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Longwave

This page gives hints on how to compute spatial dispersion properties (e.g. flexoelectric tensor or dynamical quadrupoles) with the longwave DFPT driver of the ABINIT package.

Introduction

In condensed-matter physics, spatial dispersion refers to the dependence of many material properties on the wavevector q at which they are probed, or equivalently on the gradients of the external field (electric, magnetic, strain…) and/or the response in real space. Remarkable examples of such gradient effects include the natural optical rotation, [Belinicher1980] whereby some crystals rotate the polarization plane of the transmitted light, or the flexoelectric tensor, [Zubko2013] which describes the polarization response to a gradient of applied strain.

Since ABINIT v9.0.2, the calculation of several spatial dispersion quantities is accessible via the longwave driver. The implementation, detailed in [Romero2020], follows the formalism developed in [Royo2019] that adapts the classic longwave method of Born and Huang [Born1954] with the modern tools of the DFPT. Technically, the driver computes analytical third-order energy derivatives (readily converted to physical spatial dispersion quantities) with respect to three of the standard perturbations (atomic displacements, electric field and strain) and to the wavevector q.

At present (November 28, 2024 ), ABINIT enables the calculation of four spatial dispersion tensors required in order to build all the contributions to the bulk flexoelectric tensor following the prescriptions exposed in [Stengel2013] and [Royo2022]. These include, the clamped-ion flexoelectric tensor (a purely electronic contribution) and the first real-space moment of three other tensors (entering the mixed and lattice mediated contributions): the polarization response to an atomic displacement, the interatomic force constants and the piezoelectric force-response tensor. The implementation also provides access, as a by-product of the flexoelectric formalism, to the dynamical quadrupoles, which can be considered as the spatial-dispersion counterpart of the Born effective charges (dynamical dipoles). After execution, the longwave routines generate a third-order derivative database that is subsequently used by ANADDB either to compute and print the different contributions to the flexoelectric tensor, or to consider quadrupolar fields within the Fourier interpolation procedure of the dynamical matrix [Royo2020].

The calculation of another spatial-dispersion tensor has been recently implemented. This is the natural optical activity tensor, which is obtained from the third-order energy derivative with respect to two electric fields and to the wavevector q.

The underlying theory of the long-wave DFPT approach [Royo2019] [Royo2022] has been developed for its application on time-reversal symmetric insulating crystals only. Therefore, the usage of the longwave driver is restricted to materials of this kind. An extension of the theoretical framework and the ABINIT implementation to magnetic insulators and/or metals will be hopefully pursued in the future.

Regarding the flexoelectric tensor that ABINIT provides, a few remarks are in order. First, recall that this is the bulk flexoelectric tensor and that a surface counterpart is still missing in order to obtain the total flexoelectric tensor of a system.[Stengel2016] Even though the implementation can be applied to slabs or low-dimensional systems (such as 2D materials), via a supercell approach, the outcome of such a calculation will not directly produce the total (i.e., bulk+surface) flexoelectric response. The procedure to incorporate surface effects from a set of quantities that are available as byproducs of the longwave calculation has been described in Ref. [Springolo2021] for the spefic case of a flexural deformation of a 2D monolayer. The same approach can be directly applied to obtain the total flexoelectric response of a bent material slab.

The interested user must be likewise aware of the physical ambiguities existing in the definition of the bulk flexoelectric tensor which inherently affect the longwave driver. One of them precludes its usage to obtain the flexoelectric tensor of non-centrosymmetric (i.e., piezoelectric) materials (see section VII.c of [Stengel2013]). The other one is related with the dependence of the bulk flexoelectric coefficients on the choice of an arbitrary reference energy. The average electrostatic potential has been taken as the energy reference within the longwave driver. Nonetheless, as illustrated in section IV.d of [Stengel2016a], other choices might lead to quantitatively and qualitatively different outcomes. On the other hand, such ambiguity is well known to disappear when the surface-specific part is accounted for, as done e.g. in [Stengel2014].

The longwave implementation is still under heavy development. To date it requires the use of norm-conserving pseudopotentials without XC nonlinear core corrections and it can be used with LDA and GGA XC functionals. The use of spherical harmonics for the nonlocal projectors is mandatory through the option useylm=1, although useylm=0 can be exceptionally used in the calculation of the natural optical activity tensor lw_natopt=1.

Since ABINIT v9.x the longwave driver has been thoroughly modified. These changes, while being mostly devoted at optimizing the internal structure of the driver, do not entail big differences to the end user in terms of execution when compared with the previous version. Nonetheless, it is strongly recommended to take a look at the input files for the longwave tests indicated below. Particularly, in what refers to the new usage of the variables rf2_dkdk, prepalw and rfstrs_ref.

The following steps are required to perform a longwave DFPT calculation of the bulk flexoelectric tensor (see, e.g., tests lw[1] to lw[3]):

  • Perform ground-state calculation.
  • Perform ddk and d2_dkdk (rf2_dkdk=3 is mandatory) response function calculations.
  • Perform response function calculations at q =Γ to atomic displacements, electric field and strain perturbations, including the option prepalw=1 and rfstrs_ref=1.
  • Perform a longwave DFTP calculation of third-order energy derivatives (optdriver=10 and lw_flexo=1).
  • Use MRGDDB to merge 1st, 2nd and 3rd order DDB files.
  • Run ANADDB with flexoflag=1.

The following steps are required to perform a phonon dispersion calculation including quadrupolar fields in the nonanalytical part of the dynamical matrix (see, e.g., tests lw[4] to lw[6]):

  • Perform ground-state calculation.
  • Perform ddk and d2_dkdk (rf2_dkdk=3 is mandatory) response function calculations.
  • Perform response function calculations at q =Γ to atomic displacements and electric field, including the option prepalw=2.
  • Perform a longwave DFTP calculation of third-order energy derivatives (optdriver=10 and lw_qdrpl=1).
  • Perform response function calculations to atomic displacements at finite q (coarse grid).
  • Use MRGDDB to merge 2nd and 3rd order DDB files.
  • Run a phonon dispersion calculation of ANADDB including dipquad=1 and/or quadquad=1.

The following steps are required to perform a longwave DFPT calculation of the natural optical activity tensor (see, e.g., test lw[8]):

  • Perform ground-state calculation.
  • Perform ddk and d2_dkdk (rf2_dkdk=3 is mandatory) response function calculations.
  • Perform response function calculations to electric field perturbation including the option prepalw=4.
  • Perform a longwave DFTP calculation of third-order energy derivatives (optdriver=10 and lw_natopt=1).

compulsory:

  • lw_flexo LongWave calculation of FLEXOelectricity related spatial dispersion tensors
  • lw_natopt LongWave calculation of NATural OPTical activity tensor
  • lw_qdrpl LongWave calculation of dynamical QuaDRuPoLes tensor
  • prepalw PREPAre LongWave calculation

basic:

  • dipquad DIPole-QUADdrupole interaction
  • dipquad DIPole-QUADdrupole interaction
  • flexoflag FLEXOelectric tensor FLAG
  • quadquad QUADdrupole-QUADdrupole interaction
  • quadquad QUADdrupole-QUADdrupole interaction

useful:

  • rfstrs_ref Response Function with respect to STRainS with the energy REFerence at the average electrostatic potential

expert:

  • ffnl_lw NonLocal Form Factors in LongWave calculation

Selected Input Files

tutorespfn:

v9:

Tutorials

A tutorial is in preparation.