PHYSICAL REVIEW B VOLUME 42, NUMBER 18 15 DECEMBER 1990-II

Simple method for calculating exciton binding energies in quantum-confined

semiconductor structures

R. P. Leavitt
U. S. Army Laboratory Command, Harry Diamond Laboratories, 2800 Powder Mill Road, Adelphi, Maryland 20783-1197

J. %. Little
Martin Marietta Laboratories, 1450 South Rolling Road, Baltimore, Maryland 21227
(Received 6 November 1989; revised manuscript received 7 June 1990)
We present a simple, general method for calculating the binding energies of excitons in quantum-
confined structures. The binding energy is given by an integral (over the electron and hole coordi-
nates perpendicular to the confining layers) of a prescribed function weighted by the squares of the
electron and hole subband envelope functions. As a test of the method, we calculate the binding en-
ergies for heavy- and light-hale excitons in a rectangular GaAs/A1Q 3GaQ ~As quantum well as func-
tions of the well width. Very good agreement with previous results is obtained over a wide range of
quantum-well widths. Also, we determine the binding energies for heavy-hole excitons as functions
of electric field in a GaAs/A1Q 35GaQ 65As asymmetric coupled-quantum-well structure. Our results
compare favorably with those obtained in a treatment in which coupling of the electron subbands
via the electron-hole Coulomb interaction is considered. Our method should be applicable to a
variety of complex quantum-confined semiconductor structures for which more rigorous ap-
proaches require extensive numerical calculations.

I. INTRODUCTION the coupling of light and heavy holes away from k~~=0re-
sulted in highly distorted subband dispersion relations
For the past several years there has been considerable (particularly for the light hole) and, consequently, in-
interest in the electronic structure and optical properties creases in the exciton binding energies on the order of 1
of quantum wells and superlattices. Much of the interest meV for heavy holes and 2 meV for light holes. Many
has centered around the properties of quantum-confined other binding-energy calculations for rectangular quan-
states, i.e., states that have a limited spatial extent along tum wells have been performed as well.
a particular crystal direction. Because of the close prox- The main feature of all these calculations is that they
imity of electrons and holes that occupy such states, exci- are variational; the accuracy of the results depends to a
tons have considerably larger bindings energies in these large degree on physical intuition used in choosing the
quantum-confined structures than in bulk semiconduc- form of the exciton wave function. Consequently, gen-
tors and can, for example, be easily observed in room- eralization to more complex systems has been difticult,
temperature absorption spectra. ' Consequently, excitons and few results on systems other than rectangular wells
have large effects on optical phenomena observed in these have been published. (Exciton binding-energy calcula-
structures, and elucidation of their properties has become tions have been performed for parabolic wells and wells
an important topic in the physics of multilayered semi- with linearly graded band gaps. ' )
conductor systems. In this work, we present a simple method for calculat-
Much of the theoretical work in this area has concen- ing exciton binding energies in quantum-confined struc-
trated on excitons in rectangular quantum wells grown tures. We were motivated to develop this method by the
along a [001] direction. Several authors' have calculat-
ed binding energies for excitons in wells with infinite bar- '
increasing interest in two particular semiconductor sys-
tems: coupled quantum wells' and Stark-localized
riers. Greene, Bajaj, and Phelps calculated the binding states in superlattices. ' In an accompanying paper, ' we
energies for excitons in GaAs/Al„Ga& „As quantum include the method as part of a theoretical description of
wells with finite barriers for various values of the Al mole the electronic and optical properties of small-period su-
fraction x using a variation approach, in which coupling perlattices in an electric field, and we defer discussion of
of the 1ight- and heavy-hole bands was neglected, and these structures until then. Here, we will treat excitons
each of the bands was assumed to have a parabolic in the coupled-well system as an example of the applica-
dispersion relation. Priester, Allan, and Lannoo extend- bility of the method.
ed the results of Greene, Bajaj, and Phelps, by allowing The main result obtained in this paper is an expression
for effective-mass mismatch between the well and barrier for the exciton binding energy as the integral (over the
materials. Both sets of authors obtained results that are coordinates of the electron and hole perpendicular to the
qualitatively in accord with experiment. Sanders and confining layers) of a prescribed function multiplied by
Chang allowed for valence-band mixing and showed that the squares of the electron and hole subband envelope

1990 The American Physical Society

 SIMPLE METHOD FOR CALCULATING EXCITON BINDING. .. ll 775

functions. For simplicity, we neglect both band nonpara- and V& respectively, which can represent the effects of
bolicity' (arising from coupling between valence and heterojunctions, doping, and external electric fields. In
conduction bands) and valence-band mixing (described by what follows, we allow for the possibility that the
off-diagonal terms in the Kohn-Luttinger Hamiltonian' ). effective masses depend on position, although we do not
We also assume that each exciton supported by the sys- express the position dependence explicitly.
tem can be associated with a specific pair of electron and Within the approximations given above, the Harnil-
hole subbands. Normally, the latter assumption would tonian for an electron-hole pair (with zero center-of-mass
imply that the exciton binding energy is small compared momentum) can be written as
with the differences between successive subband-edge en-
ergies. However, in our approach the exciton envelope
function naturally rnanifests correlated electron-hole
fi a 1 a a'a 1 a
motion along the quantization direction. Consequently, a 2 az m az 2 az)» m)») az)»

single term for the envelope function is sufficient to give

reasonably accurate exciton binding energies for a wide + V, (z, )+ Vh(zh) —
)» (»
(»)'— e
range of quantum-well widths, as opposed to previous ap-
proaches. ' Because of this, and because details of the
structure enter only through the subband envelope func- where (M)l=m, mhll/(m, +m„ll), m, is the conduction-
tions, our method can be applied to a wide variety of band effective mass, m» and m&~~ are the valence-band
structures in which the electron and hole states are local-
effective masses perpendicular and parallel to the layers,
ized.
respective1y, r, and r& are the electron and hole position
In Sec. II we derive our results for the exciton binding
vectors, z, and zh are their z components (perpendicular
energy. The binding energy for an exciton in which both
to the layers), and V'll is the component of the gradient
electron and hole motions are confined to spatially
(with respect to the relative coordinate p=r, — r„ll)
separated planes is of central importance in evaluating ll

the quantum-confined binding energies, and we treat the parallel to the layers.
solution of this problem in Sec. III. In Sec. IV we com- We choose the following form for the wave function of
an electron-hole pair:
pare results obtained with our method for rectangular
GaAs jAIQ 3Gao 7As quantum wells with corresponding
results obtained by Greene, Bajaj, and Phelps, and by f„(r„rh ) =P(z„z)» }g„(p;z,—z)» }, (2)
Priester, Allan, and Lannoo. 5 In Sec. V we apply the for-
malism to the coupled-quantum-well system, and we
show that it gives results that compare well with a more where n =1,2, . . . labels the eigenstates in order of in-
complete treatment that includes coupling of electron creasing energy (i.e., n =1 is the excitonic ground state),
subbands by the electron-hole Coulomb interaction. and where g„ is an eigenfunction of a two-dimensional
Harniltonian, representing an exciton in which the elec-
tron is confined to the plane z =z„and the hole is
II. EXCITON BINDING ENERGY confined to the plane z =zz. This quantity is obtained by
solving the radial Schrodinger equation:
We consider a general multilayer semiconductor sys-
tem. Although valence-band coupling has been
shown to contribute to some extent to exciton binding
energies in quantum wells, we will, for simplicity, assume 2pll p dp dp
that this coupling can be neglected. We also neglect cou-
pling of the valence and conduction bands. Therefore,
E' '(Z)g (p'Z)—
both conduction and valence bands are assumed to have
parabolic dispersion relations. The valence-band anisot- where Z =z, — zz, subject to the boundary conditions
ropy is described by using different hole masses in that g„(p;Z) — +0 as p~ao and g„(0;Z) is bounded,
different directions. The hole mass along the quantiza- where E„' '(Z) is the —corresponding eigenvalue. The
tion direction (i.e., perpendicular to the layers comprising function g„ is normalized as follows:
the structure) is mhh()h)J and the hole mass in directions
orthogonal to the quantization direction (i.e., parallel to
the layers) is mhh()h)(l, where hh(lh) stands for heavy f "pdp~g„(p;Z)~'=1. (4)

(light) holes. Generally, the perpendicular and parallel

hole effective masses are distinct. If the quantization Both E„' '(Z) and g„(p;Z) depend parametrically on the
direction is parallel to a [001] axis, then we obtain z coordinates through the Coulomb potential.
hh()h)). mo» (} 1+ 2Y2) and ™0»
mhh()h)ll (71— 1 2)»
' We as-
The form of Eq. (2) is similar to that used in the Born-
where y& and y2 are the Luttinger parameters. Oppenheirner separation' ' of electronic and nuclear
sume that the average dielectric constant for the struc- coordinates. Following Messiah, ' we derive an effective
ture is e, and we neglect image-charge effects in our treat- Hamiltonian that acts on (() only. Consider the expecta-
ment of the electron-hole Coulomb interaction. The elec- tion value of the Hamiltonian, Eq. (1), using the trial
tron and the hole are subject to confining potentials V, wave function given by Eq. (2). We have
11 776 R. P. LEAVING l AND J. W. LIl iLE 42

a a
&a)= f dz dz„. f pdp
2me ()Ze 2mhy ()zh
gn

+[V,(z, )+Vh(zh) —E„' '(z, —zh)]$2

'

f dZ, dzhp

where we have used Eq. (3). This expression may be recast into the form of

R (){)(t R2 (){t)
(H ) d d
2m, Bz, 2m h) {)zh

+ [ V, (z, ) + Vh(zh ) —E„' '(z, —zh )+ W„'(z, —zh )]P

f dz, dzh P, (6)

where

()g„(p; Z}
W'(Z)=
2py
f 0
pdp .

If we require that Eq. (6) be stationary with respect to small changes in ((), we obtain the following effective Hamiltonian
for state n of the electron-hole pair:
~'

'(z,—
—zh)+ W„'(z, —zh) .
fi {3 1 {3 fi () 1 ()
2 Bze 2 Bzh
+ V, (z, )+ Vh(zh) E„' (8)
11ke Bze mh) {)Zh

In the standard treatment of the electronic structure of the difFerences between successive subband-edge energies;
molecules and solids, the adiabatic approximation' is consequently, each exciton can be associated with a pair
used to separate the electronic and nuclear degrees of
freedom. Born and Oppenheimer introduce a small pa-
j
(i, ) of electron and hole subbands. (This assumption is
generally valid for quantum wells less than about 200 A
rameter y=(m/M)'~, where m and M are the electron in thickness. The coupled-quantum-well system is an ex-
and nuclear masses, respectively; in their treatment, the ception, since, in that case, an electric field can tune two
quantity analogous to W„'(Z) is of order y and is there- electron subbands into resonance. We shall see in what
fore negligible compared with the other terms. In the ex- follows that our formalism gives an adequate description
citonic problem, there is no corresponding small parame- of these systems well beyond the range in which this as-
ter, and dropping the W„'(Z) term at this stage is not sumption is strictly valid. } Hence, we can treat the quan-
justified. Nevertheless, as we shall see below, this term tities E„' '(Z) and W„'(Z) in Eq. (8) as perturbations.
contributes negligibly to the exciton binding energy. Consequently, we can evaluate the expectation value of
In the semiconductor structures of interest, it is as- Eq. (8} using the unperturbed wave function:
sumed that a number mph' of discrete subbands exist in both
{t)(z„zh ) = f,"(z, ) J"'(zh f . (10)
the valence and conduction bands. The subband-edge en-
ergies (i.e., the energies for which k{)=0) are labeled E ' The result is
)

and E'"' for the electron and hole bands, respectively,

where (i, j)=1, 2, 3, . . . label the states in order of in- (H'n ) =E"
i j E'nij '+ W'n, ij
+E'h' "— "
f'
creasing energies. The states are described by single-band
envelope functions and '"', satisfying f, where E„',J ' and W„';1 are matrix elements of E„'2D'(Z)
and W„'(Z) calculated using the unperturbed wave func-
f2 d 1 d f(e) E(e)f(e)(z tion, Eq. (10).
+V ( }
The binding energy is defined as the difference in ener-
2 dz, m, dz,
gy between the bottom of the electron-hole-pair continu-
(9a} um and the lowest excitonic bound state. In terms of the
quantities defined above, we obtain the binding energy as
and
Wc —
fi
2 dzg de
+V (z } f (h) E(h)f (h)(z—EB ij
E(2D)
l, , ij l, ij 1 (E(2D)
n ij
Wc
n, ij ) (12}
I

(9b)
Equation (12) may be simplified further as follows. Note
first that lim„„E„' '(Z)=0 for all Z. [Continuum
We now assume that the binding energies of the exci- solution to Eq. (3) first appear at zero energy. ] Also, al-
tons supported by the system are small compared with though the functions W„'(Z) are comparable to the two-
42 SIMPLE METHOD FOR CALCULATING EXCITON BINDING. .. 11 777

dimensional binding energies E„' '(Z), our numerical 2/(u +v )' -2/v —u /v; thus Eq. (14) reduces to the
calculations show that the difference between W;(Z} and two-dimensional harmonic oscillator equation, and hence
W„'(Z) for n & 1 is significantly difFerent from zero only
for arguments whose magnitudes are on the order of 0.01
times the exciton Bohr radius or smaller. This result sug-
G(u, v »1)= v2 3
exp( —u /v ), (16)
gests that the W' terms in Eq. (12) can be neglected. To
estimate the error arising from neglecting these terms, with w(v =2/v —2/v . Therefore, exact results
»1)
the functions W„'(Z) were calculated numerically using a can be obtained in both limits.
25-parameter variational wave function (similar to that A reasonably accurate variational wave function can be
described in Sec. III below) for g„. While the results ob- chosen that gives correct results for both v — +0 and
tained are not very accurate (and depend to some degree v))1. We choose
on the choice of basis), they can be used to establish a G(u, v)=%exp[ —A[(u +v )' —v]J, (17)
rough upper bound on the contribution of the W' terms
to the binding energy. For systems such as those con- where A, ~2as v~0, and ~v ' as v~~.
A, The ex-
sidered below, where the spatial extents of the electron pectation value of the binding energy can be obtained as
and hole subband envelope functions are not considerably
smaller than the exciton Bohr radius, we obtain an upper w(v) = — + 4A. + 4k, v E &
(2A v )exp(2A v )
(18)
bound on the 8"
contributions of about 0.5 meV for
A,
I+ 2K, v
heavy holes and 0.9 meV for light holes for excitons in where E&(x ) is the exponential integral. ' Treating A, as a
GaAs/Al„Ga, „As quantum wells. Therefore we can variational parameter, we choose the value that maxim-
safely neglect these terms in the remainder of this paper, izes ( w(v) ) (i.e., minimizes the total energy). This varia-
and we obtain the following simple result for the exciton tional approach is referred to below as approximation 1:
binding energy:
w(v)=max&(w(v)) . (19}
z~ E z z
Since the limiting values of A, are known for both v-+0
(13) and v » 1, it seems reasonable to evaluate ( w( v) ) with a
simple functional form for )L, (v) that describes both limits
Equation (13) represents the binding energy for a two-
correctly. A good choice is
dimensional exciton, with the electron confined to z =z„
and the hole confined to z =zz, weighted by the probabili- Av(v)=2/(1+2&v ) . (20)
ty of finding an electron at z, and a hole at zz, and aver-
aged over the (z„zz } configuration space. This approach has the desirable feature that Eq. (18) can
be evaluated directly without finding its minimum. We
III. SOLUTION TO THE TWO-DIMENSIONAL refer to this result below as approximation 2:
EXCITON PROBLEM w(v)=(w(v))z (21)
All that remains to complete the model is to obtain an
For comparison, we solved the eigenvalue equation,
expression for the two-dimensional exciton binding ener-
Eq. (14}, using a 15-parameter variational wave function.
gy, EP '(Z), that occurs in Eq. (13). Here, we obtain For the purpose of this work, this 15-term variational
three approximate forms for this quantity. In addition,
solution can be considered exact. We developed also an
we obtain a 15-parameter variational solution which, for
expression for the binding energy as a ratio of two poly-
the purpose of the present work, can be considered exact.
nomials by fitting to the 15-parameter result over the
The value of the approximate forms is that they can be
easily implemented by others to calculate exciton binding
(
range 0 v & 1000. The result is approximation 3:
energies without the need of repeating the calculations 4+ci v +catv
described below. w(v)= (22)
1+d]V +dpV +d3V
It is convenient to case Eq. (3) into dimensionless form.
If we define u =p/av, v =Z/av, EP '(Z)=Evw(v), and with
as=eh' /p~~e, and Ev

+,
G(u;v)=aug(p;Z), where
=p~~e /2e A, we obtain the differential equation
c, =12.97,
cz =0. 7180,
G—
1 d dG 2
u w(v)6=0 . (14)
u du du (u~+v~)'i~ d) =9.65, (23)

If v =0, Eq. (14) reduces to the difFerential equation for dq =9.24,

the two-dimensional hydrogen atom, whose ground-state
solution is d3=0. 3706 .
G(u;0) =4e (15) In Fig. 1 we compare our approximation 1, the one-
parameter variational result (dashed curve), with the 15-
and we obtain w(0)=4. On the other hand, if v»1, parameter "exact" result (solid curve). Agreement is ex-
the potential energy term can be expanded as cellent everywhere; the maximum deviation in m is 0.014,
11 778 R. P. LEAVITT AND J. %'. LITTLE

C)
turn wells, the corresponding maximum error in the cal-
culated exciton binding energy is on the order of 0.2 meV
for approximation 2, and much less than this for the oth-
er approximations. This error is considerably less than
the error in measuring the binding energy and on the or-
der of the errors expected in the approximations made in
developing Eq. (13).
In terms of the dimensionless quantities discussed
above, Eq. (13}may be rewritten as

E;, = J «, dz» Eo IfI

'I'I f,'"'I'w [(z, —z„)/ao],
(24)

where we have included the factor Eo under the integral

o sign since it depends on the position-dependent effective
0.0 0.2 0.4 0.6 0.8 masses.

FIG. 1. Exciton binding energy w(v) (normalized to the bulk IU. EXCITONS IN RECTANGULAR GaAs
binding energy Eo) for a two-dimensional system in which the
QUANTUM WELLS
electrons and holes are confined to planes separated by
z, — zj, =aov. The solid curve corresponds to the ful) 15- As a simple check on the validity and accuracy of our
parameter variational calculation; the dashed curve is the one- method, we have calculated the binding energy of the
parameter variational result (approximation 1; see text). ground-state heavy- and light-hole excitons in GaAs!
A1II 3Ga07As quantum wells as functions of the well
width, where the quantization direction is taken as (001).
which occurs at U -0.
1. To further compare our approx- Since we compare these results with the results of
Greene, Bajaj, and Phelps and Priester, Allan, and Lan-
imations, we show in Fig. 2 the relative errors b, w/w,
where hw is the difference between the exact result and noo, we use the same effective-mass parameters. They
an approximate result, plotted as functions of U on a loga-
are as follows: For GaAs, m, =0.067, m&hj =0.08, and
rithmic scale. Since approximations 1 and 2 are varia- m hhJ 0.4S; for Alo. 3Gao. 7As m = 0.092, m u j = 0. 102,
tional lower bounds, the corresponding relative errors are and mhh~=0. 51. The conduction- and valence-band en-
always positive. The error in approximation 3, on the ergy discontinuities are h, =322. 8 meV, and 3, =57.0
other hand, oscillates in sign. The maximum absolute meV, respectively. The dielectric constant e is 12.5. Our
value of the errors for approximations 2 and 3 are calculations of the exciton binding energies via Eq. (24)
~b, w~ =0.063 and 0.039, respectively. For GaAs quan- are made using the density-of-states reduced mass, given
by

&p~~
') = I «, ~f!'(z, )~'/m,
o
o + I «» If,'"'(z» )I'/m»~, , (25)

in place of the z-dependent reduced masses.

o
CV

In Fig. 3(a) we show the results for the ground-state

o
heavy-hole exciton binding energies. Our results (solid
line} are in very good agreement with the results of Ref. 4
(dot-dashed line) for wide wells, and with those of Ref. 5
(dashed line) for narrow wells. These results can be un-
derstood simply in terms of the approximations made by
these authors. In the Greene-Bajaj-Phelps calculation, a
o flexible form (with several variational parameters) is
oI chosen for the exciton envelope function. This function
contains some dependence on the difference in electron
o
o I I I I IIII( I I I IIIII} I I I IIIII) I I I I IIII) I I I I IIII( I I I I IIII) I I I I IIII
and hole z coordinates, and therefore allows for correlat-
10 10 10 10 10 10 10 10 ed electron and hole motion along the quantization direc-
tion. This correlation is expected to be important for
FIG. 2. Relative errors hw{v)/w(v) in the exciton binding wide wells, in which the exciton retains some three-
energy for a two-dimensional system, in which the electrons and dimensional character. The fact that our formalism gives
holes are confined to planes separated by z, — zI, =aov. Curves good results for wide wells implies that the assumption
labeled 1, 2, and 3 correspond to approximations 1, 2, and 3, re- made following Eq. (9) is not as restrictive as it might ap-
spectively (see text). pear.
42 SIMPLE METHOD FOR CALCULATING EXCITON BINDING. . . 11 779

As the well width decreases, both electron and hole en- On the other hand, the exciton envelope function
velope functions become compressed, and the Coulomb chosen by Priester, Allan, and I.annoo is a simple ex-
attraction between the electron and the hole results in an ponential function of the radial coordinate; correlated
increase in the binding energy. However, for very narrow electron-hole motion along z is neglected, and as a result
wells, the envelope functions "spill" over into the barrier, they obtain less accurate results for wider wells. Since
and their spatial extents actually begin to increase as the our results account for the effective-mass mismatch and
well width is decreased further. There is a maximum in also contain some z-correlation effects [through the
the calculated binding energy corresponding to the onset dependence of the envelope function g(p;Z) on the z
of this spreading. Because Greene, Bajaj, and Phelps coordinates], they reproduce accurately the expected be-
have neglected the effective-mass mismatch between the havior of the exciton binding energy throughout a wide
well and barrier materials, the well width at which their range of quantum-well widths.
binding energy is a maximum is larger than what would Figure 3(b) gives corresponding results for the ground-
result if the mismatch were included, as it is in our results state light-hole exciton. The well width for which the
and in those of Priester, Allan, and Lannoo. binding energy is a maximum is about the same in our re-
sults as in those of Priester, Allan, and Lannoo. For wid-
er wells, our calculated binding energies are closer to
those obtained by Greene, Bajaj, and Phelps, although
the agreement is not as favorable as for the heavy-hole
CO exciton. Generally, the difference between our results
O—
and the results obtained elsewhere is between 0.2 and 0.5
meV for the heavy-hole exciton, and between 0.4 and 1
+o meV for the light-hole exciton, in accord with the error
estimates made at the end of Sec. II.
(3
~
Z oo-
LLJ V. KXCITONS IN ASYMMETRiC COUPLED
Z g QUANTUM WELLS

z
CQ
Structures containing coupled quantum wells are of in-
C)
creasing interest because of new electroabsorption and
CD

O
'
electrorefraction phenomena that occur in these struc-
tures. ' '
Here, we use a system of asymmetric, cou-
pled quantum wells as a test of our method by comparing
0.0 100.0
I

200.0 300.0 400.0 calculated exciton binding energies with those deter-
WELL WIDTH (A) mined from a treatment (see the Appendix) in which cou-
pling of the two electron subbands by the electron-hole
Coulomb interaction is included. In particular, this corn-
parison provides a measure of the effects of violating the
C)
assumption [made following Eq. (9)] that the separations
0— of electron and hole subband energies are large compared
with the exciton binding energy.
Consider the coupled-well configuration shown in Fig.
4. Figure 4(a) shows an energy-level diagram for the
configuration at zero field. If an external field is applied
& o
LLJ g)— that points to the left, the energies of the lowest-energy
LLJ electron states associated with each of the wells approach
(3 one another, whereas the lowest-energy hole state associ-
0
Oi ated with the wide well remains well isolated in energy
CQ from other hole states for all field magnitudes. In Fig.
4(b), the field magnitude is equal to the resonance field F„,
for which these two electron states are in resonance, lead-
ing to a minimum splitting of the two states' energies. In
I each state, the electron is delocalized, i.e., has a
0.0 100.0 200.0 300.0 400.0 significant probability of being in either of the wells.
WELL WIDTH (A) Beyond resonance, as in Fig. 4(c), electrons once again
FIG. 3. Binding energies for (a) heavy-hole and (b) light-hole become localized in the individual wells.
excitons in rectangular GaAs/A1030ao7As quantum wells as We take the two GaAs quantum wells as having widths
functions of the well width. Parameter values used in the calcu- of 100 and 50 A, respectively, bounded by A1Q 35GaQ 65As,
lation are given in the text. The solid curves were calculated us- and with a 25-A A1Q 35GaQ 65As potential barrier between
ing Eq. (24); the dot-dashed curves are from Greene, Bajaj, and the wells. Energy gaps and effective masses used in the
Phelps (Ref. 4), and the dashed curves are from Priester, Allan, calculations were taken from Refs. 23 and 24 (with linear
and Lannoo (Ref. 5). interpolation of the effective-mass parameters for
11 780 R. P. LEAVITT AND J. W. LIi ILE 42

Alp 35Gap $5As). They are as follows: For GaAs, o

00

m, =0.067, m&hz =0.08, and mhz~ =0.353; for

Alp 3sGap ssAsq rB& =0. 1206' mt I =0.0993' and
mhhj =0.370. The conduction- and valence-band energy o
discontinuities are 5, =306 meV, and 6, =131 meV, re-
spectively. The dielectric constant e is 13.1. The heavy-
hole —to — electron transitions, shown in Fig. 4, are the ~o
E
)- cci—
lowest energy electron-hole transitions supported by the CE
Lal
coupled-well system when the external field points from
w
the narrow to the wide well, and they are well isolated C3 uj—
from other optical transitions. For the well and barrier
C)
parameters given above, the avoided crossing between the 2'-
two electron energies occur at a field F„ofabout 47 CQ

kVicm; the splitting between electron subbands at this

field is 6.6 meV.
Electron and hole subband-edge energies and envelope o
functions were calculated by using the scattering phase- 0.0 20.0 40.0 60.0 80.0
shift method in conjunction with the exact Airy-function FIELD (kV/cm)
solutions of the effective-mass Schrodinger equations.
In Fig. 5 we show the binding energies of the excitons as- FIG. 5. Exciton binding energies vs electric field for the
asymmetric coupled-quantum-well structure shown in Fig. 4.
As the field is tuned through resonance, e& becomes localized in
the narrow well and e2 in the wide well, with a corresponding
reversal in the exciton binding energies. The solid lines result
from Eq. (24), and the dashed lines were obtained by using the
coupled-subband treatment (with two electron subbands and
one hole subband retained).
(a) —e 1

sociated with the two heavy-hole transitions depicted in

Fig. 4, as functions of the external field. Solid and dashed
1ines in this figure correspond to results obtained with Eq.
(24), our result, and Eq. (A2), the coupled-subband treat-
ment, respectively. The hh, -e, exciton has the electron
(b)
in the wide well for fields well below F„, and in the nar-
row well for fields well above F, . For the hh&-e2 exciton,
the situation is reversed. For all fields away from reso-
nance, the exciton having both electron and hole in the
same (i.e. , wide) well has the highest binding energy and
is the one observed optically. Near resonance, electrons
associated with both excitons are likely to be found in ei-
ther well; thus the binding energies for the two states are
comparable. From the figure it is apparent that our ap-
proach gives a good description of these phenomena; the
difference between binding energies calculated with Eq.
(24) and with the coupled-subband method is less than 0.5
meV over the entire range of electric fields considered, in
agreement with the error estimates given above.
Although the binding energies of the two excitons
cross near the resonance field, the net electron-hole tran-
sition energies do not. In Fig. 6 we show these transi-
FIG. 4. Energy-band diagram for a GaAs/Alo»Gao6, As
asymmetric coupled-quantum-well system for three values of
tions energies plotted as functions of electric field with
the applied field: (a) zero field, (b) resonance field, and (c) a field (solid lines) and without (dashed lines) the exciton bind-
beyond the resonance field. The lowest-energy pair of electron ing energies. It is apparent from the figure that inclusion
levels (e&, e2) and the lowest-energy heavy-hole level (hh&) are of the field-dependent exciton binding energies changes
shown; solid lines indicate regions where the wave functions are the apparent resonance fie1d, i.e., the field of closest ap-
large and dashed lines where the wave functions are small. The proach of the two transition energies. Thus it follows
hh& state remains localized in the wide well for all electric fields. that the exciton binding energies should be included in
Arrows show aHo wed electron-hole transitions. The well order to obtain an accurate interpretation of experimen-
widths are 100 and SO A, and the barrier width is 2S A. tal data in these structures.
42 SIMPLE METHOD FOR CALCULATING EXCITON BINDING. .. 11 781

D which one of our assumption (that subband separations

00
are large compared with exciton binding energies) is not
strictly valid. Hence we feel that this assumption is not
overly restrictive, and that the model, because of its gen-
erality, will be extremely useful for calculating binding
energies in complex quantum-confined structures, such as
superlattices, which we consider in the following paper. '
The main advantage of our approach is its computa-
LJJ
tional simplicity. The result, Eq. (24), is very easy to in-
LLJ
corporate into a computer code that calculates the elec-
0+o~ tron and hole subband energies and envelope functions.
Three different approximations (all quite accurate) for the
prescribed function m(u) have been given that allow rap-
id computation of this function. No additional differ-
ential equations need to be solved, nor variational minim-
C)
C)
izations performed. As an example of the computational
advantage, the calculation reported in Sec. V for the
0.0 20.0 40.0 60.0 80.0 asymmetric coupled-well system, using the coupled-
subband approach, took well in excess of 1000 times the
F}ELD (kV/cm)
computer time needed to obtain the corresponding results
using Eq. (24). Thus the method should prove extremely
FIG. 6. Electron-hole transition energies for the structure of useful for parametric studies, in which well and barrier
Fig. 4 with (solid lines) and without (dashed lines) the exciton thicknesses and compositions, as well as external parame-
binding-energy correction. The e6'ect of subtracting the binding ters, such as electric 6elds, are varied to optimize some
energies is to push the apparent crossing to higher electric desired effect, such as electroabsorption.
fields. Generally speaking, other methods for calculating exci-
ton binding energies fall into two categories. In the first,
typified by Bastard et al. , a specific form for the wave
VI. DISCUSSION AND CONCLUSIONS function is chosen, and the binding energy is calculated
variationally. This method has the disadvantage that it is
not general; the form of the wave function must be al-
We have presented a simple, general method for calcu- tered if the structure is changed. In the second category
lating exciton binding energies in quantum-confined semi- (typified by the coupled-subband treatment of the
conductor structures. Our main result is an expression coupled-well system given in the Appendix), one in-
for the binding energy as the integral of a prescribed tegrates out the z coordinates to obtain an equation (or
function (the exciton binding energy for a system in set of equations) for the radial component of the exciton
which electrons and holes are confined to spatially envelope function. This approach is more general than
separated planes) weighted by the squares of the electron the first, but it suffers from the disadvantage that one
and hole subband envelope functions. This result was ob- must solve a relatively complicated eigenvalue problem to
tained by choosing a form for the wave function similar obtain each binding energy. Also, in this approach, the
to that used in the Born-Oppenheimer separation of elec- exciton envelope function does not contain any depen-
tron and nuclear coordinates. Although in the latter case dence on z other than that contained in the subband en-
it is necessary to assume the existence of a small parame- velope functions. As a result, correlated motion of the
ter (the ratio of electron to nuclear mass), such an as- electron and hole along the quantization direction is
sumption is not required in our treatment of the exciton neglected, and the partially three-dimensional nature of
problem because of a near cancellation of the terms the exciton in wider wells is not properly characterized.
W„(Z) in the expression for the exciton binding energy, Our method overcomes these deficiences in that it (a) is
Eq. (12). As a consequence, the simple result, Eq. (13) general, i.e., the result for the binding energy depends on
[or, equivalently, Eq. (24)], gives accurate results when the details of the structure only through the subband en-
applied to a variety of physical systems. velope functions, and (b) includes correlated electron-hole
As examples of the applicability of the method, we cal- motion along the quantization direction in a natural
culated the binding energies for light- and heavy-hole ex- manner.
citons in rectangular GaAs/Ala 3Ga07As quantum wells In a study of the e5ects of strain on quantum-well opti-
as functions of the well width. The results compare "
cal spectra, Lee et al. determined the binding energy
favorably with the corresponding results of Refs. 4 and 5. via an equation similar to Eq. (A2) (with a single term), in
%'e also calculated the binding energies of excitons in a which the effective radial potential U,'j(p) was replaced
GaAs/A1035Ga065As asymmetric coupled-quantum-well by — e /e[p +((z, — z&) )]'~, where ((z, —zz) ) is
system. Very good agreement is obtained in comparison averaged over the z coordinates. They then proceeded to
with results calculated using a coupled-subband ap- solve the resulting second-order differential equation, for-
proach. Both of these calculations have shown that the mally identical to Eq. (14), using a variational wave func-
validity of our method extends well into a regime in tion similar to Eq. (17). Their approach is similar to
11 782 R. P. LEAVI i I' AND J. W. LITTLE 42

ours, in that the binding energy can be simply evaluated Kohn-Luttinger Hamiltonian is not included. It has been
as an integral of some quantity over the z coordinates and shown that including valence-band coupling adds
is therefore computationally simple. However, there is about 1 —2 meV to the binding energies of excitons in iso-
no stated justification for replacing the e5'ective radial po- lated, rectangular wells grown along a [001] direction.
tential (which depends on the details of the structure, %e feel that the lack of this correction is the dominant
electric fields, strain, etc.) by the simple form used. Also, error in our calculations. The correction could be added
Lee et al. do not give separately the results for the bind- perturbatively to the model in a manner analogous to the
ing energies, nor do they compare these results with those treatment of Ekenberg and Altarelli. Even without it,
obtained using other methods. Therefore it is difficult to we feel that our model, because of its generality, is a use-
assess the accuracy of their method. (This method also ful tool for examining the dependences of exciton binding
does not account for correlated electron-hole motion energies on system parameters (such as well widths and
along z. ) electric fields) in complex quantum-confined structures.
Our study of the asymmetric quantum-well structure
has provided a stringent test that extends the applicabili-
ty of our method to a more complex system than the rec- ACKNOWLEDGMENTS
tangular quantum well. It has revealed interesting exci-
tonic phenomena that occur in these types of structures. %e thank John D. Bruno of Harry Diamond Labora-
As the field is tuned through the electron-subband reso- tories for a careful and critical reading of the manuscript.
nance, the electron states become increasingly delocalized A portion of this work was supported by Rome Air De-
(until at resonance a given state's charge density is equal- velopment Center (RADC/ESOC), under Contract No.
ly distributed between both wells) and then localized F19628-86-C-0059.
again, but with their roles reversed. Beyond resonance,
the lowest-energy electron state is localized again, but APPENDIX: COUPLED-SUBBAND TREATMENT
with their roles reversed. Beyond resonance, the lowest- OF EXCITON BINDING ENERGIES
energy electron state is localized to the narro~ well, and
the next-lowest-energy state is in the wide well. Binding The coupled-subband treatment of excitons in quantum
energies of excitons formed from the lowest-energy hole wells is similar to that described in Refs. 8-10, where it is
state (always in the wide well) and this pair of electron applied to coupling of subbands by the electron-hole
states show very strong field dependences as the system Coulomb interaction in rectangular wells. %e assume an
passes through resonance. The method presented in this excitonic wave function of the form:
paper reproduces the results of the more sophisticated
(A 1)
coupled-subband treatment very well, although the latter
treatment is necessary to quantitatively describe certain
aspects of this system (such as the electric field at which The radial functions g,"(p) are to be determined, and the
the binding energies of the two excitons become equal). sum in Eq. (A 1) is over all subbands that couple
These results are of general significance since they show significantly with the subbands of interest.
that our method is accurate for systems in which the elec- By calculating the expectation value of Eq. (1) and re-
tron and hole are spatially separated. quiring that the set of g; give a variational lower bound
The main drawback in the present treatment is that to the energy, we obtain the following set of effective ra-
valence-band coupling by o8'-diagonal terms in the dial equations:

(A2)

where (p';~J ) ' and U&'(p) are integrals over z, and zh of the inverse parallel reduced mass and the electron-hole
Coulomb interaction multiplied by the appropriate sets of electron and hole envelope functions, i.e. ,
(p'Jz') '= f dz, f"(z, )f (z, )/m, + f dzhf("'(zh)f. ")(zh)/mh((, (A3)

and
2
j' — f dz dz f'e (z )f(~){z )f(h){z )f(h)(z„)/[p +{z z„) ]

For the coupled-well system, we considered two terms bound state, and also, as a check, by using a variational
in Eq. (Al) (i.e. , the two electron states of interest and a set of wave functions, consisting of superpositions of ex-
single-hole state localized in the wide well). Properties of ponential functions. In all cases, the calculated ground-
the ground-state exciton were calculated both by numeri- state energies agreed to within a tenth of a percent. For
cally integrating the set of Eqs. (A2) and searching for a the specific problem considered here, the exciton associ-
42 SIMPLE METHOD FOR CALCULATING EXCITON BINDING. .. 11 783

ated with the higher of the two electron subbands is ton feature appears naturally as a Fano resonance in the
"buried" in the continuum absorption associated with the continuum absorption; the exciton energy was chosen as
lowest electron subband. Continuum solutions to Eqs. the energy for which the calculated absorption was a
(A2} are obtained by numerical integration, and the exci- maximum.

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