The Quantum Electrostatic Heterostructure (QEH) model: Theory

We follow the notation of Ref [1]. For each monolayer in a heterostructure, the monolayer response function \(\widetilde{\chi}(\mathbf{r}, \mathbf{r}', q_\parallel, \omega)\) is first calculated. We assume here that \(\widetilde{\chi}\) is isotropic, i.e. only a function of \(q_\parallel = |\mathbf{q}_\parallel|\), and independent of the direction of \(\mathbf{q}_\parallel\). The response function is averaged over the in-plane coordinates, and we define

(1)\[ \widetilde{\chi}(z, z', q_\parallel, \omega) = \frac{1}{A} \int_A \int_A \mathrm{d} \mathbf{{r}}_\parallel \mathrm{d} \mathbf{{r}}'_\parallel \widetilde{\chi}(\mathbf{r}, \mathbf{r}', q_\parallel, \omega),\]

where the integration is over the in-plane coordinates, and \(A\) is the in-plane area of the supercell. The \(z\)-dependence can be approximated in a monopole-dipole basis, in which we express \(\widetilde{\chi}\) as a \(2 \times 2\) matrix \(\chi_{\alpha \alpha'}\), where \(\alpha=0\) corresponds to a monopole component, while \(\alpha = 1\) corresponds to a dipole component, and likewise for \(\alpha'\). These components are given by

(2)\[ \widetilde{\chi}_{\alpha \alpha'} (q_\parallel, \omega) = \int \int \mathrm{d}z \mathrm{d}z' (z-z_c)^\alpha \widetilde{\chi}(z, z', q_\parallel, \omega) (z'-z_c)^{\alpha'},\]

where each integral runs over the interval \([z_c - \frac{L}{2}, z_c + \frac{L}{2}]\), where \(L\) is the thickness of the layer, and \(z_c\) the position of the middle of the layer. To make explicit the monopole/dipole structure, we label the components of the \(\chi_{\alpha \alpha'}\) matrix as \(\alpha \in {M, D}\), where \(M\) corresponds to \(\alpha=0\) and \(D\) to \(\alpha = 1.\) This corresponds to the naming convention used in the GPAW implementation.

Expressed in a plane-wave basis, we have

\[\widetilde{\chi}(\mathbf{r}, \mathbf{r}', q_\parallel, \omega) = \frac{1}{\Omega} \sum_{\mathbf{G} \mathbf{G}'} e^{i(\mathbf{q}_\parallel + \mathbf{G})\cdot \mathbf{r}} \widetilde{\chi}_{\mathbf{G}\mathbf{G}'}(q_\parallel, \omega) e^{-i(\mathbf{q}_\parallel + \mathbf{G'})\cdot \mathbf{r}'},\]

\(\Omega\) being the volume of the supercell. Integrating over the plane corresponds to taking \(\mathbf{G}_\parallel = \mathbf{G}_\parallel' = 0\), such that equation (1) becomes

\[\widetilde{\chi}(z, z', q_\parallel, \omega) = \frac{1}{L} \sum_{G_z G_z'} e^{iG_z z} \widetilde{\chi}_{G_z G_z'}(q_\parallel, \omega) e^{-iG_z' z'}\]

The integrals over \(z\) in equation (2) can then be carried out analytically, and we find

\[\begin{split}\begin{aligned} &\widetilde{\chi}_M(q_\parallel, \omega) = L \widetilde{\chi}_{G_z = 0, G_z' = 0} \\ &\widetilde{\chi}_{MD}(q_\parallel, \omega) = \sum_{G_z' \neq 0} \widetilde{\chi}_{0,G_z'} z_F^*(G_z') \\ &\widetilde{\chi}_{DM}(q_\parallel, \omega) = \sum_{G_z \neq 0} z_F(G_z) \widetilde{\chi}_{G_z,0} \\ &\widetilde{\chi}_{D}(q_\parallel, \omega) = \frac{1}{L} \sum_{G_z \neq 0, G_z' \neq 0} z_F(G_z) \widetilde{\chi}_{G_z G_z'}z_F^*(G_z'), \end{aligned}\end{split}\]

where the so-called z-factor \(z_F\) is

\[z_F(G_z) = \int_{z_c - \frac{L}{2}}^{z_c + \frac{L}{2}} e^{i G_z z} z \mathrm{d}z = -\frac{i e^{i G_z z_c}}{G_z^2} \left[G_z L \cos\left(\frac{G_z L}{2}\right) - 2 \sin\left(\frac{G_z L}{2}\right)\right],\]

and \(z_F^*\) is the complex conjugate of \(z_F\).

For systems with mirror symmetry in the out of plane (\(z\)) direction, the off-diagonal elements \(\chi_{MD}\) and \(\chi_{DM}\) must vanish. This can be seen from the following: the mirror symmetry implies that \(\chi(z,z') = \chi(-z, -z')\), where we have set \(z_c = 0\) for simplicity, and we have then for e.g. \(\chi_{DM}\) that

\[\chi_{DM} = \int z \chi(z,z') \mathrm{d}z\mathrm{d}z' = \int z \chi(-z,-z') \mathrm{d}z \mathrm{d}z' = \int (-z) \chi(z,z') \mathrm{d}z \mathrm{d}z' = - \chi_{DM}\]

where for the last equality we made the substitution \(z \rightarrow-z\) and \(z' \rightarrow- z'\). A similar result holds for \(\chi_{MD}\). Therefore one only needs to calculate the off-diagonal elements for materials that do not have mirror symmetry.