Surface Diffusion and Elasticity in SiGe Heterostructures: Continuum Approach

1
Surface Diffusion and Elasticity in SiGe
Heterostructures Continuum Approach

• Martin Burger
• UCLA

Quantum Dots Surface Diffusion and Elasticity
Collaboration with Günther Bauer, Institute of
Semiconductor Physics Johannes Kepler University
Linz
2
Quantum Dot Growth
Quantum Dots Surface Diffusion and Elasticity
3
SiGe Heterostructures
Quantum dots form when a germanium film is
deposited on a silicon substrate (multi-layer
further silicon on germanium etc.)

• Basic mechanism of growth
• Asaro-Grinfeld-Tiller instability
• Lattice mismatch (4,2 ) causes misfit strain -
  reliefed by surface roughening

Quantum Dots Surface Diffusion and Elasticity
Under appropriate conditions, 3D islands on
wetting layer Stranski-Krastanow mode cf.
Shukin-Bimberg for phase diagram
4
SiGe Heterostructures
Typical sizes wetting layer 2 nm, dot height
5-10 nm, dot width 100-160 nm
Quantum Dots Surface Diffusion and Elasticity
5
PbSe/PbEuTe Heterostructures
Typical sizes dot height 10 nm,
dot width 20 nm
Quantum Dots Surface Diffusion and Elasticity
6
InAs/InGaAs/GaAs Heterostructures
Typical sizes dot height 10 nm,
dot width 30-40 nm
Quantum Dots Surface Diffusion and Elasticity
7
Equilibrium Shapes
Equilibrium shapes can be obtained by minimizing
the total energy, i.e., E Eelastic
Einterface Esurface under volume constraint.
Quantum Dots Surface Diffusion and Elasticity
In the Stranski-Krastanow mode, interface and
consequently interface Energy are constant (top
of substrate), therefore second term can be
ignored.
8
Continuum Model
Domain W with free boundary G
Energy terms
Quantum Dots Surface Diffusion and Elasticity
9
Chemical Potential

• Chemical potential is determined by energy
  gradient

Quantum Dots Surface Diffusion and Elasticity
Equilibrium is obtained for constant chemical
potential, constant determined by volume
constraint
10
Surface Diffusion
Ignoring intermixing (or alloying) effects,
driving force of the surface instability is
surface diffusion (cf. Freund-Kukta)
Quantum Dots Surface Diffusion and Elasticity
Vn is normal velocity of the surface, DS the
Laplacian with respect to surface variables, M
kinetic coefficient, D diffusion coefficient.
11
Equation for Film Height
At least for silicon, a representation of the
form zh(x,y) is reasonable for the surface
Surface diffusion is 4th order parabolic equation
for h . 2D
Quantum Dots Surface Diffusion and Elasticity
Relastic is elastic energy density
12
Numerical Simulation
Semi-implicit finite element method for surface
diffusion (cf. Bänsch-Morin-Nocchetto 2003)
Coupled to bulk elasticity by solving elastic
equlibrium equations at each time step, using
adaptive finite element method
Quantum Dots Surface Diffusion and Elasticity
Boundary element seems more favorable for this
problem, but will not be able to include alloying
in the bulk
13
Numerical Solution, 2 D
Free surface represented by piecewise linear
elements, elasticity equations discretized on the
arising grid
Quantum Dots Surface Diffusion and Elasticity
Values for surface energy, diffusion and kinetic
coefficients from literature vary by several
magnitudes, therefore parametric study with
respect to surface energy (diffusion and kinetic
coefficient can be incorporated into time scaling)
14
Simulation, 2 D
Single initial dot no deposition
Quantum Dots Surface Diffusion and Elasticity
15
Simulation, 2 D
Random initial surface random deposition
Quantum Dots Surface Diffusion and Elasticity
16
Simulation, 2 D
Random initial surface random deposition
Quantum Dots Surface Diffusion and Elasticity
17
Simulation, 2 D
Random (rough) initial surface no deposition
Quantum Dots Surface Diffusion and Elasticity
18
Simulation, 2 D
No deposition, later stage
Quantum Dots Surface Diffusion and Elasticity
19
Simulation, 2 D
No deposition, evolution of energy
Quantum Dots Surface Diffusion and Elasticity
20
Numerical Solution, 3 D
Free surface represented by piecewise linear
elements, elasticity equations discretized on a
larger cube not resolving the moving boundary
Quantum Dots Surface Diffusion and Elasticity
For approximation of elasticity system, weak
material formulation, i.e., Cijkl(x) Cijkl
for x in the film (or substrate) Cijkl(x) e
Cijkl for x above the film, e ltlt 1 Fine
resolution needed for reasonable results !
21
Simulation, 3 D
Random initial surface random deposition
Quantum Dots Surface Diffusion and Elasticity
22
Numerical Solution, 3 D
Numerical solution can possibly improved using
small island approximation (Shanahan-Spencer
2002, 2D isotropic) asymptotic expansion of the
elasticity for islands with small height / width
ratio
Quantum Dots Surface Diffusion and Elasticity
This approximation is reasonable for Si-Ge
or Si-SixGe1-x systems, doubtfull for InAs-GaAs
Asymptotic expansion yields elasticity system on
fixed cube with nonhomogeneous boundary
conditions - allows efficient solution
23
Intermixing Effects
In general, silicon diffuses into germanium
layer. Not well-studied, theory not well
understood.
Equilibrium conditions can be obtained by
continuum model (additional homoge-nization of
the elasticity equations compositional energy)
Quantum Dots Surface Diffusion and Elasticity
Minimization of total energy with respect to
displacement, free boundary and concentration
24
Dynamic Intermixing
Effects of alloys are difficult to model in
continuum setting
Spencer, Tersoff, Vorhees (2001) give transport
equation on the surface, if alloy is deposited,
but ignore bulk diffusion
Quantum Dots Surface Diffusion and Elasticity
Gurtin derives diffusion equation in bulk from
simple assumptions
Constitutive relation for flux jc is missing