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Molecular-dynamics simulation of threshold displacement energies in BaTiO3
Nuclear Instruments and Methods in Physics Research B 358 (2015) 142–145
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Molecular-dynamics simulation of threshold displacement energies in BaTiO3 E. Gonzalez ⇑, Y. Abreu, C.M. Cruz, I. Piñera, A. Leyva Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), 30 St, # 502, Miramar, Playa, Havana City, Cuba
a r t i c l e
i n f o
Article history: Received 18 February 2015 Received in revised form 12 June 2015 Accepted 17 June 2015
Keywords: Radiation damage Threshold displacement energy Molecular dynamics Perovskite Defects
a b s t r a c t Molecular-dynamics simulations were used to calculate threshold displacement energies for each atom type in BaTiO3 perovskite. A primary knock-on atom with an energy range between 10 and 300 eV in principal crystallographic directions at 300 K was introduced. A statistical approach has been applied calculating displacement probability curves along main crystallographic directions. For each sublattice, the simulation was repeated from different initial conditions to estimate the uncertainty in the threshold displacement energy calculated values. The threshold displacement energies vary considerably with crystallographic direction and sublattice. The weighted average threshold displacement energies are 40 eV for oxygen, 64 eV for barium and 97 eV for titanium atoms. These values are comparable to ab initio calculated and experimentally derived values in perovskites. These results are proposed as threshold displacement energies, ideal for simulation programs that use atomic displacement calculation algorithms. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction ABO3 type crystals with the perovskite structure have been extensively studied due to their technological importance in electronic and optical devices [1,2]. In addition to borosilicate glass, ceramic assemblages of titanate have been proposed as a high level radioactive waste immobilization form [3]. Although with a relatively simple structure, the perovskite materials exhibit a broad range of properties such as superconductivity, ferroelectricity, magnetism, etc [4]. The BaTiO3 is of great importance in producing ferroelectric memories. These ferroelectric memories possess high radiation resistance [5], which makes them especially suitable for applications in aeronautic and astronautic engineering [6]. With the trend of decreasing size of electronic devices, there is increased interest on the response of these devices in a radiation environment. Radiation effects in ferroelectric materials has been studied, using classical and ab initio methods. There are studies about the defects formation in doped and undoped BaTiO3 [7], about the radiation-induced structural changes in BaTiO3 [4]. Threshold displacement energies have been calculated for each atom type in SrTiO3 perovskite using molecular dynamics (MD) simulations [8], in which case Td values were calculated depending on the crystallographic direction and also were reported a weighted average for each atom type (50 eV for oxygen, 70 eV for strontium and ⇑ Corresponding author. http://dx.doi.org/10.1016/j.nimb.2015.06.015 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
140 eV for titanium atoms). In [9] threshold displacement energies are determined experimentally for the 2O ion in some perovskites, reporting a single value of 45 ± 4 eV. Also in [10] displacement energies are determined experimentally for the ions in CaTiO3 reporting 82 ± 11 eV for calcium and 69 ± 9 eV for titanium. In this case, it is necessary to perform a detailed study of the threshold displacement energies in BaTiO3 due to its proved importance as a material used in radiation environments. An important process in the radiation damage formation in materials theory is the atomic displacements production, in which an atom is displaced from its lattice original site forming a stable vacancy or a Frenkel pair [11]. As an important parameter in this process, the threshold displacement energy for each atom type, is defined as the required energy to permanently displace an atom from its lattice site, forming a defect on that sublattice. These energy values are used in simulating radiation interaction with matter, in programs like MCSAD [12], which allow to obtain the atomic displacement in-depth distribution in irradiated materials and also in TRIM-type calculations [13,14], in order to convert experimental ion irradiation doses into the standard units of displacements per atom. This parameter is rather difficult to measure, since it is complicate, both, to distinguish among displacements of different atomic species, and to determine the threshold irradiation energy value producing the atom displacement. Also there are numerical studies using molecular dynamics that lead to the calculation of displacement probability curves which depend on
E. Gonzalez et al. / Nuclear Instruments and Methods in Physics Research B 358 (2015) 142–145
primary knock-on atom (PKA) energy and crystallographic direction [15–20]. Most of the radiation interaction with matter simulation programs use threshold displacement energies as an input parameter, being desirable that its value becomes as realistic as possible. However, in many cases, either an approximated value of 25 eV for all atoms based on Seitz’s suggestion [21] for materials with strong bonds or approximated threshold displacement energy (Td) values measured in other oxides are used, even when there exists studies that suggest Td values completely different to the assumed formal ones [22,8,23,19]. Therefore, radiation damage evaluation results in good agreement with reality relies in the use of legitimate threshold displacement energies. Consequently, in the present work atom displacement processes and the related threshold energies are simulated and calculated in BaTiO3 using molecular dynamics simulation methods at 300 K. The statistical approach and the Td calculation methodology based in the algorithms introduced in [8] for SrTiO3 were applied for threshold displacement energies calculations in the present study. A comparison of the obtained Td values with those ones reported for materials with similar crystalline structure is presented, as well as, the behavior of the radiation damage induced on BaTiO3 under 57Mn implantation process at 250 keV is discussed. 2. Calculations methods Taking into the account the BaTiO3 perfect crystal structure with parameters a = b = 3.9998 Å and c = 4.0180 Å [24] corresponding to the tetragonal phase, molecular dynamics simulations were performed. Using the DL_POLY code [25] the simulation cell contained 5000 ions (10 10 10 unit cells) with a side length of 40 Å. Periodic boundary conditions were used and the simulation cell was initially equilibrated at 300 K and 101325 Pa. This temperature was maintained for about 0.5 ps and all the simulation was produced under the constant NVE ensemble. A variable timestep of 0.8 fs was used, with a maximum integration distance of a time step of 0.01 Å. Simulations are run for 1.5 ps before the introduction of the PKA. Like in SrTiO3 [8], it is extremely rare for a Frenkel pair to recombine between 2 and 10 ps. Then the criteria of formation of an stable defect, as one that has a lifetime of at least 2 ps was used. The empirical model used is the ionic one with integral charges and consists of a short range Buckingham potential and a Coulomb term, in general it has the following form:
U ¼ Aexpðr ij =qÞ b=r 6ij þ
qi qj 4p0 rij
ð1Þ
Here, qi and qj are the charges of ions i and j; r ij is the interatomic distance, and A; q and b are adjustable parameters. The cationcation interaction in the Buckingham terms have very little effect and can be omitted [26]. The potential parameters are listed in Table 1, and were taken from previous works [7], in which, this set of parameters were used to propose a defect model consistent with experimental studies in doped and undoped BaTiO3. In this work it was neglected the ion polarization by way of the core and shell model.
Table 1 Potential parameters for BaTiO3, used in the present work. Interaction
A ðeVÞ
q ðÅÞ
6
b ðeVðÅÞ Þ
Ba2þ O2
1214.4
0.35220
8.0
Ti4þ O2
877.2
0.38096
9.0
O2 O2
22764.0
0.1490
43.0
143
To calculate the threshold displacement energy for each atomic species in the simulated crystallographic directions a methodology presented in previous work [8] was used. There, the defect formation probability (DFP) is defined as the probability of forming a stable defect (one that has a lifetime of at least 2 ps) on the same sublattice as the PKA. For testing the statistical variability n = 10 simulations are run for each PKA energy and direction. For each simulation the initial conditions were changed. The mathematical definition of DFP used in this work is:
DFP ¼
# of simulations with stable defects n
ð2Þ
The obtained results of DFP for different PKA energies in the same direction were fitted with their corresponding errors to a Fermi function. The Td values were calculated as the PKA energy where the DFP is equal to 0.5. There, the Td for a given atomic species in the simulated crystallographic directions, h0 0 1i; h1 1 0iand h1 1 1iis found. Also a weighted average value is calculated for each atomic species, using the displacement direction multiplicity. 3. Results and discussion In total, more than 2000 simulations were made, 10 simulations with the same PKA energy and crystallographic direction (changing the initials conditions), also ranges from 5 eV to 200 eV of the PKA energy with an step of 2.5 eV were simulated. In general, the DFP increases with PKA energy, as shown in Fig. 1. For some atomic species in some directions there is a decrease in the DFP after an initial increase, as in the cases of Ba, Ti in the direction h1 1 1i and O in h0 0 1iand h1 1 1i. This behavior were also found in previous works [23,8] in materials like TiO2 and SrTiO3. It may have to do with a higher mobility of the displacement atoms as the local temperature increases with PKA energy. 3.1. Threshold displacement energies in BaTiO3 Different threshold displacement energies reported and the ones calculated in this work, using molecular dynamics and experimental methods are listed in Table 2. It was found as the most radiation resistance crystallographic direction in BaTiO3 the h1 1 1i, of the three simulated, this direction has the biggest Td values for each atomic species. Due to the higher oxygen atoms concentration in the structure and for some simulated directions, the oxygen PKAs strike directly with other oxygen ions, favoring the formation of oxygen stable defects. Then it is normal to find smaller values of Td due to the used parameter definition. The Ba ions are aligned in the h0 0 1iand symmetrically equivalent directions. Promoting displacements in this directions via replacement sequence, increasing the probability of formation of an stable defect at relative low PKA energies (18.9 eV). In other directions the Ba ions collide with oxygen in h1 1 0i and titanium in h1 1 1i. In the same manner titanium ions strike directly with other titanium ions in the h1 1 0i direction, but we have found a smaller Td value in the PKA direction h0 0 1i. It can be related that in this direction the PKA strikes with an oxygen ion, and this oxygen behaves like a minor perturbation, due to its smaller Td value and higher mobility in comparison with titanium atoms. There, titanium PKAs in the direction h0 0 1icould interact directly with other titanium ion. The Td values found in this work are in good agreement with experimental and molecular dynamics calculated Td ones for similar materials. In the case of SrTiO3 it can be observed some differences in the reported Td values for each crystallographic directions in comparison with the BaTiO3 ones here presented. The weighted
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atoms in the other hand for the crystal A sites. In the case of titanium there are differences up to 40 eV. The three atomic species of the SrTiO3 have higher Td values than the BaTiO3 ones, respectively. These results allow establishing Td values ranges for the different crystallographic sites of materials with ABO3 perovskite structure, but also, differences in their Td values relaying on their chemical composition. The work [9] reports a Td value for the oxygen atoms determined using an experimental method, which agrees very well with the here calculated one. The calculated weighted Td values for each atomic species for BaTiO3 are also presented in Table 2. In the case of the Td values reported in [10] there is great difference principally for the Ti ions. This work uses the maximum energy Tm (eV) transferable from an incident electron of energy E (MeV) to a lattice ion of mass number A as given in 3 to calculate the Td.
T m ¼ 2147:7EðE þ 1:022Þ=A
(a)
ð3Þ
The difference in the Td between Ti and Ca reported in [10] is mainly because the ion mass number. Our calculated values are more reliable due to the used methodology, and that cation displacement energies are more difficult to measure experimentally than oxygen displacement energies. 3.2. Td and defect formation dependence
(b)
Simulations with SRIM code, modeling +Mn implantation in a BaTiO3 target, at 250 keV were made. All the simulations were performed on the same calculation conditions but only changing the Td values for the different atomic species. In one simulation trial it was used the average Td values calculated in this work and in a second simulation trial the values reported for SrTiO3 in the work [8](using the Td value of Sr for Ba atoms). The obtained results are shown in Fig. 2. Comparing the results of the simulation with the SrTiO3 Td values and the ones calculated in this work, there are significant differences in the quantity of vacancies formed as a function of the target depth, these differences are up to 40% in the case of the total amount of vacancies induced. 4. Conclusions Molecular dynamics simulations and DFP concept were applied to calculate the Td values for the BaTiO3 atomic species, the obtained values are about 40 eV for oxygen, 64 eV for barium
Table 2 Threshold atomic displacement energies (eV) for some materials, calculated using different methods, in several crystallographic directions, and reported Td for each atomic species. Direction
Material-Method
O
Ba/Sr/Ca
Ti
h0 0 1i
BaTiO3-MDa SrTiO3-MDb BaTiO3-MDa SrTiO3-MDb BaTiO3-MDa SrTiO3-MDb
37.5 ± 8.6 30 29.3 ± 7.8 60 52.1 ± 14.9 60
18.9 ± 2.7 35 40.3 ± 9.4 70 122.2 ± 22.5 100
53.2 ± 18.8 115 69.2 ± 18.2 90 172.3 ± 45.3 225
40.3 ± 7.6 50 45 ± 4 –
64.3 ± 8.9 70 – 82 ± 11
97.2 ± 16.9 140 – 69 ± 9
h1 1 0i h1 1 1i
Reported as Td BaTiO3 -MDa⁄ SrTiO3-MDb⁄ BaTiO3 -OEIDc CaTiO3 -HARECXSd
(c) Fig. 1. PKA defect formation probability for: (a) oxygen , (b) barium and (c) titanium in three principal crystallographic directions, as a function of PKA energy. Lines represent fitted Fermi functions for each dataset.
a
This work. Ref [8]. Values are quoted within ±10 eV. c Ref [9] OEID (optical emission due to ionic displacements). d Ref [10] HARECXS (high angular resolution electron channelling X-ray spectroscopy). ⁄ Weighted average. b
average Td values of the two similar structures are relative close in the cases of the oxygens, on one hand, and barium and strontium
E. Gonzalez et al. / Nuclear Instruments and Methods in Physics Research B 358 (2015) 142–145
145
found that the SrTiO3 is more resistive to radiation than BaTiO3. These results allow the systematization of the Td value range for materials with perovskite ABO3 crystal structures, but also show that materials with close crystalline properties own differences in their Td values. Simulations with SRIM code, modeling Mn+ implantation in a BaTiO3 target at 250 keV using different values of Td were made. It was shown that it is necessary to calculate the Td values for each atomic species in these type of materials if a more exact radiation interaction with matter simulation is required. References
(a)
(b) Fig. 2. Formed vacancies as a function of the target depth. (a) Using the Td values found in this work. (b) Using the Td values reported in [8] for the SrTiO3.
and for titanium atoms 97 eV. These values are different than the ones used until this moment for this material, and are a better approximation of Td values, which are here proposed as threshold displacement energies for this material. These values could be used as reference for perovskite type of materials and other oxides of the same family, in which this parameters are unknown. The obtained results were compared with those reported for materials with similar crystal structure like SrTiO3, and it was
[1] M.E. Lines, A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford, 2001. 633–662. [2] A. Kornev, L. Bellaiche, P. Bouvier, P.-E. Janolin, B. Dkhil, J. Kreisel, Phys. Rev. Lett. 95 (2005) 19684. [3] K.L. Smith, M. Colella, R. Cooper, E.R. Vance, Measured displacement energies of oxygen ions in titanates and zirconates, J. Nucl. Mater. 321 (2003) 19–28. [4] Y. Ma, Y. Chen, Y. Zhou, Molecular dynamics simulations of the radiationinduced structural changes in BaTiO3, Radiat. Eff. Defects 163 (2008) 189197. [5] J.F. Scott, Ferroelectric Memories, Vol. 3, 2000. [6] C. Sansoe, M. Tranchero, Use of FRAM memories in spacecrafts, Ferroelectrics – Appl. InTech 10 (2011) 214–230. [7] G.V. Lewis, C.R.A. Catlow, Defect studies of doped and undoped barium titanate using computer simulation techniques, Phys. Chem. Solids 47 (1986) 89–97. [8] B. Thomas, N. Marks, B. Begg, Defects and threshold displacement energies in SrTiO3 perovskite using atomistic computer simulations, Nucl. Instrum. Method. Phys. Res. B 254 (2007) 211–218. [9] R. Cooper, K. Smith, M. Colella, E. Vance, M. Phillips, Optical emission due to ionic displacements in alkaline earth titanates, J. Nucl. Mater. 289 (2001) 199– 203. [10] K. Smith, N. Zaluzec, The displacement energies of cations in perovskite (CaTiO3), J. Nucl. Mater. 336 (2005) 261–266. [11] R. Stoller, Primary radiation damage formation, Comprehensive Nucl. Mater. 1 (2004) 293–332. [12] C.M. Cruz, I. Piñera, C. Correa, Y. Abreu, A. Leyva, MCSAD: Monte Carlo simulation of atom displacements induced by fast electrons in solids, IEEE Nucl. Sci. Symp. Conf. Rec. (2011) 4622–4626. [13] J.F. Ziegler, J.P. Biersack, M.D. Ziegler, The Stopping and Range of Ions in Solids (SRIM), 2010. [14] J.F. Ziegler, M.D. Ziegler, J.P. Biersack, SRIM – the stopping and range of ions in matter (2010), Nucl. Instr. Meth. B 268 (2010) 1818–1823. [15] D. Delgado, R. Vila, Statistical molecular dynamics study of displacement energies in diamond, J. Nucl. Mater. 419 (2011) 32–38. [16] N. Juslin, K. Nordlund, J. Wallenius, L. Malerba, Simulation of threshold displacement energies in FeCr, Nucl. Instrum. Method. Phys. Res. B 255 (2007) 75–77. [17] H.Y. Xiao, F. Gao, X.T. Zu, W.J. Weber, Threshold displacement energy in gan: Ab initio molecular dynamics study, J. Appl. Phys. 105 (2009) 123527(1–5). [18] H. Tsuchihira, T. Oda, S. Tanaka, Molecular-dynamics simulation of threshold displacement energies in lithium aluminate, Nucl. Instrum. Method. Phys. Res. B 269 (2011) 1707–1711. [19] B. Park, W.J. Weber, L.R. Corrales, Molecular dynamics study of the threshold displacement energy in MgO, Nucl. Instrum. Method. Phys. Res. B 166–167 (2000) 357–363. [20] G. Lucas, L. Pizzagalli, Ab initio molecular dynamics calculations of threshold displacement energies in silicon carbide, Phys. Rev. B 72 (2005) 161202(1–4). [21] F. Seitz, Disc. Faraday Soc. 5 (1949) 271. [22] P.A.F.P. Moreira, R. Devanathan, J. Yu, W.J. Weber, Molecular-dynamics simulation of threshold displacement energies in zircon, Nucl. Instrum. Method. Phys. Res. B 267 (2009) 3431–3436. [23] B. Thomas, N. Marks, L. Corrales, R. Devanathan, Threshold displacement energies in rutile TiO2:a molecular dynamics simulation study, Nucl. Instrum. Method. Phys. Res. B 239 (2005) 191–201. [24] R.H. Buttner, E.N. Maslen, Acta Crystllogr. B 48 (1992) 764. [25] I. Todorov, W. Smith, The DLPOLY4 User Manual (2012). [26] B. Thomas, N. Marks, B. Begg, Developing pair potentials for simulating radiation damage in complex oxides, Nucl. Instrum. Method. Phys. Res. B 228 (2005) 288–292.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Molecular-dynamics simulation of threshold displacement energies in BaTiO3 E. Gonzalez ⇑, Y. Abreu, C.M. Cruz, I. Piñera, A. Leyva Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), 30 St, # 502, Miramar, Playa, Havana City, Cuba
a r t i c l e
i n f o
Article history: Received 18 February 2015 Received in revised form 12 June 2015 Accepted 17 June 2015
Keywords: Radiation damage Threshold displacement energy Molecular dynamics Perovskite Defects
a b s t r a c t Molecular-dynamics simulations were used to calculate threshold displacement energies for each atom type in BaTiO3 perovskite. A primary knock-on atom with an energy range between 10 and 300 eV in principal crystallographic directions at 300 K was introduced. A statistical approach has been applied calculating displacement probability curves along main crystallographic directions. For each sublattice, the simulation was repeated from different initial conditions to estimate the uncertainty in the threshold displacement energy calculated values. The threshold displacement energies vary considerably with crystallographic direction and sublattice. The weighted average threshold displacement energies are 40 eV for oxygen, 64 eV for barium and 97 eV for titanium atoms. These values are comparable to ab initio calculated and experimentally derived values in perovskites. These results are proposed as threshold displacement energies, ideal for simulation programs that use atomic displacement calculation algorithms. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction ABO3 type crystals with the perovskite structure have been extensively studied due to their technological importance in electronic and optical devices [1,2]. In addition to borosilicate glass, ceramic assemblages of titanate have been proposed as a high level radioactive waste immobilization form [3]. Although with a relatively simple structure, the perovskite materials exhibit a broad range of properties such as superconductivity, ferroelectricity, magnetism, etc [4]. The BaTiO3 is of great importance in producing ferroelectric memories. These ferroelectric memories possess high radiation resistance [5], which makes them especially suitable for applications in aeronautic and astronautic engineering [6]. With the trend of decreasing size of electronic devices, there is increased interest on the response of these devices in a radiation environment. Radiation effects in ferroelectric materials has been studied, using classical and ab initio methods. There are studies about the defects formation in doped and undoped BaTiO3 [7], about the radiation-induced structural changes in BaTiO3 [4]. Threshold displacement energies have been calculated for each atom type in SrTiO3 perovskite using molecular dynamics (MD) simulations [8], in which case Td values were calculated depending on the crystallographic direction and also were reported a weighted average for each atom type (50 eV for oxygen, 70 eV for strontium and ⇑ Corresponding author. http://dx.doi.org/10.1016/j.nimb.2015.06.015 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
140 eV for titanium atoms). In [9] threshold displacement energies are determined experimentally for the 2O ion in some perovskites, reporting a single value of 45 ± 4 eV. Also in [10] displacement energies are determined experimentally for the ions in CaTiO3 reporting 82 ± 11 eV for calcium and 69 ± 9 eV for titanium. In this case, it is necessary to perform a detailed study of the threshold displacement energies in BaTiO3 due to its proved importance as a material used in radiation environments. An important process in the radiation damage formation in materials theory is the atomic displacements production, in which an atom is displaced from its lattice original site forming a stable vacancy or a Frenkel pair [11]. As an important parameter in this process, the threshold displacement energy for each atom type, is defined as the required energy to permanently displace an atom from its lattice site, forming a defect on that sublattice. These energy values are used in simulating radiation interaction with matter, in programs like MCSAD [12], which allow to obtain the atomic displacement in-depth distribution in irradiated materials and also in TRIM-type calculations [13,14], in order to convert experimental ion irradiation doses into the standard units of displacements per atom. This parameter is rather difficult to measure, since it is complicate, both, to distinguish among displacements of different atomic species, and to determine the threshold irradiation energy value producing the atom displacement. Also there are numerical studies using molecular dynamics that lead to the calculation of displacement probability curves which depend on
E. Gonzalez et al. / Nuclear Instruments and Methods in Physics Research B 358 (2015) 142–145
primary knock-on atom (PKA) energy and crystallographic direction [15–20]. Most of the radiation interaction with matter simulation programs use threshold displacement energies as an input parameter, being desirable that its value becomes as realistic as possible. However, in many cases, either an approximated value of 25 eV for all atoms based on Seitz’s suggestion [21] for materials with strong bonds or approximated threshold displacement energy (Td) values measured in other oxides are used, even when there exists studies that suggest Td values completely different to the assumed formal ones [22,8,23,19]. Therefore, radiation damage evaluation results in good agreement with reality relies in the use of legitimate threshold displacement energies. Consequently, in the present work atom displacement processes and the related threshold energies are simulated and calculated in BaTiO3 using molecular dynamics simulation methods at 300 K. The statistical approach and the Td calculation methodology based in the algorithms introduced in [8] for SrTiO3 were applied for threshold displacement energies calculations in the present study. A comparison of the obtained Td values with those ones reported for materials with similar crystalline structure is presented, as well as, the behavior of the radiation damage induced on BaTiO3 under 57Mn implantation process at 250 keV is discussed. 2. Calculations methods Taking into the account the BaTiO3 perfect crystal structure with parameters a = b = 3.9998 Å and c = 4.0180 Å [24] corresponding to the tetragonal phase, molecular dynamics simulations were performed. Using the DL_POLY code [25] the simulation cell contained 5000 ions (10 10 10 unit cells) with a side length of 40 Å. Periodic boundary conditions were used and the simulation cell was initially equilibrated at 300 K and 101325 Pa. This temperature was maintained for about 0.5 ps and all the simulation was produced under the constant NVE ensemble. A variable timestep of 0.8 fs was used, with a maximum integration distance of a time step of 0.01 Å. Simulations are run for 1.5 ps before the introduction of the PKA. Like in SrTiO3 [8], it is extremely rare for a Frenkel pair to recombine between 2 and 10 ps. Then the criteria of formation of an stable defect, as one that has a lifetime of at least 2 ps was used. The empirical model used is the ionic one with integral charges and consists of a short range Buckingham potential and a Coulomb term, in general it has the following form:
U ¼ Aexpðr ij =qÞ b=r 6ij þ
qi qj 4p0 rij
ð1Þ
Here, qi and qj are the charges of ions i and j; r ij is the interatomic distance, and A; q and b are adjustable parameters. The cationcation interaction in the Buckingham terms have very little effect and can be omitted [26]. The potential parameters are listed in Table 1, and were taken from previous works [7], in which, this set of parameters were used to propose a defect model consistent with experimental studies in doped and undoped BaTiO3. In this work it was neglected the ion polarization by way of the core and shell model.
Table 1 Potential parameters for BaTiO3, used in the present work. Interaction
A ðeVÞ
q ðÅÞ
6
b ðeVðÅÞ Þ
Ba2þ O2
1214.4
0.35220
8.0
Ti4þ O2
877.2
0.38096
9.0
O2 O2
22764.0
0.1490
43.0
143
To calculate the threshold displacement energy for each atomic species in the simulated crystallographic directions a methodology presented in previous work [8] was used. There, the defect formation probability (DFP) is defined as the probability of forming a stable defect (one that has a lifetime of at least 2 ps) on the same sublattice as the PKA. For testing the statistical variability n = 10 simulations are run for each PKA energy and direction. For each simulation the initial conditions were changed. The mathematical definition of DFP used in this work is:
DFP ¼
# of simulations with stable defects n
ð2Þ
The obtained results of DFP for different PKA energies in the same direction were fitted with their corresponding errors to a Fermi function. The Td values were calculated as the PKA energy where the DFP is equal to 0.5. There, the Td for a given atomic species in the simulated crystallographic directions, h0 0 1i; h1 1 0iand h1 1 1iis found. Also a weighted average value is calculated for each atomic species, using the displacement direction multiplicity. 3. Results and discussion In total, more than 2000 simulations were made, 10 simulations with the same PKA energy and crystallographic direction (changing the initials conditions), also ranges from 5 eV to 200 eV of the PKA energy with an step of 2.5 eV were simulated. In general, the DFP increases with PKA energy, as shown in Fig. 1. For some atomic species in some directions there is a decrease in the DFP after an initial increase, as in the cases of Ba, Ti in the direction h1 1 1i and O in h0 0 1iand h1 1 1i. This behavior were also found in previous works [23,8] in materials like TiO2 and SrTiO3. It may have to do with a higher mobility of the displacement atoms as the local temperature increases with PKA energy. 3.1. Threshold displacement energies in BaTiO3 Different threshold displacement energies reported and the ones calculated in this work, using molecular dynamics and experimental methods are listed in Table 2. It was found as the most radiation resistance crystallographic direction in BaTiO3 the h1 1 1i, of the three simulated, this direction has the biggest Td values for each atomic species. Due to the higher oxygen atoms concentration in the structure and for some simulated directions, the oxygen PKAs strike directly with other oxygen ions, favoring the formation of oxygen stable defects. Then it is normal to find smaller values of Td due to the used parameter definition. The Ba ions are aligned in the h0 0 1iand symmetrically equivalent directions. Promoting displacements in this directions via replacement sequence, increasing the probability of formation of an stable defect at relative low PKA energies (18.9 eV). In other directions the Ba ions collide with oxygen in h1 1 0i and titanium in h1 1 1i. In the same manner titanium ions strike directly with other titanium ions in the h1 1 0i direction, but we have found a smaller Td value in the PKA direction h0 0 1i. It can be related that in this direction the PKA strikes with an oxygen ion, and this oxygen behaves like a minor perturbation, due to its smaller Td value and higher mobility in comparison with titanium atoms. There, titanium PKAs in the direction h0 0 1icould interact directly with other titanium ion. The Td values found in this work are in good agreement with experimental and molecular dynamics calculated Td ones for similar materials. In the case of SrTiO3 it can be observed some differences in the reported Td values for each crystallographic directions in comparison with the BaTiO3 ones here presented. The weighted
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E. Gonzalez et al. / Nuclear Instruments and Methods in Physics Research B 358 (2015) 142–145
atoms in the other hand for the crystal A sites. In the case of titanium there are differences up to 40 eV. The three atomic species of the SrTiO3 have higher Td values than the BaTiO3 ones, respectively. These results allow establishing Td values ranges for the different crystallographic sites of materials with ABO3 perovskite structure, but also, differences in their Td values relaying on their chemical composition. The work [9] reports a Td value for the oxygen atoms determined using an experimental method, which agrees very well with the here calculated one. The calculated weighted Td values for each atomic species for BaTiO3 are also presented in Table 2. In the case of the Td values reported in [10] there is great difference principally for the Ti ions. This work uses the maximum energy Tm (eV) transferable from an incident electron of energy E (MeV) to a lattice ion of mass number A as given in 3 to calculate the Td.
T m ¼ 2147:7EðE þ 1:022Þ=A
(a)
ð3Þ
The difference in the Td between Ti and Ca reported in [10] is mainly because the ion mass number. Our calculated values are more reliable due to the used methodology, and that cation displacement energies are more difficult to measure experimentally than oxygen displacement energies. 3.2. Td and defect formation dependence
(b)
Simulations with SRIM code, modeling +Mn implantation in a BaTiO3 target, at 250 keV were made. All the simulations were performed on the same calculation conditions but only changing the Td values for the different atomic species. In one simulation trial it was used the average Td values calculated in this work and in a second simulation trial the values reported for SrTiO3 in the work [8](using the Td value of Sr for Ba atoms). The obtained results are shown in Fig. 2. Comparing the results of the simulation with the SrTiO3 Td values and the ones calculated in this work, there are significant differences in the quantity of vacancies formed as a function of the target depth, these differences are up to 40% in the case of the total amount of vacancies induced. 4. Conclusions Molecular dynamics simulations and DFP concept were applied to calculate the Td values for the BaTiO3 atomic species, the obtained values are about 40 eV for oxygen, 64 eV for barium
Table 2 Threshold atomic displacement energies (eV) for some materials, calculated using different methods, in several crystallographic directions, and reported Td for each atomic species. Direction
Material-Method
O
Ba/Sr/Ca
Ti
h0 0 1i
BaTiO3-MDa SrTiO3-MDb BaTiO3-MDa SrTiO3-MDb BaTiO3-MDa SrTiO3-MDb
37.5 ± 8.6 30 29.3 ± 7.8 60 52.1 ± 14.9 60
18.9 ± 2.7 35 40.3 ± 9.4 70 122.2 ± 22.5 100
53.2 ± 18.8 115 69.2 ± 18.2 90 172.3 ± 45.3 225
40.3 ± 7.6 50 45 ± 4 –
64.3 ± 8.9 70 – 82 ± 11
97.2 ± 16.9 140 – 69 ± 9
h1 1 0i h1 1 1i
Reported as Td BaTiO3 -MDa⁄ SrTiO3-MDb⁄ BaTiO3 -OEIDc CaTiO3 -HARECXSd
(c) Fig. 1. PKA defect formation probability for: (a) oxygen , (b) barium and (c) titanium in three principal crystallographic directions, as a function of PKA energy. Lines represent fitted Fermi functions for each dataset.
a
This work. Ref [8]. Values are quoted within ±10 eV. c Ref [9] OEID (optical emission due to ionic displacements). d Ref [10] HARECXS (high angular resolution electron channelling X-ray spectroscopy). ⁄ Weighted average. b
average Td values of the two similar structures are relative close in the cases of the oxygens, on one hand, and barium and strontium
E. Gonzalez et al. / Nuclear Instruments and Methods in Physics Research B 358 (2015) 142–145
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found that the SrTiO3 is more resistive to radiation than BaTiO3. These results allow the systematization of the Td value range for materials with perovskite ABO3 crystal structures, but also show that materials with close crystalline properties own differences in their Td values. Simulations with SRIM code, modeling Mn+ implantation in a BaTiO3 target at 250 keV using different values of Td were made. It was shown that it is necessary to calculate the Td values for each atomic species in these type of materials if a more exact radiation interaction with matter simulation is required. References
(a)
(b) Fig. 2. Formed vacancies as a function of the target depth. (a) Using the Td values found in this work. (b) Using the Td values reported in [8] for the SrTiO3.
and for titanium atoms 97 eV. These values are different than the ones used until this moment for this material, and are a better approximation of Td values, which are here proposed as threshold displacement energies for this material. These values could be used as reference for perovskite type of materials and other oxides of the same family, in which this parameters are unknown. The obtained results were compared with those reported for materials with similar crystal structure like SrTiO3, and it was
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