Phonon-Derived Transport Properties

The Phonon-derived transport properties module provides post-processing tools for three phonon-based descriptors:

  • phonon group velocity;

  • mode Gruneisen parameter;

  • Debye temperature.

It is available from:

VQkit
 -> Post-processing
 -> [7] Application-oriented analysis
 -> [4] Vibrational / phonon analysis
 -> [1] Phonon-derived transport descriptors

The current menu contains:

[1] Phonon group velocity
    Calculate v_g from the slope of phonon dispersion

[2] Gruneisen parameter
    Calculate mode gamma from volume-dependent phonon frequencies

[3] Debye temperature
    Automatic band.dat + structure workflow or maximum-frequency estimate

These are post-processing analyses of already calculated phonon data.

VQkit does not perform the underlying lattice-dynamics calculation in this module.

Input phonon data can come from:

  • VASP + Phonopy;

  • Quantum ESPRESSO + Phonopy;

  • another phonon workflow exported in a supported numerical format.

[1] Phonon group velocity

Choose:

[1] Phonon group velocity

VQkit calculates phonon group velocity from the slope of the phonon dispersion.

For a phonon mode with ordinary frequency \(\nu\):

\[v_g = \frac{d\nu}{dq}\]

with the unit conversion appropriate to the supplied reciprocal-distance convention.

For native Phonopy band.dat output, VQkit follows the Phonopy path-distance convention directly.

No additional \(2\pi\) factor is applied.

Supported dispersion-table formats

VQkit accepts three layouts.

Phonopy band.dat

Recommended:

phonopy-bandplot --gnuplot band.yaml > band.dat

In this format, every phonon branch contains multiple path-segment blocks.

Blank lines separate segments.

The header may contain:

# End points of segments:

followed by cumulative reciprocal-space distances.

VQkit reconstructs the segments into:

  • one common q-distance grid;

  • one frequency column per branch.

Rectangular table

VQkit also accepts:

q_distance   f1   f2   f3   ...

where frequencies are in THz.

Simple branch-block table

A two-column format is also supported:

q_distance   frequency

with each complete phonon branch separated by a blank line.

Source-unit selection

VQkit asks:

Phonon data source / reciprocal-distance convention:

[1] VASP / Phonopy              -> Angstrom^-1
[2] Quantum ESPRESSO / Phonopy  -> bohr^-1
[3] Custom data                 -> choose unit manually

The source unit matters because the q-distance derivative must be converted into SI velocity units.

For VASP/Phonopy:

q-distance unit = Angstrom^-1

For QE/Phonopy:

q-distance unit = bohr^-1

Group-velocity conversion

For a VASP/Angstrom workflow:

\[1\ \mathrm{THz\cdot\AA} = 100\ \mathrm{m/s}\]

The conversion is therefore applied directly to the numerical slope.

For bohr-based reciprocal coordinates, VQkit uses the Bohr length internally.

Segment-aware differentiation

Phonopy commonly repeats a segment endpoint as the first point of the next path segment.

A naive central difference would mix two different propagation directions.

VQkit therefore uses:

  • backward differentiation at the final point of one segment;

  • forward differentiation at the first point of the next segment.

This keeps directional slopes from neighboring path segments separate.

Branch-resolved velocities

VQkit computes a velocity for every phonon branch at every valid path point.

The main output is:

VQkit_GROUP_VELOCITY.dat

The file contains:

q_distance
f1_THz
v1_m_s
f2_THz
v2_m_s
...

Acoustic-mode estimate near Gamma

VQkit also attempts to identify the three acoustic branches near the first Gamma-like point.

The three branches with frequencies closest to zero are selected.

They are accepted as acoustic branches only when their starting frequencies are sufficiently close to zero.

The current tolerance is approximately:

0.20 THz

VQkit then fits the first:

5 points

of the first path segment.

The fitted slopes give acoustic-mode velocities.

Representative output

VQkit may report:

Acoustic-1 branch 1 : ...
Acoustic-2 branch 2 : ...
Acoustic-3 branch 3 : ...

and then calculates an acoustic RMS velocity.

Acoustic RMS velocity

For three acoustic speeds:

\[v_1,\ v_2,\ v_3\]

VQkit uses the root-mean-square value:

\[v_{\mathrm{RMS}} = \sqrt{ \frac{ v_1^2+v_2^2+v_3^2 }{3} }\]

The result can be reused automatically by the Debye-temperature workflow in the same VQkit session.

Group-velocity cautions

The near-Gamma acoustic velocity estimate assumes:

  • the first q point is Gamma-like;

  • the acoustic branches are identifiable;

  • the first segment is sufficiently linear near Gamma;

  • the first few q points sample the acoustic regime.

If these conditions are not satisfied, VQkit does not force an acoustic summary.

[2] Gruneisen parameter

Choose:

[2] Gruneisen parameter

The mode Gruneisen parameter measures phonon-frequency sensitivity to volume.

It is defined as:

\[\gamma_{qj} = - \frac{ \partial \ln\omega_{qj} }{ \partial \ln V }\]

where:

  • \(\omega_{qj}\) is the frequency of mode j at q point q;

  • \(V\) is cell volume.

VQkit supports several Gruneisen workflows.

Gruneisen menu

The current menu contains:

[1] Read gruneisen_band.dat and report TA, TA', LA RMS G
[2] Calculate mode gamma from two phonon calculation folders
[3] Select two phonon tables + structure files
[4] Manual two phonon tables + manual volumes

[1] gruneisen_band.dat

For a Phonopy-style file:

gruneisen_band.dat

VQkit expects rows of the form:

q_a q_b q_c Distance gamma_1 freq_1 gamma_2 freq_2 ...

The first three phonon branches are treated as:

TA
TA'
LA

for the acoustic RMS summary.

Acoustic RMS Gruneisen parameter

For one branch:

\[\gamma_{\mathrm{RMS}} = \sqrt{ \left< \gamma(q)^2 \right> }\]

VQkit calculates this independently for:

  • TA;

  • TA’;

  • LA.

The summary is written to:

VQkit_GRUNEISEN_ACOUSTIC_RMS.txt

The file includes:

  • source file;

  • number of q points;

  • number of branches;

  • RMS Gruneisen value for each acoustic branch.

Two-volume Gruneisen calculation

VQkit can also calculate mode Gruneisen parameters directly from two volume-dependent phonon datasets.

For lower and higher volumes:

\[V_- \quad\mathrm{and}\quad V_+\]

with corresponding frequencies:

\[\omega_- \quad\mathrm{and}\quad \omega_+\]

VQkit evaluates the logarithmic finite difference:

\[\gamma = - \frac{ \ln\omega_+ - \ln\omega_- }{ \ln V_+ - \ln V_- }\]

The two datasets must have:

  • the same q grid;

  • the same number of branches;

  • the same branch ordering.

Non-positive frequencies

The logarithmic derivative is undefined for:

  • zero frequencies;

  • imaginary / negative frequencies.

VQkit therefore returns:

NaN

for those mode points.

They are excluded from RMS statistics.

Folder-based workflow

In automatic folder mode, each volume directory should contain:

  • one recognizable phonon band.dat or *_band.dat file;

  • a readable structure file.

VQkit searches for:

POSCAR
CONTCAR

or a readable QE input.

It then obtains:

  • the phonon dispersion;

  • the cell volume;

  • the q grid;

  • branch frequencies.

If multiple candidate band files exist in one folder, VQkit stops rather than selecting one silently.

Volume-span warning

VQkit reports the approximate half-volume span around the two-point midpoint.

If the volume difference is large, it warns that a smaller, more symmetric volume perturbation is generally preferable for a local derivative.

Gruneisen outputs

For the full mode-resolved calculation, VQkit writes:

VQkit_GRUNEISEN.dat

The file contains:

q_distance
gamma1
gamma2
gamma3
...

VQkit also reports:

  • per-branch RMS Gruneisen parameter;

  • overall RMS Gruneisen parameter.

Scientific interpretation

Large absolute Gruneisen values can indicate stronger anharmonic sensitivity of phonon frequencies to volume.

However, Gruneisen parameters alone do not determine lattice thermal conductivity.

Thermal transport also depends on quantities such as:

  • heat capacity;

  • phonon lifetimes;

  • scattering phase space;

  • group velocities;

  • isotope scattering;

  • boundary scattering.

[3] Debye temperature

Choose:

[3] Debye temperature

The current menu contains:

[1] Automatic from phonon band file + POSCAR/CONTCAR
[2] From a maximum phonon frequency
[3] Manual acoustic velocities

Automatic workflow is recommended.

Automatic Debye workflow

VQkit first locates a phonon band file.

Preferred names include:

band.dat
*_band.dat

If one candidate exists, it is selected automatically.

If several are present, VQkit asks the user to choose.

Acoustic velocity

The automatic workflow uses the near-Gamma acoustic analysis described in the group-velocity section.

If acoustic velocities were already calculated earlier in the same VQkit session, VQkit can reuse them instead of fitting them again.

Structure detection

VQkit then searches for:

POSCAR
CONTCAR

and can also read a suitable QE input.

From the structure VQkit obtains:

  • lattice vectors;

  • lattice lengths;

  • number of atoms;

  • cell volume.

Velocity-based Debye temperature

The implemented relation is:

\[\Theta_D = \frac{h}{k_B} v_g \left( \frac{ 3N }{ 4\pi V } \right)^{1/3}\]

where:

  • \(h\) is Planck’s constant;

  • \(k_B\) is Boltzmann’s constant;

  • \(v_g\) is the acoustic RMS velocity;

  • \(N\) is the number of atoms in the cell;

  • \(V\) is cell volume.

VQkit converts:

Angstrom^3 -> m^3

internally before evaluating the expression.

Automatic Debye output

VQkit writes:

VQkit_DEBYE_TEMPERATURE.txt

The file includes:

  • phonon band source;

  • structure source;

  • reciprocal-distance unit;

  • near-Gamma fit-point count;

  • three acoustic branch indices;

  • three acoustic velocities;

  • acoustic RMS velocity;

  • number of atoms;

  • cell volume;

  • Debye temperature.

Maximum-frequency estimate

VQkit also supports the simplified estimate:

\[\Theta_D = \frac{ h\nu_{\max} }{ k_B }\]

where:

nu_max

is entered in THz.

This is a branch-frequency-based estimate rather than the acoustic-density Debye expression.

The result is written to:

VQkit_DEBYE_TEMPERATURE.txt

Manual acoustic-velocity mode

If automatic acoustic extraction is unavailable, VQkit allows manual entry of:

Acoustic-1 velocity
Acoustic-2 velocity
Acoustic-3 velocity

all in:

m/s

It calculates their RMS velocity and then evaluates the same velocity-based Debye relation.

The structure information can be:

  • read from VASP/QE input;

  • entered manually as atom count and volume.

Relationship among the three descriptors

The three quantities are related but measure different aspects of phonon behavior.

Group velocity

Describes the propagation speed associated with the phonon-dispersion slope.

High acoustic group velocities can support efficient heat transport when phonon lifetimes are also sufficiently long.

Gruneisen parameter

Measures the sensitivity of phonon frequencies to volume.

It is commonly used as a descriptor of anharmonicity.

Debye temperature

Provides an effective characteristic phonon-energy scale.

In VQkit it can be estimated from either:

  • acoustic RMS velocity and atomic density;

  • maximum phonon frequency.

These two routes are not mathematically identical and should be identified clearly when reported.

Scientific cautions

Phonon stability

Imaginary phonon frequencies can indicate:

  • structural instability;

  • insufficient convergence;

  • inadequate supercell size;

  • numerical noise;

  • genuine soft modes.

Transport descriptors should not be interpreted blindly for an unstable structure.

q-path dependence

The group velocity reported from a band path is directional.

A high-symmetry path does not sample the full Brillouin zone.

Therefore path-derived velocities are useful descriptors, but they are not a substitute for full Brillouin-zone transport integration.

Acoustic fitting

The automatic acoustic velocity uses the first five points of the first Gamma-starting segment.

The result depends on:

  • q-point spacing;

  • linearity near Gamma;

  • branch identification;

  • dispersion quality.

Volume perturbation

Gruneisen parameters are local derivatives.

Use sufficiently small volume changes while keeping numerical phonon noise under control.

Branch correspondence

The two volume-dependent phonon datasets must use the same branch ordering.

Branch crossings can complicate naive mode matching.

Debye-model limitation

Debye temperature is a model descriptor.

It does not contain complete information about:

  • optical phonons;

  • phonon lifetimes;

  • anharmonic scattering;

  • full thermal conductivity.

Thermal conductivity

The current VQkit module does not calculate full lattice thermal conductivity.

For quantitative lattice thermal conductivity, a full Boltzmann transport or equivalent anharmonic phonon workflow is required.

Generated files summary

Group velocity

VQkit_GROUP_VELOCITY.dat

Acoustic Gruneisen RMS

VQkit_GRUNEISEN_ACOUSTIC_RMS.txt

Mode-resolved Gruneisen parameter

VQkit_GRUNEISEN.dat

Debye temperature

VQkit_DEBYE_TEMPERATURE.txt

Summary

The current VQkit phonon-derived transport module provides:

Group velocity
-> branch-resolved dnu/dq
-> acoustic velocities near Gamma
-> acoustic RMS velocity

Gruneisen parameter
-> Phonopy gruneisen_band.dat analysis
-> two-volume finite-difference gamma
-> TA / TA' / LA RMS gamma
-> branch and overall RMS gamma

Debye temperature
-> automatic band.dat + structure
-> acoustic-velocity relation
-> maximum-frequency estimate
-> manual acoustic-velocity fallback

These quantities provide useful phonon transport descriptors while keeping the underlying phonon calculation and physical interpretation under the researcher’s control.