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.
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
