PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
The plane wave method can be used to solve the Schrödinger equation for an electron moving in a periodic potential, $U(\vec{r})$,
$$-\dfrac{\hbar^2}{2m} \, \nabla^2 \psi_{\vec{k}} + U(\vec{r}) \psi_{\vec{k}} = E \psi_{\vec{k}}. $$Since the potential $U(\vec{r})$ is periodic, it can be written as a Fourier series,
$$U(\vec{r}) = \sum_{\vec{G}} U_{\vec{G}} e^{i \vec{G} \cdot \vec{r}}.$$Felix Bloch showed that the solutions to the Schrödinger equation for an electron in a periodic potential have the form,
$$\psi_{\vec{k}}(\vec{r}) = e^{i\vec{k}\cdot\vec{r}}u_{\vec{k}}(\vec{r}) = e^{i\vec{k}\cdot\vec{r}}\sum\limits_{\vec{G}'}C_{\vec{G}'}e^{i\vec{G}'\cdot\vec{r}},$$where the periodic function $u_{\vec{k}}(\vec{r})=\sum\limits_{\vec{G}'}C_{\vec{G}'}e^{i\vec{G}'\cdot\vec{r}}$ has been expressed as a Fourier series. The reciprocal lattice vectors have been relabeled as running over $\vec{G}'$ instead of $\vec{G}$. It does not matter how they are labeled since we sum over all of the reciprocal lattice vectors, $\sum\limits_{\vec{G}}C_{\vec{G}}e^{i\vec{G}\cdot\vec{r}}=\sum\limits_{\vec{G}'}C_{\vec{G}'}e^{i\vec{G}'\cdot\vec{r}}$. This form for the wavefunction is substituted into the Schrödinger equation,
$$\sum\limits_{\vec{G}'}\dfrac{\hbar^2(\vec{k}+\vec{G}')^2}{2m} C_{\vec{G}'}e^{i(\vec{k}+\vec{G}')\cdot\vec{r}} + \sum_{\vec{G}}\sum\limits_{\vec{G}''} U_{\vec{G}} C_{\vec{G}''}e^{i(\vec{k}+\vec{G}+\vec{G}'')\cdot\vec{r}} = E \sum\limits_{\vec{G}'}C_{\vec{G}'}e^{i(\vec{k}+\vec{G}')\cdot\vec{r}}. $$In the middle term with the double sum, a third labeling $\vec{G}''$ has been introduced $\sum\limits_{\vec{G}'}C_{\vec{G}'}e^{i\vec{G}'\cdot\vec{r}}=\sum\limits_{\vec{G}''}C_{\vec{G}''}e^{i\vec{G}''\cdot\vec{r}}$ Again, it does not matter that the label has changed since the sum is over all of the states, but we need a way to keep track of the product terms in the double sum. 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 $\vec{k} +\vec{G}'= \vec{k} + \vec{G}+\vec{G}''$ can be equal to each other. This results in the condition $\vec{G}'=\vec{G} + \vec{G}''$ and for each $\vec{G}'$ there is an equation,
$$\dfrac{\hbar^2(\vec{k}+\vec{G}')^2}{2m}C_{\vec{G}'} + \sum_{\vec{G}} U_{\vec{G}} C_{\vec{G}'-\vec{G}} = E C_{\vec{G}'}. $$This set of equations is called the central equations. The Schrödinger equation, a differential equation for $\psi_{\vec{k}}$, has been replaced with the central equations, which are algebraic equations for the coefficients $C_{\vec{G}'}$. The algebraic equations can be put in the form of an eigenvalue problem. These equations written out in the one-dimensional case for $-G_0, 0, G_0$ are,
$$\dfrac{\hbar^2(k-G_0)^2}{2m}C_{-G_0} + (\cdots + U_{-2G_0}C_{G_0} + U_{-G_0}C_{0} + U_{0}C_{-G_0} + \cdots ) = E C_{-G_0}. $$ $$\dfrac{\hbar^2k^2}{2m}C_{0} + (\cdots + U_{-G_0}C_{G_0} + U_{0}C_{0} + U_{G_0}C_{-G_0} + \cdots ) = E C_{0}. $$ $$\dfrac{\hbar^2(k+G_0)^2}{2m}C_{G_0} + (\cdots + U_{0}C_{G_0} + U_{G_0}C_{0} + U_{2G_0}C_{-G_0} + \cdots ) = E C_{G_0}. $$If the terms indicated by $\cdots$ are neglected, this can be written in matrix form as an eigenvalue problem that can be solved for the energies.
$$\textbf{M} \vec{C} = E \vec{C}.$$When more equations for $\vec{G}'$ are included, the matrix gets bigger, and since the coefficients $U_{\vec{G}}$ typically get smaller as $\vec{G}$ gets larger, it becomes more justified to neglect the terms indicated by $\cdots$. An $N\times N$ matrix will have $N$ solutions for the energy $E$ at each value of $\vec{k}$. The different solutions correspond to different bands.
The periodic potential could be a periodic Coulomb potential, a muffin tin potential, or a pseudopotential. To calculate the dispersion relation for a simple cubic, bcc, or fcc lattice with a periodic potential not listed below, use the code for the Coulomb potential and it will only be necessary to change the line that specifies the off-diagonal elements of the matrix $\textbf{M}$ for that potential.
One way to check if the plane wave method has been coded correctly is to run the calculation for a periodic potential with zero amplitude. This should result in the empty lattice approximation. Below are some examples where the amplitude of the potential is slowly increased.
To make an accurate calculation with the plane wave method, typically hundreds of reciprocal lattice vectors are included in the sum. The minimum number of reciprocal lattice vectors that can be used is two. This is a crude approximation, but it has the advantage that there is an analytic solution at the Brillouin zone boundary. In the case of only two reciprocal lattice vectors, the central equations written in matrix form are,
\begin{equation} \left[\begin{matrix} \frac{\hbar^2\vec{k}^2}{2m}-E & U \\ U & \frac{\hbar^2(\vec{k} -\vec{G})^2}{2m}-E \end{matrix}\right] \left[\begin{matrix} C_0 \\ C_{-\vec{G}} \end{matrix}\right] =0. \end{equation}At the Brillouin zone boundary, $k=\frac{G}{2}$, and the equations become,
\begin{equation} \left[\begin{matrix} \frac{\hbar^2(G/2)^2}{2m}-E & U \\ U & \frac{\hbar^2(G/2)^2}{2m}-E \end{matrix}\right] \left[\begin{matrix} C_0 \\ C_{-\vec{G}} \end{matrix}\right] =0, \end{equation}and the energies are,
$$E = \frac{\hbar^2\left(\frac{G}{2}\right)^2}{2m} \pm U.$$The result is that the size of the band gap at the Brillouin zone boundary is approximately $2U.$ In other words, the size of the band gap at the Brillouin zone boundary is approximately the amplitude of the periodic potential.
Comparisons can be made with DFT calculations: Li, Na, Mg, Al, Cu.