The One-Dimensional Kronig-Penney Model

The influence of the periodic potential of the crystal lattice on the electronic structure is introduced through the one-dimensional Kronig-Penney model that illustrates the essential features of band theory of solids. 

It is easy to see that the left-hand side of (6.12) is the expression of the plane wave, and fe is the wave vector, fe = InjX. Expression exp(ikna) is a phase multiplier. 

Equation (6.12) is the usual statement of the Bloch theorem in one dimension, see (6.5). 

the translational symmetry of the potential leads to the eigenfunctions being characterized by a wave vector k (the Bloch vector). It is only defined modulo 2nfa since k + p (2 Jt/a) results in the same phase vector in (6.12) as k alone (p is an integer). It is, therefore, customary to label the eigenfunction fe(x) by restricting k to lie within the first Brillouin zone that is defined by 

In a one-dimensional case na is the magnitude of a direct lattice vector, whereas a value of p [In/a) is the magnitude of a reciprocal lattice vector. In the first Brillouin zone one has n = p = 1, and the product of the direct and reciprocal vectors equals to 2jt. 

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Figure 6.3 The one-dimensional Kronig-Penney potential. The rectangular wells model of the interionic distances, whereas the barriers of a Vq height model the one-dimensional ion lattice.

An important simple model that demonstrates some of the properties of electron states in periodic solids is the so called Kronig-Penney model. In this model, a particle of mass m experiences a one-dimensional periodic potential with period a ... 

Figure 3.7. (a) Potential energy of an electron in a one-dimensional crystal (b) Kronig-Penney model of the potential energy of an electron in a one-dimensional crystal (square-well periodic potential model). 

Figure 5.4 shows the one-dimensional potential V(x) of the Kronig-Penney model, which comprises square wells that are separated by barriers of height,... 

Let us consider this simple but significant enough physical model of the electron movement in a field of periodic potential. Figure 6.3 illustrates the distribution of the one-dimensional potential V x) in the Kronig-Penney model. This potential comprises square wells that are separated by barriers of height Vo and thickness b. The potential is periodic with the period a so that... 

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