Shape transitions of metastable surface nanostructures
D. J. Vine, D. E. Jesson,* and M. J. Morgan
School of Physics, Monash University, Victoria 3800, Australia
V. A. Shchukin
†
and D. Bimberg
Institut für Festkörperphysik, Technische Universität Berlin, D-10623 Berlin, Germany
Received 15 August 2005; revised manuscript received 21 September 2005; published 9 December 2005
A shape transition between surface nanostructures which, as a function of island size, are associated with
minima in formation energy per atom is modeled using a Fokker-Planck equation. We find that metastable
states, associated with positive gradients in island chemical potential, can dominate the dynamics of the
transition. The resulting bimodal island size distribution function is metastable to Ostwald ripening which has
important implications for the self-organization of quantum dots.
DOI: 10.1103/PhysRevB.72.241304 PACS numbers: 81.07.Ta, 68.35.Md, 68.65.-k, 81.16.-c
The self-assembly and self-organization of nanostructures
on surfaces can be utilized to produce quantum dot arrays for
device applications.
1,2
This can be readily achieved, for ex-
ample, by depositing thin films using deposition techniques
such as molecular beam epitaxy.
3–5
The resulting islands, or
dots, can be overgrown by appropriate layers to form the
basis for devices such as semiconductor lasers. However,
size uniformity is critical for many applications which has
led to significant efforts to understand the key factors gov-
erning the coarsening of quantum dot arrays.
Surface nanostructures that possess a minimum in forma-
tion energy per atom MEA as a function of island size
MEA systems are particularly attractive candidates for de-
vice applications because they are associated with a thermo-
dynamically favored size. By simply annealing such struc-
tures, one might anticipate the creation of arrays with good
size uniformity. Although it is not possible to identify MEA
systems a priori, theoretical studies have shown that coher-
ently strained two-dimensional 2D islands,
6–8
three-
dimensional 3D islands with surface stress discontinuities
at their edges
9,10
or 3D islands with strain renormalized sur-
face energy
9,11
are potential candidates for MEA systems.
A feature of particular interest in the case of 3D nano-
structures is the possibility that surface islands can undergo
shape transitions.
12–23
This can result in a multimodal island
size distribution function during the self-organization of
quantum dot systems that can deleteriously influence device
performance. Understanding the dynamics of shape transi-
tions is therefore of critical importance to control island size
distributions and obtain good size uniformity.
Theoretical descriptions of quantum dot systems undergo-
ing shape transitions can be broadly classified as being ther-
modynamic or kinetic in nature. Kinetic models have empha-
sized the discontinuity in island chemical potential as islands
attain a critical transition size.
12
Upon transformation to a
new shape, the lower chemical potential islands grow rap-
idly, resulting in a bimodal size distribution. In contrast, ther-
modynamic models associate peaks in the island size distri-
bution function with minima in formation energy per atom
for different island shapes.
13,15,17
However, as discussed by
Rudd et al.,
15
the dynamics of the transition between “stable”
states in thermodynamic models and the consequences for
self-organization are relatively unexplored.
In this Rapid Communication, we therefore develop a the-
oretical description of shape transition dynamics in MEA
systems. Surprisingly, metastable states, associated with
positive gradients in chemical potential, are found to domi-
nate the dynamics of the transition. These states are distinct
from energy per atom minima and have important implica-
tions for self-organization of MEA systems and the applica-
tion of thermodynamic models to interpret experimental
data.
13,15,17
To model a shape transition between nanostructures ex-
hibiting MEA behavior we consider the specific case of 3D
semiconductor islands with strain-renormalized surface en-
ergy which are assumed to exhibit MEA properties.
9,11
The
dimensionless formation energy E
s
N of a faceted quantum
dot as a function of the number of atoms N it contains is
given by
9,11
E
s
N =-
s
N +
s
N
2/3
-2N
1/3
ln e
1/2
N
1/3
. 1
The first term is the island relaxation energy, the second
term incorporates the change in renormalized surface energy
due to island formation, and the third term combines the
positive short range energy of the island edges with the nega-
tive surface stress induced elastic relaxation energy at the
edges, E
elastic
edges
. The parameter
s
is the ratio of the volume
relaxation energy to |E
elastic
edges
| and
s
is the ratio of the renor-
malized surface energy to |E
elastic
edges
|. The subscript s =1, 2 of
the parameters
s
,
s
, refers to two different nanostructure
shapes. We assume that the island array is sufficiently dilute
so that the elastic interaction between islands can be ne-
glected.
Quantitative values of the parameters
s
,
s
are presently
unknown. For the purpose of this simulation we assume the
arbitrary but physically reasonable values of
1
= 4.2,
1
=-0.8,
2
=5.2 and
2
=0.0. In Fig. 1a we plot the island
formation energy per atom
s
N = E
s
N / N as a function of
island size for two shapes 1 and 2. Each shape is associated
with a minimum in
s
N at
1
N
1
E
and
2
N
2
E
. For small
island sizes, shape 1 is energetically favorable. However,
PHYSICAL REVIEW B 72, 241304R2005
RAPID COMMUNICATIONS
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