Low-field magnetic anomalies in single crystals of the A-type square-lattice antiferromagnet ${\mathrm{EuGa}}_4$

The body-centered-tetragonal antiferromagnet ${\mathrm{EuGa}}_4$ was recently identified as a Weyl nodal-line semimetal that exhibits the topological Hall effect below its reported antiferromagnetic (AFM) ordering temperature $T_{\mathrm{N}}=15–16.5$ K which we find to be $T_{\mathrm{N}}=16.4\left(2\right)$ K. The ${\mathrm{Eu}}^{+2}$ ions are located at the corners and body centers of the unit cells. ${\mathrm{EuGa}}_4$ exhibits A-type antiferromagnetic order below $T_{\mathrm{N}}$, where the ${\mathrm{Eu}}^{2+}$ spin-7/2 moments are ferromagnetically aligned in the $ab$ plane with the Eu moments in adjacent Eu planes along the $c$ axis aligned antiferromagnetically. Low-field magnetization versus field $M\left(H_{ab}\right)$ data at $T=2$ K with the field aligned in the $ab$ plane are reported that exhibit anomalous positive curvature up to a critical field $H_{\mathrm{c}1}$ at which a second-order transition occurs with $H_{c1}\approx 0.85$ kOe for $\mathbf{H}\parallel [1,1,0]$ and $\approx 4.8$ kOe for $\mathbf{H}\parallel [1,0,0]$. For larger fields, the linear behavior $M_{ab}=\chi \left(T_{\mathrm{N}}\right)H_{ab}$ is followed until the critical field $H_{\mathrm{c}}$ is reached at which all moments become aligned with the applied field. A theory is formulated for $T=0$ K that fits the observed $M\left(H_{ab}\right)$ behavior at $T=2$ K well, where domains of $\mathrm{A}$-type AFM order with fourfold rotational symmetry occur in the AFM state in zero field. The moments in the four domains reorient to become almost perpendicular to ${\mathbf{H}}_{ab}$ at $H_{c1}$, followed by increasing canting of all moments toward the field with increasing field up to $H_{\mathrm{c}}$ which is reported to be 71 kOe. A first-order transition in $M\left(H_{ab}\right)$ at $H_{ab}=H_{\mathrm{c}1}$ is predicted by the theory for $T=0$ K when ${\mathbf{H}}_{ab}$ is at a small angle from the [1,0,0] direction.

Figure 1

Body-centered-tetragonal crystal structure of ${\mathrm{EuGa}}_4$. The Eu atoms form square lattices in the $ab$ plane.

Figure 2

Left ordinate: Temperature dependence of the magnetic susceptibility $\chi \left(T\right)$ of ${\mathrm{EuGa}}_4$ for $H=0.1$ kOe with $H\parallel ab\parallel [1,0,0]$ (filled blue triangles), $H\parallel ab\parallel [1,1,0]$ (open black circles), and $H\parallel c$ (filled red circles). The corresponding susceptibility ratio $\chi \left(T\right)/\chi \left(T_{\mathrm{N}}\right)$ is shown on the right ordinate.

Figure 3

Inverse magnetic susceptibility ${\chi }^{-1}\left(T\right)$ of ${\mathrm{EuGa}}_4$ with $H=0.1$ T for (a) $H\parallel ab$ and (b) $H\parallel c$.

Figure 4

(a) Out-of-plane magnetic susceptibility ${\chi }_c\left(T\right)$ ($H\parallel c$) of ${\mathrm{EuGa}}_4$ for different applied magnetic fields. In-plane magnetic susceptibility for different magnetic fields for (b) $H\parallel ab\parallel [1,1,0]$ and (c) for $H\parallel ab\parallel [1,0,0]$. The field responses in these two $ab$-plane symmetry directions are seen to be significantly different.

Figure 5

Low-field $M\left(H\right)$ data measured at $T=2$ K for $H\parallel [1,0,0], H\parallel [1,1,0]$, and $H\parallel [0,0,1]$ (left ordinate). The corresponding field derivatives are also plotted (open symbols, right ordinate). Although $M_c\left(H\right)$ for $H\parallel [0,0,1]$ is linear, distinct nonlinearities in $M_{ab}\left(H\right)$ for $H\parallel [1,0,0]$ and [1,1,0] are observed in this low-field region.

Figure 6

In-plane $M\left(H\right)$ data measured at different temperatures for (a) $H\parallel [1,1,0]$ and (b) $H\parallel [1,0,0]$. The respective $dM\left(H\right)/dH$ vs $H$ data are shown in (c) and (d). The data for $T=18$ K are about 2 K above $T_{\mathrm{N}}$.

Figure 7

In-plane field-cooled $M\left(H\right)$ behavior at $T=2$ K for $H\parallel [1,0,0]$ in the hysteresis $H$ cycle 10 kOe $\to$ 0 $\to$ 10 kOe. No hysteretic behavior is observed. The corresponding $dM\left(H\right)/dH$ behavior is also shown in the right ordinate.

Figure 8

Fourfold $ab$-plane rotational anisotropy energy normalized by the anisotropy constant $K_4, E_{\mathrm{anis}}/K_4$, vs the $ab$-plane angle $\varphi /\pi$ rad of a moment in tetragonal ${\mathrm{EuGa}}_4$.

Figure 9

(a) Schematic diagram of the nearly-locked moment orientations in the $ab$ plane of adjacent antiparallel layers of moments along the $c$ axis in the four collinear A-type AFM domains A, B, C, and D and their magnetic field evolution with increasing $x$-axis field $H$ at low fields $H\le H_{\mathrm{c}1}$ shown by arrows. (b) For $H_x>H_{\mathrm{c}1}$, each moment increasingly cants towards the increasing field as shown, until at the critical field $H_{\mathrm{c}}$ all moments are aligned with the field with ${\mu }_x={\mu }_{\mathrm{sat}}=gS{\mu }_{\mathrm{B}}=7{\mu }_{\mathrm{B}}$.

Figure 10

The angle $\mathrm{\Delta }\varphi$ in Fig. 9 vs $h_x={\chi }_{\perp }{H}_x^2/K_4$ for ${\varphi }_H=0$ rad as defined at the bottom of Fig. 9.

Figure 11

The experimental $ab$-plane $M\left(H\right)$ behavior for two applied field directions (a) $H\parallel [1,0,0]$ and (b) $H\parallel [1,1,0]$ measured at $T=2$ K along with theoretical predictions for $T=0$ K with $H_{c1}\approx 4.8$ kOe and 0.85 kOe, respectively, as shown by the respective solid red curves. For $H>H_{\mathrm{c}1}, M\left(H\right)=\chi \left(T_{\mathrm{N}}\right)H$ is plotted according to molecular-field theory [30] which agrees very well with the slope of the data and is not a fit to the slope.

Figure 12

The angles $\mathrm{\Delta }{\varphi }_1$ and $\mathrm{\Delta }{\varphi }_2$ in Eqs. (15) vs ${\chi }_{\perp }{H}_x^2/K_4$ for ${\varphi }_H=1/100$ rad. $\mathrm{\Delta }{\varphi }_1=\pi /4$ for ${\chi }_{\perp }{H}_x^2/K_4>7.1$, whereas $\mathrm{\Delta }{\varphi }_2$ eventually asymptotes to $\pi /4$ rad at much larger fields.