Consider an intrinsic semiconductor with the following density of states:
- Conduction band: \(g_c(ϵ) = C_1 (ϵ - ϵ_c)^{1/2}\)
- Valence band: \(g_v(ϵ) = C_2 (ϵ_v - ϵ)^{1/2} \)
An expression for n , the number of electrons in the conduction band, in terms of kB , T , C1 , ϵc , ϵf is
1
\( n = C_1 (k_B T)^{3/2} e^{-(\epsilon_c - \epsilon_f) / k_B T} \int_0^\infty e^{x} x^{-1/2} \, dx \)
2
\( n = C_1 (k_B T)^{3/2} e^{-(\epsilon_c - \epsilon_f) / k_B T} \int_0^\infty e^{-x} x^{1/2} \, dx \)
3
\( n = C_1 (k_B T)^{3/2} e^{-(\epsilon_c - \epsilon_f) / k_B T} \int_0^\infty e^{-x} x^{3/2} \, dx \)
4
\( n = C_1 (k_B T)^{3/2} e^{-(\epsilon_c - \epsilon_f) / k_B T} \int_0^\infty e^{-x} x^{-1/2} \, dx \)