Category:Dielectric properties: Difference between revisions

Introduction

Optics - response to a varying E-field (absorption, reflectance, MOKE)

Methods based on screened interaction in solids (MBPT)

Most general definition of epsilon D=epsilon E

Methods for computing ${\displaystyle \epsilon}$

Static response: Density functional perturbation Theory (DFPT) and Finite differences based methods

LEPSILON

By setting LEPSILON=.True., VASP uses DFPT to compute the static ion-clamped dielectric matrix with or without local field effects. Derivatives are evaluated using Sternheimer equations, avoiding the explicit computation of derivatives of the periodic part of the wave function. This method does not require the inclusion of empty states via the NBANDS parameter.

At the end of the calculation the both the values of ${\displaystyle \epsilon}$ including (LRPA=.True.) or excluding (LRPA=.False.) local-field effects are printed in the OUTCAR file. Users can perform a consistency check by comparing the values with no local field to the zero frequency results for ${\displaystyle \epsilon}$ obtained from a calculation with LOPTICS=.True..

LCALCEPS

With LCALCEPS=.True., the dielectric tensor is computed from the derivative of the polarization, using

${\displaystyle \epsilon^\infty_{ij}=\delta_{ij}+ \frac{4\pi}{\epsilon_0}\frac{\partial P_i}{\partial \mathcal{E}_j} \qquad {i,j=x,y,z}. }$

However, here the derivative is evaluated explicitly by employing finite-differences. The direction and intensity of the perturbing electric field has to be specified in the INCAR using the EFIELD_PEAD variable. As with the previous method, at the end of the calculation VASP will write the dielectric tensor in the OUTCAR file. Control over the inclusion of local-field effects is done with the variable LRPA.

Dynamical response

LOPITCS

The variable LOPTICS allows for the calculation of the frequency dependent dielectric function once the ground state is computed. It uses the explicit expression to evaluate the imaginary part of ${\displaystyle \epsilon}$

${\displaystyle \epsilon^{(2)}_{\alpha \beta}\left(\omega\right) = \frac{4\pi^2 e^2}{\Omega} \mathrm{lim}_{q \rightarrow 0} \frac{1}{q^2} \sum_{c,v,\mathbf{k}} 2 w_\mathbf{k} \delta( \epsilon_{c\mathbf{k}} - \epsilon_{v\mathbf{k}} - \omega) \times \langle u_{c\mathbf{k}+\mathbf{e}_\alpha q} | u_{v\mathbf{k}} \rangle \langle u_{v\mathbf{k}} | u_{c\mathbf{k}+\mathbf{e}_\beta q} \rangle, }$

while the real part is evaluated using the Kramers-Kroning relation. At this level there are no effects coming from local fields.

This method requires a relatively large number of empty states, controlled by the variable NBANDS in the INCAR and should be checked for convergence.

Furthermore, the INCAR should also include values for CSHIFT (the broadening applied to the Lorentzian function which replaces the ${\displaystyle \delta}$-function), and NEDOS (the frequency grid for ${\displaystyle \omega}$).

ALGO = TDHF

Time-dependent HF/DFT calculation. Follows the Casida equation. Uses a FT of the time-evolvind dipoles to compute epsilon Also requires CSHIFT, NBANDS(not necessarily as many as LOPTICS, depends on how many peaks you need to converge), and NEDOS. Choice of time-dependent kernel controllled by AEXX and HFSCREEN

ALGO = TIMEEV

Uses a delta-pulse to probe all transitions with a time-evolution equation. FT the dipoles also gives you epsilon

ALGO = CHI

ALGO = BSE

Setting ALGO=BSE computes epsilon by solving the Bethe-Salpeter equation. Here the electron-hole pairs are treated as interacting particles and a new Hamiltonian is built to deal with their interaction

${\displaystyle H_{cv\mathbf k,c'v'\mathbf k'} = (e_{c\mathbf k} - e_{v\mathbf k})\delta_{cc'}\delta_{vv'}\delta_{\mathbf k\mathbf k'} + \mathrm i \left[2V_{cv\mathbf k,c'v'\mathbf k'} - W(\omega=0)_{cv\mathbf k,c'v'\mathbf k'}\right], }$

where ${\displaystyle W}$ is the screened direct interaction and ${\displaystyle V}$ the exchange interaction.

This calculation builds the BSE Hamiltonian and obtains its eigenvalues (${\displaystyle E_\lambda}$) and eigestates (${\displaystyle A^\lambda_{cv\mathbf k}}$). The full BSE Hamiltonian is not Hermitian, so in the general case the dielectric tensor is built using

${\displaystyle \epsilon_{\alpha \beta}\left(\omega\right) = \delta_{\alpha\beta} - \frac{4\pi^2 e^2}{\Omega} \sum_{\lambda\lambda '}\sum_{c,v,\mathbf{k}}\sum_{c',v',\mathbf k'} 2 w_\mathbf{k} \frac{A^\lambda_{cv\mathbf k}A^{\lambda',*}_{c'v'\mathbf k'}S^{-1}_{\lambda\lambda'}}{ \omega - E_\lambda +\mathrm i \delta} \times \langle c\mathbf{k+q}|r_\alpha | v\mathbf{k} \rangle \langle v'\mathbf{k'} |r_\beta | c'\mathbf{k'+q} \rangle, }$

where ${\displaystyle S_{\lambda\lambda'}}$ is the overlap between exciton states of indices ${\displaystyle \lambda}$ and ${\displaystyle \lambda'}$.

Level of approximation

Micro-macro connection - Including local fields (GG' vs GG or 00, to confirm)

Inhomogeniety - Long wavelength vs short

Local fields in the Hamiltonian

Ion-clamped vs relaxed/dressed dielectric function

Static vs dynamic

Density-density versus current-current response functions

Relation to observables

Polarizability

Optical conductivity

Optical absorption

Reflectance

MOKE

Combination with other perturbations

Atomic displacements

Strain