Topological phase transition in a magnetic Weyl semimetal

Topological Weyl semimetals (TWSs) are exotic crystals possessing emergent relativistic Weyl fermions connected by unique surface Fermi arcs (SFAs) in their electronic structures. To realize the TWS state, certain symmetries (such as the inversion or time reversal symmetry) must be broken, leading to a topological phase transition (TPT). Despite the great importance in understanding the formation of TWSs and their unusual properties, direct observation of such a TPT has been challenging. Here, using a recently discovered magnetic TWS ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$, we were able to systematically study its TPT with detailed temperature dependence of the electronic structures by angle-resolved photoemission spectroscopy. The TPT with drastic band structure evolution was clearly observed across the Curie temperature $(T_{\mathrm{C}}=177\mathrm{K})$, including the disappearance of the characteristic SFAs and the recombination of the spin-split bands that leads to the annihilation of Weyl points with opposite chirality. These results not only reveal important insights on the interplay between the magnetism and band topology in TWSs, but also provide a method to control their exotic physical properties.

Figure 1

Topological phase transition across $T_{\mathrm{C}}$ of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. (a) Illustration of the topological phase transition with band structure evolution across $T_{\mathrm{C}}$ (Curie temperature) in ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$. Two sets of Weyl points at different energies are labeled WP1 and WP2, respectively. Blue and red arrows indicate the spin-down and spin-up bands, respectively. Magenta and green color of the Weyl points represent positive (+) and negative (–) chirality, respectively. FM: ferromagnetism; PM: paramagnetism; DP: Dirac point. (b) Schematics of the bulk and surface electronic structure in the vicinity of Weyl points for the FM (i) and PM (ii) states. Bulk band dispersions across the WPs (i) and gapped bulk bands (ii) are also illustrated. Solid yellow lines connecting two WPs in (i) are surface Fermi arc (SFAs). Note the SFAs disappear in the PM state in (ii). (c) Calculated bulk band dispersions across a pair of Weyl points [along the $A{A}^{\prime }$ $A{A}^{\prime }$ direction as indicated by the blue planes in (b)] in the FM state (i) and PM state (ii), respectively. An energy gap opens in the PM state, indicating a topological insulator phase. (d) Calculated Fermi surfaces from both bulk and surface states in the FM (i) and PM (ii) states. Note that the SFAs vanish in the PM state, as illustrated in (b).

Figure 2

Comparison of the general electronic structure below and above $T_{\mathrm{C}}$. (a) Stacking plot of constant energy contours at different binding energies below (i) and above (ii) $T_{\mathrm{C}}$. The data have been symmetrized according to the crystal symmetry (see Part XII in the Supplemental Material [30]). (b) Comparison of the Fermi surface topology below (i) and above (ii) $T_{\mathrm{C}}$. The triangle-shaped SFAs near the $\overline{\mathrm{K}}/{\overline{\mathrm{K}}}^{\prime }$ points are clearly observed below $T_{\mathrm{C}}$ (i); they vanish above $T_{\mathrm{C}}$ (ii). (c) Comparison of the experimental (i) and calculated (ii) band dispersions in the FM state along different high-symmetry directions across the whole Brillouin zone (BZ), showing good agreement. Note the calculated bandwidth was renormalized by a factor of 1.43, and the energy position was shifted to match the experiment (same below). Data were collected using photons at 125 eV with linear horizontal (LH) polarization. (d) Same as (c) for the PM state; the experimental and calculated results again show good agreement. The integration window of the calculations is $0-0.2k_z^{\mathrm{BZ}}$.

Figure 3

Temperature evolution of SFAs across $T_{\mathrm{C}}$. (a) Comparison of the calculated Fermi surfaces of bulk and surface states in the FM (i) and PM (viii) states to the experimental results at different temperatures (ii–vii). The triangle-shaped SFAs [indicated by red arrows in (ii)] gradually shrink in size with increasing temperature and eventually vanish above $T_{\mathrm{C}}$ (vii), leaving only the bulk Fermi surface pocket. The data have been symmetrized according to the crystal symmetry. Here we use all three $\overline{\mathrm{M}}$ points in (i) and (viii) for simplicity as $\overline{\mathrm{M}}$/${\overline{\mathrm{M}}}^{\prime }$ points are equivalent for the surface states. (b) Comparison of the calculated dispersions of topological surface state (TSS) in the FM (i) and PM (viii) states to experimental results at different temperatures (ii–vii). The TSSs (indicated by red arrows) gradually move up and eventually disappear upon increasing temperature to above $T_{\mathrm{C}}$, agreeing well with the calculations.

Figure 4

Temperature evolution of the Weyl band dispersions. (a) Schematic of the two sets of Weyl points from different spin-polarized bands with different energies. The blue plane indicates the dispersion direction of the bands illustrated in (b,d). Definition of WP1 and WP2 is the same as in Fig. 1. (b)(i) Calculated band dispersions along the plane indicated in (a), in the FM states. Red and blue curves represent the spin-up and spin-down bands, respectively; the two Weyl points (WP1, WP2) are marked in both bands. (ii) The two spin-split bands in (i) merge in the PM states (shown as magenta curves); the two Weyl points merge and annihilate, forming an energy gap as indicated. (c) Calculated energy position [at the momentum indicated by the orange arrow in (b)(i)] for spin-up and spin-down bands (i) and their separation (ii) as a function of temperature. (d) Temperature dependence of the dispersion across the Weyl points. Red and blue dashed curves are the guidelines of two sets of Weyl dispersions. The definition of red, blue, and magenta color is the same as in (b). Note that the experimental data were obtained through dividing the Fermi-Dirac function of the raw data to highlight the unoccupied state above $E_{\mathrm{F}}$ and then symmetrization with respect to $k_x=0$ was applied. (e) Side by side comparison of the experimental results and the calculations. (f) Temperature evolution of the experimental energy distribution curves [EDCs, extracted from $k_x=0.4{\AA }^{\text{–}1}$ as indicated by the orange arrow in (d)(i)]. The red, blue, and magenta triangles mark the spin-up, spin-down, and merged bands’ energy loci. (g) Temperature dependent plot of the spin-up, spin-down, and merged bands’ energy loci (i) and their energy separation (ii), showing excellent agreement with the calculations in (c).