In Kronig-Penny model, it is assumed that the potential energy of an electron in a linear array of positive nuclei has the form of a periodic array of square wells (Fig. 5.10). Let the period of the periodic potential be (x + b). For -b < x < 0 the potential energy is V0 while for 0 < x < a the potential energy is zero. The Schrodinger equations for two cases are-
If V0 > E and α, β are real quantities then equations (xxx) and (xxxi) will become-
According to Bloch, the Solutions of the Schrodinger equations are of the form
Ψ(x) = uk(x) eikx
where uk(x) is a periodic function with the periodicity of lattice i.e.,
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uk(x) = uk(x + a + b)
Therefore, equations (xxx) and (xxxi) can be written as-
The solutions of these equations are,
where A, B, C and D are constants.
These constants can be determined in such a way that the wave function Ψ and its derivative dΨ/dx are single valued and continuous.
Using these boundary conditions in equations (xxxii) and (xxxiii) we get,
These equations have non-zero solutions if the determinant of the coefficients A, B, C and D vanishes. That is-
On expanding this determinant and after simplification, we get-
β2 – α2/2αβ – sinh βb sin αa + cosh βb cos αa = cos k (a + b) … (xxxiv)
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In order to express the relation in a more simplified form Kronig and Penny suggested delta functions such that V0 → ∞ and b → 0 but the product V0 b or β2b remains finite. Within such limit, equation (xxxiv) reduces to-
β2b/2α sin αa + cos αa = cos ka … (xxxv)
Because when b → 0, sinh βb – βb and cosh βb → 1
Equation (xxxv) can be written as-
Where the quantity P is defined as,
Since V0 → ∞ i.e. P is a measure of the quantity V0 b which is the area of the potential barrier. When P is increased, the area of the potential barrier is increased and the given electron is bound more strongly to a particular potential well. When P → 0, the potential barrier becomes very weak which means that electrons are free electrons. In this case, from equation (xxxvi), we get,
If we plot a graph between cos ka and αa for P 3/2, we get the curve as shown in Fig. 5.11. Since α = √2mE/h2, the abscissa αa is a measure of energy and by finding the value of αa at any point, the energy represented by the function at that point is calculated.
The values of αa satisfying equation (xxxvi) are obtained by drawing a line parallel to αa-axis at a distance cos ka from it. If we change αa continuously from 0 to i.e., cos ka = ± 1, we obtain all possible values of α a and hence that of energy.
From Fig. 5.11 following conclusions are drawn:
1. The energy spectrum consists of an infinite number of allowed energy band separated by intervals in which there are no energy levels. These are known as forbidden regions.
The boundaries of the allowed energy levels corresponding to the values of cos ka = ± 1
or ka = nπ
or k = nπ/a
2. As αa increases, the first term on L.H.S. of equation (xxxvi) decreases, so the width of allowed energy bands increases and hence forbidden regions become narrower.
3. The width of allowed band decreases with increasing value of P (i.e., with increasing the binding energy of electrons). When P → ∞, the allowed energy bands are infinitely narrow and are independent of k.
We find equation (xxxvi) has solutions in this case only if,
i.e., E is independent of k.
From equation (xxxvii) it is also possible to find the energy E as a function of wave number k. The graph between E and k is shown in Fig. 5.12.
The discontinuities in E vs k occur at-
k = nπ/a
where, n = 1, 2, 3, …
The k values define boundaries of 1st, 2nd, … Brillouin zones.
From k = π/a to – π /a, there exists the first Brillouin zone. The second zone consists of two parts, one form + π/a to + 2π/a and second from – π/a to 2π/a. These zoned boundaries represent the maximum energies that the elector can have without any discontinuity.
A further boundaries conclusion may be drawn from equation (xxxvi). Within a given energy band, the energy is a periodic function of k. For example, if we replace k by k + 2πn/a, where n is an integer, the R.H.S. of equation (xxxvi) remains the same. In other words, k is not uniquely determined. Therefore, it is convenient to introduced the reduced wave vector which limits to the region-
– π/a ≥ k ≥ ± π/a
The energy versus reduced wave vector is represented in Fig. 5.13.
So far, we have assumed the crystal to be infinite, but now it will be necessary to investigate the consequences of imposing boundary condition. For a linear crystal of length L, the boundary condition may be taken as-
Ψ (x + L) = Ψ(x)
Making use of Bloch functions,
eik(x + L) uk(x + L) = eikx uk(x)
Because of the periodicity of uk, we have uk(x + L) = uk(x). The boundary conditions, thus require-
k = 2πx/L … (xxxix)
Where, n = ± 1, ±2, ±3 …
The number of possible wave functions (k-values) in the ring dk is therefore
dn = L/2π dk
Since k = nπ/a, it follows that the maximum value of n in equation (xxxix) is L/2a = N/2,
Where, N is the number of unit cells. Hence we conclude that the total number of possible wave functions in any energy band is equal to the number of unit cells N.
We have concluded in the Kronig-Penny model that the energy discontinuities in a monatomic one-dimensional lattice occur when k = (nπ/a), where n is any positive or negative integer. In one-dimensional monatomic lattice, a line representing the value of k is divided up by energy discontinuities into segments of length π/a as shown in Fig. 5.14(a).
These line segments are known as Brillouin zones. The segment –π/a ≤ k ≤ + π/a is the first Brillouion zone. The segments -2π/a ≤ k ≤ -π/a and π/a ≤ k ≤ 2π/a form the second Brillouin zone. Each crystal structure gives rise to its own characteristics Brillouin zones. The first zone for two dimensional square lattices will be a square ABCD, the boundaries of which will be given by the relations.
kx = ± π/a; ky = ± π/a
The boundaries of the zone are given by-
±kx = ± ky = 2π/a
This zone is represented by EFGH as shown in Fig. 5.14(b). Note that the right half of the second Brillouin zone is similar to the left region of the first zone. Similarly, the regions of third and higher zones match with those of the first zone.