Step line tension and step morphological evolution on the Si(111) „1 Ã 1… surface
A. B. Pang, K. L. Man, and M. S. Altman*
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
T. J. Stasevich,
†
F. Szalma, and T. L. Einstein
Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
Received 25 July 2007; revised manuscript received 6 December 2007; published 17 March 2008
The temperature dependence of the step line tension on the Si1111 1 surface is determined from a
capillary wave analysis of two-dimensional island edge fluctuations and straight step fluctuations that are
observed with low energy electron microscopy. The line tension decreases by nearly 20% with a linear
temperature coefficient of -0.14 meV / Å K between 1145 and 1233 K. Temporal correlations of step fluctua-
tions exhibit the distinctive signature in the wavelength dependence of the relaxation time of a terrace
diffusion-limited mechanism for step motion. We also find that the role of desorption in island decay increases
dramatically in the temperature range 1145–1380 K that island decay is studied. Consequently, we general-
ize the current quasistatic model of island decay to take account of desorption. The evaluation of the island
decay time with this model referenced to the temperature-dependent line tension accurately determines acti-
vation energies that are relevant to mass transport and sublimation.
DOI: 10.1103/PhysRevB.77.115424 PACS numbers: 68.35.Md, 68.43.Jk, 68.37.Nq, 68.35.Fx
I. INTRODUCTION
In simple phenomenological models of crystalline sur-
faces, the step stiffness can play a prominent role in defining
equilibrium step configurations and in governing step mor-
phological evolution. The step stiffness is defined,
˜
=
+
2
/
2
, in terms of the step line tension , where is the
azimuthal angle, and as such is a measure of the tendency of
a step to remain straight.
1
One important way in which the
step stiffness may exert its influence on step morphology is
through its presence in the Gibbs-Thomson GT relation,
which figures generally in descriptions of curved surfaces.
2
In the context of crystalline surfaces, the GT relation has
frequently been used to express the dependence of the ada-
tom concentration in equilibrium with an atomic step upon
step curvature.
3–14
This dependence plays an important role
in several phenomena that affect surface morphology, such
as the response of a step to shape perturbations
8
including
step flow instabilities
9
and island coarsening and
decay.
3–7,10–12,15
Therefore, accurate knowledge of step stiff-
ness or line tension, including their temperature dependence,
should contribute to the understanding of many step morpho-
logical phenomena.
Steps on the Si1111 1 surface, which are the subject
of the investigations described here, have been studied
widely
3,8,15–33
due to their intriguing phenomenology. The
Si111 surface undergoes a structural phase transition be-
tween 7 7 and 1 1 configurations at a transition tem-
perature of T
c
=1133 K. The step line tension is expected to
be nearly isotropic on the Si1111 1 surface above T
c
.
16
Under this condition, the step stiffness will be equal to and
can be used interchangeably with the line tension. Several
values of the step stiffness and line tension have been re-
ported for the Si1111 1 surface at a few temperatures
based on measurements that were made using reflection elec-
tron microscopy REM. These values cover a fairly large
range. Step stiffness was originally derived from measure-
ments of the mean-square displacement of steps during equi-
librium fluctuations.
17
Stiffness values of 69 and 38 meV / Å
were determined from the fluctuation behavior of two differ-
ent steps at 1173 K.
17
These values were later revised up-
ward by a factor of 2,
18
and finally a single lower value of
46 meV / Å was settled on after further corrections were
made to the analysis.
19
This is a little larger than the stiffness
of 30 meV / Å that was determined earlier at 1173 K from an
evaluation of the time correlation functions for the different
Fourier modes of equilibrium step fluctuations.
20
The stiff-
ness was determined from the mean-square fluctuation dis-
placement at 1323 K to be 3.2 meV / Å.
21
The line tension
was also reported to be 18.8 meV / Å at this temperature
based on an evaluation of the equilibrium Si crystal shape.
22
At 1373 K, stiffness was determined to be
16.3 1.8 meV / Å from step diffusivity via measurement of
the spatial correlation function.
23
A slightly smaller value of
12 meV / Å was determined at this temperature in the same
work from the mean-square displacement due to
fluctuations.
23
The stiffness at 1373 K was later reported to
be between 22.8 and 31.9 meV / Å.
24
It was noted in that
work that desorption is significant at 1373 K. Therefore, care
was taken to deposit a replenishing flux of Si atoms that
compensated the desorption flux.
23,24
It is not clear if the
surface is in equilibrium or just in steady state under these
conditions. On the other hand, values of a quantity called the
dynamical step edge stiffness that were determined under
dynamical conditions of sublimation between 1230 and
1380 K are orders of magnitude larger.
8
If we disregard the
dynamical stiffness for the moment and focus only on the
most recent results obtained under equilibrium or steady-
state conditions at 1173 K Ref. 19 and 1373 K Ref. 24,
then the decrease of the step stiffness with increasing tem-
perature is found to be in qualitative agreement with theoret-
ical expectations.
16
However, the variation of the reported
values and the limited number of temperatures that were con-
sidered in separate experiments suggest that the issue of step
PHYSICAL REVIEW B 77, 115424 2008
1098-0121/2008/7711/11542412 ©2008 The American Physical Society 115424-1