Behaviour of an Electron in a Periodic Potential -The Kronig Penney Model (Qualitative Treatment)

BEHAVIOUR OF AN ELECTRON IN A PERIODIC POTENTIAL -THE KRONIG PENNEY MODEL (QUALITATIVE TREATMENT)

Kronig and Penney treated a simplest example for one dimensional periodic potential. In this model it is assumed that the potential energy of an electron has the form of a periodic array of square wells as shown in Fig. 1.17.

Here we have two regions. viz.

Region (i) In this region, between the limits 0 < x <a, the potential energy is zero and hence the electron is assumed to be a free particle.

The one dimensional Schroedinger wave equation for a free particle is

............(1)

The one dimensional Schroedinger wave equation is

............(2)

For both the region, the appropriate solution suggested by Bloch is of the form,

Ψ (x) = e^ikx u_x (x) ......(3)

Differentiating equation (3) and substituting it in equation (1) and (2) and then further solving it under the boundary conditions, we get

where  is called as Scattering power of the potential barrier, whichis the measure of the strength with which the electrons are attracted by the positive ions.

In equation (4), there are only two variables (i.e) α and k. We know cos ka can take values only from - 1 to 1, Therefore the left hand side of equation (4) must also fall in this range. A plot is made between the LHS of equation (4) and aa for a value of P = 3 π/2 (arbitrary) as shown in Fig. 1.18.

Conclusions

From the Fig. 1.18 the following conclusions can be made.

(i) The energy spectrum has a number of allowed energy bands denoted by solid horizontal line separated by forbidden band gaps denoted by dotted lines.

(ii) The width of allowed energy band (shaded portion) increases with the increase in αa.

(iii) When P is increased, the binding energy of the electrons with the lattice points is also increased. Therefore the electron will not be able to move freely and hence the width of the allowed energy band is decreased. Especially for P→∞, then allowed energy band becomes infinitely narrow and the energy spectrum becomes a line spectrum as shown in Fig. 1.19.

 (iv) When P is decreased, the binding energy of the electron decreases and thus it moves freely over the lattice points and hence we get a wide range of allowed energy levels as shown in Fig. 1.19.

(v) Thus by varying P from zero to infinity we get the energy spectra of all ranges.