Electron and phonon interactions and transport in the ultrahigh-temperature ceramic ZrC

We have simulated the ultrahigh-temperature ceramic zirconium carbide (ZrC) in order to predict electron and phonon scattering properties, including lifetimes and transport. Our predictions of heat and charge conductivity, which extend to 3000 K, are relevant to extreme-temperature applications of ZrC. Mechanisms are identified on a first-principles basis that considerably enhance or suppress heat transport at high temperature, including strain, anharmonic phonon renormalization, and four-phonon scattering. The extent to which boundary confinement and isotope scattering effects lower thermal conductivity is predicted.

Figure 1

Interaction broadened harmonic phonon dispersion at (a) 10 K and (b) 300 K. FWHM ${\mathrm{\Gamma }}_{\lambda }^{{}^{\prime \prime },\text{ph-ph}}(\omega ,\mathbf{q},T)$ line broadening is Lorentzian and specified by the contour brightness scale on the right axis, which ranges from 0–10 GHz for $10\text{K}$ to 0–100 GHz for $300\text{K}$. (c) Linewidth vs temperature at selected points (labeled) in the Brillouin zone.

Figure 2

(a) Electron relaxation time and associated imaginary self-energy ${\mathrm{\Sigma }}_{\lambda }^{{}^{\prime \prime },\text{el-ph}}(\varepsilon ,T)$ (inset). (b) Imaginary phonon self-energy ${\mathrm{\Pi }}^{{}^{\prime \prime },\text{el-ph}}(\omega ,T)$ for each of the three acoustic and optic bands.

Figure 3

(a) Electrical conductivity ${\sigma }_{\text{el}}$ predictions (solid line). (b) Thermal conductivity predictions for ${\kappa }_{\text{total}}={\kappa }_{\text{ph}}+{\kappa }_{\text{el}}$ and ${\kappa }_{\text{el}}$. Dashed and dotted lines denote the experimental measurements [4, 6, 10, 48] and computed results of Crocombette [11].

Figure 4

Phonon thermal conductivity enhancement and suppression mechanisms in ZrC. (a) Thermal expansion lowers conductivity by ${\kappa }_{\text{qha}}^{\text{ph}}\left[V\left(T\right)\right]/{\kappa }_{}^{\text{ph}}\left(V_0\right)$. (b) Zero-point motion lowers conductivity by ${\kappa }_{}^{\text{ph}}\left(V_{\text{ZP}}\right)/{\kappa }_{}^{\text{ph}}\left(V_0\right)$. (c) Isotope purity enhances conductivity by a factor of ${\kappa }_{\text{pure,qha}}^{\text{ph}}\left[V\left(T\right)\right]/{\kappa }_{\text{iso,qha}}^{\text{ph}}\left[V\left(T\right)\right]$ compared to mass scattering at the natural isotopic abundance. (d) Anharmonic phonon renormalization enhances conductivity by ${\kappa }_{\text{prn}}^{\text{ph}}\left(V\right)/{\kappa }_{\text{qha}}^{\text{ph}}\left(V\right)$ relative to a crystal with quasiharmonic frequency dependence. (e) Additionally including four-phonon scattering suppresses conductivity by ${\kappa }_{\text{4ph}}^{\text{ph}}\left(V\right)/{\kappa }_{\text{prn}}^{\text{ph}}\left(V\right)$. (f) Isobaric heat effect enhances conductivity ${\kappa }^{\text{ph}}\left(p_0\right)/{\kappa }_{}^{\text{ph}}\left(V\right)$.

Figure 5

Comparison of the ZrC phonon thermal conductivity computed with the PBE and LDA exchange-correlation functionals at $a=4.667\AA$.

Figure 6

Comparison of conductivity from compressive sensing [18] and third-order lattice dynamics (phono3py [12]) at zero-temperature equilibrium and high-temperature volumes.

Figure 7

Comparison of computational approaches to determine electrical conductivity. (1) LDA + Wannier functions, (2) LDA + DFPT linear response, (3) LDA + Bloch functions, and (4) PBE + snapshots from classical potential MD [11].

Figure 8

Grain-size effects estimated by geometric restrictions on phonon conductivity at a series of temperatures in perfect ZrC. Phonon cutoff lengths span the interval $[{10}^{-2},{10}^3]\mu \text{m}$.

Figure 9

Accumulated phonon thermal conductivity as a function of mean free path, shown by solid lines (left axis). The percentage of accumulated phonon thermal conductivity is shown by dashed lines (right axis).

Figure 10

Phonon dispersion at (a) 300 K, (b) 1500 K, and (c) 3800 K at the quasiharmonic and quasiharmonic + renormalization levels of theory. Three- and four-phonon scattering at (d) 300 K, (e) 1500 K, and (f) 3800 K.