$A$-type antiferromagnetic order and magnetic phase diagram of the trigonal Eu spin-$\frac{7}{2}$ triangular-lattice compound ${\mathrm{EuSn}}_2{\mathrm{As}}_2$

The trigonal compound ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ was recently discovered to host Dirac surface states within the bulk band gap and orders antiferromagnetically below the Néel temperature $T_{\mathrm{N}}=23.5\left(2\right)$ K from our neutron-diffraction measurements. Here the magnetic ground state of single-crystal ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ and the evolution of its properties versus temperature $T$ and applied magnetic field $H$ are reported. Included are the zero-field single-crystal neutron diffraction measurements versus $T$, magnetization $M(H,T)$, magnetic susceptibility $\chi (H,T)=M(H,T)/H$, heat capacity $C_{\mathrm{p}}(H,T)$, and electrical resistivity $\rho (H,T)$ measurements. The neutron-diffraction and $\chi \left(T\right)$ measurements both indicate a collinear A-type antiferromagnetic (AFM) structure below $T_{\mathrm{N}}$, where the ${\mathrm{Eu}}^{2+}$ spins $S=7/2$ in a triangular $ab$-plane layer (hexagonal unit cell) are aligned ferromagnetically in the $ab$ plane, whereas the spins in adjacent Eu planes along the $c$ axis are aligned antiferromagnetically. The $\chi (H_{ab},T)$ and $\chi (H_c,T)$ data together indicate a smooth crossover between the collinear AFM alignment and an unknown magnetic structure at $H\approx 0.12$ T. Dynamic spin fluctuations up to 60 K are evident in the $\chi \left(T\right)$, $C_{\mathrm{p}}\left(T\right)$ and $\rho (H,T)$ measurements, a temperature that is more than twice $T_{\mathrm{N}}$. The $\rho (H,T)$ is consistent with a low-carrier-density metal with strong magnetic scattering and does not reflect a contribution of the topological state of the material as reported earlier by ARPES measurements. This observation is consistent with previous ones for other topological insulators where the chemical potential is above the Dirac point so that ARPES readily detects the surface states, whereas resistivity measurements do not. The magnetic phase diagrams for both $H\parallel c$ and $H\parallel ab$ in the $H\text{−}T$ plane are constructed from the $T_{\mathrm{N}}\left(H\right)$, $\chi (H,T)$, $C_{\mathrm{p}}(H,T)$, and $\rho (H,T)$ data.

Figure 1

Three projections of the unit cell and crystal structure of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$. The Eu atoms form a hexagonal layer situated between two buckled SnAs honeycomb layers. The Eu atoms are colored dark purple, the Sn atoms blue, and the As atoms green. The crystal structure was drawn using the vesta program [43].

Figure 2

Room-temperature XRD pattern of powdered ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ crystals C along with the refinement of the data using the fullprof software. The red circles are the experimental data points, the black solid line is the calculated refinement based on the ${\mathrm{NaSn}}_2{\mathrm{As}}_2$ crystal structure (space group $R\overline{3}m$), the blue solid curve is the difference between the experimental and calculated diffraction patterns, and the green vertical bars represent the allowed Bragg positions. The fit-quality parameters are given in the figure.

Figure 3

(a) Diffraction patterns along $\left(00L\right)$ reciprocal-lattice units of single-crystal ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ (crystal C) at 4 and 35 K as indicated. (b) Difference between the $\left(00L\right)$ patterns at 4 and 35 K only showing Bragg-diffraction reflections at $L=1.5+3n(n=1,2,3,...)$.

Figure 4

(a) Diffraction patterns along $\left(HH0\right)$ of single-crystal ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ at 4 and 35 K as indicated. The extra peaks are from the aluminum sample holder. (b) The difference between the $\left(HH0\right)$ patterns taken at 4 and 35 K. The absence of emergent magnetic Bragg reflections at low temperature indicates that $ab$-plane layers of Eu moments are aligned ferromagnetically in the $ab$ plane and are stacked antiferromagnetically along the $c$ axis.

Figure 5

Integrated intensity as a function of temperature $T$ of the (0 0 4.5) rlu magnetic Bragg reflection. The solid curve is a phenomenological power-law fit by $I\left(T\right)=C|1-T/T_{\mathrm{N}}{|}^{2\beta }$, yielding $T_{\mathrm{N}}=(23.5\pm 0.2)$ K and $\beta =0.33\left(1\right)$. The magnetic structure of the ${\mathrm{Eu}}^{2+}$ ordered moments depicts half of the AFM unit cell. The neutron-diffraction data are not sufficient for determining the direction of the ferromagnetically-aligned moment in each $ab$ plane.

Figure 6

Inverse magnetic susceptibility ${\chi }^{-1}\left(T\right)$ of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ for (a) $H=0.1$ and (b) 1 T, each with $H\parallel ab$ and $H\parallel c$, along with the respective fits by the Curie-Weiss law (1).

Figure 7

Temperature $T$ dependence of the in-plane ($H\parallel ab$) magnetic susceptibility $\chi \equiv M/H$ measured in $H=0.01$ T for three crystals (crystals A, B, and C) grown in three different ways. As discussed in Sec. 2, the concentration of magnetic defects is expected to decrease from crystal A to crystal B to crystal C, as observed.

Figure 8

(a) Temperature dependence of the zero-field-cooled in-plane (${\chi }_{ab},H\parallel ab$) and out-of-plane (${\chi }_c,H\parallel c$) magnetic susceptibilities ${\chi }_{\alpha }=M_{\alpha }/H$ of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ (Crystal C) measured in a magnetic field $H=0.01$ T. The values of the anisotropic susceptibilities at $T_{\mathrm{N}}=24$ K are ${\chi }_{ab}\left(T_{\mathrm{N}}\right)=0.904{\mathrm{cm}}^3/\mathrm{mol}$ and ${\chi }_c\left(T_{\mathrm{N}}\right)=0.762{\mathrm{cm}}^3/\mathrm{mol}$. (b) The Heisenberg susceptibility ${\chi }_J\left(T\right)$ in the paramagnetic state ($T>T_{\mathrm{N}}$) calculated by taking the spherical average of the data in panel (a) at $T>T_{\mathrm{N}}$ as shown by the green symbols on a scale normalized to that at $T=T_{\mathrm{N}}$. ${\chi }_{ab}\left(T\right)$ (filled black circles) and ${\chi }_c\left(T\right)$ (filled red squares) with $T\le T_{\mathrm{N}}$ are obtained by vertical translations of the respective $\chi$ data in panel (a) to match the ${\chi }_J(T=T_{\mathrm{N}})$ data. The solid green curve gives the fit by Eq. (6) to the ${\chi }_{ab}(T\le T_{\mathrm{N}})$ data to obtain the low-$T$ magnetic-impurity Curie-Weiss contribution. The open circles give ${\chi }_{ab}/\chi \left(T_{\mathrm{N}}\right)$ vs $T$ after subtraction of the Curie-Weiss contribution which is consistent with A-type AFM ordering with the ordered moments ferromagnetically aligned in the $ab$ plane with the moments in adjacent planes along the $c$ axis aligned antiferromagnetically.

Figure 9

Temperature dependence of the magnetic susceptibilities (a) ${\chi }_{ab}\left(H\right)\equiv M_{ab}\left(H\right)/H$ and (b) ${\chi }_c\left(H\right)\equiv M_c\left(H\right)/H$ for crystal C normalized by the respective values at $T=T_{\mathrm{N}}\left(H\right)$ in applied magnetic fields $H$ from 0.01 T to 4 T. The ${\chi }_{ab}\left(H\right)$ data are strongly dependent on $H$, while the ${\chi }_c\left(H\right)$ data are almost independent of $H$. $T_{\mathrm{N}}$ is seen to decrease by about 40% at the highest field. The data suggest a smooth evolution of the magnetic structure on increasing $H$ from 0.01 to $\sim 0.15\mathrm{T}$.

Figure 10

Four-quadrant magnetization vs field $M\left(H\right)$ hysteresis data for crystal C with $ab$-plane and $c$-axis magnetic-field orientations at $T=2$ K. No hysteresis is observed for either field direction.

Figure 11

[(a) and (b)] Magnetization $M$ isotherms for crystal C as a function of applied magnetic field $H$. Low-field $H$-dependences of [(c) and (d)] $M/H$, and [(e) and (f)] $dM/dH$ at different temperatures for $H\parallel ab$ and $H\parallel c$, respectively.

Figure 12

Temperature dependence of zero-field heat capacity $C_{\mathrm{p}}$ for crystal C along with a fit by Eqs. (7) in the temperature range 50–300 K. The data are fitted well for $T\ge 60$ K. The bump in the data at $\approx 290$ K is an experimental artifact.

Figure 13

Temperature $T$ dependence of (a) the magnetic component of the heat capacity $C_{\mathrm{mag}}$, (b) $C_{\mathrm{mag}}/T$, and (c) the magnetic entropy $S_{\mathrm{mag}}$ obtained from the $C_{\mathrm{mag}}/T$ data in (b) using Eq. (10) for ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ (crystal C). The solid black lines in (a) and (b) represent the predictions of MFT in Eq. (9) for $S=7/2$ and $T_{\mathrm{N}}=24$ K. $S_{\mathrm{mag}}$ attains its saturation value $Rln(2S+1)=Rln8$ by $T\approx 60$ K, demonstrating the presence of AFM correlations above $T_{\mathrm{N}}=24$ K.

Figure 14

$C_{\mathrm{p}}\left(T\right)$ of crystal C measured for different applied magnetic fields with $H\parallel c$ showing the suppression of the AFM ordering temperature $T_{\mathrm{N}}$ with increasing $H$. Inset: Expanded plot of the data between 15 and 27 K.

Figure 15

In-plane electrical resistivity of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$. The $T$ dependence over the whole $T$ range in $H=0$ is shown in the top-left inset. The main panel zooms in on the $T$ range of the main AFM transition, with measurements made in $H\parallel ab\perp I$ of 0 (black, top curve), 1 (red), 2 (green), 3 (blue), 4 (cyan), and 6 T (magenta). Note the negative magnetoresistance in the $T$ range below 40 K and the crossover to positive magnetoresistance above. The bottom-right inset shows a zoom of the second feature in the $T$-dependent resistivity, a slight decrease below $T_L\sim 6$ K in $H=0$ (black curve). This feature broadens in $H=0.05$ T (red curve) and is suppressed with $H=0.1$ (green curve) and 0.2 T (blue curve).

Figure 16

Temperature $T$ derivative $d\left[\rho \right(T)/\rho (300\mathrm{K}\left)\right]/dT$ of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ up $T=50$ K. The derivative shows a jump at $T_{\mathrm{N}}\approx 23$ K in zero field, with the maximum denoted by a filled red circle. The jump remains sharp in a magnetic field $H\parallel ab$ of 1 T (maroon curve), and an additional sharp peak occurs for 2 T and a step at 3 T with a broad crossover background, while the sharp peak increases in magnitude. The feature at $T_{\mathrm{N}}$ further broadens in 4 T. The inset shows a suppression of the zero-field anomaly in the derivative at 6 K with magnetic field, and magenta diamonds show the position of the feature onset.

Figure 17

Temperature dependence of the resistivity $\rho \left(T\right)\mathrm{of}{\mathrm{EuSn}}_2{\mathrm{As}}_2$ measured in magnetic fields $H\parallel c$ as indicated. The data for successive fields are offset from each other for clarity, where the bottom curve is for $H=0$ and the top one is for $H=9$ T.

Figure 18

Temperature dependence of the normalized resistivity derivative $d\left[\rho \right(T)/\rho (300\mathrm{K}\left)\right]/dT$ of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ measured in magnetic fields $H\parallel c$ as indicated. The filled diamonds and open pentagons at temperatures $T_{\mathrm{A}}$ represent potential transitions of unknown origin. The filled blue triangles represent maxima in $d\left[\rho \right(T)/\rho (300\mathrm{K}\left)\right]/dT$ at $T_{\mathrm{N}}\left(H\right)$.

Figure 19

(a) Magnetic $c$-axis field $H_c$ vs $T$ phase diagram of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$ constructed from the $C_{\mathrm{p}}(H,T)$ (filled green circles), $\chi (H,T)$ (filled black squares), and $\rho (H,T)$ (filled blue triangles) data. The solid red curve is a fit of the critical field $H_c^{\mathrm{c}}\left(T\right)$ data obtained from the $C_{\mathrm{p}}\left(T\right)$ data in Fig. 14 by Eq. (11) that separates the antiferromagnetic (AFM) phase from the paramagnetic (PM) phase. We suggest that the filled magenta diamonds near zero field near 5 and 6 K identified from the $\rho (H,T)$ measurements are due to lock-in transitions of the Eu moments in three AFM domains becoming perpendicular to the applied field. The unfilled pentagons at temperatures $T_{\mathrm{A}}$ are unidentified transitions suggested from the $\rho (H,T)$ measurements in Fig. 18. (b) $ab$-plane magnetic field $H_{ab}$ vs $T$ phase diagram of ${\mathrm{EuSn}}_2{\mathrm{As}}_2$. The red filled circles represent $H_{ab}\left(T_{\mathrm{N}}\right)$ obtained from Fig. 16. The violet diamonds near $H_{ab}=0$ correspond to a feature in the $T$-dependent resistivity derivative which we suggest reflects a lock-in transition at which the Eu moments in the $ab$ plane become perpendicular to the applied field. The solid magenta diamonds at 3–7 K correspond to a small anomaly observed in $\rho \left(T\right)$ of unknown origin. The critical fields $H^{\mathrm{c}}$ vs $T$ in (a) and (b) (half-filled boxes) from Table 2 were determined as the intersection of the extrapolated lines from larger $H$ and smaller $H$ with respect to the approximate saturation field.