Biexciton-biexciton interaction in semiconductors

Abstract

For a semiconductor with single isotropic conduction and valence bands, the effective biexciton-biexciton interaction is derived starting from the second-quantization representation for the Hamiltonian of the system of interacting excitons and for the biexciton wave function within the framework of the adiabatic approximation. The interaction is found to be an average of the sum of effective interactions between excitons forming interacting biexcitons over the envelope functions of these biexcitons. Depending on the momentum transfer and on the difference between the momenta of interacting biexcitons, the interaction admits an analytical study when the first vanishes. In this case, the interaction has in the r space the form of a function of the interbiexciton distance consisting of a strong repulsive and a weak attractive part. At low temperatures the function describes main features in the behavior of the interaction among biexcitons, in which repulsion predominates over attraction. It is shown that while biexcitons remain stable quasiparticles they weakly attract each other for any value of the distance. A quantitative analysis of obtained results for the CuCl crystal shows that the biexciton system in this model substance is a weakly nonideal Bose gas with positive scattering length as≃3ax, which closely approaches the ideal one at excitation densities n⩽1018 cm-3.

Full text

Biexciton-biexciton interaction in semiconductors
Hoang Ngoc Cam*
Department of Optics, P. N. Lebedev Physical Institute, Russian Academy of Sciences,
Leninski Prospect 53, 117924 Moscow, Russia
~Received 21 September 1995; revised manuscript received 29 October 1996!
For a semiconductor with single isotropic conduction and valence bands, the effective biexciton-biexciton
interaction is derived starting from the second-quantization representation for the Hamiltonian of the system of
interacting excitons and for the biexciton wave function within the framework of the adiabatic approximation.
The interaction is found to be an average of the sum of effective interactions between excitons forming
interacting biexcitons over the envelope functions of these biexcitons. Depending on the momentum transfer
and on the difference between the momenta of interacting biexcitons, the interaction admits an analytical study
when the first vanishes. In this case, the interaction has in the rspace the form of a function of the interbiex-
citon distance consisting of a strong repulsive and a weak attractive part. At low temperatures the function
describes main features in the behavior of the interaction among biexcitons, in which repulsion predominates
over attraction. It is shown that while biexcitons remain stable quasiparticles they weakly attract each other for
any value of the distance. A quantitative analysis of obtained results for the CuCl crystal shows that the
biexciton system in this model substance is a weakly nonideal Bose gas with positive scattering length
as.3ax, which closely approaches the ideal one at excitation densities n<1018 cm23.
@S0163-1829~97!02211-X#
I. INTRODUCTION
Since that time, when Moskalenko1and Lampert2inde-
pendently predicted the existence of the biexciton ~or exci-
tonic molecule!in semiconductors, the study of this bound
complex of two electrons and two holes has rapidly devel-
oped. The biexciton has been theoretically3–7 as well as
experimentally8–14 shown to be the most stable entity of the
lowest energy per electron-hole (e-h) pair at low tempera-
tures in many semiconductors.15 In these semiconductors, the
role of the biexciton in the nonlinear optical phenomena in
the spectral vicinity of the band edge has long been
recognized.13–25 The electronic excitation at intermediate ex-
citation levels forms a dense gas of charge carriers, excitons
and biexcitons.26 As a quasiparticle, the exciton has boson-
like character when the number of excited e-hpairs is neg-
ligibly small compared to the total number of valence elec-
trons. For a finite density, a hypothetical boson space may be
imagined, where excitons appear as ideal boson and their
deviation from ideal bosons is described by effective inter-
action between these bosons.27 It is exactly this exciton-
exciton interaction that ultimately determines almost all ex-
citonic optical nonlinearities including the biexciton
formation and related phenomena. During the past decades,
the interaction between excitons has been a subject of inten-
sive studies within different approaches by many research
groups.5,6,27–33 In the meantime, because of calculational dif-
ficulty of the four-exciton problem, very little is known
about the interaction between biexcitons which are approxi-
mately considered as bound states of two excitons. Brinkman
and Rice have been the first and the only so far, to our
knowledge, who considered the problem of the interaction
between biexcitons.34 Following closely the similarity be-
tween the biexciton and the hydrogen molecule, the authors
have calculated the quantum mechanical interbiexciton inter-
action potential as a function of the interhole distance and
pointed out that the biexciton-biexciton interaction is pre-
dominantly repulsive. Their result provides the first insight
into the biexciton-biexciton interaction strength though it is
certainly not sufficient to understand many features in the
optical property of the biexciton. Moreover, the bosonlike
nature of excitonic particles has been the basis for expecting
them to undergo Bose-Einstein condensation ~BEC!.6,29,35–37
Except for the special case of the Cu2O crystal, where a
weak ~due to nearly equal electron and hole masses!biexci-
ton creation is suppressed by strong e-hexchange interaction
38 and quantum-statistical properties have been displayed in
an exciton gas,39–41 in most other semiconductors BEC is
expected to occur inside the biexciton system. Physically, the
biexciton condensation can be considered as a result of pair
condensation,42 or coherent pairing43 in an exciton gas with
mutual attraction. The observation of Bose condensates in
CuCl was reported many years ago.10,44 Recently more
promising results were obtained by a different experimental
approach.45 Nevertheless, clear evidence for the BEC of
biexcitons has not appeared yet. In theoretical respect, the
interaction between particles is one of most important as-
pects of the problem of BEC in a system of weakly interact-
ing bosons. The property of such a system depends critically
on the small-momentum scattering amplitudes, among which
the only significant one is the amplitude for an ‘‘s-wave’’
collision. The determination of this quantity has therefore
been a major issue for the study of the possibility of a con-
densate formation and its properties. Consequently, an elabo-
rate theory of the biexciton-biexciton interaction is of impor-
tance not only to the fundamental exciton physics, but as
well in connection with experimental efforts to observe BEC
in excitonic systems.
The aim of the present paper is to derive the effective
biexciton-biexciton interaction in a semiconductor with
PHYSICAL REVIEW B 15 APRIL 1997-IIVOLUME 55, NUMBER 16
55
0163-1829/97/55~16!/10487~11!/$10.00 10 487 © 1997 The American Physical Society

simple band structure; the biexciton we refer to here is the
lowest one of the G1symmetry. Certainly we limit ourselves
to excitation densities nsatisfying the condition nabx
3!1,
abx is the biexciton radius, where the biexciton ~and the more
so the exciton with radius ax,abx) remains a stable quasi-
particle. Throughout the paper, subscripts x and bx will de-
note exciton and biexciton, respectively. Most of the biexci-
tons are assumed to be formed through the interaction among
excitons injected into the semiconductor from an incoherent
light source with frequency very close to the 1s-exciton en-
ergy level. Under the resonant condition, all of the excitons
in the system can be considered ground-state ones. The ef-
fective interaction between these excitons, especially that
among paraexcitons in the Cu2O crystal, and the form of the
biexciton wave function are discussed in Sec. II. Section III
is devoted to the derivation of the effective biexciton-
biexciton interaction, the analysis of a particular case allow-
ing an analytical study of the interaction and illustrations on
the CuCl crystal. Finally, discussions and conclusions are
given in Sec. IV. Throughout the paper we set \51.
II. EXCITON EFFECTIVE HAMILTONIAN
AND BIEXCITON WAVE FUNCTION
We consider a direct cubic semiconductor with isotropic
single conduction and valence bands at low temperatures. In
such a semiconductor, ground-state (1s-!excitons are of two
kinds: the paraexciton G2with angular momentum Jx50
and the orthoexciton G5with Jx51 which is higher in en-
ergy by a value Dof the e-hexchange interaction. Denoting
three basis vectors of the irreducible representation G5of the
crystal symmetry group by G5X,G5Y,G5Z, we write the
effective Hamiltonian of the system of interacting
1s-excitons within the framework of the boson description as
follows:
Hx5(
j
,kEx
j
~k!a
j
k
†a
j
k11
2(
q,k1,k2
H
F
Ud~q!
11
2Uex~q,k1,k2!
G
(
j
a
j
k11q
†a
j
k22q
†a
j
k2a
j
k11@Ud~q!
1Uex~q,k1,k2!#
3(
j
1Þ
j
2a
j
1k11q
†a
j
2k22q
†a
j
2k2a
j
1k121
2Uex~q,k1,k2!
3(
j
1Þ
j
2C
j
1C
j
2a
j
1k11q
†a
j
1k22q
†a
j
2k2a
j
2k1,~1!
where the sums over
j
,
j
1,
j
2go over all possible symmetry
states of the 1s-exciton: G2,G5X,G5Y,G5Zwhich for the
sake of brevity will be hereby denoted by O,X,Y,Z, re-
spectively. a
j
k
†(a
j
k) is the creation ~annihilation!operator
for the exciton of the symmetry
j
with momentum kand
energy Ex
j
(k) obeying Bose-Einstein statistics,
@a
j
8k8,a
j
k
†#5
d
j
8
j
d
k8k,
@a
j
8k8
†,a
j
k
†#5@a
j
8k8,a
j
k#50. ~2!
The coefficient C
j
in the last sum on the right-hand side ~rhs!
of ~1!is defined as follows:
C0521, C
j
51 for
j
5X,Y,Z.~3!
Udand Uex are, respectively, the direct and the exchange
exciton interaction function parametrically depending on the
e-hmass ratio
s
[(me/mh) through
a
5@
s
/(11
s
)#and
b
5@1/(11
s
)#. A general formula for Udand Uex is well
known.5,31–33 The function Uddescribes the direct Coulomb
interaction between two excitons and depends on the mo-
mentum transfer,
Ud~q!51
V
E
d3rexp~iqr!Ud~r…,~4!
where Vis the crystal volume and Ud(r) is a function of the
distance between interacting excitons5
Ud~r!52R(
t
5
a
,
b
1
t
exp
S
22r
t
ax
D
3
H
F
6
t
4
~2
t
21!224
t
6
~2
t
21!321
G
t
ax
r
12
t
4
~2
t
21!2211
823
4
r
t
ax
21
6
S
r
t
ax
D
2
J
,~5!
with Rdenoting the exciton binding energy ~exciton Ryd-
berg!,ethe electron charge, and
e
0the static dielectric con-
stant of the semiconductor.
On the contrary, the function Uex describes the effect due
to Pauli exclusion principle acting between constituent par-
ticles belonging to different interacting excitons. Except for
the momentum transfer, Uex depends on the difference be-
tween the momenta of interacting excitons and therefore in
the rspace cannot be presented as a function of only the
distance between these excitons. The nonlocality of
Uex(q,k1,k2) depends on
ab
and on values of vectors qand
k12k2. It can be neglected if either the mass of hole is much
larger than that of the electron, or all momenta are very
small. Here we find it convenient to present the exchange
function in the following way:
Uex~q,k1,k2!5Uex~q,k12k21q!,
Uex~q,k1q!51
V
EE
d3rd3r8
exp@iqr1i~k1q!r8#
3Uex~r,r8!,~6!
where
10 488 55HOANG NGOC CAM

Uex~r,r8!52R
a
3
b
3
E
d3r1
H
f~r1!f
S
r12r
a
2r8
b
D
3f
S
r12r
a
D
f
S
r12r8
b
D
3
F
ax
U
r12r
a
2r8
b
U
2
a
ax
r
G
1f~r1!
3f
S
r12r8
a
2r
b
D
f
S
r12r8
a
D
f
S
r12r
b
D
3
F
ax
U
r12r8
a
2r
b
U
2
b
ax
r
G
J
,~7!
with fdenoting the 1shydrogenlike function,
f~r!51
A
p
ax
3exp
S
2r
ax
D
,
describing the relative motion of e-hpair in an exciton. As a
sum of four-center integrals, Uex cannot be calculated ana-
lytically. Considering particular cases of Eq. ~6!,
Uex~k,0!51
V
E
d3rexp~ikr!
F
E
d3r8Uex~r,r8!
G
,
and
Uex~0,k!51
V
E
d3rexp~ikr!
F
E
d3r8Uex~r8,r!
G
,~8!
we find that functions
U1
ex~r![
E
d3r8Uex~r,r8!
and
U2
ex~r![
E
d3r8Uex~r8,r!~9!
can be evaluated. As Ud(r), these functions depend only on
the distance between interacting excitons,
U1
ex~r!52R(
t
5
a
,
b
1
t
3
F
S
11r
t
ax
D
exp
S
2r
t
ax
D
2
t
ax
rS
S
r
t
ax
D
G
S
S
r
t
ax
D
,
U2
ex~r!52R(
t
5
a
,
b
1
t
3
H
S
11r
t
ax
D
exp
S
2r
t
ax
D
S
S
r
t
ax
D
2
F
5
8223
20
r
t
ax
23
5
S
r
t
ax
D
2
21
15
S
r
t
ax
D
3
G
exp
S
22r
t
ax
D
26
5
t
ax
r
$
@
g
1ln~r/
t
ax!#S~r/
t
ax!21S~2r/
t
ax!2Ei~24r/
t
ax!22S~2r/
t
ax!S~r/
t
ax!Ei~22r/
t
ax!
%
J
,~10!
where Ei(z)5
*
z
`@exp(2y)/y#dy is the exponent integral,
g
.0.577 the Euler’s constant, and
S~lr![
F
11lr11
3~lr!2
G
exp~2lr!.~11!
Using the expressions for Ud(r), U1
ex(r), and U2
ex(r), the
effective interaction potential between excitons of different
symmetries at vanishing momentum transfer can be de-
scribed. To shed some light on the problem of BEC in an
exciton gas, we consider a particular case of interaction be-
tween paraexcitons which probably are elementary excita-
tions of the lowest energy in the Cu2O crystal. From the
formula for the interaction potential between two paraexci-
tons with momenta k1,k2and momentum transfer q50,33
Ueff
p2p~k!51
2
F
Ud~k!11
2Uex~0,k!11
2Uex~k,0!
G
,~12!
k[k12k2and using Eqs. ~4!,~8!, and ~9!we find that the
function
Ueff
p2p~r!5R
F
Ud~r!11
2U1
ex~r!11
2U2
ex~r!
G
~13!
describes the potential in the rspace. The variation of this
function with the distance is shown in Fig. 1 for a semicon-
ductor with
s
50.7. One can observe that the repulsion be-
tween paraexcitons decreases rapidly at r.2axand vanishes
at r>3ax:Ueff
p2p(3ax),0.005R. Thus the system of paraex-
citons in the Cu2O crystal (
s
50.7, ax57Å!is a nearly
ideal Bose gas even at densities as high as 1019
cm23which has been indicated in experimental works on
this crystal.40,41 Because at low temperatures most particles
in such a gas have small momentum @k!(1/ax)#,46 the char-
acter of the interaction between paraexcitons in a general
case (qÞ0,k12k2Þ0) is expected to be similar to that rep-
resented by the curve in Fig. 1. By the use of Ueff
p2p(r), the
‘‘s-wave’’ scattering length ~or the hard-core radius!of
paraexcitons in Cu2O is estimated to be approximately
2.2ax.47 The result is in agreement with the experimental
upper bound provided by the authors of Ref. 48.
Further we assume for the considered semiconductor the
orthopara splitting Dto be much smaller than the biexciton
55 10 489BIEXCITON-BIEXCITON INTERACTION IN SEMICONDUCTORS

binding energy Ebx
b~the situation is realized in CuCl where
D56.2 meV and Ebx
b530 meV15!. Then, in the formation of
a biexciton participate two excitons in a superposition of
zero-angular-momentum states which are made up of two
ortho- and two paraexcitons, respectively.15 Since Ebx
bis
typically an order of magnitude smaller than the correspond-
ing exciton binding energy,49 the traditionally used adiabatic
approximation5,16,28 for the biexciton wave function is legiti-
mate. In the approximation the biexciton state with momen-
tum Kcan be written in the following form:
u
K
&
51
A
8V(
p
f
~p!(
j
5O,X,Y,ZC
j
a
j
~K/2!2p
†a
j
~K/2!1p
†
u
0
&
~14!
where
f
(p) is the Fourier transform of normalized spheri-
cally symmetric function F(r) describing the relative motion
of two excitons in a biexciton. The equation for
f
(p) can be
obtained from the Heisenberg equation of motion for a biex-
citon with the aid of the Hamiltonian ~1!. It appears as fol-
lows:
F
Ex
S
K
22p
D
1Ex
S
K
21p
D
G
f
~p!
1(
q@Ud~q!2Uex~q,2p2q!#
f
~p2q!
5Ebx~K!
f
~p!,~15!
where Ebx(K) is the energy of the biexciton with momentum
K.InEq.~15!the exciton energy is taken to be the same for
both ortho- and paraexcitons,
Ex~k!5ET1k2
2mx,~16!
with ETthe transverse orthoexciton energy at k50 and mx
the exciton translational mass. Equation ~15!admits no rig-
orous solution even in the case
s
50 when it presents the
Fourier transform of the Schro
¨dinger equation for the energy
and wave function of nuclei relative motion in a hydrogen
molecule within the Heitler-London approximation. For any
s
Þ0, Eq. ~15!corresponds to an integro–differential equa-
tion for F(r) which can be reduced to a differential one only
if the nonlocality of Uex(r,r8) is neglected. However, this
equation will not be dealt with in the present paper. The
biexciton envelope function is expected to play some aver-
aging role in the problem of biexciton-biexciton interaction
and its extension appears to have more influence upon the
interaction rather than its form. In our study of the biexciton-
biexciton interaction for a particular case, the hydrogenlike
form
F~r!51
A
p
abx
3exp
S
2r
abx
D
~17!
will be used for the biexciton envelope function. The extent
of the biexciton then is defined by the radius abx , which is
connected with the binding energy Ebx
bby the following re-
lation:
Ebx
b51
V
p
abx
3
F
EE
d3rd3r8
exp
S
2
u
r1r8
u
abx
2
u
r2r8
u
abx
D
Uex~r,r8!2
E
d3r
3exp
S
22r
abx
D
Ud~r!
G
,~18!
resulting from Eqs. ~15!and ~17!.In~18!the biexciton bind-
ing energy Ebx
bis meant to be greater than the dissociation
energy D52ET2Ebx(0) by the energy of the so called zero-
point vibration.50 By an acceptable approximation for
Uex(r,r8) from ~18!a formula relating a function of the
biexciton-exciton radius ratio a[abx /axto a value of
E[Ebx
b/Rtaken from experiment can be derived. In this way
the biexciton radius may be estimated. The two quantities
aand Eare the most important individual characteristics of
the biexciton. Along with
s
, they determine the picture of
the biexciton-biexciton interaction as one can see in the next
section.
III. EFFECTIVE BIEXCITON-BIEXCITON INTERACTION
Since in the adiabatic approximation excitons are as-
sumed to remain themselves binding into biexcitons, the ef-
fective interaction between biexcitons has as the origin the
interaction between their component excitons. Then, we may
expect the interaction between two biexcitons to be the sum
of the effective interactions between constituent excitons av-
eraged over wave functions of their relative motions. Intro-
FIG. 1. Variation of the no-momentum-transfer interaction po-
tential between two paraexcitons in Cu2O with the distance.
s
50.7.
10 490 55
HOANG NGOC CAM

ducing the creation operator bK
†for the biexciton with mo-
mentum K, we write the biexciton wave function as follows:
u
K
&
5bK
†
u
0
&
.~19!
Comparing ~19!with ~14!, we get biexciton operators ex-
pressed in terms of exciton ones
bK
†51
A
8V(
p
f
~p!(
j
5O,X,Y,ZC
j
a
j
~K/2!2p
†a
j
~K/2!1p
†.~20!
From the bosonlike character of exciton operators and the
normalization of the biexciton envelope function, we obtain
the following commutation relation:
@bK8,bK
†#5
d
K8K11
2V(
p
f
~p!
f
S
p1K82K
2
D
3(
j
5O,X,Y,Za
j
~K/2!1p
†a
j
K82~K/2!2p,~21!
with bKthe annihilation operator for the momentum Kbiex-
citon. Equation ~21!indicates that strictly speaking the biex-
citon is not a boson. However, it is easy to see that matrix
elements of the last term on the rhs of ~21!are of order
nabx
3. So, in the considered interval of excitation densities,
the biexciton may be treated as a boson. Then two-biexciton
wave function can be constructed as a direct product of
single-biexciton wave functions for each biexciton,
u
K1K2
&
5NK1K2bK1
†bK2
†
u
0
&
,~22!
where K1and K2are the momenta of two biexcitons, and
NK1K2511
S
1
A
221
D
d
K1K2~23!
ensures the orthonormalization of two-biexciton wave func-
tion ~22!,
^
K1
8K2
8
u
K1K2
&
5
d
K1
8K1
d
K2
8K21
d
K1
8K2
d
K1K2
8
2
d
K1K2
d
K1
8K2
8
d
K1
8K1.~24!
By the use of ~20!,~22!can be written as follows:
u
K1K2
&
5NK1K2
1
8V(
p1
f
~p1!
3(
j
15O,X,Y,ZC
j
1a
j
1~K1/2!2p1
†a
j
1~K1/2!1p1
†
3(
p2
f
~p2!
3(
j
25O,X,Y,ZC
j
2a
j
2~K2/2!2p2
†a
j
2~K2/2!1p2
†
u
0
&
.
~25!
Putting the Hamiltonian ~1!of the exciton system between
two-biexciton wave functions in the form ~25!, we arrive
after intricate calculations with the use of Eqs. ~2!,~15!,~24!
at the following result:
^
K2
8K1
8
u
Hx
u
K1K2
&
5@Ebx~K1!1Ebx~K2!#
^
K2
8K1
8
u
K1K2
&
1
d
K1
81K2
8,K11K2
3P~K1
82K1,K1
82K2!,~26!
where P(K1
82K1,K1
82K2) is the effective interaction po-
tential between two biexcitons with momenta K1,K2and
with momentum transfer Q[K1
82K1. The potential is found
to depend on Qand on the difference K5K12K2between
the momenta of two interacting biexcitons. It can be written
as follows:
P~Q,K1Q!51
2@W~Q,K1Q!1W~K1Q,Q!#,~27!
where Wis the effective biexciton-biexciton interaction first
introduced by Hanamura for the boson description of a biex-
citon gas by means of the model Hamiltonian,
Hbx5(
KEbx~K!bK
†bK
11
2(
Q,K1,K2W~Q,K1,K2!bK11Q
†bK22Q
†bK2bK1.
~28!
We obtain here the expression for Win the following
form:
FIG. 2. Diagrams showing the nature of the biexciton-biexciton
interaction. ~a!the direct and ~b!and ~c!the exchange processes.
Single solid lines denote excitons. Two close parallel exciton lines
denote a biexciton. A crossing of lines denotes an exchange of
excitons belonging to different biexcitons. Dashed lines represent
effective interactions between excitons depending on the momen-
tum transfer, the momenta difference of interacting excitons and on
their symmetry
j
,
j
1,or
j
2
.
55 10 491BIEXCITON-BIEXCITON INTERACTION IN SEMICONDUCTORS

W~Q,K1Q!51
V2(
p1,p2
F
4Ud~Q!17
2Uex
S
Q,p22p11Q
21Q1K
2
D
G
f
~p1!
f
S
p12Q
2
D
f
~p2!
f
S
p21Q
2
D
11
V2(
p1,p2
F
Ud~p1!11
2Uex~p1,Q!
G
f
~p2!
f
S
p21Q
2
D
f
S
p22p11Q1K
2
D
f
S
p22p11Q
21Q1K
2
D
11
V2(
p,q@Ud~q!2Uex~q,2p1Q2q!#
f
S
p2q1Q
2
D
f
~p!
f
S
p1Q1K
2
D
f
S
p1Q
21Q1K
2
D
.~29!
The first term on the rhs of ~29!is the contribution of the
biexciton interaction process of the direct character, when
each of interacting biexcitons after the interaction consists of
the same excitons that form the biexciton before the interac-
tion @see Fig. 2~a!#. In contrast, the two last terms on the rhs
of ~29!represent exchange processes when interacting biex-
citons exchange with each other one of the constituent exci-
ton particles @see Figs. 2~b!,~c!#. As expected, the effective
biexciton-biexciton interaction is found to be the sum of ef-
fective interactions Ueff
j
12
j
2between component excitons av-
eraged over biexciton envelope functions. Besides their de-
pendence on the momentum transfer and the momenta
difference, the functions Ueff
j
12
j
2depend on the symmetry
j
1,
j
2of interacting excitons. Thus, in the process of Fig.
2~b!as well as of Figs. 2~a!and 2~c!for the case
j
15
j
2we
have Ueff
j
2
j
5Ud(q)11
2Uex(q,k1q), while for
j
1Þ
j
2,
Ueff
j
12
j
25Ud(q)1Uex(q,k1q) in the process of Fig. 2~a!
and Ueff
j
12
j
252 1
2Uex(q,k1q) in that of Fig. 2~c!. It is inter-
esting to note that the direct biexciton-biexciton interaction
schematically shown in Fig. 2~a!in reality is the interaction
between two excitons belonging to different interacting biex-
citons and therefore includes both the direct and the ex-
change exciton interactions. Thus not only the exchange
biexciton-biexciton interaction which certainly is nonlocal,
but the direct one contains a nonlocal term as well. For this
reason, it is convenient to present the biexciton-biexciton
interaction as a sum of a local and a nonlocal part,
W~Q,K1Q!5Wl~Q!1Wnl~Q,K1Q!,~30!
where the local one consists in the net direct interaction,
when not only excitons in biexcitons but also the electrons
and holes in those excitons interact directly. Wldepends only
on the momentum transfer,
Wl~Q!54Ud~Q!
F
1
V(
p
f
~p!
f
S
p2Q
2
D
G
2,~31!
whereas the whole dependence of the biexciton-biexciton in-
teraction on the difference between momenta of interacting
biexcitons enters the nonlocal part Wnl,
Wnl~Q,K1Q!51
2V2(
p1,p2
H
7Uex
S
Q,p11Q
21Q1K
2
D
f
~p22p1!
f
S
p22p12Q
2
D
1@2Ud~p1!1Uex~p1,Q!#
3
f
S
p22p11Q1K
2
D
f
S
p22p11Q
21Q1K
2
D
J
f
S
p21Q
2
D
f
~p2!2Ebx
b1
V2(
p
f
~p!
f
S
p1Q
2
D
3
f
S
p1Q1K
2
D
f
S
p1Q
21Q1K
2
D
,~32!
the last term of which is written applying Eq. ~15!to the sum
over qin the last term on the rhs of Eq. ~29!.
Since Ud(0)50 the local part of the biexciton-biexciton
interaction vanishes when no momentum is transferred be-
tween biexcitons. Estimations show that for QÞ0,Wlis as
well negligible for all values of
s
and possible forms of
f
.
So the biexciton-biexciton interaction is mostly nonlocal
and, in general, cannot be evaluated analytically. To see this
more clearly, we go to the rspace by the following transfor-
mations:
Wl~Q!51
V
E
d3rexp
S
iQ
2r
D
Wl~r!~33!
Wnl~Q,Q8!51
V
EE
d3rd3r8
3exp
S
iQ
2r1iQ8
2r8
D
Wnl~r,r8!,~34!
where Wl(r) and Wnl(r,r8) are obtained from ~31!,~32!,
taking into account ~4!and ~6!,
Wl~r!54
E
d3r1Ud~r1!
E
d3r2F~r2!2F~2r11r22r!2,
~35!
10 492 55HOANG NGOC CAM

Wnl~r,r8!5
E
d3r1F~r1!F~r12r8!
H
F~r12r2r8!
3F~r12r!@Ud~r8!2Ebx
b#
11
2
E
d3r2F~r112r22r2r8!F~r112r22r!
3@Uex~r8,r2!17Uex~r2,r8!#
J
.~36!
It can be seen from ~35!,~36!that while Wl(r…can be
evaluated for any possible form of Fusing bipolar variables,
multicenter integrals in the expression for Wnl(r,r8) cannot
be calculated. The essential problem resides as well in the
fact that full knowledge of Uex(r,r8) is lacking. For this rea-
son, even in particular cases K50and K1Q50, Eq. ~36!
remains hard to deal with and has no practical value. An
exception is the case Q50when in place of Uex appears
U1
ex and U2
ex and multicenter integrals on the rhs of ~36!turn
to products of two- and single-center ones. From now in this
section we focus our attention on this case to extract from it
general features in the behavior of the biexciton-biexciton
interaction.
Putting Q50in Eqs. ~30!,~31!,~34!and using ~36!,we
find the no-momentum-transfer biexciton-biexciton interac-
tion in the form
W~0,K!51
V
E
d3rexp
S
iK
2r
D
W0~r!,~37!
with W0(r) being a product of the sum of no-momentum-
transfer effective interactions between excitons which form
interacting biexcitons and squared overlap integral of enve-
lope functions of these biexcitons,
W0~r!5
E
d3r8Wnl~r,r8!
5
F
Ud~r!11
2U1
ex~r!17
2U2
ex~r!2Ebx
b
G
3
F
E
d3r8F~r8!F~r82r!
G
2.~38!
The overlap integral takes the unit value at r50due to the
normalization of Fand decreases with the increase of the
distance rat sufficiently large ras Fdoes. For Fof the
hydrogenlike form ~17!the integral is equal to S(r/abx)@see
Eq. ~11!#. Thus the binding of excitons into biexcitons some-
what decreases the magnitude of the effective interaction be-
tween them and the influence depends on the binding degree,
namely, on the relative size of the biexciton and the exciton.
The sum of effective interactions between constituent ex-
citons consists of a linear combination of exciton interaction
functions and a small constant inverse in sign to the biexci-
ton binding energy. With the smooth weighting factor
S(r/abx)2, the first
F
Ud~r!11
2U1
ex~r!17
2U2
ex~r!
G
S
S
r
abx
D
2
~39!
describes the main strong repulsive part of the biexciton-
biexciton interaction. Due to the exchange exciton-exciton
interaction, this part varies drastically near r50, then takes
very large positive values giving the biexciton an incom-
pressible ‘‘core.’’ From distances of few axthe behavior of
~39!is characterized by factors exp(22r/
a
ax)S(r/
abx)2and exp(22r/
b
ax)S(r/abx)2which drop with the in-
crease of rmuch more rapidly than S(r/abx)2does. Then, as
the distance increases, there comes a turning point r0where
the biexciton-biexciton interaction reverses its sign, after
which the weak attractive part described by the term
2Ebx
bS
S
r
abx
D
2
~40!
becomes dominant. The position of r0is determined by the
e-hmass and biexciton-exciton binding energy ratios
s
and
E. In the theory of intermolecular interaction, r0is called the
effective diameter of the molecule: when the distance be-
tween molecules becomes greater than this diameter, they
begin to attract each other. We see in the case of biexciton-
biexciton interaction that the attraction is due to exchange
interaction between excitons of the same symmetry resulting
in the biexciton interaction process of Fig. 2~c!. It has noth-
ing in common with the van der Waals attraction between
neutral complex particles which exists at distances a few
times larger than the biexciton diameter. That attraction is
important only at low densities and is not under consider-
ation in this paper.
While the range of the repulsive core of the biexciton-
biexciton interaction scales in the exciton radius, that of the
attractive part is defined by the radius of the biexciton. The
attractive part is therefore relatively extended. We suppose,
however, that it does not lead to pair bound states and the
biexciton remains the fundamental unit in the considered
semiconductor. Then, in the considered range of excitation
densities (nabx
3!1), the biexciton system can be treated as a
weakly nonideal Bose gas,51 with most particles having
small momentum k,kabx!1 at low temperatures. In this
connection, it is worthwhile to calculate the small-
momentum value of the biexciton-biexciton interaction by
which the interaction is often replaced in practical
estimations.6,32 From ~37!,~38!we have
W~0,0!5
E
d3r
F
Ud~r!11
2U1
ex~r!17
2U2
ex~r!
G
S
S
r
abx
D
2
2Ebx
b
E
d3rS
S
r
abx
D
2.~41!
A reasonable estimation may be made for the first integral
of W(0,0): Since S(r/abx)2is a very smooth function in
comparison with exciton interaction functions, it can be ap-
proximated by one over the interval where the last noticeably
differ from zero. Taking into account Eqs. ~4!,~8!,~9!and
the fact Ud(0)50 , we find this integral equal to
55 10 493BIEXCITON-BIEXCITON INTERACTION IN SEMICONDUCTORS

4Uex~0,0!5426
p
3Rax
3.
The last integral of W(0,0)in~41!can be easily calcu-
lated. Its value is proportional to the effective volume of the
biexciton. As a result,
W~0,0!.104
p
3Rax
3233
p
2Ebx
babx
3.~42!
As in practice Ebx
babx
3is usually of the order of Rax
3, we have
W(0,0)'2Uex(0,0).
The first term of W(0,0)in~41!can be obtained from the
result ~4.11!of Ref. 34 by performing the Fourier transfor-
mation and taking into account the energy and length units
used there. The last one has not been considered in Ref. 34.
As a matter of fact, in the sum of effective interactions be-
tween excitons @see Eq. ~38!#, the negative term at first sight
may be neglected because Ebx
bis typically a tenth of R, while
the average value of the exchange interaction functions in the
range of their maximum (0,r,ax) are of order of tens
R. However, this small negative term gives rise to a fairly
extended attractive part of the biexciton-biexciton interaction
which not only considerably decreases the small-momentum
value but also essentially changes the picture of the interac-
tion.
For a thorough evaluation of the first integral of W(0,0)
in ~41!we write it as follows:
E
d3r
F
Ud~r!11
2U1
ex~r!17
2U2
ex~r!
G
S
S
r
abx
D
2
532
p
I
s
~a!Rax
3.~43!
From the above estimation we know that for abx@ax, the
parametric dependence of Ion
s
vanishes, I
s
(`)513
12.By
the use of Eqs. ~5!and ~10!we get I
s
(a) in the following
form:
I
s
~a!5
E
dtt2
4exp~22t!(
t
5
a
,
b
t
2S
S
t
at
D
2
H
F
6
t
4
~2
t
21!2
24
t
6
~2
t
21!321
G
1
t12
t
4
~2
t
21!2211
823
4t21
6t2
J
1
E
dt t
8
H
S
29
81481
20 t1223
15 t2147
15t3
D
t
3exp~22t!242
5
F
S
5
421
g
1lnt
D
S~t!2
1S~2t!2Ei~24t!22S~t!S~2t!Ei~22t!
G
J
3(
t
5
a
,
b
S
S
t
at
D
2.~44!
From explicit expressions for Sand for the exponent in-
tegral we can see that I
s
(a) can be calculated analytically.
However, as the result is found to be very awkward and the
dependence of Ion aand
s
is too intricate to follow, we
shall not present it here. We wish to note only that I
s
(a)
slightly decreases as adecreases. For practical estimation, it
is much more simple to compute ~44!for definite
s
and a,
which we shall do for the CuCl crystal afterwards.
With the aid of ~43!, we have
W~0,0!5
F
32
p
I
s
~a!233
p
2Ea3
G
Rax
3.~45!
It is certain that a proper calculation of I
s
(a) which gives for
the integral on the lhs of ~43!a result a little smaller than
4Uex(0,0) does not change the relation between the contri-
butions of two parts of the biexciton-biexciton interaction to
W(0,0). The last is a positive quantity: the attraction is pre-
dominated over by the repulsion which is extremely strong at
small distances. The repulsion that comes from a quantum-
statistical principle working between identical component
particles of interacting biexcitons ~both between excitons and
between electrons and holes of the excitons!makes the biex-
citons impenetrable to each other. Thus each biexciton like
any molecule may be imagined as a hard sphere of diameter
r0surrounded by an attractive potential of the range rw~of
order of few abx) and maximum depth «w~of order of tenth
of Ebx
b). It is well known from quantum mechanics that the
relationship between rw,«wand the particle mass deter-
mines the possibility of a bound state formation by the at-
tractive potential.52 We shall return to this question while
considering the example of CuCl crystal later in this section.
Here no bound state is assumed. Then the biexciton-
biexciton ‘‘s-wave’’ scattering length asis positive,53 and as
it was mentioned earlier, the biexciton system is a weakly
nonideal Bose gas in the excitation range nabx
3!1. At low
temperatures the last condition means that kabx!1 too.
Hence the no-momentum-transfer biexciton-biexciton inter-
action ~38!, although not exact, possesses not only main
qualitative but as well quantitative features of the interaction
among biexcitons. It can therefore be used for the computa-
tion of the scattering length asand in this way an important
problem of the theory of biexciton Bose condensates be-
comes solvable: According to the standard procedure of de-
scribing a weakly nonideal Bose gas with positive scattering
length — the Bogoliubov approximation — properties of the
ground state of such a gas ~the condensate!are described in
terms of first two powers of the small parameter (nas
3)1/2.46
To interpret the results discussed above , we consider the
CuCl crystal, a prototypical material for the study of biexci-
tons on which most extensive experimental work has been
carried out. The following parameters of CuCl are used:
s
50.25, Ebx
b530 meV, R5200 meV, ax57 Å. According
to Eq. ~18!, the values of
s
and Ecorrespond to a.1.7 if we
neglect the terms proportional to the second and higher pow-
ers of
a
in the expansion of Uex in terms of U1
ex and U2
ex . The
computation of the integral ~44!gives I0.25(1.7)50.9234 and
from ~45!the corresponding value of W(0,0)is
17.4
p
Rax
3. So, for CuCl the assumption W(0,0)
.2Uex(0,0) used in Ref. 6 is in a very close agreement with
our calculation.
To characterize a density of the biexciton gas in CuCl,
following the authors of Ref. 34, we use the radius rof the
sphere of volume n21as the mean distance between quasi-
particles. The dependence of the biexciton-biexciton interac-
10 494 55HOANG NGOC CAM

tion W0(r) on this distance is plotted in Fig. 3. We see that
the biexciton system in CuCl may be considered as a nearly
ideal Bose gas at densities n<1018 cm23, when attraction
among particles almost vanishes:
u
W0(r)
u
,0.01Ebx
bfor
r.8ax. As the density increases, the attraction between
biexcitons rises and its magnitude mounts up to about eighth
of Ebx
bin the interval r.4ax–4.5ax. With further increase of
the density, the biexciton-biexciton interaction reaches the
turning point at n.1.531019 cm23(r0.3.6ax), beyond
which it goes over into the region of a rapidly increasing
repulsion. For n>231019 cm23the magnitude of the repul-
sion becomes comparable with Ebx
band causes the biexciton
to break up. Strictly speaking, the excitation densities corre-
sponding to r,r0is out of the range of validity of the model
used in this paper. At such densities the biexciton is no
longer a stable quasiparticle. In this way, in the range of their
stability (n<1.531019 cm23for CuCl!, biexcitons attract
each other at any distance. The question of whether the at-
traction leads to the formation of biexciton pair bound state
becomes essential. To answer to it, we approximate the
biexciton-biexciton interaction in CuCl by the following
Morse’s formula:
W0~r!.«w
$
exp@22
d
~r2r1!#22exp@2
d
~r2r1!#
%
,
~46!
with «w5Ebx
b/8, r154.36ax, and
d
5ln2/0.76ax. Equation
~46!does not represent the behavior of W0(r)atr,r
0ex-
actly, but describes very well main features of the shallow
potential well of Fig. 3. According to a result of Ref. 52 ~see
problem 4!,if
A
2m
x
«
w
d
,
1
2~47!
bound state cannot be formed in the system with pair inter-
action ~46!. Using the relation between mxand mand be-
tween mand Rand ax, we gain ~47!in the form
0.38~11
s
!
ln2
A
E
2
s
,1
2
which is satisfied for CuCl with
s
50.25 and E50.15. Defi-
nitely, the biexciton system in CuCl is a weakly nonideal
Bose gas with positive scattering length. By the variational
method,53 asis estimated to be approximately 3ax'2 nm,
smaller than the experimental value as'3 nm provided by
Hasuo et al. recently.45
At last, it is interesting to note that the expression for the
biexciton-biexciton interaction in the form ~38!places an
upper limit on possible values of the biexciton radius. In-
deed, for a given semiconductor with definite
s
and E, the
range r0of the repulsive core of the biexciton-biexciton in-
teraction is determinate and r0>2abx . For CuCl, we see
from foregoing analysis that abx<1.8ax. This result may
partially remove the existing discrepancy in the estimation of
the value of the biexciton radius in the model crystal.
IV. DISCUSSION AND CONCLUSION
For simplicity, the effective biexciton-biexciton interac-
tion in the present paper is derived for a simple model of
isotropic single bands. In principle, the derivation can be
performed for any direct cubic two-band semiconductor. The
effective bosonic Hamiltonian for an exciton system can be
written by the use of general formula from Ref. 33, and the
biexciton wave function constructed by group-theoretical
methods. Qualitative features of the biexciton-biexciton in-
teraction in the general case of a direct cubic semiconductor
may be expected to be the same as obtained for the simple
model. Being an intricate function of the momentum transfer
and of the difference between momenta of interacting biex-
citons, the interaction allows an analytical study when the
first vanishes. In this case the biexciton-biexciton interaction
in the r-space depends only on the interbiexciton distance
and consists of a short-range strong repulsive core and a
short-range weak attractive part ~which at far distances may
join up with the very weak long-range van der Waals tail!.
As stable quasiparticles, biexcitons attract each other at any
distance. If the attraction of exchange nature does not lead to
pair bound state formation, biexciton system can be consid-
ered as a weakly nonideal Bose gas with positive
‘‘s-wave’’ scattering length. The last can be estimated using
the expression obtained for no-momentum-transfer
biexciton-biexciton interaction.
The behavior of the biexciton-biexciton interaction versus
distance is characteristic of the intermolecular interaction. In
this respect, the biexciton closely resembles the hydrogen
molecule. The result obtained in the present paper in the
limit
s
50 describes the interaction between H2molecules
within the Heitler-London approximation. For
s
50, the
computation of the interaction can be carried out for general
case of QÞ0since the nonlocality in Uex vanishes and the
exchange interaction function in the rspace can be analyti-
cally evaluated. The attraction between H2is certainly more
strong than that among biexcitons in CuCl since the param-
eter Eof H2is more than twice greater, E.0.35 for
s
50.
In a molecular system, attractive pair interaction leads to an
abrupt transformation from gaseous to a liquid state at low
temperatures. It was shown by Keldysh that because of the
FIG. 3. No-momentum-transfer effective biexciton-biexciton in-
teraction in CuCl as a function of the distance.
s
50.25, E50.15,
and a51.7.
55 10 495BIEXCITON-BIEXCITON INTERACTION IN SEMICONDUCTORS

smallness of the mass of excitonic particles, such a gas-
liquid transition is not expected neither in a biexciton, nor in
an exciton gas.54 An N-body bound state in semiconductors
is formed at very high densities, where not only biexcitons
but excitons as well are destroyed and the system of charge
carriers exists in the form of a dense nonequilibrium e-h
liquid. Such densities which are above the Mott threshold are
not considered in this paper. Hence, despite the resemblance
to H2, the biexciton system remains a gas down to absolute
zero. From the interparticle interaction point of view, the
formation of a stable Bose condensate with a sufficiently
high condensation fraction ~owing to relative smallness of
as) is possible at low temperatures. In this way the biexciton
system in semiconductors is the prime candidate for the ob-
servation of such an important phase transition as BEC.
ACKNOWLEDGMENTS
The problem described in this paper was suggested by
Professor S.A. Moskalenko during the stay of the author at
the Department of Semiconductor Theory and Quantum
Electronics, Institute of Applied Physics in Kishinev. It is a
great pleasure for the author to thank him and Professor A.I.
Bobrysheva for their direction and encouragement in that
period. Enlightening discussions with Dr. S.P. Cojocaru, his
help in the numerical calculation and critical reading of the
manuscript are gratefully acknowledged. The author is also
greatly indebted to Professors L.A. Shelepin and V.S.
Gorelic for their assistance in continuing the study of the
problem.
*Permanent address: Institute of Physics, National Centre of Natu-
ral Sciences and Technology of Vietnam, P.O. Box 429 BoHo,
10000 Hanoi, Vietnam.
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55 10 497BIEXCITON-BIEXCITON INTERACTION IN SEMICONDUCTORS