Large lattice thermal conductivity, interplay between phonon-phonon, phonon-electron, and phonon-isotope scatterings, and electrical transport in molybdenum from first principles

We describe an ab initio phonon Boltzmann transport equation (BTE) approach accounting for phonon-electron scattering in addition to the well-established phonon-phonon and isotope scatterings. The phonon BTE is linearized and can be exactly solved beyond the relaxation time approximation (RTA). We use this approach to study the lattice thermal conductivity (${\ensuremath{\kappa}}_{\text{ph}}$) of molybdenum (Mo). ${\ensuremath{\kappa}}_{\text{ph}}$ of Mo is found to possess several anomalous features: (1) like in another group VI element tungsten (W), ${\ensuremath{\kappa}}_{\text{ph}}$, with a large value of 37 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature, follows weak temperature dependence due to interplay between phonon-phonon (ph-ph), phonon-electron (ph-el), and phonon-isotope (isotope) scatterings; and (2) compared with W, though Mo is much lighter in mass, Mo has a smaller ${\ensuremath{\kappa}}_{\text{ph}}$. This is attributed to weaker interatomic bonding, larger isotope mixture, and larger density of states at Fermi level in Mo. In isotopically pure samples, ${\ensuremath{\kappa}}_{\text{ph}}$ increases from 37 to 48 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature. Considering the similarity of the phonon dispersion, our work suggests that chromium should also have a large ${\ensuremath{\kappa}}_{\text{ph}}$, which, rather than the complexity of the electronic band structure argued in the literature, accounts for the significant deviation of measured Lorenz number $L$ from the expected Sommerfeld value. The electrical conductivity ($\ensuremath{\sigma}$) and electronic thermal conductivity (${\ensuremath{\kappa}}_{\text{e}}$) of Mo are also calculated by using an ab initio electron BTE approach. $\ensuremath{\sigma}$ and the total thermal conductivity ($\ensuremath{\kappa}$) agree with the experimental data reasonably. These results demonstrate that the ab initio calculations can quantify the lattice and electronic contributions to $\ensuremath{\kappa}$. We also look into the cumulative $\ensuremath{\sigma}$ and ${\ensuremath{\kappa}}_{\text{ph}}$ with respect to electron and phonon mean free paths (MFPs), respectively, in order to reveal the size effect in Mo. The MFPs of electrons contributing to conductivity range from 5 to 22 nm, whereas the MFPs of phonons primarily distribute between 5 and 73 nm with more than 80% contribution to ${\ensuremath{\kappa}}_{\text{ph}}$. This suggests that a reduced Lorenz number can be observed in Mo nanostructures when the relevant size goes below $\ensuremath{\sim}70$ nm.