Quantum Theory of Fermi Gas | Conductivity | Electrical Engineering

In this article we will discuss about the possible electronic states of electrons in the potential energy box (one dimensional as well as three dimensional) and their distribution using quantum theory.

The Fermi-Dirac distribution allows only a fraction of the total number of free electrons to gain energy and the values of heat capacity and paramagnetic susceptibility thus obtained coincide with the observed ones.

Sommerfeld treated the problem quantum mechanically using Fermi-Dirac distribution and considered the corresponding electron gas as Fermi gas (Fermi gas obeys Pauli’s exclusion principle). Now we will compare the classical gas with Fermi gas at absolute zero. Classically, all the electrons of a gas at zero temperature have zero kinetic energy i.e. all are at rest.

On the other hand, electrons of Fermi gas at zero temperature cannot all have zero kinetic energy because if so they will contradict the Pauli’s exclusion principle. Instead, at zero temperature the electrons of Fermi gas will occupy the available states of lowest energy i.e. two electrons with opposite spins in each orbital state.

Fermi Gas in One Dimensional Box:

Consider an electron of mass m confine to remain within a one dimensional crystal of length L by infinite barriers at the ends of the length. Also assume that the potential energy everywhere within this crystal is constant and is equal to zero. At the two ends of the crystal, the electron is prevented from leaving the crystal by an infinite potential energy barrier.

i.e., V(x) = 0; 0 ≤ x ≤ L

= ∞; x ≤ 0, x > L            … (x)

Since the potential energy inside the crystal is zero, the Schrodinger equation is in the form of;

Where, E_n is the energy of the electron in the state n. As the probability of finding the electron across the wall is zero, so we have the following boundary conditions,

Ψ_n (0) = Ψ_n (L) = 0      … (xii)

These boundary conditions are automatically satisfied if the wave function is sine-like with an integral number, n, of half wavelength between 0 and L.

Hence equation (xiii) becomes-

Hence equation (xi) becomes-

∴ The energy is a quadratic function of the quantum number as shown in Fig. 5.3. The probability of finding the electron somewhere on the line is unity.

Let, suppose we want to accommodate N electrons on the line. According to Pauli Exclusion Principle, no two electrons can have all their quantum numbers identical, i.e., each quantum state can be occupied by at most one electron. For a given value of n there are two quantum states of the electrons m_s = ±1/2. Each energy level of quantum number n can accommodate two electrons, one with spin up and one with spin down.

If there are five electrons, e.g., Boron, then the ground state (n = 1) of the system has 2, next state n = 2 has two and third state n = 3 has only one electron and the higher states are empty (Fig. 5.4).

Let the top most filled energy level be represented by n_f. If N, i.e., the total number of electrons to be accommodated is even, then-

2n_f = N          … (xviii)

determines the value of n_f.

The Fermi energy E_f is defined as the energy corresponding to the top most filled energy level in the ground state of the N electron system. From equation (xvi)-

The total energy E₀ of N electrons in the lowest energy state of the entire system can be obtained by adding the individual energies E_n between n = 1 to n_f = N/2

The factor 2 is due to fact that we are taking account of two electrons in each level. Substituting, the value of E_N from equation (xvi) in equation (xx), we have-

Hence equation (xxi) becomes,

Using equation (xix), we have-

Thus, the average kinetic energy in the ground state is one third of the Fermi energy. From equation (xvi), we have-

which gives the number of energy levels.

Density of Fermi Fermi Gas:

Density of States in One Dimensional:

As there are two quantum states for each energy level (Pauli Exclusion Principle), therefore, the density of states (number of electronic states per unit energy range) of a free electron gas in one dimension is-

From equation (xvi), the above equation reduces to-

The variation of density of states D (E) with energy E is shown in Fig. 5.5.

It shows that the levels are filled from 0 to E_f beyond which all the states are empty.

Density of States in Three Dimensions:

In three dimensional cases, the Schrodinger wave equation for a free particle is-

In the present case, let us consider a parallelepiped of volume V. the shape of Fermi surface in momentum space (k-space) for the free electron gas model is perfectly spherical (see Fig. 5.6), whose Fermi radius-

is k_f when E = E_f i.e. E_f = ħ² k²_f/2m.

In k-space, the volume occupied by solid is 8π³/V.

Therefore, the numbers of energy states inside the Fermi sphere are-

The total number of electrons to be accommodated are even i.e. N = 2n_f

Thus, Fermi energy E_f is given by-

The Fermi energy E_f is corresponding to Fermi velocity v_f and is given by-

Moreover, the Fermi temperature is given by-

The density of states D (E) can be obtained by the fact that, in the ground state of the system, all the energy states below E_f are occupied and the total number of states is equal to the total number of electrons i.e.

If we express the above relation in an indefinite form, we get-

Therefore, the electron density C (E) is given by-

Effect of Temperature on Fermi Energy:

The kinetic energy of the electron gas increases as the temperature is increased; some energy levels are occupied which were vacant at OK and some levels are vacant which were occupied at absolute zero.

Since the electron density (number of electrons per unit volume) is very large and motion of the electron is random, the energy levels occupied by electrons at a given temperature can be determined statistically. As the electrons obey Pauli Exclusion Principle, therefore, according to Fermi-Dirac statistics, the number of electrons occupying energy states between E and E + dE in a unit volume is given by-

F(E) gives the probability of a state corresponding to energy E (Fig. 5.7) occupied by an electron at temperature TK. K_B = (8.62 × 10^-5 eV/K = 1.38 × 10^-23 J/K) is the Boltzmann constant and E_f is the Fermi energy. F (E) is also known as Fermi function.

The total number of electrons per unit volume n in the free electron gas is determined by integrating the equation (xxvi) i.e.,

Thus, the Fermi energy may be calculated simply by knowing the electron concentration n. The calculated values for a number of metals are given as-

At T > 0 K and E = E_f, F (E) = ½ i.e., energy state is half filled.

Hence, we conclude that at absolute zero, all the states above Fermi level E_f are empty and when the temperature is raised above absolute zero, and then there is greater probability of electrons being found above Fermi level, with equal probability of finding holes (deficiency of electrons) below the Fermi level.