(inter_band_transport)= # Inter-band transport The standard lattice thermal conductivity calculation of phono3py (selected by `--br` or `--lbte`) treats phonons as particles and accumulates only the diagonal (intra-band) contributions of the heat flux. In systems with many phonon bands that are close in frequency (e.g. crystals of large unit cells or glass-like), the off-diagonal (inter-band) terms of the heat-flux operator between such distinct bands provide an additional contribution to the thermal conductivity. The `--tt` (`--transport-type`) option selects which transport formulation is used to evaluate these contributions: ``` phono3py ... --br --tt njc23 phono3py ... --lbte --tt ibdb19 ``` - `--tt` is combined with `--br` (RTA) or `--lbte` (direct solution). Without `--tt`, the standard particle-like formulation is used. - The built-in variants `njc23`, `ibdb19`, and `smm19` are **experimental**. Their interface and behavior may change without notice. - `wte` invokes the external [phono3py-wte](https://github.com/MSimoncelli/phono3py-wte) plugin, which must be installed separately (see {ref}`wigner_solution`). In the following, the phonon frequency, eigenvector, Bose-Einstein distribution, and scattering half-linewidth (HWHM) of mode $(\mathbf{q}, j)$ are denoted by $\omega_{\mathbf{q}j}$, $e_{\mathbf{q}j}$, $n_{\mathbf{q}j}$, and $\Gamma_{\mathbf{q}j}$, respectively. $T$ is the temperature and $D(\mathbf{q})$ is the dynamical matrix. (velocity_matrix)= ## Velocity matrix The off-diagonal contributions are built from the velocity matrix, which generalizes the group velocity to pairs of bands $(j, j')$ at the same q-point. For the Cartesian direction $\alpha$, $$ v^{\alpha}_{\mathbf{q}jj'} = \frac{1}{2\sqrt{\omega_{\mathbf{q}j}\omega_{\mathbf{q}j'}}} \langle e_{\mathbf{q}j} | \frac{\partial D(\mathbf{q})}{\partial q_{\alpha}} | e_{\mathbf{q}j'} \rangle . $$ The diagonal element ($j = j'$) reduces to the ordinary group velocity $v_{\mathbf{q}j}$. (mode_conductivity)= ## Mode thermal conductivity All inter-band variants build the thermal conductivity from the velocity matrix and a heat capacity matrix $C_{\mathbf{q}jj'}$. They differ only in the definition of this matrix, given in the sections below. The mode contribution to the thermal conductivity tensor for the band pair $(j, j')$ at q-point $\mathbf{q}$ is $$ \kappa^{\alpha\beta}_{\mathbf{q}jj'} = C_{\mathbf{q}jj'}\, v^{\alpha}_{\mathbf{q}jj'} v^{\beta *}_{\mathbf{q}jj'} \frac{\Gamma_{\mathbf{q}j} + \Gamma_{\mathbf{q}j'}} {(\omega_{\mathbf{q}j} - \omega_{\mathbf{q}j'})^{2} + (\Gamma_{\mathbf{q}j} + \Gamma_{\mathbf{q}j'})^{2}} . $$ Each heat capacity matrix reduces on the diagonal ($j = j'$) to the ordinary mode heat capacity $C_{\mathbf{q}j}$, so the diagonal terms reproduce the standard particle-like (populations) conductivity, while the off-diagonal terms ($j \ne j'$) give the additional inter-band contribution. The thermal conductivity tensor is obtained by summing the mode contributions over the Brillouin zone and all band pairs (with the usual normalization by the cell volume and the number of grid points). Of the built-in variants, `njc23` and `ibdb19` arise from the Green-Kubo formulation and `smm19` from the Wigner / unified-theory picture. All reduce to the formula above with a variant-specific heat capacity matrix. The `njc23` and `ibdb19` heat capacity matrices are closely related. `ibdb19` evaluates the main-text Eq. (9) of Isaeva, Barbalinardo, Donadio, and Baroni (2019), a further approximation of a more detailed formula derived in the Supplementary Information of the same paper. The expression of Ndour, Jund, and Chaput (`njc23`) was obtained independently and turns out to coincide with the first term of Eq. (19) of that Supplementary Information, so `njc23` is the more accurate of the two. The sections below are ordered from the more accurate (`njc23`) to the more approximate (`ibdb19`), followed by `smm19`. ## `--tt njc23` This variant evaluates the expression of Ndour, Jund, and Chaput. Its heat capacity matrix, generalizing the scalar mode heat capacity to a band pair, is $$ C_{\mathbf{q}jj'} = -\frac{\hbar\,(\omega_{\mathbf{q}j} + \omega_{\mathbf{q}j'})^{2}} {4 T (\omega_{\mathbf{q}j} - \omega_{\mathbf{q}j'})} (n_{\mathbf{q}j} - n_{\mathbf{q}j'}) . $$ - M. Ndour, P. Jund, and L. Chaput, "Practical approach to thermal conductivity calculations of small SiO2 samples", J. Non-Cryst. Solids **621**, 122618 (2023). DOI: [10.1016/j.jnoncrysol.2023.122618](https://doi.org/10.1016/j.jnoncrysol.2023.122618) ## `--tt ibdb19` This variant evaluates the main-text Eq. (9) of Isaeva, Barbalinardo, Donadio, and Baroni. Its heat capacity matrix is $$ C_{\mathbf{q}jj'} = -\frac{\hbar\,\omega_{\mathbf{q}j} \omega_{\mathbf{q}j'}} {T (\omega_{\mathbf{q}j} - \omega_{\mathbf{q}j'})} (n_{\mathbf{q}j} - n_{\mathbf{q}j'}) . $$ The two heat capacity matrices differ only in the frequency prefactor: `njc23` uses the arithmetic-mean-square $(\omega_{\mathbf{q}j} + \omega_{\mathbf{q}j'})^{2} / 4$, while `ibdb19` uses the geometric-mean-square $\omega_{\mathbf{q}j} \omega_{\mathbf{q}j'}$. Since $\omega_{\mathbf{q}j} \omega_{\mathbf{q}j'} \le (\omega_{\mathbf{q}j} + \omega_{\mathbf{q}j'})^{2} / 4$ with equality only at $\omega_{\mathbf{q}j} = \omega_{\mathbf{q}j'}$, the two coincide on the diagonal (and thus give the same particle-like conductivity), while the off-diagonal contribution of `ibdb19` is slightly smaller than that of `njc23`. - L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, "Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach", Nat. Commun. **10**, 3853 (2019). DOI: [10.1038/s41467-019-11572-4](https://doi.org/10.1038/s41467-019-11572-4) ## `--tt smm19` This variant evaluates a Wigner / unified-theory term in the spirit of the formulation by Simoncelli, Marzari, and Mauri. The original formulation and its reference implementation are due to those authors and are provided in the phono3py-wte plugin, invoked with `--tt wte` (see {ref}`wigner_solution`). That plugin should be used for results consistent with the published Wigner formulation. The built-in `smm19` here is an experimental, simplified in-tree variant that adopts the velocity matrix defined in the {ref}`Velocity matrix ` section above (the same definition as the other variants) instead of the original velocity-operator definition. Its off-diagonal results are therefore not identical to those of the original Wigner formulation. This in-tree `smm19` shares the same velocity matrix and Lorentzian linewidth factor as the other variants. The difference is its heat capacity matrix, an effective matrix built from the scalar mode heat capacities $C_{\mathbf{q}j}$ in a symmetrized form, $$ C_{\mathbf{q}jj'} = \frac{1}{4} (\omega_{\mathbf{q}j} + \omega_{\mathbf{q}j'}) \left( \frac{C_{\mathbf{q}j}}{\omega_{\mathbf{q}j}} + \frac{C_{\mathbf{q}j'}}{\omega_{\mathbf{q}j'}} \right) , $$ which reduces on the diagonal ($j = j'$) to the ordinary mode heat capacity $C_{\mathbf{q}j}$. - M. Simoncelli, N. Marzari, and F. Mauri, "Unified theory of thermal transport in crystals and glasses", Nat. Phys. **15**, 809 (2019). DOI: [10.1038/s41567-019-0520-x](https://doi.org/10.1038/s41567-019-0520-x) ## Example The lattice thermal conductivity is reported split into the intra-band part `K_intra`, the inter-band part `K_inter`, and their sum `K_TOT`. As an example, NaMgF3 (an orthorhombic perovskite with 20 atoms per unit cell, mp-2955 in the [phonondb LTC list](https://github.com/atztogo/phonondb/blob/main/mdr/phonondb_LTC/README.md)) at 300 K in the RTA with `--tt njc23 --mesh 50` gives roughly ``` # T(K) xx yy zz yz xz xy K_intra 300.0 3.022 3.230 3.234 0.000 0.000 0.000 K_inter 300.0 0.702 0.746 0.740 0.000 0.000 0.000 K_TOT 300.0 3.724 3.976 3.975 0.000 0.000 0.000 ``` For this system the three built-in variants `njc23`, `ibdb19`, and `smm19` give `K_inter` values that agree to within about 1%. ## `--tt wte` The solution of the Wigner transport equation is provided as a separate plugin, phono3py-wte, and is invoked with `--tt wte`. The plugin must be installed separately (it is not bundled with phono3py). See {ref}`wigner_solution` for installation and usage. ```{toctree} :hidden: wigner-solution ```