Solid State Electronic Devices, 7th Edition

Appendix IV Derivation of the Density of States in the Conduction Band

In this derivation we shall consider the conduction band electrons to be essentially free. Constraints of the particular lattice can be included in the effective mass of the electron at the end of the derivation. For a free electron, the three-dimensional Schrödinger wave equation becomes

$$-\frac{{\hslash }^2}{2m}{\nabla }^2\text{ψ}=E\text{ψ}$$ (IV–1)

where ψ is the wave function of the electron and E is its energy. The form of the solution to Eq. (IV–1) is

$$\text{ψ}=(\text{const}\text{.})e^{jk\cdot r}$$ (IV–2)

We must describe the electron in terms of a set of boundary conditions within the lattice. A common approach is to use periodic boundary conditions, in which we quantize the electron energies in a cube of material of side L. This can be ...