Abstract
Ab initio molecular dynamics simulations have been carried out to study low energy recoil events in SrTiO3. The threshold displacement energies are shown to be strongly dependent on both the orientation and the corresponding atomic arrangement. The minimum threshold displacement energies are 13 eV for an O recoil along the 〈100〉 O–O chain, 25 eV for a Sr recoil along the 〈100〉 Sr–Sr chain and 38 eV for a Ti recoil along the 〈110〉 Ti–Ti chain. The weighted average threshold displacement energies along the primary crystallographic directions are 35.7, 53.5 and >64.9 eV for O, Sr and Ti, respectively. The interstitial configurations produced by the recoil events are 〈100〉 and 〈111〉 split interstitials for O and Sr, respectively, together with a Ti interstitial occupying a distorted bridge position between two Sr sites. It is found that the recoil events in SrTiO3 are partial-charge transfer assisted processes, and the partial-charge transfer plays an important role in these recoil events.
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1. Introduction
Ceramics with the perovskite-type structure (general formula ABO3) consist of a network of corner-linked BO6 octahedra enclosing large cavities, which form higher-coordination sites for the A-cations in a roughly cubic array. This structure can accommodate a wide variety of chemical compositions and atomic defects [1]. The perovskite structure allows incorporation of both fission products and actinides, and as a result, SrTiO3 is one of the materials proposed for the immobilization of high-level nuclear wastes [2]. In addition, SrTiO3 thin films are used as insulating layers in dynamic random-access memories [3], ferroelectric thin-film structures [4], and high-Tc superconductor devices [5], as well as potential gate oxide candidates [6]. In many of these applications, knowledge of defect accumulation from ion implantation, dynamic defect recovery and microstructural evolution under irradiation is critical.
SrTiO3 has received much attention both experimentally and theoretically with respect to its defect chemistry and radiation resistance. Experimentally, different low energy ion species have been used for implantation and irradiation in order to understand the changes in electrical, optical and mechanical properties of SrTiO3 [7–13]. Several of these studies focused on irradiation-induced amorphization and subsequent recrystallization of SrTiO3. Experimental work has shown that radiation damage in SrTiO3 progresses primarily via accumulation and interaction of point defects [9–13], rather than via direct impact amorphization. Thermal annealing of these defects occurs over a broad temperature range, with significant defect recovery below 400 K [12] and epitaxial recrystallization around 800 K [7]. Characterization of radiation-induced defects therefore becomes an important aspect of understanding the radiation response of this material. Meanwhile, theoretical investigations of SrTiO3 have been performed to study defect formation and migration energies and stable defect configurations, using both empirical atomic simulations and first-principles calculations [14–18]. However, the dynamics of atomic processes in SrTiO3 under irradiation are not yet well understood.
During irradiation, numerous defects are produced from elastic energy transfers to atoms in the crystal structure, but many of these defects undergo simultaneous recombination. The accumulation of immobile defects results in amorphization, while the migration and recombination of mobile surviving defects at higher temperatures can lead to the evolution of microstructure, such as dislocation loops and voids, that can lead to degradation of thermal and mechanical properties [19]. This microstructural evolution depends on defect production rate and temperature. Experimentally, the defect production rate under irradiation is determined from the local rate of atomic displacements, often given in the unit of displacements per atom per second (dpa s−1). The displacement rate is often determined using the SRIM code [20], which requires values of the threshold displacement energy, Ed. The threshold displacement energy is defined as the minimum amount of transferred kinetic energy necessary to permanently displace an atom from its original lattice site to form a stable defect, such as a Frenkel pair. While accurate values of Ed are important in determining the defect production rate (dpa s−1) and cumulative dose (dpa), computational modeling of threshold displacement events provides, more importantly, critical insights into the fundamental mechanisms of defect production and nature of the defects produced. The experimental measurement of Ed in ceramics is difficult, partly due to lack of experimental facilities and its dependence on multiple sub-lattices and directions [21]. Classical molecular dynamics (MD) has been widely used to study atomic level elastic collision processes in order to obtain Ed, including in SrTiO3 [1, 22–24]. However, the results of classical MD simulations highly depend on the quality of the empirical interatomic potentials employed and may both overestimate threshold displacement energies and incorrectly predict final defect states because the effects of partial-charge transfer are not considered [25, 26]. Recently, ab initio molecular dynamics (AIMD) simulations have been demonstrated to be a preferable choice, and have been used to study the low energy radiation response of semiconductors and ceramics, such as SiC [27, 28], pyrochlore-structured oxides [25, 29] and fluorite-structured oxides [26, 30]. Because the forces between atoms are obtained from electronic structure calculations, AIMD simulations can predict atomic dynamics with ab initio accuracy, as well as the effect of partial-charge transfer on recoil processes. For instance, after considering the partial-charge transfer, a recent AIMD investigation of Gd2Zr2O7 and Gd2Ti2O7 obtained smaller threshold displacement energies and revealed new defect creation mechanisms and new interstitial configurations, as compared with classical MD simulations [25].
In this paper, the AIMD method is employed to study low energy recoil events in SrTiO3. To shed light on the effects of inclusion of electronic structure on the radiation response, we have determined the threshold displacement energies, the associated defect configurations and the mechanisms for defect generation. The partial-charge transfer and its effect on the threshold displacement energies are clarified. Our results are compared with available classical MD simulations and experiments.
2. Computational details
The AIMD calculations were performed using the modified SIESTA code [28, 31]. Norm-conserving Troullier–Martins pseudopotentials [32] factorized in the Kleinman–Bylander form [33] were employed to describe the interaction between ions and electrons. The exchange–correlation functional was determined within the generalized gradient approximation (GGA) parameterized by Perdew et al [34]. The reference electronic configurations are 5s25p0 for Sr, 4s24p03d2 for Ti and 2s22p4 for O. A cutoff energy of 120 Ryd for the basis set and a K-point sampling of 1 × 1 × 1 in the Brillouin zone were employed. The valence wavefunctions were expanded in a basis set of localized atomic orbitals, and double-ζ basis sets plus polarization orbitals were used.
In these AIMD simulations, the NVE ensemble and the Nose–Hoover thermostat were used. All the simulations were conducted with a supercell containing 180 atoms (3 × 3 × 4) or 240 atoms (3 × 4 × 4) at a constant particle number and volume with periodic boundary conditions imposed along three directions. The cell size was chosen to ensure that all displacements occurred in the cell, and based on previous work [30], further increasing the cell size was not expected to significantly affect the results. The initial temperature was set to be 0 K in order to eliminate the influence of random velocities. Significantly increasing the temperature is known to increase the threshold displacement energy because the PKA needs to be displaced farther to avoid thermal-induced recombination [35]. However, exploring the temperature dependence of threshold displacement events is not the goal of the present work. Such studies require longer simulation times and larger simulation cells for thermal effects to be observed and are best carried out by classical MD methods. A fixed time step of 1 fs was used for all simulations. The time step of 1 fs was validated by testing an O PKA along [100] with a time step of 0.5 fs, and the same Ed and final defect configurations were observed. To ensure that the irradiated system converges to thermal equilibrium states, a maximum time duration of 1 ps for each run was employed. To initiate a recoil event, an atom (Sr, Ti or O) was selected as the primary knock-on atom (PKA) and given a specific kinetic energy along a specific direction. If the PKA returned to its original site, the simulation was restarted at higher recoil energy with an energy increment of 6 eV. Once the PKA was permanently displaced from its lattice site, additional runs were performed to improve the precision to 1 eV.
3. Results and discussion
3.1. Ground state properties
Figure 1 shows the variation of total energy with volume. The lattice parameter a0 corresponding to the minimum energy is determined to be 3.92 Å, as compared to the experimental measurement of a0 = 3.89 Å [36] and theoretical calculation values of 3.86–3.98 Å [37, 38]. Fitting the energy–volume data in figure 1 to the Murnaghan equation of state yields a bulk modulus of 212.6 GPa, in fairly good comparison to the values of 200 GPa [37], 214 GPa [38] and 179 GPa [39] reported in the literature.
Figure 1. Total energy versus volume.
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Standard image High-resolution image3.2. Threshold displacement energies
To calculate Ed, we provide kinetic energy to an atom along one of the three main crystallographic directions, i.e., 〈100〉,〈110〉 and 〈111〉. As a consequence of the adopted 
symmetry (figure 2(a)), any of the three types of atoms (Sr, Ti and O) has only one crystallographic inequivalent site. If an oxygen is selected, its 〈100〉 and 〈110〉 direction groups are divided into two subgroups. As indicated in figure 2(b), the nearest neighbor atom along the [100] direction for the selected oxygen is O (O–O chain) but that along [010] is Ti (O–Ti chain), which are denoted as OO〈100〉 and OTi〈100〉, respectively. The superscript indicates the first nearest neighbor atom in the chosen direction. Similarly, as shown in figure 2(c), the nearest neighbor atom along the [110] direction for the selected oxygen is O (O–O chain) but that along [101] is Sr (O–Sr chain), which are denoted as OO〈110〉 and OSr〈110〉, respectively. In contrast, all 〈111〉 directions are equivalent, comprising O–O chains, and denoted as OO〈111〉. In the cases of Sr and Ti, all the directions are equivalent within the 〈100〉,〈110〉 and 〈111〉 direction groups. Therefore, the recoil events along representative crystallographic directions [100], [110] and [111] are selected. According to their atomic arrangements along these directions, corresponding recoil events are denoted as SrSr〈100〉,SrO〈110〉 and SrTi〈111〉, and TiO〈100〉,TiTi〈110〉 and TiSr〈111〉 for Sr and Ti, respectively.
Figure 2. (a) Schematic view of SrTiO3 structure. The different neighbor atoms along 〈100〉 and/or 〈110〉 direction groups are indicated in (b) and (c), respectively.
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Standard image High-resolution imageAfter the recoil events, the final defect configurations are identified. The structures of atomic vacancies (VO,VSr and VTi) and cation antisite defects (SrTi and TiSr) are straightforward since there is only one crystallographic inequivalent site for each atom. In contrast, the structures of the interstitial configurations are more complex. The interstitial atom can be located at an unoccupied position to form a single-interstitial defect or bond to the same type of lattice atom to form a split-interstitial defect occupying a single lattice site. Figure 3 schematically illustrates the interstitial configurations for O, Sr and Ti that are observed in corresponding recoil events. Oxygen favors a split-interstitial configuration along the 〈100〉 direction (figure 3(a)), with two atoms sharing a single lattice site. Similarly, Sr forms a split interstitial but along the 〈111〉 direction (figure 3(b)). The titanium interstitial (figure 3(c)) is different and occupies a bridge position between two Sr atoms along a channel with a slight 〈110〉 deviation and in-plane with four nearest neighboring Ti and/or O atoms (simply called a distorted bridge position).
Figure 3. Interstitial configurations in SrTiO3, view along 〈010〉: (a) oxygen split interstitial along 〈100〉 (in purple), (b) strontium split interstitial along 〈111〉 (in blue) and (c) titanium interstitial occupying the distorted bridge position between two Sr sites (in brown).
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Standard image High-resolution imageThe threshold displacement energies (Ed) along the main crystallographic directions, the associated PKA displacement distance (dPKA) and the resulting defect configurations are summarized in table 1. The displacement energies for cations and anions are strongly dependent on the crystallographic directions along which the atom is displaced and its corresponding atomic arrangement. For Sr, Ti and O, the minimum threshold displacement energies are 13 eV for OO〈100〉, 25 eV for SrSr〈100〉 and 38 eV for TiTi〈110〉. To better understand the defect generation, the detailed mechanisms for all recoil events are discussed below.
Table 1. Calculated threshold displacement energies (Ed), weighted average values (Eave), the associated PKA displacement distance (dPKA), final defects and increase in system energy, ΔE, for final defect state in SrTiO3.
| PKA direction | Ed (eV) | dPKA (Å) | Defect state | ΔE (eV) |
|---|---|---|---|---|
| SrSr〈100〉 | 25 | 3.92 | VSr + Sri | 13.5 |
| SrO〈110〉 | 50 | 5.02 | VSr + Sri | 26.6 |
| SrTi〈111〉 | 80 | 2.93 | VSr + SrTi + TiSr + SrTi + Tii | 45.9 |
|
53.5 | |||
| TiO〈100〉 | 72 | 3.94 | VTi + Tii | 38.2 |
| TiTi〈110〉 | 38 | 3.43 | VTi + Tii | 21.1 |
| TiSr〈111〉 | >100 | — | — | |
|
>64.9 | — | — | |
| OO〈100〉 | 13 | 2.70 | VO + Oi | 4.6 |
| OTi〈100〉 | 54 | 2.59 | VO + Oi | 32.5 |
| OO〈110〉 | 29 | 2.06 | VO + Oi | 12.9 |
| OSr〈110〉 | 64 | 3.02 | VO + Oi | 33.3 |
| OO〈111〉 | 35 | 5.36 | VO + Oi | 16.9 |
|
35.7 | — | — | |
In the case of oxygen, all recoil events create an O Frenkel pair, but experience different dynamic mechanisms. When a kinetic energy of 13 eV is transferred to an O atom along the [100] O–O chain (OO〈100〉), the O PKA moves 2.7 Å from its equilibrium location along this direction and forms a [100] split interstitial with the nearest O along this direction. In the case of OTi〈100〉, the O PKA moves along the [100] O–Ti chain at first, but is subsequently scattered along the [110] O–O chain due to repulsion from the Ti and other neighbor atoms; a displacement and replacement sequence along this direction is observed. The damage end state consists of a VO and a [100] O split interstitial. The threshold displacement energy for this recoil event is 54 eV, which is much higher than that of OO〈100〉. For OO〈110〉 with an Ed of 29 eV, the O PKA moves 2.06 Å along the [110] O–O chain to collide with an O atom (denoted as a secondary recoil atom, SRA). This process transfers sufficient kinetic energy to the O SRA atom such that the PKA replaces this atom. The O SRA continues moving along the [110] O–O chain and finally forms a [100] split interstitial with its neighboring O along this direction. If the O recoil event occurs along the 〈110〉 O–Sr chain (OSr〈110〉), a much higher Ed of 64 eV is found. The O PKA initially moves along the [110] O–Sr chain, whereas its trajectory is subsequently scattered by the strong repulsive interaction between the O and the neighbor Sr and other atoms. This causes the O atom to move along the [001] direction and eventually to form a [100] split interstitial with the nearest O along the [001] direction. For OO〈111〉, the threshold displacement energy is determined to be 35 eV. In this case, the O PKA moves 3.02 Å and forms a [010] split interstitial with the nearest O along the [111] direction. The fact that the threshold displacement energies of OTi〈100〉 and OSr〈110〉 are higher than those of OO〈100〉,OO〈110〉 and OO〈111〉 indicates that oxygen displacements occur much more easily along single type atomic (O–O) chains than along the mixed atomic chains (O–Sr and O–Ti).
Similar to oxygen, the threshold displacement energies for cation recoil events along a single type atomic chain (SrSr〈100〉 and TiTi〈110〉) are much lower than along the corresponding mixed atomic chain (SrO〈110〉 and SrTi〈111〉, and TiO〈100〉 and TiSr〈111〉). Namely, the minimum threshold displacement energy is observed along the 〈100〉 direction for Sr and along the 〈110〉 direction for Ti. For SrSr〈100〉, giving a kinetic energy of 25 eV to a Sr atom along the [100] Sr–Sr chain will create permanent defects. The Sr PKA moves 3.96 Å along this direction to replace a Sr (SRA), and the Sr SRA continues moving along [100] and forms a ![$[\bar {1}1 1]$](https://content.cld.iop.org/journals/0953-8984/25/48/485003/revision1/cm486392ieqn122.gif)
split interstitial with the nearest Sr along this direction. For the SrO〈110〉 recoil, the threshold displacement energy is determined to be 50 eV. In this case, the Sr PKA moves along the [110] Sr–O chain and collides with an O atom (SRA), which is knocked out, and the Sr PKA continues moving along [110] to form a ![$[\bar {1}\bar {1}1]$](https://content.cld.iop.org/journals/0953-8984/25/48/485003/revision1/cm486392ieqn124.gif)
split interstitial with the nearest Sr along this direction. The O SRA collides with neighbor atoms and rebounds to its initial position. For SrTi〈111〉, continuous displacements and replacements of Sr and Ti are observed after giving a kinetic energy of 80 eV to a Sr atom along the [111] direction. In the end, two SrTi and one TiSr, plus one VSr and Tii are produced. For Ti, the minimum threshold displacement energy is determined to be 38 eV for the TiTi〈110〉 recoil, in which the Ti PKA moves 3.25 Å along the [110] Ti–Ti chain and occupies a distorted bridge interstitial position. When a kinetic energy of 72 eV is given to a Ti along the [100] Ti–O chain, the Ti PKA moves along this direction and first collides with an O atom (O SRA). After the collision, the Ti PKA continues moving along the [100] direction and replaces a Ti (Ti SRA). Because of the combined effect of transferred kinetic energy and strong repulsion from neighbor atoms, the Ti SRA moves along the ![$[\bar {3}2 2]$](https://content.cld.iop.org/journals/0953-8984/25/48/485003/revision1/cm486392ieqn131.gif)
direction and forms a bridge interstitial. The O SRA and its neighbor O along the [101] direction exchange their positions with each other after a series of displacements and collisions with their neighbor atoms. In the case of the TiSr〈111〉, no permanent defect forms even when giving a very high initial kinetic energy up to 100 eV.
3.3. Discussions
In order to compare the present results with other results, the weighted average Ed values for O, Sr and Ti are calculated, as listed in table 1. The weighted average threshold displacement energies are 35.7 eV for O, 53.5 eV for Sr and >64.9 eV for Ti. Experimentally, only an Ed of 45 ± 4 eV for oxygen in SrTiO3 has been reported, and it was suggested that this value may overestimate the minimum threshold displacement energy [40]. Recent work by Robinson et al [35] on the temperature dependence of threshold displacement energies for O in TiO2 supports a lower value of Ed for O in SrTiO3 than that measured at room temperature. For Sr and Ti, the experimental values are unavailable. However, the oxygen and cation threshold displacement energies for CaTiO3, which has the same structure as SrTiO3, have been determined experimentally [41, 42]. The Ed value for O in CaTiO3 is the same as that in SrTiO3 [41], and the cation values are 82 ± 11 eV for Ca and 69 ± 9 eV for Ti [42]. In CaTiO3, the measured cation displacement energies are larger than those for oxygen, similar to the case of SrTiO3. Because of different mass and ionic radii, direct comparison of Ca and Sr displacements energies is not considered; however, the threshold displacement energy for Ti in CaTiO3 is comparable to that calculated in the present work. Finally, the system energy increases due to the final defect configurations for each recoil event, which is the potential energy difference between the final defective structure after the cascade and the initial perfect structure, are also included in table 1.
Our simulation results are also compared quantitatively with recent Ed calculations from classical MD simulations [1], as shown in figure 4. For oxygen recoil events along the 〈100〉 and 〈110〉 directions, the reported MD simulations did not consider the different atomic arrangements that are distinguished in this work. As shown in figure 4, the oxygen recoil events along the same type atomic chains (OO〈100〉 and OO〈110〉) in the present study exhibit smaller values of Ed than corresponding MD values along the O〈100〉 and O〈110〉 directions, but the oxygen recoil events along the mixed type atomic chains (OTi〈100〉 and OSr〈110〉) in the present study show larger values of Ed than the O〈100〉 and O〈110〉 directions from classical MD. Along 〈111〉, the AIMD Ed is smaller than that of classical MD. For three Sr and two Ti recoil events, the AIMD Ed are also significantly lower than those from classical MD. The phenomenon that the Ed from AIMD are generally lower than those from classical MD in SrTiO3 has also been observed for Y2Ti2O7 [29], Gd2Zr2O7 [25] and GaN [43]. The discrepancy in Ed between the AIMD and the classic MD is attributed to the partial-charge transfer during the dynamic process captured in AIMD but not in classical MD as discussed in what follows.
Figure 4. Comparison of Ed obtained by AIMD and MD [1].
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Standard image High-resolution imageDuring each recoil event, the system potential energy increases as the kinetic energy is partially converted to potential energy during the screened-Coulomb interactions of the PKA with atomic nuclei in the structure, and may exhibit a peak before relaxing to a stable structure. At the same time, the PKA and struck atoms have overlapping orbitals that result in continuous partial-charge transfer. Figure 5 shows variation in effective charges of PKAs with time for OO〈100〉,SrO〈110〉 and TiTi〈110〉, in which the charges are relative to the effective charge of each atom in bulk SrTiO3. It is shown that partial-charge transfer from and to the recoil atom takes place during the whole dynamic process. The Sr recoil is prone to lose electronic charge to neighbor atoms, while O and Ti may either gain or lose electronic charge from or to atomic neighbors. Such partial-charge transfer often extends beyond the first neighboring atoms. This result clearly indicates that continuous partial-charge transfer occurs during recoil events, which must contribute to the recoil processes.
Figure 5. Variation in effective charges of PKAs with time for (a) OO〈100〉, (b) SrO〈110〉 and (c) TiTi〈110〉.
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Standard image High-resolution imageThe system energy peaks reflect the point of maximum potential energy and most probably represent the maximum screened ion–ion interactions. Gao et al have demonstrated that the various potential energy peaks correspond to different partial-charge transfer states in the SiC [28]. This relationship was further strengthened by identifying the important role of partial-charge transfer in the response of thoria, not only in single recoils but also in overlapping recoils [30]. In a recent study of Gd2Zr2O7 and Gd2Ti2O7, the partial-charge transfer was also found to contribute to the threshold displacement energies [25]. To clarify the role of charger transfer on low energy recoil events in SrTiO3, figure 6 shows the dependence of threshold displacement energies on the total amount of partial-charge transfer at the system potential energy peak, 
, which is the difference in the effective Mulliken charge of all atoms that lose (or gain) electrons between the structure at the system potential energy peak and the initial structure. It is found that there is a nearly linear relationship with acceptable deviations between the threshold displacement energies and amount of partial-charge transfer no matter what atom type is the PKA, which is consistent with previous studies mentioned above [28, 30]. Such partial-charge transfer, however, is not considered in classical MD studies, which may be partly the source for the large discrepancy in threshold displacement energies between AIMD and classical MD. These results also demonstrate that the AIMD method provides a better predictive understanding of such partial-charge transfer assisted defect evolution in nuclear materials under irradiation and may provide guidance on the development of charge-transfer interatomic potentials for classic MD simulations.
Figure 6. Threshold displacement energies (Ed) versus partial-charge transfer at the energy peaks 
.
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4. Conclusions
Low energy recoil events in SrTiO3 have been investigated by ab initio molecular dynamics simulations based on density functional theory. The threshold displacement energies are shown to be strongly dependent on both the direction and the corresponding atomic arrangement. No matter the atom type of the PKA, displacements along single type atomic chains occur more easily than those along mixed atomic chains. The minimum Ed of 13 eV for O recoil and 25 eV for Sr recoil are both along 〈100〉 single type atomic chains. For Ti, the minimum Ed of 38 eV is along the 〈110〉 Ti–Ti chain. The weighted average threshold displacement energies along the primary crystallographic directions are 35.7 eV, 53.5 eV and >64.9 eV for O, Sr and Ti, respectively.
The O and Sr interstitial types are determined to be 〈100〉 and 〈111〉 split interstitials, respectively, while the interstitial Ti prefers a distorted bridge position between two Sr atoms. Compared with classical MD methods, the AIMD simulations agree qualitatively but not quantitatively. It is found that all the studied recoils of SrTiO3 are partial-charge transfer assisted processes and partial-charge transfer plays the important role in these recoil events.
Acknowledgments
This work was supported by the US Department of Energy, Basic Energy Sciences, Materials Science and Engineering Division. The theoretical calculations were performed using the supercomputer resources at the National Energy Research Scientific Computing Center located at Lawrence Berkeley National Laboratory.