Unit –III
Free Electron Theory
Engineering Physics
1. Introduction
The electron theory of metals aims to explain the structure and properties of solids
through their electronic structure. The electron theory is applicable to all solids i.e., both
metals and non metals. It explains the electrical, thermal and magnetic properties of
solids etc. The theory has been developed in three main stages.
The classical free electron theory
Drude and Lorentz proposed this theory in 1900. According to this theory, the
metals containing the free electrons obey the laws of classical mechanics.
The quantum free electron theory
Somerfield developed this theory in 1928. According to this theory the free
electrons obey quantum laws. According to this theory the free electrons are moving in a
constant potential.
The zone theory
Bloch stated this theory in 1928. According to this theory, the free electrons move
in a periodic field provided by the lattice. According to this theory the free electrons are
moving in a constant potential.
2. Definitions and relations
Drift velocity It is defined as the average velocity acquired by the free electrons of a metal in a
particular direction by the application of electric field.
Relaxation time It is defined as the time taken by the free electrons to reach its equilibrium position
from the disturbed position in the presence of electric field.
Collision time (τ )
It is defined as the average time taken by the free electrons between two successive
collisions.
Current density ( j )
It is defined as the magnitude of current passing through unit area.
I
or I = ∫ J • ds
J=
A
Expression for Relaxation time (τ r )
When the metal is subjected to an external electric field, the electrons move opposite to
the applied field. After removal of electric field the drift velocity decays exponentially as
−t
vd = vo e
τr
Where vo is the initial velocity of an electron, before application of electric field and τ r is
the relaxation time.
Dr. P.Sreenivasula Reddy
M.Sc, (PhD)
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Unit –III
Free Electron Theory
If t = τ r then
vd = vo e −1
or
vd =
Engineering Physics
vo
e
Thus the relaxation time may be stated as the time taken for the drift velocity to decay
to 1 of its original initial value.
e
Mean free path (λ )
Free electrons in a metal are continuously moving in all directions and with various
speeds. They frequently collide with one another. Therefore, they move in straight line
with constant speeds between two successive collisions. The distance traveled by the
electron between two successive collisions is called as free path and their mean is called
the mean free path.
The average distance traveled by the electron between two successive collisions is
called mean free path.
Or
The mean free path is the average distance traveled by an electron between two
successive collisions with other free electrons.
λ = cτ
Where c is the mean square velocity of electron.
3K BT
c=
m
Expression for Mean collision time:The average time taken by the electrons between two consecutive collisions of
electron with the lattice points is called mean collision time.
If ‘ v ’ be the total velocity of electron i.e. thermal and drift velocity, then the mean collision
time is given by
λ
τ=
v
v = vd + vth
If vd << vth then v = vth
∴τ =
λ
vth
Expression for drift velocity
When electric field is applied on an electric charge ‘ e ’, then it moves in opposite
direction to the field with a velocity vd . This velocity is known as drift velocity.
The Lorentz force acting on the electron is FL = −eE
m vd
The resistance force acting on the electron is Fr = τ r
When the system is in steady state Fr = − FL
m vd
τr
vd =
= − (− eE )
eEτ r
m
Dr. P.Sreenivasula Reddy
M.Sc, (PhD)
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Unit –III
Free Electron Theory
Engineering Physics
Mobility of electrons
The mobility of electrons is defined as the magnitude of drift velocity acquired
by the electron in a unit field.
µ=
vd
E
We know σ =
∴µ =
σ
ne
=
neAvd nevd
J
I
=
=
=
= neµ
E AE
AE
E
n e 2τ eτ
=
mne m
Qσ =
Q I = n e A vd
ne2τ
m
3. The classical free electron theory of metals (Drude –Lorentz theory of metals)
Drude and Lorentz proposed this theory in 1900. According to this theory, the
metals containing the free electrons obey the laws of classical mechanics.
Assumptions ( or salient features) in classical free electron theory
The classical free electron theory is based on the following postulates.
1. The valence electrons of atoms are free to move about the whole volume of the
metal, like the molecules of a perfect gas in a container.
2. The free electrons move in random direction and collide with either positive ions
fixed to the lattice or the other free electrons. All the collisions are elastic in nature
i.e., there is no loss of energy.
3. The momentum of free electrons obeys the laws of the classical kinetic theory of
gases.
4. The electron velocities in a metal obey classical Maxwell-Boltzman distribution of
velocities.
5. When the electric field is applied to the metal, the free electrons are accelerated in
the direction opposite to the direction of applied electric field.
6. The mutual repulsion among the electrons is ignored, so that they move in all the
directions with all possible velocities.
7. In the absence of the field, the energy associated with an electron at temperature
3
1
3
2
T is given by
kT . It is related to the kinetic energy equation kT = m vth
2
2
2
Here
vth
represents the thermal velocity.
Success of classical free electron theory
1.
2.
3.
4.
It verifies ohm’s law
It explains electrical conductivity of metals.
It explains thermal conductivity of metals.
It derives Widemann – Franz law. (I.e. the relation between electrical and thermal
conductivity.
Draw backs of classical free electron theory.
1. It could not explain the photoelectric effect, Compton Effect and black body radiation.
2. Electrical conductivity of semiconductors and insulators could not be explained.
K
3. Widemann – Franz law (
= constant) is not applicable at lower temperatures.
σT
Dr. P.Sreenivasula Reddy
M.Sc, (PhD)
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Unit –III
Free Electron Theory
Engineering Physics
4. Ferromagnetism could not be explained by this theory. The theoretical value of
paramagnetic susceptibility is greater than the experimental value.
5. According to classical free electron theory the specific heat of metals is given by 4.5 R
where as the experimental value is given by 3R
3
6. According to classical free electron the electronic specific heat is equal to
R while
2
the actual value is 0.01R
4. Classical free electron theory –electrical conductivity
The classical free electron theory was proposed by Drude and Lorentz. According to this
theory the electrons are moving freely and randomly moving in the entire volume of the
metal like gas atoms in the gas container. When an electric field is applied the free
electrons gets accelerated.
When an electric field is applied between the two ends of a metal of area of cross
section A
Force acting on the electron in the electric field
From Newton’s second law
The acceleration of electron
The average velocity acquired (i.e. drift velocity) by the electrons by the application of
electric field is
1
.
2
Where = RMS velocity.
The relation between current and drift velocity is
3
Substituting the value of in from equation (1) into equation (3), we get
! Conductivity "
"
Resistivity #
#
According to kinetic theory of gasses $
"
Dr. P.Sreenivasula Reddy
M.Sc, (PhD)
!
%
!
&'(
√3
!
!
!
*+
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Unit –III
Free Electron Theory
√3
#
Mobility ,
,
"
!
!
Engineering Physics
*+
,
5. Sources of electrical resistance in metals
According to quantum free electron theory, the free electrons always collides with the
positive ions or electrons present in the metal. The scattering of conduction electrons are
due to
1. Effect of temperature
2. Defect, e.g. impurities, imperfections, etc.
Temperature effect:
The positive ions are always in oscillating (or vibrating) state about their mean
position; even the substance is present at 0k temperature. The vibrating amplitude of
ions is always depends the temperature. The mean free path λ of the electrons is
inversely proportional to the mean square of amplitude of ionic vibrations ..
0
/ 3
(1)
12
The energy of lattice vibrations is proportional to . ! and increases linearly with
temperature T.
. ! / T
(2)
From equations (1) and (2)
0
(3)
/
(
The resistivity # of the metal is inversely proportional to mean free path of electrons.
0
#/
(4)
5
From equations (3) and (4)
#/T
(5)
From equation (5) we observe that the resistivity of metal is linearly increases with
temperature.
The conductivity " is defined as the reciprocal of resistance.
0
0
"
(6)
6
(
From equation (6) we observe that the conductivity of metal is inversely proportional to
their temperature.
The variation of resistivity of metal with temperature is shown in figure.
Unit –III
Free Electron Theory
Engineering Physics
Defect dependence:
The variation of resistivity of copper- nickel alloys as a function of temperature is shown in
figure
At a particular temperature the ideal resistivity of alloy can be written as
#7 AX1 : X
Where X is the concentration and A is a constant which depends upon the base metal and
impurity.
The total resistivity change is generally given by the Matthiessen’s rule
#;<;=> #?@A B #C?@A
Where #?@A is temperature dependent resistivity due to thermal vibrations of the positive
ions while #C?@A is caused by the scattering of electrons by impurity atoms. #?@A is
dominated at higher temperature and #C?@A is dominated at low temperature.
6. Distribution laws introduction
Three statistical distributions have been developed in explaining the distribution of atoms
or molecules or electrons in energy levels.
1. Maxwell Boltzmann distribution
2. Bose Einstein distribution
3. Fermi Dirac distribution
Maxwell Boltzmann distribution
In Maxwell Boltzmann distribution all the particles in the system are distinguishable
and no more restrictions on filling the particles in the energy levels. This is mainly
applicable for atoms and molecules. According to Maxwell Boltzmann distribution, the
probability of occupying an energy level DC is
EDC 1
FG H FI J(
Unit –III
Free Electron Theory
Engineering Physics
Bose Einstein distribution
In Bose Einstein distribution all the particles in the system are indistinguishable and no
more restrictions on filling the particles in the energy levels. This is mainly applicable for
bosons. (Bosons are the particles with zero or integral spin). According to Bose Einstein
distribution, the probability of occupying an energy level DC is
EDC 1
FG H FI J(
:1
Fermi Dirac distribution
In Fermi Dirac distribution all the particles in the system are indistinguishable and it
obeys Pauli’s exclusive principle (i.e., not more than two electrons can occupy the
same energy level) on filling the particles in the energy levels. This is mainly
applicable for fermions (Fermions are the particles with odd half integral
0 & M
spins 0, , , , … … . ). According to Fermi - Dirac distribution, the probability of
! ! !
occupying an energy level DC is
EDC 1
FG H FI J(
B1
Fermi energy
The energy of the highest occupied level at 0K is called the
Fermi energy and the energy level is referred as the Fermi
level. The Fermi energy is denoted by EF
Or
The Fermi level is that energy level for which the probability of
1
occupation is or 50% at any temperature.
2
Dr. P.Sreenivasula Reddy
M.Sc, (PhD)
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Unit –III
Free Electron Theory
Engineering Physics
7. Fermi Dirac distribution
In Fermi Dirac distribution all the particles in the system are indistinguishable and it
obeys Pauli’s exclusive principle (i.e., not more than two electrons can occupy the same
energy level) on filling the particles in the energy levels. This is applicable for fermions
(Fermions (e.g. electrons, protons, neutrons, etc.) are the particles with odd half integral
0 & M
spins0, , , , … … . ).
! ! !
. According to Fermi - Dirac distribution, the probability of electron occupying an
energy level DC is
1
EDC FG H FI J(
B1
th
DC is the energy of i level
D is the energy of Fermi level,
k is a Boltzmann constant
T is the absolute temperature and
E DC is the Fermi function
The variation of Fermi function with temperature is shown in figure.
Fϵ
1
0.5
0
ϵF
ϵ
Effect of temperature on Fermi Dirac distribution
When the material is at a temperature ‘T’ 0K, below the Fermi energy levels are
totally fulfilled and above the energy levels are empty. When it receives thermal energy
from surroundings, the electrons are thermally excited into higher energy levels. The
occupation of electrons in energy levels obeys a Fermi Dirac statistical distribution law.
According to Fermi - Dirac distribution, the probability of electron occupying an
energy level DC is
E DC Dr. P.Sreenivasula Reddy
M.Sc, (PhD)
1
FG H FI J(
B1
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Unit –III
Free Electron Theory
Engineering Physics
The variation of Fermi function with temperature is shown in figure.
Fϵ
ϵ
ϵ
F
From this curve we can observe the probability of finding an electron at different
temperatures.
Case I. At + Q 0R E DC
1
EDC F H F E DC Case II. At +
E DC G
J(
1
2
I
E DC B1
.
T U
J(
!
.
B1
0R E DC Q D , S EDC 1
FG H FI .
1
∞B1
Case III. At +
0
D , S E DC 1
B1
1
∞B1
1
FG H FI .
1
∞
1
1
H∞ B 1
EDC 1
1
B1
: WY
1
1
1
0B1 1
0
EDC 1
B1
]
0
When + Q 0 EDC Z 1 T^Y_ Q U E`a DC Z D
!
E DC Q 0 T^Y_ Z U E`a DC Q D
!
0
B WY
0
B WX
∞
] DC Z D DC – D From figure (1) we illustrate that
0
1
2
V DC : D B1
H\ \=>@
.
1
1B1
0
0R E DC Z D , S EDC FG – FI .
1
B1
H∞
1
∞
1
∞
0
Unit –III
Free Electron Theory
0
D , S E DC When + Q 0R E DC
Engineering Physics
So the Fermi level can be defined as
!
The energy level for which the probability of occupation is
temperature.
0
!
`a 50% at any
Properties of Fermi function
1.
It is applicable for all insulators, semiconductors and metals.
In semiconductor the probability of electron occupying an energy level DC is
1
E DC FG H FI J(
B1
In semiconductor the probability of hole occupying an energy level DC is
1
1
Ed DC 1 : E DC 1 : F H F e Ff HFI g
I
G
J(
B1
J(
B1
2.
3.
D , S EDC At + Q 0R E DC
4.
0R E DC Q D , S EDC 0R E DC Z D , S EDC At +
At +
5.
6.
0
!
0
1
From the Fermi Dirac distribution we understood that, below the Fermi levels are totally
fulfilled and above the levels are totally empty
8. Kroning and Penney model
According to Kroning and Penney the electrons move in a periodic square well
potential. This potential is produced by the positive ions (ionized atoms) in the lattice.
The potential is zero near to the nucleus of positive ions and maximum between the
adjacent nuclei. The variation of potential is shown in figure.
V=0
I
V=V0
II
-b
h 7 0
h 7 E`a
h.
E`a
0
a
i a j`
0Z7Z
ii a j` : ^ Z 7 Z 0
The corresponding Schrödinger wave equations for I region is
k3 Ψ
k
l3
B
k3Ψ
k l3
mn 3 d3
Ψ
B o !Ψ
pq a
o!
0
] h7
0
0
E`a
i a j`
(1)
mn 3 d3
Dr. P.Sreenivasula Reddy
(2)
M.Sc, (PhD)
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Unit –III
Free Electron Theory
Engineering Physics
The corresponding Schrödinger wave equations for II region is
k3 Ψ
kl
B
3
k3 Ψ
k l3
k3Ψ
k l3
mn 3 :
d3
: h. Ψ
mn 3 d3
h. : Ψ
: r!Ψ
pq a
0
] h7
0
h.
E`a
ii a j`
] h. Q 0
mn 3 r!
d3
(3)
h. : (4)
The general solutions of the equations (1) and (3) are of the form
Ψ0 x
AeC u l B BeHC u l
Ψ! x
(5)
Ae w l B BeH w l
(6)
Solving the above equations (5) and (6) by applying boundary conditions, we get
mn 3 x2
!d3 u
y
u=
^. Vo B $`o Vo B $`o Where z
mn 3 x2 = {
The plot of
u=
y
! d3
$`R $`R potential barrier strength
Vo B $`o verses o is shown in figure.
From the above spectrum we observe that
1. The energy spectrum of the electron consists of a large number of allowed energy
bands and forbidden energy bands
2. The width of the allowed energy bands increases with increasing of o i.e. with
increases energy.
3. As ˆ ‰ ∞ the allowed regions becomes infinity narrow and the energy spectrum
becomes a line spectrum as shown in figure (2)
Unit –III
4. For ˆ
Free Electron Theory
Engineering Physics
0 the energy spectrum is continuous as shown in figure (3)
9. Introduction about Band Theory and definitions
The origin of an energy gap is explained by considering the formation of energy
bands in solids. The concentration of atoms in a gaseous medium is very low compared
to the concentrations of atoms in a solid medium. As a result, the interaction between
any two atoms in a gaseous substance is very weak, since the interatomic distance is
very large. In case of solid substance interatomic distance is very small, and hence there
is an interaction between any two successive atoms. Due to this interaction the energy
levels of all atoms overlap with each other and hence bands are formed.
Band
A set of closely packed energy levels is called as band
Valence band
A band which is occupied by the valence electrons is called as valence band. The
valence band may be partially or completely filled up depending on the nature of the
material.
Conduction band
The lowest unfilled energy band is called as conduction band. This band may be
empty of partially filled. In conduction band the electrons can move freely.
Unit –III
Free Electron Theory
Engineering Physics
Forbidden energy gap or Forbidden gap
The energy gap between valence band and conduction band is called forbidden energy
gap or forbidden gap or band gap.
10. Origin of bands in solids
According to quantum free electron theory, the free electrons move in constant
potential and the atoms have independent energy levels (discrete energy levels). Pauli’s
exclusive principle is applied in filling the energy levels.
According to Kroning and
Penney model the free electrons moves in a square well periodic potential and predicts
(tell in advance) the existence of allowed energy band and forbidden energy gaps.
The origin of an energy gap is explained by considering the formation of energy
bands in solids. The concentration of atoms in a gaseous medium is very low compared
to the concentrations of atoms in a solid medium. As a result, the interaction between
any two atoms in a gaseous substance is very weak, since the interatomic distance is
very large.
In case of solid substance interatomic distance is very small, and hence there is an
interaction between any two successive atoms. Due to this interaction the energy levels
of all atoms overlap with each other and hence bands are formed.
11. Conductors, Semi conductors and Insulators
The solids are classified into three types based on the forbidden gap. Those are
Conductors, Semiconductors and Insulators.
Conductors
In case of conductors, the valence and conduction bands are overlap to each other
as shown in figure 1.
Unit –III
Free Electron Theory
Engineering Physics
In conduction band, plenty of free electrons are available for conduction. The energy gap
in conductors is zero. The charge carriers in conductors are electrons. Conductors have
negative temperature coefficient. All metals are the examples of conductors.
Semiconductors
In case of semiconductors the forbidden gap is very small as shown in figure.
Germanium and Silicon are the best examples for semiconductors. In Germanium the
forbidden gap is 0.7eV. In silicon the forbidden energy gap is 1.1eV. At 0K the valence
band is completely fulfilled and the conduction band is totally empty. When a small
amount of energy is supplied, the electrons can easily jump from valence band to
conduction band. The charge carriers in semiconductors are both electrons and holes.
Insulators
In case of insulators, the forbidden energy gap is very wide as shown in figure.
In insulators the valence electrons are bound vary tightly to their parent atoms. In case of
insulators the forbidden energy gap is always >6eV. Due tot this fact electrons cannot
jump from valence band to conduction band.
Unit –III
Free Electron Theory
Engineering Physics
12. Quantum free electron theory-electrical conductivity
To overcome the draw backs of classical free electron theory, Sommerfeld
proposed quantum free electron theory. He treated electron as a quantum particle. The
electrons follow quantum laws. Under this, the velocities of the electrons are plotted in the
velocity space with dots as shown in figure. The Fermi level electrons have maximum
velocity and are known as Fermi velocity and is represented by the Fermi sphere.
If electric field E is applied along 7 direction, then the electrons in the sphere
experiences a force along – 7. only those electrons present near the Fermi surface can
take electrical energy and occupy higher vacant energy levels. For the rest of the
electrons, the energy supplied by electrical force is too small so that they are unable to
occupy higher vacant energy levels.
The relation between momentum and wave vector is
z
ŠR
ŠR
R
ŠR
2‹
1
Differentiating w.r.t. ‘t’ we get
Œ Š ŒR
2
Œ_
Œ_
Force applied on the electron in the applied field is given by
Š ŒR
Œ_
ŒR
Integrating the above the equation, we get
M.Sc, (PhD)
Œ_
Š
R; : R.
Dr. P.Sreenivasula Reddy
_
Š
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Unit –III
On Fermi surface _
Free Electron Theory
and R; : R.
∆R
Š
∆R
The current density
3
Ž
∆
From equation (1)
Š ∆R
∆
Ž
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Š ∆R
4

Substituting the value of ∆R from equation (3) into equation (4), we get
Ž
Ž
!

"
Comparing the above equations
"
!

The above equation represents the electrical conductivity.
Merits of quantum free electron theory
1. It successfully explains the electrical and thermal conductivity of metals.
2. We can explain the Thermionic phenomenon.
3. Temperature dependence of conductivity of metals can be explained by this theory.
4. It can explain the specific heat of metals.
5. It explains magnetic susceptibility of metals.
Demerits of quantum free electron theory
1. It is unable to explain the metallic properties of exhibited by only certain crystals
2. It is unable to explain why the atomic arrays in metallic crystals should prefer
certain structures only
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M.Sc, (PhD)
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