Giant anomalous thermal Hall effect in tilted type-I magnetic Weyl semimetal ${\text{Co}}_3{\text{Sn}}_2{\text{S}}_2$

The recent discovery of magnetic Weyl semimetal ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ opens up new avenues for research into the interactions between topological orders, magnetism, and electronic correlations. Motivated by the observations of a large anomalous Hall effect because of large Berry curvature, we investigate another Berry curvature-induced phenomenon, namely the anomalous thermal Hall effect in ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. We study it with and without strain, using a Wannier tight-binding Hamiltonian derived from first-principles density functional theory calculations. We first identify this material as a tilted type-I Weyl semimetal based on the band-structure calculation. Within the quasiclassical framework of Boltzmann transport theory, a giant anomalous thermal Hall signal appears due to the presence of large Berry curvature. Surprisingly, the thermal Hall current changes and even undergoes a sign-reversal upon varying the chemical potential. Furthermore, applying about 13 GPa stress, an enhancement as large as 33% in the conductivity is observed; however, the tilt vanishes along the path connecting the Weyl nodes. In addition, we have confirmed the validity of the Wiedemann-Franz law in this system for anomalous transports. We propose specific observable signatures that can be directly tested in experiments.

Figure 1

Crystal structure of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. The left part of the figure shows the rhombohedral primitive unit cell of space group R-3m. The cobalt and tin atoms form a ferromagnetic kagome lattice, which is shown at the right.

Figure 2

Electronic structure of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. (a) The spin-polarized (red for spin-up and blue for spin-down channel) and SOC-included (dotted green curves) band structure plotted along the high-symmetry direction $W\text{−}T\text{−}U\text{−}L\text{−}\mathrm{\Gamma }$. The spin-down channel has a gap, whereas the spin-up channel is gapless and has two band-crossings along $U\text{−}L\text{−}\mathrm{\Gamma }$, which are parts of a nodal ring. (b) The orbital-projected density of states. The large area under the red curve implies that the Co-$d$ orbital dominates near the Fermi level with negligible contributions from other orbitals. The $+\mathrm{ve}$ and $-\mathrm{ve}$ values correspond to spin-up and spin-down channels, respectively. (c) The Brillouin zone of the system with a truncated octahedral shape. The green line shows the high-symmetry direction for the band-structure plot. (d),(e) Fermi surface for the valence and conduction bands, respectively. Both bands cross Fermi level.

Figure 3

Tilted Weyl cones. Band structure showing Weyl nodes of opposite chiralities is plotted along their connecting line. Their locations are $(0,0.425,0.063)$ and $(0,-0.425,-0.063)$, respectively, in fractional coordinates. One of them has been zoomed-in to show the tilting of the cone with respect to the vertical line via the red line.

Figure 4

Berry curvature and anomalous thermal Hall conductivity. (a) The strengths of the Berry curvature ($z$-component) corresponding to the bands forming Weyl nodes are shown by color variation of the bands along the high-symmetry path $W\text{−}T\text{−}U\text{−}L\text{−}\mathrm{\Gamma }$. (b) The total Berry curvature is shown on the mirror-symmetric plane ($k_y=0$). The sharp peaks correspond to the Weyl nodes present on this plane. (c) Anomalous thermal Hall conductivity (ATHC) plotted with respect to chemical potential $\mu$ at $T=50$ K. The heat current can be controlled as well as reversed by changing the chemical potential of the system. However, the maximum value is achieved at $\mu =E_f$. (d) The relationship between ATHC and temperature. Note that ${\kappa }_{xy}$ varies linearly with $T$ at low temperatures up to 80 K, after which it gradually deviates from linearity with increasing $T$. The chemical potential was set equal to the Fermi level.

Figure 5

Strain-induced band dispersion. (a)–(c) SOC-included band structures for compressive stresses of 0, 5.6, and 12.9 GPa along the $z$-axis. The valence band starts to flatten as the stress is increased. The local band gap at the L point shows an expanding nature throughout the process. (d)–(f) The band dispersion around two of the Weyl nodes along their connecting line in the Brillouin zone. The nodes move towards the Fermi level with increasing stress, and finally at 12.9 GPa they fall directly on the Fermi level. The tilting of the cones decreases with stress and almost vanishes at 12.9 GPa. The red line in the inset indicates tilting with respect to the vertical.

Figure 6

The anomalous thermal Hall conductivity plotted against chemical potential for different values of compressive stress along the $z$-axis. At $\mu ={\mathrm{E}}_f$, ATHC increases by about 33% for a stress of 12.9 GPa.