Kronig-Penney model

One of the first models to describe electronic states in a periodic potential was the Kronig-Penney model [1]. This model is commonly used to illustrate the fundamental features of Bloch s theorem and solutions of the Schrodinger... 

Figure Al.3.7. Evolution of energy bands in the Kronig-Penney model as the separation between wells, b (figure A 1,3.61 is deereased from (a) to (d). In (a) the wells are separated by a large distanee (large value of b) and the energy bands resemble diserete levels of an isolated well. In (d) the wells are quite elose together (small value of b) and the energy bands are free-eleetron-like.

The Kronig-Peimey solution illustrates that, for periodic systems, gaps ean exist between bands of energy states. As for the ease of a free eleetron gas, eaeh band ean hold 2N eleetrons where N is the number of wells present. In one dimension, tliis implies that if a well eontains an odd number, one will have partially occupied bands. If one has an even number of eleetrons per well, one will have fully occupied energy bands. This distinetion between odd and even numbers of eleetrons per eell is of fiindamental importanee. The Kronig-Penney model implies that erystals with an odd number of eleetrons per unit eell are always metallie whereas an even number of eleetrons per unit eell implies an... 

Expressing (k) is complicated by the fact that k is not unique. In the Kronig-Penney model, if one replaced k by k + lTil a + b), the energy remained unchanged. In tluee dimensions k is known only to within a reciprocal lattice vector, G. One can define a set of reciprocal vectors, given by... 

The Kronig-Penney model, although rather crude, has been used extensively to generate a substantial amount of useful solid-state theory [73]. Simple free-electron models have likewise been used to provide logical descriptions of a variety of molecular systems, by a method known in modified form as the Hiickel Molecular Orbital (HMO) procedure [74]. 

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). 

Thus, the free-electron model is not valid when Eq. (3.7) applies since the wave is reflected. The E,k) curve constructed on this basis is like that obtained from the Kronig-Penney model bands of allowed and forbidden energy regions. 

Fermi-level DOS 115 Jellium model 92—97 failures 97 schematic 94 surface energy 96 surface potential 93 work function 96 Johnson noise 252 Kohn-Sham equations 113 Kronig-Penney model 99 Laplace transforms 261, 262, 377 and feedback circuits 262 definition 261 short table 377 Lateral resolution... 

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,... 

Consider the simplest possible case, a monatomic crystalline solid. The potential at each lattice site is represented by a single square well in the Kronig-Penney model (Kronig and Penney, 1931) by Ralph Kronig (1904-1995) and William G. Penney (1909-1991). For a perfect monatomic crystaHine array (Fig. 7.3a), all the potential... 

The properties of electrons in a periodic potential are demonstrated with the use of a simple square well potential, the Kronig-Penney model. The potential is zero for 0 < x < a and Vq for —b < x < 0, i.e. it has a period (a + b). The Schrodinger equations for the two regions follow directly and lead on substitution of a Bloch function to ... 

Fig. 4.5 Plot of the solution to the Kronig-Penney model for <5 function barriers the shaded regions correspond to allowed values of aa (see text).

The potential utilised in the Kronig-Penney model is not a very realistic representation of the potential function of an atomic lattice. A model that can incorporate more realistic potentials is the tight-binding model. The lattice... 

Fig. 4 Structure of type II heterolayer superlattice and emission peak shift as a function of layer thickness, (solid line is estimated from 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. 

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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