DEPARTMENT OF HUMANITIES AND SCIENCE

PH23111 – PHYSICS FOR INFORMATION SCIENCE

UNIT I ELECTRICAL PROPERTIES OF MATERIALS

Classical free electron theory - Expression for electrical conductivity -
Matthiessen rule, Thermal conductivity, expression - Wiedemann-Franz law -
Success and failures - electrons in metals - Fermi- Dirac function - Density of
energy states - Electron in periodic potential - Energy bands in solids - tight
binding approximation - Electron effective mass - concept of hole.

FACULTY HOD Dean

 ELECTRICAL PROPERTIES OF MATERIALS
1 .1 INTRODUCTION
Engineering properties are important in our day to day due to immense applications
with versatile structural properties. It is also playing a vital role because of its interesting
physical properties.
 Prime physical properties of materials include: electrical properties; thermal properties;
magnetic properties; and optical properties.
 The electrical behaviors of engineering materials are diverse, and so are their uses in
electrical applications.

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. Positive and negatively charged
particles are driven in the direction opposite to the applied electric field directions. Solid
materials allowing these charged particles to move freely is called conducting materials. These
materials have high electrical and thermal conductivity. The conducting property of a solid is
not a function of total number of the electrons in the metal, but it is due to the number of FREE
ELECTRONS or conduction electrons which are not bounded to the nuclear attraction. Hence in
metals the electrical conductivity depends on the number of free electrons available.
Electrons have similar behaviour in conductors, semiconductors, dielectric, magnetic and
super conducting materials. They also have same behaviour in all applications, such as
computer, television, radiography, electrical conduction etc... Thus, the electron theory of
metals explains the following concepts viz.

(i) Structural, electrical and thermal properties of materials

(ii) Elasticity, cohesive force and binding in solids

(iii) Behaviour of conductors, semiconductors, insulators etc.

So far three free electron theories have been proposed.

(i) Classical free electron theory: It is a macroscopic theory, proposed by Drude and
Lorentz in 1900. This theory explains the free electrons in lattice and it obeys the laws
of classical mechanics.
 (ii) Quantum free electron theory: It is a microscopic theory, proposed by
Sommerfeld in 1928. This theory is explained with the concept that the electron moves
in a constant potential and it obeys the Quantum laws.

(iii) Brillouin Zone theory (or) Band theory: This theory is proposed by Bloch in
1928. It is explained with the concept that the electron moves in a periodic potential.
This theory also explains the mechanism of semi conductivity, based on the bands and
hence called band theory.

1.2 CLASSICAL FREE ELECTRON THEORY

It was suggested by the P. Drude in 1900 that the metallic crystals consists of free
electrons that travelling among the positive ion cores. The free electrons are moving in the
potential created by the positive ion (metal ions) cores. The Coulomb force of attraction does
not allow these electrons to escape the metal surface. The free electrons carry the kinetic
energy which are strong enough hence the potential energy of the lattice is ignored and hence
the mutual repulsion between the electrons is neglected.

1.2.1 Postulates of free electron theory

In the absence of electric field
It has been suggested that the free electrons in the metals behave like atoms and
molecules behave in a perfect gas. Hence these electrons are treated as free electron gas or
Fermi gas or free electron cloud. (Fig. 1.1)

Fig.1.1. Random motion of electrons in a crystal lattice.

The force between the conduction electrons and the ion core is neglected and the total
energy of the electron is assumed to be purely kinetic energy (potential energy is zero).
The free electrons move randomly in all directions. The free electron collides with each
other and also with the lattice elastically, without any loss in the energy.
In the presence of electric field
In the presence of non-zero electric field the electrons get some amount of energy from
the applied field and get accelerated in the direction of the positive potential applied electric
field. Fig. 1.2.

Fig. 1.2. Ordered motion of free electrons in an applied external electric field (E).

Since the electrons are assumed to be a perfect gas they obey classical kinetic theory of gases
and the electron velocities in the metal obey the Maxwell-Boltzmann statistics.

Drift Velocity: (Vd): The average velocity acquired by the free electron in a particular

direction by the application of electric field is called drift velocity

Collision Time (τc): The average time taken by a free electron between any two successive

collisions is known as collision time

Mean free path (λ): The average distance travelled by a free electron between any two
successive collisions in the presence of an applied field is known as mean free path (λ). It is
the product of drift velocity of electrons (Vd) and collision time (τc).

λ = Vd x τc

Mobility (µ): Mobility is defined as the drift velocity acquired by the free electron per unit

electric field (E) applied to it.

Relaxation time (τ): It is the time taken by the free electron to reach its equilibrium position
from its disturbed position, in the presence of applied field. It is approximately equal to the
10-14 s.

1.2.2 CONDUCTION IN METALS-MOBILITY AND CONDUCTIVITY

We know in metals the electrical conductivity depends on the number of charge
carriers (free electrons) present in the material.
Let us consider a solid material (S- metal) of length 'l ' and area of cross-section 'A' as
shown in Fig. 1.3. Let 'n' number of charge carriers (free electrons) be present in it. Then,

Fig. 1.3. Solid material with length l, applied voltage V and current I.

Total number of electrons in the solid (metal) (1)

We know, Total charge Q = Total number of electrons × Charge of one electron
Total charge present in the solid (2)
The - sign indicates that the charge of the electron is negative.
Substituting equation (1) in (2) we get
Total charge present in the solid (3)
Now, when the solid is applied with a voltage 'V', then the electrons starts moving with an
average velocity called drift velocity from one end to the other i.e., along a length 'Ɩ' (or)
through a distance 'l', giving rise to conduction of current in the solid.

The current through the solid (4)

Substituting equation (3) in (4) we get

Current (5)

The current density i.e., the current, flowing through the solid per unit area is given by

Current density (6)

Substituting equation (5) in equation (6) we can write,

(7)

We can write equation (7) as

Current density (8)

Current density from Ohms Law

We know that the conduction in solids (metals) obeys Ohms law,

(9)

where R is the electrical resistance i.e., opposition offered by the solid (metal) for the
movement of electrons, given by

(10)

where ρ is the resistivity of the solid (metal).

Substituting equation (10) in (9) we get

Current

Current density (11)

Since the conductivity is the reciprocal of the resistivity i.e.

& Electric field

We can write equation (11) as

(12)
Comparing equations (8) and (12) we can write

(13)
where 'μ' is the mobility of the electron.
Mobility: Mobility of the electron is defined as the drift velocity acquired by the electron per
unit electric field applied to it.

Mobility (14)

1.3 EXPRESSION FOR ELECTRICAL CONDUCTIVITY

We know in the absence of external electric field, the motion of electrons in a metal moves
randomly in all directions. When an electric field (i.e.,) potential difference is maintained
between the two ends of a metallic rod, as shown in Fig. 1.4 the electrons will move towards
the positive field direction and produces the current in the metallic rod.

If 'n' is the free electron density and ‘e' is the charge of electron then the current density (i.e.,)
the current flowing through unit area is given by

(1)

Fig 1.4. Free electrons in an external applied electric field.

The - ve sign implies that the charge of the electron is negative and it also indicates that the
conventional direction of current is in the opposite direction to the electron movement.

Due to the electric field applied, the electron gains the acceleration 'a'
 (or) (2)

If E is the electric field intensity and 'm' is the mass of the electron, then

The force experienced by the electron is (3)

From Newton's second law of motion,

The force on the electron (4)

Equating equation (3) and equation (4) we have

(5)

Substituting equation (5) in equation (2) we have

(6)

Substituting equation (6) in equation (1) we have

Current density

(7)

Here the number of electrons flowing per second through unit area (i.e.,) the current density
depends on the field applied. Thus if the field (E) applied is more, current density (J) will also
be more.

We can write

(8)
Definition: Coefficient of electrical conductivity (σ) which is defined as the quantity of
electricity flowing per unit area per unit time maintained at unit potential gradient. Unit: Ω -
1m-1

Comparing equation (7) and equation (8) we can write

(9)

1.4 THERMAL CONDUCTIVITY

In metals, the conduction takes place not only by thermal motion of free electrons but
also by thermally excited lattice vibrations, called phonons.

The total thermal conductivity

KTotal = Kelectron + KPhonon

In metals thermal conductivity due to free electrons is predominant.

KTotal =Kelectron

In insulators the thermal conductivity due to atomic (or) molecular vibrations of the lattice is
more predominant.

KTotal = KPhonon

In Semiconductors both K electron and Kphonon will contribute for thermal conduction KTotal =
Kelectron +KPhonon

Definition: Thermal conductivity of the material is defined as the amount of heat conducted
per unit area per unit time maintained at unit temperature gradient. Unit: Wm-1 K-1

EXPRESSION FOR THERMAL CONDUCTIVITY

Let us consider a uniform rod AB with the temperatures T1 (Hot) at end 'A' and T2 (cold) at
end B. Heat flows from hot end 'A' to the cold end 'B'. Let us consider a cross sectional area 'C'
which is at a distance equal to the mean free path (λ) of the electron between the ends A and
B of the rod as shown in Fig. 1.5.
 Fig. 1.5. Heat flow from hot end to cold end in a uniform rod with length λ.

The amount of heat (Q) conducted by the rod from the end A to B of length 2 λ is given by

(1)

where K→ Coefficient of thermal conductivity, A→ Area of cross section of the rod

T1-T2 → Temperature difference between the ends A and B, t → Time for conduction
2 λ → Length of rod (A to B)
From equation (1) we can write
Coefficient of thermal conductivity per unit area per unit time

2λ (2)

Let us assume that there is equal probability for the electrons to move in all the six directions
as shown in Fig. 1.6. Since each electron travels with thermal velocity v, if 'n' is the free
electron density, then on an average (1/6)nv electrons will travel in any one direction.
 Fig. 1.6. Motion of electrons in 6- directions.

The number of electrons crossing unit area per unit time at

(3)

According to kinetic theory of gas, [Since free electrons are assumed to be gas molecules
which are freely moving]

The average kinetic energy of an electron at hot end 'A' of temperature (T1) =

The average kinetic energy of an electron at cold end 'B' of temperature (T2)= =

where kB→ Boltzmann constant = 1.380 × 10-23 J/K

The heat energy transferred per unit Number of electrons X Average K.E of
area per unit time from end A to B across C = electrons moving from A to B

(4)
Similarly, the heat energy transferred per unit area per unit time from end B to A across

(5)

The net heat energy transferred from end A to B per unit area per unit time across 'C' can be
got by subtracting equation (5) from equation (4)

(6)

Substituting equation (6) in equation (2) we have

Thermal conductivity
 (or) (7)

We know for metals.

Relaxation time (τ) = Collision time (τc)

(i.e.,)

(8)

(8)
Substituting equation (8) in equation (7) we have

Thermal conductivity

(9)

Equation (9) is the classical expression for thermal conductivity. It is evident from the above
Eq. (9) the thermal conductivity depends on relaxation time, velocity and also the carrier
concentration.

1.5 WIEDEMANN - FRANZ LAW AND LORENTZ NUMBER

Law:
The ratio between the thermal conductivity and electrical conductivity of a metal is directly
proportional to the absolute temperature of the metal.

where L is a constant called as Lorentz number whose value is

2.44 × 10-8 W Ω K-2 (Quantum mechanical value) at temperature T = 293 K.
Proof:
(i) By Classical theory
We know electrical conductivity (from classical theory)
Thermal conductivity (from classical theory)

(1)

We know kinetic energy of an electron=

Substituting this in equation (1) we can write

Where L=

Substituting the value of Boltzmann constant KB = 1.38 × 10-23 JK-1

charge of electron e = 1.6021 x 10-19 Joules, we get

Lorentz number * +

L = 1.12 x 10-8 WΩK-2

It is found that the classical value of Lorentz number, is only one half of the experimental
value (i.e.,) 2.44 × 10-8 W Ω K-2.
This Discrepancy in the experimental and theoretical value of 'L' is the failure of classical
theory. This discrepancy can be rectified by quantum theory.

(ii) By Quantum theory

In quantum theory the mass of the electron (m) is replaced by the effective mass m*.

Rearranging the expression for thermal conductivity and substituting the electronic specific
heat, the thermal conductivity can be written as

* +

[ ]

Where L=

Substituting the values for Boltzmann constant (KB) and the charge of the electron e we get

Lorentz number L=2.44 × 10-8 W Ω K-2

Thus quantum theory verifies Wiedeman-Franz law and has good agreement with the
experimental value of Lorentz number.

1.6 SIMILARITIES BETWEEN ELECTRICAL AND THERMAL CONDUCTIVITY OF METALS

(i) The electrical and thermal conductivity decreases with the increase in temperature and
impurities.
(ii) The electrical and thermal conductivity is very high at low temperature.
(iii) For non-metals, the electrical and thermal conductivity is very less.

1.6.1 SUCCESS OF CLASSICAL FREE ELECTRON THEORY OF METALS

(i) It verifies Ohms law.
(ii) It explains the electrical and thermal conductivity of metals.
(iii) It is used to derive Wiedemann - Franz law.
(iv) The optical properties of metals can be explained using this theory.

1.6.2 FAILURES OF CLASSICAL FREE ELECTRON THEORY

(i) Classical theory states that all free electrons will absorb energy, but quantum theory states
that only few electrons will absorb energy.
(ii) This theory cannot explain the Compton, photo-electric effect, paramagnetism,
ferromagnetism, etc.
(iii) The theoretical and experimental values of specific heat and electronic specific heat are
not matched.
(iv) Dual nature cannot explained.
(v) By classical theory constant for all temperatures but by Quantum theory constant for all
temperatures.
(vi)The Lorentz number by classical theory does not have good agreement with the
experimental value and is rectified by quantum theory.
1.7 MATTHIESSEN’S RULE
Measurement of electrical resistance and its dependence on temperature is
commonly used for the detection of structural changes and phase transformations in the
material. To be able to successfully understand processes, which occur in the material by
measuring electrical resistance, basic theoretical knowledge of electrical conductivity is
necessary. The electrical conductivity (σ) is the ability of a substance to conduct an electric
current and it is the inverse of the resistivity (ρ): σ = 1/ρ
The resistivity is a material property, whilst resistance (R) refers only to a specific sample.
The relationship between these two quantities is defined as: ρ = (S/l) R, where S stands for
cross-section area of the sample and l for its length.
In most materials (e.g. metals), the current is carried by electrons. The resistivity ρ is then
defined by the scattering of the electrons due to the imperfections and thermal vibrations.
Matthiessen rule says that the total resistivity ρ total can be approximated by adding up
several different terms:
ρtotal = ρthermal + ρimpurity + ρdeformation,
where the components stand for different types of electron scattering: ρthermal – on thermal
vibrations, ρimpurity - on impurities, ρdeformation - on deformation-induced defects.

1.8 QUANTUM THEORY OF ELECTRONS IN METALS

The drawbacks of classical theory can be rectified using quantum theory. In classical
theory the properties of metals such as electrical and thermal conductivities are well
explained on the assumption that the electrons in the metal freely moves like the particles of a
gas and hence called "free-electron gas."
According to classical theory, the particles (electrons) of a gas at zero kelvin will have zero
kinetic energy (Since E = (3/2) KBT) and hence all the particles are found to be in rest. But
according to quantum theory when all the particles are at rest, all of them should be filled
only in the ground state energy level, which is impossible and is controversial to the Pauli's
exclusion principle.
Thus in order to fill the electrons in a given energy level, we should know the following.
(i) Energy distribution of electrons
(ii) Number of available energy states
(iii) Number of filled energy states
(iv) Probability of filling an electron in a given energy state, etc.
As the "free-electron gas" obeys Fermi-Dirac statistics, all the above can be very easily
determined using it.

1.8.1 ENERGY DISTRIBUTION OF ELECTRONS IN METALS

We know according to Quantum free electron theory the energy levels are discrete
(microscopically). But since the spacing between any two energy levels is very less (10-
6 eV), the distribution of energy levels seems to be continuous. (macroscopically).
Proof:

We know the energy Eigen value in three dimension is .

This equation represents the permissible energy values of the valence electrons in a cubical
metal piece. If the cubical metal piece has a dimension say 1 cm3, then the ground state energy
is given by . Since, the Energy Eigen value is inversely proportional to

the square of the width of the box, more the value of the physical dimension reduces the
energy spacing the consecutive energy level.
Also, the maximum spacing between the consecutive energy level is very less, say in the order
of 10-6 eV.
If a plot is made between the number of energy levels N (E) that are filled with electrons per
unit energy and Energy E, it is found that the number of energy levels N(E) increases
parabolically with the increase of energy 'E' as shown in Fig. 1.8.

Fig. 1.8. Energy versus number of energy levels.

In figure 1.8, the dotted line shows the change in energy of electron at room temperature.
Here each energy level can provide only two states, one for spin up and the other for spin
down. Hence only 2 electrons can be filled in a given energy state as shown in Fig. 1.9. Thus
the Pauli's exclusion principle is satisfied.

Fig 1.9
At T=0, if there are 'N' number of electrons (N being even), then we have N/2 number of filled
energy levels and other higher energy levels will be completely empty.
This (N/2)th level is the highest filled energy level, called Fermi Energy level (EF).

1.9 FERMI-DIRAC STATISTICS

In the case of Fermi-Dirac statistics the following assumptions are considered.
(i) The particles (electrons) are identical and indistinguishable.
(ii) The particles which obey the Fermi-Dirac statistics are called as fermions (electron,
proton, neutron, 3He, neutrino).
(iii) Spin of the particles should be half integer spin etc.

(iv) Each quantum state can have only one particle.

(v) There is no restriction in choosing the particle, (i.e.) we can choose any particle to fill in
any state.
(vi) The total energy of the system = Sum of all the energies of the particles.
Fermi Energy and its importance
Fermi energy level (EF): Fermi energy level is the maximum energy level upto which the
electrons can be filled at OK.
Importance
(i) Thus it acts as a reference level which separates the vacant and filled states at OK.
(ii) It gives the information about the filled electrons states and the empty states.
(iii) At OK, below Fermi level all the energy levels are completely filled by the electrons and
above Fermi level are empty.
(iv) When the temperature is increased, few electron gains the thermal energy and it goes to
higher energy levels.
Conclusions
(i) In the quantum free electron theory, though the energy levels are discrete
(microscopically), the spacing between consecutive energy level is very less and thus the
distribution of energy levels seems to be continuous.
(ii) The number of energy levels N(E) that are filled with electrons per unit energy increases
parabolically with the increase of energy E as shown in Fig. 1.8.
(iii)Each energy level can provide only two states, namely, one for spin up and the other for
spin down and hence only 2 electrons can be occupied in a given energy state i.e., Pauli’s
exclusion principle.
(iv) At T=0, If there are 'N' number of electrons (N being even), then we have N/2 number of
filled energy levels and the other higher energy levels will be completely empty.
(v) This (N/2)th level is the highest filled energy level is known as Fermi energy level (EF0).
(vi) The electrons are filled in a given energy level, according to Pauli's exclusion principle
(i.e.) no two electrons will have the same set of four quantum numbers.
(vii) At Room temperature, the electrons within the range of KBT [where KB → Boltzmann
constant] below the Fermi energy level will absorb KBT and goes to higher energy states with
energy of EFo + KBT as shown in Fig. 1.8.

1.9.1 FERMI DISTRIBUTION FUNCTION

Definition: Fermi function F(E) represents the probability of an electron occupying a given
energy state.
To find out the energy states actually occupied by the free electron at any temperature (T), we
can apply the Fermi-Dirac statistics. The Fermi-Dirac statistics deals with the particles
(electrons) having half integral spin, named as Fermions.
Thus we can write the Fermi distribution function (i.e.) the probability of an electron
occupying a given energy state as

(1)

where EF is the Fermi energy, KB is the Boltzmann constant.

1.9.2 EFFECT OF TEMPERATURE ON FERMI FUNCTION

The effect of temperature on Fermi function F(E) can be discussed with respect to equation
(1)
(I) At 0 Kelvin
At 0 Kelvin, the electrons can be filled only up to a maximum energy level called Fermi energy
(EF0), above EF0 all the energy levels will be empty. It can be proved from the following
conditions.
(i) When E<EF, equation (1) becomes

i.e., 100% probability for the electron to be filled below the Fermi energy level
(ii) When E> EF, equation (1) becomes

i.e., 0% probability for the electron not to be filled above the Fermi energy level
(iii) When E=EF, and T>0K equation (1) becomes

i.e., 50% probability for an electron to be filled and not to be filled within the Fermi energy
level.
This clearly shows that at 0 Kelvin all the energy states below EF0 are filled and all those
above it are empty.

Fig 1.10, Energy versus Fermi energy at T=0K

The Fermi function at 0 K (EF0) can also be represented graphically as shown in Fig. 1.10. It is
seen from the figure that the curve has step-like character at 0 K.
(i.e) F (E) = 1 below EF0 and F (E) = 0 above EF0
(iv) At any temperature T K
When the temperature is raised slowly from absolute zero, the Fermi distribution function
smoothly decreases to zero as shown in Fig. 1.11.

Fig. 1.11, Energy versus Fermi energy at T>0K.

Explanation: Due to the supply of thermal energy the electrons within the range of KBT
below the Fermi level (EF0) alone takes the energy KBT and goes to higher energy state. Hence
at any temperature (T), empty states will also be available below EF0.
Note: Electrons lying very deep (far below) the Fermi energy level (E F0) will not go to excited
state
Also under classical limit
When KBT<< EF; F(E) is said to be degenerate function.
KBT>>EF; F(E) is said to be non-degenerate function.

1.10 DENSITY OF ENERGY STATES AND CARRIER CONCENTRATION IN METALS

The Fermi function F(E) gives only the probability of filling up of electrons in a
given energy state, it does not gives the information about the number of electrons that can be
filled in a given energy state. To know that we should know the number of available energy
states so called density of states.
Definition: Density of energy states Z(E)dE is defined as the number of available electron
states per unit volume in an energy interval E and E+dE.
Explanation: In order to fill the electrons in an energy state we have to first find the number
of available energy states within a given energy interval.
We know that a number of available energy levels can be obtained for various combinations
of quantum numbers nx, ny and nz ((i.e) n2 = nx2+ny2+nz2)
Therefore, let us construct a three dimensional space of points which represents the quantum
numbers nx, ny, and nz as shown in Fig. 1.12. In this space each point represents an energy
level.

Fig 1.12. Representation of Energy levels

(E) in the quantum number space (n) for a cubic
metal piece.
Number of energy levels in a cubical metal piece
To find the energy levels in a cubical metal piece and to find the number of
electrons that can be filled in the given energy level, let us construct a sphere of radius 'n' in
the space.
 The sphere is further divided into many shells and each of this shell represents a
particular combination of quantum numbers (nx, ny & nz) and therefore represents a particular
energy value.
Let us consider two energy values E and E+dE. The number of energy states
between E and E+dE can be found by finding the number of energy states between the shells
of radius n and n+Δn, from the origin.
The number of energy states within the sphere of radius n =(4/3)πn3. Since
nx, ny and nz will have only positive values, we have to take only one octant of the sphere (i.e.)
(1/8)th of the sphere volume.
The number of available energy states within the sphere of radius

⌊ ⌋

Similarly, the number of available energy states within the sphere of radius

⌊ ⌋

The number of available energy states between the shells of radius n and n+dn (or) between
the energy levels

⌊ ⌋

(i.e.) The number of available energy states between the energy interval dE is

⌊ ⌋

Since the higher powers of dn is very small, dn2 and dn3 terms can be neglected.

⌊ ⌋

⌊ ⌋ (1)

We know the energy of the electron in a cubical metal piece of sides

(2)

* + (3)
Differentiating equation (2) we get

(4)

Equation (1) can be written as

[ ]

Substituting equation (3) and (4) in the above equation we have

[ ] [ ]

= [ ] [ ]

= * +

* +

Here represents the volume of the metal piece.

If , then we can write that

The number of available energy states per unit volume (i.e.,) Density of States

* + (5)

Since each energy level provides 2 electron states one with spin up and another with spin
down (Pauli's exclusion principle), we have

[ ]

* + (6)
Carrier Concentration in metals
Let N (E)dE represents the number of filled energy states between the interval
of energy dE. Normally all the energy states will not be filled. The probability of filling of
electrons in a given energy state is given by Fermi function F (E).
N(E)dE=Z (E) dE . F (E) (7)
Substituting equation (6) in equation (7), we get
Number of filled energy states per unit volume

* + (8)

N(E) is known as carrier distribution function (or) Carrier concentration in metals

Fermi energy at 0 Kelvin
We know at OK maximum energy level that can occupied by the electron is called Fermi
energy level (EF0)
(i.e.,) at 0 Kelvin for E <EF and Therefore F (E) = 1
Integrating equation (8) within the limits 0 to EF0 we can get the number of energy states
electrons (N) within the Fermi energy EF0

∫ ( ) ∫

( )

Number of filled energy states at zero Kelvin is

( ) (9)

( )

Fermi Energy ( ) ( ) (10)

Average energy of an electron at OK

Average energy of an electron

(11)
Here the total number of electrons at 0K = (No, of energy state at 0K Energy of an electron

( )

( ) (12)

Substituting equation (9) and (12) in (11) we get

⁄
( )

⁄
( )

⁄ ⁄

The average energy of an electron at 0 K is

At room temperature

( )

where KB→ Boltzmann constant

T→ Temperature
EF0→ Fermi energy at 0 Kelvin.

1.11 ELECTRON IN PERIODIC POTENTIAL

In quantum free electron theory of metals the electrons were assumed to be moving in a
region of constant potential and hence it moves freely about the crystal. But it fails to explain
why some solids behaves as conductors, some as insulators and some an semiconductors etc.

Therefore instead of considering an electron to move in a constant potential, In Zone

theory of solids the electrons are assumed to move in a field of periodic potential.

In a metal piece, the positive ions are arranged in a regular and proper order,
therefore a periodic potential (i.e) the potential field which varies periodically with the same
period as the lattice, exists in the metal. Also the potential is minimum near the centre of
positive ions and is maximum between the centres of ions as shown in Fig. 1.13
 Fig. 1.13

Therefore the potential energy of the electron near by the centre of positive ion is
maximum and will not be able to move freely, but the electrons which are above these
potential peaks are free to move inside the metal and hence they are termed as free electrons.

1.12 ENERGY BANDS IN SOLIDS

To picture the energy spectra in atoms, molecules and solids let us consider a metal
say sodium, which consists of 11 electrons with electronic configuration of 1s2 2s2 2p6 3s1.
The energy spectrum of a single atom is as shown in Fig. 1.14. When two sodium atoms and
assembled to form a sodium molecule, the energy spectrum of the molecule is as shown in Fig.
1.15. It is found that for a sodium molecule each atomic levels are splitted into two closely
spaced levels.

Fig. 1.16
Similarly if 'N' number of atoms are assembled to form a solid, then we have N number of
very closely spaced sub levels so called as energy band as shown in Fig. 1.16.
It can be found that each energy band is separated by the gaps and are known as Energy gap
(or) forbidden band gap energy.
It can be seen that the electrons present in the outermost energy band (3s1) are mixed
together and they are free to move over the metals. These free electrons are responsible for
the conduction to occur.
Conclusion
Thus we can conclude that the electron moving in a periodic potential lattice will have
discontinuous energy values (i.e.,) they are separated as allowed and forbidden zones (or)
bands.
Note: In the case of electron moving in constant potential lattice (quantum theory) it has quasi
continuous energy values.

1.13 TIGHT BINDING APPROXIMATION

Free electron approximation

We know in solids, there exist the Ionic cores which are tightly bounded to the lattice location,
while the electrons are free to move here and there throughout the solid. This is called the
free-electron approximation.

In free electron approximation the following points are observed, viz.

1. The potential energy of the electron is assumed to be lesser than its total energy.

2. The width of the forbidden bands (Eg) are smaller than the allowed bands as shown in fig
1.20(a).

3. Therefore, the interaction between the neighbouring atoms will be very strong.

4. As the atoms are closer to each other, the inter atomic distance decreases and hence the
wave functions overlap with each other as shown in fig 1.20.
 Fig, 1.20

Tight binding approximation

Tight binding approximation is exactly an opposite approach of discussing the atomic

arrangements, when compared to free electron approximation.

Here instead of beginning with the solid core, we begin with the electrons, i.e., all the
electrons are bounded to the atoms. In other way we can say that the atoms are free, while the
electrons are tightly bounded. Hence, this is called tight binding approximation.

The following points are observed in tight binding approximation, viz.

1. The potential energy of the electrons is nearly equal to the total energy.

2. The width of the forbidden bands (Eg) are larger than the allowed bands, as shown in fig
1.20(b).

3. Therefore, the interactions between the neighbouring atoms will be weak.

4. As the atoms are not closer, the interatomic distance increases and hence the wave
functions will not overlap, as shown in fig 1.20.

Explanation
 Let us consider the atoms with larger inter atomic distance (a2) as shown in fig 1.20.
Here the atoms are far apart, and all the bounded electrons have fixed energy levels.
Therefore when a solid is formed by using the same element, then the energy levels occupied
by the electrons in each atom will be identical, which lead to tight binding approximation.

Now, when we bring the atoms closer to each other to form the solid, then inter atomic
distance (a1), decreases. Therefore, the outer shell electrons begin to overlap and the energy
levels also splits as shown in fig 1.20.

If the inter atomic distance is further reduced, then the splitting of energy level happens for
the inner shall electrons also, which lead to free electron approximation.

Note: For, better understanding about the width of allowed energy bands and forbidden band
gaps, the flat band diagrams corresponding to various inter atomic distances is also shown in fig
1.20(a) and 1.20(b), for free electron approximation and tight binding approximation,
respectively.

1.14 EFFECTIVE MASS OF AN ELECTRON

Definition: Effective mass of an electron is the mass of the electron when it is accelerated in a
periodic potential and is denoted by m*.

Explanation: When an electron of mass 'm' (9.11 x 10-31 Kg) placed in a periodic potential and
if it is accelerated with the help of an electric (or) magnetic field, then the mass of the electron
will not be a constant, rather it varies with respect to the field applied. That varying mass is
called as effective mass (m*).

Expression: To study the effect of electric field on the motion of an electron in one
dimensional periodic potential, let us consider the Brillouin Zone which contains only one
electron (1st Brillouin Zone) of charge 'e' in the state k, placed in an external field 'E'. Due to
the field applied the electron gains a group velocity (Vg) [Quantum mechanically] and
therefore the acceleration changes.

The group velocity with which the electron can travel is

(1)

where k → wave vector, ω→ Angular velocity of the electron

We know ω= 2πv
 , Since E=hv (2)

Substituting equation (2) in equation (1) we get

Group velocity

(3)

If the field (E) is applied to the electron for a time say dt seconds then
Change in field (or) Work done = Force × distance
dE = Force × Velocity × Time
dE = eEvg dt [Since Force = eE] (4)
Substituting equation (3) in equation (4) we get

(5)

We know acceleration a =
Substituting for vg from equation (3) we get

(6)

Substituting equation (5) in equation (6) we get

Acceleration =

(7)
⁄

Equation (7) resembles with the newtons force equation

(i.e) F= eE= m*a (8)
where m* is the effective mass of the electron.
Comparing equation (7) and (8), we can write
 ………………….(9)
⁄

Equation (9) represents the effective mass of an electron in a periodic potential, which
depends on d2E/dk2
Special Cases
Case (i) If d2E/dk2 is+ve, then effective mass m* is also + ve.
Case (ii) If d2E/dk2 is - ve, then effective mass m* is also -ve.
Case (iii) If d2E/dk2 is 0, then effective mass m* becomes ∞.
Thus we can say that it is not so the effective mass (m*) should always be greater than real
mass (m), it may also have negative value.
1.5 Negative effective mass (or) Concept of hole
To show that the effective mass has negative value, let us take the Energy - wave vector (E-k)
curve of a single electron in a periodic potential i.e., consider the 1st Brillouin Zone (allowed
energy band) alone as shown in Fig. 1.24.

Fig. 1.24
In the E-k curve, the band (1st Brillouin Zone) can be divided into two bands viz, upper band
and lower band with respect to a point (P) called as Point of inflection.
From the E-k curve (fig. 1.24) we can say that
(i) In the Lower band the value of d2E/dk2 is a decreasing function (Indicated by arrow mark)
from the point of inflection.
d2E/dk2 is +ve. and hence m* should be + ve in the lower band.
If a plot is made between m* and k for various values of we get the curve as shown in Fig.1.25.
In which we can see that m* has +ve curve.
 Fig 1.25
(ii) In the Upper band of E-k curve (Fig. 1.24) the value of d2E/dk2 is found to be an increasing
function from the point of inflection.
d2E/dk2 is negative and hence m* should also be negative in the upper band.
If a plot is made between m and k (Fig. 1.26) we can see that, if d2E/dk2is -ve, the effective
mass (m*) has negative value.
(iii) At the point of inflection d2E/dk2 =0 [Fig. 1.24] and hence in m*-k plot, effective mass
goes to ∞ as shown in Fig. 1.26.
Physically speaking we can say that, In the upper band [Fig. 1.24], the electron has negative
effective mass.
Hole: The electron with the negative effective mass is called Hole, in other words the electron
in the upper band which behaves as a positively charged particle is called hole. It has the
same mass as that of an electron but with positive charge.
Conclusion:
If a single electron is taken in a one dimensional periodic potential, we get the 1st Brillouin
Zone (i.e.,) only one allowed energy band.
If the energy band is divided into two bands (i.e) upper band and lower band. The electron is
found to exist with positive effective mass in the lower band and with negative effective mass
(hole) in the upperband as shown in Fig. 1.26.
 Fig. 1.26

Fig. 1.27
Therefore, the advantage of the concept of hole is, for a nearly filled band with 'n'
number of empty states as shown in Fig. 1.28 'n' number of holes (empty states) arises.
In other words we can say that the presence of hole is attributed to an empty state, for an
electron to be filled. Thus, based on the hole concept several phenomenon like Thomson
effect, Hall effect etc., are well explained.