Effective Mass of Electron and Hole

EFFECTIVE MASS OF ELECTRON AND HOLE

Effective mass of electron

The mass acquired by an electron when it is accelerated in a periodic potential is called effective mass of an electron. It is denoted by

Explanation:

When an electron is accelerated by an electrical magnetic field in a periodic potential, the mass of the electron is not a constant. But, it varies with respect to the field applied. This varying mass is called effective mass

Derivation of effective mass of electron

Consider a crystal subjected to an electric field of intensity 'E'. Due to this applied field, the electron gains a velocity which can be described by a wave vector k.

According to wave mechanics, a particle moving with a velocity v is equivalent to a wave packet moving with a group velocity vg. 

⸫ Group velocity with which the electron can travel

where ω angular frequency of the electron

k → wave vector

We know that E  = hv

Substituting (2) in (1),

The acceleration 'a' experienced by the electron is given by

Momentum (p) of an electron inside the crystal.

Differentiating the equation (5) with respect to 'ť

Substituting equation (6) in (4)

When an electrical field is applied, acceleration of the electron due to this field

Comparing the equations (7) and (8), we have

The above equation indicates that the effective mass of an electron is not a constant, but depends on the value of

Special cases:

Variation of m* with k:

Figure 2.21 shows the variation of m* with k. Near k = 0, the effective mass approaches m. As the value of k increases, m* increases reaching its maximum value at the point of inflection on the E - k curve.

Above the point of inflection, m* is negative and as k tends to π/a, it decreases to a small negative value.