Theory of Ferromagnetism in Reduced ZrO2-x Nanoparticles.

Bulk ZrO2 is both nonreducible and nonmagnetic. Recent experimental results show that dopant-free, oxygen-deficient ZrO2-x nanostructures exhibit a ferromagnetic behavior at room temperature (RT). Here, we provide a comprehensive theoretical foundation for the observed RT ferromagnetism of zirconia nanostructures. ZrO2 nanoparticles containing up to 700 atoms (3 nm) have been studied with the help of density functional theory. Oxygen vacancies in ZrO2 nanoparticles form more easily than in bulk zirconia and result in electrons trapped in 4d levels of low-coordinated Zr ions. Provided the number of these sites exceeds that of excess electrons, the resulting ground state is high spin and the ordering is ferromagnetic. The work provides a general basis to explain magnetism in intrinsically nonmagnetic oxides without the help of dopants.

Introduction

Ferromagnetism in semiconductor metal oxides finds important applications in spintronics and optoelectronics.[1,2] Many efforts have been dedicated to the study of ferromagnetism (FM) induced by transition metal (TM) doping of semiconductor oxides such as ZnO,[3,4] TiO2,[5−7][5−7] and ZrO2,[8,9][8,9] also called diluted magnetic semiconductors (DMSs). However, it remains unclear whether FM is an intrinsic property of the system or is due to the aggregation of magnetic metal dopants or other magnetic impurities.[10,11] Oxides doped by light 2p elements (e.g., C- and N-ZnO,[12,13] C- and N-TiO2,[14,15] N-ZrO2,[16] and N-BaTiO3[17]) have been also considered and predicted as ferromagnets at room temperature (RT), but the origin of the magnetic behavior is a matter of debate.[18−20] In other cases, staying with the material of interest for this study, tetragonal zirconia, no ferromagnetism has been observed by doping the oxide with Mn.[21]

Encouraging results have been recently obtained for undoped thin films of TiO2, ZnO, In2O3, and HfO2,[22−26] which are FM at RT without any type of dopant element. Also, thin films of ZrO2 exhibit RT FM. It has been suggested that this is related to the presence of intrinsic defects. The crystallographic phase also appears to play a role, with tetragonal ZrO2 that seems to be essential.[27−29] In thin films, however, the necessity to precisely control the nature of the interface with the support is a critical issue.

For this reason, metal oxide nanoparticles (NPs) and nanostructures represent ideal systems to obtain FM at high Curie temperatures (TC > RT).[30] Recently, a FM behavior at RT has been reported for undoped ZrO2 nanostructures.[2] The phenomenon has been attributed to the presence of surface oxygen vacancies (VO). However, there is a conceptual problem with this hypothesis. A neutral oxygen vacancy in the bulk and on the surface of zirconia consists of two electrons trapped in a cavity, with a diamagnetic ground state.[31,32] This means that removing oxygen from the bulk or the surface of zirconia does not necessarily lead to the appearance of magnetic states.

Another problem is the low reducibility of the material. The formation of VO centers in the bulk or on the surface of ZrO2 is thermodynamically very unfavorable, with vacancy formation energies of about 6 eV (computed with respect to 1/2 O2).[33] In fact, ZrO2 belongs to the class of nonreducible oxides. However, density functional theory (DFT) calculations have shown that the cost to remove oxygen in ZrO2 NPs is substantially lower and that the excess electrons associated with a VO defect can lead in some cases to the formation of Zr3+ at the position of low-coordinated (LC) Zr ions.[33] This suggests a special role of nanostructuring in turning the nonreducible ZrO2 into a reducible material.

But under which conditions the Zr3+ centers can lead to a FM ordering? Is this ordering stable at RT? What size of NP is required to reach these conditions? What is the level of reduction requested? To answer these questions, we have performed an extensive theoretical investigation of nonstoichiometric ZrO2– NPs characterized by increasing the size and level of oxygen deficiency. For the calculations, we have used both DFT + U and hybrid (PBE0) functionals. Hybrid functionals, in particular, are needed to provide a correct description of the electronic structure[34] and to accurately reproduce the strength of the magnetic coupling and of the resulting Curie temperature (TC) of the finite systems.[35−37]

We show that the high-spin (HS) FM solution is, beyond a given size of the NP, the ground state of the system. This work provides a solid theoretical basis for the observed magnetic behavior of nonstoichiometric ZrO2 NPs and nanostructures.

Results and Discussion

We model ZrO2– NPs following the relative stabilities of the orientation of the lattice planes in bulk tetragonal zirconia (t-ZrO2). This is analogous to a Wulff construction at the nanoscale.[33,38] In particular, bulk t-ZrO2 was cleaved along the O-terminated (101) surface, the most stable crystallographic face (see Supporting Information). This structure has also been observed experimentally for CeO2 NPs, which have similar characteristics.[39] For a discussion on the stability of oxide nanostructures, see also ref (40). We considered tetragonal ZrO2 because FM has been observed for this specific polymorph.[27−29]

We built four nonstoichiometric (oxygen-deficient) octahedral nanostructures: Zr44O80, Zr85O160, Zr146O280, and Zr231O448 (see Figure ). In the smallest NP, Zr44O80, eight O atoms are missing with respect to the stoichiometric composition. In the larger particles, the numbers of missing O atoms are 10, 12, and 14, respectively.

Figure 1

Optimized structures of ZrO2 nanoparticles. Red spheres represent the O atoms, and light blue spheres, Zr atoms.

The size of the NPs goes from 1.4 nm to almost 3.0 nm, which is the same range as that of synthesized octahedral-based zirconia NPs.[41] The NPs are characterized by the presence of six 4-coordinated Zr corners (Zr4c) and by a size-dependent number of 6-coordinated Zr edges (Zr6c): from two in Zr44O80 up to five in Zr231O448.

Nonmagnetic versus Magnetic Ground State

The systems considered in this study are finite in size, and as such, their electronic structure should be discussed in terms of discrete levels and molecular orbitals. However, at the sizes considered, bands start to form; furthermore, the study aims at comparing NPs with the extended solid (bulk). For these reasons, we will make use of the language of the band structure to address the electronic structure of the zirconia NPs. The relative stability of diamagnetic and high-spin (HS) solutions of ZrO2 NPs has been evaluated by means of PBE + U exploratory calculations. For the smallest NP considered, Zr44O80, the ground state is a singlet closed shell. The lowest HS solution, characterized by 16 unpaired electrons, lies 1.69 eV higher in energy. In the ground state (singlet), the excess electrons are localized on the d orbital of the six Zr4c corner ions, which are reduced from Zr4+(4d0) to Zr2+(4d2) (see Figure ). This is because the 4d states of the low-coordinated Zr ions (ZrLC) are lower in energy than those of the fully coordinated Zr ones and appear as mid-gap states. In Zr44O80, they lie about 1 eV below the conduction band (CB) (see Figure S2).[38] The remaining excess of charge is distributed inside the NP. In the HS model, the excess electrons are localized on the ZrLC ions at corners and edges, which are reduced from Zr4+(4d0) to Zr3+(4d1). The electron localization on the Zr ions leads to a remarkable deformation of the Zr first coordination sphere, corresponding to the formation of a small polaron. The small number of Zr6c ions along each edge (only two in Zr44O80) implies a dense distribution of the unpaired electrons, with a higher cost for the polaronic distortion. This is the origin of the low stability of the HS state.

Figure 2

(a) Spin densities of three ZrO2– NPs in the high-spin solution; (b) charge density plots of the excess electrons of the singlet NPs. ρ = 0.01 e–/Å3.

On increasing the NP size, the nature of the ground state changes. In the medium-size NP considered, Zr85O160, there are 20 excess electrons (10 missing O atoms) and the ground state is HS and FM, with the singlet closed-shell state lying 0.86 eV higher in energy (see Table ). The order of diamagnetic and magnetic solutions is thus reversed with respect to the smaller NP. The distribution of the 20 unpaired electrons is similar to that in Zr44O80, but now the ZrLC3+ ions are more uniformly and homogeneously distributed over the entire surface due to the higher number of Zr6c sites along the edges (see Table ). No evidence of clustering of Zr3+ sites is found. This is an important conclusion. The switch from the diamagnetic to the magnetic ground state depends on the ratio between the number of excess electrons and the number of ZrLC sites available (see Table ). If ZrLC ≫ n(e–), the magnetic ground state is preferred. In the absence of a sufficient number of low-lying acceptor levels, in fact, electrons are forced to doubly occupy these states, leading to a nonmagnetic ground state.

Table 1

Number of Excess Electrons (e–) and of Zr Edge and Zr Corner Atoms and Relative Stabilities (eV) of the Singlet Closed Shell (S) vs Those of High-Spin (HS) Solutions of ZrO2– NPsa

	e–	Zr4c	Zr6c	S	HS
Zr44O80	16	6	24	0.0	+1.69
Zr85O160	20	6	36	+0.86	0.0
Zr146O280	24	6	48	not conv	0.0

PBE + U results.

This interpretation is further corroborated by the results for the larger Zr146O280 NPs. Here, only the FM solution is obtained; various attempts to converge on a diamagnetic state failed because of the intrinsic instability of the singlet closed-shell state compared to that of the HS one (see Table ).

To summarize, a larger size of the ZrO2– NPs implies that an increase of the number of ZrLC ions is able to stably trap excess electrons, favoring a single occupancy of the Zr 4d levels (Zr3+). The results clearly show that an O-deficient ZrO2– NP has a magnetic ground state, at variance with the extended surface or the bulk of reduced zirconia. The spontaneous magnetization is thus an intrinsic property of nanostructured, O-deficient zirconia.

Magnetic Ordering and the Origin of Magnetic Behavior in Zirconia NPs

To study the existence of RT FM in ZrO2– NPs, we have compared the stabilities of FM and antiferromagnetic (AFM) solutions using a broken-symmetry solution (the calculations described in what follows are based on the PBE0 hybrid functional and CRYSTAL14 code; all structures have been geometrically optimized).

The first-principles AFM solution has been computed for Zr146O280 starting from the optimal FM geometry and then fully relaxing the structures (see Figure ). However, the two structures are virtually identical. In Zr146O280, the cost to flip 12 spins, going from 24 up ↑ electrons (FM) to 12 up ↑ and 12 down ↓ electrons (AFM), is 13.3 meV, confirming that the FM ordering is the ground state (see Table ).

Figure 3

Spin density plots of (a, c) high-spin ferromagnetic (FM) and (b, d) antiferromagnetic (AFM) solutions of Zr146O280 and Zr231O448 NPs, respectively. Yellow contours represents spin up ↑ and blue contour represents spin down ↓ electrons. ρ = 0.01 e–/Å3.

Table 2

Energy Difference between AFM and FM Solutions, ΔE (AFM – FM), for Zr146O280 and Zr231O448 NPsa

NP	ΔE (AFM – FM), meV
Zr146O280	+13.3
Zr231O448	+30.4

PBE0 results.

We considered an even larger zirconia NP, Zr231O448 (see Figure ). Notice that this particle has nearly 700 atoms and a total of 28 unpaired electrons. For this large NP, the FM state is more stable than the AFM one by 30.4 meV (see Table ). The trend is then confirmed: a higher number of missing O atoms in the NPs leads to a more stable FM state.

The FM coupling can be explained by bound magnetic polaron (BMP) theory.[11,42] The model works for transition metal (TM) dopants and implies an exchange interaction between shallow electrons associated with defects and localized d electrons of the TM dopants. This approach has been reconsidered by taking into account only the oxygen vacancy (VO) as the defect responsible for the observed FM in undoped-TiO2 nanoribbons,[43] and the polaron considered is associated with oxygen vacancies. Once the VO concentration reaches a certain limit, an overlap among the BMPs is established, thus enhancing the FM behavior. However, it should be mentioned that the model cannot explain the absence of RT FM in some oxide films, such as monoclinic HfO2 and ZrO2, characterized by a large number of oxygen vacancies.[27]

Recently, the charge transfer ferromagnetism (CTF) model has been proposed.[44] According to this model, the presence of defects introduces an impurity band below the CB. The CTF model involves the presence of a charge reservoir that facilitates the hopping of electrons between the impurity band and the CB, leading to the splitting of spin states (see the inset of Figure ).

Figure 4

Projected density of states of Zr146O280 NP. The contributions of O (red), Zr4+ 4d0 (blue), Zr3+ 4d1 corner (black), and Zr3+ 4d1 edge (purple) atoms are reported. The Fermi energy is set to the highest occupied level. The schematically proposed mechanism of the FM interaction is due to the hybridization of the Zr3+ states with the CB (composed by Zr4+ 4d and 5s orbitals), which leads to the splitting of up and down states, as shown in the inset. A simplified Zr146O280 model with excess electrons in the FM configuration is also shown.

Here, we propose a model that is a hybrid of these two theories to explain the FM behavior of undoped ZrO2 and possibly other oxides.[2,43] The model implies the formation of a BMP among electrons localized on two nearby Zr centers. As shown in Figure , at high vacancy concentrations, these defects lead to the formation of a band in the gap just below the CB. The defect states, composed by the Zr 4d1 states, can merge with the CB, leading to a splitting of up and down spin states that favors the FM ordering. The charge reservoir is represented by the Zr3+ ions. The stability of the FM state depends on the hybridization degree between the impurity band and the CB. Because the nature of the CB in turn depends on the polymorph and the crystallographic facets involved, this model can explain the effect of the structure on the FM alignment.

As reported in Table , the FM ordering in Zr146O280 is more stable than the AFM one by about 13 meV. This means that the AFM state is not expected to survive at RT because the value of 13.3 meV is well below the thermal energy (26 meV). In contrast, the higher magnetic coupling obtained for Zr231O448 (30 meV) suggests that the ferromagnetism could exist at higher temperatures and could be stable against thermal fluctuations.[45] In fact, increasing the number of BMPs, the total magnetization and then the TC are also expected to increase.

Magnetic Coupling in Zirconia NPs

Finally, we have computed the exchange coupling (J) of a pair of isolated Zr ions. This requires us to compare the stabilities of FM and antiferromagnetic (AFM) solutions using a broken-symmetry solution (PBE0 results, CRYSTAL14 code). An approximated approach has been adopted. We considered the stoichiometric Zr40O80 NP, where a LC oxygen has been removed (Zr40O80-VO) from one of the most favorable sites (low O vacancy formation energy) to study with first principles the exchange interaction between just two magnetic moments. This leaves two electrons that can give a FM (triplet), an AFM (singlet open shell), or a diamagnetic (singlet closed shell, S) solution. In the open-shell solutions, one of the two excess electrons is localized on a Zr4c corner ion and the other in the vacancy (Figure ). The singlet closed-shell configuration with electron localization in the vacancy is favored by the presence of the void formed in the position of the missing oxygen atom (as for zirconia surface and bulk). This further shows that the magnetic behavior appears only in the presence of several vacancies and the related excess of charge.

Figure 5

Spin density plot of high-spin (a) ferromagnetic and (b) antiferromagnetic solutions for a Zr40O80 NP with VO at the surface. Yellow represents up ↑ and blue represents down ↓ electrons. ρ = 0.008 e/Å3.

Then, the exchange coupling (J) of two electrons localized on two low-coordinated Zr atoms and considered as an isolated pair (ij), J = ΔE (AFM – FM), is computed. The FM solution is 1.0 meV more stable than the AFM one (J = 1.0 meV). Of course, this value may slightly change if the vacancy is formed in another position or if it is created in a particle of different size. However, the final goal of this model is to compute a realistic J value for two magnetic Zr sites with similar environment and separated by more or less the same average distance observed among magnetic Zr ions in Zr146O280 and Zr231O448 nanoparticles. From this point of view, we believe that the J value computed is representative of the magnetic interaction on larger zirconia reduced nanoparticles. Various FM and AFM solutions with different geometries have been considered (see Figure S3 and Table S3). It turns out that the FM state is always lower in energy. The preference for a FM coupling seems to be an intrinsic property of the system.

To estimate the Curie temperature, one can use the mean-field approximation (MFA) and the Heisenberg model, according to the following equation[46]The MFA method does not deal with the statistical problem of the local magnetic moments and tends to overestimate TC.[47−49] Nevertheless, it has given acceptable results.[49] The J value is that obtained for a pair of Zr ions as discussed above, 1.0 meV, and c is the Zr3+ concentration. Although this is well defined in the bulk of a magnetic solid, it is much more complex to define on a nanoparticle where the magnetic moments localize on low-coordinated ions at the surface of the nanoparticle. The value of TC depends critically on this quantity. If we define c as the ratio between the number of Zr3+ ions in the NP and the total number of Zr ions, for Zr231O448 (c = 28/231 = 0.12), we estimate a Curie temperature TC = 64 K. If we assume a hypothetical concentration of 2% of Zr3+ ions (c = 0.02), we obtain TC = 387 K. Clearly, it is impossible to provide a quantitative estimate of TC without knowing the exact concentration of Zr3+ ions.

Conclusions

Bulk ZrO2 is both nonreducible and nonmagnetic. Recent experimental results show that dopant-free, oxygen-deficient ZrO2– nanostructures exhibit FM behavior at RT. We have recently shown that ZrO2 NPs are easy to reduce via O2 or H2O desorption from low-coordinated sites.[50] In this work, we provide a theoretical foundation for the observed magnetic character of zirconia nanostructures.

In ZrO2– NPs, the presence of low-coordinated Zr ions favors the localization of the excess electrons originated from the oxygen deficiency in the low-lying Zr 4d levels (Zr3+). The magnetic Zr3+(4d1) defects can form only if a sufficient number of ZrLC ions exist. When this is not the case, the excess of charge is accommodated in doubly occupied 4d levels, Zr2+(4d2), leading to an overall diamagnetic ground state. Thus, there is a clear relationship among the maximum possible level of reduction (number of excess electrons), the surface area of the NP, and the nature of the ground state (magnetic or diamagnetic).

If ZrLC sites exceed the number of excess electrons, the ground state is high spin and the preferred ordering is ferromagnetic. Increasing the number of magnetic defects also leads to a stronger exchange coupling and higher Curie temperatures.

This work provides a solid theoretical ground to explain the completely different magnetic behavior of oxygen-deficient nanostructured zirconia compared to that of the bulk material.

Computational Details

For the study of ZrO2– NPs, we used two different approaches. Both are based on the use of codes that use periodic boundary conditions. A NP is contained in a supercell and separated by sufficiently large distances from their replicas in adjacent supercells. The structures of ZrO2– have been geometrically optimized by means of spin-polarized DFT calculations with the Vienna Ab-initio Simulation Package (VASP 5.3)[51−53] (plane wave basis set with a kinetic energy cutoff of 400 eV; core electrons described by the projector-augmented wave method).[54,55] The generalized gradient approximation (GGA) for the exchange functional was used within the Perdew, Burke, and Ernzerhof (PBE) formulation.[56] The self-interaction error of GGA functionals was partly corrected using the PBE + U approach[57,58] with the onsite Coulomb correction (Ueff = U – J) for the Zr(4d) states set to 4 eV.[31] Geometry optimizations of the NP were carried out at the Γ-point, with all atoms free to relax until the ionic forces become smaller than |0.01| eV/Å (self-consistency of the electron density set to 10–5 eV).

The magnetic nature of the finite nanoparticles has been evaluated with the hybrid DFT functional PBE0[59,60] using the CRYSTAL14 code.[61] The structures have been fully optimized at the PBE0 level. Crystalline orbitals are represented as linear combinations of Bloch functions (BFs). Each BF is built from linear combinations of Gaussian-type orbitals. An all-electron basis set has been used for O, 8-411(d1), whereas for Zr, the Hay and Wadt small-core effective core potential[62] associated with a 311(d2) basis set was selected. For the numerical integration of the exchange-correlation term, 75 radial points and 974 angular points (XLGRID) in a Lebedev scheme in the region of chemical interest were adopted.

The accuracy of the integral calculations was increased with respect to the default value by setting the tolerances to 10, 10, 10, 10, and 20. The self-consistent field iterative procedure converged to a tolerance in total energy of ΔE = 1 × 10–8 au. The threshold for the maximum and the root-mean-square forces were set to 0.00045 and 0.0003 au, respectively.