What is the effect of periodic potential on the energy in a metal? Explain it on
the basis of Kronig-Penney model and explain the formation of energy bands.
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Answer:
According to quantum free electron theory of metals, a conduction electron in a metal experiences constant (or zero) potential and free to move inside the crystal but will not come out of the metal because an infinite potential exists at the surface. This theory successfully explains electrical conductivity, specific heat, thermionic emission and paramagnetism. This theory is fails to explain many other physical properties, for example: (i) it fails to explain the difference between conductors, insulators and semiconductors, (ii) positive Hall coefficient of metals and (iii) lower conductivity of divalent metals than monovalent metals.
To overcome the above problems, the periodic potentials due to the positive ions in a metal have been considered. shown in Fig. (a), if an electron moves through these ions, it experiences varying potentials. The potential of an electron at the positive ion site is zero and is maximum in between two ions. The potential experienced by an electron, when it passes along a line through the positive ions is as shown in Fig. (b).
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It is not easy to solve Schrödinger’s equation with these potentials. So, Kronig and Penney approximated these potentials inside the crystal to the shape of rectangular steps as shown in Fig. (c). This model is called Kronig-Penney model of potentials.
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The energies of electrons can be known by solving Schrödinger’s wave equation in such a lattice. The Schrödinger time-independent wave equation for the motion of an electron along X-direction is given by:
...............(1)
The energies and wave functions of electrons associated with this model can be calculated by solving time-independent one-dimensional Schrödinger’s wave equations for the two regions I and II as shown in Fig
The Schrödinger’s equations are:
for 0<x<a.............(2)
for -b<x<0.............(3)
We define two real quantities (say) α and β such that:
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Hence, Equations (5.41) and (5.42) becomes:
for 0<x<a
for -b<x<0
According to Bloch's theorem, the wavefunction solution of the Schrödinger equation when the potential is periodic and to make sure the function u(x) is also continuous and smooth, can be written as:
\psi (x)=e^{ikx}u(x).
Where u(x) is a periodic function which satisfies u(x + a) = u(x).
Using Bloch theorem and all the boundary conditions for the continuity of the wave function the solution of Schrodinger wave equation obtained as
where
This equation 5.59 shows the relation between the energy (through α) and the wave-vector, k, and as you can see, since the left hand side of the equation can only range from −1 to 1 then there are some limits on the values that α (and thus, the energy) can take, that is, at some ranges of values of the energy, there is no solution according to these equation, and thus, the system will not have those energies: energy gaps. These are the so-called band-gaps, which can be shown to exist in any shape of periodic potential (not just delta or square barriers). This has been plotted in Fig.
1. The permissible limit of the term
lies between +1 to -1. By varying αa, a wave mechanical nature could be plotted as shown in Fig, the shaded portion of the wave shows the bands of allowed energy with the forbidden region as unshaded portion.
2. With increase of αa, the allowed energy states for a electron increases there by increasing the band width of the bands, i.e., the strength of the potential barrier diminishes. This also leads to increase of the distance between electrons and the total energy possessed by the individual electron.
3.Conversly if suppose the effect of potential barrier dominate i.e., if P is large, the resultant wave obtained in terms of shows a stepper variation in the region lies between +1 to -1. This results in the decrease of allowed energy and increase of forbidden energy gap. Thus at extremities,
Case (i) when This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program., the allowed energy states are compressed to a line spectrum.
Case (ii) when This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program. the energy band is broadened and it is quasi continuous.