arvau

Wikipedia / Particle in a one-dimensional lattice

In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal 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.

Problem definition

When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is , the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period . According to Bloch's theorem, the wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as:

where is a periodic function which satisfies . It is the Bloch factor with Floquet exponent 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. The solutions can be given with the help of the Mathieu functions.

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 is the length of the lattice so that

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

Kronig–Penney model

The Kronig–Penney model (named after Ralph Kronig and William Penney) 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 is also continuous and smooth.

Considering a single period of the potential: We have two regions here. We will solve for each independently: Let E be an energy value above the well (E>0)

To find u(x

And that and

For energy values inside the well (E<.

with the same formula for P.

Band gaps in the Kronig–Penney model

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 of both sides. Thus, the expression gives rise to the dispersion relation.

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 means there are certain values of E for which there are no eigenfunctions of the Schrödinger equation. These values constitute the band gap.

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

Kronig–Penney model: alternative solution

An alternative treatment to a similar problem is given. Here we have a delta

is some constant, and is the lattice constant (the spacing between each site). Since this potential is periodic, we could expand it as a Fourier series

We sum this last equation over all values of

Now, is a reciprocal lattice vector, which means that a sum over

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

We use the sum of and then, the product of (which is part of the formula for the sum of

This equation shows the relation between the energy (through ) and the wave-vector, , and as you can see, since the left hand side of the equation can only range from to then there are some limits on the values that (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).

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

Finite lattice

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

The eigenstates of the finite system can be found in terms of the Bloch states of an infinite system with the same periodic potential. 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. For each energy band. These states are standing waves is chosen so that the wavefunction vanishes at the boundaries. The energies of these states match the energy bands of the infinite system.

. The energy of such a state can lie either at the band edge or within the band gap. If the energy is within the band gap, the state is a surface state localized at one end of the lattice, but if the energy is at the band edge, the state is delocalized across the lattice.

See also

References