The nearly free electron theory is a model that explains the behaviour of electrons in a crystal lattice. It assumes that the electrons in the crystal lattice are not tightly bound to individual atoms but are instead free to move throughout the lattice. However, they are not entirely free, but instead, they experience periodic potential energy barriers due to the periodic arrangement of atoms in the lattice.

The periodic potential energy barriers cause the electrons to form energy bands, which are regions of allowed energy levels. The energy bands can be either filled with electrons or empty, and the separation between them is called the band gap. In a solid material, the electrons occupy the lowest energy bands, and the higher energy bands are empty.

The origin of the band gap can be understood in terms of the electronic structure of the crystal lattice. The band gap arises from the constructive and destructive interference of the electron waves within the crystal lattice. When the waves interfere constructively, they form energy bands, and when they interfere destructively, they create energy gaps.

The size of the band gap depends on the spacing of the atoms in the crystal lattice. If the atoms are spaced close together, the electrons experience a larger potential energy barrier, resulting in a larger band gap. Conversely, if the atoms are spaced farther apart, the electrons experience a smaller potential energy barrier, resulting in a smaller band gap.

According to free electron theory, the energy of the free particle of mass 'm' in terms of wave vector k is given by

E = ℏ²k² / 2m ......1

This equation gives the energy that varies continuously from zero to infinity. The corresponding free electron wavefunction in 1D is

The nearly free electron theory, according to which band electrons are believed to move in a periodicity potential of an ion cored, can be used to explain the band structure of solids. As seen in the following image, the potential energy of electrons in the field of positive ions is attracting, i.e. negative.

The low energy electrons with wavelength greater than lattice spacing 'a ' can travel freely through the crystal while more energetic electrons with wavelengths similar to the lattice spacing experience Bragg's reflection in accordance with the relation

2asin θ = nλ ; θ = Bragg's angle

n= order

λ = wavelength of electron

with k = ± 2π / λ

2asin θ = ± 2πn /k

In 1 -D, θ = π/2

a = ± nπ/k

=> k = ± nπ/a .....3

At zone boundary i.e at k = ± π/a , Bragg's diffraction is satisfied that the travelling to the right is reflected to travel to the left and vice versa resulting in a stationary wave. We can form two stationary waves from travelling wave

The wavefunction 𝜓 (+) and 𝜓 (-) pile up the electrons at different position so that they have different potential energies. This gives the origin of band gap. According to quantum mechanics, the probability density of a particle is given by

This is maximum when x =0, a , 2a, 3a, ..... .This means this wave function accumulates electrons at the position of ions.

For 𝜓 (-) , 𝜌 (-) ∝ sin ² π/a x which is maximum at x = a/2, 3a/2 , 5a/2, 7a/2.....This means this wavefunction gathers electrons midway between ions as shown in figure below.

If we calculate the expectation value of potential energy over these three distributions, we can find that potential energy 𝜌 (+) is lower than that of travelling wave and potential energy of 𝜌 (-) is greater than that of travelling wave. The difference between these energies gives the energy of band gap as shown in figure below.

The curve is symmetrical about energy axis having point of inflexion at M and N where dE/dx is maximum. Just below the point AA', the wavefunction is 𝜓 (+) and just below the point BB', the wavefunction is 𝜓 (-).

The nearly free electron theory provides a useful framework for understanding the electronic properties of materials. It is the basis for many important concepts in solid-state physics, including conductivity, semiconductors, and superconductivity.

Note:

1.Variation of v with k:

According to quantum mechanical theory, the particle velocity is equal to the group velocity of wave representing the particle.

v = dw/dk; w = angular velocity of wave packet

with E = ℏw

w = E/ ℏ

∴ v = 1/ℏ dE/dk ....1

2. Variation of effective mass (m*) with k:

Let E be the electric field applied to an electron of mass m for time dt. Then gain in kinetic energy of an electron

dE = (eE) (vdt)

with equation 1,

Differentiating equation 1

Acceleration (a) = dv/dt

Comparing this equation with acceleration of free particle

a = eE/m

We see that electron behaves as if it has an effective mass m* is given by

This indicates that effective mass of an electron is not constant but depends on value of d²E/ dk². m* - k curve is shown in fig below. Here m* is positive in the lower half of the energy band and negative in the upper half of the energy band. m* is infinite at inflexion point of E(k) curve. It is convenient to introduce of factor f_k = m/m* = m/ℏ² (d²E/dk²) f_k determines extent upto which an electron in the k state behaves as a free electron. If f_k =1, then electron behaves as a free electron.

This note is a part of the Physics Repository.

Explore

posts on Naya and its networks

Collections