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Page 1: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Chapter 5

Junctions

Page 2: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

5.1 Introduction (chapter 3)5.2 Equilibrium condition

5.2.1 Contact potential5.2.2 Equilibrium Fermi level5.2.3 Space charge at a junction

5.3 Forward bias5.3.1

Page 3: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

(negative) Photoresist

Silicon Oxide

Silicon

Mask/Shield/Pattern

Irradiation

Metal

Oxide

Lift off

Develop

Page 4: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Fermi Gas and Density of State

m

pmvE

22

1 22

m

k

m

pmvE F

F 222

1 2222

kp

2/h

h

p

2

kFE

Fk

Page 5: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Particle in a Infinite Well

n

L2

L

nhhp

2

2

2222

822

1

mL

hn

m

pmvEn

2

223

22

21

8

)(

mL

hnnnE

For three-dimensional box

Page 6: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Electron Energy Density

2

223

22

21

8

)(

mL

hnnnE

xn

yn

zn

knjninn zyx

Page 7: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Density of State ρ(E)

xn

yn

zn

knjninn zyx

2

223

22

21

8

)(

mL

hnnnE

Page 8: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Properties Dependent on Density of States

3/)( 22 TkEDCheatSpecific BFel

)(2FBel EDlitySusceptibi

Experiment provide information on density of state

spectrumEDSyspectoscopionPhotoemiss

effectSeebeckorsemionductinionconcentratCarrier

constantdielectricofiondeterminatabsorptionOpticalNMRintermcontactFermi

effectAlphenvanHaasde gapenergyctingSupercondu

ctorssuperconduintunnelingjunctionJosephson

Page 9: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 10: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 11: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 12: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 13: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

)(EN )(Ef )()( EFEN 0 0.5 1

cE

vE

cE

vE

N(E)[1-f(E)]

N(E)f(E)

(a) Intrinsic

FE

N(E): Density of state f(E): Probability of occupation (Fermi-Dirac distribution function)

)(EN )(Ef ionconcentratCarrier0 0.5 1

cE

vE

cE

vE

Holes(a) Intrinsic

FE

N= N(E)dE: Total number of states per unit volume N= N(E)f(E)dE: Concentration of electrons in the conduction band

Page 14: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

)(EN )(Ef ionconcentratCarrier0 0.5 1

cE

vE

cE

vE

(c) p-type

FE

cE

vE

cE

vE

(b) n-type

FE

cE

vE

cE

vE

Holes

Electrons

(a) Intrinsic

FE

Page 15: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

212322

)2

(2

1)()( // E

mENE

This density of state equation is derived from assumption of electron in the infinite well with vacuum medium, where the E is proportional to k2.

FE

Fk

m

kE F

F 2

22

FE

FkgE

We found that the free electron in the conduction band of semiconductor has local minimum of energy E versus wave number k. We can approximate the bottom portion of the curve as if E is still proportional to k2 and write down the similar energy-wave number equation as

*n

FF m

kE

2

22

to describe the behavior of the free electrons, where mn* is the

equivalent electron mass, which account for the electron accommodation to medium change.

212322

)()2

(2

1)()( //

*

cn EE

mENE

If we prefer to the energy at the bottom of the conduction band as a nun-zero value of Ec instead of Ec = 0, The density of state equation can be further modified as

Page 16: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

kTEEkTEE

F

Fe

eEf /

/)(

)( 1

1)(

212322

)()2

(2

1)()( //

*

cn EE

mENE

0

21)(23220

)2

(2

1)()( dEeEe

mdEEfENn kTEkTEE cF ////

kTEE cFeh

mkTn // )(23

2)

2(2

)2

(0

21

aadxexgiven ax

/

kTEEc

kTEEno

FccF eNeh

kTmn ///

*)(-)(23

2)

2(2

232

)2

(2 /*

h

kTmN n

c

232

)2

(2 /*

h

kTmN p

v

kTEE

vkTEEp

ovFvF eNe

h

kTmp ///

*)(-)(-23

2)

2(2

Page 17: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Nc: Effective density of state at bottom of C.B.Nv: Effective density of state at top of V.B.no: Concentration of electrons in the conduction bandpo: Concentration of holes in the valence bandEc: Conduction band edgeEv: Valence band edgeEF: Fermi levelEi: Fermi level for the undoped semiconductor (intrinsic)

kTEEco

FceNn /)-( kTEE

vovFeNp /)-(

kTEEci

iceNn /)-( kTEE

vivieNp /)-(

)(general )(intrinsic

kTEEio

iFenn /)( kTEE

ioFiepp /)(

ii pnwhere

iioo pnpnand

Page 18: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Fermi Level and Carrier Concentration of Intrinsic Semiconductor

kTEEci

iceNn /)-( kTEE

vivieNp /)-(

ii pnand

*

*

ln

ln

n

pvc

c

vvci

m

mkTEE

N

NkTEEE

4

3

2

22

kTEgnpi emm

h

kTn 23/423

2)()

2(2 /**/

Page 19: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 20: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 21: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Example 3-5

A Si sample is doped with 1017 As atoms/cm3. What is the equilibrium hole concentration po at 300K? Where is EF relative to Ei?

Page 22: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

5.1 Introduction5.2 Equilibrium condition

5.2.1 Contact potential5.2.2 Equilibrium Fermi level5.2.3 Space charge at a junction

5.3 Forward bias5.3.1

Page 23: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Electric field

Page 24: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 25: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Einstein relationship(explained later)

Electric field

Page 26: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 27: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

Einstein Relationship

dx

xdpqDxxpqxJ pnp

)()()()(

drift diffusion

• At equilibrium, no net current flows in a semiconductor. Jp(x) = 0• Any fluctuation which would begin a diffusion current also sets up an electric

field which redistributes carriers by drift.• An examination of the requirements for equilibrium indicates that the diffusion

coefficient and mobility must be related.

Page 28: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

cl

ldxdp

lp

)(21

0

cl

ldxdp

lp

)(21

0

dx

dpD

dx

dpldx

dpll

dxdp

l

ppc

cp

cllx

v

v

l l0

x

l

l�

dx

dpqDqxJ pxp )(

cppathfreemeanl v:

)( lD pp v

Einstein Relationship

ppc mq v

)(p

cp m

q

pp

cp m

q vhole

Drift

Diffusion

Page 29: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

cpl v

kTm thp 2

1

2

1 2 v

)( lD pp v)(p

cp m

q

drift diffusion

q

kTD

p

p

Einstein Relationship

Drift and diffusion

diffusion

Page 30: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 31: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 32: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 33: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 34: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 35: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 36: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 37: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 38: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 39: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 40: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

The derivation of Poisson's equation in electrostatics follows. SI units are used and Euclidean space is assumed.

Starting with Gauss' law for electricity (also part of Maxwell's equations) in a differential control volume, we have:

is the divergence operator.

is the electric displacement field.

is the free charge density (describing charges brought from outside).

Assuming the medium is linear, isotropic, and homogeneous (see polarization density), then:

is the permittivity of the medium.

is the electric field.

By substitution and division, we have:

fD

D

f

ED E

fE

http://en.wikipedia.org/wiki/Poisson's_equation

Poisson's equation

Page 41: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

2r

kQqqEF

2r

kQE

r

kQEdV

r

kQqdEqqVU

Page 42: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 43: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 44: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 45: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 46: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 47: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 48: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at
Page 49: Chapter 5 Junctions. 5.1 Introduction (chapter 3) 5.2 Equilibrium condition 5.2.1 Contact potential 5.2.2 Equilibrium Fermi level 5.2.3 Space charge at

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