Introduction to Monte Carlo Simulation

1. Introduction to Monte Carlo Simulation

2. What is a Monte Carlo simulation? • In a Monte Carlo simulation we attempt to follow the `time dependence’ of a model for which change, or growth, does not proceed in some rigorously predefined fashion (e.g. according to Newton’s equations of motion) but rather in a stochastic manner which depends on a sequence of random numbers which is generated during the simulation.

3. Details of the Method • Random Walk: Markov chain is a sequence of events with the condition that the probability of each succeeding event is uninfluenced by prior events • Choosing from Probability Distribution: Any random variable has a probability distribution for its occurrence. We need to choose a random variable which mimics that probability distribution • Best way to relate random number to a random variable is to use cumulative probability distribution and equating it to the random nuber

4. Random Numbers • Uniformly distributed numbers in [0,1] • Most useful method for obtaining random numbers for computer use is a pseudo random number generator • How random are these pseudo random numbers? Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. John von Neumann (1951)

5. Application to Microscale Heat Transfer • Boltzmann Transport Equation (BTE) for phonons best describes the heat flow in solid nonmetallic thin films • difficult to solve analytically or even numerically using deterministic approaches • alternative is to solve the BTE using stochastic or Monte Carlo techniques

6. Boltzmann Transport Equation for Particle Transport Distribution Function of Particles: f= f(r,p,t) --probability of particle occupation of momentum p at location rand time t Equilibrium Distribution: f0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons, Plank for photons, etc. Non-equilibrium, e.g. in a high electric field or temperature gradient: Relaxation Time Approximation t Relaxation time

7. Monte Carlo Solution Technique • Phonons are drawn from the six individual stochastic spaces, including three wave-vector components and the three position vector components • Phonons are then allowed to drift (or unrestrained motion) and scatter in time, and their statistics is collected at various points in time and space, and processed to extract the necessary information

8. Initial Conditions • number of phonons per unit volume and polarization (p) is usually an extremely large number • a scaling factor is used to simulate only a fraction of the phonons • A series of random numbers properly distributed to match the equilibrium distribution are drawn to initialize the positions, frequencies, polarizations, and wavevectors of the ensemble of phonons chosen for the simulation

9. Initial conditions • Mazumdar and Majumdar developed a numerical scheme to obtain the number of phonons within the ith frequency interval Dw as:

10. Boundary Conditions • Isothermal boundary condition: Phonons incident on the wall are removed from the computation domain and a new phonon is introduced in the system which depends on the wall temperature • Adiabatic boundary condition: reflects all the phonons that are incident on the wall

11. Drift • During the drift phase, phonons move linearly from one location to another and their positions are tracked using an explicit first-order time integration • phonons are tallied within each spatial bin, and the energy of each spatial bin is computed and stored

12. Scattering • Three-Phonon Scattering (Normal and Umklapp Processes): need to know scattering time-scales, probability of 3-P scattering is given by PNU = 1-exp(-Dt/tNU) • A random number is chosen and compared to the probability, if less then it is scattered • If scattered then the new phonon is generated based on the pseudo temperature of the cell

13. Scattering • Scattering by Impurities: Scattering by impurities, defects and dislocations are treated in the Monte Carlo scheme in isolation from normal and Umklapp scattering • The time-scale for scattering due to impurities,ti is given by • where a is a constant of the order of unity, r is the defect density per unit volume, and s is the scattering cross-section

14. Temperature profile for ballistic transport

15. 2-D Temperature profile Mazumder et al. 2001

16. Monte Carlo Simulation of Silicon Nanowire Thermal Conductivity • Boundary scattering play an important role in thermal resistance as the structure size decreases to nanoscale

17. Heat Generation in Electronic Nanostructure Pop E. et al. 2002

18. Statistical Error • Monte Carlo simulation is a stochastic sampling process, hence have inherent statistical error • errors depend primarily on the number of stochastic samples used in the simulation and the number of scattering events that occur

19. Reference • Mazumder, S. and Majumdar, A., “Monte Carlo study of phonon transport in solid thin films including dispersion and polarization,” J. of Heat Transfer, vol. 123, pp. 749-759, 2001 • Pop E., Sinha S., Goodson K. E., “Monte Carlo modeling of heat generation in electronic nanostructures”, 2002 ASME International Mechanical Engineering Congress and Exposition • Jacoboni, C. and Reggiani., L., “The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials,” Reviews of Modern Physics, vol. 55, pp. 645-705, 1983