Question

For an intrinsic semiconductor with a band gap of 0.75eV, calculate the effective density of states for the electrons and the concentration of electrons in the conduction band at T=300K, given that the effective masses of the electron and hole are equal to the free mass of the electron.

Answer 1

The effective density of states for the electron in the conduction band (take

$$m_e^*=m_0$$

):

$$N_C=2[\frac{2\pi m_e^*kT}{h^2}]^{3/2}=$$ $$=2[\frac{2\pi\cdot 9.11\cdot 10^{-31}\cdot 1.38\cdot 10^{-23}\cdot 300}{(6.63\cdot 10^{-34})^2}]^{3/2}=2.5\cdot 10^{25} \text{ m}^{-3}.$$

The same will be for the valence band according to the condition:

$$N_V=2[\frac{2\pi m_h^*kT}{h^2}]^{3/2}=N_C=2.5\cdot 10^{25} \text{ m}^{-3}.$$

The concentration of electrons in the conduction band:

$$n_i=\sqrt{N_C N_V}\cdot e^{-E_g/2kT}=$$ $$=2.5\cdot 10^{25}\cdot \text{exp}(-0.75\cdot 1.6\cdot 10^{-19}/(2\cdot 1.38\cdot 10^{-23}\cdot 300))=$$ $$=1.25\cdot 10^{19}\text{ m}^{-3}.$$