Oscillator strengths of the optical transitions in a semiconductor superlattice under an electric field

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Oscillator strengths of the optical transitions in a semiconductor superlattice under an electric field

P. Tronc

To cite this version:

P. Tronc. Oscillator strengths of the optical transitions in a semiconductor superlattice under an electric field. Journal de Physique I, EDP Sciences, 1992, 2 (4), pp.487-499. �10.1051/jp1:1992158�.

�jpa-00246501�

Classific3tion

Physics

Abstracts

73.00 72.40 78.55

Oscillator strengths of the optical transitions in

_a

semiconductor superlattice under

_an

electric field

P. Tronc

Laboratoire

d'optique Physique,

Ecole

Supdrieure

de

Physique

et Chimie Industrielles, 10 _rue

Vauquelin,

75231 Pads Cedex 05, France

(Received 7

May

1991, revised 21 November 1991, accepted 6December1991)

R4sum4.-Les forces d'oscillateur des transitions

optiques

dans _un

superr6seau

semiconducteur soumis h un

champ £lectrique parallble

h la direction de croissance, peuvent ^%tre calcu16es h l'aide d'un modme de

perturbation

avec des fonctions

enveloppes

de Bloch. Le

champ 61ectrique appliqu£

ainsi que l'interaction dlectron-trou,

qui

induit la formation d'excitons indirects,

entrdment _une

asym£trie

entre les forces d'oscillateur des transitions ₊ p et p dons l'dchelle de Wannier-Stark. Certaines

caract6ristiques

des spectres ^de

photocourant enregistrds

h basse

temp6rature

peuvent ^%tre

pr6vues

d'une manikre trks

simple.

Abstract. The oscillator strengths of the

optical

transitions in _a semiconductor

superlattice

under _an electric field

parallel

to the

growth

axis _can be calculated

using

_a

perturbative

model with Bloch

envelope

functions. The

applied

electric field and the electron-hole interaction

inducing

formation of indirect excitons both induce strength asymmetry ^{between the}

oblique

+p and -p transitions of the Wannier-Stark ladder. Features of the photocurrent spectra recorded at low temperature can be accounted for

by

the present ^{model in} a very

simple

_manner.

1. Introduction.

Due to the lack of resonant

coupling

between

adjacent

quantum ^{Wells when} an electric field is

applied along

the

growth

axis of _a

superlattice (SL),

the miniband

spectrum

converges towards the

evenly spaced

_« Stark ladder _»

iii.

^In a

type

^I

SL,

the

peak corresponding

to the

transition between _a hole and _an electron with _a wavefunction whose maximum is

p

periods

_away from the hole state is labelled ₊ p or p

(Fig. I),

the

sign indicating

that its energy is

respectively greater

or smaller than the energy of the

peak corresponding

to an

electron and _a hole localized in the _same well

(peak

labelled

0).

Photoluminescence

[2, 3]

and

photoconduction [2,4] experiments

show that the +p

transitions _are

generally

weaker than their -p

counterparts.

^{The ratio of the} +p to

p transition intensities

depends

on the value of p and on the value of the

applied

electric field F. The

photocurrent

is

proportional

to the

optical absorption

in the structure while the

photoluminescence

_{spectra are}

strongly dependent

on the radiative and non-radiative lifetimes of the carriers. It is

widely

known

[5]

that the matrix element for _an

optical

transition is

proportional

to the

overlap

of the electron and hole

envelope

functions.

/ /

/ _/

/ /

' /

,'

,

',

' i /

-2 -1 O+1+2

, , ^>,

~ , II,

' 'j

,, ,

Fig.

I. Sketch of the conduction and valence band

potential profiles

in _a type ^{I SL under} an

electric field.

Dignam

_et al.

[6]

have fitted the measurements of

photocurrent

made

by Agullo-Rueda

et al.

[4] using

Wannier functions and variational methods but

imposing

at the

beginning

of their calculations _an asymmetry ^{of the} wavefunction

along

the

growth

direction _z in every well.

2. The model.

Before

developing

_our model it _can

readily

be

shown,

in the

special

_case ^of the functions

proposed by

Bleuse _et al.

[I],

that the oscillator

strengths

of the ₊ p and p transitions in _a type ^{I SL} are

equal

if the electron-hole interaction which induces the formation of indirect

excitons is not considered.

After Bleuse's

model,

the SL

eigenfunction

of _a carrier centered _on the

q-th

^well is _:

nq

~

z cn,q

_9~(Z

nd)

=

z cn,

q

9~n(Z) (1)

where _n is

expanded

_over all the N wells of the

SL,

d

being

the

period

and _~

(z )

the

envelope eigenfunction

of the

ground

state of _an isolated

quantum

^{well centered} at the

origin.

Vfhen the Nefd total

potential drop

in the SL is

larger

than the miniband A,

C~,~

^is

^{given by}

:

C~

~ ⁼

J~

~

(A/2 eFd~ (2)

where

J~(x)

is the Bessel function of

integer

index _m, with _:

J-

m

(x

=

(-

_{i )m}

Jm (x )

and

~

(

^~

Jm (x

)2 ₌

(3)

In this

theory,

the

approximation

is made that the

~~(z)

functions _are

orthogonal.

The

same

approximation

is made if two functions

corresponding respectively

to an electron and _a hole located in two different wells _are considered.

The

overlap

^{of the} electron and hole wavefunctions centered _at the

origin

is then _:

Trill nl)

~

z Cl,

0

Cl, _0(~S ~l) (4)

m

m runs over the wells of the SL

(~$ ~(

does not

depend

on m and is taken to be

equal

to

S.

By translating

the hole

envelope

^function

by

± d

along

the _z

axis,

_one

gets

^the

overlap corresponding

to the _± I transition

(mini) =SZCS,oC$,1

~

(5)

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

m

It _can

readily

be shown

(Fig. 2)

that C^~

~, i ⁼

(- 1)~

⁺ C

(,

i.

Neglecting

the effect of the limited extension of the SL since the wavefunctions _are localized

by

the electric

field,

_one

gets

:

(nil ^n~ j)

=

(trim)) (6)

In the _{same manner} it _can be shown that _:

(all _{n~ p)}

=

(- IT (all n)) ₍₇₎

,'

J i

I i

I j

i

/ ,-, I

f

, i

/

I I

',) electron hole

,

+ d hole

,

d

Fig. 2.- Sketches of the electron wavefunction centered at the

origin

and of the

heavy-hole

wavefunctions translated by ± d

along

the _z axis.

In

conclusion,

in the Bleuse's

model,

the ₊ p and p

optical

transitions between free carriers have the _same oscillator

strengths

when _a

type

^{I SL is} considered. This

result,

which could also be deduced from the

optical absorption

coefficient calculated in reference

[II,

has

to be considered

only

_as an

approximation

because the

~~(z)

functions _{are not}

actually

strictly orthogonal. Nevertheless,

it _can be noticed that the localization of the carriers induced

by

the electric

field,

which is _an

important

feature of the Bleuse's

theory,

has been

unambiguously

verified in all

reported experiments [2-4, 7].

Let _{us now tum to our more}

general theory.

^We shall _{use a}

perturbative

model

operating

with the

superlattice envelope

functions of the carriers introduced

by

Bastard

[8].

The

unperturbed superlattice envelope

functions are Bloch functions

$r~(z),

the allowed values of k

being

determined

by

_use of

cyclic boundery

conditions.

Let _us consider _a SL with _an added

perturbative potential Q (z ). Moreover,

if _one considers the electron-hole interaction which induces formation of

excitons,

_one notices that in _a type ^{I SL the} system ^{built from} an electron centered _on the

0~

_{well and}

a hole centered _on the

p~

well is the mirror

image

of the system ^{built from the} same electron and _a hole centered _on the

(-p)~

_{well. The} mirror is

parallel

to the

layers

and located at the center of the

ll~

well. The interaction

potential

induced

by

the electron

(hole)

_on the hole

(electron) fP~l~~

is

changed

into

f~P~l~~

when _one goes from the first

system

to the second.

An

important

_step is _now to prove the

validity

of

using

_a

perturbative

model _to take into account the

applied

^electric

potential Q(z)

=

efz and the excitons formation.

We have _seen that the

applied

electric field F localizes the carriers in the SL. In III-V SLS the

heavy

hole is

strongly

^localized due to its

large

effective _mass. If N is the number of wells

over which the electron wavefunction is

spread,

N is limited because the

potential drop

over

the whole

spread

of the electron wavefunction cannot exceed the electron first minibandwidth A~

(if

_not the

probability

for _an electron to tunnel _over N wells would be

zero)

:

Nefd = A~

(8)

N is odd from symmetry

arguments.

^{This value of N} can be checked _on the Bleuse's functions which

provide

_a

good approximation

to the

spread

of the electron

[3]

_; table I

displays

the number of wells for these

functions,

with the value of _x

=

A~/2

^{efd which has} been used to truncate the

expansion

_over the

J~(x)

and the

corresponding

values of

Nefd/A~,

which _are all

close to I. The squares of the moduli of these functions _are shown in

figure

3 for

N

=

3 and N

=

5. The

voltage drop along

N

wells, being

of the order of A~,

justifies

the _use of _a

perturbative

model since A~ is

generally widely

smaller than the

heights

of the SL barriers for the electrons and the holes. Moreover the

Rydbergs

of the indirect excitons _are small when

compared

to A~

[9] showing

that the

perturbation arising

from the electron hole

interaction is

widely

weaker than the

perturbation

induced

by

the electric field.

The

spread

^{of the electron} wavefunction and hence of the electron-hole system

being

limited _to N wells _we have _{to use, to} calculate the

perturbed eigenenergies

and the

expansions

of the

perturbed eigenfunctions,

the

corresponding integration

interval over z

(of

extension

Nd~,

the allowed values of k

being

those of _a

periodic

_array of N wells.

Indeed,

after the electric field has been switched _on, the electron wavefunction is the _same whether the SL has

only

N wells _{or more.} This

drastically

reduces the number of allowed values of

k to N

(including

the 0

value)

and

drastically

increases the energy difference between the

Table I. Values, _versus

N, ofx

used _to truncate the

expansion ofthe

Bleuse's

fiznctions

_over

the

J~(x)

and

corresponding

values

of Nefd/A~.

N _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

(a)

>

« ~

cc

i~

z

(b)

>

j ~

~

~

z

z

Fig.

3.

Square

of the M modulus of the electron wavefunction after Bleuse's model [II a) with N

= 3, b) with N

=

5.

states

corresponding

to two consecutive allowed values of k

(note that,

in the Bleuse's

model,

the

unperturbed

_energy

E(k)

is _a cosine _curve

(Fig. 4)).

The electric field

being

constant, the

envelope

function of _a carrier centered _on the

n~

_{well is deduced from the}

envelope

^{function of the} same carrier when centered _on the

(n p)~

well

by

the

pd

translation.

Obviously

the electric field

splits

the

energies

under the

pd

and ₊

pd

translations

(Stark ladder).

It is also clear that the

fP~l~~ potential

is invariant under _a md translation of the exciton

along

the _z

axis,

_m

being

_an

integer.

The

voltage drop along

N wells

being

of the order of the first electron miniband A~, ^{makes it} necessary in _a

perturbative model,

to

diagonalize

the electric field

perturbation

over, at

least,

the

subspace corresponding

to the first miniband both for the electrons and the holes.

Higher

minibands will _not be considered because their difference in energy is

large

when

compared

to A~, even for the holes. Moreover the contributions _to the

overlap

of the electron and hole

envelope

functions is _zero for the

envelope

functions from minibands with indexes of

opposite parities

and very weak for indexes of _same

parities

but different from

[5].

Let _us consider _a SL with _an

applied

electric field but without

taking

into _account the electron-hole interaction. Before

switching

_on the electric

field,

the bottom of the first

'

nld -4nlsd 2nlsd o 2nlsd 4n/5d Rid

k

Fig. 4. Allowed values of k and

corresponding

values of the electron energy at N

=

5 after Bleuse's model.

electron miniband and the top ^of the

heavy

hole first miniband

correspond

to a

k wavevector

equal

to zero; both

eigenenergies

_are not

degenerate.

On the contrary

E

(k )

with k _# 0 is twice

degenerate

since E

(k )

₌ E

(-

k for the SL has _even symmetry. ^The

perturbation arising

from the electric field modifies the

eigenenergies

of the carriers. The allowed values of k _are very few and the energy difference between

unperturbed eigenstates

is

large

due to the localization

(see above)

_; nevertheless it is not

possible

to assume in every

case that the lowest

perturbed eigenenergy corresponds

to the

perturbed

state

originating

from the

unperturbed

_one with k=0

(specially

for the holes since the

heavy-hole

minibandwidth is very small when

compared

to the

potential drop,

which is of the order of the first electron

miniband).

We shall therefore consider in

i)

the situation where the

previous assumption

is valid and in

2)

the situation where the lowest

perturbed eigenenergy corresponds

to a state

originating

from _an

unperturbed

_one with _a wavevector

ko

^different from _zero at least for the electrons _or the holes. In both situations _we shall _assume that the

radiative recombination takes

place

between the electron and the hole with the lowest

energies.

The effect of the electron-hole interaction

(exciton formation)

which is weak when

compared

to the electric field will be

finally

introduced in both situations.

1)

It is assumed that the

envelope

functions of the electron and the hole which

radiatively

recombine _are the Bloch

envelope

functions

xi (z

=

$r(

_o

(z )

and

[X((z))

=

$r)~o(z)) perturbed by

the electric

potential Q(z).

These

perturbed

functions _are

respectively

written _as

(A)~~[c(X(+x[)

and

(~ )~ cl X(

_{+ X}

)) Xi

^~~ is

expanded

_over the Bloch

envelope

functions

$r((( _{(z ),}

A and _{~ are} normalization coefficients and

c(l~~

^{is the}

weight

of

x(l~~

in the

perturbed envelope

function.

For

example,

the

overlap

of the

envelope

functions of the electron and the hole centered _on the

0~

_{well is}

:

~~~~~~~~

=

~~

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

^~~~

with _:

lAol~

₌

(Clxl+ xllclxl

₊

xl)

=

(Cl(~

+

(xllxl)

(~ol~

=

(c(x(+ x)lc(x(+ x))

=

(c((~

+

(x)lx)) (lo)

(xllxl)

=

(xllxl)

=1.

The matrix element of the electric

potential

is _:

~l'$I"

_~

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

with _:

$r(l~~(z)

=

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

^~~~~

where

u(l~~(z)

is the

periodic

_part of the electron

(hole)

Bloch

envelope

function. The

potential

of the barriers

being

_even with

respect

to z, ^one gets :

uif~(z)

=

Ui~~~(- z) (12)

We have _seen that

E(k)

^is

equal

to

E(- k)

and it _can

easily

be shown that _:

~((i"

~

(~l'(I' )

^{* a~d} ~- _{k', k"} ~

~k',

_k"

therefore

qjl()~

=

qjj>

ar~d

q~jh)

= o

(13)

q_~, _~ = q~, _{_~} ^is

purely imaginary

and q~,_~ ^{is real.}

The coefficient c~ of the

expansion

of

xi(z)

over

$r~,o(z)

^is a function of the

q~,~ ^{matrix elements.}

At the first

order,

_{c~ is}

[E(0) E(k)]~

_qo

~. The second order _term is _:

iE (o)

E

(k )i-

ⁱ

z iE (o)

E

(k') i-

ⁱ _q~,

~, q~,,_o

(14)

~,, o

At the first order c_~ is

equal

to -c~. The second order term is _even when

k is

changed

into k. c~ has _no defined

parity

and _can therefore be written _{as :}

c~ = a~ + b~ ^with a_~ = a~ k # 0

b~ =

b_

~

a~ is of first

order,

_b~ of second order.

Moreover _one has _:

pj(h)j _~/j,(h))

=

§~,

($~(( $~")

~ ~k

~kk' (Ao(~

₌

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

~'°

~

[~o(~= _[c([~+ z (a)+b)[ (15)

k*0

The

overlap

for the _± p transition is calculated

by operating

the _±

pd

translation

along

the

growth

direction _on the

$r)= o(z)

function

perturbed by

the

Q(z) potential

:

(x~(x~)

=

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

*P *P

(16)

(x

_x

II

=

Z la(

₊

b(

*

[exp (

_±

ikpd ) (at

₊

b) _{) ]}

_tr~

~~

k#0

j2 ej2~ ~

~e _~

bej2~ _h~j2

with:

(A±P

~°

~,~

~ ~

(17) j2 hj2~ ~ ^~h

_~

bhj2- _j~

ll±p

~0 _{k k °}(~

k+0

~k~~-k.

The numerator of the

overlap

is then _:

N

+ p ^#

(C~)

^* _{C~ ~0} + 2

I

_CDS

kpd [(a()

*

a'

₊

_(b()

*

b'l

_~k

k»°

(18)

± 2

z

I sin

kpd (b()

*

at

₊

_(al)

*

b)I

_trk

k>0

One goes from

(x~(X~)

_to

(x~(x~) _by changing

_a, into a~.

P -P

In the above

formulae,

the main contribution _comes from the smallest allowed values of

k for the term with k has _a contribution

proportional

to

lE~(o) E~(k )j~ jE~(o) E~(k) _j~

One

readily

sees that

iApi

=

ii-pi

and

i~pi

=

i~-pi

and it _can be

predicted

that the rank of the transitions with the

strongest asymmetry

^between

the oscillator

strengths

of the +p and -p transitions _are those which _are such that

sin

kpd

m I and therefore _cos

kpd

« where k has its lowest finite allowed value which is

equal

to 2 ar/Nd. This leads _to the criterion _:

p = NM

(19)

which will be called _«

asymptotic

^criterion _» for _reasons

appearing

when the exciton formation

is considered. This relation

provides

_an

integer

value for p on each side of NM

(N

is

odd).

If the formation of excitons is _now taken into account, the coefficient of the

expansion

^of

x_i

(z )

over $r~

, o

(z )

becomes

d~

~k

~ Ck ⁺ ~k

e~ is _a function of the matrix elements of the electron hole interaction _:

g~ejh)_ j~ejh)(~~j

_{',k" k'}

~rpejh)~~~j ~ejh)~~~j

_k"

^(~Q)

The symmetry ^{of the SL}

provides

_:

~-i'~~~/"

_~

~f'))~~ (21)

We have _seen that the electron-hole interaction

perturbation

is weak when

compared

to the electric field _one. It is the _reason

why

_we

keep only

the first order term.

ef

^~l~) is then

equal

to

[E(0) E(k)]~ ef[lh)

The formulae _are

changed

into _:

(X(( X))

~

i [El

~~ +

al

+

b(]

*

[e/P

^~ ₊

exp(± ikpd ) (at

₊

b))]

_tr~

(22)

~~

k*0

with _:

[A±~(~

=

[d([~+ z [e/P~+a(+ _b([~

~ ~

~'°

~

(23)

(~±~(

=

[d([

+

z [e/Ph+ exp(±ikpd)(a)+ b))[

k#0

~±p

~

(d~)* _d~

_{~0 +}

I [(~f~)* Ef~

+

(b()* Ef~

_~

(a()* Ef~l

_"k + k+0

+

z b)[cos kpd(sf~

+

sP[)*

₊ I sin

kpd(sf~ eP[)*]

_tr~

~"°

(24)

±

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

_{+ I sin}

_{kpd(sf~+ eP[)*]}

_tr~

k~0

+ 2

~j (cos kpd [(a()

^*

at

₊

_(b()

*

b)]

_± I sin

kpd [(b()

*

at

₊

_(a()

*

b)])

_tr~

k>0

and _:

'~±p'~

~

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

k~0

[~±~(~= [d([~+ _~j ([eP( +exp(- ikpd)(Ta)+ _{b))[~ (25)}

k>0

+

ei

^h + exp

(ikpd ) (± at

₊

b)) _~)

Again

the main contribution _comes from the smallest allowed value of

(k(.

The criterion for the maximal contribution _to the asymmetry ^is more

complicated

because of the

at

_cos

kpd

term and because A and _{~ are} different for the + p and p transitions.

Nevertheless it _can be _seen that

e(

is real and that

ei

is _a continuous function of

k, making (sf~)*- (sP[)*

to be _zero at k= 0. For the smallest allowed values of

(k [,

which

provide

the main contribution to the

overlap, (sf~)* (eP[)*

is therefore small when

compared

to

(ef~)*

₊

(eP[)*.

Moreover

fP(z),

which is small when

compared

_to

Q(z)

even at p= I, decreases with

increasing

values of p,

making

it clear that

ef

is

always

small when

compared

to a~ which is of first order with respect to the electric

potential

and _even very small

except perhaps

for the first values of p

(the

values for which

pd

is

equal

to _or smaller than the Bohr radius of the three-dimensional

exciton).

A and _{~ are} therefore

approximately equal

at the first values of p and very close for

transitions at upper values of p. These considerations show that the

asymptotic

criterion remains still valid with

perhaps

_some

discrepancies

for the first values of p.

In _a

type

^II

SL,

similar conclusions _to those drawn in

type

^{I still hold but}

kpd

in the above formulae has to be

changed

into

kp'd,

with

p'= 1/2, 3/2,

2)

It is _now assumed that the radiative recombination involves at least _one carrier with _a wavefunction

originating

from _an

unjerturbed

_one with _{a wavevector}

ko

^{different fkom} zero.

The

corresponding unperturbed eigenenergy

is twice

degenerate

^since

E(ko)

₌

E(- ko).

At the lowest order in

perturbation,

the

eigenenergies

_{are :}

l~

(k0)

t

[(~k~

_ko)~ ⁺

~k~

_{ko ~l~~~}

(26)

and the ratio c~~/c_~~ ^{is :}

1* [(~k~

_ko)~ +

~k~

_ko ^~l~~~

~k~

ko) (~ko, ko)

(27)

The first miniband

originating

fkom _{an even}

single quantum

^well

wavefunction,

q~~,_~~ is _zero

[10].

The lower

perturbed eigenenergy

and the

corresponding

ratio c~~/c_~~ ^are

respectively equal

to E

(ko) _q~~

~~[ and I

(if

_{one assumes}

iq~~

_~~ ^{to be}

negative).

Even _at the lowest

order,

_c~ has therefore _no defined

parity

_versus k

(at

the next order it _can be also

readily

checked that _no symmetry property ^exists versus k between

q~~

_~ ^and

q~~ _~).

As _a consequence

asymmetry generally

does exist between the ₊ p and p transitions oscillator

strengths.

In the

special

situation where

only

_one

perturbed eigenfunction (electron

_or

hole) originates

from _an

unperturbed

_one with _a

ko

wavevector different from _zero, the _numerator

N±~

of the electron-hole wavefunctions

overlap depends

_{on p} at the lowest order

by

_:

2~

~'~[E~(0) E~(ko)]~ ~(q(

~

)* [cos ko pd(I I)

± I sin

ko pd(I

+

I)]

_tr~

(28)

This leads to the

following

criterion for the maximal asymmetry

sin

ko pd

=

(29)

which coincides with the

asymptotic

criterion when

ko

^{is the} lowest finite allowed value of k

(which

is necessary the _case when N

=

3).

Another

special

situation _can be

imagined

where the electron and hole wavefunctions

originate

from

unperturbed

wavefunctions with the _same wavevector

ko

^{different from} zero.

At the lowest order N

±~

depends

on p

by

± tr~~ ^sin

ko pd

if

iq(~,

_~~ ^and

iq(~

_~~ ^{have the} same

sign

and

by itr~~

^cos

^ko pd

in the

opposite

_case. The first _case leads to the criterion

(29).

The second does not present

asymmetry.

This result is inverted if the

unperturbed

wavefunctions have

opposite

wavevectors

ko

and

ko.

If the excitonic interaction is taken into account, it _can be _seen that

ef~,~ has _no defined

parity

_{versus p} and k. The excitonic interaction induces

asymmetry

in any situation. In the

same

special

situations _as mentioned

above,

the

(29)

criterion remains

probably approxi- mately

^valid since the excitonic interaction is weak when

compared

to the electric field.

3. Discussion.

Table II

displays

the values of sin

kpd

and _cos

kpd

_versus N for the smallest allowed finite values of

(k(,

in _a

type

^I

SL,

and the square of the ratio

Ei/E~

of the electron

(hole) eigenenergies

measured from the bottom

(top)

of the miniband and

corresponding

_respec-

tively

to the first and second finite allowed values of

(k[

in the Bleuse's

theory.

Are also

displayed

in table II features of the

photocurrent spectra

recorded at 5 K in the

experiments by Agullo-Rueda

et al.

[4]

with the

corresponding

values of the electric field and the

R_~ Rydberg

of the p exciton calculated

[9]

in _a SL with

technological

_parameters close _to those of reference

[4].

The electron minibandwidth is assumed to be

equal

to 65 mev. The values of sin

kpd

which may induce _a strong

asymmetry

after the

asymptotic

criterion _are underlined. The fit between the

experimental

results and the

asymptotic

criterion is

good, except

^{for the} strong ^measured

asymmetry

between the +2 and -2 transitions when N

=

5,

which does not

correspond

to a value of sin 2 kd close to I. It may be

imagined (see (2.I))

that the

asymptotic

criterion does not work well at N

=

5 because the difference between the two allowed values of k with the smallest finite modulus

(I.e.

^k ₌ ² ar/5 d and k

= 2 ar/5

d)

is maximal at N

= 5 and

equal

to 4 ar/5 d.

Indeed,

at N

=

3,

the difference between the two

corresponding

allowed values of k

(I,e.

k

=

2 ar/3 d and k

=

2 ar/3

d~

^is

only equal

to 2 ar/3 d because of the

cyclic boundary

conditions. Moreover it has to be noticed that

e(j~

=

eP

~j~. On the other hand

(Ei/E~)~,

which,

according

to the Bleuse's

theory,

tends _to

1/2i

= 0.0625 for

large

values of N, is maximal and

equal

to 0.15 _at N

= 5

(Tab. II).

This,

perhaps, gives

_a role to the second finite allowed value of

[k (I.e.

^{4 ar/5}

d~,

_a value which

corresponds

to sin 2

~pd

= 0.97

(Tab. II).

It is not

possible,

at this

stage,

to separate ^{the roles}

Table II. Values

of

sin

kpd

and _cos

kpd,

_versus

N, for

the smallest

finite

allowed values

of

k in _a ~ype ^{I SL}

and, after

the Bleuse's

model, of

the square

of

the ratio

Ei/E~ of

the electron

eigenenergies corresponding respectively

_to the

first

and second

finite

allowed values

of [k ^{features of}

^the

photocurrent

spectra ⁱⁿ the

experiments by Agullo-Rueda

et al. with the

values

of

the F electric

field

and

of

the

R_~ ^{Rydberg of}

^the p exciton.

kd sin kd sin 2kd sin 3kd sin 4kd sin sin 6kd Features F R_p

(cos kd~ 2kd~ (mev)

2 w/3 0.87 asymmetry 36.I

j ⁼

(0.5) 1/+

2 w/5 0.95 -1/+ 21.7 4

(0.31) asymmetry ₂ "

2/+ 2

4 w/5 0.59

2 w/7 0.78 asymmetry ^15.5 =

(0.62) 1/+

asymmetry

2 ^"

2/+ 2

w/9 0.87 asymmetry ¹² =

(- 0.5) 1/+

asymmetry 2/+ 2 & 3/+ 3

VIII @ Strong _asymmetry 9.85

=

42) (- 0.14) 2/+ 2 & 3/+ 3

w/13 Q69 Strong asymmetry ^8.83

3/+ 3 & -4/+ 4

played by

indirect excitons and the electric

potential. Nevertheless,

_at N

=

5,

the deviation from the

asymptotic

criterion _can

probably

be understood _as the

signature

of indirect

excitons.

Another

important

feature appears in the

reported

measurements

[2-4, 7]

: the transitions with the

largest

oscillator

strengths

_are the 0 and the -p

(when compared

to the

corresponding

₊

p).

^No

argument

seems to be drawn from the thermalization of the carriers to

explain

the

photocurrent experiments

results

[2, 4]

since the _current is

just proportional

to the

joint density

of states between the electron and

heavy-hole

minibands. On the other

hand,

we can consider that _an exciton

originates

from the electric field induced

by

the electron

(hole)

_on the hole

(electron).

The _mean value of this electric field F~~~

is,

in the _case of _an

indirect

exciton, parallel

to the

growth

axis. _For the p

exciton,

_F~~~ and the

applied

electric field F have

opposite

directions

(Fig. 5),

therefore

decreasing

the effect of

F,

whereas

they

have the _same direction for the ₊ p exciton. In this

simplified

model

fPhl~)

is taken _as

equal

to zero, but the a~ and b~ coefficients _are then _no

longer equal

for the _{+ p} and _p excitons since the total F ₊ F~~~ ^{electric field has} not the _same value _;

they

will be labelled

respectively

a(

and

al'

and

b(

and

bl'.

F F

Fexc

Fexc

(a)

(b)

Fig. 5. Sketches of the conductions and valence band

potential profiles

for _a type ^{I SL under} an

electric field F with the electric field F~~~ ^induced on the hole (electron) by the electron (hole) a) in

a I exciton b) in

a + I exciton.

We have _seen that the difference between

a(

and

al'

and

b(

and

bl' originates

from the F~~~ ^{field. One} may suppose ^that

jai al'(

and

b( bl'(

increase with

R~, R~ being

^{the value}

of the

Rydbergs,

that _{we assume} to be

approximately equal [9],

of the +p and p excitons. In the

experiments by Agullo-Rueda

_et al.

[4], R~,

which

corresponds

to _a

strong

2 _{versus +} 2

asymmetry

^{when N} ₌

5, 7,

9 _or II is

certainly

smaller than the

Rydberg

^{of the}

GaAs bulk exciton

(4.6 mev) [9].

Moreover _one would expect ^{that the}

asymmetry

^decreases when

R~

^decreases

^(I.e.

^for

increasing

values of

N),

which is _{not true} from the measured

photocurrent

spectra

[4].

It is _not therefore

possible

to conclude _at

large

value of p

~pm3)

that

mainly

the indirect excitons _are

responsible

for the oscillator

strength

asymmetry. ^{On the} contrary, ^the strong asymmetry ^which does exist at

large

values of N _can be considered _as the

signature

^of the electric

potential applied

to the SL. A

complete

conclusion could be drawn

only

from the

computation

of the a~,

b~,

e~ and tr~ coefficients.

Conclusion.

We have calculated the oscillator

strengths

of the various

optical

transitions in _a semiconduc- tor

superlattice

under _an electric field

using

a

perturbative theory

with the Bloch

envelope

functions. The asymmetry ^{between the} p and ₊ p transitions arises fkom both the

applied

electric field and the electron-hole interaction

(indirect excitons).

The rank of the transitions with the strongest asymmetry can be

predicted

_versus the

applied

field in _a very

simple

manner.

Acknowledgments.

thank G. Bastard, P. Voisin and A. Sibille for very

helpful

discussions.

References

[II BLEUSE J., BASTARD G. and VOISIN P., Phys. Rev. Lett. 60 (1988) 220.

[2] MENDEz E. E., AGULLO-RUEDA F. and HONG J. M.,

Phys.

Rev. Lett. 60 (1988) 2426.

[3] TRONC P., CABANEL C., PALMIER J. F. and ETIENNE B., Solid State Comman. 75 (1990) 825.

[4] AGULLO-RUEDA F., MENDEz E. E. and HONG J. M.,

Phys.

Rev. B 40 (1989) 1357.

[5] BASTARD G., Wave Mechanics

Applied

to Semiconductor Heterostructures (Les ^Editions de

Physique,

Les Ulis, France, 1988) _p. 246.

[6] DIGNAM M. M, and SIPE J. E.,

Phys.

Rev. Lett. 64 (1990) 1797.

[7] BLEUSE J., ^VOISIN P., ALLOVON M. and QUILLEC M.,

Appl.

Phys. Lett. 53 (1988) 2632.

[8] BASTARD G., in reference [5] _p. 63.

[9] BLUM J. A. and AGULLO-RUEDA F.,

Surf

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[10] BASTARD G., in reference [5] _p. 18.