Carrier multiplication in semiconductor nanocrystals via intraband optical transitions involving
virtual biexciton states
Valery I. Rupasov*
ALTAIR Center LLC, Shrewsbury, Massachusetts 01545, USA
and Landau Institute for Theoretical Physics, 117940 Moscow, Russia
Victor I. Klimov
†
Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Received 27 June 2007; published 25 September 2007
We propose and analyze a physical mechanism for photogeneration of multiexcitons by single photons
carrier multiplication in semiconductor nanocrystals, which involves intraband optical transitions within the
manifold of biexciton states. In this mechanism, a virtual biexciton is generated from nanocrystal vacuum by
the Coulomb interaction between two valence-band electrons, which results in their transfer to the conduction
band. The virtual biexciton is then converted into a real, energy-conserving biexciton by photon absorption on
an intraband optical transition. The proposed mechanism is inactive in bulk semiconductors as momentum
conservation suppresses intraband transitions. However, it becomes highly efficient in the case of zero-
dimensional nanocrystals, where quantum confinement results in relaxation of momentum conservation, which
is accompanied by the development of strong intraband absorption. Our calculations show that the efficiency
of the carrier multiplication channel mediated by intraband optical transitions can be comparable to or even
greater than that for impact-ionization-like processes mediated by interband transitions.
DOI: 10.1103/PhysRevB.76.125321 PACS numbers: 78.67.Bf, 73.21.La, 78.47.+p
I. INTRODUCTION
Carrier multiplication CM is a process in which absorp-
tion of a single photon produces not just one but multiple
electron-hole e-h pairs excitons. CM can benefit a num-
ber of technologies, especially photovoltaics and photoca-
talysis. In bulk semiconductors, CM is typically explained by
impact ionization.
1
In this process, a conduction-band elec-
tron or a valence-band hole of sufficiently high energy inter-
acts with a valence-band electron promoting it across the
energy gap E
g
. An important characteristic of CM is the e-h
pair creation energy defined as the energy lost by the
ionizing particle in a single e-h pair creation event. On the
basis of energy conservation, the minimal value of is E
g
.
However, because of restrictions imposed by momentum
conservation and significant phonon losses, the values of
measured for bulk semiconductors significantly exceed this
energy-conservation-defined limit. In particular, is approxi-
mately 3E
g
for wide-gap semiconductors and even greater
for narrow-gap materials.
2,3
Recent experimental studies of zero-dimensional 0D
semiconductor nanocrystals
4–6
NCs indicate that in these
structures, approaches its ultimate limit of E
g
, which re-
sults in extremely high CM efficiencies.
6
The mechanism for
this enhancement is not well understood. For example,
Zunger and co-workers adopted a traditional impact-
ionization model for treating CM in NCs.
7,8
On the other
hand, recent calculations by Allan and Delerue
9
indicate that
impact ionization is not enhanced by quantum confinement,
suggesting the existence of either additional or alternative
mechanisms for CM in NCs.
Two such mechanisms have been discussed by Efros and
co-workers
5,10
and Schaller et al.
11
Their common motif is
that CM occurs via direct Coulomb coupling of a single-
exciton state, generated in a NC via an interband transition,
to a biexciton state corresponding matrix element U
II
is the
same as in the impact-ionization models. In Ref. 11, this
process was treated using second-order perturbation theory
as a transition via intermediate virtual single-exciton states,
while Ref. 10 utilized a time-dependent density-matrix ap-
proach. In the limit of weak Coulomb coupling U
II
xx
;
xx
is the biexciton dephasing rate, both models lead to a
similar result for the ratio of the biexciton N
xx
to the single-
exciton N
x
populations. Specifically, in the case of a single
intermediate exciton state dephasing rate
x
coupled to a
single biexciton state, N
xx
/ N
x
= |U
II
|
2
/
x
xx
. This expres-
sion shows that N
xx
/ N
x
can be greater than unity only if
U
II
x
, which was regarded as a necessary condition for
efficient CM in Ref. 10. However, as was pointed out in Ref.
11, the available experimental data indicate that Coulomb
coupling in NCs is likely smaller than
x
. Therefore, it was
suggested that an important factor contributing to the high
efficiency of CM in NCs is a large density of biexciton states
g
xx
, which can greatly exceed that of single-exciton states
g
x
of similar energies.
11
Recent pseudopotential
calculations
8
indeed indicate fast increase in the g
xx
-to-g
x
ratio with increasing energy above the lowest biexciton reso-
nance.
In this paper, we use second-order perturbation theory to
analyze an unexplored aspect of the direct biexciton photo-
generation model associated with the possibility of two dif-
ferent time orderings of the interaction terms responsible for
photoexcitation and the Coulomb interaction. One time or-
dering, considered in Ref. 11, involves first interband optical
excitation of an intermediate exciton state virtual exciton,
which is then converted into a final real biexciton via the
impact-ionization term, H
II
, of the Coulomb interaction Fig.
1; left of the thick gray arrow. However, another possibility
PHYSICAL REVIEW B 76, 125321 2007
1098-0121/2007/7612/1253216 ©2007 The American Physical Society 125321-1