README

Introduction

This repository provides a simple function to perform the Kramers-Kronig transformation on the frequency-dependant imaginary dielectric function, to obtain the real part of the dielectric function. The transformation is defined as: ^[1]

$$\epsilon _{{\alpha \beta }}^{{(1)}}(\omega )=1+{\frac {2}{\pi }}P\int _{0}^{{\infty }}{\frac {\epsilon _{{\alpha \beta }}^{{(2)}}(\omega ')\omega '}{\omega '^{2}-\omega ^{2}+i\eta }}d\omega '$$

Where $\epsilon _{{\alpha \beta }}^{{(1)}}(\omega )$ is the real part of the dielectric constant at energy, $\omega$. $P$ is the principle value, $\epsilon _{{\alpha \beta }}^{{(2)}}(\omega ')$ is the imaginary part of the dielectric constant and $\eta$ is a small complex shift. A larger $\eta$ results in a slight smoothing of the real dielectric function.

The dielectric constant can be used to calculate the optical absorption via the complex refractive index.

References

[1] M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, "Linear optical properties in the PAW methodology", Phys. Rev. B 73, 045112 (2006) DOI: 10.1103/PhysRevB.73.045112

Usage

The kkr.py contains the function to perform the Kramers-Kronig transformation.

The kkr function expects the dielectric function, eps_imag, to be provided as a nx3x3 numpy array, containing the dielectric tensor on a regular energy (photon frequency) grid.

The energy grid spacing, de, is also required.

A jupyter notebook has been provided, which gives a tutorial on how the script should be used.

Requirements

This script is currently compatible with Python 2.7 and Python 3.4. Numpy is required for matrix operations.

License