Supplemental Material
1. Descriptions of Kinetic Data
a. Nucleation
File values used in the calculations in this manuscript for thermal nucleation rates for
GexSi1-x/Si(100) are fit to an expression of the form Nt’ = N’(ex/r)m exp[-E’/kT] cm-2
s-1 (eqn. 13 of manuscript) where r = 1 Pa. The raw data for this fitting come from
references 6 and 21 of the manuscript. Fit values used from earlier analysis are N’ = e-69.7
cm-2s-1, m = 5, E’ = 1.84 eV. These values are obtained by fitting to data in the range 0.10
< x > 0.40.
It is not proposed that these fit values have a specific fundamental origin – indeed they are
likely averaging over several different mechanisms - but rather that they are an effective
fit to the temperature and excess stress dependent data considered. They are provided as
a basis for initiating simulations, but are expected to be variable depending on growth
method, chamber history, deposition conditions etc. and should be evaluated for each
specific application. Each parameter in the nucleation equation can be modified by the
user through the program interface.
For figures 2 and 4 of the manuscript a heterogeneous source density of 103 cm-2 was used.
This is a rather low value, but is chosen to emphasize the transition from blocking to nonblocking regimes in Figure 2 (which signature is lost at more realistic heterogeneous source
densities, e.g. 105 cm-2).
b. Propagation
Misfit dislocation velocities are fit to an equation derived from v = vo (ex/r)n e-Ev/kT
(eqn. 6 of the manuscript), where r = 1 Pa and n=1 throughout the calculations presented
in this paper. The activation energy Ev is assumed to vary linearly with x in GexSi1-x
between endpoint values of 1.6 eV for pure Ge and 2.2 eV for pure Si, i.e. E v(x) = 2.20.6x eV. All velocities from different structures are then compared to each other by
normalizing to an equivalent velocity at 1 Pa in pure Si according to the equation:
vn = vm e-0.6x (eV)/kT / (ex/r)
where vn is the normalized velocity and vm is the measured velocity. Casting vn in the
standard form of vo (ex/r) e-Ev/kT then yields the following values for the systems
considered here.
For GexSi1-x/Si(100) films during annealing: vo = e-7.8 m/s, Ev = 2.03 eV, based on extensive
in-situ TEM measurements from annealing of structures grown with metastable strain
(Reference 5 in manuscript).
For uncapped GexSi1-x/Si(100) films during growth: vo = e-14.5 m/s, Ev = 1.74 eV (this
description is based on a more limited set of data than the description above, as derived
from in-situ TEM measurements during thin film growth, Reference 20 in manuscript)
In each of the above cases, the propagating dislocations have Burgers vectors of a/2<110>.
For uncapped GexSi1-x/Si(110) films with 60o a/2<110> dislocations during annealing: vo
= e-9.0 m/s, Ev = 2.07 eV. For uncapped GexSi1-x/Si(110) films with 90o a/6<112>
dislocations during annealing: vo = e-12.1 m/s, Ev = 2.00 eV. We use these values for the
calculations of Figures 4, 6 and Table 1. For the (110) interface, no separate measurements
exist during growth.
c. Interactions
C
h
ho
D
Epitaxial
Layer
B
A
Substrate
Figure S1: Geometry of dislocation interactions. A propagating misfit dislocation (AB)
with its threading arm (BC) intersects an orthogonal pre-existing dislocation (D, coming
out of page). The interaction stress between D and BC exceeds the excess stress up to a
distance ho from the interface. If the remaining thickness h-ho > hc then the dislocation
ABC can continue to propagate, otherwise its motion is blocked.
The interaction stress between a propagating threading/misfit dislocation and a preexisting orthogonal dislocation, Figure 1, is described by the equation:
(h)i = G b cos / [ rh (1-) 2]
Where rh is the distance from a segment on the threading dislocation and the closest
part of the misfit dislocation. This is clearly a simplification of a complex geometry
but captures the essential physics.
Near the interface this will exceed the excess stress driving the propagating
dislocation, up to a coordinate ho. If the unblocked thickness, h-ho in Figure 1 is
greater than the critical thickness, hc, for dislocation motion, then the film is assumed
to be in a non-blocking regime. If h-ho < hc, then the film is assumed to be in a blocking
regime.
2. Materials and Geometrical Parameters
Shear modulus, G, 64 GPa
Poisson ratio, , 0.28
Strain,  = 0.042x (in GexSi1-x)
Burgers vector, b (60o a/2<110> dislocation), 0.384+0.016x nm
Burgers vector, b (90o a/6<211> dislocation), 0.222+0.009x nm
Dislocation energy parameter, : 2 (default value, can be changed by user)
Stacking fault energy, : 65 mJ/m2 ((default value, can be changed by user)
Angular factors;
60o a/2<110> dislocation at (100) interface: cos = 0.82, cos  = 0.50, cos  = 0.50
60o a/2<110> dislocation at (110) interface: cos = 0.58, cos  = 0.71, cos  = 0.50
90o a/2<110> dislocation at (110) interface: cos = 0.58, cos  = 0.82, cos  = 0
Epilayer growth rate for Figs 2,3,4,6: 1 Angstrom/sec
3. System State Equations
The simulator explicitly calculates the excess stress, total length of misfit dislocation,
and total number of dislocations (each normalized to unit area) at time t, (t), L(t),
and N(t) respectively.
From these two parameters, the following additional parameters cam be calculated
and selected as output variables:
i)
ii)
iii)
The average dislocation line length lave(t) = L(t)/N(t)
The average misfit dislocation spacing p(t) = 2/L(t) (the factor 2 accounts
for the two orthogonal sets of interfacial dislocations)
The remaining strain in the system (t) = o – [bcos/ p(t)] where o is
the initial lattice mismatch strain
4. Use of the Simulator
The
GexSi1-x/Si(100)
simulator
is available in compiled form at
http://www.predictdislocationsin3n.com.Instructions for use are provided at that site.
The simulator is coded in Visual Basic 5; an updated version in C++ will be posted at
the site soon.
Important notes: This program runs on Windows. You will need to create a
directory C:\data for the output files to be written to. You will need a file
msvbvm50.dll in directory C:\Windows\SysWOW64 (link available at
http://www.predictdislocationsin3n.com).
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