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Monte Carlo Simulation
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ECE-656: Fall 2011
Lecture 34a:
Monte Carlo Simulation
Mark Lundstrom Purdue University
West Lafayette, IN USA
1 11/16/11
Lundstrom ECE-656 F11 2
Damocles
SAT
Lundstrom ECE-656 F11 3
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
Lundstrom ECE-656 F11 4
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+ =
Lundstrom ECE-656 F11
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:
Lundstrom ECE-656 F11
Lundstrom ECE-656 F11 7
total scattering rate
http://www.research.ibm.com/DAMOCLES/html_files/sirates.html#history
typical scattering rates vs. energy
Lundstrom ECE-656 F11 8
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
Lundstrom ECE-656 F11 9
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
Lundstrom ECE-656 F11 11
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
Lundstrom ECE-656 F11 12
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
Lundstrom ECE-656 F11 13
free flight
( )E
E
0 01 =
Lundstrom ECE-656 F11 14
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
Lundstrom ECE-656 F11 17
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!
Lundstrom ECE-656 F11 19
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=
Lundstrom ECE-656 F11 20
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?
Lundstrom ECE-656 F11 21
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
Lundstrom ECE-656 F11
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