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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 J x = σ E x σ S dT L dx J x q = T L σ S E x κ 0 dT L dx E x = ρ J x + S dT L dx J x q = π J x κ e dT dx (diffusive transport) σ E ( ) = 2q 2 h λ E ( ) ME ( ) A f 0 E κ 0 = T L k B q 2 E E F k B T L 2 σ E ( ) dE σ = σ E ( ) dE π = T L S S = k B q E E F k B T L σ E ( ) dE σ E ( ) dE

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### Transcript of 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

Lundstrom ECE-656 F11

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

Lundstrom ECE-656 F11

5

outline

Lundstrom ECE-656 F11

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

6

quantum vs. semi-classical transport

Lundstrom ECE-656 F11

particle or wave?

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

7

semi-classical transport

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8

semi-classical transport

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“free flight” (followed by scattering)

particle

9

semi-classical transport

Lundstrom ECE-656 F11

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

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 F11

no effective mass!

11

exercise: equations of motion for m*(x)

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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

Lundstrom ECE-656 F11

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

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

Lundstrom ECE-656 F11

14

Boltzmann Transport Equation (BTE)

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15

Boltzmann Transport Equation (BTE)

∂f∂t

+υ•∇r f +

Fe •∇ p f = 0

Lundstrom ECE-656 F11

16

result

optical absorption, impact ionization, etc. and carrier scattering

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17

Boltzmann Transport Equation (BTE)

2) neglected generation-recombination

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

neglected scattering!

Lundstrom ECE-656 F11

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

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19

scattering operator

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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

Lundstrom ECE-656 F11

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

Lundstrom ECE-656 F11

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.”

Lundstrom ECE-656 F11

23

meaning of the RTA

Perturbations decay away exponentially with a characteristic time, τm

Lundstrom ECE-656 F11

Assume spatial uniformity, no E-field.

24

Lundstrom ECE-656 F11

RTA near-equilibrium

no B-fields for now

25

outline

Lundstrom ECE-656 F11

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

26

near eq., s.s BTE

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27

BTE solution

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28

BTE solution

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29

generalized force

“generalized force”

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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

Lundstrom ECE-656 F11

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

31

another look at the solution…

Lundstrom ECE-656 F11

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…

Lundstrom ECE-656 F11

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?

Lundstrom ECE-656 F11

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∑

Lundstrom ECE-656 F11

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

Lundstrom ECE-656 F11

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

recall lecture 4

36

outline

Lundstrom ECE-656 F11

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

37

summary

Lundstrom ECE-656 F11

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

Lundstrom ECE-656 F11

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

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