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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
ecut
AlAs
with the inclusion of LO-TO splittingWe 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:
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.
Before starting, we need to import the python modules and the functions we will need in the notebook:
# Use this at the beginning of your script so that your code will be compatible with python3
from __future__ import print_function, division, unicode_literals
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
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 Sites (2) # SP a b c --- ---- ---- ---- ---- 0 Al 0 0 0 1 As 0.25 0.25 0.25
scf_input.structure.plot();
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 using ixc
in the input file.
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 essentialy 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 repetive 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 frequenty 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=345794, 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 pertubations.
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")
[1, 1]
Control the range of atoms for which displacements will be considered in phonon calculations (atomic polarizations), using the 2n+1 theorem. 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:
For the calculation of a full dynamical matrix, use rfatpol(1)=1 and rfatpol(2)=natom, together with rfdir 1 1 1. 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 [help:respfn|the response function help file]).
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.
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.
There are several output files located inside the outdata
directories:
!find flow_alas_ecut_conv/ -name "*_DDB"
flow_alas_ecut_conv//w0/t0/outdata/out_DDB flow_alas_ecut_conv//w1/outdata/out_DDB flow_alas_ecut_conv//w2/t0/outdata/out_DDB flow_alas_ecut_conv//w3/outdata/out_DDB flow_alas_ecut_conv//w4/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
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))
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 extremly 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
an 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:
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 |
w2_t0 | 1 | [1, 1] | [1, 0, 0] | [0.5, 0.0, 0.0] | 3 | PhononTask |
w2_t1 | 1 | [1, 1] | [0, 1, 0] | [0.5, 0.0, 0.0] | 3 | PhononTask |
w2_t2 | 1 | [2, 2] | [1, 0, 0] | [0.5, 0.0, 0.0] | 3 | PhononTask |
w2_t3 | 1 | [2, 2] | [0, 1, 0] | [0.5, 0.0, 0.0] | 3 | PhononTask |
w3_t0 | 1 | [1, 1] | [1, 0, 0] | [0.5, 0.5, 0.0] | 3 | PhononTask |
w3_t1 | 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.
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: /Users/gmatteo/git_repos/abitutorials/abitutorials/dfpt/flow_alas_phonons/outdata Size: 28.11 kb Access Time: Sun Aug 12 00:43:48 2018 Modification Time: Thu Oct 12 22:34:40 2017 Change Time: Thu Oct 12 22:34:40 2017 ================================= Structure ================================= Full Formula (Al1 As1) Reduced Formula: AlAs abc : 3.970101 3.970101 3.970101 angles: 60.000000 60.000000 60.000000 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) 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
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.combiplot();
A 16x16x16 $q$-mesh with the tethraedron 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 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 bands with matplotlib:
phbands.plot();
and the high-symmetry q-path:
phbands.qpoints.plot();
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.plot();
To plot bands and DOS on the same figure:
phbands.plot_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.plot_pjdos_type(title="AlAs type-projected phonon DOS");
and it is even possible to plot fatbands and type-projected DOSes on the same figure with:
phbands.plot_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?
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_emacro_and_becs()
emacro
DielectricTensor([[ 1.03961652e+01 -0.00000000e+00 -0.00000000e+00] [-0.00000000e+00 1.03961652e+01 1.19611765e-15] [-0.00000000e+00 1.19611765e-15 1.03961652e+01]])
becs
element | frac_coords | xx | yy | zz | yz | xz | xy | |
---|---|---|---|---|---|---|---|---|
0 | Al | [0.0, 0.0, 0.0] | 2.168053 | 2.168053 | 2.168053 | -2.087488e-19 | -2.087488e-19 | 0.0 |
1 | As | [0.25, 0.25, 0.25] | -2.168053 | -2.168053 | -2.168053 | 2.087488e-19 | 2.087488e-19 | 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 frac_coords xx yy zz yz xz xy 0 Al [0.0, 0.0, 0.0] 2.168053 2.168053 2.168053 -2.087488e-19 -2.087488e-19 0.0 1 As [0.25, 0.25, 0.25] -2.168053 -2.168053 -2.168053 2.087488e-19 2.087488e-19 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_emacro_and_becs(chneut=0)
print(becs_chneut0)
Born effective charges in Cartesian coordinates (Voigt notation) element frac_coords xx yy zz yz xz xy 0 Al [0.0, 0.0, 0.0] 2.127295 2.127295 2.127295 8.199282e-20 8.199282e-20 0.0 1 As [0.25, 0.25, 0.25] -2.208811 -2.208811 -2.208811 4.994904e-19 4.994904e-19 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]]
The thermodynamic properties of an ensemble of non-interacting phonons can be expresses in terms of integrals of the DOS.
\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}\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}\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}\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.plot_harmonic_thermo();
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)?
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