Semiconductor Physics

Introduction

• Semiconductors are materials whose electronic properties are intermediate between those of Metals and Insulators.

• They have conductivit ies in the range of 10 - 4 to 10 + 4S/m.

• The interest ing feature about semiconductors is that they are bipolar and current is transported by two charge carriers of opposite sign.

• These intermediate properties are determined by

1.Crystal Structure bonding Characterist ics. 2.Electronic Energy bands.

• Silicon and Germanium are elemental semiconductors and they have four valence electrons which are distributed among the outermost S and p orbital's.

• These outer most S and p orbital's of Semiconductors involve in Sp 3 hybridanisation.

• These Sp 3 orbital ' s form four covalent bonds of equal angular separation leading to a tetrahedral arrangement of atoms in space results tetrahedron shape, resulting crystal structure is known as Diamond cubic crystal structure

Semiconductors are mainly two types

1. Intrinsic (Pure) Semiconductors

2. Extrinsic (Impure) Semiconductors

Intrinsic Semiconductor

• A Semiconductor which does not have any kind of impurit ies, behaves as an Insulator at 0k and behaves as a Conductor at higher temperature is known as Intrinsic Semiconductor or Pure Semiconductors.

• Germanium and Sil icon (4 t h group elements) are the best examples of intrinsic semiconductors and they possess diamond cubic crystal l ine structure.

Si

Si

SiSiSi

Valence Cell

Covalent bonds

Intrinsic Semiconductor

E

Ef

Ev

Valence band

Ec

Conduction band

Ec

Electronenergy

Distance

KE ofElectron = E - Ec

KE of Hole = Ev - E

Fermi energy level

Carrier Concentration in Intrinsic Semiconductor

When a suitable form of Energy is supplied to a Semiconductor then electrons take transit ion from Valence band to Conduction band.

Hence a free electron in Conduction band and simultaneously free hole in Valence band is formed. This phenomenon is known as Electron - Hole pair generation.

In Intrinsic Semiconductor the Number of Conduction electrons wil l be equal to the Number of Vacant si tes or holes in the valence band.

)1......(..........)()(

)()(band theof top

∫=

=

cE

dEEFEzn

EFdEEZdn

Calculation of Density of Electrons

Let ‘dn’ be the Number of Electrons available between energy interval ‘E and E+ dE’ in the Conduction band

Where Z(E) dE is the Density of states in the energy interval E and E + dE and F(E) is the Probability of Electron occupancy.

dEEEmh

dEEZ ce2

1

2

3

3)()2(

4)( −= ∗π

Since the E starts at the bottom of the Conduction band E c

dEEmh

dEEZ e2

1

2

3

3)2(

4)( ∗= π

We know that the density of states i .e. , the number of energy states per unit volume within the energy interval E and E + dE is given by

dEEmh

dEEZ 2

1

2

3

3)2(

4)(

π=

)exp()(exp)(

)exp(

1)(

res temperatupossible allFor

)exp(1

1)(

kT

EE

kT

EEEF

kT

EEEF

kTEEkT

EEEF

FF

f

F

f

−=−−=

−=

>>−

−+

=

Probability of an Electron occupying an energy state E is given by

)2.....()exp()()exp()2(4

)exp()()2(4

)exp()()2(4

)()(

2

1

2

3

3

2

12

3

3

2

1

2

3

3

band theof top

∫

∫

∫

∫

∞∗

∞∗

∞∗

−−=

−−=

−−=

=

c

c

c

c

E

cF

e

E

Fce

E

Fce

E

dEkT

EEE

kT

Em

hn

dEkT

EEEEm

hn

dEkT

EEEEm

hn

dEEFEzn

π

π

π

Substitute Z(E) and F(E) values in Equation (1)

)3.....()(exp)()exp()2(4

)(exp)()exp()2(4

)exp()()exp()2(4

0

2

1

2

3

3

0

2

1

2

3

3

0

2

1

2

3

3

∫

∫

∫

∞∗

∞∗

∞∗

−−=

+−=

−−=

=+==−

dxkT

xx

kT

EEm

hn

dxkT

xEx

kT

Em

hn

dEkT

EEE

kT

Em

hn

dxdE

xEE

xEE

cFe

cFe

cF

e

c

c

π

π

π

To solve equation 2, let us put

)exp()2

(2

}2

){()exp()2(4

2

3

2

2

1

2

3

2

3

3

kT

EE

h

kTmn

kTkT

EEm

hn

cFe

cFe

−=

−=

∗

∗

π

ππ

)3(

2)()exp()(

2

1

2

3

0

2

1

equationinsubstitute

kTdEkT

xxthatknowwe

π=−∫∞

The above equation represents Number of electrons per unit volume of the Material

Calculat ion of density of holes

)1......(..........)}(1){(

)}(1{)(

band theof bottom∫ −=

−=Ev

dEEFEzp

EFdEEZdp

Let ‘dp’ be the Number of holes or Vacancies in the energy interval ‘E and E + dE’ in the valence band

Where Z(E) dE is the density of states in the energy interval E and E + dE and 1-F(E) is the probability of existence of a hole.

dEEmh

dEEZ h2

1

2

3

3)2(

4)( ∗= π

Density of holes in the Valence band is

Since E v is the energy of the top of the valence band

dEEEmh

dEEZ vh2

1

2

3

3)()2(

4)( −= ∗π

)exp()(1

exp

)}exp(1{1)(1

})exp(1

1{1)(1

1

kT

EEEF

valuesThigherfor

ansionaboveintermsorderhigherneglectkT

EEEF

kT

EEEF

f

f

f

−=−

−+−=−

−+

−=−

−

Probabil i ty of an Electron occupying an energy state E is given by

)2....()exp()()exp()2(4

)exp()()2(4

)}(1){(

2

1

2

3

3

2

1

2

3

3

band theof bottom

∫

∫

∫

∞−

∗

∞−

∗

−−=

−−=

−=

v

v

E

vF

h

E

Fvh

Ev

dEkT

EEE

kT

Em

hp

dEkT

EEEEm

hp

dEEFEzp

π

π

Substitute Z(E) and 1 - F(E) values in Equation (1)

∫

∫

∫

∞∗

∞

∗

∞−

∗

−−=

−−−=

−−=

−=−==−

0

2

1

2

3

3

0

2

1

2

3

3

2

1

2

3

3

)exp()()exp()2(4

))(exp()()exp()2(4

)exp()()exp()2(4

dxkT

xx

kT

EEm

hp

dxkT

xEx

kT

Em

hp

dEkT

EEE

kT

Em

hp

dxdE

xEE

xEE

Fvh

vFh

E

vF

h

v

v

v

π

π

π

To solve equation 2, let us put

)exp()2

(2

2))(exp()2(

4

2

3

2

2

1

2

3

2

3

3

kT

EE

h

kTmp

kTkT

EEm

hp

Fvh

Fvh

−=

−=

∗

∗

π

ππ

The above equation represents Number of holes per unit volume of the Material

Intrinsic Carrier ConcentrationIn intrinsic Semiconductors n = pHence n = p = n i is called intrinsic Carrier Concentration

)2

exp()()2

(2

)2

exp()()2

(2

)}exp()2

(2)}{exp()2

(2{

4

3

2

3

2

4

3

2

3

2

2

3

22

3

2

2

kT

Emm

h

kTn

kT

EEmm

h

kTn

kT

EE

h

kTm

kT

EE

h

kTmn

npn

npn

ghei

cvhei

FvhcFei

i

i

−=

−=

−−=

=

=

∗∗

∗∗

∗∗

π

π

ππ

Fermi level in intrinsic Semiconductors

sidesboth on logarithms taking

)exp()()2

exp(

)exp()2

()exp()2

(

)exp()2

(2)exp()2

(2

pn torssemiconduc intrinsicIn

2

3

2

3

22

3

2

2

3

22

3

2

kT

EE

m

m

kT

E

kT

EE

h

kTm

kT

EE

h

kTm

kT

EE

h

kTm

kT

EE

h

kTm

cv

e

hF

FvhcFe

FvhcFe

+=

−=−

−=−

=

∗

∗

∗∗

∗∗

ππ

ππ

E

Ef

Ev

Valence band

Ec

Conduction band

Ec

Electronenergy

Temperature

**eh mm =

Thus the Fermi energy level EF is located in the middle of the forbidden band.

)2

(

that know tor wesemiconduc intrinsicIn

)2

()log(4

3

)()log(2

32

cvF

he

cv

e

hF

cv

e

hF

EEE

mm

EE

m

mkTE

kT

EE

m

m

kT

E

+=

=

++=

++=

∗∗

∗

∗

∗

∗

Extrinsic Semiconductors

• The Extrinsic Semiconductors are those in which impurities of large quantity are present. Usually, the impurities can be either 3rd group elements or 5th group elements.

• Based on the impurities present in the Extrinsic Semiconductors, they are classified into two categories.

1. N-type semiconductors 2. P-type semiconductors

When any pentavalent element such as Phosphorous,

Arsenic or Antimony is added to the intrinsic Semiconductor , four electrons are involved in covalent bonding with four neighboring pure Semiconductor atoms.

The f if th electron is weakly bound to the parent atom. And even for lesser thermal energy i t is released Leaving the parent atom posit ively ionized.

N - type Semiconductors

N-type Semiconductor

Si

Si

SiPSi

Free electron

Impure atom (Donor)

The Intrinsic Semiconductors doped with pentavalent impurit ies are called N-type Semiconductors. The energy level of f i f th electron is called donor level. The donor level is close to the bottom of the conduction band most of the donor level electrons are excited in to the conduction band at room temperature and become the Majority charge carriers.

Hence in N-type Semiconductors electrons are Majority carriers and holes are Minority carriers.

E Ed

Ev

Valence band

Ec

Conduction band

Ec

Electronenergy

Distance

Donor levelsEg

Carrier Concentration in N-type Semiconductor

• Consider N d is the donor Concentration i .e. , the number of donor atoms per unit volume of the material and E d is the donor energy level.

• At very low temperatures all donor levels are fi l led with electrons.

• With increase of temperature more and more donor atoms get ionized and the density of electrons in the conduction band increases.

)exp()2(2 2

3

2 kT

EE

h

kTmn cFe −=

∗π

The density of Ionized donors is given by

)exp(

)}(1{)(

kT

EEN

EFdEEZ

Fdd

d

−=

−=

At very low temperatures, the Number of electrons in the conduction band must be equal to the Number of ionized donors.

)exp()exp()2(2 2

3

2 kT

EEN

kT

EE

h

kTm Fdd

cFe −=−∗π

Density of electrons in Conduction band is given by

Taking logarithm and rearranging we get

2

)(

0.,

)2(2

log22

)(

)2(2

log)(2

)2(2loglog)()(

2

3

2

2

3

2

2

3

2

cdF

e

dcdF

e

dcdF

ed

FdcF

EEE

kath

kTm

NkTEEE

hkTm

NkTEEE

h

kTmN

kT

EE

kT

EE

+=

++=

=+−

−=−−−

∗

∗

∗

π

π

π

At 0k Fermi level l ies exactly at the middle of the donor level and the bottom of the Conduction band

Density of electrons in the Conduction band

kT

EE

hkTm

N

kT

EE

hkTm

N

kT

EE

kT

EE

kT

E

hkTm

N

kT

EE

kT

EE

kT

E

hkTm

NkTEE

kT

EE

kT

EE

h

kTmn

cd

e

dcF

e

dcdcF

c

e

dcdcF

c

e

dcd

cF

cFe

2

)(exp

])2(2[

)()exp(

}

])2(2[

)(log

2

)(exp{)exp(

}

])2(2[

)(log

2

)(exp{)exp(

}

}

)2(2

log22

)({

exp{)exp(

)exp()2(2

2

1

2

1

2

1

2

3

2

2

1

2

3

2

2

1

2

3

2

2

1

2

3

2

2

3

2

−=−

+−=−

−++=−

−++

=−

−=

∗

∗

∗

∗

∗

π

π

π

π

π

kT

EE

h

kTmNn

kT

EE

h

kTm

N

h

kTmn

kT

EE

h

kTmn

cded

cd

e

de

cFe

2

)(exp)

2()2(

}2

)(exp

])2(2[

)({)

2(2

)exp()2(2

4

3

22

1

2

3

2

2

1

2

3

2

2

3

2

2

1

−=

−=

−=

∗

∗

∗

∗

π

ππ

π

Thus we f ind that the density of electrons in the conduction band is proportional to the square root of the donor concentration at moderately low temperatures.

Variat ion of Fermi level with temperature

To start with ,with increase of temperature E f

increases sl ightly.

As the temperature is increased more and more donor atoms are ionized.

Further increase in temperature results in generation of Electron - hole pairs due to breading of covalent bonds and the material tends to behave in intrinsic manner.

The Fermi level gradually moves towards the intrinsic Fermi level E i .

P - type Semiconductors

• When a trivalent elements such as Al, Ga or Indium have three electrons, added to the Intrinsic Semiconductor all the three electrons of these are involved in Covalent bonding with the three neighboring Si atoms.

• These like compound accepts one extra electron, the energy level of this impurity atom is called Acceptor level and this acceptor level lies just above the valence band.

These type of trivalent impurities are called acceptor impurit ies and the semiconductors doped with the acceptor impurities are called P-type Semiconductors .

Si

Si

SiInSi

HoleCo-Valent bonds

Impure atom (acceptor)

E

Ea

Ev

Valence band

Ec

Conduction band

Ec

Electronenergy

distance

Acceptor levels

Eg

• Even at relat ively low temperatures, these acceptor atoms get ionized taking electrons from valence band and thus giving rise to holes in valence band for conduction.

• Due to ionization of acceptor atoms only

holes and no electrons are created.

• Thus holes are more in number than electrons and hence holes are majority carriers and electros are minority carriers in P-type semiconductors.

Then current density

Then conductivity )1.........(E

nevE

J

EJ

d=

=

=

σ

σ

σ

)2........(EvE

v

nd

d

µ

µ

=

=As we know that mobility of electrons.

Drift CurrentThe moment of electron in the presence of electric field.

Substitute the drift velocity value in equation 1

EnedriftJ

ne

nn

n

µµσ

==

)(

In case of semiconductor, the drift current density due to holes is given by

eEpdriftJpP

µ=)(

Then the total drift current density

)()()( driftJdriftJdriftJpn

+=

eEpeEn pn µµ +=

pn

pn

epenE

driftJdrift

pneEdriftJ

µµσ

µµ

+==

+=)(

)(

)()(

For an intrinsic Semiconductor, n = p = ni, then

)()(pnii

endrift µµσ +=

Diffusion:

Due to non-uniform carrier concentration in a semiconductor, the charge carriers moves from a region of higher concentration to a region of lower concentration. This process is known as diffusion of charge carriers.

Diffusion of charge carriers

Drifting of charge carriers

x

Diffusion of charge carriers in a Semiconductor

Let Δn be the excess of electron concentration. Then according to Fick’s law, the rate of diffusion of electrons

x

nD

x

n

n ∂∆∂−=

∂∆∂−∝

)(

)(

)(

)]([

nx

eD

nx

De

n

n

∆∂∂=

∆∂∂−−=

The diffusion current density due to holes

)]([)( px

DediffusionJ pP ∆∂∂−+=

)( px

eDp ∆∂∂−=

Where Dn is the diffusion of electrons, the diffusion current density due to electrons is given by Jn(diffusion)

The total current density due to electrons is the sum of the current densities due to drift and diffusion of electrons

)()( diffusionJdriftJJ nnn +=

)(

)(

px

eDEpeJ

Similarly

nx

eDEneJ

ppp

nnn

∆∂∂−=

∆∂∂+=

µ

µ

Direct band gap and indirect band gap Semiconductors

• We known that the energy spectrum of an electron moving in the presence of periodic potential f ield is divided into al lowed and forbidden zones.

• In Crystals the inter atomic distances and the internal potential energy distr ibution vary with direction of the crystal .

• Hence the E-k relationship and hence energy band formation depends on the orientation of the electron wave vector to the Crystal lographic axes.

• In few crystals l ike gal l ium arsenide, the maximum of the valence band occurs at the same value of k as the minimum of the conduction band as shown in below, this is cal led direct band gap semiconductor.

Valence band

Conduction band

gE

k

E

k

E

gE

Valence band

Conduction band

• In few semiconductors l ike Sil icon the maximum of the valence band does not always occur at the same k value as the Minimum of the conduction band as shown in f igure. This we call indirect band gap semiconductor.

• In direct band gap semiconductors the direction of motion of an electron during a transit ion across the energy gap remains unchanged.

• Hence the eff iciency of transit ion of charge carriers across the band gap is more in direct band gap than in indirect band gap semiconductors.

Hall Effect When a Magnetic f ield is applied perpendicular to a current Carrying Conductor or Semiconductor, Voltage is developed across the specimen in a direction perpendicular to both the current and the Magnetic field. This phenomenon is called the Hall effect and voltage so developed is called the Hall voltage. Let us consider, a thin rectangular slab carrying Current in the X-direction. If we place it in a Magnetic field B which is in the y-direction.

Potential difference Vpq will develop between the faces p and q which are perpendicular to the z-direction.

i

B

X

Y

Z

VH

+

-

__ _

__ _

_

__ __

_ _

_

_

_

_ _ P

Q

N – type Semiconductor

Magnetic deflecting force

citydrift velo is vWhere

)(

)(

d

BvE

qEBvq

dH

Hd

×==×

Hall eclectic deflecting force

HqEF =

When an equilibrium is reached, the Magnetic deflecting force on the charge carriers are balanced by the electric forces due to electric Field.

)( BvqF d ×=

ne

Jvd =

The relat ion between current density and drif t velocity is

Where n is the number of charge carriers per unit volume.

BJ

E

netcoefficienHallR

BJRE

BJne

E

Bne

JE

BvE

HH

HH

H

H

dH

×⇒=

×=

×=

×=

×=

1),.(

)(

)1

(

)(

)(

If VH be the Hall Voltage in equil ibrium , the Hall Electric f ield.

IB

LVR

BdA

IRV

A

IJ

JBdRVd

V

JBR

JB

ER

d

VE

HH

HH

HH

HH

HH

HH

=

=

=

=

×=

=

=

sample, theof thickness theis L If

)(

density current area sectional cross isA If

1

slab. theof width theis d Where

• Since all the three quantit ies E H , J and B are Measurable, the Hall coeff icient R H and hence the carrier density can be find out.

• Generally for N-type material s ince the Hall f ield is developed in negative direction compared to the f ield developed for a P-type material, negative sign is used while denoting hall coeff icient R H .