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PHY.K02UF Molecular and Solid State Physics

## Intrinsic semiconductors with a split-off band

Many common semiconductors such as Si, Ge, and GaAs have a split-off band just below the valence band. The states in the split-off band change the temperature dependence of the concentration of holes. In the Boltzmann approximation, the density of states of a semiconductor with a split-off band just below the valence band is,

$D(E) = \begin{cases} \frac{(2m_h^*)^{3/2}}{2\pi^2\hbar^3}\sqrt{E_v-E}+\frac{(2m_{so}^*)^{3/2}}{2\pi^2\hbar^3}\sqrt{E_{so}-E}, & \mbox{for } E\lt E_{so} \\\frac{(2m_h^*)^{3/2}}{2\pi^2\hbar^3}\sqrt{E_v-E}, & \mbox{for } E_{so} \lt E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{(2m_e^*)^{3/2}}{2\pi^2\hbar^3}\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$

Here $m_e^*$, $m_h^*$, and $m_{so}^*$ are the 'density of states effective masses'. Often in the literature, effective density of states at 300 K is given instead of the 'density of states effective masses'. The relationship between the two is,

$\begin{array}{arr} m_{so}^* =\frac{\pi\hbar^2}{300k_B}\left(\sqrt{2}N_{so}(300)\right)^{2/3} \\m_h^* =\frac{\pi\hbar^2}{300k_B}\left(\sqrt{2}N_v(300)\right)^{2/3} \\ m_e^* =\frac{\pi\hbar^2}{300k_B}\left(\sqrt{2}N_c(300)\right)^{2/3} \end{array}$

The density of states can therefore also be written as,

$D(E) = \begin{cases} \frac{2N_v(300)}{\sqrt{\pi}}\left( \frac{1}{300k_B}\right)^{3/2}\sqrt{E_v-E}+ \frac{2N_{so}(300)}{\sqrt{\pi}}\left( \frac{1}{300k_B}\right)^{3/2}\sqrt{E_{so}-E}, & \mbox{for } E\lt E_{so} \\ \frac{2N_v(300)}{\sqrt{\pi}}\left( \frac{1}{300k_B}\right)^{3/2}\sqrt{E_v-E}, & \mbox{for } E_{so} \lt E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{2N_c(300)}{\sqrt{\pi}}\left( \frac{1}{300k_B}\right)^{3/2}\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$

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.

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

By setting the concentration of electrons equal to the concentration of holes,

$n=N_c(300)\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{\mu-E_c}{k_BT}\right)=p=N_v(300)\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{E_v-\mu}{k_BT}\right)+N_{so}(300)\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{E_{so}-\mu}{k_BT}\right)$,

it is possible to solve for the chemical potential,

$$$\mu=\frac{E_v+E_c}{2}+\frac{k_BT}{2}\ln \left( \frac{N_v(300)}{N_c(300)}+\frac{N_{so}(300)}{N_c(300)}\exp\left(\frac{E_{so}-E_v}{k_BT}\right)\right).$$$
 $\mu$ [eV] $T$ [K]
 Nc(300 K) = 1/cm³ Semiconductor Nv(300 K) = 1/cm³ Nso(300 K) = 1/cm³ Eg = eV Ev-Eso = eV T1 = K T2 = K
 $\log_{10}$ $n_i$ [cm-3] $T$ [K]
 $\log_{10}$ $n_i$ [cm-3] $1/T$ [K-1]

See www.ioffe.rssi.ru/SVA/NSM/Semicond/index.html for the temperature dependence of the bandgaps of various semiconductors.