Crystal growth modeling and nanotechnology Research educational by ays20225


									 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 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

                    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

   Island         Layer + Island       Layer
Volmer-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
 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:

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
 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     C     C   2
                               C        2
                    = DX 2 + DY 2
 solution to     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 Dt is:
      exp(-2Dt(DX(1-cos(2p(X-1)/kx))/a2 +
    Program Structure (MatLab 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

                                 height = 5
                                 time = 90
                                 Dt = 0.1
                                 64*64 grid

                                 grows to

                                 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: web-
  based, web-enhanced courses, book
 See for details
 Opportunities for undergraduate (REU), and
  graduate projects as part of M.S, Ph.D
   Pattern formation: magnetic wires

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 Magnetic superlattice Ag/Co
Murray 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 for details
 Opportunities for undergraduate (REU), and
  graduate projects as part of M.S, Ph.D

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