(auxiliary_tools)=

# Auxiliary tools

```{contents}
:depth: 3
:local:
```

(auxiliary_tools_kaccum)=

## `phono3py-kaccum`: Cumulative lattice thermal conductivity and related properties

Cumulative physical properties with respect to frequency or mean free path are
calculated using this command.

For example, cumulative thermal conductivity is defined by

$$
\kappa^\text{c}(\omega) =
   \int^\omega_0 \frac{1}{N} \sum_\lambda
\kappa_\lambda \delta(\omega_\lambda - \omega') d\omega'
$$

$\kappa_\lambda$ is the contribution to $\kappa$ from the phonon mode $\lambda$,
which is defined as

$$
\kappa_\lambda = \frac{1}{V_0}
C_\lambda \mathbf{v}_\lambda \otimes \mathbf{v}_\lambda
\tau_\lambda.
$$

(The notations are found in http://arxiv.org/abs/1501.00691.)

### How to use `phono3py-kaccum`

Let's compute lattice thermal conductivity of Si using the `Si-PBEsol` example
found in the example directory.

```bash
% phono3py --mesh="11 11 11" --fc-symmetry --br
```

Then using the output file, `kappa-m111111.hdf5`, run `phono3py-kaccum` as
follows:

```bash
% phono3py-kaccum kappa-m111111.hdf5 |tee kaccum.dat
```

`kappa-m111111.hdf5` is the output file of thermal conductivity calculation,
which is passed to `phono3py-kaccum` as the first argument. `phono3py_disp.yaml`
or `phono3py.yaml` is necessary to be located under the current directory to
read the crystal structure.

The format of the output is as follows: The first column gives frequency in THz,
and the second to seventh columns give the cumulative lattice thermal
conductivity of 6 elements, xx, yy, zz, yz, xz, xy. The eighth to 13th columns
give the derivatives. There are sets of frequencies, which are separated by
blank lines. Each set is for a temperature. There are the groups corresponding
to the number of temperatures calculated.

To plot the output by gnuplot at temperature index 30 that may correspond to 300
K,

```bash
% gnuplot
...
gnuplot> p "kaccum.dat" i 30 u 1:2 w l, "kaccum.dat" i 30 u 1:8 w l
```

The plot like below is displayed.

```{image} Si-kaccum.png
:width: 50%
```

With $19\times 19\times 19$ mesh:

```{image} Si-kaccum-m191919.png
:width: 25%
```

That calculated by QE with $19\times 19\times 19$ mesh:

```{image} Si-kaccum-pwscf.png
:width: 25%
```

### General options

#### `--temperature`

Pick up one temperature point. For example, `--temperature=300` for 300 K, which
works only if thermal conductivity is calculated at temperatures including 300
K.

#### `--nsp`

Number of points to be sampled in the x-axis.

### Options for tensor properties

For cumulative thermal conductivity, the last value is given as the thermal
conductivity in W/mK. For the other properties, the last value is effectively
the sum of values on all mesh grids divided by number of mesh grids. This is
understood as normalized for one primitive cell. Before version 1.11.13.1, the
last value for `gv_by_gv` (--gv option) was further divided by the primitive
cell volume.

Number of columns of output data is 13 as explained above. With `--average` and
`--trace` options, number of columns of output data becomes 3.

#### `--mfp`

Mean free path (MFP) is used instead of frequency for the x-axis. MFP is defined
in the single-mode RTA by a vector

$$
\mathbf{l}_\lambda = \mathbf{v}_\lambda \tau_\lambda.
$$

The MFP cumulative $\kappa^\text{c}(l)$ is given by

$$
\kappa^\text{c}(l) =
   \int^l_0 \frac{1}{N} \sum_\lambda
\kappa_\lambda \delta(l_\lambda - l') dl'
$$

where $l_\lambda = |\mathbf{l}_\lambda|$ and $\kappa_\lambda$ is the
contribution to $\kappa$ from the phonon mode $\lambda$ in the single-mode RTA,
which is defined as

$$
\kappa_\lambda = \frac{1}{V_0} C_\lambda \mathbf{v}_\lambda \otimes
\mathbf{v}_\lambda \tau_\lambda = \frac{1}{V_0} C_\lambda
\mathbf{v}_\lambda \otimes \mathbf{l}_\lambda.
$$

The physical unit of MFP is Angstrom.

The figure below shows the results of Si example with the $19\times 19\times 19$
and $11\times 11\times 11$ sampling meshes used for the lattice thermal
conductivity calculation. They look differently. Especially for the result of
the $11\times 11\times
11$ sampling mesh, the MFP seems converging but we can see
it's not true to look at that of the $19\times 19\times 19$ sampling mesh. To
show this type of plot, be careful about the sampling mesh convergence.

```{image} Si-kaccum-MFP.png
:width: 50%
```

(This plot is based on the `Si-PBEsol` example.)

#### `--gv`

Outer product of group velocities $\mathbf{v}_\lambda \otimes
\mathbf{v}_\lambda$
divided by primitive cell volume (in $\text{THz}^2 /
\text{Angstrom}$).

#### `--average`

Output the traces of the tensors divided by 3 rather than the unique elements.

#### `--trace`

Output the traces of the tensors rather than the unique elements.

### Options for scalar properties

For the following properties, those values are normalized by the number of full
grid points. This is understood as normalized for one primitive cell.

Number of columns of output data is three, frequency, cumulative property, and
derivative of cumulative property such like DOS.

#### `--gamma`

$\Gamma_\lambda(\omega_\lambda)$ (in THz).

#### `--tau`

Lifetime $\tau_\lambda = \frac{1}{2\Gamma_\lambda(\omega_\lambda)}$ (in ps)

#### `--cv`

Modal heat capacity $C_\lambda$ (in eV/K).

#### `--gv-norm`

Absolute value of group velocity $|\mathbf{v}_\lambda|$ (in
$\text{THz}\cdot\text{Angstrom}$).

#### `--pqj`

Averaged phonon-phonon interaction $P_{\mathbf{q}j}$ (in $\text{eV}^2$).

#### `--dos`

Constant value of 1. This results in phonon DOS.

(auxiliary_tools_kdeplot)=

## `phono3py-kdeplot`: Distribution of phonon lifetime

**This script is under the development and may contain bugs.** But a feature is
briefly introduced below since it may be useful. Scipy is needed to use this
script.

This script draws density of phonon modes in the frequency-lifetime plane. Its
density is estimated using Gaussian-KDE using
[scipy](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gaussian_kde.html).
Then (frequency, lifetime)-data points are superimposed on the density plot.

`phono3py-kdeplot` reads a result of the thermal conductivity calculation as the
first argument:

```bash
% phono3py-kdeplot kappa-m111111.hdf5
```

This calculation takes some time from minutes to hours depending on mesh numbers
and nbins. Therefore it is recommended to start with smaller mesh and gradually
to increase mesh numbers and nbins up to satisfaction.

After finishing the calculation, the plot is saved in `lifetime.png`. The black
dots show original data points. The drawing area is automatically set to make
the look good, where its higher lifetime side is not drawn if all density beyond
a lifetime value is smaller than some ratio (see {ref}`kdeplot_density_ratio`)
of the maximum density.

The following plot is drawn with a $19 \times 19 \times 19$ mesh and nbins=200
and the `Si-PBEsol` example is used to generate the data. The colormap of 'jet'
in matplotlib is used.

```{image} Si-kdeplot.png
:width: 50%
```

### Options

#### `--temperature`

Pick up one temperature point. For example, `--temperature=300` for 300 K, which
works only if thermal conductivity is calculated at temperatures including 300
K.

Without specifying this option, the 31st temperature index is chosen. This often
corresponds to 300 K if phono3py ran without setting temperature range and step.

#### `--nbins`

This option controls the resolution of the density plot. The default value
is 100. With larger nbins, the resolution of the plot becomes better, but the
computation will take more time.

```bash
% phono3py-kdeplot --nbins=200 kappa-m111111.hdf5
```

#### `--cutoff`, `--fmax`

The option `--cutoff` (`--fmax`) sets the maximum value of lifetime (frequency)
to be included as data points **before** Gaussian-KDE. Normally increasing this
value from the chosen value without specifying this option does nothing since
automatic control of drawing area cuts high lifetime (frequency) side if the
density is low.

#### `--xmax` and `--ymax`

Maximum values of drawing region of phonon frequency (x-axis) and lifetime
(y-axis) are specified by `--xmax` and `--ymax`, respectively.

`--ymax` switches off automatic determination of maximum value of drawing region
along y-axis, therefore as a side effect, the computation will be roughly twice
faster.

```bash
% phono3py-kdeplot --ymax=60 kappa-m111111.hdf5
```

#### `--zmax`

Maximum value of the density is specified with this option. The color along
colorbar saturates by choosing a smaller value than the maximum value of density
in the data.

(kdeplot_density_ratio)=

#### `--dr`, `--density-ratio`

The density threshold is specified by the ratio with respect to maximum density.
Normally smaller value results in larger drawing region. The default value is
0.1. When `--ymax` is specified together, this option is ignored.

```bash
% phono3py-kdeplot --dr=0.01 kappa-m111111.hdf5
```

#### `--cmap`

Color map to be used for the density plot. It's given by the name presented at
the matplotlib documentation, https://matplotlib.org/users/colormaps.html. The
default colormap depends on your matplotlib environment.

```bash
% phono3py-kdeplot --cmap="OrRd" kappa-m111111.hdf5
```

The following figures are those drawn with `jet`, `bwr`, `seismic`, `gnuplot`,
`hsv`, and `OrRd` colormaps.

```{image} Si-kdeplot-jet.png
:width: 25%
```

```{image} Si-kdeplot-bwr.png
:width: 25%
```

```{image} Si-kdeplot-seismic.png
:width: 25%
```

```{image} Si-kdeplot-gnuplot.png
:width: 25%
```

```{image} Si-kdeplot-hsv.png
:width: 25%
```

```{image} Si-kdeplot-OrRd.png
:width: 25%
```
