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

3

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

7

derivation

Lundstrom ECE-656 F15

d kx( )dt

= Fe = −dEC (x)

dx

E = EC x( ) + E k( )

dEdt

= 0

8

“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!

9

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

10

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

11

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

12

in and out-scattering

dfdt coll

= Cf = in-scattering - out-scattering

Lundstrom ECE-656 F15

∂ f∂t

+υ•∇r f +

Fe •∇ p f = Cf