UNIT 1

FREE ELECTRON THEORY

INTRODUCTION

• The electron theory aims to explain the structure and properties of solids through their electronic structure.

• According to this theory a metal can be considered to consist of ion cores having the nucleus and electrons other than valence electrons.

• These valence electrons form an electron gas surround the ion cores and are free to move anywhere within the metals.

• The electron theory of solids has been developed in three main stages

1) Classical free electron theory : Drude and Lorentz developed this theory in 1900 . According to this theory metal containing free electron obey the laws of classical mechanics.

2) Quantum free electron theory: Sommerfeld developed this theory during 1928. According to this theory, the free electrons obey quantum laws.

iii). 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. This theory is also called “Band theory of solids”.

Drude –Lorentz theory

postulates :(a). In an atom electrons revolue around the

nucleus and a metal is composed of such atoms.(b). The valence electrons of atoms are free to

move about the whole volume of the metals like the molecules of a perfect gas in a container. The collection of valence electrons

from all the atoms in a given piece of metal forms electrons gas. It is free to move throughout the volume of the metal

c) These free electrons move in random directions and collide with either positive ions fixed to the lattice or other free electrons. All the collisions are elastic i.e., there is no loss of energy.

(d). The movements of free electrons obey the laws of the classical kinetic theory of gases.

(e). The electron velocities in a metal obey the classical Maxwell – Boltzmann distribution of velocities.

f). The electrons move in a completely uniform potential field due to ions fixed in the lattice.

(g). 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.

Successfully

( 1) It verifies Ohm’s law.

(2). It explains the electrical and thermal conductivities of metals.

(3) It derives Wiedemann – Franz law. (i.e., the relation between electrical conductivity and thermal conductivity)

(4). It explains optical properties of metalsl.

Limitations of classical theory:

1. The phenomena such a photoelectric effect, Compton effect and the black body radiation couldn’t be explained by classical free electron theory.

2. According to the classical free electron theory the value of specific heat of metals is not recognized and the variation in temperature is unexplained.

For information

• 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.

• 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.

Quantum free electron Theory

• According to quantum theory of free electrons energy of a free electron is given by

En = n2h2/8mL2

• According to quantum theory of free electrons the electrical conductivity is given by

σ = ne2T/m

Fermi Level

• “The highest energy level that can be occupied at 0 degree Kelvin ” is called Fermi level.

• At 0 K, when the metal is not under the influence of an external field, all the levels above the Fermi level are empty, those lying below Fermi level are completely filled.

• Fermi energy is the energy state at which the probability of electron occupation is ½ at any temperature above 0 k.

Fermi-Dirac statistics

According to Fermi Dirac statistics, the probability of electron occupation an energy level E is given by

F(E) = 1 / 1+exp (E-EF/kT

Application of Schrodinger’s wave equation

Particle confined in one dimensional box or a particle in the infinite square well:

A particle in this potential is completely free, except at the two ends (x = 0 and x = L), where an infinite force prevents it from escaping.

v

m

0U

U U

Inside is just like the free particle, no potential

Walls are like an infinitely steep hill- no way the particle can escape

0x x L

Outside the well, if (x) = 0 (the probability of finding the particle there is zero). Inside the well, where U(x) = 0, the time-independent Schrodinger equation

or

where

Equation 1 is the (classical) simple harmonic oscillator equation; the general solution is

2

2 2

20

mE

x

( ) 0, 0U x if x L

( ) ,U x otherwise

22

20......(1)

x

2

2

2mE

sin cos .......(2)A x B x

where A and B are arbitrary constants. Typically, these constants are fixed by the boundary conditions of the problem.

What are the appropriate boundary conditions for (x)?

Ordinarily, both (x) and d/dx are continuous, but where the potential goes to infinity only the first of these applies.

Continuity of (x) requires that

Which is known as boundary conditions.

First take:

Put in eqn (2), we get

so

0 0 0ASin BCos 0B

...........(3)ASin x

Now apply:

So

because then wave function will be zero everywhere.

where n = 1,2,3,…….

or

now either A=0 or SinL=0

0 ASin L

0Sin L

L n

............(4)n x

ASinL

n

L

The eqn (4) gives the wave function of a particle confined in a one dimensional box of length L.

Now the condition for the normalized wave function is 2

0

1,L

dx 2

0

1L n xASin dx

L

22

0

1L n x

A Sin dxL

22

0

2 12

LA n xSin dx

L

2

0

21 1

2

LA n xCos dx

L

Putting this in eqn (4) we get the normalized wave function

This is the normalized wave function.

2

0

21

2 2

LA L n xx Sin

n L

2

( 0) ( 2 0) 12 2

A LL Sin n

n

2

12

AL Since Sin 2nπ=0

2 2A

L

2A

L

2 n xSinL L

So, energy of the particle:

Put value of ,

Or

Where n = 1, 2, 3, 4…….

Eqn (5) gives the energy of the particle confined in a one dimensional box of length L.

22

2mE

2 2

2 2

2m nE

L

2 2 2

2........(5)

2n

nE

mL

U U 0U

|Ψ|2

Finding prob. |Ψ|2 at different energy Level n=4

n=3

n=2

n=1

Wave fn. Ψ at different energy Level

Classification of Solids

• Based on ‘band theory’, solids can be classified into three categories, namely

1. insulators, 2. semiconductors & 3. conductors.

INSULATORS

• Bad conductors of electricity • Conduction band is empty and valence band

is full, and these band are separated by a large forbidden energy gap.

• The best example is Diamond with

Eg=7ev.

SEMI CONDUCTORS

• Forbidden gap is less• Conduction band an d valence band are

partially filled at room temperature.• Conductivity increases with temperature as

more and more electrons cross over the small energy gap.

• Examples Si(1.2ev) & Ge(0.7ev)

CONDUCTORS

• Conduction and valence bands are overlapped

• Abundant free electrons already exist in the conduction band at room temperature hence conductivity is high.

• The resistively increases with temperature as the mobility of already existing electrons will be reduced due to collisions.

• Metals are best examples.