8. Solid Electronic material PPT - April 2020-converted

Solid Electronic Materials

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

➢ Electron in periodic potential
➢ Kronig-Penny model (only basic to introduce origin of band gap)
➢ E-k diagram
➢ Electron conduction, Conductivity, Drift velocity,
➢ Energy bands in solids, Direct and indirect bandgaps
➢ Types of electronic materials: metals, semiconductors, and

insulators
➢ Occupation probability, Fermi level
➢ Effective mass, Density of states and energy band diagrams

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Free Electron Theory

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.

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

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.

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Electron in periodic potential ➢ In solids one deals with a large number of interacting particles

and hence the problem of determining the electron wave functions
and the energy levels is extremely complicated.
➢ A simple quantum mechanical picture of an electron in a crystal
can, however, be obtained by assuming that the atomic nuclei are
at rest in the crystalline state and that the electron is in a periodic
potential which has periodicity of the lattice.
➢ The periodic potential may be considered to be caused by a fixed
nuclei plus some average potential due to all other electrons.
➢ The solution of Schrodinger equation for this potential gives a set
of state or levels, which may be occupied by the electrons
according to Pauli’s exclusive principle.

➢ A crystalline solid consists of a lattice which is composed of a large number of ionic cores at
regular intervals, and, the conduction electrons can move freely throughout the lattice.
➢ Let the lattice be in only one-dimension i.e., only an array of ionic cores along x-axis is considered.
➢ The potential energy of an electron at a distance r from an atomic nucleus of charge Ze is given by

V = Ze2 / 4⌅εo r
➢ Since the lattice is a repetitive structure of the ion arrangement in a crystal.
➢ If a is the interionic distance, then , as we move in x direction, the value of V will be same at all
points which are separated by a distance equal to a.

ie V(x) = V( x + a ) where, x is distance of the electron from the core.
Such a potential is said to be a periodic potential.

Kronig-Penny model

According to quantum free electron theory of metals, a conduction electron in a
metal experiences constant (or zero) potential and free to move inside the
crystal but will not come out of the metal because an infinite potential exists at
the surface. This theory successfully explains electrical conductivity, specific
heat, thermionic emission and paramagnetism. This theory is fails to explain
many other physical properties, for example: (i) it fails to explain the difference
between conductors, insulators and semiconductors, (ii) positive Hall
coefficient of metals and (iii) lower conductivity of divalent metals than
monovalent metals.
To overcome the above problems, the periodic potentials due to the positive
ions in a metal have been considered. shown in Fig. (a), if an electron moves
through these ions, it experiences varying potentials. The potential of an
electron at the positive ion site is zero and is maximum in between two ions.
The potential experienced by an electron, when it passes along a line through
the positive ions is as shown in Fig. (b).

It is not easy to solve Schrödinger’s equation with these potentials. So, Kronig and Penney approximated these
potentials inside the crystal to the shape of rectangular steps. This model is called Kronig-Penney model of
potentials. The Kronig-Penney model demonstrates that a simple one-dimensional periodic potential yields energy
bands as well as energy band gaps. While it is an over simplification of the three-dimensional potential and band
structure in an actual crystal, it is an instructive tool to demonstrate how the band structure can be calculated for a
periodic potential, and how allowed and forbidden energies are obtained when solving the Schrodinger equation.

The energies of electrons can be known by solving Schrödinger’s wave equation in such a lattice. The Schrödinger
time-independent wave equation for the motion of an electron along X-direction is given by:

The energies and wave functions of electrons associated with this model can be calculated by solving time-
independent one-dimensional Schrodinger’s wave equations for the two regions I and II as shown in Fig
The Schrodinger’s equations are:

We solve the Schrodinger wave equation for electron for Kronig Penney potential under the condition that ѱ and
dѱ/dx are continuous at the boundaries of the well. A complicated expression for the allowed energies in terms of k
shows that gaps in energy are obtained at values such that

(4)

The solution of Schrodinger’s wave equations for free electrons results in the energy values given by

(5)

E-k diagram

An E-k diagram shows characteristics of a particular semiconductor
material. It shows the relationship between the energy and
momentum of available quantum mechanical states for electrons in
the material.
First, consider a basic E-k band diagram like this one (the x-axis can
be either momentum, pp, or wave number, kk, since p=ℏkp=ℏk):
The band gap (EG), which is the difference in energy between the
top of the valence band and the bottom of the conduction band.
The effective mass of electrons and holes in the material. This is
given by the curvature of each of the bands.
This diagram indicates (diagramatically) how the actual electron
states are equally spaced in k-space. Which means that the density
of states in E (ρ(E)ρ(E)) depends on the slope of the E-k curve.

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Electron conduction If a potential difference V is applied across a solid, it

establishes an electric field in the solid.

E is given by E = V / L (1)

Where L is length along which charge carries move

in the solid. The electric field accelerates the mobile

charges and causes a flow of electric current

through the solid, The current I passing across an

area A is defined as the net charge Q transported

through the area per unit time. Thus

I=Q/t (2)

When an electric current flows through a material ,it

is said to be conducting electricity. Any material can

conduct electricity if it contains mobile charge

carriers. A free electron is an example of charge

carrier. Other carriers include mobile positive or

negative ions, holes etc. The magnitude of the

current flowing through a material is governed by
Ohm’s law.

I=V/R (3)

Where R is known as electrical resistance.

When electrons travel in a vacuum , they are not impeded in their motion. They travel along
straight-line paths and acquires kinetic energy which is equal to the work done by the
accelerating electric field.

The resistance R is given by R = ρ L / A (4)

Where ρ is proportionality constant of the material . It is ρ = RA/L (5)

ρ is independent of the physical dimensions of material and is known as resistivity of the

material.

The unit of resistivity is ohm meter.

The reciprocal of resistance is known as conductance (G) and reciprocal of resistivity is

conductivity σ .

G = 1/ R = A / ρ L (6)

σ = 1/ ρ = L/RA (7)

The unit of conductance is mho , also called siemens. The unit of conductivity is mho/m or S/m .

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur



Relation (10) is known as point form or microscopic form of Ohm’s Law .

By combining the equation (9) and (10) , we obtain (11)
σ = neυd /E (12)

Or
μ = υd /E

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Drift Velocity

If a particle moves in space in such a
manner that it randomly changes its
directions and velocities, the resultant of
these random motions as a whole is
called drift velocity.
When a current `I` is passed through a
metallic wire, the electrons undergo
multiple scattering with the + ve ions and
move in a direction opposite to the
current direction.

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Energy band formation in solids

➢ In an isolated atom, the electrons are tightly bound and have discrete, sharp energy levels. When two identical atoms are
brought closer, the outermost orbits of these atoms overlap and interact.

➢ When the wave functions of the electrons of the different atoms begin to overlap considerably, the energy levels split into
two.

➢ If more atoms are bought together, more levels are formed and for a solid of N atoms, each of the energy levels of an atom
splits into N levels of energy.

➢ The levels are so close together that they form an almost continuous band. The width of this band depends on the degree of
overlap of the electrons of adjacent atoms and is largest for the outermost atomic electrons.

➢ In a solid, many atoms are brought together that the split energy levels form a set of energy bands of very closely spaced
levels with forbidden energy gaps between them. Overlapping of these atoms occurs for smaller equilibrium spacing r.

➢ The band corresponding to outermost orbit is called conduction band and the next band is called valence band. The gap
between these two allowed bands is called forbidden energy gap or band gap. According to the width of the gap between the
bands and band occupation by electrons all solids can be classified broadly into three groups namely, conductors,
semiconductors and insulators.



Direct and Indirect Band Gap Semiconductors

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

DIRECT BAND GAP INDIRECT BAND GAP

A direct band-gap (DBG) is one in which the maximum An Indirect band-gap (IBG) is one in which the maximum

energy level of the valence band aligns with the minimum energy level of the valence band and the minimum energy

energy level of the conduction band with respect to level of the conduction band are misaligned with respect to

momentum. momentum.

Example, Gallium Arsenide (GaAs). Example, Silicon and Germanium

In DBG , a direct recombination takes place with the release In IBG , due to a relative difference in the momentum, first,

of the energy equal to the energy difference between the the momentum is conserved by release of energy and only

recombining particles. after the both the momenta align themselves, a recombination

occurs accompanied with the release of energy.

The probability of a radiative recombination is high. The probability of a radiative recombination is comparatively
low.

Classification of Metals, Semiconductors and Insulators



Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Fermi Energy The Fermi energy is a concept in quantum mechanics usually
referring to the energy difference between the highest and
lowest occupied single-particle states in a quantum system of
non-interacting electrons(Fermions) at absolute zero
temperature.
● The Fermi energy is only defined at absolute zero
● The Fermi energy is an energy difference (usually
corresponding to a kinetic energy)
● The Fermi energy can only be defined for non-interacting
particles ( Fermions where the potential energy or band edge
is a static, well defined quantity)

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Physical Significance of Fermi Level

• Fermi Level is an abstract reference level that indicates the charge carrier
concentration in a semiconductor. It does not exist physically.

• In an intrinsic semiconductor Fermi level exists in the middle of the energy bandgap
at 0°K, as there are equal number of holes and electron. It shifts according to the
charge carrier variation. It shifts towards conduction band in case of n-type
semiconductor, in which majority carriers are electrons, which resides in conduction
band and towards valence band in case of p type semiconductor which has holes in
the valence band as the majority carriers.

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Occupation probability and Fermi level

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur



Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

f(E)

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Density of State

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Effective Mass

An electron has a well defined mass, and it obey Newtonian Mechanics when accelerated by an
electric field E.
It is found that the mass of and electron in a crystal appears, in general, different from the free
electron mass, and is usually referred to as the effective mass.

(1)

From motion of electrons υh= dE/dk (2)
From equ. 1 and 2

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

From eqn. (5) and (6)

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur

Dr.Shubhra Mishra and Dr. D.S. Kshatri SSIPMT, Raipur