Interplay between Magnetic and Orbital Ordering in the Strongly

Nov 22, 2010 - In opposition, the Jahn−Teller effect is predominant for Ueff > 5 eV, leading to a tetragonal structure. View: ACS ActiveView PDF | P...
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Interplay between Magnetic and Orbital Ordering in the Strongly Correlated Cobalt Oxide: A DFT + U Study A.-L. Dalverny, J.-S. Filhol, F. Lemoigno, and M.-L. Doublet* Institut Charles Gerhardt-UniVersite´ Montpellier 2 and CNRS, Place Euge`ne Bataillon, 34095 Montpellier, France ReceiVed: September 9, 2010; ReVised Manuscript ReceiVed: October 20, 2010

The strongly correlated CoO is investigated by means of DFT + U calculations to elucidate the origin of the cubic-to-monoclinic distortion observed below TN for that system. An effective Ueff value between 4 and 5 eV is required to reproduce, in the meanwhile, the monoclinic symmetry, the band gap, and the type-II antiferromagnetic ordering of CoO. The extraction of the exchange coupling constants combined with full structural relaxations allows the interpretation of the low-temperature structure of CoO. An interplay between magnetic and orbital ordering is shown to be responsible for the peculiar structural behavior of CoO. For Ueff < 4 eV, the magnetic driving force dominates over the Jahn-Teller effect, leading to a rhombohedral structure. In opposition, the Jahn-Teller effect is predominant for Ueff > 5 eV, leading to a tetragonal structure. I. Introduction Cobalt oxide has attracted a lot of attention for decades because of its numerous applications, for example, as gas sensors,1 magnetodevice applications,2,3 and, more recently, as electrode material for energy storage applications.4 From a fundamental point of view, CoO is a prototype for strongly correlated systems. The failure of conventional DFT to reproduce the insulating behavior of transition-metal oxides is due to a poor description of the Coulombic repulsions between electrons in the 3d-like metallic shell and leads to the so-called self-interaction error:5 similar to other transition-metal oxides, such as NiO, LDA and GGA calculations predict the CoO electronic ground state to be metallic, whereas it is experimentally observed as being insulating.6 Several alternatives have been proposed to partially correct the self-interaction error of the conventional DFT method, among which are the SIC-LDA,7 the DFT + U,8 the hybrid DFT/HF (Hartree-Fock) B3LYP9 and PBE0,10 or the range-separated HSE11 and RSHX approaches.12 In the particular case of CoO, efforts have been devoted to the extraction of the appropriate DFT + U effective parameter or DFT/HF mixing parameter required to properly describe the insulating ground state of CoO, as well as various experimental data, such as band gaps, photoemission spectra, magnetic moments, oxidation energies, etc.5,8,13-19 Nevertheless, these experimental-to-theoretical mappings have yielded controversial results for CoO: following their work on NiO, Terakura et al. were among the first authors to point out the inability of DFT to account for the strongly correlated character of CoO.5 This assumption was confirmed by Anisimov et al. in their original paper on the LDA + U method, showing that a value of Ueff ) 6.9 eV (U ) 7.8 eV, J ) 0.92 eV) was required to properly describe the electronic structure of CoO.8 More recently, Archer et al. claimed that the rock-salt structure of CoO is well-reproduced using the LDA + U formalism with a correlation parameter of Ueff ) 4 eV.13 Using the GGA + U formalism, Wdowik and Parlinski have proposed Ueff ) 6.1 eV * To whom correspondence should be addressed. E-mail: [email protected].

as the proper value to describe the lattice parameters, magnetic moment, and band gap of CoO,14 whereas a significantly smaller value of 3.3 eV15 was extracted by Ceder et al. for the CoO oxidation energy.15 Depending on the HF/DFT mixing or rangeseparation parameter used in the calculations, hybrids16,17 and RS18,19 functionals also led to ambiguous results: in particular, the energy band gap of CoO, in addition to being systematically overestimated, ranges from 3.4 to 6.2 eV.16-19 Although CoO has already been widely studied, there are still interesting points to discuss, in particular, its structural properties. Below TN ) 291 K, CoO undergoes a paramagnetic-to-type-II antiferromagnetic transition. Meanwhile, a structural distortion from the cubic to the monoclinic symmetry is also observed.20 This symmetry is, however, not consistent with the magnetic ordering and results from the combination of a tetragonal distortion (a ) b * c and R ) β ) γ ) 90°) and a small rhombohedral distortion. In contrast, MnO, FeO, and NiO show the same type-II antiferromagnetic order as CoO, but all undergo a rhombohedral distortion below TN (a ) b ) c and R ) β ) γ * 90°).21 Experimentally, the origin of the low-temperature transition of CoO has long been discussed. Using high-resolution synchrotron powder diffraction and neutron powder diffraction data, Jauch et al.20 have suggested that a magnetostriction phenomenon is responsible for the structural and magnetic transition of CoO. However, Ding et al. have shown that the magnetic ordering can be induced by pressure at room temperature without any tetragonal distortion, suggesting a Jahn-Teller driving force for the tetragonal distortion of CoO.22 From a theoretical point of view, little can be found in the literature about this distortion. Most studies have focused on the electronic properties, assuming that the structural distortion is insignificant. To our knowledge, the only crystallographic studies reported on CoO16,23 have led to another distortion (rhombohedral) than the one experimentally observed (monoclinic). Using first-principles calculations, the reproduction of the CoO distortion is challenging: first, because of its low amplitude and, second, because it is subject to a good description of the electronic structure of CoO. The goal of this paper is thus to elucidate the apparent contradiction between the magnetic and structural symmetry of CoO below TN and to

10.1021/jp108599m  2010 American Chemical Society Published on Web 11/22/2010

A DFT + U Study of the Strongly Correlated CoO System

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determine the driving forces of the transition. The DFT + U formalism is chosen for its tunable Ueff parameter and also for its less time-demanding cost compared to that of hybrid functionals. The paper is organized as follows: the technical and computational details of the calculations are given in the next section. The results obtained for the structural, magnetic, and electronic properties of CoO are presented in section 3 before the Discussion and the Conclusion in sections 4 and 5, respectively. 2. Computational Methods All calculations were performed using the plane-wave density functional theory (DFT) code available in the Vienna Ab initio Simulation Package (VASP)24,25 within the generalized gradient approximation (GGA). Two different exchange and correlation potentials have been tested, namely, the PBE26 and its revised version, PBEsol,27 recently implemented in the VASP code to improve equilibrium properties of densely packed solids.28 With cobalt ions being in a formal +II oxidation state in CoO, a particular attention was given to their local magnetic moment and to the resulting magnetic structures of the material. The structural properties (bulk parameters and local chemical bonds) and the electronic properties (local magnetic moments, band structures, band gaps, exchange integrals) of CoO were then computed within the spin-polarized DFT + U framework, using the rotationally invariant approximation of Dudarev.29 The study was performed from the DFT limit (Ueff ) 0 eV) to the strongly correlated limit (Ueff ) 10 eV) in order to check the influence of the effective Ueff value on the different parameters of interest. The electron wave functions were described in the projected augmented wave formalism (PAW),30 and a real-space projection was further used for the total wave function analysis. The planewave energy cutoff was set to 600 eV, and the Brillouin zone integration was done in a k-point grid distributed as uniformly as possible, using Monkhorst-Pack meshes of 7 × 12 × 6 (254 irreducible k-points) in the starting monoclinic cell, consisting of four CoO per unit cell. Atom coordinates and lattice parameters were independently relaxed to avoid Pulay stress, prior to a full relaxation, using conjugate gradient energy minimization, until the forces acting on each atom were less than 5 × 10-3 eV/Å. 3. Results 3.1. Magnetic Structures of CoO. At ambient temperature, CoO crystallizes in a conventional cubic cell (ac ) 4.2614(3) Å) and shows a paramagnetic behavior.20 Below TN ) 291 K, it undergoes a magnetic transition to a type-II antiferromagnetic state in which ferromagnetic (111) planes are antiferromagnetically stacked along the [111] direction (see Figure 1). Meanwhile, a slight cubic-to-monoclinic transition is observed. As shown in Figure 1a, the use of the CoO conventional cubic cell would require a 2 × 2 × 2 supercell (32 formula units) to compute the type-II antiferromagnetic structure, whereas a 1 × 1 × 1 unit (4 formula units) is sufficient for both the ferromagnetic (FM) and the type-I antiferromagnetic (AFI) structures. To decrease the computational time and the numerical errors arising from different k-point meshes associated with different unit cell sizes,31 it is more convenient to choose the monoclinic cell represented by red lines in Figure 1b, and, hereafter, referred to as the reduced unit cell. In this cell (with parameters am ) 5.218 Å, bm ) 3.013 Å, cm ) 6.026 Å, Rm ) γm ) 90°, and βm ) 125.2644°), all of the magnetic structures can be computed using equivalent k-point and density grids for numerical integrations and a reduced number of atoms (four

Figure 1. (a) Ferromagnetic (FM) and type-I and type-II antiferromagnetic (AFI and AFII) structures of the rock-salt CoO structure. (b) AFII structure represented in the 2 × 2 × 2 cubic cell (in black) and in the 1 × 1 × 1 monoclinic unit cell (in red). The Co spin-up and spin-down and oxygen are illustrated by gray, light gray, and red balls, respectively. For the sake of clarity, the oxygen atoms are represented only for the FM structure.

formula units). The ferromagnetic (FM) and type-I and type-II antiferromagnetic structures (AFI and AFII) were then fully relaxed in this reduced cell within the GGA + U formalism with an effective parameter Ueff varying from the DFT limit (Ueff ) 0 eV) to the strongly correlated limit (Ueff ) 10 eV) with a constant exchange parameter of J ) 1 eV. The results are listed in Table 1. In agreement with experiments, the AFII state is the most stable magnetic arrangement in the whole range of Ueff, independent of the functional used in the calculations. Its stabilization over the two other magnetic orders decreases with Ueff so that the energy difference between AFII and FM approaches the room-temperature thermal energy (kBT ≈ 25 meV) above the critical Ueff value of 5 eV for PBE and 6 eV for PBEsol. Provided that the energy of the broken symmetry AFII state (typical multideterminant state here treated as a single-determinant in the DFT formalism) is computed within a negligible error with respect to the FM state energy, this result is consistent with the room-temperature paramagnetism of CoO. Below Ueff ) 5 eV, the strong stabilization of the AFII structure with respect to FM and AFI (from ∼200 to 50 meV) is no longer consistent with the paramagnetic behavior of CoO at room temperature. As shown in Figure 2a, the Co local magnetic moments (µCo in µB) obtained for the three magnetic structures differ by, at most, 0.2 µB for any Ueff values and are strictly greater than 2.5 µB. Note that equivalent results were obtained with the PBEsol functional. This can be considered as a reasonable agreement with the value expected for high-spin Co2+ (d7). Their increase with Ueff is linked to the loss of Co 3d-O 2p orbital interactions (i.e., to the decrease of the O 2p contribution in the metallic-like orbitals) resulting from the relocalization of the Co 3d orbitals. In addition to the increase of the Co magnetic moment, the orbital relocalization should directly affect the amplitude of the magnetic coupling between Co2+ ions and, in particular, those involving nonorthogonal orbitals on each interacting atom. Because this question is central to determining the driving forces of the CoO distortion, we have extracted the Co-Co exchange integrals from a direct mapping of an Ising-type model (see eq 1) onto the relative energies

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Dalverny et al.

TABLE 1: Relative Energies ∆EMag (in meV) Computed within the GGA + U Formalism Using PBE and PBEsol as a Function of Ueff ) U - J (J ) 1.0 eV) for the Ferromagnetic (FM) and Type-I Antiferromagnetic (AFI) Structures with Respect to the Reference AFII Structure Set to Zero GGA + U (Ueff ) U - J (J ) 1.0 eV))

GGA ∆EMag

PBE PBEsol

FM AFI FM AFI

0.0

1.0

2.0

3.0

4.0

5.0

6.0

9.0

10.0

224 279 257 341

135 229

118 258 190 235

95 235 120 235

96 209 65 207

44 170 52 185

28 159 30 169

2 146 4 146

-5 147

Figure 2. (a) Local magnetic moment of Co (in µB) for the three magnetic structures and (b) J1 (black lines) and J2 (grey lines) exchange integrals extracted from the DFT-Ising mapping as a function of Ueff using |sz| ) 3/2 (solid lines) and the local magnetic moment extracted from the calculations (dotted line). The inset shows the J1 and J2 magnetic interactions occurring in the CoO rock-salt structure.

computed for the different CoO magnetic structures, following the general method described in refs 32 and 33

HIsing ) -

1 2

∑ JijSˆzi · Sˆzj

DFT + U framework with equivalent crystal structures (no ionic relaxation), the J1 and J2 exchange integrals can then be extracted from the formulation

(1)

J1 )

i,j

In the rock-salt CoO structure, the main interactions are the nearest Co-Co interactions at d1 ) ac/2 (ac being the cubic lattice parameter) and the next-nearest Co-Co interactions at d2 ) ac. The former correspond to direct metal-metal interactions, whereas the latter correspond to oxygen mediated Co-O-Co interactions. Note that interactions beyond the nextnearest neighbors can be reasonably neglected here because they correspond to Co-Co distances greater than 5.2 Å with no oxygen mediation in between the two metallic ions. The ferromagnetic structure (FM) thus corresponds to 12 nearest Co-Co interactions at d1 ) ac/2 (ac being the cubic lattice parameter) and 6 next-nearest Co-Co interactions at d2 ) ac. For the antiferromagnetic structure (AF), the spin frustration arising from the lattice symmetry leads to two different magnetic orders: in the AFI structure, each Co is surrounded by eight Co with an opposite spin and four Co with a parallel spin at d1 and six Co with a parallel spin at d2. In the AFII structure, the 12 neighboring Co at d1 are distributed with half parallel and half opposite spins and the 6 Co at d2 with an opposite spin. Considering the Ising model of eq 1, where Sˆzi and Sˆzj are the spin operators on sites i and j, and Jij the exchange integrals between neighboring sites i and j, the expectation values for the magnetic structures can be written in a general form as

1 EIsing ) + {(12 - n)J1 |sz | 2 - nJ1 |sz | 2 + 2 (6 - m)J2 |sz | 2 - mJ2 |sz | 2}

(2)

where n and m refer to the number of ferromagnetic interactions among the nearest and next-nearest neighbors, respectively, and where sz corresponds to the spin projection on the metallic ions, here, Co2+ (d7) in the high-spin configuration. Using the energies of the different magnetic structures computed within the

J2 )

1 (E(AFI) - E(FM)) 8|sz | 2

-1 (3E(AFI) + E(FM) - 4E(AFII)) 24|sz | 2

(3)

In these expressions, the local spin projection of Co was set to sz ) (3/2. Note that using the sz values obtained in the calculations does not affect the results (see Figure 2a,b). The nearest- and next-nearest- neighboring exchange integrals, i.e., J1 and J2, computed as a function of Ueff are reported in Figure 2b. Their absolute magnitudes (|J2| > |J1|) reflect the antiferromagnetic ground state of CoO and are qualitatively comparable with those previously reported by Archer et al. using the LDA + U approximation with Ueff ) 4 eV.13 As expected, the negative (AF) contribution to the exchange integral is more sensitive to the Ueff parameter than the positive (FM) contribution. This is directly related to the nature of the metallic interactions in the structure. As already mentioned, J1 corresponds to direct Co-Co interactions at d1 ) 3.02 Å, whereas J2 corresponds to oxygen-mediated Co-O-Co interactions, implying a Co-O overlap at d2/2 ) 2.13 Å (see the inset of Figure 2b). The exchange integral being directly related to the transfer energy (overlap) between adjacent sites,34,35 a larger Ueff dependence is observed for magnetic interactions, implying nonorthogonal orbital interactions (J2), than for magnetic interactions implying orthogonal orbital interactions (J1). 3.2. Crystal Structure of CoO. Dealing with magnetic structures, one has to keep in mind that the spin symmetry imposed in the calculations may differ from the crystal symmetry. This is the case for the AFI and AFII broken-symmetry states for which (i) the spin arrangement no longer fits the crystal symmetry of CoO and (ii) the DFT ground-state wave function no longer is a spin-eigenfunction of the time-independent electronic Hamiltonian (i.e., not an eigenfunction of the spin-

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TABLE 2: Crystallographic Parameters of CoO Obtained from Experiments (ref 20) at T ) 10 and 293 K and Expressed in the Conventional Unit Cell and in the Reduced Unit Cell T (K)

exptl sym

unit cell

a (Å)

b (Å)

c (Å)

R (°)

β (°)

γ (°)

10

monoclinic

reduced conventional reduced conventional

5.182 4.268 5.218 4.261

5.182 4.268 3.013 4.261

6.037 4.215 6.026 4.261

90.00 89.97 90.00 90.00

125.58 89.97 125.26 90.00

90.00 89.98 90.00 90.00

293

cubic

Figure 3. Relaxed unit cell parameters of CoO for the three magnetic structures (FM, AFI, and AFII) using the PBE potential for exchange and correlation and plotted as a function of Ueff (Ueff ) U - J; J ) 1.0 eV). Results with PBEsol (Ueff ) 0 eV) are also reported for the sake of comparison, together with the room-temperature (T ) 293 K) and low-temperature (T ) 0 K) experimental structures.

Figure 4. Relaxed unit cell angle and local Co-O distances obtained for the fully relaxed AFII magnetic structure of CoO as a function of Ueff (Ueff ) U - J; J ) 1.0 eV) using the PBE potential for exchange and correlation. Results with PBEsol (Ueff ) 0 eV) are also reported for the sake of comparison, together with the room-temperature (T ) 293 K) and low-temperature (T ) 0 K) experimental data.

operator Sˆ2). If magnetic interactions were responsible for the structural distortion observed in CoO below TN, then a rhombohedral distortion of the cubic lattice would be expected for the AFII structure, a tetragonal distortion along the cubic cc axis for the AFI structure, and no structural distortion for the FM structure. Because these two distortions affect either the angle (rhombohedral) or the unit cell parameter (tetragonal) of the conventional unit cell, whereas they affect both the angles and the parameters of the reduced cell, the results of our structural relaxations (as computed in the reduced unit cell) were transformed into the conventional cell, whose parameters are linked, as shown in Table 2. The rhombohedral distortion of the AFII structure is confirmed in the DFT limit (Ueff ) 0 eV), which is slightly more pronounced for PBEsol compared with PBE (see Figures 3 and 4). It is characterized by a homogeneous contraction of the Co-Co distances within the (111) planes and an elongation of the Co-Co interplane distances, resulting in a deviation of 6° and 7° from the conventional angles for PBE and PBEsol, respectively, with no change in the unit cell parameters. When Ueff is switched on, the rhombohedral distortion significantly decreases down to 0.03° (see Figure 4). Although comparable with experimental data (i.e., 0.04°),20 this value falls in the accuracy limit of the calculations and does not allow us to conclude on its relevance. CoO is thus found to be nearly cubic in the range of 0 < Ueff e 4 eV prior to undergoing a tetragonal distortion for Ueff > 4 eV (see Figures 3 and 4). Compared to experiments, the magnitude of the distortion matches the cc/ac ratio obtained by Jauch et al. at T ) 10 K20 for Ueff ∼ 5.5 eV. This tetragonal distortion corresponds to a shortening of the c parameter with respect to a and b, which is not consistent with the symmetry imposed by the AFII magnetic order. This suggests another driving force than the spin ordering for the transition. In Figure 4, we have reported the evolution of the local Co environment in the AFII structure as a function of Ueff. With cobalt ions being located on an

inversion center in the cubic structure, three pairs of Co-O distances can be considered as the pertinent parameters to characterize the local distortion around the CoO6 octahedra. The results show that, regardless of the functional used in the calculations, all Co-O distances are homogeneous for Ueff strictly smaller than 5 eV and fit to the experimental Co-O distances very well for the room-temperature structure. For Ueff g 5 eV, the discontinuity observed in the cell parameters corresponds to a small compression of the axial Co-O bond lengths and a small elongation of the equatorial Co-O bond lengths. This corresponds to an Oh-to-D4h symmetry lowering around Co, which is consistent with a Jahn-Teller driving force, as suggested by Ding et al.22 The same structural investigation was done for the FM and AFI structures. In agreement with the tetragonal symmetry of the AFI spin arrangement, a significant elongation of the cc parameter is observed, regardless of the Ueff parameter considered in the calculations (see Figure 3), leading to Co-O local distances very different from the experimental distances. In that case, the spin arrangement imposed in the calculation obviously forces the tetragonal distortion, following the antiferromagnetic interactions along the cc axis. Given its energy, this AFI state should not contribute in a large manner to the total electronic ground state of CoO. For the FM structure, while the spin arrangement imposed in the calculation no longer breaks the crystal symmetry, a tetragonal distortion is also observed in the range of 3 < Ueff e 6 eV (see Figure 3), which corresponds to Co-O distances, in quite good agreement with experiments. Although associated with an elongation of the cc parameter (and not a contraction as experimentally observed), this distortion lowers the symmetry of the system and suggests that a Jahn-Teller driving force prevails over the magnetic one in the range of 3 < Ueff e 6 eV for the FM state of CoO. Beyond this range, Co adopts an almost

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Figure 5. Energy band gap Eg ()EA - PI in eV) as obtained from the ionization potential (PI ) E(n) - E(n - 1)) and electron affinity (EA ) E(n + 1) - E(n)) of CoO and computed for the AFII structure using the PBE functional for exchange-correlation. The horizontal gray line refers to the experimental gaps (2.5-2.7 eV).

perfect octahedral environment with computed Co-O distances in very good agreement with the experimental room-temperature structure. 3.3. Electronic Structure of CoO. As mentioned in the Introduction, the electronic properties of CoO have been widely studied using various flavors of the DFT method. Nevertheless, the evolution of the CoO band gap with Ueff displays interesting features that deserve to be discussed. The energy band gaps of CoO, computed at T ) 0 K for the fully relaxed AFII structure as a function of Ueff, are reported in Figure 5. As expected, the PBE + U method restores the insulating character of CoO, as does the PBEsol + U (not shown here). Given the roomtemperature thermal energy, a simple mapping of our computed gap with the experimental ones (i.e., EgRT ) 2.5-2.7 eV)36-39 would yield a value of Ueff close to 4 eV. The Ueff dependence of our computed gaps displays a global slope that is lower than Ueff. This is typical for systems whose behavior is between a charge-transfer insulator and a Mott-Hubbard insulator,40 as confirmed by van Elp et al. using XAS/BIS experiments.36 Interestingly, a small discontinuity is observed around Ueff ) 3-4 eV, which could be consistent with the global feature of the CoO band structure. Indeed, as shown in the band structure plots of Figure 6a, the two bands lying below and above the Fermi level exhibit a major contribution of the metallic 3dorbitals for Ueff ) 2 eV. When Ueff is increased, a progressive change of the electronic states around the Fermi level is observed, as a consequence of the lowering of the metallic 3dlike levels in the occupied band as well as the rising of the metallic 3d-like levels in the unoccupied band: whereas the top portion of the valence band shows a Co 3d/O 2p hybridization typical for a charge-transfer insulator, the bottom portion of the conduction band shows a surprising majority Co 4s character. As shown in Figure 6b, the Co 4s states come from the widely dispersive s-type band of Co. While their contribution to the total density of states is still negligible compared to the Co 3d contribution at Ueff ∼ 5 eV, their occurrence in the bottom portion of the conduction band is questionable. Note that an equivalent tail that is consistent with this pattern has already been observed on several band structure plots reported for both the CoO41,42 and the FeO43 rock-salt structures, using both hybrid and DFT + U formalisms. 4. Discussion The results presented above show that CoO is a complex system in which magnetic interactions and Jahn-Teller instability compete. From the structural point of view, three different domains have been clearly identified depending on the effective Ueff parameter used in the calculations. In the DFT limit (Ueff
4 eV, the magnetic interactions almost vanish and the structure is now controlled by the orbital ordering. A cubic-to-tetragonal distortion is then observed along the c axis, which results from a Jahn-Teller effect. Eventually, in the intermediate regime of 2 e Ueff e 4 eV, the two effects compete. The magnetic interactions are not strong enough to induce the rhombohedral distortion, but sufficient to maintain the structure in the cubic symmetry, that is, to prevent the tetragonal Jahn-Teller distortion. Hence, although the type-II antiferromagnetic ground state of CoO is confirmed by our calculations in the whole range of Ueff < 9 eV, different crystal symmetries arise from the interplay between (i) the magnetic interactions, which are the driving force for the rhombohedral distortion, and (ii) the Jahn-Teller instability of the Co2+ (d7) electronic configuration, which induces the tetragonal distortion. At room temperature, the partial screening of the on-site electronic repulsion expected from thermal activation should favor the paramagnetic behavior of CoO. At this temperature, none of the AFII or FM spin arrangements should prevail in the total electronic wave function of the paramagnetic CoO; that is, they would both contribute to the total electronic ground state of CoO in a many-body scheme. It is thus likely that the spin fluctuations occurring at room temperature in the electronic structure of CoO induce local distortions in the crystal structure. This is at least consistent with the opposite tetragonal distortion obtained for the fully relaxed FM (cc/ac > 1) and the AFII (cc/ac < 1) magnetic structures, leading to a nearly cubic average structure when both magnetic states participate in the total electronic wave function. Work is in progress to check the possible influence of spin-orbit coupling on the relative amplitudes of the magnetic and orbital ordering. From the electronic point of view, two interesting results have been pointed out around Ueff ) 4 eV. Below this critical value, the energy difference between the AFII and FM structures is too high to explain the room-temperature paramagnetic behavior of CoO and the energy band gap is underestimated. Above Ueff ) 5 eV, the room-temperature paramagnetic behavior of CoO is well-reproduced, but the energy band gap is overestimated and the lowest unoccupied band shows a surprising Co 4s character. Given the DFT + U formalism, this feature could be thought of as a nonphysical consequence of the introduction of the Ueff correction over the restricted subset of Co 3d orbitals, leading to an over destabilization of the Co 3d band above the widely dispersive Co 4s band. Although virtual states have no physical meaning in the DFT formalism, the shift of the Co 3d orbitals above the low-lying tail of the Co 4s band may appear as an artificial consequence of an overestimated Ueff value. In that sense, hybrid Hartree-Fock/DFT functionals are often thought as an alternative to DFT+U because the “exact” HF exchange energy is not restricted only to the M d-orbitals subset. However, in their recent work, Tran et al.44 have clearly shown that the Co 4s contribution to the conduction band increases with the weight of HF exchange used in hybrid functionals. Also stressed by Alfredsson et al.43 for the FeO system, these results suggest that the HF mixing parameter used in hybrids has qualitatively the same impact on the Co 3d versus Co 4s energy than the Ueff parameter in the DFT + U formalism. As already mentioned in the Introduction, range-separated functionals lead to even larger band gaps for CoO, that is, 3.4 eV18

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Figure 6. (a) Electronic band structures projected on the Co 4s, Co (t2g-like), and Co (eg-like) atomic levels for the AFII structure of CoO computed within the GGA(PBE) + U formalism for Ueff ) 2.0, 5.0, and 10.0 eV and the (b) partial density of states for Ueff ) 5.0 eV.

for HSE03 and 6.2 eV for RSHXDLA, and would probably not solve this issue.19 To what extent this Co 4s feature may indirectly affect the nature/shape of the total electronic groundstate wave function is a crucial question for one being interested in material properties, such as redox or catalysis reactivity, because these properties are primarily governed by the nature/ shape of the crystal orbitals in the vicinity of the Fermi level. Likely, methods accounting for quantum spin fluctuations, such as the DMFT method,45 for instance, should provide more accurate redistribution of the electron density into the strongly correlated M 3d orbitals and address this remaining question. About the ground-state properties of CoO, we show that the DFT + U formalism is able to reproduce even slight structural distortions, provided that the origin of the distortion is fully interpreted. Given that the tetragonal (cc/ac) and the rhombohedral (δβm) distortions experimentally observed for CoO at T ) 10 K correspond to minimal deviations from the perfectly cubic structure (i.e., 1.2% and 0.3°, respectively), the agreement of our results with experiments is quite impressive. Moreover, because our computed structures agree with experiments with relative errors less than 2% in the whole range of Ueff presently investigated, it is not surprising that a large number of Ueff values have already been reported in the literature for the CoO system (from Ueff ) 3.315-6.1 eV14 from GGA + U calculations and from Ueff ) 413-6.9 eV8 from LDA + U calculations). Here, we show that the apparent agreement with experiments is fortuitous for Ueff < 4 eV and Ueff > 5 eV. Indeed, although the computed crystal structures are pretty close to the experimental ones for Ueff < 4 eV, the magnetic and electronic properties of CoO are not well-reproduced. In other words, although the computed crystal structure, as obtained from GGA + U structural relaxations at Ueff ) 2 eV, is very close to the experimental structure at T ) 293 K, it is shown to result from a fair competition between the magnetic and orbital interactions, rather than from an equivalent contribution of the AFII and FM states in the total ground state of CoO at room temperature. On the other hand, for Ueff > 5 eV, although the tetragonal distortion

and the room-temperature paramagnetic behavior of CoO are well reproduced by our calculations, the energy band gap is overestimated and an ambiguous contribution of the Co 4s atomic levels appears in the bottom portion of the conduction band. The crystal, electronic, and magnetic ground state of the strongly correlated CoO are thus accurately reproduced for an effective parameter between Ueff ) 4 to 5 eV. This value is consistent with the Ueff ) 6.9 eV initially proposed by Anisimov and co-workers8 using the LDA + U formalism because a larger self-interaction error is expected within the local density approximation, especially for transition-metal-based compounds. 5. Conclusion The present study shows that the structural, magnetic, and electronic properties of the strongly correlated CoO are accurately reproduced for an effective parameter of Ueff ) 4-5ev. Both the PBE and the PBEsol functionals for the exchangecorrelation potential give equivalent results. The Ueff dependence of both the crystal structure and the magnetic coupling constants of CoO has allowed us to distinguish different regimes for which either a magnetic or an orbital control governs. Such interplay between orbital and magnetic interactions is shown to be responsible for the peculiar structural behavior of CoO with respect to other transition-metal oxides. In the narrow Ueff ) 4-5 eV domain, the low-temperature monoclinic structure of CoO is fully interpreted as resulting from a small magnetic driving force favoring the rhombohedral distortion (δβm ) 0.3°, δβc ) 0.04°) and a larger Jahn-Teller driving force favoring the tetragonal distortion (cc/ac < 0). References and Notes (1) Wo¨llenstein, J.; Burgmair, M.; Plescher, G.; Sulima, T.; Hildenbrand, J.; Bo¨ttner, H.; Eisele, I. Sens. Actuators, B 2003, 93, 442. (2) Jin, S.; Tiefel, Th.; McCormarck, M.; Fastnacht, R.; Rchen, R. Science 1994, 264, 413. (3) Raquet, B.; Mamy, R.; Ousset, J. C.; Ne`gre, N.; Goiran, M.; GuerretPie´court, C. J. Magn. Magn. Mater. 1998, 184, 41.

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