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Phonopy

Optional dependency

PhonopyAnalyzer requires the phonopy extra.

uv add "materialsframework[phonopy]"

pip install "materialsframework[phonopy]"

Computes harmonic phonon frequencies, density of states, and thermal properties from finite-displacement force constants.

Overview

PhonopyAnalyzer builds a supercell (default 2x2x2), displaces one atom at a time by a small distance (default 0.01 Å), and evaluates the resulting forces with the provided calculator. Phonopy uses these forces to build the harmonic force constants, then diagonalizes the dynamical matrix on a q-point mesh to get phonon frequencies everywhere in the Brillouin zone. From there it derives the total phonon density of states (DOS), the site-projected DOS (PDOS), and thermodynamic quantities (free energy, entropy, heat capacity) under the harmonic approximation.

The harmonic approximation treats phonons as non-interacting: it is exact only to second order in atomic displacement, so it ignores phonon-phonon scattering. That's sufficient for vibrational thermodynamics and DOS/band structure, but not for finite thermal conductivity. See Phono3py for the anharmonic (third-order) extension.

Theory

Force constants and the dynamical matrix

The harmonic force constants are the second derivatives of the potential energy with respect to atomic displacements:

\[ \Phi_{\alpha\beta}(l\kappa, l'\kappa') = -\frac{\partial F_\alpha(l\kappa)}{\partial u_\beta(l'\kappa')} \]

where \(F_\alpha(l\kappa)\) is the force on atom \(\kappa\) in cell \(l\) along Cartesian direction \(\alpha\), and \(u_\beta(l'\kappa')\) is a displacement of atom \(\kappa'\) in cell \(l'\) along \(\beta\). PhonopyAnalyzer estimates \(\Phi\) numerically: it displaces each inequivalent atom by distance and reads off the force response from the calculator, then uses the supercell's symmetry to fill in the remaining components.

The mass-weighted, Fourier-transformed force constants give the dynamical matrix at wavevector \(\mathbf{q}\):

\[ D_{\alpha\beta}(\kappa\kappa', \mathbf{q}) = \frac{1}{\sqrt{m_\kappa m_{\kappa'}}} \sum_{l'} \Phi_{\alpha\beta}(0\kappa, l'\kappa')\, e^{i\mathbf{q}\cdot\mathbf{r}(l')} \]

Diagonalizing \(D(\mathbf{q})\) gives the phonon frequencies \(\omega_s(\mathbf{q})\) and eigenvectors (polarization vectors) for each band \(s\):

\[ D(\mathbf{q})\, \mathbf{e}_s(\mathbf{q}) = \omega_s(\mathbf{q})^2\, \mathbf{e}_s(\mathbf{q}) \]

Sampling \(\omega_s(\mathbf{q})\) on the mesh grid and histogramming (optionally Gaussian-smeared by sigma) gives the total DOS; resolving each eigenvector's atomic-site weight on pdos_mesh gives the site-projected DOS.

Thermal properties

For each phonon mode \(\lambda = (\mathbf{q}, s)\) with frequency \(\omega_\lambda\), the harmonic oscillator partition function gives the vibrational free energy, entropy, and constant-volume heat capacity at temperature \(T\):

\[ F(T) = \frac{1}{2}\sum_\lambda \hbar\omega_\lambda + k_B T \sum_\lambda \ln\left(1 - e^{-\hbar\omega_\lambda / k_B T}\right) \]
\[ S(T) = \sum_\lambda \left[ \frac{\hbar\omega_\lambda / k_B T}{e^{\hbar\omega_\lambda / k_B T} - 1} - \ln\left(1 - e^{-\hbar\omega_\lambda / k_B T}\right) \right] k_B \]
\[ C_V(T) = k_B \sum_\lambda \left(\frac{\hbar\omega_\lambda}{k_B T}\right)^2 \frac{e^{\hbar\omega_\lambda / k_B T}}{\left(e^{\hbar\omega_\lambda / k_B T} - 1\right)^2} \]

These sums run over every mode on the mesh grid and are evaluated at each temperature between t_min and t_max in steps of t_step.

References