The LAPW basis for thin film systems

Calculations on thin films can be performed in different ways. A popular approach to this task is to define a bulk unit cell in which the film is provided along the xy-plane and the width of the film in z direction only covers a small fraction of the unit cell's length in that direction. Such a setup defines periodic repetitions of the film in z direction and one has to make sure that the vacuum in between the films is large enough to decouple them.

On the basis of the FLAPW method a more elegant treatment of thin film systems is possible. As sketched in the following figure one can set up unit cells with semi-infinite vacuum regions above and below the film.

An LAPW basis function for such a setup is defined as

$$\begin{equation} \renewcommand{\vec}[1]{\mathbf{#1}} \phi_{\vec{k}_\parallel \vec{G}}(\vec{r})=\left\lbrace \begin{array}{l l l} {\frac{1}{\sqrt{\Omega}} e^{i(\vec{k}_\parallel + \vec{G})\vec{r}}} & \text{for } \vec{r} \in \text{IR} \\ \sum\limits_{L} \left[ a_{\vec{k}_\parallel\vec{G}}^{L\alpha} u_{l}^\alpha(r_\alpha,E_l^\alpha) + b_{\vec{k}_\parallel\vec{G}}^{L\alpha} \dot{u}_{l}^\alpha(r_\alpha,E_l^\alpha) \right] Y_L(\vec{\hat{r}_\alpha}) & \text{for } \vec{r} \in \text{MT}^\alpha\\ \left[a_{\vec{k}_\parallel\vec{G}}^{\text{vac}} u_{\vec{k}_\parallel \vec{G}_\parallel}^\text{vac}(z,E^\text{vac}) + b_{\vec{k}_\parallel\vec{G}}^{\text{vac}} \dot{u}_{\vec{k}_\parallel \vec{G}_\parallel}^\text{vac}(z,E^\text{vac})\right] \times \frac{1}{\sqrt{A}}e^{i(\vec{k}_\parallel+\vec{G}_\parallel)\vec{r}_\parallel} & \text{for } \vec{r} \in \text{VR}^\text{vac} \\ \end{array}\right. \end{equation}$$

where the extension in the vacuum regions consists of the two functions $u_{\vec{k}_\parallel \vec{G}_\parallel}^\text{vac}(z,E^\text{vac})$ and $\dot{u}_{\vec{k}_\parallel \vec{G}_\parallel}^\text{vac}(z,E^\text{vac})$ are solutions and energy derivatives to the Schrödinger equation in the respective vacuum region

$$\begin{equation} \renewcommand{\vec}[1]{\mathbf{#1}} \left[-\frac{1}{2}\frac{\partial^2}{\partial z^2} + \frac{1}{2}\left(\vec{k}_\parallel + \vec{G}_\parallel\right)^2 + V_\text{eff}^\text{avg,vac}(z) - E^\text{vac}\right] u_{\vec{k}_\parallel \vec{G}_\parallel}^\text{vac}(z,E^\text{vac}) = 0 \end{equation}$$

at the energy parameter $E^\text{vac}$. The coefficients $a_{\vec{k}_\parallel\vec{G}}^{\text{vac}}$ and $b_{\vec{k}_\parallel\vec{G}}^{\text{vac}}$ are determined by enforcing continuity of value and slope of the basis function at the vacuum boundary defined by the parameter $D$. The z components of the plane waves in the interstitial region are constructed such that they feature a periodicty in agreement to the parameter $\tilde{D}$. Note that $\tilde{D}$ is larger than $D$ to avoid a kink of the implicitely periodic interstitial representation of the wave functions at the interstitial-vacuum boundaries.