Phonon calculations¶

Module for calculating vibrational normal modes for periodic systems using the so-called small displacement method (see e.g. [Alfe]). So far, space-group symmetries are not exploited to reduce the number of atomic displacements that must be calculated and subsequent symmetrization of the force constants.

For polar materials the dynamical matrix at the zone center acquires a non-analytical contribution that accounts for the LO-TO splitting. This contribution requires additional functionality to evaluate and is not included in the present implementation. Its implementation in conjunction with the small displacement method is described in [Wang].

Example¶

Simple example showing how to calculate the phonon dispersion for bulk aluminum using a 7x7x7 supercell within effective medium theory:

from ase.build import bulk
from ase.calculators.emt import EMT
from ase.phonons import Phonons

# Setup crystal and EMT calculator
atoms = bulk('Al', 'fcc', a=4.05)

# Phonon calculator
N = 7
ph = Phonons(atoms, EMT(), supercell=(N, N, N), delta=0.05)
ph.run()

# Read forces and assemble the dynamical matrix
ph.read(acoustic=True)
ph.clean()

path = atoms.cell.bandpath('GXULGK', npoints=100)
bs = ph.get_band_structure(path)

dos = ph.get_dos(kpts=(20, 20, 20)).sample_grid(npts=100, width=1e-3)

# Plot the band structure and DOS:
import matplotlib.pyplot as plt  # noqa

fig = plt.figure(1, figsize=(7, 4))
ax = fig.add_axes([.12, .07, .67, .85])

emax = 0.035
bs.plot(ax=ax, emin=0.0, emax=emax)

dosax = fig.add_axes([.8, .07, .17, .85])
dosax.fill_between(dos.get_weights(), dos.get_energies(), y2=0, color='grey',
                   edgecolor='k', lw=1)

dosax.set_ylim(0, emax)
dosax.set_yticks([])
dosax.set_xticks([])
dosax.set_xlabel("DOS", fontsize=18)

fig.savefig('Al_phonon.png')

Mode inspection:

# from ase.io.trajectory import Trajectory
# from ase.io import write

# Write modes for specific q-vector to trajectory files:
L = path.special_points['L']
ph.write_modes([l / 2 for l in L], branches=[2], repeat=(8, 8, 8), kT=3e-4,
               center=True)


# Generate gif animation:
# XXX Temporarily disabled due to matplotlib writer compatibility issue.
# with Trajectory('phonon.mode.2.traj', 'r') as traj:
#     write('Al_mode.gif', traj, interval=50,
#           rotation='-36x,26.5y,-25z')

[Alfe]

D. Alfe, PHON: A program to calculate phonons using the small displacement method, Comput. Phys. Commun. 180, 2622 (2009)

[Wang]

Y. Wang et al., A mixed-space approach to first-principles calculations of phonon frequencies for polar materials, J. Phys.: Cond. Matter 22, 202201 (2010)

List of all Methods¶

class ase.phonons.Phonons(*args, **kwargs)[source]¶

Class for calculating phonon modes using the finite displacement method.

The matrix of force constants is calculated from the finite difference approximation to the first-order derivative of the atomic forces as:

             2             nbj   nbj
 nbj        d E           F-  - F+
C     = ------------ ~  -------------  ,
 mai     dR   dR          2 * delta
           mai  nbj

where F+/F- denotes the force in direction j on atom nb when atom ma is displaced in direction +i/-i. The force constants are related by various symmetry relations. From the definition of the force constants it must be symmetric in the three indices mai:

 nbj    mai         bj        ai
C    = C      ->   C  (R ) = C  (-R )  .
 mai    nbj         ai  n     bj   n

As the force constants can only depend on the difference between the m and n indices, this symmetry is more conveniently expressed as shown on the right hand-side.

The acoustic sum-rule:

            _ _
 aj         \    bj
C  (R ) = -  )  C  (R )
 ai  0      /__  ai  m
           (m, b)
             !=
           (0, a)

Ordering of the unit cells illustrated here for a 1-dimensional system (in case refcell=None in constructor!):

    m = 0        m = 1        m = -2        m = -1
-----------------------------------------------------
|            |            |            |            |
|        * b |        *   |        *   |        *   |
|            |            |            |            |
|   * a      |   *        |   *        |   *        |
|            |            |            |            |
-----------------------------------------------------

Example:

>>> from ase.build import bulk
>>> from ase.phonons import Phonons
>>> from gpaw import GPAW, FermiDirac

>>> atoms = bulk('Si', 'diamond', a=5.4)
>>> calc = GPAW(mode='fd',
...             kpts=(5, 5, 5),
...             h=0.2,
...             occupations=FermiDirac(0.))
>>> ph = Phonons(atoms, calc, supercell=(5, 5, 5))
>>> ph.run()
>>> ph.read(method='frederiksen', acoustic=True)

Initialize with base class args and kwargs.

acoustic(C_N)[source]¶: Restore acoustic sumrule on force constants.

apply_cutoff(D_N, r_c)[source]¶

Zero elements for interatomic distances larger than the cutoff.

Parameters:

D_N: ndarray: Dynamical/force constant matrix.
r_c: float: Cutoff in Angstrom.

band_structure(path_kc, modes=False, born=False, verbose=True)[source]¶

Calculate phonon dispersion along a path in the Brillouin zone.

The dynamical matrix at arbitrary q-vectors is obtained by Fourier transforming the real-space force constants. In case of negative eigenvalues (squared frequency), the corresponding negative frequency is returned.

Frequencies and modes are in units of eV and 1/sqrt(amu), respectively.

Parameters:

path_kc: ndarray: List of k-point coordinates (in units of the reciprocal lattice vectors) specifying the path in the Brillouin zone for which the dynamical matrix will be calculated.
modes: bool: Returns both frequencies and modes when True.
born: bool: Include non-analytic part given by the Born effective charges and the static part of the high-frequency dielectric tensor. This contribution to the force constant accounts for the splitting between the LO and TO branches for q -> 0.
verbose: bool: Print warnings when imaginary frequncies are detected.

Returns:

If modes=False: Array of energies

If modes=True: Tuple of two arrays with energies and modes.

check_eq_forces()[source]¶: Check maximum size of forces in the equilibrium structure.

compute_dynamical_matrix(q_scaled: ndarray, D_N: ndarray)[source]¶

Computation of the dynamical matrix in momentum space D_ab(q).: This is a Fourier transform from real-space dynamical matrix D_N for a given momentum vector q.

q_scaled: q vector in scaled coordinates.

D_N: the dynamical matrix in real-space. It is necessary, at least

currently, to provide this matrix explicitly (rather than use self.D_N) because this matrix is modified by the Born charges contributions and these modifications are momentum (q) dependent.

Result:

D(q): two-dimensional, complex-valued array of: shape=(3 * natoms, 3 * natoms).

dos(kpts=(10, 10, 10), npts=1000, delta=0.001)[source]¶

Calculate phonon dos as a function of energy.

Parameters:

kpts: tuple: Shape of Monkhorst-Pack grid for sampling the Brillouin zone.
npts: int: Number of energy points.
delta: float: Broadening of Lorentzian line-shape in eV.

Returns:: Tuple of (frequencies, dos). The frequencies are in units of eV.

Deprecated since version 3.23.1: Please use the .get_dos() method instead, it returns a proper RawDOSData object.

get_band_structure(path, modes=False, born=False, verbose=True)[source]¶

Calculate and return the phonon band structure.

This method computes the phonon band structure for a given path in reciprocal space. It is a wrapper around the internal \(band_structure\) method of the \(Phonons\) class. The method can optionally calculate and return phonon modes.

Frequencies and modes are in units of eV and 1/sqrt(amu), respectively.

Parameters:

pathBandPath object: The BandPath object defining the path in the reciprocal space over which the phonon band structure is calculated.
modesbool, optional: If True, phonon modes will also be calculated and returned. Defaults to False.
bornbool, optional: If True, includes the effect of Born effective charges in the phonon calculations. Defaults to False.
verbosebool, optional: If True, enables verbose output during the calculation. Defaults to True.

Returns:

BandStructure or tuple of (BandStructure, ndarray)

If modes is False, returns a BandStructure object containing the phonon band structure. If modes is True, returns a tuple, where the first element is the BandStructure object and the second element is an ndarray of phonon modes.

If modes are returned, the array is of shape (k-point, bands, atoms, 3) and the norm-squared of the mode is \(1 / m_{eff}\), where \(m_{eff}\) is the effective mass of the mode.

Example:

>>> from ase.dft.kpoints import BandPath
>>> path = BandPath(...)  # Define the band path
>>> phonons = Phonons(...)
>>> bs, modes = phonons.get_band_structure(path, modes=True)

get_dos(kpts=(10, 10, 10), indices=None, verbose=True)[source]¶

Return a phonon density of states.

Parameters:

kpts: tuple: Shape of Monkhorst-Pack grid for sampling the Brillouin zone.
indices: list: If indices is not None, the amplitude-weighted atomic-partial DOS for the specified atoms will be calculated.
verbose: bool: Print warnings when imaginary frequncies are detected.

Returns:: A RawDOSData object containing the density of states.

get_force_constant()[source]¶: Return matrix of force constants.

read(method='Frederiksen', symmetrize=3, acoustic=True, cutoff=None, born=False, **kwargs)[source]¶

Read forces from json files and calculate force constants.

Extra keyword arguments will be passed to read_born_charges.

Parameters:

method: str: Specify method for evaluating the atomic forces.
symmetrize: int: Symmetrize force constants (see doc string at top) when symmetrize != 0 (default: 3). Since restoring the acoustic sum rule breaks the symmetry, the symmetrization must be repeated a few times until the changes a insignificant. The integer gives the number of iterations that will be carried out.
acoustic: bool: Restore the acoustic sum rule on the force constants.
cutoff: None or float: Zero elements in the dynamical matrix between atoms with an interatomic distance larger than the cutoff.
born: bool: Read in Born effective charge tensor and high-frequency static dielelctric tensor from file.

read_born_charges(name='born', neutrality=True)[source]¶

Read Born charges and dieletric tensor from JSON file.

The charge neutrality sum-rule:

_ _
\    a
 )  Z   = 0
/__  ij
 a

Parameters:

neutrality: bool: Restore charge neutrality condition on calculated Born effective charges.
name: str: Key used to identify the file with Born charges for the unit cell in the JSON cache.

Deprecated since version 3.22.1: Current implementation of non-analytical correction is likely incorrect, see issue: #941

symmetrize(C_N)[source]¶: Symmetrize force constant matrix.

write_modes(q_c, branches=0, kT=0.02585199101165164, born=False, repeat=(1, 1, 1), nimages=30, center=False)[source]¶

Write modes to trajectory file.

Parameters:

q_c: ndarray of shape (3,): q-vector of the modes.
branches: int or list: Branch index of modes.
kT: float: Temperature in units of eV. Determines the amplitude of the atomic displacements in the modes.
born: bool: Include non-analytic contribution to the force constants at q -> 0.
repeat: tuple: Repeat atoms (l, m, n) times in the directions of the lattice vectors. Displacements of atoms in repeated cells carry a Bloch phase factor given by the q-vector and the cell lattice vector R_m.
nimages: int: Number of images in an oscillation.
center: bool: Center atoms in unit cell if True (default: False).

To exaggerate the amplitudes for better visualization, multiply kT by the square of the desired factor.