dr. p. sagayaraj state physics unit … · space of the boundaries of the zones; these positions...

Solid State Physics Module-II

THEORY OF METALS AND SEMICONDUCTORS

E- Content study material compiled by

Dr. P. Sagayaraj

Associate Professor of Physics, Loyola

College (Autonomous), Chennai - 34

Unit-2 Theory of metals and semiconductors, Dr P Sagayaraj 2.1

⁄

BRILLOUIN ZONES

In the propagation of any type of wave motion through a crystal lattice, the

frequency is a periodic function of wave vector k. In order to simplify the treatment of

wave motion in a crystal, a zone in k -space is defined which forms the fundamental

periodic region, such that the frequency or energy for a k outside this region may be

determined from one of those in it. This region is known as the Brillouin zone (sometimes

called the first or the central Brillouin zone). The concept of Brillouin zone provides a

way to understand the origin of allowed and forbidden bands in solids.

The energy discontinuities in the monatomic one-dimensional lattice occur

when the wave number is

= (1)

where n is any positive or negative integer.

The equation determining the position of the energy discontinuities which actually

occur will depend somewhat on the type of the crystal lattice.

Fig. 1Brillouin zones of a linear monatomic lattice with lattice constant a


ting

/

the

as

/

sho

<

w Fig

/

.

nes

/

; the

<

s t −

/ an /

<

< the

/

first

for

ne

+

d by t he

=

soluti ons

(

of t

+

e equ

)/

ation

tin

= ±

± 1

⁄

, n2

ing

= ± ⁄

In the one-dimensional monatomic lattice a line represen value of is

divided up by the energy discontinuities into segments of length n in

1. The line segments are known as Brillouin zo egmen

is Brillouin zone; the two segments −2 < − d <

2 m the second Brillouin zone, etc. The zone description was introduced by

Brillouin, who pointed out that many important and characteristic features of electron

propagation in periodic structures could be described by considering the positions in k-

space of the boundaries of the zones; these positions are independent of the details of

the electron lattice interaction, being determined instead by the crystal struct ure. The

Brillouin zones of a simple square lattice in two dimensions are shown in Fig. 2 The

zone boundaries are determi

(2)

Here n1, n2 are integers and a is the lattice constant. The equation essentially expresses

the Bragg law for reflection of a wave by a periodic lattice. We find the boundaries of the

first zone by first set g n1 = = 0, obtaining

(3a)

and then setting n1 = 0, n2 = ±1, obtain

(3b)

The four lines of the above two equations determine the boundary of the first zone.


in

± = 2 /

es

Fig. 2 The first three Brillouin zones for a two-dimensional square lattice

The outer boundary of the second zone is determined by setting n1 = ± 1, n2 = ±1,

obtaining the equations of the four l

± (4)

where the signs are independent.

ELECTRONS IN A PERIODIC POTENTIAL

An electron passing through a crystal structure experiences a periodic variation in

potential energy, caused in a metal by the positive cores of the metal ions. In sodium, for

example, the ion cores are singly charged, with 10 electrons in the configuration ls22s22p6,

while the outer electron, which in the free atom is the 3s valence electron, becomes in the

metal a conduction electron. The periodic nature of the potential has far-reaching

consequences for the behavior of the conduction

electrons:


.

or t e ri dic p

.

Nature of the wave functions: The plane wave solutions for the wave functions of

the free electron model go over f h pe o otential to solutions of the form

= ( )

Where, uk(r) has the periodicity of the lattice. Wave functions of this form are called

Bloch functions and are basic to the theory of metals.

Allowed and forbidden bands: On the free electron model all values of the energy were

allowed; but in a periodic potential there are forbidden ranges of energy (Figs. 1 and 2)

where solutions representing an electron moving through the crystal do not exist.

Fig. 1 Dependence of energy levels upon lattice constant, for a line of 6 hydrogen atoms,

showing the incipient formation of allowed and forbidden energy bands


lly a qu

= (

adr

⁄2

tic fu

)

nction

fo ion

⁄ = ⁄ ∗. I

Fig.2 Allowed and forbidden energy bands. The energy is plotted as a function of

the wave number k; the dashed line shows the connection with the case of

entirely free electrons

Fig. 2 shows the allowed and forbidden vibrational frequencies of a diatomic crystal in

one dimension. We know also that x-rays do not propagate through crystals at certain

frequencies and orientations, but instead are reflected. In fact, the Bragg equation

determines the occurrence of the forbidden electronic bands, as waves satisfying the

Bragg condition are so strongly reflected that they cannot propagate in the crystal.

EFFECTIVE MASS

Near the top or bottom of a band the energy is genera a of

the wave numbers, so that by analogy with the express r free

electrons we may define an effective mass m* such that t may be

shown that the motion of a wave packet in applied electric or magnetic fields is

characterized by using m* as the mass. Near the top of a band m* is negative, so that here

the motion corresponds to that of a positive charge.


ecti

⁄

ve m

. Due

The effective mass concept does not imply that Newton's second law =

fails for metals. The effective mass applies to the acceleration of electrons relative to the

crystal lattice. Newton's law holds when applied to the entire system of electrons plus

lattice, thereby allowing for momentum transfer between the accelerated electron and the

lattice.

The variation of effective mass with k is shown in Fig.1

Fig.1 The variation of energy, velocity and effective mass with k

The concept of the negative eff ass may be understood in terms of the Bragg’s

reflection when k is close to ± to the Bragg’s reflection, a force applied in

one direction lead to a gain of momentum in the opposite direction which results in the

negative effective mass.


+ ( − ħ

)

sol tions

(

of t

)

he

.

+ 2 + ħ

( − − ) = 0

The effective mass is a new concept and arises because of the interaction of the electron

wave packet with the periodic lattice. It also has a physical significance. It provides a

satisfactory description of the charge carriers in crystals.

KRONIG-PENNEY MODEL

We demonstrate some of the characteristic features of electron propagation in

crystals by considering the periodic square-well structure in one dimension (Fig. 1).

Fig. 1 Kronig and Penney one-dimensional periodic potential

The wave equation of the problem is

(1)

The running wave solutions will be of the form of a plane wave modulated with the

periodicity of the lattice. We obtain u form

(2)

where uk(x) is a periodic function in x with the period (a + b) and is determined by

substituting (2) into (1):

where, Ek = ħ2k2 /2m.

(3)

= 0


uation

=

has t

(

he

) +

tion

( )

= ⁄ ) ⁄ .

< < + the

= ( ) + ( )

= ( − )⁄ ) ⁄

inear ho

+

mog

=

eneou

+ ;

s eq

( ) − ( ) = ( − ) − ( + ) ; −

( )

+

( ) ( ) ( )

( ) ( ) (

+

) ( ) −

+

) ( )

;

+ ) ( )

si n

ly if the d

sin

eterm

+ co

nt of the c

cos

oeffic

= c

nts v

(

ani

+

shes

)

ina ie

sh os

In the region 0 < x < a the eq

provided that

(4)

(5)

In the region

Provided that

(6)

(7)

The constants A, B, C, D are to be chosen so that u and du/dx are continuous at x = 0 and

x = a, and by the periodicity required of u(x) the values at x = a must equal those at x =

- b. Thus we have the four l uations:

=

− + = ( − (

(8)

These have a solution on , or

(9)

solu

(2

solution is

(2


→ →

=

mes

+ c o = c

+ c o

ig. 2

, for = ⁄2

2

3

ħ f

.Th

or

rbidden

(

r

=

= /

lim

os

In order to obtain a handier equation we represent the potential by a periodic delta

function, passing to the limit where b = 0 and Vo = ∞ in such a way that β2b stays finite.

We set

(10)

So that the condition (9) beco

(11)

This transcendental equation must have a solution for a in order that wave functions of

the form (1) should exist.

F

Fig. 2 Plot of the function s e allowed values of the

energy W are given by those ranges of α = which the function lies

between +1 and -1.

If P is small, the fo anges disappear. If P —» ∞: the allowed ranges of

αa reduce to the points ±1, ±2, ±3 … . ). The energy spectrum becomes

discrete, and the eigen values

8

are those of an electron in a box of length a.


uation c

8 +

be writt

[

en as

− ( ) ]

( ) m

( ) = ( + )

( ) = ( ) ( )

( ( . )

respe

(

ctiv

) an

y. h ent th

(

e fr

) or

ee le

) = ( )

BLOCH THEOREM

In order to understand the difference between the conductors and insulators, it is

necessary to incorporate the variation of potential inside the crystal due to the presence

of positive ion cores in the free electron model. The potential is minimum at the positive

ion sites and maximum between the two ions. The corresponding one dimensional

Schrödinger eq an

= 0 (1)

Where the periodic potential ay be defined by means of the lattice constant “a”

as

(2)

Employing the periodic potential the one dimensional solution of the Schrödinger

takes the form

(3)

In three dimensions it is given by

(4)

The eqns (3) and (4) are known as Bloch function in one and three dimensions,

el T ey repres e ctron wave modulated by the periodic function d

( ), where ( ) is periodic with the periodicity of the lattice

in one and three dimensions, respectively. Therefore, considering only the one

dimensional case and suppose if we have N (even) number of atoms in a linear chain of

atoms of length L, then we can write,


( ) = ( + )

an

(

depe

+

nds on th

) =

e e

(

xac

+

t nature o

) [

f the pot

( +

e ntial fie

)]

( ) (x)exp(ik ( ) ( ) (6)

as

∗ + ) = (− ) ∗ ( )

ives

(

us

+ ) ∗ ( + ) = ( ) ∗ )

(6) gives us

( ) = 1

= 2 × = 2

2 =

2 = (9)

, we here

This is frequently referred to as Bloch condition. Similarly, the complex conjugate of

eqn (6) can be written

( (7)

The eqs. (6) and (7) g

( (8)

This indicates that the probability of finding the electron is same everywhere in the

whole chain of atoms, i.e, it is not localized around any particular atom but is shared

by all atoms in the chain. Thus eqn.

This will be true only if

W n = 0,±1, ±2, … and L is the length of chain of atoms. When n =

have k = , which is the edge of the first Brillouin zone. When L is large, the allowed

values of k would come close together and their distribution along k-axis becomes

quasi- continuous. The total number of allowed k values in the first zone is

(5)

ld . From eqn (3) and (5) we have

x) =


2 ⁄ = =

2 ⁄ =

nction

( )

is zero. i.e

= 0

u

∫

aves,

) = exp( )

( ) = 8

(10)

This is equal to the total number of atoms in the chain of atoms (or in the unit cell in

three dimensions).

THE NEARLY FREE ELECTRON MODEL

In this model, the crystal potential is assumed to be very weak as compared to the

electron kinetic energy so that the electrons behave essentially like free particles. The

weak periodic potential introduces only a small amount of perturbing effect on the free

electrons in the solid. Therefore this model demands the application of very elementary

perturbation theory. It is convenient to choose the zero of energy so that the mean free

value of the potential f .,

(1)

where a is the periodicity of the lattice. The unperturbed wave functions

corresponding to V= 0 are the plane w

∅ ( (2)

where the wave functions are normalized over a microcrystal containing N atoms. The

unperturbed electron energies are

(3)

This is free electron case where k can take any value. Now suppose we take in to

account periodicity of the lattice keeping V = 0 and restricting k to lie in the first

Brillouin zone only. Then we have,


( ) = exp( ′ )

′ = + = + ,

( ′) = 2

+

≤ = 0 ± 1

iodi

(

c pote

) =

l, the

) +

ave fun

′ (

cti

)∅

′ ( ) ar

are giv

′ ( )

en

=

′| |

) −

>

( ′

<

( )

′| | > ∗ ∅

d to sec

( )

ond

=

orde

( )

r

+

s

′| |

) −

> |

(

|<

( ′)

he lattic

( ) =

rms of

ex

he Fourier series,

) and ∗ =

it ca

′| | ≥

p(

if − ′ =

(4)

Where and

(5)

Where , ±2, … … …

As we introduce the per ntia real w on becomes

∅ ( ∑ ′ ′ (6)

where e constants and are found to be small from perturbation theory. Their

values corrected to the first order by

(7)

Where < = ∫ ∅ (8)

The perturbed energy, correcte i

(9) ′

Because of eqn (2), the first order term becomes zero.

Now let us take in to account the real periodic potential which may have the same

periodicity as that of t e. In te n be expressed as,

∑

Then the integral (8) becomes <

= 0 ; otherwise (10)


ly, t

(

e wav

) =

nction becomes

) ∑

√ ( ) ex ) exp

= = ( )

( ) de ghtly de

( )

cr

−

eased o

( )

r

,

incre

which he diffe

( )

re

−

nce

( ) =

8 [ −

i

+

s giv

) ]

−

(

2 + =

4 + −

2

2 +

y

= / = ′ = /2

is theo

an

neve

).

r the

Near

sary

(

to write

) = exp )+ exp ( + )

an are

4)

ts. Th

=

eqn

. ear the

( )

b

−

oundar

( )

y

+ ( )

±

]

Su

ex

stituting eqn (14

) + [ ( +

) in eq

) −

n (1) w

( )

e

( ) ] exp ( + ) =

Consequent h e fu

( p(− (11)

With, 1 and (12)

The energy is sli ased slightly with respect to pending

on t en by

=

8

= − (13)

At or near the zone boundar

− − −

Where an integer. The perturbation ry there for break down whe wave

vector is an integral multiple of ± d the wave is reflected (k= − a

zone boundary, it is neces

( (1

Where d constan e (14) means that one Fourier coefficient

will be large n b obtain

[ p(

+ 0 (15)


) or by exp ( + ) ] an

ultaneo

( )

us

−

eqns.

( )] + = [ ∗

+[ ( + ) − ( )]

0

= 0

ic eqn h

( ) =

two so

(

lu

+

tion

)] ( ) − ( + )} + | ]

=

[{

an ( ) = ( +

4|

). T

= + | |

= − | |

= − =

If we multiply eqn (15) either with exp( [− d integrate,

we obtain the following sim

(16)

From a non trial solution, the determinant of coefficient must vanish. The resulting

quadrat as s

2 [ ± (17)

At the zone boundary, ± d herefore the two energies

are

Thus the energy gap at the first zone boundary (n=1) is ∆ 2

Fig. 1 Energy band in the NFE model (a) the extended zone scheme, (b) the

reduced zone scheme


d the s

⁄

econ

| =

ndarie

=

n

( )- ( + ) ≫ , ) ≅ ( ) ( + ), i

The Fig (1) shows the energy gap at the first an d zone bou s i

extended and reduced zone schemes. Also in the case,| 1, so that at ±

the wave function is a superposition of two plane waves travelling opposite directions

and thus represents a standing wave.

Near k = 0, the difference in the unperturbed energies is so large that the

perturbation effect is negligible and hence there is no effect on the curve in this region.

When then ( .e., away

from Bragg reflection, the energy values are practically plane waves, the same as that of

free electrons.

ZONE SCHEMES

The wave vector representation of a plane wave Eigen state is simple and

unambiguous while the representation of the Bloch states are not so because the Bloch

function is not a simple plane wave but a modulated plane wave. To represent such sates,

three different schemes are commonly used. They are called the extended zone scheme

(or the Brillouin zone scheme), the reduced zone scheme and the periodic zone scheme

(or the repeated zone scheme). All the three schemes represent the identical physical

behavior.

Extended Zone Scheme

Consider a one dimensional lattice in which the energy of an electron is being

slowly increased so that the value of k is also increased. When the value of k becomes


hat th

=

e Bragg

=

large enough, the wavelength becomes small enough as k = 2π/λ, the electron will

suffer a Bragg reflection following the Bragg condition 2d sin θ = nλ.

For a one dimensional lattice, d= a so t reflection will occur at

where ±1, ±2, …..

As a result of these reflections energy gaps are developed in the free electron

parabola as shown in the Fig.1. This representation of energy as a function of k is known

as extended zone scheme. Physically this scheme is very close to the free electron scheme

and differs only at the zone boundaries. However, like free electron case, the k-space is

infinite in the extended zone scheme but is dissected by the planes of energy discontinuity

in to segments called Brillouin zones. The extended zone scheme represents various

Brillouin zones in k-space.

Fig.1 Extended zone scheme


y fr

(

ee ele

) =

tron wave is g

( )

iven by

ssion i

( )

s

=

8

te eq

(

n (1)

) =

n the form,

( ) = ( + ) exp( )

′ ) ex

−

) = exp( p(−

Reduced Zone Scheme

Fig.2 Reduced zone scheme

For a free electron case, the ordinar c

(1)

And the corresponding energy expre

(2)

A plot between E versus k gives the free electron parabola (Fig.3).

Let us now wri i

(3)

Where ′= k+g is the reduced value of the original wave vector k. Since, the second term

on the right hand side of eqn (3) is periodic function in the lattice. It is the limiting case

of a vanishingly small periodic potential under this assumption one must


parab

=

la

⁄

in k-

(Fi e energ

( ′) =

y express

( +

ion i

) =

he uced zone

2 +

sche me

= 0

( ) i s

place

=

the part

/ as

t red

take in to account the symmetry requirement of the periodicity. The general demand

of the periodicity implies that the possible electron states are not restricted to a simple

o space, but can be found equally well on parabolas shifted by a vector

g. 3b). Th n becomes

(4)

Where , ±1, ±2, … ..

Since, the behavior of periodic in k-space and hence follows the

symmetry properties, it is sufficient to represent this in the first zone only. In order to

achieve this, let us dis of the parabola of interest linearly by the

appropriate multiple of 2 shown by arrows in Fig. 3b. The E versus k

curves for several values of n reduced into the first zone for a simple cubic lattice with

vanishing potential are shown in Fig.3c. Each value of n defines a Brillouin zone in the

reduced zone scheme representation.


versus plo

porta

val

elec

app

a crystal

/ .

be

Fig. 3 (a) Energy as a function of wave vector for a free electron (b) Dispersion

curve with the translational symmetry and the various bands (c) The same

dispersion curve in the first zone only

Periodic Zone Scheme

In the preceding section, we obtained the t in reduced zone

scheme by translating the desired portion of the neighbouring parabolas into the first zone

as the energies obey translational symmetry. A reverse process is equally probable.

Accordingly, we can translate the desired portion of the first zone to any or every other

zone, we can obtain a periodically repeated zone as shown in Fig. 4. This construction is

known as periodic or repeated zone scheme. This representation leads us to a very im

nt result that the trons in have like free electrons

for most of the ues except when roaches


ril are

;

one given b

2 4 ± ; …

y:

… ± = 0;

/

±

= / , f o We cut th

/

e series

= − /

of

is n nted

/

as an

. Th

nnecte

, th

to

With

in

Fig. 4 Periodic Zone Scheme

Band model of metal, semiconductor and insulator

Consider a linear crystal constructed of an even number N of primitive cells of

lattice constant a. In order to count states we apply periodic boundary conditions to the

wave functions over the length of the crystal. The allowed values of the electron wave

vector k in the first B louin z

f at r this is the zone boundary. The point

− ot to be cou independent point because it is co d

by a reciprocal lattice vector with e total number of points is exactly e number

of primitive cells.

Each primitive cell contributes exactly one independent value of each

energy band. This result carries over into three dimensions. account taken of the two

independent orientations of the electron spin, there are 2 dependent orbitals in

each energy band. If there is a single atom of valence, one in each primitive cell, the


band can be half filled with electrons. If each atom contributes two valence electrons to

the band, the band can be exactly filled. If there are two atoms of valence, one in each

primitive cell, the band can also be exactly filled.

Metals and Insulators

If the valence electrons exactly fill one or more bands, leaving others empty, the

crystal will be an insulator. An external electric field will not cause current flow in an

insulator. (We suppose that the electric field is not strong enough to disrupt the electronic

structure.) Provided that a filled band is separated by energy gap from the next higher

band, there is no continuous way to change the total momentum of the electrons if every

accessible state is filled. Nothing changes when the field is applied. A crystal can be an

insulator only if the number of valence electrons in a primitive cell of the crystal is an

even integer.

If a crystal has an even number of valence electrons per primitive cell, it is

necessary to consider whether or not the bands overlap in energy. If the bands overlap in

energy, then instead of one filled band giving an insulator, we can have two partly filled

bands giving a metal (Fig. 1) the alkali metals and the noble metals have one valence

electron per primitive cell, so that they have to be metals. The alkaline earth metals have

two valence electrons per primitive cell; they could be insulators, but the bands overlap

in energy to give metals, but not very good metals. Diamond, silicon, and germanium

each have two atoms of valence four, so that there are eight valence electrons per

primitive cell; the bands do not overlap, and the pure crystals are insulators at absolute

zero.


is co

(th

Fig. 1 Occupied states and band structures giving (a) an insulator, (b) a metal or a semimetal

because of band overlap, and (c) a metal because of electron concentration. In (b) the

overlap need not occur along the same directions in the Brillouin zone. If the overlap is

small, with relatively few states involved, we speak of a semimetal.

INTRINSIC SEMICONDUCTORS

Intrinsic semiconductors are those materials having an energy gap of the order of

1 eV. Hence the conduction process in these materials is achieved as a result of thermal

excitation of electrons across the energy gap. Pure chemical substances such as silicon

and germanium (elements of group IV of the periodic Table), GaAs, InSb (III-V

compounds), SiC (a IV-IV compound) and PbS (a IV-VI compound) are some

examples of intrinsic semiconductors. At absolute zero, the valence band of an intrinsic

semiconductor mpletely filled and the conduction band which is

separated by a distance e band gap) from the valence band is empty. For this

reason, at absolute zero, an intrinsic semiconductor behaves as an insulator and has zero

conductivity. Fig. la shows a simplified schematic diagram of an intrinsic

semiconductor.


electrons

availab

However, as the temperature is gradually increased, the of the

valence band get excited. For some electrons, the thermal energy le at the

room temperature is sufficient to surmount the forbidden energy gap and move into the

conduction band (Fig. 1b). The excitation of electrons from the valence band to the

conduction band leaves an equal number of vacancies (called holes) in the valence

band. Both, electrons in the conduction band and holes in the valence band serve as

charge carriers and contribute to the electrical conductivity.

Fig. 1 Intrinsic semiconductor: (a) at absolute zero the valence band is completely

filled by electrons and the conduction band completely empty; (b) at temperature

above absolute zero some of the electrons from the valence band are excited to the

conduction band; holes appear in the valence band and free electrons in the

conduction band

EXTRINSIC SEMICONDUCTORS

A real semiconducting crystal grown in a laboratory always contains some

impurity atoms (in addition to other imperfections), which create their own energy


is

levels termed as impurity levels. These levels may occupy positions anywhere in the

allowed and forbidden bands of the semiconductors. However, frequently some known

impurities (in terms of both qualities and quantities) are introduced intentionally to

impart specific properties to the host material. The resulting semiconducting material is

known as extrinsic semiconductor, whose properties depend on the type of the impurity

introduced into the host material. For this purpose, normally a few parts per million of

suitable impurity material is added to the melt during the growth of semiconductor

crystal. For silicon (and germanium) two types of impurities are used. These are from

group Ill and V of the periodic Table, i.e. trivalent elements such as born and indium

and pentavalent elements such arsenic and phosphorus.

CARRIER CONCENTRATION IN SEMICONDUCTORS

Intrinsic Semiconductors

As we know that in semiconductors the magnitude of the energy gap,

very small, therefore some of the electrons lying on the top of the valence band gain

sufficient thermal energy (which is the most usual source, other energy sources may be

electrical or radiant) even at moderate temperatures and get excited into the conduction

band leaving behind equal number of holes in the valence band (in a pure

semiconductor under thermal equilibrium the number of electrons and holes are equal

since they are produced as electron-hole pairs). These holes attract electrons, in the

process one of the neighboring electrons will jump to fill the vacancy.

When an electric field is applied, the electrons drift towards the positive


he sam

are p l l ec

an

m co

( )

ndi

d tempe

( ) (w

r

f ma

( )

ter

. T

( ) = ( ) =

and to int

take

= . S

( )

o tha

=

t,

( ) =

electrode and constitute a current. Similarly, the holes move towards the negative

electrode and constitute a current. However, since the electrons and holes have opposite

charges, the current due to both is in t e direction. Their drift

velocities under the action of an ap ied e tric field

= μ (1)

= μ

where the subscripts d refer to electrons and holes, respectively. The

proportionality factor µ is known as the mobility (the freedom of charge movement) and

is a function of thermal vibration of the atoms and the crystal structure. Under equilibriu

tions, in intrinsic semiconductors, the rate of generation of electron hole

pair epends on the nature of material and the ature. To maintain a

constant concentration of pairs, the rate of recombination hich depends on the

properties o ial and is proportional to the concentrations of two charges) must be

equal to hen

(2)

where , refer rinsic electron and hole densities, respectively. The

proportionality constant s into account the charge densities and properties of the

semiconductor material.

Now for pair generation,

= (3)


an are

o

=

Since the densities d inherent property of a semiconductor at a given

temperature, then the product r the square of the intrinsic charge density, also has a

fixed value at a given temperature and depends only on the nature of material. This result

is analogous to the constant solubility product in chemistry.

Extrinsic Semiconductors

In the preceding section, we observed that when a small quantity of suitable

impurity element from group V of the periodic Table is added to pure material, n-type

semiconductor results, where electrons act as majority charge carriers. Similarly, the

addition of suitable group III element produces p-type semiconductor, where holes are

majority charge carriers.

In both n and p type semiconductors, the electrons which create the ions (donor

impurity atoms become positively ionized after donating electrons and acceptor atoms

become negatively ionized) receive relatively higher energy than needed to create

electron-hole pairs. Therefore, it is possible to establish conditions for conduction by

holes in p-types semiconductors, and conduction by electrons in n-type semiconductors

at temperature much below than the temperature at which intrinsic conduction by holes

and electrons occur. Since no charge is added or removed during the above processes, the

crystal as a whole is assumed to be electrically neutral.

Because of the ease with which the electrons from the donor impurity can be excited

to the conduction band, it is reasonable to assume that these (donor impurity) atoms are

totally ionized at usual ambient temperatures. Hence, the density of conduction


h ge neu rality

+

of th

= an

ar

i

electrons, ND can be assumed equal to the density of donor impurity atoms. Likewise, the

acceptor states may assume to be filled completely in a p-type semiconductor at usual

temperatures, and so the density of holes in the valence band, NA is taken as equal to the

density of acceptor atoms.

As a matter of fact, the concentration of impurity atoms is very low (usually one

atom per 106-1010 host atoms) as compared to the parent semiconductor atom and

therefore impurity doping does not appreciably affect the thermal generation and

recombination process. Making use of the argument that the recombination is

proportional to concentrations of two charges, for p-type material, the eqn (3) becomes

= (4)

It is to be noted that when the concentration of majority charges is increased over its

intrinsic value by doping, the concentration of minority charges is found to decrease to

maintain the concentration product, ni2, constant. Based on above considerations, the

amount of reduction in the minority charges can be determined by the use of an

equation expressing the overall c e material as

Now, making use of the equality np = n 2 we have

(5)

So that from eqn (5) we have

d = (6)


− + = − +

− − ) + =

uation are

−

2 ±

[( −

4

) ] +

= −

2 ±

[( −

4

) ] +

0, he ho

+

le den sity

1 + (

2

/ )

ms, th

+

2

appr

2

1 +

ate h

2

le den

= 2

+ 2

1 + 2

+ ≅

The roots of the above eq

(7)

Similarly

(8)

For an intrinsic semiconductor, where NA = ND = 0, from eqs. (8) and (9), we

obtain n = p = ni, as they should be. These are general equations for charge densities and

let us see how they change under special cases:

Case I: n-type Semiconductors

In this case, ND >> ni and NA ≅ then t is

− 4

= (9)

The second term under the radical is less than unity and may be expanded as a power

series. Neglecting the higher power ter e oxim o sity can be found as

= − = (10)

Similarly, eqn (9) provides the electron density in n-type material as

(11)


then the

≅

ch

≅ + ≅

Case II: p-type Semiconductors

In this case, NA >> ni and ND ≅0, arge densities can be found as

(12)

(13)

The density of majority carriers is thus approximately equal to the density of

impurity atom (as in eqs. 11 and 13) at usual ambient temperatures. On the other hand,

because of recombination process the density of minority carriers remains significantly

below the intrinsic ni, or pi, level.

THE P-N JUNCTION

The pn junction consists of an abrupt discontinuity or a graded distribution of the

two types of doping impurity across a region of a particular semiconductor sample. The

electrostatic conditions at the junction can be obtained if we consider the special case

where a piece of p-type material is brought into intimate contact with a piece of n-type

material. The energy-band diagram at the moment of contact is given in Fig. 1 (a).

Fig. 1 Energy-band diagram for an abrupt pn junction: (a) before contact (b) at thermal

equilibrium

brup

( ) = 0

t jun ion, N

(

a an

) =

d Nd are constants, inde

− < <

( ) =

,

, ( ) = 0 < <

0

ities a

=

pressed in

=

t

1

er ms of the condition, if the internal field is ex electrostatic potential:

Owing to the concentration gradient, electrons diffuse from the conduction band

of the n-type region into that of the p-type region and recombine with the free holes by

dropping into the valence band. The space charge, produced by the negatively charged

acceptors in the p-type side and the positively charged donors left behind in the n-type

side, exerts a repulsive force on further charges crossing the junction, and inhibits further

flow. In terms of the energy-band diagram, the Fermi energy of the p-type region is raised

with respect to that of the n-type region until the Fermi levels become equal, as in

Fig.1(b). The transferred charge which terminates on the immobile impurity atoms in the

transition region produces an electric field which is directed from the n-type side to the

p-type side. This built-in potential barrier Vint corresponds to the difference between the

Fermi levels on the n and p sides before contact was made.

Let Nd and Na be the concentrations of charged donor and acceptor impurities, and

xd, xa be the thicknesses of the space charge regions on each side of the junction

respectively. For an a ct pendent of x:

0

and we may assume that all impurities are ionized, so that thermal equilibrium carrier

dens t the edges of the space charge regions will be taken as n0 (xd) = Nd and p0

− Na. The built-in potential barrier results from the thermal equilibrium



= ( ) − ( −

a

)

x = x

l

d,

n

one o

(

(−

btai

)

) =

(− ) . Henc e

= l n

o

: terial of

=

diele c Ԑ

< <

= for <

0

ions of zer o f

=

ld at both extre

=

=

On integration over the transition region, from x = -x to ns:

(1)

We can take n0 (xd) = Nd and n0 (-xa) = = , the thermal

equilibrium potential barrier, whose n-type side is positive with respect to the p-type

side can be written as:

(2)

The variation of the potential in the depletion region is btained by solving the

Poisson equation within the ma tric constant

for −

− 0 < (3)

subject to the boundary condit ie mities:

0 (4)

Since the pn junction is made from separate pieces of p-type and n-type material, we must

have the same charge per unit area on each side of the transition region, due to charge

separation:

(5)

On the p-type side of the junction (-xa <x <0), integrating Eq.(3) and bearing in mind the

boundary condition (4), we obtain the field distribution:


= ( + )

( ) = ( + 2

) + (7

nt ca

( )

n b

=

e eliminate

( +

2

d by

) f

ing th

<

zero

<

(− ) (0)

or −

2

n sid

( )

e (

=

x < xd):

( −

2 ) +

ontinuity at x

( )

=0

=

allo

2

ws us to

+

e v

2

a lua

−

te the

(

2

const

− )

ant, g iving:

0 < < (1

( ) = 2

+ 2

= 2

( + )

(6)

Integrating Eq.(6) gives:

)

where the arbitrary consta defin e of potential at - xa :

0 (8)

= 0 and = (9)

Similarly we obtain on the 0<

−

and the c

0)

From Eq.(10) it follows that:

The built-in potential barrier Vint = V(xd) -V(-xa) = V(xd) can be written, using the

condition (5), as:

(11)


= + for

= +

= +

we obt

=

2 ( + )

= 2

at therm

+

al eq

From Eq.(5), introducing the total width of the depletion region, we

have

(12)

On substitution into Eq. (11) ain:

(13)

so that the width of the depletion region uilibrium can be written as:

(14)

Fig. 2 Energy-band diagrams for a pn junction: (a) forward bias (b) reverse bias

A pn junction is in a state of dynamic equilibrium and can be described, in terms

of the energy-band diagram shown in Fig.1, as a balance between majority carriers,

moving from n-type to p-type side and climbing the potential barrier, and an equal number

of minority carriers, moving down the barrier from the p-type to n-type side. When a

potential difference V is applied across the junction, so that the p-type


side is biased positively with respect to the n-type side (V >0 or forward bias), the barrier

to diffusion is lowered, as in Fig. 2 (a), resulting in large forward currents. Reversing the

polarity of the applied voltage (V <0 or reverse bias), the barrier to diffusion is raised, as

in Fig. 2(b), so that there are only the minority carriers near the transition region that

produce a small reverse current.

Energy-band diagrams of inhomogeneous and homogeneous semiconductors

It is often convenient to represent inhomogenous semiconductors by energy- band

diagrams which visualize the electrical behaviour described by the carrier transport

equations. Most phenomena can be understood if we use a one-dimensional effective

energy-band model, shown in Fig.1, where Ec is the lowest energy in the conduction band

and Ev is the highest energy in the valence band for any possible crystallographic

direction. In other words, Eg = Ec - Ev is the minimum thermal energy required to excite

electrons from the valence band into the conduction band and the x axis represents an

averaged direction through the crystal. This convention is illustrated in Fig.1for a

homogenous semiconductor.

Fig.1 The band structure in a homogenous semiconductor


tion for

(∆

th

) =

e exce

−

s elec

+ (

∆

ron co

)µ

ncentration

+ µ

∆

(∆ ) +

( ∆ )

, µ rep

he mot

(∆ ) =

of the

−

xc

∆

ol oncentration

+ µ

∆p

(∆ ) +

( ∆ )

, µ rep

For an inhomogenous semiconductor at thermal equilibrium, the Fermi energy is

always represented in such diagrams by a horizontal line, as in Fig. 2. Any local increase

or decrease in the carrier potential energy with respect to the average effective value

will be represented by an upward or downward bending of the bands. It is apparent that,

in an energy-band representation, the electrons move towards the bottom of the

conduction band, while the holes move towards the top of the valence band.

Fig. 2 The energy-band diagram for an inhomogenous semiconductor

at thermal equilibrium

Steady-state diffusion of excess minority carriers

Equation of mo s t

∆ (1)

Where, , and resent the generated electrons, mobility, diffusion time

and diffusion constant of electrons.

Equation for t ion e ess h e

+ (∆ )µ (2)

Where, , and resent the generated holes, mobility, diffusion time and

diffusion constant of holes.


les in

=

ion e

(∆

q

) ∆ − =

usion length f

=

or

+

ctron

= nd E=0

(∆

in eq

) −

(1)).

=

∆

= +

= i

n.

For the steady-state motion of excess ho a n-type semiconductor we must take

d(∆p)/dt = 0 in Eq.(2). If we assume that 0 and that the applied field is E

= 0, Eq.(2) reduces to the diffus uation:

0 (3)

where it is usual to introduce a diff holes as:

(4)

Eqn (3) has the general solution:

∆ = (5)

Where, a and b are arbitrary constants. In a similar manner, the steady-state diffusion

equation for excess ele s in a p-type semiconductor may be derived from eqn (1) as:

(take d(∆n)/dt = 0; 0 a

0 (6)

The solution is:

∆ (7)

where, s the diffusion length for electrons and c and d are arbitrary

constants.

dr. p. sagayaraj state physics unit … · space of the boundaries of the zones; these positions...

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