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<html><head><title>Tutorial "eph"</title></head>
<body bgcolor="#ffffff">

<hr>
<h1>ABINIT, eph lesson of the tutorial: </h1>
<h2>Electron-Phonon interaction and superconducting properties of Al.
</h2>
<hr>
<p>This lesson demonstrates how to obtain
the following physical properties, for a metal :
</p><ul>
<li>the phonon linewidths (inverse lifetimes) due to the electron-phonon interaction</li>
<li>the Eliashberg spectral function</li>
<li>the electron-phonon coupling strength</li>
<li>the McMillan critical temperature</li>
<li>the resistivity and electronic part of the thermal conductivity</li>
</ul>
Here you will learn to use the electron-phonon coupling part of the anaddb
utility. This implies a preliminary calculation of the electron-phonon matrix
elements and phonon frequencies and eigenvectors, from a standard ABINIT
phonon calculation, which will be reviewed succinctly.

<p>This lesson should take about 1 hour.

</p><h5>Copyright (C) 2005-2014 ABINIT group (MVer)
<br> This file is distributed under the terms of the GNU General Public License, see
~abinit/COPYING or <a href="http://www.gnu.org/copyleft/gpl.txt">
http://www.gnu.org/copyleft/gpl.txt </a>.
<br> For the initials of contributors, see ~abinit/doc/developers/contributors.txt .
</h5>

<script type="text/javascript" src="list_internal_links.js"> </script>

<br>&nbsp;
<h3><b>Content of lesson Elphon</b></h3>

<ul>
	<li><a href="lesson_eph.html#E1">1</a> Calculation of the ground state and phonon structure of fcc Al.</li>
	<li><a href="lesson_eph.html#E2">2</a> Merging of the 2DTE DDB files using MRGDDB.</li>
	<li><a href="lesson_eph.html#E3">3</a> Merging of the electron-phonon matrix elements using MRGGKK.</li>
	<li><a href="lesson_eph.html#E4">4</a> Basic ANADDB calculation of electron-phonon quantities.</li>
	<li><a href="lesson_eph.html#E5">5</a> Convergence tests of the integration techniques.</li>
	<li><a href="lesson_eph.html#E6">6</a> Transport quantities within Boltzmann theory.</li>
</ul>

<hr>

<p><a name="E1"></a><br> </p>

<h3><b>1. Calculation of the ground state and phonon structure of fcc Al.</b> </h3>
<h3>&nbsp;</h3>

<p> <i>Before beginning, you might consider making a different subdirectory to
	work in.
Why not create "Work_eph" in ~abinit/tests/tutorespfn/Input ?
</i> <br> </p>

<p> It is presumed that the user has already followed the Tutorials
      <a href="lesson_rf1.html" target="kwimg">RF1</a>
and <a href="lesson_rf2.html" target="kwimg">RF2</a>, and understands the calculation of
ground state and response function (phonon) properties with ABINIT.</p>

<p> The file ~abinit/tests/tutorespfn/Input/teph_1.files lists the file names and root names for the first run
(GS+perturbations). You can copy it to the working directory.
You can also copy the file ~abinit/tests/tutorespfn/Input/teph_1.in to your working
directory.  This is your input file.</p>

<li><i>cp ../teph_1.files . </i></li>
<li><i>cp ../teph_1.in . </i></li>

<p>You can immediately start this run - the input files
will be examined later ... </p>

<pre>
../../abinit < teph_1.files > tmp-log &
</pre>

<h3>Dataset Structure and Flow</h3>
<p>
The teph_1.in file contains a number of datasets (DS). These will perform the
ground state (DS 1), then the phonon perturbations (DS 2-4), for a cell of FCC
aluminium. The DDK perturbation (DS 5) is also calculated, and will be used
to obtain the Fermi velocities for the transport calculations in 
<a href="lesson_eph.html#E6">Section 6</a>.
</p>
<p>
Once these are done, abinit calculates the wave functions on the full grid of
k-points (using <a href="../input_variables/varbas.html#kptopt">kptopt 3</a>)
in DS 6: these will be used to calculate the electron-phonon matrix elements.
In a full calculation the density of k-points should be increased significantly
here and for the following datasets.  For DS 1-5 only the normal convergence of
the phonon frequencies should be ensured. In DS 7-10 only the matrix elements
are calculated, for the electron-phonon coupling and for the DDK
(position/momentum matrix elements), on the dense and complete grid of k-points
from DS 6. Note that the separation of the matrix element calculation is new from
version 7.6.
</p>
<p>
The important variable for electron-phonon coupling calculations is <a
	href="../input_variables/varfil.html#prtgkk"
	target="kwimg">prtgkk</a>. This prints out files suffixed GKK,
which contain the electron-phonon matrix elements. The matrix elements only
depend on the self-consistent perturbed density (files suffixed 1DENx), which
we get from DS 2-4 (linked by variable <a
        href="../input_variables/varfil.html#get1den"
	target="kwimg">get1den</a>). These can therefore be calculated on
arbitrarily dense k-point meshes. Even better, only a single step is needed,
since no self-consistency is required. To enforce the calculation of all the
matrix elements on all k-points, symmetries are disabled when <a
        href="../input_variables/varfil.html#prtgkk"
	target="kwimg">prtgkk</a> is set to 1, so be sure not to use it during
the normal self-consistent phonon runs DS 2-4. Again this is very different
from versions before 7.6.
</p>


<h3>Convergence</h3>
<p>
The calculation is done using minimal values of a number of parameters, in
order to make it tractable in a time appropriate for a tutorial. The results
will be completely unconverged, but by the end of the lesson you should know
how to run a full electron phonon calculation, and be able to improve the
convergence on your own.
</p>
<p>
Edit the file teph_1.in. We now examine several variables.
The kinetic energy cutoff <a
	href="../input_variables/varbas.html#ecut"
	target="kwimg">ecut</a> is a bit low, and the number of k-points
(determined by <a href="../input_variables/varbas.html#ngkpt"
	target="kwimg">ngkpt</a>) is much too low. Electron-phonon calculations
require a very precise determination of the Fermi surface of the metal. This
implies a very dense k-point mesh, and the convergence of the grid must be
checked. In our case, for Al, we will use a (non-shifted) 4x4x4 k-point grid,
but a converged calculation needs more than 16x16x16 points. This will be
re-considered in section <a href="lesson_eph.html#E5">5</a>. The q-point grid
will be 2x2x2.  It must be a sub-grid of the full k-point grid, and must
contain the &Gamma; point.
</p>

<p>
The value of <a
	href="../input_variables/varbas.html#acell"
	target="kwimg">acell</a> is fixed to a rounded value from experiment.
It, too, should be converged to get physical results (see <a
	href="lesson_base3.html"
	target="kwimg"> Tutorial 3</a>).
</p>

<p> 
Note that the value of 1.0E-14 for <a
	href="../input_variables/varbas.html#tolwfr"
	target="kwimg">tolwfr</a> is tight, and should be even lower (down to
1.0E-20 or even 1.0E-22). This is because the wavefunctions will be used later
explicitly in the matrix elements for ANADDB, as opposed to only energy values
or densities, which are averages of the wavefunctions and eigenenergies over
k-points and bands. Electron-phonon quantities are delicate sums of a few of
these small matrix elements (those near the Fermi surface), so each matrix
element must be accurate. You can however set <a
	href="../input_variables/varfil.html#prtwf"
	target="kwimg">prtwf</a> to 0 in the phonon calculations,
and avoid saving huge perturbed wavefunction files to disk (you only need to keep
the ground state wave functions, with prtwf1 1).
</p>

<h3>Execution</h3>
<p>
Run the first input (a few seconds on a recent PC), and you should obtain a
value of <pre> etotal -2.0828579336121 Ha </pre> for the energy at the end of
DATASET 1. The following datasets calculate the second order energy variations
under atomic displacement in the three reduced directions of the fcc unit cell.
This is done for three different phonons, Gamma, (1/2,0,0), and X=(1/2,1/2,0),
which generate the 2x2x2 q-point grid (take care with the reduced coordinates
of the reciprocal space points! They are not along cartesian directions, but
along the reciprocal space lattice vectors.). The whole calculation follows the
same lines as <a
	href="lesson_rf1.html"
target="kwimg"> Tutorial RF1</a>. As an example, DATASET 3 calculates the
perturbed wavefunctions at k+q, for k in the ground state k-point mesh, and q=(1/2,0,0).
Then, DATASET 3 calculates <pre> 2DTE 0.80951882353353 </pre> for the
second-order energy variation for movement of the (unique) atom along the first
reduced direction for q=(1/2,0,0). The main differences with <a
	href="lesson_rf1.html"
	target="kwimg"> Tutorial RF1</a> are that
Given we are dealing with a metal, no perturbation wrt electric fields is
considered ; However, if you want to do transport calculations, you need the
ddk calculation anyway, to get the electron band velocities. This is added in
dataset 5.

In the standard case, ABINIT uses symmetry operations and non-stationary
expressions to calculate a minimal number of 2DTE for different mixed second
derivatives of the total energy. In our case we use the first derivatives, and
they must all be calculated explicitly. </p>

<p> You are now the proud owner of 9 first-order matrix element files (suffixed
_GKKx), corresponding to the three directional perturbations of the atom at
each of the three q-points. The _GKK files contain the matrix elements of the electron-phonon
interaction, which we will extract and use in the following. Besides the _GKK
files there are the _DDB files for each perturbation which contain the 2DTE for
the different phonons wavevectors q.</p>

<p></p>

<hr>
<p><a name="E2"></a><br></p>

<h3><b>2. Merging of the 2DTE _DDB files using MRGDDB.</b> </h3>
<h3>&nbsp;</h3>

<p> You can copy the following content to a file teph_2.in within your working directory:</p>
<pre>
teph_2.ddb.out
Total ddb for Al FCC system
3
teph_1o_DS2_DDB
teph_1o_DS3_DDB
teph_1o_DS4_DDB
</pre>
<p> This is your input
file for the <a href="../users/mrgddb_help.html"
	target="kwimg">MRGDDB</a> utility, which will take the different _DDB
files and merge them into a single one which ANADDB will use to determine the
phonon frequencies and eigenvectors. teph_2.in contains the name of the final
file, a comment line, then the number of _DDB files to be merged and their
names.</p>

</p><a href="../users/mrgddb_help.html"
target="kwimg">MRGDDB</a> is run with the command <pre> mrgddb < teph_2.in </pre>
It runs in a few seconds.<p>

<p>
</p><hr>



<p><a name="E3"></a><br>

</p><h3><b>3. Extraction and merging of the electron-phonon matrix elements using MRGGKK.</b> </h3>
<h3>&nbsp;</h3>

<p> A merge similar to that in the last section must be carried out for the
electron-phonon matrix elements. This is done using the MRGGKK utility, and its
input file is ~abinit/tests/tutorespfn/Input/teph_3.in, shown below
<pre>
teph_3o_GKK.bin   # Name of output file
0                    # binary (0) or ascii (1) output
teph_1o_DS1_WFK   # GS wavefunction file
0  9 9               # number of 1WF files, of GKK files, and of perturbations in the GKK files
teph_1o_DS2_GKK1  # names of the 1WF then the (eventual) GKK files
teph_1o_DS2_GKK2
...</pre>
The matrix element sections of all the _GKK files will be
extracted and concatenated into one (binary) file, 
here named teph_3o_GKK.bin.  The following lines in teph_3.in give the output
format (0 for binary or 1 for ascii), then the name of the ground state wavefunction file. The
fourth line contains 3 integers, which give the number of _1WF files
(which can also be used to salvage the GKK), the
number of _GKK files, and the number of perturbations in the _GKK files. Thus,
MRGGKK functions very much like <a
	href="../users/mrgddb_help.html"
	target="kwimg">MRGDDB</a>, and can merge _GKK files
which already contain several perturbations (q-points or atomic displacements).
Finally, the names of the different _1WF and _GKK files are listed.</p>

<p> MRGGKK will run on this example in a few seconds. In more general cases, the
runtime will depend on the size of the system, and for a large number of bands or
k-points can extend up to 20 minutes or more. </p>
</p>

<hr>

<p><a name="E4"></a><br>

</p><h3><b>4. Basic ANADDB calculation of electron-phonon quantities.</b> </h3>
<h3>&nbsp;</h3>

<p>The general theory of electron-phonon coupling and Eliashberg
superconductivity is reviewed in <i>Theory of Superconducting Tc, P.B. Allen
	and B. Mitrovic in Sol. State Phys. <b>37</b> (1982) ed. Ehrenreich,
	Seitz, and Turnbull</i>. The first implementations similar to that in
ABINIT are those in <i>S.Y. Savrasov and D.Y. Savrasov, Phys. Rev. B
	<b>54</b> 16487 (1996)</i> and <i>A.Y. Liu and A.A. Quong, Phys.
	Rev. B <b>53</b> R7575-R7579 (1996)</i>.</p>

<p> File ~abinit/tests/tutorespfn/Input/teph_4.in contains the input needed by ANADDB to carry out the
calculation of the electron-phonon quantities. ANADDB takes a files file,
just like ABINIT, which tells it where to find the input, ddb, and gkk files,
and what to name the output, thermodynamical output, and electron phonon
output files. ~abinit/tests/tutorespfn/Input/teph_4.files is your files file for ANADDB. You can
edit it now.</p>

<p> The new variables are at the head of the file:
<pre>
# turn on calculation of the electron-phonon quantities
elphflag 1

# Path in reciprocal space along which the phonon linewidths
#  and band structure will be calculated
nqpath 7
qpath
 0.0 0.0 0.0
 1/2 1/2 0.0
 1   1   1
 1/2 1/2 1/2
 1/2 1/2 0.0
 1/2 3/4 1/4
 1/2 1/2 1/2

# Coulomb pseudopotential parameter
mustar 0.136</pre> <a
href="../users/anaddb_help.html#elphflag"
target="kwimg"> elphflag</a> is a flag to turn on the calculation of the
electron-phonon quantities. The first quantities which will be calculated are
the phonon linewidths along a path in reciprocal space (exactly like the band
structure in <a
	href="lesson_base3.html#35"
	target="kwimg">Lesson 3.5</a>). The path is specified by the variable
<a href="../users/anaddb_help.html#qpath"
	target="kwimg">qpath</a> giving the apexes of the path in reciprocal
space, which are usually special points of high symmetry. The number of points
is given by <a href="../users/anaddb_help.html#nqpath"
	target="kwimg">nqpath</a>. Note that qpath can be used in normal phonon
band structure calculations as well, provided that <a href="../users/anaddb_help.html#qpath"
        target="kwimg">q1phl</a> is omitted from the
input file (the latter overrides qpath). The phonon linewidths are printed to a file
suffixed _LWD.</p>

<p>The phonon linewidths are proportional to the electron phonon coupling, and
still depend on the phonon wavevector q. The other electron-phonon calculations
which are presently implemented in ANADDB, in particular for superconductivity,
determine isotropic quantities, which are averaged over the Fermi surface and
summed over q-points.  Integrating the coupling over reciprocal space, but
keeping the resolution in the phonon mode's energy, one calculates the
Eliashberg spectral function &alpha;&#178;F. The &alpha;&#178;F function is
similar to the density of states of the phonons, but is weighted according to
the coupling of the phonons to the electrons. It is output to a file with
suffix _A2F, which is ready to be represented using any graphical software
(Xmgr, matlab, OpenDX...). The first inverse moment of &alpha;&#178;F gives the
global coupling strength, or mass renormalization factor, &lambda;. From
&lambda;, using the McMillan formula (<i>Phys. Rev. <b>167</b> 331-344
(1968)</i>) as modified by Allen and Dynes (<i>Phys. Rev. B <b>12</b> 905
(1975)</i>), ANADDB calculates the critical temperature for superconductivity.
The formula contains an adjustable parameter &mu;* which approximates the effect
of Coulomb interactions, and is given by the input variable <a
	href="../users/anaddb_help.html#mustar"
	target="kwimg">mustar</a>. For Al with the k-point grid given and a
value of &mu;=0.136 the ANADDB output file shows the following values
<pre>
 mka2f: lambda &lt;omega^2&gt; =     8.890935E-07
 mka2f: lambda &lt;omega^3&gt; =     7.653347E-10
 mka2f: lambda &lt;omega^4&gt; =     8.752205E-13
 mka2f: lambda &lt;omega^5&gt; =     1.124640E-15
 mka2f: isotropic lambda =     1.005830E+01
 mka2f: omegalog  =     1.706719E-04 (Ha)     5.389384E+01 (Kelvin)
 mka2f: input mustar =     1.360000E-01
 mka2f: MacMillan Tc =     4.004565E-05 (Ha)     1.264540E+01 (Kelvin)
</pre>
As expected, this is a fairly bad estimation of the experimental value of 1.2
K. The coupling strength is severely overestimated (experiment gives 0.44), and
the logarithmic average frequency is too low, but not nearly enough to
compensate &lambda;. Aluminum is a good case in which things can be improved,
easily because its Fermi surface is isotropic and the coupling is weak.</p>

</p><hr>



<p><a name="E5"></a><br>

</p><h3><b>5. Convergence tests of the integration techniques.</b> </h3>
<h3>&nbsp;</h3>

<p> In section <a href="lesson_eph.html#E4">4</a>, we used the default
method for integration on the Fermi surface, which employs a smearing of the
DOS and attributes Gaussian weights to each k-point as a function of its
distance from the Fermi surface. Another efficient method of integration in
k-space is the tetrahedron method, which is also implemented in ANADDB, and can
be used by setting <a
	href="../users/anaddb_help.html#telphint"
	target="kwimg">telphint</a> = 0. In this case the k-point grid must be
specified explicitly in the input, repeating the variable <a
	href="../users/anaddb_help.html#kptrlatt"
	target="kwimg">kptrlatt</a> from the ABINIT output, so that ANADDB can
re-construct the different tetrahedra which fill the reciprocal unit cell. In
the Gaussian case, the width of the smearing can be controlled using the input
variable <a href="../users/anaddb_help.html#elphsmear"
	target="kwimg">elphsmear</a>. </p>

<p> To test our calculations, they should be re-done with a denser k-point grid
and a denser q-point grid, until the results (&alpha;&#178;F or &lambda;) are
converged. The value of <a
	href="../users/anaddb_help.html#elphsmear"
	target="kwimg">elphsmear</a> should also be checked, to make sure that
it does not affect results. Normally, the limit for a very small <a
	href="../users/anaddb_help.html#elphsmear"
	target="kwimg">elphsmear</a> and a very dense k-point grid is
the same as the value obtained with the tetrahedron method (which usually converges
with a sparser k-point grid). </p>

<p>Edit input file ~abinit/tests/tutorespfn/Input/teph_5.in and you will see
the main difference with teph_4.in is the choice of the tetrahedron
integration method. If you are patient, save the output _LWD and _A2F files and
run the full lesson again with a denser k-point grid (say, 6x6x6) and you will
be able to observe the differences in convergence.</p>

<hr>
<p><a name="E6"></a><br>
<h3><b>6. Transport quantities within Boltzmann theory.</b> </h3>
<h3>&nbsp;</h3>

<p>
The electron-phonon interaction is also responsible for the resistivity of
normal metals and related phenomena.  Even in a perfect crystal, interaction
with phonons will limit electron life times (and vice versa). This can be
calculated fairly simply using the Boltzmann theory of transport with first order
scattering by phonons (see, e.g., "Electrons and Phonons" by Ziman).
</p>
<p>
The additional ingredient needed to calculate transport quantities (electrical
resistivity, heat conductivity limited by electron-phonon coupling) is the
Fermi velocity, ie the group velocity of a wavepacket of electrons placed at
the Fermi surface. This is the "true" velocity the charge will move at, once
you have displaced the Fermi sphere a little bit in k space (see, e.g. Ashcroft
and Mermin as well).  The velocity can be related simply to a commutator of the
position, which is also used for dielectric response, using a DDK calculation
(see <a href="./lesson_rf1.html#5">the Gamma point phonon tutorial</a>).  The
phonon calculation at Gamma need not include the electric field (this is a
metal after all, so the effect on the phonons should be negligible), but we
need an additional dataset to calculate the 3 DDK files along the 3 primitive
directions of the unit cell. To be more precise, just as for the el-ph matrix
elements, we do not need the perturbed wavefunctions, only the perturbed
eigenvalues. Calculating the DDK derivatives with <a
	href="../input_variables/varfil.html#prtgkk" target="kwimg">prtgkk</a>
set to 1 will output files named _GKKxx (xx=3*natom+1 to 3*natom+3) containing
the matrix elements of the ddk perturbation (these are basically the first part
of the normal DDK files for E field perturbation, without the wave function
coefficients). 
</p>
<p>
The anaddb "files" file must specify where the ddk files are, so anaddb can
calculate the Fermi velocities. It actually reads:
<pre>
teph_6.in
teph_6.out
teph_2.ddb.out
moldyn
teph_3o_GKK.bin
teph.ep
teph_6.ddk
</pre>
where the last line is the name of a small file listing the 3 DDK files to be used:
<pre>
teph_1_DS5_GKK4
teph_1_DS5_GKK5
teph_1_DS5_GKK6
</pre>
The abinit input file teph_1.in already obtained the DDK files from the
additional dataset, DS5, with the following lines of teph_1.in:
<pre>
tolwfr5 1.0d-14
qpt5 0 0 0
rfphon5 0
rfelfd5 2
</pre>
Copy the additional .ddk file from the tests/tutorespfn/Inputs directory, and run
anaddb with the new "files" file. The input for teph_6 has added to
teph_5.in the following 2 lines:
<pre>
ifltransport 1
ep_keepbands 1
</pre>
and has produced a number of additional files:
<ul>
<li> *_A2F_TR* contain the equivalent Eliashberg spectral functions with Fermi
speed factors (how many phonons do we have at a given energy, how much do they
couple with the electrons, and how fast are these electrons going). 
Integrating with appropriate functions of the phonon energy, one gets:</li>
<li>the resistivity as a function of temperature (teph_6.out_ep_RHO and
figure) and</li>
<li>the thermal conductivity as a function of temperature
(teph_6.out_ep_WTH) but ONLY the electronic contribution. You are still
missing the phonon-phonon interactions, which are the limiting factor in
the thermal conductivity beyond a few 100 K. For metals at even higher temperature
the electrons will often dominate again as they contain more degrees of freedom.</li>
</ul>
</p>
<img style="width: 640px; height: 480px;" alt="Resistivity of Al as a function of T" src="images/teph_6_RHO.png" />
<p>
The high T behavior is necessarily linear if you include only first order e-p
coupling and neglect the variation of the GKK off of the Fermi surface.
The inset shows the low T behavior, which is not a simple
polynomial (with simple models it should be T^3 or T^5 - see Ashcroft and
Mermin). See the Savrasov paper above for reference values in simple metals
using well converged k- and q- point grids.
</p>
<p>
Finally, note that the _RHO and _WTH files contain a series of tensor
components, for the resistivity tensor (2 1 = y x or the current response along
y when you apply an electric field along x). In most systems the tensor should
be diagonal by symmetry, and the value of off-diagonal terms gives
an estimate of the numerical error.
</p>


<br>
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