# EC 102 Lecture 1 Semiconductor Theory

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Autumn 2011-12

EC 102: Fundamentalsof Electronics

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Name of Books/Authors Year of publication

Electronic Devices and Circuit Theory,9th edition by R.L. Boylestad and L.

Nashelsky

PearsonEducation,

Asia, 2006

Integrated Electronics, 2nd ed. J.Millman and C. Halkias, C. D. Parikh.

Tata McGraw-Hill, 2010

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Marks Distribution:

MTE 1: 15

MTE 2: 15

ETE: 40

CWS: 15(10 + 5*)

PRS: 15(10 + 5*)

*Common Quiz

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Lecture - 1

Introduction toSemiconductor Theory

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Introduction to Semiconductor Theory

Semiconductors, as the name implies, are a group of materials whose electricalconductivity is greater than that of insulators but less than that of metals.

Semiconductor materials are found in Group IV and neighbouring columns of the

periodic table. The ones in group IV (C, Si, Ge) are called elemental semiconductors

because they are composed of the pure element.

Insulators MetalsSemiconductors

Increasing conductance

II III IV V VI

B C N

Al Si P S

Zn Ga Ge As Se

Cd In Sb Te

Semiconductor Materials

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These compounds are widely used in various electronic and optical applications.

Elemental IV compunds

Binary III-V

compounds

Binary II-VI

compunds

Ternary

compounds

Quaternary

compounds

Si SiC AlP ZnS GaAsP InGaAsP

Ge SiGe AlAs ZnSe AlGaAs

AlSb ZnTe

GaN CdS

GaP CdSe

GaAs CdTe

GaSb

InP

InAs

InSb

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Covalent Bond Structure of Semiconductors

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Charge Carriers and Energy Levels

Two types of charge carriers exist in semiconductors, namely: electrons and holes

A hole is the term used to describe an atom with a missing electron. An electron may beshaken loose from a bond structure by lattice vibration caused from thermal heating. Theremaining atom (ion) now has a positive charge. The loose electron is usually called a freeelectron.

The mechanism of conduction in semiconductors is best explained via energy leveldiagrams Specific energy levels are always associated with each shell of orbiting electrons in atomic

structures. While the energy of each shell is different, the further away an electron is from theparent nucleus, the higher its energy state.

In order to break the covalent bond, a valence electron must gain a minimum energy, Eg,

called the bandgap energy.

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Notice that : Eg(Ge) < Eg(Si) < Eg(GaAs)

Energy level diagrams

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Types of Semiconductors: IntrinsicExtrinsic

Intrinsic Semiconductors This is the term given to near perfect semiconductors crystals with no

impurities or lattice defects.

At 00 Kelvin there are no free charge carriers, but as temp increases a fewelectron-hole pairs (EHP) are generated due to valence electrons getting

thermally excited to have enough energy to jump over the bandgap into theconduction band.

Since electron-holes are always created in pairs, then

n = p = ni

where n and p are the electrons and holes concentration (per cm3),

respectively

Recombination occurs when an electron in the conduction band makes atransition to the valence band to recreate a complete covalent bond structurewith the hole At steady state, EHPs recombine at the same rate as they are generated

Generation and recombination of EHP are dependent on temperature

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Extrinsic Semiconductors

It is usually desirable to have greater number of available charge carriers that

will not be subjected to recombination. The bond structures of pure

semiconductors may be manipulated in such a way that an excess of freeelectrons or Holes may be generated in predetermined and controllable

manner. The process used to generate these excess charge carriers is called

DOPING. Any semiconductor that has been subjected to the doping process is

called an extrinsicsemiconductor.

Doping is the process of adding specific impurities to a pure semiconductorin such a way that the newly formed covalent bonding creates excess charge

carriers within the crystal lattice

Semiconductors with predominantly excess Electrons are called n-Type

semiconductors

Semiconductors with predominantly excess Holes are called p-Typesemiconductors

When impurities or lattice defects are introduced into an otherwise perfect crystal additional energy

levels are created, usually within the bandgap

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n-Type SemiconductorAn n-type material is created by introducing impurity element thathave 5 valence electrons (e.g. arsenic, antimony and phosphorus)into pure Si or Ge.

Diffused impurities with 5 valence electrons are calleddonor atoms

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p-Type Semiconductor

The p-type semiconductor is formed by doping pure Si or Ge with impurity atoms

having 3 valence electrons. Typically Boron, gallium, or indium is use as the dopant.

Diffused impurity with 3 valence electrons are called acceptor atoms

In n-type material, the electron is the majority charge carrier and holes are minority.

In p-type material, the holes are the majority charge carrier and electrons are minority

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Under thermal equilibrium, the product of the free negative

and positive concentration is a Constant independent of the

amount of donor or acceptor impurity doping.

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In n-type semiconductor: NA = 0, nn >>pn

Similarly, in p-type semiconductor:

p NAA

ip

N

nn

2

=

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A fundamental relationship between the electron and hole concentration in a semiconductor inthermal equilibrium is given by:

where,

nois the thermal equilibrium concentration of free electrons,

po is the thermal equilibrium concentration of holes, and

niis the intrinsic carrier concentration.

At room temperature (T = 300 K), each donor atom donates a free electron to the semiconductor. If

the donor concentration Ndis much larger than the intrinsic concentration, we can approximate

therefore

Similarly, at room temperature, if each acceptor atom accepts a valence electron, creating a hole. Ifthe acceptor concentration, Na, is much larger than the intrinsic concentration, we can approximate

making

2

ioo

npn =

do Nn d

io

N

np

2

=

ao Np a

i

oN

nn

2

=

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Introduction to Semiconductor Theory

The Fermi-Dirac Distribution Function

A very important property of the density of electrons in a crystal lattice is their

distribution among the allowed states at thermal equilibrium. Electrons have integer

spin that obey Paulis exclusion principle. The occupational probability of an energylevel E by an electron is given by the Fermi-Dirac distribution function

where Ef is the reference energy called the Fermi Level.

Note that when E = Ef, f(E) =

Throughout any semiconductor structure at thermal equilibrium, the Fermi

Level is always constant. This may be expressed as:

The Fermi Level is one of the principal quantities which is used to describe the

behavior of semiconductor materials and devices

kT

EE f

e

Ef)(

1

1)(

+= Tis temp in Kelvin

kis Boltzman constant

0=dx

dEf

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Charge concentration in semiconductor

At absolute zero temperature,

Ef is the maximum energy that an electron may possess at abs.

zero temperature.

Let density of energy states be N(E)

>=

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Charge concentration in semiconductor

Probability of electrons in conduction band

Probability of holes in valence band

Concentration of electrons in conduction band

( )c

kTEEEEeEf f >

for,)(

( )V

kTEEEEeEf f

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Charge concentration in semiconductor

Similarly, concentration of holes in valence band

We can derive

( )( )

32

32

3

21 per1082.4

)(1)(

mTm

mN

eNdEEfENp

p

V

kTEE

V

E

Vf

V

=

==

( )

np

NNkTEEE

N

NkTEEE

VcVcG

V

cVcf

ln

ln

22

1

==

+=

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Drift and Diffusion Currents

The two basic processes which cause electronsand holes to move in a semiconductor are:

(b) diffusion, which is the flow caused by variations

in the concentration, that is, concentration gradients.Such gradients can be caused by a non-

homogeneous doping distribution, or by the injection

of a quantity of electrons or holes into a region.

(a) drift, which is the movement caused by electric

fields; and

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When a stedy electric field E is applied to a

semiconductor sample, each electrons willexperience a forceqEfrom the field and will beaccelerated in the opposite direction of the field.This is called the drift velocity and willsuperimpose upon the usual random thermal

motion of the electrons.

Each Hole will also be similarly affected by thefield, but drift will occur along the direction of thefield.

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Current Density

Drift velocity of electrons

Current

Current density

( )

L

vNqI

L

Nqv

T

NqI

=== Ampere

Ev

=

( )

( )

( )

tyconductivi,

conc.charge,

conc.electron,

===

====

==

EE

v

nvnqLA

vNq

A

I

J

L

Av

E

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Current Density

Units

Mobility: m2

/V-s Current density: A/m2

Charge density: Coulomb/m3

Conductivity: = Coulomb/m3 m2/V-s = (Ohm-m)1

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Drift velocities for electrons and holes are:

Ev nn = Ev pp =

wheren is called the electron mobility andp is the holemobility, both with units ofcm2 / V-s.

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Drift CurrentsWhen an electric field Eis applied across a length of uniformly doped semiconductor,

of cross sectional area, A, the electron current density Jn flowing in the sample can be

found by summing the product of the charge (-q) on each electron times the electron

velocity over all electrons per unit volume (n):

where In is the electron current.

A similar argument applies to holes:

The total drift current may now be written as the sum

=

====n

i

nni

n

n EqnqnvqvA

IJ

0

)(

EqpqpvJ ppp ==

EqpqnJ pn )( +=

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Conductivity and ResistivityThe quantity in parenthesis from the previous equation is known as the conductivity:

The electron and hole contribution to conduction is simply additive. The

corresponding resistivity of the semiconductor is:

Because of the many orders of magnitude difference between the majority carriers in

extrinsic semiconductor, the resistivity reduces to

and

)( pn qpqn +=

)(11

pn qpqn

+==

)(

1

nqn =

)(

1

pqp =

for n-type for p-type

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At room temperature (3000K),EG = 1.1 eV

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For Germanium:

EG(T) = 0.785 2.23 104T

At room temp EG(T) = 0.72 eV

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ForSi, m = 2.5 for electrons

= 2.7 for holes

ForGe, m = 1.66 for electrons= 2.33 for holes

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Diffusion CurrentsAs stated earlier, if there is a spatial variation of carrier concentration in the

semiconductor material, carriers will move from a region of high concentration to a

region of low concentration. This current component is called diffusion current. The

electron and hole diffusion current may be expressed as

where Jn and Jp are the electron and hole diffusion density in units ofcm

2

/s, andDnand Dp are the electron and hole diffusivity.

The Current Density Equation:

When Eis present in addition to carrier concentration gradients, both drift and

diffusion currents will flow

dx

dnqDJ nn =

dx

dpqDJ pp =

pncond

ppp

nnn

JJJ

dx

dpqDpEqJ

dxdnqDnEqJ

+=

=

+=

For large E field

TheEterms must

Be replaced by field

Dependent carrier

velocity

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dx

dpqDpEqJ

dx

dn

qDnEqJ

ppp

nnn

=

+=