Electron Theory of Metals Explained

The Free Electron of Metal
G. Beyene (Ph.D) (ASTU) Chapter One Cover Page - 1/22 March - 2022
Chapter One
Gashaw Beyene (Ph.D)
March 2022
ASTU

G. Beyene (Ph.D) (ASTU) Chapter One The Free Electron… - 2/22 March - 2022
 The electron theory of solids aims to explain the
structures and properties of solids through their
electronic structure.
 The electron theory of solids has been developed in
three main stages.
1. The classical free electron theory:
Drude and Lorentz developed this theory in 1900. According to this theory,
the metals containing free electrons obey the laws of classical mechanics.
2. The quantum free electron theory:
Sommerfeld developed this theory during 1928. According to this theory,
the free electrons obey quantum laws.
2. The zone theory:
Bloch states this theory in 1928. According to this theory, the free electrons move in a periodic
field provided by the lattice. This theory is also called ‘Band theory of Solids’.

 The classical free electron theory of metals (Drude-Lorentz theory of metals)
Postulates
In an atom electrons revolve around the nucleus and a metal is
composed of such atoms.
a.
The valence electrons of atoms are free to move
about the whole volume of the metals like the
molecules of a perfect gas in container. The
collection of valance electrons from the all
atoms in a given pieces of metal forms electron
gas. It is free to move throughout the volume of
the metal.
b.

These free electros move in random directions and collide with
either positive ions to the lattice or other free electrons. All the
collisions are elastic i.e., there is no loss of energy.
c.
The movement of free electrons obey the laws of the classical kinetic
theory of gases.
d.
The electron velocities in the metal obey the classical Maxwell-
Boltzmann distributions of velocities.
e.
The electrons move in a completely uniform potential field
due to its ions fixed in the lattice.
f.
When an electric field is applied to the metal, the free electrons
are accelerated in the direction opposite to the direction of
applied electric field.
g.

1
2
𝑚𝑣𝐷
2
=
3
2
𝑘𝐵𝑇 𝑣𝐷 =
3𝑘𝐵𝑇
𝑚 At RT 𝑣𝐷 ≈ 105𝑚𝑠−1
 Mean free path
 The average distance traveled by an electron between two successive collisions inside a metal in the
presence of applied field is known as mean free path (λ).
 Relaxation time
 The time taken by the electron to reach equilibrium position from its disturbed position in the
presence of an electric field is called relaxation time (τ).
λ = τ𝑣𝐷 λ ≈ 1𝑛𝑚 τ ≈ 1 × 10−14
𝑠
 The average velocity (𝑣𝐷) of the free electrons with which they move towards the positive terminal
under the influence of the electrical field.

 Success of classical free electron theory:
1. It verifies Ohm’s law
2. It explains the electrical and thermal conductivity of metals
3. It drives Wiedemann-Franz law (i.e., the relation between
electrical conductivity and thermal conductivity).
4. It explains the optical properties of metals
 Drawbacks of classical free electron theory:
 The phenomena such a photoelectric effect, Compton effect and the black body radiation couldn’t
be explained by classical free electron theory.
 According to classical free electron theory the value of specific heat of metals is given by 4.5𝑅𝑢 is the
universal gas constant whereas the experimental value nearly equal to 3𝑅𝑢. Also according to this
theory the value of electric specific heat is equal to 1.5𝑅𝑢 while the actual value is about 0.01𝑅𝑢.

 Electrical conductivity of semiconductor or insulators couldn’t explained using this model.
 Through 𝑘 𝜎𝑇 is a constant (Wiedemann-Franz law) according to classical free electron theory, it is
not a constant at low temperature.
 Ferromagnetism couldn’t be explained by this theory. The theoretical value of paramagnetic
susceptibility is greater than the experimental value.
 Electrical Conductivity
 In the presence of an electric field E, the electrons acquire a
drift velocity 𝑣𝐷 which is superimposed on the thermal motion.
 Drude assumed that the probability that an electron collides with
an ion during a time interval 𝑑𝑡 is simply proportional to 𝑑𝑡 τ ,
where τ is called the collision time or relaxation time.

 Then Newton’s law gives 𝑚
𝑑𝑣𝐷
𝑑𝑡
+
𝑣𝐷
τ
= −𝑒𝐸
 The behavior of 𝑣𝐷(𝑡) as a function of time is given by
𝑣𝐷 𝑡 = 𝑣𝐷(0)𝑒−𝑡/τ  In the steady-state (where 𝑑𝑣𝐷 𝑑𝑡 = 0), 𝑣𝐷 is given by
𝑣𝐷 = −
𝑒𝐸τ
𝑚
 The quantity 𝑒τ 𝑚 , the drift velocity per unit electric field, is called 𝜇, the drift mobility. The velocity
of an electron including both thermal and drift components is
𝑣 = 𝑣𝑇 −
𝑒𝐸τ
𝑚
 The current density caused by the electric
field 𝐸 is simply
𝐽 = 𝑉−1
𝑎𝑙𝑙
𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠
(𝑒𝑣)

But
𝑎𝑙𝑙
𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠
𝑣𝑇 = 0
so that
𝐽 = 𝑉−1𝑁 −𝑒 −
𝑒𝐸τ
𝑚
= 𝜎𝐸
 Here 𝜎, the electrical conductivity, is equal to 𝑛0𝑒2
τ 𝑚 where 𝑛0 = 𝑁 𝑉 is the electron concentration.
 Wiedemann–Franz Law
 The ratio of 𝑘 to 𝜎 is given by
𝑘
𝜎
=
1
3
𝑣𝑇
2τ𝐶𝑣
𝑛0𝑒2τ 𝑚
 Now Drude applied the classical gas laws to evaluation of 𝑣𝑇
2 and 𝐶𝑣, viz.,
1
2
𝑚𝑣𝑇
2
=
3
2
𝑘𝐵𝑇 and 𝐶𝑣 = 𝑛0(
3
2
𝑇𝐵).
This gave
𝑘
𝜎
=
3
2
𝑘𝐵
𝑒
2
𝑇 = 𝐿𝑇 where 𝐿 =
3
2
𝑘𝐵
𝑒
2
is a constant and it is known as Lorentz number.

 Quantum Free Electron Theory:
 He developed this theory by applying the principles of quantum mechanics.
 It overcomes many of the drawbacks of classical theory.
 Sommerfeld explained them by choosing Fermi- Dirac statistics instead of Maxwell-Boltzmann
statistics.
 Assumptions of Quantum Free Electron Theory
 Valence electrons move freely in a constant potential within the boundaries of metal and is prevented from
escaping the metal at the boundaries (high potential). Hence the electron is trapped in a potential well.
 The distribution of electrons in various allowed energy levels occurs as per Pauli Exclusion Principle.
 The attraction between the free electrons and lattice ions and the repulsion between electrons themselves
are ignored.

 To find the possible energy values of electron Schrodinger time independent wave equation is applied. The
problem is similar to that of particle present in a potential box.
 The distribution of energy among the free electrons is according to Fermi-Dirac statistics.
 The energy values of free electrons are quantized.
energy of electron is 𝐸𝑛 =
𝑛2ℎ2
8𝑚𝐿2 , where 𝑛 = 1, 2, 3, …
 Merits of quantum free electron theory
 It successfully explains the electrical and thermal conductivity of metals.
 It can explain the Thermionic phenomenon.
 It explains temperature dependence of conductivity of metals.

 It gives the correct mathematical expression for the thermal conductivity of metals.
 It can explain the specific heat of metals.
 It explains magnetic susceptibility of metals.
 It can explain photo electric effect, Compton Effect and block body radiation etc.
 It is unable to explain the metallic properties exhibited by only certain crystals.
Demerits of quantum free electron theory
 It is unable to explain why the atomic arrays in metallic crystals should prefer certain structures only.
 This theory fails to distinguish between metal, semiconductor and Insulator.
 It also fails to explain the positive value of Hall Co efficient.

 According to this theory, only two electrons are present in the Fermi level and they are responsible for
conduction which is not true.
 Fermi Dirac Distribution:
 As we know that for a metal containing N atoms, there will be N number of energy levels in each band.
 According to Pauli’s exclusion principle, each energy level can accommodate a maximum of two
electrons, one with spin up (+
1
2
) and the other with spin down (−
1
2
).
 At absolute zero temperature, two electrons with opposite spins will occupy the lowest available energy
level.
 The next two electrons with opposite spins will occupy the next energy level and so on. Thus, the top most energy
level occupied by electrons at absolute zero temperature is called Fermi-energy level.

 The energy corresponding to that energy level is called Fermi energy.
 The energy of the highest occupied level at zero degree absolute is called Fermi energy, and the energy
level is referred to as the Fermi level. The Fermi energy is denoted as 𝐸𝐹.
 At temperatures above absolute zero, the electrons get thermally
excited and move up to higher energy levels.
 As a result there will be many vacant energy levels below as
well as above Fermi energy level.

 The function 𝑓(𝐸) is called Fermi-factor and this gives the probability of occupation of a given energy level
under thermal equilibrium. The expression for 𝑓(𝐸) is given by
 Under thermal equilibrium, the distribution of electrons among various energy levels is given by statistical
function 𝑓(𝐸).
𝑓 𝐸 =
1
1 + 𝑒
𝐸−𝐸𝐹
𝐾𝑇
 where 𝑓(𝐸) is called Fermi-Dirac distribution function or
Fermi factor, 𝐸𝐹 is the Fermi energy, 𝐾 is the Boltzmann
constant and 𝑇 is the temperature of metal under thermal
equilibrium.
Note:
1. The Fermi-Dirac distribution function 𝑓(𝐸) is used to calculate the probability of an electron occupying
a certain energy level.
2. The distribution of electrons among the different energy levels as a function of temperature is known as
Fermi-Dirac distribution function.

 Variation of Fermi factor with Energy and Temperature
Case 1:
𝒇(𝑬) for 𝑬 < 𝑬𝑭 at 𝑻 = 𝟎𝑲; from the probability function 𝑓(𝐸) we have
𝑓 𝐸 =
1
𝑒−∞ + 1
=
1
0 + 1
= 1
 This implies that at absolute zero temperature, all the
energy levels below 𝐸𝐹 are 100% occupied which is
true from the definition of Fermi energy.
Case 2:
𝒇(𝑬) for 𝑬 > 𝑬𝑭 at 𝑻 = 𝟎𝑲; from the probability function 𝑓(𝐸) we have
𝑓 𝐸 =
1
𝑒∞ + 1
=
1
∞ + 1
=
1
∞
= 0
 This implies that at absolute zero temperature, all the
energy levels above 𝐸𝐹 are unoccupied (completely
empty) which is true from the definition of Fermi
energy.

Case 3:
𝒇(𝑬) for 𝑬 = 𝑬𝑭 at 𝑻 = 𝟎𝑲; from the probability function 𝑓(𝐸) we have
𝑓 𝐸 =
1
𝑒0/0 + 1
= 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒
𝒇 𝑬 = ∞ for 𝑬 = 𝑬𝑭 at 𝑻 = 𝟎𝑲
 Hence, the occupation of Fermi level at 𝑇 = 0𝐾 has an
undetermined value ranging between zero and unity (0 &
1). The Fermi-Dirac distribution function is
discontinuous at 𝐸 = 𝐸𝐹 for 𝑇 = 0𝐾.
Case 4:
𝒇(𝑬) for 𝑬 = 𝑬𝑭 at 𝑻 > 𝟎𝑲; from the probability function 𝑓(𝐸) we have
𝑓 𝐸 =
1
𝑒0 + 1
=
1
1 + 1
=
1
2
 If 𝐸 < 𝐸𝐹, the probability starts decreasing from 1 and reaches 0.5
(1 2) at 𝐸 = 𝐸𝐹 and for 𝐸 > 𝐸𝐹, it further falls off as shown in
figure. In conclusion, the Fermi energy is the most probable or
average energy of the electrons in a solid.

 Homework
1. Find the probability of an electron occupying an energy level 0.02 eV above the Fermi level at 200 K
and 400 K in a material.
2. Find the temperature at which there is 1 % probability that a state with 0.5 eV energy above the Fermi
energy is occupied.

 Density of states:
 Assignment 1, #1
1. Show that the sum of the probability of occupancy of an energy state at ∆E above the Fermi level and
that at ∆E below the Fermi level is unity.

End
G. Beyene (Ph.D) (ASTU) Chapter One End of chapter one - 22/22 March - 2022
Assignment I
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Electron Theory of Metals Explained

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