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Lecture 17: Hall Effect, Diffusion

Muhammad Ashraful Alamalam@purdue.edu

Alam  ECE‐606 S09 1

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Outline

1) Measurement of  mobility

 

3) Physics of  diffusion

u

REF: ADF,

 Chapter

 5,

 pp.

 190

‐202

Alam  ECE‐606 S09 2

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Problem of  mobility measurement …

V

 ρ = J E 

( )μ μ = +n pq n p J  E 

( )n p

q n p ρ 

μ μ =

+

1

-n D

q N μ …

1= -

Alam  ECE‐606 S09 3

  p A

q N μ …

?

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Outline

1) Measurement of 

 mobility

 3) Physics of  diffusion

Alam  ECE‐606 S09 4

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Set up for Hall Measurement

z

VH

B

w

-+

LI

UIC system: 4‐300K, 0‐1.5 T

Alam  ECE‐606 S09 5

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Mobility Measurement

Magnetic FieldI z

VHxy

Magneto‐electric effect

Alam  ECE‐606 S09 6

v   z   ….

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Drude Model

*

−− −m υ *

υ

Weak B field …

τ  0τ 

− − ≈qE υ

'τ −q E 

*

τ τ = − − ×υ B

m q qE 

*

τ τ τ 

⎛ ⎞≈ − − − ×⎜ ⎟

⎝ ⎠B

qq

mq

E E 

*m

2 2

* *1c

q B q Bτ τ τω = ≡ <<

 

….*

τ τ = − + ×Bqqm

E E 

Alam  ECE‐606 S09 7

m q mτ 

* *2

τ τ υ  = − + ×B

q q

m m

E 

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Drude Model & Hall Effect …

−n

B Fieldz2 2τ τ τ 

= −q n q n q

I VH

y

−

* * *m m m

0 0

0 0

0 0

σ σ μ 

σ μ σ −=⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦

 x

 y z

 x z

 y B J  E 

Alam  ECE‐606 S09 8

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Hall Resistance

0 0 x x J E  Bσ σ μ −⎡ ⎤ ⎡ ⎤⎡ ⎤B Fieldz

0 0 y y z J E  Bσ μ σ 

=

⎣ ⎦⎣ ⎦ ⎣ ⎦ VH

0

0 0

0 x x

 y y z

 J E 

 J E  B

σ 

σ μ σ 

⎡ ⎤≈⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦

0 0 x xE  J  σ  ⎡ ⎤⎡ ⎤⎡ ⎤

= ⎢ ⎥

 E B

0 0 y zσ μ σ 

between 0.5 and 

Alam  ECE‐606 S09 9

 y z

 H 

 x

 R

  J qn

= = −  

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Temperature‐dependent Concentration

Freezeout

Extrinsic 

Intrinsic 

 D

n

 N 

1

in

Temperature

 D

Alam  ECE‐606 S09 10

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Outline

1) Measurement of 

 mobility

 

3) Physics of  diffusion

Alam  ECE‐606 S09 11

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Now the Diffusion term…

1   N N N  n r gt q

∂ = ∇ • − +∂J

 I 

1∂

J μ = + ∇  N N N  qn E qD n

= − • − +∂

P P Pr gt q

P P Pqp E qD pμ = − ∇J

( )+ −∇ • = − + − D A D q p n N N  

Alam  ECE‐606 S09 12

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Diffusion Flux …

x

(0) (0)(0 ( )) 0+⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟

= − × + ×⎢ ⎥⎜ ⎟ ⎜ ⎟

+ − +dp

 p ldx

 p pdp

q q x l

 p

 J l

l

ld 

υ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦th

υ  d d l

Alam  ECE‐606 S09 132

= ≡q q

dx dx

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Flat Fermi‐level defines Equilibrium …

0 μ = = +n n n

dn  J qn qDE 

n2

n1

n

1 μ ⇒ = − n

n

 x

dn

n dx D

E 

F  E C  E  1C  E 

2C 

0

μ  μ −

∫ = =

 L

n n

n n

V 

 D  D

E 

V  E 

2

2 1

( ) / ( ) /  / 2

− −− −

C F 

C C 

  E E kT    E E kT   qV kT C 

 N en

L

1

1( ) / − −C  F  E  E  kT 

C  N n e

V V 

Alam  ECE‐606 S09 15

n n

n nkT D kT D= ⇒ =  

cn and en discussed in Chapter 5

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Conclusion

1) Measurement 

of  

mobility 

and 

carrier 

concentration 

is 

particularly important for analysis of  semiconductor 

devices. 

‐,  , 

the elemental processes in semiconductor device physics.

e w

 pu

 e p eces

 oge er

 n 

e nex

 c ass.

 

Alam  ECE‐606 S09 16