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Electronics Devices and Circuits

In metals, electric current flows due to the flow of electrons.

In semiconductors the current flows due to the flow of charged particles ( holes / electrons )

under the influence of electric and magnetic fields.

Electron mass =

charge =

Proton mass =

charge =

mass of an electron =

mass of an proton

ElectronVolt

It is the amount of energy gained or work done by an electron if an electron falls through a pot.

of 1 volt.

( )( )

The negatively charged particles move around the nucleus in orbits in particular energy levels.

Each electron will have a different energy level.

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Energy bands

Insulators, Semi conductors, and Metals classification using Energy Band

Diagrams

The range of energies that an electron may possess in an atom is known as the energy band.

There Important energy bands are,

i) Valence Band ii) Conduction Band iii) Forbidden Band

The lowest energy level in conduction band is Ec

The highest energy level in the valence band is Ev

Valence Band :- The Range of Energy possessed by valence electrons is known as valence

Bands. At 0 deg C, when no energy is applied, valency band is full of electrons and the

conduction band is empty. The energy of holes in the valence band is extended from -∞ to EV

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Conduction Band :-

The valence electrons are less tightly bound with the nucleus. So that even an appication of small

elctric field some of the valence electrons detached from the nucleus and it becomes free

electrons. The electrons move from low energy level to high energy level. These fee elctrons are

responsible for the conduction of current in good conductors. The electrons are also called

conduction electrons. The Range of energy possed by these elctrons is known as conduction

band. The electrons in the conduction band has energy in the range of EC to +∞.

Forbidden Band (or) Energy Gap :-

The energy band in between the condition band and the valence band is called forbidden Band.

The forbidden band gap is given by ( )

Classification of Materials (or) Solids according to Energy Bands :-

Solids are classified into there types

i)Insulators ii) Conudctors iii) Semi conductors

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Insulators

→ The materials in which the condition band and valence bands are separeated by a wide

energy gap (≈ 15 eV) as shown in figure.

→ A wide energy gap means that a large amount of energy is required, to free the

electrons, by moving them from the valence band into the condition band ;

→ Thus insulators have very high resistively (or extremely low conductivity) at room

temperatures.

→ However if the temperature is raised, some of the valence electrons may acquire energy and

jump in to the conduction band. It causes the resistively of insulators to decrease.

→ Therefore an insulator have negative temperature co-efficient of resistance. As temp increases

the resistance decreases.

.For a diamond, which is an insulator, the forbidden gap is about 6 eV. Such conduction is rare

and is called breakdown of an insulator. The other insulating materials are glass, wood, mica,

paper etc.

Conductors

The materilas in which conduction and valence bands overlap as shown in figure are called

conductors. There is no forbidden gap between valence band and conduction band.

→ The overlapping indicates a large number of electrons available for conduction.

→ Hence the application of a small amount of voltage results a large amount of current.

→ Conductor have positive temperature co-efficient of resistance. As temp increases, the

overlapping increases the collision of particles takes place and the resistance increases.

For example, copper has 8.5 x 1028

free electrons per cubic metre which is a very large

number. Hence copper is called good conductor. Hence even at room temperature, a large

number of electrons are available for conduction. So without any additional energy, such metals

contain a large number of free electrons and hence called good conductors.

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Semiconductors :-

→ The materials, in which the conduction and valence bands are separeated by a small

energy gap (1eV) as shown in figure are called semiconductors.

→ Silicon and germanium are the commonly used semiconductors.

→ A small energy gap means that a small amount of energy is required to free the

elctrons by moving them from the valence band in to the conduction band.

→ The semiconductors behave like insulators at 0K, because no electrons are available

in the conduction band. Hence at 0o K, the semiconductor materials behave as perfect insulators.

→ If the temperature is further increased, more valence elctrons will acquire energy to

jump into the conduction band.

→ Thus like insulators, semiconductors also have negative temperature co-efficient of

resistance.

→ It means that conductivity of semiconductors increases with the increases tempertature

In case of semiconductors, forbidden gap energy depends on the temperature. For silicon and

germanium, this energy is given by,

( ) , where temp coefficient,

= (for Silicon) &

= (for Germanium)

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EG decreases for every degree rise in temperature.

EG = (for Silicon)

EG = (for Germanium)

Where T = Absolute temperature in K

Ex : Calculate the value of forbidden gap for silicon and germanium at the temperature of 35o C.

Solution : Forbidden gap for silicon is given by,

EG =

EG =

While forbidden gap for germanium is given by,

EG =

Why silicon is most widely used?

Looking at the structure of silicon and germanium atom, it can be seen that valence shell of

silicon is 3rd

shell while valence shell of germanium is 4rd

shell. Thus valence electrons of

germanium can easily escape from the atom, due to very small additional energy imparted to

them. So at high temperature, germanium becomes unstable than silicon and hence silicon is

widely used semiconductor material. Silicon is having more energy gap (EG) than Germanium

and hence more stable.

(i) Smaller ICBO. (ii) Smaller variation of ICBO with temperature.

(iii) Greater working temperature. (iv) Higher PIV rating

Leakage current

It is also known as minority saturation current or thermally generated current.

doubles for every C and for every 10C, increases by 7%.

( ) ( ) [ ( ) ; where ( > )

Io for and for , Si is suitable for high temperature application.

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Classification of Semiconductors

Charge Carriers in Semiconductors

When an electron leaves the valence band to go to the conduction band, it is essentially breaking

free of its bond, leaving behind an empty state available for another electron in the valence band

energy levels. This empty state is called a “hole” and effectively behaves like a positively

charged particle.

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Intrinsic Semiconductor

An intrinsic semiconductor is ideally a perfect crystal. When an electron in an intrinsic

semiconductor gets enough energy, it can go to the conduction band and leave behind a hole.

This process is called “electron hole pair (EHP) creation”. For the intrinsic material, since

electrons and holes are always created in pairs,

Where is the symbol for “intrinsic carrier concentration.”

At room temperature, relatively few electrons have enough thermal energy to make this jump.

Law of Mass Action

If n is the concentration of free electrons and p is the concentration of holes, then the law of

mass action states that the product of concentration of electrons and holes is always constant at a

fixed temperature. ni = Intrinsic charge carrier conc. For semiconductor

p = ;

;

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Without any applied electric field, in a solid, electrons (or, in the case of semiconductors, both

electrons and holes) move around randomly. Therefore, on average there will be no overall

motion of charge carriers in any particular direction over time.

Drift velocity in an electric field

When an electric field is applied, each electron is accelerated by the electric field. It moves with

a finite average velocity, called the drift velocity. In a semiconductor the two charge carriers,

electrons and holes, will typically have different drift velocities for the same electric field.

.The electron mobility is defined by the equation:

( )

E is the magnitude of the electric field applied to a material,

is the magnitude of the electron drift velocity caused by the electric field, and µ is the

electron mobility.

The electron mobility characterizes how quickly an electron can move through a metal or

semiconductor, when pulled by an electric field. In semiconductors, there is an analogous

quantity for holes, called hole mobility. The term carrier mobility refers in general to both

electron and hole mobility in semiconductors.

Electron mobility is almost always specified in units of cm2 / (V·s). Semiconductor mobility

depends on the impurity concentrations (including donor and acceptor concentrations), defect

concentration, temperature, and electron and hole concentrations.

Effective mass of hole is greater than mass of electron, hence the mobility of holes is less than

electron. Mobility decreases with the increase in temperature due to thermal vibration. Mobility

decreases with increasing doping concentration

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The drift velocity of electrons in silicon increases linearly with electric field at low values of

electric field and gradually saturates at higher values of electric field called the saturation

velocity VSAT.

Relation between mobility and Electric field

DRIFT CURRENT DENSITY

Let us consider conductor of length „L‟ having „N‟ number of electrons

and having a area of cross section „A‟. So number of electrons crossing

the area in unit time is

Thus the total charge per second passing any area, which, by definition, is the current I in

amperes,

=

; where v = drift velocity (

)

The current density J for the conductor is current per unit cross sectional area of conductor

Ampere/m2

Where LA = volume of the conductor; n = nu mber of electrons per unit volume; n = N/LA

but ;

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Therefore nqµE (amphere/m2)

For n type semiconductor and for p type semiconductor

The total current density in a semiconductor is given by J= nqµnE+ pqµpE,

( ) ( )

Conductivity is proportional to the product of mobility and carrier concentration. For example,

the same conductivity could come from a small number of electrons with high mobility for each,

or a large number of electrons with a small mobility for each.

There is a simple relation between mobility and electrical conductivity. Let n be the number

density of electrons, and let μe be their mobility. The current density is related to electric field E

by relation, J = σE;

where σ = conductivity of the material in (Ω-m) -1

.

If a semiconductor has both electrons and holes, the total conductivity is

( )

( ) ( ) or ( )

Resistivity =

( ) ( ) ( )

Where n is the electron concentration per cubic centimeter

p is the hole concentration per cubic centimeter

is the mobility of electrons in

; is the mobility of holes in

E is the electric field intensity in

q is the charge of hole per electron

In case of metals, increase in temperature won't affect the electric field strength. But it will

decrease the drift velocity because as the temperature increases, the atomic vibrations will

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increase, which will cause more collisions of the electrons with the crystal lattice. Hence the drift

velocity will decrease.

Effect of temperature on semiconductors

Lastly, let's consider what will happen to ni for semiconductors as temperature increases. The

electrons in the valance band will gain energy and go into the higher energy levels in the

conduction band where they become charge carriers! It will it increase exponentially!

Bonds in Semiconductors

The atoms of every element are held together by the bonding action of valence electrons. In

semiconductors, bonds are formed by sharing of valence electrons. Such bonds are called co-

valent bonds. In the formation of a co-valent bond, each atom contributes equal number of

valence electrons and the contributed electrons are shared by the atoms engaged in the formation

of the bond.

Commonly Used Semiconductors

The two most frequently used materials are germanium (Ge) and silicon (Si). It is because the

energy required to break their co-valent bonds (i.e. energy required to release an electron from

their valence bands) is very small; being about 0.7 eV for germanium and about 1.1 eV for

silicon. Therefore, we shall discuss these two semiconductors in detail.

Germanium: Ge: 32=2, 8, 18, 4

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The atomic number of germanium is 32. It is a tetravalent element. Figure shows how the various

germanium atoms are held through co-valent bonds. As the atoms are arranged in an orderly

pattern, therefore, germanium has crystalline structure.

Silicon: Si: 14 = 2, 8, 4

The atomic number of silicon is 14. It is clear that silicon atom has four valence electrons i.e. it is

a tetravalent element. Like germanium, silicon atoms are also arranged in an orderly manner.

Therefore, silicon has crystalline structure.

Property Ge Si

Atomic Number 32 14

Atoms per cm3 4.4 x 10

22 5 x 10

22

EGO at 0deg K 0.785 eV 1.21 eV

Intrinsic conc.(ni) at 300 deg k per cm3 2.5 x 10

13 1.5 x 10

10

Intrinsic resistivity at 300 deg K in (Ω cm) 45 2.3 x 105

µn cm2/Volt.sec 3800 1800

µp cm2/Volt.sec 1300 500

Leakage current µA nA

Temperature range 600 to 75

0C 60

0 to 175

0C

For 10C conductivity Increases by 6% Increases by 8%

Power handling capacity Less High

Dn cm2/sec 99 34

Dp cm2/sec 47 13

Applications High conductivity

and high frequency

Switching applications

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Direct Band Gap Semiconductor

During the recombination the falling e – from the conduction band will be releasing energy in the

form of light. Momentum and direction of e – will remain same.

Ex: GaAs, InP, ZnS

Indirect Band gap Semiconductor

Most of the falling e – will directly releasing energy in the form of heat.

Momentum and direction of e – will change.

Direct band gap materials having higher carrier lifetime and are used for fabrication of LED,

Laser, tunnel diode, photodiode.

Ex: Ge and Si

Extrinsic Semiconductor

The intrinsic semiconductor has little current conduction capability at room temperature. This is

achieved by adding a small amount of suitable impurity to a semiconductor. It is then called

impurity or extrinsic semiconductor. The process of adding impurities to a semiconductor is

known as doping. Generally, for 108 atoms of semiconductor, one impurity atom is added.

The purpose of adding impurity is to increase either the number of free electrons or holes in the

semiconductor crystal.

(i) n-type semiconductor

(ii) p-type semiconductor

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N-type Semiconductor

When a small amount of pentavalent impurity is added to a pure semiconductor, it is known as

n-type semiconductor. The addition of pentavalent impurity provides a large number of free

electrons in the semiconductor crystal. Typical examples of pentavalent impurities are

phosphorous, arsenic and antimony such impurities which produce n-type semiconductor are

known as donor impurities because they donate or provide free electrons to the semiconductor

crystal.

The fifth valence electron of arsenic atom finds no place in co-valent bonds and is thus free.

Therefore, for each arsenic atom added, one free electron will be available in the germanium

crystal. The addition of pentavalent impurity has produced a number of conduction band

electrons i.e., free electrons.

Thermal energy of room temperature still generates a few hole-electron pairs. However, the

number of free electrons provided by the pentavalent impurity far exceeds the number of holes. It

is due to this predominance of electrons over holes that it is called n-type semiconductor

(n stands for negative).

In an n-type material the electron is the majority carrier and the hole is the minority carrier.

Minority carrier noise is called thermal noise or white noise or Johnson noise while majority

carrier will contribute less noise. N type semiconductor as a whole is electrically neutral.

So if ND is the donor impurity concentration, for an n-type material at equilibrium:

=

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If n is the density of electrons and µn, is the electron mobility, then the conductivity is

for n type semiconductor and current density for n type semiconductor

Resistivity for n type semiconductor =

P-type Semiconductor

When a small amount of trivalent impurity is added to a pure semiconductor, it is called p-type

semiconductor. The addition of trivalent impurity provides a large number of holes in the

semiconductor. Typical examples of trivalent impurities are boron, Aluminum gallium and

indium (At. Such impurities which produce p-type semiconductor are known as acceptor

impurities because the holes created can accept the electrons.

When a small amount of trivalent impurity like gallium is added to germanium crystal, there

exist a large number of holes in the crystal. Each atom of gallium fits into the germanium crystal

but now only three co-valent bonds can be formed. In the fourth co-valent bond, only germanium

atom contributes one valence electron while gallium has no valence electron to contribute as all

its three valence electrons are already engaged in the co-valent bonds with neighboring

germanium atoms. This missing electron is called a hole. Therefore, for each gallium atom

added, one hole is created. A small amount of gallium provides millions of holes. It is due to the

predominance of holes over free electrons that it is called p-type semiconductor ( p stands for

positive). In a p-type material the hole is the majority carrier and the electron is the minority

carrier.

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So if NA is the acceptor concentration, for a p-type material at equilibrium:

If p is the density of holes and μh is the hole mobility, then the conductivity is

for p type semiconductor and current density for p type semiconductor

Resistivity for p type semiconductor =

Law of electric neutrality

Total positive charges=Total negative charges

For intrinsic semiconductor ND = 0, NA = 0, hence n = p

N type SC NA = 0, n = ND + p; n = ND

P type SC ND = 0, p= NA + n; p = NA

Drift and diffusion currents:-

The flow of charge (i.e.) current through a semiconductor material is of two types namely drift &

diffusion. The net current that flows through a (PN junction diode) semiconductor material has

two components (i) Drift current (ii) Diffusion current

Drift Current:-

When an electric field is applied across the semiconductor material, the charge carriers attain a

certain drift velocity Vd, which is equal to the product of the mobility of the charge carriers and

the applied Electric Field intensity E ;

Drift velocity Vd = mobility of the charge carriers X Applied Electric field intensity.

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Thus the drift current is defined as the flow of electric current due to the motion of the charge

carriers under the influence of an external electric field. Drift current density Jn , due to free

electrons is given by

Drift current density JP, due to holes is given by

( )

Diffusion current

A concentration gradient exists if the number of either elements or holes is greater in one region

of a semiconductor as compared to the rest of the Region. Thus the movement of charge carriers

takes place resulting in a current called diffusion current. Current flow due to mobile charge

diffusion is proportional to the carrier concentration gradient.

Diffusion current density =charge × concentration gradient

For example, consider an electron concentration that varies as a function of distance x, as shown

the diffusion of electrons from a high-concentration region to a low-concentration region

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produces a flow of electrons in the negative x direction. Since electrons are negatively charged,

the conventional current direction is in the positive x direction. where e, in this context, is the

magnitude of the electronic charge, dn/dx is the gradient of the electron concentration, and Dn is

the electron diffusion coefficient.

diffusion current density due to electrons.

dp/dx is the gradient of the hole concentration, and Dp is the hole diffusion coefficient.

diffusion current density due to holes.

The total current density = drift current density + diffusion current density

due to electrons

due to holes.

The total current density due to holes and electrons = (drift + diffusion) current density

Einstein relation

At room temperature: ⇒ the relationship between Diffusion D and mobility μ

In semiconductors:

=

(volt)

= thermal voltage,

K is Boltzmann‟s constant, q is charge of particle, is in deg K.

=

=

=

=

(

) ( )

( )

HALL EFFECT & APPLICATIONS

If a specimen (metal or semiconductor) carrying a current I is placed in a transverse magnetic

field B, an electric field E is induced in the direction perpendicular to both I and B. This

phenomenon is known as the Hall Effect.

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If the semiconductor is n-type material, so that the current is carried by electrons, these

electrons will accumulate on side 1, and this surface becomes negatively charged with respect to

side 2. Hence a potential, called the Hall voltage (VH), appears between surfaces 1 and 2.

On the other hand, terminal 1 becomes charged positively with respect to terminal 2, the

semiconductor must be p-type.

Experimental Determination of Mobility

In the equilibrium state the electric field intensity E due to the Hall Effect must exert a force on

the carrier which just balances the magnetic force,

Where q is the magnitude of the charge in coulomb, ν is the drift speed in m/s and B is the

magnetic field in

VH is Hall voltage and d is the distance between surfaces 1 and 2

I =

=

; where v = drift velocity (L/T)

The current density J for the conductor is current per unit cross sectional area of conductor

J =

ampere/m2 J =

;

(n =

; is charge density per m

3 ,

= nq)

But current density J =

J =

(

)

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Hall coefficient

is defined as

For conduction is due primarily to charges of one sign, the conductivity is related to the mobility

as .

If the conductivity is measured together with the Hall coefficient, the mobility can be determined

from

is the mobility of carriers in

By Hall experiment, mobility is given by

Hall voltage is positive for n type semiconductors and metals, negative for p type semiconductor

Hall voltage is zero for intrinsic semiconductors.

RH increases with temperature for metals and extrinsic semiconductors and decreases with

temperature for intrinsic semiconductor.

Hall effect is used to find out the concentration of charge carriers and mobility of charge carriers.

Fermi Level

Fermi energy also called characteristic energy is expressed in eV

Fermi energy is defined as the maximum energy possessed by electron at 00K.

Fermi energy is also defined as the energy possessed by the fastest moving electron at 00K

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Fermi Function

Fermi-Dirac distribution function. f(E) is the probability that a level with energy E will be filled

by an electron, and the expression This number has a value between zero and unity and is a

function of temperature and energy

The electron in the conduction band has energy in the range of Ec to +∞ and the energy of holes

in the valence band is extended from -∞ to EV.

( )

(( ) ( ))

( )

(( ) ( ))

( )

Fermi level for the intrinsic semiconductor like germanium, lies in the middle of forbidden band

E is the energy possessed by electron in eV at T=00K

If E > EF, then (E-EF) >0, F (E) =

= , probability of finding an electronis0%

If E = EF, then (E-EF) =0, F (E) =

=

, probability of finding an electron is50%

If E < EF, then (E-EF) <0, F (E) =

= , probability of finding an electron is100%

Then, in the conduction band, all we have to do to find the number of electrons per unit volume

is to count this number of states at the band edge per unit volume (NC) and multiply it with the

probability that a state at that level will be filled (f(EC))

( )

(( ) ( )) ( ( ) ( ))

If the assumption |EC − EF | > kBT holds,

Where KB is the Boltzmann constant

T is the absolute temperature of the intrinsic semiconductor

Nc is the effective density of states in the conduction band.

Nv is the effective density of states in the valence band.

NC and NV are the material constant and function of temperature

In an n-type semiconductor, if we know the doping level ND, we know we can say

n0 = ND; therefore, using the Boltzmann approximation:

( ( ) ( ))

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Similarly, for the holes in the valence band, we take the effective density of states (NV) and

multiply it with the probability that a state at that level will be empty (1 − f(EV ))

( )

(( ) ( ))

( ) (( ) ( ))

(( ) ( ))

( ) ( ( ) ( ))

[

( ) ( ( ) ( ))

The hole-conc. in the valence band and electron conc. in conduction band is given as

( ( ) ( )) ( ( ) ( )

The number of electrons in the conduction band is depends on effective density of states in the

conduction band and the distance of Fermi level from the conduction band. The number of holes

in the valence band is depends on effective density of states in the valence band and the distance

of Fermi level from the valence band.

In an intrinsic semiconductor, n = p. implies that there is an equal chance of finding an electron

at the conduction band edge as there is of finding a hole at the valence band edge. Fermi level EF

must be in the middle of the band gap for an intrinsic semiconductor, as seen in

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FERMİ LEVEL IN INTRINSIC SEMICONDUCTOR( EF)

For an intrinsic semiconductor n = p;

Where p = hole-carrier concentration, n = electron-carrier concentration

Where EF is the Fermi level, EC is the conduction and EV is the valence band, n = p;

( ( ) ( ) ( ( ) ( ))

( ( ) ( ))

( )

Taking logarithm of both sides, we obtain the Fermi level at any temp T deg K

( )

If the effective masses of hole and free electron are the same, Nc = Nv,

Then at T = 0 deg K

( )

Therefore, the Fermi level in an intrinsic semiconductor lies in the middle of the forbidden gap.

In intrinsic semiconductor the Fermi level depends only upon temp. EF on the band diagram is a

function of temperature and carrier concentration. This can be seen from the equations used to

determine the position of the Fermi level.

( ( ) ( ) ( ( ) ( ))

( ) ( ))

( ) ( ))

If EF at the centre of energy gap Eg ( conductivity is zero)

If EF moves away from centre of Eg ( conductivity increases)

If EF moves towards the centre of Eg ( conductivity decreases)

Intrinsic concentration

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FERMILEVEL IN EXTRINSIC SEMICONDUCTOR

For an n-type semiconductor, there are more electrons in the conduction band than there are

holes in the valence band. This also implies that the probability of finding an electron near the

conduction band edge is larger than the probability of finding a hole at the valence band edge.

Therefore, the Fermi level is closer to the conduction band in an n-type semiconductor:

In n type semiconductor n = ND

( ( ) ( )

( ( ) ( )

( ) ( )

( )

To find the Fermi level with respect to the intrinsic Fermi level, we use the expression that links

Electron concentration to Ei and ni:

( ) ( )

( )

If the temperature is varied, the Fermi level will also vary. Fermi energy is a function of temp

At T=0 deg K, Ec = EF

i) With increase in temp, the Fermi level for N type semiconductor will move downward or it

will move away from the conduction band towards centre.

( )

if T increases then ( ) > 0

ii) Effect of doping, the Fermi level will move upward towards the conduction band.

( )

ND > NC, then ( ) < 0

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For a p-type semiconductor, there are more holes in the valence band than there are electrons in

the conduction band. This also implies that the probability of finding an electron near the

conduction band edge is smaller than the probability of finding a hole at the valence band edge.

Therefore, the Fermi level is closer to the valence band in an n-type semiconductor.

Similarly for p-type material EF moves closer to the valence band and the equation is given as

In p-type p = NA where

( ( ) ( ))

( ) ( )

( ) ( )

( )

To find the Fermi level with respect to the intrinsic Fermi level, we use the expression that links

Electron concentration to Ei and ni:

( ) ( )

( )

If the temperature is varied, the Fermi level will also vary. Fermi energy is a function of temp

At T=0 deg K, EV = EF

i) With increase in temp, the Fermi level for P type semiconductor will move upward or it will

move away from the valence band.

( )

if T increases then ( ) > 0

ii) Effect of doping, the Fermi level will move downwards towards the valence band.

( )

NA > NV, then ( ) < 0

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Ei = EF = EC − Eg / 2 = EV + Eg / 2

Carrier Lifetime

In a pure semiconductor the number of holes is equal to the number of free electrons. Due to

thermal agitation, new electron hole pairs are produced, while other hole-electron pairs

disappeared as a result of recombination. On an average a hole (an electron) will exist for Гp,

(Гn) sec before recombination. This is called the mean lifetime for hole and electron

respectively. Carrier lifetimes range from nanoseconds to hundreds of microseconds.

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Diffusion Length L

The average distance covered by an charge carrier while diffusion during its life time is called

the diffusion length of the charge carrier. It is denoted by (Гn, Гp – mean life time for electron

and hole)

Ln = Dn Гn for free electron

Lp = Dp Гp for free hole

Continuity Equation

The continuity equation describes a basic concept, namely that a change in carrier density over

time is due to the difference between the incoming and outgoing flux of carriers plus the

generation and minus the recombination.

The carrier concentration in the body of a semiconductor is a function of time and distance. Such

an equation is called continuity equation. The equation is based on the fact that charge can

neither be created nor destroyed.

Consider the infinitesimal n type element of volume of area A and length dx as shown. The

current entering the volume at x is I and leaving at ( ) is ( ). This change in current is

because of diffusion = dI

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Now due to diffusion the concentration of charge carriers decreases exponentially with the distance.

Hence,dl = Number of coulombs per second decreased within the volume ... (1)

Now if Гp , is the mean life time of the holes then,

= holes per second lost by recombination

per unit volume. Due to recombination, number of coulombs per second decreased within the

given volume is,

(Charge on Holes) x (Holes /Sec per unit volume) x (Volume)

( )

While let g is the rate at which electron hole pairs are generated by thermal generation per unit

volume. Due to this, number of coulombs per second increases with the volume

(Charge on Holes) x (Rate of generation) x (Volume) = q g A dx

Thus the total change in number of coulombs per second is because of three factors as indicated

by the equations (1), (2) and (3).

Total change in holes per unit volume per second is the total change in coulombs per second

within the given volume =

( )

According to law of conservation of charges,

But J =

Prepared by I Krishna Rao (Associate Professor ECE Dept) Page 30

The total current density is due to drift and diffusion current.

where E = Electric field intensity within the volume

If the semiconductor is in thermal equilibrium and subjected to no external electric field then

hole density will attain a constant value po. Under this condition I = 0 i.e. J = 0 and dp/dt= 0 due

to equilibrium.

;

;

The equation indicates the thermal equilibrium i.e. the rate at which holes are thermally

generated just equal to the rate at which holes are lost due to the recombination.

[

( )

This is called equation of conservation of charge or the continuity equation.

As holes in n type material are considered, let us use the suffix n. And as concentration is a

function of both time t and distance x, let us use partial differentiation. Hence the final continuity

equation takes the form as,

( )

( )

Similarly the continuity equation for the electrons in p type material can be written as,

( )

( )