Creating and controlling Dirac fermions, Weyl fermions, and nodal lines in the magnetic antiperovskite ${\mathrm{Eu}}_3\mathrm{PbO}$

The band topology of magnetic semimetals is of interest both from the fundamental science point of view and with respect to potential spintronics and memory applications. Unfortunately, only a handful of suitable topological semimetals with magnetic order have been discovered so far. One such family that hosts these characteristics is the antiperovskites, $A_{3}B\mathrm{O}$, a family of 3D Dirac semimetals. The $A={\mathrm{Eu}}^{2+}$ compounds magnetically order with multiple phases as a function of the applied magnetic field. Here, by combining band structure calculations with neutron diffraction and magnetic measurements, we establish the antiperovskite ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ as a new topological magnetic semimetal. This topological material exhibits a multitude of different topological phases with ordered Eu moments which can be easily controlled by an external magnetic field. The topological phase diagram of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ includes an antiferromagnetic Dirac phase, as well as ferro- and ferrimagnetic phases with both Weyl points and nodal lines. For each of these phases, we determine the bulk band dispersions, the surface states, and the topological invariants by means of ab initio and tight-binding calculations. Our discovery of these topological phases introduces ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ as a new platform to study and manipulate the interplay of band topology, magnetism, and transport.

Figure 1

Magnetic phase diagram of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$. (a) Magnetization of a single crystal of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ for several applied magnetic field directions at both 5 K and 20 K. The arrows show the location of the phase transitions with the roman numerals denoting the phases. The highest field transition to phase V is outside the field range at 5 K, but within it at 20 K. (b) The magnetic phase diagram of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ based on powder magnetization data, both as a function of temperature (purple) and field (blue), and specific heat measurements (orange) on powder samples. Phase I is antiferromagnetic, phases II and III ferrimagnetic, phase IV likely ferrimagnetic, and phase V fully polarized. The star-shaped markers indicate the position of our neutron scattering experimental datasets. The inset depicts the crystal structure of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$.

Figure 2

(a) Schematic representation of the AFM-I magnetic structure with the magnetic space group $P_{I}a\overline{3}$. (b) Schematic representation of the ferrimagnetic Ferri-II structure with the $P4/m{m}^{\prime }{m}^{\prime }$ magnetic space group as a superposition of two components, antiferromagnetic (top) and ferromagnetic (bottom). (c) Schematic representation of the ferrimagnetic Ferri-III structure with the $P{m}^{\prime }{m}^{\prime }m$ magnetic space group as a superposition of two components, antiferromagnetic (top) and ferromagnetic (bottom). In all panels, three Eu sites associated with the 3c Wyckoff position of the paramagnetic cubic $Pm\overline{3}m$ structure are shown as red, blue, and green spheres, respectively, and the paramagnetic unit cell is shown for clarity alongside the magnetic cells.

Figure 3

Electronic band structures, magnetic orders, and Brillouin zones of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$. The Fermi energies are indicated by the dashed lines at 0 eV. The positions of the Dirac points, nodal lines, and Weyl points are labeled by D, L, and W, respectively, see Table 1. (a) Paramagnetic phase with Brillouin zone shown in (g). (b) Antiferromagnetic phase. The paramagnetic zone is folded along $\mathrm{\Gamma }\text{−}X$, $\mathrm{\Gamma }\text{−}Y,$ and $\mathrm{\Gamma }\text{−}Z$, giving an eightfold increase in unit cell size. The cubic symmetry is retained and the bands are Kramers degenerate. (c) Ferri-II phase. The blue and black arrows indicate the orientations of the magnetic moments on the Eu atoms for clarity. The parent zone is backfolded halfway along $\mathrm{\Gamma }\text{−}X$ and $\mathrm{\Gamma }\text{−}Y$. The symmetry is lowered to tetragonal. The bands are now singly degenerate except at the Weyl points and nodal lines. (d) Ferri-III phase with Brillouin zone shown in (f). The bands are back folded along $\mathrm{\Gamma }\text{−}Z$ and the symmetry is lowered to orthorhombic. The degeneracy loss is the same as for the Ferri-II phase as described above. (e) Fully polarized with field along [110]. This phase has no zone folding but loss of Kramers degeneracy and splitting of Dirac cones into Weyl points. With the field direction along [110] the symmetry is lowered to orthorhombic. Note that while in (e) the field is along [110], in all other panels the field is along [100].

Figure 4

Surface states of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ and their spin polarization. Calculated surface density of states (SDOS) and spin polarization for a [001] slab of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ with oxygen and lead termination. The color code represents the SDOS on a linear scale for the five outermost layers, while the in-plane spin polarization is indicated by the red arrows. (a) and (b) show the SDOS for the paramagnetic and antiferromagnetic phases at the energy of the upper Dirac points $E=0.017$ and 0.05 eV, respectively (cf. Table 1). The position of the Dirac points is marked by the label D1. (c) and (d) display the SDOS for the ferromagnetic phase with magnetization in [110] and [100] directions at the energy of the Weyl points W1 and W2, respectively. The chiralities and positions of the Weyl points W1 and W2 are indicated in all panels by the green and blue dots and numbers.

Figure 5

Intrinsic anomalous Hall conductivity. Calculation of the intrinsic anomalous Hall conductivity ${\sigma }_{yz}$ in the ferromagnetic phase as a function of chemical potential $\mu$, for different magnetization directions at a temperature of 10 K. Shaded regions highlight broad features in ${\sigma }_{yz}$ that are attributed to different Weyl points.

Figure 6

Magnetic susceptibility (top), specific heat (middle), and resistivity (bottom) of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$. For more details see text.

Figure 7

The paramagnetic (50 K) and antiferromagnetic (1.5 K) phase data from banks 3 and 8 of WISH. The substantial background is from the quartz ampoule used to contain the sample.

Figure 8

The zero-field magnetic structure of ${\mathrm{Eu}}_3\mathrm{Pb}\mathrm{O}$ (top) and spin-only representation (bottom).

Figure 9

A portion of the Rietveld refinement for the $P_{I}a\overline{3}$ structure (90${}^{\circ }$ banks 3 and 8 WISH, magnetic only subtracted pattern) with the data in blue, calculated pattern in black, difference in gray, magnetic reflections indicated by black tickmarks.

Figure 10

Peak intensity changes as a function of applied field at $T=1.5\mathrm{K}$. Reflection labels are in the notation of the parent unit cell.

Figure 11

A portion of the Rietveld refinement for the 1.5K 5T structure in $P4/m{m}^{\prime }{m}^{\prime }$ (90${}^{\circ }$ bank 3 WISH). The unindexed peak at 2.14 Å is from vanadium, the substantial background is from the quartz ampoule used to contain the sample with the data in blue, calculated pattern in black, difference in gray, nuclear reflections indicated by red tickmarks and magnetic in black.

Figure 12

A portion of the Rietveld refinement for the 1.5K 8T structure in $P{m}^{\prime }{m}^{\prime }m$ (90${}^{\circ }$ bank 3 WISH). The unindexed peak at 2.14 Å is from vanadium, the substantial background is from the quartz ampoule used to contain the sample with the data in blue, calculated pattern in black, difference in gray, nuclear reflections indicated by red tickmarks and magnetic in black.

Figure 13

Brillouin zone with nodal points (W1–W4) and the nodal line (L1) of the ferromagnetic phase for magnetic moments pointing along the [110] direction. The light blue plane corresponds to the mirror plane normal to direction of magnetic moments, in which the nodal line L1 is situated.