ECE-656: Fall 2011

Lecture 34a:

Monte Carlo Simulation

Mark Lundstrom Purdue University

West Lafayette, IN USA

1 11/16/11

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Damocles

SAT

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outline

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/

(Reference: Chapter 6, Lundstrom, FCT)

1. Introduction 2. Review of carrier scattering 3. Simulating carrier trajectories 4. Free flight 5. Collision 6. Update after collision 7. Putting it all together 8. Summary

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carrier scattering

(0)p

[ ](0)E p

1) Scattering events are assumed to be instantaneous - scattering changes the carriers momentum (and energy) but not position.

2) Scattering events are treated quantum mechanically. Carrier motion between scattering events is treated semi-classically

(0)r

( )Cp t

( )CE p t

( )Cr t

( )Cp t+

( )CE p t+

( ) ( )C Cr t r t+ =

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5

carrier scattering

( )Cp t

( )Cp t+

( )CE p t

( )CE p t+

( ) ( )C CE p t E p t+ =

Elastic scattering:

Inelastic scattering:

( ) ( )C CE p t E p t+

Isotropic scattering: -no preferred direction

Scattering rate:

( )1 ~( ) fi

D EE

Anisotropic scattering: -preferred direction

6

total scattering rate

1

1 1 ( )( ) ( )

N

iki k i

EE E =

= =

ionized impurity acoustic phonon emission acoustic phonon absorption optical phonon ems optical phonon abs intervalley electron-electron electron-hole polar optical phonon surface roughness etc.

scattering mechanisms:

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total scattering rate

http://www.research.ibm.com/DAMOCLES/html_files/sirates.html#history

typical scattering rates vs. energy

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outline

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/

(Reference: Chapter 6, Lundstrom, FCT)

1. Introduction 2. Review of carrier scattering 3. Simulating carrier trajectories 4. Free flight 5. Collision 6. Update after collision 7. Putting it all together 8. Summary

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carrier trajectories in 2D

x x=EEdp qdt

=

E

x

y

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carrier trajectories in phase space

x x=EE

t

( )xp t

xx

dp qdt

= E

0ydpdt

=

0

( ) (0 ) ( )Ct

xx t x t dt= +

free-flight ( )x t

t

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MC algorithm

1) free flight for tC seconds.

2) update E(tC-) and r (tC-)

3) identify collision

4) update E(tC+) and p (tC+)

5) Set t = 0 and repeat

1r

2r

3 4,r r

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outline

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/

(Reference: Chapter 6, Lundstrom, FCT)

1. Introduction 2. Review of carrier scattering 3. Simulating carrier trajectories 4. Free flight 5. Collision 6. Update after collision 7. Putting it all together 8. Summary

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free flight

( )E

E

0 01 =

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free flight

( )0CFN

( )CFN t

( )0p ( )p t

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free flight time

( ) 0( ) ( )CF CF CFN t dt N t N t dt+ =

How many carriers survive until time t + dt without scattering?

( )0( )

CFCF

dN tN t

dt=

0( ) (0) tCF CFN t N e=

0( )(0)

tCF

CF

N t eN

=

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free flight time

What is the probability that a carrier survives until time, t, and then scatters between t and t + dt ?

00 0

( )( )(0)

tCF

CF

N tt dt dt e dtN

= =

P

If we have a random number generator that produces random numbers with a probability distribution, P(r), how do we use it to choose free flight times?

00( ) ( )

tt dt e dt r dr= =P P

free-flight time probability density function

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free flight times

Assume a random number generator that produces random numbers uniformly distributed between 0 and 1.

00( )

tr dr e dt= P

00

tdr e dt=

( )0 0 00 00 0

1C C

C C

r ttt t t

cdr r e dt e e = = = =

( )0

1 ln 1C Ct r=

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free flight distribution

( )10

1 lnCt r= ( )CtP

Ct

00

Cte

0

1Ct =

but, the scattering rate is not constant with energy!

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free flight

( )E

E

0

0 ( )SELF E =

( )Cp t ( )Cp t+

1

01

1( )

N

k k E

+

=

=

1

1( )( )

N

k k

EE=

=

( )10

1 lnCt r=

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update at end of free-flight

( ) (0)C Cp t p q t = E

( ) ( )C CE t E p t =

( ) ( )(0) 0C Cr t r t = +

0t =

Ct t=

1) what collision terminated the free flight?

2) how does the collision affect the carriers momentum and energy?

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L32 outline

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/

(Reference: Chapter 6, Lundstrom, FCT)

1. Introduction 2. Review of carrier scattering 3. Simulating carrier trajectories 4. Free flight 5. Collision 6. Update after collision 7. Putting it all together 8. Summary

22

collision

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To be continued next lecture

ECE-656: Fall 2011Lecture 34a:Monte Carlo SimulationMark LundstromPurdue UniversityWest Lafayette, IN USADamoclesoutlinecarrier scatteringcarrier scatteringtotal scattering ratetotal scattering rateoutlinecarrier trajectories in 2Dcarrier trajectories in phase spaceMC algorithmoutlinefree flightfree flightfree flight timefree flight timefree flight timesfree flight distributionfree flightupdate at end of free-flightL32 outlinecollision