Modeling Facet Nucleation and Growth
of hut clusters on Ge/Si(001)
John Venables1,3 Mike McKay2,4 and Jeff Drucker1,2
1) Physics, 2) Materials, Arizona State University, Tempe
3) LCN-UCL, London, 4) Lawrence Semiconductor, Tempe
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Ge/Si(001) hut clusters: Annealing in STM
2D Modeling of facet nucleation & growth
Model details: Dm, i, DG(i), reconstruction
Conclusion and References
Explanation for NAN 546: April 09
This talk was given at two conferences in 2008-09: first at
MRS Boston, 12/2/08 as a contributed 15 minute talk (13
slides), and then at the Surface Kinetics International (SKI)
meeting at University of Utah, 3/20/09, as an invited 25
min talk (15 slides #1, 3-16 here).
Slide #1 is an Agenda slide to guide one through the talk. The
hyperlinks on slide #1 connect to Custom Shows, available
under the Slide Show dropdown menu. The remaining
slides #17-30 are in reserve for questions, and in practice
were not used at the time. But they can be useful at a
conference if one chooses to get into more detail.
This material is copyright of the Authors. A summary paper is
published in PRL 101 (2008) 216104 (M.R. McKay, J.A.
Venables and J. Drucker) Reprints are available on request
Ge/Si(001) STM Movies:
watching paint dry at 450 OC
gas-source MBE from Ge2H6
Ge = 5.0ML, 0.1 ML / min
T = 450 °C, 26 min/frame
62 hrs total elapsed time
first frame after 33min anneal
Field of view 600nm x 600 nm
Ge = 5.6ML, 0.2 ML / min
T = 500 °C, 7 min/frame
14 hrs total elapsed time
first frame after 160min anneal
Field of view 400nm x 400 nm
Mike McKay, John Venables and Jeff Drucker, 2007-08
7
4
30 nm
6
8
5
2
9
3
1
10
33
7
4
6
5
2
8
9
3
10
1,255
1
Ge/Si(001) hut clusters:
Annealing at T = 450 oC
9
500
450
400
350
300
250
200
150
100
30
Volume
25
Length
20
15
Width
0
1000
2000
3000
10
4000
Anneal Time (min)
4
6
5
2
7
8
9
3
1
10
2,503
7
4
6
5
8
9
2
3
1
3,751
10
8
500
450
400
350
300
250
200
150
100
30
Volume
25
Length
20
15
Width
0
1000
2000
3000
10
4000
Anneal Time (min)
Most islands static, smallest island grows (8).
Volume, Length data and model result
Volume, Density
data for all huts in
video field of view
at selected times
Density is constant
Model Length
increase DL(t),
Average for 34 huts
from STM video
Background for real Ge/Si(001)
• Wetting layer ~ 3+ ML; supersaturation on and in the
WL, source of very mobile ad-dimers (Ed2 ~ 1 eV)
for rapid growth eventually of dislocated islands
• Low dimer formation energy (Ef2 ~ 0.3 eV) gives
large i, even though condensation is complete
• Stress grows with island size, sx decreases
• Interdiffusion, and diffusion away from high stress
regions around islands, reduces stress at higher T and
lower F (e.g. at 600, not 450 oC for F ~1-3 ML/min.)
• Specifics of {105} hut clusters, reconstruction, etc
Chaparro et al. JAP 2000, McKay, Shumway & Drucker, JAP 2006.
Outline of Si/Ge(001) hut growth model
(105) facet nucleation or dissolution, nucleation at
the apex, surface vacancy nucleation at the base
Finite but low edge (or ridge) energy on the facet favors
2D facet nucleation case 1);
Ad-dimers move everywhere,
but perimeter barrier impedes
replenishment on the hut
Barrier height increases with
hut size but almost saturates
Mike McKay, John Venables and Jeff Drucker, 2008
2D Facet nucleation with perimeter barrier
Elastic energy of Ge adatom on huts
on s. c. model Si(001) substrate
width 2r, length 2(s+r), 0 < s < (40-r)
.
Relative occupation
of boundary sites at
T= 450oC (723 K)
J. Drucker and J. Shumway
2D Critical nuclei: i and DG(i)
i = (X/(2Dm')2 and DG(i) = X2/(4Dm')
How long does it take to nucleate a facet?
Variables: Eb on the facet; Internal bias Vi; Barrier height Ep
Values: 0.2-0.3 eV
zero to -0.1 eV
0.3 - 0.7? eV
very sensitive to supersaturation Dm/kT (10-30 meV at T = 673K)
Additional elements in the model
• Strain dependent adsorption energy at facet apex
E = ahr2/2 + bhr3cos(g)/3s, g = 11.3o
with a = 0.7 eV/nm3 and b = 0.81 eV/nm3
from Finite element elastic calculations
• Extra "un-reconstruction" energy of {105} faces
may increase DG(i): DFT calculations, ~ 0.5 eV for
i > 3.5 dimers Cereda & Montalenti (2007)
• Couple growth of facets to reduced dimer density n1
via Dm = kTln(n1/n1e) and nucleation rate Un to get
dn1/dt = C.Un with known material constant C.
McKay, Venables & Drucker, PRL 101, 216104 (2008).
Effects of {105} reconstruction
MD simulations with Tersoff potential + DFT (VASP):
3 Dimers + 2 Vacancies/2*2.5 unit cell; 1.23 dimers/nm2
Reconstruction P. Ratieri et al. PRL 88 (2002) 256103
Un-reconstruct S. Cereda & F Montalenti PRB 75 (2007) 195321
Coupling of 2D nucleation and annealing
Supersaturation S = (n1/n1e); Dm = kT log (S)
Nucleation rate/facet Un = AfZsiDn1ni, (1)
with ni = n1exp(-DG(i)/kT), Af = area/facet
Annealing reduces n1 and hence S, Dm via
dn1/dt = -2NAfsdUn
(2)
with N huts/unit area, sd = facet dimers/area.
Finding the critical nucleus size i and DG(i),
the energy associated with Un is
En = 2(Dm - L2) - [Ed + DG(i)]
Length increase (t) data and model result
Evolution of Dm,
i and DG(i) with
annealing time, t
Model Length
increase DL(t),
Average for 34 huts
from STM video
Conclusions: Facet nucleation & hut growth
• Growth rate limited by facet nucleation at hut apex
• Energy gradient on the facet modifies 2D nucleation
formulae, biases towards larger critical nucleus size.
• Consumption of the adsorbed layer reduces the
supersaturation Dm(t) and slows the growth rate:
• Quantitative agreement with hut length L(t) annealing
data for reasonable Dm and parameter values
• Undoing the {105} reconstruction may account for a
substantial part (~ 0.5 eV) of the critical nucleus energy
• Ostwald ripening is suppressed (at 450 oC) because
dimer supersaturation stays positive during annealing
References
Ge/Si(001) hut growth AFM and STM experiments
S.A. Chaparro et al: JAP 87, 2245-2254 (2000)
M.R. McKay, J. Shumway & J. Drucker: JAP 99, 094305 (2006)
F. Montalenti et al: PRL 93, 216102 (2004)
STM movies online at http://physics.asu.edu/jsdruck/stmanneal.htm
Previous Ge/Si(001) {105} hut growth and step energy models
D.E. Jesson et al.: PRL 80, 5156-5169 (1998)
M. Kästner & B. Voightländer: PRL 82, 2745-2748 (1999)
S. Cereda, F Montalenti & L. Miglio Surf. Sci. 591, 23-31 (2005)
Details of {105} reconstruction and un-reconstructions
Reconstruction P. Ratieri et al. PRL 88, 256103 (2002)
Un-reconstruct S. Cereda & F. Montalenti PRB 75, 195321 (2007)
Anisotropic elastic energy calculations
fig 5a
Approximately linear energy density on facet
fig 5b
Alternative approaches to modeling
1) Rate and rate-diffusion equations
2) Kinetic Monte Carlo simulations
3) Level-set and related methods
plus
4) Correlation with ab-initio calculations
Issues: Length and time scales, multi-scale;
Parameter sets, lumped parameters;
Ratsch and Venables, JVST A S96-109 (2003)
Potentials due to strain, e
Ovesson, D constant
In general, D not constant,
a2 depends on direction
Demonstrate with 1D model & Lennard-Jones potential
DFT calculations for Si, Ge/Si(001), and Si/Ge(001)
D.J. Shu, X.G. Gong, L. Huang, F. Liu (2001) JCP 114,
10922; PRB 64, 245410; (2004) PRB 70, 155320
Transition rates in a 2D potential field
- b Ei j
0
i j
Wi  j  W
e
i, j on lattice
s saddle point
if
if
then
but unfortunately this isn't true in general....
S. Ovesson PRL 88, 116102 (2002)
Mean-field equations from microscopic dynamics
Strain dependent Diffusion D and Drift velocity V
as deduced by Grima, DeGraffenreid, Venables 2007
From Shu, Liu, Gong et al:
For Ge/Si(001): a1 = -1.75 eV; at lattice sites
a2 -a1 = 0.75 eV fast diffusion direction
Ge/Si(001) concentration profiles
a2= a1= -1.75 eV
a2- a1= 0.75 eV
a2- a1= 1.50 eV
R. Grima, J. DeGraffenreid and J.A. Venables, 2007, PRB
Visualization: Discussion points
• 1D & 2D Graphics and Movies are excellent complement to RateDiffusion Equations; ideal for projects/talks, not so easy for papers
Annealing, deposition, direct impingement, individual surface/edge
processes. Nanowire systems using Ge/Si(001) model parameters.
• Approximate solutions that concentrate on "Events" are great for
understanding, and answer questions like: "What happens when and
where?" So, how far do we want to go in the "realism" direction?
• Hybrid FFT uses constant D in k-space + difference terms in real
space and is more stable, but perhaps less accurate. Comparison of
MED, hybrid FFT and multigrid methods for speed/accuracy done,
But what are the general computational lessons to be drawn?
• Tests on strongly non-linear problems (e.g. high-i* nucleation +
growth) and "real systems", e.g. Ge/Si(001), work in progress. Need
to include reconstructions, fluctuations, local environment, long
timescales, etc: very complicated! But should we expect otherwise?
Nanotechnology, modeling & education
Interest in crystal growth, atomistic models
and experiments in collaboration
Interest in graduate education: web-based,
web-enhanced courses, book
See http://venables.asu.edu/ for details
New Professional Science Masters (PSM) in
Nanoscience degree program at ASU at
http://physics.asu.edu/graduate/psm/overview
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height = 5
time = 90
Dt = 0.1
64*64
grid
• (5*11)
island
• grows to
• (19*33)
• Dx = 5
• Dy = 10
MatLab Movie as *.avi (Quick time)
Sizes and shapes in Ge/Si(001)
TEM, AFM: Chaparro, Zhang, Drucker, Smith J. Appl. Phys (2000)
Size distributions and alloying
T = 600 °C
Number of islands / cm2 / 2.5 nm
bin
1.5 x109
Strain relief via
1) interdiffusion
2) change of shape
(b
)
X2
1 x109
5
x108
Hut-dome transitions
reversible via alloying at
high T > 500 oC
0
T = 450 °C (d)
4.8
x109
5 ML
6.5 ML
8.0 ML
9.5 ML
11.0 ML
12.5 ML
3.2 x109
1.6 x109
S. Chaparro, Jeff Drucker
et al. PRL 1999, JAP 2000
0
0
40
80
120
Diameter (nm)
16
0
Nucleation of new facets
- hut growth controlled by nucleation of new {105} planes on small facets
- nucleation rate is how fast critical nuclei become supercritical
nj = number density of nascent facets comprised of j dimers
n i  n1e-DG(i) kT
number density of critical nuclei
nucleation rate (number of new stable clusters per small {105} facet per second)
facet area
U n  AZs i Dn1n i  AZs i Dn12e-DG(i) kT
Zeldovich factor
(typically 0.1-0.5)
dimer diffusion
coeff. on {105}
capture number
(~# perimeter sites)
D

4N o
e Ed
kT
dimer sublimation energy
from step edges (~0.3eV)

Dm kTln n1 n1e  & n1e  Noe-L2
Un 
1
[
AZ
s

N
e
4
i
o
2Dm-L 2 -E d DG(i) kT
kT
Why do smaller islands grow?
• facet nucleation and growth depletes ad-dimer concentration on island
• ‘refilling’ rate controlled elastic potential barrier at hut perimeter
cf = dimer concentration on hut after facet nucleation event
cn = dimer concentration required to nucleate stable facet
co = dimer concentration outside of island
V (r )  elastic potential energy at position r
Vp = potential at hut perimeter. Vi = Vo.


A new facet forms at t=0, depleting the dimer concentration on the hut surface to cf.
How long is required for the island to refill to cn so that another stable facet can form?
dc
Island concentration, c, obeys A dt  c o - c-   c o - c  . Solution for c is
c  co  c f - co e-
A t


Use barrier form for boundary capture number, sB:
time for hut to ‘refill’ to cn is
tr 
p -V p -Vo  kT a 2 -E d
  s B D   e
 e
a
 4
4 A E d V p -Vo  kT c f - c o 
e
ln 

pa
c
n - c o 
so, large huts grow ~20 times slower than small huts
kT


