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Electronic structure and thermodynamic stability of uranium-doped yttrium iron garnet

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 J. Phys.: Condens. Matter 25 495502 (http://iopscience.iop.org/0953-8984/25/49/495502) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 25 (2013) 495502 (10pp)

doi:10.1088/0953-8984/25/49/495502

Electronic structure and thermodynamic stability of uranium-doped yttrium iron garnet Zs R´ak1 , R C Ewing2 and U Becker2 1

Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7907, USA 2 Department of Earth and Environmental Sciences, University of Michigan, Ann Arbor, MI 48109-1005, USA E-mail: [email protected]

Received 11 September 2013, in final form 12 October 2013 Published 1 November 2013 Online at stacks.iop.org/JPhysCM/25/495502 Abstract The electronic and thermodynamic properties of yttrium iron garnet (Y3 Fe5 O12 , YIG), as a possible uranium-bearing phase, have been investigated using first-principles and semi-empirical methods. The electronic structures of pure and U-doped YIG were calculated and compared in order to obtain a fundamental understanding of the incorporation mechanism and stability of U in a YIG matrix. Uranium at the A-site is in 4+ oxidation state, acting as a single donor and introducing a localized defect state in the band gap. The ionic relaxations show U at the A-site is an off-center impurity. At the B-site, uranium is in 5+ oxidation state giving rise to two localized defect states in the middle of the band gap. At thermodynamic equilibrium the incorporation of U is limited by (i) the relatively narrow stability domain of the host YIG and (ii) the precipitation of uranium oxides as secondary phases. Under Y-rich growth conditions, YIG is unstable with respect to competing phases such as the iron oxides, Y2 O3 and YFeO3 . Under O-rich conditions, the incorporation U is obstructed by the formation of uranium-oxide precipitates. Under Fe-rich growth conditions, the formation energies of UY (U at the A-site) and UFe (U at the B-site) become negative for 0 ≤ EF ≤ 0.62 eV and 0 ≤ EF ≤ 0.77 eV, respectively, indicating that U might be incorporated in p-type YIG. (Some figures may appear in colour only in the online journal)

burning MOX fuel in a reactor, generates more 239 Pu (238 U + 1 n → 239 U → 239 Np → 239 Pu), and at the end of its lifetime, 0 the spent MOX fuel also requires disposal. Other countries, such as the USA, Canada, Finland, and Sweden have currently opted for the direct disposal of the spent fuel. In order to ensure safe, long-term disposal, long-lived radionuclides, such as actinides, must be incorporated into durable solid matrices that allow the radioactivity to decay while the material maintains its physical and chemical integrity [3, 4]. This is important because after the disposal of the solid nuclear waste form, interaction with subsurface water can cause the release of radionuclides to the environment. In addition to physical and chemical durability, the matrix must have high capacity for the incorporation of

1. Introduction One of the most critical environmental issues related to the nuclear energy industry is the future of the discharged fuel from the nuclear reactors. There is a world-wide consensus that a safe way of dealing with the radioactive wastes is to isolate them in deep geological repositories [1, 2]. Spent fuel contains substantial quantities of fissile (235 U and 239 Pu), fertile (238 U), and other radioactive materials. Presently, countries such as France, Russia, Japan, India, and China reprocess most of their spent fuel and use the extracted Pu, combined with UO2 , as a mixed-oxide fuel (MOX) in light water reactors (LWR). This approach, however, does not eliminate the need for a geological repository, because 0953-8984/13/495502+10$33.00

1

c 2013 IOP Publishing Ltd Printed in the UK & the USA

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Zs R´ak et al

waste form. The choice of YIG was based on previous results which indicated that the incorporation energies of An decrease with the Fe-concentration in the garnet. Here, we have carried out electronic structure calculations to investigate the incorporation mechanism of U at the A- and B-sites and used first-principles-based thermodynamics to analyze the stability of U inside the YIG structure.

actinides and, possibly, other waste components. One matrix that recently has received considerable attention as an actinide (An) waste form is garnet [5–19]. Based on a variety of experimental and computational studies, Fe-rich matrices, with garnet structure, can incorporate large amounts of actinides (e.g. up to 30 wt% of U) [5–10]. Moreover, a natural uranium-rich garnet, called elbrusite-Zr, has recently been discovered in the Northern Caucasus, and it contains up to 22 wt% uranium [20]. The high Fe- and low Si-content of elbrusite-Zr is similar to synthetic, actinide-bearing garnets. The radiation response of garnet has been investigated by means of ion beam irradiation (1 MeV Kr, 1.5 MeV Xe) and α-decay in 244 Cm-doped samples [11–13]. Radiation-induced amorphization occurs at the dose of ∼0.4 displacements/atom (dpa) and, in contrast to other actinide-bearing matrices, such as pyrochlore, the dose required for amorphization is not affected by composition. As regards the chemical stability of the garnet structure, it has been observed that in neutral and alkaline solutions the leach rate of actinides is negligible (less than 10−4 g m−2 d−1 ); the actinides are essentially not released to solution [14, 15]. Acidic media, however, promote the dissolution of garnet matrices, suggesting that the site of the geological repository must be carefully chosen to avoid leaching of actinides into the biosphere. More importantly, despite the relatively low amorphization dose, the damage to the garnet structure does not affect its chemical stability [14]. Therefore, garnet is an attractive candidate for incorporation and immobilization of actinides. The atomic structure of garnet (Ia3d) is complex. The general formula is A3 B2 X3 O12 (Z = 8), where A is a large cation occupying a dodecahedral site, B and X are smaller cations in octahedral and tetrahedral coordination, respectively. The only free structural parameters belong to the oxygen site, while all the cations occupy special crystallographic positions. Due to the presence of the three cation sites, garnet can incorporate large concentrations of different elements, with a variety of valences and sizes. In recent publications, we have described the electronic structure of Ca3 (Ti, Zr, Hf, Sn)2 Fe3+ 2 O12 series as possible actinide-bearing structures, and we have investigated the mechanism and energetics of actinide (U, Np, and Pu) incorporation at the A- and B-sites [17–19]. To understand the stability of actinide-bearing phases, we examined the electronic interactions between the An 5f and Fe 3d shells [18]. We determined that the antiferromagnetic (AFM) coupling of the An and Fe spins through An–Fe electron transfer lowers the energy of the system and stabilizes actinides at the A-site. Furthermore, to elucidate the role of Fe in the An-incorporation mechanism, the calculations have been performed on several different atomic configurations where the number of Fe atoms in the vicinity of the An was systematically varied. We found that the presence of Fe is crucial, because it accommodates the extra charges introduced by the An, by switching between the 2+ and 3+ oxidation states; therefore, Fe contributes to the stabilization of the An elements in the garnet structure. In the present paper, we investigate the properties of yttrium iron garnet, Y3 Fe5 O12 , as a possible actinide-bearing

2. Methodology The calculations have been performed using the projector augmented wave (PAW) [21, 22] method within density functional theory (DFT) [23, 24] as implemented in the Vienna ab initio simulation package (VASP) [25–28]. The exchange–correlation potential was approximated by the generalized gradient approximation (GGA), as parameterized by Perdew, Burke, and Ernzerhof (PBE) [29]. The standard PAW potentials, supplied with the VASP package, were employed in the calculations [21, 22]. The 4s, 4p, 4d, and 5s states of Y, 3d and 4s states of Fe, 2s and 2p states of O, and 6s, 6p, 5f, and 7s states of U 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 500 eV, and the convergence of self-consistent cycles was assumed when the energy difference between two consecutive cycle was less than 10−4 eV. All calculations involving the garnet structure used a 2 × 2 × 2 Monkhorst–Pack k-point mesh [30] and a Gaussian smearing of 0.1 eV. These parameters were chosen based on convergence tests and they guarantee an accuracy of less than 10 meV/atom in the electronic structure calculations. The internal structural parameters were relaxed until the total energy and the Hellmann–Feynman forces on ˚ −1 . The calculations on each nuclei were less than 0.02 eV A the U-containing garnet structures were completed using the cubic unit cell of YIG containing 160 atoms. The theoretical ˚ was used in the defect lattice constant of YIG (a = 12.49 A) calculations, in order to avoid spurious elastic interaction between the U ions in neighboring supercells [31]. In order to describe the behavior of the localized Fe 3d and U 5f states, we have included the orbital-dependent, Coulomb potential (Hubbard U) and the exchange parameter J in the calculations within the DFT + U method [32]. We used the rotationally invariant approach introduced by Liechtenstein et al [33]. The value of the Hubbard U parameter can be estimated from band-structure calculations in the supercell approximation with different d and f occupations [34] or from calculations base on constrained random-phase approximation [32, 35]. Here, we treat U and J as adjustable parameters using the following values: U(Fed ) = 4.8 eV with the corresponding J(Fed ) = 0.5 eV, and U(Uf ) = 4.5 eV with J(Uf ) = 0.5 eV. These values are physically reasonable and are within the range of the previous values in the literature [32, 36, 37]. In order to deal with the metastable states inherent in a calculation involving explicitly orbital-dependent Hamiltonian (such as DFT + U), we performed the calculations using the, so called, U-ramping method, proposed by Meredig et al [38]. 2

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Table 1. Calculated and measured interatomic distances in YIG. The experimental values are taken from Geller and Gilleo [41]. All ˚ distances are given in A. Feoct –O

Fetetr –O

Y–O

Feoct –Y

Fetetr –Y

Theor.

2.04

1.88

3.49

Exp.

2.00

1.88

2.37, 2.46 2.37, 2.43

3.12, 3.82 3.09, 3.79

3.46

The PAW potentials supplied with the VASP package have been generated from fully relativistic calculations of the atomic or ionic references. Therefore in a PAW-VASP calculation the core electrons are treated fully relativistically, while the valence electrons are treated using a scalar relativistic approximation (this includes the mass-velocity and the Darwin terms but no spin–orbit interaction) [39]. Although the spin–orbit interaction (SOI) is important for the heavy elements, such as U, because the properties investigated here are based on total energy differences, we expect that the errors introduced by neglecting the SOI will mostly cancel out [40].

Figure 1. (a) The total DOS of the pure YIG. (b) DOS projected on the 3d orbitals of the octahedral and (c) tetrahedral Fe.

value of 2.85 eV obtained from high-temperature conductivity measurements [45]. The main features in the electronic structure can be easily identified in the DOS plot shown in figure 1. The occupied Fe 3d states are located between −6 and −7 eV below the Fermi level (EF ) and their localized nature is well-captured by the on-site Coulomb interaction of U(Fed ) = 4.8 eV. The corresponding unoccupied portion of the Fe 3d states is located in the conduction band (CB) between 2 and 4 eV, and they overlap with the empty Y 4d and Y 5s bands. The valence band (VB), between EF and −5 eV, is dominated by O 2p states, with small Y 4d and Y 5s contribution. Figures 1(b) and (c) show the DOS projected on the 3d orbitals of the octahedral and tetrahedral Fe. The spin-up states of the Fe3+ ions located at the octahedral site (16c) are occupied and the spin-down states are unoccupied. The situation is reversed for the Fe3+ ions located at the tetrahedral site (24d), giving rise to ferrimagnetism in YIG. The calculated spin magnetic moments are 4.20 and −4.12 µB for the tetrahedral and octahedral site, respectively. This leads to a magnetization of 3.96 µB per formula unit, in fairly good agreement with the measured magnetization of 3.8 µB in YIG with magnetic structure refined in the trigonal R3 space group [46].

3. Results and discussions In order to understand the electronic interaction of U with the host YIG, we analyzed the electronic structures of the pure and U-doped YIG, where U was treated as a substitutional impurity located successively at the A- and B-sites. The thermodynamic stability of U inside the structure was studied using a first-principle-based, equilibrium thermodynamics in which we combined experimental enthalpies of formation with calculated DFT energies. 3.1. Pure YIG Yttrium iron garnet (Y3 Fe2 Fe3 O12 ) is a ferrimagnetic oxide with the garnet structure. The dodecahedral sites (A-site, 24c) are occupied by the Y3+ ions, while both the octahedral and tetrahedral sites (B- and X-sites, 16a and 24d) are occupied by Fe3+ ions. The ferrimagnetism in this material is due to the antiferromagnetic coupling between the moments of the tetrahedral and octahedral Fe3+ ions. As listed in table 1, the calculated structural parameters compare well with the experimental values of Geller and Gilleo [41]. For example, the theoretical unit cell constant ˚ is approximately 1% larger than the experimental (12.49 A) ˚ This is due to one, measured at room temperature (12.38 A). the well-know limitation of the GGA functional. Although the physical properties of YIG make it very useful for optical, magneto-optical, and microwave applications, only a few theoretical studies have been carried out that address the fundamental electronic properties of this material [42–44]. In this paper, because we do not focus on device applications of YIG, we only give a brief description of the calculated electronic structure. Figure 1 shows the total electronic density of states (DOS) and the partial Fe d DOS for YIG. The calculated band gap is 2.18 eV, which is ∼24% smaller than the experimental

3.2. Uranium at the A-site To describe the electronic structure of U-doped YIG, we treat U as a substitutional impurity. Because the U atom (5d3 6d1 7s2 ) has more valence electrons than Y (4d1 5s2 ) or Fe (3d6 4s2 ), the substitutional U behaves as a donor impurity that provides extra electrons to the system. When at the A-site, three of the extra electrons introduced by U participate in bonding, just like the valence electrons of Y, and the rest (up to three electrons per unit cell) can be donated to the conduction band or can be trapped in localized defect state in the band gap. As discussed in [18], the magnetic coupling via electron transfer between the U 5f and Fe 3d orbitals plays a major role in the incorporation mechanism. The U–Fe electron transfer 3

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Figure 2. (a) The local atomic structure of the A-site, showing the off-centering of the substitutional U. (b) Charge density distribution associated with the UY -induced defect state.

is energetically more favorable if the 5f and 3d spins are antiferromagnetically aligned. This is because the 3d shell of the Fe3+ ion being half-filled, with all the spins of the unpaired electrons pointing in the same direction, it can only accommodate electrons with spins oriented in the opposite direction. As a result, the electrons will be able to keep their spin orientation as they travel from An to Fe, only if the spins of the f shell and the d shell are anti-parallel. In the present investigation we only consider the spin configuration that, according to our previous work, is most favorable for uranium incorporation (i.e. the U 5f and Fe 3d spins are anti-parallel). Figure 2(a) illustrates the local atomic structure of the dodecahedral site (A-site): it is coordinated by eight O atoms that are bonded to octahedral and tetrahedral Fe’s. Figure 2(a) shows the two tetrahedral Fe atoms that are located closest to the A-site. The equilibrium position of the substitutional U is slightly shifted away from the regular crystallographic site, as indicated in figure 2(a). The mechanism that causes this ionic displacement is related to the charge transfer between the U and the neighboring Fe atoms. As mentioned earlier, since U has six valence electrons (5d3 6d1 7s2 ), it can provide up to three extra electrons to the system when it substitutes for Y (4d1 5s2 ). However, according to these calculations, the most stable oxidation state of the substitutional U at the A-site (UY ) is U4+ (5d2 ), which indicates that only one extra electron is given up by U. This electron, instead of being equally shared by the two nearest neighbor (NN) Fe ions (as it would happen if U occupied the center of the site), is rather transferred to one single Fe. Consequently, the covalency of the chemical bonds between U and its NN ions will be dissimilar, resulting in the displacement of U toward the Fe ion that receives the extra electron. In order to describe the incorporation mechanism of U at the A-site, in figure 3 we plot the total DOS of the U-doped YIG, along with the projected DOS (PDOS) associated with the f and d states of U, as well as the d states of the two Fe ions from the proximity of U. From the comparison of the DOS plots shown in figures 1(a) and 3(a), we observe that the presence of U slightly perturbs the host VB and CB and introduces a defect state in the band gap. The origin

Figure 3. (a) Total DOS of YIG with U at the A-site (substituting Y) and the partial DOS associated with (b) U 5f states, (c) U 6d states, (d) 3d states of the Fe atom that is NN to U, and (e) 3d states of an Fe that is next NN to U.

of this state can be identified from the inspection of the PDOS shown in figures 3(b) and (d). We observe that the 5f orbitals of U (figure 3(b)) and the 3d orbitals of one of the Fe atoms (figure 3(d)) contribute significantly to this state. The 3d states of the other Fe, shown in figure 3(e), remain almost unperturbed as compared with the Fe 3d in the pure YIG (figure 1(c)). This is consistent with the idea that the extra electron introduced by U is transferred to one of its NN Fe creating a Fe2+ ion. As a result of this charge transfer one spin-up Fe 3d orbital becomes occupied and pulled down in energy, giving rise to the defect state in the band gap. The charge density distribution associated with this state, illustrated in figure 2(b), also shows a strong O p contribution. This implies that the U–Fe charge transfer takes place via the closed 2p shells of the O2− ions, consistent with 4

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Figure 4. (a) Local atomic structure of the B-site. (b) The band decomposed charge density associated with the two gap state introduced by UFe .

the double exchange interaction observed earlier for similar systems [18]. The presence of U at the A-site also introduces several conduction band states with energies located between 3 and 7 eV above EF . As shown in figures 3(b)–(d) these states originate from the empty U 5f and 6d orbitals, as well as from the perturbed 3d orbital of the Fe that accommodates the extra electron. 3.3. Uranium at the B-site Figure 4(a) illustrates the local atomic environment of the octahedral site (B-site). In pure YIG this site is occupied by Fe3+ that is coordinated by eight O2− ions. Each O is connected to tetrahedrally coordinated Fe3+ . When the large U ion is placed at the octahedral site (B-site), we observe a strong outward relaxation of the six NN O atoms, as compared with the tetrahedral Fe–O distances in pure YIG. The relaxation is not identical for all the six O atoms: two of ˚ as compared to the them remain slightly closer to U (2.12 A) ˚ As explained below, the origin rest of the O atoms (2.18 A). of this non-uniform ionic relaxation can be related to the local electronic interactions between U and its NN O and Fe ions. When U (5f3 5d1 7s2 ) substitutes for Fe (3d6 4s2 ), it can provide up to three extra valence electrons to the system, depending on the oxidation state of U. According to our calculations, the most stable oxidation state of the substitutional U at the B-site (UFe ) is U5+ (5f1 ), suggesting that two electrons are transferred from U to the system. To further analyze the interaction between U and its environment, we calculate the electronic band structure of the U-doped YIG. As shown in figure 5, U at the B-site introduces two localized, almost degenerate states in the band gap (see the inset in figure 5). The origin of these two states can be identified by inspecting the charge density distribution shown in figure 4(b). The most significant contribution to the charge density comes from the 3d orbitals of two Fe atoms. These are precisely the two Fe’s that are connected to the two O’s located closest to the U, indicating that the two extra electrons provided by U are localized on these two Fe ions. As a result,

Figure 5. The electronic band structure of YIG with U at the B-site (substituting Fe). There are two localized defect states in the band gap, as shown in the inset.

two spin-up 3d orbitals (one from each Fe) become occupied and pulled down in energy giving rise to the defect states in the band gap. In addition, the U 5f and two O 2p orbitals also contribute to the gap states, indicating that the U–Fe electron transfer takes place through the closed shell O2− ion, consistent with the double exchange mechanism described earlier. 3.4. Thermodynamics of U incorporation To identify the most stable location of U inside the YIG structure and to determine the thermodynamic limits of U incorporation, we calculate the defect formation energies of substitutional U at the A- and B-sites. In general, the creation of a defect in a crystalline material can be regarded as a process by which particles (atoms and electrons) are exchanged between the host and chemical reservoirs. Thus, the formation energy of a defect D in charge state q can be calculated as [47, 48]: X 1Hf (Dq ) = E(Dq ) − E0 + ni (µi + Ei ) i

+ q(EF + EVBM ). 5

(1)

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In (1) E(Dq ) and E0 are the total energies of the defect-containing and defect-free solids. The third term in (1) represents the change in energy due to the exchange of particles between the host compound and the chemical reservoir. µi are the chemical potential of the atomic species i (i = Y, Fe, O, and U) referenced to the elemental solid/gas with energy Ei and ni are the number of atoms added to (ni < 0) or removed from (ni > 0) the supercell. EF is the Fermi energy referenced to the energy of the valence band maximum (VBM), EVBM . This value is calculated as the VBM energy of the pure YIG corrected by aligning the core potential of atoms far away from the defect in the defect-containing supercell with that in the defect-free supercell [48]. q represents the charge state of the defect, i.e. the number of electrons exchanged with the electron reservoir with chemical potential EF . Equation (1) shows that, in principle, by adjusting the atomic chemical potential of the constituents and by tuning the electronic Fermi energy, one can control the defect formation energy and consequently the solubility of the dopant in the host structure. Under thermodynamic equilibrium, the achievable values of the chemical potentials are limited by several conditions:

differences between insulating or semiconducting compounds and their elemental constituents in the standard state. The theoretical enthalpy (heat) of formation of the compound An Bm . . . can be calculated as: 1H theor (An Bm . . .) = E(An Bm . . .) − nEA − mEB − · · · (8) where E(An Bm . . .) is the total energy per formula unit (f.u.) of the compound and EA , EB , . . . are the total energies per atom of the elements in their standard state. In our case the compounds under consideration are the insulating or semiconducting YIG, Fen Om , and Un Om while the elemental phases include the metallic form of the cations (Y, Fe, U) and the gaseous anion (O2 ). These are chemically and physically dissimilar systems, where the cancelation of the DFT errors is known to be incomplete [49–51]. Furthermore, to reproduce the correct electronic structures of YIG, iron oxides, and uranium oxides it is essential to include the Hubbard U correction for the Fe 3d and U 5f states. The values of the U parameters used for the non-metal compounds, however, are different than those used for metallic phases, leading to large errors in the formation enthalpy calculations. One way to overcome these issues is to utilize the experimental enthalpies of formation (1H exp ) in equations (3)–(7). However, we cannot employ this approach because, to our knowledge, the experimental value of the YIGs formation enthalpy from the elemental constituents is not available. To compute the formation enthalpies we use the approach proposed by Lany [52], in which the elemental energies EA , EB , . . . are approximated from the system of linear equations:

(a) to avoid elemental precipitations, the chemical potentials are bound by: µY ≤ 0;

µFe ≤ 0,

µO ≤ 0,

and

µU ≤ 0

(2)

(b) to maintain a stable YIG host the µi ’s must satisfy:

1H exp (An Bm . . .) = E(An Bm . . .) − nEA − mEB − · · · . (9)

3µY + 5µFe + 12µO = 1H (Y3 Fe5 O12 ) = 1H(YIG), (3)

We calculate the DFT energies of 16 binary compounds that can be formed from six elements (F, Fe, N, O, U, and Y), for which the 1H exp values are available [53–55]. Then, we solve the overdetermined system of equations given by (9) in a least-squares approach. This way we compute the elemental energies, EA , EB , . . ., without directly calculating the DFT energies of the elements in their standard metallic or gaseous state. The obtained values are used as the elemental reference energies in (1) and for the calculation of the enthalpies of formation required in (3)–(7). The experimental and theoretical values of the formation enthalpies, along with the DFT total energies of the compounds are listed in table 2, and the fitted elemental reference energies are listed in table 3. To estimate the uncertainty of the theoretical enthalpies we calculate the difference between 1H exp and 1H theor using the values listed in table 2. Since the average discrepancy between 1H exp and 1H theor is approximately 0.4 eV we estimate that the calculated values are determined with and accuracy of ±0.4 eV.

where 1H(Y3 Fe5 O12 ) = 1H(YIG) is the formation enthalpy of YIG, (c) to avoid formation of competing phases, such as iron oxides (wustite, hematite and magnetite) and yttrium oxide, the following conditions must apply: nµFe + mµO ≤ 1H(Fen Om ), where (n, m) = (1, 1), (2, 3), and (3, 4) 2µY + 3µO ≤ 1H(Y2 O3 ).

(4) (5)

Here we also consider YFeO3 as a competing phase, so that: µY + µFe + 3µO ≤ 1H(YFeO3 ).

(6)

(d) Further constraints on the chemical potential are posed by avoiding formation of uranium oxides (UO2 and UO3 ) secondary phases nµU + mµO ≤ 1H(Un Om ), where (n, m) = (1, 2) and (1, 3).

(7)

3.6. Uranium incorporation energies The dark area in figure 6 shows the calculated chemical potential domain where YIG is stable, according to conditions (a)–(c). The vertices of the triangle in figure 6 represent the achievable limits of the Fe and Y chemical potentials under

3.5. Formation enthalpies and elemental reference energies To predict the enthalpies of formation (1H) required for equations (3)–(7), it is necessary to compute total energy 6

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Table 2. The calculated DFT energies/f.u. of the 16 binary and the two ternary compounds used in (1)–(9), along with the experimental and theoretical formation enthalpies (per f.u.) of the same compounds. The DFT-optimized structural parameters are also shown. ˚ Structural parameters (A) Compound

DFT energy/f.u. (eV)

1H

exp

(eV)

1H

theor

(eV)

a

b

c

Binaries FeO (monoclinic) Fe2 O3 (hexagonal) Fe3 O4 (cubic) FeF2 (tetragonal) FeF3 (cubic) UO2 (cubic) UO3 (tetragonal) UF3 (hexagonal) UF4 (monoclinic) UF5 (tetragonal) UF6 (orthorhombic) UN (cubic) UN2 (cubic) Y2 O3 (cubic) YF3 (cubic) YN (cubic)

−13.63 −34.00 −48.12 −17.87 −21.49 −29.51 −34.31 −30.02 −35.80 −39.19 −41.99 −19.91 −28.22 −45.56 −27.72 −18.18

−2.73 −8.53 −11.49 −7.33 −10.80 −11.24 −12.75 −15.62 −19.89 −21.47 −22.78 −3.05 −4.52 −19.74 −17.81 −3.10

−2.72 −8.37 −11.57 −7.99 −10.23 −11.50 −12.50 −15.46 −19.87 −21.87 −23.29 −2.67 −4.16 −19.88 −16.44 −4.21

5.21 5.09 8.46 4.74 9.84 5.55 7.02 7.21 12.99 6.84 10.24 5.18 5.33 10.71 5.39 4.92

3.21 5.09 8.46 4.74 9.84 5.55 7.02 7.21 10.98 6.84 9.26 5.18 5.33 10.71 5.39 4.92

3.21 13.91 8.46 3.36 9.84 5.55 20.33 7.27 8.50 4.56 5.42 5.18 5.33 10.71 5.39 4.92

−39.83



−14.17

5.65

7.70

5.34

−154.51



−51.90

12.49

12.49

12.49

β = 52.02◦

β = 53.73◦

Ternaries YFeO3 (orthorhombic) Y3 Fe5 O12 (YIG) (cubic)

Table 3. The fitted elemental reference energies of the six elements used in (9). All values are given in eV. Element Fitted elemental energies (eV)

Y

Fe

O

−7.14

−7.11

−3.80

U −10.40

F

N

−1.38

−6.83

−19.88 eV. At O-rich condition these give µFe ≤ −4.18 eV and µY ≤ −9.94 eV. To make the incorporation of U at the Y-site (i.e. U at the A-site) favorable we have to minimize µY and maximize µFe . Therefore, we set µFe = −4.18 eV, which gives, through (3), µY = −10.33 eV. With these values the formation of the secondary phase YFeO3 (6) is also avoided. To eliminate precipitation of uranium oxides (UO2 and UO3 considered here) as secondary phases, the chemical potentials must satisfy µU + 2µO ≤ −11.50 eV and µU + 3µO ≤ −12.50 eV, which lead to µU ≤ −12.50 eV. To incorporate as much U as possible into the YIG structure, we have to maximize the chemical potential of U: µU = −12.50 eV. Using (1) and the values described above, the formation energy of a neutral U defect at the A-site is given by 1Hf (U0Y ) = 0.91 + µY − µU = 3.08 eV. In the case of the charged defects, the formation energy depends also on the Fermi level, because in order to ionize a defect, electrons must be taken from or added to an electronic reservoir with energy EF . Therefore, using the chemical potentials specified above, 2+ (1) gives 1Hf (U+ Y ) = 1.56+EF and 1Hf (UY ) = 1.26+2EF . The above formalism can also be used to describe the case of U at the B-site (i.e. U substituting for an octahedral Fe), with the difference that in order to make the incorporation favorable we have to minimize µFe (or maximize µY ). Therefore, we have µY = −9.94 eV, which gives through

thermodynamic equilibrium: A corresponds to the Fe-rich, Y-rich (µFe = 0, µY = 0) limit, B represents the Fe-poor, Y-rich (µFe = −10.38 eV, µY = 0) limit, and C is the Fe-rich, Y-poor (µFe = 0, µY = −17.30 eV) limit. As shown in figure 6, the area of the allowed chemical potentials for a stable YIG is relatively narrow. In the white regions YIG is unstable with respect to competing phases: the upper part of the triangle represents the region where Y2 O3 and YFeO3 will form, while the lower right corner is excluded due to precipitation of FeO, Fe2 O3 , and Fe3 O4 . To calculate the incorporation energy of U into the YIG matrix, we use (1), which shows that the energy depends on the chemical potentials of the constituents and the Fermi level. Since under Y-rich (µY = 0) condition the host YIG is not stable (figure 6), we will describe the incorporation energy calculations under O-rich (µO = 0) and Fe-rich (µFe = 0) conditions. 3.6.1. O-rich condition. The calculated enthalpy of formation of YIG is 1H(YIG) = −50.91 eV, therefore the chemical potentials must satisfy 3µY + 5µFe = −51.90 eV. Because Fe and Y can form oxides as secondary phases, as described in condition (c), the chemical potentials are further limited by µFe + µO ≤ −2.27 eV, 2µFe + 3µO ≤ −8.37 eV, 3µFe + 4µO ≤ −11.57 eV, and 2µY + 3µO ≤ 7

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Fe-rich condition, U can be stabilized at the A-site. We will discuss more details about this possibility later in this section. According to (1), the formation energy of UFe does not depend explicitly on the chemical potential of O. However, within the limits of condition (d), a reduced value of µO , allows for a larger µU . A large µU , in turn, decreases the formation energy of UFe . Therefore, to make the incorporation of U at the B-site favorable, we have to minimize the chemical potential of O. The lowest possible value of µO is −2.94 eV, which results in µU ≤ −5.62 eV. With these values the incorporation energies of U at the B-site, under Fe-rich conditions, are 1Hf (U0Fe ) = 2.01 eV, 1Hf (U+ Fe ) = 0.63 + EF , and 1Hf (U2+ ) = −1.54 + 2E . F Fe Figures 7(a) and (b) show the calculated incorporation energies of UY and UFe as a function of EF , under Fe-rich conditions, for different charge states. We observe that for almost all values of EF above the middle of the bandgap (n-type material), the formation energies are relatively high. If EF is below the midgap (p-type semiconductor) UY and UFe become positively charged, and the formation energies decrease significantly. Moreover, when EF is tuned closer to the VBM (EF ≤ 0.62 eV for UY and EF ≤ 0.77 eV for UFe ) both defects are positively charged, and their formation energies become negative. This suggests that the incorporation of U into the YIG structure is possible in the p-type YIG. This could be achieved by introducing divalent dopants, such as Ca, Cd or Zn, into the YIG structure. We note that even though the intervals where the formation energies are negative are reasonably wide, we cannot exclude the possibility that in these domains other uranium oxides, that are not considered in this work (such as U3 O8 , U4 O9 , etc) could precipitate, which would hinder U incorporation. By comparing figures 7(a) and (b), we observe that the range of EF for which 1Hf (UFe ) is negative is larger than the range for which 1Hf (U2+ Y ) is negative. Also, at the VBM (EF = 0) the value of 1Hf (U2+ Fe ) = −1.54 eV is smaller as 2+ compared with 1Hf (UY ) = −0.92 eV. This indicates that U is more stable at the B-site than at the A-site, therefore it is more likely that U substitutes for octahedral Fe in YIG.

Figure 6. The calculated stability domain (dark area) of the YIG in the (µFe, µY) plane. The white areas are regions where secondary phases precipitate.

(3) µFe = −4.42 eV. In this case, the formation energy of the neutral UFe is 1Hf (U0Fe ) = −3.61 + µFe − µU = 4.47 eV, while the formation energies of the charged defects 2+ are 1Hf (U+ Fe ) = 3.09 + EF and 1Hf (UFe ) = 0.92 + 2EF . The calculated formation energies are relatively high, suggesting that, under O-rich condition, the incorporation of U into the YIG structure is unlikely. This is because under O-rich condition the incorporation is limited by the formation of binary oxides as secondary phases. 3.6.2. Fe-rich condition. In order to maintain a stable host under Fe-rich (µFe = 0) condition, µY and µO must satisfy 3µY + 12µO = −51.90 eV. Combining this equation with conditions (a) and (c), we obtain the domains of the chemical potentials in which the formation of iron oxides, Y2 O3 , and YFeO3 are avoided: −5.73 ≤ µY ≤ −5.54 eV and −2.94 ≤ µO ≤ −2.89 eV. By tuning the values of the chemical potentials within these intervals, we can stabilize U at the A- or B-sites inside the YIG structure. To make the incorporation of U at the A-site as easy as possible we minimize chemical potential of Y: µY = −5.73 eV. This gives through (3) µO = −2.89 eV. By imposing conditions (c), to avoid precipitation of UO2 and UO3 , we obtain µU ≤ −5.72 eV. Using these values the defect formation energies of U at the A-site, under Fe-rich conditions, can be calculated as: 1Hf (U0Y ) = 0.90 eV, 1Hf (U+ Y ) = −0.62 + EF , 2+ and 1Hf (UY ) = −0.92+2EF . We observe that the formation energy of the neutral defect is positive and relatively high, while the energies of the charged defects can be negative for certain values of EF . This suggests that by tuning EF , under

4. Summary Using first-principles methods within the DFT, we have investigated the electronic structure of the pure and U-doped YIG. We have analyzed the incorporation mechanism of U at the A- and B-sites and combined experimental data with theoretical results to examine the thermodynamic limits to U incorporation. The most stable oxidation state of the substitutional U at the A-site (UY ) is U4+ (5d2 ), which indicates that only one extra electron is given up by U. This electron is transferred to a NN tetrahedral Fe. As a result of the charge transfer, one empty Fe 3d orbital becomes occupied and pulled down in energy, giving rise to a defect state in the band gap of YIG. Another consequence of the U–Fe electron transfer, is that the covalency of the chemical bonds between U and its NN ions will be dissimilar, resulting in the off-centering of the U. 8

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Figure 7. The formation energies of (a) UY and (b) UFe under Fe-rich conditions, as a function of the Fermi energy. The slope of the line indicates the charge state of the defect and the value of EF where the slope changes represents the charge transition level (ionization level). In the case of UFe , the 1+ charge state is not stable with respect to the neutral or doubly charged state.

When U is at the B-site, substituting for a tetrahedral Fe3+ (UFe ), the most stable oxidation state is U5+ . The two extra electrons introduced by U are transferred to NN tetrahedral Fe ions. Consequently, two spin-up 3d states become occupied resulting in two localized, energetically almost degenerate gap states. Under thermodynamic equilibrium U can only be incorporated into YIG under Fe-rich conditions. The main factors that limit the U incorporation are: (i) the narrow stability domain of the host YIG and (ii) the precipitation of uranium oxides as secondary phases. Under Y-rich growth conditions, YIG is unstable with respect to competing phases such as the iron oxides, Y2 O3 and YFeO3 , while under O-rich condition the uranium oxides are easily formed. The incorporation energies depend sensitively on the electron chemical potential (EF ) and the charge state of the defect. Under Fe-rich conditions, in p-type YIG both UY and UFe become positively charged, and the incorporation energies decrease significantly. The formation energies of UY and UFe become negative for 0 ≤ EF ≤ 0.62 eV and 0 ≤ EF ≤ 0.77 eV, respectively, indicating that U might be incorporated in p-doped YIG. The p-type doping can be achieved by divalent substitutional impurities, such as Ca, Cd, and Zn. Comparison of the calculated formation energies of U2+ Y and U2+ suggest that in the YIG structure U prefers the octahedral Fe site.

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Acknowledgments This research was supported as part of the Materials Science of Actinides, an Energy Frontier Research Center, funded by the Office of Basic Energy Sciences under Award No. DESC0001089. The computational work has been performed at the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231.

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Electronic structure and thermodynamic stability of uranium-doped yttrium iron garnet.

The electronic and thermodynamic properties of yttrium iron garnet (Y3Fe5O12, YIG), as a possible uranium-bearing phase, have been investigated using ...
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