2011.11.18-ece656-l34b

ECE-656: Fall 2011

Lecture 34b:

Monte Carlo Simulation: II

Mark Lundstrom Purdue University

West Lafayette, IN USA

1 11/18/11

Lundstrom ECE-656 F11 2

Damocles

SAT


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


free flight

( )E

E

0 01Ct =

Given this scattering rate, how long does a carrier travel before it scatters for the first time?

( )10

1 lnCt r=


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=


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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8

collision As an example, assume that there are 3 scattering mechanisms, plus self-scattering.

1 2 3 0( ) ( ) ( ) ( ) ( )SELFE E E E E = + + + =

( )E

E

0

1( )E

1 2( ) ( )E E +

1 2 3( ) ( ) ( ) ( )selfE E E E + + +

1 2 3( ) ( ) ( )E E E + +

( )CE tLundstrom ECE-656 F11

9

collision

1 2 3 0( ) ( ) ( ) ( ) ( )SELFE E E E E = + + + =

0

( )1 CE t

1 2 +

1 2 3 + +

0 1 2 3 SELF = + + +

( )1 0CE t

( )1 2 0 +

( )1 2 3 0 + +

0

1.0

choose random number between 0 and 1

10

collision

1 2 3 0( ) ( ) ( ) ( ) ( )SELFE E E E E = + + + =

0

( )1 0CE t

( )1 2 0 +

( )1 2 3 0 + +

1.0

choose random number uniformly distributed between 0 and 1

2 1 00 r< < process 1

( )1 0 2 1 2 0r < < + process 2

( ) ( )1 2 0 2 1 2 3 0r + < < + + process 3

( )1 2 3 0 2 1.0r + + < < self-scattering


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update after scattering

( ) ?Cp t+ =

( ) ( )C C iE t E t E+ = +

( ) ( )C Cr t r t+ =

0t =

Ct t=

We know the magnitude of p from the known energy after the collision, but how do we determine the direction?

Ct t+=

00, ,etciE =


update after scattering: 1D

( )Cp t

( )Cp t+i) forward scattering by phonon absorption

ii) backward scattering by phonon absorption

( )Cp t+

If scattering is isotropic, then forward and backward scattering are equally probable.

( ) ( )30 0.5 : C Cr p t p t+ < =

( ) ( )20.5 1.0 : C Cr p t p t+ < =



( )Ctp

assume isotropic scattering (e.g. optical phonon abs)

( )Ct+p

32 r =

r3 is a random number uniformly distributed between 0 and 1

x

y



( )Cp t( )Cp t+

y

z

( )Cp t+

x

( )Cp t

rx

ry

rz

need two random numbers to select ( ) ( ), ,



( )Cp t( )Cp t+

rx

ry

rz

32 r =

r3 is a random number uniformly distributed between 0 and 1



( )Cp t( )Cp t+

rx

ry

rz( )

( )2

2

0 0 0

1 ,

, sin

ipi

S p p

C d S p p d p dp

= =

=

( )

( )

22

0 02

2

0 0 0

, sin( )

, sin

C d S p p d p dpd

C d S p p d p dp

=

P



( )Cp t( )Cp t+

rx

ry

rz ( )

( )

2

0

2

0 0

, sin( )

, sin

S p p d p dpd

S p p d p dp

=

P

example: isotropic scattering

0

0

sin sin( )cossin

d ddd

= =

P



( )Cp t( )Cp t+

rx

ry

rz example: isotropic scattering

sin( ) ( )2

dd r dr = =P P

4

400 0

sin cos2 2

r ddr r

= = =

42 1 cosr =

4cos 1 2r =


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21

MC simulation of a bulk semiconductor

E

initialize ( )0p t =

1) select: 10

1 lnCt r=

2) update: ( ) ( ) ( ), ,C C Cr t p t E t

3) identify scattering event: 2r

4) update: ( ) ( ), , ,C CE t p t + + 3 4,r r

5) repeat

Lundstrom ECE-656 F11


MC simulation of a device

1) Load up device with N super electrons

TOT DQ qALN+=

example: 1 m 1 mA = 10 mL = 18 -310 cmDN =

( )710TOTQ q=

( )electronssupersim

NQ qN

=


23

MC simulation of a device

1) load up device with N super electrons

2) track all electron trajectories for T (replace electrons that leave the device

3) collect statistics

4) solve Poisson equation for new self-consistent potential

5) repeat until steady-state


Monte Carlo simulation and the BTE

r p

f f q f Cft

+ =

E a six-dimensional, time-dependent problem in 3D

curse of dimensionality

One can show (see Lundstrom, Sec. 6.8) that Monte Carlo simulation provides a solution to the BTE -

Except when short range e-e scattering is included in which case it goes beyond the BTE by treating particle-particle correlations.


short-range e-e scattering

1) long range forces due to

the average charge in the device (from Poissons equation)

2) short-range (Coulomb) forces

1 22

124q qF

r=


near-equilibrium

x0

( )CE x

( )VE x

CE

VE

IEFE

184 10bi BqV k Te

also: delicate balance between drift and diffusion


27

Damocles


L32 outline





summary

1) Monte Carlo simulation is a technique to solve the BTE (can go beyond by treating correlations).

2) Details of bandstructure and scattering are readily included.

3) But rare events can be hard to treat.

4) When applicable, MC simulation typically provides the most accurate solutions of the BTE.

5) For a description of a state-of-the-art MC simulator:

http://www.research.ibm.com/DAMOCLES/home.html


questions

ECE-656: Fall 2011Lecture 34b:Monte Carlo Simulation: IIMark LundstromPurdue UniversityWest Lafayette, IN USADamoclesoutlinefree flightfree flightupdate at end of free-flightL32 outlinecollisioncollisioncollisionL32 outlineupdate after scatteringupdate after scattering: 1Dupdate after scattering: 2Dupdate after scattering: 3Dupdate after scattering: 3Dupdate after scattering: 3Dupdate after scattering: 3Dupdate after scattering: 3DL32 outlineMC simulation of a bulk semiconductorMC simulation of a deviceMC simulation of a deviceMonte Carlo simulation and the BTEshort-range e-e scatteringnear-equilibriumDamoclesL32 outlinesummaryquestions

2011.11.18-ece656-l34b

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