Kinetic lattice Monte Carlo simulations of diffusion processes in Si and SiGe alloys

Transcription

1 Kinetic lattice Monte Carlo simulations of diffusion processes in Si and SiGe alloys, Scott Dunham Department of Electrical Engineering

2 Multiscale Modeling Hierarchy Configuration energies and transition rates. Induced strains, binding/migration energies. Calibration/testing of empirical potentials Configuration energies and transition rates Behavior during regrowth, atomistic vs. global stress

3 KLMC Flow Chart Start Initialize rates k ij Generate r 1, select transition Perform transition Generate r 2, advance time by Update rates k ij t=-ln(r 2 )/k tot Conditions Satisfied? No Yes Terminate

4 Building the rate catalogue Task: find migration barriers for all allowed transitions at each step Surrounding conditions modify migration barriers Presence of other defects (binding energy) Presence of strain (additional strain energy) Solution: Store an unbiased energy barrier E m0 Calculate change of barriers due to change of formation energies of initial and final states. Assume transition state scales linearly with initial and final state E m = E m (ΔE fi ΔE in ) Unbiased Modified Δ in Dunham et al., J. Appl. Phys. 78, 2362 (1995) in (Δ in + Δ fi )/2 fi Δ fi

5 Outline Interstitial-mediated self-diffusion in Si Vacancy mediated inter-diffusion in SiGe

6 Silicon Self Diffusion Mediated by point defects (V, I) V I One of the few directly measurable point defect properties. Quantified by diffusion of an isotropic tracer (T) D T = f I D I C I f I, f V : correlation factors C s + f V D V C V Point defects perform uncorrelated random walks. Tracer atoms perform correlated random walks. C s

7 Correlation of Self Diffusion Tracer atoms perform correlated random walk The previous direction affects the direction of the next step V A tracer, once exchanges with a V, has a higher probability of reexchange with the V. I A tracer, once kicks an atom out, has a higher probability of being kicked back. Correlation of interstitial diffusion (f I ) via indirect process depends on mechanism/hopping network

8 Methodology Interstitial self diffusion on diamond lattice: KLMC: Simulates diffusion of various hopping mechanisms on different lattice networks, find f I by: f I = Δr2 /N tracer Δr 2 /N interstitial N: number of steps Δr 2 : square displacement A B C A B C Compaan et al., Trans. Faraday Soc Interstitial self diffusion in silicon DFT: Find dominant hopping mechanisms KLMC: Simulates diffusion in Si and find f I.

9 Kick-out Mechanism Tetrahedral network: Hexagonal network: I tet I tet f I = ± c.f [Compaan et. al.] I hex I hex f I = ±0.0001

10 Stable-split Mechanism Assuming the split configuration is stable Tetrahedral network: Hexagonal network: I tet I sp I tet f = ± I hex I sp I hex f = ±0.0001

11 DFT: Migration Events I sp I h I sp I t I t I sp I sp I h I h

12 KLMC Simulations Dominant mechanisms: The direct (uncorrelated) mechanism: I h I h via I t. Forward barrier:0.17 ev, reverse barrier: 0.17 ev. The indirect mechanism: I sp I h. Forward barrier:0.34 ev, reverse barrier: 0.25 ev. Probability to follow the direct path (P d ) depends on the barriers and temperature. Ω d exp ( E d m kt ) P d = Ω d exp E d m kt +Ω iexp ( E i m kt ) Effective correlation factor is temperature dependent P i (indirect) I sp P d (direct) I h I h

13 Correlation of Si Self Diffusion Effective correlation factor as a function of the probability of the direct mechanism. f at C calculated to be Experimental value is 0.6 [Voronkov et. al. 2005] Possible sources of inaccuracy: Uncertainties of DFT predicted barriers Neglect of entropy difference

14 Outline Interstitial-mediated self-diffusion in Si Vacancy mediated inter-diffusion in SiGe

15 Simulational perspective SiGe Inter-diffusion dominated by vacancy mechanism Si Ge V V + Si Si + V V + Ge Ge + V Change of formation energies due to Ge-V binding energy (DFT: 0.31 ev) Strain energy (determined by induced strains of defects) Energy barrier scales as: E m = E m (ΔE f fi ΔE f in ) Si V Unbiased Modified Δ in 0.34eV Ge V 0.17eV Unbiased barrier E m 0 (DFT) in (Δ in + Δ fi )/2 fi Δ fi

16 Determining Strain Energy Hooke s Law Strain energy C e 1 2 e E Strain dependence T E E0 xd C xd 2 1 ai a0 Dei x a e e e e e 0 stress strain x : defect concentration 1 C11 C12 C13 e1 C C C e e C C C 3 Induced strain (De) and elastic stiffness tensor (C) can be extracted from DFT calculations for different strain conditions

17 Simulation Set-up Simulation set-ups are chosen to mimic experimental conditions (Si layers on relaxed Si 1-x Ge x layers, 920 º C). Simulation results scaled to the condition of equilibrium vacancy concentration Experimental profile [Xia et al] Simulation domain [Blue: Ge, Yellow: Si]

18 Extracting interdiffusivity Ge affects inter-diffusivity via Stress effect: strain energy compensation of Ge and V Alloy effect: Ge-V binding lowers V formation energy. Interdiffusivity increases exponentially with Ge content D x Ge ~exp(bx Ge ) Extracting B from KLMC B alloy 5.8 alloy B stress 7.6 B total 13.4 B total (Expt*) 8.1 * [Xia et al] alloy + stress Sources of error: Inaccurate parameters for diffusion and formation energies Small lateral dimension that results in large fluctuations Insufficient sampling of Monte Carlo simulations

19 Conclusion KLMC combined with DFT calculations for Self diffusion in Si Interdiffusion in SiGe Future Research More accurate ways to calculate migration barriers based on local environment Fully implement strain effect

20 Acknowledgements Special thanks to Scott T. Dunham (Advisor) EMSL, PNNL Financial support