Energy Spectrum of Electrons in Heavily Doped Semiconductors

Abstract

Schrödinger’s Equation. In an isolated atom, such as hydrogen, only the valence electron and the nucleus interact. Therefore, Schrödinger’s equation, which expresses essentially the law of conservation of energy, has the following simple form:

$$\frac{{\hbar ^2 }}{{2m}}\frac{{d^2 \psi \left( r \right)}}{{dr^2 }} + V_0 \left( r \right)\psi \left( r \right) = E\psi \left( r \right).$$

(1.1.1)

The first term on the left-hand side of this equation represents the kinetic energy of the interaction between the electron and the nucleus and the second term represents the potential energy. The sum of these energies is equal to the total energy of the system Eψ(r). The solution of Eq. (1.1.1) yields plane waves of the type

$$\psi \left( r \right) = e^{ \pm ikr} $$

(1.1.2)

where k is the wave vector of the electron, equal to k = p/h, where p is the electron momentum.