Magnetization tunable Weyl states in $\mathrm{Eu}{\mathrm{B}}_6$

The interplay between magnetism and topological band structure offers extraordinary opportunities to realize rich exotic magnetic topological phases such as axion insulators, Weyl semimetals, and quantum anomalous Hall insulators, and therefore has attracted fast growing research interest. The rare-earth hexaboride $\mathrm{Eu}{\mathrm{B}}_6$ represents an interesting magnetic topological phase with tunable magnetizations along different crystallographic directions, while the correlation with the topological properties remains scarcely explored. In this work, combining magnetotransport measurements and first principles calculations, we demonstrate that $\mathrm{Eu}{\mathrm{B}}_6$ exhibits versatile magnetic topological phases along different crystallographic directions, which tightly correlate with the varied magnetizations. Moreover, by virtue of the weak magnetocrystalline anisotropy and the relatively strong coupling between the local magnetization and the conduction electrons, we show that the magnetic ground state of the system can be directly probed by the anisotropy in the magnetotransport properties. Our work thus introduces an excellent platform to study the rich topological phases that are tunable by magnetic orders.

Figure 1

(a) Temperature dependent longitudinal resistivity ${\rho }_{xx}$. Inset: Schematic crystal structure of $\mathrm{Eu}{\mathrm{B}}_6$. (b) Isothermal magnetizations measured at 2 K between 0 and 7 T for $B$ along the [001], [110], and [111] directions, respectively. Inset: Temperature dependent magnetic susceptibility measured at $B=0.01\mathrm{T}$ and an image of a typical $\mathrm{Eu}{\mathrm{B}}_6$ single crystal. (c) Schematic spin structures for three different spin-polarized states, which could result in the illustrated Weyl state.

Figure 2

(a)–(c) Isothermal magnetizations at various temperatures for $M//\left[001\right]$, $M//\left[110\right]$, and $M//\left[111\right]$, respectively.

Figure 3

(a) Spin-resolved band structure for the FM state of $\mathrm{Eu}{\mathrm{B}}_6$ from $\mathrm{GGA}+U\left(U=8\mathrm{eV}\right)$ calculations. Band structures calculated using $\mathrm{GGA}+U+\mathrm{SOC}\left(U=8\mathrm{eV}\right)$ for FM states with the $M$ orientations along the (b) [001], (c) [110], and (d) [111] directions. Insets in [(b)–(d)] illustrate the projections of the nodal line (indicated by the cyan lines) configurations and Weyl point (labeled by red/blue dots with different chiralities) distribution on the (001) surface Brillouin zone.

Figure 4

The band structures around the $X$ and $Z$ points calculated by using $\mathrm{GGA}+U+\mathrm{SOC}\left(U=8\mathrm{eV}\right)$ for FM states with spin orientation along the (a) [001], (b) [110], and (c) [111] directions.

Figure 5

(a)–(c) the longitudinal resistivity at different temperatures with $M//\left[001\right]$, $M//\left[110\right]$, and $M//\left[111\right]$, respectively.

Figure 6

(a) Longitudinal resistivity ${\rho }_{xx}$ versus magnetic field $B$ at different angles and at the temperature of 2 K. (b) Schematic geometry for the magnetic field rotation measurements. The blue plane is (110), the orange plane is (–110), and $\theta$ highlighted by red is the angle between $\mathit{B}$ and [110]. (c) Difference between ${\rho }_H$ and ${\rho }_L$ versus angles signifying the scattering between carriers and magnetism during the spin-reorientation transition.

Figure 7

(a)–(c) Anomalous Hall resistivities ${\rho }_{xy}^A$ versus magnetic field at various temperatures. (d)–(f) ${\rho }_{xy}^A/\left(M{\rho }_{xx}\right)$ versus ${\rho }_{xx}$, where the red dashed line denotes the linear fit. [(g)–(i)] Anomalous Hall conductivities ${\sigma }_{xy}^A$ versus magnetic field at 2 K for $M//\left[001\right]$, $M//\left[110\right]$, and $M//\left[111\right]$, respectively.

Figure 8

(a)–(c) Transverse resistivity at different temperatures with $M//\left[001\right]$, $M//\left[110\right]$, and $M//\left[111\right]$, respectively. (d)–(f) The fitting results by using the two-band model at 2 K. The large dashed frame zone is the fit part and the small dashed frame shows a deviation.

Figure 9

The anomalous Hall resistivities with $M//\left[001\right]$, $M//\left[110\right]$, and $M//\left[111\right]$ in the temperature range of $2\text{–}20\mathrm{K}$.

Figure 10

The intrinsic anomalous Hall conductivities as a function of magnetization with $M//\left[001\right]$, $M//\left[110\right]$, and $M//\left[111\right]$ at 2 K.