Superconducting properties within the isotropic Eliashberg formalism

This tutorial explains how to compute the phonon linewidths induced by the electron-phonon (e-ph) interaction in metallic systems and how to use the Eliashberg spectral function $\alpha^2F$ and the McMillan equation to estimate the superconducting critical temperature $T_c$ within the isotropic Eliashberg formalism.

We start by presenting the basic equations implemented in the code and their connection with the ABINIT variables. Then we discuss how to run isotropic $T_c$-calculations and how to perform typical convergence studies for MgB$_2$, a well-known phonon-mediated superconductor with $T_c$ = 39 K. For a more complete theoretical introduction, see [Giustino2017] and references therein.

It is assumed the user has already completed the two tutorials RF1 and RF2, and that he/she is familiar with the calculation of ground state and vibrational properties in metals. The user should have read the fourth lesson on Al as well as the introduction page for the EPH code before running these examples.

This lesson should take about 1.5 hour.

Formalism and connection with the implementation

Due to the interaction with electrons, vibrational energies get renormalized and phonons acquire a finite lifetime. These many-body effects are described by the phonon-electron self-energy:

$$\Pi_{\qq\nu}(\ww, T) = \dfrac{2}{N_\kk} \sum_{mn\kk} |g_{mn\nu}(\kk, \qq)|^2 \dfrac{f_\nk - f_{m\kq}}{\ee_{\nk} - \ee_{m\kq} -\ww - i\eta^+}$$

where only contributions up to the second order in the e-ph vertex $g$ have been included. The sum over the electron wavevector $\kk$ is performed over the first BZ, $\eta$ is a positive real infinitesimal and $\gkkp$ are the e-ph matrix element discussed in the EPH introduction. The self-energy depends on the temperature via the Fermi-Dirac distribution function $f(\ee, T)$ and the factor two accounts for spin degeneracy (henceforth we assume a non-magnetic system with scalar wavefunctions i.e. nsppol == 1 and nspinor == 1 thus the electron energies $\ee$ and the e-ph matrix elements $g$ do not depend on the spin).

The real part of $\Pi_\qnu$ gives the correction to the vibrational energies due to e-ph interaction while the phonon linewidth $\gamma_{\qq\nu}$ (full width at half maximum) is twice the imaginary part $\Pi^{''}_\qnu$ evaluated at the “bare” phonon frequency $\ww_\qnu$ as computed with DFPT:

$$\gamma_{\qq\nu}(T) = 2\, \Pi^{''}_\qnu(\ww=\ww_\qnu, T).$$

Using the Sokhotski–Plemelj theorem:

$$\lim_{\eta \rightarrow 0^+} \dfrac{1}{x \pm i\eta} = \mp i\pi\delta(x) + \mathcal{P}(\dfrac{1}{x})$$

the imaginary part of $\Pi$ can we rewritten as:

$$\Pi^{''}_{\qq\nu}(\ww_\qnu, T) = \dfrac{2\pi}{N_\kk} \sum_{mn\kk} |g_{mn\nu}(\kk, \qq)|^2 (f_\nk - f_{m\kq})\,\delta(\ee_{\kk n} - \ee_{m\kpq} -\ww_\qnu)$$

where the delta function enforces energy conservation in the scattering process. Because phonon energies are typically small compared to electronic energies, the energy difference $\ee_{\kk n} - \ee_{m\kpq}$ is also small hence it is possible to approximate the difference in the occupation factors with:

$$f_\nk - f_{m\kq} \approx f'(\ee_\nk) (\ee_{\kk n} - \ee_{m\kpq}) = -f'(\ee_\nk) \ww_\qnu$$

At low T, the derivative of the Fermi-Dirac occupation function is strongly peaked around the Fermi level thus we can approximate it with:

$$f'(\ee_\nk) \approx -\delta(\ee_\nk - \ee_F)$$

By neglecting $\ww_\qnu$ in the argument of the delta, we obtain the so-called double-delta approximation (DDA) for the phonon linewidth [Allen1972]:

$$\begin{equation}\label{eq:DDA_gamma} \gamma_{\qq\nu} = \dfrac{4\pi \ww_{\qq\nu}}{N_\kk} \sum_{mn\kk} |g_{mn\nu}(\kk, \qq)|^2 \delta(\ee_{\kpq m} -\ee_F) \delta(\ee_{\kk n} -\ee_F) \end{equation}$$

For a given phonon wavevector $\qq$, the DDA restricts the BZ integration to electronic transitions between $\kk$ and $\kq$ states on the Fermi surface (FS) hence very dense $\kk$-meshes are needed to converge $\gamma_{\qq\nu}$. The convergence rate is expected to be slower than that required by the electron DOS per spin at $\ee_F$:

$$g(\ee_F) = \dfrac{1}{N_\kk} \sum_{n\kk} \delta(\ee_{n\kk} - \ee_F)$$

in which a single Dirac delta is involved.

Note

The DDA breaks down at small $\qq$ where the phonon frequency cannot be neglected. Note also that the DDA predicts the linewidths to not depend on T.

It is also worth stressing that there is another important contribution to the phonon lifetimes induced by non-harmonic terms in the Taylor expansion of the Born-Oppenheimer energy surface around the equilibrium point. Within the framework of many-body perturbation theory, these non-harmonic terms lead to phonon-phonon scattering processes that can give a substantial contribution to the lifetimes, especially at “high” T. In the rest of the tutorial, however, non-harmonic terms will be ignored and we will be mainly focusing on the computation of the imaginary part of the phonon self-energy $\Pi$ in the harmonic approximation.

At the level of the implementation, the eph_intmeth input variable selects the technique for integrating the double delta over the FS: eph_intmeth == 2 (default) activates the optimized tetrahedron scheme [Kawamura2014] as implemented in the libtetrabz library while eph_intmeth == 1 replaces the Dirac distribution with a Gaussian function of finite width. In the later case, one can choose between a constant broadening $\sigma$ specified in Hartree by eph_fsmear or an adaptive Gaussian scheme (activated if eph_fsmear < 0) in which a state-dependent $\sigma_\nk$ is automatically computed from the electron group velocities $v_\nk$ [Li2015]. Note that the adaptive Gaussian scheme is internally used by the code also when the optimized tetrahedron method is employed since the DDA is ill-defined for $\qq = \Gamma$.

The value of the Fermi level $\ee_F$ is automatically computed from the KS eigenvalues stored in the input WFK file according to the two input variables occopt and tsmear. These variables are usually equal to the ones used for the GS/DFPT calculations although it is possible to change the value of $\ee_F$ during an EPH calculation using three (mutually exclusive) input variables: eph_fermie, eph_extrael and eph_doping. This may be useful if one wants to study the effect of doping within the rigid band approximation.

The sum over bands in Eq.\ref{eq:DDA_gamma} is restricted to the Bloch states within a symmetric energy window of thickness eph_fsewin around $\ee_F$. This value should be chosen according to the FS integration scheme eph_intmeth and the value of eph_fsmear (if standard Gaussian). Additional details are given in the tutorial. The $\kk$-mesh for electrons is defined by the input variables ngkpt, nshiftk and shiftk. These parameters must be equal to the ones used to generate the input WFK file passed to the EPH code.

The code computes $\gamma_{\qq\nu}$ for all the $\qq$-point in the IBZ associated to the eph_ngqpt_fine $\qq$-mesh and the DFPT potentials are interpolated starting from the ab-initio ddb_ngqpt $\qq$-mesh associated to the input DVDB/DDB files. If eph_ngqpt_fine is not specified, eph_ngqpt_fine == ddb_ngqpt is assumed and no interpolation of the scattering potentials in $\qq$-space is performed.

Once the phonon linewidths $\gamma_{\qq\nu}$ are known in the IBZ, ABINIT computes the isotropic Eliashberg function defined by:

$$\alpha^2F(\ww) = \dfrac{1}{N_\qq\, N_F} \sum_{\qq\nu} \dfrac{\gamma_{\qq\nu}}{\ww_{\qq\nu}} \delta(\ww - \ww_{\qq \nu})$$

where $N_F$ is the density of states (DOS) per spin at the Fermi level. The Eliashberg function can be equivalently expressed as:

$$\alpha^2F(\ww) = \dfrac{1}{N_\qq N_\kk\, N_F} \sum_{\kk\qq mn \nu} |g_{mn\nu}(\kk, \qq)|^2 \delta(\ee_{\kpq m} -\ee_F) \delta(\ee_{\kk n} -\ee_F) \delta(\ww - \ww_{\qq \nu}).$$

From a physical perspective, $\alpha^2F(\ww)$ gives the strength by which a phonon of energy $\hbar\ww$ scatters electronic states on the FS (remember that ABINIT uses atomic units hence $\hbar = 1$). This quantity is accessible in experiments and experience has shown that $\alpha^2F(\ww)$ is qualitatively similar to the phonon DOS $F(\ww)$:

$$F(\ww) = \dfrac{1}{N_\qq} \sum_{\qq\nu} \delta(\ww - \ww_{\qq \nu}).$$

This is not surprising as the equation for $\alpha^2F(\ww)$ resembles the one for $F(\ww)$, except for the weighting factor $\frac{\gamma_{\qq\nu}}{\ww_{\qq\nu}}$. This also explains the $\alpha^2F$ notation: the Eliashberg function can be seen as the phonon DOS times the positive frequency-dependent prefactor $\alpha^2(\ww)$.

Warning

Converging $\alpha^2F(\ww)$ usually requires very fine $\kk$- and $\qq$-meshes. This is especially true if the FS significantly deviates from the free electron picture (spherical FS) and/or the e-ph coupling is strongly anisotropic as in MgB$_2$. The $\kk$-mesh affects the quality of the individual $\gamma_\qnu$ linewidths whereas dense $\qq$-meshes may be needed to resolve the fine detauls and/or sample regions in $\qq$-space where the coupling is strong.

The technique used to compute $\alpha^2F(\ww)$ is defined by ph_intmeth (note the ph_ prefix instead of eph_). Both the Gaussian (ph_intmeth = 1 with ph_smear smearing) and the linear tetrahedron method by [Bloechl1994] (ph_intmeth = 2, default) are implemented. The $\alpha^2F(\ww)$ function is evaluated on a linear mesh of step ph_wstep covering the entire range of phonon frequencies. The total e-ph coupling strength $\lambda$ (a dimensionless measure of the average strength of the e-ph coupling) is defined as the first inverse moment of $\alpha^2F(\ww)$:

$$\lambda = \int \dfrac{\alpha^2F(\ww)}{\ww}\dd\ww = \sum_{\qq\nu} \lambda_{\qq\nu}$$

where we have introduced the mode-dependent contributions:

$$\lambda_{\qq\nu} = \dfrac{\gamma_{\qq\nu}}{\pi N_F \ww_{\qq\nu}^2}$$

Warning

Due to the $1/\omega$ factor, “low-energy” modes are expected to contribute more to $\lambda$ than “high-energy modes”. On the other hand, $\alpha^2F(\ww)$ should go to zero as $\ww^2$ for $\ww \rightarrow 0$ so that the integrand function is finite at the acoustic limit. Obviously, if the system is not dynamically stable (“negative” frequencies in the BZ) $\lambda$ will explode but this does not mean you have found a room-temperature superconductor!

In principle, $T_c$ can be obtained by solving the isotropic Eliashberg equations for the superconducting gap (see e.g. [Margine2013]) but many applications prefer to bypass the explicit solution and estimate $T_c$ using the semi-empirical McMillan equation [McMillan1968] in the improved version proposed by [Allen1975]:

$$T_c = \dfrac{\ww_{log}}{1.2} \exp \Biggl [ \dfrac{-1.04 (1 + \lambda)}{\lambda ( 1 - 0.62 \mu^*) - \mu^*} \Biggr ]$$

where $\mu^*$ describes the screened electron-electron interaction and $\ww_{\text{log}}$ is the logarithmic average of the phonon frequencies defined by:

$$\ww_{\text{log}} = \exp \Biggl [ \dfrac{2}{\lambda} \int \dfrac{\alpha^2F(\ww)}{\ww}\log(\ww)\dd\ww \Biggr ]$$

In pratical applications, $\mu^*$ is treated as an external parameter, typically between 0.1 and 0.2, that is adjusted to reproduce experimental results. The default value of eph_mustar is 0.1.

Before concluding this brief theoretical introduction, we would like to stress that the present formalism assumes a single superconducting gap with a weak dependence on $\kk$ so that it is possible to average the anisotropic equations over the FS. This approximation becomes valid in the so-called dirty-limit (metals with impurities) as impurities tend to smear out the anisotropy of the superconducting gap. A more detailed discussion about isotropic/anisotropic formulations can be found in [Margine2013].

Getting started

Note

Supposing you made your own installation of ABINIT, the input files to run the examples are in the ~abinit/tests/ directory where ~abinit is the absolute path of the abinit top-level directory. If you have NOT made your own install, ask your system administrator where to find the package, especially the executable and test files.

In case you work on your own PC or workstation, to make things easier, we suggest you define some handy environment variables by executing the following lines in the terminal:

export ABI_HOME=Replace_with_absolute_path_to_abinit_top_level_dir # Change this line
export PATH=$ABI_HOME/src/98_main/:$PATH      # Do not change this line: path to executable
export ABI_TESTS=$ABI_HOME/tests/             # Do not change this line: path to tests dir
export ABI_PSPDIR=$ABI_TESTS/Psps_for_tests/  # Do not change this line: path to pseudos dir

Examples in this tutorial use these shell variables: copy and paste the code snippets into the terminal (remember to set ABI_HOME first!) or, alternatively, source the set_abienv.sh script located in the ~abinit directory:

source ~abinit/set_abienv.sh

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

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

Most of the tutorials do not rely on parallelism (except specific tutorials on parallelism). However you can run most of the tutorial examples in parallel with MPI, see the topic on parallelism.

Before beginning, you might consider to work in a different subdirectory as for the other tutorials. Why not create Work_eph4isotc in $ABI_TESTS/tutorespfn/Input?

cd $ABI_TESTS/tutorespfn/Input
mkdir Work_eph4isotc
cd Work_eph4isotc

In this lesson we prefer to focus on e-ph calculations and the associated convergence studies. For this reason, we rely on pre-computed DEN.nc, DDB and DFPT potentials to bypass both the GS and the DFPT part. The DEN.nc file will be used to perform NSCF computations on arbitrarily dense $\kk$-meshes while the DFPT POT.nc files will be merged with the mrgdv utility to produce the DVDB database required by the EPH code.

Note that these files are not shipped with the official ABINIT tarball as they are relatively large. In order to run the examples of this tutorial, you need to download the files from this github repository. If git is installed on your machine, you can easily fetch the entire repository with:

git clone https://github.com/abinit/MgB2_eph4isotc.git

Alternatively, use wget:

wget https://github.com/abinit/MgB2_eph4isotc/archive/master.zip

or curl:

curl -L https://github.com/abinit/MgB2_eph4isotc/archive/master.zip -o master.zip

or simply copy the tarball by clicking the “download button” in the github web page, then unzip the archive and rename the directory with:

unzip master.zip
mv MgB2_eph4isotc-master MgB2_eph4isotc

Warning

The directory with the precomputed files must be located inside Work_eph4isotc and must be named MgB2_eph4isotc. The reason is that all the input files and examples of this tutorial read data from external files specified in terms of relative paths.

The AbiPy script used to perform the GS + DFPT steps is available here. In order to facilitate comparison with previous studies, we use norm-conserving pseudopotentials with a cutoff energy ecut of 38 Ha and the experimental parameters for hexagonal MgB$_2$ (a = 5.8317 and c/a= 1.1416). All the calculations are performed with a 12x12x12 ngkpt Gamma-centered $\kk$-grid for electrons (too coarse), and the Marzari smearing (occopt = 4) with tsmear = 0.02 Ha. The DFPT computations is done for 12 irreducible $\qq$-points corresponding to a $\Gamma$-centered 4x4x4 $\qq$-mesh (again, too coarse as we will see in the next sections).

Please keep in mind that several parameters have been tuned in order to reach a reasonable compromise between accuracy and computational cost so do not expect the results obtained at the end of the lesson to be fully converged. It is clear that, in real life, one should start from convergence studies for lattice parameters and vibrational properties as a function of the $\kk$-mesh and tsmear before embarking on EPH calculations.

Merging the DFPT potentials

To merge the DFPT potential files, copy the first input file in Work_eph4isotc with:

cp $ABI_TESTS/tutorespfn/Input/teph4isotc_1.abi .

and execute the mrgdv tool using:

mrgdv < teph4isotc_1.abi

The first line in teph4isotc_1.abi specifies the name of the output DVDB file, followed by the number of partial DFPT POT files and the full list of files we want to merge:

View tests/tutorespfn/Input/teph4isotc_1.abi

This step produces the teph4isotc_1_DVDB file that will be used in the next examples. Executing:

mrgdv info teph4isotc_1_DVDB

shows that all the independent atomic perturbations are available:

 The list of irreducible perturbations for this q vector is:
    1)  idir= 1, ipert=   1, type=independent, found=Yes
    2)  idir= 2, ipert=   1, type=symmetric, found=No
    3)  idir= 3, ipert=   1, type=independent, found=Yes
    4)  idir= 1, ipert=   2, type=independent, found=Yes
    5)  idir= 2, ipert=   2, type=symmetric, found=No
    6)  idir= 3, ipert=   2, type=independent, found=Yes
    7)  idir= 1, ipert=   3, type=symmetric, found=No
    8)  idir= 2, ipert=   3, type=symmetric, found=No
    9)  idir= 3, ipert=   3, type=symmetric, found=No

 All the independent perturbations are available
 Done

Analyzing electronic and vibrational properties

Before proceeding with e-ph calculations, it is worth spending some time to analyze the structural, electronic and vibrational properties of MgB$_2$ in more detail. We will be using the AbiPy scripts to post-process the data stored in the precomputed netcdf files.

To visualize the crystalline structure with e.g. vesta , use the abiview.py script and the structure command.

abiview.py structure MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/run.abi -a vesta

Full Formula (Mg1 B2)
Reduced Formula: MgB2
abc   :   3.086000   3.086000   3.523000
angles:  90.000000  90.000000 120.000000
Sites (3)
  #  SP           a         b    c
---  ----  --------  --------  ---
  0  Mg    0         0         0
  1  B     0.333333  0.666667  0.5
  2  B     0.666667  0.333333  0.5

Visualizing structure with: vesta
Writing data to: .xsf with fmt: xsf
Executing MacOSx open command: open -a vesta --args   /Users/gmatteo/git_repos/abinit_rmms/_build/tests/.xsf

Tip

Other graphical applications are supported, see abiview.py structure --help. Note that AbiPy assumes that these tools are already installed and the binaries can be found in $PATH. The same AbiPy command can be used with ABINIT input files as well as output files in netcdf format e.g. GSR.nc.

MgB$_2$ crystallizes in the so-called AlB$_2$ prototype structure with Boron atoms forming graphite-like (honeycomb) layers separated by layers of Mg atoms. This structure may be regarded as that of completely intercalated graphite with C replaced by B. Since MgB$_2$ is formally isoelectronic to graphite, its band dispersion is expected to show some similarity to that of graphite and graphite intercalation compounds.

Now use the abiopen.py script to plot the electronic bands stored in the GSR file produced by the NSCF calculation (the second task of the first Work i.e. w0/t1):

abiopen.py MgB2_eph4isotc/flow_mgb2_phonons/w0/t1/outdata/out_GSR.nc -e

This command produces the following figures:

The electronic properties of MgB$_2$ are intensively discussed in literature, see e.g. [Mazin2003]. The dispersion is quite similar to that of graphite with three bonding $\sigma$ bands corresponding to in-plane $sp^2$ hybridization in the boron layer and two $\pi$ bands (bonding $\pi$ and anti-bonding $\pi^*$) formed by B $p_z$ orbitals. The lowest band in the band plot is a fully occupied $\sigma$ band. The other two $\sigma$ bands are incompletely filled and correspond to the relatively flat states located slightly above $\ee_F$ along the $\Gamma-A$ segment. Ab-initio calculations showed that these states contribute the most to the e-ph coupling.

The other two bands crossing at the K point at around 1eV are $\pi-\pi^*$ bands.

To compute the so-called fatbands, one can use perforn a NSCF calculation with a $\kk$-path and prtdos 3. This is left as optional exercise. Fatbands plots produced with AbiPy starting from the FATBANDS.nc file are available here.

Tip

The high-symmetry $\kk$-path has been automatically computed by AbiPy when generating the Flow. To get the explicit list of high-symmetry $\kk$-point, use the abistruct.py script with the kpath command:

abistruct.py kpath mgb2_DEN.nc

...

# K-path in reduced coordinates:
 ndivsm 10
 kptopt -11
 kptbounds
    +0.00000  +0.00000  +0.00000  # $\Gamma$
    +0.50000  +0.00000  +0.00000  # M
    +0.33333  +0.33333  +0.00000  # K
    +0.00000  +0.00000  +0.00000  # $\Gamma$
    +0.00000  +0.00000  +0.50000  # A
    +0.50000  +0.00000  +0.50000  # L
    +0.33333  +0.33333  +0.50000  # H
    +0.00000  +0.00000  +0.50000  # A
    +0.50000  +0.00000  +0.50000  # L
    +0.50000  +0.00000  +0.00000  # M
    +0.33333  +0.33333  +0.00000  # K
    +0.33333  +0.33333  +0.50000  # H

The value of the DOS at the Fermi level plays a very important role when discussing superconducting properties. Actually this is one of the quantities that should be subject to convergence studies with respect to the $\kk$-grid before embarking on DFPT/EPH calculations. To compute the DOS with the tetrahedron method, use the ebands_edos command of abitk

abitk ebands_edos MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_GSR.nc

to read the eigenvalues in the IBZ from a GSR.n file (WFK.nc files are supported as well). This command produces a text file named out_GSR.nc_EDOS that can be visualized with:

abiopen.py out_GSR.nc_EDOS -e

Note that the phonon linewidths have a geometrical contribution due to Fermi surface since $\gamma_\qnu$ is expected to be large in correspondence of $\qq$ wave vectors connecting two portions of the FS. Strictly speaking this is true only if the e-ph matrix elements are constant. In real materials the amplitude of $g_{mn\nu}(\kk, \qq)$ is not constant and this may enhance/suppress the value $\gamma_\qnu$ for particular modes. Yet visualizing the FS is rather useful when discussing e-ph properties in metals. For this reason, it is useful to have a look at the FS with an external

Graphical tools for the visualization of the FS usually require an external file with electronic energies in the full BZ whereas ab-initio codes usually take advantage of symmetries to compute $\ee_\nk$ in the IBZ only. To produce a BXSF that can be used to visualize the FS with e.g. XCrysDen , use the ebands_bxsf command of the abitk utility located in src/98_main and provide a GSR.nc (WFK.nc) file with energies computed on a $\kk$-mesh in the IBZ:

abitk ebands_bxsf MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_GSR.nc

This command reconstructs the KS energy in the full BZ by symmetry and produces the out_GSR.nc_BXSF file that can be opened with XCrysDen using:

xcrysden --bxsf out_GSR.nc_BXSF

Other tools such as fermisurfer or pyprocar can read data in the BXSF format as well. For a minimalistic matplotlib-based approach, use can use the abiview.py script with the fs command:

abiview.py fs MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_GSR.nc

to visualize the FS in the unit cell of the reciprocal lattice.

Tip

The BXSF file can be produced at the end of the GS calculation or inside the EPH code by setting prtfsurf to 1 but abitk is quite handy if you already have a file and you don’t want to write a full Abinit input file and rerun the calculation from scratch.

We now focus on the vibrational properties of MgB$_2$. In principle, one can compute phonon frequencies either with anaddb or with the EPH code. However, for many applications, it is much easier to automate the entire process by invoking anaddb directly from the abiview.py script. To compute the phonon band structure using the DDB file produced on the 4x4x4 $\qq$-mesh, use:

abiview.py ddb MgB2_eph4isotc/flow_mgb2_phonons/w1/outdata/out_DDB

that produces the following figures:

The Projected Phonon DOS (right panel) shows that the low-energy vibrations (below 40 meV) are mainly of Mg character whereas the higher energy modes mainly involve motions of B atoms. Of particular interest for us is the two-fold degenerate band located at ~ 60 meV along the $\Gamma-A$ segment. These are the so-called $E_{2g}$ vibrational modes that couple strongly with the two $\sigma$ bands crossing $\ee_F$.

Tip

A more detailed analysis reveals that $E_{2g}$ involve in-plane stretching of the B-B bonds. To visualize the phonon modes, one can use the phononwebsite web app by H. Miranda.

abiview.py ddb MgB2_eph4isotc/flow_mgb2_phonons/w1/outdata/out_DDB

Opening URL: http://henriquemiranda.github.io/phononwebsite/phonon.html?json=http://localhost:8000/__abinb_workdir__/Mg1B2136lmnep.json
Using default browser, if the webpage is not displayed correctly
try to change browser either via command line options or directly in the shell with e.g:

     export BROWSER=firefox

Press Ctrl+C to terminate the HTTP server

If the page does not work, upload the json file directly.

The vibrational spectrum just obtained looks reasonable: no vibrational instability is observed and the acoustic modes tend to zero linearly for $|\qq| \rightarrow 0$ (note that the acoustic sum-rule is enforced by default via asr = 1). Yet this does not mean that our results are fully converged. This is clearly seen if we compare the phonon bands computed with a 4x4x4 and a 6x6x6 ab-initio $\qq$-mesh. Also in this case, we can automate the process by using the ddb command of the abicomp.py script that takes in input an arbitrary list of DDB files, calls anaddb for all the DDB files and finally compare the results:

abicomp.py ddb \
   MgB2_eph4isotc/flow_mgb2_phonons/w1/outdata/out_DDB \
   MgB2_eph4isotc/666q_DDB -e

The figure reveals that the vibrational spectrum interpolated from the 4x4x4 $\qq$-mesh underestimates the maximum phonon frequency. Other differences are visible above 40 meV, especially in the $E_{2g}$ modes.

Note

The 666q_DDB file was produced with the same AbiPy script by just changing the value of the ngqpt variable. The reason why we do not provide files obtained with a 6x6x6 $\qq$-sampling is that the size of the git repository including all the DFPT potentials is around 200 Mb. For this reason, we continue using the 4x4x4 DDB file but we should take this into account when comparing our results with previous works.

Our first computation of the isotropic Tc

For our first example, we use a relatively simple input file that allows us to introduce the most important variables and the organization of the results. Copy teph4isotc_2.abi in the working directory and run the code using:

abinit teph4isotc_2.abi > log 2> err

Tip

If you prefer, you can run it in parallel with e.g two MPI processes using:

mpirun -n 2 abinit teph4isotc_2.abi > log 2> err

without having to introduce any input variable for the MPI parallelization as the EPH code can automatically distribute the workload over k-points and spins. Further details concerning the MPI version are given in the last section of the tutorial

We now discuss the meaning of the different variables in more detail.

View tests/tutorespfn/Input/teph4isotc_2.abi

To activate the computation of $\gamma_{\qq\nu}$ in metals, we use optdriver = 7 and eph_task = 1. The location of the external DDB, DVDB and WFK files is specified via getddb_filepath getdvdb_filepath getwfk_filepath, respectively.

getddb_filepath  "MgB2_eph4isotc/flow_mgb2_phonons/w1/outdata/out_DDB"

getwfk_filepath  "MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_WFK.nc"

getdvdb_filepath "teph4isotc_1_DVDB"

The DDB and the WFK files are taken from the git repository while for the DVDB we use the file produced by mrgdv in the previous section (remember what we said about the use of relative paths in the input files). Note also the use of the new input variable structure (added in Abinit v9) with the abifile prefix

structure "abifile:MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_DEN.nc"

to read the crystalline structure from an external file so that we do avoid repeating the unit cell in every input file.

Next, we have the variables defining the coarse and the fine $\qq$-mesh (ddb_ngqpt and eph_ngqpt_fine, respectively):

ddb_ngqpt 4 4 4        # The ab-initio q-mesh (DDB, DVDB)
eph_ngqpt_fine 6 6 6   # Activate interpolation of DFPT potentials
                       # gamma_{q,nu} are computed of this fine grid.
dipdip 0               # No treatment of the dipole-dipole part. OK for metals

dipdip is set to zero as we are dealing with a metal and the inter-atomic force constants are short-ranged provided we ignore possible Kohn-anomalies [Kohn1959].

The $\kk$-point integration is performed with the Gaussian smearing (eph_intmeth == 1 and 0.1 eV for eph_fsmear).

eph_intmeth 1       # Gaussian method for double-delta integration.
eph_fsmear 0.1 eV   # Constant broadening in Gaussian function.
eph_fsewin 0.3 eV   # Energy window for wavefunctions.

Note

In order to accelerate the calculation, we have decreased eph_fsewin from its default value of 1.0 eV to 0.3 eV. This is possible when the Gaussian method is used since states whose energy is 3-4 standard deviation from the Fermi level give negligible contribution to the double delta integral. In other words, for the Gaussian technique, an optimal value of eph_fsewin can be deduced from eph_fsmear Unfortunately, this is not possible when the tetrahedron method is used and, to some extent, also when the adaptive broadening is used. Hopefully, the next versions of the code will provide … 0.04 Ha ~ 1 eV

Then we activate two tricks to make the calculation faster:

mixprec 1
boxcutmin 1.1

As explained in the documentation, using mixprec = 1 and 2 > boxcutmin >= 1.1 should not have significant effects on the final results yet these are not the default values as users are supposed to compare the results with/without these tricks (especially boxcutmin) before running production calculations.

Output files

We now discuss in more detail the main output file produced by the EPH run.

View tests/tutorespfn/Refs/teph4isotc_2.abo

After the standard section with info on the unit cell and the pseudopotentials, we find info on the electronic DOS:

 Linear tetrahedron method.
 Mesh step:  10.0 (meV) with npts: 2757
 From emin:  -5.1 to emax:  22.5 (eV)
 Number of k-points in the IBZ: 133
 Fermi level:   7.63392108E+00 (eV)
 Total electron DOS at Fermi level in states/eV:   7.36154553E-01
 Total number of electrons at eF:    8.0

- Writing electron DOS to file: teph4isotc_2o_DS1_EDOS

Then the code outputs some basic quantities concerning the Fermi surface and the method for the BZ integration of the double delta:

 ==== Fermi surface info ====
 FS integration done with adaptive gaussian method
 Total number of k-points in the full mesh: 1728
 For spin: 1
    Number of BZ k-points close to the Fermi surface: 291 [ 16.8 %]
    Maximum number of bands crossing the Fermi level: 3
    min band: 3
    Max band: 5

Then, for each $\qq$-point in the IBZ, the code outputs the values of $\ww_\qnu$, $\gamma_\qnu$ and $\lambda_\qnu$:

 q-point =    0.000000E+00    0.000000E+00    0.000000E+00
 Mode number    Frequency (Ha)  Linewidth (Ha)  Lambda(q,n)
    1        0.000000E+00    0.000000E+00    0.000000E+00
    2        0.000000E+00    0.000000E+00    0.000000E+00
    3        0.000000E+00    0.000000E+00    0.000000E+00
    4        1.444402E-03    4.898804E-07    3.731164E-03
    5        1.444402E-03    4.898804E-07    3.731164E-03
    6        1.674636E-03    2.955247E-15    1.674492E-11
    7        2.426616E-03    1.025439E-03    2.767185E+00
    8        2.426616E-03    1.025439E-03    2.767185E+00
    9        3.127043E-03    3.508418E-08    5.701304E-05

The frequencies of the acoustic modes at $\Gamma$ are zero (as they should be) since asr is automatically set to 1 so the acoustic rule is automatically enforced. Also, the linewidths of the acoustic modes are zero. In this case, however, the code uses a rather simple heuristic rule:

abiview.py ddb_asr MgB2_eph4isotc/flow_mgb2_phonons/w1/outdata/out_DDB

Finally, we have the value of the isotropic $\lambda$:

 lambda=   0.4286

The calculation has produced the following output files:

$ ls teph4isotc_2o_DS1*

teph4isotc_2o_DS1_A2F.nc       teph4isotc_2o_DS1_NOINTP_A2FW    teph4isotc_2o_DS1_PHDOS.nc
teph4isotc_2o_DS1_A2FW         teph4isotc_2o_DS1_NOINTP_PH_A2FW teph4isotc_2o_DS1_PHGAMMA
teph4isotc_2o_DS1_EBANDS.agr   teph4isotc_2o_DS1_PHBANDS.agr    teph4isotc_2o_DS1_PH_A2FW
teph4isotc_2o_DS1_EDOS         teph4isotc_2o_DS1_PHBST.nc

The files without the .nc extension are text files that can be plotted with e.g. gnuplot or xmgrace . More specifically,

Apple

Pomaceous fruit of plants of the genus Malus in the family Rosaceae.

The A2F.nc netcdf file stores all the results of the calculation in a format that can can be visualized with abiopen.py :

abiopen.py teph4isotc_2o_DS1_A2F.nc -e

that produces the following plot:

Our first results should be interpreted with a critical eye due to coarse sampling used, yet our $\alpha^2F(\ww)$ already shows features that are observed in other (more accurate) calculations reported in the literature: a broad peak at ~70 meV that gives the most important contribution to the total $\lambda$ and a second smaller peak at ~70 meV. The sharp delta-like peaks are an artifact of the linear interpolation and will hopefully disappear if denser $\qq$-meshes are used. Our initial values for $\lambda$ and $\ww_{log}$ are:

a2f(w) on the [4 4 4] q-mesh (ddb_ngqpt|eph_ngqpt)
Isotropic lambda: 0.39, omega_log: 0.067 (eV), 780.042 (K)k

If we compare our results for $\lambda$ with those reported in [Margine2013],

we see that our first calculation underestimates $\lambda$ by almost a factor two. Note, however, that all the previous studies used much denser samplings and all of them use at least a 6x6x6 $\qq$-mesh for phonons. It is clear that we need to densify the BZ sampling to get more reliable results yet our results are not that bad considering that the calculation took less than XXX minutes in sequential.

Using the tetrahedron method

In this section, we repeat the calculation done in teph4isotc_2.abi but now with the optimized tetrahedron scheme by [Kawamura2014]. This test will show that (i) phonon linewidths are very sensitive to the $\kk$-mesh and the integration scheme and that (ii) one has to test different values of eph_fsewin to find the optimizal one.

The reason is that the optimized tetrahedron scheme constructs a third-order interpolation function with 20 $\kk$-points so states that are relatively far from the Fermi level contributes to the integrand. If eph_fsewin is too small, part of the weigt is lost. To find an optimal value, we can use:

ndtset 4
eph_intmeth 2   # Tetra
eph_fsewin: 0.4 eV
eph_fsewin+ 0.4 eV

We do not provide an input file to perform such test as one can easily change teph4isotc_2.abi. and the run the calculation in parallel. To compare the results, we use the a2f command of the abicomp.py script and pass the list of A2F.c files:

abicomp.py a2f teph4isotc_2o_DS*_A2F.nc -e

Preparing the convergence study wrt the k-mesh

The NSCF computation of the WFK becomes quite CPU-consuming and memory-demanding if dense $\kk$-meshes are needed. Fortunately, we can optimize this part since the computation of $\gamma_\qnu$ requires the knowledge of Bloch states inside a relatively small energy window around $\ee_F$. Similarly to what is done in the eph4mob tutorial, we can therefore take advantage of the star-function SKW interpolation to find the $\kk$ wavevectors whose energy is inside the sigma_erange energy window around the Fermi level. The choice of an optimal window is discussed afterwards.

First of all, we strongly recommend to test whether the SKW interpolation can reproduce the ab-initio results with reasonable accuracy by comparing the ab-initio band structure with the interpolated one by using abitk with the skw_compare command:

abitk skw_compare \
    MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_GSR.nc \
    MgB2_eph4isotc/flow_mgb2_phonons/w0/t1/outdata/out_GSR.nc  --is-metal

The fist argument is a GSR.nc (WFK.nc) file with the energies in the IBZ that will be used to build the star-function interpolant. The quality of the interpolant obviously depends on the density of this $\kk$-mesh. The second file contains the ab-initio eigenvalues along a $\kk$-path computed with a standard NSCF run. In this example, we are using precomputed GSR.nc files with a 12x12x12 $\kk$-mesh for the SKW interpolation.

Note

Note the use of the --is-metal option else abitk will complain that it cannot compute the gaps as the code by default assumes a semiconductor.

The skw_compare command has produced two netcdf files (abinitio_EBANDS.nc and skw_EBANDS.nc) that we can be compared with the abicomp.py script and the ebands command:

abicomp.py ebands abinitio_EBANDS.nc skw_EBANDS.nc -p combiplot

to produce the following plot:

The figure shows that the SKW interpolant nicely reproduces the ab-initio dispersion around $\ee_F$. Discrepancies between the ab-initio results and the interpolated values are visible in correspondence of band-crossings. This is expected since SKW is a Fourier-based approach and band-crossing leads to a non-analytic behaviour of the signal that cannot be reproduced with a finite number of Fourier components. Fortunately, these band crossings are relatively far from the Fermi level hence they do not enter into play in the computation of superconducting properties.

Tip

Should the fit be problematic, use the command line options:

abitk skw_compare IBZ_WFK KPATH_WFK [--lpratio 5] [--rcut 0.0] [--rsigma 0]

to improve the interpolation and take note of the SKW parameters as the same values should be used when calling ABINIT with the einterp input variable. The worst case scenario of band crossings and oscillations close to the Fermi level can be handled by enlarging the sigma_erange energy window.

At this point, we are confident that the SKW interpolation is OK and we can use it to locate the $\kk$-wavevectors of a much denser $\kk$-mesh whose energy is inside the sigma_erange window around the Fermi level. This is done in the teph4isotc_4.abi input file:

View tests/tutorespfn/Input/teph4isotc_3.abi

The most important section of the input file is reproduced below:

optdriver 8
wfk_task "wfk_kpts_erange"
getwfk_filepath  "MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_WFK.nc"

# Define fine k-mesh for the SKW interpolation
sigma_ngkpt   24 24 24
sigma_nshiftk 1
sigma_shiftk  0 0 0

sigma_erange -0.3 -0.3 eV  # Select kpts in the fine mesh within this energy window.
einterp 1 5 0 0            # Parameters for star-function interpolation (default values)

The first part (optdriver and wfk_task) activates the computation of the KERANGE.nc file with ab-initio energies in the IBZ taken from getwfk_filepath. The three variables sigma_ngkpt, sigma_nshiftk and sigma_shiftk define the final dense $\kk$-mesh. Finally, sigma_erange defines the energy window around the Fermi level while einterp lists the parameters passed to the SKW routines In this example, we are using the default values. You may need to change einterp if you had to use different options for the SKW interpolation when using abitk skw_compare.

Important

When dealing with metals, both entries in sigma_erange must be negative so that the energy window is refered to $\ee_F$ and not to the CBM/VBM used in semiconductors. Remember to use a value for the window that is reasonably large in order to account for possible oscillations and/or inaccuracies of the SKW interpolation around $\ee_F$.

View tests/tutorespfn/Refs/teph4isotc_3.abo

View tests/tutorespfn/Refs/teph4isotc_4.abo

Once we have the KERANGE.nc file, we can use it to perform a NSCF calculation to generate a customized WFK file on the dense $\kk$-mesh: This is done in the teph4isotc_4.abi input file:

View tests/tutorespfn/Input/teph4isotc_4.abi

to perform a NSCF calculation with kptopt 0 to produce a new WFK file on the dense $\kk$-mesh.

iscf  -2
tolwfr 1e-18
kptopt 0                        # Important!

with the getkerange_filepath:

Note

Note how we use getden_filepath and the syntax:

getkerange_filepath "teph4isotc_3o_DS1_KERANGE.nc"

# Read DEN file to initialize the NSCF run.
getden_filepath "MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_DEN.nc"

# Init GS wavefunctions from this file (optional).
getwfk_filepath "MgB2_eph4isotc/flow_mgb2_phonons/w0/t0/outdata/out_WFK.nc"

to start the NSCF run from the precomputed GS DEN file, initialize the trial wafefunctions from getwfk_filepath to accelerate a bit the calculation.

TODO: Discussion about energy range and integration scheme.

Convergence study wrt to the k/q-mesh

At this point, we can use the WFK file to perform EPH calculations with denser $\kk$-meshes. We will be using settings similar to the ones used in teph4isotc_2.abi except for the use of eph_ngqpt_fine, getwfk_filepath and ngkpt:

View tests/tutorespfn/Input/teph4isotc_5.abi

Notes on the MPI parallelism

EPH calculations performed with eph_task = 1 support 4 different levels of MPI parallelism and the number of MPI processes for each level can be specified via the eph_np_pqbks input variable. Note that we wrote 4 MPI levels instead of 5 because, for isotropic $T_c$ calculations, the band parallelism is not supported, only atomic perturbations, $\qq$-points, $\kk$-points and spins (if any).

The eph_np_pqbks variable is optional in the sense that whatever number of MPI processes you use, the EPH code will be able to parallelize the calculation by distributing over $\kk$-points and spins (if any). This distribution, however, may not be optimal, especially if you have lot of CPUs available and/or you need to decrease the memory requirements per CPU.

To decrease memory, we suggest to activate the parallelism over perturbations (up to 3 x natom) to make the memory for $W(\rr, \RR, \text{3 x natom})$ scale. If memory is not of concern, activate the $\qq$-point parallelism for better efficiency.

Important

Note that the code is not able to distribute the memory for the wavefunctions although only the states in the eph_fsewin energy window are read and stored in memory.