MnNiO₃ revisited with modern theoretical and experimental methods Free

Transition metal oxides (TMOs) show a wide range of useful and tunable physical properties that include magnetism, superconductivity, or ferroelectricity. As such, they form the building blocks and find applications in most technologically important devices. Therefore, a fundamental understanding of TMO properties is crucial, and this is where computational modeling plays an important role. MnNiO₃ is one such TMO material that has recently been theoretically investigated for its potential application as an oxygen-evolution photocatalyst.¹ A very interesting feature of MnNiO₃ is that it has two transition metal ions, Mn and Ni, both of which have partially filled d orbitals. Therefore, MnNiO₃ is at the crossroads of two very important families of highly correlated materials that have received intense attention: the nickelates and manganites.

An accurate theoretical study of this material requires an approach beyond that of the standard band approaches. The broadly applied density functional theory (DFT) could in principle be exact; however, common approximations of the density functional, e.g., the local density approximation (LDA)² and semi-local generalized gradient approximation (GGA)³ do not account for all experimental observations without adjustable parameters. In general, those approximations fail to describe the electronic gap of strongly correlated systems (see Ref. 4 and reference therein) because of the so-called self-interaction or delocalization error. The DFT+U method is commonly used to correct for these errors with a Hubbard-type model.^5–7 The theoretical calculations of MnNiO₃ show large variations in the computed band gaps depending on the nature of the functional used for the calculation. For example, the computed band gap using the Perdew-Burke-Ernzerhof PBE+U functional (where U is 3.9 eV and 6.2 eV for Mn and Ni, respectively) was found to be 1.75 eV, while the Heyd-Scuseria-Ernzerhof (HSE) functional yielded a gap of 2.98 eV.¹ Furthermore, the use of functionals such as the HSE requires an adequate choice of the mixing parameter α which needs to be determined empirically if accurate values are to be obtained. The same applies for the PBE+U functional, where the choice of the U parameter could be somewhat arbitrary.⁸ Hence this reliance on empiricism poses limitations on the quantitative prediction of the band gap of any material whose experimental value is unknown. While usual DFT approximations encounter problems in TMOs, quantum Monte Carlo (QMC) solutions of the full many-electron problem—without adjustable parameters—have recently become practical for bulk TMOs and have provided accurate values for defect formation energies in multiple oxides.^9–13 We discuss in more detail the accuracy of DFT compared with QMC in Secs. III and IV.

Previous experimental reports suggest that although MnNiO₃ crystallizes in the ilemenite-type structure (space group R-3), oxygen vacancies are known to exist:¹⁴ the activation energy of MnNiO_2.98 was estimated using conductivity and thermopower results as 1.4 eV.¹⁴ In rhombohedral MnNiO₃, both the Mn and Ni have an octahedral environment, and Mn and Ni are expected to be in +4 and +2 oxidation states, respectively.^15–17 The ground state magnetic structure is ferrimagnetic, where the Mn and Ni sublattices are antiferromagnetically aligned with respect to each other. Mn and Ni have different resulting moments, which result in a net magnetic moment in the material.

In this report, we calculate some relevant properties of MnNiO₃ with an ab initio many-body method: diffusion Monte Carlo (DMC).^18,19 We also revisit the synthesis of MnNiO₃, in order to assess its potential use as a photocatalyst. We report the details of its crystal structure and bulk magnetic susceptibility. We use DMC to investigate the equilibrium volume, bulk modulus, the quasiparticle band gap, and the formation energy of the antisite pair defect theoretically. Experimentally, although stoichiometric composition is suggested by the refinement of X-ray diffraction patterns, the ordering of the Mn and Ni cations along the crystallographic c-direction may not be perfect, presenting antisite defects. The gray fine-powder produced is presumably a bad metal. The Tyndall effect and off-stoichiometry presumably hinder the determination of the band gap with reflectivity measurements. We find good agreement between the DMC-computed value for the equilibrium volume and the experimental value. The rest of this paper is organized as follows: in Sec. II, we describe the experimental and theoretical methods used in this study. Results are presented in Sec. III, and we summarize our conclusions in Sec. IV.

MnNiO₃ was synthesized following similar routes to Ref. 14. The purity of the material was checked by collecting X-ray powder diffraction on an X’Pert PRO MPD diffractometer (Cu K_α1 radiation); the lattice parameters, atomic sites, and their occupancies were determined by least-squares fitting within the program package WinCSD. The chemical and particle homogeneity of the powder was checked using a Hitachi S3400 scanning electron microscope operating at 20 kV and by the use of energy-dispersive X-ray spectroscopy (EDS). The temperature- and field-dependence of the magnetization were performed with a Quantum Design magnetometer using the normal and oven settings, under an applied field of 100 G; the data were collected on the powders in zero-field-cooled (zfc) and field-cooled (fc) modes.

In this section, we provide a brief description of the basic steps followed in a DMC calculation. A detailed description of the methodology can be found elsewhere.^20,21 To achieve accurate DMC ground state energies, variational Monte Carlo (VMC)²² is first performed to optimize a many-body trial wave function, ψ_T(R) as

where $e−J1R$ and $e−J2R$ are the one-body and two-body Jastrow²³ terms, respectively. These represent the electron-ion and electron-electron correlation terms. To further improve the description of the correlation between electrons, a three-body Jastrow term is also used, $e−J3R$⁠, which represents the electron-electron-ion correlation. The determinants $D↑$ and $D↓$ are for the up and down spin sates, respectively. In this work, we obtain determinants $D↑$ and $D↓$ using orbitals from LDA+U calculations performed with Quantum ESPRESSO.²⁴ The important step here is to optimize the Jastrow factors by minimizing the variance of the VMC energy. We next find the value of U which minimizes the DMC energy.

The resulting optimized trial wave function serves as the starting point for the subsequent production of DMC calculations. The need for DMC arises from the fact that VMC is limited by the functional form of the trial wave function, ψ_T(R), and the ability to optimize this trial wave function. This is where DMC proves to be vital. Within DMC, the wave function is evolved according to the Schrödinger equation which is transformed into a diffusion equation in imaginary time. When the wave function evolves in imaginary time, the ground state is projected out as the higher-energy excited states decay exponentially in time. The space of electronic configurations is explored with an ensemble of configurations, which follow quasi-independent random walks. In DMC, the random walk is guided by an importance sampling approach^18,25,26 which makes the process more efficient. After the configurations are equilibrated, the average of their energies is statistically analyzed.

The fixed-node approximation is used to allow DMC to be applied to fermions, where the well-known sign problem is controlled by using fixed nodes from a many-body trial wave function. The nodal surface of these wave functions is determined by the zeros of the Slater determinants $D↑$ and $D↓$ $(D↑D↓=0)$ as indicated in Eq. (1). This approximation is variational where nodal errors only increase the DMC energy. Another approximation arises from the use of pseudopotentials. All-electron calculations are computationally very demanding and often impractical for QMC calculations. In Sec. II C, we describe the pseudopotentials used in our calculations and the various tests that are performed for validation.

The accuracy of DMC is also affected by errors introduced by a finite simulation cell. The simulation cell is subject to periodic boundary conditions, and finite size errors are present in all many-body calculations of extended systems. The finite size errors can be broadly classified into one-body and two-body errors. One-body errors arise due to the discrete sampling of the Brillouin zone. The two-body errors arise due to spurious correlation effects with an electron and its image in the neighboring periodic cell. In Sec. II D, we describe the corrections used in this work.

The effects of the pseudopotential approximation on the final DMC results can be significant,³¹ and it is therefore necessary to carefully test the pseudopotentials. Previous studies have found that the effects of semi-core states are pronounced for transition metal elements.^32,33 Therefore, we use a Ne-core for Ni and Mn and a He-core for O. Our pseudopotentials are norm-conserving³⁴ and generated within the LDA with the OPIUM code.³⁵ Scalar relativistic effects are included in the pseudopotential construction. The nonlocal cutoff radius is constrained to be small (0.8 a.u.) to improve transferability, and the resulting potentials are quite hard. To soften the potentials to the extent possible, we use the optimization method of Rappe et al.³⁶ resulting in plane wave cutoff energies of 280 Ry and 269 Ry for Ni and Mn, respectively, to achieve an accuracy of 1 meV/electron.

These pseudopotentials were validated with DMC for (i) atomic ionization potentials for manganese, nickel, and oxygen and (ii) equilibrium distances and dissociation energies of MnO, NiO, and O₂ molecules. DMC yields ionization potentials and dissociation energies within 0.2 eV of experimental values.³⁸ The DMC equilibrium distance for MnO of 1.668(2) Å compares well with the experimental value of 1.648 Å, similar to the DMC equilibrium distance for NiO of 1.625(2) Å compared with the experimental value of 1.6271 Å. We have recently studied the magnitude of the locality and fixed-node errors in this pseudopotential family and their Jastrow sensitivities.³⁷ Additional details of our Mn, Ni, and O pseudopotentials can be found in Refs. 10 and 39.

All DMC calculations were performed using the QMCPACK code.³⁹ Single-determinant Slater-Jastrow trial wave functions were generated with one-, two-, and three-body Jastrow factors [as represented in Eq. (1)]. All DMC calculations used the variational T-move scheme.⁴⁰ The single particle orbitals were generated within LDA+U^5,41 using a plane wave cutoff energy of 300 Ry using Quantum ESPRESSO.²⁴ Such a high cutoff energy is needed due to the hard pseudopotentials used (see Sec. II B). Unlike many studies where the value of U is chosen empirically, here we scan over U values to find those that minimize the DMC energy. In this way, we use U as a variational parameter to improve the nodal structure of the wave function since all nodal errors increase the DMC energy. When optimizing the nodes by varying the U, the magnetic primitive cell (Mn₂Ni₂O₆) was used. A larger 2 × 2 × 1 40-atom supercell (4× the primitive magnetic cell) was used for all subsequent calculations to minimize finite size errors. DMC energies for various combinations of U values in the magnetic primitive cell are shown in Table I. From our scan, U values of 6.2 eV (Mn) and 2.9 eV (Ni) yield the lowest DMC energy and are used for all subsequent calculations. All energies are presented per formula unit.

To evaluate the equation of state (EOS) of MnNiO₃, we use a time step of 0.0025 Ha⁻¹, which converges the DMC total energy of MnNiO₃ to 9 mHa/formula unit. Residual time step energy differences are expected to be about an order of magnitude smaller than this. Figure 1 shows a time step convergence study for the 40-atom 2 × 2 × 1 supercell of MnNiO₃. We use twist-averaged boundary conditions to correct for the one-body finite-size effects,²⁷ and we use the model periodic Coulomb (MPC) interaction^28–30 to correct the potential energy, while the applied S(k) correction⁴² corrects the kinetic energy. The simulation workflow was managed by Nexus.⁴³ 

The MnNiO₃ crystal structure was refined using the R-3 space group with refined lattice parameters of a = 4.9076(2) Å and c = 13.5966(6) Å. Ni and Mn were refined in atomic positions with z = 0.3455(6) and z = 0.1461(6) and oxygen site was refined with occupancy of 1.0(2). According to EDS, there is a uniform distribution of Mn, Ni, and O elements and uniform particle sizes averaging less than 2 μm. However, the EDS averaging over a spot of sample of 753 μm diameter hints at a slightly (∼5%) Mn-rich material that potentially means off-stoichiometric Ni_0.95Mn_1.05O₃, and with Mn being smaller than Ni in terms of atomic radius, this can easily cause antisite defects. The Curie temperature (T_c) is evident from divergent zfc and fc data and the rise in the magnetic susceptibility (χ) signal below T_c = 430 K (Fig. 2). This value is similar to that reported in Ref. 14. The moment does not fully saturate at 4 K and 6 T (Fig. 2, inset), and the magnetization reaches a small value of ∼0.3 μ_B/MnNiO₃ formula unit, which hints at ferrimagnetic ordering. Field-dependent magnetization is linear at 750 K.

Diffuse reflection measurements were performed on a powder sample of MnNiO₃, which may be slightly Mn-rich (up to ∼5%) from EDS measurements. We observed an abrupt transition in reflectance in the infrared near 1600 nm (=0.78 eV), where the reflectivity at 1900 nm was 0.63 and was 0.136 at 1100 nm. According to the Kubelka-Munk theory,⁴⁴ the ratio of the absorption to scattering coefficient (K/S) equals 0.068 at 1900 nm and 0.373 at 1100 nm. However, the abrupt reflectance at 1600 nm is consistent with the size of the MnNiO₃ particles observed in a Hitachi S3400 scanning electron microscope (not shown); therefore we cannot rule out Tyndall effects to be the source of the observed reflectance.

To calculate the bulk modulus and equilibrium volume of solid MnNiO₃, we compute the total energy as a function of volume as shown in Fig. 3. The resulting plot is then fitted to Murnaghan’s equation of state,⁴⁵ 

where E_T(V) is the energy at volume V, and V₀, B₀, and $B0′$ are the volume, bulk modulus, and its pressure derivative, respectively, at equilibrium. Upon fitting the computed DMC values to Eq. (2), we obtain an equilibrium volume of 92.8 ų compared with the experimental value of 94.4 ų. We further predict a bulk modulus of 217 GPa. Corresponding LDA+U calculated values are 90.5 ų and 228 GPa.

We see an improvement in the calculated equilibrium volume using DMC (1.7% error) compared with LDA+U (4.1% error). A study on NiO¹¹ has shown a similar geometric improvement, where the error in the lattice constant went from 1.7% error (LDA+U) to 0.3% error (DMC). This can be attributed to the consistent description of the electronic correlation made in DMC, and also that geometric properties for some transition metal oxides depend strongly on the value of U.⁸ However, errors remain, presumably due to the pseudopotential, localization, and fixed-node approximations.

The quasiparticle energy gap E_qp can be computed as the difference between the addition energy (E_A) and the removal energy (E_R) as follows:

where E_g, $Eq−$⁠, and $Eq+$ are the DMC energies of the ground state, negatively charged state, and positively charged state of bulk MnNiO₃, respectively. This method is commonly used to calculate the quasiparticle energy gap in DMC.^9,10,46–48 For a more complete discussion, the reader is referred to Williamson et al.⁴⁹ 

As mentioned earlier, MnNiO₃ stabilizes in the ferrimagnetic ground state. Within LDA+U, we find the computed magnetic moments on Mn and Ni atoms to be −2.9 μ_B and +1.7 μ_B, respectively. Considering the electronic configurations of Mn and Ni atoms are 3d⁵4s² and 3d⁸4s², respectively, the oxidation states of Mn and Ni atoms are 4+ with three d-electrons in the spin down channel while those of Ni atoms are 2+ with eight d-electrons, 5 spin up and 3 spin down. In determining the quasiparticle gaps, one can therefore create the charged states by adding or removing electrons for two separate spin channels. In Table II, we present the results for both the DMC and LDA+U calculations. We perform calculations on three supercell sizes, and the LDA+U results show a variation of about 0.2 eV between the smallest (10 atoms) and the largest supercell (40 atoms). Within DMC, we obtain quasiparticle gaps of 2.0(5) eV and 3.8(6) eV for majority and minority spin channels, respectively.

It has proven to be challenging to synthesize pure MnNiO₃. Calculations were performed to rationalize which impurities in MnNiO₃ are more likely. Using previously calculated energies for MnO₂⁵⁰ and NiO,⁵¹ the reaction

gives a formation energy of −0.997 eV (LDA+U) and −1.857(1) eV (DMC) per formula unit. The formation of an antisite pair, where one Mn swaps location with one Ni, costs +2.021 eV (LDA+U) and +2.8(1) eV (DMC). The formation of an oxygen vacancy by the reaction

is found to be +3.751 eV (LDA+U). This corresponds to approximately a 4% concentration of oxygen vacancies (not extrapolated to the low-density limit). In oxygen-rich conditions, we have

In oxygen-poor conditions, we have

We find $ΔμOrich−μOpoor$ to be +2.624 eV (LDA+U). We also compare the gaps between bulk MnNiO₃, MnNiO₃ containing an antisite pair, and the Mn-rich conditions Mn_1.06Ni_0.94O₃ that more closely reflect the experimental stoichiometry. We find that while there is a well-defined gap for the bulk MnNiO₃, MnNiO₃ with an antisite pair and Mn antisite Mn_1.06Ni_0.94O₃ are both metallic in LDA+U.

In this work, we present new experimental data and ab initio results for MnNiO₃. High-level diffusion quantum Monte Carlo calculations resolve the functional-dependent discrepancies in previous reports using approximations of density functional theory. DMC reproduces the experimental results available accurately and provides insight into the experimentally observed imperfect stoichiometry.

We have theoretically and experimentally revisited a relatively unexplored material—MnNiO₃—and considered its properties as a potential photocatalyst. We performed highly accurate diffusion quantum Monte Carlo calculations of the equation of state and the quasiparticle band gaps for minority and majority spins of MnNiO₃. We synthesized MnNiO₃ as evidenced by the lattice structure. Experimentally, the ferrimagnetism found in MnNiO₃ (T_c = 430 K) is consistent with previous reports. A defective material seems to form with potential oxygen vacancies and Ni and Mn off-stoichiometries, which cause antisite defects. We found that the theory and experiment agree very well for the structural properties and noted an improvement of 2.4% in the calculated equilibrium volume using DMC compared with LDA+U due to the accurate treatment of the electron correlation in DMC. Despite this improvement, the DMC-calculated volume is still not reaching the experimental value. Common sources of suspected errors are the fixed-node approximation, the pseudopotentials, and locality errors in the pseudopotential evaluations. Locality errors can be reduced in the future since our Jastrow factors are within an atomic picture. Adding electron-ion-ion terms and/or electron-electron-ion-ion terms to the wave function could reduce locality errors. However, Jastrow terms of this form are not yet available in QMCPACK.

The calculated excitation band gap obtained with DMC without adjustable parameters was found to be 2.0(5) eV, somewhere in between the calculated values obtained with empirically adjusted functionals [LDA+U (1.75 eV) and HSE (2.98 eV)].¹ Calculated DMC band gaps can be used to determine which functional is more appropriate for this class of materials. The experimental determination of the optical properties was hindered by the difficulty to control the Mn/Ni ratio and to simultaneously produce large enough (mm) insulating crystals. We find theoretically that the formation energy of an antisite pair, an exchange of two neighboring atoms of Ni and Mn, costs approximately 1 eV less than the formation of an oxygen vacancy. We find that these antisite pairs, as well as any departure from the perfect stoichiometry turn the material metallic-like.

The authors are grateful to Michael McGuire for helpful discussions. This work was supported by the Materials Science and Engineering Division of the U.S. Department of Energy Office of Science (BES). This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Lab, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).