Structure of atom :

Resistance :
• measure of the opposition to current flow in an electrical circuit
• Unit (Ω).

Conductance :
• Reciprocal of resistance
• Measure of the ease with which current is established
• Unit (siemens)

Ohm’s Law
• electric current is proportional to voltage and inversely proportional to resistance.

Comparison

Success of classical free electron
theory:

•It verifies Ohm’s Law.

•It explains the electrical and thermal conductivities of metals

•It derives Wiedemann – Franz Law .

Failure of classical free electron
theory:
1.Lorentz number

2.Heat capacity
-Metal specific heat :
•C contributed by electron is much smaller than the calculated results.
•Drude model fails to predict C for metal

3.Failure to correctly account for the T dependence of resistivity in metals

•At 0 K→ Fermi energy EF is represented as EF0

As temperature of the metal is increased from 0K to TK :
•The electrons below KBT from Fermi level, will not take thermal energies → they will not find vacant electron states.

•The electrons above KBT from Fermi level,may take thermal energies equal to KBT and occupy higher energy levels

At T = 0 K

• All states having energies less than the Fermi energy are occupied
• All states having energies greater than the Fermi energy are vacant

At T > 0 K
• As T increases, the distribution rounds off slightly
• Due to thermal excitation, states below EF lose population & states above EF gain population.

From

Temperature dependence of Fermi-Dirac distribution

Quantum-based free-electron theory of metals
• The outer-shell electrons are free to move through the metal but trapped within a 3D box formed by metal surfaces.
• Each electron represented as a particle in a box
• Particles in a box are restricted to quantized energy levels.
• By quantum statistics- each state of the system can be occupied by only 2 electrons (1 spin up & 1 spin down)

3D region of space
•φ(x, y, z) = 0 at the boundaries of the metal.
•Energy for electron

•me is the mass of the electron and nx, ny, and nz are quantum numbers
•Energy are quantized.

Energy of particle

Density of states function
• We treat quantum numbers as continuous variables

Electronic properties
• Drude model- applied kinetic theory to the electrons in a solid.
• It’s able to explain electrical and thermal conductivity although it overestimated the electronic heat capacity
• Electrons are modelled as a Fermi gas, which obey the quantum mechanical Fermi-Dirac statistics.
• Improvement to the Drude’s model- Applied Pauli’s exclusion principle
• Electrons cannot all be in the lowest energy state

The equation can be decomposed into 3 independent equations involving x, y or z by setting (reduction of 3D to 1D case): (

The total wave function is now expressed as a linear combination of eight different plane waves:

Free electron in a box
•An application of the boundary condition 𝜓(0) = 0 and 𝜓(L) = 0 to Schrodinger’s equation for a single free electron

• Without e-core /e-e interactions, the conduction electrons behave like independent free electrons
• An extension to 3D space leads to wave function

• Energy of free electron confined in a cube with edge length L

The Fermi sphere

•The surface of the Fermi sphere represent the boundary between occupied and unoccupied k states at absolute zero for the free electron gas.
•Fermi radius kF:

•Energy of a free electron with Fermi radius kF

Addition of the periodic potential
• PE of the electron around an isolated atom

• When N atoms are arranged to form the crystal, then there is an overlap of individual electron PE functions

• PE of the electron, V(𝑥), inside the crystal is periodic with a period 𝑎.
• The electron PE, V(𝑥), inside the crystal is periodic with the same periodicity as that of the crystal, 𝑎.

Bloch’s Waves
• The mathematical representation of the potential is a periodic function with a period 𝑎
•The wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as Ψ(𝑥) = 𝑒^(𝑖𝑘𝑥)𝑢(𝑥) where 𝑢(𝑥) is a periodic function which satisfies 𝑢(𝑎 + b) = 𝑢(𝑥).

Plot of E vs k
• Discontinuities occur at k = ±π/a, ±2π/a, ±3π/a,
• Extended zone representation
• -π/a<k< +π/a is defined as BZ
• Since k’= k+G
• P→0 ;Weak barrier

• E vs k follow free electron theory→parabolic

• P→∞;large barrier

• At large P,electron are tightly bound
• The difference in energy between these two standing waves leads to a discontinuity in energy at k=±nπ/a

Insulators
• Fermi energy EF is at the midpoint between the valence band and the conduction band.
•At T=0, the valence band is filled and the conduction band is empty.
•The band gap energy Eg between the two is relatively large (~ 10 eV)
•Very few electrons in the conduction band and the electrical conductivity is low

Semiconductors
• Fermi energy EF is at the midpoint between the valence band and the conduction band.
•At T=0, the valence band is filled and the conduction band is empty.
•The band gap energy is relatively small (1-2 eV)
•the electrical conductivity of semiconductors is poor at low T but increases rapidly with temperature.

Metals
• The Fermi energy EF is inside the conduction band
•At T=0, all levels in the conduction band below EF are filled with electrons while those above EF are empty.
•For T>0, some electrons can be thermally excited to energy levels above EF
•Electrons can be easily excited to levels above EF by applying a (small) electric field to the metal.
• Metals have high electrical conductivity

Effective Mass

Mass electron:

m from eq above is effective mass

• If one electron in the band

• If the bands are full, then the total effective electron is zero
• If one of the electron marked as j, is jump to the upper bands.

• This phenomena is referred as current
• this current is caused by a positive charged particle known as “hole”