Intrinsic semiconductors

Near the top of the valence band and the bottom of the conduction band the density of states of a semiconductor can be approximated as,

$\large D(E) = \begin{cases} D_v\sqrt{E_v-E}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ D_c\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$

Where $D_v$ and $D_c$ are constants that describe the form of the density of states near the band edges. Often in the literature, these constants are given in terms of the 'density of states effective masses' $m_h^*$ and $m_e^*$ or the 'effective density of states at 300 K' $N_v(300)$ and $N_c(300)$. To calculate $m_h^*$ and $m_e^*$ you determine $D_v$ and $D_c$ from a band structure calculation and then calculate the effect mass that would be needed to produce the same density of states from a dispersion relation consisting of a single isotropic parabola $E=\frac{\hbar^2k^2}{2m^*}$.

$\large D_v = \frac{(2m_h^*)^{3/2}}{2\pi^2\hbar^3}\qquad D_c = \frac{(2m_e^*)^{3/2}}{2\pi^2\hbar^3} $

In the Boltzmann approximation, the concentration of holes in the valence band $p$ and electrons in the conduction band $n$ are given by,

$\large p = \frac{2D_v}{\sqrt{\pi}}(k_BT)^{3/2}\exp\left(\frac{E_v-\mu}{k_BT}\right), \qquad n = \frac{2D_c}{\sqrt{\pi}}(k_BT)^{3/2}\exp\left(\frac{\mu-E_c}{k_BT}\right).$

The temperature dependent prefactors to the exponential functions are called the effective density of states of the valence band $N_c(T)$ and the effective density of states of the conduction band $N_c(T)$.

$\large N_v(T) = \frac{2D_v}{\sqrt{\pi}}(k_BT)^{3/2}\qquad N_c(T) = \frac{2D_c}{\sqrt{\pi}}(k_BT)^{3/2}$

In an intrinsic semiconductor, the density of electrons equals the density of holes. The intrinsic carrier concentration, $n_i$, depends exponentially on the bandgap, $E_g$. For most semiconductors the bandgap is a function of temperature. The plots on this page use the temperature dependence specified in the form below.

$\large n=p=n_i=\sqrt{N_c(300)\left(\frac{T}{300}\right)^{3/2}N_v(300)\left(\frac{T}{300}\right)^{3/2}}\exp\left(\frac{-E_g}{2k_BT}\right)$.

By setting the concentration of electrons equal to the concentration of holes, it is possible to solve for the chemical potential.

$\large n=N_c(300)\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{\mu-E_c}{k_BT}\right)=p=N_v(300)\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{E_v-\mu}{k_BT}\right)$.

$\large \mu=\frac{E_v+E_c}{2}+\frac{k_BT}{2}\ln\left(\frac{N_v(300)}{N_c(300)}\right)$.

$\mu$ [eV]
	$T$ [K]

$\log_{10}$ $n_i$ [cm^-3]
	$T$ [K]

$\log_{10}$ $n_i$ [cm^-3]
	$1/T$ [K^-1]

Once the chemical potential has been determined, it is straightforward to calculate the other thermodynamic properties. The electronic contribution to the internal energy density is,

$\large u = u(T=0)+\frac{\sqrt{2\pi}}{2\pi^2\hbar^3}(m_h^*m_e^*)^{3/4}\exp\left(\frac{-E_g}{2k_BT}\right)(k_BT)^{3/2}(E_g+3k_BT)\text{ [J/m}^3\text{].}$

$u-u(T=0)$ [J/m³]
	$T$ [K]

The specific heat is the derivative of the internal energy density with respect to temperature, $ c_v= \frac{du}{dT}$. The expression for the specific heat is cumbersome to work with when the band gap is temprature dependent. Below the specific heat is plotted by differentiating the internal energy numerically.

$c_v$ [J K^-1 m^-3]
	$T$ [K]

The Helmholtz free energy is,^ref

$\large f = u(T=0) - \frac{\sqrt{2\pi}}{\pi^2\hbar^3}(m_h^*m_e^*)^{3/4}\exp\left(\frac{-E_g}{2k_BT}\right)(k_BT)^{5/2}\text{ [J m}^{-3}\text{].}$

$f-u(T=0)$ [J m^-3]
	$T$ [K]

The entropy is minus the derivative of the Helmholtz free energy with respect to the temperature, $s= -\frac{df}{dT}$. The expression for the entropy is cumbersome to work with when the band gap is temprature dependent. Below the entropy is plotted by differentiating the Helmholtz free energy numerically.

$s$ [J K^-1 m^-3]
	$T$ [K]

The electronic contribution to the thermodynamic properties is usually much smaller than the phonon contribution. For this reason the electronic contribution is often neglected. The electronic contribution becomes more important for small bandgap semiconductors.

See http://www.matprop.ru/semicond for the temperature dependence of the bandgaps of various semiconductors.