Rare-earth magnetic nitride perovskites

The applications of perovskite materials are unparalleled among the different families of compounds. In fact, many materials with exceptional properties (some of them of enormous technological relevance) exhibit a perovskite-like structure [1–12]. As a few examples, perovskite oxides have been found to be conductive and even high-temperature superconducting [9, 10], they can be used as thermoelectrics [3], for lasing [2, 13] and magnetism [14]. This wide range of applicability arises from the large number of elements that are possible to accommodate in a rather simple crystal structure. Furthermore, this structure can easily tolerate distortions from intercalations, dopants and defects [15].

Perovskites have the generic chemical formula ABX₃, where A and B are cationic sites, with B located in the center of a characteristic octahedra of X anions and A occupying the center of the space left between eight of them. While the A and B sites can be occupied by species from a large portion of the periodic table, X is commonly occupied by oxygen or, less frequently, by other chalcogens and some halogens [16]. Other more complex phases also exist such as the oxynitride perovskites, in which oxygen and nitrogen occupy the same sublattice. In fact, these materials are being extensively studied due their interesting optical, photocatalytic, dielectric, and magnetoresistive properties [17]. The O/N ratio in these structures, which allows to tune their electronic properties, can be as low as 0.25% as in LaWO_0.6N_2.4 [17–19]. However, the synthesis of pure nitride perovskites resulted to be a very challenging task, and only one compound (TaThN₃ [20]) has been reported up to now.

Indeed, crystals with the ABN₃ stoichiometry are extremely rare regardless of the crystal structure. A simple rule of thumb to understand the singularity of this stoichiometry is to think in terms of the standard oxidation states. In order to achieve overall charge neutrality, a −3 oxidation state of nitrogen requires that the cations must be able to assume very high oxidation states to reach a total cationic valence of +9. Such high oxidation states are rarer than those required for oxide perovskites. It is therefore not surprising that the cations in TaThN₃ correspond to the region of the periodic table where the highest oxidation states can be found.

Recently, three novel nitride perovskites (along with other ABN₃ non-perovskite low-symmetry crystals) were obtained theoretically [21] with a combination of high-throughput techniques and global structure prediction [22, 23]. The three perovskites are LaReN₃, YReN₃ and LaWN₃. As could be expected, all these cations are capable of high oxidation states.

The presence of La in the cationic site A in two of these systems opens up the tantalizing possibility of finding other thermodynamically stable compounds by simply replacing it for other lanthanide cations: it is well known that the strong localization of the 4f orbitals of the lanthanides leads to a reduced orbital interaction with other atoms. This implies that there is essentially no effect other than steric in replacing one lanthanide by another, resulting in chemically similar materials but with different electronic and magnetic properties. It is therefore reasonable to expect that LnReN₃ and LnWN₃ compositions, with Ln = Ce, La, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu, can likewise be stable compounds with perovskite structures.

It has been reported experimentally that oxide perovskites with lanthanide cations show interesting magnetic and electronic properties [24–26]. For instance LaNiO₃ and LaFeO₃, which are isostructural to CaTiO₃, are stable structures with exceptional magnetic and electronic properties [27]. These lanthanide perovskites exhibit temperature-depending physical properties like metal-insulator or paramagnetic-ferromagnetic transitions [24, 27]. Other lanthanide-transition metal oxide perovskites such as LnCrO₃, LnVO₃, LnRhO₃ and LnAlO₃ have been scarcely studied. However, similar materials such as perovskite LaCoO₃ show Bose–Einstein condensation of excitons into the lowest energy state (also called the excitonic magnet) [28]. This effect stems from the unique structure-electron-host configuration, making this type of materials a new playground to explore further technological applications.

In this work we study via first-principles calculations whether nitrides including rare-earth metals are chemically stable and if they form perovskite structures. We report that intriguing magnetic properties can arise in these novel and unique materials.

Crystal structure and stability

We consider Re and W cations in the B-site, while for the A-site the entire lanthanide series is explored. Our study includes three perovskite prototypes (with space group $R\bar{3}c,R3c$ and Pnma), which allow for the usual octahedra tilting and distortions. Two other prototypes included in our study, and that do not exhibit the characteristic anion octahedra, are an orthorhombic and a monoclinic crystal structure with space group Ama2 and C2/c. These structures were chosen as they turned out to be the the lowest in energy in our previous study of nitride perovskites [21]. We then fully relaxed all structures allowing for spin-polarization, and calculated the energies of formation and other properties.

The distances to the ternary convex hulls of stability for the resulting structures show that all stoichiometries considered in this work are thermodynamically stable. All the values are tabulated in the supplemental information (SI) available online at stacks.iop.org/JPMATER/2/025003/mmedia. Moreover, the (negative) distances to the convex hull are rather large (mostly between 100 and 300 meV atom⁻¹), and therefore, larger than the typical errors of the theoretical framework used for the calculations on this work. This make us confident that these compositions can indeed be synthesized experimentally under certain thermodynamic conditions.

Tables 1 and 2 summarize the crystal types and symmetries of the lowest-energy configuration found for each composition. Interestingly, many of the compounds adopt a perovskite structure as their most stable configuration, which is particularly evident when Re occupies the B-site, with the Pnma structure dominating for 13 rare-earth elements. For W, on the other hand, only La, Ce and Yb crystallize in a perovskite structure, with the remaining 12 cases adopting a monoclinic (C2/c) phase. The most relevant structures are depicted in figure 1, and the crystallographic details are available in the supplemental information.

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Table 1.  Calculated saturation magnetization (M_s) for LnReN₃ using the LSDA + U approximation and magnetic spin and orbital moment per Ln site. The structure type (P = perovskite, M = monoclinic, O = orthorhombic), space group (Spg) and electronic character (m: metal, s: semiconductor) information is also included.

Material	Type	Spg		$2\langle s\rangle$	$\langle l\rangle$	Ms
				(μB)	(μB)	(MA m−1)
LaReN3	P	Pnma	m	0.0008	0.000 07	0.0001
CeReN3	P	Pnma	m	0.0642	0.043 20	0.0016
PrReN3	P	Pnma	m	0.8204	3.332 64	0.6221
NdReN3	P	Pnma	m	2.1762	4.156 73	0.9574
PmReN3	P	Pnma	m	3.7600	4.241 44	1.2210
SmReN3	P	Pnma	m	4.6322	3.724 93	1.2832
EuReN3	O	Cmcm	s	6.7896	0.012 61	0.7338
GdReN3	P	Pnma	m	6.6995	0.182 87	1.0708
TbReN3	P	Pnma	m	5.8292	1.743 32	1.1903
DyReN3	P	Pnma	m	4.5819	3.628 36	1.2985
HoReN3	P	Pnma	m	3.1787	4.710 87	1.2558
ErReN3	P	Pnma	m	2.5378	4.439 85	1.1178
TmReN3	P	Pnma	m	0.6853	1.394 49	0.3355
YbReN3	M	C2/c	m	0.2270	0.802 24	0.1225
LuReN3	P	Pnma	m	0.0008	0.000 03	0.0001

Table 2.  Calculated saturation magnetization (M_s) for LnWN₃ using the LSDA + U approximation and magnetic spin and orbital moment per Ln site. The structure type (P = perovskite, M = monoclinic, O = orthorhombic), space group (Spg) and electronic character (m: metal, s: semiconductor) information is also included.

Material	Type	Spg		$2\langle s\rangle$	$\langle l\rangle$	Ms
				(μB)	(μB)	(MA m−1)
LaWN3	P	R3c	s	0.0000	0.000 00	0.0000
CeWN3	P	Pnma	m	0.0756	0.061 55	0.0021
PrWN3	M	C2/c	s	0.7862	3.302 20	0.4533
NdWN3	M	C2/c	s	2.2069	4.115 46	0.7099
PmWN3	M	C2/c	s	3.7260	4.332 33	0.9154
SmWN3	M	C2/c	s	4.8440	3.358 19	0.9029
EuWN3	P	R3c	s	4.9425	2.351 47	0.7990
GdWN3	M	C2/c	s	6.7944	0.063 84	0.7978
TbWN3	M	C2/c	s	5.8173	1.696 20	0.8827
DyWN3	M	C2/c	s	4.7049	3.275 75	0.9437
HoWN3	M	C2/c	s	3.2564	4.778 50	0.9554
ErWN3	M	C2/c	s	2.5439	4.967 83	0.8979
TmWN3	M	C2/c	s	1.3418	3.100 13	0.5352
YbWN3	P	R3c	s	0.0019	0.015 11	0.0025
LuWN3	M	C2/c	s	0.0001	0.000 05	0.0000

The volume per formula unit for each system is shown in figure 2. The experimental volume per atom of the corresponding lanthanide elemental metal [29] is shown as well for comparison. Clearly, the volume for each nitride compound follows the trend of the elemental substances: volume decreases with atomic number for both LnReN₃ and LnWN₃. Striking to the eye are the outlier cases of Eu and Yb. This is due to the fact that these two elements are known to assume divalent states [30], in contrast to the other lanthanides that usually form trivalent ions. We would like to point out that the volume is fundamental in understanding the electronic and magnetic properties of these materials (see next section).

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Electronic structure

Figures 3 and 4 show the computed spin-polarized density of states for all LnReN₃ and LnWN₃ compounds. Interestingly, EuReN₃ is a semiconductor while the remaining 14 LnReN₃ systems are metallic. W, in contrast, shows the tendency to form insulating structures, with the sole exception of CeWN₃. There is clearly a correlation between volume and the electronic character: small volumes lead to a metallic character while more open frameworks tend to yield semiconducting systems.

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Most systems exhibit a large magnetic moment, as can be seen from the difference between the up and down densities of states. The exceptions happen at the beginning and the end of the Ln series, where this difference vanishes. In these figures the black lines correspond to the Ln partial density of states, allowing us to observe how the hybridization between 5d-states (Re/W), p-states (N) and 4f–Ln takes place. As expected, all large peaks in the density of states correspond to the localized 4f–Ln states. Moreover, another important consequence of having rare-earth atoms occupying the perovskite A-sites, is that they interact, electronically, very little with the surrounding environment. Clearly, the correlated (atomic-like) 4f electrons remain localized in the vicinity of the valence band, so we can expect that their magnetic properties closely resemble those in the free atoms.

Magnetic properties

It is well known that standard Kohn–Sham DFT with semi-local approximations to the elusive exchange-correlation functional includes a self-interaction error and, since this error increases with the electron localization, its contribution has substantial effects for 4f-electrons. This situation is particularly severe for systems with partially occupied d or f shells and may lead, e.g. to incorrect metallic ground states for insulating systems. Hybrid functionals, which incorporate a fraction of exact-exchange to the Kohn–Sham energy, partially correct this self-interaction error. Another existing approach to alleviate this problem, is to apply locally a Hubbard U to certain atomic sites. We decided therefore to use the LSDA + U formalism in our calculations of magnetic properties. For the Hubbard term we used the most general version that includes independent U and J parameters [31]. We also used the fully localized limit for double-counting [32, 33], and the Perdew–Wang parametrization [34] for the LSDA.

Unfortunately, a remaining concern is the dependence of the results on the on-site Coulomb repulsion U and the exchange parameter J. Extensive studies have shown that in principle the dependence on such adjustable parameters can be attenuated by considering how ground state properties like the electronic gap, magnetic moment and charge distribution, are affected by U [35], and compared to well established experimental values (see for instance [36]). Since we are dealing here with the prediction of novel materials, we can not compare to experiments. We decided therefore to consider the dependence of the electronic band gap and the magnetic moments on U for two systems as measure to determine a suitable value for U. Figure 5 shows the dependence of the total magnetic moment on the value of U for two representative perovskites belonging to the LnReN₃ and the LnWN₃ families. From the results we can observe that a value of U = 8 eV can provide a realistic description of these systems, as the magnetic moment does not changes anymore upon increase of U. Note that this value has also been reported for similar systems (LaNiO₃) [37].

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Figure 6 summarizes the calculated saturation magnetization (M_s), defined as the total magnetic moment per volume, as a function of the lanthanide species. Numerical data is also summarized in tables 1 and 2. The total magnetic moment for both families of compounds increases according to the number of unpaired-electrons in the 4f-shell of the Ln atom. The maximum can therefore be found for GdReN₃ and GdWN₃, as Gd has the largest number of unpaired electrons. It is important to mention that the entire magnetization is provided solely by the Ln cation, as there are no induced moments on the 5d atoms (Re and W).

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The lanthanide series can be considered as divided in two blocks according to their magnetic properties: first, La, Ce, Pr, Nd, Sm and Eu, which are known as the light rare-earth elements, are expected to show weak ferromagnetism. The heavy rare-earths, i.e. elements from Gd to Yb, on the other hand, having more than half of their 4f-electron shell occupied, are ferromagnetic at low temperatures in their elementary phases. This also seems to be the case in nitrides, as seen in figure 6 and in tables 1 and 2.

To evaluate the potential applicability of these systems as hard magnets we need to estimate the required energy to rotate the magnetization from its ground-state direction, called the easy axis, to the direction of maximum energy, the hard axis. This quantity is called the magnetocrystalline anisotropy energy (MAE). Here we estimated the first moment of the uniaxial magnetic anisotropy defined as

$$\begin{eqnarray}&&{K}_{1}=\left|{E}_{[100]}^{\mathrm{tot}}-{E}_{[001]}^{\mathrm{tot}}\right|,\end{eqnarray} \tag{ 1 }$$

where ${E}_{[100]}^{\mathrm{tot}}$ and ${E}_{[001]}^{\mathrm{tot}}$ are the total energies calculated with the spin-orientation along the [100] and [001] directions. The obtained values for selected perovskite systems are condensed in table 3. For Re-nitrides, Nd and Dy create a large anisotropy while maintaining good levels of saturation magnetization. For W-nitrides, Pm, Sm and Ho are examples of compounds that if synthesized could become hard-magnets. The same table includes values of benchmarked compounds in order to test our MAE estimation (see the appendix). As we can see, these materials have values of the magnetization and MAE of magnitude comparable to the best known hard-magnets [38]. The comparison with the state-of-the-art hard-magnet, Nd₂Fe₁₄B is striking. The compound has M_s = 1.28 MA m⁻¹ at 300 K and K₁ ∼ 4900 K J m⁻³ [39], the values of the systems presented in this work are, if not superior, at least comparable to the Nd-based magnets.

Table 3.  Calculated saturation magnetization (M_s) and first-moment uniaxial magnetic anisotropy for outstanding nitride perovskites and selected known hard magnets, experimental values are taken from page 377 of [39].

	Calculated		Experimental
Material	Ms	K1	Ms	K1	Easy
	(M A m−1)	(K Jm−3)	(MA m−1)	(K Jm−3)	axis
SmCo5	1.2	21 800	0.86	17 200	[001]
YCo5	0.95	7071	0.85	6500	[001]
Co	1.55	250	1.44	410	[001]
NdReN3	0.95	54 300	—	—	[001]
DyReN3	1.29	36 525	—	—	[100]
PmWN3	0.91	39 173	—	—	[001]
HoWN3	0.95	28 580	—	—	[100]

As we are dealing with the prediction of magnetic materials, it is pertinent to check the magnetic order of the phases. In order to compare the total energies of ferromagnetic and antiferromagnetic configurations, we created supercells [40] with antiferromagnetic AF1 spin configurations for NdReN₃, DyReN₃, PmWN₃ and HoWN₃ (see the supercell configurations in figure 7). The results of calculations of saturation magnetization, magnetic anisotropy, and easy axis of magnetization for the antiferromagnetic configurations are summarized in table 4. In all these cases we found that the ferromagnetic configuration was energetically lower (see table 4). These calculations were performed using the full-potential linearized augmented plane-wave (FP-LAPW) setup as implemented in the ELK code. We used a fully relativistic scheme (i.e. including spin–orbit coupling and non-collinear spin), together with a on-site Coulomb repulsion U and an exchange parameter J, as defined above.

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Table 4.  Calculated saturation magnetization (M_s), first-moment uniaxial magnetic anisotropy and easy axis of magnetization for selected structures. Energy difference in mHa atom⁻¹ between ferromagnetic and antiferromagnetic (AF1) configurations (negative values means ferromagnetic configuration is the lowest), see figure 7 for AF details.

	MAE			Energy
Material	Ms	K1	Easy	difference
	(M A m−1)	(K Jm−3)	axis	(mHa atom−1)
NdReN3	0.95	54 300	[001]	−3.584
DyReN3	1.29	36 525	[100]	−0.015
PmWN3	0.91	39 173	[001]	−0.707
HoWN3	0.95	28 580	[100]	−1.271

Several important conclusions can be drawn from our results. First let us discuss the possibility of the synthesis of these systems. As indicated before, many of the compounds (and in particular with the light rare-earth elements) are well below the respective convex hull of stability. Taking into account the typical errors of energy of formation that can be expected in DFT calculations [41, 42], we can affirm that a large majority of these compositions (if not all) can indeed be produced under attainable thermodynamic conditions.

Moreover, we performed phonon calculations in the frozen phonon approximation, using phonopy [43] and forces from vasp. This calculation resulted quite challenging. We restricted the convergence of the supercell to a maximum of 600 atoms per cell (approximately 1000 electron system). Furthermore, to alleviate the computational burden, we had to resort to spin-unpolarized calculations. For this reasons we decided to focus only on two interesting materials. Figure 8 display the phonon band dispersion for selected material predictions. The two compounds containing W (C2/c space group symmetry) are dynamically stable and both have phonon frequencies in a similar range.

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From the electronic and magnetic point of view, several of our systems are extremely interesting and could find their way into several technological applications. Let us give a few examples.

Eu is at the limit of weak magnetism, where spin-fluctuations increase and make all Eu-compounds very interesting for many electronic applications. Our nitrides (both Re- and W-based) with Eu have considerable values of saturation magnetization, in contrast to Eu-oxides, which have the tendency to form antiferromagnetic configurations. For instance, the calculated saturation magnetization for EuReN₃ is 0.75 MA m⁻¹. Since EuReN₃ has a non-cubic crystal symmetry and Eu has large values of angular momentum, there is a considerable spin–orbit coupling and rather large magnetic anisotropy energy. This is especially true when compared to the widely used oxide-ferrite-based magnets, such as BaFe₁₂O₁₉ with M_s = 0.38 MA m⁻¹ and K₁ ∼ 330 KJ m⁻³. The compound EuReN₃ could therefore have potential uses as a semiconducting hard magnet, or as a spin filter [44].

The material TbReN₃, on the other hand, exhibits outstanding values of anisotropy and magnetization comparable to early 3d-row elements. Other systems such as NdReN₃and DyReN₃show as well extremely high values for these quantities. It is not only interesting to further study these systems to understand how such values are attained (especially of the record magnetic anisotropy energy), but also as they may have practical applications as metallic permanent magnets. As for possible applications, it is important to mention one possible obstacle: although we did not carried out calculations to estimate the Curie temperature (T_c), we expect it to be rather low (well below room temperature) for the compounds predicted in this work. However, there are suitable ways to increase the T_c for these compounds such as incorporating other magnetic elements or doping with magnetic impurities.

Nitrides containing mid-series lanthanides such as Gd, Tb and Dy together with Re, show values of saturation magnetization comparable to strong ferromagnets. As for these elements the orbital moment vanishes, the magnetic anisotropy energy is extremely low, becoming easy to demagnetize. A possible niche of applications for this class of materials lies in exploiting its strong ferroelectricity and piezoelectric effects [45].

In summary, we predicted novel perovskite nitrides with magnetic lanthanide cations in their sub-lattices. These compounds are thermodynamically stable and, due to the intrinsic diversity of the lanthanide 4f-electrons, they exhibit a variety of electronic and magnetic properties. It is clear that their successful synthesis would enlarge the palette of applications of perovskites. Finally, we certainly expect that properties of these materials could be modified by doping or by creating solid solutions between the different rare-earth nitrides. This opens numerous possibilities to further tune and enhance their electronic and magnetic properties and with that push further the limits of nitride materials.

All structural relaxations were evaluated within density functional theory (DFT) with the Perdew–Burke–Erzernhof (PBE) [46] approximation to the exchange-correlation functional. A plane wave basis-set with cutoff energy of 820 eV was used to expand the wave-functions together with the projector augmented wave (PAW) method as implemented in the Vienna ab initio Simulation Package VASP [47]. Formation energies and distances to the convex hulls of stability (at 0 K) were calculated with the pymatgen python package [48]. For compatibility with the Materials Project database, we used the PAW data sets of version 5.2 supplied with VASP. Geometry relaxations were performed allowing for spin-polarization and starting from a ferromagnetic alignment with tight tolerance criteria for the convergence (forces on the atoms less than 5 meV Å⁻¹). Once the structures were relaxed, all magnetic calculations were performed using the full-potential linearized augmented plane-wave (FP-LAPW) method as implemented within the ELK code [49, 50]. We took into account spin–orbit coupling and performed fully non-collinear magnetic calculation. Finally, the potential and density were expanded in plane-waves with a cutoff of $| {\bf{G}}| =20/{a}_{0}$, and we set ${R}_{{\rm{\min }}}\times | {\bf{G}}+{\bf{k}}|$ to 8, where ${R}_{{\rm{\min }}}$ is the smallest muffin-tin radius. The maximum angular momentum l for the expansion of the wave-function inside the muffin-tins was set to 10. The muffin-tin radii and k-point grids are automatically determined by the code. All the above mentioned parameters are set automatically by invoking 'highq = true' in the elk input file.

JAF-L acknowledges computational resources under the project (s752) from the Swiss National Supercomputing Center (CSCS) in Lugano and resources in sciCORE, scientific computing center at University of Basel. MALM acknowledges partial support from the DFG through project SFB-762. This research was supported by the NCCR MARVEL, funded by the Swiss National Science Foundation.

Magnetic anisotropy benchmark calculations were conducted for SmCo₅, YCo₅ (two important magnets showing high coercitivity) and hcp-Co in order to estimate the numerical accuracy of our approximations. The first two systems require the use of a Coulomb on-site U for Sm and Co. For SmCo₅ the fully-localized limit double counting correction method and a on-site value of 8 eV and a J of 0.5 eV was used. For YCo₅, the around mean filed correction for the double counting was used, together with a U of 2 eV and J of 0.8 eV for Co. Figure A1 shows the number of k-points necessary to converge the MAE for the above mentioned compounds using their experimental volume. As alluded above, MAE is defined as the difference energy between the hard and easy axes of magnetization.

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For hcp-Co, the average converged value is 250 K J m⁻³ using the GGA-PBE functional while the experimental value is 400 K J m⁻³ . This large difference has been already well documented in the works by Daalderop [51], Halilov [52] and Burkert [53]. Cobalt, due to its small anisotropy value represents a very good example to test the numerical convergence with respect to k-points.

For YCo₅ the converged value is ∼7071 K J m⁻³ (3.7 meV) and the experimentally reported value is 6500 K J m⁻³ . Our calculated number is in agreement with the works of Daalderop [54] and Nguyen [55] which reportMAE values between 3 and 4 meV. For SmCo₅ we obtain a MAE of ∼21 800 K J m⁻³ (∼12 meV). Experimentally the reported value oscillate between 13 and 16 meV [56–59]. Other groups report values between 12.6 and 21.6 meV, including the study of Daalderop [54]. In sharp contrast to our results, the previous work by Larson–Mazin [60] not only wrongly determined the easy axis but overestimating MAE. This is due to the relatively poor k-sampling of only 180 k-points. In fact, as one can see from figure A1, more than 1000 k-points are needed to achieve convergence.