Chapter 4 The semiconductor in equilibriumocw.nctu.edu.tw/course/physics/semiconductorphysics...W.K....

W.K. Chen Electrophysics, NCTU 1

Chapter 4The semiconductor in equilibrium


Equilibrium (Thermal equilibrium)No external forces such as voltages, electric fields, magnetic fields, or temperature gradients are acting on the semiconductor

All properties of the semiconductor will be independent of time at equilibrium

Equilibrium is our starting point for developing the physics of the semiconductor. We will then be able to determine the characteristics that result when deviations from equilibrium occur


Outline

Charge carriers in semiconductor

Dopant atoms and energy levels

The extrinsic semiconductor

Statistics of donars and acceptors

Charge neutrality

Position Fermi energy level


4.1 Charge carriers in semiconductors

Current is the rate at which charge flow

Two types of carriers can contribute the current flowElectrons in conduction band

Holes in valence band

The density of electrons and holes is related to the density of state function and Fermi-Dirac distribution function

dEEfEgdEEn Fc )()()( =

dEEfEgdEEp F )](1)[()( −= υ

n(E)dE: density of electrons in CB at energy levels between E and E+dE

p(E)dE: density of holes in VB at energy levels between E and E+dE


Carriers concentration in intrinsic semiconductor at equilibrium

dEEfEgdEEn Fc )()()( =

dEEfEgdEEp F )](1)[()( −= υ


no equation

∫∫ ==top

Ec Fc

top

Eco dEEfEgdEEnn )()()(

The thermal equilibrium concentration of electrons in CB

kT

EEEf

fF )(

exp1

1)(

−+

=Q

For electrons in CB,

⇒

>>−⇒> kTEEEE fC )(

on)aproximati (Boltzmann )(

exp)(

exp1

1)(

kT

EE

kT

EEEf f

fF

−−≈

−+

=Q

cn

c EEh

mEg −=

3

2/3* )2(4)(

π


∫−−

⋅−=top

Ec

fc

no dE

kT

EEEE

h

mn

)(exp

)2(43

2/3*π

kT

EE

h

kTmn fCn

o

)(exp

22

2/3

2

* −−⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

We define effective density of states in CB,

The thermal-equilibrium electron concentration in CB

2/3

2

*22 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

h

kTmN n

C

π

kT

EENn fC

Co

)(exp

−−=


Example 4.1 electron concentration

31519 cm108.1)0259.0

25.0exp()108.2(

)(exp

−×=−

×=

−−≈

kT

EENn fc

co

300Kat CBin ion concentratelectron theFind

below eV 0.25 is

cm108.2)300( 319

⇒

×= −

cf

c

EE

KN

Solution


po equation

∫∫ −==top

Ec F

top

Eco dEEfEgdEEpp ))(1)(()( υ

The thermal equilibrium concentration of holes in VB

kT

EE

kT

EEEf

ffF )(

exp1

1)(

exp1

11)(1

−+

=−

+−=−Q

For holes in VB,

⇒

>>−⇒< kTEEEE f )( υ

on)aproximati (Boltzmann )(

exp)(

exp1

1)(1

kT

EE

kT

EEEf f

fF

−−≈

−+

=−Q

EEh

mEg p −= υυ

π3

2/3* )2(4)(


∫−−

⋅−= υ

υ

πE fpo dE

kT

EEEE

h

mp

0 3

2/3* )(exp

)2(4

kT

EE

h

kTmp fp

o

)(exp

22

2/3

2

*υπ −−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

We define effective density of states in CB,

The thermal-equilibrium electron concentration in CB

2/3

2

*22 ⎟

⎟⎠

⎞⎜⎜⎝

⎛=

h

kTmN pπ

υ

kT

EENp f

o

)(exp υ

υ

−−=


2/3

2

*22 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

h

kTmN n

C

π

Nc & Nυ values

Nc & Nυ are determined by the parameters of effective masses and temperature. The effective mass is nearly a constant value ( a slight function of temperature) for a semiconductor, which implies the Nc & Nυ will increase in values with power of 3/2 with increasing temperature

Assume mn*=mo, then the value of the density of the effective density of state at 300K is

which is lower the density of atoms in semiconductor

The effective mass of the electron in semiconductor is larger or smaller than mo, but still of the same order of magnitude

, if cm105.2 *319onC mmN =×= −

2/3

2

*22 ⎟

⎟⎠

⎞⎜⎜⎝

⎛=

h

kTmN pπ

υ


Example 4.2 hole concentration

31519 cm1043.6)03453.0

27.0exp()1060.1(

)(exp

−×=−

×=

−−=

kT

EENp f

oυ

υ

3192/3

192/3

cm1060.1300

400)1004.1()400(

300

400

)300(

)400( −×=⎟⎠⎞

⎜⎝⎛×=⇒⎟

⎠⎞

⎜⎝⎛= KN

KN

KNυ

υ

υ

Solution:400Kat VBin ion concentrat hole theFind

above eV 0.27 is

cm1004.1)300( 319

⇒

×= −

υ

υ

EE

KN

f


4.1.3 Intrinsic carrier concentrationFor intrinsic semiconductor

The concentration of electrons in CB is equal to the concentration of holes in VB


kT

EENnn ifC

Cio

)(exp

−−==

kT

EENpp if

io

)(exp υ

υ

−−==

torsemiconduc intrinsicfor ii pn =

kT

EENN

kT

EEN

kT

EENn

CC

ffCCi

ii

)(exp

)(exp

)(exp2

υυ

υυ

−−=

−−⋅

−−=

kT

ENNn g

Ci

−= exp2

υ

The intrinsic carrier concentration is a function of bandgap, independent of Fermi level


Example 4.3 intrinsic carrier concentration

36

122

cm1026.2)300(

1009.5)0259.0

42.1exp()300()300()300(

−×=

×=−

=

Kn

KNKNKn

i

ci υ

Solution:

400K and300K at ion concentratcarrier intrinsic theFind

eV 42.1

cm100.7)300(

cm107.4)300( GaAs,For 318

317

⇒

=×=

×=−

−

g

c

E

KN

KN

υ

kT

ENNn g

Ci

−= exp2

υ

310

212

cm1085.3)400(

1048.1)03885.0

42.1exp()400()400()400(

−×=

×=−

=

Kn

KNKNKn

i

ci υ


kT

ENNn g

Ci

−= exp2

υ

The intrinsic carrier concentration is nearly increase exponentially with temperature


4.1.4 The intrinsic Fermi-level position

torsemiconduc intrinsicfor ii pn =

2/3

*

*

ln2

1)(

2

1

ln2

1)(

2

1

)(exp

)(exp

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

−−=

−−

p

pcf

ccf

ffCC

m

mkTEEE

N

NkTEEE

kT

EEN

kT

EEN

i

i

ii

υ

υυ

υυ

2/3

*

*

ln4

3⎟⎟⎠

⎞⎜⎜⎝

⎛+=

p

pmidgapf m

mkTEE

i

midgapEcE

υE

fiE

)(2

1υEEE cmidgap +=


Example 4.3 Intrinsic Fermi level

Solution:

level Fermi intrinsic theFind

56.0

08.1*

*

⇒

=

=

op

on

mm

mm

2/3

*

*

ln4

3⎟⎟⎠

⎞⎜⎜⎝

⎛+=

p

pmidgapf m

mkTEE

i

meV8.12eV0128.008.1

56.0ln)0259.0(

4

32/3

−=−=⎟⎠⎞

⎜⎝⎛+= midgapf EE

i


4.2 Dopant atoms and energy levelsThe intrinsic semiconductor may be an interesting material, but the real power of semiconductor is extrinsic semiconductor, realized by adding small, controlled amounts of specific dopant, or impurity atom.

n-type semiconductor

A group V element, such as P atom is added into Si. 5 valence electrons, 4 of them contribute to the covalent bonding, leaving the 5th electron loosely bound to P atom, referred as a donor electron



The energy level Ed is the energy state of donor impurity

If a small amount of energy, such as thermal energy, is added to the donor electron, it can be elevated into CB, leaving behind a positively charged P ion.

This type of impurity donates an electron to CB and so is called donor impurity atoms, which add electrons to contribute the CB current, without creating holes in VB


p-type semiconductor

For silicon, a group III element, such as B atom is added. 3 valence electrons are all taken up in covalent bonding, one covalent bonding position appears to be empty. (Fig. a)

The valence electrons in the VB (Fig. b) may gain a small amount of thermal energy and move to the empty state of group III element, forming empty state in VB

The energy level Ea is the energy state of group III element in Si..

VB

Acceptor energy level



The empty positions in the VB are thought of as holes.

The group III atom accepts an electron from the VB and so is referred to as an acceptor

The holes in the VB, formed due to the adding of acceptor impurity atoms without creating electrons in the CB, can move through the crystal generating a current

hole

acceptor

B


4.2.2 Ionization energyIonization energy

The energy required to elevate the donor electron into the conduction band

nnc r

m

r

eF

2*

2

2

4

υπε

==

The coulomb force of attraction between the electron and ion equal to the centripetal force of the orbiting electron

+

Due to the quantization of angular momentum

)( *υmrn n ⋅== hl

oo

rn am

mn

em

nr ⎟

⎠⎞

⎜⎝⎛==

*2

2*

224 επε hao: Bohr radius

o

oo em

a A53.04

2

2

==hπε

2nrn ∝

2

2

*υ=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅mr

n

n

h

*3

222

*

*

2

2

4 mr

n

mr

n

r

m

r

e

nnnn ⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

=hh

πε


Consider the lowest energy state of donor impurity in silicon material, in which n=1

oo

rn am

mn

em

nr ⎟

⎠⎞

⎜⎝⎛==

*2

2*

224 επε h

o

o

o

or

aar

m

mn

>>==⇒

===

A9.2345

26.0 ,7.11)Si( ,1

1

*ε

.

Because of the difference in the dielectric constant and effective electron mass, the electron orbit in a semiconductor is much larger than that in a free atom and the ionization energy of a donor state is considerably smaller than that of hydrogen atom

The radius of lowest energy state in Si is about 4 lattice constant of Si, so the radius of the orbiting donor electron encompasses many silicon atoms

The energy required to elevate the donor electron into the conduction band


VTE +=

22

4*2*

)4()(22

1

πευ

hn

emmT ==

22

4*2

)4()(4 πεπε hn

em

r

eV

n

−=

−=

22

4*

)4()(2Energy Ionization

πεhn

emVTE

−=+=

eV m8.25)Si( :silicon ofenergy Ionization

eV 6.13)H( :hydrigen ofenergy Ionization

−=−=

E

E


22

4*

)4()(2 πεhn

emVTE

−=+=

Ordinary impurity

For ordinary impurities, such as P, As, and Sb in Si and Ge, the hydrogen model works quite well.

Amphoteric impurity

For III-V semiconductor, the group IV such as Si and Ge is amphotericimpurity. If a silicon atom replace a Ga atom, The Si impurity will act as a donar, but if the Si atom replaces an As atom, then the Si impurity will act as an acceptor


4.3 Extrinsic semiconductorIntrinsic semiconductor

A semiconductor with no impurity atoms present in the crystal

Extrinsic semiconductor

A semiconductor in which controlled amounts of specific dopant or impurity atoms have been added so that the thermal-equilibrium electron and hole concentrations are different the intrinsic carrier concentrations.


4.3.1 Equilibrium distribution of electrons & holesAdding donor or acceptor will change the distribution of electrons and holes in semiconductor

Since the Fermi level is related to the distribution function, the Fermi level will change as dopant atoms are added

kT

EENn fC

Co

)(exp

−−=

kT

EENp f

o

)(exp υ

υ

−−=



When the density of electrons in CB is greater than the density of holes in VB


When the density of holes in VB is greater than the density of electrons in CB

kT

EENn fC

Co

)(exp

−−=

kT

EENp f

o

)(exp υ

υ

−−=

fifoo

fifoo

EEnp

EEpn

<⇒>

>⇒>

type-p

type-n


Example 4.5 thermal equilibrium concentrations

3-419

3-1519

cm107.20259.0

87.0exp)1004.1(

cm108.10259.0

25.0exp)108.2(

×=−

×=

×=−

×=

o

o

p

n

kT

EENp f

o

)(exp υ

υ

−−=

ionsconcentrat hole &electron equlibrium thermal theFind

cm1004.1 cm108.2

band conduction below eV 0.25 isenergy Fermi

eV 1.12 is bandgap Si,For

319319 −− ×=×= υNNC

Solution:

kT

EENn fC

Co

)(exp

−−=

The electron and hole concentrations change by order of magnitude as the Fermi energy changes by a few tenths of an electron-volt


Another form of equations for no and po

⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡ −−=

⎥⎦

⎤⎢⎣

⎡ −+−−=⇒

−−==

−−=

kT

EE

kT

EENn

kT

EEEENn

kT

EENnn

kT

EENn

fiFfiCCo

fiFfiCCo

fCCio

fCCo

i

)(exp

)(exp

)()(exp

)(exp intrinsic

)(exp type-n

)(

exp ⎥⎦

⎤⎢⎣

⎡ −=

kT

EEnn fiF

io ) , type-(n fifio EEnn >>

)(

exp ⎥⎦

⎤⎢⎣

⎡ −−=

kT

EEnp fiF

io ) , type-(p fifio EEnn >>


4.3.2 The nopo product

kT

EENn fC

Co

)(exp

−−=

kT

EENp f

o

)(exp υ

υ

−−=

kT

EEN

kT

EENpn ffC

Coo

)(exp

)(exp υ

υ

−−⋅

−−=

2exp ig

Coo nkT

ENNpn =

−⋅= υ

The product of no and po is always a constant for a given semiconductor material at a given temperature, no matter it is intrinsic or extrinsic semiconductors,

The above equation is valid only when Boltzmann approximation is valid

The constant nopo product is one of the fundamental principles of the semiconductor in thermal equilibrium


4.3.3 The Fermi-Dirac IntegralIf the Boltzmann approximation does not hold, the thermal equilibrium carrier concentrations is expressed as Fermi-Dirac distribution function

on distributi Dirac-Fermi elsewhere

holdsion approximatBoltzmann 3 typep

3 typen

⇒

⇒⎪⎭

⎪⎬⎫

>−−

>−−

kTEE

kTEE

f

fC

υ

cE

υE

fEkT3

kT3

Boltzmannapproximation

Fermi-Dirac

Fermi-Dirac


∫ −+

−=

cEf

cno dE

kT

EEEE

mh

n)exp(1

)()2(

4 2/12/3*

3

π

2/3

2

*22 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

h

kTmN n

C

π

kT

EE

kT

EE cFF

c −=

−= ηη and let

∫∞

−+=

0

2/12/3

2

*

)exp(1)

2(4 η

ηηηπ d

h

kTmn

F

no

∫∞

−+=

0

2/1

2/1 )exp(1)( η

ηηηη dF

FF

)(2

2/1 Fco FNn ηπ

=

)(2

2/1 Fo FNp ηπ υ=

2/3

2

*22 ⎟

⎟⎠

⎞⎜⎜⎝

⎛=

h

kTmN pπ

υ


4.3.4 Degenerate and nondegeneratesemiconductorsNondegenerate semiconductor

If the doping is low, the impurity atoms are spread far enough apart so that there is no interaction between donor (acceptor) electrons (holes), the impurities introduced discrete, noninteracting energy states


Degenerate semiconductor

If the doping is high, the impurity atoms are close enough so that donor/acceptor electrons/holes will begin to interact with each other. When this occur, the single discrete energy state will split into a band of energy. When the concentration of donor (acceptor) electrons (holes) exceeds the effective density of states (Nc or Nυ), the Fermi level lies with the band. This type of semiconductors are called degenerate semiconductors.

f

cfc

EENp

EENn

>>

>>

υυ ,tor semiconduc degenerate type-p

,tor semiconduc degenerate type-n

o

o


4.4 Statistics of donors and acceptorsSuppose we have Ni electrons and gi quantum states at ith impurity energy levelEach donor level at least has two possible spin orientations for donor electron, thus each donor has at least two quantum statesThe distribution function of donor electrons in the donor energy states is slightly different than the Fermi-Dirac function

bands allowedfor )exp(1

1)(

kT

EEEf

fF −

+=

statesdonor for )exp(

21

1

1)(

kT

EEN

nEf

fd

dF −

+==

nd: the density of electrons occupying the donor levelNd: the concentration of donor Nd

+ : the concentration of ionized donorEd: the energy level of the donor

+−= ddd NNn


statesacceptor for )exp(

11

1)(

kT

EE

gN

pEf

afa

aF −

+==

+−= aaa NNp

g is normally taken as four for acceptor level in Si and Ge because of the detailed band structure


4.4.2 Complete ionization and free-out

⎥⎦

⎤⎢⎣

⎡ −−=

−≈

−+

==

>>−

kT

EEN

kT

EEN

kT

EEN

EfNn

kTEE

fdd

fd

d

f

dFdd

fd

)(exp2

)exp(21

)exp(

21

1)(

If

For non-degenerate semiconductor,

kT

EENn fC

Co

)(exp

−−=

The total number of electrons in and near the conduction band is

do nn +

The electron concentration in the conduction band is

type-nfor )( ddio nNnn −+≈


The ratio of electrons in the donor states to the total number of electrons in and near the conduction band is

kT

EEN

kT

EEN

kT

EEN

nn

n

fCC

fdd

fdd

od

d

)(exp

)(exp2

)(exp2

−−+⎥

⎦

⎤⎢⎣

⎡ −−

⎥⎦

⎤⎢⎣

⎡ −−

=+

kTEE

NNnn

n

dC

d

Cod

d

)(exp

21

1−−

+=

+

cE

υE

fEkT3

dE + + + + +


Example 4.7

%41.00041.0

0259.0)045.0(

exp)10(2

108.21

1

16

19 ==−×

+=

+ od

d

nn

n

3-16 cm10

doping sphosphorou 300K,T tor,semiconduc Si

=

=

dN

Solutions:

Nondegenerate semiconductor

For shallow doped semiconductor, only a few electrons are still in the donor states, essentially all of the electrons from the donor states are in the conduction band. In this case, the donor states are said to be completely ionized


kT

EE

N

Nnn

n

dC

d

Cod

d

)(exp

21

1−−

+=

+kT

EEN

Npp

p

a

a

oa

a

)(exp

41

1

υυ −−+

=+

At T=300K


At T=0 K

All electrons are in their lowest possible energy states

dffd

fdd

dF EE

kT

EE

kT

EEN

nEf >⇒−∞==

−⇒=

−+

== )exp(0)exp( 1)exp(

21

1

1)(

At T= 0 K, all the energy states below the Fermi level are full, and all the states above the Fermi level are empty. Since the donor states is fully occupied by donor electrons, the Fermi level must be well above the donor energy state.

At T= 0K, no electrons from the donor state are thermally elevated into conduction band ; this effect is called free-out.


Example 4.8 90% ionization temperature

kTEE

NNpp

p

a

a

oa

a

)(exp

41

1

υυ −−+

=+

ionized are acceptors of 90%at which re temperatu theFind

cm10

(B)boron with dpoedtor semiconduc Si type-p3-16=aN

Solution:

K 193

)300

(0259.0

045.0exp

)10(4

)300

)(1004.1(1

11.0

16

2/319

=⇒

−×+

==+

T

T

Tpp

p

a

oa

a


4.5 Charge neutralityCharge neutrality

In thermal equilibrium, the semiconductor crystal is electrically neutral

The charge-neutrality condition is used to determine the thermal equilibrium electron and hole concentration as a function of impurity dopingconcentration

Compensated semiconductor

A semiconductor contains both donor and acceptor impurity atoms in the same region

N-type compensated semiconductor ( Nd>Na)

P-type compensated semiconductor (Na>Nd)

Completely compensated semiconductor (Na=Nd)


4.5.2 Equilibrium electron and hole concentration

+− +=+ doao NpNn

)()( ddoaao nNppNn −+=−+

Charge neutrality:

Under complete ionization

0 ,0 == ad pn

0)( 22

2

=−−−

+=+

+=+

ioado

do

iao

doao

nnNNn

Nn

nNn

NpNn

22

22

)( type-n i

adado n

NNNNn +⎟

⎠⎞

⎜⎝⎛ −

+−

=


o

io

idada

o

p

nn

nNNNN

p

2

22

22

)(tor semiconduc type-p

=

+⎟⎠⎞

⎜⎝⎛ −

+−

=

o

io

iadad

o

n

np

nNNNN

n

2

22

22

)(tor semiconduc type-n

=

+⎟⎠⎞

⎜⎝⎛ −

+−

=


Example 4.9 thermal equilibrium carrier conc.

3416

2102

316210

21616

cm1025.210

)105.1(

carrierminority

cm10)105.1(2

10

2

10

carriermajority

−

−

×=×

==

≈×+⎟⎟⎠

⎞⎜⎜⎝

⎛+=

o

io

o

n

np

n

Solution:

The thermal equilibrium majority carrier concentration is essentially equal to impurity concentration

3-103-16 cm105.1 and 0 ,cm10

doping sphosphorou 300K,T tor,semiconduc Si type-n

×===

=

iad nNN


Redistribution of electrons when donors are added

When add shallow donorsMost of donor electrons will gain thermal energy and jump into the CB

A few of the donor electrons will fall into the empty states in the VB and will annihilate some of the intrinsic holes


Electron concentration vs. temperature

type-nfor )( ddio nNnn −+≈

Freeze-out (0 K)

Partial ionization (0- ∼100K)

Extrinsic region (∼100K-400K)

Intrinsic region (>∼400 K)

kTEE

NNnn

n

dC

d

Cod

d

)(exp

21

1−−

+=

+

kT

ENNn g

Ci

−⋅= exp2

υ

0 , == odd nNn

The above regions are determined by type of semiconductor and doping concentration


Example 4.11 Compensated p-semiconductor.

3415

2102

315210

215161516

cm1021.3107

)105.1(

carrierminority

)(

cm107)105.1(2

10310

2

)10310(

carriermajority

−

−

×=×

×==

−≈

×≈×+⎟⎟⎠

⎞⎜⎜⎝

⎛ ×−+

×−=

o

io

da

o

p

nn

NN

p

Solution:

Because of complete ionization at 300K, the majority carrier hole concentration is just the difference between the acceptor and donor concentrations

3-103-153-16 cm105.1 and cm105.1 ,cm10

300K,T tor,semiconduc Si type-p

×=×==

=

ida nNN

22

22

)(i

dadao n

NNNNp +⎟

⎠⎞

⎜⎝⎛ −

+−

=


4.6 Position of Fermi energy level

kT

EENn fC

Co

)(exp

−−= kT

EENp f

o

)(exp υ

υ

−−=

When Boltzmann approximation holds,

The position of Fermi energy level moves through the bandgap energy, depending on the electron and hole concentrations and temperature

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

o

CfC n

NkTEE ln type-n ⎟⎟

⎠

⎞⎜⎜⎝

⎛=−

oF p

NkTEE υ

υ ln type-p


iaodada

iadaa

f

nNpNNNN

NkT

nNNNN

NkTEE

>>=−⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

>>>>⎟⎟⎠

⎞⎜⎜⎝

⎛=−

, when ln

, when ln type-p

υ

υυ

idoadad

C

idadd

CfC

nNnNNNN

NkT

nNNNN

NkTEE

>>=−⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

>>>>⎟⎟⎠

⎞⎜⎜⎝

⎛=−

, when ln

, when ln type-n


Example 4.13 Fermi energy

3161616

31619

cm1024.2101024.1

cm1024.1)0259.0

20.0exp(108.2

carriermajority

−

−

×=+×=

−=

×=−

×=

d

ado

o

N

NNn

n

Q

Solution:

edge band conduction thebelow eV 0.20 isenergy Fermi

cm105.1 and cm10

300KTtor,semiconduc Si type-n3-103-16 ×==

=

ia nN

kT

EENn fC

Co

)(exp

−−=


Variation of Fermi level with doping conc at 300K

For silicon, at 300K, Nd, Na>>ni = (1.5x1010cm-3 )

The carrier concentrations are mainly determined by the impurityconcentration.

As the doping levels increase, the Fermi level moves closer to the conduction band for n-type material and closer to valence band for the p-type material

kT

EENn fC

Co

)(exp

−−=

kT

EENp f

o

)(exp υ

υ

−−=


Ef versus temperature

The intrinsic carrier concentration ni is a strong function of temperature, so the Ef is a function of temperature also

As high temperature, the semiconductor material begins to lose its extrinsic characteristics and begins to behave more like an intrinsic semiconductor

At the very low temperature, free-out occurs; the Fermi level goes above Ed for the n-type material and below Ea for the p-type material

kT

ENNn g

Ci

−⋅= exp2

υ


υE

fE

ifE

cEdE

Free-out

υE

fEif

E

cEdE

Extrinsic

υEfEif

E

cEdE

Intrinsic


4.6.3 Relevance of the Fermi energy

Thermal equilibrium occurs when the distribution of electrons, as a function of energy, is the same in the two materials.

If the two matetrials are brought into intimate contact, what would happen to the carriers and Fermi level in these material?

The electrons will tend to seek the lowest possible energy

In this example, the electrons in material A will flow into the lower energy states of material B

Chapter 4 The semiconductor in equilibriumocw.nctu.edu.tw/course/physics/semiconductorphysics...W.K....

Documents

Transcript of Chapter 4 The semiconductor in equilibriumocw.nctu.edu.tw/course/physics/semiconductorphysics...W.K....