2. Ohm’s law relates the current I to the applied voltage V as follows:
where R - resistance of the material.
Units: V : volts (j/c)
I : amperes (c/s)
R : ohms (V/A )
The resistivity ‘ρ’ is independent of specimen geometry.
l : distance between 2 points at which voltage is measured
A : cross sectional area perpendicular to the direction of
units : ρ ohm meter (Ω-m)
3. It is defined as the reciprocal of resistivity
It indicates the case with which a material is capable of conducting
an electric current.
units : σ ( ohm meter (1 ̸Ωm ) )
ohm’s law can be expressed as
where J – current density [I/A]
E – Electric field density ( v/l )
solid materials are classified according to ease with whivh they conduct
electric current . They are classified as
1) conductors : conductivity order
3) semi conductors
4. An electric current results from the motion of electrically charged
particles in response to forces that act on them from an externally
applied electric field.
Within most solid materials a current arises from the flow of
electrons, which is termed electronic conduction.
For ionic materials a net motion of charged ions is possible that
produces a current; such is termed ionic conduction.
5. ENERGY BAND STRUCTURES IN SOLIDS:
In all conductors, semiconductors, and many insulating materials,
Conduction exists which is strongly dependent on number of
electros available to participate in the conduction process.
These electrons are related to the arrangement of electron states or
For each individual atom there exist discrete energy levels that may
be occupied by electrons, arranged into shells and subshells.
Shells are designated by integers (1, 2, 3, etc.), and for each
subshells represented by letters (s, p, d, and f ) there exists one,
three, five, seven states respectively.
Pauli exclusion principle states that the electrons fill in the states
with lower energy with two electrons of opposite spins.
7. A solid may be thought of as consisting of a large number of atoms(say
N) that are brought together and bounded to form the ordered atomic
Each atom has atomic levels that split into a series of closely spaced
electrons in solid to form electron energy band.
The extent of splitting depends on interatomic separation.
Within each band, the energy states are discrete, yet the difference
between adjacent states is exceedingly small.
The number of states within each band will equal the total of all states
contributed by the N atoms.
8. each energy state accommodate two electrons, which must have
oppositely directed spins.
Four different types of band structures are possible at 0 K.
a. In the first one outermost band is only partially filled with
electrons.The energy corresponding to the highest filled state at 0
K is called the Fermi energy Ef.
b. For the second band structure, there is an overlap of an empty
band and a filled band. Eg:Magnesium.
c. The final two band structures are similar; one band (the valence
band) that is completely filled with electrons is separated from an
empty conduction band, and an energy band gap lies between
For the insulators band gap is relatively wide whereas for
semiconductors it is narrow.
10. The energy corresponding to the highest filled state at 0k is
called Fermi energy.
Electrons with energies greater than the Fermi energy acted on
them can be accelerated in the presence of an electric field.
The electrons that participate in the conduction process are
known as free electrons.
Electrical conductivity is a direct function of the numbers of
free electrons and holes.
11. For an electron to become free, it must be excited or
promoted into one of the empty and available energy
states above Fermi energy.
The energy band gap for metals is almost zero for
metals,so they are very good conductors.
13. For insulators and semiconductors, empty states adjacent to the top of the
filled valence band are not available.
To become free, electrons must be promoted across the energy band gap.
The number of electrons excited thermally (by heat energy) into the
conduction band depends on the energy band gap width as well as
Larger the band gap, the lower is the electrical conductivity at a given
For insulators the width of the band gap is narrow whereas for insulators, it
is relatively wide.
Increasing the temperature of either a semiconductor or an insulator results
in an increase in the thermal energy that is available for electron excitation.
14. For electrically insulating materials, interatomic
bonding is ionic or strongly covalent.
Thus, the valence electrons are tightly bound to or
shared with the individual atoms.
The bonding in semiconductors is covalent and
relatively weak, which means that the valence
electrons are not as strongly bound to the atoms.
So, these electrons are more easily removed by
thermal excitation than they are for insulators.
16. When an electric field is applied, a force is brought to bear on
the free electrons;so, they all experience an acceleration in a
direction opposite to that of the field, by virtue of their
All the free electrons accelerate as long as the electric field is
applied, which will give rise to an electric current that
continuously increases with time.
Current reaches a constant value the instant that a field is
applied, indicating that there exist “frictional forces,” that
counter this acceleration from the external field.
17. These frictional forces result from the scattering of electrons by
imperfections in the crystal lattice, including impurity atoms vacancies,
interstitial atoms, dislocations, and even the thermal vibrations of the atoms
Each scattering event causes an electron to lose kinetic energy and to
change it direction of motion.
18. The drift velocity represents the average electron velocity in
the direction of the force imposed by the applied field. It is
directly proportional to the electric field as
The constant of proportionality µe is called the electron
mobility, which is an indication of the frequency of scattering
Units are square meters per volt-second(m2/V-s).
19. The conductivity of most materials is
n is the number of free or conducting electrons per unit
volume (e.g., per cubic meter), and |e| is the absolute
magnitude of the electrical charge on an electron .
Thus, the electrical conductivity is proportional to both
the number of free electrons and the electron mobility.
20. Most metals are extremely good conductors of electricity.
Metals have high conductivities because of the large numbers
of free electrons that have been excited into empty states
above the Fermi energy.
Crystalline defects serve as scattering centers for conduction
electrons in metals, increasing their number raises the
resistivity (or lowers the conductivity).
The concentration of these imperfections depends on
temperature, composition,and the degree of cold work of a
21. The total resistivity of a metal is the sum of the contributions
from thermal vibrations, Impurities, and plastic deformation.
Where ρt,ρi,ρd represent the individual thermal, impurity, and
deformation resistivity contributions, respectively.
The above is sometimes known as Mat-thiessen’s rule.
22. For the pure metal and all the copper–nickel alloys, the
resistivity rises linearly with temperature:
where ρo and a are constants for each particular metal.This
dependence of the thermal resistivity component on
temperature is due to the increase with temperature in thermal
vibrations and other lattice irregularities (e.g., vacancies),
which serve as electron-scattering centers.
23. For additions of a single impurity that forms a solid
solution, the impurity resistivity ρi is related to the
impurity concentration ci in terms of the atom fraction
(at%100) as follows:
A is a composition-independent constant that is a function of
both the impurity and host metals.
24. For a two-phase alloy consisting of and phases, a rule-of-
mixtures expression is used to approximate the resistivity as:
where the V’s and ’s represent volume fractions and individual
resistivities for the respective phases.
27. Copper render it the most widely used metallic conductor.
Oxygen-free high-conductivity (OFHC) copper, having
extremely low oxygen and other impurity contents, is
produced for many electrical applications.
Both solid-solution alloying and cold working improve
strength at the expense of conductivity.
For some applications, such as furnace heating elements
materials must have not only a high resistivity, but also a
resistance to oxidation at elevated temperatures and,a high
Nichrome, a nickel–chromium alloy, is commonly employed
in heating elements
28. Electrical conductivity of semiconductors less than
that of metals
have unique useful electrical characteristics
sensitive to the presence of even minute
concentrations of impurities
Two types : 1) Intrinsic – pure materials
2) Extrinsic –impurities added
29. band structure
relatively narrow forbidden band
gap (<2 eV)
Eg : Si (1.1eV) and Ge(0.7eV)
Also compounds composed of
Groups IIIA and VA – GaAs, In Sb
Groups IIB and VIA- Cd S, Zn Te
30. In above compounds ,
-> atomic bonding becomes more ionic
-> band gap energy increases—more
31. a vacant electron state in the VB created because of
e- from VB excited to CB
Treating a missing electron from the valence band
as a positively charged particle called
Charge of hole= +1.6x10^-19 C
In the presence of an electric field, excited
electrons and holes move in opposite directions
33. Two types of charge
carrier (free electrons
p is the number of
holes per cubic meter
µh is the hole mobility
ni is known as the
34. impurity concentration determines the
electrical behavior of semiconductors
One atom in 10^12 is sufficient to render
silicon extrinsic at room temperature.
Types : 1) n-type
35. Si has 4 e-s : covalent bonded with other
If we add a Grp VA impurity , 4e-s form
bonds , there will be 1 e- free .
It is free e- / conducting e-
For these free e-s , energy state is in
forbidden band gap just below CB
If energy is given,e- get excited to CB
Hence this type impurity is termed a ‘donor’
No hole in VB
36. At room temp, we can easily excite
large no. of e-s from donor states
No.of e-s in CB >>no.of holes in
This type material is n-type
e- are majority carriers
Holes are minority charge carriers
n-type : Fermi level shifted
upward in band gap
39. If we add a Group IIIA such as Al, Ga, B three electrons form bonds.
One covalent bond is deficient in electrons. This is called Hole. It
participates in conduction process.
Impurity atom introduces energy close to the top of valence band in
No free electrons are created either in valence band or conduction band.
This type of impurity is called acceptor.
41. For p>>n material is called P type(positively charged)
Holes are majority carries ,electronns are minority.
For P type, fermi level with bond gap near to the acceptor level.
Extrinsic semiconductors (both n- and p-type) Controlled
concentrations of specific donors or acceptors are then intentionally
This alloying process in
semiconducting materials is termed doping.
At relatively high room-temperature electrical conductivities are
obtained in extrinsic semiconductors
43. Figure,plots the logarithm of the intrinsic carrier
concentration versus temperature
for both silicon and germanium.
The concentrations of electrons and holes increase
with temperature because, with rising temperature
At all temperatures, carrier concentration in Ge is
greater than for Si . This effect is due to germanium’s
smaller band gap thus, for Ge, at any given
temperature more electrons will be excited across its
45. The conductivity (or resistivity) of a semiconducting material, in addition
to being dependent on electron and/or hole concentrations, is also a
function of the charge carriers’ mobilities.
Influence of Dopant Content:
46. At dopant concentrations less than about both
carrier mobilities are at their maximum levels
and independent of the doping concentration.
Influence of Temperature:
47. The temperature dependences of electron and
hole mobilities for silicon are presented in
These plots, note that, for dopant
concentrations of 10−24
and below, both
electron and hole mobilities decrease in
magnitude with rising temperature.