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First-principles investigation of boron defects in nickel ferrite spinel

The accumulation of boron within the porous nickel ferrite (NiFe2O4, NFO) deposited on fuel rods is a major technological problem with important safety and economical implications. In this work the electronic structure of nickel ferrite is investigated using first-principles methods, and the results are combined with experimental data to analyze B incorporation into the NFO structure. Under thermodynamic equilibrium the calculations predict that the incorporation of B into the NFO structure is unfavorable. The main limiting factors are the narrow stability domain of NFO and the precipitation of B2O3, Fe3BO5, and Ni3B2O6 as secondary phases. In n-type NFO, the most stable defect is Ni vacancy while in p-type material lowest the formation energy belongs to tetrahedrally coordinated interstitial B . Because of these limiting conditions it is more thermodynamically favorable for B to form secondary phases with Fe, Ni and O than it is to form point defects in NFO


Introduction
Because of the increasing cost of fossil fuel based energy and the pressure to reduce greenhouse gases, energy policy in the United States currently encourages research and development in nuclear energy [1]. To allow longer lifetimes and higher power output from a 2 nuclear reactor, a major technical issue that has to be conquered is the corrosion and corrosionrelated failure of nuclear fuel. CRUD (Chalk River Unidentified Deposits) is the name given to corrosion products that accumulate on the hot surface of the fuel cladding. In a pressurized water reactor (PWR) CRUD is produced from dissolved metal cations and particulate corrosion products originating from the surfaces of the reactor coolant system, and consists mainly of nickel ferrite spinel (NiFe2O4, NFO), nickel oxide (NiO). [2,3] Although a thin layer of CRUD on the cladding surface can enhance heat transfer, thicker deposits have a negative effect on fuel performance and generate operational challenges. Thick CRUD, usually deposited on the upper part of the fuel rods, reduces the heat transfer in the reactor core and raises the local surface temperature of the cladding, which increases the corrosion rate. Furthermore, boron (B) can accumulate in thick CRUD, which can trigger fluctuations in the neutron flux and cause a shift in the power output from the top half to the bottom half of the core. This phenomenon, known as the axial offset anomaly (AOA), has important safety implications and could lead to down-rating of a power plant with significant economic consequences. Therefore, it is crucial to understand and predict the mechanisms of CRUD formation and B accumulation within the CRUD.
Even though it is clear that the key factors necessary for AOA to manifest are (i) subcooled nucleate boiling, (ii) thick CRUD, and (iii) sufficient B in the coolant, many details about the root cause of AOA are not fully understood. For example, little is known about the role played by the coolant chemistry in AOA development or about the precise mechanism of B deposition. Within the EPRI/WEC (Electric Power Research Institute/Westinghouse Electric Company) simulation models B uptake by CRUD is considered to take place via lithium borates in the form of LiBO2, Li2BO7, and Li2B4O7. [4][5][6][7] However, because these compounds exhibit retrograde solubility with respect to temperature, they are difficult to observe in PWR CRUD scrapes, but they have been observed in simulated CRUD. [8] Sawicki analyzed the corrosion deposits found on the fuel assemblies in various PWRs and identified the formation of Ni2FeBO5 (mineral bonnacordite) as a new mechanism for B retention. [9,10] The results of Sawicki's work have triggered new modeling efforts to describe the chemistry of borated fuel CRUD. [11] Mesoscale CRUD models under development within the CASL program (Consortium for Advanced Simulation of Light Water Reactors) assume precipitation of boron oxide (B2O3) as a possible B deposition mechanism in the porous CRUD. [12] 3 Although current research is moving toward a better description of CRUD properties and its behavior in PWRs, more work is needed to elucidate the mechanisms through which B or boron-containing solids are deposited on the fuel rods. In this work we carry out electronic structure calculations to investigate the possibility of B incorporation into crystalline NFO. To do this, we treat B as either a substitutional or interstitial point defect in the NFO structure and use first-principles-based thermodynamics to analyze the stability of B inside the structure.

Crystal and defect modeling
The spinel atomic arrangement is shared by many transition metal oxides with formula AB2O4 (Fd3m), where A and B are divalent (A 2+ ) and trivalent (B 3+ ) ions, respectively. In the normal structure the oxygen ions form a face centered cubic array and the A 2+ and B 3+ ions sit in tetrahedral (1/8 occupied) and octahedral (1/2 occupied) sites in the lattice, giving a unit cell with 8 A's, 16 B's and 32 O's. The inverse spinel is an alternative arrangement where the divalent ions swap with half of the trivalent ions so that the A 2+ ions occupy octahedral sites. Therefore, the general formula of an inverse spinel can be written as B 3+ (A 2+ B 3+ )O4.
Nickel ferrite crystallizes in the inverse spinel structure, with half of the Fe 3+ ions occupying the tetrahedral sites while Ni 2+ and the remaining Fe 3+ are randomly distributed over the octahedral sites. Because of the periodic boundary conditions employed in our calculations, any atomic distribution within a supercell corresponds to an array of cations with long-range order. In theory, the cation distribution could be modeled using a special quasi-random structure; [13,14] however, the large unit cells required for such simulation would lead to prohibitively expensive computational efforts. In this work we employ the structural model used by Fritsch and Ederer, where the Ni and Fe cations are distributed over the octahedral sites such that the symmetry is reduced from cubic (Fd3m) to orthorhombic (Imma). [15] The same structure has been used by O'Brien et al. [16] to investigate the thermodynamic properties of NFO surfaces under conditions typical to PWR coolant. Assuming a random Ni distribution, all unoccupied octahedral sites are equivalent, while there are two types of unoccupied tetrahedral sites. The three possible interstitial atomic configurations are illustrated in Fig. 1, where the two tetrahedral sites are denoted T1 and T2.

Computational parameters
The calculations were performed using the projector augmented wave (PAW) [17,18] method within density functional theory (DFT) [19,20] as implemented in the Vienna Ab initio Simulation Package (VASP). [21][22][23][24] The exchange-correlation potential was approximated by the generalized gradient approximation (GGA), as parameterized by Perdew, Burke, and Ernzerhof (PBE). [25]. The standard PAW potentials, supplied with the VASP package, were employed in the calculations. [17,18] The 3d and 4s states of Ni and Fe as well as the 2s and 2p states of B and O are considered as valence states while the rest are treated as core states. The cut-off energy for the plane wave basis was set to 550 eV and the convergence of self-consistent cycles was assumed when the energy difference between two consecutive cycle was less than 10 -4 eV. The Brillouin-zone was sampled by the Г-point in all calculations and a Gaussian smearing of 0.1 eV was used. The internal structural parameters were relaxed until the total energy and the Hellmann-Feynman forces on each nucleus were less than 0.02 eV/Å. To minimize the interaction between periodic images of defects, all defect calculations have been performed on 2×2×2 supercells containing 448 atoms, using the calculated lattice constant of pure NFO (a = 8.41 Å). [26] To describe the behavior of the localized Fe 3d and Ni 3d states, the orbital-dependent, Coulomb potential (Hubbard U) and the exchange parameter J were included in the calculations using the DFT+U formalism. [27] The simplified, rotationally invariant approach introduced by Dudarev et al [28] was used. The value of the Hubbard U parameter can be estimated from bandstructure calculations in the supercell approximation with different d and f occupations [29] or 5 from calculations based on a constrained random-phase approximation. [27,30] Here U and J are

Thermodynamics of B incorporation intro nickel ferrite
The formation of a defect in a crystalline solid can be regarded as an exchange of atoms and electrons between the host material and chemical reservoirs. Therefore the formation energy of a defect D in charge state q can be written as: [34,35] where In Eqs. (1)  (i) to avoid elemental precipitations, the chemical potentials are bound by 0, 0, 0, and 0 is the formation enthalpy of NiFe2O4, (iii) to avoid formation of competing phases, such as iron oxides (Wüstite, hematite and magnetite) and nickel oxides, the following conditions must apply Further constraints on the chemical potential are posed by the possibility of forming secondary phases between boron and the host elements. In this work we consider the following compounds as possible secondary phases: B2O3, NiB, Ni2B, Ni4B3, FeBO3, Fe3BO5, Fe3BO6, NiB2O4, Ni3B2O6, and Ni2FeBO5. To avoid the formation of these compounds, the chemical potentials of B, Ni, Fe, and O must satisfy conditions similar to (5) and (6).

Formation enthalpies and elemental reference energies
The theoretical enthalpy (heat) of formation of a compound ...

nm AB
can be calculated as: is the total energy per formula unit (f. u.) of the compound and , ,... Eq. (7) yields very accurate results because the DFT errors will cancel out when calculating the total energy differences. In our case the compounds are the insulating or semiconducting materials, such as NFO, FenOm, or NinOm, while the elemental phases include the metallic form of the cations (Ni, Fe, and semimetallic B) and the gaseous anion (O2). These are chemically and physically dissimilar systems, where the cancellation of the DFT errors is known to be incomplete. [36,37] Furthermore, to reproduce the correct electronic structures of the Ni-and Fecontaining systems it is essential to include the Hubbard U correction for the Ni and Fe 3d states.
The values of the U parameters used for the non-metal compounds, however, are different from those used for metallic phases, which can lead to large errors in the formation enthalpy calculations. One way to overcome these issues is to utilize the experimental enthalpies of . However, this approach cannot be used because the experimental value of the formation enthalpy of the ternary compounds from the elemental constituents is not available.
To compute the formation enthalpies we use an approach, [16,32,[38][39][40] Table 1, along with the DFT total energies of the compounds and the fitted elemental reference energies are listed in Table 2.

Results and discussions
To assess the possibility of B incorporation into NFO, we calculate the electronic structure of NFO, evaluate the formation energies of B-related defects and use the thermodynamic scheme described above to estimate the defect stability with reference to the formation of Ni-Fe-B compounds as secondary phases.
6.1 Pure NFO 8 Nickel ferrite spinel is a ferrimagnetic insulator with high Curie temperature (870K) [43] and, therefore, it has a great potential for technological applications especially in the area of spintronics. [44] However, because in this work we do not focus on device applications of NFO, we only give a brief description of the calculated electronic structure of NFO, comparing our results with data available in the literature.   Table III, are consistent with this magnetic structure and are in fairly good agreement with earlier experimental [45] and theoretical values. [15,46] The calculated band gap measured from the VBM to the conduction band minimum (CBM) is 1.3 eV, somewhat larger than the gaps of 0.97, 0.99. and 1.1 eV, obtained with the DFT+U method by Fritsch and Ederer [15], Antonov et al. [46], and Sun et al. [47], respectively.

Fe and Ni vacancies
According to Eq. (1) and (2) the formation energy of a defect D in charge state q is equal to the difference between the energies of the defect-containing and defect free supercells, corrected by the chemical potentials of the atomic and electronic reservoirs with which the system exchanges particles (atoms and electrons). In the case of Fe or Ni vacancies (VFe/Ni) the formation energies are given by The first term on the right-hand side represents the energy difference between the vacancy-containing and pristine systems, corrected by the elemental reference energies and VBM energy (Eq. (2)). The In the case of the charged defects, the formation energy depends on the Fermi level, because to ionize a defect, electrons must be added to or taken from an electron reservoir with energy EF. Figure 4 illustrates the vacancy formation energies as a function of EF, calculated for different charge states, under Fe-poor and Ni-poor conditions. We observe that the Ni and Fe vacancies display similar behavior; when EF is tuned closer to the VBM or CBM, the vacancies become charged and the formation energies decrease considerably. This effect is stronger in ntype NFO (EF closer to CBM), where the formation energy of Ni vacancy drops to approximately 0.5 eV. Figure 4 also illustrates that for all values of EF within the band gap, the formation energy of Ni vacancy is the lowest, suggesting that under appropriate conditions, Ni vacancy could be the dominant intrinsic defect in NFO. 11

Substitutional B defects
There are three types of substitutional B defects in NFO; boron can occupy a tetrahedral or octahedral Fe site (BFe T or BFe O ) or it can substitute for an octahedral Ni ion (BNi O ). Given that the methodology of the formation energy calculations are the same for the Fe-and Ni-site defects, here we describe the details of Fe-site defect calculations while for the Ni-site defect we only present the results.
The formation energy of substitutional B at the Fe site is given by  or Fe3BO5 as secondary phases. The compositions of these phases are very close to those considered by Sawicki [9] in a recent Mössbauer analysis of CRUD scrapes from fuel assemblies exhibiting AOA; the compound include iron borate (FeBO3), iron orthoborate (Fe3BO6) and bonnacordite (Ni2FeBO5).

Interstitial B defects
As illustrated in Fig. 1 Fig. 3, we find that the maximum possible μB is -3.63 eV and it is limited by the formation of Fe3BO5. Figure 6 illustrates the formation energies of the interstitial B defects in NFO as a function of EF. The most stable position for B is the tetrahedral interstitial site (T2), with a formation energy that is close to 0.15 eV as the EF approaches the VBM. Even though this value is considerably lower than all other formation energies, the calculations indicate that it is energetically more favorable for B to form Fe3BO5 instead of entering the NFO structure as an interstitial impurity.

Summary
Using first-principles methods, we have investigated the electronic structure of NFO and combined thermodynamical data with theoretical results to analyze the possibility of B