Crystal growth modeling and nanotechnology:Research & educational opportunities for MatLab

programming

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

John A.VenablesArizona State University and University of Sussex

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

• 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

John A.VenablesArizona State University and LCN-UCL

Growth modes

Island Layer + Island LayerVolmer-Weber Stranski-Krastanov 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/ 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 = iDn1ni , with i , x as capture numbers

–1 = iD(i+1)ni + xDnx nucleation, growth

Competitive capture

dn1/dt = R (or F) – n1/; an

c…

Venables PRB 36 4153-62 (1987)

Differential equations versus Algebra

Using cluster shape, assumed or measured, express

nx(Z) (Z). f1(Rpexp(E/kT))

t(Z) (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 (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/ n1(t), single terrace adatoms

dn1t/dt = 1tDn1nte - n1tdexp(-(Et+Ed)/kT) n1t(t)

.... empty traps trapped adatomsdnj/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 densitydnxt/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)

Haas et al. 2000 PRB; Venables and Harding 2000 JCG

Defect nucleation, i = 3 at high T

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)/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)]V(r) radial current capture number

Diffusion and attachment limits

a) B=2exp(-EB)

b) BV=2exp(-V0)

Diffusion solution, at r = rk+ r0

D = 2Xk0.(K1(Xk0)/ K0(Xk0)),

with Xk0 = (rk+ r0)/D11/2

Attachment (barrier) solution:B = 2(rk+ r0)exp(-EB)

= B(rk+ r0) or BV(rk+ r0)

They combine inversely ask

-1 = B-1 + D

-1 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)

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

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

Interpolation scheme for annealing: i = 1

dn1/d(D1t) = -21n12 -xn1nx, dnx/d(D1t) = 1n1

2,

with k = init ft + kd(1-ft), init = Bft;

ft = K0(Xd)/K0(Xk0); Xd = (rk+r0+rd)/(D1)1/2

with time-dependent rd = (0.5D1t)1/2BV/2.

Full lines: Attachment limit

Dashed lines:Diffusion limit

Previous slides:Interpolation

Extrapolation to higher temperatures

Compare KMC-STM: 10 < V0 < 14 meV; Venables & Brune 2002

REs: integrate to 2 or 20 min. anneal with given V0.

KMC: hexagonal lattice simulations (1000 x 1155) sites with EB = V0.

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, x 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,,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

2 2

X Y2 2D + D

C C C

t x y

on a rectangular grid (X,Y) (kx, ky points) is C = Field = ifft2(fft2(Field).*Pmat)where the propagator Pmat for time t is:

unitmat.*exp(-2t(DX(1-cos(2(X-1)/kx))/a2 +

DY(1-cos(2(Y-1)/ky)))/b2)

Program Structure (MatLab 6.5)

• Initialize, set up Field and Island Masks

• Calculate Propagator, Pmat, for t

• 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

height = 5time = 90 t = 0.164*64 grid

(5*11) grows to(19*33)

Dx = 5Dy = 10

Venables & Yang 2004

Annealing: rectangular islands

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: web-based, web-enhanced courses, book

See http://venables.asu.edu/ for detailsOpportunities for undergraduate (REU), and

graduate projects as part of M.S, Ph.D

Pattern formation: magnetic wires

Sugawara et al. (1997) APL, JAP

Fe/SiO/NaCl(110):Anisotropic magnet, MOKE, Holography

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 Magnetic superlattice Ag/CoMurray et al., 1995++, Science Sun & Murray, 1999+, JAP

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 detailsOpportunities for undergraduate (REU), and

graduate projects as part of M.S, Ph.D