Magnetic, thermal, and electronic-transport properties of EuMg₂Bi₂ single crystals

The interplay between magnetism and band topology has generated immense interest recently due to the discovery of nontrivial phenomena such as the quantum anomalous Hall effect (QAHE) Qi2006_QAHE ; Liu2008_QAHE ; Yu2010_QAHE ; Chang2013_QAHE ; Liu2016_QAHE , axion electrodynamics Essin2009_AED ; Li2010_AED ; Lee_AED ; Wu2016_AED , and realization of relativistic particles like Majorana fermions Do2017_Majorana ; Akhmerov2009_Majorana ; Cook2011_Majorana ; Xu2015_Majorana . A topological state manifests topologically-protected electronic surface states that are different in nature from the bulk states Fu2007_TI ; Hasan2010_TI ; Moore2010_TI ; Qi2011_TI ; Ando2013_TI ; Tokura2019_TI . In addition to the parabolic bulk band, depending upon symmetry preservation, a unique gapless/gapped surface state with a linear energy versus crystal momentum $E$-$k$ relation is also observed in these materials, resulting in massless surface electrons with ultrahigh mobility.

Time reversal symmetry (TRS) invariance in topological insulating states and its breaking in the presence of magnetism play a key role in discoveries in this field. Exotic quantum phenomena such as QAHE, chiral Majorana modes, and the topological magnetoelectric effect (TME) have been predicted in topological materials based on TRS breaking by magnetic order, with experimental support Liu2008_QAHE ; Yu2010_QAHE ; Chang2013_QAHE ; Essin2009_AED ; Li2010_AED ; Do2017_Majorana ; Fu2007_TI ; Qi2008_TRB ; Garate2010_TRB ; Xu2012_TRB . In a topological material exhibiting the QAHE, electrons can carry a dissipationless current, which is thus promising for use in energy-efficient electronic devices and also for fast computing. The experimental realization of QAHE in magnetic topological insulators has paved the way for researchers to discover other novel phenomena by studying various magnetic topological systems. In order to fully understand the phenomena triggered by TRS breaking in such magnetic topological systems, it is essential to understand the magnetic structure and symmetry of those materials.

Rare-earth-based intermetallic compounds have been of significant interest for many years due to their complex properties such as superconductivity, heavy fermion behavior, valence fluctuations, giant magnetocaloric effect, Kondo behavior, and quantum criticality Fertig1977_IC ; Nagarajan1994_IC ; Ghosh1995_IC ; Curro2000_IC ; Hundley2001_IC ; Sakai2011_IC ; Yamaoka2014_IC ; Pecharsky1997_IC ; Pakhira2016_IC ; Pakhira2017_IC ; Buschow1969_IC ; Gignoux1984_IC ; Ishida2002_IC ; Arndt2011_IC . Several rare-earth-based intermetallic compounds have also been discovered recently to exhibit topological states coupled with magnetic interactions Hirschberger2016_GdPtBi ; Wang2016_YbMnBi2 ; May2014_EuMnBi2 ; Soh2019_EuMnSb2 ; Schellenberg2011_EuCd2As2 ; Jo2020_EuCd2As2 ; Xu2019_EuIn2As2 . These discoveries reveal the subtle importance of magnetism in controlling the electronic surface states in the magnetic topological materials.

Recently, multiple Dirac states at different energies with respect to the Fermi energy were reported in a new magnetic topological material EuMg₂Bi₂ Kabir2019 . The magnetic ground state of the compound was suggested to be AFM in nature below $T_{\rm N}\approx 7$ K, the details of which are unclear as yet Kabir2019 ; May2011 ; Ramirez2015 . Based on the magnetic susceptibility behavior, where the $c$-axis susceptibility $\chi_{c}$ decreases somewhat and the $ab$-plane susceptibility $\chi_{ab}$ is almost temperature independent below $T_{\rm N}$, it was suggested that the moments are aligned along the $c$ axis Kabir2019 , presumably in a collinear AFM structure. In order to understand the mechanism of topological surface states and also to tune its nature it is important to clarify the magnetic structure and its evolution with applied magnetic field in EuMg₂Bi₂.

The primary purpose of this work was to investigate the magnetic structure and to explore the magnetic phase diagram of EuMg₂Bi₂ as a function of temperature and applied magnetic field $H$ to provide an experimental basis for relating the properties to the topology of the theoretical band structure. We present detailed measurements of the anisotropic $\chi$ versus temperature $T$ and magnetization $M$ versus $H$ isotherms for EuMg₂Bi₂ single crystals, together with complementary heat-capacity and electrical-resistivity measurements. We find that the magnetic susceptibilities for fields applied both in-plane and out-of-plane are almost temperature independent below $T_{\rm N}$, which is very unusual in magnetic Eu⁺²-based systems. Using a recently-developed molecular-field theory (MFT) Johnston2012 ; Johnston2015 , we deduce that the nearly $T$-independent $\chi(T)$ data below $T_{\rm N}$ indicate that the magnetic structure is either a $c$-axis helix in which the Eu moments are ferromagnetically aligned within each $ab$ plane layer, with a turn angle $kd\approx 120^{\circ}$ between the moments in adjacent layers as generically illustrated in Fig. 1 Johnston2012 ; Johnston2015 , or a stacked coplanar structure with a $120^{\circ}$ angle between adjacent Eu moments in each triangular-lattice $ab$-plane layer as in the 120^∘ cycloidal structure in Fig. 2. The in-plane electrical resistivity $\rho(T)$ data indicate metallic character with a mild and disorder-sensitive upturn below $T_{\rm min}=23$ K. An anomalous rapid drop on cooling below $T_{\rm N}$ as found in zero field is replaced by a two-step decrease in magnetic fields. The $\rho(T)$ measurements also reveal an additional transition below $T_{\rm N}$ in applied fields of unknown origin that is not observed in the other measurements and may be associated with an incommensurate to commensurate AFM transition.

The experimental details and the crystal structure studies are discussed in Sec. II. The $\chi(T)$ and $M(H)$ isotherm data and analyses are given in Sec. III. Our heat capacity $C_{\rm p}(H,T)$ data and analyses are presented in Sec. IV, and the electrical resistivity $\rho(H,T)$ data and analyses in Sec. V. The paper concludes with a summary in Sec. VI.

Single crystals of EuMg₂Bi₂ and YbMg₂Bi₂ were grown with the flux method in two different ways using high purity elements Eu (Ames Laboratory), Mg (Alfa Aesar, 99.98%), and Bi (Alfa Aesar, 99.9999%). One batch of crystals was obtained from the nominal flux composition EuMg₄Bi₆ and YbMg₄Bi₆ as described elsewhere May2011 . Another batch of crystals was grown from the starting composition EuMg₂Bi₇. The starting elements were loaded into alumina crucibles and sealed in silica tubes under $\approx 1/4$ atm of Ar gas. A sealed tube was heated to 900 ^∘C at a rate of 50 ^∘C/h and held at that temperature for 12 h. Then it was cooled to 500 ^∘C over 200 h and the single crystals were then obtained by decanting the excess flux using a centrifuge. Most of the crystals obtained are three dimensional in shape with clear trigonal facets. In this work measurements were carried out on crystals obtained using the first growth process.

Room-temperature x-ray diffraction (XRD) measurements were carried out using a Rigaku Geigerflex x-ray diffractometer using Cu-K_α radiation. Structural analysis was performed by Rietveld refinement using the Fullprof software package Carvajal1993 . The chemical composition and homogeneity of the crystals were checked using a JEOL scanning-electron microscope (SEM) equipped with an EDS (energy-dispersive x-ray spectroscopy) analyzer.

The $T$- and $H$-dependent magnetization measurements were performed using a magnetic-properties measurement system (MPMS, Quantum Design, Inc.) in the $T$ range 1.8–300 K and $H$ up to 5.5 T (1 T $\equiv 10^{4}$ Oe). A Quantm Design, Inc., physical-properties measurement system (PPMS) was used to measure $C_{\rm p}(H,T)$ and $\rho(H,T)$. Contacts to the samples were soldered with In in standard four-probe configuration, similar to the technique used for FeSe FeSeresistivity . The contact resistance was in the sub-Ohm range.

The reported CaAl₂Si₂-type trigonal crystal structure of EuMg₂Bi₂ (space group $P\bar{3}m1$) May2011 ; Zheng1986 is shown in Fig. 3. The structure consists of stacked planar triangular lattices of Eu in the $ab$ plane separated along the $c$ axis by two ordered MgBi layers. The room-temperature powder x-ray diffraction (XRD) pattern collected on crushed EuMg₂Bi₂ single crystals along with our Rietveld refinement are shown in Fig. 4. The refinement confirms that EuMg₂Bi₂ crystallizes in the CaAl₂Si₂-type crystal structure with space group $P\bar{3}m1$. The refined parameters are summarized in Table 1. The lattice parameters obtained are in agreement with previously reported values May2011 ; Ramirez2015 ; Kabir2019 . The SEM-EDS measurements confirm the homogeneity of the grown crystals with an average composition EuMg_1.98(2)Bi_2.01(2), which agrees with the stoichiometric composition EuMg₂Bi₂ to within the errors.

In this section we use the heat capacity $C_{\rm p}(T)$ of the nonmagnetic compound YbMg₂Bi₂ as a reference for the lattice heat capacity of EuMg₂Bi₂. The data for single-crystal YbMg₂Bi₂ at low $T$ are shown in Fig. 11, plotted as $C_{\rm p}/T$ versus $T^{2}$. For most nonmagnetic materials, the behavior at low temperatures is described by the expression

where $\gamma$ is the Sommerfeld coefficient associated with degenerate itinerant charge carriers and $\beta$ is the coefficient of the $T^{3}$ term in the low-$T$ limit of the Debye lattice heat capacity. However, from Fig. 11 this two-term fit does not fit the data at all. Furthermore, it yields an unphysical negative value for $\gamma$. In order to obtain a good fit to the data below 10 K we added two additional lattice heat capacity terms according to

An excellent fit to the data was obtained by this expression as illustrated in Fig. 11 where

The value of $\Theta_{\rm D}$ is obtained from $\beta$ according to

where $n$ is the number of atoms per formula unit ($n=5$ here) and $R$ is the molar gas constant, yielding $\Theta_{\rm D}=255(15)$ K. It is notable that $\gamma=0$ to within its error. Therefore below we assume $\gamma=0$ when evaluating the heat capacity of EuMg₂Bi₂.

The zero-field single-crystal heat capacities $C_{\rm p}(T)$ of EuMg₂Bi₂ and the nonmagnetic analogue YbMg₂Bi₂ measured in the temperature range 1.8–300 K are plotted in Fig. 12(a). The $C_{\rm p}$ of EuMg₂Bi₂ exhibits a pronounced peak at $T_{\rm N}$ = 6.7 K as evident from the inset of Fig. 12(a). The $C_{\rm p}(T)$ attains values of $\approx 122.5$ J/mol K and $\approx 121$ J/mol K at $T=300$ K for EuMg₂Bi₂ and YbMg₂Bi₂, respectively. These values are close to the classical Dulong-Petit limit due to acoustic phonon modes $C_{\rm V}=3nR=124.7$ J/mol K with $n=5$ being the number of atoms per formula unit of the above compounds.

We fitted the $C_{\rm p}(T)$ data in the temperature regions 25–300 K for EuMg₂Bi₂ and 2–300 K for YbMg₂Bi₂ by the Debye lattice heat capacity prediction

where $\Theta_{\rm D}$ is the Debye temperature. The Padé approximant for the Debye function in Ref. Goetsch_2012 was used for the fits. As seen from dashed curves in Figs. 12(b) and 12(c), the lattice heat capacity is not described well by the Debye model. Much better fits were obtained by including an Einstein lattice contribution to the fits according to

where

with $\Theta_{\rm E}$ being the Einstein temperature. The parameter $\alpha$ determines the fraction of the Einstein contribution to the total lattice heat capacity. Very good fits of the $C_{\rm p}(T)$ data by Eq. (11) were achieved, as depicted by the solid curves in Figs. 12(b) and 12(c). The fitted parameters are listed in Table 3. The Debye temperatures of $\sim 300$ K are much larger than the values of 207 K and 211 K previously reported from Debye fits to the data from 20 to 200 K for YbMg₂Bi₂ and EuMg₂Bi₂, respectively May2011 . The low Einstein temperatures of $\sim 75$ K suggest the presence of low-frequency optic modes associated with the heavy Bi and/or Yb or Eu atoms, respectively, as also suggested from Fig. 11.

The magnetic contribution $C_{\rm{mag}}(T)$ to the heat capacity of EuMg₂Bi₂ was obtained by subtracting the $C_{\rm p}(T)$ data of the nonmagnetic analogue YbMg₂Bi₂ in Fig. 12(a) after correcting for the difference in formula weights, where the temperature scale $T^{*}$ of YbMg₂Bi₂ was obtained as

where $T$ is the experimentally-measured temperature. The $C_{\rm{mag}}(T)$ was also estimated by subtracting the lattice contribution obtained from the fit of the $C_{\rm p}(T)$ data by Eq. (11). Figure 13(a) shows $C_{\rm{mag}}(T)$ obtained using the two different lattice heat capacity estimates. The two calculations of $C_{\rm mag}(T)$ agree very well below $T_{\rm N}$, but the data above $T_{\rm N}$ obtained using the measured lattice contribution of YbMg₂Bi₂ is physically more likely.

According to MFT, the discontinuity in $C_{\rm{mag}}$ at $T=T_{\rm N}$ is given by Johnston2015

The jump $\Delta C_{\rm mag}$ in $C_{\rm{mag}}$ for EuMg₂Bi₂ at $T_{\rm N}$ = 6.7 K expected from Eq. (14) is 20.14 J/mol K for $S=7/2$. The experimentally-observed jump $\Delta C_{\rm{mag}}(T_{\rm N})\approx 25$ J/mol K is significantly larger than the MFT prediction. This difference arises because the observed $\lambda$ shape of $C_{\rm mag}(T)$ below $T_{\rm N}$ is different from step shape of the MFT prediction shown as the solid curve in Fig. 13(a) obtained as follows.

The magnetic contribution $C_{\rm mag}(T,H=0)$ to the heat capacity according to MFT is given by Johnston2015

where $\bar{\mu}_{0}(t)\equiv\mu_{0}(t)/\mu_{\rm sat}$ is the reduced ordered moment versus $t$ in $H=0$, $\mu_{\rm sat}=gS\mu_{\rm B}$ is the saturation moment of the spin, $B_{S}(y)$ is Brillouin function, and $B^{\prime}_{S}(y_{0})\equiv\frac{dB_{S}(y)}{dy}\big{|}_{y=y_{0}}$. The solid black curve in Fig. 13(a) represents the $C_{\rm{mag}}(T)$ predicted by MFT for $T_{\rm N}=6.7$ K and $S=7/2$. The disagreements with the data in Fig. 13(a) arise from neglect of dynamic magnetic fluctuations and correlations in MFT.

The zero-field magnetic entropy $S_{\rm{mag}}(T)$ for EuMg₂Bi₂ in Fig. 13(c) was obtained from the zero-field $C_{\rm{mag}}(T)/T$ data in Fig. 13(b) using the relation

The $C_{\rm{mag}}(T)$ data in the $T$ range 0–1.8 K for which we have no data was estimated as described in the caption to Fig. 13 and shown as the dashed blue line in Fig. 13(b). The $S_{\rm{mag}}(T)$ calculations for EuMg₂Bi₂ using the above two methods of calculating $C_{\rm{mag}}(T)$ are shown in Fig. 13(c). The $S_{\rm{mag}}(T)$ calculated using the Debye-Einstein lattice contribution saturates at high $T$ to a value of 18.2 J/mol K, which is slightly larger than the theoretical high-$T$ limit $S_{\rm mag}=R{\rm ln}(2S+1)=17.29$ J/mol K for $S=7/2$. On the other hand, when using the YbMg₂Bi₂ lattice contribution, $S_{\rm{mag}}$ saturates to a value of 16.7 J/mol K which is slightly smaller than the theoretical prediction. The $S_{\rm{mag}}$ reaches to 94% and 91% of $R\ln(8)$ at $T_{\rm N}$ in these two calculations, respectively. The nonzero contributions to $C_{\rm mag}(T)$ and $S_{\rm mag}(T)$ above $T_{\rm N}$ arise from dynamic short-range magnetic order. This feature has also been observed in other spin-7/2 Eu⁺² helical AFM systems Sangeetha_EuCo2P2_2016 ; Sangeetha_EuCo2As2_2018 ; Sangeetha_EuNi2As2_2019 .

The variation of $C_{\rm p}(T)$ with magnetic field $H_{\perp}$ defined to be the field applied along the $c$ axis, perpendicular to the zero-field $ab$ ordering plane, is shown in Fig. 14. Here we define the field along the $c$ axis as $H_{\perp}$ so as not to confuse the notation with the critical field $H_{\rm c}$ and the $c$-axis field $H_{c}$. Due to the constraints of the PPMS used to measure $C_{\rm p}$, it was not possible to measure $C_{\rm p}(T)$ with $H\parallel ab$. As seen from Fig. 14, $T_{\rm N}$ shifts to lower temperature with increasing $H_{\perp}$ and also the jump in $C_{\rm p}$ at $T_{\rm N}$ decreases, where $T_{\rm N}$ is defined as the temperature of the peak in the heat capacity because of the contribution of dynamic short-range magnetic ordering to $C_{\rm mag}$ above $T_{\rm N}$.

From the data in Fig. 14, one could plot $T_{\rm N}$ versus the field $H_{\perp}$ or the critical field $H_{\rm c\perp}$ versus $T$. We use the latter scaling because for a helical antiferromagnet, $H_{c\perp}(t)$ is given theoretically by MFT as Johnston2015

where $\bar{\mu}_{0}(t)$ is calculated using Eqs. (15). A fit of the $H_{c\perp}$ versus $T$ data in Fig. 15 by Eq. (17) yields $T_{\rm N}=6.64(2)$ K and $H_{c\perp}(T=0)=4.0(1)$ T, as shown by the red curve in the figure. This curve divides the phase diagram in the $(H_{\perp},T)$ plane into helical AFM and PM regions as shown.

The temperature dependence of the $ab$-plane electrical resistivity measured in the temperature range 2 to 300 K in zero magnetic field is plotted in Fig. 16(a). The data are presented using a normalized $\rho(T)/\rho(300~\rm K)$ scale. The resistivity shows a metallic decrease on cooling down to a shallow minimum at $T_{\rm min}\sim 23$ K, followed by a slight increase on further cooling. A sharp resistivity decrease below $T=6.75$ K (transition midpoint) indicates the onset of AFM ordering in the system.

The observed $\rho(T)$ behavior is similar to that reported in similarly-grown crystals, blue triangles in Fig. 16(a) May2012 , but notably different from that reported in the previous study Kabir2019 in which a Sn-flux-grown single crystal was used for the measurements. The shallow minimum in $\rho(T)$ as observed in our study at about 23 K is not observed in similarly-grown samples with higher RRR and is observed at a much higher temperature of $\sim 125$ K, in samples with notably lower RRR. Using the value of resistivity above $T_{N}$ as a proxy for $\rho(0)$, the extrapolated residual-resistivity ratio RRR $\equiv\rho(300$ K)/$\rho(T_{\rm N})$ increases from $\approx 1.14$ in the Sn-flux-grown crystal, to $\approx 1.33$ in our crystals and to $\approx 1.39$ in the data of Ref. May2012 . Progressively the position of the resistivity minimum shifts to zero. In previous studies of polycrystalline EuMg₂Bi₂ May2012 the minimum was much more pronounced and located at about 100 K. The slight resistivity increase on cooling is accompanied by a temperature-independent Hall constant May2012 arguing against an activated character of charge transport below the minimum. The minimum was also observed in nonmagnetic CaMg₂Bi₂, which makes a possible contribution of the Kondo effect Kondoeffect unlikely. We conclude that the most likely reason for the minimum is an onset of Anderson localization Anderson due to the strong effect of disorder in a low carrier-density metal.

The decrease of resistivity below $T_{\rm N}$ in both previous studies is notably smaller than in our crystals, and is significantly less sharp. The origin of this discrepancy is unclear and deserve further study. The resistivity decrease below $T_{\rm N}$ is governed by a decrease of magnetic entropy Paglione2005 and the sharp feature may be suggestive of the first-order character of the transition. The evolution of the temperature-dependent resistivity with $H\parallel c$ is shown in Fig. 16(b). Of special note is the evolution between $H=0$ T (black) and 0.5 T (red). While the transition with onset at $T_{\rm N}=6.93$ K in zero field is accompanied by a pretty sharp (full width of 0.4 K) resistivity decrease, the feature at $T_{\rm N}=6.85$ K [which does not change much in temperature compared to the 0 T curve, as indicated by simple vertical black and red arrows in Fig. 16(b)], the decrease in $\rho(T)$ at 0.5 T is quite small. On further cooling $\rho(T)$ for 0.5 T reveals a second feature [we label it $T_{2}=5.1$ K and indicate with a cross arrow in Fig. 16(b)], below which the main resistivity decrease happens. A similar $T_{2}$ feature is found for the curve in 0.75 T field. The third feature at $T_{3}=3.8$ K (up-triangle arrow) leads to flattening of the $\rho(T)$ in $H_{c}=0.5$ T for $T\to 0$ K. The $T_{3}$ is not found for 0.75 T in the temperature range studied. A magnetic field of 1 T suppresses the $T_{2}$ feature, but the feature at $T_{\rm N}$ remains clearly discernible. With further magnetic field increase to 1.25 T the shape of the anomaly in $\rho(T)$ changes qualitatively, with initial flattening and eventual increase of resistivity on cooling for $H_{c}=2.5$ T. A monotonically decreasing $\rho(T)$ as found at 3 T suggests the suppression of $T_{\rm N}$ to below the temperature range studied.

We summarize the anomalies in $\rho(T,H)$ in the above phase diagram in Fig. 15. For fields below 1 T the feature in $\rho(T,H)$ at $T_{\rm N}$ (solid blue circles) is in good agreement with the heat capacity (black solid squares) and magnetization (green down-triangles) measurements. For $H_{\perp}=1.25$ T the position of the highest-temperature feature starts to go to zero notably faster than suggested by the magnetization and heat capacity determinations of $T_{\rm N}$. Note that the shape of $\rho(T)$ changes in the same field range.

The position of the $T_{2}$ feature (blue crosses in Fig. 15) seems to be smoothly connected to the zero-field $T_{\rm N}$ (blue dashed line). The position of the $T_{3}$ feature (solid blue up-triangle) does not seem to be connected to any special point in the phase diagram.

The sharpness of the feature at $T_{\rm N}$ in zero-field resistivity measurements suggest a first-order character of the transition, and its splitting into two second-order transitions with application of magnetic field. However, this interpretation is not supported by either the $\chi(T)$ or $C_{\rm p}(T)$ data in Fig. 5 and the inset of Fig. 12(a), respectively.

The fact that the low-temperature features at $T_{2}$ and $T_{3}$ are not observed in heat capacity and magnetization measurements may suggest that the features are due to lock-in transitions (transformations of helical order in magnetic field between incommensurate and commensurate on cooling), particularly difficult to observe in thermodynamic measurements due to the minute entropy changes involved Jensen .

In this work, we have investigated the detailed magnetic, thermal, and electronic transport properties of single crystals of the trigonal compound EuMg₂Bi₂, which has recently been reported to host multiple Dirac states. The magnetic susceptibility shows that the compound undergoes AFM ordering below $T_{\rm N}=6.7$ K associated with the Eu⁺² spins 7/2, as reported earlier.

The magnetic susceptibilities are found to be almost independent of temperature below $T_{\rm N}$ for both $H\parallel c$ and $H\parallel ab$, where the hexagonal setting of the trigonal structure has lattice parameters $a=b$ and $c$. According to molecular field theory (MFT), this behavior strongly suggests that the magnetic structure of EuMg₂Bi₂ below $T_{\rm N}$ is a $c$-axis helix, where ferromagnetically-aligned moments in $ab$ planes rotate in a $c$ axis helical structure by a turn angle of $\approx 120^{\circ}$ from plane to plane along the $c$ axis. Another possible but less probable magnetic structure is a planar structure with in-plane nearest-neighbor Eu spins aligned at $\approx 120^{\circ}$ with respect to each other, with such planes stacked along the $c$ axis. The latter structure is less likely because the calculated value of the net in-plane magnetic exchange interaction $J_{0}$ derived from MFT is positive (FM), rather than negative (AFM) as would be expected for the latter structure, and because the Weiss temperature in the Curie-Weiss law is positive (FM-like) for fields in the $ab$ plane.

According to MFT, the $c~(\perp)$-axis magnetization $M_{c}$ of a $c$-axis helix is linear for applied fields from $H=0$ to the critical field $H_{{\rm c\perp}}$ at which the magnetization approaches the saturation magnetization and a second-order transition to the paramagnetic state occurs. For fields applied in the $ab$ plane of a helix with a $120^{\circ}$ turn angle, $M_{ab}$ is also predicted to be linear from $H=0$ to the critical field $H_{{\rm c}\,ab}$ at $T=0$, even though a smooth crossover from a helix to fan phase occurs if the spins are confined to the $ab$ plane. On the full scale of our $M(H)$ measurement field, these predictions are verified. However, on closer examination, we find that $M_{ab}(H)$ shows a subtle nonlinearity at fields below about 500 Oe. It would be interesting in future work to determine experimentally what change in the magnetic structure this nonlinearity is associated with.

The zero-field heat capacity $C_{\rm p}(T)$ measurement reveals a $\lambda$ anomaly at $T_{\rm N}$ that shifts to lower temperature with increasing $H$. The zero-field magnetic contribution $C_{\rm mag}(T)$ to $C_{\rm p}(T)$ obtained using two different background subtractions reveals the presence of short-range dynamic magnetic fluctuations both below and above $T_{\rm N}$ that contribute to the high-temperature limit of the magnetic entropy. This limit is close to the value expected for Eu spins $S=7/2$.

A sharp drop in the electrical resistivity $\rho(T)$ is observed on cooling below $T_{\rm N}$ in zero field. It is replaced by a two-stage resistivity decrease in the smallest applied magnetic fields. This behavior is contrary to the previous studies and deserves further investigation. This behavior is not reflected in the $\chi(T)$ or $C_{\rm p}(T)$ data, and to our knowledge has not been observed previously on cooling below $T_{\rm N}$ in any other antiferromagnetic material. The drop is linear in temperature in nonzero $c$-axis magnetic fields with the temperature width of the drop increasing with increasing field. A resistivity minimum above $T_{\rm N}$ was also observed in the $\rho(T)$ data at $\sim 23$ K, which is a significantly lower temperature than that reported earlier in Ref. Kabir2019 . Interestingly, although only one magnetic transition at $T_{\rm N}$ is observed in our magnetic and heat capacity data, in addition to the above-noted feature at $T_{\rm N}$ the $\rho(T)$ data also reveal another distinct field-dependent anomaly in the magnetic-field range $0.5~{\rm T}\leq H\leq 0.75$ T for $T=4.3$ to $5.2$ K.

On the basis of the magnetic, thermal, and electronic transport studies, a magnetic phase diagram in the $H$-$T$ plane for fields parallel to the $c$ axis was constructed that includes the antiferromagnetic and paramagnetic regions. The phase boundary between these two phases is fitted satisfactorily by MFT.

It would be interesting to theoretically investigate the degree and manner to which one or more of our measured properties of single-crystalline EuMg₂Bi₂ are influenced or even caused by topological features of the band structure. Of particular interest is the origin of the rapid drop in the resistivity on cooling below $T_{\rm N}$ in zero field.