Principles of Semiconductor Devices-16

8/8/2019 Principles of Semiconductor Devices-16

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Lecture 16: Carrier Transport

Muhammad Ashraful [email protected]

Alam ECE‐606 S09 1



Outline

1) Overview

3) Physics of Mobility

5) Conclusion

REF: Advanced Device Fundmentals, Pages 175‐ 192




Current Flow Through Semiconductors

I I G V= ×

V q n Av= × × ×

Depends

on

chemical

composition,

cr stal structure tem erature do in etc.

arr er

Densityvelocity

Quantum Mechanics + Equilibrium Statistical Mechanics ⇒ Encapsulated into concepts of effective masses

Transport with scattering, non‐equilibrium Statistical Mechanics

and occupation

factors

(Ch.

1‐4)


⇒ Encapsulated into drift‐diffusion equation with

recombination‐generation (Ch. 5 & 6)



Non‐equilibrium Systems

Chapter 6Chapter

5

vs.

I

V

Alam ECE‐606 S094



Summary of Transport Equations …

( )+ −∇ • = − + − D A D q p n N N

I 1

N N N

nr g

t q= ∇ • − +

∂J

1∂= − ∇ • − +J

pr

μ = + N N N qn q n

∂t q

P P Pqp E qD pμ = − ∇J




Outline

1) Overview


5) Conclusion




Meaning of Effective Mass …

0m

2 2

2

0

( ) (

2

) ψ ψ − + =⎜ ⎟

⎝ ⎠

+

ecry xt sU U d

x x E

m dx

*m

n

2 2

*)( φ φ

⎛ ⎞− + =⎜ ⎟

ext U x

d E


0 xm



Drift by Electric field ….

n n=

* *( )υ υ

τ

= − −n n

n

d m mq

dt

E

*( ) 1

τ τ υ

−⎡ ⎤= − −⎢ ⎥n

t

nq

t eE x

n x x




Drift by Electric field ….

τ τ −⎡ ⎤− − n

t

nq

*

⎢ ⎥⎣ ⎦nm

x

x x x

*( , 1-2 ps)

τ = − → ∞n

n

t q

m

E

friction

μ ≡ nE

1( )υ E

n n J qnμ = E υ 2υ


(Theory valid once t > 1‐2 ps)



Mobility and Physics of Scattering Time

x x

x

*

τ

μ = n

n

q

m0m

*

n

2

*2π ∞

−


Fermi’s Golden rule … ψ ψ τ −

−∞

∼

n x x x x



Phonon and Ionized Impurity Scattering

Ionized impurity

0m3 2

~n

T τ

D

*

nm

,

more phonon scattering

3 2~n T τ −




Multiple Scattering Events

* Ionized impurity

n Phonon scattering

others ….

1 1 1 1

ph II n s

= + + +

τ τ τ τ

1 1 1

p II n hμ μ

= +

*1n

m

qμ τ =

II

n

II

ph

phμ

⇒ =+

ph II μ μ ⎛ ⎞

min min I h I p μ μ

−+⎝ ⎠

0μ ⎛ ⎞

= + ⎜ ⎟


Matthession Rule ….)01

I N N +⎝ ⎠



Model for Ionized impurity Scattering

*0,nμ ⎛ ⎞

=n

( )

,mi

0,

n

1

nn

I n

n

N N

α ⎜ ⎟+⎝ ⎠

,minnμ




Temperature‐dependent Mobility

3 2

~n n T μ τ

−∼




Outline

1) Overview


5) Conclusion




Mobility at High Fields?

0τ μ

υ = ≡ ⇒ N q

E E E *

1⎡ ⎤⎛ ⎞⎢ ⎥+ ⎜ ⎟⎢ ⎥⎝ ⎠

n N n

C

m

E

E

x xx x

( )υ E

What causes velocity

saturation at high fields?

Where does all the mobility formula


c



Velocity Saturation in Si/Ge

10 0 J J J −+= = − =E

2 1cJ J J J

−+= − >E E

3 2cJ J J J −+≈ = − >E E

−+

Alam ECE‐606 S0918

4 3c −



Velocity Overshoot & Inter‐valley Transfer

Larger m*

*


What type of scattering would you need for inter‐valley transfer?



Doping dependent Resistivity

V

ρ = J E

( )μ μ = +

n pq n p J E

( )n p

q n p ρ

μ μ =

+

1-

n Dq N μ …

1= -


p A

q N μ …



Conclusion

1) Poisson and drift‐diffusion equations form a

complete

semi‐

classical

transport

model

that

can

explain wide variety of device phenomena.

2) Drift current results from response of electrons/holes

. material dependent.

3) Constancy

of

low

‐field

mobility

can

be

checked

by

experiments.


Principles of Semiconductor Devices-16

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