Abstract
Ab initio calculations have been performed on titanate pyrochlores A2Ti2O7 (A = Dy, Ho, and Y) to investigate their electronic structures. The generalized gradient approximation (GGA) + formalism has been used to correct the strong onsite Coulomb repulsion between the localized 4f electrons. The effects of effective values on the structural and electronic properties of A2Ti2O7 (A = Dy, Ho, and Y) have been discussed. It is shown that Dy2Ti2O7 and Ho2Ti2O7 exhibit different electronic structures from Y2Ti2O7. The strong interaction between Dy and Ho 4f electrons and O 2p orbitals may increase the covalency of and bonds and decrease their irradiation resistance.
1. Introduction
Materials with the A2B2O7 pyrochlore structure have wide ranges of composition that lead to remarkable properties and wide variations in ionic and electronic conductivity, catalytic activity, and electrooptic and piezoelectric behavior [1]. Because they can be used to immobilize actinides [2–5], the pyrochlores have attracted significant attention both theoretically and experimentally [6–21]. In A2B2O7 pyrochlore structure, the A and B cations occupy the 16 (0.5, 0.5, 0.5) and 16 (0, 0, 0) sites, respectively, and the oxygens are in the 48 (, 0125, 0.125) and 8 (0.375, 0.375, 0.375) positions (using the Wyckoff notation) [22]. Single crystals of the A2Ti2O7 pyrochlores (A = Sm to Lu and Y) have been irradiated by 1 MeV Kr+ ions, and their microstructural evolutions as a function of increasing radiation dose have been characterized [7]. A slight deviation from the monotonic trend of critical amorphization temperature versus the ionic radius was observed for Y2Ti2O7. is frequently used as a measurement of the resistance of a material to amorphization, and lower values of are the result of substantial dynamic annealing occurring during irradiation, which allows the materials to remain in the crystalline state. Generally, pyrochlores that are closer to the ideal fluorite structures are more susceptible to the radiation-induced pyrochlore-to-defect fluorite structural transition. The defect fluorite structure results from disordering of the A- and B-site cations, as well as the anion vacancies. Thus, pyrochlore compositions that are more easily disordered to the defect fluorite structure are more “resistant” to ion-beam-induced amorphization [7]. On the other hand, theoretical investigations reported by Sickafus et al. [18] have demonstrated that compounds with very dissimilar cationic radii (e.g., closer to the ideal pyrochlore structure, ) should exhibit the greatest susceptibility to structural destabilization (e.g., amorphization), whereas compounds with more similar radii (e.g., closer to the ideal fluorite structure) should behave more robustly in a radiation environment. For A2Ti2O7 (A = Dy, Ho, and Y), the ionic radius ratio () decreases monotonically. According to Sickafus’ point of view, the compounds should become more radiation resistant as the cation A varies from Dy to Ho. However, the critical amorphization temperatures were measured to be 910, 850, and 780 K for Dy2Ti2O7, Ho2Ti2O7, and Y2Ti2O7 compounds, respectively [7], meaning that of these three compounds, Y2Ti2O7 is the most radiation resistant. It is clear that cation radius ratio alone cannot be used to explain the different responses of these compositions to radiation.
To resolve the discrepancy between experimental observations and theoretical predictions, it is important to fundamentally understand the electronic structures of A-site elements and their effects on the stability of the pyrochlore structure. In the present study, ab initio total energy calculations [17, 19, 23] based on density functional theory have been performed on A2Ti2O7 (A = Dy, Ho, and Y) pyrochlores. GGA + formalism has been used to account for the strong on-site Coulomb repulsion among the 4f electrons in Dy2Ti2O7 and Ho2Ti2O7. How the structural and electronic properties of A2Ti2O7 (A = Dy, Ho, and Y) pyrochlores are affected by the choice of is discussed. These calculations will provide significant insight into the effects of electronic configuration on the thermochemical stability of pyrochlore of different compositions.
2. Calculational Method
All the calculations have been completed using the VASP code [24] with spin-polarized effects taken into account. A primitive unit cell containing 22 atoms was used for the present investigations, with a Monkhorst-Pack -point mesh. The ion-electron interaction was described by PAW pseudopotentials with the following atomic valence configurations: Ti (3s2, 3p6, 3d2, and 4s2), Y (4s2, 4p6, 4d1, and 5s2), Dy (5s2, 5p6, 4f10, and 6s2), and Ho (5s2, 5p6, 4f11, and 6s2). The PBE functional within the generalized gradient approximation was used to describe the exchange-correlation potential energy [25, 26], with the basis set for valence electrons consisting of plane waves with a cut-off energy of 400 eV. The calculations were performed based on ferromagnetic ordering of the magnetic moments and antiferromagnetic ordering, and spin ice model [27, 28] is not considered in the present work. The Hubbard correction was introduced using the method proposed by Dudarev et al. [29], in which the Hubbard parameter reflecting the strength of onsite Coulomb interaction and parameter adjusting the strength of exchange interaction are combined into a single parameter .
3. Results and Discussion
3.1. Atomic and Electronic Structure of Dy2Ti2O7
The pyrochlore structure can be completely described by the a-cell edge, , and the 48 oxygen positional parameter, . The dependence of the lattice parameter , , and band gap of the Dy2Ti2O7 on is shown in Figures 1(a), 1(b), and 1(c), respectively. As shown in Figure 1, the structural parameter decreases slightly for eV. Above this value, the parameter changes more significantly. A of 2.6 eV yields a lattice constant of 10.12 Å, in excellent agreement with experiments [7]. For O48f positional parameter , it deviates from experimental value of 0.3275 with increasing values; that is, introducing makes the deviation from the experiment larger. Concerning the band structure of Dy2Ti2O7, a sharp increase of band gap value with growing is observed. For a certain value of around 3.5 eV, the band gap value of 2.85 eV matches the calculated optical band gaps performed by Nemoshkalenko et al. [30]. It is shown that the band gap of Dy2Ti2O7 presents stronger dependence on the value than lattice parameters.
(a)
(b)
(c)
Figure 2 displays the total density of state (DOS) distribution of Dy2Ti2O7 at , 2.0, and 3.0 eV. The Fermi level is set to 0 eV. At the pure GGA level, the f band does not split but shows a large peak around the Fermi level, leading to a metallic ground state, which disagrees with experiments [31]. Obviously, the pure GGA calculation without modifying the intra-atomic Coulomb interaction gives wrong results. It is suggested that introduction of a penalty function which corrects the intraband Coulomb interaction by the Hubbard parameter is necessary for strongly correlated systems [32] such as Dy2Ti2O7. If the is increased to 2 eV, the f bands undergo splitting, and a semiconducting solution with a finite separation of the occupied and unoccupied f band is found. The obtained band gap between the valence band edge (contributed by Dy 4f) and the bottom of the conduction band (contributed by Dy 4f) is 1.53 eV. In the case of eV, the unoccupied f bands shift toward higher energy level, resulting in a larger band gap value of 2.36 eV. A notable difference between the cases of eV and eV is that there is no mixture between the occupied f bands and the O 2p orbitals in the valence region at eV, whereas the occupied f bands become hybridized with the O 2p orbitals in the valence region at eV.
| (a) eV |
| (b) eV |
| (c) eV |
3.2. Atomic and Electronic Structure of Ho2Ti2O7
Figure 3 shows the lattice parameters and band gaps of the Ho2Ti2O7 as a function of . Generally, Ho2Ti2O7 shows similar dependence on the effective values to the case of Dy2Ti2O7. The lattice constants decrease with increasing , and the reason for this behavior is a slight hybridization of Ho 4f and O 2p orbitals. The experimental lattice constant [7] of 10.104 Å is obtained for eV. For the O48f positional parameter, it deviates from experimental value of 0.3285 with increasing values. As compared with Dy2Ti2O7, the band gap value increases less significantly with . At eV, the band gap of Ho2Ti2O7 is still 0.82 eV smaller than experimental measurement of 3.2 eV [33].
(a)
(b)
(c)
Figure 4 presents the total DOS of Ho2Ti2O7 at , 2.0, and 3.0 eV. At eV, the pure GGA calculation without modifying the intra-atomic Coulomb interaction yields a metallic ground state, in contrast to experiments [33]. As the is increased to 2 eV, the f bands splits, and the occupied and unoccupied f band contributes significantly to the valence bands and conduction bands, respectively. The corresponding band gap between the valence band edge and the bottom of the conduction band is 1.65 eV. Different from the total DOS distribution of Dy2Ti2O7 at eV, the occupied f bands hybridize with O 2p orbitals in the valence region. At eV, it is noted that the main effect of increasing value is to push the unoccupied f bands toward higher energy level, resulting in a larger band gap value of 2.38 eV.
| (a) eV |
| (b) eV |
| (c) eV |
3.3. Atomic and Electronic Structure of Y2Ti2O7
Yttrium titanate pyrochlore is an important member of pyrochlore family, and it is often served as a model system for pyrochlores because of its simple electronic structure [12]; that is, no f electrons exist in Y2Ti2O7. For this composition, GGA + method has also been employed to study if the intra-atomic electron correlations are important for Y-4d states. In Figure 5, we plot the equilibrium lattice parameters and band gap value of Y2Ti2O7 as functions of . As shown in the figure, the lattice constant and O48f positional parameter deviate from experimental values of 10.1 Å and 0.33 [7] with increasing values. Specially, the choice of values has almost no effects on the band gap values. It is indicated that the intra-atomic electron correlations are negligible for Y-4d states in Y2Ti2O7.
(a)
(b)
(c)
The total DOS distribution of Y2Ti2O7 at eV is presented in Figure 6. The valence bands mainly consist of O 2p orbitals with small contribution from Ti 3d states, and the conduction bands are mainly composed of Ti 3d states hybridized with O 2p orbitals. The corresponding band gap is 2.82 eV, in excellent agreement with the calculated value of 2.84 eV reported by Jiang et al. [34]. This value is 0.32 eV larger than our previous work on Y2Ti2O7 [10], as a result of different pseudopotential of Y employed.
3.4. Comparison of the Electronic Properties of Dy2Ti2O7, Ho2Ti2O7, and Y2Ti2O7
The following discussions are based on the results obtained by eV for Dy, eV for Ho, and eV for Y. Comparison of the partial DOS distribution of these three compositions is presented in Figure 7. It is noted that the Dy2Ti2O7 and Ho2Ti2O7 have similar DOS results; in contrast, Y2Ti2O7 shows a very different character. This is probably due to the fact that 4f electrons play an important role in Dy2Ti2O7 and Ho2Ti2O7. The electronic structures of a series of titanate oxides A2Ti2O7 (A = Sm-Er, Yb, and Lu) have been studied by Nemoshkalenko et al. [30] using X-ray photoelectron, emission spectroscopy, as well as the first-principles band structure calculations, where the lanthanide 4f states are assumed not to be hybridized with the other states. The calculations in the present work show that A-site 4f electrons do take part in the chemical bonding. For Dy2Ti2O7 and Ho2Ti2O7, the most striking features are the hybridization of Dy 4f and Ho 4f orbitals with O 2p orbitals, as shown in Figure 7. Especially, a strong hybridization occurs for Dy2Ti2O7 in its upper valence region. The 4f electrons also contribute greatly to the lower conduction bands of Dy2Ti2O7 and Ho2Ti2O7. Unlike Dy2Ti2O7 and Ho2Ti2O7, interaction is more significant in Y2Ti2O7, since its valence and conduction bands are mainly contributed by O 2p states hybridized with Ti 3d states and Ti 3d orbitals hybridized with O 2p orbitals, respectively.
(a)
(b)
(c)
Under irradiation, Y2Ti2O7 is the most radiation resistant and Dy2Ti2O7 is the least [7]. Since the pyrochlore compositions that are more easily disordered to the defect fluorite structure are more “resistant” to ion-beam-induced amorphization [18], Dy2Ti2O7 is the least probable to transform into defect-fluorite structure and Y2Ti2O7 is the most. This means that of the three compositions investigated, Dy2Ti2O7 is the most stable thermodynamically and Y2Ti2O7 is the least. The cation radius ratio criteria, as proposed by Sickafus et al. [18], clearly cannot be used to explain the different responses of these compositions to radiation. The radiation tolerance of nonmetallic solids has been correlated with the nature of the chemical bond in earlier work [35–38]. They demonstrated that the more covalently bonded materials are more readily amorphized at lower temperatures under heavy ion irradiation. For pyrochlores, the less covalently bonded compositions are more easily disordered to defect-fluorite structures [7], which are highly radiation resistant and remain crystalline at extreme radiation dose. In the present work, the strong interaction between Dy and Ho 4f electrons and O 2p orbitals may increase the covalency of and bonds and decrease the irradiation resistance of Dy2Ti2O7 and Ho2Ti2O7.
4. Conclusions
The electronic structures of A2Ti2O7 (A = Dy, Ho, and Y) have been investigated using GGA + method. The effects of effective values on the structural and electronic properties of pyrochlores have been studied. It is shown that for strongly correlated systems such as Dy2Ti2O7 and Ho2Ti2O7, it is necessary to correct the intraband Coulomb interaction by the Hubbard parameter. We suggest that the electronic structure can be reasonably described with of eV for Dy2Ti2O7 and Ho2Ti2O7.
Dy2Ti2O7 and Ho2Ti2O7 have similar DOS distribution; in contrast, Y2Ti2O7 shows a very different character. The DOS distributions of Dy2Ti2O7 and Ho2Ti2O7 show that A-site 4f electrons hybridize significantly with O 2p orbitals in the valence region. Since Dy2Ti2O7 and Ho2Ti2O7 are less radiation resistant than Y2Ti2O7, it is suggested that the strong interaction between Dy and Ho 4f electrons and O 2p electrons may increase the covalency of and bonds and decrease the irradiation resistance of Dy2Ti2O7 and Ho2Ti2O7.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11004023) and by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.