Addis Ababa Science and Technology University

College of Electrical & Mechanical Engineering

Electrical & Computer Engineering Department

Electrical Materials and Technology (EEEg-3093)

Chapter Two
Conducting Materials
Conducting Materials

Outline:

 Introduction

 Electron Theory of Conductors

 Common Elemental Conductors

 Fermi Energy of Metals

 Influence of Frequency on Conductivity

 Factors Influencing Resistivity

 Thermal Conductivity of Conductors

Sem. I, 2019/20 Chapter 2- Conducting Materials 2

Introduction

 Electrical conduction occurs through transport of electric

charge in response to an applied electric field.

 Electric charge is carried by electrons, holes and ions.

 When a conductor is subjected to an external field, there is

a flow of electrons in the direction of the electric field.

 Conductors have small bandgap energy (Eg).

Sem. I, 2019/20 Chapter 2- Conducting Materials 3

Introduction…….

 The charge transport in pure metals is caused by the drift

of free electrons.

 Free electrons have comparatively high velocities and

relatively long mean free paths until they collide with ions
constituting the crystal lattice. This process is called
scattering.

 In a perfect periodic lattice structure, no collisions would

occur and the resistivity would be zero.

Sem. I, 2019/20 Chapter 2- Conducting Materials 4

Introduction…….
 Mean free paths of the electrons are limited by:

i. ionic vibrations due to thermal energy

ii. the crystal defects such as vacancies and dislocations

iii. the random substitution of impurity atoms for pure

metal atoms on the pure metal lattice sites.
 As the temperature increases, the amplitudes of the ionic
vibrations grow larger and scattering of the electrons
increases.
 This offers more resistance to the flow of electrons.

Sem. I, 2019/20 Chapter 2- Conducting Materials 5

Electron Theory of Conductors

 According to Drude-Lorentz theory, the motion of electrons

in a conductor is random when no electric field is applied.

 The number of electrons moving from left to right at any

time is the same as that moving from right to left. This
shows that no net current flows through the conductor.

 However, if we apply an electric field across a conductor,

the electrons move in the positive direction of the field and
current is produced.

Sem. I, 2019/20 Chapter 2- Conducting Materials 6

Electron Theory of Conductors……

 When an electric field E is applied across the conductor,

the force experienced by an electron due to the applied
electric field is given by: F  eE (1)

 Due to this force, the electron moves with an average

acceleration given by:
dv d 2 x
a  2 (2)
dt dt
 The force with which the electron moves is then:
dv
F  ma  m (3)
dt

Sem. I, 2019/20 Chapter 2- Conducting Materials 7

Electron Theory of Conductors……

 From equations (1) and (3), we obtain:

dv eE
m  eE   dv   dt
dt m
eE
 v t  K (4)
m
where K is constant of integratio n, which represents the random

velocity of the electrons.

 The average value of random velocity must be zero,

otherwise there will be a flow of current even in the
absence of external field. Thus, K  0
Sem. I, 2019/20 Chapter 2- Conducting Materials 8
Electron Theory of Conductors……

 Therefore, the average velocity of the electron is given by:

eE
v t (6)
m
where t is the collision time (average time between
two successive collisions )

 Equation (6) is called the equation of motion of an electron

under the applied electric field.

 The average velocity is also known as drift velocity because

the drift in electrons is due to applied field (E).

Sem. I, 2019/20 Chapter 2- Conducting Materials 9

Resistivity and Conductivity

 The property or ability of a conductor due to which it

opposes the flow of current through it is called resistivity
(ρ). The unit of resistivity is Ohm-meter (Ω-m).

 The reciprocal of resistivity is termed as conductivity (σ).

The unit of conductivity is Siemens/meter (S/m).

 Electrical conductivity is very low in insulators σ<10-15 S/cm,

intermediate in semiconductors, σ=10-5 to 103 S/cm, very
high in conductors, σ=104 to 106 S/cm and infinite in
superconductors.

Sem. I, 2019/20 Chapter 2- Conducting Materials 10

Current Density in Conductors

 Let us consider N number of free electrons distributed

uniformly throughout a conductor of length L and cross-
sectional area A as shown in the figure below.

 If an electric field E is applied to such a conductor, the

electrons travel a distance L meters in T seconds.

Sem. I, 2019/20 Chapter 2- Conducting Materials 11

Current Density in Conductors……

 Now, the number of electrons passing through any area

per second is equal to N/T.

 Total charge passing through any area per second called

current and is given by:

N eN eN L eNv
I  e*   I 
T T L T L

 The current per unit area is known as the current density

I eNv
and is given by: J  
A LA

Sem. I, 2019/20 Chapter 2- Conducting Materials 12

Current Density in Conductors……

 Here LA is the volume of the conductor containing N

electrons and therefore the concentration of electrons per
unit volume is:
N eEt
n  J  env , but v 
LA m

 eEt  e nEt
2
 J  en 
 m  m

 In the above equation, e, n, t and m are all constants for

any conductor.

Sem. I, 2019/20 Chapter 2- Conducting Materials 13

Current Density in Conductors……
2
 Therefore, the term nt is a constant and is equal to
e
m
conductivity of a metal, i.e.

e 2 nt
  J  E
m

 From the above equation, it is obvious that the current

density in a conductor is directly proportional to the applied
electric field (E). Now, substituting

v et nev
  
E m E

Sem. I, 2019/20 Chapter 2- Conducting Materials 14

Mobility

 It is observed that the average velocity of the electrons in a

conductor is directly proportional to the applied electric
field, i.e.,
v  E  v  E (7)

 The constant of proportionality, µ is called mobility of the

electrons. It is usually expressed in m2/Volt-sec or
cm2/Volt-sec. The magnitude is given by:
et
 (8)
m
where t is collision time
Sem. I, 2019/20 Chapter 2- Conducting Materials 15
Mobility……
v
 Now, substituting   in equation (8), we obtain:
E

nev e 2 nt
  ne 
E m

 Sometimes, one requires to determine the concentration of

free electrons per unit volume (n).

 The number of free electrons per unit volume is given by:

x * Avogadro’ s number x*N
n  n
M olar volume Vm

x : is the number of free electrons per atom.

Sem. I, 2019/20 Chapter 2- Conducting Materials 16

Mobility……

 The molar volume is given by:

Atomic weight M
Molar Volume   Vm 
Density d

 Thus, the number of free electrons per unit time is:

x * N * Density x* N *d
n  n
Atomic weight M

 The number of charge carriers, n, remains constant for

metallic conductors with increasing temperature , but
increases exponentially for semiconductors and insulators.
Sem. I, 2019/20 Chapter 2- Conducting Materials 17
Examples on Electron Theory of Conductors……

 Thus, at very high temperatures some insulators become

semiconducting, while at low temperatures some
semiconductors become insulators.

Example-1:

There are 1019 electrons/m3, which serves as carriers in a

material. The conductivity of material is 0.01 S/m. Find the
drift velocity of these carriers, when 0.17 V potential
difference is applied across 0.27 mm distance within the
material. Given: e = 1.602 x 10-19 C and m = 9.11 x 10-31 kg.

Sem. I, 2019/20 Chapter 2- Conducting Materials 18

Examples on Electron Theory of Conductors……

Solution:
V 0.17 V
E  3
 630 V / m
d 0.27 x10 m

Let v be the drift velocity.Then conductivity is given by :

nev E
  v
E ne

630 * 0.01
 v 19 19
 3.93
10 * 1.602 *10

 v  3.93 m / s

Sem. I, 2019/20 Chapter 2- Conducting Materials 19

Examples on Electron Theory of Conductors……

Example-2:

Find the conductivity of copper at 300 K. The collision time

for electron scattering in copper at 300 K is 2 x10-14 sec.
Given that density of copper = 8960 kg/m3, atomic weight of
copper = 63.54 amu and mass of an electron = 9.11 x10-31 kg.

Solution:
We know that 63.54 gm of copper contains 6.02 x 10 23 free electrons

(i.e., Avogadro’ s number as the one atom contributes one electron).

The volume of 63.54 gm of copper is 8.9 cm 3 .

Sem. I, 2019/20 Chapter 2- Conducting Materials 20
Examples on Electron Theory of Conductors……

The electron density is given by :

6.023 *10 23
n  8.5 *10 22 electrons / cm 3
63.54 / 8.9

 8.5 *10 22 * 10 6  8.5 * 10 28 electrons / m 3

The conductivity  is then :

e 2 nt (1.6023 *10 19 ) 2 * 8.5 * 10 28 * 2 * 10 14

 
m 9.11 * 10 31

   4.78 *10 7 S / m

Sem. I, 2019/20 Chapter 2- Conducting Materials 21

Examples on Electron Theory of Conductors……

Example-3:

The mean free time between the collisions is 10-14 sec. Find
the mobility of electrons?

Solution:
The mobility of electrons is given by :

et 1.602 *10 19 *10 14

    1.76 * 10 3

m 9.11 *10 31

1.602 *10 19 * 10 14

  31
 1.76 * 10 3
m 2
/ V .s
9.11 *10
Sem. I, 2019/20 Chapter 2- Conducting Materials 22
Examples on Electron Theory of Conductors……

Example-4:

The conductivity of silver is 6.5 x 107 S/m and number of

conduction electrons is 6 x 1028 electrons/m3. Find the
mobility of conduction electrons and the drift velocity in an
electric field of 1 V/m.

Given m = 9.11 x 10-31 kg and e = 1.602 x 10-19 C.

Sem. I, 2019/20 Chapter 2- Conducting Materials 23

Examples on Electron Theory of Conductors……

Solution:
M obility of conduction electrons,


  ne   
ne

6.5 * 10 7
   6.76 * 10 3

6 *10 23 *1.602 *10 19

   6.76 *10 3 m 2 / V .s

Drift velocity,

v  E  6.76 * 10 3 *1  6.76 * 10 3 m / s

Sem. I, 2019/20 Chapter 2- Conducting Materials 24

Examples on Electron Theory of Conductors……

Example-5:

The density of silver is 10.5 gm/cm3 and its atomic weight is

107.9 amu. Assuming that each silver atom provides one
conduction electron, find the number of free electrons/cm3.
Take conductivity of the silver as 6.8 x 107 mhos/m. Also
calculate the mobility of electrons.

Given e = 1.6 x 10-19 C.

Sem. I, 2019/20 Chapter 2- Conducting Materials 25

Examples on Electron Theory of Conductors……

Solution:
The number of free electrons,

Nd 6.02 *10 23 * 10.5

n  n  5.86 *10 22
Atomic Weight 107.9

 n  5.86 * 10 22 electrons / cm 3

M obility of free electrons,

 6.8 * 10 5
  ne     22 19
 72.42
ne 5.86 * 10 * 1.6 * 10

   72.42 cm 2 / V .s

Sem. I, 2019/20 Chapter 2- Conducting Materials 26

Common Elemental Conductors

 Conducting materials can, in general, be classified as low

resistivity and high resistivity materials.

 Low resistivity materials include:

 common metals such as copper, aluminum and steel

 copper alloys such as brass and bronze

 some special metals such as silver, gold and platinum

 High resistivity materials include:

 manganin, constantin, Nichrome, mercury, carbon and
tungsten

Sem. I, 2019/20 Chapter 2- Conducting Materials 27

Fermi Energy of Metals

 Fermi Energy ( EF) is the energy of the state at which the

probability of electron occupation is ½ at any temperature
above 0 K.

 It is also the maximum kinetic energy that a free electron

can have at 0 K.

 Thus, the energy of the highest occupied level at absolute

zero temperature is called the Fermi Energy or Fermi Level.

Sem. I, 2019/20 Chapter 2- Conducting Materials 28

Fermi Energy of Metals……

 The Fermi energy at 0 K for metals is given by:

 h2 
2/3
 3N   
E F (0)   
   8me 
 When temperature increases, the Fermi level or Fermi
energy also slightly decreases.

 The Fermi energy at non–zero temperature is given by:

  2  kT  2 
E F (T )  E F (0) 1    
 12  E F (0)  

Sem. I, 2019/20 Chapter 2- Conducting Materials 29

Fermi Energy of Metals……

 The probability of an electron occupying an energy level E

is described by the Fermi-Dirac distribution function, f(E).

 The Fermi-Dirac distribution function, f(E), is defined as:

1
f (E) 
1  e ( E  EF ) / kT

 If the level is certainly empty, then f(E) = 0.

 Generally f(E) has a value in between zero and unity.

Sem. I, 2019/20 Chapter 2- Conducting Materials 30

Fermi Energy of Metals……
 When E< EF , i.e., for energy levels lying below EF,
(E –EF) is a negative quantity and hence,

1 1
f (E)  
 1
1 e 1 0
 That means all the levels below EF are occupied by the
electrons.
 When E > EF , i.e., for energy levels lying above EF,
(E – EF) is a positive quantity and hence,
1 1
f (E)  
 0
1 e 1 
Sem. I, 2019/20 Chapter 2- Conducting Materials 31
Fermi Energy of Metals……
 This equation indicates all the levels above EF are vacant.
 At absolute zero, all levels below EF are completely filled
and all levels above EF are completely empty.
 This level, which divides the filled and vacant states, is
known as the Fermi energy level.
 The probability of finding an electron with energy equal to
the Fermi energy in a metal is ½ at any temperature, i.e., at
all temperatures, when E=EF :
1 1 1
f (E)   
1 e 0
11 2
Sem. I, 2019/20 Chapter 2- Conducting Materials 32
Fermi Energy of Metals……

 At T = 0 K, all the energy level up to EF are occupied and

all the energy levels above EF are empty .

 When T > 0 K, some levels above EF are partially filled

while some levels below EF are partially empty.

 The average Fermi energy is given by:

3
E ave  EF
5

Sem. I, 2019/20 Chapter 2- Conducting Materials 33

Fermi Energy of Metals……

Fig. Fermi-Dirac distribution function at different temperatures

Sem. I, 2019/20 Chapter 2- Conducting Materials 34

Fermi Energy of Metals……

Mean Free Path:

 The mean free path is an average distance which an

electron covers in its wavelike pattern without any reflection
or deflection.

 Mathematically, the mean free path is defined as:

  vt

where :

v: is velocity of an electron and

t: is collision time or mean free time

Sem. I, 2019/20 Chapter 2- Conducting Materials 35
Fermi Energy of Metals……

 For metals, the velocity of an electron corresponds to that

of Fermi energy (EF) and is given by the relation:
2E F
vF 
m

where :

E F : is Fermi energy in Joules

m  9.11 * 10 31 kg

Sem. I, 2019/20 Chapter 2- Conducting Materials 36

Examples on Fermi Energy of Metals……

Example-1:
The density of Ga atoms is 1028 atoms/m3.
Given : me  9.11 *10 31 Kg , h  6.626 *10 34 J .s and

k  1.38 *10 -23 J / K

a. Calculate the Fermi energy and the averaged energy
of the conduction electrons in a solid piece of Ga in
units of eV at a temperature of 0 K.
b. Find the value of the Fermi energy at room
temperature (300 K).

Sem. I, 2019/20 Chapter 2- Conducting Materials 37

Examples on Fermi Energy of Metals……

Solution:
First, indicate the electronic configurat ion for

gallium (Ga) Z  31.

1s 2 2s 2 2 p 6 3s 2 3 p 6 4 s 2 3d 10 4 p 1

Thus, there are three conduction electrons per atom.

The electron density is then, N  3 * 10 28 electrons / m 3

This is because each Ga atom offers three electrons

for conduction.
Sem. I, 2019/20 Chapter 2- Conducting Materials 38
Examples on Fermi Energy of Metals……

a. The Fermi level at 0 K can be calculated as :

2/3
h  3N 2
EF   
8m e   


6.626 * 10  34 2
 3 * 3 * 10

26


2/3

8 * 9.11 * 10 31   

 5.64 * 10 -19 J

 E F  3.525 eV

Sem. I, 2019/20 Chapter 2- Conducting Materials 39

Examples on Fermi Energy of Metals……

a. The average energy of the conduction electrons is :

3
E ave  E F  0.6 E F  2.115 eV
5
b. The Fermi level at 300 K can be calculated as :

 2  kT 
2

E F (T )  E F (0)1    
 12  E F (0)  
 

 2  0.025 eV  
2

 3.525 * 1      3.5248 eV
 12  3.525 eV  


Sem. I, 2019/20 Chapter 2- Conducting Materials 40

Examples on Fermi Energy of Metals……

Example-2:

Evaluate the temperature at which there is 1 % probability

that a state with an energy 0.5 eV above the Fermi energy
will be occupied by an electron.

Solution:
1 1 E  EF
f (E)   f (E)  , where x 
 E  EF 
  1 e x
kT
1 e  kT 

E  E F  0.5 eV and f ( E )  1%  0.01

Sem. I, 2019/20 Chapter 2- Conducting Materials 41

Examples on Fermi Energy of Metals……

1
 0.01   1  e x
 100  e x
 99
1 e x

0.5
 x  2.303 log 99   2.303 log 99
kT

0.5
 kT   0.109 eV  0.109 *1.6 * 10 19 J
2.303 log 99

0.109 0.109 * 1.6 * 10 19

T    23
 1264
k 1.38 *10

 T  1264 K

Sem. I, 2019/20 Chapter 2- Conducting Materials 42

Examples on Fermi Energy of Metals……

Example-3:

a. Estimate the maximum velocity of an electron in a metal

in which Fermi energy has a value of 3.75 eV.
Given e = 1.602 x 10-19 C and m = 9.1 10-31 kg.

b. What will be the mobility of electrons when the mean free

time between the collisions is 10-14 sec?

Sem. I, 2019/20 Chapter 2- Conducting Materials 43

Examples on Fermi Energy of Metals……

Solution:

2E F
a. v F 
m

E F  3.75 *1.602 * 10 19  6 *10 19 J

2 * 6 *10 19
 vF  31
m / s  1. 76 * 10 6
m/s
9.11 * 10

et 1.602 * 10 19 * 10 14

b.    31
 1.76 * 10 3
m 2
/ V .s
m 9.11 * 10

Sem. I, 2019/20 Chapter 2- Conducting Materials 44

Influence of Frequency on Conductivity

 A good conductor is one in which σ>>ωε so that σ/ωε goes

to infinity.

 This means, as frequency increases, the conductivity of a

material decreases.

 The reverse is true for insulators/dielectric materials.

 A good insulator/dielectric material is one in which σ<<ωε

so that σ/ωε goes to zero.

 This means, as frequency increases, the insulation

(dielectric nature) of a material increases.

Sem. I, 2019/20 Chapter 2- Conducting Materials 45

Factors Influencing Resistivity

 Materials in general have a characteristic behavior of

opposing the flow of electric charge.

 This opposition is due to the collisions between electrons

that make up the materials.

 This physical property, or ability to resist current, is known

as resistance and is represented by the symbol R.

 Thus, the resistance R of an element denotes its ability to

resist the flow of electric current and it is measured in
ohms (Ω).

Sem. I, 2019/20 Chapter 2- Conducting Materials 46

Factors Influencing Resistivity……

 The resistance of any material is influenced by four factors:

1. Material property: each material will oppose the flow of
current differently.
2. Length: the longer the length , the more is the probability of
collisions and, hence, the larger the resistance.
3. Cross-sectional area: the larger the area A, the easier it
becomes for electrons to flow and, hence, the lower the
resistance.
4. Temperature: typically, for metals, as temperature
increases, the resistance increases.

Sem. I, 2019/20 Chapter 2- Conducting Materials 47

Factors Influencing Resistivity……

 Thus, the resistance R of any material with a uniform cross-

sectional area A and length l is directly proportional to the
length and inversely proportional to its cross-sectional area.
 In mathematical form:

l
R
A

 Resistivity is a physical property of the material and is

measured in ohm-meters (Ω-m).

Sem. I, 2019/20 Chapter 2- Conducting Materials 48

Factors Influencing Resistivity……

 The cross section of an element can be circular, square,

rectangular, and so on.

 Because most conductors are circular in cross-section, the

cross-sectional area may be determined in terms of the
radius r or diameter d of the conductor as:

 d2
2
d 
A   r    
2

2 4

 The resistivity ρ varies with temperature and is often

specified for room temperature.

Sem. I, 2019/20 Chapter 2- Conducting Materials 49

Factors Influencing Resistivity……

Fig. Resistivities of some common materials

Sem. I, 2019/20 Chapter 2- Conducting Materials 50

Factors Influencing Resistivity……

 The above table presents the values of ρ for some

common materials at room temperature (200C).

 Of those materials shown in the table, silver is the best

conductor.

 However, a lot of wires are made of copper because

copper is almost as good and is much cheaper.

 In general, the resistance of a conductor increases with a

rise in temperature.

Sem. I, 2019/20 Chapter 2- Conducting Materials 51

Factors Influencing Resistivity……

Examples-1:

a. Calculate the resistance of an aluminum wire that is 2m

long and of circular cross section with a diameter of 1.5mm.

b. Consider a copper bus bar shown in the figure below.

Calculate the length of the bar that will produce a
resistance of 0.5Ω.

Sem. I, 2019/20 Chapter 2- Conducting Materials 52

Factors Influencing Resistivity……

Solution:

a. We first calculate the cross - sectional area :

 d2  (1.5 *10 3 ) 2
A   1.767 * 10 6 m 2
4 4
From the above table, we are given the resistivity of

aluminum to be   2.8 *10 8   m

Thus,

l
2.8 *10 8 * 2
R  6
 31.69 m
A 1.767 *10

Sem. I, 2019/20 Chapter 2- Conducting Materials 53

Factors Influencing Resistivity……

b. The cross - sectional area of the bus bar is given by :

  
A  Width * Breadth  2 * 10 3 * 3 *10 3 
 6 * 10 6 m 2  6  m 2
From the above table, we are given the resistivity of

copper tobe   1.72 * 10 8   m

Thus,

l RA 0.5 * 6 *10 6
R  l   174.4 m
A  1.72 *10 8

Sem. I, 2019/20 Chapter 2- Conducting Materials 54

Thermal Conductivity of Conductors

 Thermal conduction is the transfer of thermal energy from a

region of higher temperature to a region of lower
temperature through direct molecular communication within
a medium or between mediums indirect physical contact
without a flow of the material medium.

 Thermal energy is transferred by conduction when adjacent

atoms vibrate against one another, or as electrons move
from atom to atom.

Sem. I, 2019/20 Chapter 2- Conducting Materials 55

Thermal Conductivity of Conductors……

 Thermal conduction is greater in solids, where atoms are in

constant close contact.

 In liquids (except liquid metals) and gases, the molecules

are usually further apart, giving a lower chance of
molecules colliding and passing on thermal energy.

 Metals (eg. copper) are usually the best conductors of

thermal energy.

 This is due to the way that metals are chemically bonded:

metallic bonding with free-moving electrons.

Sem. I, 2019/20 Chapter 2- Conducting Materials 56

Thermal Conductivity of Conductors……

 The thermal conductivity of a material is given by:

Q T ( x, t )
 A
dt x
where : Q is infinitesi mal quantity of thermal energy

A is conduction surface area

x is infinitesi mal conduction thickness

 is thermal conductivity constant

 Thermal conductivity is a material property that is primarily

dependent on the medium's phase, temperature, density,
and molecular bonding.
Sem. I, 2019/20 Chapter 2- Conducting Materials 57
Exercises
1. The density of Zn is 1.8*1028 atoms/m3.

a. Calculate the Fermi energy and the averaged energy

of the conduction electrons in a solid piece of Zn in
units of eV at a temperature of 0 K.

b. Find the value of the Fermi energy at room

temperature (300 K).

Sem. I, 2019/20 Chapter 2- Conducting Materials 58

Exercises……

2. Two wires are made of the same material. The first wire
has a resistance of 0.2Ω. The second wire is twice as
long as the first wire and has a radius that is half of the
first wire. Determine the resistance of the second wire.

3. Two wires have the same resistance and length. The first
wire is made of copper, while the second wire is made of
aluminum. Find the ratio of the cross-sectional area of
the copper wire to that of the aluminum wire.

Sem. I, 2019/20 Chapter 2- Conducting Materials 59

Exercises……

4. High-voltage power lines are used in transmitting large

amounts of power over long distances. Aluminum cable
is preferred over copper cable due to low cost. Assume
that the aluminum wire used for high-voltage power lines
has a cross-sectional area of 4.7*10-4 m2. Find the
resistance of 20 km of this wire.

Sem. I, 2019/20 Chapter 2- Conducting Materials 60