Lecture 32: Balance Equation Approach: II

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ECE-656: Fall 2011 Lecture 32: Balance Equation Approach: II Mark Lundstrom Purdue University West Lafayette, IN USA 1 11/14/11 Lundstrom ECE-656 F11 Lundstrom ECE-656 F11 2 general approach f t + υ x f x q E x f k x = ˆ Cf n φ t = −∇ F φ + G φ R φ R φ n φ n φ 0 τ φ G φ = q E 1 L d p φ p ( ) f p F φ r , t ( ) 1 L d φ p ( ) υ f r , p, t ( ) p n φ ( r , t ) = 1 L d φ ( p) f r , p, t ( ) p

Transcript of Lecture 32: Balance Equation Approach: II

Page 1: Lecture 32: Balance Equation Approach: II

ECE-656: Fall 2011

Lecture 32:

Balance Equation Approach: II

Mark Lundstrom Purdue University

West Lafayette, IN USA

1 11/14/11

Lundstrom ECE-656 F11

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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∑

Page 2: Lecture 32: Balance Equation Approach: II

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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 ≡

φ p( ) υ f r , p,t( )p∑ =

pz

υ f r , p,t( )

p∑

nφ (r ,t) = Pz (

r ,t) = n pz

Fφi =

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

Page 4: Lecture 32: Balance Equation Approach: II

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drift-diffusion equation

assume:

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DD equation

W ≈ n 3

2kBTe

assume:

Te = constant

Dn =

kBTe

qµn

Page 5: Lecture 32: Balance Equation Approach: II

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2nd moment of the BTE

Fφ =

E p( ) υ f r , p,t( )p∑ =

FW

Gφ =Jn •

E

Rφ ≡nφ − nφ

0

τφ

=W −W0

τ E

Lundstrom ECE-656 F11 10

recap

0th moment

1st moment

2nd moment

τ FW

∂FWr ,t( )

∂t+FW = −µEW

E - 2µE

W •

E − τ FW

∇ •X 3rd moment

Page 6: Lecture 32: Balance Equation Approach: II

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

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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.

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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 =

E( p)p∑ f =

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

Page 8: Lecture 32: Balance Equation Approach: II

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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 =

E( p)υx fp∑

FWx =

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”

Page 10: Lecture 32: Balance Equation Approach: II

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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

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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

Lundstrom ECE-656 F11 22

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

Page 12: Lecture 32: Balance Equation Approach: II

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review: pn homojunctions

potential must decrease

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review: pn homojunctions

Page 13: Lecture 32: Balance Equation Approach: II

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reference for the energy bands field-free vacuum level

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local vacuum level

Page 14: Lecture 32: Balance Equation Approach: II

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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

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“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”

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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

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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.

Page 19: Lecture 32: Balance Equation Approach: II

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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

Lundstrom ECE-656 F11 38

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

Page 20: Lecture 32: Balance Equation Approach: II

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questions?

1) Review of L31 2) Carrier temperature and heat flux 3) Heterostructures 4) Summary