Particle in a one-dimensional lattice

The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice.

It is a generalization of the free electron model, which assumes zero potential inside the lattice.

When talking about solid materials, the discussion is mainly around crystals – periodic lattices.

Assuming the spacing between two ions is a, the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period a.

According to Bloch's theorem,[1] the wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as:

which gives rise to the band structure of the energy spectrum of the Schrödinger equation with a periodic potential like the Kronig–Penney potential or a cosine function as it was shown in 1928 by Strutt[2].

When nearing the edges of the lattice, there are problems with the boundary condition.

Therefore, we can represent the ion lattice as a ring following the Born–von Karman boundary conditions.

If L is the length of the lattice so that L ≫ a, then the number of ions in the lattice is so big, that when considering one ion, its surrounding is almost linear, and the wavefunction of the electron is unchanged.

If N is the number of ions in the lattice, then we have the relation: aN = L. Replacing in the boundary condition and applying Bloch's theorem will result in a quantization for k:

The Kronig–Penney model (named after Ralph Kronig and William Penney[3]) is a simple, idealized quantum-mechanical system that consists of an infinite periodic array of rectangular potential barriers.

The potential function is approximated by a rectangular potential: Using Bloch's theorem, we only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function u(x) is also continuous and smooth.

We will solve for each independently: Let E be an energy value above the well (E>0) To find u(x) in each region, we need to manipulate the electron's wavefunction:

To complete the solution we need to make sure the probability function is continuous and smooth, i.e.:

In the previous paragraph, the only variables not determined by the parameters of the physical system are the energy E and the crystal momentum k. By picking a value for E, one can compute the right hand side, and then compute k by taking the

The right hand side of the last expression above can sometimes be greater than 1 or less than –1, in which case there is no value of k that can make the equation true.

, that means there are certain values of E for which there are no eigenfunctions of the Schrödinger equation.

Thus, the Kronig–Penney model is one of the simplest periodic potentials to exhibit a band gap.

Since this potential is periodic, we could expand it as a Fourier series:

is a function that is periodic in the lattice, which means that we can expand it as a Fourier series as well:

To save ourselves some unnecessary notational effort we define a new variable:

We can juggle this expression a little bit to make it more suggestive (use partial fraction decomposition):

If we use a nice identity of a sum of the cotangent function (Equation 18) which says:

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).

For a different and detailed calculation of the gap formula (i.e. for the gap between bands) and the level splitting of eigenvalues of the one-dimensional Schrödinger equation see Müller-Kirsten.

[5] Corresponding results for the cosine potential (Mathieu equation) are also given in detail in this reference.

In some cases, the Schrödinger equation can be solved analytically on a one-dimensional lattice of finite length[6][7] using the theory of periodic differential equations.

If there is a band gap between two consecutive energy bands of the infinite system, there is a sharp distinction between two types of states in the finite lattice.

bulk states whose energies depend on the length

The energies of these states depend on the point of termination