Phonons and Born effective charges

This lesson discusses how to compute phonon band structures, DOS and Born effective charges with Abinit and AbiPy. The discussion closely follows the second lesson on DFPT available on the Abinit web site. More specifically, we will discuss how to

• Perform a convergence study for the phonon frequencies at \(\Gamma\) as function of ecut

• Compute the full phonon band structure of AlAs with the inclusion of LO-TO splitting

• Obtain thermodynamic properties within the harmonic approximation

We assume that you have read the references mentioned in the first Abinit lesson on DFPT. You might find additional material, related to the present section, in the following references:

• Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory

• Phonons and related crystal properties from density-functional perturbation theory

If you are already familiar with python and AbiPy-Abinit are already installed and configured, you may want to use directly the command line interface. See the README.md file in the directory of this lesson explaining how to analyze the data from the shell using ipython and matplotlib.

Note

AbiPy provides two different APIs to produce figures either with matplotlib or with plotly. In this tutorial, we will use the plotly API as much as possible although it should be noted that not all the plotting methods have been yet ported to plotly. AbiPy uses a relatively simple rule to differentiate between the two plotting libraries: if an object provides an obj.plot method producing a matplotlib plot, the corresponding plotly version (if any) is named obj.plotly. Note that plotly requires a web browser hence the matplotlib version is still valuable if you need to visualize results on machines in which only the X-server is available.

Phonon frequencies at \(\Gamma\) as function of ecut

Before starting, we need to import the python modules and the functions we will need in the notebook:

import numpy as np
import warnings

warnings.filterwarnings("ignore") # to get rid of deprecation warnings

from abipy import abilab
abilab.enable_notebook() # This line tells AbiPy we are running inside a notebook
import abipy.flowtk as flowtk

# This line configures matplotlib to show figures embedded in the notebook.
# Replace `inline` with `notebook` in classic notebook
%matplotlib inline

# Option available in jupyterlab. See https://github.com/matplotlib/jupyter-matplotlib
#%matplotlib widget

and an useful function from the lesson_dfpt module that will be used to generate our DFPT flows:

from lesson_dfpt import make_scf_input
abilab.print_source(make_scf_input)

def make_scf_input(ecut=2, ngkpt=(4, 4, 4)):
    """
    This function constructs an `AbinitInput` to perform a GS-SCF calculation in crystalline AlAs.

    Args:
        ecut: cutoff energy in Ha.
        ngkpt: 3 integers specifying the k-mesh for the electrons.

    Return:
        `AbinitInput` object
    """
    # Initialize the AlAs structure from an internal database. Use the pseudos shipped with AbiPy.
    gs_inp = abilab.AbinitInput(structure=abidata.structure_from_ucell("AlAs"),
                                pseudos=abidata.pseudos("13al.981214.fhi", "33as.pspnc"))

    # Set the value of the Abinit variables needed for GS runs.
    gs_inp.set_vars(
        nband=4,
        ecut=ecut,
        ngkpt=ngkpt,
        nshiftk=4,
        shiftk=[0.0, 0.0, 0.5,   # This gives the usual fcc Monkhorst-Pack grid
                0.0, 0.5, 0.0,
                0.5, 0.0, 0.0,
                0.5, 0.5, 0.5],
        ixc=1,
        nstep=25,
        diemac=9.0,
        tolvrs=1.0e-10,
        #iomode=3,
    )

    return gs_inp

The function makes some assumptions for important parameters such as the crystalline structure and the pseudos. This is done on purpose to keep the code as simple as possible. It should not be so difficult to generalize the implementation to take into account other cases. Let’s start to play with our new function:

scf_input = make_scf_input()
scf_input

##############################################
#### SECTION: basic
##############################################
nband 4
ecut 2
ngkpt 4 4 4
nshiftk 4
shiftk
0.0 0.0 0.5
0.0 0.5 0.0
0.5 0.0 0.0
0.5 0.5 0.5
ixc 1
nstep 25
tolvrs 1e-10
##############################################
#### SECTION: files
##############################################
indata_prefix indata/in
tmpdata_prefix tmpdata/tmp
outdata_prefix outdata/out
pp_dirpath /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/abipy/data/pseudos
pseudos 13al.981214.fhi 33as.pspnc
##############################################
#### SECTION: gstate
##############################################
diemac 9.0
##############################################
#### STRUCTURE
##############################################
natom 2
ntypat 2
typat 1 2
znucl 13 33
xred
0.0000000000 0.0000000000 0.0000000000
0.2500000000 0.2500000000 0.2500000000
acell 1.0 1.0 1.0
rprim
0.0000000000 5.3050000000 5.3050000000
5.3050000000 0.0000000000 5.3050000000
5.3050000000 5.3050000000 0.0000000000

print(scf_input.structure)

Full Formula (Al1 As1)
Reduced Formula: AlAs
abc   :   3.970101   3.970101   3.970101
angles:  60.000000  60.000000  60.000000
pbc   :       True       True       True
Sites (2)
  #  SP       a     b     c
---  ----  ----  ----  ----
  0  Al    0     0     0
  1  As    0.25  0.25  0.25

scf_input.structure.plot();

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/IPython/core/formatters.py:343, in BaseFormatter.__call__(self, obj)
    341     pass
    342 else:
--> 343     return printer(obj)
    344 # Finally look for special method names
    345 method = get_real_method(obj, self.print_method)

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/IPython/core/pylabtools.py:170, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
    167     from matplotlib.backend_bases import FigureCanvasBase
    168     FigureCanvasBase(fig)
--> 170 fig.canvas.print_figure(bytes_io, **kw)
    171 data = bytes_io.getvalue()
    172 if fmt == 'svg':

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2172     # we do this instead of `self.figure.draw_without_rendering`
   2173     # so that we can inject the orientation
   2174     with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175         self.figure.draw(renderer)
   2176 if bbox_inches:
   2177     if bbox_inches == "tight":

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
   3159             # ValueError can occur when resizing a window.
   3161     self.patch.draw(renderer)
-> 3162     mimage._draw_list_compositing_images(
   3163         renderer, self, artists, self.suppressComposite)
   3165     renderer.close_group('figure')
   3166 finally:

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    130 if not_composite or not has_images:
    131     for a in artists:
--> 132         a.draw(renderer)
    133 else:
    134     # Composite any adjacent images together
    135     image_group = []

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/mpl_toolkits/mplot3d/axes3d.py:441, in Axes3D.draw(self, renderer)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
--> 441 for artist in sorted(collections_and_patches,
    442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

File /usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/mpl_toolkits/mplot3d/axes3d.py:442, in Axes3D.draw.<locals>.<lambda>(artist)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
    441 for artist in sorted(collections_and_patches,
--> 442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

AttributeError: 'Arrow3D' object has no attribute 'do_3d_projection'

<Figure size 640x480 with 1 Axes>

We are using the same pseudopotentials than the official tutorial. Note that the xc functional is LDA in both pseudos but with a different parametrization. This is the reason why we are specifying ixc in the input file. This is not needed if you are using pseudos generated with the same ixc.

for pseudo in scf_input.pseudos:
    print(pseudo, "\n")

<NcAbinitPseudo: 13al.981214.fhi>
  summary: Aluminum, fhi98PP : Hamann-type, LDA CA PerdewWang, l=2 local
  number of valence electrons: 3.0
  maximum angular momentum: d
  angular momentum for local part: d
  XC correlation: PW
  supports spin-orbit: False
  radius for non-linear core correction: 0.0
  hint for low accuracy: ecut: 0.0, pawecutdg: 0.0
  hint for normal accuracy: ecut: 0.0, pawecutdg: 0.0
  hint for high accuracy: ecut: 0.0, pawecutdg: 0.0 

<NcAbinitPseudo: 33as.pspnc>
  summary: Troullier-Martins psp for element  As        Thu Oct 27 17:37:14 EDT 1994
  number of valence electrons: 5.0
  maximum angular momentum: p
  angular momentum for local part: p
  XC correlation: LDA_XC_TETER93
  supports spin-orbit: False
  radius for non-linear core correction: 2.0573171556401
  hint for low accuracy: ecut: 0.0, pawecutdg: 0.0
  hint for normal accuracy: ecut: 0.0, pawecutdg: 0.0
  hint for high accuracy: ecut: 0.0, pawecutdg: 0.0 

As you can see, we essentially have a standard input to perform a GS calculation. This object will represent the building block for our DFPT calculation with AbiPy.

It this is not your first time you use the DFPT part of Abinit, you already know that phonon calculations require an initial GS run to produce the WFK file followed by a DFPT run that reads the WFK file and solves the Sternheimer equations for \(N_{\text{irred}}(q)\) atomic perturbations where \(N_{\text{irred}}\) is the number of independent atomic displacements (assuming \(q\) belongs to the k-mesh).

If you try to do a convergence study wrt ecut without multi-datasets, you will likely start from an initial GS input file with a given value of ecut, use it as a template to generate the DFPT input files, create symbolic links to the WFK file produced in the first GS step and then instruct Abinit to read this file with irdwfk. Once you have a set of input files that work for a particular ecut, one can simply replicate the set of directories and files and use a script to change the value of ecut in the input files. Then, of course, one has to run the calculations manually, collect the results and produce nice plots to understand what is happening.

This approach is obviously boring and error-prone if you are a human being but it is easy to implement in an algorithm and machines do not complain if they have a lot of repetitive work to do! There are also several technical advantages in using this task-based approach vs multi-datasets but we discuss this point in more details afterwards.

If the machine could speak, it will tell you: give me an object that represents an input for GS calculations, give me the list of q-points you want to compute as well as the parameters that must be changed in the initial input and I will generate a Flow for DFPT calculations. This logic appears so frequently that we decided to encapsulate it in the flowtk.phonon_conv_flow factory function:

from lesson_dfpt import build_flow_alas_ecut_conv
abilab.print_source(build_flow_alas_ecut_conv)

def build_flow_alas_ecut_conv(options):
    """
    Build and return Flow for convergence study of phonon frequencies at Gamma as function of ecut.
    """
    scf_input = make_scf_input()
    return flowtk.phonon_conv_flow("flow_alas_ecut_conv", scf_input, qpoints=(0, 0, 0),
                                   params=["ecut", [4, 6, 8]])

Let’s call the function to build our flow:

flow = build_flow_alas_ecut_conv(options=None)

flow.show_info()

<Flow, node_id=60, workdir=flow_alas_ecut_conv>
Number of works: 6, total number of tasks: 9
Number of tasks with a given class:

Task Class      Number
------------  --------
ScfTask              3
PhononTask           6

and call the get_graphviz method to visualize the connection among the Tasks:

flow.get_graphviz()

# matplotlib version based on networkx
#flow.plot_networkx(with_edge_labels=True);

The flow contains three independent groups of tasks, one group per each value of ecut specified in params.

for work in flow:
    for task in work:
        print(task.pos_str, "uses ecut:", task.input["ecut"])

w0_t0 uses ecut: 4
w1_t0 uses ecut: 4
w1_t1 uses ecut: 4
w2_t0 uses ecut: 6
w3_t0 uses ecut: 6
w3_t1 uses ecut: 6
w4_t0 uses ecut: 8
w5_t0 uses ecut: 8
w5_t1 uses ecut: 8

flow.get_vars_dataframe("ecut")

	ecut	class
w0_t0	4	ScfTask
w1_t0	4	PhononTask
w1_t1	4	PhononTask
w2_t0	6	ScfTask
w3_t0	6	PhononTask
w3_t1	6	PhononTask
w4_t0	8	ScfTask
w5_t0	8	PhononTask
w5_t1	8	PhononTask

Each group represents a Workflow and consists of one ScfTask(red circle) that solves the KS equations self-consistently producing a WFK file that will be used by the two children (PhononTasks - blue circles) to compute the first-order change of the wavefunctions due to one of the irreducible atomic perturbations.

Note that phonon_conv_flow invokes Abinit under the hood to get the list of irreducible perturbations and uses this information to build the flow. This explains why we have two PhononTasks per q-point instead of the total number of phonon modes that equals \(3*N_{atom}=6\).

Perhaps a table with the values of the input variables associated to the DFPT perturbation will help. None means that the variable is not defined in that particular input.

flow.get_vars_dataframe("rfphon", "rfatpol", "rfdir", "qpt", "kptopt")

	rfphon	rfatpol	rfdir	qpt	kptopt	class
w0_t0	None	None	None	None	None	ScfTask
w1_t0	1	[1, 1]	[1, 0, 0]	[0.0, 0.0, 0.0]	2	PhononTask
w1_t1	1	[2, 2]	[1, 0, 0]	[0.0, 0.0, 0.0]	2	PhononTask
w2_t0	None	None	None	None	None	ScfTask
w3_t0	1	[1, 1]	[1, 0, 0]	[0.0, 0.0, 0.0]	2	PhononTask
w3_t1	1	[2, 2]	[1, 0, 0]	[0.0, 0.0, 0.0]	2	PhononTask
w4_t0	None	None	None	None	None	ScfTask
w5_t0	1	[1, 1]	[1, 0, 0]	[0.0, 0.0, 0.0]	2	PhononTask
w5_t1	1	[2, 2]	[1, 0, 0]	[0.0, 0.0, 0.0]	2	PhononTask

If the meaning of these variables is not clear, you can consult the Abinit documentation e.g. the documentation of the rfatpol input variable or access the documentation directly from python with:

abilab.docvar("rfatpol")

Default value:

[1, 'natom']

Description

Control the range of atoms for which displacements will be considered in phonon calculations (atomic polarizations). These values are only relevant to phonon response function calculations. May take values from 1 to natom, with rfatpol(1)<=rfatpol(2). The atoms to be moved will be defined by the do-loop variable iatpol: - do iatpol=rfatpol(1),rfatpol(2) For the calculation of a full dynamical matrix, use rfatpol(1)=1 and rfatpol(2)=natom, together with rfdir 1 1 1, both being the default values. For selected elements of the dynamical matrix, use different values of rfatpol and/or rfdir. The name 'iatpol' is used for the part of the internal variable ipert when it runs from 1 to natom. The internal variable ipert can also assume values larger than natom, denoting perturbations of electric field or stress type (see [the DFPT help file](/guide/respfn)). As a side technical information, the value rfatpol(1)=-1 is admitted, and transformed immediately to rfatpol(1)=1, while rfatpol(2)=-1 is transformed to rfatpol(2)=natom, while the default input values are actually rfatpol=-1 .

Note

For your convenience the links to the doc of the different variables are listed below:

• qpt

• rfphon

• rfatpol

• rfdir

• kptopt

Now we can generate the flow_alas_ecut directory with the input files by executing the lesson_dfpt.py script. Then use the abirun.py script to launch the entire calculation with:

abirun.py flow_alas_ecut_conv scheduler

You will see that all PhononTasks will be executed in parallel on your machine.

Warning

Please make sure that AbiPy is properly configured by running abicheck –with flow

If you prefer to skip this part, you may want to jump to the next section, that presents the post-processing of the results. Note that the output files are already available in the repository so it is also possible to try the AbiPy post-processing tools without having to run the flow.

Convergence study at \(\Gamma\)

There are several output files located inside the outdata directories:

!find flow_alas_ecut_conv/ -name "*_DDB"

flow_alas_ecut_conv/w2/t0/outdata/out_DDB
flow_alas_ecut_conv/w4/t0/outdata/out_DDB
flow_alas_ecut_conv/w1/outdata/out_DDB
flow_alas_ecut_conv/w3/outdata/out_DDB
flow_alas_ecut_conv/w0/t0/outdata/out_DDB
flow_alas_ecut_conv/w5/outdata/out_DDB

Remember that our goal is to analyze the convergence of the phonon frequencies at \(\Gamma\) as function of ecut. So we are mainly interested in the DDB files located in the outdata directories of the PhononWorks (w0/outdata, w1/outdata, w2/outdata). These are indeed the DDB files with all the information needed to reconstruct the dynamical matrix at \(\Gamma\) and to compute the phonon frequencies (AbiPy calls mrgddb to merge the DDB files when all the perturbations in the PhononWork have been computed).

The code below tells our robot that we would like to analyze all the DDB files located in the output directories of the works:

robot = abilab.DdbRobot.from_dir_glob("./flow_alas_ecut_conv/w*/outdata/")
robot

0. flow_alas_ecut_conv/w1/outdata/out_DDB
1. flow_alas_ecut_conv/w3/outdata/out_DDB
2. flow_alas_ecut_conv/w5/outdata/out_DDB

For more examples on the use of DDB and robots, see the DDB notebook. Now we ask the robot to call anaddb to compute the phonon frequencies at \(\Gamma\) for all DDBs and return a pandas DataFrame:

data_gamma = robot.get_dataframe_at_qpoint((0, 0, 0))

Calculation completed.
Anaddb results available in dir: /tmp/tmp0v7hudop

Calculation completed.
Anaddb results available in dir: /tmp/tmpuf71i84y

Calculation completed.
Anaddb results available in dir: /tmp/tmpokkuifqq

The DataFrame is a dict-like object whose keys are the name of the colums in the table

print(data_gamma.keys())

Index(['mode0', 'mode1', 'mode2', 'mode3', 'mode4', 'mode5', 'nkpt', 'nsppol',
       'ecut', 'tsmear', 'occopt', 'ixc', 'nband', 'usepaw', 'formula',
       'natom', 'alpha', 'beta', 'gamma', 'a', 'b', 'c', 'volume',
       'abispg_num', 'spglib_symb', 'spglib_num', 'spglib_lattice_type'],
      dtype='object')

where mode-i is the frequency in eV of the i-th phonon mode.

We are mainly interested in the convergence of the phonon frequencies versus ecut so we filter these columns with:

data_gamma = data_gamma[["ecut"] + [k for k in data_gamma if k.startswith("mode")]]
data_gamma

	ecut	mode0	mode1	mode2	mode3	mode4	mode5
flow_alas_ecut_conv/w1/outdata/out_DDB	4.0	0.0	0.0	0.0	0.043641	0.043641	0.043641
flow_alas_ecut_conv/w3/outdata/out_DDB	6.0	0.0	0.0	0.0	0.044485	0.044485	0.044485
flow_alas_ecut_conv/w5/outdata/out_DDB	8.0	0.0	0.0	0.0	0.044625	0.044625	0.044625

and we get some statistics about our data with:

data_gamma.describe()

	ecut	mode0	mode1	mode2	mode3	mode4	mode5
count	3.0	3.0	3.0	3.0	3.000000	3.000000	3.000000
mean	6.0	0.0	0.0	0.0	0.044250	0.044250	0.044250
std	2.0	0.0	0.0	0.0	0.000533	0.000533	0.000533
min	4.0	0.0	0.0	0.0	0.043641	0.043641	0.043641
25%	5.0	0.0	0.0	0.0	0.044063	0.044063	0.044063
50%	6.0	0.0	0.0	0.0	0.044485	0.044485	0.044485
75%	7.0	0.0	0.0	0.0	0.044555	0.044555	0.044555
max	8.0	0.0	0.0	0.0	0.044625	0.044625	0.044625

Pandas tables are extremely powerful and the describe method already gives some useful info about the convergence of the phonon modes. Sometimes, however, we would like to visualize the data to have a better understanding of what’s happening:

data_gamma.plot(x="ecut", y="mode3", style="-o");

Let’s plot all the modes in different subplots with:

data_gamma.plot(x="ecut", y=[k for k in data_gamma if k.startswith("mode")], subplots=True, style="-o");

This convergence study at \(\Gamma\) thus reveals that our pseudos require ecut >= 6 Ha to get reasonably converged phonon frequencies at \(\Gamma\). In what follows, we assume that also the modes at the other \(q\)-points present a similar convergence behaviour and we use ecut = 6 Ha to keep the computational cost low.

For a quick introduction to Pandas, see:

• Pandas cookbook

• Pandas cookbook Chapter 1 Reading data from a csv file

Phonon band structure of AlAs

Now we are finally ready for the calculation of the vibrational spectrum of \(AlAs\). We already managed to run DFPT calculations at \(\Gamma\) with different values of ecut and the steps required to get a full band structure are not that different, provided that the following differences are taken into account:

• we need the dynamical matrix \(D(q)\) on a homogeneous mesh so that it is possible to calculate \(D(R)\) in anaddb via Fourier transform and then phonon frequencies for arbitrary q-points via Fourier interpolation

• \(AlAs\) is a polar semiconductor so we need to include the LO-TO splitting for \(q \rightarrow 0\) that, in turns, requires the DFPT computation of the Born effective charges and of the dielectric constant.

In AbiPy, these concepts are translated in an easy-to-use API in which you pass an initial AbinitInput object, you specify the q-mesh for phonons in terms of ph_nqpt and activate the computation of the Born effective charges with the boolean flag with_becs.

Let’s have a look at the code (as usual there are more comments than lines of code):

from lesson_dfpt import build_flow_alas_phonons
abilab.print_source(build_flow_alas_phonons)

def build_flow_alas_phonons(options):
    """
    Build and return a Flow to compute the dynamical matrix on a (2, 2, 2) qmesh
    as well as DDK and Born effective charges.
    The final DDB with all perturbations will be merged automatically and placed
    in the Flow `outdir` directory.
    """
    scf_input = make_scf_input(ecut=6, ngkpt=(4, 4, 4))
    return flowtk.PhononFlow.from_scf_input("flow_alas_phonons", scf_input,
                                            ph_ngqpt=(2, 2, 2), with_becs=True)

We can finally construct the flow with:

flow_phbands = build_flow_alas_phonons(options=None)

and visualize the connections with:

flow_phbands.get_graphviz()
#flow_phbands.plot_networkx();

Note that there are a lot of things happening under the hood here.

First of all, AbiPy generates PhononTasks only for the \(q\)-points in the irreducible wedge of the Brillouin zone corresponding to ph_ngqpt. Moreover, for a given \(q\)-point, only the irreducible atomic perturbations are explicitly computed since the other atomic perturbations can be reconstructed by symmetry. Fortunately you do not have to care about all these technical details as AbiPy and Abinit will automate the whole procedure.

Remember that the \(q\)-point mesh cannot be chosen arbitrarily since all \(q\) wavevectors should connect two \(k\) points of the grid used for the electrons.

It is also worth stressing that the computational cost of the DFPT run depends on the q-point since only those symmetries that preserve the q-point as well as the direction of the perturbation can be employed (calculations at \(\Gamma\) are therefore much faster than other q-points).

flow_phbands.get_vars_dataframe("rfphon", "rfatpol", "rfdir", "qpt", "kptopt")

	rfphon	rfatpol	rfdir	qpt	kptopt	class
w0_t0	None	None	None	None	None	ScfTask
w1_t0	None	None	(1, 0, 0)	(0, 0, 0)	2	DdkTask
w1_t1	None	None	(0, 1, 0)	(0, 0, 0)	2	DdkTask
w1_t2	None	None	(0, 0, 1)	(0, 0, 0)	2	DdkTask
w1_t3	1	[1, 1]	[1, 0, 0]	(0, 0, 0)	2	BecTask
w1_t4	1	[2, 2]	[1, 0, 0]	(0, 0, 0)	2	BecTask
w1_t5	1	[1, 1]	[1, 0, 0]	[0.5, 0.0, 0.0]	3	PhononTask
w1_t6	1	[1, 1]	[0, 1, 0]	[0.5, 0.0, 0.0]	3	PhononTask
w1_t7	1	[2, 2]	[1, 0, 0]	[0.5, 0.0, 0.0]	3	PhononTask
w1_t8	1	[2, 2]	[0, 1, 0]	[0.5, 0.0, 0.0]	3	PhononTask
w1_t9	1	[1, 1]	[1, 0, 0]	[0.5, 0.5, 0.0]	3	PhononTask
w1_t10	1	[2, 2]	[1, 0, 0]	[0.5, 0.5, 0.0]	3	PhononTask

Now we can generate the directories and the input files of the Flow. Change lesson_dfpt.py so that the build_flow_alas_phonons function is called in main instead of build_flow_alas_ecut_conv. Run the script to generate the directory with the flow. Finally, use

abirun.py flow_alas_phonons scheduler

to launch the entire calculation.

Post-processing the results

Our flow is completed and we have the final DDB file with all the \(q\)-points and all the independent atomic perturbations. Let’s open this DDB file with:

ddb = abilab.abiopen("flow_alas_phonons/outdata/out_DDB")
print(ddb)

================================= File Info =================================
Name: out_DDB
Directory: /home/runner/work/abipy_book/abipy_book/abipy_book/dfpt/flow_alas_phonons/outdata
Size: 28.11 kB
Access Time: Sun Oct 27 17:41:29 2024
Modification Time: Sun Oct 27 17:38:46 2024
Change Time: Sun Oct 27 17:38:46 2024

================================= Structure =================================
Full Formula (Al1 As1)
Reduced Formula: AlAs
abc   :   3.970101   3.970101   3.970101
angles:  60.000000  60.000000  60.000000
pbc   :       True       True       True
Sites (2)
  #  SP       a     b     c
---  ----  ----  ----  ----
  0  Al    0     0     0
  1  As    0.25  0.25  0.25

Abinit Spacegroup: spgid: 0, num_spatial_symmetries: 24, has_timerev: True, symmorphic: False

================================== DDB Info ==================================

Number of q-points in DDB: 3
guessed_ngqpt: [2 2 2] (guess for the q-mesh divisions made by AbiPy)
ecut = 6.000000, ecutsm = 0.000000, nkpt = 128, nsym = 24, usepaw = 0
nsppol 1, nspinor 1, nspden 1, ixc = 1, occopt = 1, tsmear = 0.010000

Has total energy: False
Has forces: False
Has stress tensor: False

Has (at least one) atomic pertubation: True
Has (at least one diagonal) electric-field perturbation: True
Has (at least one) Born effective charge: True
Has (all) strain terms: False
Has (all) internal strain terms: False
Has (all) piezoelectric terms: False
Has (all) dynamical quadrupole terms: False

The DdbFile object provides an easy-to-use interface that invokes anaddb to post-process the data stored in the DDB file.

anacompare_phdos, for example, computes the phonon DOS with different \(q\)-meshes. Each mesh is defined by a single integer, nqsmall, that gives the number of divisions used to sample the smallest reciprocal lattice vector. The number of divisions along the other directions are chosen so that proportions are preserved:

c = ddb.anacompare_phdos(nqsmalls=[8, 10, 12, 14, 16])

c.plotter.combiplotly();

A 16x16x16 \(q\)-mesh with the tetrahedron method gives a well converged phonon DOS.

To function anaget_phbst_and_phdos_files allows one to compute the phonon band structure on an automatically defined \(q\)-path as well as the phonon DOS:

phbst_file, phdos_file = ddb.anaget_phbst_and_phdos_files(ndivsm=10, nqsmall=16, lo_to_splitting=False)

# Extract the phonon bands and the phonon DOS from phbst_file and phdos_file
phbands = phbst_file.phbands
phdos = phdos_file.phdos

Let’s plot the phonon bands with plot:

phbands.plotly(template="plotly_dark");

as well as the high-symmetry q-path:

phbands.qpoints.plotly();

Do you see the two strange dips for the highest phonon band, at the \(\Gamma\) point? They are due to the lack of LO-TO splitting for the ANADDB treatment of the first list of vector. See also the discussion in the second DFPT lesson.

For years, Abinit users had to patch manually the output frequencies to include the LO-TO splitting. These days are finally gone and we can plot the LO-TO splitting with AbiPy by just setting lo_to_splitting=True`:

phbst_file, phdos_file = ddb.anaget_phbst_and_phdos_files(ndivsm=10, nqsmall=16, lo_to_splitting=True)

# Extract the phonon bands and the phonon DOS from phbst_file and phdos_file
phbands = phbst_file.phbands
phdos = phdos_file.phdos

#phbst_file.phbands.qpoints.plot();

The band structure plot now correctly shows the non-analytical behaviour around \(\Gamma\):

phbands.plotly();

Warning

lo_to_splitting=True works only when the DDB contains the Born effective charges and the dielectric constant, that must be computed in the Abinit run.

To plot bands and DOS on the same figure:

phbands.plotly_with_phdos(phdos);

The PhdosFile contains the phonon frequencies, the displacement vectors as well as the decomposition of the total DOS in terms of the contributions due to the different types of atom in the unit cell. This means that one can plot the type-projected phonon DOS with:

phdos_file.plotly_pjdos_type(title="AlAs type-projected phonon DOS");

and it is even possible to plot fatbands and type-projected DOS on the same figure with:

phbands.plotly_fatbands(phdos_file=phdos_file);

The highest frequency modes have a strong Al-character while the low frequency modes originate from As. This behaviour is somehow expected. Could you explain it in terms of a simple physical model?

Macroscopic dielectric tensor and Born effective charges

Our calculations includes the response of the system to an external electric field. The code below extracts the macroscopic dielectric tensor (emacro) and the Born effective charges (becs) from the DDB file:

emacro, becs = ddb.anaget_epsinf_and_becs()

emacro

	x	y	z
x	10.396165	0.000000	0.000000
y	0.000000	10.396165	0.000000
z	0.000000	0.000000	10.396165

becs

/usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/spglib/spglib.py:115: DeprecationWarning:

dict interface (SpglibDataset['equivalent_atoms']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead

/usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/spglib/spglib.py:115: DeprecationWarning:

dict interface (SpglibDataset['wyckoffs']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead

/usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/spglib/spglib.py:115: DeprecationWarning:

dict interface (SpglibDataset['equivalent_atoms']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead

/usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/spglib/spglib.py:115: DeprecationWarning:

dict interface (SpglibDataset['wyckoffs']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead

	element	site_index	frac_coords	cart_coords	wyckoff	xx	yy	zz	yz	xz	xy
0	Al	0	[0.0, 0.0, 0.0]	[0.0, 0.0, 0.0]	1a	2.168053	2.168053	2.168053	0.0	0.0	0.0
1	As	1	[0.25, 0.25, 0.25]	[1.40364, 1.40364, 1.40364]	1d	-2.168053	-2.168053	-2.168053	0.0	0.0	0.0

As explained in the references, the Born effective charges must fulfill the charge neutrality sum-rule. This rule is usually broken due to the discretization introduced by the FFT mesh, and anaddb will enforce it if chneut is set to 1 (default behaviour). Let’s check it out!

print(becs)

Born effective charges in Cartesian coordinates (Voigt notation)
  element  site_index         frac_coords                  cart_coords wyckoff        xx        yy        zz   yz   xz   xy
0      Al           0     [0.0, 0.0, 0.0]              [0.0, 0.0, 0.0]      1a  2.168053  2.168053  2.168053  0.0  0.0  0.0
1      As           1  [0.25, 0.25, 0.25]  [1.40364, 1.40364, 1.40364]      1d -2.168053 -2.168053 -2.168053  0.0  0.0  0.0

Born effective charge neutrality sum-rule with chneut: 1

[[0.00000000e+00 0.00000000e+00 0.00000000e+00]
 [0.00000000e+00 0.00000000e+00 0.00000000e+00]
 [2.40741243e-35 2.40741243e-35 0.00000000e+00]]

Let’s repeat the same calculation but without enforcing the sum-rule:

emacro, becs_chneut0 = ddb.anaget_epsinf_and_becs(chneut=0)
print(becs_chneut0)

Born effective charges in Cartesian coordinates (Voigt notation)
  element  site_index         frac_coords                  cart_coords wyckoff        xx        yy        zz   yz   xz   xy
0      Al           0     [0.0, 0.0, 0.0]              [0.0, 0.0, 0.0]      1a  2.127295  2.127295  2.127295  0.0  0.0  0.0
1      As           1  [0.25, 0.25, 0.25]  [1.40364, 1.40364, 1.40364]      1d -2.208811 -2.208811 -2.208811  0.0  0.0  0.0

Born effective charge neutrality sum-rule with chneut: 0

[[-8.15163891e-02  0.00000000e+00  0.00000000e+00]
 [ 0.00000000e+00 -8.15163891e-02  0.00000000e+00]
 [ 5.81483171e-19  5.81483171e-19 -8.15163891e-02]]

/usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/spglib/spglib.py:115: DeprecationWarning:

dict interface (SpglibDataset['equivalent_atoms']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead

/usr/share/miniconda/envs/abipy/lib/python3.12/site-packages/spglib/spglib.py:115: DeprecationWarning:

dict interface (SpglibDataset['wyckoffs']) is deprecated.Use attribute interface ({self.__class__.__name__}.{key}) instead

Thermodynamic properties within the harmonic approximation

The thermodynamic properties of an ensemble of non-interacting phonons can be expresses in terms of integrals of the DOS.

(8)\[\begin{equation} %\label{eq:helmholtz} \Delta F = 3nNk_BT\int_{0}^{\omega_L}\text{ln}\left(2\text{sinh}\frac{\hbar\omega}{2k_BT}\right)g(\omega)d\omega \end{equation}\]

(9)\[\begin{equation} %\label{eq:free_en} \Delta E = 3nN\frac{\hbar}{2}\int_{0}^{\omega_L}\omega\text{coth}\left(\frac{\hbar\omega}{2k_BT}\right)g(\omega)d\omega \end{equation}\]

(10)\[\begin{equation} %\label{eq:c_v} C_v = 3nNk_B\int_{0}^{\omega_L}\left(\frac{\hbar\omega}{2k_BT}\right)^2\text{csch}^2\left(\frac{\hbar\omega}{2k_BT}\right)g(\omega)d\omega \end{equation}\]

(11)\[\begin{equation} %\label{eq:entropy} S = 3nNk_B\int_{0}^{\omega_L}\left(\frac{\hbar\omega}{2k_BT}\text{coth}\left(\frac{\hbar\omega}{2k_BT}\right) - \text{ln}\left(2\text{sinh}\frac{\hbar\omega}{2k_BT}\right)\right)g(\omega)d\omega, \end{equation}\]

where \(k_B\) is the Boltzmann constant.

This should represent a reasonable approximation especially in the low temperature regime in which anharmonic effects can be neglected.

Let’s plot the vibrational contributions thermodynamic properties as function of \(T\):

phdos.plotly_harmonic_thermo();

Exercises

• Our first phonon band structure has been computed with a (4, 4, 4) \(k\)-mesh for the electrons and a (2, 2, 2) \(q\)-mesh for phonons. You may try to increase the density of \(k\)-points/\(q\)-points to see if this change affects the final results.

• Why do you get an error from AbiPy if you try ngkpt = (4, 4, 4,) and ngqpt = (3, 3, 3)?