Kinetic Lattice Monte Carlo Simulations of Dopant Diffusion/Clustering in Silicon

1. Kinetic Lattice Monte Carlo Simulations of Dopant Diffusion/Clustering in Silicon Zudian Qin and Scott T. Dunham Department of Electrical Engineering University of Washington SRC Review February 25-26,2002

2. Outline • Introduction to KLMC Simulations • High Concentration Arsenic Diffusion • Acceleration of KLMC Simulations • Fermi Energy Level Modeling in Atomistic Scale • Random dopant fluctuations (initial results) • Summary

3. Kinetic Lattice Monte Carlo Simulations Set up a silicon lattice structure (10-50nm)3 Defects (dopant and point defects) initialized - based on equilibrium value - or imported from implant simulation - or user-defined

4. Kinetic Lattice Monte Carlo Simulations Fundamental processes are point defect hop/exchanges. Vacancy must move to at least 3NN distance from the dopant to complete one step of dopant diffusion in a diamond structure.

5. Kinetic Lattice Monte Carlo Simulations Simulations include B, As, I, V, Bi, Asi and interactions between them. Hop/exchange rate determined by change of system energy due to the event. Energy depends on configuration and interactions between defects with numbers from ab-initio calculation (interactions up to 9NN).

6. Kinetic Lattice Monte Carlo Simulations • Calculate rates of all possible processes. • At each step, Choose a process at random, weighted by relative rates. • Increment time by the inverse sum of the rates. Perform the chosen process and recalculate rates if necessary. Repeat until conditions satisfied.

7. High Concentration Arsenic Diffusion Experiments found strong enhancement of diffusivity above 1020 cm-3. Dunham/Wu found strong D increase using KLMC simulations.

8. High Concentration Arsenic Diffusion List et al. found reduced D in long term of simulations with fixed number of Vs in system. The reason for the discrepancy is the formation of AsnV clusters during the simulation---number of free V drops. Dunham/Wu did a relatively short simulation before clusters can form. ---Possible transient effects. Solution: Long term simulations tracking free V concentration. Problem: Computationally demanding for good statistics.

9. Acceleration of KLMC Simulations Once a cluster is formed, the system can spend a long time just making transitions within a small group of states.

10. Acceleration of KLMC Simulations The solution is to consider the group of states as a single effective state. States inside the group are near local equilibrium.

11. Acceleration of KLMC Simulations Comparison of time that a vacancy is free as a function of doping concentration via simulations and analytic function Both simulations with/without acceleration mechanism agree with the analytic prediction, but acceleration saves orders of magnitude in CPU time.

12. High Concentration Arsenic Diffusion Equilibrium vacancy concentration increased significantly since the formation energy is lowered due to presence of multiple arsenic atoms. At high concentration, vacancy likely interacts with multiple dopant atoms. The barrier is lowered due to attraction of nearby dopant atoms.

13. High Concentration Arsenic Diffusion --- KLMC Results As seen experimentally, simulations show arsenic diffusivity has strong increase with doping level: polynomial or exponential form. Effect stronger at lower T, critical for As modeling (Pavel Fastenko).

14. Fermi Energy Level Modeling The concentration of charged point defect is a function of Fermi level. Dopant atoms are ionized (e.g. As+, B-) and exposed to the field.

15. Fermi Energy Level Modeling Continuous models derive Fermi level from dopant profiles. At atomistic scale, dopant atoms are discrete. Each donor (acceptor) contributes an electron (hole) cloud around itself.

16. Fermi Energy Level Modeling Contributions of all charged dopants and defects add to give the total electron density. Simulation of in a nonuniform background. Residence time follows electron density, as predicted by continuous model.

17. Fermi Energy Level Modeling Example of KLMC simulations with incorporated field effect.

18. Random Dopant Fluctuations Initial simulations show like dopant atoms tend to repel each other, resulting in a more uniform potential.

19. Summary • As diffusion at high concentrations shows a strong increase with doping level that is consistent with experimental measurements. • Acceleration mechanism improves simulation efficiency, significantly reducing CPU time. • Developed a Fermi level model and incorporated into LAMOCA KLMC simulation code. • Initial dopant fluctuation simulations give more uniform Fermi level than random distribution (dopant/dopant repulsion).