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Lesson on optical properties

Frequency-dependent linear and second order nonlinear optical response.

This lesson 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 discover the optic help file.

This lesson should take about 1 hour.

1 Computing the momentum matrix elements

Before beginning, you might consider to work in a different subdirectory as for the other lessons. Why not create “Work_optic” in ~abinit/tests/tutorespfn/Input?

In order to use the Optic utility, you should first have some theoretical background. Could you read the first two sections of the optic help file?

Now, you are ready to run ABINIT and prepare the needed file.

Copy the files ~abinit/tests/tutorespfn/Input/toptic_1.files and ~abinit/tests/tutorespfn/Input/ in “Work_optic”. Issue:

abinit < toptic_1.files >& log

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 worth to take some time to examine the other file Edit 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 printed.
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 might be skipped but is included for later CPU time saving).
The fourth to sixth datasets correspond to the computation of the ddk matrix elements. Note that the number of bands is the same as for dataset 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 derivative of the wavefunction with respect to the wavevector, while only the matrix elements are needed. Such a larger value of nstep 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 ridiculously small. Instead of 4x4x4 FCC (256 k points), it should be something like 28x28x28 FCC (about 100000 k points). Also, the cut-off energy (2 Ha) is too small. As often, convergence studies are the responsibility of the user.

The run is 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 three latter ones contain the matrix elements.

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

2 Computing the linear and nonlinear optical response

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 material under study.

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

Copy the files ~abinit/tests/tutorespfn/Input/toptic_2.files and ~abinit/tests/tutorespfn/Input/ in “Work_optic”.

The is your input file. You should edit it, read it carefully. For help on various input parameters in this file please see the help file.

When you have read the input file, you can run the code, as usual, using the following command (assuming optic can be accessed thanks to “optic” - copy the executable in the current directory if needed)

optic < toptic_2.files > log &

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

The header contains the information about the calculation. See the section 4 of the Optic help file. These files can be plotted in xmgrace, please do so using the following command and look at the results. (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

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, that are much smoother. The origin of this discrepancy is to be found in the very bad k point sampling that we used in order to be able to perform the tutorial. 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

It might be the perfect time to finish with 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:

xmgrace ~abinit/doc/tutorial/lesson_optic/toptic_ref_0001_0001-linopt.out

for the linear optics, obtained with a 28x28x28 grid (keeping everything else fixed), and

xmgrace ~abinit/doc/tutorial/lesson_optic/toptic_ref_0001_0002_0003-ChiTotIm.out

as well as

xmgrace ~abinit/doc/tutorial/lesson_optic/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, “Electronic Structure and Optical Properties of Semiconductors” (Springer-Verlag, Berlin, 1988) presents a comparison of experimental data with the empirical pseudopotential method spectrum. If you do not have acess to this book, you can see an experimental spectrum in e.g. H.R. Phillips, H. Ehrenreich, Phys. Rev. 127, 1550 (1963), and a theoretical spectrum in e.g. M.-Z. Huang and W. Y. Ching, Phys. Rev. B 47, 9449 (1993) (well, there are numerous papers that present such data).

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 different 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 the Cohen-Chelikowsky book), 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 present additional deficiencies with respect to the other ones, mostly due to a still too rough sampling of the k space (there are too many wiggles in the spectrum), and to an 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 states should be 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.

3 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 in “Work_optic”:

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


abinit < toptic_3.files >& log

Now, examine the file . There are two important changes with respect to the file

  • 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 in “Work_optic”:

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

Examine the file: only the linear optic spectra will be build.

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

optic < toptic_4.files > log &

Then, you can vizualize 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
 #broadening:    0.000000E+00    2.000000E-03
 #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 part slightly differ (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 with 28x28x28). For information, we give some timings of the corresponding ABINIT run for a 2.8 GHz PC:

k-point grid     CPU time
4x4x4            8 secs
6x6x6            20 secs
8x8x8            43 secs
10x10x10         80 secs
12x12x12         138 secs
16x16x16         338 secs
20x20x20         702 secs
24x24x24         1335 secs
28x28x28         2633 secs

For grids on the order of 16x16x16, the treatment by optics also takes several minutes, due to IO (30 minutes for the 28x28x28 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.