# Tutorial on optical properties¶

## Frequency-dependent linear and second order nonlinear optical response.¶

This tutorial aims at showing how to get the following physical properties, for semiconductors:

• Frequency dependent linear dielectric tensor
• Frequency dependent second order nonlinear susceptibility tensor

in the simple Random-Phase Approximation or Sum-over-states approach. This tutorial will help you to understand and make use of optic. Before starting, you should first have some theoretical background. We strongly suggest that you first read the first two sections of the optic help file.

Important

All the necessary input files to run the examples can be found in the ~abinit/tests/ directory where ~abinit is the absolute path of the abinit top-level directory.

To execute the tutorials, you are supposed to create a working directory (Work*) and copy there the input files and the files file of the lesson.

The files file ending with _x (e.g. tbase1_x.files) must be edited every time you start to use a new input file. You will discover more about the files file in section 1.1 of the help file.

To make things easier, we suggest to define some handy environment variables by executing the following lines in the terminal:

export ABI_HOME=Replace_with_the_absolute_path_to_the_abinit_top_level_dir

export ABI_TESTS=$ABI_HOME/tests/ export ABI_TUTORIAL=$ABI_TESTS/tutorial/           # Files for base1-2-3-4, GW ...
export ABI_TUTORESPFN=$ABI_TESTS/tutorespfn/ # Files specific to DFPT tutorials. export ABI_TUTOPARAL=$ABI_TESTS/tutoparal/         # Tutorials about parallel version
export ABI_TUTOPLUGS=$ABI_TESTS/tutoplugs/ # Examples using external libraries. export ABI_PSPDIR=$ABI_TESTS/Psps_for_tests/       # Pseudos used in examples.

export PATH=$ABI_HOME/src/98_main/:$PATH


The examples in this tutorial will use these shell variables so that one can easily copy and paste the code snippets into the terminal (remember to set ABI_HOME first!)

The last line adds the directory containing the executables to your PATH so that one can invoke the code by simply typing abinit in the terminal instead of providing the absolute path.

Finally, to run the examples in parallel with e.g. 2 MPI processes, use mpirun (mpiexec) and the syntax:

mpirun -n 2 abinit < files_file > log 2> err


The standard output of the application is redirected to log while err collects the standard error (runtime error messages, if any, are written here).

This tutorial should take about 1 hour.

## Computing the momentum matrix elements¶

Before beginning, you might consider working in a different subdirectory. Why not create Work_optic?

We also need to copy toptic_1.files and toptic_1.in from $ABI_TUTORIAL/Input to Work_optic. cd$ABI_TUTORIAL/Input
mkdir Work_optic
cd Work_optic
cp ../toptic_1.files .
cp ../toptic_1.in .


Now, you are ready to run Abinit and prepare the files needed for Optic. Issue:

abinit < toptic_1.files > log 2> err


We now examine the files.

The computation concerns a crystal of GaAs, in the zinc-blende structure (2 atoms per primitive cell).
The toptic_1.files is a typical Abinit files file (nothing special). By contrast, it is worthwhile to take some time to examine the input file toptic_1.in. Examine it, it has six datasets.

The first dataset is a quite standard self-consistent determination of the ground state for a fixed geometry. Only the occupied bands are treated. The density is output. The second dataset is a non-self-consistent calculation, where the number of bands has been increased to include unoccupied states. The k points are restricted to the Irreducible Brillouin Zone.
The third dataset uses the result of the second one to produce the wavefunctions for all the bands, for the full Brillouin Zone (this step could be skipped, but is included for later CPU time saving).

The fourth to sixth datasets correspond to the computation of the ddk matrix elements, that is, matrix elements of the $\partial/\partial k$ operators. Note that the number of bands is the same as for datasets 2 and 3. Note also that these are non-self-consistent calculations, moreover, restricted to nstep = 1 and nline = 0. Indeed, only the matrix elements between explicitly computed states are required. Using a larger nstep would lead to a full computation of the derivative of the wavefunction with respect to the wavevector, while in Optic, only the matrix elements are needed. Thus a value of nstep larger than one would not lead to erroneous matrix elements, but would be a waste of time.

In order to have a sufficiently fast tutorial, the k point sampling was chosen to be extremely small. Instead of a $4\times 4\times 4$ FCC lattice (256 k points), it should be something like $28\times 28\times 28$ FCC (about 100000 k points). Also, the cut-off energy (2 Ha) is too small. As usual, convergence studies are the responsibility of the user. Moreover, we emphasize that in general the results of a sum-over-states approach, as is used in Optic, typically converges quite slowly with the k point mesh. Thus it is of extra importance to test convergence carefully.

The run takes less than one minute on a 2.8 GHz PC. The files toptic_1o_DS3_WFK, toptic_1o_DS4_1WF7, toptic_1o_DS5_1WF8 and toptic_1o_DS6_1WF9 are the four files requested for the Optic run. The first file contains the wavefunctions for the filled and empty states in the entire Brillouin zone, while the latter three contain the matrix elements of the $\partial/\partial k$ operators, one file for each Cartesian direction.

Real preparation runs (with adequate k point sampling and cut-off energy) can last several hours (or even days) on such a PC.

## Computing the linear and nonlinear optical response¶

The next step is to compute the linear and nonlinear optical response: once the momentum matrix elements are available, you are ready to determine the optical response (up to second order in the current implementation) for the material under study.

First, read the section 3 of the Optic help file.

Copy the files toptic_2.files and toptic_2.in from $ABI_TUTORIAL/Input to Work_optic: cp ../toptic_2.files . cp ../toptic_2.in .  The toptic_2.in is your input file. You should edit it and read it carefully. For help on various input parameters in this file, please see the optic help file. When you have read the input file, you can run the code, as usual, using the following command (assuming optic is in$PATH - copy the executable in the current directory if needed):

optic < toptic_2.files > log 2> err &


It will take a few seconds to run. You have produced numerous output files. Now, you can examine some of these output files.

The headers contains information about the calculation. See the section 4 of the Optic help file. These files can be plotted in xmgrace or gnuplot . If you do not have xmgrace installed on your computer, please get it from the Web, and install it, or alternatively, use your preferred plotting package.

We will first have a look at the linear optic file.

xmgrace toptic_2_0001_0001-linopt.out


This file contains the xx component of the dielectric tensor, and includes, as a function of energy, the magnitude, real, and imaginary parts of the tensor element. On the graph, you should see three curves. One of them is positive, and always larger than the two others. It is the modulus of the dielectric function. Another one is also always positive, it is the imaginary part of the dielectric function. The last one is the real part.
There are a large number of peaks. This is at variance with the experimental spectra, which are much smoother. The origin of this discrepancy is to be found in the very sparse k point sampling that we used in order to be able to perform the tutorial quickly. In the next section, we will improve this sampling, and start a convergence study.

Concerning the non-linear optics, the graphs for the xyz components are also quite bad, with many isolated (but broadened) peaks. However, the yyy ones are perfect. Indeed, they vanish due to symmetry reasons! Visualize the imaginary part with:

xmgrace toptic_2_0002_0002_0002-ChiTotIm.out


and the Real part with:

xmgrace toptic_2_0002_0002_0002-ChiTotRe.out


Tip

If AbiPy is installed on your machine, you can use the abiopen script with the --expose option to visualize the results stored in the OPTIC.nc file:

abiopen.py toptic_2_OPTIC.nc --expose -sns=paper


This would be a good time to review section 5 of the optic help file.

For comparison, we have included in the tutorial, three files that have been obtained with a much better k point sampling (still with a low cut-off energy and a number of bands that should be larger). You can visualize them as follows:

xmgrace $ABI_HOME/doc/tutorial/optic_assets/toptic_ref_0001_0001-linopt.out  for the linear optics, obtained with a 28x28x28 grid (keeping everything else fixed), and xmgrace$ABI_HOME/doc/tutorial/optic_assets/toptic_ref_0001_0002_0003-ChiTotIm.out


as well as

xmgrace \$ABI_HOME/doc/tutorial/optic_assets/toptic_ref_0001_0002_0003-ChiTotRe.out


for the non-linear optics, obtained with a 18x18x18 grid (keeping everything else fixed).

Concerning the linear spectrum, we will now compare this (underconverged) result toptic_ref_0001_0001-linopt.out with experimental data and converged theoretical results.

The book by Cohen M.L. and Chelikowsky [Cohen1988] presents a comparison of experimental data with the empirical pseudopotential method spectrum. If you do not have access to this book, you can see an experimental spectrum in [Philipp1963], and a theoretical spectrum in [Huang1993], as well as other sources.

We discuss first the imaginary spectrum. Prominent experimental features of this spectrum are two peaks located close to 3 eV and 5 eV, both with the same approximate height. The spectrum is zero below about 1.5 eV (the direct band gap), and decreases with some wiggles beyond 5.5 eV. Converged theoretical spectra also show two peaks at about the same location, although their heights are markedly different: about 10 for the first one (at 3 eV), and 25 for the second one (at 5 eV). Other features are rather similar to the experimental ones. In the linear optic spectrum of toptic_ref_0001_0001-linopt.out, we note that there is a shoulder at around 3 eV, and a peak at 4.2 eV, with respective approximate heights of 7 and 25. Some comments are in order:

• The main difference between experimental and converged theoretical spectra is due to the presence of excitons (electron-hole bound states), not treated at all in this rather elementary theoretical approach: excitons transfer some oscillator strength from the second peak (at 5 eV) to the first one (at 3 eV). Going beyond the Sum-Over-State approach, but still keeping the independent-electron approximation, e.g., in the framework of the TDDFT (adiabatic LDA) will not correct this problem. One needs to use the Bethe-Salpeter approximation, or to rely on fancy exchange-correlation kernels, to produce an optical spectrum in qualitative agreement with the experimental data. Still, trends should be correct (e.g. change of the peak positions with pressure, comparison between different semiconductors, etc.).

• In many early theoretical spectra (including the ones in [Cohen1988]), the agreement between the theoretical and experimental band gap is artificially good. In straight DFT, one cannot avoid the band gap problem. However, it is possible to add an artificial “scissor shift”, to make the theoretical band gap match the experimental one.

• Our theoretical spectrum presents additional deficiencies with respect to the other ones, mostly due to a still too coarse sampling of the k space (there are too many wiggles in the spectrum), and to a rather inaccurate band structure (the cut-off energy was really very low, so that the first peak only appears as a shoulder to the second peak).

The real part of the spectrum is related by the Kramers-Kronig relation to the imaginary part. We note that the deficiencies of the imaginary part of the spectrum translate to the real part: the first peak is too low, and the second peak too high, while the spectrum correctly changes sign around 5 eV, and stays negative below 8 eV.

In our simulation, more empty states are needed to obtain a better behaviour. Also, the limiting low-frequency value is only 4.3, while it should be on the order of 10. This can be corrected by increasing the cut-off energy, the k point sampling and the number of unoccupied states.

Similar considerations apply to the non-linear spectra.

## Faster computation of the imaginary part of the linear optical response¶

In the case of the imaginary part of the linear optical response, there are several points that make the calculation easier:

• The time-reversal symmetry can be used to decrease the number of k points by a factor of two (this is also true for the computation of the real spectrum);

• The number of unoccupied bands can be reduced to the strict minimum needed to span the target range of frequencies.

We will focus on the energy range from 0 eV to 8 eV, for which only 5 unoccupied bands are needed.

Copy the files toptic_3.files and toptic_3.in in Work_optic:

cp ../toptic_3.files .
cp ../toptic_3.in .


Issue:

abinit < toptic_3.files > log 2> err &


Now, examine the file toptic_3.in. There are two important changes with respect to the file toptic_1.in:

• the number of unoccupied bands has been reduced, so that the total number of bands is 9 instead of 20
• when applicable, the value of kptopt 3 in our previous simulation has been changed to 2, in order to take advantage of the time-reversal symmetry

When the run is finished (it is only 8 secs on a 2.8 GHz PC), you can process the WFK files and obtain the linear optic spectra. Copy the files toptic_4.files and toptic_4.in in Work_optic:

cp ../toptic_4.files .
cp ../toptic_4.in .


Examine the toptic_4.in file: only the linear optic spectra will be built.

When you have read the input file, you can run the code, as usual using the following command

optic < toptic_4.files > log 2> err &


Then, you can visualize the files toptic_2_0001_0001-linopt.out and toptic_4_0001_0001-linopt.out using xmgrace and compare them. The spectra looks completely identical. However, a careful look at these files, by editing them, show that indeed, the imaginary part is very similar:

 # Energy(eV)         Im(eps(w))
#calculated the component:  1  1  of dielectric function
#scissors shift:    0.000000E+00
#energy window:    3.982501E+01eV    1.463542E+00Ha
8.163415E-03    7.204722E-04
1.632683E-02    1.441005E-03
2.449025E-02    2.161659E-03
3.265366E-02    2.882494E-03
....


But the real parts differ slightly (this is seen at lines 1007 and beyond):

 # Energy(eV)         Re(eps(w))
8.163415E-03    1.186677E+01
1.632683E-02    1.186693E+01
2.449025E-02    1.186720E+01
3.265366E-02    1.186758E+01
...


for toptic_2_0001_0001-linopt.out and

 # Energy(eV)         Re(eps(w))
8.163415E-03    1.177773E+01
1.632683E-02    1.177789E+01
2.449025E-02    1.177816E+01
3.265366E-02    1.177854E+01
...


for toptic_4_0001_0001-linopt.out. This small difference is due to the number of bands (nband 20 for toptic_2_0001_0001-linopt.out and nband 9 for toptic_4_0001_0001-linopt.out).

Then, you can increase the number of k points, and watch the change in the imaginary part of the spectrum. There will be more and more peaks, until they merge, and start to form a smooth profile (still not completely smooth even with $28\times 28\times 28$). For your information, we give some timings of the corresponding Abinit run for a 2.8 GHz PC:

k-point grid     CPU time
4 x 4 x 4            8 secs
6 x 6 x 6            20 secs
8 x 8 x 8            43 secs
10 x 10 x 10         80 secs
12 x 12 x 12         138 secs
16 x 16 x 16         338 secs
20 x 20 x 20         702 secs
24 x 24 x 24         1335 secs
28 x 28 x 28         2633 secs


For grids on the order of $16\times 16\times 16$, the treatment by optics also takes several minutes, due to IO (30 minutes for the $28\times 28\times 28$ grid).
You might note how the first peak slowly develop with increasing number of k points but nevertheless stays much smaller than the converged one, and even smaller than the experimental one.