ME 595M: Monte Carlo Simulation - Purdue Universitytsfisher/ME595M/Monte_Carlo.pdf · 8 Scattering Probability and Time • Probability that an electron survives until time t without

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ME 595M: Monte Carlo Simulation

T.S. FisherPurdue University


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Monte Carlo Simulation

• BTE solutions are often difficult to obtain♦ Closure issues force the use of BTE moments

• Inherent approximations♦ Ballistic transport is often not described well by perturbed

equilibrium distributions

• Instead, seek to track representative carriers and use statistical averaging to predict transport behavior♦ Monte Carlo methods are essentially averaging or integration tools

• In neither case can we easily incorporate quantum (wave) effects


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Scattering

• Treatment of scattering underlies almost all MC methods

• Consider electron scattering mechanisms♦ Acoustic deformation potential (ADP)

♦ Intervalley scattering by phonon absorption

♦ Others (τ3-1, τ4

-1, …)• Combine with Mathiessen’s rule

( )13

1

1 2 10 ( ) /E p q= ×τ

( )13

2

1 1.5 10 ( ) / 0.050E p q= × +τ 1( )

( )

(Effective Scattering Rate)ii

pp

Γ =τ∑


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Combining Scattering Events

Lundstrom Fig. 6.1


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Equation of Motion

• The most complex transport equation employed in most MC simulations is Newton’s law, F=ma

• Collision times are typically much smaller than free-flight times♦ Collisions are treated as instantaneous events

• Carrier position can be expressed as an integral of velocity

d qdt

= = −epF E

0( ) (0) ( ') '

tt t dt= + ∫r r v


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Free Flight

• Consider an electron under the action of an electric field directed along the z-axis

• Position relations become

( ) (0)( ) (0)

( ) (0) ( )

x x

y y

z z z

p t pp t p

p t p q t

=

=

= + − E

*

*

(0)( ) (0)

(0)( ) (0)

( ) (0)( ) (0)( )

x

y

z

px t x tm

py t y t

mE t Ez t z

q

= +

= +

−= +

− E

last equality derivesfrom the parabolic energy band assumption

E in numeratoris energy


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Ensemble Scattering

• Consider a collection of carriers of concentration nCF that have not undergone scattering since time t=0

• Assume a constant scattering rate Γ0 (for now)• The time evolution of nCF is expressed

mathematically as

♦ solution

0CF

CFdn n

dt= −Γ

0( ) (0) tCF CFn t n e−Γ=


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Scattering Probability and Time

• Probability that an electron survives until time t without scattering is

• Probability that a carrier undergoes its first collision between t and t+dt is the scattering rate times the survival probability

0( )(0)

tCF

CF

n t en

−Γ=

00( ) tP t dt e dt−Γ= Γ


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Random Selection• Now we wish to choose a random number such that the

probability of choosing a number between r and r+dr equals the probability of selecting a collision time between t and t+dt

• Note that r1 is also a uniformly distributed random number between 0 and 1

• Foregoing equation relates random numbers to collision times

0

0

0

10 0

( ) ( )let P(r) = 1 for a random number generator between 0 and 1

11 1ln(1 ) ln( )

c

t

tc

c c

P r dr P t dt

dr e dt

r e

t r r

−Γ

−Γ

=

→ = Γ

→ = −

→ = − − = −Γ Γ


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Self-Scattering

Lundstrom Fig. 6.4


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Collision Distributions

Lundstrom Fig. 6.5

tc based on all scattering events (includingself-scattering)

tc based on only realscattering events (not includingself-scattering)

0

1ct ≈

Γ

1( )ct p

≈Γ


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Identification of Scattering Events

• After selecting the duration of free flight (tc), we must properly choose the type of scattering event that occurs

• Each scattering mechanism can alter the particle’s momentum (direction and magnitude) differently

• Ultimately, the choices must be proportional to the relative likelihoods of occurrence


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Cumulative Scattering Probabilities• Consider k possible scattering mechanisms

• Imagine a cumulative bar chart that includes a sum of the fractional probabilities of each type

• Choose a uniformly distributed random number r2 between 0 and 1

• Select scattering event l if the following holds1

1 12

0 0

1 1( ) ( )

l l

i ii ip pr

−

= =τ τ≤ <

Γ Γ

∑ ∑


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Graphical Interpretation

• Lundstrom, Fig. 6.8


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Selecting a Final State• Generally the most expensive part of the computation

• Must select the final magnitude of momentum (energy) andits direction

• Consider particle states immediately before (tc-) and after (tc+) scattering

♦ assumes parabolic energy bands♦ ΔE is a fundamental characteristic of the scattering event (e.g.,

ΔE=0 for elastic scattering, ΔE ~ ħω for events involving phonons)

( )*( ) ' 2c cp t p m E t E+ −⎡ ⎤≡ = + Δ⎢ ⎥⎣ ⎦


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Selecting a Final Direction• Align local coordinate system with initial momentum

vector p• Assume azimuthal invariance

• Thus, we can select azimuthal angle from a uniformly distributed random number r3 as β=2πr3

LundstromFig. 6.9

2

0

1( ) 1 ( )2

P d P d dπ

β β = → β β = βπ∫


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Determination of the Polar Angle

• Scattering often depends on the polar angle; hence, its treatment is more complicated

• If we assume a simple delta-function scattering mechanism [S~δ(E’-E)], then

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0 02

20

0 0 0

sin ( ) ' '

( ) 1 ( )

( ) sin '

d S d p dp

P d P d

S d d p dp

∞ π

π

∞ π π

α α β

α α = → α α =

α α β

∫ ∫∫

∫ ∫ ∫

p, p'

p, p'

sin( )2

dP d α αα α =


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Final Polar Angle• Now we seek another uniformly distributed random

number r4 between 0 and 1 to satisfy

• Finally, we find p’ in the local coordinate system as

• Last step involves transformation from local to global coordinates

( )4

40 0

sin( ) ( ) ( ) 12

1 1sin ' ' 1 cos cos 1 22 2

r

dP r dr P d P r

dr d rα

α α= α α = → =

→ = α α = − α → α = −∫ ∫

' sin cos' sin sin' cos

ppp

α β⎛ ⎞⎜ ⎟= α β⎜ ⎟⎜ ⎟α⎝ ⎠

p'


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Other Monte Carlo Topics• Ensemble vs. incident flux approaches

♦ Ensemble follows particles in parallel in a time-stepping procedure

• Uses “superelectron” approximation♦ Incident flux follows each particle sequentially from

beginning to end• Treatment of Coulomb effects

♦ Charged particles alter the local field through Coulomb interactions

♦ Can be handled by summing individual contributions♦ If foregoing is too onerous, then particle-in-cell-type

methods can be applied♦ Must ensure that the size of the time step does not

exceed natural fluctuations at the plasma frequency


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Relation between Monte Carlo Simulation at the BTE• Consider a 1D slab on infinitesimal thickness that

contains a single particle trajectory

• Apply the chain rule

( , ) ( ), and( , ) ( ( )) ( ( ))i

i

n tf t t tδ = δ −

δ = δ − δ −i

i i

r r rr, p r r p p

, ,but we notice that

and

now sum over all trajectories

i i

i i

i i

i

ir i p i

r i p icoll

r p r p

ip i

coll coll

r pcoll

f d df ft dt dt

f f qt

f f

fft t

f ff q ft t

∂δ= ∇ δ ⋅ + ∇ δ ⋅

∂⎛ ⎞∂

= ∇ δ ⋅ + ∇ δ ⋅ − +⎜ ⎟∂⎝ ⎠∇ δ = −∇ δ

⎛ ⎞ ∂δ∂∇ δ ⋅ =⎜ ⎟∂ ∂⎝ ⎠

∂ ∂+ ∇ ⋅ − ⋅∇ =

∂ ∂

i i

i

i

r p

pv E

p

v E


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Homework Problem

1. Two parallel infinite plates2. Particles leave the same spot on the

source plate at a fixed initial velocity3. Assume no interactions among particles,

i.e. neglect the force between particles4. Particles scatter with a constant cross

section5. Post-collision velocity will be determined

by the model provided6. The spacing is so small that, for each

particle, at most one collision would occur

7. Determine the radial and angular distributions on the collector, and the total number of collisions

Emitter

Collector x


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Code Breakdown

• Data structure:♦ What do we use to represent a particle?

• Something to bind together all the interesting kinetic properties of particles

• A “particle” class in C++• A “particle” structure in C• A n by 7 array in C/C++, matlab, or Fortran 77

zyx3

zyx2

zyx1yVxV zV

xV

xV yV

yVzV

zV


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Algorithm


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Algorithm Breakdown

• Generate the particles at the emitter♦ Set the Vx according to the initial energy, assuming y, z

velocity components of particles are neligible

♦ Set X,Y,Z to zero

0,0,/2 === zyx VVmEV

0,0,0 === ZYX


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Move particles

• What time step to use?

• Recommendation: ~ 1 femtosecond or less

1.0)exp(1)( <Δ−−=Δ tntP συ

tL Δ>υ


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Position Updates

dtzz

dtyy

dtxx

znn

ynn

xnn

υ

υ

υ

=−

=−

=−

+

+

+

1

1

1


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Collision

• Calculate the probability:

• Select a random number• If r<P, then a collision occurs• See notes about polar scattering angle and

azimuthal angle• See notes about post-collision velocity

)exp(1)( tntP Δ−−=Δ συ[0,1]r∈


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Parameters

• Number of particles: N=50,000• Initial kinetic energy of all particles: E=5eV

• Gap: L = 10 nm• Cross-section, σ = 10-18 m2

• Number density of media (target particles), n = 2.5x1025 m-3

• Assume that all scattering is elastic

221 υemE =


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Record the particle information

• When the particle hits the collector or the emitter, we assume it is absorbed without other side effects

• The final particle positions should be stored only for particles that have collided

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ME 595M: Monte Carlo Simulation - Purdue Universitytsfisher/ME595M/Monte_Carlo.pdf · 8 Scattering Probability and Time • Probability that an electron survives until time t without