Tutorial on optical properties¶
Frequencydependent 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 RandomPhase Approximation or Sumoverstates 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 toplevel 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 base1234, 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.
toptic_1.in toptic_1.out toptic_1i toptic_1o toptic_1 ../../../Psps_for_tests/31ga.pspnc ../../../Psps_for_tests/33as.pspnc
# Prepare the computation of linear and nonlinear optic properties # of GaAs crystal : groundstate with few bands, # then nonSCF with a larger number of bands, then ddk for different directions # Note that the k point sampling shoud be finer for significant results. The cutoff energy is also too low. ndtset 6 #First dataset : SC run with kpoints in the IBZ nband1 4 nstep1 25 kptopt1 1 nbdbuf1 0 prtden1 1 getden1 0 getwfk1 0 # Usual file handling data #Second dataset : NSC run with large number of bands, and points in the IBZ iscf2 2 nband2 20 # This number of bands might be too low for nonlinear optics and real part of linear optics nstep2 25 kptopt2 1 getwfk2 1 getden2 1 # Usual file handling data #Third dataset : NSC run with large number of bands, and points in the the full BZ iscf3 2 nband3 20 # This number of bands might be too low for nonlinear optics and real part of linear optics nstep3 25 kptopt3 3 getwfk3 2 getden3 1 # Usual file handling data #Fourth dataset : ddk response function along axis 1 iscf4 3 nband4 20 # This number of bands might be too low for nonlinear optics and real part of linear optics nstep4 1 nline4 0 prtwf4 3 kptopt4 3 nqpt4 1 qpt4 0.0d0 0.0d0 0.0d0 rfdir4 1 0 0 rfelfd4 2 getwfk4 3 #Fifth dataset : ddk response function along axis 2 iscf5 3 nband5 20 # This number of bands might be too low for nonlinear optics and real part of linear optics nstep5 1 nline5 0 prtwf5 3 kptopt5 3 nqpt5 1 qpt5 0.0d0 0.0d0 0.0d0 rfdir5 0 1 0 rfelfd5 2 getwfk5 3 #Sixth dataset : ddk response function along axis 3 iscf6 3 nband6 20 # This number of bands might be too low for nonlinear optics and real part of linear optics nstep6 1 nline6 0 prtwf6 3 kptopt6 3 nqpt6 1 qpt6 0.0d0 0.0d0 0.0d0 rfdir6 0 0 1 rfelfd6 2 getwfk6 3 #Data common to all datasets nshiftk 4 shiftk 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5 ngkpt 4 4 4 # This is much too low : should be at least 24x24x24 acell 3*10.60 amu 69.72 74.9216 diemac 10.0 ecut 2.00 # This is also too low ixc 3 natom 2 nbdbuf 2 ntypat 2 rprim 0 .5 .5 .5 0 .5 .5 .5 0 xred 3*0.00d0 3*0.25d0 tnons 72*0.0 typat 1 2 tolwfr 1.e20 znucl 31 33 ## After modifying the following section, one might need to regenerate the pickle database with runtests.py r #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% test_chain = toptic_1.in, toptic_2.in #%% need_cpp_vars = !HAVE_MPI_IO_DEFAULT #%% [files] #%% files_to_test = #%% toptic_1.out, tolnlines= 0, tolabs= 0.000e+00, tolrel= 0.000e+00, fld_options = easy #%% psp_files = 31ga.pspnc, 33as.pspnc #%% [paral_info] #%% max_nprocs = 2 #%% [extra_info] #%% authors = S. Sharma, X. Gonze #%% keywords = NC, DFPT #%% description = #%% Prepare the computation of linear and nonlinear optic properties #%% of GaAs crystal : groundstate with few bands, #%% then nonSCF with a larger number of bands, then ddk for different directions #%% Note that the k point sampling shoud be finer for significant results. The cutoff energy is also too low. #%%<END TEST_INFO>
The computation concerns a crystal of GaAs, in the zincblende 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 selfconsistent
determination of the ground state for a fixed geometry. Only the occupied bands are treated.
The density is output.
The second dataset is a nonselfconsistent 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 nonselfconsistent 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 cutoff 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 sumoverstates 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 cutoff 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.
&FILES ddkfile_1 = 'toptic_1o_DS4_1WF7', ddkfile_2 = 'toptic_1o_DS5_1WF8', ddkfile_3 = 'toptic_1o_DS6_1WF9', wfkfile = 'toptic_1o_DS3_WFK' / &PARAMETERS broadening = 0.002, domega = 0.0003, maxomega = 0.3, scissor = 0.000, tolerance = 0.002 / &COMPUTATIONS num_lin_comp = 1, lin_comp = 11, num_nonlin_comp = 2, nonlin_comp = 123,222, num_linel_comp = 0, num_nonlin2_comp = 0, / ## After modifying the following section, one might need to regenerate the pickle database with runtests.py r #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = optic #%% test_chain = toptic_1.in, toptic_2.in #%% need_cpp_vars = !HAVE_MPI_IO_DEFAULT #%% [files] #%% files_to_test = #%% toptic_2_0001_0001linopt.out, tolnlines= 0, tolabs= 0.000e+00, tolrel= 0.000e+00, fld_options = easy; #%% toptic_2_0001_0002_0003ChiTotRe.out, tolnlines= 16, tolabs= 7.000e04, tolrel= 7.000e04, fld_options = easy; #%% toptic_2_0001_0002_0003ChiTotIm.out, tolnlines= 16, tolabs= 4.000e04, tolrel= 2.000e04, fld_options = easy #%% [paral_info] #%% max_nprocs = 4 #%% [extra_info] #%% authors = S. Sharma, X. Gonze #%% keywords = #%% description = Input file for optic code. #%%<END TEST_INFO>
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_0001linopt.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 nonlinear 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_0002ChiTotIm.out
and the Real part with:
xmgrace toptic_2_0002_0002_0002ChiTotRe.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 cutoff 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_0001linopt.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_0003ChiTotIm.out
as well as
xmgrace $ABI_HOME/doc/tutorial/optic_assets/toptic_ref_0001_0002_0003ChiTotRe.out
for the nonlinear optics, obtained with a 18x18x18 grid (keeping everything else fixed).
Concerning the linear spectrum, we will now compare this (underconverged) result toptic_ref_0001_0001linopt.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_0001linopt.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 (electronhole 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 SumOverState approach, but still keeping the independentelectron approximation, e.g., in the framework of the TDDFT (adiabatic LDA) will not correct this problem. One needs to use the BetheSalpeter approximation, or to rely on fancy exchangecorrelation 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 cutoff 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 KramersKronig 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 lowfrequency value is only 4.3, while it should be on the order of 10. This can be corrected by increasing the cutoff energy, the k point sampling and the number of unoccupied states.
Similar considerations apply to the nonlinear 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 timereversal 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 .
toptic_3.in toptic_3.out toptic_3i toptic_3o toptic_3 ../../../Psps_for_tests/31ga.pspnc ../../../Psps_for_tests/33as.pspnc
# Prepare the computation of linear optic properties (for the imaginary spectrum only) # of GaAs crystal : groundstate with few bands, # then nonSCF with a larger number of bands, then ddk for different directions # Note that the k point sampling shoud be finer for significant results. The cutoff energy is also too low. ndtset 6 #First dataset : SC run with kpoints in the IBZ nband1 4 nstep1 25 kptopt1 1 nbdbuf1 0 prtden1 1 getden1 0 getwfk1 0 # Usual file handling data #Second dataset : NSC run with large number of bands, and points in the IBZ iscf2 2 nband2 9 # Minimal number of bands for linear optics (imaginary part of the spectrum) nstep2 25 kptopt2 1 getwfk2 1 getden2 1 # Usual file handling data #Third dataset : NSC run with large number of bands, and points in the the full BZ iscf3 2 nband3 9 # Minimal number of bands for linear optics (imaginary part of the spectrum) nstep3 25 kptopt3 2 # Timereversal symmetry can be used in the present implementation for linear optics getwfk3 2 getden3 1 # Usual file handling data #Fourth dataset : ddk response function along axis 1 iscf4 3 nband4 9 # Minimal number of bands for linear optics (imaginary part of the spectrum) nstep4 1 nline4 0 prtwf4 3 kptopt4 2 # Timereversal symmetry can be used in the present implementation for linear optics nqpt4 1 qpt4 0.0d0 0.0d0 0.0d0 rfdir4 1 0 0 rfelfd4 2 getwfk4 3 #Fifth dataset : ddk response function along axis 2 iscf5 3 nband5 9 # Minimal number of bands for linear optics (imaginary part of the spectrum) nstep5 1 nline5 0 prtwf5 3 kptopt5 2 # Timereversal symmetry can be used in the present implementation for linear optics nqpt5 1 qpt5 0.0d0 0.0d0 0.0d0 rfdir5 0 1 0 rfelfd5 2 getwfk5 3 #Sixth dataset : ddk response function along axis 3 iscf6 3 nband6 9 # Minimal number of bands for linear optics (imaginary part of the spectrum) nstep6 1 nline6 0 prtwf6 3 kptopt6 2 # Timereversal symmetry can be used in the present implementation for linear optics nqpt6 1 qpt6 0.0d0 0.0d0 0.0d0 rfdir6 0 0 1 rfelfd6 2 getwfk6 3 #Data common to all datasets nshiftk 4 shiftk 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.5 ngkpt 3*4 # This is much too low : should be at least 24x24x24 acell 3*10.60 amu 69.72 74.9216 diemac 10.0 ecut 2.00 # This is also too low ixc 3 natom 2 nbdbuf 2 ntypat 2 rprim 0 .5 .5 .5 0 .5 .5 .5 0 xred 3*0.00d0 3*0.25d0 tnons 72*0.0 typat 1 2 tolwfr 1.e20 znucl 31 33 ## After modifying the following section, one might need to regenerate the pickle database with runtests.py r #%%<BEGIN TEST_INFO> #%% [setup] #%% executable = abinit #%% test_chain = toptic_3.in, toptic_4.in #%% need_cpp_vars = !HAVE_MPI_IO_DEFAULT #%% [files] #%% files_to_test = #%% toptic_3.out, tolnlines= 0, tolabs= 0.000e+00, tolrel= 0.000e+00, fld_options = easy #%% psp_files = 31ga.pspnc, 33as.pspnc #%% [paral_info] #%% max_nprocs = 2 #%% [extra_info] #%% authors = S. Sharma, X. Gonze #%% keywords = NC, DFPT #%% description = #%% Prepare the computation of linear optic properties (for the imaginary spectrum only) #%% of GaAs crystal : groundstate with few bands, #%% then nonSCF with a larger number of bands, then ddk for different directions #%% Note that the k point sampling shoud be finer for significant results. The cutoff energy is also too low. #%%<END TEST_INFO>
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 timereversal 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_0001linopt.out and toptic_4_0001_0001linopt.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 2.000000E03 #scissors shift: 0.000000E+00 #energy window: 3.982501E+01eV 1.463542E+00Ha 8.163415E03 7.204722E04 1.632683E02 1.441005E03 2.449025E02 2.161659E03 3.265366E02 2.882494E03 ....
But the real parts differ slightly (this is seen at lines 1007 and beyond):
# Energy(eV) Re(eps(w)) 8.163415E03 1.186677E+01 1.632683E02 1.186693E+01 2.449025E02 1.186720E+01 3.265366E02 1.186758E+01 ...
for toptic_2_0001_0001linopt.out and
# Energy(eV) Re(eps(w)) 8.163415E03 1.177773E+01 1.632683E02 1.177789E+01 2.449025E02 1.177816E+01 3.265366E02 1.177854E+01 ...
for toptic_4_0001_0001linopt.out. This small difference is due to the number of bands (nband 20 for toptic_2_0001_0001linopt.out and nband 9 for toptic_4_0001_0001linopt.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:
kpoint 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.