Category:Dielectric properties: Difference between revisions

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The dielectric function <math>\epsilon(\mathbf q,\omega)</math> descrives the electric response of a system when perturbed by an external electric field.
==Introduction==


Depending on the origin of the perturbing electric field, <math>\epsilon(\mathbf q,\omega)</math> 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.
Optics - response to a varying E-field (absorption, reflectance, MOKE)


Methods based on screened interaction in solids (MBPT)


== Dielectric response from atomic oscillations (Finite differences and DFPT)==
Most general definition of epsilon D=epsilon E
=== Static response ===
The total energy of a system can be expanded expressed as <math>E = E(u,\eta,\mathcal E)</math>, i.e. a function of the atomic displacements, <math>u_m</math> (<math>m=1, ..., N_{atoms}</math>), homogeneous strains <math>\eta_j</math> (<math>j=1, ..., 6</math>), and the static electric field <math>\mathcal E_a</math> (<math>a=1, ..., 3</math>). Under a weak electric field one can write


::<math>
==Methods for computing <math>\epsilon</math>==
E(u,\eta,\mathcal E) = E_0 - \Omega\mathbf{\mathcal E}\cdot \mathbf {P},
===Static response: Density functional perturbation Theory (DFPT) and Finite differences based methods===
====LEPSILON====
 
By setting {{TAG|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 {{TAG|NBANDS}} parameter.
 
At the end of the calculation the both the values of <math>\epsilon</math> including ({{TAG|LRPA}}=.True.) or excluding ({{TAG|LRPA}}=.False.) local-field effects are printed in the {{TAG|OUTCAR}} file. Users can perform a consistency check by comparing the values with no local field to the zero frequency results for <math>\epsilon</math> obtained from a calculation with {{TAG|LOPTICS}}=.True..
 
====LCALCEPS====
With {{TAG|LCALCEPS}}=.True., the dielectric tensor is computed from the derivative of the polarization, using
:<math>
\epsilon^\infty_{ij}=\delta_{ij}+
\frac{4\pi}{\epsilon_0}\frac{\partial P_i}{\partial \mathcal{E}_j}
\qquad
{i,j=x,y,z}.
</math>
</math>
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 {{TAG|EFIELD_PEAD}} variable. As with the previous method, at the end of the calculation VASP will write the dielectric tensor in the {{TAG|OUTCAR}} file. Control over the inclusion of local-field effects is done with the variable {{TAG|LRPA}}.
===Dynamical response===
====LOPITCS====
The variable {{TAG|LOPTICS}} allows for the computation of the frequency dependent dielectric function
Requires CSHIFT (\detla), NBANDS(empty states), and NEDOS(frequency grid).


where <math>E_0</math> is the energy of the cell under no electric field. If the perturbing field is weak enough <math>E</math> can be expressed in a Taylor expansion keeping only the terms up to second order
====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


::<math>
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 + ...
</math>


While each derivative has its own significance and can be computed with VASP, for the dielectric constant the quantities which deserve our focus are:
====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====
Computes epsilon by solving the BSE eigenstates and eigenenergies.


1) The polarisation <math>P_\alpha = -\frac{1}{\Omega_0}\frac{\partial E}{\partial \mathcal{E}_\alpha} </math>


2) The ion-clamped dielectric susceptibility <math>\overline{\chi}_{\alpha\beta} =  
==Level of approximation==
- \frac{1}{\Omega_0}\frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \mathcal{E}_\beta} |_{u,\eta}</math>
===Micro-macro connection -  Including local fields (GG' vs GG or 00, to confirm)===


3) The force constants <math>\Phi_{mn} =
\frac{\partial^2 E}{\partial u_m \partial u_n} |_{\mathcal{E},\eta}</math>


4) The Born effective charges <math>Z_{m\alpha} =
===Inhomogeniety - Long wavelength vs short===
-\frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} |_{\eta} </math>


Note that <math>\overline{\chi}_{\alpha\beta}</math> 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


::<math>
===Local fields in the Hamiltonian===
\chi_{\alpha\beta} = \overline{\chi}_{\alpha\beta} +
\Omega_0^{-1} Z_{m\alpha} (\Phi)^{-1}_{mn} Z_{n\beta}
</math>


== 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 <math>\epsilon(\mathbf q,\omega)</math> can be computed, which are described bellow.


=== Non-interacting electrons and holes ===
===Ion-clamped vs relaxed/dressed dielectric function===
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


::<math>
\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}.
</math>


Here, <math>c</math> and <math>v</math> are electron and hole indices, respectively, and <math>\varepsilon_{c\mathbf k}</math> and <math>\varepsilon_{v\mathbf k - \mathbf q}</math> are their respective energies. These can come from either DFT, Hartree-Fock (HF), or GW.
===Static vs dynamic===


In order to obtain the dielectric function at this level, the INCAR file must have the line {{TAG|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.
===Density-density versus current-current response functions===




=== Random Phase approximation (RPA) ===
==Relation to observables==
===Polarizability===
===Optical conductivity===
===Optical absorption===
===Reflectance===
===MOKE===


=== Bethe-Salpeter equation (BSE) ===
==Combination with other perturbations==
===Atomic displacements===
===Strain===

Revision as of 14:37, 17 October 2023

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

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 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 obtained from a calculation with LOPTICS=.True..

LCALCEPS

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

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 computation of the frequency dependent dielectric function Requires CSHIFT (\detla), NBANDS(empty states), and NEDOS(frequency grid).


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

Computes epsilon by solving the BSE eigenstates and eigenenergies.


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

Subcategories

This category has the following 2 subcategories, out of 2 total.

B

X

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