W.K. Chen Electrophysics, NCTU 1

Chapter 4

The semiconductor in

equilibrium

W.K. Chen

Electrophysics, NCTU

2

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

W.K. Chen

Electrophysics, NCTU

3

Outline

Charge carriers in semiconductor

Dopant atoms and energy levels

The extrinsic semiconductor

Statistics of donars and acceptors

Charge neutrality

Position Fermi energy level

W.K. Chen

Electrophysics, NCTU

4

4.1 Charge carriers in semiconductors

Current is the rate at which charge flow

Two types of carriers can contribute the current flow

Electrons 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

W.K. Chen

Electrophysics, NCTU

5

Carriers concentration in intrinsic semiconductor

at equilibrium

dEEfEgdEEn

Fc

)()()( =

dEEfEgdEEp

F

)](1)[()( −=

υ

W.K. Chen

Electrophysics, NCTU

6

n

o

equation

∫∫

==

top

Ec

Fc

top

Ec

o

dEEfEgdEEnn )()()(

The thermal equilibrium concentration of electrons in CB

k

T

EE

Ef

f

F

)(

exp1

1

)(

−

+

=Q

For electrons in CB,

⇒

>>−⇒> kTEEEE

fC

)(

on)aproximati (Boltzmann

)(

exp

)(

exp1

1

)(

kT

EE

k

T

EE

Ef

f

f

F

−

−

≈

−

+

=Q

c

n

c

EE

h

m

Eg −=

3

2/3*

)2(4

)(

π

W.K. Chen

Electrophysics, NCTU

7

∫

−−

⋅−=

top

Ec

f

c

n

o

dE

kT

EE

EE

h

m

n

)(

exp

)2(4

3

2/3*

π

kT

EE

h

kTm

n

fC

n

o

)(

exp

2

2

2/3

2

*

−−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

π

We define effective density of states in CB,

The thermal-equilibrium electron concentration in CB

2/3

2

*

2

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

h

kTm

N

n

C

π

kT

EE

Nn

fC

Co

)(

exp

−−

=

W.K. Chen

Electrophysics, NCTU

8

Example 4.1 electron concentration

31519

cm108.1)

0259.0

25.0

exp()108.2(

)(

exp

−

×=

−

×=

−−

≈

kT

EE

Nn

fc

co

300

K

at CBin ion concentratelectron theFin

d

below eV 0.25 is

cm108.2)300(

319

⇒

×=

−

cf

c

EE

KN

Solution

W.K. Chen

Electrophysics, NCTU

9

p

o

equation

∫∫

−==

top

Ec

F

top

Ec

o

dEEfEgdEEpp ))(1)(()(

υ

The thermal equilibrium concentration of holes in VB

k

T

EE

k

T

EE

Ef

ff

F

)(

exp1

1

)(

exp1

1

1)(1

−

+

=

−

+

−=−

Q

For holes in VB,

⇒

>>

−

⇒

<

kTEEEE

f

)(

υ

on)aproximati (Boltzmann

)(

exp

)(

exp1

1

)(1

kT

EE

k

T

EE

Ef

f

f

F

−

−

≈

−

+

=−

Q

EE

h

m

Eg

p

−=

υυ

π

3

2/3*

)2(4

)(

W.K. Chen

Electrophysics, NCTU

10

∫

−−

⋅−=

υ

υ

π

E

fp

o

dE

kT

EE

EE

h

m

p

0

3

2/3*

)(

exp

)2(4

kT

EE

h

kTm

p

fp

o

)(

exp

2

2

2/3

2

*

υ

π −−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

We define effective density of states in CB,

The thermal-equilibrium electron concentration in CB

2/3

2

*

2

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

h

kTm

N

p

π

υ

kT

EE

Np

f

o

)(

exp

υ

υ

−−

=

W.K. Chen

Electrophysics, NCTU

11

2/3

2

*

2

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

h

kTm

N

n

C

π

N

c

& N

υ

values

N

c

& N

υ

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 N

c

& N

υ

will increase

in values with power of 3/2 with increasing temperature

Assume m

n

*=m

o

, 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

*319

onC

mmN =×=

−

2/3

2

*

2

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

h

kTm

N

p

π

υ

W.K. Chen

Electrophysics, NCTU

12

Example 4.2 hole concentration

31519

cm1043.6)

03453.0

27.0

exp()1060.1(

)(

exp

−

×=

−

×=

−−

=

kT

EE

Np

f

o

υ

υ

319

2/3

19

2/3

cm1060.1

300

400

)1004.1()400(

300

400

)300(

)400(

−

×=

⎟

⎠

⎞

⎜

⎝

⎛

×=⇒

⎟

⎠

⎞

⎜

⎝

⎛

= KN

KN

KN

υ

υ

υ

Solution:

400Kat VBin ion concentrat hole theFin

d

above eV 0.27 is

cm1004.1)300(

319

⇒

×=

−

υ

υ

EE

KN

f

W.K. Chen

Electrophysics, NCTU

13

4.1.3 Intrinsic carrier concentration

For intrinsic semiconductor

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

W.K. Chen

Electrophysics, NCTU

14

kT

EE

Nnn

i

fC

Cio

)(

exp

−−

==

kT

EE

Npp

i

f

io

)(

exp

υ

υ

−−

==

torsemiconduc intrinsicfor

ii

pn =

kT

EE

NN

kT

EE

N

kT

EE

Nn

C

C

ffC

Ci

ii

)(

exp

)(

exp

)(

exp

2

υ

υ

υ

υ

−−

=

−−

⋅

−−

=

kT

E

NNn

g

Ci

−

= exp

2

υ

The intrinsic carrier concentration is a function of bandgap, independent

of Fermi level

W.K. Chen

Electrophysics, NCTU

15

Example 4.3 intrinsic carrier concentration

36

122

cm1026.2)300(

1009.5)

0259.0

42.1

exp()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

E

NNn

g

Ci

−

= exp

2

υ

310

212

cm1085.3)400(

1048.1)

03885.0

42.1

exp()400()400()400(

−

×=

×=

−

=

Kn

KNKNKn

i

ci υ

W.K. Chen

Electrophysics, NCTU

16

kT

E

NNn

g

Ci

−

= exp

2

υ

The intrinsic carrier concentration is

nearly increase exponentially with

temperature

W.K. Chen

Electrophysics, NCTU

17

4.1.4 The intrinsic Fermi-level position

torsemiconduc intrinsicfor

ii

pn =

2/3

*

*

ln

2

1

)(

2

1

ln

2

1

)(

2

1

)(

exp

)(

exp

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++=

−−

=

−−

p

p

cf

c

cf

ffC

C

m

m

kTEEE

N

N

kTEEE

kT

EE

N

kT

EE

N

i

i

ii

υ

υ

υ

υ

υ

2/3

*

*

ln

4

3

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+=

p

p

midgapf

m

m

kTEE

i

midgap

E

c

E

υ

E

fi

E

)(

2

1

υ

EEE

cmidgap

+=

W.K. Chen

Electrophysics, NCTU

18

Example 4.3 Intrinsic Fermi level

Solution:

level Fermi intrinsic theFin

d

56.0

08.1

*

*

⇒

=

=

op

on

mm

mm

2/3

*

*

ln

4

3

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+=

p

p

midgapf

m

m

kTEE

i

meV8.12eV0128.0

08.1

56.0

ln)0259.0(

4

3

2/3

−=−=

⎟

⎠

⎞

⎜

⎝

⎛

+=

midgapf

EE

i

W.K. Chen

Electrophysics, NCTU

19

4.2 Dopant atoms and energy levels

The 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

W.K. Chen

Electrophysics, NCTU

20

n-type semiconductor

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

W.K. Chen

Electrophysics, NCTU

21

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 E

a

is the energy state of group III element in Si..

VB

Acceptor energy level

W.K. Chen

Electrophysics, NCTU

22

p-type semiconductor

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

W.K. Chen

Electrophysics, NCTU

23

4.2.2 Ionization energy

Ionization energy

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

nn

c

r

m

r

e

F

2*

2

2

4

υ

πε

==

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

)(

*

υ

mrn

n

⋅== hl

o

o

rn

a

m

m

n

em

n

r

⎟

⎠

⎞

⎜

⎝

⎛

==

*

2

2*

22

4

ε

πε h

a

o

: Bohr radius

o

o

o

em

a

A53.0

4

2

2

==

hπε

2

nr

n

∝

2

2

*

υ=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⋅ mr

n

n

h

*3

22

2

*

*

2

2

4 mr

n

mr

n

r

m

r

e

nnnn

⋅

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⋅

=

hh

πε

W.K. Chen

Electrophysics, NCTU

24

Consider the lowest energy state of donor impurity in silicon material, in

which n=1

o

o

rn

a

m

m

n

em

n

r

⎟

⎠

⎞

⎜

⎝

⎛

==

*

2

2*

22

4

ε

πε h

o

o

o

o

r

aar

m

m

n

>>==⇒

===

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

W.K. Chen

Electrophysics, NCTU

25

VTE +=

22

4*

2*

)4()(22

1

πε

υ

hn

em

mT ==

22

4*2

)4()(4 πεπε

h

n

em

r

e

V

n

−

=

−

=

22

4*

)4()(2

Energy Ionization

πεhn

em

VTE

−

=+=

eV m8.25)Si( :silicon ofenergy Ionization

eV 6.13)H( :hydrigen ofenergy Ionization

−=

−

=

E

E

W.K. Chen

Electrophysics, NCTU

26

22

4*

)4()(2

πεh

n

em

VTE

−

=+=

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 amphoteric

impurity. 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

W.K. Chen

Electrophysics, NCTU

27

4.3 Extrinsic semiconductor

Intrinsic 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.

W.K. Chen

Electrophysics, NCTU

28

4.3.1 Equilibrium distribution of electrons & holes

Adding 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

EE

Nn

fC

Co

)(

exp

−−

=

kT

EE

Np

f

o

)(

exp

υ

υ

−−

=

W.K. Chen

Electrophysics, NCTU

29

n-type semiconductor

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

p-type semiconductor

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

kT

EE

Nn

fC

Co

)(

exp

−−

=

kT

EE

Np

f

o

)(

exp

υ

υ

−−

=

fifoo

fifoo

EEnp

EEpn

<⇒>

>⇒>

type-p

type-n

W.K. Chen

Electrophysics, NCTU

30

Example 4.5 thermal equilibrium concentrations

3-419

3-1519

cm107.2

0259.0

87.0

exp)1004.1(

cm108.1

0259.0

25.0

exp)108.2(

×=

−

×=

×=

−

×=

o

o

p

n

kT

EE

Np

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 −−

×=×=

υ

NN

C

Solution:

kT

EE

Nn

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

W.K. Chen

Electrophysics, NCTU

31

Another form of equations for n

o

and p

o

⎥

⎦

⎤

⎢

⎣

⎡

−

⎥

⎦

⎤

⎢

⎣

⎡

−−

=

⎥

⎦

⎤

⎢

⎣

⎡

−+−−

=⇒

−−

==

−−

=

kT

EE

kT

EE

Nn

kT

EEEE

Nn

kT

EE

Nnn

kT

EE

Nn

fiFfiC

Co

fiFfiC

Co

fC

Cio

fC

Co

i

)(

exp

)(

exp

)()(

exp

)(

exp intrinsic

)(

exp type-n

)(

exp

⎥

⎦

⎤

⎢

⎣

⎡

−

=

kT

EE

nn

fiF

io

) , type-(n

fifio

EEnn >>

)(

exp

⎥

⎦

⎤

⎢

⎣

⎡

−−

=

kT

EE

np

fiF

io

) , type-(p

fifio

EEnn >>

W.K. Chen

Electrophysics, NCTU

32

4.3.2 The n

o

p

o

product

kT

EE

Nn

fC

Co

)(

exp

−−

=

kT

EE

Np

f

o

)(

exp

υ

υ

−−

=

kT

EE

N

kT

EE

Npn

ffC

Coo

)(

exp

)(

exp

υ

υ

−−

⋅

−−

=

2

exp

i

g

Coo

n

kT

E

NNpn =

−

⋅=

υ

The product of n

o

and p

o

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 n

o

p

o

product is one of the fundamental principles of the semiconductor

in thermal equilibrium

W.K. Chen

Electrophysics, NCTU

33

4.3.3 The Fermi-Dirac Integral

If 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

c

E

υ

E

f

E

kT3

kT3

Boltzmann

approximation

Fermi-Dirac

Fermi-Dirac

W.K. Chen

Electrophysics, NCTU

34

∫

−

+

−

=

c

E

f

c

no

dE

kT

EE

EE

m

h

n

)exp(1

)(

)2(

4

2/1

2/3*

3

π

2/3

2

*

2

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

h

kTm

N

n

C

π

kT

EE

kT

EE

cF

F

c

−

=

−

= ηη and let

∫

∞

−+

=

0

2/1

2/3

2

*

)exp(1

)

2

(4 η

ηη

η

π d

h

kTm

n

F

n

o

∫

∞

−+

=

0

2/1

2/1

)exp(1

)( η

ηη

η

η dF

F

F

)(

2

2/1 Fco

FNn

η

π

=

)(

2

2/1 Fo

FNp

η

π

υ

=

2/3

2

*

2

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

h

kTm

N

p

π

υ

W.K. Chen

Electrophysics, NCTU

35

4.3.4 Degenerate and nondegenerate

semiconductors

Nondegenerate 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

W.K. Chen

Electrophysics, NCTU

36

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 (N

c

or N

υ

), 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

W.K. Chen

Electrophysics, NCTU

37

4.4 Statistics of donors and acceptors

Suppose we have Ni electrons and gi quantum states at ith impurity energy

level

Each donor level at least has two possible spin orientations for donor

electron, thus each donor has at least two quantum states

The distribution function of donor electrons in the donor energy states is

slightly different than the Fermi-Dirac function

bands allowedfor

)exp(1

1

)(

k

T

EE

Ef

f

F

−

+

=

statesdonor for

)exp(

2

1

1

1

)(

k

T

EE

N

n

Ef

f

d

d

F

−

+

==

n

d

: the density of electrons occupying the donor level

N

d

: the concentration of donor N

d

+

: the concentration of ionized donor

E

d

: the energy level of the donor

+

−=

ddd

NNn

W.K. Chen

Electrophysics, NCTU

38

statesacceptor for

)exp(

1

1

1

)(

kT

EE

g

N

p

Ef

af

a

a

F

−

+

==

+

−=

aaa

NNp

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

detailed band structure

W.K. Chen

Electrophysics, NCTU

39

4.4.2 Complete ionization and free-out

⎥

⎦

⎤

⎢

⎣

⎡

−−

=

−

≈

−

+

==

>>−

kT

EE

N

kT

EE

N

kT

EE

N

EfNn

kTEE

fd

d

fd

d

f

d

Fdd

fd

)(

exp2

)exp(

2

1

)exp(

2

1

1

)(

If

For non-degenerate semiconductor,

kT

EE

Nn

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 −+≈

W.K. Chen

Electrophysics, NCTU

40

The ratio of electrons in the donor states to the total number of electrons

in and near the conduction band is

kT

EE

N

kT

EE

N

kT

EE

N

nn

n

fC

C

fd

d

fd

d

od

d

)(

exp

)(

exp2

)(

exp2

−−

+

⎥

⎦

⎤

⎢

⎣

⎡

−−

⎥

⎦

⎤

⎢

⎣

⎡

−−

=

+

kT

EE

N

N

nn

n

dC

d

C

od

d

)(

exp

2

1

1

−−

+

=

+

c

E

υ

E

f

E

kT3

d

E

+

+

+

+

+

W.K. Chen

Electrophysics, NCTU

41

Example 4.7

%41.00041.0

0259.0

)045.0(

exp

)10(2

108.2

1

1

16

19

==

−×

+

=

+

od

d

nn

n

3-16

cm10

doping sphosphorou 300K,T tor,semiconduc Si

=

=

d

N

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

W.K. Chen

Electrophysics, NCTU

42

kT

EE

N

N

nn

n

dC

d

C

od

d

)(

exp

2

1

1

−−

+

=

+

kT

EE

N

N

pp

p

a

a

oa

a

)(

exp

4

1

1

υυ

−−

+

=

+

At T=300K

W.K. Chen

Electrophysics, NCTU

43

At T=0 K

All electrons are in their lowest possible energy states

df

fd

fd

d

d

F

EE

kT

EE

kT

EE

N

n

Ef >⇒−∞==

−

⇒=

−

+

== )exp(0)exp( 1

)exp(

2

1

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.

W.K. Chen

Electrophysics, NCTU

44

Example 4.8 90% ionization temperature

kT

EE

N

N

pp

p

a

a

oa

a

)(

exp

4

1

1

υυ

−−

+

=

+

ionized are acceptors of 90%at which re temperatu theFind

cm10

(B)

b

oron with dpoedtor semiconduc Si t

y

pe-

p

3-16

=

a

N

Solution:

K 193

)

300

(0259.0

045.0

exp

)10(4

)

300

)(1004.1(

1

1

1.0

16

2/319

=⇒

−

×

+

==

+

T

T

T

pp

p

a

oa

a

W.K. Chen

Electrophysics, NCTU

45

4.5 Charge neutrality

Charge 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 doping

concentration

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)

W.K. Chen

Electrophysics, NCTU

46

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

d

o

i

ao

doao

nnNNn

N

n

n

Nn

NpNn

2

2

22

)(

type-n

i

adad

o

n

NNNN

n +

⎟

⎠

⎞

⎜

⎝

⎛

−

+

−

=

W.K. Chen

Electrophysics, NCTU

47

o

i

o

i

dada

o

p

n

n

n

NNNN

p

2

2

2

22

)(

tor semiconduc type-p

=

+

⎟

⎠

⎞

⎜

⎝

⎛

−

+

−

=

o

i

o

i

adad

o

n

n

p

n

NNNN

n

2

2

2

22

)(

tor semiconduc type-n

=

+

⎟

⎠

⎞

⎜

⎝

⎛

−

+

−

=

W.K. Chen

Electrophysics, NCTU

48

Example 4.9 thermal equilibrium carrier conc.

34

16

210

2

316210

2

1616

cm1025.2

10

)105.1(

carrierminority

cm10)105.1(

2

10

2

10

carrie

r

ma

j

orit

y

−

−

×=

×

==

≈×+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+=

o

i

o

o

n

n

p

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

W.K. Chen

Electrophysics, NCTU

49

Redistribution of electrons when donors are

added

When add shallow donors

Most 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

W.K. Chen

Electrophysics, NCTU

50

Electron concentration vs. temperature

type-nfor )(

ddio

nNnn −+≈

Freeze-out (0 K)

Partial ionization (0- ∼100K)

Extrinsic region (∼100K-400K)

Intrinsic region (>∼400 K)

kT

EE

N

N

nn

n

dC

d

C

od

d

)(

exp

2

1

1

−−

+

=

+

kT

E

NNn

g

Ci

−

⋅= exp

2

υ

0 ,

=

=

odd

nNn

The above regions are determined by

type of semiconductor and doping

concentration

W.K. Chen

Electrophysics, NCTU

51

Example 4.11 Compensated p-semiconductor.

34

15

210

2

315210

2

15161516

cm1021.3

107

)105.1(

carrierminority

)(

cm107)105.1(

2

10310

2

)10310(

carrie

r

ma

j

orit

y

−

−

×=

×

×

==

−≈

×≈×+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

×−

+

×−

=

o

i

o

da

o

p

n

n

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

2

2

22

)(

i

dada

o

n

NNNN

p +

⎟

⎠

⎞

⎜

⎝

⎛

−

+

−

=

W.K. Chen

Electrophysics, NCTU

52

4.6 Position of Fermi energy level

kT

EE

Nn

fC

Co

)(

exp

−−

=

kT

EE

Np

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

C

fC

n

N

kTEE ln type-n

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=−

o

F

p

N

kTEE

υ

υ

ln type-p

W.K. Chen

Electrophysics, NCTU

53

iaoda

da

iada

a

f

nNpNN

NN

N

kT

nNNN

N

N

kTEE

>>=−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

=

>>>>

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=−

, when ln

, when ln type-p

υ

υ

υ

idoad

ad

C

idad

d

C

fC

nNnNN

NN

N

kT

nNNN

N

N

kTEE

>>=−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

=

>>>>

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=−

, when ln

, when ln type-n

W.K. Chen

Electrophysics, NCTU

54

Example 4.13 Fermi energy

3161616

31619

cm1024.2101024.1

cm1024.1)

0259.0

20.0

exp(108.2

carrierma

j

orit

y

−

−

×=+×=

−=

×=

−

×=

d

ado

o

N

NNn

n

Q

Solution:

edge band conduction thebelow eV 0.20 isenergy Fermi

cm105.1 and cm10

300KTtor,semiconduc Si t

y

pe-n

3-103-16

×==

=

ia

nN

kT

EE

Nn

fC

Co

)(

exp

−−

=

W.K. Chen

Electrophysics, NCTU

55

Variation of Fermi level with doping conc at 300K

For silicon, at 300K, N

d

, N

a

>>n

i

= (1.5x10

10

cm

-3

)

The carrier concentrations are mainly determined by the impurity

concentration.

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

EE

Nn

fC

Co

)(

exp

−−

=

kT

EE

Np

f

o

)(

exp

υ

υ

−−

=

W.K. Chen

Electrophysics, NCTU

56

E

f

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

E

NNn

g

Ci

−

⋅= exp

2

υ

W.K. Chen

Electrophysics, NCTU

57

υ

E

f

E

i

f

E

c

E

d

E

Free-out

υ

E

f

E

i

f

E

c

E

d

E

Extrinsic

υ

E

f

E

i

f

E

c

E

d

E

Intrinsic

W.K. Chen

Electrophysics, NCTU

58

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

## Comments 0

Log in to post a comment