mirror of https://github.com/abinit/abinit.git
374 lines
17 KiB
Markdown
374 lines
17 KiB
Markdown
---
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authors: SS, XG, YG, NAP, FG
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---
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# Tutorial on optical properties
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## Frequency-dependent linear and second-order nonlinear optical response.
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This tutorial aims at showing how to get the following physical properties, for semiconductors:
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* Frequency-dependent linear dielectric tensor
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* Frequency-dependent second-order nonlinear susceptibility tensor
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* Frequency-dependent electro-optical susceptibility tensor
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in the simple *Random-Phase Approximation* or *Sum-over-states* approach.
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This tutorial will help you to understand and make use of *optic*.
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Before starting, you should first have some theoretical background.
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We strongly suggest that you first read the first two sections of the [[help:optic]].
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This tutorial should take about 1 hour.
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[TUTORIAL_README]
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## Computing the momentum matrix elements
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*Before beginning, you might consider working in a different subdirectory.
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Why not create Work_optic?*
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We also need to copy *toptic_1.abi* from
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*$ABI_TESTS/tutorial/Input* to *Work_optic*.
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```sh
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cd $ABI_TESTS/tutorespfn/Input
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mkdir Work_optic
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cd Work_optic
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cp ../toptic_1.abi .
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```
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Now, you are ready to run Abinit and prepare the files needed for Optic.
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Issue:
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abinit toptic_1.abi > log 2> err
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We now examine the input file.
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{% dialog tests/tutorespfn/Input/toptic_1.abi %}
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The computation concerns a crystal of GaAs, in the zinc-blende structure (2 atoms per primitive cell).
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It has six datasets.
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The first dataset is a quite standard self-consistent determination of the ground state for a fixed geometry.
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Only the occupied bands are treated. The density is output and used in later datasets.
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The second dataset is a non-self-consistent calculation, where the number of bands has been increased to include unoccupied states.
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The k points are restricted to the Irreducible Brillouin Zone.
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The third dataset uses the result of the second one to produce the
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wavefunctions for all the bands, for the full Brillouin Zone
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(this step could be skipped for linear optic, but is included for later CPU time saving - for nonlinear optic, checks of symmetry behaviour are ongoing currently as of 23 June 2024, better to perform set three).
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If only the linear optical response is computed, then time-reversal symmetry can be used, and the computation
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might be restricted to the half Brillouin zone ([[kptopt]]=2).
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The fourth, fifth, and sixth datasets correspond to the computation of the *ddk* matrix elements,
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that is, matrix elements of the $\partial H / \partial k$ operators where $H$ is the Hamiltonian.
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Note that the number of bands is the same as for datasets 2 and 3.
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Note also that these are non-self-consistent calculations, moreover, restricted to [[nstep]] = 1 and [[nline]] = 0.
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Indeed, only the matrix elements between explicitly computed (unperturbed) states are required.
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This also is why [[prtwf]]=3 is used.
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Using a larger [[nstep]] would lead to a full computation of the derivative of the wavefunction with respect to
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the wavevector, while in Optic, only the matrix elements between unperturbed states are needed.
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Thus a value of [[nstep]] larger than one would be a waste of time.
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A useful alternative is to use [[wfk_task]]="wfk_optics_fullbz" after the calculation of the WFK file for
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just the irreducible Brillouin zone: this postprocessing step calculates the matrix elements needed for optic,
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and is much more memory efficient.
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In order to have a sufficiently fast tutorial, the k point sampling was chosen to be extremely dense.
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Instead of a $4\times 4\times 4$
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FCC lattice (256 k points), it should be something like $28\times 28\times 28$ FCC (about 100000 k points).
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Also, the cut-off energy (2 Ha) is too small. As usual, convergence studies are the responsibility of the user.
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Moreover, we emphasize that in general the results of a sum-over-states approach, as is used in Optic,
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typically converges quite slowly with the k point mesh. Thus it is of uttermost importance to
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test convergence carefully.
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The run takes less than one minute on a 2.8 GHz PC. The files *toptic_1o_DS3_WFK*,
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*toptic_1o_DS4_1WF7*, *toptic_1o_DS5_1WF8* and *toptic_1o_DS6_1WF9* are the four
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files requested for the Optic run.
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The first file contains the wavefunctions for the filled and empty states in the entire
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Brillouin zone, while the latter three contain the matrix elements of the
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$\partial/\partial k$ operators, one file for each Cartesian direction.
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Real preparation runs (with adequate k point sampling and cut-off energy) can
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last several hours (or even days) on such a PC.
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## Computing the linear and nonlinear optical response
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The next step is to compute the linear and nonlinear optical response: once the
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momentum matrix elements are available, you are ready to determine the optical
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response (up to second order in the current implementation) for the material under study.
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First, read the [[help:optic#input|section 3]] of the Optic help file.
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Copy the *toptic_2.abi* input file from *$ABI_TESTS/tutorial/Input* to *Work_optic*:
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```sh
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cp ../toptic_2.abi .
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```
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The *toptic_2.abi* is your input file. You should edit it and read it carefully. For
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help on various input parameters in this file, please see the [[help:optic]].
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{% dialog tests/tutorespfn/Input/toptic_2.abi %}
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When you have read the input file, you can run the code, as usual, using the
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following command (assuming optic is in $PATH - copy the
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executable in the current directory if needed):
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optic toptic_2.abi > log 2> err &
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It will take a few seconds to run. You have produced numerous output files.
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Now, you can examine some of these output files.
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The headers contains information about the calculation.
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See the [[help:optic#output|section 4]] of the Optic help file.
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These files can be plotted in |xmgrace| or |gnuplot|.
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If you do not have xmgrace installed on your computer, please get it from the Web, and install it,
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or alternatively, use your preferred plotting package.
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We will first have a look at the linear optic file.
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xmgrace toptic_2_0001_0001-linopt.out
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This file contains the *xx* component of the dielectric tensor, and includes,
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as a function of energy, the magnitude, real, and imaginary parts of the tensor element.
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On the graph, you should see three curves. One of them is positive, and always
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larger than the two others. It is the modulus of the dielectric function.
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Another one is also always positive, it is the imaginary part of the
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dielectric function. The last one is the real part.
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There are a large number of peaks. This is at variance with the experimental
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spectra, which are much smoother. The origin of this discrepancy is to be found
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in the very sparse k point sampling that we used in order to be able to perform
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the tutorial quickly.
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In the next section, we will improve this sampling, and start a convergence study.
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Concerning the non-linear optics, the graphs for the *xyz* components are also
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quite bad, with many isolated (but broadened) peaks. However, the *yyy* ones are
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*perfect*. Indeed, they vanish due to symmetry reasons!
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Visualize the imaginary part with:
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xmgrace toptic_2_0002_0002_0002-ChiTotIm.out
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and the Real part with:
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xmgrace toptic_2_0002_0002_0002-ChiTotRe.out
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!!! tip
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If |AbiPy| is installed on your machine, you can use the |abiopen| script
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with the `--expose` option to visualize the results stored in the OPTIC.nc file:
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abiopen.py toptic_2_OPTIC.nc --expose -sns=paper
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This would be a good time to review [[help:optic#troubleshooting|section 5]] of the optic help file.
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For comparison, we have included in the tutorial directory of the ABINIT package (ask your administrator to have access
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to the full ABINIT package if you do not have your own installation), three files that have been
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obtained with a much better k point sampling (still with a low cut-off energy
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and a number of bands that should be larger). You can visualize them as follows:
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xmgrace $ABI_HOME/doc/tutorial/optic_assets/toptic_ref_0001_0001-linopt.out
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for the linear optics, obtained with a 28x28x28 grid (keeping everything else fixed), and
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xmgrace $ABI_HOME/doc/tutorial/optic_assets/toptic_ref_0001_0002_0003-ChiTotIm.out
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as well as
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xmgrace $ABI_HOME/doc/tutorial/optic_assets/toptic_ref_0001_0002_0003-ChiTotRe.out
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for the non-linear optics, obtained with a 18x18x18 grid (keeping everything else fixed).
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Concerning the linear spectrum, we will now compare this (underconverged)
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result *toptic_ref_0001_0001-linopt.out* with experimental data and converged
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theoretical results.
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The book by Cohen M.L. and Chelikowsky [[cite:Cohen1988]] presents a
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comparison of experimental data with the empirical pseudopotential method
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spectrum. If you do not have access to this book, you can see an experimental
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spectrum in [[cite:Philipp1963]],
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and a theoretical spectrum in [[cite:Huang1993]], as well as other sources.
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We discuss first the imaginary spectrum. Prominent experimental features of
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this spectrum are two peaks located close to 3 eV and 5 eV, both with the same
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approximate height. The spectrum is zero below about 1.5 eV (the direct band
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gap), and decreases with some wiggles beyond 5.5 eV. Converged theoretical
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spectra also show two peaks at about the same location, although their heights
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are markedly different: about 10 for the first one (at 3 eV), and 25 for the
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second one (at 5 eV). Other features are rather similar to the experimental
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ones. In the linear optic spectrum of *toptic_ref_0001_0001-linopt.out*, we note
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that there is a shoulder at around 3 eV, and a peak at 4.2 eV, with respective
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approximate heights of 7 and 25. Some comments are in order:
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* The main difference between experimental and converged theoretical spectra is due
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to the presence of excitons (electron-hole bound states), not treated at all in
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this rather elementary theoretical approach: excitons transfer some oscillator strength
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from the second peak (at 5 eV) to the first one (at 3 eV). Going beyond the Sum-Over-State approach,
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but still keeping the independent-electron approximation, e.g., in the framework of the TDDFT (adiabatic LDA)
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will not correct this problem. One needs to use the Bethe-Salpeter approximation, or to rely on
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*fancy* exchange-correlation kernels, to produce an optical spectrum in qualitative agreement with the experimental data.
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Still, trends should be correct (e.g. change of the peak positions with pressure,
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comparison between different semiconductors, etc.).
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* In many early theoretical spectra (including the ones in [[cite:Cohen1988]]),
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the agreement between the theoretical and experimental band gap is artificially good.
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In straight DFT, one cannot avoid the band gap problem. However, it is possible to
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add an artificial "[[scissor@optic|scissor shift]]", to make the theoretical band gap match the experimental one.
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* Our theoretical spectrum presents additional deficiencies with respect to the other ones,
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mostly due to a still too coarse sampling of the k space (there are too many wiggles in the spectrum),
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and to a rather inaccurate band structure (the cut-off energy was really very low, so that
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the first peak only appears as a shoulder to the second peak).
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The real part of the spectrum is related by the Kramers-Kronig relation to the imaginary part.
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We note that the deficiencies of the imaginary part of the
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spectrum translate to the real part: the first peak is too low, and the
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second peak too high, while the spectrum correctly changes sign around 5 eV,
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and stays negative below 8 eV.
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In our simulation, more empty states are needed
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to obtain a better behaviour. Also, the limiting low-frequency value is only
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4.3, while it should be on the order of 10. This can be corrected by
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increasing the cut-off energy, the k point sampling and the number of unoccupied states.
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Similar considerations apply to the non-linear spectra.
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## Faster computation of the imaginary part of the linear optical response
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In the case of the imaginary part of the linear optical response, there are
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several points that make the calculation easier:
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* The time-reversal symmetry can be used to decrease the number of k points
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by a factor of two (this is also true for the computation of the real spectrum);
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* The number of unoccupied bands can be reduced to the strict minimum needed
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to span the target range of frequencies.
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We will focus on the energy range from 0 eV to 8 eV, for which only 5 unoccupied bands are needed.
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Copy the input file *toptic_3.abi* in *Work_optic*:
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cp ../toptic_3.abi .
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{% dialog tests/tutorespfn/Input/toptic_3.abi %}
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Issue:
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abinit toptic_3.abi > log 2> err &
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Now, examine the file *toptic_3.abi*. There are two important changes with respect to the file *toptic_1.abi*:
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* the number of unoccupied bands has been reduced, so that the total number of bands is 9 instead of 20
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* when applicable, the value of [[kptopt]] 3 in our previous simulation has been changed to 2,
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in order to take advantage of the time-reversal symmetry
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When the run is finished (it is only 8 secs on a 2.8 GHz PC), you can process
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the WFK files and obtain the linear optic spectra.
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Copy the *toptic_4.abi* input file in *Work_optic*:
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cp ../toptic_4.abi .
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{% dialog tests/tutorespfn/Input/toptic_4.abi %}
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Examine *toptic_4.abi* file: only the linear optic spectra will be built.
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When you have read the input file, you can run the code, as usual using the following command
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optic toptic_4.abi > log 2> err &
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Then, you can visualize the files *toptic_2_0001_0001-linopt.out* and
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*toptic_4_0001_0001-linopt.out* using xmgrace and compare them. The spectra
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looks completely identical. However, a careful look at these files, by editing
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them, show that indeed, the imaginary part is very similar:
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#calculated the component: 1 1 of dielectric function
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#broadening: 0.000000E+00 2.000000E-03
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#scissors shift: 0.000000E+00
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#energy window: 3.897388E+01eV 1.432264E+00Ha
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# Energy(eV) Im(eps(w))
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8.163415E-03 1.463528E-04
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1.632683E-02 2.927094E-04
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2.449025E-02 4.390740E-04
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3.265366E-02 5.854504E-04
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...
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But the real parts differ slightly (this is seen at lines 1007 and beyond):
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# Energy(eV) Re(eps(w))
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8.163415E-03 6.623599E+00
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1.632683E-02 6.623632E+00
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2.449025E-02 6.623686E+00
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3.265366E-02 6.623763E+00
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...
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for *toptic_2_0001_0001-linopt.out* and
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# Energy(eV) Re(eps(w))
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8.163415E-03 6.518576E+00
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1.632683E-02 6.518608E+00
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2.449025E-02 6.518663E+00
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3.265366E-02 6.518740E+00
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for *toptic_4_0001_0001-linopt.out*.
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This small difference is due to the number
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of bands ([[nband]] 20 for *toptic_2_0001_0001-linopt.out* and [[nband]] 9 for *toptic_4_0001_0001-linopt.out*).
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Then, you can increase the number of k points, and watch the change in the
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imaginary part of the spectrum. There will be more and more peaks, until they
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merge, and start to form a smooth profile (still not completely smooth even with $28\times 28\times 28$).
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For your information, we give some timings of the corresponding Abinit run for a 2.8 GHz PC:
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k-point grid CPU time
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4 x 4 x 4 8 secs
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6 x 6 x 6 20 secs
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8 x 8 x 8 43 secs
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10 x 10 x 10 80 secs
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12 x 12 x 12 138 secs
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16 x 16 x 16 338 secs
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20 x 20 x 20 702 secs
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24 x 24 x 24 1335 secs
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28 x 28 x 28 2633 secs
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For grids on the order of $16\times 16\times 16$, the treatment by optics also takes several
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minutes, due to IO (30 minutes for the $28\times 28\times 28$ grid).
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You might note how the first peak slowly develop with increasing number of k
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points but nevertheless stays much smaller than the converged one, and
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even smaller than the experimental one.
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## Computing the linear electro-optical susceptibility
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Calculations of the linear electro-optical susceptibility follows the same inital calculations
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as those described in the first two sections of this tutorial.
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To calculate the coefficients of the linear electro-optical susceptibility
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one needs to modify the optic input file with two additional keywords.
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Copy the *toptic_5.abi* input file in *Work_optic*:
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cp ../toptic_5.abi .
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{% dialog tests/tutorespfn/Input/toptic_5.abi %}
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For *toptic_5.abi*, only the linear electro-optic susceptibility will be calculated.
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When you have read the input file, you can run the code, as usual using the following command
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optic toptic_5.abi > log 2> err &
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The calculation should run in a few seconds on a modern PC.
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The resulting calculation produces a number of files ending in *ChiEO* and are related to different parts of the linear electro-optical tensor:
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* *ChiEOAbs.out* gives the absolute value of the linear electro-optical susceptibility
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* *ChiEOIm.out* gives the imaginary components of the calculated linear electro-optical susceptibility
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* *ChiEORe.out* gives the real components of the calculated linear electro-optical susceptibility
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* *ChiEOTotIm.out* gives the total imaginary part of the calculated linear electro-optical susceptibility
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* *ChiEOTotRe.out* gives the total real part of the calculated linear electro-optical susceptibility
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Generally, the low energy (or frequency) range of the linear electro-optical susceptibility
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is linear and of experimental importance. Here, low energy means energies much less than the band gap energy.
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