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J. Phys. Soc. Jpn. 92, 114703 (2023) [12 Pages]
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Magnetic and Fermi Surface Properties of Semimetals EuAs3 and Eu(As1−xPx)3

Yoshichika Ōnuki1,2, Kosuke Karube1, Dai Aoki3, Ai Nakamura3, Yoshiya Homma3, Tatsuma D. Matsuda2, Yoshinori Haga4, and Tetsuya Takeuchi5
+ Affiliations
1RIKEN, Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan2Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan3Institute for Materials Research, Tohoku University, Oarai, Ibaraki 311-1313, Japan4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan5Core Facility Center, Osaka University, Toyonaka, Osaka 560-0043, Japan
Received July 5, 2023; Accepted August 10, 2023; Published October 4, 2023

A Eu-divalent antiferromagnet EuAs3 exhibits six magnetic phases in the magnetic field vs temperature (H–T) phase diagram for the magnetic field along the monoclinic b*(b)-axis. The existence of such multiple magnetic phases is rare among Eu compounds because the magnetism is governed by the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction and the corresponding localized 4f-electrons only exhibit the spin angular momentum S. To clarify characteristics in these magnetic phases of EuAs3 and Eu(As1−xPx)3 (0.7 < x < 0.97), we measured the magnetization, magnetoresistance, and Hall resistivity using single-crystal samples. A large anomalous Hall resistivity was found in these magnetic phases, especially in Eu(As0.97P0.03)3. In addition, we carried out de Haas–van Alphen (dHvA) experiments to clarify the Fermi surface properties of the semimetal EuAs3, and it was revealed that there are small and closed electron and hole Fermi surfaces with equal volumes.

©2023 The Physical Society of Japan
https://doi.org/10.7566/JPSJ.92.114703
1. Introduction

The antiferromagnets EuAs3 and Eu(As\(_{1-x}\)Px)3 are semimetals and exhibit various magnetic phases in the magnetic field vs temperature (H–T) phase diagram.1–7) EuAs3 crystallizes in a monoclinic structure (\(C2/m\), No. 12, \(a=9.471\) Å, \(b=7.598\) Å, \(c=5.778\) Å, and \(\beta=112.53\)°)8) and shows an antiferromagnetic order below the Néel temperature \(T_{\text{N}}=11.2\) K. At lower temperatures, a first-order magnetic transition appears at \(T_{\text{L}}=10.3\) K. The magnetic structure for \(T_{\text{L}}<T<T_{\text{N}}\) was clarified to be an incommensurate (IC) sine-wave amplitude-modulated phase with the ordered moment along the [010] direction [\(b^{*}\)(b)-axis]. The IC phase locks into a commensurate phase below \(T_{\text{L}}\). The magnetic structure in the low-temperature collinear antiferromagnetic phase (AF1) for \(T<T_{\text{L}}\) consists of the ferromagnetic spin arrangement in the \((\bar{2}01)\) plane, and ferromagnetic \((\bar{2}01)\) planes are stacked antiferromagnetically along the \([\bar{2}01]\) direction. Note that the corresponding Eu ions are divalent, namely, Eu2+ (\(4f^{7}\): \(S=J=7/2\), \(L=0\), and \(g=2\)), and the ordered moments of \(\mu_{\text{s}}=6.05\) \(\mu_{\text{B}}\)/Eu at \(T=5\) K are oriented along the [010] direction [\(b^{*}\)(b)-axis], where S, L, J, and g are the spin, orbital, and total angular momenta, and the Landé g-factor, respectively. We show the monoclinic crystal and magnetic structures of the AF1 phase in Figs. 1(a) and 1(b), respectively, where the opposite spin directions along the \(b^{*}\)(b)-axis are indicated on the Eu atoms (\(\odot\) and \(\otimes\)) in Fig. 1(b). The previously obtained H–T phase diagrams of EuAs3 and Eu(As0.02P0.98)3 are shown in Figs. 1(c) and 1(d), respectively.5)


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Figure 1. (Color online) (a) Monoclinic structure in EuAs3 and (b) magnetic structure, cited from Ref. 4, where the \(b^{*}\)(b)-directions of the Eu spins are indicated on the atoms by up (\(\odot\)) and down (\(\otimes\)) spin states in the \(b^{*}\)(b)-plane. The magnetic phase diagrams of (c) EuAs3 and (d) Eu(As0.02P0.98)3 for \(H\parallel \text{$b^{*}$-axis}\) cited from Ref. 5.

The magnetic phase diagram of EuAs3 in Fig. 1(c) is composed of six magnetic phases, which are surprisingly complex. This is rare in Eu compounds. This is because the magnetism is governed by the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction and the corresponding localized \(4f\)-electrons only exhibit the spin angular momentum S, as mentioned above. These magnetic phases might be produced by anisotropic magnetic exchange interactions. For example, the distances between the Eu ion named A and the Eu ions named B, C, and D in Fig. 1(b) are 3.91, 4.24, and 4.29 Å, respectively. Here, the a-, b-, and c-axes are used in the crystal structure. The \(a^{*}\)-, \(b^{*}\)-, and \(c^{*}\)-axes are used here in the reciprocal lattice, which correspond to the directions of the Laue pattern. In the Brillouin zone, the crystal structure of EuAs3 belongs to the base-centered monoclinic lattice, and then \(a_{0}{}^{*}\), \(b_{0}{}^{*}\), and \(c_{0}{}^{*}\) are used in the Brillouin zone, following the previous report of the de Haas–van Alphen (dHvA) experiments for SrAs3 with the same crystal structure as in EuAs3.9)

The SF1 phase in Fig. 1(c) is an IC spiral arrangement of the magnetic moments in the \(b^{*}\)- or \(ac\)-plane, where the magnetic field is directed along the \(b^{*}\)-axis. The corresponding magnetization indicates a spin–flop (SF) transition at \(H_{\text{SF1}}\). Next, the commensurate collinear SF2 phase with the propagation vector \((0, 1, 0.25)\) appears and is followed by another incommensurate phase (SF3) with increasing magnetic field. The spin-canted phase (SF4) finally enters into the field-induced ferromagnetic phase (paramagnetic phase) at around \(\mu_{0}H_{\text{c}}=6{\text{–}}7\) T.

Eu(As\(_{1-x}\)Px)3 (\(x<0.98\)) also crystallizes in the same crystal structure as in EuAs3 and exhibits two magnetic transitions at \(T_{\text{N}}\) and \(T_{\text{L}}\). Arsenic-rich compounds of Eu(As\(_{1-x}\)Px)3 (\(x < 0.5\)) indicate similar magnetic properties to EuAs3. On the other hand, phosphorus-rich compounds of Eu(As\(_{1-x}\)Px)3 (\(0.5 < x < 0.98\)) are different from EuAs3 in magnetic properties. Figure 1(d) represents the magnetic phase diagram of Eu(As0.02P0.98)3.5) The IC sine-wave amplitude-modulated phase for \(T_{\text{L}}<T<T_{\text{N}}\) is the same as that in EuAs3, but the helical phase (IC-P1) with magnetic moments modulated in the \(b^{*}\)-plane appears below \(T_{\text{L}}\). This phase is reported to be similar to the antiferromagnetic component of the SF1 phase of EuAs3. There appear two field-induced magnetic phases named SF-P1 and SF-P2 below \(H_{\text{c}}\), which are IC phases. The magnetic moments are also modulated in the \(b^{*}\)-plane in SF-P1 and SF-P2. Later, the helimagnetic structure of Eu(As0.2P0.8)3 at zero fields was studied again by zero-field neutron polarimetric measurements.7) The previous magnetic structure model in which Eu moments are confined to the \(b^{*}\)-plane is slightly corrected, exhibiting a small spin component parallel to the \(b^{*}\)-axis.7)

Here, we note that there exist two types of compound for EuP3. One is the semimetal α-EuP3 with \(T_{\text{N}}=8.2\) K, which is isostructural to EuAs3.10) This compound is synthesized under high pressure and/or is thermodynamically stable above 960 K. The other compound is an insulator of β-EuP3 with \(T_{\text{N}} = 10\) K and is slightly different from EuAs3 in crystal structure. This compound can be grown under ambient pressure. β-EuP3 is, however, changed to EuAs3 in the crystal structure upon alloying a very small amount of As into β-EuP3. Indeed, the crystal structure of Eu(As0.02P0.98)3 in Fig. 1(d) is the same as that of EuAs3.

As mentioned above, the H–T phase diagram of EuAs3 is surprisingly complex and is composed of several collinear and noncollinear phases. The present monoclinic structure is, however, centrosymmetric, and thus, there is no Dzyaloshinsky–Moriya interaction. The magnetism is governed by the RKKY interaction, involving anisotropic magnetic interactions, as mentioned above. In addition, EuAs3 and Eu(As\(_{1-x}\)Px)3 (\(x<0.98\)) compounds are semimetals, and a carrier number of \(5\times 10^{20}\) cm−3 was simply estimated by the Hall effect measurements of EuAs3.6)

The purpose of the present study is to clarify the characteristics of the magnetic phases of EuAs3 and Eu(As\(_{1-x}\)Px)3 (\(0.7<x<0.97\)) from the measurements of magnetization, magnetoresistance, and Hall resistivity in magnetic fields. These measurements are helpful, for example, to find new magnetic phases such as magnetic skyrmions in ecent studies of EuPtSi11,12) and Gd2PdSi3,13,14) or the cycloidal magnetic ordering in EuIrGe3.15,16) The second purpose is to clarify the Fermi surface properties of the semimetal EuAs3 by dHvA experiments to understand conduction electrons.

2. Experimental Procedure

We grew single crystals of EuAs3 by both the Bi-flux and Bridgman methods. In the case of the Bi-flux method, the starting materials of 3N (99.9% pure)–Eu, 5N–As, and 5N–Bi with the constitution of \(1.2:3:30\) were inserted into an alumina crucible. The crucible was encapsulated in a quartz tube, heated to 500 °C for 72 h, kept at 500 °C for 48 h, heated again to 700 °C for 72 h, kept at 700 °C for 24 h, cooled gradually to 400 °C for 192 h, and cooled quickly to room temperature. The excess Bi flux was removed by spinning the ampoule in a centrifuge. Single crystals are small, \(0.7\times 0.8\times 1.5\) mm3, as shown in Fig. 2, where the long side direction corresponds to the \(b^{*}\)(b)-axis.


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Figure 2. (Color online) EuAs3 ingot with a size of \(0.7\times 0.8\times 1.5\) mm3, which was grown by the Bi-flux method. The long side direction corresponds to the \(b^{*}\)(b)-axis.

In the case of the Bridgman method, first, we synthesized polycrystals of EuAs3. Namely, the Eu and As materials with the constitution of \(1:3\) were encapsulated in the quartz tube, which was heated to 750 °C over 144 h, kept at 750 °C for 96 h, and cooled to room temperature. The obtained polycrystals of EuAs3 were inserted into the alumina crucible in a glove box in Ar atmosphere and encapsulated in the quartz tube. The quartz tube was heated to 840 °C over 18 h, cooled gradually to 750 °C over 72 h, and finally cooled to room temperature. Note that the melting point of EuAs3 is 800 °C.17) Single crystals of EuAs3 were thus obtained by the Bi-flux and Bridgman methods. Both compounds have high quality, but large single-crystal ingots with a diameter of about 5 mm and a length of 30 mm were obtained by the Bridgman method and used in the present magnetization, magnetoresistance, and Hall resistivity measurements. The chemical transport method using I2 was not used for growing single crystals of EuAs3.

The directions of single crystal samples were determined by the X-ray Laue method. Lattice parameters of the present EuAs3 single crystals were determined by X-ray analysis at room temperature using a tiny single-crystal sample in order to minimize the absorption and secondary extinction effects. The determined values are \(a=9.4723\) Å, \(b=7.6050\) Å, \(c=5.7893\) Å, \(\alpha=90\)°, \(\beta=112.56\)°, and \(\gamma=90\)°, which are in good agreement with the previous results mentioned in Introduction.

Note that the single crystals of EuAs3 and Eu(As\(_{1-x}\)Px)3 are oxidized rather quickly in air. We determined the direction of each sample using a cleaved \(c^{*}\)-plane. Important in the present experiments is the \(b^{*}\)-direction of the magnetic field. Namely, the antiferromagnetic easy-axis corresponds to the \(b^{*}\)-direction. Therefore, the current direction in the magnetoresistance and Hall resistivity measurements is either the a- or \(c^{*}\)-axis, as shown in Fig. 1, which was determined by the cleaved plane of ingots.

Similarly, we grew polycrystals of β-EuP3. In the case of the Bridgman method for β-EuP3, the highest temperature was 930 °C. Single crystals of Eu(As0.2P0.8)3, for example, were obtained from EuAs3 and EuP3 single crystals with the constitution of \(2:8\). These single crystals were crushed into powder, homogeneously mixed, inserted into the alumina crucible, and encapsulated in the quartz tube. The highest temperature in the Bridgman method was the same as that for β-EuP3, namely, 930 °C. The lattice parameters of these single-crystal samples of Eu(As\(_{1-x}\)Px)3 (\(x = 0.7\), 0.75, 0.8, and 0.97) are summarized in Table I. For example, the measured constitution x for the nominal constitution \(x = 0.7\) is roughly in the range from 0.64 to 0.76. The samples of Eu(As0.03P0.97)3 are surprisingly homogeneous, and the determined constitution ranges from 0.96 to 0.97. The obtained volumes V (\(= abc\sin\beta\)) of Eu(As\(_{1-x}\)Px)3 samples are plotted in Fig. 3 as a function of the measured x. V follows Vegard's law. Hereafter, we describe Eu(As\(_{1-x}\)Px)3 compounds using the nominal constitution x. Note that α-EuP3 with the same crystal structure as EuAs3 can be obtained in metastable form by quenching the ingots from about 750 °C, where the melting point of EuP3 is higher than that of EuAs3 (800 °C) and close to 900 °C. It is, however, interesting to note that the α-phase is stabilized at room temperature by adding a small amount of As (2%), as shown in Fig. 1(d).


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Figure 3. Constitution x dependence of unit cell volume V of Eu(As\(_{1-x}\)Px)3.

Data table

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Table I. Lattice parameters, nominal and measured constitutions of Eu(As\(_{1-x}\)Px)3 single crystals at room temperature.

Magnetic susceptibility and magnetization were measured using a commercial superconducting quantum interference device (SQUID) magnetometer. The Hall resistivity and magnetoresistance measurements were carried out by the AC/DC method. dHvA experiments were carried out by the standard field modulation method.

3. Experimental Results and Analyses
Magnetic properties

Figures 4(a) and 4(b) show the temperature dependences of the electrical resistivity ρ and the Hall resistivity \(\rho_{yx}\) for the \(J\parallel \text{$a$-axis}\) and \(H\parallel \text{$b^{*}$($b$)-axis}\), and the \(J\parallel \text{$b^{*}$($b$)-axis}\) and \(H\parallel [\bar{2}01]\) at the magnetic field of \(\mu_{0}H=1\) T, respectively. Almost T-linear temperature dependences of the electrical resistivities in Fig. 4(a) are characteristic of Eu-divalent compounds.18) This is because of the absence of \(4f\)-level splitting by the crystalline electric field for the Eu2+-localized \(4f\)-electrons. The resistivity data at low temperatures exhibit two magnetic transitions at the Néel temperature \(T_{\text{N}}=11.2\) K and the first-order magnetic phase transition temperature \(T_{\text{L}}=10.4\) K, as indicated by arrows in the inset of Fig. 4(a). These transitions are also reflected in the Hall resistivity measurement, as shown in Fig. 4(b). The Hall resistivity differs in sign and magnitude for two configurations of the current and magnetic field, indicating the existence of electrons and holes as the carriers of conduction electrons. We can estimate the carrier numbers from the Hall resistivity at room temperature, following the simple one-carrier model: \(5.9\times 10^{20}\) electrons/cm3 from \(\rho_{yx}=-1.06\) µΩ·cm and \(3.7\times 10^{20}\) holes/cm3 from \(\rho_{yx}=1.7\) µΩ·cm at 1 T. These results are approximately consistent with the previous results.6)


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Figure 4. (Color online) Temperature dependences of the (a) electrical resistivity ρ for \(J\parallel a\)- and \(b^{*}\)-axes and (b) Hall resistivity \(\rho_{yx}\) for \(J\parallel \text{$a$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\), and \(J\parallel \text{$b^{*}$-axis}\) and \(H\parallel [\bar{2}01]\) at \(\mu_{0}H=1\) T in EuAs3.

Next, we measured the magnetization M, magnetoresistance \(\rho_{xx}\), and Hall resistivity \(\rho_{yx}\) for two configurations: \(H\parallel \text{$b^{*}$-axis}\) and \(H\parallel [\bar{2}01]\). As mentioned in Introduction, the magnetic moments are antiferromagnetically oriented along the \(b^{*}\)-axis; therefore, the field direction of the \(H\parallel \text{$b^{*}$-axis}\) corresponds to the antiferromagnetic easy-axis. On the other hand, \(H\parallel [\bar{2}01]\) corresponds to the magnetic hard-axis. Namely, Eu atoms are situated along the \((\bar{2}01)\) plane, forming a distorted hexagon, as shown in Fig. 5, and stacked antiferromagnetically along the \([\bar{2}01]\) direction. Note that the 4th layer in Fig. 5 is the same as the 1st layer. The \([\bar{2}01]\) direction corresponds to the hard-axis of magnetization, and the magnetic moments are expected to be simply canted toward \([\bar{2}01]\) for \(H\parallel [\bar{2}01]\).


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Figure 5. (Color online) Crystal structures of EuAs3 in (a) \(b^{*}\)(\(b\))-plane and (b) \((\bar{2}01)\) plane, where the 4th layer is the same as the 1st layer.

We show in Figs. 6(a)–6(d) the easy-axis magnetization M and field derivative of the magnetization \(dM/dH\) for the \(H\parallel \text{$b^{*}$-axis}\), and the corresponding magnetoresistance \(\rho_{xx}\) and Hall resistivity \(\rho_{yx}\) for the \(J\parallel \text{$a$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at several selected temperatures. The M and \(dM/dH\) curves at \(T=2\) K show the metamagnetic transitions at \(H_{\text{SF1}}\), \(H_{\text{SF2}}\), \(H_{\text{SF3}}\), and \(H_{\text{SF4}}\), and M saturates at \(H_{\text{c}}\), as indicated by arrows in Figs. 6(a) and 6(b). Note that the M curves indicate hystereses at \(H_{\text{SF1}}\), \(H_{\text{SF2}}\), and \(H_{\text{SF3}}\). The corresponding data of the magnetoresistance \(\rho_{xx}\) and Hall resistivity \(\rho_{yx}\) indicate steps and/or dips, as shown in Figs. 6(c) and 6(d), respectively. The Hall conductivity \(\sigma_{xy}\) and the magnetic phase diagram, which approximately corresponds to Fig. 1(c), are shown in Figs. 6(e) and 6(f), respectively. Here, the Hall conductivity \(\sigma_{xy}\) was calculated using \(\rho_{xx}\) and \(\rho_{yx}\), following the relation \(\sigma_{xy}=\rho_{yx}/(\rho_{xx}^{2}+\rho_{yx}^{2})\).


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Figure 6. (Color online) Magnetic field dependences of the (a) magnetization M and (b) field derivative of magnetization \(dM/dH\) for \(H\parallel \text{$b^{*}$-axis}\), and (c) magnetoresistance \(\rho_{xx}\) and (d) Hall resistivity \(\rho_{yx}\) for \(J\parallel \text{$a$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at selected temperatures. (e) Corresponding Hall conductivity \(\sigma_{xy}\) and (f) magnetic phase diagram for \(H\parallel \text{$b^{*}$-axis}\) in EuAs3.

Since EuAs3 is the semimetal, a simple \(H^{2}\) dependence of the transverse magnetoresistance is expected at high magnetic fields. The anomalous Hall resistivity is proportional to the magnetization M, together with the normal Hall resistivity, which is proportional to H. The M curve at \(T=2\) K shows steplike increases at \(H_{\text{SF1}}\) and \(H_{\text{SF2}}\), revealing the metamagnetic transitions. Correspondingly, the \(\rho_{xx}\) and \(\rho_{yx}\) curves indicate steplike increase and decrease at \(H_{\text{SF1}}\) and \(H_{\text{SF2}}\), respectively, as shown in Figs. 6(c) and 6(d). It is very interesting that the M curve at \(T=2\) K shows only small anomalies at \(H_{\text{SF3}}\) and \(H_{\text{SF4}}\), as observed in \(dM/dH\) in Fig. 6(b), whereas the corresponding \(\rho_{xx}\) and \(\rho_{yx}\) curves indicate distinct steplike decrease and increase, respectively, at these transitions. These are characteristics for the \(H\parallel \text{$b^{*}$-axis}\), especially emphasizing the existence of the phase SF3.

Similarly, we measured the hard-axis magnetization M for \(H\parallel [\bar{2}01]\) and the magnetoresistance \(\rho_{xx}\) and Hall resistivity \(\rho_{yx}\) for the \(J\parallel \text{$b^{*}$-axis}\) and \(H\parallel [\bar{2}01]\) at several selected temperatures, as shown in Figs. 7(a), 7(c), and 7(d), respectively.


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Figure 7. (Color online) Magnetic field dependences of the (a) magnetization M and (b) field derivative of magnetization \(dM/dH\) for \(H\parallel [\bar{2}01]\), and (c) magnetoresistance \(\rho_{xx}\) and (d) Hall resistivity \(\rho_{yx}\) for \(J\parallel \text{$b^{*}$-axis}\) and \(H\parallel [\bar{2}01]\) at selected temperatures. (e) Corresponding Hall conductivity \(\sigma_{xy}\) and (f) magnetic phase diagram for \(H\parallel [\bar{2}01]\) of EuAs3. The inset of (c) shows the Shubnikov–de Haas oscillations of branch δ at \(T=2\) K.

The field derivative of the magnetization \(dM/dH\), the Hall conductivity \(\sigma_{xy}\), and the magnetic phase diagram are shown in Figs. 7(b), 7(e), and 7(f), respectively. For \(H\parallel [\bar{2}01]\), there exist three transitions at \(H^{*}\), \(H_{\text{L}}\), and \(H_{\text{c}}\). A small change in M is observed at \(H^{*}\), and a clear metamagnetic transition occurs at \(H_{\text{L}}\), which is connected to \(T_{\text{L}}\) at zero magnetic field. Note that the Shubnikov–de Haas oscillations for the branch δ are observed in \(\rho_{xx}\) at \(T=2\) K, which are shown in the inset in Fig. 7(c) with enlarged scales. In the later section, we show the results of dHvA experiments. The present magnetoresistance for the hard-axis direction of \(H\parallel [\bar{2}01]\) in Fig. 7(c) indicates the usual \(H^{2}\) dependence, which is characteristic of semimetals. On the other hand, the magnetoresistance for the easy-axis direction of \(H\parallel b^{*}\) in Fig. 6(c) indicates a huge drop of resistivity in the \(H_{\text{FS3}}\)–\(H_{\text{FS4}}\) region, which is characteristic of the magnetic SF3 phase.

Here, we note the Hall conductivity. In the case of the anomalous Hall effect, the Hall conductivity \(\sigma_{xy}\) is proportional to the magnetization M, and the upper limit of \(|\sigma_{xy}|\) is on the order of \(10^{3}\) (Ω·cm)−1.19) However, the field dependence of \(\sigma_{xy}(H)\) in Fig. 7(e) shows a significantly large negative peak at \(\mu_{0}H = 0.5\) T; \(\sigma_{xy}\ (0.5\,\text{T}) = -17\times 10^{-2}\ \text{($\unicode{x00B5}{}\Omega{\cdot}$cm)$^{-1}$}= -1.7\times 10^{5}\ (\text{$\Omega{\cdot}$cm)$^{-1}$}\) at 2 K, which is different from the magnetization curve in Fig. 7(a), and thus cannot be understood from the conventional anomalous Hall effect. Instead, the peak structure with an extremely large magnitude of \(\sigma_{xy}\) can be well explained by the one-carrier Drude model with high mobility, \(\sigma_{xy}(\mu_{0}H) = -\mu ne\cdot\frac{\mu (\mu_{0}H)}{1+[\mu (\mu_{0}H)]^{2}}\), where μ and n represent the carrier mobility and carrier number, respectively, as shown in Fig. 8. Note that \(\sigma_{xy}\) is proportional to \(\mu_{0}H\) at low fields and decreases in the form of \(1/(\mu_{0}H)\) at high fields. Here, the peak position is given as follows: \(\sigma_{xy} = -\mu ne/2\) at \(\mu_{0}H =\mu^{-1}\). Using this model, we can estimate the mobility and carrier number of EuAs3 to be \(\mu\sim 2\times 10^{4}\) cm2 v−1 s−1 and \(n\sim 1.1\times 10^{20}\) cm−3, respectively. The obtained carrier number is not far from the value estimated from the Hall resistivity at 1 T and room temperature (see Fig. 4). Therefore, the Hall conductivity of EuAs3 at a low temperature is governed by a normal Hall component with high mobility. However, some additional kinks in the \(\sigma_{xy}\) of EuAs3 \((J\parallel a, H\parallel b^{*})\), as shown in Fig. 6(e) at the field-induced phase boundaries cannot be simply explained by the above Drude model probably owing to changes in electronic states.


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Figure 8. (Color online) Hall conductivity based on single-carrier Drude model for EuAs3 in Fig. 7(e).

Next, we measured the electrical resistivity (ρ) values of Eu(As0.3P0.7)3, Eu(As0.25P0.75)3, and Eu(As0.2P0.8)3 for the \(J\parallel \text{$c^{*}$-axis}\) and the Hall resistivity (\(\rho_{yx}\)) values of Eu(As0.2P0.8)3 for the \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\), as shown in Fig. 9. The ρ values are from one to two orders of magnitude larger than the resistivity of EuAs3. This is mainly due to the small carrier number of these compounds. The carrier number was estimated to be \(8\times 10^{18}\)/cm3 from the Hall resistivity at room temperature for Eu(As0.2P0.8)3, as shown in Fig. 9(d), using the one-carrier model. The Néel and lock-in temperatures for Eu(As0.2P0.8)3 are \(T_{\text{N}}=9.2\) K and \(T_{\text{L}}=7.8\) K, respectively, as shown in the inset of Fig. 9(c), which are approximately the same as the previous data for \(T_{\text{N}}=9.0\) K and \(T_{\text{L}}=7.7\) K.3)


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Figure 9. Temperature dependences of the electrical resistivities ρ of (a) Eu(As0.3P0.7)3, (b) Eu(As0.25P0.75)3, and (c) Eu(As0.2P0.8)3 for \(J\parallel \text{$c^{*}$-axis}\) and (d) Hall resistivity \(\rho_{yx}\) of Eu(As0.2P0.8)3 for \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at \(\mu_{0}H=0.5\) T.

We also measured the magnetization M, magnetoresistance \(\rho_{xx}\), and Hall resistivity \(\rho_{yx}\) for the \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at several selected temperatures in Eu(As0.2P0.8)3, as shown in Figs. 10(a), 10(c), and 10(d), respectively. It is remarkable that \(\rho_{xx}\) at \(T=2\) K drops steeply at \(H_{\text{SF-P1}}\) and increases slightly above \(H_{\text{c}}\). Correspondingly, an anomalously large peak structure of \(\rho_{yx}\) is observed in the field range from \(H_{\text{SF-P1}}=1.29\) T to \(H_{\text{SF-P2}}=2.05\) T at \(T=2\) K, which corresponds to the peak structure of \(dM/dH\) shown in Fig. 10(b). Figures 10(e) and 10(f) show the Hall conductivity \(\sigma_{xy}\) and the magnetic phase diagram, respectively. The observed phase diagram is similar to that shown in Fig. 1(d).


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Figure 10. (Color online) Magnetic field dependences of (a) the magnetization M and (b) field derivative of magnetization \(dM/dH\) for \(H\parallel \text{$b^{*}$-axis}\), and (c) magnetoresistance \(\rho_{xx}\) and (d) Hall resistivity \(\rho_{yx}\) for \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at several selected temperatures. (e) The corresponding Hall conductivity \(\sigma_{xy}\) and (f) magnetic phase diagram for \(H\parallel \text{$b^{*}$-axis}\) of Eu(As0.2P0.8)3.

Similar experiments were carried out for Eu(As0.03P0.97)3. We show in Figs. 11(a)–11(d) the temperature dependences of the electrical resistivity ρ for the \(J\parallel \text{$c^{*}$-axis}\), the Hall resistivity \(\rho_{yx}\) for the \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at \(\mu_{0}H=0.5\) T, the magnetic susceptibility χ and inverse susceptibility \(1/\chi\) for the \(H\parallel \text{$b^{*}$-axis}\) at \(\mu_{0}H=1\) T, respectively. The electrical resistivity at room temperature is \(\rho_{\text{RT}}=12\) mΩ·cm, which is larger than \(\rho_{\text{RT}}=4.2\) mΩ·cm for Eu(As0.3P0.7)3 shown in Fig. 9(a), although the compound Eu(As0.03P0.97)3 is metallic, revealing the monotonic decrease in ρ with decreasing temperature, and the residual resistivity \(\rho_{0}=2.5\) mΩ·cm is almost the same between two compounds. The characteristic feature is a sharp peak of ρ at \(T_{\text{N}}=8.1\) K and \(T_{\text{L}}=7.3\) K, as shown in the inset of Fig. 11(a). Note that the resistivity of β-EuP3, which can be grown under the same conditions as Eu(As0.03P0.97)3, is \(\rho_{\text{RT}}=0.1\) Ω·cm, which increases with decreasing temperature, revealing an insulating feature.1) The Hall resistivity in Fig. 11(b) indicates a sharp dip at around \(T_{\text{N}}\). The χ data show the Néel temperature \(T_{\text{N}}=8.0\) K and the lock-in temperature \(T_{\text{L}}=7.2\) K, as shown in the inset of Fig. 11(c). We estimated \(\mu_{\text{eff}}=7.89\) \(\mu_{\text{B}}\)/Eu and \(\Theta_{\text{P}}=10.7\) K from a fitting of the Curie–Weiss law to the \(1/\chi\) data, as shown in Fig. 11(d).


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Figure 11. Temperature dependences of (a) the electrical resistivity ρ for \(J\parallel \text{$c^{*}$-axis}\), (b) Hall resistivity \(\rho_{yx}\) for \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\), (c) magnetic susceptibility, and (d) invserse magnetic susceptibility for \(H\parallel \text{$b^{*}$-axis}\) of Eu(As0.03P0.97)3. The inset of (c) is the magnetic susceptibility at \(\mu_{0}H=0.3\) T in enlarged scales. The solid line in (d) is a fitting result of the Curie–Weiss law.

We also measured the magnetization M, the field derivative of the magnetization \(dM/dH\), the magnetoresistance \(\rho_{xx}\), and the Hall resistivity \(\rho_{yx}\) at several selected temperatures for Eu(As0.03P0.97)3, as shown in Fig. 12. A large hysteresis of M is observed at \(H_{\text{SF-P1}}\) and \(T=2\) K, as shown in Figs. 12(a) and 12(b). Interestingly, a steep drop of ρ is observed at \(H_{\text{SF-P1}}\), as shown in Fig. 12(c), together with a huge peak structure of \(\rho_{yx}\) between \(H_{\text{SF-P1}}\) and \(H_{\text{SF-P2}}\). These characteristics are the same as those of Eu(As0.2P0.8)3 shown in Fig. 10, but the magnitudes are extremely large for Eu(As0.03P0.97)3. The field dependences of the Hall conductivity \(\sigma_{xy}\) and the magnetic phase diagram are shown in Figs. 12(e) and 12(f), respectively.


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Figure 12. (Color online) Magnetic field dependences of the (a) magnetization M and (b) field derivative of magnetization \(dM/dH\) for \(H\parallel \text{$b^{*}$-axis}\), and (c) magnetoresistance \(\rho_{xx}\) and (d) Hall resistivity \(\rho_{yx}\) for \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at several selected temperatures. (e) Corresponding Hall conductivity \(\sigma_{xy}\) and (f) magnetic phase diagram for \(H\parallel \text{$b^{*}$-axis}\) of Eu(As0.03P0.97)3.

It is interesting to compare the present data for Eu(As0.03P0.97)3 with those for EuAs3, as shown in Figs. 13 and 14. The temperature dependences of the electrical resistivities of the two compounds in Fig. 13 are almost the same, but the magnitudes are markedly different between these compounds. The characteristic observations are the marked changes in the magnetoresistance \(\rho_{xx}\) and Hall resistivity \(\rho_{yx}\) between \(H_{\text{SF-P1}}\) and \(H_{\text{SF-P2}}\) for Eu(As0.03P0.97)3 and between \(H_{\text{SF3}}\) and \(H_{\text{SF4}}\) for EuAs3, as shown in Fig. 14.


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Figure 13. (Color online) Temperature dependences of the electrical resistivities of EuAs3 for \(J\parallel \text{$a$-axis}\) and Eu(As0.03P0.97)3 for \(J\parallel \text{$c^{*}$-axis}\).


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Figure 14. (Color online) Magnetic field dependences of the (a) electrical resistivity \(\rho_{xx}\) and (b) Hall resistivity \(\rho_{yx}\) of EuAs3 for \(J\parallel \text{$a$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\), and those of Eu(As0.03P0.97)3 for \(J\parallel \text{$c^{*}$-axis}\) and \(H\parallel \text{$b^{*}$-axis}\) at \(T=2\) K.

dHvA effect

We carried out the dHvA experiments using a small single crystal with the size of \(0.7\times 0.8\times 1.5\) mm3, which was grown by the Bi-flux method. Figures 15(a) and 15(b) show typical dHvA oscillations for the \(H\parallel \text{$b^{*}$-axis}\) and the corresponding fast Fourier transformation (FFT) spectrum in the field range from 7.2 to 14.7 T, respectively. The main dHvA branches are named α with the dHvA frequency \(F_{\alpha}\simeq 1.1\) kT and branch γ with \(F_{\gamma}\simeq 0.4\) kT. Here, the dHvA frequency is proportional to the maximum or minimum cross-sectional area \(S_{\text{F}}\) of the Fermi surface, \(F=c\hbar S_{\text{F}}/2\pi e\). Branch \(2\alpha\) is a second harmonic.


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Figure 15. (a) dHvA oscillations and (b) the corresponding FFT spectrum in EuAs3 grown by the Bi-flux method. (c) Corresponding angular dependences of dHvA frequencies.

Figure 15(c) shows the angular dependences of dHvA frequencies in the field direction from the \(H\parallel \text{$b^{*}$-axis}\) to the \(c^{*}\)-axis. Branch α is spin-split because the dHvA experiments were carried out in the field-induced ferromagnetic (paramagnetic) state of \(H>H_{\text{c}}\). Each spin-split branch α is split further into two branches when the field is tilted from the \(b^{*}\)-axis to the \(c^{*}\)-axis. It is speculated that there appear maximum and minimum cross sections in the Fermi surface for this field angle process. Namely, the Fermi surface has a dumbbell-like or peanut-like shape.

We also carried out the same dHvA experiments using a large single crystal sample with the size of \(1.4\times 1.3\times 4\) mm3, which was fit to the sample space of dHvA detecting coil 2 mm in diameter and 5 mm in length. Figure 16 shows the typical dHvA oscillations and the corresponding FFT spectrum for the \(H\parallel \text{$b^{*}$-axis}\). The angular dependences of the dHvA frequencies for the \(H\parallel \text{$b^{*}$-axis}\) to the \(c^{*}\)-axis and for the \(b^{*}\)-axis to the \(a^{*}\)-axis are shown in Figs. 17(a) and 17(d).


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Figure 16. (a) dHvA oscillations and (b) the corresponding FFT spectrum in EuAs3 grown by the Bridgman method.


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Figure 17. (Color online) (a, d) Angular dependence of dHvA frequency, and the proposed (b) electron and (e) hole Fermi surfaces of EuAs3. The red and blue solid lines in (a) and (d) indicate the angular dependences of the dHvA frequencies based on the (b) electron and (e) hole Fermi surfaces, respectively. The electron Fermi surfaces are assumed to have a peanut-like shape in (b′), which theoretically locate at the T point in (c), as reported in Ref. 23.

We considered that FFT branches α, β, γ, and δ are fundamental and the others are harmonics such as the sum of γ and δ. This is because the dHvA frequency and cyclotron effective mass for the \(H\parallel \text{$b^{*}$-axis}\) are approximately consistent with the sum of branches γ and δ: \(F_{\gamma +\delta}=0.493\) kT and \(m_{\text{c}}^{*}=0.79m_{0}\) were obtained for branch \(\gamma +\delta\) by considering 0.406 kT (\(0.23m_{0}\)) and 0.368 kT (\(0.26m_{0}\)) for branch γ, and 0.106 kT (\(0.69m_{0}\)) for branch δ. We summarize in Table II the fundamental dHvA frequencies and corresponding cyclotron masses for the \(H\parallel \text{$b^{*}$-axis}\). Here, the cyclotron mass \(m_{\text{c}}^{*}\) was determined from the temperature dependence of dHvA amplitude.

Data table

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Table II. dHvA frequencies F and the corresponding cyclotron effective masses \(m_{\text{c}}^{*}\) for \(H\parallel \text{$b^{*}$-axis}\) in EuAs3.

From our previous studies, we found that the Fermi surface of Eu compounds is well explained by the corresponding Sr compounds, namely, SrBi3 for EuBi320) and SrPtSi for EuPtSi.21) The Fermi surfaces of EuAs3 are thus similar to those of SrAs3. From the recent results of band calculation for SrAs322) and EuAs3,23) we considered that branches α and \(\alpha'\) correspond to the electron Fermi surfaces and branches β and γ are hole Fermi surfaces. Here, we do not discuss the small spherical Fermi surface named branch δ. Branches α and \(\alpha'\) in Fig. 17(b) are approximately ellipsoids of revolutions in the Fermi surfaces whose ellipsoidal axes are perpendicular to the \(a_{0}{}^{*}\)–\(b_{0}{}^{*}\) plane.

We analyzed the topologies of the Fermi surfaces on the basis of the experimental angular dependences of dHvA frequencies, as performed in the studies of PbTe and PbS.24) Our proposed angular dependences of the dHvA frequencies are shown as red and blue solid lines in Figs. 17(a) and 17(d), respectively. The corresponding Fermi surfaces are illustrated in Figs. 17(b), 17(b′), and 17(e). The volume of electron Fermi surfaces, \(V_{\text{e}}\), is equal to the volume of hole Fermi surfaces, \(V_{\text{h}}\), which corresponds to 1.2% of the volume of the Brillouin zone \(V_{\text{BZ}}\), namely, \(V_{\text{e}}=V_{\text{h}}=0.012V_{\text{BZ}}\). The theoretical electron Fermi surfaces named α and \(\alpha'\) are located at the T point in Fig. 17(c).

Here, we explain the details of our proposed Fermi surfaces on the basis of the results of dHvA experiments. Two red lines in Fig. 17(a) correspond to the spin-split electron Fermi surfaces named α and \(\alpha'\) shown in Fig. 17(b). The volumes are \(V_{\alpha}=0.013V_{\text{BZ}}\) and \(V_{\alpha'}=0.012V_{\text{BZ}}\). The volume of the electron Fermi surface is \(V_{\text{e}}=0.012V_{\text{BZ}}\) in the paramagnetic state. On the other hand, we simply assumed that the hole Fermi surfaces are flat cylinders. Two blue lines in Fig. 17(d) correspond to two spin-split Fermi surfaces named β and γ, which consist of two separated hole Fermi surfaces. The volume of two separated hole Fermi surfaces named β is \(V_{\beta}=2\) (\(0.0084V_{\text{BZ}}\)), and that named γ is \(V_{\gamma}=2\) (\(0.0035V_{\text{BZ}}\)). The volume of the hole Fermi surface is thus \(V_{\text{h}}=0.012V_{\text{BZ}}\) in the paramagnetic state. The volumes of the present electron and hole Fermi surfaces are \(V_{\text{e}}=V_{\text{h}}=0.012V_{\text{BZ}}\). Note that exact electron Fermi surfaces are peanut-like shape in Fig. 17(b′).

We estimated the carrier number of the present Fermi surfaces in Figs. 17(b) and 17(e). A unit cell contains four molecules of EuAs3 (\(z=4\)), and then the unit cell volume V is \(V=abc\sin\beta\). EuAs3, however, belongs to the base-centered monoclinic lattice whose volume is \(V_{0}=(1/2)abc\sin\beta\). The corresponding volume of the Brillouin zone is \(V_{\text{BZ}}=(2\pi)^{3}/V_{0}\). The present volumes of electron and hole Fermi surfaces, which are estimated from the angular dependences of the dHvA frequencies, are \(V_{\text{e}}=V_{\text{h}}=0.012V_{\text{BZ}}\). The carrier number is thus estimated to be \(n_{\text{e}}=n_{\text{h}}=0.012V_{\text{BZ}}/4\pi^{3}=1.2\times 10^{20}\)/cm3, which is roughly consistent with the carrier number estimated from the Hall coefficient.

4. Summary

We grew single crystals of EuAs3 by the Bi-flux and Bridgman methods, together with Eu(As\(_{1-x}\)Px)3 (\(x=0.3\), 0.25, 0.2, and 0.03) by the Bridgman method, and measured the magnetic susceptibility, magnetization, electrical resistivity, magnetoresistance, and Hall resistivity, as well as the dHvA oscillations. Experimental results are summarized as follows:

1) The antiferromagnet EuAs3 with the Néel temperature \(T_{\text{N}}=11.2\) K and the first-order magnetic phase transition, which is called the lock-in transition at \(T_{\text{L}}=10.4\) K, exhibits six magnetic phases. The magnetization shows four metamagnetic transitions at \(H_{\text{SF1}}\), \(H_{\text{SF2}}\), \(H_{\text{SF3}}\), and \(H_{\text{SF4}}\) and saturates at \(H_{\text{c}}\). The characteristic drop of the manetoresistance \(\rho_{xx}\) and a step-like increase in the Hall resistivity \(\rho_{yx}\) are observed between \(H_{\text{SF3}}\) and \(H_{\text{SF4}}\), namely, in the SF3 phase.

2) Similar antiferromagnets Eu(As\(_{1-x}\)Px)3 also exhibit four magnetic phases. The extremely large drop of the magnetoresitance \(\rho_{xx}\) at \(H_{\text{SF-P1}}\) and the marked dip structure between \(H_{\text{SF-P1}}\) and \(H_{\text{SF-P2}}\) in the Hall resistivity \(\rho_{yx}\) measurement are marked characteristics, especially in Eu(As0.03P0.97)3. Note that the lowest IC-P1 phase at zero magnetic field is helical in the magnetic structure. It is necessary to study further the present SF-P1 phase in Eu(As0.03P0.97)3 and the SF3 phase in EuAs3 by, for example, neutron scattering experiments.

3) We clarified the Fermi surfaces of EuAs3 by the dHvA experiments. In the field-induced ferromagnetic (paramagnetic) state, the detected branches α and \(\alpha'\) correspond to peanuts-like electron Fermi surfaces. Branches β and γ correspond to two-separated hole Fermi surfaces. The volume of electron Fermi surfaces, \(V_{\text{e}}\), is the same as that of hole Fermi surfaces, \(V_{\text{h}}\), revealing \(V_{\text{e}}=V_{\text{h}}=0.012V_{\text{BZ}}\), where \(V_{\text{BZ}}\) is the volume of the Brillouin zone. The carrier number is \(n_{\text{e}}=n_{\text{h}}=1.2\times 10^{20}\)/cm3. EuAs3 is indeed a semimetal.

Acknowledgments

One of the authors (Y.O.) would like to thank to Masashi Kakihana, Akiko Kikkawa, Yasujiro Taguchi, and Yoshinori Tokura for helpful support. This work was financially supported by KAKENHI (JP22K03517, JP22K03522, JP23H01841, JP23H04870, JP22H04933, JP20K20889).

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