A Level-Set Method for Modeling Epitaxial Growth and Self-Organization of Quantum Dots Christian Ratsch, UCLA, Department of Mathematics Russel Caflisch Xiabin Niu Max Petersen Raffaello Vardavas Collaborators $$$: NSF and DARPA Santa Barbara, Jan. 31, 2005 Outline Introduction The basic island dynamics model using the level set method Include Reversibility Ostwald Ripening Include spatially varying, anisotropic diffusion self-organization of islands

What is Epitaxial Growth? o  –  = “on” – “arrangement”

Why do we care about Modeling Epitaxial Growth? Methods used for modeling epitaxial growth: KMC simulations: Completely stochastic method Continuum Models: PDE for film height, but only valid for thick layers New Approach: Island dynamics model using level sets Many devices for opto-electronic application are multilayer structures grown by epitaxial growth. Interface morphology is critical for performance Theoretical understanding of epitaxial growth will help improve performance, and produce new structures.

KMC Simulation of a Cubic, Solid-on-Solid Model E S : Surface bond energy E N : Nearest neighbor bond energy  0 : Prefactor [O(10 13 s -1 )] Parameters that can be calculated from first principles (e.g., DFT) Completely stochastic approach But small computational timestep is required D =  0 exp(-E S /kT) F D det = D exp(-E N /kT) D det,2 = D exp(-2E N /kT)

KMC Simulations: Effect of Nearest Neighbor Bond E N Large E N : Irreversible Growth Small E N : Compact Islands Experimental Data Au/Ru(100) Ni/Ni(100) Hwang et al., PRL 67 (1991)Kopatzki et al., Surf.Sci. 284 (1993)

440°C Ml/s 20 min anneal 380°C Ml/s 60 min anneal KMC Simulation for Equilibrium Structures of III/V Semiconductors Experiment (Barvosa-Carter, Zinck) KMC Simulation (Grosse, Gyure) Problem: Detailed KMC simulations are extremely slow ! Similar work by Kratzer and Scheffler Itoh and Vvedensky F. Grosse et al., Phys. Rev. B66, (2002)

Outline Introduction The basic island dynamics model using the level set method Include ReversibilityOstwald Ripening Include spatially varying, anisotropic diffusion self-organization of islands

The Island Dynamics Model for Epitaxial Growth Atomistic picture (i.e., kinetic Monte Carlo) F D v Treat Islands as continuum in the plane Resolve individual atomic layer Evolve island boundaries with levelset method Treat adatoms as a mean-field quantity (and solve diffusion equation) Island dynamics

The Level Set Method: Schematic Level Set Function  Surface Morphology t  =0  =1 Continuous level set function is resolved on a discrete numerical grid Method is continuous in plane (but atomic resolution is possible !), but has discrete height resolution

The Basic Level Set Formalism for Irreversible Aggregation Governing Equation:  =0 Diffusion equation for the adatom density  (x,t): Velocity: Nucleation Rate: Boundary condition: C. Ratsch et al., Phys. Rev. B 65, (2002)

Typical Snapshots of Behavior of the Model   t=0.1 t=0.5

Numerical Details Level Set Function 3 rd order essentially non-oscillatory (ENO) scheme for spatial part of levelset function 3 rd order Runge-Kutta for temporal part Diffusion Equation Implicit scheme to solve diffusion equation (Backward Euler) Use ghost-fluid method to make matrix symmetric Use PCG Solver (Preconditioned Conjugate Gradient)

Essentially-Non-Oscillatory (ENO) Schemes ii-1 i+1i+2 Need 4 points to discretize with third order accuracy This often leads to oscillations at the interface Fix: pick the best four points out of a larger set of grid points to get rid of oscillations (“essentially-non-oscillatory”) i-3i-2 i+3i+4 Set 1Set 2Set 3

Numerical Details Level Set Function 3 rd order essentially non-oscillatory (ENO) scheme for spatial part of levelset function 3 rd order Runge-Kutta for temporal part Diffusion Equation Implicit scheme to solve diffusion equation (Backward Euler) Use ghost-fluid method to make matrix symmetric Use PCG Solver (Preconditioned Conjugate Gradient)

Solution of Diffusion Equation Standard Discretization: Leads to a symmetric system of equations: Use preconditional conjugate gradient method Problem at boundary: i-2i-1 i i+1 Matrix not symmetric anymore : Ghost value at i “ghost fluid method” ; replace by:

Nucleation Rate: Fluctuations need to be included in nucleation of islands Probabilistic Seeding weight by local  2   max C. Ratsch et al., Phys. Rev. B 61, R10598 (2000)

A Typical Level Set Simulation

Outline Introduction The basic island dynamics model using the level set method Include ReversibilityOstwald Ripening Include spatially varying, anisotropic diffusion self-organization of islands

So far, all results were for irreversible aggregation; but at higher temperatures, atoms can also detach from the island boundary Dilemma in Atomistic Models: Frequent detachment and subsequent re- attachment of atoms from islands Significant computational cost ! In Levelset formalism: Simply modify velocity (via a modified boundary condition), but keep timestep fixed Stochastic break-up for small islands is important Extension to Reversibility Velocity: Nucleation Rate: Boundary condition:

For islands larger than a “critical size”, detachment is accounted for via the (non-zero) boundary condition For islands smaller than this “critical size”, detachment is done stochastically, and we use an irreversible boundary condition (to avoid over-counting) Details of stochastic break-up calculate probability to shrink by 1, 2, 3, ….. atoms; this probability is related to detachment rate. shrink the island by this many atoms atoms are distributed in a zone that corresponds to diffusion area Note: our “critical size” is not what is typical called “critical island size”. It is a numerical parameter, that has to be chosen and tested. If chosen properly, results are independent of it.

Sharpening of Island Size Distribution with Increasing Detachment Rate Experimental Data for Fe/Fe(001), Stroscio and Pierce, Phys. Rev. B 49 (1994) Petersen, Ratsch, Caflisch, Zangwill, Phys. Rev. E 64, (2001).

Scaling of Computational Time Almost no increase in computational time due to mean-field treatment of fast events

Ostwald Ripening Verify Scaling Law Slope of 1/3 M. Petersen, A. Zangwill, and C. Ratsch, Surf. Science 536, 55 (2003).

Outline Introduction The basic island dynamics model using the level set method Include ReversibilityOstwald Ripening Include spatially varying, anisotropic diffusion self-organization of islands

Nucleation and Growth on Buried Defect Lines Growth on Ge on relaxed SiGe buffer layer Dislocation lines are buried underneath. Lead to strain field This can alter potential energy surface: Anisotropic diffusion Spatially varying diffusion Hypothesis: Nucleation occurs in regions of fast diffusion Results of Xie et al. (UCLA, Materials Science Dept.) Level Set formalism is ideally suited to incorporate anisotropic, spatially varying diffusion without extra computational cost

Modifications to the Level Set Formalism for non-constant Diffusion Velocity: Nucleation Rate: Replace diffusion constant by matrix: Diffusion in x-directionDiffusion in y-direction Diffusion equation: drift no drift Possible potential energy surfaces

What we have done so far Assume a simple form of the variation of the potential energy surface (i.e., sinusoidal) For simplicity, we look at extreme cases: only variation of adsorption energy, or only variation of transition energy (real case typically in- between)

Isotropic Diffusion with Sinusoidal Variation in x-Direction fast diffusionslow diffusion Islands nucleate in regions of fast diffusion Little subsequent nucleation in regions of slow diffusion Only variation of transition energy, and constant adsorption energy

Comparison with Experimental Results Results of Xie et al. (UCLA, Materials Science Dept.) Simulations

Anisotropic Diffusion with Sinusoidal Variation in x-Direction In both cases, islands mostly nucleate in regions of fast diffusion. Shape orientation is different

Isotropic Diffusion with Sinusoidal Variation in x- and y-Direction

Comparison with Experimental Results Results of Xie et al. (UCLA, Materials Science Dept.) Simulations

Anisotropic Diffusion with Variation of Adsorption Energy Spatially constant adsorption and transition energies, i.e., no drift small amplitudelarge amplitude Regions of fast surface diffusion Most nucleation does not occur in region of fast diffusion, but is dominated by drift What is the effect of thermodynamic drift ? E tran E ad

Transition from thermodynamically to kinetically controlled diffusion But: In all cases, diffusion constant D has the same form: D x Constant adsorption energy (no drift) Constant transition energy (thermodynamic drift)

What is next with spatially varying diffusion? So far, we have assumed that the potential energy surface is modified externally (I.e., buried defects), and is independent of growing film Next, we want to couple this model with an elastic model (Caflisch et al., in progress); Solve elastic equations after every timestep Modify potential energy surface (I.e., diffusion, detachment) accordingly This can be done at every timestep, because the timestep is significantly larger than in an atomistic simulation

Conclusions We have developed a numerically stable and accurate level set method to describe epitaxial growth. The model is very efficient when processes with vastly different rates need to be considered This framework is ideally suited to include anisotropic, spatially varying diffusion (that might be a result of strain): Islands nucleate preferentially in regions of fast diffusion (when the adsorption energy is constant) However, a strong drift term can dominate over fast diffusion A properly modified potential energy surface can be exploited to obtain a high regularity in the arrangement of islands. More details and transparencies of this talk can be found at