Question

Please use MATLAB or EXCEl

Schrodinger’s equation for a periodic-potential (representing a condensed-phase matter) have solutions, subject to the satisfaction of the Kronig-Penney condition. Use MATLAB, Excel, or any other tool to graphically show the solutions, and the formation of Allowed-Energy Bands and Forbidden Energy Gaps. [Hint: i) Use the Kronig-Penney condition given in the posted Appendix from the book by D. Navon. Ii) Select a reasonable value for P to clearly show the solution of the K-P condition.]

Answer 1

Given, the equation : p(sin(i±a)/ſ±a) + cos(i±a) cos(ka),À Â Â .... (1)

where, p = 3 Ï€/2 and cos(ka) = cos(±ÏC), k = Ï€/a where, n

- ±1, ±2, ..

The equation (1) shows the relation between the energy (through I±) and the wave-vector (k).

Since the LHS of the equation can only range from â^'1 to +1, there are some limits on the values that It (and thus, the energy) can take. It means, at some ranges solution according to this equation, and thus, the system will not have those energies: energy band-gaps.

The energy band-gaps can be shown to exist in any shape of periodic potential (not just delta or square barriers). This has been plotted in given figure-1

(fig-1)

1. The permissible limit of the term p(sin(αa)/αa) + cos(αa) lies between +1 to -1.

By varying I±a, a wave mechanical nature is plotted as shown in fig-2. The shaded portion of the wave shows the bands of allowed energy with the forbidden region as unshaded portion.

(fig-2)

2.     With increase of αa, the allowed energy states for an electron increases there by increasing the band width of the bands. It means, the strength of the Potential barrier diminishes. This also leads to increase of the distance between electrons and the total energy possessed by the individual electron.

3. Â Â Â Â Conversely, if the effect of potential barrier dominate i.e., if P is large, the resultant wave obtained in terms of shows a stepper variation in the region lies between +1 to -1. This results in the decrease of allowed energy and Increase of forbidden energy gap. Thus at extremities.

• Case 1:

When Pât' â^ž, the allowed energy states are compressed to a line spectrum. (fig-3)

(fig-3)

• Case 2:

When Pât' 0, the energy band is broadened and it is quasi continuous. (fig-4)

(fig-4)