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 m^*.

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 (m^*).

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 v_g

Group velocity with which the electron can travel

v_g = dw / dk ....(1)

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.

p = h / λ (from De-Broglie’s equation)

= h/2π 2π/λ

= hk     [k = 2π/λ]....(5)

Differentiating the equation (5) with respect to 't'

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

α = eE/ m^*= F/m^*                                [F= eE]

or = m^*a...(8)

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 d²E/dk²

Special cases:

case (i) If d²E/dk² is positive, then m^* is also positive

case (ii) If d²E/dk² is negative, then m^* is also negative

case (iii) If d²E/dk²  is more, then the electrons behave as light particle

case (iv) If d²E/dk² is very small, then the electrons behave dk as heavy particle.

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 π to it decreases to a small negative value.