Category:Dielectric properties: Difference between revisions

The dielectric function ${\displaystyle \epsilon(\mathbf q,\omega)}$ descrives the electric response of a system when perturbed by an external electric field.

Depending on the origin of the perturbing electric field, ${\displaystyle \epsilon(\mathbf q,\omega)}$ can be computed in different ways. At the present, VASP is able to compute the dielectric function from atomic oscillations using methods based on Finite Differences and Density Functional Perturbation Theory (DFPT), or from the changes in the electronic charge density due to an external electric field at different levels of interaction between electrons and holes (e.g. Random Phase approximation and Bethe-Salpeter equations). Despite their differences, these methods are all based on the concept of linear response, meaning that changes in quantities with an external density or current are computed within first or second order. A summary of the different methods is described below.

Dielectric response from atomic oscillations (Finite differences and DFPT)

Static response

The total energy of a system can be expanded expressed as ${\displaystyle E = E(u,\eta,\mathcal E)}$, i.e. a function of the atomic displacements, ${\displaystyle u_m}$ (${\displaystyle m=1, ..., N_{atoms}}$), homogeneous strains ${\displaystyle \eta_j}$ (${\displaystyle j=1, ..., 6}$), and the static electric field ${\displaystyle \mathcal E_a}$ (${\displaystyle a=1, ..., 3}$). Under a weak electric field one can write

${\displaystyle E(u,\eta,\mathcal E) = E_0 - \Omega\mathbf{\mathcal E}\cdot \mathbf {P}, }$

where ${\displaystyle E_0}$ is the energy of the cell under no electric field. If the perturbing field is weak enough ${\displaystyle E}$ can be expressed in a Taylor expansion keeping only the terms up to second order

${\displaystyle E(u,\eta,\mathcal E) = E_0 + \frac{\partial E}{\partial u_m} u_m + \frac{\partial E}{\partial \mathcal{E}_\alpha} \mathcal{E}_\alpha+\frac{\partial E}{\partial \eta_j} \eta_j + \frac{1}{2} \frac{\partial^2 E} {\partial u_m \partial u_n } u_m u_n + \frac{1}{2} \frac{\partial^2 E} {\partial \mathcal{E}_\alpha \partial \mathcal{E}_\beta } \mathcal{E}_\alpha \mathcal{E}_\beta + \frac{1}{2} \frac{\partial^2 E} {\partial \eta_j \partial \eta_k} \eta_j \eta_k + \frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} u_m \mathcal{E}_\alpha + \frac{\partial^2 E}{\partial u_m \partial \eta_j} u_m \eta_j + \frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \eta_j} \mathcal{E}_\alpha \eta_j + ... }$

While each derivative has its own significance and can be computed with VASP, for the dielectric constant the quantities which deserve our focus are:

1) The polarisation ${\displaystyle P_\alpha = -\frac{1}{\Omega_0}\frac{\partial E}{\partial \mathcal{E}_\alpha} }$

2) The ion-clamped dielectric susceptibility ${\displaystyle \overline{\chi}_{\alpha\beta} = - \frac{1}{\Omega_0}\frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \mathcal{E}_\beta} |_{u,\eta}}$

3) The force constants ${\displaystyle \Phi_{mn} = \frac{\partial^2 E}{\partial u_m \partial u_n} |_{\mathcal{E},\eta}}$

4) The Born effective charges ${\displaystyle Z_{m\alpha} = -\frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} |_{\eta} }$

Note that ${\displaystyle \overline{\chi}_{\alpha\beta}}$ is computed at fixed ionic positions and one should take into account the ionic relaxation due to the external perturbation. It is possible to do so by taking the Taylor expansion starting from the configuration where the energy is minimal, which leads to the following expression to the total dielectric susceptibility

${\displaystyle \chi_{\alpha\beta} = \overline{\chi}_{\alpha\beta} + \Omega_0^{-1} Z_{m\alpha} (\Phi)^{-1}_{mn} Z_{n\beta} }$

Dielectric response from optics (non interacting case, RPA, and BSE)

Here the system is assumed to be under a weak electric field, which can be often referred to as a probe field in experiments. The external field is weak enough that it does not create enough excitations so that the system is driven into an excited state. This means that quantities such as the optical absorption can be measured. There are several levels of theory at which ${\displaystyle \epsilon(\mathbf q,\omega)}$ can be computed, which are described bellow.

Non-interacting electrons and holes

At this level there is no interaction between electrons and holes after the former are promoted to higher energy levels. The dielectric function is computed from the dipolar moments using

${\displaystyle \epsilon(\mathbf q,\omega) = 1 + \frac{4\pi^2 e^2}{\Omega}\frac{1}{q^2}\lim_{\delta\to 0^+}\sum_{c,v,\mathbf k}\frac{|\langle c\mathbf k |\mathbf r| v\mathbf{k-q}\rangle|^2}{\omega - \varepsilon_{c\mathbf k} + \varepsilon_{v\mathbf k - \mathbf q} + \mathrm i \delta}. }$

Here, ${\displaystyle c}$ and ${\displaystyle v}$ are electron and hole indices, respectively, and ${\displaystyle \varepsilon_{c\mathbf k}}$ and ${\displaystyle \varepsilon_{v\mathbf k - \mathbf q}}$ are their respective energies. These can come from either DFT, Hartree-Fock (HF), or GW.

In order to obtain the dielectric function at this level, the INCAR file must have the line LOPTICS=.TRUE. during a DFT or GW run. However, since the electrons and holes are taken as being non-interacting, effects such as the formation of excitons cannot be included.

Random Phase approximation (RPA)

Bethe-Salpeter equation (BSE)