Chapter 5

Junctions

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

(negative) Photoresist

Silicon Oxide

Silicon

Mask/Shield/Pattern

Irradiation

Metal

Oxide

Lift off

Develop

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

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

Electron Energy Density

2

223

22

21

8

)(

mL

hnnnE

xn

yn

zn

knjninn zyx

Density of State ρ(E)

xn

yn

zn

knjninn zyx

2

223

22

21

8

)(

mL

hnnnE

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

)(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

)(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

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

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

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

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 /**/

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?

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

Electric field

Einstein relationship(explained later)

Electric field

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.

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

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

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

2r

kQqqEF

2r

kQE

r

kQEdV

r

kQqdEqqVU