Elastic Properties of Binary d-Metal Oxides Studied by Hybrid Density Functional Methods

: Detailed understanding of the elastic properties and mechanical durability of ceramic materials is crucial for their utilization in advanced microelectronic or micro-electromechanic devices. We have systematically investigated the elastic properties of 97 binary d-metal oxides using hybrid density functional methods. We report the polycrystalline and single-crystal bulk moduli and the symmetrized elastic constants of the studied oxides and compare the elastic properties with experimental information where available. We discuss the periodic trends of several key structure types, namely, rutile, corundum, and rocksalt, in detail. The calculated bulk moduli and elastic constants of the nonmagnetic and magnetic d-metal oxides are in reasonable overall agreement with experiment, but some materials show relatively large discrepancies between the calculated and experimental bulk moduli. In several cases, such as MnO, CoO, NiO, ReO 3 , and ZrO 2 ( tP 6), some of the elastic constants calculated for ideal single crystals at 0 K are clearly different from the experimentally determined elastic constants.


■ INTRODUCTION
−7 Understanding the elastic properties of d-metal oxides is thus relevant for engineering them toward new applications and devices.Elastic constants of a material are usually represented in a tensor form using Hooke's law, which constitutes a relationship between the stress and the strain and second-order elastic constants. 1,8rystal symmetry reduces the elastic tensor to 21 independent components that are further reduced according to the Laue class of the crystal.
The elastic constants can be measured experimentally by using several different techniques. 9Mechanical tests can be used to find out the relation between the applied stress and strain.In dynamic techniques, the resonant frequencies of a material are measured, and a relationship between them and the elastic constant is established.Furthermore, the time-offlight of an elastic wave in a material can be determined and used to obtain elastic constants.In addition to elastic constants, bulk modulus is a key quantity when considering the elastic properties of materials.Bulk modulus (K) represents the resistance of a material to uniform compression and is defined as the ratio of pressure change (dP) and change (dV) of volume (V), K = −V(dP/dV).−12 Second-order elastic constants can be determined quantum chemically by studying the derivatives of total energy as a function of crystal deformation.This can be achieved either numerically or analytically, yet most current ab initio approaches are based on numerical techniques.For example, the implementation in the periodic CRYSTAL code is based on strain second derivatives obtained from analytical first derivatives of the energy for both lattice parameters and atomic positions, coupled with relaxation of atomic positions. 13The acquired elastic compliance tensor can be used to obtain the elastic constants.From these the bulk moduli, Young's moduli, and shear moduli are acquired using the Voigt-Reuss-Hill approximation 14 and correspond to values for a polycrystalline sample.In addition to the approach of obtaining the polycrystalline bulk modulus via elastic constants, the CRYSTAL code provides an equation-of-state procedure that scans over the volume in order to compute energy versus volume curves that are used to fit various equations of state such as Murnaghan's or Birch−Murnaghan's. 15The equationof-state approach, in turn, yields values of the bulk moduli corresponding to a single-crystal sample.The CRYSTAL code and hybrid density functional (DFT) methods have recently been used to study the elastic properties of highly correlated cerium oxides CeO 2 and CeO 3 , 16 wurtzite-type ZnO under pressure, 17 and of various silicate garnet minerals. 18Other recent examples of using DFT for the investigation of elastic properties include a large database of full elastic information on inorganic crystalline compounds using the VASP code. 19lastic properties can also be studied with many other codes such as CASTEP, ABINIT, and CP2K. 20Typically DFT calculations of elastic properties assume a perfect crystal.While from a materials engineering point of view the presence of defects in small atomic concentrations might not be relevant, they might still affect the measurement of elastic properties when high-precision methods are used. 21This can, in addition to temperature effects, explain some of the deviations between calculated and experimentally determined values of elastic constants.
We have recently carried out a systematic computational study on the crystal structures, magnetic ground states, and fundamental electronic properties of 100 binary d-metal oxides known at atmospheric pressure by using a hybrid DFT-PBE0 method. 22Hybrid DFT methods generally treat the valence bands of d-metal oxide materials near the Fermi level more accurately compared to nonhybrid methods, 23 resulting in an improved description of the localization of the d electrons.Structural properties of the studied d-metal oxides were accurately described by the DFT-PBE0 method.Here we use hybrid density functional methods to investigate the elastic constants and bulk moduli of both magnetic and nonmagnetic d-metal oxides.The structure−property correlations within different structure types are investigated in detail.We systematically compare the calculated elastic properties with available experimental and computational data in the literature.

■ COMPUTATIONAL DETAILS
All quantum-chemical calculations have been carried out using the CRYSTAL17 program package 24 and hybrid PBE0 density functional method (DFT-PBE0, 25% exact exchange). 25,26ocalized Gaussian-type triple-ζ-valence + polarization (TZVP) basis sets, based on Karlsruhe def2 basis sets, 27 were used for all elements.All basis sets used here are available in ref 22.The same Monkhorst−Pack-type 28 k-meshes and magnetic ground states were used as in ref 22.For magnetic oxides, the space group used in the calculations might differ from the crystallographic space group, but the crystallographic space group is used here for labeling of the crystal structures.Full structural details are reported in ref 22. Tight tolerance factors (TOLINTEG) of 8, 8, 8, 8, and 16 were used for the evaluation of the Coulomb and exchange integrals (for Rh 2 O 3 (oP40) and WO 3 (mP16), even tighter values of 10, 10, 10, 10, 20 had to be used).Default CRYSTAL17 geometry optimization convergence criteria and DFT integration grids were used.For RuO 2 , the calculations with the larger TZVP basis set had to be carried out in space group P1 to avoid selfconsistent field (SCF) convergence issues.The second-order elastic constants, together with bulk moduli corresponding to a Reuss model for polycrystalline samples for the optimized geometries reported in ref 22, were obtained with the fully automated procedure implemented in CRYSTAL (ELAST-CON keyword). 13,15For systems crystallizing in the rutile structure type, the elastic constants were also evaluated with an alternative scheme with an analytical evaluation of the nuclearrelaxation term using an interatomic force constant Hessian matrix (keywords NUCHESS and HESSNUM2). 29,30Re-Figure 1. Bulk moduli for the studied d-metal oxides (GPa).DFT-PBE0/TZVP results in blue, experimental values in yellow (with error bars, where available).The oxides are ordered by the group of the d-metal (groups 3 to 12).References for the experimental data are in Table 1.
ported elastic properties have been obtained for three points and a second-order fit.Bulk moduli were also calculated with the equation-of-state (EOS) fitting approach implemented in CRYSTAL 15 to obtain values corresponding to experimental single-crystal measurements.The default settings of CRYS-TAL17 were used for the EOS calculations (unit cell volume range ±8%).For some materials, the EOS calculations had to be carried out with reduced range (unit cell volume range ±5%) to avoid SCF convergence issues, and in some cases TOLINTEG had to be increased to 10 10 10 10 20, and the geometry was reoptimized before the EOS calculation.These calculations were carried out with CRYSTAL23. 31Full technical details for all elastic property calculations are openly available as CRYSTAL output files at a Github repository. 32onstochiometric rutile-type oxide TaO 2 not previously reported in ref 22 was also included, the calculations being performed with a k-mesh of 8 × 8 × 12.

■ RESULTS
We studied the elastic properties of almost all d-metal oxides studied in ref 22. TiO (hP6) was not included in this study due to its nonstoichiometry.Mn 2 O 3 (cI80) and Mn 2 O 3 (oP80), which are effectively the same structure with different magnetic ordering, were not included due to technical problems with elastic constant calculations on the fairly large unit cells with complex magnetic ordering.
We first discuss the calculated bulk moduli and investigate periodic trends within structure types important for d-metal oxides.In the case of second-order elastic constants, we focus on comparisons with available experimental data.The full elastic tensors for the studied materials are available as Supporting Information (as CRYSTAL output files). 32The densities of the optimized structures were also employed in the analysis of the elastic properties and are included as Supporting Information.
Bulk Moduli.The DFT-PBE0/TZVP bulk moduli of all the studied compounds along with the available experimental data are presented in Figure 1 and Table 1.The reported calculated bulk moduli correspond to the Reuss model.Bulk moduli of some binary d-metal oxides have also been reported in previous computational studies, but here we focus on comparisons to experimental bulk moduli (see, e.g., TiO 2 , 33 NiO, 34 and other d-metal monoxides 35 ).
The bulk moduli for the rutile, distorted rutile, corundum, and rocksalt structure types are discussed in detail below due to interest in these structure types and abundance of experimental data.A comprehensive list of all calculated bulk moduli of the investigated oxides is given in the Supporting Information.
Rutile and Distorted Rutile Structure Types.Rutile is the thermodynamically most stable polymorph of TiO 2 and gives its name to the rutile-type structure of AB 2 oxides and fluorides in the tetragonal space group P4 2 /mnm. 75Distorted versions of the rutile structure type are also common for d-metal oxides such as MoO 2 or WO 2 . 76Typically, in this distortion, the metal atoms appear in pairs along the c axis of the rutile pseudocell, and the resulting structures belong to monoclinic space groups (P2 1 /c for MoO 2 -type).Of these rutile-structured oxides, IrO 2 , RuO 2 , and RhO 2 show metallic character. 77,78n addition to exhibiting the highest bulk moduli of the oxides investigated here, the rutile structure type is characterized by high anisotropic compressibility with the a axis experimentally observed to be approximately twice as compressible as c. 69 While both the prototypical rutile (tP6) and anatase (tI12) polymorphs of TiO 2 consist of TiO 6 The oxides are ordered according to periodic rows (3d, 4d, and 5d).

Crystal Growth & Design
octahedra, the packing is different, explaining the higher bulk modulus and compressible anisotropy of rutile. 33In rutile, the octahedra share edges along the c direction and corners along the ab planes.Corners share just one Ti−O bond while edges share two, making the c direction more compressible.In anatase, the octahedra are arranged in a zigzag motif with the sharing of four edges, but the edges do not lie at opposite ends like in rutile, allowing for more uniform compressibility of the structure.
The calculated and experimental values of the bulk moduli for rutile-type oxides are summarized in Figure 2. A general trend of the bulk moduli slightly increasing with the density is observed both experimentally and in our computational results.Likewise, an increase in the bulk moduli is associated with the increasing value of the Pauling electronegativity of the d-metal for corresponding oxides (see Supporting Information, Figure S1).For ReO 2 , both a distorted rutile (MoO 2 -type, mP12) and a rutile structure (tP6) are reported. 79,80In our geometry optimization we observed a large difference of about 14% for the lattice constant c of rutile-type ReO 2 (tP6). 22However, despite this, the rutile-type structure shows a similar bulk modulus compared to the other rutile and distorted rutile structures.The distorted rutile polymorph of ReO 2 exhibits a smaller bulk modulus than the other similar oxides.
Rutile-type VO 2 existing at elevated temperatures has been experimentally found to be metallic, while our calculations yield a band gap of 2.8 eV. 22This could indicate nonstoichiometry, which would explain the larger discrepancy between the experimental and calculated value of the bulk modulus.
In order to compare the two implemented approaches for calculation of elastic properties in CRYSTAL, the NUCHESS approach was tested for all the rutile-structured oxides that crystallize in the ideal P4 2 /mnm space group (see Computational Details).The initial reason for this test was that the standard ELASTCON approach failed for CrO 2 , RhO 2 , and RuO 2 .For VO 2 , MnO 2 , IrO 2 , TaO 2 , and ReO 2 the values of the two different approaches are effectively the same, but for TiO 2 , the NUCHESS method yielded a value of 230 GPa that is 7 GPa smaller compared to the standard approach.
Corundum Structure Type.The corundum-type structures of d-metal oxides in trigonal space group R3̅ c derive their name from the structure of Al 2 O 3 . 81Ti 2 O 3 V 2 O 3 , Cr 2 O 3 , Rh 2 O 3 , and α-Fe 2 O 3 crystallize in the corundum structure type.For Ti 2 O 3 , we previously observed an imaginary frequency upon harmonic frequency calculations, 22 and the structure (C1c1) resulting from subsequent distortion along the imaginary mode has been used for the calculation of elastic properties.
The calculated and experimental values are summarized in Figure 3. Like for rutile-structured oxides, the increase of the dmetal Pauling electronegativity generally coincides with an increase of the bulk modulus of the oxide (see Supporting Information, Figure S2), albeit for Fe 2 O 3 this value is clearly below the trend of other 3d oxides.Unlike observations in the experimental measurements of Sato et al., 45 the bulk modulus of V 2 O 3 is not noticeably smaller compared to other corundum-structured oxides.Instead, the value for Ti 2 O 3 is smaller, likely due to the further lowering of the symmetry due to the magnetic configuration and distortion along the imaginary mode.This smaller value is also observed experimentally. 37he bulk modulus of corundum-structured Rh 2 O 3 has previously been reported to be smaller by about 1 GPa than that of rutile-structured RhO 2 at the DFT-PBE/FPLAPW level of theory. 82This is also corroborated by our results, with the difference being 8 GPa and bulk modulus values of Rh 2 O 3 and RhO 2 being 256 and 264 GPa, respectively.
Rocksalt Structure Type.The cubic rocksalt structure type (Fm3̅ m) derives its name from NaCl and is adopted by a number of d-metal oxides with divalent metal cations; MnO, CoO, NiO, and CdO. 83FeO would also adopt a similar structure, but it is excluded in our studies due to nonstoichiometry.
The calculated and experimental values are summarized in Figure 4.The antiferromagnetic ordering of MnO, CoO, and NiO lowers the space group from cubic to trigonal in the calculations, but the values of the bulk moduli are in reasonable agreement with the experimental values.For the nonmagnetic oxide CdO, a good agreement is also observed.Our reported values have a trend of being larger than previously reported values at the DFT-PBE0/FP-(L)APW+lo level of theory (143 GPa for MnO, 167 GPa for CoO, and 187  GPa for NiO) 35 but offer a slightly better agreement with the experimental results.For CdO, the reported calculated value is almost identical to the previously reported DFT-LDA computational value of 158 GPa. 84The difference in bulk moduli of the 3d rocksalt oxides and 4d CdO has previously been attributed to the increased electronegativity compared to ionic radius in CdO. 52olycrystalline Versus Monocrystalline Bulk Moduli.In order to compare the difference between polycrystalline and monocrystalline bulk moduli, we also calculated monocrystalline bulk moduli using the Birch−Murnaghan equation-of-state (EOS).The unit cell volume changes included in the EOS procedure resulted in serious SCF convergence issues for some of the studied metal oxides, especially metallic rutile-type oxides.In the end, we were able to obtain monocrystalline bulk moduli for 86 d-metal oxides.The data are listed in the Supporting Information (Tables S1−S3), and both the polycrystalline and monocrystalline bulk moduli are compared to experimental bulk moduli in Figure S4.For most of the studied materials, the differences of the polycrystalline and monocrystalline bulk moduli are small (0−2%), but larger differences of more than 5% were obtained for some triclinic, monoclinic, and orthorhombic oxides such as ZrO 2 (mP12) with 6% difference.The largest deviations were obtained for the polymorphs of WO 3 , especially for oP32 (+22%).
Elastic Constants and Moduli.Symmetrized elastic constants and elastic moduli compliance tensors were also obtained for all the structures discussed above.As experimental data is scarcer than for bulk moduli, the results here introduce just a few selected oxides for which experimental information could be found.Full elastic constants and compliance tensors of all investigated oxides are given in the associated GitHub repository. 32While computational studies have also been performed with DFT for a number of d-metal oxides including ZrO 2 , 85 ZnO, 86 and Cu 2 O, 87 we discuss here only oxides for which the experimental values of elastic constants are available.
Rutile TiO 2 .The calculated and experimental values of the elastic constants of rutile-type TiO 2 are presented in Table 2.The calculated values are generally larger than the experimental ones, with C 33 showing the largest difference.However, the experimental values also vary quite considerably.Employing a DFT-PBE approach, Iuga et al. report a better fit with the experimental values, especially for C 33 . 88Our C 33 value is roughly 1.8 times larger than C 11 , corresponding with the observation of Hazen et al. 69 that, for rutile-type oxides, the a axis is close to twice more compressible than the c axis.The ratio is similar for some rutile-type oxides such as OsO 2 (1.9) and RhO 2 (1.8) but notably lower for CrO 2 (1.2), TaO 2 (1.2), and NbO 2 (1.1).
Harmonic frequency calculations for TiO 2 at the PBE0-TZVP level of theory yield an imaginary vibration of 76i cm −1 .It is well-known that with DFT despite of Hamiltonian choice the anatase phase is found to be more stable than rutile, while experimental studies indicate the opposite. 91Despite this instability indicated by frequency calculations, the calculated elastic constants indicate that rutile TiO 2 is mechanically stable since the Born stability criteria are satisfied C 11 > |C 12 |, 2C 13 < C 33 (C 11 + C 12 ), C 44 > 0, C 66 > 0. 92 Wurtzite-type ZnO.Zinc oxide ZnO exists in the hexagonal wurtzite and cubic zinc blende structures, 86,93 of which the wurtzite polymorph is thermodynamically more stable.The structure type is also adopted by various semiconductor materials such as GaN.The calculated elastic constants of wurtzite-type ZnO, presented in Table 3, show good agreement with measured values for both bulk 94 and thinfilm samples. 95Our values are also comparable with the previous results obtained with DFPT 93,96 and DFT-LDA. 86he nonindependent C 66 constant, defined as (C 11 − C 12 )/2, is usually hard to determine experimentally. 95irconia ZrO 2 .Zirconia ZrO 2 is a material of interest due to its optimal mechanical, chemical, and dielectric properties. 85It also exhibits considerable polymorphism under temperature and pressure variations.The room-temperature polymorph is monoclinic.The elastic tensor of monoclinic zirconium oxide takes a more complex form due to the low symmetry of the compound in the P2 1 /c (14) space group.The calculated and experimental values tabulated in Table 4 are in good agreement for the diagonal elements (C 11 to C 66 ), but some of the other constants, such as C 15 and C 25 , show larger discrepancies.Similar performance is observed with DFT-PBE/PAW by Fadda et al. 85 Rocksalt Structure Type.Cubic symmetry limits the number of experimentally measurable elastic moduli compo-

Crystal Growth & Design
nents to three, C 11 , C 12 , and C 44 . 98For antiferromagnetic MnO, NiO, and CoO calculations have been carried out in the trigonal space group R3̅ m to reproduce the antiferromagnetic ordering.Due to this lowering of symmetry from the cubic symmetry in space group Fm3̅ m, the elastic constants and moduli cannot be directly compared with experimental data.
To compare the calculated and experimental elastic moduli components we utilized a projection method previously utilized for Ti 0.5 Al 0.5 N and CrN 99,100 5.The full elastic moduli components in the trigonal symmetry are available in the Supporing Information (Table S4).
Despite the symmetry breaking, the bulk moduli of the studied cubic oxides show a good agreement with the experimental data.It is notable that there is an increasing trend in the bulk moduli and elastic moduli components when moving along the periodic row from MnO to NiO, observable both in the experimental and calculated elements.Like in the bulk modulus, the elastic tensor component values for 4d oxide CdO is smaller than for the antiferromagnetic 3d oxides.Even with averaging, the DFT-PBE0 values of the elastic moduli component C 11 remain higher than the experimentally measured ones, for MnO, CoO, and NiO.A similar trend was reported in a recent study with the DFT-PBEsol functional, where the C 11 elastic constants of NiO and CoO were overestimated compared to the experimental values. 101A previous DFT study on antiferromagnetic refractory material CrN reported the elastic constants of the magnetically ordered system to be overestimated compared to experimental values and the values obtained for a paramagnetic model system. 100her Cubic and Tetragonal Oxides.Finally, we present the elastic constants calculated for other cubic and tetragonal oxides, for which experimental data are available.The calculated and experimentally determined elastic constants of the cubic rhenium(VI) oxide ReO 3 , yttria Y 2 O 3 , and copper(I) oxide Cu 2 O are given in Table 6 74 ).ReO 3 is a highly anisotropic material also according to experimental measurements, and the experimental anisotropy factor A = 2C 44 /(C 11 − C 12 ) = 0.24 is also reproduced in our calculations, even though the calculated value of C 12 is larger the experimentally observed one.
The calculated and experimentally determined elastic constants of the tetragonal modifications of NbO 2 and ZrO 2 are given in Table 7.The calculated and experimental values for NbO 2 are in reasonable agreement, but for ZrO 2 the calculated values are clearly overestimated in comparison to experiment.Our calculated bulk modulus of ZrO 2 is also over 50 GPa larger than experimentally determined.

■ CONCLUSION
We investigated the elastic properties of almost all binary dmetal oxides known at ambient pressure.The DFT-PBE0/ TZVP level of theory has previously been shown to describe the structural parameters of magnetic d-metal oxides with good accuracy, and for elastic properties the overall agreement with experiment is also at a reasonable level.However, in several cases there are relatively large discrepancies between the calculated and experimental bulk moduli and elastic constants.The calculated bulk moduli and elastic constants are for ideal perfect crystals at 0 K, not taking into account defects or temperature effects.More extensive comparisons between the experimental and calculated data are also complicated by the variance of the experimental data and measurement conditions.Especially for high-temperature applications, it is necessary to also take into account temperature effects, as demonstrated for refractory materials such as CrN 100 and TiN. 105The elastic properties of ordered 0 K ground state can clearly differ from the elastic properties of higher-temperature paramagnetic ground state.Overall, the elastic properties of both nonmagnetic and magnetic d-metal oxides can be studied systematically with hybrid DFT methods, but in many cases, methodological improvements are required to reach a quantitative agreement with experimentally determined elastic constants.In line with a previous study focusing on cubic nonoxide semiconductors and insulators, the C 11 elastic constants, in particular, are overestimated by DFT-PBE0 in comparison to experimental values. 106Possible routes toward improvement are the incorporation of defects such as nonstoichiometry in the structural models and considering the effect of phonon anharmonicity and finite temperature, especially for magnetic oxides where the elastic constants have been measured for a paramagnetic ground state.

■ ASSOCIATED CONTENT
* sı Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/

Figure 2 .
Figure 2. Bulk moduli of the studied rutile and distorted rutile-type dmetal oxides as a function of density.Values calculated (DFT-PBE0) with the ELASTCON approach in blue circles, values calculated with NUCHESS+HESSNUM approach in red triangles, and experimental values with error bars in yellow circles and blue lines.

Figure 3 .
Figure 3. Bulk moduli of the studied corundum-type d-metal oxides as a function of density.Calculated DFT-PBE0 values in blue and experimental values with error limits in yellow.

Figure 4 .
Figure 4. Bulk moduli of the studied rocksalt-type d-metal oxides as a function of density.Calculated DFT-PBE0 values in blue and experimental values with error limits in yellow.

Table 1 .
DFT-PBE0/TZVP Bulk Moduli for d-Metal Oxides with Reported Experimental Data (GPa) a
. The calculated and experimental values for Y 2 O 3 and Cu 2 O are in reasonable agreement, although the C 11 values are overestimated in comparison to the experimental values.For ReO 3 , the calculated and experimental values of C 11 and C 12 differ considerably: the calculated C 11 is 701 GPa, while the experimentally determined value of 517.8 GPa is clearly smaller.The value of the bulk modulus for ReO 3 differs as well (274 GPa DFT-PBE0 vs 211 GPa experimental

Table 5
aFor antiferromagnetic MnO, CoO, and NiO the calculated elastic moduli of the trigonal magnetic structure have been averaged for comparison with measurements in the cubic symmetry.