Multiferroics, materials that exhibit ferroelectric and magnetic ordering, have been the focus of great attention over the last two decades. Cross-coupling of the electric and magnetic degree of freedom is quite interesting not only for applications but also for basic science[1-3].

Among multiferroic materials, hexagonal manganites R MnO₃ (R = rare earth) were found as research of multiferroic materials in the early stages and were classified as type I multiferroics. The symmetry of the crystal structure is reported as non-centrosymmetric P63cm. Under this symmetry, there are six irreducible representations of the magnetic structures. These are Γ₁ (P63cm), Γ₂ (P63c′m′), Γ₃ (P6′3cm′), Γ₄ (P6′3c′m), Γ₅ (P6′3) and Γ₆ (P6′3). Among them, Γ1−4 are one-dimensional irreducible representations and Γ5−6 are two-dimensional irreducible representations. Γ₁ (P63cm), Γ₂ (P63c′m′), Γ₃ (P6′3cm′), Γ₄ (P6′3c′m) have been used to describe the magnetic structures of hexagonal RMnO₃ with some exceptions.

A key parameter of this system is the atomic position of Mn. It is because the magnetic atoms Mn occupy (xMn,0,0), xMn 13 and equivalent sites, that is, they form a triangular lattice and interactions between Mn and Mn in the plane are antiferromagnetic, so magnetic frustration in this system is expected. If the position was exactly xMn=13, the intralayer exchange interactions are J1=J2 and the interlayer interactions J1c and J2c cancel out. If xMn,Fe>13, there are 4 J₁ and 2J₂ in the intralayer interactions while 2 J1c and 4 J2c in the interlayer interactions. On the other hand, if xMn,Fe<13, there are 2 J₁ and 4 J₂ in the intralayer interactions, while 4 J1c and 2 J2c in the interlayer interactions[4].

Interesting physics appears for this delicate parameter x_Mn. One is the so-called trimerization[5, 6]. The atomic position xMn 13 shows a large shift that can induce the coupling with the electric dipole moment at T_N in RMnO₃. Lee et al.[5] discussed that magneto-elastic coupling is the source of magnetoelectric coupling. The other is spin reorientation. HoMnO₃ and ScMnO₃ show spin reorientation at low temperature, whereas YbMnO₃ does not. The phenomenon occur at the crossover of xMn=13. x_Mn increases below T_N in HoMnO₃, while it decreases in ScMnO₃. The temperature of crossing xMn=13 matches the spin reorientation temperature. YbMnO₃ shows no spin reorientation and x_Mn remains lower than 13 over the whole temperature down to 1.5 K.

In this paper we focus on the hexagonal RFeO₃ (R = Lu, Sc, and Yb) which are also multiferroic materials and exhibit the same 2D-triangular lattice Heisenberg antiferromagnetic system (2D-THA) with P63cm. Hexagonal RFeO₃ compounds are rare, with only a few examples reported, such as Lu_0.5Sc_0.5FeO₃[7-9] and Yb_0.42Sc_0.58FeO₃[10]. The magnetic moment of Fe corresponds to S=52 while that of Mn is S=2, and comparing these helps to better understand the 2D-THA. Yb_0.42Sc_0.58FeO₃ shows successive magnetic phase transitions at TN 165 K, the spin reorientation temperature TSR 40 K, and the Yb³⁺ ordering temperature TYb 15 K[10]. The magnetic structure of Yb_0.42Sc_0.58FeO₃ was reported as Γ₁ + Γ₂ below T = 25 K. The weighting coefficients of the two irreducible representations were not reported but they are necessary. It is essential to accurately determine the magnetic structure of this system in order to accurately model inelastic neutron scattering data and capture magnetoelastic excitations, magnon-phonon interactions, and other related phenomena[11] Here, we report the magnetic structural analyses of Yb_0.42Sc_0.58FeO₃ and discuss the multiferroics of these two 2D-THAs, RMnO₃ and RFeO₃.

The Yb_0.42Sc_0.58FeO₃ polycrystalline samples were synthesized using the solid-state reaction method. High-purity Yb₂O₃, Sc₂O₃, and Fe₂O₃ powders were stoichiometrically mixed and ground. The mixed powders were pelletized and sintered at 1200 °C for 24 h and then at 1400 °C for 24 h in air with two intermediate grindings. The neutron powder diffraction patterns were collected on the high resolution powder diffractometer Echidna[12] installed on the Opal Research Reactor at ANSTO, Australia. The incident neutron wavelength of 2.4395 Å was selected by Ge (331) reflection at the take-off angle of 140 degrees. The sample was loaded in a 9 mm diameter vanadium can. The sample temperatures were at T = 3.5, 10, 25, 50, 100, 125, 150, 250, and 300 K achieved by using a closed cycle refrigerator with top loading.

We have performed a series of neutron scattering studies of Lu_1-yY_yMnO₃, Lu_1-ySc_yFeO₃, and Yb_1-ySc_yFeO₃ including inelastic neutron scattering experiments by using large single crystal samples of Lu_0.3Y_0.7MnO₃ and Lu_0.47Sc_0.53FeO₃. The magnetic structures of Lu_1-yY_yMnO₃ have been reported[13] and inelastic neutron scattering data of Lu_0.3Y_0.7MnO₃ and Lu_0.47Sc_0.53FeO₃ will be reported elsewhere.

This paper presents the powder neutron diffraction data and the magnetic structural analysis of Yb_0.42Sc_0.58FeO₃. The determined crystal structural parameters at 300 K are summarized in Table 1. The lattice parameters determined were a = 5.85912(2) Å and c = 11.69126(7) Å with the space group P63cm at T = 300 K. Yb and Sc are not distinguishable in neutron powder diffraction, so the composition was fixed. We assume that Yb and Sc are uniformly distributed. The sample contains 1.78(11)% ScFeO₃.

Table 1 . Refined structural parameters of Yb_0.42Sc_0.58FeO₃ at 300 K. The lattice parameters are a = 5.85912(2) Å and c = 11.69126(7) Å.

Atom (Wyck.)	x	y	z	occ.	B-iso
Yb1(2a)	0	0	0.2930(7)	0.42	0.39(3)
Sc1(2a)	0	0	0.2930(7)	0.58	0.39(3)
Yb2(4b)	1/3	2/3	0.2643(6)	0.42	0.39(3)
Yb2(4b)	1/3	2/3	0.2643(6)	0.58	0.39(3)
Fe1(6c)	0.3350(10)	0	0	1	0.24(4)
O1(6c)	0.3128(9)	0	0.1552(4)	1	0.19(4)
O2(6c)	0.6530(10)	0	0.3209(4)	1	0.19(4)
O3(2a)	0	0	0.510(2)	1	0.19(4)
O4(4b)	1/3	2/3	0.048(1)	1	0.19(4)

Figure 1 shows four representative powder neutron diffraction patterns of Yb_0.42Sc_0.58FeO₃. There are three magnetic phases. Magnetic R factors and χ2 values are shown in these figures.

Figure 1. (Color online) Four representative neutron powder diffraction patterns (a) T = 3.5 K (b) 50 K (c) 125 K (b) 300 K. Experimental data are shown as filled red circles, the fitted profile as a black line, and the difference between these two as a blue line. The peak positions are shown as tick marks.

Figure 2 shows the temperature dependence of the magnetic moment and the atomic position of Fe. The temperature dependence of the magnetic peak showed up below 200 K and reached the 3.7 μB / Fe atom at base temperature. The atomic position remains xFe=13 at most temperatures. Below T = 25 K, there is a slight change from xFe=13 but it changes again to xFe=13 by further decreasing the temperature. The weak trimerization appears at TYb= 25 K, which implies that interactions between R³⁺ and Fe³⁺ could be the cause of the trimerization in the Yb_0.42Sc_0.58FeO₃.

Figure 2. (a) Temperature dependence of magnetic moment of Fe (b) Temperature dependence of Fe atomic position (xFe,0,0).

The magnetic structures are fitted using the model below and shown in Fig. 3,

Figure 3. (Color online) Temperature dependence of coefficients of the magnetic structure of Yb_0.42Sc_0.58FeO₃.

Here, Γ₁ is P63cm, Γ₂ is P63c′m′ in the plane, and Γ2z has the magnetic moment along the c-axis. The irreducible representation Γ₂ accounts for all the magnetic peaks in the magnetically ordered phase below T_N = 150 K. In hexagonal RFeO₃ (R = Lu, Sc, Yb), a weak ferromagnetic moment is reported. The compatible irreducible representation is Γ₂. The magnetic (100) Bragg peak developed at T = 50 K, and another irreducible representation Γ₁ was introduced to solve the magnetic structure. Either Γ₁ or Γ₃ could contribute to the (100) peak, but only Γ₁ fits the pattern well. We observed rotations of magnetic moment in the plane by changing the ratio of Γ₁ and Γ₂. Spin reorientation is not as drastic as reported in the literature[4]. Interestingly, the magnetic moment of Fe with Γ2z is stable below T_N = 150 K to the base temperature. The magnetic moment of Yb orders below T = 25 K. The best fit showed that the magnetic moments of both Yb(2a) and Yb(4b) are along the c-axis but they are in opposite directions. The results are qualitatively consistent with[10].

In the hexagonal RFeO₃ (R = Lu, Sc, Yb), a weak ferromagnetic moment is reported[7, 8, 10]. So, the only compatible irreducible representation So, the only is Γ₂. In addition to Γ₂, Γ₁ is necessary to describe for powder neutron diffraction below T = 100 K. C1Γ1 + C2Γ2 is the correct magnetic structure of RFeO₃. Lu_1-yY_yMnO₃ adopts the Γ₄ because there is no ferromagnetic moment observed by the magnetization measurement. If we used Γ₄, the magnetic structure fit to the neutron powder diffraction data should be C3Γ3 + C4Γ4. In fact, the neutron diffraction and polarimetric study by Brown and Chatterji[14] reported the magnetic structure of YMnO₃ with Γ₆ (P6'3) with angle deviation from the axial directions being Ψ = 11 degrees from Γ₃ (P6'3cm'). Neutron scattering data show Lu_1-yY_yMnO₃ is C3Γ3 + C4Γ4[13] and Yb_0.42Sc_0.58FeO₃ is C1Γ1 + C2Γ2.

To uniquely determine the magnetic structure, there are several difficulities. As Katsufuji[6] notes, if there is no trimerization distortion, the nuclear (100) Bragg peak is forbidden. On the contrary, if there is trimerization distortion, both nuclear and magnetic scattering could contribute to the Bragg (100) peak. That makes it difficult to determine the crystal and magnetic structure precisely by neutron diffraction alone. A simple polarized neutron diffraction should show whether the Bragg (100) peak is a magnetic peak or not. By investigating the (h0l) plane with polarized neutrons, one can observe that the magnetic moment out of the plane to demonstrate that the Γ₂ phase exists. Magnetization measurement also supports the existence of weak ferromagnetism. However, to uniquely determine magnetic structures, even neutron polarimetry with cryopad[14] is inconclusive, since Γ₅ and Γ₆ are compatible with the results[14]. We would require a combination of measurements such as magnetization, second harmonic generation,[17] and polarized neutron scattering to be conclusive.

Determining the magnetic structure is an important issue because only Γ₁ and Γ₂ allow magnetoelectric coupling, while Γ₃ and Γ₄ do not[18]. In the case of Lu_1-yY_yMnO₃, neither Γ₃ nor Γ₄ should have magnetoelectric coupling, so multiferroicity should be through magnetoelastic coupling. On the other hand, Γ₁ or Γ₂ can have a magnetoelectric coupling that is consistent with[19,20]. Therefore, even though the crystal structure is similar in RFeO₃ and RMnO₃, there could be different mechanisms for multiferroicity. If the ferromagnetic component Γ2z is the source of multiferroicity of RFeO₃, the magnetic field might be able to control it by stabilizing the domain in the material. For example, the ferromagnetic moment of Lu_0.5Sc_0.5FeO₃ decreases while the sample is cooled, which means that the component or domain of Γ₂ is decreasing[7-9]. Further investigation of the weak ferromagnetic component and multiferroic properties is needed.

In another point, Fabréges[4] discussed the interlayer interactions that cause spin reorientation in the RMnO₃ system by showing the curvature of spin wave dispersion along (1,0,Ql) where the value of J1c−J2c is critical. However, the spin dynamics of (Lu,Sc)FeO₃[21] do not show a clear dispersion along the (1,0,Ql) even though a clear spin reorientation is observed in the RFeO₃. The spin reorientation of the RFeO₃ system could have a different origin than the the interlayer interaction of J1c and J2c. This could be because the Fe magnetic moment is larger than the Mn magnetic moment, so intralayer interactions J₁ and J₂ are 2.5 meV in RMnO₃ while J₁ and J₂ are 4.0 meV in RFeO₃. In addition, the ratio of the lattice parameters ca of RFeO₃ is typically larger compared to those of RMnO₃[9]. Interlayer interactions are less important to have spin reorientation in RFeO₃.

We show the coefficients of the magnetic structure differnet orderd sites of using two irreducible representations in this paper. The optimized magnetic structures of 2D-THA are necessary to understand magnetic excitations and multiferroic properties. They should be discussed based on the correct magnetic structures to extract precise parameters such as exchange parameters, magnetoelastic coupling, or magnetoelectric coupling in these RMnO₃ and RFeO₃ systems[19-21].

We used powder neutron diffraction to analyze Yb_0.42Sc_0.58FeO₃ and identified three magnetic phases. The structure requires two irreducible representations, Γ₁ and Γ₂, for explanation. These representations also apply to hexagonal-RMnO₃ and hexagonal-RFeO₃, but differences in magnetic symmetries suggest distinct origins for spin reorientation and multiferroicity. The weak ferromagnetic moment of Fe may be pivotal for ferroelectricity.

S.Y. is financially supported by the National Science and Technology Council, Taiwan, with Grants No.110-2112-M-213-013, 111-2112-M-213-023, and 112-2112-M-213-019. The work at NJIT was supported by the US Department of Energy under Grant No. DOE: DE- SC0021188. The experiments were carried out under the ANSTO user program (MI17467:Echidna).