The free electron model implies that potential inside the solid is uniform in metals. A more accurate model allows for variation in potential energy due to the fixed positively charged lattice ions. In Figure 11.5, a one-dimensional lattice with ions separated by lattice parameter a, is presented.
FIGURE 11.5 One-Dimensional lattice
The highest potential is half way between the atom and goes to – as position of the ion is approached. This potential distribution is quite complicated and for mathematical solution of Schrodinger’s equation a simpler model known as “Kronig-Penney model” is used. The Kronig-Penney model still displays the following essential features, where the function:
- has the same period as lattice.
- potential is higher between ions and lower near the lattice ion.
FIGURE 11.6 Geometrical arrangement for Kronig-Penney Model
The ions are located at x = 0, a, 2a, 3a etc. The potential wells are separated from each other by potential barrier of height V0 and width ω. The Schrodinger’s equation should be solved for potential distribution separately for V = + V0/2 and V = – V0/2 and compare the solutions at the boundary conditions.
The assumed wave function has the form
Here the relationship between E and k is not as easy as that of free electron. To find the exact E-k curve, we must first of all plot the right hand side of equation (2) as a function of α a and then by plotting the left hand side and using the fact that this always must be between + 1 to – 1, a solution will be found for the value of E for which right side is within these limits.
FIGURE 11.7 Solution for Kronig-Penney Model
The solution is defined only in the shaded region, i.e., only certain values are allowed.
This implies that there are only certain allowed energy bands in crystal solids.
For the solution of Kronig-Penney model mentioned above, if the value of P is large i.e., V0 W is large, the function is given by right hand side of equation (2) crosses at a steep slopes. This means that the allowed bands are narrower and forbidden bands Arc wider. Let us observe some specific conditions.
- As P→, the band reduces to a single energy level, i.e., discrete energy spectrum for isolated atoms, which is the actual case. So P is proportional to the potential barrier of the given system.
Below E vs. k2 curve is plotted for one-dimensional lattice structure from Kronig-Penney model. At the boundaries, we can see discontinuities in energy levels as the value of n changes. By transferring the zone curves by an appropriate multiple π/a to the axis E we can see the separation between energy levels very clearly. Each of these energy levels is termed as Brillouin zone.
From the Figure 11.8, we can say that in the middle of the Brillouin zones the E-k curve is identical to the free electron. But the boundaries of the zones, i.e., at k=nπ/a, their behavior is totally different.
Electrons that occupy Brillouin zone can move freely inside a crystal. All the information for the Brillouin zones are contained in the first Brillouin in zone if We use the reduced zone plot. So, the Brillouin zones are the allowed energy regions in k.