Lundstrom ECE-656 F15
ECE 656: Electrothermal Transport in Semiconductors Fall 2015
Boltzmann Transport Equation
Professor Mark Lundstrom
Electrical and Computer Engineering Purdue University
West Lafayette, IN USA
11/12/2015
2
f(r, k, t)
xk
x
( ), ,xf x k t
f0 x,kx( ) = 1
1+ e E−EF( ) kBT
Lundstrom ECE-656 F15
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distribution function
f0 x,kx( ) = 1
1+ e E−EF( ) kBT
Lundstrom ECE-656 F15
What is the probability that a state is occupied?
Answer: in equilibrium:
ECE 606 Answer: f x,kx ,t( ) = 1
1+ e E−Fn t( )( ) kBT
Generally: f x,kx ,t( )
4
from the distribution function
n x,t( ) = 1
Ωf x,!k ,t( )!
k∑
Lundstrom ECE-656 F15
Electron density:
Kinetic energy per electron: f x,kx ,t( ) = 1
1+ e E−Fn t( )( ) kBT
etc…. f x,kx ,t( )
Electron current density: Jnx x,t( ) = 1
Ω−q( )υx f x,
!k ,t( )!
k∑
5
goals
1) Find an equation for f(r, p, t) out of equilibrium
2) Learn how to solve it near equilibrium
3) Relate the results to our Landauer approach results – in the diffusive limit
4) Add a B-field and show how transport changes
Lundstrom ECE-656 F15
6
semi-classical transport
Lundstrom ECE-656 F15 x
k
( )E k
E
x
( )CE x
( )E k
k0k
“free flight” (followed by scattering)
0( )E k particle 1( )E k
1k
1 0k k>
d kx( )dt
= Fe = −dEC (x)
dx
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derivation
Lundstrom ECE-656 F15
d kx( )dt
= Fe = −dEC (x)
dx
E = EC x( ) + E k( )
dEdt
= 0
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“semi-classical transport”
d k( )
dt= −∇r EC (r ) = −q
E (r )
dpdt
=Fe
υg (t) = 1
∇k E
k t( )⎡⎣ ⎤⎦
r t( ) = r 0( ) + υg
0
t
∫ ( ′t )d ′t
k t( ) = k 0( ) + −q
E ( ′t )
0
t
∫ d ′t equations of motion for “semi-classical transport”
EC varies slowly on the scale of the electron’s wavelength.
Lundstrom ECE-656 F15
no effective mass!
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trajectories in phase space
px = kx
x
[ ]( ) ( ), ( )xt x t p tΤ =
υx (t) =dE
d kx( ) k (t )
x t( ) = x 0( ) + υx0
t
∫ ( ′t )d ′t kx t( ) = kx 0( ) + −qE x ( ′t )
0
t
∫ d ′t
Lundstrom ECE-656 F15
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Boltzmann Transport Equation (BTE) px = kx
x
[ ]( ) ( ), ( )xt x t p tΤ =
( ), ,xf x p t
( ), ,x x ef x dt p F dt t dtυ− − −
( ) ( ), , , ,x x x ef x p t f x dt p F dt t dtυ= − − −
0dfdt
=Lundstrom ECE-656 F15
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Boltzmann Transport Equation (BTE)
Fe = −q
E − q
υ ×B
ˆ ˆ ˆrf f ff x y zx y z∂ ∂ ∂∇ = + +∂ ∂ ∂
ˆ ˆ ˆp x y zx y z
f f ff p p pp p p∂ ∂ ∂∇ = + +∂ ∂ ∂
p =
k
∂ f∂t
+υ•∇r f +
Fe •∇ p f = 0
Lundstrom ECE-656 F15
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in and out-scattering
dfdt coll
= Cf = in-scattering - out-scattering
Lundstrom ECE-656 F15
∂ f∂t
+υ•∇r f +
Fe •∇ p f = Cf