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Lesson on static non-linear properties

Electronic non-linear susceptibility, non-resonant Raman tensor, electro-optic effect.

This lesson aims at showing how to get the following non-linear physical properties, for an insulator:

  • The non-linear optical susceptibilities
  • The Raman tensor of TO and LO modes
  • The electro-optic coefficients

We will work on AlAs. During the preliminary steps needed to compute non-linear properties, one will also obtain several linear response properties:

  • The Born effective charges
  • The dielectric constant
  • The proper piezoelectric tensor (clamped and relaxed ions)

Finally, we will also compute the derivative of the susceptibility tensor with respect to atomic positions (Raman tensor) thanks to finite differences.

The user should have already passed through several advanced lessons of the tutorial: the lesson Response-Function 1, the lesson Response-Function 2, the lesson on Polarization and finite electric field, and the lesson on Elastic properties

This lesson should take about 1 hour and 30 minutes.

1 Ground-state properties of AlAs and general parameters

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

In order to save some time, you might immediately start running a calculation. Copy the file ~abinit/tests/tutorespfn/Input/ in Work-NLO. Copy also ~abinit/tests/tutorespfn/Input/tnlo_x.files in Work-NLO, and modify it so that all occurrences of tnlo_x are replaced by tnlo_2 , then run abinit with these data. This calculation might be one or two minutes on a PC 3GHz.

In this tutorial we will assume that the ground-state properties of AlAs have been previously obtained, and that the corresponding convergence studies have been done. Let us emphasize that, from our experience, accurate and trustable results for third order energy derivatives usually require a extremely high level of convergence. So, even if it is not always very apparent here, careful convergence studies must be explicitly performed in any practical study you could start after this tutorial. As usual, convergence tests must be done on the property of interest (i.e. it is wrong to determine parameters giving proper convergence on the total energy, and to use them blindly for non-linear properties)

We will adopt the following set of generic parameters (the same than in the lesson on Polarization and finite electric field):

   acell               10.53
   ixc                 3
   ecut                2.8     (results with ecut = 5 are also reported
                               in the discussion)
   ecutsm              0.5
   dilatmx             1.05
   nband               4 (=number of occupied bands)
   nbdbuf              0
   ngkpt               6 6 6
   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

   pseudopotentials    13al.pspnc

In principle, the acell to be used should be the one corresponding to the optimized structure at the ecut, and ngkpt combined with nshiftk and shiftk, chosen for the calculations. Unfortunately, for the purpose of this tutorial, in order to limit the duration of the runs, we have to work at an unusually low cutoff of 2.8 Ha for which the optimized lattice constant is unrealistic and equal to 7.45 Bohr (instead of the converged value of 10.64). In what follows, the lattice constant has been arbitrarily fixed to 10.53 Bohr. For comparison, results with ecut=5 are also reported and, in that case, were obtained at the optimized lattice constant of 10.64 Bohr. For those who would like to try later, convergence tests and structural optimizations can be done using the file ~abinit/tests/tutorespfn/Input/ Before going further, you might refresh your mind concerning the other variables : ixc, ecutsm, dilatmx, nbdbuf.

2 Linear and non-linear responses from density functional perturbation theory (DFPT)

As a theoretical support to this section of the tutorial, you might consider reading the following article:
M. Veithen, X. Gonze, and Ph. Ghosez,
Nonlinear optical susceptibilities, Raman efficiencies, and electro-optic tensors from first-principles density functional perturbation theory
Phys. Rev. B 71, 125107 (2005).

In the first part of this tutorial, we will describe how to compute various linear and non-linear responses directly connected to second-order and third- order derivatives of the energy, using DFPT. From the (2n+1) theorem, computation of energy derivatives up to third order only requires the knowledge of the ground-state and first-order wavefunctions, see X. Gonze and J.-P. Vigneron, PRB 39, 13120 (1989) and X. Gonze, Phys. Rev. A 52, 1096 (1995). Our study will therefore include the following steps : (i) resolution of the ground-state problem, (ii) determination of the first-order wavefunctions and construction of the related databases for second and third- order energy derivatives, (iii) combination of the different databases and analysis to get the physical properties of interest.

This is closely related to what was done for the dielectric and dynamical properties, except that an additional database for third-order energy derivatives will now be set up during the run. Only selected third-order derivatives are implemented at this stage and concern responses to electric field and atomic displacements:

  • non-linear optical susceptibilities (related to a third-order derivative of the energy with respect to electric fields at clamped nuclei positions)

  • Raman susceptibilities (mixed third-order derivative of the energy, twice with respect to electric fields at clamped nuclei positions, and once with respect to atomic displacement)

  • Electro-optic coefficients (related to a third-order derivative of the energy with respect to electric fields, two of them being optical fields -clamped nuclei positions- and one of them being a static field -the ions are allowed to move-)

Many different steps can be combined in one input file. For the sake of clarity we have decomposed the calculation into individual inputs that are now described.

Responses to electric fields and atomic displacements.

Let us examine the file . Its purpose is to build databases for second and third energy derivatives with respect to electric fields and atomic displacements. You can edit it. It is made of 5 datasets. The first four data sets are nearly the same as for a usual linear response calculation : (1) self-consistent calculation in the IBZ; (2) non self-consistent calculations to get the wave-functions over the full BZ; (3) ddk calculation, (4) derivatives with respect to electric field and atomic displacements. Some specific features must however be explicitly specified in order to prepare the non-linear response step (dataset 5). First, from dataset 2 it is mandatory to specify:

        nbdbuf  0
        nband   4 (= number of valence bands)

Also, in dataset 4, it is required to impose prtden, and prepanl

        prtden4    1
        prepanl4   1

The purpose for this is (i) to constrain kptopt = 2 even in the computation of phonons where ABINIT usually take advantages of symmetry irrespective of kptopt and (ii) compute the electric field derivatives in the 3 directions of space, irrespective of the symmetry.

If it was not done at the beginning of this tutorial, you can now make the run. You can have a quick look to the output file to verify that everything is OK. It contains the values of second and third energy derivatives. It has however no immediate interest since the information is not presented in a very convenient format. The relevant information is in fact also stored in the database files (DDB) generated independently for second and third energy derivatives at the end of run steps 4 and 5. Keep these databases that will be used later for a global and convenient analysis of the results using ANADDB.

Responses to strain. We combine the above-mentioned computation of the response to electric field and atomic displacements with the response to strain. This is not at all mandatory for the computation of the presently accessible non-linear response coefficients. However, this was used in the above-mentioned paper by Veithen et al, to add corrections corresponding to free boundary conditions, thanks to a further finite difference calculation on top of linear response calculations. The DFPT implementation of the computation of this correction is not available at present.

You can now copy the file ~abinit/tests/tutorespfn/Input/ in Work- NLO, and modify the tnlo_x.files accordingly (or create a file tnlo_3.files - in any case, this new file should contain tnlo_3 instead of tnlo_x or tnlo_2). You can launch the calculation, it might last about 1 minute on a PC 3 GHz. The purpose of this run is to build databases for second energy derivatives with respect to strains. You can edit . It is made of 4 datasets : (1) self-consistent calculation in the IBZ; (2) non self-consistent calculations to get the wave-functions over the full BZ; (3) ddk calculation; (4) strain perturbation. The ddk calculations has been included in order to access to the piezoelectric tensor.

You can have a quick look to the output tnlo_3.out, when it is ready. It contains rigid ions elastic and piezoelectric constants as well as the internal strain coupling parameters. This information is also stored in a database file (DDB) for further convenient analysis with ANADDB.

Merge of the DDB.

At this stage, all the relevant energy derivatives have been obtained and are stored in individual databases. These must be combined with the MRGDDB merge utility in order to get a unique database tnlo_4.ddb.out. Explicitely, you should merge the files tnlo_2o_DS4_DDB, tnlo_3o_DS4_DDB, and tnlo_2o_DS5_DDB . You might have a look at the input file for MRGDDB named , and use it to perform the merge. You already used MRGDDB previously. It might be located in ~abinit/src/98_main or another (build) directory. You might copy it, or make an alias.

Analysis of the DDB.

We are now ready for the analysis of the results using ANADDB. You can copy the files ~abinit/tests/tutorespfn/Input/ and ~abinit/tests/tutorespfn/Input/tnlo_5.files in Work-NLO. You already used ANADDB previously. It is located in ~abinit/src/98_main or another (build) directory. You might copy it, or make an alias. The present input is in principle very similar to the one you have used for the analysis of dynamical and dielectric responses except that some new flags need to be activated.

For the strain perturbation you need to specify elaflag, piezoflag, and instrflag:

        elaflag 3
        piezoflag  3
        instrflag  1

For the non-linear responses you need

        nlflag  1
        ramansr 1
        alphon  1
        prtmbm  1

nlflag=1 activates the non-linear response.

ramansr=1 will impose the sum rule on the first-order change of the electronic dielectric susceptibility under atomic displacement, hereafter referred to as dchi/dtau. It is a condition of invariance of chi under translation of the whole crystal, similar to the acoustic sum rules for phonons at Gamma or the charge neutrality sum rule on Z*.

prtmbm=1 will allow to get the mode by mode phonon contributions of the ions to the electro-optic coefficients.

alphon=1 will allow to get the previous mode by mode contribution when aligning the eigenvectors with the cartesian axis of coordinates (in the input, the principal axis should always be aligned with z for a convenient analysis of the results).

Finally, the second list of phonon, specified with nph2l and qph2l, must also be explicitely considered to obtain the Raman efficiencies of longitudinal modes (in a way similar to the computation of frequencies of longitudinal mode frequencies at Gamma):

        # Wave vector list no. 2
               nph2l  1
               qph2l  1.0 0.0 0.0 0.0

You can now run the code ANADDB. The results are in the file tnlo_5.out. Various interesting physical properties are now directly accessible in this output in meaningful units. You can go through the file and look in order to identify the results mention hereafter. Note that the order in which they are given below is not the same than the order in which they appear in the tnlo_5.out. You will have to jump between different sections of tnlo_5.out to find them.

For comparison, we report in parenthesis (…) the values obtained with ecut = 5, and for nonlinear responses in brackets […] the fully converged result as reported in PRB 71, 125107 (2005).

  • Born effective charge of Al:

    Z*_Al = 2.043399E+00 (2.105999E+00)
  • Optical phonon frequencies at Gamma :

    w_TO (cm^-1) = 3.694366E+02 (3.602635E+02)

    w_LO (cm^-1) = 4.031189E+02 (3.931598E+02)

  • Linear optical dielectric constant :

    Electronic dielectric tensor = 9.20199931 (9.94846084)
  • Static dielectric constant :

    relaxed ion dielectric tensor = 10.95642097 (11.84823634)

Some other quantities, as the piezoelectric coefficients, are related to the strain response as it is more extensively discussed in the tutorial on the strain perturbation.

  • Proper piezoelectric coefficients :
    clamped ion (Unit:c/m^2) = -0.65029623 (-0.69401363)

    relaxed ion (Unit:c/m^2) = 0.03754602 (-0.04228777)

Finally, different quantities are related to non-linear responses.

  • Nonlinear optical susceptibility :
    They are directly provided in the output in pm/V. As you can see the value computed here is far from the well converged result as reported in PRB 71, 125107 (2005).

    d_36 (pm/V)  = 21.175523 (32.772254) [fully converged :35]
  • Electro-optic coefficients:
    As we asked for mode by mode decomposition the output provides individual contributions. We report below a summary of the results. It concern the clamped r_63 coefficient.

            Electronic EO constant (pm/V): -1.000298285 (-1.324507791) [-1.69]
        Full Ionic EO constant (pm/V):  0.543837671 (0.533097548)  [0.64]
        Total EO constant (pm/V):      -0.456460614 (-0.791410242) [-1.05]
  • Raman properties
    The code directly report the Raman susceptibilities for both transverse (TO) and longitudinal (LO) optic modes at Gamma:

            alpha(TO) = -0.008489212 (-0.009114814)
        alpha(LO) = -0.011466211 (-0.013439375)

The basic quantity to get the Raman susceptibilities are the dchi/dtau that are also reported separately :

    dchi_23/dtau_1 (Bohr^-1) of Al = -0.094488281 (-0.099889084)

In cubic semiconductors, it is usual to report the Raman polarizability of optical phonon modes at Gamma which is defined as

    a = Omega_0 * dchi/dtau = Sqrt[mu * Omega_0] alpha

where Omega_0 is the primitive unit cell volume (i.e. one quarter of the cubic unit cell volume, to be expressed here in Ang) and mu is the reduced mass of the system (1/mu = 1/m_Al + 1/m_As). From the previous data, we get :

            a(TO) (Unit: Ang^2)= -7.7233 (-8.4222112)  [-8.48]
        a(LO) (Unit: Ang^2)= -10.4317 (-12.418168) [-12.48]

3 Finite difference calculation of the Raman tensor

For comparison with the DPFT calculation, we can compute dchi/dtau for the Al nucleus from finite differences. In practice, this is achieved by computing the linear optical susceptibility for 3 different positions of the Al nucleus. This is done with the file ~abinit/tests/tutorespfn/Input/, however with the unrealistic cutoff of 2.8 Ha. The calculation is about 2 or 3 minutes on a PC 3 GHz). For those who want to do it you anyway, you can copy ~abinit/tests/tutorespfn/Input/ in your working directory. If you have time, you should modify the cutoff to ecut=5 Ha, in order to obtain realistic results. So, you might as well start the run after this modification (the run is about two times more time-consuming than with 2.8 Ha).

You can have a look at this input file. It contains 8 datasets. We need to compute the linear optical susceptibility (4 datasets for SC calculation in the IBZ, NSC calculation in the full BZ, ddk, ddE) for different atomic positions. We will do this for 2 sets of atomic positions, the reference symmetric structure (referred to as tau=0), and a distorted structure (referred to as tau= +0.01), for which the Al atom has been displaced to the right by 0.01 Bohr (look at xcart to identify the differences). In the first case, the dielectric tensor must be diagonal, isotropic, while in the second case, a off-diagonal yz component will appear, that is an odd function of the Al atomic displacement.

Supposing you are running the calculation, you have now time for a Belgian Beer, why not a Gouyasse ?! … Or you can look at the results as summarized below.

For tau = 0:

Dielectric tensor, in cartesian coordinates,
     j1       j2             matrix element
  dir pert dir pert     real part    imaginary part

   1    4   1    4         9.2020015668        -0.0000000000
   1    4   2    4         0.0000000000        -0.0000000000
   1    4   3    4         0.0000000000        -0.0000000000

   2    4   1    4         0.0000000000        -0.0000000000
   2    4   2    4         9.2020015668        -0.0000000000
   2    4   3    4         0.0000000000        -0.0000000000

   3    4   1    4         0.0000000000        -0.0000000000
   3    4   2    4         0.0000000000        -0.0000000000
   3    4   3    4         9.2020015668        -0.0000000000

For tau = +0.01 :

Dielectric tensor, in cartesian coordinates,
     j1       j2             matrix element
  dir pert dir pert     real part    imaginary part

   1    4   1    4         9.2023220436        -0.0000000000
   1    4   2    4        -0.0000000000        -0.0000000000
   1    4   3    4        -0.0000000000        -0.0000000000

   2    4   1    4        -0.0000000000        -0.0000000000
   2    4   2    4         9.2021443491        -0.0000000000
   2    4   3    4        -0.0123700617        -0.0000000000

   3    4   1    4        -0.0000000000        -0.0000000000
   3    4   2    4        -0.0123700617        -0.0000000000
   3    4   3    4         9.2021443491        -0.0000000000

Note that the following results would have been obtained for tau = -0.01 (with obvious even / odd behaviour with respect to tau of the different components, and some very small numerical noise):

Dielectric tensor, in cartesian coordinates,
     j1       j2             matrix element
  dir pert dir pert     real part    imaginary part

   1    4   1    4         9.2023220610        -0.0000000000
   1    4   2    4        -0.0000000000        -0.0000000000
   1    4   3    4         0.0000000000        -0.0000000000

   2    4   1    4         0.0000000000        -0.0000000000
   2    4   2    4         9.2021443663        -0.0000000000
   2    4   3    4         0.0123700529        -0.0000000000

   3    4   1    4         0.0000000000        -0.0000000000
   3    4   2    4         0.0123700529        -0.0000000000
   3    4   3    4         9.2021443663        -0.0000000000

You can extract the value of dchi_23/dtau_1 for Al from the dielectric tensor (hereafter called eps) above using the following finite-difference formula [unit of bohr^-1] :

 dchi_23/dtau_1= (1/4 pi) (eps_23[tau=+0.01] +eps_23[tau=0.00])/tau
               = (1/4 pi) (-0.0123700 -0.0)/(0.01)
               = -0.098437

This value is close to that obtained at ecut=5 from DFPT (-0.099889084). When convergence is reached (beware, the k point convergence is extremely slow, much slower than for other properties), both approaches allow to get the right answer. You might therefore ask which approach is the most convenient and should be used in practice.

As a guide, we can mention that the finite-difference approach give results very similar to the DFPT ones for a similar cutoff and k-point grid. It is however more tedious because, individual atomic displacement must be successively considered (heavy for complex crystals) and the results must then be converted into appropriate units with risk of error of manipulations.

The DFPT approach is the most convenient and avoid a lot of human work. Everything is reported together (not only dchi/dtau but also the full Raman polarizability tensors) and in appropriate units. It should therefore be considered as the best choice (when available, as in ABINIT).