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Crystal growth modeling and nanotechnology: Research & educational opportunities for MatLab programming. John A.Venables Arizona State University and Universit y of Sussex. 1) Growth modes and surface processes 2) Rate equations and algebraic solutions 3) Extensions to defect nucleation

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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
Explanation for NAN 546: April 09

This talk was given at several departmental seminars in the period 2003-05. These seminars typically require about 40-50 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
Crystal growth modeling & nanotechnology:Research & educational opportunities for MatLab programming

John A.Venables

Arizona State University and LCN-UCL

  • 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
Growth modes

IslandLayer + Island Layer

Volmer-Weber Stranski-Krastanov Frank-VdM

atomic level processes
Atomic-level processes

Variables: R (or F), T, time sequences (t)

Parameters: Ea, Ed, Eb, mobility, defects…

early tem pictures au nacl 001
Early TEM pictures: Au/NaCl(001)

Donohoe and Robins (1972) JCG

alternative approaches to modeling
Alternative approaches to modeling

Rate and diffusion equations

Kinetic Monte Carlo simulations

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
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 = ®å 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+ sxDnxnucleation, growth

competitive capture
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
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
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
Nucleation on point and line defects

(a) Point defects (vacancies) (b) Line defects (steps)

extension to defect nucleation parameters n t e t
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 samei, i’

....

dnx/dt = ®å dnj/dt = Ui - ...®nx(t),

(j > i +1) terrace cluster density

dnxt/dt = ®å dnj’t/dt = Ui’t - ...®nxt(t),

(j’ > i’ +1) trapped cluster density

point defects and nanofabrication
Point defects and Nanofabrication?

For Fe/CaF2(111):Heim et al. 1996, JAP; Venables 1999

a particular case pd mgo 001
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
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
Capture numbers:1D radial rate-diffusion equations

dn1(r,t)/dt = G(r,t) –n1(r,t)/t(r,t) +Ñ.[D(r)Ñn1(r,t)]

G(r,t),generation rate ®n1(r, t), adatom profile

dnx(r,t)/dt = å 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)Ñn1(r) – [n1(r)D*(r)]bÑV(r)

radial current ® s capture number

diffusion and attachment limits
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
Delayed onset of nucleation

Reduced capture numbers: longer transient regime (nx)

Venables and Brune 2002

repulsive adsorbate interactions cu cu 111
Repulsive adsorbate interactions: Cu/Cu(111)

Annealing, low T (16.5K),Cu/Cu(111) Rate equations, full lines as f (rd); KMC, squares with error bars.

Cu/Cu(111): STM, 0.0014 ML, preferred spacing

Knorr et al. PRB 65 (2002) 115420; Venables & Brune (2002) PRB

interpolation scheme for annealing i 1
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
Extrapolation to higher temperatures

REs:integrate to 2 or 20 min. anneal with givenV0.

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
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
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
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
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
FFT Method for time-dependent x-y Diffusion Fields

The general

solution to

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
Program Structure (MatLabâ 6.5)
  • 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
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

1d and 2d results and conclusions
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 nanowiresBennett
  • Anisotropic 2D nucleation and growth: AFM/ LEEM/HREM Ag/Si nanowiresLi & 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
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
  • Opportunities for undergraduate (REU), and graduate projects as part of M.S, Ph.D
pattern formation magnetic wires
Pattern formation: magnetic wires

Sugawara et al. (1997) APL, JAP

Fe/SiO/NaCl(110):

Anisotropic magnet, MOKE, Holography

optical properties of ingaas gaas
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
Size and position uniformityin stacked PbSe/PbEuTe QDSL’s

AFM, almost hexagonal ordering, see FFT insert:

Raab, Lechner, Springholz, APL ’02 & refs quoted

nanotechnology alternative routes
Nanotechnology: alternative routes

Optical superlattice: CdSe Magnetic superlattice Ag/Co

Murray et al., 1995++, Science Sun & Murray, 1999+, JAP

nanotechnology modeling education37
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
  • Opportunities for undergraduate (REU), and graduate projects as part of M.S, Ph.D