Lecture 14: The Boltzmann Transport EquationL... · Boltzmann Transport Equation (BTE) 2) neglected...
of 20
/20
Embed Size (px)
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
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
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
Lundstrom ECE-656 F11
8
semi-classical transport
Lundstrom ECE-656 F11
“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)
Lundstrom ECE-656 F11
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
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/
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)
Lundstrom ECE-656 F11
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
Lundstrom ECE-656 F11
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
Lundstrom ECE-656 F11
19
scattering operator
Lundstrom ECE-656 F11
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
steady-state BTE in 1D
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
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/
26
near eq., s.s BTE
Lundstrom ECE-656 F11
27
BTE solution
Lundstrom ECE-656 F11
28
BTE solution
Lundstrom ECE-656 F11
29
generalized force
“generalized force”
Lundstrom ECE-656 F11
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
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/
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
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/
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
