Off-Lattice KMC Simulation of Quantum Dot Formation T. P. Schulze University of Tennessee Peter Smereka & Henry Boateng University of Michigan Research supported by NSF-DMS-0854920
Heteroepitaxy and Quantum Dots G. Medeiros-Ribeiro, M. Bratkovski, T. I. Kamins, D. A. A. Ohlberg and R. S. Williams, Shape transition of germanium nanocrystals on a silicon (001) surface from pyramids to domes. Science 279, 353-355 (1998).
KMC with Elastic Interactions Snapshots of film growth: a) Flux = 1 ML/sec, Misfit =0.07, Growth of 0.5 monolayers. b) Annealed shape with misfit = 0.04, flux=0.0.
KMC Models Off Lattice Effects QEKMC OLKMC WOLKMC SBCKMC EBCKMC Interaction Cutoff r i j = Kexp[ E B /kt] Off-Lattice KMC: E B = E sad E i Simple Bond-Counting KMC: E B = E off E i = NE n Metropolis: E B = E i j E i
KMC Model with Elastic Interactions Reference: B.G. Orr, D.A. Kessler, C.W. Snyder, and L.M. Sander, Europhysics Lett. 19, 33-38 (1992). 2+1 SOS with height h ij Z. Nearest, next & 3rd nearest neighbor bonds.
Elastic Energy Barrier Hopping rate: r i j = Kexp[ E B /kt], Hopping barrier is modeled as: E B = (E 0 + U + W), U = (B 11 +B 22 +B 12 ), ( ) B αβ = an (1) αβ +bn(2) αβ +cn(3) αβ γ αβ, W = W(with atom) W(without atom), W = 1 2 (i,j,k) Ω w i,j,k Equilibrium configuration: W u i,j,k = 0, W v i,j,k = 0, W w i,j,k = 0.
KMC Algorithm w/o Rejection 1. Calculate all of the rates {r i } N i=1 2. Choose a random number r [0,R = N i=1 r i) 3. Select the corresponding event (e.g. use partial sums to partition [0,R) and locate the subinterval P i 1 r < P i. 4. Move atom; relax atoms in new configuration. 5. Repeat This requires N+1 elastic solves to update the rates.
Methods and Approximations We have implemented a number of improvements to the computational scheme for the basic model: 1. SS 2009: Rejection using estimates for upper bounds on rates Energy localization method Expanding box method 2. SS 2011: Coarse-grained random walks Local Energy Method 3. SS 2012: Two scale domain decomposition
KMC Algorithm w/ Rejection Change in Elastic Energy 4 3 2 1 0 0 1 2 3 4 Upper and lower bounds for elastic energy content W : CLwij < W < CU wij,
Understanding of the Bounds on W J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. Lond. A, 241 376-396 (1957). Let w = U +V,where U = 1 18κ (t ll) 2, V = 1 4µ (t ij 1 3 δ ijt ll ) 2, κ is the bulk modulus, and µ is a Lamé coefficient. One can rework Eschelby s result to give where A = 3(1 σ) 2 4σ It is convenient to define max t ij AU +BV U +V W = τ(au +BV), max(a,b) = S U and B = 15(1 σ) 7 5σ. and min t ij AU +BV U +V min(a,b) = S L.
Understanding of the Bounds on W Since U and V are both positive it follows that S L W τw S U. In the figure on the left, the bounds S U (blue) and S L (red) are plotted as a function of the Poisson ratio. As a result, the relationship between the change in elastic energy W and the local energy density w ij is bounded between the linear curves shown on the right. 3 2.8 2.6 2.4 2.2 2 1.8 Change in Elastic Energy 6 5 4 3 2 1 1.6 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 Poisson Ratio 0 0 1 2 3 Initial Energy Content
Two-Scale Domain Decomposition 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 140 140 120 120 140 100 120 80 100 60 80 60 40 40 20 20 0 0 140 100 120 80 100 60 80 60 40 40 20 20 0 0
Two-Scale Domain Decomposition 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 0 0.05 0.1 0.15 0.2 0.25 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 0 0.05 0.1 0.15 0.2 0.25
Validation I(u, v) = Z Z h (x u, y v)h (x, y)dxdy, 1 g (R) = 2πR Z Z I (u(r, θ), v(r, θ)) δ(r R)drdθ,
Validation number density 300 250 200 150 100 50 0 5 10 15 20 effective radius g(r) 15 10 5 0 5 0 10 20 30 R 40 50 60 Island size distributions (ISD) and autocorreclation curves for Local Energy Method (red); Surface Decomposition Method (blue). For comparison, the ISD for growth without elastic effects is also shown (black).
Stranski-Krastonov Growth The film is shown after 0.5 monolayer of deposition. At this stage in the growth only two dimensional islands have formed. The presence of blue colored atoms within the two dimensional islands is due to intermixing.
Stranski-Krastanov Growth The film is shown after 1.0 monolayer of deposition. The two dimensional islands have grown into each other to completely cover the surface. While this single layer of yellow colored atoms is strained, the surface forces prevent the formation of three dimensional islands.
Stranski-Krastanov Growth The film is shown after 1.5 monolayers of deposition. There are two dimensional islands, several pre-pyramids, and the start of a fully faceted pyramid.
Stranski-Krastanov Growth The film is shown after 3.0 monolayers of deposition. The two dimensional islands have mostly been consumed. Most of the film is covered in three dimensional, fully faceted islands.
Quantum Dot Alignment The amount of material deposited is.02 ML,.05ML and.1 ML. 10 20 30 20 40 60 80 100 120
Quantum Dot Alignment Film profiles after 1.4 monolayers of deposition.
Capping of Quantum Dots The 3.0 monolayer film is capped with an additional 0.6 ML of blue material. The quantum dots have been reduced in size due to dot material disolving onto the wetting layer.
Capping and Quantum Rings 50 100 150 200 250 50 100 150 200 250 There is now 4.5 monolayers of capping (blue/substrate) material. The high strain energy inside the dots provides a driving force for the yellow material to leave. If it can leave before it is capped, a crater/ring is formed.
Quasi-Equilibrium KMC We want to define a simple KMC model with off-lattice effects. To this end, we use the local minima/minimizers of an empirical potential: U(x) = N φ(r ij ), where i<j [ (σij ) 12 φ(r ij ) = 4ǫ ij r ij ( σij r ij ) 6 ] The potential is also used to locate these states on-the-fly. The allowed transitions are associated with surface atoms. Rates are defined using E off and approximated using local binding energy.
2D Quasi-Equilibrium KMC Four monolayers of film with misfits of 2%, 4% and 10%. Growth behavior changes from layer-by-layer to SK to VW.
2D Off-Lattice Quasi-Equilibrium KMC Edge dislocation: Extra half plane 10.6 Tensile region Compressive region 10.4 10.2 10 9.8 9.6 9.4 9.2 9 8.8 Close up of a dislocation (roted view) with atoms colored by average distance to neighbors.
2D Off-Lattice Quasi-Equilibrium KMC 10.5 10 9.5 9 An annealed system of 4ML of film on 40ML of substrate with η = 0.04. The two leftmost islands (I and II) develop over a vacancy while the rightmost island develops over a dislocation. Atoms are colored by average distance to neighbors.
Summary Presently, our best weakly off-lattice method combines: Local relaxation A Local Energy Approximation for rates Two scale domain decomposition This code is simple-cubic, 3D and can exhibit dots, stacked dots, rings, etc., but not dislocations. Our current off-lattice method combines: Local relaxation A Local Energy Approximation for rates This code is off-lattice, 2D, uses the Lennard-Jones potential and can exhibit dislocations.