The review of modern physics has given us a description of nature. Waves are described with a wave equation. Particles are described with particle equations.

The review of modern physics has given us a description of nature.

• Waves are described with a wave equation.• Particles are described with particle equations.• Experiments indicated that particle phenomena

could be explained with a wave theory.• Experiments indicated that wave phenomena

could be described with particle theory.• Schrodinger equation evolved whose solutions

yielded a probability density of finding a state.• Fermi function determined whether the state was

filled.

Materials could now be described.

• Materials were understood in terms of an energy band diagram.

• Conduction band – valence band – Fermi energy.

• Electrons being promoted from the valence band into the conduction band left a vacancy (“hole”) in the valence band.

• “n type” semiconductor and “p type” semiconductor.

Basic model

p n

aN

dNhole diffusion electron diffusion

Homogeneous doping model

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

Homogeneous doping model

p n+-

space charge region -- uniform distribution

negative chargeaN positive chargedN

Electrons diffuse into the p region and holes diffuse into the n region. This creates an electric field.

px nx

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

Homogeneous doping model

p n

space charge region -- uniform distribution

negative chargeaN positive chargedN

px nx

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

Forces on electrons and holes

p n

negative chargeaN positive chargedN

px nxdiffusion forceon electrons

diffusion forceon holes E field force

on holes

E field forceon electrons

Basic model – thermal equilibrium

FE

- - - - - -

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- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

p n+-

cE

cE

vE

vE

FiEbi biE eV

FiEbi biE eV

bi biE eV

FneVFpeV

= is a "built in" potentialbi Fp FnV V V

electrons and holes face a barrier

FE

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

p n+-

cE

cE

vE

vE

FiEbi biE eV

FiE

+

-

Electron density in the conduction band of the n type material

- - - - - -

- - - - - -

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- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

p n+-

cE

vE

FiE

c F0 c

B

E En N exp

T

or F Fi

0 iB

E En n exp

T

Hole density in the p type semiconductor material

- - - - - -

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- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

p n+-

cE

vE

FiE

Fi F0 i

B

E Ep n exp

T

F v0 v

B

E Ep N exp

T

Barrier potential can now be defined

a dBbi 2

i

N NTV ln

e n

=bi Fp FnV V V

F Fi0 i

B

E En n exp

T

Fni

B

eVn exp

T

dBFn

i

NTV ln

e n

aBFp

i

NTV ln

e n

d 0N n

a 0N p

Potential distribution in the depletion region

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- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

p n+-

0( )

0

a p

d n

eN x xx

eN x x

1( )

a a

s s

eN eNE x dx x A

0E 10

ap

s

eNx A

( )

a

s

eNdE x

dx

0( )

0

a p

d n

eN x xx

eN x x

2( )

d d

s s

eN eNE x dx x A

0E 20

dn

s

eNx A

( )

d

s

eNdE x

dx

0

( )

0

ap p

s

dn n

s

eNx x x x

E xeN

x x x x

a dp n

s s

eN eNx x

a p d nN x N x

Epx nx

0

( )

0

ap p

s

dn n

s

eNx x x x

E xeN

x x x x

1( ) ( ) V x E x dx B

Epx nx

1( )

a

ps

eNV x x x dx B

a

ps

eNx x

V = 0

2

102

pa

s

xeNB

0

( )

0

ap p

s

dn n

s

eNx x x x

E xeN

x x x x

2( ) ( ) V x E x dx B

Epx nx

2( )

d

ns

eNV x x x dx B

Voltage is continuous

2

22

02

( )

02 2

ap p

s

d an p n

s s

eNx x x x

V xeN eNx

x x x x x

2d nN x

2 2

2bi d n a ps

eV N x N x

22( )

2 2d a

n ps s

eN eNxV x x x x

0 nx x

a p d nN x N x2 1s bi a

nd d a

V Nx

e N N N

2 1s bi d

pa d a

V Nx

e N N N

2 1s bi an

d d a

V Nx

e N N N

2 1s bi d

pa d a

V Nx

e N N N

2 s bi d a

d a

V N NW

e N N

( ) tanh( )V x x

( ) tanh( )V x x

( )( )

dV xE x

dx

2

2

( )( )

d V xx

dx

Reverse biased PN junction- - - - - -

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- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

p n

bias RV Vtotal bi RV V V

WW

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

p n

2 s bi R a d

a d

V V N NW

e N N

bi Rmax

2 V VE

W

Reverse biased PN junctionenergy diagram

cE

cEvE

vE

FiE

FiE

FnE

FpE

ReV

total bi ReV e V V

Voltage-dependent capacitor

px

nx

with RV

with R RV dV

Voltage-dependent capacitor

px

nx

with RV

with R RV dV

dQ

dQ

differential R

dQC'

dV

nd

R R

dxdQC' eN

dV dV

deN

aeN

Simple three-dimensional unit cell

a

b

c