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PHY.K02UF Molecular and Solid State Physics

## Boltzmann approximation

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,

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

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 $N_c(300)$. The relations to $D_v$ and $D_c$ are,

$$D_v = \frac{(2m_h^*)^{3/2}}{2\pi^2\hbar^3}=\frac{\sqrt{\pi}N_v(300)}{2(k_B300)^{3/2}},\qquad D_c = \frac{(2m_e^*)^{3/2}}{2\pi^2\hbar^3}=\frac{\sqrt{\pi}N_c(300)}{2(k_B300)^{3/2}}.$$

Data for different semiconducting materials can be found in the NSM Archive.

The plot below shows the density of states of various semiconductors in this approximation. The Fermi function is plotted as well. At low energies the value of the Fermi function is 1 and those states are occupied. At high energies the Fermi function goes to zero and those states are unoccupied. In the limit of low temperture, the Fermi energy is in the middle of the band gap, $\mu = E_g/2$. As the temperature increases, the Fermi energy moves towards the band with the lower density of states.

 $D(E)$ [eV-1 cm-3]× 10-20 $E$ [eV]
 Dc = eV-3/2 cm-3 Semiconductor Dv = eV-3/2 cm-3 Eg = eV T = K

Ev = 0,  Ec = Eg = eV,  μ = eV
Nc(300) = cm-3, Nv(300) = cm-3

Electrons in the conduction band
The concentration of electrons in the conduction band is the integral over the conduction band of the Fermi function times the density of states,

$$n = \int\limits_{E_c}^{\infty}D(E)f(E)dE.$$

This integral cannot be performed analytically but in the conduction band the Fermi function can be aproximated as an exponential function,

$$f(E) = \frac{1}{1+\exp\left(\frac{E-\mu}{k_BT}\right)}\approx \exp{\left(\frac{\mu-E}{k_BT}\right)}.$$

The plot below shows that the Fermi function can be approximated by $\exp{\left(\frac{\mu-E}{k_BT}\right)}$ for energies $E > \mu + 3k_BT$. This is known as the Boltzmann approximation.

 $D(E)$ [eV-1 cm-3]× 10-20 $E$ [eV]
 Semiconductor

In the Boltzmann approximation, the concentration of electrons in the conduction band is,

$$n \approx D_c \int\limits_{E_c}^{\infty} \exp{\left(\frac{\mu-E}{k_BT}\right)}\sqrt{E-E_c}\,dE.$$ $$= D_c \exp{\left(\frac{\mu-E_c}{k_BT}\right)}\int\limits_{E_c}^{\infty} \exp{\left(\frac{E_c-E}{k_BT}\right)}\sqrt{E-E_c}\,dE.$$

The substitution of $x = E-E_c$ leads to an integral of the form,

$$\int\limits_0^{\infty} \sqrt{x} \exp{\left(\frac{-x}{k_BT}\right)}dx = \frac{\sqrt{\pi}}{2}(k_BT)^{3/2}.$$

Using this result, the density of electrons in the conduction band can be expressed as,

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

This is often written in terms of a temperature dependent 'effective density of states in the conduction band' $N_c(T)$,

$$n = N_c(T)\exp{\left(\frac{\mu-E_c}{k_BT}\right)}, \qquad \text{where}\qquad N_c(T) =\frac{\sqrt{\pi}D_c}{2}(k_BT)^{3/2}=2\left[\frac{m_e^{*}k_BT}{2\pi\hbar^2}\right]^{3/2}.$$

Holes in the valence band
The concentration of holes in the valence band is given by the integral, $$p = \int\limits_{-\infty}^{E_v}D(E)(1-f(E))dE.$$

Here $1-f(E)$ is the probability that there is a hole at energy $E$. The plot below shows that $1-f(E)$ can be approximated by $\exp{\left(\frac{E-\mu}{k_BT}\right)}$ for energies $E < \mu - 3k_BT$.

 $D(E)$ [eV-1 cm-3]× 10-20 $E$ [eV]
 Semiconductor

Using the Boltzmann approximation to calculate the concentration of holes in the valence band we find,

$$p \approx D_v \int\limits_{-\infty}^{E_v} \exp{\left(\frac{E-\mu}{k_BT}\right)}\sqrt{E_v-E}\,dE.$$

Performing the integral similarly to the integral for electrons in the conduction band results in the expression,

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

Often this is written in terms of a temperature dependent 'effective density of states in the valence band' $N_v(T)$,

$$p = N_v(T)\exp{\left(\frac{E_v-\mu}{k_BT}\right)}, \qquad \text{where}\qquad N_v(T) =\frac{\sqrt{\pi}D_v}{2}(k_BT)^{3/2}=2\left[\frac{m_h^{*}k_BT}{2\pi\hbar^2}\right]^{3/2}.$$

Fermi energy of an intrinsic semiconductor
In an intrinsic semiconductor, the density of electrons equals the density of holes, $n=p$,

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

This can be solved for the Fermi energy.

$\large \mu=\frac{E_v+E_c}{2}+\frac{k_BT}{2}\ln\left(\frac{D_v}{D_c}\right)$.

At $T=0$, the Fermi energy is in the middle of the band gap. As the temperature increases, the Fermi energy moves towards the band with the lower density of states.

The Boltzmann approximation assumes that the Fermi energy is at least $3k_BT$ from the band edges. This is not true at high temperatures. When the Boltzmann approximation is no longer valid, the Fermi energy can be calculated numerically, see: Temperature dependence of the Fermi energy.