Introduction to Monte Introduction to Monte Carlo SimulationCarlo Simulation

What is a Monte Carlo simulation?What is a Monte Carlo simulation?

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

Details of the MethodDetails of the Method Random Walk: Markov chain is a sequence of Random Walk: Markov chain is a sequence of

events with the condition that the probability of events with the condition that the probability of each succeeding event is uninfluenced by prior each succeeding event is uninfluenced by prior eventsevents

Choosing from Probability Distribution: Any Choosing from Probability Distribution: Any random variable has a probability distribution for random variable has a probability distribution for its occurrence. We need to choose a random its occurrence. We need to choose a random variable which mimics that probability distributionvariable which mimics that probability distribution

Best way to relate random number to a random Best way to relate random number to a random variable is to use cumulative probability variable is to use cumulative probability distribution and equating it to the random nuberdistribution and equating it to the random nuber

Random NumbersRandom Numbers

Uniformly distributed numbers in Uniformly distributed numbers in [0,1][0,1]

Most useful method for obtaining Most useful method for obtaining random numbers for computer use is random numbers for computer use is a pseudo random number generatora pseudo random number generator

How random are these pseudo How random are these pseudo random numbers?random numbers?

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.John von Neumann (1951)

Application to Microscale Heat Application to Microscale Heat TransferTransfer

Boltzmann Transport Equation (BTE) Boltzmann Transport Equation (BTE) for phonons best describes the heat for phonons best describes the heat flow in solid nonmetallic thin films flow in solid nonmetallic thin films

difficult to solve analytically or even difficult to solve analytically or even numerically using deterministic numerically using deterministic approaches approaches

alternative is to solve the BTE using alternative is to solve the BTE using stochastic or Monte Carlo techniques stochastic or Monte Carlo techniques

Boltzmann Transport Equation for Particle Transport

Distribution Function of Particles: f = f(r,p,t)--probability of particle occupation of momentum p at location r and time t

scatp t

fff

t

f

Fv r

Relaxation Time Approximation

t

off

t

e

Equilibrium Distribution:

f0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons, Plank for photons, etc.

Relaxation time

Non-equilibrium, e.g. in a high electric field or temperature gradient:

pr,pppppp,p

pppp

fffWfW

t

f o

scat

,

Monte Carlo Solution Technique Monte Carlo Solution Technique

Phonons are drawn from the six individual Phonons are drawn from the six individual stochastic spaces, including three wave-stochastic spaces, including three wave-vector components and the three position vector components and the three position vector components vector components

Phonons are then allowed to drift (or Phonons are then allowed to drift (or unrestrained motion) and scatter in time, unrestrained motion) and scatter in time, and their statistics is collected at various and their statistics is collected at various points in time and space, and processed to points in time and space, and processed to extract the necessary information extract the necessary information

Initial Conditions Initial Conditions

number of phonons per unit volume and number of phonons per unit volume and polarization (polarization (pp) is usually an extremely large ) is usually an extremely large numbernumber

a scaling factor is used to simulate only a a scaling factor is used to simulate only a fraction of the phonons fraction of the phonons

A series of random numbers properly A series of random numbers properly distributed to match the equilibrium distributed to match the equilibrium distribution are drawn to initialize the distribution are drawn to initialize the positions, frequencies, polarizations, and positions, frequencies, polarizations, and wavevectors of the ensemble of phonons wavevectors of the ensemble of phonons chosen for the simulation chosen for the simulation

Initial conditionsInitial conditions

Mazumdar and Majumdar developed a Mazumdar and Majumdar developed a numerical scheme to obtain the number of numerical scheme to obtain the number of phonons within the phonons within the iith frequency interval th frequency interval as: as:

Boundary Conditions

Isothermal boundary condition: Isothermal boundary condition: Phonons incident on the wall are Phonons incident on the wall are removed from the computation removed from the computation domain and a new phonon is domain and a new phonon is introduced in the system which introduced in the system which depends on the wall temperaturedepends on the wall temperature

Adiabatic boundary condition: Adiabatic boundary condition: reflects all the phonons that are incident on the wall

Drift 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

ScatteringScattering

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 A random number is chosen and compared to the probability, if less then it is to the probability, if less then it is scatteredscattered

If scattered then the new phonon is If scattered then the new phonon is generated based on the pseudo generated based on the pseudo temperature of the celltemperature of the cell

ScatteringScattering

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,i is given by

where is a constant of the order of unity, is the defect density per unit volume, and is the scattering cross-section

Temperature profile for ballistic Temperature profile for ballistic transporttransport

2-D Temperature profile2-D Temperature profile

Mazumder et al. 2001

Monte Carlo Simulation of Monte Carlo Simulation of Silicon Nanowire Thermal Silicon Nanowire Thermal

ConductivityConductivity Boundary scattering play an important role in Boundary scattering play an important role in

thermal resistance as the structure size thermal resistance as the structure size

decreases to nanoscaledecreases to nanoscale

0 50 100 150 200 250 300 350

0

10

20

30

40

50

60

70T

herm

al c

ondu

ctiv

ity (

W/K

.m)

Temperature (K)

115nm 37nm 22nm

Heat Generation in Electronic Heat Generation in Electronic NanostructureNanostructure

Pop E. et al. 2002

Statistical ErrorStatistical Error

Monte Carlo simulation is a Monte Carlo simulation is a stochastic sampling process, hence stochastic sampling process, hence have inherent statistical errorhave inherent statistical error

errors depend primarily on the errors depend primarily on the number of stochastic samples used number of stochastic samples used in the simulation and the number of in the simulation and the number of scattering events that occur scattering events that occur

ReferenceReference 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