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Modeling Facet Nucleation and Growth of hut clusters on Ge/Si(001)
• Ge/Si(001) hut clusters: Annealing in STM• 2D Modeling of facet nucleation & growth• Model details: Dm, i, DG(i), reconstruction• Conclusion and References
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
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
Mike McKay, John Venables and Jeff Drucker, 2007-08
gas-source MBE from Ge2H6
Ge = 5.0ML, 0.1 ML / minT = 450 °C, 26 min/frame62 hrs total elapsed timefirst frame after 33min annealField of view 600nm x 600 nm
Ge = 5.6ML, 0.2 ML / minT = 500 °C, 7 min/frame14 hrs total elapsed timefirst frame after 160min annealField of view 400nm x 400 nm
Ge/Si(001) hut clusters:Annealing at T = 450 oC
1
4
2
103
56
7
89
1,255
30 nm
1
4
2
103
56
7
89
33
1
4
2
103
56
7
89
2,503
1
4
2
103
56
7
89
3,751
100
150
200
250
300
350
400
450
500
10
15
20
25
30
0 1000 2000 3000 4000
9
Anneal Time (min)
Width
Volume
Length
100
150
200
250
300
350
400
450
500
10
15
20
25
30
0 1000 2000 3000 4000
8
Anneal Time (min)
Width
Volume
Length
Most islands static, smallest island grows (8).
Volume, Length data and model result
Volume, Densitydata 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, etcChaparro 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
Mike McKay, John Venables and Jeff Drucker, 2008
Ad-dimers move everywhere, but perimeter barrier impedes replenishment on the hut
Barrier height increases withhut size but almost saturates
Finite but low edge (or ridge) energy on the facet favors 2D facet nucleation case 1);
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
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 = hr2/2 + hr3cos()/3s, = 11.3o
with = 0.7 eV/nm3 and = 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
ReferencesGe/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)
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
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
2 1
2 1
2 1
( )
( / 2)( );
i j d s i
s i j i i
E E V V
V V e e e
Ovesson, D constant
In general, D not constant, 2 depends on direction
Transition rates in a 2D potential field
but unfortunately this isn't true in general.... S. Ovesson PRL 88, 116102 (2002)
0 i jE
i j i jW W e
if
i j s s i iE E V E V ( ) / 2
( ) / 2
s i j
i j d j i
V V V
E E V V
if
then
i, j on lattice
s saddle point
Mean-field equations from microscopic dynamics
From Shu, Liu, Gong et al:
For Ge/Si(001): 1 = 1.75 eV; at lattice sites
2 1 = .75 eV fast diffusion direction
1 2 1
1 1
( , ) exp( ( ) ( , ))
( ) ( )ˆ ˆ( , ) ( , )
D x y D x y
x y D x y x yx y
e
e e
V
Strain dependent Diffusion D and Drift velocity Vas deduced by Grima, DeGraffenreid, Venables 2007
Ge/Si(001) concentration profiles
R. Grima, J. DeGraffenreid and J.A. Venables, 2007, PRB
2= 1= 1.75 eV
2 1= 0.75 eV
2- 1= 1.50 eV
Visualization: Discussion points• 1D & 2D Graphics and Movies are excellent complement to Rate-
Diffusion 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 detailsNew Professional Science Masters (PSM) in
Nanoscience degree program at ASU at http://physics.asu.edu/graduate/psm/overview
MatLab Movie as *.avi (Quick time)
• height = 5
• time = 90
• Dt = 0.1
• 64*64 grid
• (5*11) island
• grows to
• (19*33)
• Dx = 5
• Dy = 10
Size distributions and alloying
T = 450 °C
1.5 x109
1 x109
0
5 x108
3.2 x109
1.6 x109
0
4.8 x109
0 40 12080 160
X 2
(d)
(b)T = 600 °C
5 ML6.5 ML8.0 ML9.5 ML11.0 ML12.5 ML
Diameter (nm)
Num
ber
of is
land
s /
cm2
/ 2.
5 nm
bin Strain relief via
1) interdiffusion 2) change of shape
Hut-dome transitionsreversible via alloying athigh T > 500 oC
S. Chaparro, Jeff Drucker et al. PRL 1999, JAP 2000
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
ni n1e DG(i) kT number density of critical nuclei
nucleation rate (number of new stable clusters per small {105} facet per second)
D m kT ln n1 n1e( ) & n1e Noe L2 kT
dimer sublimation energyfrom step edges (~0.3eV)
Un 14 AZs inNoe
2 D m L2( ) Ed DG( i)( ) kT
capture number(~# perimeter sites)
Un AZs iDn1ni AZs iDn12e DG(i) kT
Zeldovich factor(typically 0.1-0.5)
dimer diffusioncoeff. on {105}
D n
4No
eEd kT
facet area
cf = dimer concentration on hut after facet nucleation eventcn = dimer concentration required to nucleate stable facetco = dimer concentration outside of island
V ( r ) elastic potential energy at position
r
Vp = potential at hut perimeter. Vi = Vo.
Why do smaller islands grow? • facet nucleation and growth depletes ad-dimer concentration on island • ‘refilling’ rate controlled elastic potential barrier at hut perimeter
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?
Island concentration, c, obeys . Solution for c is
Adc
dtcoG cG G co c( )
c co c f co( )e G A( )t
G sB D p
ae Vp Vo( ) kT
na2
4e Ed kT
Use barrier form for boundary capture number, sB:
time for hut to ‘refill’ to cn is
tr 4A
npae Ed Vp Vo( ) kT ln
c f co
cn co
so, large huts grow ~20 times slower than small huts