Magnetic structures, spin-flop transition, and coupling of Eu and Mn magnetism in the Dirac semimetal ${\mathrm{EuMnBi}}_2$

In recently emerging correlated topological materials, such as magnetic Dirac/Weyl semimetals, additional tunabilities of their transport and magnetic properties may be achieved by utilizing possible interaction between the exotic relativistic fermions and magnetic degree of freedom. The two-dimensional antiferromagnetic (AFM) Dirac semimetal ${\mathrm{EuMnBi}}_2$, in which an intricate interplay between multiple magnetic sublattices and Dirac fermions was suggested, provides an ideal platform to test this scenario. We report here a comprehensive study of the AFM structures of the Eu and Mn magnetic sublattices as well as the interplay between Eu and Mn magnetism in this compound by using both polarized and nonpolarized single-crystal neutron diffraction. Magnetic susceptibility, specific heat capacity measurements, and the temperature dependence of magnetic diffractions suggest that the AFM ordering temperatures of the Eu and Mn moments are at 22 and 337 K, respectively. The magnetic moments of both Eu and Mn ions are oriented along the crystallographic $c$ axis, and the respective magnetic propagation vectors are ${\mathbf{k}}_{\mathrm{Eu}}=(0,0,1)$ and ${\mathbf{k}}_{\mathrm{Mn}}=(0,0,0)$. With proper neutron absorption correction, the ordered moments are refined at 3 K as 7.7(1) and 4.1(1) ${\mu }_B$ for the Eu and Mn ions, respectively. In addition, a spin-flop (SF) phase transition of the Eu moments in an applied magnetic field along the $c$ axis was confirmed to take place at a critical field of $H_c\approx$ 5.3 T. The AFM exchange interaction and magnetic anisotropy parameters ($J=0.81$ meV, $K_u=0.18$ meV, $K_e=-0.11$ meV) are determined based on a subsequent quantitative analysis of the SF transition. The evolution of the Eu magnetic moment direction as a function of the applied magnetic field in the SF phase was also determined. Clear kinks in both field and temperature dependences of the magnetic reflections ($\pm 1$, 0, 1) of Mn were observed at the onset of the SF phase transition and the AFM order of the Eu moments, respectively. This unambiguously indicates the existence of a strong coupling between Eu and Mn magnetism. The interplay between two magnetic sublattices could bring new possibilities to tune Dirac fermions via changing magnetic structures by applied fields in this class of magnetic topological semimetals.

Figure 1

X-ray diffraction (XRD) of single-crystal ${\mathrm{EuMnBi}}_2$ at 300 K. XRD pattern shows sharp (0,0,L) peaks. The left inset is a photograph of single-crystal samples of ${\mathrm{EuMnBi}}_2$, showing a typical size of around $5$ mm with a thickness of 1.5 mm (grid width is 1 mm) and clear rectangular natural edge; the right inset is an x-ray Laue pattern of the (H,K,0) reciprocal plane, and a fourfold symmetry can be clearly seen.

Figure 2

(a) ZFC/FC magnetization curves measured under applied magnetic fields along the $a$ and $c$ axes. Inset shows the enlarged plot of the high-temperature range. (b) The inverse susceptibility of the ZFC curves with Curie-Weiss fitting around the selected temperature range. Dashed lines are the extension of the linear fitting.

Figure 3

The specific heat capacity ($C_p$ vs $T$ curve) of ${\mathrm{EuMnBi}}_2$ single crystal. The inset shows the field dependence of the anomaly peak, which indicate a AFM phase transition of the Eu moments.

Figure 4

(a) Temperature dependence of the ($1,0,-2$) magnetic Bragg peak intensity of Eu. (b) Temperature dependence of the ($-1,0,1$) magnetic Bragg peak intensity of Mn. The empty circles and stars with error bar are experimental data. The solid lines are the fittings of the experimental data by the formula $I=I_0+A{(1-T/T_N)}^{2\beta }$. The insets are the corresponding log-scale plots of the peak intensity vs the reduced temperature $T_N-T$ where $T_N$ is the critical temperature for Eu and Mn respectively. The slop represents the power parameter $2\beta$.

Figure 5

Polarized neutron diffraction patterns of single crystal ${\mathrm{EuMnBi}}_2$. [(a), (b)] Nuclear reflections in the (H,0,L) plane at 30 and 4 K. (c) Magnetic reflections of Mn magnetic sublattice at 30 K ($T_N^{\mathrm{Eu}}<30\mathrm{K}<T_N^{\mathrm{Mn}}$). (d) Magnetic reflections of both Mn and Eu sublattice at 4 K (4 $\mathrm{K}<T_N^{\mathrm{Eu}}<T_N^{\mathrm{Mn}}$).

Figure 6

[(a), (b)] Integrated intensities of the Bragg reflections collected at room temperature 300 K and low-temperature 3 K are plotted against the calculated values, respectively. Panels (c) and (d) show the corresponding magnetic structure models generated by vesta [49].

Figure 7

(a) Field dependence of the integrated intensities of the selected nuclear and magnetic (contributed by Eu) reflections taken at 1.5 K. Blue lines are the fitting results. The insets are the schematic plots of the Eu magnetic moment directions in AFM (left), spin-flop (middle), and spin-flip (right) phases, respectively. Blue arrows inside circles represent the spin configuration in different phases. $H_c$ denotes the spin-flop critical field, and $H_s$ denotes the spin-flip (saturation) critical field. (b) Calculated evolution of the corresponding tilting angle $\theta$ of Eu magnetic moments in the spin-flop process. Light green zone corresponds to the measured field range in panel (a). (c) Calculated weighted factor as a function of the in-plane azimuth angle $\varphi$ of Eu magnetic moments at various applied fields. Line profiles are shifted along the vertical axis with a step of 20 from the one at 5.4 T. Panel (d) shows the best $\varphi$ angles with the smallest $R_w$ factor at different applied fields. Red dashed line is a reference line for $\varphi ={45}^{\circ }$. The corresponding inset in panel (d) is a schematic diagram of the spatial shape for the in-plane anisotropy.

Figure 8

(a) Comparison of the observed and calculated squared structure factors of neutron diffraction data taken at 1.5 K with 11.5-T magnetic field along $c$ axis. (b) In the corresponding magnetic structure models under 11-T field, the magnetic moment direction of Eu atoms are tilted by about ${60}^{\circ }$ away from $c$ axis with in-plane AFM component along the [1,1,0] direction.

Figure 9

Field and temperature dependences of the selected magnetic reflections. [(a), (b)] Rocking curve scans of the magnetic reflection (1,0,1) under various applied fields along the $c$ axis at 2 and 25 K respectively. (c) The extracted field dependence of integrated intensities of (1,0,1) at 1.5 and 25 K. The data at 25 K are shifted up by 0.4 for an easy comparison with that at 1.5 K. Solid blue line is the quadratic function fitting for 25 K and solid green line is a piecewise fitting of a quadratic function and a constant for 1.5 K. (d) Temperature dependence of the magnetic reflection (1,0,1) with $H$ = 0 and 9 T. Red solid lines are the fittings of step function. A clear kink is observed at around 22 K for magnetic reflection (1,0,1) under zero field. Dashed line shows the ordered temperature $T_N^{\mathrm{Eu}}$. Rocking curve scans of the nuclear reflection (e) (1,1,2) and magnetic reflection (f) (1,0,1) at 2 and 25 K under zero field.

Figure 10

Temperature dependence of polarized neutron diffraction at DNS. (a) Intensities of the magnetic reflection ($-1,0,1$) of Mn as a function of temperature near $T_N^{\mathrm{Eu}}$. Solid red line is a fitting of step function; dashed line with arrow shows the AFM phase transition temperature of Eu. (b) Rocking curve scans of the magnetic reflection ($-1,0,1$) measured at 4 and 30 K. (c) The difference of polarized neutron diffraction patterns between 4 and 30 K in the (H,0,L) scattering plane. (d) Enlarged plot of the red rectangle part in panel (c). Red arrows show the extra intensity of magnetic reflections (1,0,1) and ($-1,0,1$). (e) The corresponding line profile extracted from panel (d) along the [0,0,L] direction at $H=-1$. Solid blue line is the multipeak Gauss fitting.

Figure 11

Polarized neutron diffraction patterns of single crystal ${\mathrm{EuMnBi}}_2$ in the (H,0,L) plane at 4 K: (a) $z$ non-spin-flip channel and (b) $z$ spin-flip channel.

Figure 12

(a) Picture of single-crystal sample on holder for HEIDI. (b) A 3D model with approximate same shape and size as our measured sample in panel (a). The purple dots inside the 3D model represent the positions to calculate the neutron paths. (c) A schematic drawing for showing the path of incident and scattered neutron beam at different positions inside the sample. (d) The plot of absorption factors $f_{abs}$ for some reflections vs the number $N$ of selected scattering points inside sample. $f_{abs}$ quickly get saturated when $N$ increase. (e) Comparison between raw and corrected intensities of some selected equivalent reflections.