Symmetry, magnetic transitions and multiferroic properties of B-site-ordered A₂MnB′O₆ perovskites (B′ = [Co, Ni])

This article is part of a focused issue on Magnetic Structures.

The presence of magnetic atoms at the A and B sites and the coupling between these two spin subsystems in perovskites gives rise to a variety of exciting effects. In particular this coupling attracts interest from the field of novel multiferroic and magnetoelectric oxides. Moreover, magnetic double perovskites presenting cationic order at the B sites incorporate an additional modulation that can favor symmetry breaking, multiferroic, magnetoelectric and polar phases. Here, we describe the magnetic structures obtained from neutron diffraction and analyze the symmetry properties of well ordered A₂MnB′O₆ double perovskites with A = Lu, Yb, Tm, Er, Ho, Y, Tb, La_0.5Tb_0.5, La and B′ = Co or Ni. A rich variety of magnetic orders is formed that have been identified and described, and their symmetry properties are discussed in relation to the multiferroic and magnetoelectric properties of the different compounds.

1. Introduction

Rare earth double perovskites of chemical formula A₂BB′O₆ (A = trivalent cations as rare earth atoms; B,B′ = transition metals with different charge and/or different ionic radius) have been investigated in recent years as potential candidates to exhibit multiferroic and magnetoelectric properties. In samples with perfect B-site cation order, first principles density functional theory calculations predicted an E-type magnetic ground state for some BB′ = NiMn double perovskites such as Y₂MnNiO₆ (Kumar et al., 2010). The so-called E-type is an important type of collinear order characterized by up-up-down-down spin chains along a particular crystallographic direction. Interestingly, it is an example of collinear order that breaks inversion symmetry and can induce magnetoelectric and multiferroic properties. It requires competing magnetic interactions [ferromagnetic (FM) nearest neighbor and antiferromagnetic (AFM) next-nearest neighbor exchanges]. The competition is enhanced in more distorted perovskites, and Monte Carlo calculations predicted that, even in absence of spin-orbit coupling, the coupling of the E-type magnetic modulation to the buckling distortions of oxygen octahedra gives rise to a polar axis perpendicular to the magnetic order direction (Sergienko et al., 2006). This magnetic ordering along with electric polarization was first observed in orthorhombic HoMnO₃ (Kimura et al., 2003). In orthorhombic multiferroic AMnO₃ manganites, the E-type AFM order is in competition with non-collinear spiral orders that also induce ferroelectricity. So, in E-type simple perovskites the exchange striction can modulate the B—O—B bond angles through O atom displacements. The polarization direction can be different in other E-type scenarios, such as, for example, the quasi-one-dimensional Co–Mn chains of Ca₃CoMnO₆, where the new modulation is due to ionic displacements of the magnetic atoms parallel to the direction of the E-type chains (Dong et al., 2015).

A₂BB′O₆ double perovskites are characterized by the possibility of competitive magnetic interactions. The most important is the superexchange interaction between the atoms located in the B position, J_BB′, and it usually leads to long-range magnetic orderings at relatively high temperatures. If the A position is occupied by magnetic rare earth atoms, exchange interactions usually appear between the atoms of the two sublattices, J_AB, with decreasing temperature and at very low temperatures direct interactions arise between the atoms in the A position, J_AA. Typically, the strength of magnetic interactions follows the sequence, J_BB′ > J_AB > J_AA. Furthermore, when the structural distortion increases greatly due to the decrease in the size of the rare earth atoms (chemical pressure), the competition between nearest-neighbor and next-nearest-neighbor interactions is enhanced, as mentioned above.

In this report, we will analyze the symmetry-related properties of well ordered A₂MnB′O₆ double perovskites with B′ = Co, Ni, deduced from neutron diffraction experiments. The observed symmetry-related properties have a direct impact on the magnetoelectric and multiferroic behavior of the compounds.

2. Experimental

A₂MnCoO₆ (A₂ = La₂, LaTb, Tb₂, Ho₂, Y₂, Er₂, Tm₂, Yb₂ and Lu₂) and A₂MnNiO₆ (A₂ = Tb₂) compounds were prepared by solid state reaction from stoichiometric amounts of A₂O₃, Mn₂O₃ and Co₃O₄ or NiO as reported elsewhere (Blasco et al., 2014; Blasco et al., 2017a; Blasco et al., 2017b; García-Muñoz et al., 2019; Blasco et al., 2016). The last sintering step at 1200°C was followed by a slow cooling at 1°C min⁻¹ for some A₂MnCoO₆ (A₂ = La₂, LaTb) samples. For the rest of the compounds, the last sintering at 1250°C was followed by a very slow cooling at 0.2°C min⁻¹. This last cooling step is crucial to improve the cationic ordering at the perovskite B site and the oxygen stoichiometry (Dass & Goodenough, 2003; Kyômen et al., 2004). The chemical composition of the samples was tested by using the wavelength dispersive X-ray fluorescence spectrometry technique. The cationic composition agreed with the nominal one for all samples within experimental errors.

In order to verify that our specimens were single phase with the expected peaks for a perovskite phase, X-ray powder diffraction (XRD) patterns were collected at room temperature using either a Rigaku D-max or a Siemens D-5000 diffractometer and Cu Kα radiation. The neutron diffraction experiments were carried out at the high-flux reactor of the Institut Laue Langevin (Grenoble, France) using the D1B (λ = 2.52 Å), D20 (λ = 2.41 Å), and D2B (λ = 1.594 Å) diffractometers. Using helium cryostats, neutron powder diffraction (NPD) patterns were collected at fixed selected temperatures and also following temperature ramps with heating ratios ranging between 0.25 and 2 K min⁻¹. Structural and magnetic Rietveld refinements were made using the FullProf program (Rodríguez-Carvajal, 1993). Crystallographic tools from the Bilbao Crystallographic Server (Perez-Mato, 2015; Perez-Mato et al., 2010) and the ISOTROPY software suite (Campbell et al., 2006) were also used to perform the symmetry analysis.

Magnetic measurements were carried out between 5 and 300 K using a Superconducting Quantum Interference Device (SQUID) and a Physical Properties Measuring System (PPMS) from Quantum Design. The dielectric characterization of the samples was carried out as a function of temperature and magnetic field in a PPMS employing a home-made coaxial-line inset and an impedance analyzer (Wayne Kerr Electronics 6500B). The electrical characterization was reported earlier in various publications (e.g. Blasco et al., 2015; Blasco et al., 2017a; Blasco et al., 2014; Blasco et al., 2016).

3. Results and discussion

The XRD patterns at room temperature agree with a single-phase perovskite for all samples. All patterns agree with a monoclinic cell with space group P2₁/n, typical of double perovskites with atomic ordering at the B site coupled with an in-phase rotation around the c axis of BO₆ octahedra and an out-of-phase tilt along the [110] direction of the same octahedra (Howard et al., 2003). The XRD technique has poor sensitivity for detecting the order of nearby atoms in the periodic table. For this purpose, we have used NPD measurements. Refined structural data at room temperature can be consulted in the supplementary information (Table S1). This technique confirms the high degree of Mn/B′ ordering (B′ = Co, Ni) in all samples prepared with a very slow cooling rate in the last sintering process (Blasco et al., 2017a). In this case, the amount of antisite defects (ASDs), i.e. the number of Mn and B′ atoms exchanging positions is typically ∼5–6% (50% implies random distribution). On the other hand, the two samples obtained with a cooling rate of 1°C min⁻¹ present a greater number of ASDs: 18% for La₂MnCoO₆ and 42% for LaTbMnCoO₆, the latter due to the superimposed effect of a strong disorder in the A position of the perovskite (Blasco et al., 2014).

The magnetic order in double perovskites is influenced by several factors such as competitive magnetic interactions, single-ion magnetic anisotropies, external magnetic field, structural disorder or the structural strain produced by the misfit between the size of the cations occupying the two crystallographic sites of the perovskite structure. All these effects determine the different magnetic and magnetoelectric properties in this family of compounds, as it is shown below.

3.1. Magnetic orders in the A₂MnCoO₆ double perovskites

Fig. 1 shows the temperature dependence of dc magnetization, M(T), for A₂MnCoO₆ compounds. The measurements were taken in a magnetic field of 100 Oe after field cooling. Clear magnetic transitions are observed for all samples but their transition temperatures decreases as the A-atom size decreases. A smaller size of the A atoms enhances the Mn(Co)O₆ tilts and weakens the ferromagnetic Mn—O—Co superexchange interaction. Ferromagnetic-like transitions are observed in the M(T) curves for A₂MnCoO₆ compounds with A³⁺ radius (Shannon, 1976) between La³⁺ [r_La³⁺(IX) = 1.216 Å] and Tm³⁺ [r_Tm³⁺(IX) = 1.052 Å]. For smaller A atoms (Yb, Lu), the M(T) curves are more typical of an antiferromagnetic ordering. In our previous studies we have disclosed several magnetic structures in the AMnCoO₆ series (Blasco et al., 2017a; Blasco et al., 2016). The paramagnetic phase can be described in terms of the P2₁/n1′ gray group whose properties are indicated below.

Figure 1 The dc magnetization under a field of 100 Oe in field cooled conditions for (a) La- and Tb-based compounds and (b) A2MnCoO6 samples with the A atom indicated for each curve. For comparison purposes, Tb2MnCoO6 is included in both panels. Labels indicate different types of magnetic ordering (see explanation in the text).

The gray group P2₁/n1′. In this phase [No. 14.76 in BNS notation (Belov et al., 1957)], and assuming a perfect 3d cationic ordering, B and B′ cations occupy, respectively, the Wyckoff positions 2c (0, ½, 0) and 2d (½, 0, 0). A cations are placed at 4e (x, y, z), as the three independent O sites. For simplicity, the P2₁/n setting chosen for the parent cell in the compounds described in Tables S1 and S2 is that whose transformation to the standard setting (P2₁/c1′) is (a, b, −a+c; 0, 0, 0) [or also (c, b, −a−c; 0, 0, 0)]. Unlike the B,B′-site symmetries (−1), the symmetry of the A positions does not contain the inversion center and is simply 1. The decomposition of the magnetic representation into irreducible representations (irreps) of the gray group P2₁/n1′ (k = 0) is shown in Table 1, the dimensions of the irreps of the little group being all 1.

Table 1
Decomposition of the magnetic representation of the gray group P2₁/n1′ (14.76) for k = 0 in A₂BB′O₆, and positions occupied by magnetic atoms in the chosen parent setting

Wyckoff position	Decomposition into irreps
2c (0, ½, 0)	3 mGM1+(1) ⊕ 3 mGM2+(1)
2d (½, 0, 0)	3 mGM1+(1) ⊕ 3 mGM2+(1)
4e (x, y, z)	3 mGM1+(1) ⊕ 3 mGM1−(1) ⊕ 3 mGM2+(1) ⊕ 3 mGM2−(1)

3.1.1. Ferromagnetic order of B sublattices (FM1 phase)

A long-range FM ordering is observed for A₂MnCoO₆ compounds (A size between La and Tm) and, according to the Goodenough–Kanamori superexchange rules (Goodenough, 1958; Kanamori, 1959), it is ascribed to the FM Mn⁴⁺–O–Co²⁺ superexchange interaction as Mn⁴⁺ ion has empty 3d–e_g orbitals while they are half-filled for high-spin Co²⁺ cations in an octahedral coordination. The crystal structure adopts the space group P2₁/n (No. 14, standard group P2₁/c). The non-standard description is often preferably used by scientific community because the β angle is closer to 90° and allows an easier association with the undistorted cubic structure. The magnetic peaks can be well reproduced by the maximal magnetic subgroup P2₁′/n′ (No. 14.79 in BNS notation) that keeps unchanged the dimensions of the magnetic cell (the magnetic propagation vector is k = 0, see Table 2). The setting used in Table 2 to describe the magnetic symmetry corresponds to the parent setting, and transforms to the standard basis P2₁′/c′ through the transformation (a, b, −a+c; 0, 0, 0). This magnetic arrangement [P2₁′/n′ (a, b, −a+c; 0, 0, 0)] transforms as the magnetic irrep mGM2+ respect to the parent nuclear structure with P2₁/n symmetry (see Table 3). This one-dimensional magnetic irrep (mirrep) has three potential modes, two of them allow a ferromagnetic alignment of the Mn⁴⁺ and Co²⁺ moments along the ac plane. The third mode allows an AFM arrangement of the moments along the b axis (A type). In almost all A₂MnCoO₆ compounds, this third mode is inactive and the magnetic order is ferromagnetic with the largest component along the c axis. This is a very common magnetic phase in this family of compounds and hereafter, we denote it as FM1 phase. A representative description of this type of order can be seen in Fig. 2(d) and its description for Ho₂MnCoO₆ is summarized in Tables 2 and 3. Similar tables for the rest of the compounds can be found in the supporting information. The only exception known at the moment (Blasco et al., 2016) corresponds to Y₂MnCoO₆, which shows a small AFM canting along the b axis as can be seen in the supporting information.

Table 2
Magnetic structure of Ho₂MnCoO₆ at 10 K (FM1)

Atomic parameters and symmetry operations are given in a non-standard setting of the MSG 14.79. The transformation to a standard setting is specified.

Compound	Ho2MnCoO6
Parent space group	P21/n (No. 14) (a, b, −a + c; 0, 0, 0)
Propagation vector(s)	(0, 0, 0)
Transformation from parent basis to the one used	(a, b, c; 0, 0, 0)
MSG symbol	P21′/n′
MSG number	14.79
Transformation from basis used to standard setting of MSG	(a, b, −a+c; 0, 0, 0)
Magnetic point group	2′/m′ (5.5.16)
Unit-cell parameters (Å, °)	a = 5.21923, b = 5.56359, c = 7.45218, β = 89.77
MSG symmetry operations	x, y, z, +1	{1 | 0, 0, 0}
	−x, −y, −z, +1	{−1 | 0, 0, 0}
	−x + ½, y + ½, −z + ½, −1	{2′010 | ½, ½, ½}
	x + ½, −y + ½, z + ½, −1	{m′010 | ½, ½, ½}
Positions of magnetic atoms	Co1 Co 0.50000 0.00000 0.00000
	Mn1 Mn 0.00000 0.50000 0.00000
Positions of nonmagnetic atoms	Ho1 Ho 0.0194 0.0724 0.2509
	O1 O 0.2991 0.3157 0.0518
	O2 O 0.3198 0.2923 0.4451
	O3 O 0.6047 0.9667 0.2550
Magnetic atoms, magnetic moments components of magnetic atoms, their symmetry constraints and moment magnitudes (μB)	Co1 0.8 (1) 0.0 −2.80 (4) (mx, my, mz) 2.92 (4)†
	Mn1 0.8 (1) 0.0 −2.80 (4) (mx, my, mz) 2.92 (4)

†Co²⁺ and Mn⁴⁺ moments are constrained to have the same value.

Table 3
Relationship between the P2₁′/n′ (No. 14.79) magnetic structures of A₂MnB′O₆ (A₂B′ = La₂Co, LaTbCo, Tb₂Co, Tb₂Ni, Y₂Co, Ho₂Co, Er₂Co, Tm₂Co) samples and their parent paramagnetic phase

As an example, the values of the amplitudes of the magnetic modes are given only for Er₂MnCoO₆ at 2 K (see also its magnetic structure in Table S5). For each magnetic mode, the base vector and observed amplitudes C_n (in μ_B, n = 1, 2, 3, …) in the ground state of Er₂MnCoO₆ are indicated.

	A2MnCoO6 (amplitudes correspond to A = Er)
Primary irrep label with dimension	mGM2+(3)
Description of the primary irrep	mGM2+:
	{1 | 0, 0, 0}: 1
	{−1 | 0, 0, 0}: 1
	{2′010 | ½, ½, ½}: −1
	{m′010 | ½, ½, ½}: −1
Description of primary mode(s) and amplitude(s) Definition of the mode(s) P21/n[0,0,0]mGM2+(a)	mGM2+, mode 1: [B]Ag_1(a) or [A]A_1(a)
	Mn (−1, 0, −1)	C1 = 1.94
	B′ (−1, 0, −1)	C1 = 1.94
	A (−1, 0, −1)	C1′ = 2.30
	mGM2+, mode 2: [B]Ag_2(a) or [A]A_2(a)
	Mn (−1, 0, 0.49)	C2 = 2.34
	B′ (−1, 0, 0.49)	C2 = 2.34
	A (−1, 0, 0.49)	C2′ = 3.45
	mGM2+, mode 3: [B]Ag_3(a) or [A]A_3(a)
	Mn (0, 1, 0)	C3 = 0.0
	B′ (0, 1, 0)	C3 = 0.0
	A (0, 1, 0)	C3′ = 0.0
Secondary irrep(s) label(s)	Not allowed
C3 = 0 (not detected) for all samples except Y2CoMnO6. Cn′ (n = 1,2,3) only active for magnetic A atoms. C3′ = 0 (not detected) for Er2MnCoO6 but active for the FM3 phase in Tb2MnNiO6, LaTbMnCoO6 or Tm2MnCoO6.

Figure 2 (a) Magnetic structure of Lu2MnCoO6 at 4 K for low external fields (H < 2.5 T). (b) and (c) Details of the field dependence of neutron intensities collected at D1B for Lu2MnCoO6 showing both the vanishing of superstructure E-type peaks and the enhancement of FM peaks with increasing magnetic field. (d) Magnetic structure of Lu2MnCoO6 at 4 K for high external fields (H > 2.5 T). Red and yellow balls (arrows) stand for Mn and Co atoms (moments), respectively. Oxygen atoms have been removed for the sake of clarity.

3.1.2. Antiferromagnetic order of B sublattices (E phase)

For heavier rare earth atoms (A = Yb or Lu), the increase in structural distortion produces a large decrease in the Mn—O—Co bond angle causing the FM interaction to nearest neighbors to compete with the AFM interaction with the next nearest neighbors. This causes the magnetic ground state to change and it is no longer FM but AFM of E-type, i.e. displaying up-up-down-down (↑↑↓↓) spin chains along a specific crystallographic direction [see Fig. 2(a)]. The magnetic peaks can be indexed using the propagation vector k = (0, 0, ½) (P2₁/n setting), and the refinements indicate that Mn and Co moments are mainly aligned along the c axis (Blasco et al., 2015). The magnetic space group (MSG) in both A₂MnCoO₆ compounds (Yb and Lu) is P_a2₁ (No. 4.10) and, using the parent-like setting (a, b, 2c; 0, 0, 0) a transformation to the standard setting P_a2₁ is given by (−c, b, a; ¼, 0, ) [or (2a−c, b, a; −¼, 0, )] (Table 4). The primary mirrep mA1 is associated with the k-point (−1, 0, ½), equivalent to (0, 0, ½), of the non-standard parent P2₁/n (a, b, −a+c; 0, 0, 0) structure. There are three active modes for the perovskite B sites in mA1. Two of them permit the spins of the up-up-down-down chains to be parallel to the ac plane while the third mode allows a b axis spin component. Tables 4 and 5 describe the E-type magnetic structure of Lu₂CoMnO₆, represented in Fig. 2(a). As can be seen, the spins in the chains are arranged in the ac plane with the largest component along the c axis. Although it is allowed by the symmetry the contribution of the third mode was below the detection limit of the powder diffraction experiment. For that reason its amplitude has been set to zero (Table 5). Similar ordering is observed in the Yb₂MnCoO₆ compound [Fig. 3(c)] although in this case there is also a partial order of the Yb sublattice as we will see in the next section. It is noteworthy that this magnetic space group is polar and allows a ferroelectric ordering along the monoclinic unique axis. The polar axis of the point symmetry 2.1′ (No. 3.2.7) is along b, and hence compatible with spontaneous P_y electric polarization. Another property of this magnetic order is its instability in the presence of an external magnetic field, giving rise to metamagnetic transitions (Blasco et al., 2015). The application of a magnetic field produces the spin flop of half of the chain moments, to yield a linear FM ordering similar to that of the A₂MnCoO₆ compounds at zero field (FM1). Fig. 2 shows the field dependence of representative parts of the NPD patterns at 4 K for Lu₂CoMnO₆. At zero field, it shows the presence of E-type superstructure peaks that vanish with increasing the external magnetic field and accordingly, the net FM contribution is enhanced, see Figs. 2(b) and 2(c). The E-type spin ordering breaks the inversion center of the crystal cell and a finite ferroelectric polarization perpendicular to the E-chains direction is allowed. Then, at the same temperature that E-type magnetic order develops, an anomaly is observed in the dielectric permittivity that indicates the occurrence of the ferroelectric phase (Blasco et al., 2017a). This dielectric anomaly vanishes in the presence of an external magnetic field, as can be seen in the supporting information (Fig. S1). The external magnetic field produces a simultaneous spin reorientation giving rise to the conventional FM phase observed in the previous A₂MnCoO₆ compounds. This metamagnetic transition leads to the negative magnetocapacitance shown in Fig. S1. Fig. 2 also shows the two types of magnetic ordering for Lu₂MnCoO₆, the E-type order in the absence of a magnetic field and the FM order (denoted as FM1) in the presence of an external field.

Table 4
Magnetic structure of Lu₂MnCoO₆ at 2 K (E type)

Atomic parameters and symmetry operations are given in a non-standard setting of the MSG No. 4.10. The transformation to a standard setting is specified.

Compound	Lu2MnCoO6
Parent space group	P21/n (No. 14) (a, b, −a + c; 0, 0, 0)
Propagation vector(s)	(0, 0, ½)
Transformation from parent basis to the one used	(a, b, 2c; 0, 0, 0)
MSG symbol	Pa21
MSG number	4.10
Transformation to standard setting of MSG	(2a − c, b, a; −¼, 0, ) or (−c, b, a; ¼, 0, )
Magnetic point group	21′ (No. 3.2.7)
Unit-cell parameters (Å, °)	a = 5.1624, b = 5.5475, c = 14.8148, β = 89.59
MSG symmetry operations	x, y, z, +1	{1|0,0,0}
	½ − x, ½ + y, ¼ − z, +1	{2010 | ½, ½, ¼}
MSG symmetry centering operations	x, y, z + ½, −1	{1′ | 0, 0, ½}
	½ − x, ½ + y,  − z, −1	{2′010 | ½, ½, }
Positions of magnetic atoms	Co1 Co 0.75000 0.00000 0.62500
	Mn1 Mn 0.75000 0.00000 0.37500
Positions of nonmagnetic atoms	Lu_1 Lu 0.2294 0.0744 0.7502
	Lu_2 Lu 0.2706 0.9256 0.4998
	O1_1 O 0.9494 0.3203 0.6519
	O1_2 O 0.5506 0.6797 0.5981
	O2_1 O 0.9244 0.2964 0.8450
	O2_2 O 0.5756 0.7036 0.4050
	O3_1 O 0.3670 0.4559 0.74555
	O3_2 O 0.1330 0.5441 0.50445
Magnetic atoms, magnetic moments components of magnetic atoms, their symmetry constraints and moment magnitudes (μB)	Co1 0.47 (3) 0.0 1.28 (2) (mx, my, mz) 1.36 (2)†
	Mn1 0.47 (3) 0.0 1.28 (2) (mx, my, mz) 1.36 (2)

†Co²⁺ and Mn⁴⁺ moments are constrained to have the same value.

Table 5
Relationship between the E-type magnetic structure of A₂MnCoO₆ (A = Yb or Lu) compounds and their parent paramagnetic phase

For each magnetic mode, the base vector and observed amplitudes Cn (in μ_B, n = 1, 2, 3, …) for each site are indicated. mA1 has dimension 3 for B sites but 6 for A sites. The nuclear A site is split into two orbits, A1 and A2, in the magnetic cell.

Compound	A2MnCoO6
Primary irrep label with dimension	mA1(3)
Description of the primary irrep	mA1:
	{1 | 0, 0, 0}: (1, 0; 0, 1)
	{2010 | ½,  ½, ¼}: (0, −1; −1, 0)
	{1′ | 0, 0, ½}: (1, 0; 0, 1)
	{2′010 | ½, ½, ¾}: (0, −1; −1, 0)
Description of primary mode(s) and amplitude(s) Definition of the mode(s) P21/n[−1, 0, ½]mA1(a,−a)	mA1, mode 1: [B]Ag_1(a) or [A]A_1(a)
	Mn (−1, 0,− ½)	C1 = 0.88 [Yb], 0.78 [Lu]
	Co (1, 0, ½)	C1 = 0.88 [Yb], 0.78 [Lu]
	A1 (−1, 0, −½)	C1′ = 0.27 [Yb]
	A2 (1, 0, ½)	C1′ = 0.27 [Yb]
	mA1, mode 2: [B]Ag_2(a) or [A]A_2(a)
	Mn (−1, 0, 0.24)	C2 = 1.42 [Yb], 1.12 [Lu]
	Co (1, 0, −0.24)	C2 = 1.42 [Yb], 1.12 [Lu]
	A1 (−1, 0, 0.24)	C2′ = 0.19 [Yb]
	A2 (1, 0, −0.24)	C2′ = 0.19 [Yb]
	mA1, mode 3: [B]Ag_3(a) or [A]A_3(a)
	Mn (0, 1, 0)	C3 = 0.0 [Yb], 0.0 [Lu]
	Co (0, −1, 0)	C3 = 0.0 [Yb], 0.0 [Lu]
	A2 (0, −1, 0)	C3′ = 0.0 [Yb]
	A1 (0, 1, 0)	C3′ = 0.0 [Yb]
	mA1, mode 4: [A]A_4(a)
	A1 (1, 0, ½)	C4′ = 0.27 [Yb]
	A2 (−1, 0, −½)	C4′ = 0.27 [Yb]
	mA1, mode 5: [A]A_5(a)
	A1 (1, 0, −0.24)	C5′ = 0.19 [Yb]
	A2 (1, 0, −0.24)	C5′ = 0.19 [Yb]
	mA1, mode 6: [A]A_6(a)
	A1 (0, 1, 0)	C6′ = 0.0 [Yb]
	A2 (0, 1, 0)	C6′ = 0.0 [Yb]
Secondary irrep(s) label(s)	Not allowed

Figure 3 Magnetic structures at 2 K of (a) LaTbMnCoO6, (b) Er2MnCoO6, and (c) Yb2MnCoO6. Red, yellow and purple balls (arrows) stand for Mn, Co and rare earth atoms (moments), respectively. Oxygen atoms have been removed for the sake of clarity.

3.1.3. Polarization and magnetic ordering of the A site

A close inspection of Fig. 1 reveals significant differences in the low-temperature behavior of the M(T) curves for ferromagnetic A₂MnCoO₆ compounds. Firstly, the compounds with nonmagnetic A atoms (A₂ = La₂, Y₂) present an almost constant magnetization evolution at low temperature. Instead, some samples with magnetic A atoms present a strong rise in magnetization at low temperature (A₂ = LaTb) while others suffer a strong decrease (A₂ = Er₂, Tm₂). This behavior must be ascribed to the mutual polarization between the two sublattices of magnetic atoms and the competition among the J_AB, J_BB′ and J_AA′ interactions. For Er- and Tm-based samples, the changes in the neutron magnetic scattering can be accounted for only by including the magnetic contribution from the Tm(Er) sublattice. The refinements indicate that either Tm³⁺ (Blasco et al., 2017a) or Er³⁺ (Blasco et al., 2017b) moments are antiferromagnetically coupled to the Mn–Co sublattice, revealing a negative J_AB interaction for these compounds with heavy rare earth atoms. The arrangement of the rare earth follows the same mirrep (mGM2+) as can be seen in the Tables S5 (Er) and S7 (Tm) so the magnetic group remains P2₁′/n′. The rare earth sublattice is not completely polarized but the main component of this antiferromagnetic arrangement is found along the c axis as seen in Fig. 3 and we denote this ferrimagnetic structure as FI. The ordering is not fully collinear disclosing a competition between the magnetic interaction and the rare earth anisotropy. Another distinct example of interaction between the two magnetic sublattices occurs in the LaTbMnCoO₆ compound (Blasco et al., 2014). In this case, the interaction J_AB is positive leading to a ferromagnetic coupling between the two sublattices. However, the strong anisotropy of Tb³⁺ cations and the constrain of local point symmetry give rise to a confinement of Tb³⁺ moments in the ab plane. Its arrangement is similar to that found in TbFeO₃ at 4.2 K (Bertaut et al., 1967) and is denoted as F_xC_y in Bertaut's notation (Bertaut, 1968). This means that the FM component of the Tb sublattice is directed along the a axis and the J_AB interaction is so strong that it forces the moments of the Mn(Co) sublattice to tilt in that direction as seen in the Fig. 3. We denote this type of ordering involving A and B moments as FM3 because of its analogy with the same type of order observed in Tb₂MnNiO₆ (see below). The ordering of the two sublattices is compatible with the same mirrep (mGM2+) as can be seen in Table S6. Interestingly, as Tb concentration increases, a third interaction, J_AA, begins to gain importance in Tb₂MnCoO₆. The direct Tb–Tb interaction, stronger at low temperatures, usually induces an A_xG_y-type order in related systems (Mareschal et al., 1968; Cuartero et al., 2013). The strong competition between J_AB and J_AA leads to magnetic frustration in the Tb sublattice and neutron patterns at 2 K only shows the FM ordering of Mn–Co sublattice (see Table S3), mainly along c axis, and a diffuse scattering typical of short-range ordering of the Tb sublattice (Blasco et al., 2014). A similar effect occurs in the Ho₂MnCoO₆ sample, where diffuse scattering associated with the short-range ordering of Ho³⁺ moments is also observed at low temperature (Blasco et al., 2017a).

Finally, the polarization of the rare earth sublattice also shows some interesting features in the E-type magnetic structure of Yb₂MnCoO₆ (see Table S8). Unlike Lu³⁺, Yb³⁺ is magnetic and experiences the internal field produced by the E magnetic arrangement of the Mn–Co sublattice (T_N = 50 K). At sufficiently low temperatures (T < 15 K), the Yb atoms split in two different orbits in the magnetic cell with P_a2₁ (−c, b, a; ¼, 0, ) [also (2a−c, b, a; −¼, 0, )] symmetry. In the Yb1 orbit, the Yb atoms experience a strong internal field from the FM arrangement of neighboring Mn/Co moments, which prevents the AF coupling between Yb atoms and keeps them disordered. In the other Yb orbit (Yb2) instead, the AFM arrangement of neighboring Mn/Co moments cancels the internal field at the Yb position and the magnetic Yb–Yb interaction produces a long-range AFM ordering in these atoms below 15 K as can be seen in the Fig. 3(c) (Blasco et al., 2017a). The magnetic space group is preserved below the Yb ordering temperature. Fig. S1 illustrates the anomalous dielectric permittivity concurrent with the formation of the E-type phase (P_a2₁) and its response to external magnetic fields. This phase also shares polar magnetic point symmetry (21′) with Lu₂MnCoO₆.

3.2. Tb₂MnNiO₆: polar ferrimagnetic ground state by incompatibility of A and B sites magnetic symmetries

In this section we present a comparison of the magnetic transitions and symmetries in the isostructural perovskites Tb₂MnMO₆ with M = Ni and Co, having monoclinic P2₁/n distortion and almost perfect ordering of the Mn⁴⁺/M²⁺ 3d metals. As for Tb₂MnCoO₆, the structure of Tb₂MnNiO₆ is detailed in Table S1 using the nonstandard P2₁/n setting. Our non-standard description of the parent structures is given in the non-conventional setting (−a−c, b, a; 0, 0, 0) of the P2₁/c group (No. 14). As in Tb₂MnCoO₆, the magnetic transitions in Tb₂MnNiO₆ are all driven by zone-center modes (k = 0) of the parent phase. Below the paramagnetic P2₁/n phase (MSG P2₁/n1′), up to four ordered regimes can be distinguished in its magnetic evolution under cooling.

(a) FM1 phase (120 K, 70 K). MSG P2₁′/n′ (a, b, −a+c; 0, 0, 0). The ferromagnetic transition temperature (T_C1 ∼ 120 K) is higher than in the homologous Tb₂MnCoO₆ compound (∼100 K) (Blasco et al., 2016), and magnetic intensities agree with the maximal subgroup P2₁′/n′ (No. 14.79 in BNS notation) (García-Muñoz et al., 2019). As in previous A₂MnCoO₆ perovskites this ferromagnetic ordering transforms as the irrep mGM2+ [compatible with a magnetic arrangement of the F_xC_yF_z type in the Bertaut's notation (Bertaut, 1968)]. The m_y component was not detected in our measurements and the observed FM spins are confined in the ac plane but almost aligned parallel to the c axis of the perovskite [Mn and Ni moments make an angle θ ≃ 23 (3)° with c].

(b) FM2 phase [70 K, 15 K]. MSG P2₁′/n′ (a, b, −a+c; 0, 0, 0). If we continue to cool, the neutron intensities disclose a new regime where the FM P2₁′/n′ order persists but, in parallel, the direction of the FM Mn⁴⁺/Ni²⁺ moments continually rotates towards the a axis upon cooling (see Fig. 4 and Table 6), so that we have named this `rotating FM phase' FM2 = FM1r (inset of Fig. 4). In this regime the single active mGM2+ irrep is preserved. Mn and Ni keeps the collinear order and the magnetization forms an angle θ with the c axis that increases from 23 (3)° (75 K) up to 46 (2)° (15 K). The Tb sublattice plays a key role on the rotating phase and on the evolution followed by the tilting angle θ. The rotating magnetization is surely driven by a ferromagnetic Tb–(Mn, Ni) interaction (J_Tb–M > 0) that attracts the 3d moments towards the ab plane. Short-range ferromagnetic correlations between Tb moments were perceived by neutron diffraction below around ≃70 K. In fact, Fig. 4 reveals that the magnetization rotates ∼50° in the short interval 5–70 K, although partial long-range ordering of Tb moments was only detected below T_N3 ≃ 15 K.

Table 6
Magnetic structure of Tb₂MnNiO₆ at 15 K (FM2)

Atomic parameters and symmetry operations are given in a nonstandard setting of the MSG No. 14.79. The transformation to a standard setting is specified.

Compound	Tb2MnNiO6
Parent space group	P21/n (No. 14) (a, b, −a+c; 0, 0, 0)
Propagation vector(s)	(0, 0, 0)
Transformation from parent basis to the one used	(a, b, c; 0, 0, 0)
MSG symbol	P21′/n′
MSG number	14.79
Transformation from basis used to standard setting of MSG	(a, b, −a+c; 0, 0, 0)
Magnetic point group	2′/m′ (5.5.16)
Unit-cell parameters (Å, °)	a = 5.2750, b = 5.5167, c = 7.5110, β = 89.84
MSG symmetry operations	x, y, z, +1	{1 | 0, 0, 0}
	−x, −y, −z, +1	{−1 | 0, 0, 0}
	−x + ½, y + ½, −z + ½, −1	{2′010 | ½, ½, ½}
	x + ½, −y + ½, z + ½, −1	{m′010 | ½, ½, ½}
Positions of magnetic atoms	Ni1 Ni 0.50000 0.00000 0.00000
	Mn1 Mn 0.50000 0.00000 0.50000
Positions of nonmagnetic atoms	Tb1 Tb 0.0177 0.0654 0.2494
	O1 O 0.2974 0.3110 0.0509
	O2 O 0. 3184 0.2922 0.4500
	O3 O 0.5978 0.9678 0.2597
Magnetic atoms, magnetic moments components of magnetic atoms, their symmetry constraints and moment magnitudes (μB)	Ni1 1.25 (4) 0.0 1.22 (4) (mx, my, mz) 1.74 (2)†
	Mn1 1.87 (5) 0.0 1.83 (5) (mx, my, mz) 2.61 (2)

†3d ordered moments are constrained by m[Ni²⁺] = 2/3·m[Mn⁴⁺].

Figure 4 Tb2MnB′O6 (B′ = Ni and Co). Comparison of the direction of 3d moments (θ) and temperature evolution. Inset: FM1 and FM2 (= FM1r [rotating]) magnetic phases of Tb2MnNiO6 (P21′/n′).

(c) FM3 phase (15 K, 3.7 K). MSG P2₁′/n′ (a, b, −a+c; 0, 0, 0). The ordering of terbium moments in Tb₂MnNiO₆ involves two successive magnetic configurations. First there is an intermediate arrangement in which the overall P2₁′/n′ symmetry is preserved and terbium moments become partially polarized with the help of the FM exchange field created by Mn/Ni sites. The refined magnetic configuration at 6 K is given in Table 7. Tb moments preserve a single independent orbit and adopt the symmetry dictated by the mGM2+ irrep, yielding two identical Tb magnetic layers parallel to ab (at z ≃ 0.25 and z ≃ 0.75) that are connected by inversion symmetry ({−1 | 0, 0, 0}). A configuration analogous to the ground state in LaTbMnCoO₆ [Fig. 3(a) and Table S6]. Tb moments are confined in the ab plane and adopt the permitted Bertaut F_xC_yF_z configuration with F_z = 0 within our detection limit. The common F_x component points to a ferromagnetic interaction between 4f and 3d ions. Since this FM3 magnetic structure conserves the spatial inversion its point magnetic symmetry [2′/m′ (No. 5.5.16)], is nonpolar.

Table 7
Magnetic structure of Tb₂MnNiO₆ at 6 K (FM3)

Atomic parameters and symmetry operations are given in a non-standard setting of the MSG No. 14.79. The transformation to a standard setting is specified.

Compound	Tb2MnNiO6
Parent space group	P21/n (No. 14) (a, b, −a + c; 0, 0, 0)
Propagation vector(s)	(0, 0, 0)
Transformation from parent basis to the one used	(a, b, c; 0, 0, 0)
MSG symbol	P21′/n′
MSG number	14.79
Transformation from basis used to standard setting of MSG	(a, b, −a + c; 0, 0, 0)
Magnetic point group	2′/m′ (No. 5.5.16)
Unit-cell parameters (Å, °)	a = 5.2750, b = 5.5165, c = 7.5113, β = 89.84
MSG symmetry operations	x, y, z, +1	{1 | 0, 0, 0}
	−x, −y, −z, +1	{−1 | 0, 0, 0}
	−x + ½, y + ½, −z + ½, −1	{2′010 | ½, ½, ½}
	x + ½,−y + ½, z + ½, −1	{m′010 | ½, ½, ½}
Positions of magnetic atoms	Ni1 Ni 0.50000 0.00000 0.00000
	Mn1 Mn 0.50000 0.00000 0.50000
	Tb1 Tb 0.0177 0.0654 0.2494
Positions of nonmagnetic atoms	O1 O 0.2974 0.3110 0.0509 0.9678 0.2597
	O2 O 0. 3184 0.2922 0.4500 O3 O 0.5978
Magnetic atoms, magnetic moments components of magnetic atoms, their symmetry constraints and moment magnitudes (μB)	Ni1 1.64 (5) 0.0 0.76 (5) (mx, my, mz) 1.81 (3)†
	Mn1 2.46 (6) 0.0 1.15 (6) (mx, my, mz) 2.72 (4)
	Tb1 0.76 (4) 0.90 (4) 0.0 (mx, my, mz) 1.18 (4)

†3d ordered moments are constrained by m[Ni²⁺] = 2/3·m[Mn⁴⁺].

(d) FM4 phase (T < 3.7 K). MSG P2₁′ (a, b, c; ¼, 0, ¼). Upon further cooling, when temperature decreases below T_N4 ≃ 3.7 K, a new magnetic transition reveals the magnetic ground state of Tb₂MnNiO₆. In this new phase (FM4, described in Table 8) the inversion symmetry operation connecting Tb magnetic ions at the two different TbO₂ layers is lost. As a consequence, during the P2₁′/n′ to P2₁′ transformation the only Tb position in P2₁′/n′ is split to give two independent Tb sublattices below the transition. Each of the Tb1 and Tb2 orbits corresponds to a different layer in the unit cell. Only the positions of the 3d metals are not split (its site symmetry changes from −1 to 1). In this ground phase each one of the three oxygen sites (O1, O2 and O3) splits in two orbits to give six independent oxygen positions (Table 8).

Table 8
Magnetic structure of Tb₂MnNiO₆ at 2 K (FM4)

Atomic parameters and symmetry operations are given in a non-standard setting of the MSG 4.9. The transformation to a standard setting is specified.

Compound	Tb2MnNiO6
Parent space group	P21/n (No. 14) (a, b, −a+c; 0, 0, 0)
Propagation vector(s)	(0, 0, 0)
Transformation from parent basis to the one used	(a, b, c; 0, 0, 0)
MSG symbol	P21′
MSG number	4.9
Transformation from basis used to standard setting of MSG	(a, b, c; ¼, 0, ¼)
Magnetic point group	2′ (No. 3.3.8)
Unit-cell parameters (Å, °)	a = 5.2745, b = 5.5164, c = 7.5102, β = 89.839
MSG symmetry operations	x, y, z, +1	{1 | 0, 0, 0}
	−x + ½, y + ½, −z + ½, −1	{2′010 | ½, ½, ½}
Positions of magnetic atoms	Ni1 Ni 0.50000 0.00000 0.00000
	Mn1 Mn 0.50000 0.00000 0.50000
	Tb1 Tb 0.0177 0.0654 0.2494
	Tb2 Tb 0.9823 0.9346 0.7506
Positions of nonmagnetic atoms	O1_1 O 0.2974 0.3110 0.0509
	O1_2 O 0.7026 0.6890 0.9491
	O2_1 O 0.3184 0.2922 0.4500
	O2_2 O 0.6816 0.7078 0.5500
	O3_1 O 0.5978 0.9678 0.2597
	O3_2 O 0.4022 0.0322 0.7403
Magnetic atoms, magnetic moments components of magnetic atoms, their symmetry constraints and moment magnitudes (μB)	Ni1 0.8 (2) 0.0 1.6 (1) (mx, my, mz) 1.9 (2)†
	Mn1 1.3 (3) 0.0 2.4 (1) (mx, my, mz) 2.7 (2)
	Tb1 5.84 (6) 4.7 (2) 0.0 (mx, my, mz) 7.5 (1)‡
	Tb2 −5.84 (6) −4.7 (2) 0.0 (mx, my, mz) −7.5 (1)

†3d ordered moments are constrained by m[Ni²⁺] = ⅔m[Mn⁴⁺].
‡4f ordered moments are constrained by m[Tb1³⁺] = −m[Tb2³⁺].

As shown in Fig. 5, the Tb–Tb interactions transform the F_xC_y magnetic configuration of Tb moments (FM3) into A_xG_y (FM4). Tb³⁺ moments in the same layer form angles close but below 90°. As occurs in other terbium perovskites, the planar magnetic arrangement of Tb in the FM3 and FM4 phases of Tb₂MnNiO₆ reveals the Ising nature of the terbium moments derived from the lowest crystal-field energy levels. The electronic configuration of Tb³⁺ ions (4f⁸) and its non-Kramers nature entails a very large local anisotropy in low-symmetry sites. Typically, an accidental quasi-doublet gives it Ising-like properties (Novák et al., 2013; Gruber et al., 2008). The drop in the exchange field produced by the flipping of Tb moments in one of the two layers is followed by a sudden strong reorientation of the FM 3d moments towards the c axis (Δθ ≃ −40°). The latter is illustrated in Fig. 5 that displays the evolution of the magnetization direction (θ) below 10 K and a view of the FM4 and FM3 magnetic structures of Tb₂MnNiO₆. In Fig. 4 we have compared the distinct evolution of the magnetization directions θ(T) in Tb₂MnNiO₆ and Tb₂MnCoO₆ in all the range below the paramagnetic boundary.

Figure 5 Tb2MnNiO6: polar/non-polar magnetic transition (FM4/FM3). Inclination angle (θ) of 3d moments (center). Magnetic structures: (left, FM4) P21′ (a, b, c; ¼, 0, ¼) and (right, FM3) P21′/n′ (a, b, −a+c ; 0, 0, 0).

Regarding the magnetic distortion modes, in Table 9 we describe the main mirreps and their associated magnetic modes active at the ground state of Tb₂MnNiO₆ (FM4), with respect to its paramagnetic phase. In addition, Fig. 6 illustrates that (according to Table 9) two primary magnetic irreps of the parent group (mGM2+ and mGM2−) are the responsible for the magnetic symmetry of Tb₂MnNiO₆ below T_N4 ≃ 4 K. First, the ferromagnetic arrangement of the two 3d (Mn/Ni) sublattices, is giving by three one-dimensional magnetic mGM2+ modes, one of which has null amplitude or below our detection limit (F_y = 0, Table 9). The isotropy subgroup of these ferromagnetic modes is P2₁′/n′. In the FM3 intermediate magnetic regime of Tb₂MnNiO₆, 4f moments become partially polarized by the exchange field from 3d atoms and Mn, Ni and Tb all transform as the irrep mGM2+ (k = 0) giving to identical TbO₂ magnetic layers (Table 3). The three mGM2+ modes for Tb have non-null amplitude and isotropy subgroup P2₁′/n′. Finally, in the FM3-to-FM4 transformation the Tb³⁺ single-ion anisotropy and Tb³⁺–Tb³⁺ energy terms force the substitution of the mGM2+ distortion modes in the Tb sublattice by the mGM2− magnetic modes that split in two orbits the Tb magnetic sites. If the mGM2+ irrep at the zone-center generates the isotropy subgroup P2₁′/n′ (No. 14.79) (now from B sites only), the new mGM2− magnetic distortion occurring at the A sites of the perovskite induces P2₁′/n (No. 14.77) symmetry. So, the final symmetry of the FM4 phase is the MSG P2₁′ (4.9), the overlay (common maximal subgroup) of these two different isotropy subgroups associated to, respectively, the magnetic arrangement in the A and B sublattices. Notice that although, separately, mGM2+ and mGM2− modes are non-polar, the final symmetry of the ground state is polar (magnetic point symmetry 2′, No. 3.3.8) and permits the appearance of ferroelectricity with spontaneous polarization parallel to the unique axis b. Moreover, as shown in Fig. 5, there is a strong reorientation of the magnetization direction (θ) concurrent with the P2₁′/n ↔ P2₁′ phase transition. If the ground state (FM4) is ferroelectric due to magnetic trilinear coupling, a cross-correlation between the voltage (magnetic field direction) and the direction of magnetization (electric polarization) is anticipated.

Table 9
Relationship between the magnetic structure of the FM4 phase of Tb₂MnNiO₆ (P2₁′ (a, b, c; ¼, 0, ¼) and its parent paramagnetic phase

For each magnetic mode, the base vector and observed amplitudes Cn (in μ_B, n = 1, 2, 3, …) for each site are indicated.

Compound	Tb2MnNiO6
Parent space group	P21/n (No. 14) (a, b, −a + c ; 0, 0, 0)
Transformation from parent basis to the one used for the magnetic structure	(a, b, c; 0, 0, 0)
Propagation vector	(0, 0, 0)
Primary irrep(s) label(s) with dimension	mGM2+ (3)
	mGM2− (3)
Description of the primary irrep(s)	mGM2+:
	{1 | 0, 0, 0}: 1
	{−1 | 0, 0, 0}: 1
	{2′010 | ½, ½, ½}: −1
	{m′010 | ½, ½, ½}: −1
	mGM2−:
	{1 | 0, 0, 0}: 1
	{−1′ | 0, 0, 0}: −1
	{2′010 | 0, ½, 0}: −1
	{m010 | ½, ½, ½}: 1
Description of primary mode(s) and amplitude(s) Definition of the mode(s) P21/n[0,0,0]mGM2+(a) P21/n[0,0,0]mGM2−(a)	mGM2+, mode 1: [B]Ag_1(a)
	Mn1 (−1, 0, −1)	C1 = 2.71
	Ni1 (−1, 0, −1)	C1 = 1.77
	mGM2+, mode 2: [B]Ag_2(a)
	Mn1 (−1, 0, 0.491)	C2 = 0.31
	Ni1 (−1, 0, 0.491)	C2 = 0.26
	mGM2+, mode 3: [B]Ag_3(a)
	Mn1 (0, 1, 0)	C3 = 0.0
	Ni1 (0, 1, 0)	C3 = 0.0
	mGM2−, mode 1: [A]A_1(a)
	Tb1 (−1, 0, −1)	C1 = 3.35
	Tb2 (1, 0, 1)	C1 = 3.35
	mGM2−, mode 2: [A]A_2(a)
	Tb1 (−1, 0, 0.491)	C2 = 4.77
	Tb2 (1, 0, −0.491)	C2 = 4.77
	mGM2−, mode 3: [A]A_3(a)
	Tb1 (0, 1, 0)	C3 = 4.72
	Tb2 (0, −1, 0)	C3 = 4.72
Secondary irrep(s) label(s)	Not allowed

Figure 6 Tb2MnNiO6: active primary magnetic irreps in the polar magnetic ground state [P21′ (a, b, c; ¼, 0, ¼)]. The overlay of the non-polar mGM2+ (B sites) and mGM2− (A sites) modes generates the polar P21′ MSG and causes a severe reorientation (Δθ ≃ 50°) of the magnetization.

The above properties related with phases FM2, FM3 and FM4 of Tb₂MnNiO₆ were not observed in the analogous Tb₂MnCoO₆ compound, which displays a single ferromagnetic FM1 ordered phase with P2₁′/n′ symmetry down to 1.5 K. The large single-ion anisotropy term of divalent cobalt anchors Co²⁺ FM spins almost perpendicular to the xy Tb layers, regardless of temperature below T_N = 100 K. This does not favor the appearance of the rotational FM2 (= FM1r) phase nor the FM3. Moreover, the FM4 order and the non-polar/polar transition were not observed in Tb₂MnCoO₆ down to our base temperature.

4. Conclusions

We have reported an exhaustive description of the magnetic structures found in the rare-earth double perovskites A₂MnB′O₆ (A = rare earth or Y: B′ = Co or Ni). The main magnetic interaction present in these compounds (Mn–O–B′ superexchange) gives rise to a FM arrangement in the ac plane that adopts the P2₁′/n′ magnetic space group. As the A-atom size decreases in A₂MnCoO₆ series, this FM interaction weakens and competes with the next-nearest-neighbor AFM interaction causing an E-type magnetic ordering for Yb- and Lu-based samples [k = (0, 0, ½)]. The magnetic structure adopts the P_a2₁ MSG with up-up-down-down spin chains along the c axis. This magnetic ordering (T_N) breaks the inversion center and electrical polarization along the monoclinic unique axis, perpendicular to the magnetic direction, is permitted. Therefore, an anomaly in the dielectric permittivity is observed at T_N but the presence of a sufficiently strong magnetic field induces a spin flop and these distorted compounds adopt the same magnetic structure as the rest of the series at zero field. In the case of magnetic A atoms, the magnetic interaction between A and B sublattices is observed at temperatures lower than the FM ordering of the B sublattices. This interaction between sublattices is positive for samples with diluted Tb and due to its strong anisotropy, a change in orientation of the Mn–Co sublattice moments occurs, approaching the a axis. In other cases (Er or Tm), the coupling between both sublattices is negative and non-collinear ferrimagnetic orderings are observed in the ac plane with the main component along the c axis. All these rare earth arrangements can be described in the same magnetic space group, P2₁′/n′. Finally, for Tb- and Ho-based Mn/Co samples, the strong J_AA interaction competes with the other two interactions and does not favor the coupling between A and B sublattices. Mn and Co keep the FM alignment, mainly along the c axis while only diffuse magnetic scattering is observed down to 2 K from frustrated moments in the A sublattice. The large single-ion anisotropy term of divalent cobalt anchors the 3d FM spins almost perpendicular to the Tb layers in the ab plane. This configuration generates an internal exchange field that precludes the successive magnetic phases observed in Tb₂MnNiO₆, where Ni²⁺ presents a much smaller single-ion anisotropy respect to Co²⁺ ions. In the ground state of Tb₂MnNiO₆ the A and B sublattices adopt, respectively, two different non-polar magnetic irreps whose intersection generates a polar P2₁′ magnetic symmetry compatible with spontaneous P_y polarization assisted by magnetic trilinear coupling. Increasing temperature, a ferrimagnetic Tb-site ordering can be stabilized in Tb₂MnNiO₆, canceling the splitting of the Tb sublattices by recovering the inversion symmetry, with the help of a severe spin reorientation (Δθ ≃ 50°) of the ferromagnetic Mn/Ni moments, which align almost parallel to the ab plane. In this material the direction of the magnetization could be used as a lever to switch the polar/non-polar (ferroelectric/antiferroelectric) transformation.