Meaning of Kronecker Product in Transport Theory behind BoltzTrap2

Question

I am somewhat confused by the use of a Kronecker Product in Madsen, Georg K. H. et al. “BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients.” Comput. Phys. Commun. 231 (2018): 140-145. (Link to paper on arXiv). The equation I'm having trouble with is equation 8 which reads

$$ \sigma(\epsilon,T)=\int \sum_b {\bf v}_{b,{\bf k}}\otimes {\bf v}_{b,{\bf k}}\tau_{b,{\bf k}} \delta(\epsilon-\epsilon_{b,{\bf k}}){{d{\bf k}}\over{8\pi^3}} $$

What is the Kronecker product supposed to do here? My understanding is that the result of such an operation should be a matrix but that makes no mathematical sense to me in this context, so what is going on?

Answer 1

Yes, that operation yields a matrix result. The charge conductivity $\sigma$ is also a matrix (often called the conductivity tensor) so the equation looks correct. It may be clearer if written out with indices: $$ \sigma^{\alpha\beta}(\epsilon,T)=\int \sum_b { v}^\alpha_{b,{\bf k}} { v}^\beta_{b,{\bf k}}\tau_{b,{\bf k}} \delta(\epsilon-\epsilon_{b,{\bf k}}){{d{\bf k}}\over{8\pi^3}}. $$ In general crystal structures, $\sigma^{\alpha\beta}$ can have off-diagonal elements. Another way of saying that is that the electric current $\mathbf{J}_e$ need not (even in the absence of a temperature gradient) be parallel to the electric field $\mathbf{E}$ unless $\sigma$ is constrained to be diagonal by symmetry.

If you need more background than the BoltzTraP2 paper provides, Boltzmann transport theory is treated in standard textbooks. But note that different books use subtly different conventions - especially when it comes to the definition of the $\mathcal{L}^{(a)}$ symbols - so be careful if using formulas from multiple sources. The convention used in the paper appears to be the same as in Marder's Condensed Matter Physics book, which is different from the convention used by e.g. Ashcroft & Mermin.