Lecture 14: The Boltzmann Transport EquationL... · Boltzmann Transport Equation (BTE) 2) neglected...

ECE-656: Fall 2011

Lecture 14: The Boltzmann Transport

Equation

Mark Lundstrom Purdue University

West Lafayette, IN USA

1 9/28/11

2

coupled current equations

Lundstrom ECE-656 F11

Jx = σE x − σS dTL dx

Jxq = TLσSE x −κ 0 dTL dx

E x = ρJx + S

dTL

dx

Jx

q = π Jx −κ e

dTdx

(diffusive transport)

′σ E( ) = 2q2

hλ E( ) M E( )

A−∂f0

∂E⎛

⎝⎜⎞

⎠⎟

κ 0 = TL

kB

q⎛

⎝⎜⎞

⎠⎟

2E − EF

kBTL

⎛

⎝⎜⎞

⎠⎟

2

′σ E( )∫ dE

σ = ′σ E( )∫ dE

π = TLS S = −

kB

qE − EF

kBTL

⎛

⎝⎜⎞

⎠⎟′σ E( )∫ dE ′σ E( )∫ dE

3

f(r, k, t)

f0 x,kx( ) = 1

1+ e E−EF( ) kBTL


Jnr( ) = 1

A−q( ) υ k( ) f r ,

k( )

k∑

4

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


5

outline


1) Introduction 2) Equation of motion 3) The BTE 4) Solving the s.s. BTE 5) Discussion 6) Summary

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/

6

quantum vs. semi-classical transport


particle or wave?

particle when EC(x) varies slowly on the scale of the electron’s wavelength

7

semi-classical transport


8



“free flight” (followed by scattering)

particle

9



ETOT = EC (x) + E(k)

dETOT x,k( )dt

= 0 =dEC (x)

dxdxdt

+dE(k)

dkx

dkx

dt

0 =

dEC (x)dx

υx +1

dEdkx

d kx( )dt

0 =

dEC (x)dx

υx +υx

d kx( )dt

d kx( )dt

= Fe = −dEC (x)

dx

10


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.


no effective mass!

11

exercise: equations of motion for m*(x)


E k, r( ) ≈

2k 2

2m*(r )

i) assume:

ii) assume that m* varies slowly with position

iii) derive the equation of motion in k-space

12

outline




13

trajectories in phase space

υx (t) = dEd kx( )

k (t ) x t( ) = x 0( ) + υx

0

t

∫ ( ′t )d ′t kx t( ) = kx 0( ) + −qE x ( ′t )

0

t

∫ d ′t


14

Boltzmann Transport Equation (BTE)


15


∂f∂t

+υ•∇r f +

Fe •∇ p f = 0


16

result

optical absorption, impact ionization, etc. and carrier scattering


17


2) neglected generation-recombination

3) neglected e-e correlations (mean-field-approximation)

neglected scattering!


assumptions: 1) semi-classical treatment of electrons in a crystal with E(k)

18

in and out-scattering “in-scattering”

“out-scattering”

position, x, does not change


19

scattering operator


in-scattering rate = S ′p → p( ) f ′p( ) 1− f p( )⎡⎣ ⎤⎦

′p∑

out-scattering rate = S p → ′p( ) f p( ) 1− f ′p( )⎡⎣ ⎤⎦

′p∑

Cf r , p,t( ) = S ′p → p( ) f ′p( ) 1− f p( )⎡⎣ ⎤⎦

′p∑ − S p → ′p( ) f p( ) 1− f ′p( )⎡⎣ ⎤⎦

′p∑

Cf r , p,t( ) = S ′p → p( ) f ′p( )

′p∑ − S p → ′p( ) f p( )

′p∑

non-degenerate scattering operator (assumes final state empty)

20

nondegenerate scattering operator


Cf r , p,t( ) = S ′p → p( ) f ′p( ) 1− f p( )⎡⎣ ⎤⎦

′p∑ − S p → ′p( ) f p( ) 1− f ′p( )⎡⎣ ⎤⎦

′p∑

probability that the state at p’ is

occupied

probability that the state at p is empty

21

conservation of carriers


We are discussing scattering mechanisms that move carriers around in k-space. They do not create or destroy carriers.

(interchange order of summation)

(interchange labels of dummy indices)

22

Relaxation Time Approximation (RTA)

in-scattering – out-scattering

See Lundstrom: pp. 139-141. The RTA can be justified when the scattering is isotropic and/or elastic in which case the proper time to use is the “momentum relaxation time.”


23

meaning of the RTA

Perturbations decay away exponentially with a characteristic time, τm


Assume spatial uniformity, no E-field.

24

steady-state BTE in 1D


RTA near-equilibrium

no B-fields for now

25

outline




26

near eq., s.s BTE


27

BTE solution


28

BTE solution


29

generalized force

“generalized force”


The two forces driving current flow are gradients in QFL and gradients in (inverse) temperature. In Lecture 4, we saw that (f1 – f2) produces current flow and that differences in Fermi level and temperature cause differences in f.

30

outline




31

another look at the solution…


δ f = τm −

∂f0

∂E⎛

⎝⎜⎞

⎠⎟υ •

F →τm −∂f0

∂E⎛

⎝⎜⎞

⎠⎟υxFx

F = −∇r Fn + TL EC + E k( ) − Fn

⎡⎣ ⎤⎦∇r

1TL

⎛

⎝⎜⎞

⎠⎟→ −

dFn

dx= −qE x

δ f = qτmE x

∂f0

∂E⎛

⎝⎜⎞

⎠⎟υx

δ f = qτmE x

∂f0

∂px

⎛

⎝⎜⎞

⎠⎟∂px

∂E⎛

⎝⎜⎞

⎠⎟υx = qτmE x

∂f0

∂px

⎛

⎝⎜⎞

⎠⎟

32

another look at the solution…


f = f0 + δ f = f0 +

∂f0

∂px

⎛

⎝⎜⎞

⎠⎟qτmE x

δ f =

∂f0

∂px

⎛

⎝⎜⎞

⎠⎟qτmE x

Recall:

dpx = qτmE x

So the distribution has been displaced by pd is a direction opposite to the electric field

“displaced Maxwellian”

δ px = −qτ 0E x τm = τ 0

33

now what?


We have solved the BTE, now what do we do with the solution?

34

moments

n r( ) = 1

Ωf0

r ,k( )

k∑ + δ f r ,

k( ) ≈ 1

Ωf0

r ,k( )

k∑

Jnr( ) = 1

A−q( ) υ k( )δ f r ,

k( )

k∑

JQr( ) = 1

AE k( ) − Fn( ) υ k( )δ f r ,

k( )

k∑


JWr( ) = 1

AEk( ) υ k( )δ f r ,

k( )

k∑

To evaluate these quantities, we need to work out sums in k-space.

35

moments


To evaluate these quantities, we need to work out sums in k-space.

recall lecture 4

36

outline




37

summary


1)  Semi-classical transport assumes a bulk bandstructure with a slowly varying applied potential.

2)  Semiclassical transport ignores quantum reflections and assumes that position and momentum can both be precisely specified.

3) The Boltzmann Transport Equation can be solved to find the probability that states in the device are occupied.

4)  In equilibrium, the solution to the BTE is the Fermi function.

38

summary


BTE:

RTA:

Solution:

∂f∂t

+υ•∇r f +

Fe •∇ p f = C f

Cf r , p,t( ) = S ′p , p( ) f ′p( ) 1− f p( )⎡⎣ ⎤⎦

′p∑ − S p, ′p( ) f p( ) 1− f ′p( )⎡⎣ ⎤⎦

′p∑

Cf = − f p( ) − f0

p( )( ) τm

F = −∇r Fn + TL EC + E k( ) − Fn

⎡⎣ ⎤⎦∇r

1TL

⎛

⎝⎜⎞

⎠⎟ δ f = τm −

∂f0

∂E⎛

⎝⎜⎞

⎠⎟υ •

F

Lundstrom ECE-656 F11 39

questions


Lecture 14: The Boltzmann Transport EquationL... · Boltzmann Transport Equation (BTE) 2) neglected...

Documents

Transcript of Lecture 14: The Boltzmann Transport EquationL... · Boltzmann Transport Equation (BTE) 2) neglected...