Understanding the thermal conductivity and Lorenz number in tungsten from first principles

Tungsten is known to have a Lorenz number $L$ larger than the Sommerfeld value ($L_0={\pi }^2{k}_B^2/3e^2=2.445\times {10}^{-8}{\mathtt{V}}^2/{\mathtt{deg}}^2$) by 30%. By performing fully first-principles calculations, we are able to calculate the electrical conductivity ($\sigma$) and quantify the electronic (${\kappa }_e$) and the lattice (${\kappa }_{\mathtt{ph}}$) contributions to the thermal conductivity with a high accuracy. We show that the deviation of $L$ is entirely due to ${\kappa }_{\mathtt{ph}}$, and ${\kappa }_e/\sigma T$ agrees with $L_0$ within 5%. At room temperature, ${\kappa }_{\mathtt{ph}}$ is 46 W/m-K, one order of magnitude larger than that in other metals even with smaller atomic mass and higher Debye temperature, and likely the largest of all metals. The large ${\kappa }_{\mathtt{ph}}$ is ascribed to the surprisingly weak anharmonic phonon scattering. Apart from the not-strong anharmonic interatomic interaction, the weak anharmonic phonon scattering is also facilitated with the large atomic mass, leading to small thermal displacement. The interplay between the phonon-phonon and electron-phonon scatterings leads to weak temperature dependence of ${\kappa }_{\mathtt{ph}}$, and signifies the importance of an accurate solution to the Boltzmann transport equation beyond the conventional relaxation time approximation. Our findings give insights into the phonon transport in metals.

Figure 1

(a) The electrical conductivity, (b) the thermal conductivity, and (c) the Lorenz number. The symbols show experimental electrical and thermal conductivity taken from Refs. [47] and [1], respectively.

Figure 2

Phonon scattering rates limited by (a) electron-phonon coupling for tungsten and aluminum and (b) anharmonic phonon-phonon interaction for tungsten, aluminum, and iridium at room temperature.

Figure 3

The phonon thermal conductivity calculated with an exact solution (solid) and RTA (dashed) of BTE by considering only phonon-phonon scatterings and both phonon-phonon and electron-phonon scatterings.