the

provides

SL

We

have

[E(0)

seen

field

one.

E(k)]~

that

why

reason

ef[lh)

The

~~

~

+

are

al

+

perturbation the

changed

b(]

*

(21)

~f'))~~

keep only

we

formulae

i [El

(X(( X))

~

interaction

electron-hole

the

It is the

(~Q)

:

~-i'~~~/"

electric

[e/P

into ~

+

first

is order

weak term.

when

compared to the then equal to

ef

~l~) is

+

b))]

:

exp(± ikpd ) (at

tr~

(22)

~~ k*0

with

:

[A±~(~

=

[d([~+ z [e/P~+a(+ b([~ ~'°

(~±~( ~

=

[d([

~

+

of

z [e/Ph+ exp(±ikpd)(a)+ b))[ ~ k#0

(23)

N° 4

~±p

(d~)* d~

~

OF

SYMMETRY

A

~0

I

+

STARK

[(~f~)* Ef~

EFFECT

+

(b()* Ef~

+

I sin

IN

SUPERLATTICES

(a()* Ef~l

~

495

+

"k

k+0

z b)[cos kpd(sf~

+

sP[)*

+

kpd(sf~

eP[)*]

tr~

(24)

~"°

z a)[coskpd(sf~- sP[)*

±

kpd(sf~+

I sin

+

eP[)*]

tr~

k~0

~j (cos kpd [(a()

2

+

at

*

(b()

+

b)]

*

kpd [(b()

I sin

±

at

*

(a()

+

b)])

*

tr~

k>0

and

:

(~~(~+ i ((E~~ +~(+ ~((~+ (Ei~~~(+ ~((~)

'~±p'~ ~

k~0

[~±~(~= [d([~+ ~j ([eP(

+exp(-

ikpd)(Ta)+ b))[~

(ikpd ) (±

at

(25)

k>0

ei

+

Again of

the

at

the

cos

Nevertheless

kpd it

be

can

which

when

Q(z)

at

even

ef is always potential and pd is equal and

~

at

still

remains In

a

when

has

It is

ei

that

0.

For

is

with

compared

to

which

a~

perhaps

than

Bohr

the

equal

that

of the

order values

radius

of

the

at

discrepancies

some

first first

to

into

kp'd,

the

radiative

those

p'=

with

the

is

of

therefore

making

it

respect

to

and

p the

clear

hold

to

that

electric

the

for

which

exciton). for

close

very

asymptotic

values

first

small

compared

when

of

that

in type I still

drawn

of

values

of p (the values three-dimensional

values show

for

small

the

first

considerations

These

conclusions

changed

is

for

function

allowed

(eP[)* p, with

because

transitions.

p

continuous

a

smallest

the

Moreover

except

and

p

+

the

to

complicated

more

the

(k(.

of

value

is

for

different and

at

allowed

asymmetry

are

k=

b)) ~)

decreases

similar

assumed

now

smallest the

real

zero

+

overlap, (sf~)* fP(z), which is increasing values of

of p. values upper valid with perhaps be

~

is

approximately

therefore

to

e(

contribution

smaller

or

and

A

(eP[)*.

+

small

very

even

to

type II SL,

formulae

2)

small

are

transitions

I,

to

be

to

main

(ef~)*

p=

the

that

seen

(sP[)* the

from

contribution

because

and

term

exp

+

comes

maximal

the

provide compared to

(k [,

A

for

(sf~)*-

making

k,

contribution

main

criterion

The

h

criterion

of p.

kpd

but

in the

above

1/2, 3/2,

recombination

involves

least

at

with

carrier

one

a

originating from an unjerturbed one with a ko different fkom zero. wavevector corresponding unperturbed eigenenergy is twice degenerate since E(ko) E(- ko). At eigenenergies are : lowest order in perturbation, the

wavefunction The

=

the

l~

and

ratio

the

c~~/c_~~ is

(k0)

q~~,

first is

~~

zero

miniband

[10].

respectively equal the

lowest

order,

The to

E

c~ has

~k~

+

(26)

ko ~l~~~

:

1* [(~k~ ko)~ The

[(~k~ ko)~

t

+

originating lower

(ko)

~k~ fkom

perturbed q~~

therefore

~~[ no

~k~ ko) (~ko,

ko ~l~~~

an

even

eigenenergy and I (if one

defined

parity

single and

the

versus

quantum

k

(at the

~~

to next

wavefunction,

well

corresponding iq~~

assumes

(27)

ko)

be

ratio

c~~/c_~~

negative). order

it

can

Even be

are

at

also

JOURNAL

496

readily

checked

that

N±~

of

In

the

situation

unperturbed

an

leads

exist

where

only

following

for

criterion

which

coincides

(which is

k

with

asymptotic

the

the

necessary

criterion

maximal

At

=

by itr~~

and

second

does result

and

ko.

ko

If

excitonic

the

parity

versus

same

special

mately

k.

since

valid

is

The

situations

opposite

the

allowed

finite

lowest

electron

The

pd

if

iq(~,

first

ko

value

of

~~

wavefunctions

different

iq(~

and

leads

case

hole

and

wavevector

same

sin ko

case.

as

into

taken

wavefunctions it

account,

excitonic

interaction

mentioned

above,

excitonic

the

the

unperturbed

the

interaction and

p

if

inverted

is

(28)

I)] tr~

+

~~

zero.

the

same

(29).

criterion

the

to

from

have

The

asymmetry.

present

not

This

ko pd in the

cos

:

asymmetry

ko is the

where

from

±~

sign

by

order

(29)

when

situation special be imagined can wavefunctions with unperturbed depends on p by ± tr~~ lowest order N

the

numerator

=

Another

originate

ko pd(I

I sin

±

lowest

hole)

or

the

zero,

the

at

p

(electron

from

a

oscillator

3).

N

when

case

I)

transitions

eigenfunction

on

q~~ _~). As

and

~

p

different

depends

ko pd

sin

q~~

and

p

+

wavevector

the

between

perturbed

one

a

N° 4

k

the

)* [cos ko pd(I

~

I

versus

between

ko overlap

with

one

E~(ko)]~ ~(q(

the

to

does

wavefunctions

electron-hole

~'~[E~(0)

2~

This

special

the

from

exists

property

generally

asymmetry

consequence

strengths. originates

symmetry

no

PHYSIQUE

DE

be

can

induces

that

seen

asymmetry

(29)

the

opposite

have

criterion

interaction

is

weak

when

kpd

cos

kpd

versus

ef~,

in

has

~

defined

no

situation.

any

In

probably

remains

compared

wavevectors

electric

the

to

the

approxifield.

Discussion.

3.

Table

displays

II

values

(k(,

of

of

values

the

in

sin

type I SL,

a

and

the

and

of

square

N

for

smallest

the

Ei/E~

ratio

the

of

allowed

the

finite

(hole)

electron

eigenenergies from the bottom (top) of the miniband and corresponding measured respecBleuse's theory. Are also tively to the first and second finite allowed values of (k[ in the displayed in table II features of the photocurrent spectra recorded at 5 K in the experiments values of the electric by Agullo-Rueda et al. [4] with the corresponding field and the exciton calculated [9] in SL with technological close R_~ Rydberg of the parameters to a p minibandwidth reference [4]. The electron is mev. those of assumed The to be equal to 65 values of sin kpd which induce a strong after the asymptotic criterion asymmetry may are experimental results and the asymptotic criterion is good, underlined. The fit between the strong except for the 5, which does not =

(2.I))

the

that

between

k

the 2 ar/5

=

between

equal

the

to

e(j~ =

1/2i

two

allowed

~j~.

because

On

the

for

large

=

perhaps, gives corresponds to

maximal

a

of

value

a

does

of k

with

role

work

not

the

+2

the

sin 2 kd

close

well

smallest

sin 2

~pd

allowed

the =

values

to

and

transitions

-2

I. It

to

5

N

at

may because

modulus

when

imagined (see

be

difference

the

=

finite

second

is finite

(I.e. k 2 ar/5 d and 3, the difference N

allowed not

at

=

possible,

to to

[k at

the 0.15

2 ar/3 d~ is

it has to be

Moreover

of

=

2 ar/3 d and k

according and equal value

Indeed,

=

conditions.

maximal

(Tab. II). It is

4 ar/5 d.

of k (I,e. k

(Ei/E~)~, which,

of N,

0.97

equal

cyclic boundary

hand

values to

and

=

of the

other

5

N

at

corresponding

two

0.0625

values

between

asymmetry

=

d) is

2 ar/3 d

eP

measured

correspond to criterion asymptotic

N

theory,

Bleuse's at

=

(I.e. 4 ar/5 d~, this

stage,

to

a

value

separate

that

tends

(Tab. II).

5

N

only

noticed

the

to

This, which

roles

N°

4

OF

SYMMETRY

A

STARK

EFFECT

IN

497

SUPERLATTICES

of sin kpd and cos kpd, versus N, for the smallest finite allowed values of k after the Bleuse's model, of the square of the ratio Ei/E~ of the electron a of values corresponding respectively to the first and second finite allowed eigenenergies features of the photocurrent [k spectra in the experiments by Agullo-Rueda et al. with the exciton. field and of the R_~ Rydberg of the electric values of the F p Values

II.

Table

and,

~ype I SL

in

sin

kd

2kd

sin

kd

2 w/3

sin

sin

3kd

4kd

sin

6kd

sin

R_p

(mev)

0.87

(0.5) 2 w/5

F

Features

2kd~

(cos kd~

asymmetry

36.I

-1/+

21.7

j

=

2

"

1/+

0.95

(0.31)

4

asymmetry

2/+ 2 4 w/5

0.59

2 w/7

0.78

15.5

asymmetry

=

1/+

(0.62)

asymmetry

"

2

2/+ 2 w/9

0.87

12

asymmetry

=

1/+

(- 0.5)

asymmetry 2/+ 2 & 3/+ 3

Strong asymmetry

@

VII I

(- 0.14)

42)

2/+ 2 &

Strong

Q69

w/13

from

indirect

excitons

asymptotic

the

and

the

criterion

can

potential. Nevertheless, probably be understood as

electric

at

the

=

8.83

asymmetry

3/+ 3 &

played by

9.85

3/+ 3

-4/+

N

4

=

5,

deviation

the

signature

indirect

of

excitons.

important feature in the reported transitions [2-4, 7] : the measurements appears largest oscillator strengths 0 the and the (when compared to the are -p corresponding + p). No argument thermalization carriers be drawn from of the the to seems explain experiments the photocurrent [2, results 4] since is just proportional the to to current between the electron and heavy-hole the joint density of states minibands. On the other hand, consider originates from the electric that exciton field induced by the electron can an we Another

with

the

(hole) F

have

the

the

have

but

zero,

total and

hole

(electron).

The

mean

al'

direction

for

the

exciton.

For

b(

and

bl'.

of an field F~~~ is, in the case applied electric exciton, and the F~~~ p decreasing the effect of F, whereas they

this the

therefore

In + p same the a~ and b~ coefficients then are no F + F~~~ electric field has the not

and

of

value

parallel to the growth axis. opposite directions (Fig. 5),

exciton,

field

a(

the

on

indirect

this

electric

simplified

fPhl~)

model

longer equal for value ; they same

the

+ p

will

be

and

is

taken p

labelled

as

equal

excitons

to

since

respectively

498

JOURNAL

PHYSIQUE

DE

I

N°

4

F

F

Fexc Fexc

(b)

(a) Fig.

Sketches

5.

field

electric

We

I

field

band

valence

induced

F~~~

the

on

hole

potential profiles for (electron) by the

that

seen

One

may

type I SL

a

under

(hole)

electron

an

a) in

exciton.

a( and al' and b( and bl'( increase with b( Rydbergs, be approximately equal to assume excitons. In the experiments by Agullo-Rueda et al. [4], R~, which when N 5, 7, 9 or II is certainly smaller asymmetry + 2 versus have

F~~~ field. of the p 2

electric +

a

and

conductions

the

b) in

exciton

I

a

of the

with

F

difference

the

between

jai

that suppose that we

al'(

and

bl' originates R~, R~ being [9],

of

from

the

the

the

value

+p

and

corresponds to a strong than the Rydberg of the

=

(4.6 mev) [9]. Moreover would expect that the asymmetry one when R~ (I.e. for increasing of N), which is not from decreases values the true photocurrent [4]. It is not therefore possible to large conclude spectra at ~pm3) that mainly the indirect excitons oscillator responsible for the p are GaAs

bulk

exciton

On

asymmetry. N

can

the

considered

be

conclusion

could

be

contrary,

the

the

signature

as

drawn

strong of

only from

asymmetry the

the

which

does

measured value

of

strength

large

values

to

the

SL.

complete

e~

and

tr~

coefficients.

a

semiconduc-

exist

potential applied computation of the a~, b~, electric

decreases

at

A

of

Conclusion.

calculated

We

have

tor

superlattice

functions.

field

electric

with

The

the

an

asymmetry and

the

strongest

strengths of field using

oscillator

the

under

electric

between

electron-hole

asymmetry

can

Voisin

and

the a

various optical transitions in perturbative theory with the

transitions arises p and + p (indirect excitons). The interaction

the

be

predicted

A.

Sibille

versus

the

applied

fkom

both

rank

of the

field

manner.

Acknowledgments. thank

G.

Bastard,

P.

for

very

helpful

discussions.

in

envelope

Bloch

a

the

applied

transitions very

simple

N° 4

A

OF

SYMMETRY

STARK

EFFECT

SUPERLATTICES

IN

499

References

[II [2] [3] [4] [5]

J.,

BLEUSE

TRONC

BASTARD

E. E.,

MENDEz

P.,

CABANEL

AGULLO-RUEDA

BASTARD

G.,

and

VOISIN

F.,

C.,

E.

J. F.

Mechanics

E.

Phys.

P., and

PALMIER

MENDEz

Wave

F.

and

HONG and

ETIENNE

HONG

Applied

Rev.

J. M.,

to

J. M.,

Lett.

60

Phys. B.,

Phys.

Semiconductor

(1988) Rev.

Solid Rev.

Les

220. 60

Lett.

State B 40

(1988)

2426.

Comman.

75

(1989)

1357.

Heterostructures

France, 1988) p. 246. DIGNAM M. M, and SIPE J. E., Phys. Rev. Lett. 64 (1990) 1797. ALLOVON M. and QUILLEC M., Appl. Phys. BLEUSE J., VOISIN P., reference [5] p. 63. BASTARD G., in BLUM J. A. and AGULLO-RUEDA F., Surf Sci. 229 (1990) 472. reference [5] p. 18. BASTARD G., in

Physique,

[6] [7] [8] [9] [10]

G.

AGULLO-RUEDA

(Les

(1990)

Editions

Ulis,

Lett.

53

(1988)

825.

2632.

de

Phys.

I

France

(1992)

2

487-499

1992,

APRIL

487

PAGE

Classific3tion

Physics

Abstracts

73.00

72.40

78.55

optical

Oscillator strengths of the semiconductor superlattice

d'optique

Laboratoire

Vauquelin,

75231

(Received

h

appliqu£

ainsi

entrdment

with

envelope

Bloch

and +p recorded

parallel

field

formation

transitions

-p

low

at

of

temperature

qui

des

excitons

optical axis

%tre

peuvent Bloch.

+

p

semiconducteur

calcu16es

h

indirects,

d'excitons dons

l'dchelle

enregistrds

h

p

et

photocourant

l'aide

61ectrique

champ

Le

formation

la

transitions

transitions

be

can

in

de basse

field

and

ladder.

by

using the

a

of

between

oblique

the

photocurrent

the

model

present

interaction

electron-hole

asymmetry

Features

the

superlattice perturbative model

semiconductor

a

calculated

strength

induce

for

superr6seau

un

de

de

electric

both

accounted

be

can

rue

simple.

trks

Wannier-Stark

the

induit des

spectres

manikre the

dans

enveloppes

d'oscillateur

of

The

indirect

10

6December1991)

croissance,

de

fonctions

growth applied

the

to

functions.

of

direction

dlectron-trou,

strengths

oscillator

The

electric

Industrielles,

Chimie

et

accepted

optiques

transitions la

caract6ristiques pr6vues d'une

%tre

1991,

November

forces

les

entre

Physique

de

France

des

l'interaction

que

peuvent

Abstract.

inducing

21

Certaines

temp6rature

an

revised

asym£trie

une

Wannier-Stark.

under

05,

d'oscillateur

forces

Supdrieure

Ecole

Cedex

champ £lectrique parallble h de perturbation des avec

un

modme

d'un

Physique,

Pads

May 1991,

7

R4sum4.-Les soumis

in

spectra

simple

very

a

manner.

Introduction.

1. Due

to

the

applied towards

lack

of

coupling

resonant

along the growth the evenly spaced

transition

between

periods

axis

»

is

labelled

state away is respectively greater or smaller energy in the electron and a hole localized same

[2, 3]

Photoluminescence are

generally

than

lifetimes

The

of the

proportional

carriers. to

the

p the

well

It is

widely

overlap

of

the

known

value to

p

or

(Fig. I), the

-p

counterparts.

of p

and

on

the

show

The value

of

that

the

hole

matrix

element

envelope

for

functions.

that

of

ratio

the

applied

the

optical absorption in the radiative dependent the on and

is

that

its

to

an

0).

experiments

[2,4]

maximum

whose

sign indicating peak corresponding the

the

[5]

electron

wavefunction

a

of energy (peak labelled

their

transition

F.

+

than

photoconduction

and

weaker

intensities depends on the photocurrent is proportional photoluminescence strongly spectra are

p field

with

electron

an

hole

the

a

superlattice

ladder

and

adjacent quantum Wells when an electric field is (SL), the miniband spectrum converges iii. In a type I SL, the peak corresponding to the

between

of

Stark

«

hole

a

from

transitions

is

an

in a field

Tronc

P.

p

under

transitions electric

an

+p

to +p electric

while

structure

and

the

the

non-radiative

optical

transition

JOURNAL

488

PHYSIQUE

DE

N°

I

4

/

/

/

/

/ '

-2

'

-1

,

,

i

'j

,

,,

Sketch

I.

electric

al.

conduction

the

valence

and

potential

band

profiles

in

type I SL

a

under

an

field.

Dignam et

of

>,

II,

, '

/

/

O+1+2

,

~

Fig.

,'

/

',

al.

et

[4] using

their

[6]

fitted

have

Wannier

calculations

the

functions

of

asymmetry

an

photocurrent

of

measurements

and

variational

along

wavefunction

the

but

methods

the

Agullo-Rueda beginning of

by

made

imposing growth

the

at

direction

z

in

every

well.

2.

The

model.

Before developing proposed by Bleuse

type I SL excitons

is

After

equal

are

not

model

our

it

[I],

et

al.

if

the

readily

can

that

the

electron-hole

model,

Bleuse's

interaction

the

eigenfunction

SL

~

z cn,q

C~ ~

is

the

function

Bessel

(x

Jm

In same

hole

=

of

in

+

p

induces

is

=

two

different

wells

are

carrier

a

=

of

case

and

formation

the

functions

the

transitions

p

centered

z cn,

on

of

in

a

indirect

well

is

(1)

9~n(Z)

well

than

the

centered

miniband

~

at

(z ) the envelope the

origin.

Vfhen

A, C~,~ is given by

(A/2 eFd~

J~

:

q

quantum

larger

q-th

the

SL, d being the period and

isolated

:

(2)

~

integer

theory, the approximation is made approximation is made if two functions

located

special

the

of the

which

nd)

of the

(- i )m Jm (x )

this

of

(Z

9~

all the N wells where n is expanded over eigenfunction of the ground state of an Nefd the total potential drop in the SL

J~(x)

in

strengths

considered.

nq

where

shown,

be

oscillator

index ~

and

with

m,

(

:

~

Jm (x )2

that the ~~(z) functions corresponding respectively

considered.

(3)

=

orthogonal.

are

to

an

electron

The and

a

N° 4

SYMMETRY

A

overlap

The

of

the

STARK

OF

electron

nl)

Trill

SUPERLATTICES

IN

wavefunctions

hole

and

EFFECT

centered

at

489

origin

the

is

then

:

z Cl, Cl, 0(~S ~l)

(4)

0

~

m

runs

m

the

over

of the

wells

By translating corresponding to

S.

envelope

hole

the the

does

not

by

function

±

d

depend along

on

m

the

z

and is

axis,

equal to gets the overlap

taken one

to

be

transition

I

±

(~$ ~(

SL

=SZCS,oC$,1

(mini)

(5)

~

(nlln~i) =SZCS,oC$,-i m

It

readily

can

shown

be

(Fig. 2)

that C

i

~,

limited

gets

extension

of

since

SL

the

the

=

wavefunctions

C

+

(,

i.

localized

are

Neglecting

the

by the

electric

effect

of the

field,

one

:

(trim))

(nil n~ j) In

(- 1)~

~

the

same

it

manner

be

can

shown

that

(all n~ p)

(6)

=

:

(- IT

=

(all n))

(7)

,'

J

I I

i

i j i

/

,-, I

f i

,

/

I

I

',)

electron hole + d

hole

,

d ,

Fig.

Sketches

2.-

wavefunctions

In

carriers could to

be

conclusion, have also

of

the

be

considered

in

electron

the

by

translated

same

only

d

along

wavefunction the

z

centered

at

the

origin

and

of

the

heavy-hole

axis.

model, the + p and transitions between free p optical strengths when a type I SL is considered. This result, which optical absorption coefficient calculated in reference [II, has approximation ~~(z) functions actually because the not are

Bleuse's

the

deduced

±

oscillator from as

the an

strictly orthogonal. Nevertheless, it can be noticed that the carriers localization of the induced by the electric field, which is an important feature of the Bleuse's theory, has been unambiguously in all reported verified experiments [2-4, 7]. Let us general We shall perturbative model operating theory. tum to now our more use a functions introduced by Bastard [8]. The with the superlattice envelope of the carriers

JOURNAL

490

k

unperturbed superlattice being determined by Let

consider

us

the

SL

a

type I SL

p~

well

the

ll~

with

is the

The

mirror

allowed

the

formation

built

parallel

from

excitons,

of

0~

the

on

the

well

and

if

hole

a

and

located

considers

one

notices

one

and

electron

same

layers

the

to

of

values

conditions.

centered

system

is

$r~(z),

functions

Bloch

induces electron

image of the

mirror

well.

which an

N° 4

I

perturbative potential Q (z ). Moreover,

added

an

from

built

system

(-p)~

the

envelope functions are of cyclic boundery use

interaction

electron-hole

PHYSIQUE

DE

centered hole

a

the

at

in

that

a

the

on

centered

on

of

the

center

potential by the electron (hole) on the hole The interaction induced (electron) changed into f~P~l~~ when one goes from the first system to the second. the validity of using a perturbative model to take into An important step is now to prove electric efz and the excitons formation. the applied potential Q(z) account electric field F localizes the carriers in the SL. In III-V SLS We have that the applied seen localized effective If N is the of wells due to its large number the heavy hole is strongly mass. is is limited because potential which the electron wavefunction spread, N the drop over over well.

fP~l~~

is

=

spread

whole

the

A~

(if

of the

wavefunction

electron

probability

the

not

for

electron

an

Nefd from

N is odd

number

of

close

and N =

would

minibandwidth

zero)

be

:

(8)

A~

of N

can

be

checked

of

the

electron

value

of

the

first

electron

spread

the

and

the

wells

N

over

x

=

functions

being

I

displays been

has

Nefd/A~,

used

which

figure

in

shown

are

of the

;

functions

Bleuse's table

which

of

values

the

on

[3]

A~/2 efd

corresponding

of of the moduli these squares The voltage drop along N wells,

5.

=

with

J~(x)

the

over

=

value the

to

functions,

these

for

The

I.

to

3

N

a

expansion

the

truncate

approximation

good

wells

This

arguments.

symmetry

provide

which

exceed

cannot

tunnel

to

of A~, justifies heights of the SL

order

are

3 the

the to

all

for use

barriers generally widely smaller than the Moreover Rydbergs of the indirect for the electrons and the excitons small are electron when compared to A~ [9] showing that the perturbation arising from the hole induced by the electric field. interaction is widely weaker than the perturbation The electron wavefunction and hence of the electron-hole spread of the system being limited to N wells we have to use, to calculate the perturbed eigenenergies and the expansions eigenfunctions, the corresponding integration interval of the perturbed (of extension over z Nd~, the allowed values of k being those of a periodic wells. Indeed, after the array of N of

perturbative

a

electric

only k

N

to

Table the

field N

I.

J~(x)

model

has

or

(including Values, and

switched

been

wells

since A~ is holes. the

the

0

versus

value)

drastically and

N, ofx used

corresponding

N

electron

the

on,

This

more.

values

wavefunction reduces

drastically

to

truncate

the

increases

the

is the number

the

energy

expansion ofthe

of Nefd/A~.

x

=

A~/2 efd

Nefd/A~

3

1.4

1.07

5

2.4

1.04

7

3

1-1?

9

4

1.12

Ii

5.5

13

6

1.08

same

of

whether

allowed difference

Bleuse's

the

SL

values between

fiznctions

has

of the

over

N° 4

SYMMETRY

A

EFFECT

STARK

OF

491

SUPERLATTICES

IN

(a)

>

~

«

cc

i~

z

(b)

>

~

j

~

~

z

z

Fig.

3.

Square

N

3, b)

with

=

the

The

is

p)~

(n

pd It is

+

pd

z

axis,

also

at

holes. when

the

wavefunction

electron

clear

it

least,

and

indexes

of

the

Bleuse's

after

a) with

[II

model

ladder). fP~l~~ potential is

being drop along

to

A~,

the

constant,

values

(note that, in

of k

(Fig. 4)). envelope function

model,

Bleuse's

the

curve

of

the

the

of

centered

when

splits

field

electric

carrier

a

carrier

same

energies

the

on

the

on

the

under

the

centered

(Stark

m

minibands

cosine

a

envelope function Obviously translation.

pd

that

is

the

in a necessary the subspace

Higher compared

electron

E(k) being

from the

allowed

consecutive

two

translations

voltage

makes

over,

by

well

the

The

A~,

field

deduced

and

along

of

=

electric

well

modulus

M

5.

corresponding to unperturbed energy

states

the

n~

of

N

an

invariant

N wells being of perturbative model, corresponding to the

will

not

for

even

under

translation

md

a

of

the

exciton

integer.

the

functions envelope opposite parities and very hole

considered

be

the

order

of

first

miniband

because

holes.

Moreover

is

zero

for

weak

for

their

the

indexes

electron

electric

field

both

for

of

to

functions same

and

in

energy

the

overlap

from

parities

miniband

perturbation

electrons

the

difference

contributions

envelope

the

first

the

the

diagonalize

to

is

the

large of

the

minibands

with

different

from

but

[5]. Let

us

electron-hole

consider

a

SL

interaction.

with

an

Before

applied electric switching on

field the

but electric

without

field,

taking the

into

bottom

account

of

the

the

first

JOURNAL

492

PHYSIQUE

DE

N°

I

4

'

nld

-4nlsd

2nlsd

2nlsd

o

Rid

4n/5d

k

Fig.

Allowed

4.

values

corresponding

and

of k

of the

values

electron

energy

after

5

N

at

Bleuse's

=

model.

miniband

electron

k

wavevector

E

(k ) with k

# 0 is

perturbation

the

and

equal

to

hole

eigenenergies

both

degenerate

twice

heavy

the

of

top

zero;

since

E

(k )

E =

first

are

not

(- k

for

correspond

miniband

degenerate. SL

the

On

has

the

to

The

symmetry.

even

a

contrary

eigenenergies of the carriers. The allowed values of k are very few and the energy difference unperturbed eigenstates is between localization (see above) ; nevertheless it is not possible to in every large due to the assume that the lowest perturbed eigenenergy corresponds to the perturbed state originating case k=0 unperturbed with holes the heavy-hole from the (specially for the since one minibandwidth is very small when compared to the potential drop, which is of the order of the therefore miniband). We shall consider in i) the situation the previous first electron where and situation assumption is valid in 2) the where perturbed eigenenergy the lowest corresponds to a state originating from an unperturbed one with a ko different wavevector from

arising

least

at

zero

from

for

the

electrons

recombination

radiative

field

electric

the

the

or

place

takes

modifies

the

In

both

holes.

between

situations

electron

the

and

shall

we

the

hole

that

assume

with

the

is

weak

the

lowest

energies.

1)

of

effect

The

compared

to

It is

[X((z))

A

perturbed ~

will

envelope

are

+

Xi ~~

is

normalization

envelope function. For example, the overlap of the 0~ well is :

formation)

introduced

functions

of

in

both

which

the

hole

xi (z

functions

when

situations.

and

electron

the

envelope

Bloch

functions

X ))

finally

be

$r)~o(z)) perturbed by

cl X(

and

the

the

are =

These

(~ )~

that

assumed

recombine

field

(exciton

interaction

electron-hole

the

electric

the

which

$r(

=

o

radiatively (z )

and

potential Q(z). (A)~~[c(X(+x[) and respectively written are as $r((( (z ), expanded Bloch functions the envelope over coefficients and c(l~~ is the weight of x(l~~ in the perturbed

the

the

electric

envelope

functions

of the

electron

and

the

hole

centered

on

~~

~~~~~~~~

)~~-~~~~~~~j~~~~~llxl)1

~~~

=

with

:

lAol~

=

(~ol~ =

(Clxl+ xllclxl + xl) (c(x(+ x)lc(x(+ x)) (xllxl) (xllxl) =1. =

(Cl(~

=

=

(c((~

+ +

(xllxl) (x)lx))

(lo)

N° 4

A

The

element

matrix

OF

SYMMETRY

of

~l'$I" with

potential

electric

the

u(l~~(z)

where

of

is

periodic being

the

barriers

the

of

part

the

with

even

seen

E(k)

that

equal

is

~((i"

k)

z,

~

q~,

=

q~,~

_~

purely

is

coefficient

The

qjj>

the

E(k)]~

[E(0)

order, c~ is

first

iE (o)

(k )i-

E

z

i

~,,

At k is

The

is

easily ~k',

~

shown

be

that

:

k"

(13)

o =

real.

xi(z)

$r~,o(z)

over

is

of

function

a

the

elements.

matrix

At

c~

k"

q~jh)

ar~d

imaginary and q~, ~ the expansion of

of

function.

:

(12) can

~- k',

=

q_~,

envelope

Bloch

gets

one

it

and

a~d

*

qjl()~

therefore

to

Ui~~~(- z)

E(-

to

(~l'(I' )

~

~~ ~~

(hole)

electron

respect =

have

:

=

uif~(z) We

493

exp(ikz) u(l~~(z)

$r(l~~(z)

potential

is

SUPERLATTICES

IN

(#i(~~~(Z)(Q~~~~(Z)( $i(~~~(Z))

~

:

EFFECT

STARK

the

first

changed

order

into

equal

is

c_~

k.

c~ has c~

a~

=

+

iE (o)

to

order, one

b~ of

has

second

(k') i-

The

i

order

q~,

is

term

q~,,

~,

parity

second

and

a_~

can

a~

=

b_

b~ a~ is of first Moreover

second

E

-c~.

with

b~

The

~.

:

(14)

o

o

defined

no

qo

order

therefore

term

written

be

is

even as

when

:

k # 0

~

=

order.

:

pj(h)j ~/j,(h)) ($~(( $~") (Ao(~

~k

~

§~,

=

~kk'

[c([~+ z (a(+b((~

=

(15)

~'°

[~o(~= [c([~+ z (a)+b)[ ~ k*0

The

growth

overlap direction

for

the on

±

the

transition

p

$r)= o(z)

(x~(x~) =

by operating the perturbed by the Q(z)

calculated

is

function

(A?~ ~+~)~~ [(c()* c(fro+

x

la(

=

+

b(

*

[exp (

±

the

(16)

*P

Z

II

along

(x((x))

*P

(x

pd translation potential :

±

ikpd ) (at

+

b) ) ]

tr~

~~ k#0

with:

(A±P j2

ej2~ ~

~°

~e ~

~

bej2~ h~j2

~

bhj2- j~

~

(17)

~,~

ll±p j2

~0hj2~ ~ k+0

~k~~-k.

~h k

k

°

(~

JOURNAL

494

The

of

numerator

overlap

the

N

(C~)

#

p

+

*

is

then

PHYSIQUE

DE

N° 4

I

:

I

C~ ~0 + 2

kpd [(a()

CDS

a'

*

(b()

+

*

b'l

~k

(18)

k»°

z

2

±

I sin

kpd (b()

at

*

(al)

+

b)I

*

trk

k>0

One

(x~(X~)

from

goes

(x~(x~)

to

by changing

P

the

In

formulae,

above for

k

lE~(o)

the

term

E~(k )j~

jE~(o)

it

and

kpd

sin

equal

to

ar/Nd.

2

that

has

ii-pi

leads

This

rank

the

+p

cos

kpd

to

the

of the

which

will

(N

i

values

of

proportional

to

called

asymptotic

«

criterion

criterion

are

its

k has

strongest

the

lowest

asymmetry

those

which

finite

allowed

between such

are

value

that

which

is

:

(19)

NM

=

for

»

provides

relation

This

with

transitions

-p where

«

i~-pi =

appearing

reasons

integer

an

for

value

when p

exciton

the

side

each

on

formation of

NM

odd).

is If

x

be

considered.

i~pi

transitions

and

p

is

allowed

smallest

contribution

a

and

=

the

of

therefore

and

I

m

k

that

sees

predicted strengths

be

can

oscillator

a~.

the

from

comes

E~(k) j~

iApi

the

contribution

main with

the

readily

One

into

a,

-P

formation

the

(z )

$r~

over

, o

of

excitons

is

(z )

becomes

d~

taken

now

~k e~ is

a

function

of

matrix

the

elements

~

of

into

+

Ck

the

coefficient

the

account,

of the

expansion

~k electron

hole

interaction

:

g~ejh)_ j~ejh)(~~j ~rpejh)~~~j ~ejh)~~~j k' ',k" k" The

symmetry

of

the

provides

SL

We

have

[E(0)

seen

field

one.

E(k)]~

that

why

reason

ef[lh)

The

~~

~

+

are

al

+

perturbation the

changed

b(]

*

(21)

~f'))~~

keep only

we

formulae

i [El

(X(( X))

~

interaction

electron-hole

the

It is the

(~Q)

:

~-i'~~~/"

electric

[e/P

into ~

+

first

is order

weak term.

when

compared to the then equal to

ef

~l~) is

+

b))]

:

exp(± ikpd ) (at

tr~

(22)

~~ k*0

with

:

[A±~(~

=

[d([~+ z [e/P~+a(+ b([~ ~'°

(~±~( ~

=

[d([

~

+

of

z [e/Ph+ exp(±ikpd)(a)+ b))[ ~ k#0

(23)

N° 4

~±p

(d~)* d~

~

OF

SYMMETRY

A

~0

I

+

STARK

[(~f~)* Ef~

EFFECT

+

(b()* Ef~

+

I sin

IN

SUPERLATTICES

(a()* Ef~l

~

495

+

"k

k+0

z b)[cos kpd(sf~

+

sP[)*

+

kpd(sf~

eP[)*]

tr~

(24)

~"°

z a)[coskpd(sf~- sP[)*

±

kpd(sf~+

I sin

+

eP[)*]

tr~

k~0

~j (cos kpd [(a()

2

+

at

*

(b()

+

b)]

*

kpd [(b()

I sin

±

at

*

(a()

+

b)])

*

tr~

k>0

and

:

(~~(~+ i ((E~~ +~(+ ~((~+ (Ei~~~(+ ~((~)

'~±p'~ ~

k~0

[~±~(~= [d([~+ ~j ([eP(

+exp(-

ikpd)(Ta)+ b))[~

(ikpd ) (±

at

(25)

k>0

ei

+

Again of

the

at

the

cos

Nevertheless

kpd it

be

can

which

when

Q(z)

at

even

ef is always potential and pd is equal and

~

at

still

remains In

a

when

has

It is

ei

that

0.

For

is

with

compared

to

which

a~

perhaps

than

Bohr

the

equal

that

of the

order values

radius

of

the

at

discrepancies

some

first first

to

into

kp'd,

the

radiative

those

p'=

with

the

is

of

therefore

making

it

respect

to

and

p the

clear

hold

to

that

electric

the

for

which

exciton). for

close

very

asymptotic

values

first

small

compared

when

of

that

in type I still

drawn

of

values

of p (the values three-dimensional

values show

for

small

the

first

considerations

These

conclusions

changed

is

for

function

allowed

(eP[)* p, with

because

transitions.

p

continuous

a

smallest

the

Moreover

except

and

p

+

the

to

complicated

more

the

(k(.

of

value

is

for

different and

at

allowed

asymmetry

are

k=

b)) ~)

decreases

similar

assumed

now

smallest the

real

zero

+

overlap, (sf~)* fP(z), which is increasing values of

of p. values upper valid with perhaps be

~

is

approximately

therefore

to

e(

contribution

smaller

or

and

A

(eP[)*.

+

small

very

even

to

type II SL,

formulae

2)

small

are

transitions

I,

to

be

to

main

(ef~)*

p=

the

that

seen

(sP[)* the

from

contribution

because

and

term

exp

+

comes

maximal

the

provide compared to

(k [,

A

for

(sf~)*-

making

k,

contribution

main

criterion

The

h

criterion

of p.

kpd

but

in the

above

1/2, 3/2,

recombination

involves

least

at

with

carrier

one

a

originating from an unjerturbed one with a ko different fkom zero. wavevector corresponding unperturbed eigenenergy is twice degenerate since E(ko) E(- ko). At eigenenergies are : lowest order in perturbation, the

wavefunction The

=

the

l~

and

ratio

the

c~~/c_~~ is

(k0)

q~~,

first is

~~

zero

miniband

[10].

respectively equal the

lowest

order,

The to

E

c~ has

~k~

+

(26)

ko ~l~~~

:

1* [(~k~ ko)~ The

[(~k~ ko)~

t

+

originating lower

(ko)

~k~ fkom

perturbed q~~

therefore

~~[ no

~k~ ko) (~ko,

ko ~l~~~

an

even

eigenenergy and I (if one

defined

parity

single and

the

versus

quantum

k

(at the

~~

to next

wavefunction,

well

corresponding iq~~

assumes

(27)

ko)

be

ratio

c~~/c_~~

negative). order

it

can

Even be

are

at

also

JOURNAL

496

readily

checked

that

N±~

of

In

the

situation

unperturbed

an

leads

exist

where

only

following

for

criterion

which

coincides

(which is

k

with

asymptotic

the

the

necessary

criterion

maximal

At

=

by itr~~

and

second

does result

and

ko.

ko

If

excitonic

the

parity

versus

same

special

mately

k.

since

valid

is

The

situations

opposite

the

allowed

finite

lowest

electron

The

pd

if

iq(~,

first

ko

value

of

~~

wavefunctions

different

iq(~

and

leads

case

hole

and

wavevector

same

sin ko

case.

as

into

taken

wavefunctions it

account,

excitonic

interaction

mentioned

above,

excitonic

the

the

unperturbed

the

interaction and

p

if

inverted

is

(28)

I)] tr~

+

~~

zero.

the

same

(29).

criterion

the

to

from

have

The

asymmetry.

present

not

This

ko pd in the

cos

:

asymmetry

ko is the

where

from

±~

sign

by

order

(29)

when

situation special be imagined can wavefunctions with unperturbed depends on p by ± tr~~ lowest order N

the

numerator

=

Another

originate

ko pd(I

I sin

±

lowest

hole)

or

the

zero,

the

at

p

(electron

from

a

oscillator

3).

N

when

case

I)

transitions

eigenfunction

on

q~~ _~). As

and

~

p

different

depends

ko pd

sin

q~~

and

p

+

wavevector

the

between

perturbed

one

a

N° 4

k

the

)* [cos ko pd(I

~

I

versus

between

ko overlap

with

one

E~(ko)]~ ~(q(

the

to

does

wavefunctions

electron-hole

~'~[E~(0)

2~

This

special

the

from

exists

property

generally

asymmetry

consequence

strengths. originates

symmetry

no

PHYSIQUE

DE

be

can

induces

that

seen

asymmetry

(29)

the

opposite

have

criterion

interaction

is

weak

when

kpd

cos

kpd

versus

ef~,

in

has

~

defined

no

situation.

any

In

probably

remains

compared

wavevectors

electric

the

to

the

approxifield.

Discussion.

3.

Table

displays

II

values

(k(,

of

of

values

the

in

sin

type I SL,

a

and

the

and

of

square

N

for

smallest

the

Ei/E~

ratio

the

of

allowed

the

finite

(hole)

electron

eigenenergies from the bottom (top) of the miniband and corresponding measured respecBleuse's theory. Are also tively to the first and second finite allowed values of (k[ in the displayed in table II features of the photocurrent spectra recorded at 5 K in the experiments values of the electric by Agullo-Rueda et al. [4] with the corresponding field and the exciton calculated [9] in SL with technological close R_~ Rydberg of the parameters to a p minibandwidth reference [4]. The electron is mev. those of assumed The to be equal to 65 values of sin kpd which induce a strong after the asymptotic criterion asymmetry may are experimental results and the asymptotic criterion is good, underlined. The fit between the strong except for the 5, which does not =

(2.I))

the

that

between

k

the 2 ar/5

=

between

equal

the

to

e(j~ =

1/2i

two

allowed

~j~.

because

On

the

for

large

=

perhaps, gives corresponds to

maximal

a

of

value

a

does

of k

with

role

work

not

the

+2

the

sin 2 kd

close

well

smallest

sin 2

~pd

allowed

the =

values

to

and

transitions

-2

I. It

to

5

N

at

may because

modulus

when

imagined (see

be

difference

the

=

finite

second

is finite

(I.e. k 2 ar/5 d and 3, the difference N

allowed not

at

=

possible,

to to

[k at

the 0.15

2 ar/3 d~ is

it has to be

Moreover

of

=

2 ar/3 d and k

according and equal value

Indeed,

=

conditions.

maximal

(Tab. II). It is

4 ar/5 d.

of k (I,e. k

(Ei/E~)~, which,

of N,

0.97

equal

cyclic boundary

hand

values to

and

=

of the

other

5

N

at

corresponding

two

0.0625

values

between

asymmetry

=

d) is

2 ar/3 d

eP

measured

correspond to criterion asymptotic

N

theory,

Bleuse's at

=

(I.e. 4 ar/5 d~, this

stage,

to

a

value

separate

that

tends

(Tab. II).

5

N

only

noticed

the

to

This, which

roles

N°

4

OF

SYMMETRY

A

STARK

EFFECT

IN

497

SUPERLATTICES

of sin kpd and cos kpd, versus N, for the smallest finite allowed values of k after the Bleuse's model, of the square of the ratio Ei/E~ of the electron a of values corresponding respectively to the first and second finite allowed eigenenergies features of the photocurrent [k spectra in the experiments by Agullo-Rueda et al. with the exciton. field and of the R_~ Rydberg of the electric values of the F p Values

II.

Table

and,

~ype I SL

in

sin

kd

2kd

sin

kd

2 w/3

sin

sin

3kd

4kd

sin

6kd

sin

R_p

(mev)

0.87

(0.5) 2 w/5

F

Features

2kd~

(cos kd~

asymmetry

36.I

-1/+

21.7

j

=

2

"

1/+

0.95

(0.31)

4

asymmetry

2/+ 2 4 w/5

0.59

2 w/7

0.78

15.5

asymmetry

=

1/+

(0.62)

asymmetry

"

2

2/+ 2 w/9

0.87

12

asymmetry

=

1/+

(- 0.5)

asymmetry 2/+ 2 & 3/+ 3

Strong asymmetry

@

VII I

(- 0.14)

42)

2/+ 2 &

Strong

Q69

w/13

from

indirect

excitons

asymptotic

the

and

the

criterion

can

potential. Nevertheless, probably be understood as

electric

at

the

=

8.83

asymmetry

3/+ 3 &

played by

9.85

3/+ 3

-4/+

N

4

=

5,

deviation

the

signature

indirect

of

excitons.

important feature in the reported transitions [2-4, 7] : the measurements appears largest oscillator strengths 0 the and the (when compared to the are -p corresponding + p). No argument thermalization carriers be drawn from of the the to seems explain experiments the photocurrent [2, results 4] since is just proportional the to to current between the electron and heavy-hole the joint density of states minibands. On the other hand, consider originates from the electric that exciton field induced by the electron can an we Another

with

the

(hole) F

have

the

the

have

but

zero,

total and

hole

(electron).

The

mean

al'

direction

for

the

exciton.

For

b(

and

bl'.

of an field F~~~ is, in the case applied electric exciton, and the F~~~ p decreasing the effect of F, whereas they

this the

therefore

In + p same the a~ and b~ coefficients then are no F + F~~~ electric field has the not

and

of

value

parallel to the growth axis. opposite directions (Fig. 5),

exciton,

field

a(

the

on

indirect

this

electric

simplified

fPhl~)

model

longer equal for value ; they same

the

+ p

will

be

and

is

taken p

labelled

as

equal

excitons

to

since

respectively

498

JOURNAL

PHYSIQUE

DE

I

N°

4

F

F

Fexc Fexc

(b)

(a) Fig.

Sketches

5.

field

electric

We

I

field

band

valence

induced

F~~~

the

on

hole

potential profiles for (electron) by the

that

seen

One

may

type I SL

a

under

(hole)

electron

an

a) in

exciton.

a( and al' and b( and bl'( increase with b( Rydbergs, be approximately equal to assume excitons. In the experiments by Agullo-Rueda et al. [4], R~, which when N 5, 7, 9 or II is certainly smaller asymmetry + 2 versus have

F~~~ field. of the p 2

electric +

a

and

conductions

the

b) in

exciton

I

a

of the

with

F

difference

the

between

jai

that suppose that we

al'(

and

bl' originates R~, R~ being [9],

of

from

the

the

the

value

+p

and

corresponds to a strong than the Rydberg of the

=

(4.6 mev) [9]. Moreover would expect that the asymmetry one when R~ (I.e. for increasing of N), which is not from decreases values the true photocurrent [4]. It is not therefore possible to large conclude spectra at ~pm3) that mainly the indirect excitons oscillator responsible for the p are GaAs

bulk

exciton

On

asymmetry. N

can

the

considered

be

conclusion

could

be

contrary,

the

the

signature

as

drawn

strong of

only from

asymmetry the

the

which

does

measured value

of

strength

large

values

to

the

SL.

complete

e~

and

tr~

coefficients.

a

semiconduc-

exist

potential applied computation of the a~, b~, electric

decreases

at

A

of

Conclusion.

calculated

We

have

tor

superlattice

functions.

field

electric

with

The

the

an

asymmetry and

the

strongest

strengths of field using

oscillator

the

under

electric

between

electron-hole

asymmetry

can

Voisin

and

the a

various optical transitions in perturbative theory with the

transitions arises p and + p (indirect excitons). The interaction

the

be

predicted

A.

Sibille

versus

the

applied

fkom

both

rank

of the

field

manner.

Acknowledgments. thank

G.

Bastard,

P.

for

very

helpful

discussions.

in

envelope

Bloch

a

the

applied

transitions very

simple

N° 4

A

OF

SYMMETRY

STARK

EFFECT

SUPERLATTICES

IN

499

References

[II [2] [3] [4] [5]

J.,

BLEUSE

TRONC

BASTARD

E. E.,

MENDEz

P.,

CABANEL

AGULLO-RUEDA

BASTARD

G.,

and

VOISIN

F.,

C.,

E.

J. F.

Mechanics

E.

Phys.

P., and

PALMIER

MENDEz

Wave

F.

and

HONG and

ETIENNE

HONG

Applied

Rev.

J. M.,

to

J. M.,

Lett.

60

Phys. B.,

Phys.

Semiconductor

(1988) Rev.

Solid Rev.

Les

220. 60

Lett.

State B 40

(1988)

2426.

Comman.

75

(1989)

1357.

Heterostructures

France, 1988) p. 246. DIGNAM M. M, and SIPE J. E., Phys. Rev. Lett. 64 (1990) 1797. ALLOVON M. and QUILLEC M., Appl. Phys. BLEUSE J., VOISIN P., reference [5] p. 63. BASTARD G., in BLUM J. A. and AGULLO-RUEDA F., Surf Sci. 229 (1990) 472. reference [5] p. 18. BASTARD G., in

Physique,

[6] [7] [8] [9] [10]

G.

AGULLO-RUEDA

(Les

(1990)

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825.

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de