Crystal Thermal Transport in Altermagnetic ${\mathrm{RuO}}_2$

We demonstrate the emergence of a pronounced thermal transport in the recently discovered class of magnetic materials—altermagnets. From symmetry arguments and first-principles calculations performed for the showcase altermagnet, ${\mathrm{RuO}}_2$, we uncover that crystal Nernst and crystal thermal Hall effects in this material are very large and strongly anisotropic with respect to the Néel vector. We find the large crystal thermal transport to originate from three sources of Berry’s curvature in momentum space: the Weyl fermions due to crossings between well-separated bands, the strong spin-flip pseudonodal surfaces, and the weak spin-flip ladder transitions, defined by transitions among very weakly spin-split states of similar dispersion crossing the Fermi surface. Moreover, we reveal that the anomalous thermal and electrical transport coefficients in ${\mathrm{RuO}}_2$ are linked by an extended Wiedemann-Franz law in a temperature range much wider than expected for conventional magnets. Our results suggest that altermagnets may assume a leading role in realizing concepts in spin caloritronics not achievable with ferromagnets or antiferromagnets.

Figure 1

(a),(b) Schematics of anomalous Nernst and anomalous thermal Hall effects in [110]- and [001]-oriented ${\mathrm{RuO}}_2$ films. $\mathit{N}$ denotes the Néel vector rotating from the [001] to [110] axis on the $(\overline{1}10)$ plane. $J$ and $J^Q$ are detectable electrical and thermal currents, respectively, induced by a temperature gradient $\nabla T$. (c),(d) Magnetic unit cell of ${\mathrm{RuO}}_2$ without and with O atoms. Red and blue balls represent two Ru atoms with antiparallel spin magnetic moments, and gray balls represent nonmagnetic O atoms. The anomalous thermal transport is prohibited by the $\mathcal{T}\mathcal{S}$ symmetry of the magnetic lattice alone; however, this symmetry is broken when taking into account the cage of O atoms, which gives rise to crystal thermal transport. Inset: parametrization of the Néel vector $\mathit{N}(\phi ,\theta )$ in spherical coordinates. (e) Reciprocal space Fermi surface featured by strong (cigar shaped) and weak (circular) spin-splitting states. (f) Real-space alternating spin density related by fourfold crystal rotation.

Figure 2

(a)–(c) Components of the anomalous Hall conductivity $\sigma$, anomalous Nernst conductivity $\alpha$, and anomalous thermal Hall conductivity $\kappa$ at $T=300\mathrm{K}$ as a function of polar angle $\theta$ when the Néel vector rotates within the $(\overline{1}10)$ plane. The lines are guides to the eye. The conductivity at $\theta +\pi$ is not plotted as it is opposite in sign to that at $\theta$ due to time-reversal symmetry. (d) Relativistic band structures of ${\mathrm{RuO}}_2$ for $\mathit{N}(\phi =45°,\theta =0°,45°,90°)$. (e)–(g) $\sigma$, $\alpha$, and $\kappa$ as a function of Fermi energy at $\mathit{N}(\phi =45°,\theta =90°)$ for different temperatures. The color scheme from light to dark corresponds to “warming” from 1 to 400 K.

Figure 3

Three distinct types of interband transitions in altermagnetic ${\mathrm{RuO}}_2$: (a) pseudonodal lines forming among the bands of the same spin, (b) altermagnetic pseudonodal surfaces forming among the bands of opposite spin driven by crystal symmetry, and (c) ladder transitions among weakly split bands of opposite spin with similar dispersion. (d) Momentum and energy distribution of topological nodal lines and Weyl points. (e) Relativistic Fermi surface (black lines) and Berry curvature ${\mathrm{\Omega }}_{zx}$ (color maps, in atomic units) on the (001) plane at the true Fermi energy. The contributions from gapped nodal lines, pseudonodal surfaces, and ladder transitions are indicated by black arrows and pink and green dashed rectangles, respectively. (f) Similar to (e) but for the anomalous Nernst conductivity ${\alpha }_{zx}$. (g),(h) Total ${\sigma }_{zx}$ and ${\alpha }_{zx}$ and their decompositions to spin-conserved ($\uparrow \uparrow +\downarrow \downarrow$) and spin-flip ($\uparrow \downarrow$) parts. In (d)–(h), the Néel vector points to $\mathit{N}(\phi =45°,\theta =90°)$.

Figure 4

Temperature dependence of (a) anomalous Hall conductivity $\sigma$, (b) anomalous thermal Hall conductivity $\kappa$, and (c) anomalous Lorenz ratio $L$ for $\mathit{N}(\phi =45°,\theta =90°)$ with different Fermi energies $ϵ$. The horizontal dashed line in (c) denotes the Sommerfeld constant $L_0$. The vertical dashed line denotes the allowed maximal temperature range of the Wiedemann-Franz law.