Behaviour of an electron in a periodic potential

Kronig Penny Model illustrates the behaviour of electrons in a periodic potential. It assumes that the potential energy of an electron is a linear array of positive nuclei which has the form of a periodic array of square well as shown below:

The Kronig-Penney model is a simplified one-dimensional model of a periodic potential, which provides a useful framework for understanding the behavior of electrons in a crystal lattice. The model consists of a series of potential wells of depth V0 and width a, separated by potential barriers of height V1 and width b (where a and b are the lattice constants). The model assumes that the electrons are free particles that move in a periodic potential created by the lattice.

The behavior of electrons in this model can be analyzed by solving the Schrödinger equation for the system. The Schrödinger equation is a differential equation that describes the time-evolution of a quantum state, and it can be used to determine the energy levels and wavefunctions of the electrons in the periodic potential.

Using the Kronig-Penney model, it is possible to show that the energy levels of the electrons form allowed energy bands separated by forbidden energy gaps. The width and location of the allowed bands depend on the values of the potential barriers and wells in the lattice.

In the regions where the potential is constant (either V0 or V1), the wavefunction can be expressed as a linear combination of plane waves. In the regions where the potential is changing, the wavefunction can be expressed as a superposition of the solutions for the constant potential regions. This results in a set of boundary conditions that must be satisfied in order to obtain the allowed energy levels and wavefunctions.

In the allowed energy bands, the electrons are free to move and can conduct electrical current. In the forbidden energy gaps, the electrons are not allowed to move and behave like insulators. The width of the energy gaps is determined by the size of the potential barriers, with larger barriers resulting in wider gaps.

Overall, the Kronig-Penney model provides a simple but powerful framework for understanding the behavior of electrons in a periodic potential, and can be used to predict the electronic and optical properties of materials.

Derivation to obtain behaviour of an electron:

Potential experienced by electron due to ion core is

V(x) = 0; 0 ≤x ≤a

= Vo ; -b≤x≤ 0

According to Schrodinger equation,

Since potential is periodic from block function is

Thus solution in region a ≤x ≤a+b must be related to equation 3 in the region -b ≤x≤a by Bloch theorem.

We have,

A + B = C + D.....5

iα (A- B) = β (C - D) ......6

At x= a,

These are four homogeneous equation with four unknown so coefficient vanishes.

On simplifying we get,

Changing periodic well type pot into delta function. For this β>>α and b-> 0

The allowed values of energy E are given by those ranges of α for which the function lies between +1 and -1.

As there are regions for αa where value for [(P/2a) sin αa + cos αa] doesn't lie between +1 and -1.

For those values of αa and E, there are no travelling wave or Bloch like solution to wavefunction so that forbidden gap in energy spectrum are form.

Thus energy spectrum of a consists of allowed energy bands separated by forbidden energy bands.

Comparison of Kronig Penny model with free electron theory

The Kronig-Penney model and the free electron theory are two important models in solid-state physics that provide insights into the behavior of electrons in crystals.

In free electron theory, the energy curve of an electron is obtained in continuous form (parabola) as shown in figure . But in Kronig Penny model, the energy curve of an electron is obtained with finite energy gap. Thus in free electron theory, there is no band structure whereas in Kronig Penny model, there is a band structure of solid.

The free electron theory assumes that the electrons in a metal are free to move and are not bound to any particular atom. This model is based on the idea that the valence electrons in a metal are only weakly bound to the nucleus and can be easily excited to higher energy levels. The free electron model provides a simple explanation for the metallic properties of materials such as high electrical conductivity and thermal conductivity.

On the other hand, the Kronig-Penney model is a more sophisticated model that takes into account the periodicity of the crystal lattice. The model assumes that the electrons are confined to a periodic potential well and can only occupy certain energy levels. This results in the formation of allowed energy bands and forbidden energy gaps, which can have a significant impact on the electronic and optical properties of materials.

The free electron model of metals gives us good insight into the heat capacity, thermal conductivity, electrical conductivity, magnetic susceptibility and electrodynamics of metal. But semi conductors and insulators, occurrence of positive value of Hall coefficient relation of conduction electrons in metal to valence electron of free atoms so to explain this Kronig Penny model was introduced.

This note is a part of the Physics Repository.