Question 3.4: Determine the number (#/cm3) of quantum states in silicon be...

Using Equation (3.72), we can write

\begin{aligned} g_{c}(E)=\frac{4 \pi\left(2 m_{n}^{*}\right)^{3 / 2}}{h^{3}} \sqrt{E-E_{c}} & \\ \end{aligned}     (3.72)

\begin{aligned}N &=\int_{E_{c}}^{E_{c}+k T} \frac{4 \pi\left(2 m_{n}^{*}\right)^{3 / 2}}{h^{3}} \sqrt{E-E_{c}} \cdot d E \\ &=\left.\frac{4 \pi\left(2 m_{n}^{*}\right)^{3 / 2}}{h^{3}} \cdot \frac{2}{3} \cdot\left(E-E_{c}\right)^{3 / 2}\right|_{E_{c}} ^{E_{c}+k T} \\ &=\frac{4 \pi\left[2(1.08)\left(9.11 \times 10^{-31}\right)\right]^{3 / 2}}{\left(6.625 \times 10^{-34}\right)^{3}} \cdot \frac{2}{3} \cdot\left[(0.0259)\left(1.6 \times 10^{-19}\right)\right]^{3 / 2} \\ &=2.12 \times 10^{25} \mathrm{~m}^{-3} \end{aligned}

or

N=2.12 \times 10^{19} \mathrm{~cm}^{-3}

Comment

The result of this example shows the order of magnitude of the density of quantum states in a semiconductor.