Question

In a semiconductor, the concentration of holes in the valence band can be given as
1. N_V exp[-(E_F – E_V)/kT]
2. N_V exp [(E_F – E_V)/kT]
3. \(\frac{{kT}}{q}{N_V}\;exp\;\left[ {\left( {{E_V}\;-\;{E_F}} \right)} \right]\)
4. \(\frac{{kT}}{q}\;exp\;\left[ { - \left( {{E_F}\;-\;{E_V}} \right)} \right]\)

Answer 1

Correct Answer - Option 1 : N_V exp[-(E_F – E_V)/kT]

Concept:

The density of electrons in a semiconductor is related to the density of available states and the probability that each of these states is occupied, i.e.

n(E) = g_c(E) f(E)

g_c(E) = Density of states in the conduction band

f(E) = Probability of electron at Energy Level, E

Since holes correspond to empty states in the valence band, the probability of having a hole equals the probability that a particular state is not filled, so that the hole density per unit energy, p(E), equals:

p(E) = g_v(E) [1 – f(E)]

g_v(E) = Density of states in the valence band

Analysis:

The carrier density in a semiconductor is obtained by integrating the product of the density of states and the probability density function, over all possible states.

For holes in the valence band, the integral is taken from the top of the valence band, labeled E_v, to the bottom of the valence band, i.e.

\(p = \mathop \smallint \limits_{ - \infty }^{{E_v}} {g_v}\left( E \right)\left[ {1 - \;f\left( E \right)} \right]dE\)    ---(1)

The probability of finding an electron at an energy level E is given by using the Fermi-Dirac Equation as:

\(f\left( E \right) = \frac{1}{{1 + \exp \left( {\frac{{E - {E_F}}}{{{K_B}T}}} \right)}}\)  

\(1 - f\left( E \right) = 1 - \frac{1}{{1 + \exp \left( {\frac{{E - {E_F}}}{{{K_B}T}}} \right)}}\)

Solving the above, we get:

\(1 - f\left( E \right) = \frac{1}{{1 + \exp \left( {\frac{{{E_F} - E}}{{{K_B}T}}} \right)}}\)

The above can be approximated as:

\(1 - f\left( E \right) = \frac{1}{{1 + \exp \left( {\frac{{{E_F} - E}}{{{K_B}T}}} \right)}} \cong \exp \left( { - \frac{{\left( {{E_F} - E} \right)}}{{{K_B}T}}} \right)\)

The concentration of holes using Equation (1) can now be written as:

\(p = \mathop \smallint \limits_{ - \infty }^{{E_v}} {g_v}\left( E \right) \exp \left( { - \frac{{\left( {{E_F} - E} \right)}}{{{K_B}T}}} \right)\)

Assuming all the density of states to be concentrated at the top of the valence band with effective density given by Nv, the above Equation is written as:

\(p = {N_v}\exp \left[ {\frac{{ - \left( {{E_F} - {E_v}} \right)}}{{KT}}} \right]\)