# Nonlinear

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

## 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 (valid when the phonon frequency is much smaller than the gap, and also, as a consequence, features smaller than the largest phonon frequency are not 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.
A first formalism (PEAD) is described in [Veithen2005].
Thanks to the 2n+1 theorem, this formulation only requires the knowledge of
the ground-state and first-order changes in the wavefunctions,
but some quantity is evaluated thanks to a finite-difference in k-space.
It is implemented only for NC pseudopotentials, within LDA.
There is another formalism, based on 2^{nd}-order Sternheimer equation,
recently made available (L. Baguet, to be published).
The latter is implemented both for NC pseudopotentials and for PAW, again only for LDA.
Both work for nsppol=1 or 2.

The PEAD 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].

## Related Input Variables¶

*compulsory:*

- nlflag Non-Linear FLAG
- optdriver OPTions for the DRIVER
- prepanl PREPAre Non-Linear response calculation

*basic:*

- alphon ALign PHONon mode eigendisplacements
- d3e_pert1_atpol 3
^{rd}Derivative of Energy, mixed PERTurbation 1: limits of ATomic POLarisations - d3e_pert1_dir 3
^{rd}Derivative of Energy, mixed PERTurbation 1: DIRections - d3e_pert1_elfd 3
^{rd}Derivative of Energy, mixed PERTurbation 1: ELectric FielD - d3e_pert1_phon 3
^{rd}Derivative of Energy, mixed PERTurbation 1: PHONons - d3e_pert2_atpol 3
^{rd}Derivative of Energy, mixed PERTurbation 2: limits of ATomic POLarisations - d3e_pert2_dir 3
^{rd}Derivative of Energy, mixed PERTurbation 2: DIRections - d3e_pert2_elfd 3
^{rd}Derivative of Energy, mixed PERTurbation 2: ELectric FielD - d3e_pert2_phon 3
^{rd}Derivative of Energy, mixed PERTurbation 2: PHONons - d3e_pert3_atpol 3
^{rd}Derivative of Energy, mixed PERTurbation 3: limits of ATomic POLarisations - d3e_pert3_dir 3
^{rd}Derivative of Energy, mixed PERTurbation 3: DIRections - d3e_pert3_elfd 3
^{rd}Derivative of Energy, mixed PERTurbation 3: ELectric FielD - d3e_pert3_phon 3
^{rd}Derivative of Energy, mixed PERTurbation 3: PHONons - ramansr RAMAN Sum-Rule
- usepead USE of PEAD formalism

*useful:*

- get1den GET the first-order density from _1DEN file
- ird1den Integer that governs the ReaDing of 1
^{st}-order DEN file - prtmbm PRinT Mode-By-Mode decomposition of the electrooptic tensor

*expert:*

- nonlinear_info Output NON-LINEAR INFOrmation

## Selected Input Files¶

*paral:*

*v3:*

*v4:*

*v6:*

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

*v8:*

## Tutorials¶

The tutorial 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.