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Surface Diffusion and Elasticity in SiGe Heterostructures UCLA

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Surface Diffusion and Elasticity in SiGe Heterostructures UCLA Powered By Docstoc
					                                   Surface Diffusion and Elasticity in
                                        SiGe Heterostructures:
                                         Continuum Approach
Surface Diffusion and Elasticity




                                                Martin Burger
       Quantum Dots:




                                                   UCLA

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




                                                  Basic mechanism of growth:
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                                                    Asaro-Grinfeld-Tiller instability
                                                  Lattice mismatch (4,2 %) causes misfit strain
                                                  - reliefed by surface roughening
                                                  Under appropriate conditions, 3D islands on
                                                  wetting layer: Stranski-Krastanow mode
                                                  cf. Shukin-Bimberg for phase diagram
                                                          SiGe Heterostructures
                                                  Typical sizes: wetting layer 2 nm, dot height
                                                  5-10 nm, dot width 100-160 nm
               Surface Diffusion and Elasticity
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                                                    PbSe/PbEuTe Heterostructures
                                                  Typical sizes: dot height 10 nm,
                                                                 dot width 20 nm
               Surface Diffusion and Elasticity
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                                                  InAs/InGaAs/GaAs Heterostructures
                                                  Typical sizes: dot height 10 nm,
                                                                 dot width 30-40 nm
               Surface Diffusion and Elasticity
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                                                            Equilibrium Shapes

                                                  Equilibrium shapes can be obtained by
                                                  minimizing the total energy, i.e.,
               Surface Diffusion and Elasticity




                                                       E = Eelastic + Einterface + Esurface
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                                                  under volume constraint.
                                                  In the Stranski-Krastanow mode, interface
                                                  and consequently interface Energy are
                                                  constant (top of substrate), therefore second
                                                  term can be ignored.
                                                          Continuum Model

                                                  Domain W with free boundary G
               Surface Diffusion and Elasticity




                                                  Energy terms:
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                                                           Chemical Potential
                                                  Chemical potential is determined by energy
                                                  gradient:
               Surface Diffusion and Elasticity
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                                                  Equilibrium is obtained for constant
                                                  chemical potential, constant determined by
                                                  volume constraint
                                                            Surface Diffusion
                                                  Ignoring intermixing (or alloying) effects,
                                                  driving force of the surface instability is
                                                  surface diffusion (cf. Freund-Kukta)
               Surface Diffusion and Elasticity
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                                                  Vn is normal velocity of the surface, DS the
                                                  Laplacian with respect to surface variables,
                                                  M kinetic coefficient, D diffusion
                                                  coefficient.
                                                        Equation for Film Height
                                                  At least for silicon, a representation of the
                                                  form z=h(x,y) is reasonable for the surface
                                                  Surface diffusion is 4th order parabolic
               Surface Diffusion and Elasticity




                                                  equation for h . 2D:
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                                                  Relastic is elastic energy density
                                                         Numerical Simulation
                                                  Semi-implicit finite element method for
                                                  surface diffusion (cf. Bänsch-Morin-
                                                  Nocchetto 2003)
               Surface Diffusion and Elasticity




                                                  Coupled to bulk elasticity by solving elastic
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                                                  equlibrium equations at each time step,
                                                  using adaptive finite element method
                                                  Boundary element seems more favorable
                                                  for this problem, but will not be able to
                                                  include alloying in the bulk
                                                         Numerical Solution, 2 D

                                                  Free surface represented by piecewise
                                                  linear elements, elasticity equations
                                                  discretized on the arising grid
               Surface Diffusion and Elasticity
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                                                  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)
                                                              Simulation, 2 D
                                                  Single initial dot + no deposition
               Surface Diffusion and Elasticity
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                                                             Simulation, 2 D
                                                  Random initial surface + random deposition
               Surface Diffusion and Elasticity
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                                                             Simulation, 2 D
                                                  Random initial surface + random deposition
               Surface Diffusion and Elasticity
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                                                              Simulation, 2 D
                                                  Random (rough) initial surface no deposition
               Surface Diffusion and Elasticity
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Surface Diffusion and Elasticity

                                   No deposition, later stage
                                                                Simulation, 2 D
                                                              Simulation, 2 D
                                                  No deposition, evolution of energy
               Surface Diffusion and Elasticity
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                                                          Numerical Solution, 3 D
                                                  Free surface represented by piecewise
                                                  linear elements, elasticity equations
                                                  discretized on a larger cube not resolving
               Surface Diffusion and Elasticity




                                                  the moving boundary
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                                                  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 << 1
                                                  Fine resolution needed for reasonable results !
                                                             Simulation, 3 D
                                                  Random initial surface + random deposition
               Surface Diffusion and Elasticity
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                                                          Numerical Solution, 3 D
                                                  Numerical solution can possibly improved
                                                  using small island approximation
                                                  (Shanahan-Spencer 2002, 2D isotropic):
               Surface Diffusion and Elasticity




                                                  asymptotic expansion of the elasticity for
                                                  islands with small height / width ratio
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                                                  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
                                                           Intermixing Effects
                                                  In general, silicon diffuses into germanium
                                                  layer. Not well-studied, theory not well
                                                  understood.
               Surface Diffusion and Elasticity




                                                  Equilibrium conditions can be obtained by
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                                                  continuum model (additional homoge-
                                                  nization of the elasticity equations +
                                                  compositional energy)
                                                  Minimization of total energy with respect to
                                                  displacement, free boundary and
                                                  concentration
                                                          Dynamic Intermixing
                                                  Effects of alloys are difficult to model in
                                                  continuum setting
                                                  Spencer, Tersoff, Vorhees (2001) give
               Surface Diffusion and Elasticity




                                                  transport equation on the surface, if alloy is
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                                                  deposited, but ignore bulk diffusion
                                                  Gurtin derives diffusion equation in bulk
                                                  from simple assumptions



                                                  Constitutive relation for flux jc is missing