Formation of self-assembled heteroepitaxial islands in elastically anisotropic films
P. Liu,
1
Y. W. Zhang,
2,
* and C. Lu
1
1
Institute of High-Performance Computing, Singapore
2
Department of Materials Science, National University of Singapore, Singapore
and Institute of Materials Research and Engineering, Singapore
Received 7 May 2002; revised manuscript received 5 September 2002; published 25 April 2003
The formation of self-assembled heteroepitaxial islands in elastically anisotropic cubic films is investigated
by a three-dimensional dynamic method. In the formulation, the 100 film surfaces evolve through surface
diffusion driven by the gradient of the surface chemical potential. Our simulations reveal that when the elastic
anisotropic strength A A is defined as 2 C
44
/( C
11
-C
12
), where C
11
, C
12
, and C
44
are elastic constants is
larger than 1 the formed islands are preferentially aligned along 100 directions, while when A is smaller than
1 the formed islands are preferentially aligned along the 110 directions. It is found that the stronger the elastic
anisotropy, the stronger the island self-assembly. In addition, the island alignment and averaged island spacing
right after island formation are found to be related to the smallest fastest surface instability wavelength of the
elastically anisotropic films.
DOI: 10.1103/PhysRevB.67.165414 PACS numbers: 81.16.Dn, 68.35.Fx, 68.55.-a
I. INTRODUCTION
During heteroepitaxial growth, a film may undergo a
growth mode transition—that is, from a layer-by-layer
growth mode to a three-dimensional growth mode.
1
Through
such a transition, the film forms a rippled structure, which
eventually breaks up into islands. Since these islands are
normally dislocation free, the self-assembly process may be
used to fabricate quantum dot arrays, which have many po-
tential applications in microelectronic and optoelectronic de-
vices. The performance of these devices requires a uniform
and regular arrangement of quantum dot arrays. Although
many attempts have been made to grow a uniform and regu-
lar array of quantum dots through self-assembly,
2–8
so far
there are no reliable procedures to do so.
The surface roughness and subsequent island formation
are caused by the competition between the strain energy and
surface energy of the system. During the surface evolution,
the total strain energy decreases while the total surface en-
ergy increases. A first-order perturbation has been carried out
to analyze the critical condition of strain-induced surface
roughening for both elastic isotropic films
9–12
and elastic an-
isotropic films.
13,14
These analyses have shown that for a
perturbed wavelength , if
c
, where
c
is the critical
wavelength, the strain energy will dominate the process, and
therefore island formation becomes energetically favorable,
while if
c
, the surface energy will dominate the pro-
cess, and therefore the surface will remain flat. Of particular
interest is the film having elastic anisotropy. In this scenario,
the critical wavelength depends not only on the surface
orientation,
14
but also on the elastic anisotropy strength.
Therefore, tuning the elastic anisotropy of the film may
change the island spacing and alignment, providing another
degree of freedom to manipulate the self-assembly of quan-
tum dot growth.
For an elastically isotropic crystal, there are only two in-
dependent elastic constants: the elastic modulus E and
Poisson’s ratio . For cubic crystals, there are three indepen-
dent elastic constants in the reference coordinate system:
namely, C
11
, C
12
, and C
44
. The elastic property may be
expressed in terms of the elastic modulus E, Poisson’s ratio
, and the elastic anisotropy strength A. The two sets of
elastic constants have the following relationships:
E = C
11
2
+C
11
C
12
-2 C
12
2
/ C
11
+C
12
,
v=C
12
/ C
11
+C
12
,
A=2 C
44
/ C
11
-C
12
. 1
Table I lists the anisotropy strength for a number of com-
monly used crystals. It is seen that the variation of the elastic
anisotropy strength is large, ranging from 0.26 for PbTe to
5.2 for InN. Therefore it is important to understand the effect
of the elastic anisotropy on the surface evolution and the
self-assembly of quantum dots.
In this paper, we will examine the effect of the elastic
anisotropy on the surface evolution through annealing pro-
cesses. Our attention is focused on the effect of the elastic
anisotropy on the island self-assembly and the relationship
between the simulation results and first-order perturbation
results.
TABLE I. Elastic anisotropy strength =2 C
44
/( C
11
-C
12
) for
a number of commonly used crystals Ref. 15.
Crystal Elastic anisotropy strength A
PbTe 0.26
PbSe 0.30
PbS 0.51
Diamond 1.21
Si 1.51
Ge 1.66
GaAs 1.83
GaN 2.9
InN 5.2
PHYSICAL REVIEW B 67, 165414 2003
0163-1829/2003/6716/1654146/$20.00 ©2003 The American Physical Society 67 165414-1