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

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

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

\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

or

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

The density of states is now

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

or

N=4.5 \times 10^{21} \text { states } / \mathrm{cm}^{3}

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.