The Kronig-Penney model

The Kronig-Penney model demonstrates that a simple one-dimensional periodic potential yields energy bands as well as energy band gaps. While it is an oversimplification of the three-dimensional potential and bandstructure in an actual semiconductor crystal, it is an instructive tool to demonstrate how the band structure can be calculated for a periodic potential, and how allowed and forbidden energies are obtained when solving the corresponding Schrödinger equation. The Kronig-Penney consists of an infinite series of rectangular barriers with potential height, V0, and width, b, separated by a distance, a-b, resulting in a periodic potential with period, a. The analysis requires the use of Bloch functions, traveling wave solutions multiplied with a periodic function, which has the same periodicity as the potential Solutions for k and E are obtained when the following equation is satisfied:

This equation can only be solved numerically. Solutions are only obtained if the function, F, is between -1 and 1 since it has to equal cos(ka). The energy, E, is plotted as function of ka/p and the function F in Figure

The corresponding band structure is shown below (black curve) as well as the energy for a free electron (gray curve). Three different forms are presented, namely the E(k) digram, the E(k diagram combined with the reduced-zone diagram as well as the reduced-zone diagram only. From Figure above.a we observe the following: The E(k) relation resembles a parabola except that only specific ranges of energies are valid solutions to Schrödinger's equation and therefore are allowed, while others are not. The range of energies for which there is no solution is referred to as an energy band gap. The transitions between allowed and forbidden energies occur at non-zero integer multiples of ka/p. These correspond to local minima and maxima of the E(k) relation. The reduced-zone diagram shown in Figure above.c contains the first three bands and energy bandgaps. For instance the second energy bandgap occurs between 1.5 and 2 eV, between the band maximum of the second band and the band minimum of the third band.