solid states electronics

8/6/2019 Solid States Electronics

1/37

Solid State Electronics

Text Book

Ben. G. Streetman and Sanjay Banerjee: Solid State

Electronic Devices, Prentice-Hall of India Private

Limited.

Chapter 3


2/37

Bonding forces in SolidsBond: The interaction of electrons in neighboring atoms is called bond.

Ionic bond: Ionic bonds are a type of chemical bond based on electrostaticforces between two oppositely-charged ions.

In ionic bond formation, a metal donates an electron, due to a low

electronegativity to form a positive ion.The bonds between the sodium (Na) and chlorine (Cl) ions are ionic bonds.

The electronic structure of Na (Z=11) is [Ne]3s1, and Cl(Z=17) has the structure

[Ne]3s23p5.

Na atom gives up its outer 3s electron to a Cl atom, so that the crystal is made upof ions with electronic structures of the inert atoms Ne and Ar.

However, the ions have net electric charges after the electron exchange.

The Na+ ion has a net positive charge, having lost an electron, and the Cl- ion has

a net negative charge, having gained an electron.Once the electron exchanges have been made between the Na and Cl atoms to

form Na+ and Cl- ions, the outer orbits of all atoms are completely filled.

Since the ions have the closed-shell configurations of the inert atoms Ne and Ar,

there are no loosely bound electrons to participate in current flow; as a result,NaCl is a good insulator.


3/37

Metallic bond: Metallic bonding is the bonding within metals. It

involves the delocalized sharing of free electrons among a lattice

(the periodic arrangement of atoms in a crystal is called the lattice)of metal atoms.

In a metal atom the outer electron shell is only partially filled,

usually by no more than three electrons.

In Na has only one electron in the outer orbit.

This electron is loosely bound and is given up easily in ion

formation.

In the metal the outer electron of each alkali atom is contributed tothe crystal as a whole, so that the solid is made up of ions with

closed shells immersed in a sea of free electrons.

The forces holding the lattice together arise from an interactionbetween the positive ion cores and the surrounding free electrons.

This is one type ofmetallic bonding.

The metals have the sea of electrons in common, and these

electrons are free to move about the crystal under the influence of

an electric field.


4/37

Covalent bond: Covalent bonding is an intermolecular form of

chemical bonding characterized by the sharing of one or more

pairs of electrons between two components, producing a mutual

attraction that holds the resultant molecule together.

Atoms tend to share electrons in such a way that their outerelectron shells are filled.

Atom in the Ge, Si, or C diamond lattice is surrounded by four

nearest neighbors, each with four electrons in the outer orbit.In these crystals each atom shares its valence electrons with its

four neighbors.

The bonding forces arise from a quantum mechanical interaction

between the shared electrons.

This is known as covalent bonding; each electron pair constitutes a

covalent bond.


5/37

Energy Bands

The electron in an isolated atom has discrete energy level butthe electron in a solid has a range, or band, of available

energies.

The discrete energy levels of the isolated atom spread intobands of energies in the solid because in the solid the wave

functions of electrons in neighboring atoms overlap, and an

electron is not necessarily localized at a particular atom.As isolated atoms are brought together to form a solid,

various interactions occur between neighboring atoms.

The forces of attraction and repulsion between atoms will finda balance at the proper interatomic spacing for the crystal.

In the process, important changes occur in the electron energy

level configuration and these changes result in the variedelectrical properties of solid.


6/37

Metals, Semiconductors, and Insulators

Insulator: A very poor conductor of electricity is

called an insulator.

In an insulator material the valance band

is filled while the conduction band is empty.The conduction band and valance band in

the insulator are separated by a large forbidden

band or energy gap (almost 10 eV).In an insulator material, the energy which

can be supplied to an electron from a applied field

is too small to carry the particle from the field

valance band into the empty conduction band.

Since the electron cannot acquire

sufficient applied energy, conduction is

impossible in an insulator.


7/37

Semiconductor: A substance whose conductivity lies between

insulator and conductor is a semiconductor.

A substance for which the width of the forbidden energy

region is relatively small (almost 1 eV) is called semiconductor.In a semiconductor material, the energy which can be supplied to

an electron from a applied field is too small to carry the particle

from the field valance band into the empty conduction band at 0

K.As the temperature is increased, some of the valance band

electrons acquire thermal energy. Thus, the semiconductors allow

for excitation of electrons from the valance band to conduction

band.These are now free electrons in the sense that they can move

about under the influence of even a small-applied field.

Metal: A metal is an excellent conductor.

In metals the band either overlap or are only partially

filled.

Thus electrons and empty energy states are intermixed within the

bands so that electrons can move freely under the influence of an

electric field.


8/37

Direct and Indirect SemiconductorsA single electron is assumed to travel through a perfectly periodic

lattice.The wave function of the electron is assumed to be in the form of a

plane wave moving.

For example, in the x-direction with propagation constant k, also calledwave vector.

The space-dependent wave function for the electron is

where the function U(kx,x) modulates the wave function according to

the periodicity of the lattice.

Depending on the transition of an electron from conduction band to

valance band with respect to the propagation constant, the

semiconductor materials are classified as follows:

(a) Direct semiconductor and(b) Indirect semiconductor

)13(),()( = xxjexxUx kkk


9/37

Direct Material: The material (such as

GaAs) in which a transition of an electron

from the minimum point of conduction band

to the maximum point of valence band takes

place with the same value ofK(propagation

constant or wave vector) is called directsemiconductor material.

According to Eq. (3-1) the energy (E) vs

propagation constant (k) curve is shown in

the figure.

A direct semiconductor such as GaAs, anelectron in the conduction band can fall to an

empty state in the valence band, giving off

the energy differenceEgas a photon of light.


10/37

Indirect Material: The material (such as Si) in

which a transition of an electron from the

minimum point of conduction band to themaximum point of valence band takes place

with the different values of K (propagation

constant or wave vector) is called indirect

material.

According to Eq. (3-1) the energy (E) vs

propagation constant (k) curve is shown in the

figure.

An electron in the conduction band minimum of an indirect

semiconductor cannot fall directly to the valence band maximum butmust undergo a momentum change as well as changing its energy.

It may go through some defect state (Et) within the band gap.

In an indirect transition which involves a change in k, the energy isgenerally given up as heat to the lattice rather than as emitted photon.


11/37

Electrons and Holes

Hole: An empty state in the valence band is referred to as a

hole.

Electron-Hole Pair: If the conduction band electron and

the hole are created by the excitation of the valence bandelectron to the conduction band, they are called an

electron-hole pair(abbreviated EHP).


12/37

If a filled band, all available energy states are occupied.

For every electron moving with given velocity, there is an equal and

opposite electron motion elsewhere in the band.If we apply an electric field, the net current is zero because for every

electron j moving velocity vj there is a corresponding electron j with

the velocity vj.

Fig. 3-8 illustrates this effect in terms of the electron vs. wave vector

plot for the valance band.

Since k is proportional to electron momentum, it is clear the two

electrons have opposite directed velocities.

Fig. 3-8

WithNelectron/cm3 in the band we express

the current density using a sum over all of

the electron velocities, and including thecharge q on each electron.

In a unit volume,

)23()bandfilled(0)( aN

iivqJ ==


13/37

Now if we create a hole by removing the jth electron, the net current

density in the valance band involves the sum over all velocities, minus

the contribution of the electron we have removed.

)23()missingelectronth(0)()( bjN

ijvqivqJ = =

Since the first term of (3-2b) is zero in accordance (3-2a), the net

current is

)electronjthofmissingthetoduecurrentNet(jqvJ=

The current contribution of the hole is equivalent to that of a positively

charged particle with the velocity vj that of the missing electron.

The charge transport is actually due to the motion of the newuncompensated electron j. Its current contribution (-q)(-vj) is

equivalent to that of a positively charged particle with the velocity vj.

Thus, it is realized that the current flow in the semiconductor can beaccount by the motion of electrons and holes of charge carriers.

I h l b d h l i i l l


14/37

In the valence band, hole energy increases oppositely to electron

energy, because the two carriers have opposite charge.

Thus hole energy increases downward in Fig. 3-8 and holes, seekingthe lowest energy state available, are generally found at the top of the

valance band.

In contrast, conduction band electrons are found at the bottom of theconduction band.

Holes are found at the top of the valence band

because the valence band electrons will

rearrange themselves so that they occupy the

lowest energy states, leaving only the highest

energy level vacant in the valance band.

Electrons in the conduction band will

similarly rearrange themselves so they

occupy the lowest energy states ofconduction band. Fig. 3-8


15/37

Effective Mass

The electrons in a crystal are not completely free, but insteadinteract with periodic potential of the lattice of a crystal.

The electron wave particle motion cannot be expected to bethe same as for electrons in free space.

And, the mass of free electron is not same as for electrons in

a solid.

The electron momentum can be written asP=mv=k. Then

22

22212

21 k

mmPmvE h===

Thus the electron energy is parabolic with wave vector k.


16/37

The electron mass is inversely related to the curvature

(second derivative) of the (E, k) relation, since

mdEd 22

2 h=k

The effective mass of an electron in a band with a given (E,

k) relationship is found as follows:

2/22*

kdEdm h=

Thus, the mass of electron which is obtained from the

curvature of the energy band of a solid is called effective

mass.


17/37

Intrinsic MaterialA perfect semiconductor with no impurities or lattice defect is

called an intrinsic material.In intrinsic material, there are no charge carrier at 0K, since the

valence band is filled with electrons and the conduction band is

empty.At high temperature electron-hole pairs are generated as valence

band electrons are excited thermally across the band gap to the

conduction band.These EHPs are the only charge carriers in intrinsic material.

Since the electrons and holes are crated in pairs, the conduction

band electron concentration n (electron/cm3) is equal to theconcentration of holes in the valence bandp (holes/cm3).

Each of these intrinsic carrier concentrations is commonly

referred to as ni. Thus for intrinsic material: n=p=ni (3-6)At a temperature there is a carrier concentration of EHPs ni.

R bi ti i h l t i th d ti


18/37

Recombination is occurs when an electron in the conduction

band makes transition to an empty state (hole) in the valence

band, thus annihilating the pair.If we denote the generation rate of EHPs as gi (EHP/cm

3-s)

and the recombination rate ri, equilibrium requires that

ri=gi (3-7a)Each of these rates is temperature dependent.

gi(T) increases when the temperature is raised, and a new

carrier concentration ni is established such that the higherrecombination rate ri(T) just balances generation.

At any temperature, the rate of recombination of electrons

and holes ri is proportional to the equilibrium concentrationof electrons n0 and the concentration of holesp0:

ri=rn0p0= rni2=gi (3-7b)

The factor r is a constant of proportionality which dependson the particular mechanism takes place.

E i i M i l


19/37

Extrinsic Material

When a crystal is doped such that the equilibrium carrier

concentrations n0 and p0 are different from carrier concentration

ni, the material is said to be extrinsic material.

In addition to the intrinsic carriers generated, it is possible tocreate carriers in semiconductors by purposely introducing

impurities into the crystal.

This process, called doping, is the most common technique for

varying conductivity of semiconductor.

There are two types of doped semiconductors, n-type (mostlyelectrons) andp-type (mostly holes).

An impurity from column V of the periodic table (P, As and Sb)

introduces an energy level very near the conduction band in Ge or

Si.

The energy level very near the conduction band is filled with electrons


20/37

The energy level very near the conduction band is filled with electrons

at 0K, and very little thermal energy is required to excite these

electrons to the conduction band (Fig. 3-12a).

Thus at 50-100K virtually all of the electrons in the impurity level are,

donated to the conduction band.

Such an impurity level is called a donor level and the column Vimpurities in Ge or Si are called donor impurities.

Semiconductors doped with a significant number of donor atoms will

have n0>>(ni,p0) at room temperature.

This is n-type material.

Fig. 3-12 (a) Donation of

electrons from donor level toconduction band.

Similarly an impurity from column III of the periodic table (B Al Ga


21/37

Similarly, an impurity from column III of the periodic table (B, Al, Ga

and In) introduces an energy level very near the valence band in Ge or

Si.

These levels are empty of electrons at 0K (Fig. 3-12b).

At low temperatures, enough thermal energy is available to excite

electrons from the valence into the impurity level, leaving behind holes

in the valence band.Since this type of impurity level accepts electrons from the valence

band, it is called an acceptor level, and the column III impurities are

acceptor impurities in the Ge and Si.

Fig. 3.12b

Doping with acceptor

impurities can create a

semiconductor with a holeconcentration p0 much greater

that the conduction band

electron concentration n0.This type isp-type material.

C i t ti


22/37

Carrier concentrationThe calculating semiconductor properties and analyzing device

behavior, it is often necessary to know the number of charge carriersper cm3 in the material.

To obtain equation for the carrier concentration, Fermi-Dirac

distribution function can be used.The distribution of electrons over a range of allowed energy levels at

thermal equilibrium is

kTFEEeEf

/)(1)(

1

+=

where, k is Boltzmanns constant (k=8.2610-5 eV/K=1.3810-23

J/K).

The function f(E), the Fermi-Dirac distribution function, gives the

probability that an available energy state at Ewill be occupied by an

electron at absolute temperature T.

The quantity EF is called the Fermi level, and it represents animportant quantity in the analysis of semiconductor behavior.

For an energy E equal to the Fermi level energy E the occupation


23/37

For an energy E equal to the Fermi level energy EF, the occupation

probability is

2

1

1

1)(

/)(=

+

= kT

FE

FEF

e

Ef

The significant of Fermi Level is that the probability of electron and

hole is 50 percent at the Fermi energy level. And, the Fermi function

is symmetrical about EF for all temperature; that is, the probabilityf(EF +E) of electron that a state EaboveEF is filled is the same asprobability [1-f(EF-E)] of hole that a state EbelowEFis empty.

At 0K the distribution takes thesimple rectangular form shown in

Fig. 3-14.

With T=0K in the denominator ofthe exponent, f(E) is 1/(1+0)=1

when the exponent is negative

(EEF).

This rectangular distribution implies that at 0K every available energy


24/37

g p y gy

state up toEFis filled with electrons, and all states aboveEFare empty.

At temperature higher than 0K, some probability exists for states above

the Fermi level to be filled.

At T=T1 in Fig. 3-14 there is

some probabilityf(E) that states

aboveEF are filled, and there is

a corresponding probability [1-

f(E)] that states below EF are

empty.

The symmetry of the

distribution of empty and filled

states aboutEFmakes the Fermilevel a natural reference point

in calculations of electron and

hole concentration insemiconductors.

For intrinsic material the concentration of holes in


25/37

Fig. 3-15(a) Intrinsic Material

For intrinsic material, the concentration of holes in

the valence band is equal to the concentration of

electrons in the conduction band.

Therefore, the Fermi levelEF must lies at the middle

of the band gap.

Since f(E) is symmetrical

about EF, the electron

probability tail if f(E)

extending into the conduction

band of Fig. 3-15a is

symmetrical with the hole

probability tail [1-f(E)] in thevalence band.

Fig. 3.15(b) n-In n-type material the Fermi level lies near


26/37

Fig. 3.15(b) n

type material

yp

the conduction band (Fig. 3-15b) such that

the value off(E) for each energy level in

the conduction band increases as EF moves

closer toEc.

Thus the energy difference (Ec- E

F) gives

measure ofn.

Fig. 3.15(c)p-

type material

In p-type material the Fermi level lies

near the valence band (Fig. 3-15c) suchthat the [1- f(E)] tail value Ev is larger

than thef(E) tail aboveEc.

The value of (EF-Ev) indicates howstrongly p-type the material is.

Example: The Fermi level in a Si sample at equilibrium is located at


27/37

Example: The Fermi level in a Si sample at equilibrium is located at

0.2 eV below the conduction band. At T=320K, determine the

probability of occupancy of the acceptor states if the acceptor statesrelocated at 0.03 eV above the valence band.

Solution:

From above figure,Ea-EF={0.03-(1.1-0.2)} eV= -0.87 eV

kT= 8.6210-5 eV/K320=2758.4 eV

we know that,

0.11

11

1)()5104.2758/(87.0/)(=

+=

+= ee

Ef kTFEaEa

Electron and Hole Concentrations at Equilibrium


28/37

Electron and Hole Concentrations at Equilibrium

The concentration of electron and hole in the conduction band and

valance are

)12.3()()](1[

)12.3()()(

0

0

bdEENEfp

adEENEfn

vE

cE

=

=

whereN(E)dEis the density of states (cm-3) in the energy range dE.

The subscript 0 used with the electron and hole concentration symbols

(n0

,p0

) indicates equilibrium conditions.

The number of electrons (holes) per unit volume in the energy range

dE is the product of the density of states and the probability of

occupancyf(E) [1-f(E)].

Thus the total electron (hole) concentration is the integral over theentire conduction (valance) band as in Eq. (3.12).

The function N(E) is proportional to E(1/2), so the density of states in

the conduction (valance) band increases (decreases) with electron(hole) energy.

Similarly, the probability of finding an empty state (hole) in the


29/37

y, p y g p y ( )

valence band [1-f(E)] decreases rapidly below Ev, and most hole

occupy states near the top of the valence band.

This effect is demonstrated for intrinsic, n-type and p-type materials

in Fig. 3-16.

Fig. 3.16 (a) Concentration of electrons and holes in intrinsic material.


30/37

Fig. 3.16 (b) Concentration of electrons and holes in n-type material.

Fig. 3.16 (a) Concentration of electrons and holes in p-type material.

The electron concentration (in terms of effective density of states Nc


31/37

c

which is located at the conduction band edge Ec) in thermal

equilibrium can also be written as follows:

)13.3()(0 cc EfNn =2/3

2

*22where,

=

h

kTmN nc

It is assumed that the Fermi levelEF lies at least several kTbelow the

conduction band.

Then and the Fermi functionf(Ec) can be simplifiedas

1/)( >> kTFEcEe

)14.3(1

)(/)( kTFEcEc

ee

Ef+

1 /)( kTFEcE ==

For this condition the concentration of electrons in the conduction

band is

)15.3(/)(0 kTFcceNn =

By similar argument, the hole concentration (in terms of effective


32/37

density of states Nv which is located at the valance band edge Ev) in

thermal equilibrium can also be written as follows:

2/3

2

*22where,

=

h

kTmN

pv

)16.3()](1[0 vv EfNp =

For EF larger than Ev by several kT, the probability of finding an

empty state atEv is

)17.3(1

11)(1

/)(

/)(

kTvEFE

kTFEvEve

eEf

=+

=

For this condition the concentration of holes in the valance band is

)18.3(/)(0 kTvFveNp =

The electron and hole concentrations predicted by Eqs. (3-15) and (3-


33/37

18) are valid whether the material is intrinsic or doped, provided

thermal equilibrium is maintained.

Thusfor intrinsic material,EF lies at some intrinsic level Ei near the

middle of the band gap, and the intrinsic electron and hole

concentrations are

)21.3(,/)(/)( kTvi

vikTic

ci eNpeNn ==

From Eqs. (3.15) and (3.18), we obtain

kTvF

v

kTFc

ceNeNpn

/)(/)(

00

=

)22.3(//)(

00

kTgvc

kTvEcEvc eNNeNNpn

==

From Eq. (21), we obtain kTviv

kTiccii eNeNpn

/)(/)( =


34/37

)23.3(//)( kTg

vc

kTvEcE

vcii

eNNeNNpn ==

vcii p

From Eqs. (3.22) and (3.23), the product ofn0 andp0 at equilibrium is

a constant for a particular material and temperature, even if the

doping is varied.

The intrinsic electron and hole concentrations are equal, ni=pi; thus

from Eq. (3.23) the intrinsic concentrations is

)24.3(2/ kTgvci eNNn

=The constant product of electron and hole concentrations in Eq. (3.24)

can be written conveniently from (3.22) and (3.23) as

)25.3(200 inpn =

At room temperature (300K) is: For Si approximatelyni=1.51010

cm-3; For Ge approximatelyni=2.51013cm-3;

/)( kTicic enN

=From Eq. (3.21), we can write as


35/37

)26.3(/)( kTvEiE

iv

ic

epN=

Substitute the value ofNc from (3.26) into (3.15), we obtain

kTFcici

kTFckTici eneenn

/)(/)(/)(0

+ ==

)27.3(/)(/)(0 kTiFikTFii enenn ==

Substitute the value ofNv from (3.26) into (3.18), we obtain

kTvFvii

kTvFkTvii eneepp

/)(/)(/)(0

+ ==

)28.3(/)(/)(

0kTFi

ikTiF

i enenp ==

It seen from the equation (3.27) that the electron concentrations n0increases exponentially as the Fermi level moves away from Eitoward the conduction band.

Similarly, the hole concentrationsp0 varies from ni to larger values as

EFmoves fromEi toward the valence band.

Example 3-5 A Si sample is doped with 1017 As atoms/cm3. What is the

ilib i h l i 300K? Wh i E l i E ?


36/37

equilibrium hole concentrationp0 at 300K? Where isEFrelative toEi?

Solution:Nd=1017atoms/cm3; kT=0.0259eV; n

i(for Si)=1.51010 cm-3;

SinceNd>>ni, we can approximate n0=Nd

33

17

210

0

2

0 cm1025.210

)105.1(

=

== n

n

pi

We know that kTiFienn

/)(0

=

The resulting band diagramis

eV407.0

105.1

10ln0259.0

ln

10

17

0

=

=

=

i

iF

iiF

EE

EE

n

nkTEE


37/37

End ofChapter 3

solid states electronics

Documents