# Raman intensities and electro-optic properties calculation

### This page gives hints on how to compute Raman intensity, and the related electro-optic coefficients with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (RC, XG)

Mentioned in topic_DFPT, topic_Optic, help_features#6.

### Table of content:

- 1. Introduction.
- 2. Related lesson(s) of the tutorial.
- 3. Related input variables.
- 4. Selected input files.
- 5. References.

### 1. Introduction.

In Raman experiments, the incident light, usually a polarized or unpolarized laser, is scattered by the sample, and the energy as well as polarization of the outgoing light is measured. A Raman spectrum, presenting the energy of the outgoing photons, will consist of rather well-defined peaks, around an elastic peak.

At the lowest order of the theory, the dominant mechanism is the absorption or emission of a phonon by a photon. The energy of the absorbed or emitted phonon corresponds to the energy difference between the outgoing and incident photons. Thus, even more straightforwardly than the IR spectrum, a Raman spectrum is directly related to the energy of phonons at the Brillouin-zone center: when the zero of the energy scale is set at the incident light energy, the absolute value of the energy of the peaks corresponds to the energy of the phonons.

The computation of phonon energies is presented in topic_Phonons. Raman intensities due to one-phonon emission or absorption are not linked to second-order derivatives of the total energy, but, within the adiabatic approximation, to derivative of the dielectric phonon with respect to atomic displacements. Moreover, when the frequency of the incident light (usually in the 1.5 eV to 2.5 eV range) is small with respect to the band gap (e.g. for gaps larger than 4 eV), the static approximation can be made, in which the Raman intensity will be linked to the third-order derivative of the total energy with respect (twice) to an homogeneous electric field and (once) with respect to atomic displacements. Thus, DFPT can be used, see below. For the case in which the incident light frequency is not negligible with respect to the gap, the DFPT cannot be used, but, if the adiabatic approximation can be used (the phonon frequency much smaller than the gap, and also features smaller than the largest phonon frequency cannot be resolved in the Raman spectrum), one can compute the Raman intensities thanks to finite differences of dielectric function, see [Gillet2013]. For the two-phonon Raman spectrum, see [Gillet2017].

Both the derivatives of the linear electronic dielectric susceptibilities with respect to atomic displacements and the non-linear electronic dielectric susceptibilities required to evaluate the Raman intensities are thus non-linear responses. In the ABINIT implementation, they are computed within the density functional perturbation theory, as described in [Veithen2005]. Thanks to the 2n+1 theorem, their formulation only requires the knowledge of the ground-state and first-order changes in the wavefunctions.

This non-linear response formalism has been successfully applied to a large variety of systems. We have so far studied the Raman spectra of ferroelectric oxides ( BaTiO3 and PbTiO3 [Hermet2009]), different minerals under pressure conditions characteristic to the interior of the Earth [Caracas2007a] or molecular solids under extreme conditions [Caracas2008]. The computation of the non-linear optical susceptibilities has also been applied to several polar dielectrics [Caracas2007].

As a by-product of the calculation of the Raman tensor and non-linear optical coefficients, it is also possible to determine directly within ABINIT the electro-optic (EO) coefficients rijγ (Pockels effect) which describe the change of optical dielectric tensor in a (quasi-)static electric field through the following expression [Veithen2005]: Δ(ε-1)ij=∑γ=1,3 rijγΕγ

The clamped (zero strain) EO coefficients include an electronic and an ionic contribution directly accessible within ABINIT. The unclamped EO coefficients include an additional piezoelectric contribution which must be computed separately from the knowledge of the elasto-optic and piezoelectric strain coefficients. This formalism was for instance applied to different ferroelectric ABO3 compounds [Veithen2005a].

The implementation is available for norm-conserving pseudopotentials, in the LDA approximation. The extension to the PAW framework is in progress.
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### 2. Related lesson(s) of the tutorial.

The lesson on static non-linear properties presents the computation of responses beyond the linear order, within Density-Functional Perturbation Theory (beyond the simple Sum-Over-State approximation): Raman scattering efficiencies (non-resonant case), non-linear electronic susceptibility, electro-optic effect. Comparison with the finite field technique (combining the computation of linear response functions with finite difference calculations), is also provided.
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### 3. Related input variables.

Compulsory input variables:

... nlflag@anaddb [Non-Linear FLAG]
... optdriver [OPTions for the DRIVER]

Basic input variables:

... alphon@anaddb [ALign PHONon mode eigendisplacements]
... d3e_pert1_atpol [3rd Derivative of Energy, mixed PERTurbation 1: limits of ATomic POLarisations]
... d3e_pert1_dir [3rd Derivative of Energy, mixed PERTurbation 1: DIRections]
... d3e_pert1_elfd [3rd Derivative of Energy, mixed PERTurbation 1: ELectric FielD]
... d3e_pert1_phon [3rd Derivative of Energy, mixed PERTurbation 1: PHONons]
... d3e_pert2_atpol [3rd Derivative of Energy, mixed PERTurbation 2: limits of ATomic POLarisations]
... d3e_pert2_dir [3rd Derivative of Energy, mixed PERTurbation 2: DIRections]
... d3e_pert2_elfd [3rd Derivative of Energy, mixed PERTurbation 2: ELectric FielD]
... d3e_pert2_phon [3rd Derivative of Energy, mixed PERTurbation 2: PHONons]
... d3e_pert3_atpol [3rd Derivative of Energy, mixed PERTurbation 3: limits of ATomic POLarisations]
... d3e_pert3_dir [3rd Derivative of Energy, mixed PERTurbation 3: DIRections]
... d3e_pert3_elfd [3rd Derivative of Energy, mixed PERTurbation 3: ELectric FielD]
... d3e_pert3_phon [3rd Derivative of Energy, mixed PERTurbation 3: PHONons]
... prepanl [PREPAre Non-Linear response calculation]
... ramansr@anaddb [RAMAN Sum-Rule]

Useful input variables:

... get1den [GET the first-order density from _1DEN file]
... ird1den [Integer that governs the ReaDing of 1st-order DEN file]
... prtmbm@anaddb [PRinT Mode-By-Mode decomposition of the electrooptic tensor]
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### 4. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v8/Input: t49.in t50.in

tests/v3/Input: t83.in

tests/v4/Input: t52.in t54.in

tests/v6/Input: t63.in t64.in t65.in t66.in t67.in

tests/paral/Input: t54.in
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### 5. References.

[Caracas2007] R. Caracas and R.E. Cohen, "Prediction of polar ordered oxynitride perovskites", J. Appl. Phys. Lett. 91, 092902 (2007).
DOI: 10.1063/1.2776370.
[Caracas2007a] R. Caracas and R.E. Cohen, "Post-perovskite phase in selected sesquioxides from density-functional calculations", Phys. Rev. B 76, 184101 (2007).
[Caracas2008] R. Caracas, "Dynamical Instabilities of Ice X", Phys. Rev. Lett. 101, 085502 (2008).
DOI: 10.1103/PhysRevLett.101.085502.
[Gillet2013] Y. Gillet, M. Giantomassi and X. Gonze, "First-principles study of excitonic effects in Raman intensities", Phys. Rev. B 88, 094305 (2013).
DOI: 10.1103/PhysRevB.88.094305.
[Gillet2017] Yannick Gillet, Stefan Kontur, Matteo Giantomassi, Claudia Draxl and Xavier Gonze, "Ab Initio Approach to Second-order Resonant Raman Scattering Including Exciton-Phonon Interaction", Scientific Reports 7, 7344 (2017).
[Hermet2009] P. Hermet, M. Veithen and P. Ghosez, "Raman scattering intensities in BaTiO(3) and PbTiO(3) prototypical ferroelectrics from density functional theory.", J. Phys. : Condens. Matter 21, 215901 (2009).
DOI: 10.1088/0953-8984/21/21/215901.
[Veithen2005] M. Veithen, X. Gonze and Ph. Ghosez, "Nonlinear optical susceptibilities, Raman efficiencies, and electrooptic tensors from first–principles density functional theory", Phys. Rev. B 71, 125107 (2005).
DOI: 10.1103/PhysRevB.71.125107.
[Veithen2005a] M. Veithen and Ph. Ghosez, "Temperature dependence of the electro-optic tensor and refractive indices of Ba Ti O 3 from first principles", Phys. Rev. B 71, 132101 (2005).
DOI: 10.1103/PhysRevB.71.132101.

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