Frequency-dependent hopping conductivity between silicon nanocrystallites:
Application to porous silicon
E. Lampin, C. Delerue, M. Lannoo, and G. Allan
IEMN-Departement ISEN (CNRS), UMR9929, 41 Boulevard Vauban, 59046 Lille Cedex, France
Received 27 July 1998
We show how it is possible to perform a full calculation of the frequency-dependent hopping conductivity
of a disordered array of semiconductor crystallites once their statistical distribution is known. We first apply
this to a weakly disordered distribution of silicon spheres connected by silicon bridges and show the impor-
tance of the topology in determining the activation energy characteristic of the temperature dependence. We
then use a model distribution to simulate the case of porous silicon and from this get a coherent description of
various related properties. Finally, we emphasize the applicability of the method to determine the hopping
conductivity of artificially built semiconductor nanostructures. S0163-18299801842-6
The availability of new fabrication tools leads to a wide-
spread interest concerning the electrical properties of semi-
conductor nanostructures. In this respect, strong emphasis is
put on silicon, which is the backbone material of the inte-
grated circuit industry. For instance, there are efforts to real-
ize silicon-based single-electron transistors
1,2
operating at
room temperature that require a feature size smaller than 10
nm. Another structure of great interest is an array of small
islands separated from each other by tunnel barriers where
the electronic conduction can be varied by controlling the
size of the objects.
3–6
In all these cases there is a need for
quantitative calculations of the conductivity not only to get a
full understanding of the experimental data but also to allow
for an optimization of the structures. Such a theory faces
several difficulties: the complexity of the level scheme of
individual crystallites, their disorder in size and shapes, and
the level of accuracy of the description of the tunneling pro-
cess. In that context the aim of this paper is to demonstrate
that this can be already achieved even for quite complex
systems. We take as a typical situation a disordered array of
silicon crystallites connected by narrow silicon bridges. This
is believed to arise in the realization of extremely thin silicon
wires,
7
but also represents the current understanding of po-
rous silicon under the form of a network of undulating wires
i.e., crystallites connected by narrow constrictions.
8
Our
central result will be the temperature and frequency depen-
dence of the conductivity. Comparison with experimental
data obtained for porous silicon
9
shows that extremely good
agreement can be obtained for a physically reasonable statis-
tical distribution of the crystallites for the condition of being
close to the percolation threshold. Our results also give in-
sights into the current injection in electroluminescent devices
made from this material and also into the mechanism of its
formation by electrochemical etching. We finally discuss
how the present theory can be adapted to deal with a variety
of physical situations involving semiconductor nanostruc-
tures.
The first step of the calculation corresponds to the tunnel-
ing process between two spherical crystallites of different
size connected by a cylindrical silicon bridge supposed to be
narrow enough for the coupling between the spheres to be
small Fig. 1. Also shown on the same figure are the empty
states of the two isolated crystallites and of the silicon bridge
showing that indeed a substantial potential barrier occurs at
these dimensions. To get this level structure as well as all
other calculations of the electronic states in this work we
have used a tight-binding TB technique with a minimal
basis set of one s and three p orbitals on each silicon atom.
The Hamiltonian includes interactions up to third-nearest
neighbors and three center terms. Their parametrization is
described in Ref. 10 and allows one to get a very good sili-
con band structure that is a necessary condition to obtain
correct predictions of the confinement energies.
11
The sur-
faces of such nanostructures are passivated by hydrogen at-
oms as in Ref. 12. This Hamiltonian is convenient for the
tunneling calculations and provides results of accuracy com-
parable to the best calculations reported for silicon quantum
dots and wires.
13,14
The hopping probability between the two
crystallites of Fig. 1 is then obtained from the Fermi golden
rule:
W
ab
=
2
i
p i
f
f | H
c
| i
2
E
f
-E
i
, 1
FIG. 1. Top: Schematic view of a typical nanostructure contain-
ing two spherical crystallites right, diameter d =2.6nm; left, d
=1.8nm connected by a silicon bridge of cylindrical shape length,
1.1 nm; diameter, 0.6 nm. Bottom: Lowest electronic states in the
conduction band of the two silicon spheres. The numbers are the
energies of the lowest level of the spheres and of the cylinder with
respect to the bottom of the bulk silicon conduction band.
PHYSICAL REVIEW B 1 NOVEMBER 1998-II VOLUME 58, NUMBER 18
PRB 58 0163-1829/98/5818/120445/$15.00 12 044 ©1998 The American Physical Society