PHY.K02UF Molecular and Solid State Physics

Plane Wave Method

The plane wave method can be used to calculate the electronic band structures of materials. In this method, the Schrödinger equation is solved for the energy eigenstates of a single electron moving in a periodic potential. Since there is only one electron in the problem, electron-electron interactions are neglected except by possibly including the average positions of the other electrons in the periodic potential that the single electron sees. The ground state and some excited states for the single electron are determined for every $\vec{k}$-vector in the first Brillouin zone. To determine the properties of the many electron system, the single electron states are filled starting with the lowest energy states and filling the electron states up to the Fermi energy. The Schrödinger equation for a single electron is,

$$-\dfrac{\hbar^2}{2m} \, \nabla^2 \psi + U(\vec{r}) \psi = E \psi. $$

Since the potential $U(\vec{r})$ and the wave function $\psi$ are periodic functions, they can be written as Fourier series,

$$U(\vec{r}) = \sum_{\vec{G}} U_{\vec{G}} e^{i \vec{G} \cdot \vec{r}},$$ $$\psi(\vec{r}) = \sum_{\vec{k}} C_{\vec{k}} e^{i \vec{k} \cdot \vec{r}}.$$

The potential $U(\vec{r})$ has the periodicity of the lattice, so the Fourier series for it sums over the reciprocal lattice vectors $\vec{G}$. The wave function can be any function that satisfies the periodic boundary conditions of a box that contains many unit cells of the crystal. The wave vectors $\vec{k}$ satisfy the periodic boundary conditions of this box. Substituting this form for the potential and the wave function into the Schrödinger equation yields,

$$\sum_{\vec{k}} \dfrac{\hbar^2 k^2}{2m} C_{\vec{k}} e^{i \vec{k} \cdot \vec{r}} + \sum_{\vec{G}} \sum_{\vec{k}'}U_{\vec{G}} C_{\vec{k}'} e^{i (\vec{G} + \vec{k}') \cdot \vec{r}} = E \sum_{\vec{k}} C_{\vec{k}} e^{i \vec{k} \cdot \vec{r}}.$$

In the middle term with the double sum, the sum over $\vec{k}$ has been relabeled as a sum over $\vec{k}'$. It does not matter that the label has changed since the sum is over all of the states. Next we collect like terms. The exponential factors can be written as $e^{i \vec{k} \cdot \vec{r}}= \cos(\vec{k} \cdot \vec{r})+ i\sin(\vec{k} \cdot \vec{r})$. Only terms that have the same wavelength can be equal to each other so only the terms where $\vec{k} = \vec{G}+\vec{k}'$ can be equal to each other. This results in the condition,

$$\frac{\hbar^2 k^2}{2m}C_{\vec{k}} + \sum_{\vec{G}} U_{\vec{G}} C_{\vec{k}-\vec{G}} = E C_{\vec{k}}.$$

This set of equations are called the central equations. The Schrödinger equation, a differential equation for $\psi$, has been replaced with the central equations which are algebraic equations for the coefficients $C_{\vec{k}}$. The algebraic equations can be put in the form of an eigen value problem.

$$\textbf{M} \vec{C} = E \vec{C}$$

It is necessary to solve an eigen value problem for every $\vec{k}$ point in the $E$ vs. $\vec{k}$ dispersion relation. Notice that the energy $E$ only depends on the coefficient of one value of $\vec{k}$ plus the coefficients of $\vec{k}+\vec{G}$. This is comes about because of the restriction that only the terms with the same wavelength can be equal to each other. This means that the wave function for energy $E$ has the form,

$$\psi_{\vec{k}}= \sum_{\vec{G}}C_{\vec{k}+\vec{G}}e^{i(\vec{k}+\vec{G})\cdot \vec{r}}.$$

This can be written in Bloch form,

$$\psi_{\vec{k}}= e^{i\vec{k}\cdot \vec{r}}\sum_{\vec{G}}C_{\vec{k}+\vec{G}}e^{i\vec{G}\cdot \vec{r}},$$

where $\sum_{\vec{G}}C_{\vec{k}+\vec{G}}e^{i\vec{G}\cdot \vec{r}}$ is the Fourier series of a periodic function with the periodicity of the lattice.

The periodic potential could be a periodic Coulomb potential, a muffin tin potential, or a pseudopotential.

Face centered cubic with a periodic Coulomb potential

Body centered cubic with a periodic Coulomb potential