semiconductors,band theory of solids,fermi-dirac probability,distribution function

1/15/2017 1Dr A K Mishra, Academic Coordinator, JIT Jahangirabad

Engineering Physics II Unit IV

Presentation By

Dr.A.K.Mishra

Associate Professor

Jahangirabad Institute of Technology, Barabanki

Email: [email protected]

[email protected]

SEMICONDUCTOR• The branch of engineering which deals with current con-duction

through a vacuum or gas or semiconductor is known as electronics.• Electronics essentially deals with electronic devices and their

utilization.• An electronic device is that in which current flows through a vacuum

or gas or semiconductor. • Principles of Electronics• The last orbit cannot have more than 8 electrons.• The last but one orbit cannot have more than 18 electrons

Structure of Elements• Atoms are made up of protons, neutrons and electrons.• The difference between various types of elements is due to the

different number and arrangement.

1/15/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 2

Continued………….

• The structure of copper atom is different from that of carbon atom and hence the two elements have different properties.

• The atomic structure can be easily built up according to atomic weight and atomic number of the element. Taking the case of copper atom,

• Atomic weight = 64• Atomic number = 29

∴ No. of protons = No. of electrons = 29No. of neutrons = 64 −29 = 35It has 29 electrons which are arranged in different orbits.


P=29N=35

Continued……………..

• An electron is a negatively charged particle having negligible mass. Some of the important properties of an electron are

• Charge on an electron= 1.602 ×10−19coulomb• Mass of an electron= 9.0 ×10−31kg• Radius of an electron= 1.9 ×10−15metre• The ratio e/m of an electron is 1.77 ×1011coulombs/kg.• This means that mass of an electron is very small as compared to its

charge• The total energy of the electron is the sum of (KE +PE ).• The energy of an electron increases as its distance from the nucleus

increases.


Continued……………

•


Energy levels increase as the distance from the nucleus increases.

vvv

vv

vv

vv

vv v

vvv

Energy level

NucleusShell 1

Valence Electrons• The electrons in the outermost orbit of an atom are known as valence

electrons.• The outermost orbit can have a maximum of 8 electrons i.e.the maximum

number of valence electrons can be 8.• The valence electrons determine the physical and chemical properties of a

material.• These electrons also determine the electrical properties of a material.• On the basis of electrical conductivity, materials are generally classified

into Conductors insulators and semi-conductors.• the number of valence electrons of an atom is less than 4 (i.e.half of the

maximum eight electrons), the material is usually a metal and a conductorExamples are sodium, magnesium and aluminum which have 1, 2 and 3 valence electrons respectively.


Band Theory of Solids

Band Theory:


No.1

No.2No.3No.4No.5

Electron energyNucleus containingPositively charge proton

v

v

Orbital electronNegatively charge

In isolated atoms the electrons are arranged in energy levels

Isolated atom

Ener

gy le

vel

Band Theory of Solids

• A useful way to visualize the difference between conductors, insulatorsand semiconductors is to plot the available energies for electrons in the materials. Instead of having discrete energies as in the case of free atoms, the available energy states form bands.

• Conduction process is whether or not there are electrons in the conduction band.

• In insulators the electrons in the valence band are separated by a large gap from the conduction band.

• in conductors like metals the valence band overlaps the conduction band, and in semiconductors there is a small enough gap between the valence and conduction bands that thermal or other excitations can bridge the gap. With such a small gap, the presence of a small percentage of a dopingmaterial can increase conductivity dramatically.


Continued……………………


Insulator(a)

Semiconductor(b)

Conductor (c)

Valance Band

Conduction Band

Large energy gapIn Insulators, at ordinaryTemperature no electronReach the conduction band

Valance Band

Conduction Band

Conduction Band

Valance Band

Energy of Electron

In semiconductor the energy gapSmall the thermal energy can bridge The gap for a small fraction. In conductor thereIs no gap in valance and conduction band, theyAlmost overlap each other.

Insulators, Conductors, and Semiconductors

materials are made up of atoms. These atoms contribute to the electrical properties of a material, including its ability to conduct electrical current.For purposes of discussing electrical properties, an atom can be represented by the valence shell and a core that consists of all the inner shells and the nucleus.This concept can be understand as Carbon is used in some types of electrical resistors. Notice that the carbon atom has four electrons in the valence shell and two electrons in the inner shell. The nucleus consists of six protons and six neutrons, so the 6 indicates the positive charge of the six protons. The core has a net charge of 4 (6 for the nucleus and for the two inner-shell electrons).


• Insulators: An insulators a material that does not conduct electrical current under normal conditions. Most good insulators are compounds rather than single-element materials and have very high resistivity's. Valence electrons are tightly bound to the atoms; therefore, there are very few free electrons in an insulator. Examples of insulators are rubber, plastics, glass, mica, and quartz.

• Conductors A conductor is a material that easily conducts electrical current. Most metals are good conductors. The best conductors are single-element materials, such as copper (Cu), silver (Ag), gold (Au), and aluminum (Al), which are characterized by atoms with only one valence electron very loosely bound to the atom. These loosely bound valence electrons become free electrons. Therefore, in a conductive material the free electrons are valence electrons.


• Semiconductors A semiconductor is a material that is between conductors and insulators in its ability to conduct electrical current. A semiconductor in its pure (intrinsic) state is neither a good conductor nor a good insulator. Single-element semiconductors are antimony (Sb), arsenic (As), astatine (At), boron (B), polonium (Po), tellurium (Te), silicon (Si), and germanium (Ge). Compound semiconductors such as gallium arsenide, indium phosphide, gallium nitride, silicon carbide, and silicon germanium are also commonly used. The single-element semiconductors are characterized by atoms with four valence electrons. Silicon is the most commonly used semiconductor.


Valence band,condunction band and Forbidden Bands

• Due intermixing of neighboring electron energy levels the permissible energy level increases. the splitting are so closely space, they form a virtual continuum which s called an energy band.

• The band of energy develop by the splitting of valence electron and occupied by the valence electrons is called valence energy band or Valence Band. may be completely/partially filled at any temperature.

• The permitted energy band next higher to the valence band is called Conduction band (lowest unfilled energy band).may be empty or partially filled.

• In between the conduction band and valence band there is a region of energy gap known as Forbidden band or Band gap. formed by a series of close levels above the top level of valence band and below he bottom of conduction band ,forbidden band is free from electrons i.e. electron can not exist in this band.


Empty or Partially filled


Valence band

Conduction band

Forbidden Energy gap0.1 - 1 eV

Fully or Partially filled

Ener

gy (e

V)

Ener

gy (e

V)

Valence band

Conduction band

Forbidden Energygap

Energy Gap semiconductors silicon germanium0.1 – 1 eV 1.1 eV 0.7 eV(1s,2s,2p,3s) (3s + 3p)

3,3,2,2,1 22622

pspss 4,4,3,3,3,2,2,1

221062622

psdpspss

Band Gap

• When an electron acquires enough additional energy, it can leave the valence shell, become a free electron, and exist in the conduction band. The difference in energy between the valence band and the conduction band is called an energy gap or band gap. This is the amount of energy that a valence electron must have in order to jump from the valence band to the conduction band. Once in the conduction band, the electron is free to move throughout the material and is not tied to any given atom.


Comparison of a Semiconductor Atom to a Conductor Atom

• Silicon is a semiconductor and copper is a conductorSilicon have +4 valance electron and copper has +1 means silicon atom want more

energy to become freeThan copper.

Since copper atom has more energyThan silicon because it is far From nucleus than silicon.easy to become free for copper Atom by acquiring additional Energy ,even at room temperature they become free.


+14 +29

Silicon(a)

Valance Valance

Copper(b)

29

Covalent Bonds • A silicon (Si) atom with its four valence electrons shares an electron with

each of its four neighbors• This effectively creates eight shared valence electrons for each atom and

produces a state of chemical stability.• this sharing of valence electrons produces the covalent bonds that hold

the atoms together. each valence electron is attracted equally by the two adjacent atoms which share it.


v----

4

4

4

4

4

--

--

--v----

----

----

----

-----

---

v----

v---- v

----

v--

--

v----

v---

v----

v----

v--

Density of states

Since maximum spacing between consecutive energy level in solid is > eV.Large number of discrete energy levels exist in a small interval of energy dE. Each energy level consist two state and each state have only one electron.The energy levels are filled as per paulis exclusion principle according to which energy level can accommodate two electrons, one with Spin up and other with Spin down.The density of state shows the number of states within a given interval of energy that the electron may possess.If f(E) denote the average number of electrons that occupy a single quantum state of energy E.Let g(E) dE be the number of quantum state of the system whose energy lies in the range dE, the the number of electrons of the system whose energy in the range dE is given by,

N(E) dE = f(E) g(E) dE


10 6

Continue…………….

• The f(E) defined as distribution function depend on the probabilities of the distribution of electron available n quantum state.

• The g(E) depend on the quantum state of the system in energy and can be calculated using Schrödinger equation.

• Let each point with integer values of coordinate represents an energy state.


nE

E + dE

n x

ny

n z

Continued……………• A spherical surface of radius n centered on origin known as radius vector.• All point have equal energy known a constant energy space.• Thus n represent the number of state is equal

to the numerical value of volume expressed in unit cubes of lattice parameter. therefore

the number of available state in this shell of octant enclosed between E and E+ dE is obtained by dividing ,the volume of octant of the shell by the volume associated with single state i.e,

Where V is the volume of single state.the correct number of possible state is twice because orbital have two electron with opposite spin ( ).


)n n n ( , , 2z

2y

2x

2y nn nn zx

)V

dn 4 (

81

2

21

Continued……….

• Two electron in orbital are in different state, therefore if g(E) denotes the density of state, then


as expressed is a side ofbox rigid ain bem,electron theof massbox.let ldimentiona onein particlen applicatio

using analyzed becan lattice ionic ithin theelectron wan

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

V

dm) (4 81

2 dE )(

n

n

2

2

dnV

Eg

Eg


• We get


dE

But

dE

dE

dEdE

E

h

h

hE

h

2

2

dE g(E)

V

2

2

2V dE g(E)

4m

2

2

V dE g(E)

get we(1)in (3) (2)equaion from values theputting

...(3).......... 4m

ndn or 8m

2ndn

givesation differenti

(2).......... E 8m

or 8m

E)8m(

a

E)a8m(

ha)a8m(

haa

hana

hn

212

3

3

21

23

2

221

2

2

2

2

2

22

2

22

Fermi-Dirac distribution• They derived an equation in 1926 independently called Fermi-

Dirac probability distribution function F(E),is govern the distribution of electron among the energy level as a function of temperature is given by


levelenergy empty for , 0 F(E)electronby filled is levelenergy if , 1 F(E)

meanselectron an by occupied E levelenergy particularat y probabilit theindicate F(E)function the

compared. becan energy otheer h the with whicreference a as taken becan

energy Fermi theis and constant,Boltzman theisk

E 1

1 (E)

Ee

F

)KT

- E( F

where

F

Continued………..• At T =0 K ,lower energy level of conduction band are occupied by electrons

wile upper level is unoccupied in Fig (a).


Distance (b)(a)

Ener

gy (E

)

Ferm

i-fu

nctio

n F(

E)

Energy(E)

T = 0 K

EF

1

0

Continued………..

• It is clear that energy level below are occupied while above areunoccupied. So is the maximum energy of the filled level. Fermi level is defined as the highest filled energy level in a conductor at o K and Fermi level is maximum energy that an electron can have in a conductor at absolute zero temperature.


EF EF

EF

EF

unfilled. levelenergy means 0 1

1 11 F(E)

therefore. toequal becomes KT /) - (Efunction theand positive is - E , Ewhen

electrons.by occupied are below levelenergy theall means 1 F(E)

1 0 1

1 11 F(E)

. - toequal becomes KT /) - (Efunction lexponentia

and negative is - E , Efor andK o T

e

EEE

e

EEE

FFF

F

-

F

FF

E

therefore

At

Charge carrier density (electrons and Holes)in intrinsic semiconductor

•Charge carrier concentration is an important tool for the knowledge of conductivity in semiconductor. In diagram intrinsic semiconductor superimposed with Fermi-Diracdistribution function.At room temperature few electronin valence band excited, cross the forbidden energy gap and enter

Into the conduction band as a resultSome of the bottom filled near .some of the state near arefilled with holes.


Valence Band

TTTa

Conduction Band

T = O KEg

Ec

EF

Ev

E c

E v

Continued………………

• If the conduction band posses infinite number of energy levels, the density of electron in conduction band whose energy lies between E + dE,is given by


constant.planks ish andelectron of mass effective theis

)2.......(..........c *e 4 (E) Z

bygiven is Z(E), statedensity energy Theelectron. ofy probabilit givesfunction fermi is (E) F

)........(1..........(E)dE..... F (E)

m

)E - (E)m 2(he

21

23

3

C

where

where

EZn e

Continued………

• The Fermi distribution function is given by


constant. is

)4.....( / x 6.82 *4but

E

1 E11 x c *4

1 E11 x c *4

get we(1), equns.in (3) and (2) equns from f(E) and Z(E)of values thePutting

......(3).................... 1

E1 )(

(eV)m10)2(h

e

)E - (E)2(h

e

)E - (E)2(

e

23

32723

3

)KT

- E(

21

23

3

)KT

- E(

21

23

3

)KT

- E(

cF

Fc

F

where

e

dEe

dEeE

EF

m

mn

mhn

e

e

Continued………

• therefore


E 1

Ec

(5).Henceequation in neglect can weso1

E , - E , EFor

.(5).................... E

1

Ec

c

F

F

c

F

e)E - (E

eEe

)E - (E

)KT - E

(21

)KT - E

(

FF

)KT - E

(

21

dE

thereforeKT

dE

n

E

n

e

e

Continued…………….

• Hence


get we(6),equation in dx KT dE and x KT

E) - ( putting

....(6).......... E

E c

EE

E

E

EEc

E

EEEc

E

e)E - (Ee

ee)E - (E

e)E - (E

c

KTE) - (

21

)KT

- ((

KT

E) - ()

KT

- ((

21

)KT

E) - ( - ((

21

c

c

cF

c

ccF

c

ccF

dE

dE

dE

n

n

n

e

e

e


• We get


tor.semiconduc intrinsican of band conductionin density electron free is

)8.(..........EE

2

*KT e 2

EE

2 *

e 4

get, we7 equnsin 4 equns from of value theputting

...(7).......... 2

EE

integral) (stadard 2

EE

x, E when and 0 x, E ,

E

EE

e)h

m2(

e)()m(2hn

e(KT)n

ex

exe(KT)n

E

e(KTx)e

)KT

- ((

23

)KT

- ((2

1

23

23

2e

)KT

- ((2

3

e

21

x-

0

21

x-

0

21)

KT

- ((2

3

e

c

21

)KT

- ((

cF

cF

cF

cF

c

cF

this

dxbut

dx

When

dx

n

KT

n

e

xKT

e

Position of Fermi level in intrinsic semiconductors

• Width of conduction band and valence band is small as compared to forbidden energy gap.

• All energy level in one band is same energy.• At o k no conduction because at o k valence band is

completely filled while conduction band empty and semiconductor behave as insulator.

let at any temperature T K, the number of electron in conduction band is and in valence band is .total no.the number of conduction band is is given by


nc nv

.....(1)..........).........NF( Ecnc

vn vc N

Continued………………• Where F (Ec) is the probability of electron having energy Ec. • According o Fermi-Dirac ,the probability distribution function F(Ec) is given

by


EE

1

N

EE 1

N N

......(4)..........

EE 1

N

is )( band in valenceelectron ofnumber thesimilarly

......(3)..........

EE 1

N

get )we (1in (2) from )F( of value thePuttingconstant.Boltzman theisK andkelvin

in e tempraturis Tlevel, fermi ofenergy theis where

..(2)....................EE 1

1 )(

eenn

en

ne

nE

Ee

E

KT

) - (

KT

) - (vc

KT) - (v

v

KT

) - (c

c

F

KT) - (c

FvFc

Fv

Fc

Fc

F

Continued………………• now


eeeee

eeee

eeee

KT

) - (

KT

) - (

KT

) - (

KT

) - (

KT

) - ( ) - (

KT) - (

KT) - (

KT) - (

KT) - (

KT

) - (

KT

) - (

KT

) - (

KT

) - (

EE

EE 2 1

EE

EE

EEEE

EE

EE 2

EE 1

EE 1

EE 1

EE 1

EE

1 EE

1 1

FvFcFvFcFvFc

FvFcFvFc

FvFc

FvFc

Continued………………• Now


band. conduction and bandlencebetween va gapenergy forbidden of middle in theexactely liestor semiconduc intrinsicin level Fermi

)5...(....................2

)

0 )2-

l log KT log)2-

get wesides,both of logarithm Taking1

EEEE

E(EE

EE(EEE(E

e

vcF

Fvc

Fvc

KT

) - - (FvFc

Hence

e

Position of Fermi level in extrinsic semiconductors

• Here the situation is different due to impurity of trivalent and pentavalent. the number of free electron depend on the impurity atom added. Extrinsic semiconductors are two types:n-type and p-typePosition due to n-type:Let Ed and Nd be the energy andnumber of electron of thedonor level respectively.


Valence Band

Donorenergy level

Conduction Band

Ed

Ec

Ev


• The number of vacancies per unit volume in the donor level is


ee

eeN

eN

1 KT

-

KT

-

1 KT

-

1- 1 KT

-

d

1 KT

- d

EE

EE donorempty ofDensity

EE

EE donor empty ofDensity

electronsby occupied be toE stateenergy of probabilty is (E) F

EE

1 - 1 EF - 1

Fd

Fd

Fd

Fd

Fd

N

N

d

d

where

Continued……………..• Now


. lawon distributi Dirac- Fermi thefollow band valencein the holes and band conduction in theelectron

)3.........(..........EE

i.e (2)equation ofr denominato eneglect thcan weso i.e , leveldonor

theabove KT few than more lies that assume be

)2......(..........EE

1

eNn

EE

eN

KT

-

dd

F

F

KT

- d

Fd

Fd

The

KT

can

E

n

d

d


• The density of the electron in conduction band can be expressed as


)5....(..........

EE

EE2

*e 2

(3) &(2) fro level,donor in eunit volumper vacanciesofnumber toequal is band conductionin eunit volumper electron of

number theband. in theelectron of mass theis

)4(....................EE

2

*e 2

eNeh

KT m2

m

eh

KT m2

KT

-

dKT

- 23

*e

KT

- 23

FdcF

cF

where

ne

• Taking log on both side, we get


hKT m2

NlogEEE

hKT m2logNlogEEE

ENEEh

KT m2

2

*e 2

KT

- - 2

2

*e 2 -

KT -

- KT

-

KT -

logKT

-

2

*e 2 log

23

de

cdF

23

edeFcF

Fd

cF

23

E

E

d

dee

Continued………..


level.donor and band conduction theof bottom ebetween thin i.e , gapenergy theof middle theabove liestorsemiconduc type-nin level Fermi clear that is (6)

)6.......(

2

*e 2

2

KT KT

hKT m2

NlogEE23

de

cd

Equation

EF

Position due to p-type


level.accepter and band Valence theof top ebetween thin i.e , gapenergy theof middle thebelow liestorsemiconduc type-pin level Fermi clear that is (7)

)7.......(

2

*h 2

2

KT KT

hKT m2

NlogEE23

ae

vd

Equation

Similarely

E F

Conductivity of semiconductors• When a potential difference is applied across the

semiconductor block of length l, then electron in conduction band and holes in valence band move in mutually opposite directions with velocity respectively.

• If be the electron and hole currents then total current


v h & v e

I h & I e

hole. & electroboth on charge theis e where

2).........()......... (A e Ior I

and then band, valence

and conductionin eunit volumper hole andelectron ofnumber thebe &

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

nnnnnInI

nII

hehe

hhee

e

he

vvvAvAvAvA

n

heheee

heee

hif

I

Continued………….

• If be the resistivity of the material then the resistance offered to flow current is, .


Al R

e e

V

A

l


• Since V=IR equation (2) becomes


)( and ) ( therefore

mobility, as defined is field electricunit per city drift velo

)4.....(..........).........( e ) ( e 1

) ( e E

) ( e V E

bygiven isblock torsemiconduc across upset field electric thetherefore

....(3) ) ( le Vor ) (A eA

l V

he

he

he

he

hehe

nn

nn

nn

nnnn

EE

butEE

l

vv

vvvv

vv

vvvv

he

he

he

he

hehe


• Where


ure.in temprat rise with decreasesy resistivit whileurein temprat rise with increasestor semiconduc ofty conductivi

theure,in temprat rise with increases &

)6........(....................e e

istor semiconduc ofty conductivi thethereforey,resistivitelectrical toreciprocal is ty conductivi electrical the

)5...(....................e e 1get we(4), eqin value theputting

lyrespective holes andelectron ofmobility theare &

nnn

nn

e

he

he

nhhe

he

he

Since

Since

Temperature dependence of conductivity in semiconductors

• In intrinsic semiconductor two type of charges (electrons & holes),let be the sum of conductivity due to free electron and holes. total conductivity


h and

n

constant.ality proportion

are & where and

ismobility holes andelectron for

constant.ality proportion is a whereA or 1

as ureon temprat depend holes andelectron ofmobility the

....(3))......... ( e

get we(1)eqn in valu his utingion.concentrat itrinsic where)2(....................

holes. of No. toequalelectron of No.tor semiconduc intrinsicin since

holes. andelecron

on charge thebe emobility are & where

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

TT

TT

n

nnnn

nn

23-

h23-

e

23-

23

heii

iihe

he

hehn

is

hei

Continued…………

• Putting he above value in equation (3)


gap.energy forbidden and x 4.83 C

)5....(*h

*e

2 2 C where C

form theof iselectron free and holes ofion concentrat intrinsic theknows we

constant.another is where

4).........( or ) (

E10

m)h

k2(eTn

TnTn

g

21

432

3

2KTEg-

23

i

2-3

i2-3

i

iswhere

ee

m

ii

Continued…………..

• Putting this value of from (5) to (4) we get


ni

)8........(..........E

as written bemay ) (6eqn P

then, Tat ty conductivi edextrapolat theis if ture.ith tempralinearly walmost decreaesty conductivi

metalsin . decreasesy resistivit lly whileexponentia increasesty conductivi theincreases e tempraturshows (7) & (6)

)7...(..........1 B where,E

B

E 1

istor semiconduc intrinsican for y resistivit

)6...(.......... e C where, E

e C

e

e

e

e

2kT

-

2kTii

2kTii

2kT

-

g

g

g

g

eqn

electricali

Continued……………

• Taking log on both side we get


loge

loge

2kT - Egslop

T1

ure.in temprat changeon based devicesgcontrollin andpower frequency microwave of

mesurment in relay, thermala as,themometryin used are which sThermistorin utilized is

urein temprat risetor with semiconduc oftyconductiviin increase ofproperty the

tor.semiconduc intrinsican of gapenergy ofiondeterminat of method esuggest th

2kT - is slop whoselinestraight

T1 and between plot

2kT -

(8)eqn sideboth of log

Elog

Eloglog

g

e

g

ee

this

is

the

taking