Role of spontaneous strains on the biphasic nature of partial B-site disorder double perovskite La₂NiMnO₆

Multiferroics are materials with simultaneous magnetic, ferroelastic, and ferroelectric ordering.^1–6 Due to their potential for applications in memory, data storage, and spintronics, they have attracted extensive attention in the recent years.⁷ These materials show rich phase diagrams combining different ferroic orders in separate phases. Transition-metal oxides with a double perovskite structure, as one kind of multiferroics, have attracted considerable attention because of their unique electrical, magnetic, and elastic properties.^8,9 Specially, double perovskite La₂NiMnO₆ (LNMO) is a rare example of a single-material platform with multiple functions, in which the spins, electric charge, and dielectric properties can be tuned by magnetic and/or electric fields.^10–13 This behavior is observed very close to room temperature, which provides optimism for practical spintronic applications.^10,12,14 Semiconductor La₂NiMnO₆ is a ferromagnet (T_c ≈ 280 K)^11,15,16 with ordered Ni²⁺ (d⁸:$t2g6eg2$⁠, S = 2/2) and Mn⁴⁺ (d³:$t2g3eg0$⁠, S = 3/2) ions occupying the metal (M) centers of corner-sharing MO₆ octahedra in a distorted perovskite structure.¹⁰ Based on Kanamori-Goodenough rules,¹⁷ the ferromagnetic order derives from the presence of 180° Ni²⁺–O–Mn⁴⁺ superexchange bonding between an empty Mn⁴⁺ e_g orbital and a half-filled d orbital on a neighboring Ni²⁺ site, which has been theoretically predicted and experimentally reported.¹⁸ Thus, its predicted saturation magnetization value at a low temperature is M_s = 5 μ_B. The structure of bulk or film La₂NiMnO₆ is monoclinic (P2₁/n) at a low temperature and then transforms to rhombohedral (R$3¯$⁠) at a high temperature. These two structures coexist over a wide temperature range, including the room temperature.^10,11,19,20

Previous studies have concluded that the coexistence of the two structures has significant effects on the physical properties of partial B-site disorder La₂NiMnO₆.^{13,15,21–24} They also predicted that the biphasic feature is consistent with the competing relative stability of the two structures.^15,23 However, the origin of the coexistence of the two structures (or relative stability) is still unclear so far. Thus, in this study, we mainly studied the origin of the biphasic nature of partial B-site disorder La₂NiMnO₆. Then, the cation deficiency or O excess was focused because it had significant influence on the ferromagnetic/structural transition temperatures and surface morphology.²⁵ We would directly investigate the cation deficiency in different regions and then build the correlation between defects and the biphasic feature. Lastly, the first principles density functional theory (DFT) within the Perdew-Burke-Ernzerhof (PBE) functional was used to calculate the static formation enthalpies and crystal structure information of La₂NiMnO₆ at different hydrostatic pressures. The structure stability of La₂NiMnO₆ was studied. Then the spontaneous shear strains would be obtained from the calculated crystal structures at different hydrostatic pressures, and this could also make comparability to the experimental diffraction data. Through this exploration, the origin of two structure coexistence would be better understood.

Due to the wide variability in physical properties and structural states prepared in different synthesized conditions, it is significant to measure various properties on this sample, as has been done here for diffraction, magnetism, microstructures, and electronic states. The LNMO sample was prepared via a conventional solid-state reaction method, and the detailed descriptions of the experimental and theoretical calculation section are demonstrated in the supplementary material.

The Rietveld refinement analysis results of the room temperature powder X-ray diffraction (PXRD) are shown in Fig. 1(a), and the inset and the right one represent the crystal structure of R$3¯$ and P2₁/n, respectively. The fit gave the coexistence of two phases [62.64% (P2₁/n) and 35.69% (R$3¯$⁠)] at room temperature. This result coincides well with that obtained by Kumar et al.²¹ [58% (P2₁/n) and 42% (R$3¯$⁠)] and Sayed et al.¹⁵ [51% (P2₁/n) and 49% (R$3¯$⁠)]. The lattice parameters of space group P2₁/n were a = 5.440 3(5) Å, b = 5.475 9(8) Å, c = 7.737 9(5) Å, and β = 88.926(7)°, and those of space group R$3¯$ were a = 5.475 1(3) Å and α = 60.259(5)°, with R_wp = 2.705% and χ² = 1.754. The shear strain (e₅ ≈ cos β), thus, of the monoclinic phase was about 0.0187. To calculate the shear strain of the rhombohedral phase, the rhombohedral angle should convert to the face centered pseudocubic cell angle, and their relationship is

where β_f represents the pseudocubic angle and the rhombohedral angle β_R is nearly 60°.²⁶ Thus, the calculated shear strain of the rhombohedral phase was about −0.0040. This illustrated that the shear strain value in the monoclinic phase was positive, while that in the rhombohedral phase was negative, and this result coincides well with the neutron diffraction data.^10,15 The Rietveld refinement results also provided the bond length, and then the tolerance factor $t≡(A−O)√2(B−O)$⁠, where (A–O) and (M–O) are the mean equilibrium A-site and B-site bond lengths in the ABO₃ perovskite, was calculated for the rhombohedral phase (0.89) and monoclinic phase (0.84), respectively. This result accorded with the description from the work of Dass et al.¹¹ that the tolerance factor of the rhombohedral phase was higher than that of the monoclinic phase. Besides, a small proportion (1.67%) impurity phase NiO was observed from the refinement result.

The Raman spectroscopy is an effective tool for the structural characterization of La₂NiMnO₆.¹⁹ Figure 1(b) shows that the high frequency peak of A_g (P2₁/n) or A_g (R$3¯$⁠) symmetry at 676 cm⁻¹ was the stretching (“breathing”) vibrations of the (Ni/Mn)O₆ octahedra. In addition, there are several modes of A_g and B_g symmetries involving anti-stretching and/or bending of the Ni/MnO₆ octahedra for the P2₁/n structure exhibited near 530 cm⁻¹, while these motions in the R$3¯$ structure are represented by two E_g modes.^19,27 Thus, the Raman spectrum clearly demonstrated the coexistence of R$3¯$ and P2₁/n symmetries.

A sharp soft magnetic hysteresis loop at 2 K with a small opening can be seen in Fig. 1(c), with M_s ≈ ∼3.87 μB/f.u. This value implies the existence of the anti-site disorder of Ni²⁺ and Mn⁴⁺ ions. Thus, the B-site disorder degree was assumed to be ∼11% according to the relationship equation M = M_s × (1 − 2AS) for saturation magnetization (M) and the anti-site (AS) defects, where M_s is the expected moment.^28–30 AS has been confirmed as the dominating reason of the reduced magnetization.^31,32 The AC magnetic susceptibility (Figs. S1 and S2 in the supplementary material) confirmed the existence of the magnetic spin-glassy phases in the LNMO compound. In addition, when the temperature increased to 295 K, the magnetization did not saturate [Fig. 1(c)]. This feature demonstrated that the long-range ferromagnetic order was completely suppressed by thermal effects at room temperature. Thermal evolution of the field cooling (FC) and zero field cooling (ZFC) DC susceptibility measured at 50 Oe also demonstrated a low-temperature saturation characteristic of spontaneous ferromagnetic ordering [Fig. 1(d)]. Quite analogous magnetization results on partial disorder La₂NiMnO₆ have been observed in the literature.^11,21,22 To further confirm the ferromagnetic transition temperature, the AC magnetic susceptibility was performed at the temperature range from 220 K to 300 K and the real and imaginary parts are shown in the inset of Fig. 1(d). From the inset, only one ferromagnetic transition temperature could be observed at ∼285 K, which is higher than the theoretical one, and this feature could be due to the local Ni/Mn cation ordering in the LNMO sample. Thus, combined the ZFC and FC curves, the Curie temperature was around 280 K of this sample.

The microtopography of the sample was investigated by scanning electron microscope (SEM), and the element distributions over the sample were studied by the energy dispersal X-ray spectroscopy (EDS) analysis. Based on the elemental mapping analysis [Fig. 2(a)], the element nickel was inhomogeneously distributed in some local regions [the second image on the right-hand side of Fig. 2(a)]. The regions where the Ni ions were bright in the element maps were defined as Ni-rich regions, and correspondingly, the regions where the La ions were bright are the La-rich regions. As shown in Fig. 2(b), two different particle sizes were observed. In addition, the distribution of Mn and O ions was uniform inside and outside the Ni-rich regions. Figure 2(c) clearly shows that the La-rich crystal grains (0.2-1 μm) grew relatively individually and angularly. Nevertheless, the Ni-rich crystal grains (1-3 μm) were intergrowths [Fig. 2(d)]. The atomic ratio of element La, Ni, Mn, and O in the La-rich crystals tested by EDS [Fig. 2(e)] approached 2:1:1:6.3. This result indicates that in the La-rich region the element oxygen was in excess. As it is impossible to bring in oxygen interstitials into the double perovskite structure, the excess oxygen can be accommodated by the introduction of A-site or B-site vacancies.¹¹ Besides, combined the refinement results, the Ni rich crystals were in the NiO phase, and in this region, the La element was also deficient, and some detailed descriptions are in the supplementary material. To study the electronic structure and oxidation states of La, Ni, and Mn in LNMO, X-ray photoelectron spectroscopy (XPS) measurements were performed as shown in Figs. S4–S6 (supplementary material). The XPS results confirmed the inhomogeneity of the electronic states in LNMO, and in addition, the SEM with EDS results implies that the sample is inhomogeneous. The inhomogeneity can result in the coexistence of the two structures, which is proved by previous studies.^33,34

The static formation enthalpies of rhombohedral and monoclinic crystal structures per formula unit at different hydrostatic pressures, which are calculated via DFT within PBE^35,36 functional at 0 K, are demonstrated in Fig. 3(a). The static formation enthalpies increase linearly with the increase in hydrostatic pressures from zero to 50 GPa. Figure 3(b) represents the enthalpy difference (monoclinic one minus rhombohedral one) from naught to 50 GPa pressures. The difference is small, and the disparity decreases when the external hydrostatic pressures increase from naught to 40 GPa. However, then a slight increase in the pressure disparity is observed from 40 GPa to 50 GPa. This feature implies that the structure stability is similar for these two structures from the aspect of static formation enthalpy and their smallest disparity is at 40 GPa in these pressure ranges. This could be one of the vital reasons why two structures generally co-exist over a wide temperature range in La₂NiMnO₆ crystals. The static formation enthalpies of two structures with ∼4% La-site vacancies are represented in Fig. 3(a). Their values are slightly higher than those of perfect La₂NiMnO₆ crystals by 0.9071 eV (Rhombohedral) and 0.9386 eV (monoclinic) per formula unit, respectively. In addition, the static formation enthalpies of these two structures with A-site vacancies are extraordinarily close [Fig. 3(a)]. The difference also decreases from −0.0776 to −0.0461 eV/f.u. with ∼4% La-site vacancies. This feature implies that the structure stability of the two structures is closer with A-site vacancies, which increases the possibility of transformation between two structures.

To compare the spontaneous shear strains (e₅) between rhombohedral and monoclinic crystal structures, the calculated cell parameters (Table S1 in the supplementary material) were obtained via DFT. As shown in Fig. 3(c) and Table S1, the positive e₅ of the monoclinic crystal structure decrease gradually from naught to 20 GPa, and the value changes to negative when the hydrostatic pressure increases to 30 GPa. Meanwhile, e₅ of the rhombohedral crystal structure are negative and their absolute values are much higher than those of the monoclinic one. e₅ of the rhombohedral crystal structure are also more sensitive to the hydrostatic pressures. The negative e₅ increases sharply when 10 GPa hydrostatic pressure is applied, while the variation of the negative e₅ is gently from 10 to 50 GPa. Calculated shear strains also have a good comparability with the experimental data as shown in Figs. 1(a) and 3(d).

Neutron and X-ray diffraction lattice parameters of La₂NiMnO₆ double perovskite data are listed by Sayed et al.¹⁵ and Rogado et al.¹⁰ Then the variations of cos β (≈e₅) (shear strain) are calculated from their data, and the temperature-dependence of e₅ is shown in Fig. 3(d). Figure 3(d) shows that cos β of R$3¯$ is approximately linear from 300 K to 1050 K from the data of Sayed et al.¹⁵ Based on straight line fitting the temperature variations of the cos β of the pseudocubic angle, we have estimated the cubic-rhombohedral transition temperature of La₂NiMnO₆ to be 1884 K with a second-order character. It should be noted that this second-order structure transition temperature could be varied because of different synthesis conditions. The data from neutron diffractions demonstrate the small shear strain (e₅ ≈ cos β) in the P2₁/n phase.

In combination, the magnetization, SEM, EDS, Raman, XRD, XPS, and DFT within the PBE functional allow a straightforward analysis of the biphasic nature of partial B-site disorder La₂NiMnO₆. To have a better understanding of the coexistence feature, a schematic diagram of Fig. 4 illustrates that the temperatures, hydrostatic pressures, and cation vacancies have significant effects on the biphasic nature of La₂NiMnO₆. As the temperature decreases, the cubic Fm$3¯$m phase [Fig. 4(a)] first transforms to the rhombohedral R$3¯$ phase [Fig. 4(b)] by rotations of octahedral structure NiO₆ and MnO₆ at a high temperature with a character of second-order. Next, part of the rhombohedral R$3¯$ phase will transform to the monoclinic P2₁/n phase [Fig. 4(c)] at about 650 K with a character of first-order. Lastly, all rhombohedral phases of La₂NiMnO₆ will transform to the monoclinic phase at an extremely low temperature. In addition, the transition temperatures are mainly based on the cation vacancies and structure stability. The inhomogeneous distribution of oxygen, which is associated with the coexistence, over a sample was confirmed by the SEM and EDS.^33,34 Our DFT calculation results also confirmed that the cation vacancies will decrease the gap of the static formation enthalpies of rhombohedral and monoclinic structures. Compared to the monoclinic phase, therefore, the relative structure stability of the rhombohedral phase in La₂NiMnO₆ is a little higher if the cation vacancies existed from the perspective of the static formation enthalpies. However, the gap of the static formation enthalpies is so small between two crystal structures, and thus the existence of the cation vacancies might facilitate the transition between two phases due to the decrease in the static formation enthalpy difference. Therefore, the relative stability structure could mainly be based on other parameters, like temperature or pressure.

Generally, the crystal structure is less stable with higher spontaneous shear strains. Figures 3(c) and 3(d) have proved that the spontaneous shear strains e₅ of rhombohedral structures are more sensitive to the hydrostatic pressures and temperatures. Simultaneously, e₅ of the monoclinic structure is more stable and smaller in low temperatures and high hydrostatic pressures. Normally, the spontaneous strains will increase with the decrease in temperature.³⁷ This in turn will reduce the stability of the rhombohedral structure. Thus, from the aspect of spontaneous strains, we prove that the monoclinic phase would be more stable in lower temperatures.

In summary, we study the origin of the biphasic nature of partial B-site disorder La₂NiMnO₆ from the aspect of the static formation enthalpies and spontaneous shear strains. The coexistence with a wide temperature range is mainly due to the small disparity of static formation enthalpies between two structures and partly because of the inhomogeneous feature. These results were confirmed by various experiment measurements and calculations. The monoclinic phase is more stable in low temperatures and high hydrostatic pressures due to smaller spontaneous shear strains. Our study will facilitate the research and applications of La₂NiMnO₆ and its analogs in the fields of spintronics as a single-material platform with multiple functions near room temperature.

See supplementary material for the detailed description of the experimental and theoretical calculation section, and some magnetism, XPS, and SEM data are also included and are described.

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51702289 and 11504325), the China Postdoctoral Science Foundation (Grant No. 2016M601963), and Natural Science Foundation of Zhejiang Province (No. LQ15A040004).