Post on 23-Jan-2022
ECE-656: Fall 2011
Lecture 32:
Balance Equation Approach: II
Mark Lundstrom Purdue University
West Lafayette, IN USA
1 11/14/11
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general approach
∂f∂t
+υx
∂f∂x
−qE x
∂f∂kx
= C f ⇒∂nφ
∂t= −∇ •
Fφ + Gφ − Rφ
Rφ ≡nφ − nφ
0
τφ
Gφ = −q
E •
1Ld ∇ pφ
p( ) fp∑
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
Fφr ,t( ) ≡ 1
Ld φ p( ) υ f r , p,t( )p∑
nφ (r ,t) = 1
Ld φ( p) f r , p,t( )p∑
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0th moment of the BTE
In steady-state, the current is constant because we have assumed that there is no generation-recomination of electrons.
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1st moment of the BTE
Fφ ≡
1Ω
φ p( ) υ f r , p,t( )p∑ =
1Ω
pz
υ f r , p,t( )
p∑
nφ (r ,t) = Pz (
r ,t) = n pz
Fφi =
1Ω
pzυ i f r , p,t( )p∑ ≡ 2Wzi
∇ •Fφ =
∂∂xi
2Wzi( )
Rφ =
Pz
τm Gφ = −q( )nE z
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1st moment of the BTE momentum balance equation
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Wij
Wij =
Wxx Wxy Wxz
Wyx Wyy Wyz
Wzx Wzy Wzz
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
Wij =W3
1 0 00 1 00 0 1
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=
W3δ ij
Wijr ,t( ) = 1
Ωpiυ j
p∑ f r , p,t( )
W =
12
m* υ 2 2=
12
m* υx2 + υ y
2 + υ z2( ) kinetic energy density
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drift-diffusion equation
assume:
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DD equation
W ≈ n 3
2kBTe
assume:
Te = constant
Dn =
kBTe
qµn
9
2nd moment of the BTE
Fφ =
1Ω
E p( ) υ f r , p,t( )p∑ =
FW
Gφ =Jn •
E
Rφ ≡nφ − nφ
0
τφ
=W −W0
τ E
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recap
0th moment
1st moment
2nd moment
τ FW
∂FWr ,t( )
∂t+FW = −µEW
E - 2µE
W •
E − τ FW
∇ •X 3rd moment
Lundstrom ECE-656 F11 11
outline
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/
(Reference: Chapter 5, Lundstrom, FCT)
1) Review of L31 2) Carrier temperature and heat flux 3) Heterostructures 4) Summary
Lundstrom ECE-656 F11 12
electron temperature • Electrons gain energy from the electric field.
• Electrons lose energy to the lattice by inelastic scattering.
• If the electric field is high, electrons will gain energy faster than they lose it to the lattice.
• Under such conditions, the electron energy > lattice energy.
• If we measure the electron energy by a “temperature,” then the electrons are hot.
13
electron temperature
random thermal motion of electrons
c ≡ 0
W = n 1
2m* υ 2
υ =
υd +
c
υ2 = υd
2 + 2ciυd + c2
W = n 1
2m*υd
2 +12
m* c2⎛⎝⎜
⎞⎠⎟
W =
1Ω
E( p)p∑ f =
1Ω
12
m*υ 2
p∑ f
14
electron temperature
W = n 1
2m*υd
2 +12
m* c2⎛⎝⎜
⎞⎠⎟
32
kBTe ≡12
m* c2
f (k) ~ e−2 k2 2m*kBTL
f (k) ~ e−2 k− kd( )2 2m*kBTL
f (k) ~ e−2 k− kd( )2 2m*kBTe
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example
(V/cm)
Wn= n 1
2m*υd
2 +32
kBTe
⎛⎝⎜
⎞⎠⎟
drift energy + thermal energy
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heat flux
FWx =
1Ω
E( p)υx fp∑
FWx =
1Ω
12
m*υ 2⎛⎝⎜
⎞⎠⎟υx f
p∑
υx = υdx + cx
υd = υdx x
FWx =
12
m*n υ 2υx
FWx =
12
m*n υ 2 υdx + cx( )
FWx =
12
m*n υ 2 υdx +12
m*n υ 2cx
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heat flux
υx = υdx + cx
FWx =
12
m*n υ 2 υdx +12
m*n υ 2cx
FWx =Wυdx +
12
m*n υdx2 + 2
υd ic + c2( )cx
FWx =Wυdx +
12
m*n υdx2 cx + 2 cx
2 υdx + c2cx⎡⎣
⎤⎦
FW =Wυdx + n m* cx
2 υdx + n 12
m* c2cx
υ =υd +
c
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heat flux
υ = υdx x + c
FWx =Wυdx + nm* cx
2 υdx + n 12
m* c2cx
kBTe ≡ m* cx
2
Qx ≡ n 1
2m* c2cx
FWx =Wυdx + nkBTeυdx + Qx
PΩ = NkBTe
“heat flux”
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balance equations
∂n∂t
= −d Jnx (−q)⎡⎣ ⎤⎦
dxelectron continuity equation
Jnx = nqµnE x +
23µn
dWdx
drift-diffusion equation
∂W x,t( )∂t
= −dFWx
dx+ JnxE x −
W −W0( )τ E
energy-balance equation
energy-flux equation FWx =Wυdx + nkBTeυdx + Qx
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heat flux
FWx =Wυdx + nkBTeυdx + Qx
Qx ≈ −κ e
dTe
dx
but more generally
Jx
q = π Jn x −κ e
dTe
dx
Qx ≡ n 1
2m* c2cx
heat flux of a Maxwellian (or displaced Maxwellian) is 0
Lundstrom ECE-656 F11 21
Q and Jq
does Qx = Jxq ?
Jxq = π Jx > 0
N-type semiconductor
Qx < 0
Mark A. Stettler, Muhammad A. Alam, and Mark S. Lundstrom, “A Critical Examination of the Assumptions Underlying Macroscopic Transport Equations for Silicon Devices,” IEEE Trans. on Electron Devices, 40, 733, 1993.
electrons
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outline
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/
(Reference: Chapter 5, Lundstrom, FCT)
1) Review of L31 2) Carrier temperature and heat flux 3) Heterostructures 4) Summary
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review: pn homojunctions
potential must decrease
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review: pn homojunctions
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reference for the energy bands field-free vacuum level
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local vacuum level
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Al0.3Ga0.7As : GaAs (Type I HJ) field-free vacuum level
EG ≈ 1.42 eV
“electron affinity rule”
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N-Al0.3Ga0.7As : p+-GaAs (Type I HJ) field-free vacuum level
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N-Al0.3Ga0.7As : p+-GaAs (Type I HJ)
“band spike”
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general, graded heterostructure
31
“quasi-electric fields”
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BTE
(constant effective mass)
These equations do not hold when the effective mass is position dependent. Lundstrom, FCT, Sec. 5.8.
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alternative approach: hole current
dEV (x)dx
=ddx
E0 − χ(x) − qV (x) − EG (x)⎡⎣ ⎤⎦ = q E (x) +E QP( )
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hole and electron currents
quasi-electric fields
“DOS effect”
35
outline
Lundstrom ECE-656 F11
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/
(Reference: Chapter 5, Lundstrom, FCT)
1) Review of L31 2) Carrier temperature and heat flux 3) Heterostructures 4) Summary
Lundstrom ECE-656 F11 36
summary
1) The four balance equations can be reduced to two continuity equations and two constitutive relations.
2) We can write them as two equations in two unknowns.
3) The unknowns are n and W or n and Te.
Lundstrom ECE-656 F11 37
the simplified equations
∂W∂t
= −∇ •FW +
Jn •
E −W −W0( )τ E
∂n∂t
=1q∇ •Jn
Jn = nqµn
E +
23µn∇W
cont. eqn. for electrons
current equation
continuity eqn. for energy
FW = ? current eqn. for energy
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energy current equation
FW = −
23µEW
E − τ FW
∇ •X
Xij ≈
53
kBTe
qµEWδ ij
First approach:
Second approach:
FW =W
υd + nkBTe
υd +
Q
Q ≈ −κ e∇Te
Lundstrom ECE-656 F11 39
questions?
1) Review of L31 2) Carrier temperature and heat flux 3) Heterostructures 4) Summary