Calculate the density of states per unit volume over a particular energy range.

Consider the density of states for a free electron given by Equation (3.69). Calculate the density of states per unit volume with energies between 0 and $1 \mathrm{eV}$ .

$g(E)=\frac{4 \pi(2 m)^{3 / 2}}{h^{3}} \sqrt{E}$ (3.69)

Question

Calculate the density of states per unit volume over a particular energy range.

Consider the density of states for a free electron given by Equation (3.69). Calculate the density of states per unit volume with energies between 0 and [latex]1 \mathrm{eV}[/latex].

[latex] g(E)=\frac{4 \pi(2 m)^{3 / 2}}{h^{3}} \sqrt{E} [/latex] (3.69)

Accepted Answer

The volume density of quantum states, from Equation (3.69), is

[latex] g(E)=\frac{4 \pi(2 m)^{3 / 2}}{h^{3}} \sqrt{E} [/latex] (3.69)

[latex] \mathrm{N}=\int_{0}^{1 \mathrm{eV}} g(E) d E=\frac{4 \pi(2 m)^{3 / 2}}{h^{3}} \cdot \int_{0}^{1 \mathrm{eV}} \sqrt{E} d E [/latex]

or

[latex] N=\frac{4 \pi(2 m)^{3 / 2}}{h^{3}} \cdot \frac{2}{3} \cdot E^{3 / 2} [/latex]

The density of states is now

[latex] N=\frac{4 \pi\left[2\left(9.11 imes 10^{-31} ight) ight]^{3 / 2}}{\left(6.625 imes 10^{-34} ight)^{3}} \cdot \frac{2}{3} \cdot\left(1.6 imes 10^{-19} ight)^{3 / 2}=4.5 imes 10^{27} \mathrm{~m}^{-3} [/latex]

or

[latex] N=4.5 imes 10^{21} ext { states } / \mathrm{cm}^{3} [/latex]

Comment

The density of quantum states is typically a large number. An effective density of states in a semiconductor, as we will see in the following sections and in the next chapter, is also a large number but is usually less than the density of atoms in the semiconductor crystal.

Question 3.3: Calculate the density of states per unit volume over a parti...

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