mirror of https://github.com/abinit/abinit.git
119 lines
8.1 KiB
Markdown
119 lines
8.1 KiB
Markdown
---
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description: How to compute spatial dispersion properties with the longwave DFPT approach.
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authors: MR and MS
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---
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<!--- This is the source file for this topics. Can be edited. -->
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This page gives hints on how to compute spatial dispersion properties
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(e.g. flexoelectric tensor or dynamical quadrupoles) with the longwave DFPT
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driver of the ABINIT package.
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## Introduction
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In condensed-matter physics, spatial dispersion refers to the dependence of many material
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properties on the wavevector **q** at which they are probed, or equivalently on the gradients of the
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external field (electric, magnetic, strain...) and/or the response in real space. Remarkable
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examples of such gradient effects include the natural optical rotation, [[cite:Belinicher1980]] whereby some crystals
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rotate the polarization plane of the transmitted light, or the flexoelectric tensor, [[cite:Zubko2013]] which
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describes the polarization response to a gradient of applied strain.
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Since ABINIT v9.0.2, the calculation of several spatial dispersion quantities is accessible
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via the longwave driver. The implementation, detailed in [[cite:Romero2020]], follows the formalism developed in
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[[cite:Royo2019]] that adapts the classic longwave method of Born and Huang [[cite:Born1954]] with the modern tools of
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the DFPT. Technically, the driver computes analytical third-order energy derivatives (readily converted to physical
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spatial dispersion quantities) with respect to three of the *standard*
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perturbations (atomic displacements, electric field and strain) and to the wavevector **q**.
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At present ( |today| ), ABINIT enables the calculation of four spatial dispersion tensors required
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in order to build all the contributions to the bulk flexoelectric tensor following the prescriptions exposed in [[cite:Stengel2013]] and [[cite:Royo2022]].
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These include, the clamped-ion flexoelectric tensor (a purely electronic contribution) and the first real-space moment of three other
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tensors (entering the mixed and lattice mediated contributions): the polarization
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response to an atomic displacement, the interatomic force constants and the piezoelectric force-response
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tensor. The implementation also provides access, as a by-product of the flexoelectric formalism, to the dynamical
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quadrupoles, which can be considered as the spatial-dispersion counterpart of the Born effective
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charges (dynamical dipoles). After execution, the longwave routines generate a third-order derivative
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database that is subsequently used by ANADDB either to compute and print the different contributions
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to the flexoelectric tensor, or to consider quadrupolar fields within the Fourier interpolation procedure of
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the dynamical matrix [[cite:Royo2020]].
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The calculation of another spatial-dispersion tensor has been recently implemented. This is the natural optical activity
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tensor, which is obtained from the third-order energy derivative with respect to two electric fields and to the wavevector **q**.
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The underlying theory of the long-wave DFPT approach [[cite:Royo2019]] [[cite:Royo2022]] has been developed for its application on
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time-reversal symmetric insulating crystals only. Therefore, the usage of the longwave driver is restricted to materials
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of this kind. An extension of the theoretical framework and the ABINIT implementation to magnetic insulators
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and/or metals will be hopefully pursued in the future.
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Regarding the flexoelectric tensor that ABINIT provides, a few remarks are in order. First, recall that
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this is the **bulk** flexoelectric tensor and that a surface counterpart is still missing in order to obtain
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the total flexoelectric tensor of a system.[[cite:Stengel2016]] Even though the implementation can be applied
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to slabs or low-dimensional systems (such as 2D materials), via a supercell approach, the outcome of such a
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calculation will not directly produce the total (i.e., bulk+surface) flexoelectric response. The procedure to
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incorporate surface effects from a set of quantities that are available as byproducs of the longwave calculation
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has been described in Ref. [[cite:Springolo2021]] for the spefic case of a flexural deformation of a 2D monolayer.
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The same approach can be directly applied to obtain the total flexoelectric response of a bent material slab.
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The interested user must be likewise aware of the physical ambiguities existing in the definition of the bulk
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flexoelectric tensor which inherently affect the longwave driver. One of them precludes its usage to obtain the
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flexoelectric tensor of non-centrosymmetric (i.e., piezoelectric) materials (see section VII.c of [[cite:Stengel2013]]).
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The other one is related with the dependence of the bulk flexoelectric coefficients on the choice of an arbitrary
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reference energy. The average electrostatic potential has been taken as the energy reference
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within the longwave driver. Nonetheless, as illustrated in section IV.d of [[cite:Stengel2016a]], other choices might
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lead to quantitatively and qualitatively different outcomes. On the other hand, such ambiguity is well known to disappear
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when the surface-specific part is accounted for, as done e.g. in [[cite:Stengel2014]].
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The longwave implementation is still under heavy development. To date it requires the use of norm-conserving
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pseudopotentials without XC nonlinear core corrections and it can be used with LDA and GGA XC functionals.
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The use of spherical harmonics for the nonlocal projectors is mandatory through the option [[useylm]]=1,
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although [[useylm]]=0 can be exceptionally used in the calculation of the natural optical activity tensor [[lw_natopt]]=1.
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Since ABINIT v9.x the longwave driver has been thoroughly modified. These changes, while being mostly devoted
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at optimizing the internal structure of the driver, do not entail big differences to the end user in terms of
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execution when compared with the previous version. Nonetheless, it is strongly recommended to take a look at the input
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files for the longwave tests indicated below. Particularly, in what refers to the new usage of the variables
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[[rf2_dkdk]], [[prepalw]] and [[rfstrs_ref]].
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The following steps are required to perform a longwave DFPT calculation of the bulk flexoelectric tensor
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(see, e.g., tests [[test:lw_1]] to [[test:lw_3]]):
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* Perform ground-state calculation.
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* Perform ddk and d2_dkdk ([[rf2_dkdk]]=3 is mandatory) response function calculations.
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* Perform response function calculations at **q** =Γ to atomic displacements, electric field and strain perturbations,
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including the option [[prepalw]]=1 and [[rfstrs_ref]]=1.
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* Perform a longwave DFTP calculation of third-order energy derivatives ([[optdriver]]=10 and [[lw_flexo]]=1).
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* Use MRGDDB to merge 1st, 2nd and 3rd order DDB files.
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* Run ANADDB with [[anaddb:flexoflag]]=1.
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The following steps are required to perform a phonon dispersion calculation including quadrupolar fields in the
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nonanalytical part of the dynamical matrix (see, e.g., tests [[test:lw_4]] to [[test:lw_6]]):
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* Perform ground-state calculation.
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* Perform ddk and d2_dkdk ([[rf2_dkdk]]=3 is mandatory) response function calculations.
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* Perform response function calculations at **q** =Γ to atomic displacements and electric field,
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including the option [[prepalw]]=2.
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* Perform a longwave DFTP calculation of third-order energy derivatives ([[optdriver]]=10 and [[lw_qdrpl]]=1).
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* Perform response function calculations to atomic displacements at finite **q** (coarse grid).
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* Use MRGDDB to merge 2nd and 3rd order DDB files.
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* Run a phonon dispersion calculation of ANADDB including [[anaddb:dipquad]]=1 and/or [[anaddb:quadquad]]=1.
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The following steps are required to perform a longwave DFPT calculation of the natural optical activity tensor
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(see, e.g., test [[test:lw_8]]):
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* Perform ground-state calculation.
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* Perform ddk and d2_dkdk ([[rf2_dkdk]]=3 is mandatory) response function calculations.
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* Perform response function calculations to electric field perturbation including the option [[prepalw]]=4.
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* Perform a longwave DFTP calculation of third-order energy derivatives ([[optdriver]]=10 and [[lw_natopt]]=1).
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## Related Input Variables
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{{ related_variables }}
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## Selected Input Files
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{{ selected_input_files }}
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## Tutorials
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A tutorial is in preparation.
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