# Phono3py API#

How to use phono3py API is described below along with snippets that work with AlN-LDA example.

## Crystal structure#

Crystal structures in phono3py are usually PhonopyAtoms class instances. When we want to obtain the PhonopyAtoms class instances from a file written in a force-calculator format, the read_crystal_structure function in phonopy may be used, e.g., in the AlN-LDA example,

In [1]: from phonopy.interface.calculator import read_crystal_structure

In [2]: unitcell, _ = read_crystal_structure("POSCAR-unitcell", interface_mode='vasp')

In [3]: print(unitcell)
lattice:
- [     3.110999999491908,     0.000000000000000,     0.000000000000000 ] # a
- [    -1.555499999745954,     2.694205030733368,     0.000000000000000 ] # b
- [     0.000000000000000,     0.000000000000000,     4.978000000000000 ] # c
points:
- symbol: Al # 1
coordinates: [  0.333333333333333,  0.666666666666667,  0.000948820000000 ]
mass: 26.981539
- symbol: Al # 2
coordinates: [  0.666666666666667,  0.333333333333333,  0.500948820000000 ]
mass: 26.981539
- symbol: N  # 3
coordinates: [  0.333333333333333,  0.666666666666667,  0.619051180000000 ]
mass: 14.006700
- symbol: N  # 4
coordinates: [  0.666666666666667,  0.333333333333333,  0.119051180000000 ]
mass: 14.006700


Otherwise, it is directly created from PhonopyAtoms class.

In [1]: from phonopy.structure.atoms import PhonopyAtoms

In [2]: a = 3.11

In [3]: a = 3.111

In [4]: c = 4.978

In [5]: lattice = [[a, 0, 0], [-a / 2,  a * np.sqrt(3) / 2, 0], [0, 0, c]]

In [6]: x = 1. / 3

In [7]: points = [[x, 2 * x, 0], [x * 2, x, 0.5], [x, 2 * x, 0.1181], [2 * x, x, 0.6181]]

In [8]: symbols = ['Al', 'Al', 'N', 'N']

In [9]: unitcell = PhonopyAtoms(cell=lattice, scaled_positions=points, symbols=symbols)


## Phono3py class#

To operate phono3py from python, there is phono3py API. The main class is Phono3py that is imported by

from phono3py import Phono3py


As written in Workflow, phono3py workflow is roughly divided into three steps. Phono3py class is used at each step. The minimum set of inputs to instantiate Phono3py class are unitcell (1st argument), supercell_matrix (2nd argument), and primitive_matrix, which are used as

ph3 = Phono3py(unitcell, supercell_matrix=[3, 3, 2], primitive_matrix='auto')


The unitcell is an instance of the PhonopyAtoms class. supercell_matrix and primitive_matrix are the transformation matrices to generate supercell and primitive cell from unitcell (see definitions). This step is similar to the instantiation of Phonopy class.

There are many parameters that can be given to Phono3py class. The details are written in the docstring, which is shown by

help(Phono3py)


## Displacement dataset generation#

The step (1) in Workflow generates sets of displacements in supercell. Supercells with the displacements are used as input crystal structure models of force calculator that is, e.g., first-principles calculation code. Using the force calculator, sets of forces of the supercells with the displacements are obtained out of phono3py environment, i.e., phono3py only provides crystal structures. Here we call the sets of the displacements as displacement dataset, the sets of the supercell forces as force sets, and the pair of displacement dataset and force sets as simply dataset for computing second and third force constants.

After instantiating Phono3py with unitcell, displacements are generated as follows:

In [4]: ph3 = Phono3py(unitcell, supercell_matrix=[3, 3, 2], primitive_matrix='auto')

In [5]: ph3.generate_displacements()

In [6]: len(ph3.supercells_with_displacements)
Out[6]: 1254

In [7]: type(ph3.supercells_with_displacements[0])
Out[7]: phonopy.structure.atoms.PhonopyAtoms


By this, 1254 supercells with displacements were generated.

The generated displacement dataset is used in the force constants calculation. Therefore, it is recommended to save it into a file if the python process having the Phono3py class instance is expected to be terminated. The displacement dataset and crystal structure information are saved to a file by Phono3py.save() method:

In [8]: ph3.save("phono3py_disp.yaml")


## Supercell force calculation#

Forces of the generated supercells with displacements are calculated by some external force calculator such as first-principles calculation code.

Calculated supercell forces will be stored in a Phono3py class instance through Phono3py.forces attribute by setting an array_like variable with the shape of (num_supercells, num_atoms_in_supercell, 3). In the above example, the array shape is (1254, 72, 3).

If the calculated force sets are stored in the FORCES_FC3 file, the numpy array of forces is obtained by

forces = np.loadtxt("FORCES_FC3").reshape(-1, num_atoms_in_supercell, 3)
assert len(forces) == num_supercells


## Force constants calculation#

The pair of the displacement dataset and force sets is required to calculate force constants. phono3py.load is the convenient function to load these data from files and to set up them in the Phono3py class instance. However, in the case when only displacement dataset is expected, the low-level phono3py-yaml parser in Phono3pyYaml is useful.

In [1]: from phono3py.interface.phono3py_yaml import Phono3pyYaml

In [2]: ph3yml = Phono3pyYaml()

In [4]: disp_dataset = ph3yml.dataset


With this ph3yml, how to compute force constants is explained. In the following, it is assumed that we have FORCES_FC3 in the current directory. The displacement dataset and force sets are set to the Phono3py class instance as follows:

In [5]: unitcell = ph3yml.unitcell

In [6]: from phono3py import Phono3py

In [7]: ph3 = Phono3py(unitcell, supercell_matrix=ph3yml.supercell_matrix, primitive_matrix=ph3yml.primitive_matrix)

In [8]: import numpy as np

In [9]: forces = np.loadtxt("FORCES_FC3").reshape(-1, len(ph3.supercell), 3)

In [10]: ph3.dataset = disp_dataset

In [11]: ph3.forces = forces


Now it is ready to compute force constants.

In [12]: ph3.produce_fc3()


## Non-analytical term correction parameters#

Users collects Born effective charges and dielectric constant tensor from experiments or calculations, and they are used as parameters for non-analytical term correction (NAC). These parameters are stored as a python dict:

'born': ndarray
Born effective charges
shape=(num_atoms_in_primitive_cell, 3, 3), dtype='double', order='C'
'factor': float
Unit conversion factor
'dielectric': ndarray
Dielectric constant tensor
shape=(3, 3), dtype='double', order='C'


Be careful that, in the case of using phono3py API, Born effective charges of all atoms in the primitive cell are necessary, whereas in the BORN file, only Born effective charges of symmetrically independent atoms are written. Some more information about NAC parameters is found here.

This NAC parameters are set to the Phono3py class instance via the Phono3py.nac_params attribute. The NAC parameters may be read from BORN file

In [13]: from phonopy.file_IO import parse_BORN

In [14]: nac_params = parse_BORN(ph3.primitive, filename="BORN")

In [15]: ph3.nac_params = nan_params


where the parse_BORN function requires the corresponding primitive cell of the PhonopyAtoms class.

## Regular grid for q-point sampling#

Phonons are normally sampled on a $$\Gamma$$ centre regular grid in reciprocal space. There are three ways to specify the regular grid.

1. Three integer values

2. One value

3. 3x3 integer matrix

One of these is set to mesh_numbers attribute of the Phono3py class. For example,

ph3.mesh_numbers = [10, 10, 10]
ph3.mesh_numbers = 50
ph3.mesh_numbers = [[-10, 10, 10], [10, -10, 10], [10, 10, -10]]


### Three integer values#

Three integer values ($$n_1$$, $$n_2$$, $$n_3$$) are specified.

The conventional regular grid is defined by the three integer values. The $$\mathbf{q}$$-points are given by linear combination of reciprocal basis vectors divided by the respective integers, which is given as

(1)#$\begin{split}\mathbf{q} = \left( \frac{\mathbf{b}^*_1}{n_1} \; \frac{\mathbf{b}^*_2}{n_2} \; \frac{\mathbf{b}^*_3}{n_3} \right) \begin{pmatrix} m_1 \\ m_2 \\ m_3 \end{pmatrix},\end{split}$

where $$m_i \in \{ 0, 1, \ldots, n_i - 1 \}$$.

### One value#

A distance like value $$l$$ is specified. Using this value, a regular grid is generated. As default, the three integer values of the conventional regular grid are defined by the following calculation.

(2)#$n_i = \max[1, \mathrm{nint}(l|\mathbf{b}^*_i|)]$

Experimentally, use of a generalized regular grid is supported. By specifying use_grg = True at the Phono3py class instantiation, the generalized regular grid is generated using the value $$l$$. First, the conventional unit cell of the primitive cell is searched. Second, Eq. (2) is applied to the reciprocal basis vectors of the conventional unit cell. Then, $$\mathbf{q}$$-points are sampled following Eq. (1) with respect to the reciprocal basis vectors of the conventional unit cell. The parallelepiped defined by $$\mathbf{b}^*_i$$ of the conventinal unit cell can be smaller than that of the primitive cell. In this case, the $$\mathbf{q}$$-points are sampled to fill the latter parallelepiped.

### 3x3 integer matrix (experimental)#

This is used to define the generalized regular grid explicitly. The generalized regular grid is designed to be automatically generated by one given value considering symmetry. However a 3x3 integer matrix (array) is accepted if this matrix follows the symmetry properly.

## Phonon-phonon interaction calculation#

Three phonon interaction strength is calculated after defining the regular grid. When we have phono3py_disp.yaml and FORCES_FC3 used above (maybe BORN, too), it is convenient to get the Phono3py class instance with settings.

In [1]: import phono3py

In [2]: ph3 = phono3py.load("phono3py_disp.yaml", log_level=1)
NAC params were read from "BORN".
Displacement dataset for fc3 was read from "phono3py_disp.yaml".
Sets of supercell forces were read from "FORCES_FC3".
Computing fc3[ 1, x, x ] using numpy.linalg.pinv with displacements:
[ 0.0300  0.0000  0.0000]
[ 0.0000  0.0000  0.0300]
[ 0.0000  0.0000 -0.0300]
Computing fc3[ 37, x, x ] using numpy.linalg.pinv with displacements:
[ 0.0300  0.0000  0.0000]
[ 0.0000  0.0000  0.0300]
[ 0.0000  0.0000 -0.0300]
Expanding fc3.
fc3 was symmetrized.
Max drift of fc3: 0.000000 (yxx) 0.000000 (xyx) 0.000000 (xxy)
Displacement dataset for fc2 was read from "phono3py_disp.yaml".
Sets of supercell forces were read from "FORCES_FC3".
fc2 was symmetrized.
Max drift of fc2: 0.000000 (yy) 0.000000 (yy)


Three phonon interaction calculation becomes ready to run by the following lines:

In [3]: ph3.mesh_numbers = 30

In [4]: ph3.init_phph_interaction()


The three phonon interaction calculation is implemented in phono3py.phonon3.interaction.Interaction class. By phono3py’s design choice, $$\mathbf{q}_1$$ of the three phonons $$(\mathbf{q}_1, \mathbf{q}_2, \mathbf{q}_3)$$ in this class is given, and calculation iterates over different $$\mathbf{q}_2$$. $$\mathbf{q}_3$$ is uniquely determined from $$\mathbf{q}_1$$ and $$\mathbf{q}_2$$. Symmetry is employed to avoid calculating symmetrically redundant $$\mathbf{q}_2$$ at the fixed $$\mathbf{q}_1$$. One Interaction class instance stores the data only for one fixed $$\mathbf{q}_1$$.

Three phonon interaction strength requires large memory space. Therefore, in lattice thermal conductivity calculation, it is calculated on-demand, and then abandoned after use of the data. Three phonon interaction strength calculation is the most computationally demanding part for usual applications. Detailed techniques are used to avoid elements of the Three phonon interaction strength if it is specified to do so.

Three phonon interaction strength calculation is the engine of more practical (or closer to macroscopic physical properties) calculation such as lattice thermal conductivity calculation. To minimize computational resources required by each purpose of use, the outer function that calls this function determines proper computational configuration of this function.

## Lattice thermal conductivity calculation#

Once three phonon interaction calculation is prepared by ph3.init_phph_interaction(), it is ready to run lattice thermal conductivity calculation. The many parameters are explained in the docstring, but even with the default parameters, the lattice thermal conductivity calculation under the mode relaxation time approximation is performed as follows:

In [5]: ph3.run_thermal_conductivity()


## Use of different supercell dimensions for 2nd and 3rd order FCs#

Phono3py supports different supercell dimensions for second and third order force constants (fc2 and fc3). The default setting is using the same dimensions. Although fc3 requires much more number of supercells than fc2, in our experience, fc3 have shorter interaction range in direct space. Therefore we tend to expect to use smaller supercell dimension for fc3 than that for fc2. To achieve this, phonon_supercell_matrix parameter exists to specify fc2 supercell dimension independently.

Using POSCAR-unitcell in the AlN-LDA example,

In [1]: from phonopy.interface.calculator import read_crystal_structure

In [2]: unitcell, _ = read_crystal_structure("POSCAR-unitcell", interface_mode='vasp')

In [3]: from phono3py import Phono3py

In [4]: ph3 = Phono3py(unitcell, supercell_matrix=[3, 3, 2], primitive_matrix='auto', phonon_supercell_matrix=[5, 5, 3])

In [5]: ph3.save("phono3py_disp.yaml")

In [6]: ph3.generate_displacements()


Phono3py.generate_fc2_displacements() is the method to generate displacement dataset for fc2. This is normally unnecessary to be called because this is called when Phono3py.generate_displacements() is called. Phono3py.generate_fc2_displacements() may be called explicitly if non-default control of displacement pattern is expected. See their docstrings for the details.

When phonon_supercell_matrix is specified, the following attributes are usable:

Phono3py.phonon_supercell_matrix
Phono3py.phonon_dataset
Phono3py.phonon_forces
Phono3py.phonon_supercells_with_displacements


The meanings of them are found in their docstrings though they may be guessed easily.

## phono3py.load#

The purpose of phono3py.load is to create a Phono3py class instance with basic parameters loaded from the phono3py-yaml-type file and also to try setting up force constants and non-analytical term correction automatically from phono3py files in the current directory.

In AlN-LDA example, the unit cell structure, supercell matrix, and primitive matrix were recorded in the phono3py_disp_dimfc2.yaml file. This is easily read a helper function of phono3py.load. Using ipython (or jupyter-notebook):

In [1]: import phono3py

In [2]: ph3 = phono3py.load("phono3py_disp_dimfc2.yaml", produce_fc=False)

In [3]: type(ph3)
Out[3]: phono3py.api_phono3py.Phono3py

In [4]: print(ph3.unitcell)
lattice:
- [     3.110999999491908,     0.000000000000000,     0.000000000000000 ] # a
- [    -1.555499999745954,     2.694205030733368,     0.000000000000000 ] # b
- [     0.000000000000000,     0.000000000000000,     4.978000000000000 ] # c
points:
- symbol: Al # 1
coordinates: [  0.333333333333333,  0.666666666666667,  0.000948820000000 ]
mass: 26.981539
- symbol: Al # 2
coordinates: [  0.666666666666667,  0.333333333333333,  0.500948820000000 ]
mass: 26.981539
- symbol: N  # 3
coordinates: [  0.333333333333333,  0.666666666666667,  0.619051180000000 ]
mass: 14.006700
- symbol: N  # 4
coordinates: [  0.666666666666667,  0.333333333333333,  0.119051180000000 ]
mass: 14.006700

In [5]: ph3.supercell_matrix
Out[5]:
array([[3, 0, 0],
[0, 3, 0],
[0, 0, 2]])

In [6]: ph3.nac_params
Out[6]:
{'born': array([[[ 2.51264750e+00,  0.00000000e+00,  0.00000000e+00],
[ 0.00000000e+00,  2.51264750e+00,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  2.67353000e+00]],

[[ 2.51264750e+00, -2.22044605e-16,  0.00000000e+00],
[ 0.00000000e+00,  2.51264750e+00,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  2.67353000e+00]],

[[-2.51264750e+00,  0.00000000e+00,  0.00000000e+00],
[ 0.00000000e+00, -2.51264750e+00,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00, -2.67353000e+00]],

[[-2.51264750e+00,  2.22044605e-16,  0.00000000e+00],
[ 0.00000000e+00, -2.51264750e+00,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00, -2.67353000e+00]]]),
'factor': 14.39965172592227,
'dielectric': array([[4.435009, 0.      , 0.      ],
[0.      , 4.435009, 0.      ],
[0.      , 0.      , 4.653269]])}