Crystal growth modeling and nanotechnology:
Research & educational opportunities for MatLab programming
John A.Venables
Arizona State University and University of Sussex
1) Growth modes and surface processes
2) Rate equations and algebraic solutions
3) Extensions to defect nucleation
4) 1D and 2D rate-diffusion problems
5) 2D diffusion-growth problems: movies
6) Nanotechnology, modeling & education
Explanation for NAN 546: April 09
This talk was given at several departmental seminars in the
period 2003-05. These seminars typically require about 4050 minutes and this one has 36 slides (#1, 3-37 here), with
none in reserve this time. The audience is usually a Materials
Department, with the level aimed at the Graduate Students
Slide #3 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. This particular talk
has a tutorial character, which explains why we have been
able to use it for sessions on growth modes, diffusion
mechanisms, rate equations, stress effects, visualization, etc
This material is copyright of John A. Venables. Author and
Journal references are typically given in green, and I do not
talk about material that has not been published in some form.
Crystal growth modeling & nanotechnology:
Research & educational opportunities for MatLab programming
John A.Venables
Arizona State University and LCN-UCL
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Growth modes and surface processes
Rate equations and algebraic solutions
Extensions to defect nucleation
1D and 2D rate-diffusion problems
2D diffusion-growth problems: movies
Nanotechnology, modeling & education
Growth modes
Island
Volmer-Weber
Layer + Island
Stranski-Krastanov
Layer
Frank-VdM
Atomic-level processes
Variables: R (or F), T, time sequences (t)
Parameters: Ea, Ed, Eb, mobility, defects…
Early TEM pictures: Au/NaCl(001)
Donohoe and Robins (1972) JCG
Alternative approaches to modeling
1) Rate and 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)
Rate Equations
(experimental variables T, F or R, t)
dn1/dt = F(or R) –n1/t  n1(t), single adatoms
....
dnj/dt = Uj-1 - Uj = 0  nj(t), via local equilibrium
....
dnx/dt =  dnj/dt = Ui - ... nx(t),
(j > i +1) stable cluster density
Ui = siDn1ni , with si , sx as capture numbers
t –1 = siD(i+1)ni + sxDnx nucleation, growth
Competitive capture
dn1/dt = R (or F) – n1/t; t-1 = ta-1 + tn-1 + tc-1…
Venables PRB 36 4153-62 (1987)
Differential equations versus Algebra
Using cluster shape, assumed or measured, express
nx(Z)  h(Z). f1(Rpexp(E/kT))
t(Z)  k(Z). f2(Rpexp(E/kT));
where p and E are functions of i, critical nucleus size
similarly for mean size ax(Z) and
condensation coefficient b(Z), not much used.
Choice of 1) integrating differential equations, or
2) evaluating near the maximum of nx(Z).
Steady state conditions (dnx/dt, etc = 0) converts a
set of ODE’s into a (nonlinear) algebraic solution.
Nucleation density predictions
• Matlab Programs
(R, T-1 and cluster
size, j)
• Input Energies
• Simultaneous
output: Densities
and critical cluster
size, i.
McDaniels et al. PRL 87 (2001) 176105, data, and work
in progress on rate equation and 2D modeling
Nucleation on point and line defects
(a) Point defects (vacancies) (b) Line defects (steps)
Extension to Defect Nucleation
(parameters nt, Et)
dn1/dt = R –n1/t
 n1(t), single terrace adatoms
dn1t/dt = s1tDn1nte - n1tndexp(-(Et+Ed)/kT)  n1t(t)
....
empty traps
trapped adatoms
dnj/dt = Uj-1 - Uj = 0  nj(t), via local equilibrium
dnj’t/dt  nj’t(t),
not necessarily same i, i’
....
dnx/dt =  dnj/dt = Ui - ... nx(t),
(j > i +1)
terrace cluster density
dnxt/dt =  dnj’t/dt = Ui’t - ... nxt(t),
(j’ > i’ +1)
trapped cluster density
Point defects and Nanofabrication?
For Fe/CaF2(111): Heim et al. 1996, JAP; Venables 1999
A particular case: Pd/MgO (001)
Defect nucleation,
i = 3 at high T
Haas et al. 2000 PRB; Venables and Harding 2000 JCG
More complex situations:
1D and 2D rate-diffusion equations
Examples so far: dn1/dt, dnx/dt  n1(t), nx(Z),
etc., all spatial averages
Next, linear or radial 1D problems:
capture numbers, step capture,
deposition past a mask, quantum wires
Capture numbers:
1D radial rate-diffusion equations
dn1(r,t)/dt = G(r,t) –n1(r,t)/t(r,t) +[D(r)n1(r,t)]
G(r,t), generation rate  n1(r, t), adatom profile
dnx(r,t)/dt =  dnj(r,t)/dt = Ui(r,t) – ... nx(r, t)
 nx(r, t) stable cluster density profile
Deals with deposition (G~F) and annealing (G~0),
plus also potential energy landscapes, V(r),
via Nernst- Einstein equation (t-dependence implicit),
j(r) = –D(r)n1(r) – [n1(r)D*(r)]bV(r)
radial current
 s capture number
Diffusion and attachment limits
Diffusion solution, at r = rk+ r0
sD = 2pXk0.(K1(Xk0)/ K0(Xk0)),
with Xk0 = (rk+ r0)/(D1t)1/2
Attachment (barrier) solution:
sB = 2p(rk+ r0)exp(-bEB)
= B(rk+ r0) or BV(rk+ r0)
They combine inversely as
sk -1 = sB-1 + sD-1
a) B=2pexp(-bEB)
b) BV=2pexp(-bV0) Venables and Brune PRB 66 (2002) 195404
Delayed onset of nucleation
Reduced capture numbers: longer transient regime (nx)
Venables and Brune 2002
Repulsive adsorbate interactions: Cu/Cu(111)
Annealing, low T (16.5K),Cu/Cu(111)
Cu/Cu(111): STM, 0.0014 Rate equations, full lines as f (rd);
ML, preferred spacing
KMC, squares with error bars.
Knorr et al. PRB 65 (2002) 115420; Venables & Brune (2002) PRB
Interpolation scheme for annealing: i = 1
Full lines:
Attachment limit
Dashed lines:
Diffusion limit
Previous slides:
Interpolation
dn1/d(D1t) = -2s1n12 -sxn1nx, dnx/d(D1t) = s1n12,
with sk = sinit ft + skd(1-ft), sinit = sBft;
ft = K0(Xd)/K0(Xk0); Xd = (rk+r0+rd)/(D1t)1/2
with time-dependent rd = (0.5D1t)1/2BV/2p.
Extrapolation to higher temperatures
REs: integrate to 2
or 20 min. anneal
with given V0.
KMC: hexagonal
lattice simulations
(1000 x 1155) sites
with EB = V0.
Compare KMC-STM: 10 < V0 < 14 meV; Venables & Brune 2002
Extension to Ge/Si(001)
stress-limited capture numbers
• Low dimer formation energy (Ef2 ~ 0.35 eV) gives
large i, even though condensation is complete
• Stress grows with island size, sx decreases
• Lengthened transient regime results, > 1 ML, source
of very mobile ad-dimers (Ed2 ~ 1 eV) for rapid
growth eventually of dislocated islands
• 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.)
Chaparro et al. JAP 2000, Venables et al. Roy. Soc. A361 (2003) 311
Conclusions: t-dependent capture numbers
1) Explicit t-dependence involves the transient
regime and a finite number of adatoms. Barriers or
repulsive potential fields reduce capture numbers,
lengthen transients and involve more adatoms.
2) Barrier capture numbers and diffusion capture
numbers add inversely. An interpolation scheme is
needed to describe t-dependence in the transient.
3) Large critical nucleus size lengthens transient.
Annealing a low T deposit with potential fields is a
very sensitive test of t-dependent capture numbers,
as small capture numbers result in little annealing.
And finally,
Area 2D (x,y,t or r,f,t) problems:
Shapes, edge diffusion, instabilities, lithography,
quantum dots, anisotropic stress effects, etc.
Geometries to consider are square or rectangular,
using an (x,y) mesh: e.g. (2x1) and (1x2) Si(001);
hexagonal, which can be approximated by a 1D
cylindrical (r) domain;
triangular lattice, applicable to reconstructed f.c.c.
metal and semiconductor (111) surfaces.
Expts: STM of Co/Si(111); and Ag/Ag layers/Pt(111)
(Bennett et al.)
(Brune et al.)
Questions for 1D and 2D modelling
1) How far can one realistically go without
becoming over-dependent on too many unknown
parameters??
2) How many types of different experiments can
one actually perform?
3) Large-scale (commercial) packages can solve
PDEs. But, is the science unique, or do multiple
inputs give the same results?
Importance of lumped parameters
FFT Method for time-dependent x-y
Diffusion Fields
The general
solution to
C
C
C
= DX 2 + DY 2
t
x
y
2
2
on a rectangular grid (X,Y) (kx, ky points) is
C = Field = ifft2(fft2(Field).*Pmat)
where the propagator Pmat for time Dt is:
unitmat.*
exp(-2Dt(DX(1-cos(2p(X-1)/kx))/a2 +
DY(1-cos(2p(Y-1)/ky)))/b2)
Program Structure (MatLab 6.5)
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Initialize, set up Field and Island Masks
Calculate Propagator, Pmat, for Dt
Loop over time steps: ktimes = [1:900]
Update Field, calculate fluxes and reset
boundary conditions on Island and Field
• Plot Field data at plottime = [1 2.. 10.. 900]
• Save calculated capture number data &
other calculations for subsequent plotting
Annealing: rectangular islands
height = 5
time = 90
Dt = 0.1
64*64 grid
(5*11)
grows to
(19*33)
Dx = 5
Dy = 10
Venables &
Yang 2004
Capture Numbers during annealing
1D and 2D results and conclusions
 Attachment-limited solutions EB, V(r) for STM
Cu/Cu(111) Brune (EPFL), Phys. Rev. B 2002;
AFM Ge/Si(001) Drucker (ASU), in progress
 Anisotropic attachment and growth: AFM/STM/
LEEM/HREM (Co,Pd) silicide nanowires Bennett
 Anisotropic 2D nucleation and growth: AFM/
LEEM/HREM Ag/Si nanowires Li & Zuo (UIUC)
Conclusion: It is worth exploring a few models with a
few defined parameters in 1 and 2 spatial dimensions
when a strong connection to experiment is available.
Nanotechnology, modeling & education
Interest in crystal growth, atomistic models
and experiments in collaboration
Interest in graduate education: webbased, web-enhanced courses, book
See http://venables.asu.edu/ for details
Opportunities for undergraduate (REU), and
graduate projects as part of M.S, Ph.D
Pattern formation: magnetic wires
Fe/SiO/NaCl(110):
Anisotropic magnet,
MOKE, Holography
Sugawara et al. (1997) APL, JAP
Optical properties of InGaAs/GaAs
Modeling, TEM, EELS, PL:
Shumway, Zunger, Catalano, Crozier et al. PRB ‘01
Size and position uniformity
in stacked PbSe/PbEuTe QDSL’s
AFM, almost hexagonal ordering, see FFT insert:
Raab, Lechner, Springholz, APL ’02 & refs quoted
Nanotechnology: alternative routes
Optical superlattice: CdSe
Murray et al., 1995++, Science
Magnetic superlattice Ag/Co
Sun & Murray, 1999+, JAP
Nanotechnology, modeling & education
Interest in crystal growth, atomistic models
and experiments in collaboration
Interest in graduate education: webbased, web-enhanced courses, book
See http://venables.asu.edu/ for details
Opportunities for undergraduate (REU), and
graduate projects as part of M.S, Ph.D