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Hole subbands in quantum wells and superlattices
88
Journal of Luminescence 40&41 (1988) 88 91 North-Holland, Amsterdam
HOLE SUBBANDS IN QUANTUM WELLS AND SUPERLATTICES~ Run HUPJ~G, Jianbai XIA, Bangfen ZHU and Hui ThNG Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China Results of theoretical investigations on hole subbands in quantum wells and superlattices are reviewed. Topic covered include: hole subband calculation by an expansion method; pseudopotential calculation by a two—step procedure; heavy and light hole mixing and Coulomb energy of excitons; other applications of the expansion method.
1. INTRODUCTION
the
It had been usual to treat holes in quantum wells as fact,
simple
effective mass
particles.
In
arguments had been advanced to the effect
heavy and
light hole
plane
solutions
of
the L—K equations, which correspond to the same energy eigenvalue, to satisfy the boundary con— dition at (1).
This mixing of heavy and light
that the degeneracy of the valence band is remo—
hole states has also the effect of invalidating
ved
the
in a
quantum well,
heavy hole Actually
resulting
in distinct
subbands and light hole
simple
arguments
will
subbands’.
show
that
in
n=O selection rule.
For the selection
rule originally follows from the orthogonality between simple sinosoidal standing waves, where—
quantum wells the heavy and lingt hole states
as mixing
are necessarily mixed to form subbands, which
results in juxtaposition of waves of two diffe—
are considerably
rent wave lengths.
more complicated
case of the lectrons. one can treat
Thus,
than in
the
as a simple model,
the hole in the framework of the
Luttinger—Kohn effective
mass theory and assume
infinite barriers for the quantum well at z The
=
± d/2.
(1)
to
tions at all pure
heavy
light hole solutions
However the hole wave func—
the subband edges reduce to either or pure
respectively
light
hole states,
having
only a +3/2 or only a ±1/2 spinor
component. though
problem reduces thus to seeking the solu—
tions
the heavy and
this
On
mixed
basis,
in nature
the
are
hole subbands
still
designated
heavy hole(HHn) or light hole(LHn) bands.
th~ L—K equations that vanish at (1).
During
the
past
years,
there
have
appeared a
poses
hole behaviors in quantum wells, covering hole
the
plane wave
solutions with .±k~(wave
number along z—direction) to form standing waves with
nodes at
(1).
This
becomes
not
possible
number of
few
In the case of simple particles, one just super—
quantitative studies
calculations 2,3,5
subband
problems 6—8
.
and
other
on
related
In the following, we shall present
with the heavy and light hole plane wave solu—
a brief review of the theoretical investigations
tions
on
of the L—K
4—component of
the —z +k
light with
hole two
equations,
spinors; solutions results components
because they are
in
fact,
for
either
in
spinor
varying
superposition
the
subject
carried
Out
at
the
Institute
of Semiconductors.
the heavy or wave
as
functions
cos(k~z) and
the other two components varying
2. HOLE SUBBAND CALCULATION BY AN EXPANSION METHOD
as sin(k z z). Subband calculation on the basis of the Hence proper solutions must be sought by mixing Luttinger—Kohn effective mass theory has been 5Work supported by the China National Nature Science Foundation.
0022 2313/88/$03.50 C Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K. Huang et a!.
/ Hole
subbands
89
2’4’7. We carried out by a number of methods have develnped a systematic expansinn methnd~91,
carried
which proves to be efficient and versatile,
is made for an average lattice, with pseudopo—
The method is developed for a superlattice,
out
by
a
two—step procedure,
in the
first step, a usual pseudopotential calculation tentials which
are
weighted
In the second step, the Bloch solutions obtained
method
energy
follows
of
bands,
essentially
the
materials.
usual
pseudopotential
calculation
with
and light
the heavy
component
the
by assuming sufficiently thick barrier regions.
approach
the
of
pseudopotentials
Our
of
averages
but quantum well results are directly obtained
of
—
I
I
hole
plane waves taking the place of the usual plane
I
~\\\j~ •~“~\j~ ~
wave basis and a Kronig—Penney type superlattice barrier potential playing the role of the atomic pseudopotentials.
Thus
the
solutions
are
ex—
___________
kxTt/W
kxO
where W is the superlattice period and n repre—
z.
and the
kx~2Jt/W
and kz are respectively
the 2D subband wave number vector in the XY—— plane
___________
\L~\~\~
light hole
sents the integers.
___________
I
superlattice wave
number along
In our calculations for typical
structures
FIGURE 1 Hole distribution al~ng z;direction 120meV; period W = BOA + 8OA).
(barrier=
modelling GaAs—AlGaAs systems, with n restricted to in
4,
—3,
3,
...,
altogether
4,
i.e.,
36 heavy and
with
20
expansions
light hole
25
plane
\
waves, quite accurate results are obtained for the lower hole subbands confined in the quantum
~15
20~
.
\KH1
IH1 15~
wells. Fig.
1
represents
dtstribiit-tnns bands wave
and
their
number
calculated
typical
calculated
hole
in three
sub—
along z—direction dependence
along
on
dipole
Fig.
2
transition
C
10
IHI HH3
the 2D subband
x direction.
squared
HH1 \~,,
shows
H112
5
HH
matrix
elements from several hole subbands to the first electron
subband,
It
is
observed that
matrix
0
0
1
elements for n ~ 0 transitions can assume considerable magnitudes away from the subband
2 k(
0
2
1
2T~/W)
edges. The calculated subband energy dispersion as exemplified by the solid curves in Fig. 3 is seen to be manifestly nonparabolic.
FIGURt Squared matrix element
10
~nn’
3. PSEUDOPOTENTIAL CALCULATIONS Pseudopotential
calculations
for have
been
= 2
dipole
CB1 (barrier
~nn’1 2/m (eV)
transitions =
2
from hole
100meV; period W
=
subbands
iooA
+
to
l50~).
90
K Huang et a!.
/ Hole subbands
0
in the first step are used as new basis in a
HH 1 gence
in
the results
is achieved
with Bloch
E
functions from bands having wave Good numbers calculation for10the superlattice. conver—
,,,,,,,_,__
c~10 covering n = —5, —4 calculated
dispersion
curves
~
(dashed
20
bands are compared with calculated on curves in the figure) for results the lowest hole sub— the
effective
valence
mass
band
model(solid
parameters that
pseudopotential
/7
4, 5. In Fig. 3, thus
energy
calculation.
curves)
with
come out of the They are
seen to
30
show close agreement.
I
__________________
1
2
~
0 k,,(27L/L)
(110)
i
2 (IOU)
4, EXCITON COULOMB ENERGY” 7, the strongly As shown by Sanders and Chang modified hole subband dispersion has the effect of strengthening the exciton binding. In fact, the
hole wave
comparable
function mixing
in
magnitude
has an
through
the
effect
Hole
subband
FIGURE 3 dispersion from
effective
mass
(solid line) and pseudopotential (dashed line) calculations.
Coulomb
energy of the exciton. The hole subband wave functions are charac—
/a(kll,zh)in_1J
terized by the 20 wave vector k and assume the
b
(k
).° ,Zh i
n—i
J
(k
i(n—l)~ in*
(k r
11 )e n ii n+lJ( i(n+l)c~ l I c~k11,z~)i2j ~kj191)e i(n+2)cp \d~ n+ 1,zh)i n+2 ~I~l)e
following general form:
/ /
/
(a(~~zh)e_~.\
I
I
b(k,1 ,zh)
J
ex p(ik11 ~
polar
angle of
c(~1,zh)e i.e 2~ d(~1,zh)e where
e
exciton
is
the
wave
multiplying
function (3)
by a k
can
be
where ~ (r11 , ~ ) represents the hole to elec— tron relative position vector in the XY—plane and S(z ) is the s—dependent function of the e electron subband concerned. The four spinor
(3)
1<11. A general constructed
by
—matched electron wave
components are clearly associated with the dis— tinct -it
,
angular momenta (n—1).tt
,
n.11
,
(n+l)
(n+2).-Fi and characterized by the correspon—
-It
function
with
undetermined coefficients
A(k),
ding
radial
distributions
expressed
by
the
then integrating over k. With A(j~ 1
)
expressed
as a Fourier series: A(j~1
)
An (k
=
n
)e mO
(4)
Bessel functions.
The important point to note
is that
have the effect
this will
the ground State ~nergy. Because for the groundexciton state, binding n should be such
and integration over ~ carried Out, the exciton
as to make
wave function is obtained in the following form
an
Imex—Il (r
,z
h’
z)
=
(2mS(z
e
) J~An (k
)k d~1
of reducing
s—state,
ponents,
the largest of the four components the
which
presence
will
be
p
of
the
or
d
other
com—
states,
will
K. Huang et a!.
14
/
91
Hole subbands
functions as in (5). It is observed that the riiixed hole save func-
-
tion reduces the binding energy roughly as much
12
as the modified dispersion raises it. For
:~
10
other
applications
of
method,12,the 13 reader is referred papers
A
the
expansion
to the original
~u 8 REFERENCES 1. R.L. Greene, K.K. Bajaj and D.E. Phelps. Phys. Rev. B29(l984), 1807.
6
2. M. Altarelli,
0
~
u~o
i~c
~
Journal of Luminescence,
30
(1985), 472.
well width(A)
3. on L. MSS—II(Japan), J. Sham, The 2nd(1985),573. International Conference 4. D.A. Broido and L.J. Sham, Phys. Rev. B34 (1986), 3917.
FIGURE 4 HH1 exciton binding energy according to three different models.
clearly lead to correspondingly higher Coulomb energies.
Fig.
assumption
calculated
of simple non—mixed
states both as regards
by
hole
subband dispersion and
Coulomb energy calculation; Model B: improved model
taking
account
of
modified subband dispersion; Model C:
7. G.D. Sanders and Y.C. Chang, Phys. Rev. B32 (1985), 5517.
4 compares the HH1—CB1 ground
state exciton binding energies three different models: Model A:
5. J.N. Shulman and Y.C. Chang, Phys. Rev. B31 (1985), 2056. 6. Y.C. Chang and J.N. Schulman, Phys. Rev. B31 (1985), 2069.
in addition
to modified
dispersion,
Coulomb energy calculated with proper hole wave
8. W.T. Masselink, Y.C. Chang and H. Morkoc, Phys. Rev. B32(1985), 519. 9. Hui Tang and Kun Huang, Chinese Journal of Semiconductors, 8(1987), 1. lO.Jian—Bai Xis and A. Baldereschi, Chinese Journal of Semiconductors, 8(1987), in print. il.Bang—Fen Zhu and Kun Huang, to be published. 12.Jian—Bai Xis and Kun Huang, to appear in Acta Physica Sinica. 13. Jian—Bai Xia and Kun Huang, Chinese Journal of Semiconductors, 8(1987), in print.
Journal of Luminescence 40&41 (1988) 88 91 North-Holland, Amsterdam
HOLE SUBBANDS IN QUANTUM WELLS AND SUPERLATTICES~ Run HUPJ~G, Jianbai XIA, Bangfen ZHU and Hui ThNG Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China Results of theoretical investigations on hole subbands in quantum wells and superlattices are reviewed. Topic covered include: hole subband calculation by an expansion method; pseudopotential calculation by a two—step procedure; heavy and light hole mixing and Coulomb energy of excitons; other applications of the expansion method.
1. INTRODUCTION
the
It had been usual to treat holes in quantum wells as fact,
simple
effective mass
particles.
In
arguments had been advanced to the effect
heavy and
light hole
plane
solutions
of
the L—K equations, which correspond to the same energy eigenvalue, to satisfy the boundary con— dition at (1).
This mixing of heavy and light
that the degeneracy of the valence band is remo—
hole states has also the effect of invalidating
ved
the
in a
quantum well,
heavy hole Actually
resulting
in distinct
subbands and light hole
simple
arguments
will
subbands’.
show
that
in
n=O selection rule.
For the selection
rule originally follows from the orthogonality between simple sinosoidal standing waves, where—
quantum wells the heavy and lingt hole states
as mixing
are necessarily mixed to form subbands, which
results in juxtaposition of waves of two diffe—
are considerably
rent wave lengths.
more complicated
case of the lectrons. one can treat
Thus,
than in
the
as a simple model,
the hole in the framework of the
Luttinger—Kohn effective
mass theory and assume
infinite barriers for the quantum well at z The
=
± d/2.
(1)
to
tions at all pure
heavy
light hole solutions
However the hole wave func—
the subband edges reduce to either or pure
respectively
light
hole states,
having
only a +3/2 or only a ±1/2 spinor
component. though
problem reduces thus to seeking the solu—
tions
the heavy and
this
On
mixed
basis,
in nature
the
are
hole subbands
still
designated
heavy hole(HHn) or light hole(LHn) bands.
th~ L—K equations that vanish at (1).
During
the
past
years,
there
have
appeared a
poses
hole behaviors in quantum wells, covering hole
the
plane wave
solutions with .±k~(wave
number along z—direction) to form standing waves with
nodes at
(1).
This
becomes
not
possible
number of
few
In the case of simple particles, one just super—
quantitative studies
calculations 2,3,5
subband
problems 6—8
.
and
other
on
related
In the following, we shall present
with the heavy and light hole plane wave solu—
a brief review of the theoretical investigations
tions
on
of the L—K
4—component of
the —z +k
light with
hole two
equations,
spinors; solutions results components
because they are
in
fact,
for
either
in
spinor
varying
superposition
the
subject
carried
Out
at
the
Institute
of Semiconductors.
the heavy or wave
as
functions
cos(k~z) and
the other two components varying
2. HOLE SUBBAND CALCULATION BY AN EXPANSION METHOD
as sin(k z z). Subband calculation on the basis of the Hence proper solutions must be sought by mixing Luttinger—Kohn effective mass theory has been 5Work supported by the China National Nature Science Foundation.
0022 2313/88/$03.50 C Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K. Huang et a!.
/ Hole
subbands
89
2’4’7. We carried out by a number of methods have develnped a systematic expansinn methnd~91,
carried
which proves to be efficient and versatile,
is made for an average lattice, with pseudopo—
The method is developed for a superlattice,
out
by
a
two—step procedure,
in the
first step, a usual pseudopotential calculation tentials which
are
weighted
In the second step, the Bloch solutions obtained
method
energy
follows
of
bands,
essentially
the
materials.
usual
pseudopotential
calculation
with
and light
the heavy
component
the
by assuming sufficiently thick barrier regions.
approach
the
of
pseudopotentials
Our
of
averages
but quantum well results are directly obtained
of
—
I
I
hole
plane waves taking the place of the usual plane
I
~\\\j~ •~“~\j~ ~
wave basis and a Kronig—Penney type superlattice barrier potential playing the role of the atomic pseudopotentials.
Thus
the
solutions
are
ex—
___________
kxTt/W
kxO
where W is the superlattice period and n repre—
z.
and the
kx~2Jt/W
and kz are respectively
the 2D subband wave number vector in the XY—— plane
___________
\L~\~\~
light hole
sents the integers.
___________
I
superlattice wave
number along
In our calculations for typical
structures
FIGURE 1 Hole distribution al~ng z;direction 120meV; period W = BOA + 8OA).
(barrier=
modelling GaAs—AlGaAs systems, with n restricted to in
4,
—3,
3,
...,
altogether
4,
i.e.,
36 heavy and
with
20
expansions
light hole
25
plane
\
waves, quite accurate results are obtained for the lower hole subbands confined in the quantum
~15
20~
.
\KH1
IH1 15~
wells. Fig.
1
represents
dtstribiit-tnns bands wave
and
their
number
calculated
typical
calculated
hole
in three
sub—
along z—direction dependence
along
on
dipole
Fig.
2
transition
C
10
IHI HH3
the 2D subband
x direction.
squared
HH1 \~,,
shows
H112
5
HH
matrix
elements from several hole subbands to the first electron
subband,
It
is
observed that
matrix
0
0
1
elements for n ~ 0 transitions can assume considerable magnitudes away from the subband
2 k(
0
2
1
2T~/W)
edges. The calculated subband energy dispersion as exemplified by the solid curves in Fig. 3 is seen to be manifestly nonparabolic.
FIGURt Squared matrix element
10
~nn’
3. PSEUDOPOTENTIAL CALCULATIONS Pseudopotential
calculations
for have
been
= 2
dipole
CB1 (barrier
~nn’1 2/m (eV)
transitions =
2
from hole
100meV; period W
=
subbands
iooA
+
to
l50~).
90
K Huang et a!.
/ Hole subbands
0
in the first step are used as new basis in a
HH 1 gence
in
the results
is achieved
with Bloch
E
functions from bands having wave Good numbers calculation for10the superlattice. conver—
,,,,,,,_,__
c~10 covering n = —5, —4 calculated
dispersion
curves
~
(dashed
20
bands are compared with calculated on curves in the figure) for results the lowest hole sub— the
effective
valence
mass
band
model(solid
parameters that
pseudopotential
/7
4, 5. In Fig. 3, thus
energy
calculation.
curves)
with
come out of the They are
seen to
30
show close agreement.
I
__________________
1
2
~
0 k,,(27L/L)
(110)
i
2 (IOU)
4, EXCITON COULOMB ENERGY” 7, the strongly As shown by Sanders and Chang modified hole subband dispersion has the effect of strengthening the exciton binding. In fact, the
hole wave
comparable
function mixing
in
magnitude
has an
through
the
effect
Hole
subband
FIGURE 3 dispersion from
effective
mass
(solid line) and pseudopotential (dashed line) calculations.
Coulomb
energy of the exciton. The hole subband wave functions are charac—
/a(kll,zh)in_1J
terized by the 20 wave vector k and assume the
b
(k
).° ,Zh i
n—i
J
(k
i(n—l)~ in*
(k r
11 )e n ii n+lJ( i(n+l)c~ l I c~k11,z~)i2j ~kj191)e i(n+2)cp \d~ n+ 1,zh)i n+2 ~I~l)e
following general form:
/ /
/
(a(~~zh)e_~.\
I
I
b(k,1 ,zh)
J
ex p(ik11 ~
polar
angle of
c(~1,zh)e i.e 2~ d(~1,zh)e where
e
exciton
is
the
wave
multiplying
function (3)
by a k
can
be
where ~ (r11 , ~ ) represents the hole to elec— tron relative position vector in the XY—plane and S(z ) is the s—dependent function of the e electron subband concerned. The four spinor
(3)
1<11. A general constructed
by
—matched electron wave
components are clearly associated with the dis— tinct -it
,
angular momenta (n—1).tt
,
n.11
,
(n+l)
(n+2).-Fi and characterized by the correspon—
-It
function
with
undetermined coefficients
A(k),
ding
radial
distributions
expressed
by
the
then integrating over k. With A(j~ 1
)
expressed
as a Fourier series: A(j~1
)
An (k
=
n
)e mO
(4)
Bessel functions.
The important point to note
is that
have the effect
this will
the ground State ~nergy. Because for the groundexciton state, binding n should be such
and integration over ~ carried Out, the exciton
as to make
wave function is obtained in the following form
an
Imex—Il (r
,z
h’
z)
=
(2mS(z
e
) J~An (k
)k d~1
of reducing
s—state,
ponents,
the largest of the four components the
which
presence
will
be
p
of
the
or
d
other
com—
states,
will
K. Huang et a!.
14
/
91
Hole subbands
functions as in (5). It is observed that the riiixed hole save func-
-
tion reduces the binding energy roughly as much
12
as the modified dispersion raises it. For
:~
10
other
applications
of
method,12,the 13 reader is referred papers
A
the
expansion
to the original
~u 8 REFERENCES 1. R.L. Greene, K.K. Bajaj and D.E. Phelps. Phys. Rev. B29(l984), 1807.
6
2. M. Altarelli,
0
~
u~o
i~c
~
Journal of Luminescence,
30
(1985), 472.
well width(A)
3. on L. MSS—II(Japan), J. Sham, The 2nd(1985),573. International Conference 4. D.A. Broido and L.J. Sham, Phys. Rev. B34 (1986), 3917.
FIGURE 4 HH1 exciton binding energy according to three different models.
clearly lead to correspondingly higher Coulomb energies.
Fig.
assumption
calculated
of simple non—mixed
states both as regards
by
hole
subband dispersion and
Coulomb energy calculation; Model B: improved model
taking
account
of
modified subband dispersion; Model C:
7. G.D. Sanders and Y.C. Chang, Phys. Rev. B32 (1985), 5517.
4 compares the HH1—CB1 ground
state exciton binding energies three different models: Model A:
5. J.N. Shulman and Y.C. Chang, Phys. Rev. B31 (1985), 2056. 6. Y.C. Chang and J.N. Schulman, Phys. Rev. B31 (1985), 2069.
in addition
to modified
dispersion,
Coulomb energy calculated with proper hole wave
8. W.T. Masselink, Y.C. Chang and H. Morkoc, Phys. Rev. B32(1985), 519. 9. Hui Tang and Kun Huang, Chinese Journal of Semiconductors, 8(1987), 1. lO.Jian—Bai Xis and A. Baldereschi, Chinese Journal of Semiconductors, 8(1987), in print. il.Bang—Fen Zhu and Kun Huang, to be published. 12.Jian—Bai Xis and Kun Huang, to appear in Acta Physica Sinica. 13. Jian—Bai Xia and Kun Huang, Chinese Journal of Semiconductors, 8(1987), in print.