Density of Energy States and Carrier Concentration in Metals

DENSITY OF ENERGY STATES AND CARRIER CONCENTRATION IN METALS

The Fermi function F (E) gives only the probability of filling up of electrons in a given energy state, it does not gives the information about the number of electrons that can be filled in a given energy state. To know that we should know the number of available energy states so called density of states.

Definition: Density of energy states Z (E) dE is defined as the number of available electron states per unit volume in an energy interval (dE).

 Explanation: In order to fill the electrons in an energy state we have to first find the number of available energy states within a given energy interval.

We know that a number of available energy levels can be obtained for various combinations of quantum numbers n_x, n_y and n_z ̧ ((i.e) n² = n_x²+n_y²+n_z²)

Therefore, let us construct a three dimensional space of points which represents the quantum numbers n, n, and n as shown in Fig. 1.12. In this space each point represents an energy level.

Number of energy levels in a cubical metal piece

To find the energy levels in a cubical metal piece and to find the number of electrons that can be filled in the given energy level, let us construct a sphere of radius 'n' in the space.

The sphere is further divided into many shells and each of this shell represents a particular combination of quantum numbers ( n_x, n_y and n_z ̧) and therefore represents a particular energy value.

Let us consider two energy values E and E+ dE. The number of energy states between E and E+ dE can be found by finding the number of energy states between the shells of radius n and n + Δn, from the origin.

The number of energy states within the sphere of radius n =4/3 π n³

Since n_x, n_y and n_z will have only positive values, we have to take only one octant of the sphere (i.e) 1/8th of the sphere volume.

The number of available energy states within the sphere of radius

Similarly, the number of available energy states within the sphere of radius

The number of available energy states between the shells of radius n and n+dn (or) between the energy levels

 (i.e) The number of available energy states between the energy interval dE is

Since the higher powers of dn is very small, dn² and dn³ terms can be neglected.

............(1)

We know the energy of the electron in a cubical metal piece of sides

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

............(3)

Differentiating equation (2) we get

............(4)

Equation (1) can be written as

Substituting equation (3) and (4) in the above equation we have

Here Ɩ³ represents the volume of the metal piece.

If Ɩ³= 1, then we can write that

The number of available energy states per unit volume (i.e) Density of States

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

Since each energy level provides 2 electron states one with spin up and another with spin down (pauli's exclusion principle), we have

Density of States

............(6)

Carrier Concentration in metals

Let N (E) dE represents the number of filled energy states between the interval of energy dE. Normally all the energy states will not be filled. The probability of filling of electrons in a given energy state is given by Fermi function F (E).

N(E) dE=Z (E) dE • F (E) ...(7)

Substituting equation (6) in equation (7), we get

Number of filled energy states per unit volume

............(8)

N(E) is known as carrier distribution function (or) Carrier concentration in metals

Fermi energy at 0 Kelvin

We know at OK maximum energy level that can occupied by the electron is called Fermi energy level (E_F0)

(i.e.,) at 0 Kelvin for E <E_F and Therfore F (E) = 1

Integrating equation (8) within the limits 0 to E_F0 we can get the numberof energy states electrons (N) within the Fermi energy E_F0

Number of filled energy states at zero Kelvin is ............(9)

............(10)

Average energy of an electron at OK

Average energy of an electron

............(11)

............(12)

Substituting equation (9) and (12) in (11) we get

The average energy of an electron at 0 K is 

Note: At room temperature

where K_B→ Boltzmann constant

T→ Temperature

E_F0→ Fermi energy at 0 Kelvin.