The table below gives the contribution of electrons in intrinsic semiconductors and insulators to some thermodynamic quantities. These results where calculated in the Boltzmann approximation where it is assumed that the chemical potential lies in the band gap more than 3kBT from the band edge. The electronic contribution to the thermodynamic quantities are usually much smaller than the contribution of the phonons and thus the electronic components are often simply ignored.
1-d | 2-d | 3-d | |
Density of states | $D(E) = \begin{cases} \frac{D_v}{\sqrt{E_v-E}}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{D_c}{\sqrt{E-E_c}}, & \mbox{for } E_c \lt E \end{cases}$ | $D(E) = \begin{cases} D_v, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ D_c, & \mbox{for } E_c \lt E \end{cases}$ | $D(E) = \begin{cases} D_v\sqrt{E_v-E}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ D_c\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$ |
Density of states | $D(E) = \begin{cases} \frac{1}{\hbar\pi}\sqrt{\frac{2m^*_h}{E_v-E}}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{1}{\hbar\pi}\sqrt{\frac{2m^*_e}{E-E_c}}, & \mbox{for } E_c \lt E \end{cases}$ | $D(E) = \begin{cases} \frac{m_h^*}{\hbar^2\pi}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{m_e^*}{\hbar^2\pi}, & \mbox{for } E_c \lt E \end{cases}$ | $D(E) = \begin{cases} \frac{(2m_h^*)^{\frac{3}{2}}}{2\pi^2\hbar^3}\sqrt{E_v-E}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{(2m_e^*)^{\frac{3}{2}}}{2\pi^2\hbar^3}\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$ |
Density of states | |||
Density of electrons | |||
Density of holes | |||
Law of mass action | |||
Intrisic carrier density | |||
Chemical potential | |||
Internal energy density | |||
Helmholtz free energy | |||
Specific heat | |||
Entropy |
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Script Output | ||