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.
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 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, the wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as:
where u(x) is a periodic function which satisfies u(x + a) = u(x). 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 in the Mathieu equation.
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. So now, instead of two boundary conditions we get one circular boundary condition:
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 value of the expression to which cos(k a) is equated in the dispersion relation, with P = 1.5. The black bars denote regions of for which k can be calculated.
The dispersion relation for the Kronig–Penney model, with P = 1.5.
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. If you pick a value of E, you can compute the right hand side, and then compute k by taking the 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 can make the equation true. Since , 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.
If we use a nice identity of a sum of the cotangent function (Equation 18) which says:
and plug it into our expression we get to:
We use the sum of cot and then, the product of sin (which is part of the formula for the sum of cot) to arrive at:
This equation 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).
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.
^Bloch, Felix (1929). "Über die Quantenmechanik der Elektronen in Kristallgittern". Zeitschrift für Physik (in German). Springer Science and Business Media LLC. 52 (7–8): 555–600. doi:10.1007/bf01339455. ISSN1434-6001.
^de L. Kronig, R.; Penney, W. G. (3 February 1931). "Quantum Mechanics of Electrons in Crystal Lattices". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. The Royal Society. 130 (814): 499–513. doi:10.1098/rspa.1931.0019. ISSN1364-5021.
^Harald J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012), 325–329, 458–477.