Role of Nanolaminated Crystal Structure on the Radiation Damage Tolerance of Ti 3 SiC 2 : Theoretical Investigation of Native Point Defects

Nanolaminated Ti 3 SiC 2 , a representative MAX phase, shows excellent tolerance to radiation damage. In this paper, first-principles calculations were used to investigate the mechanism of intrinsic point defects in order to explain this outstanding property. Formation energies of intrinsic point defects are calculated and compared; and the results establish a low-energy disorder mechanism in Ti 3 SiC 2 . In addition, the migration energy barriers of Si vacancy, Si interstitial, and TiSi antisite yield very low values: 0.9, 0.6, and 0.3 eV, respectively. The intercalation of Si atomic plane between Ti 3 C 2 nanotwinning structures dominates the formation and migration of intrinsic native point defects in Ti 3 SiC 2 . The present study also highlights a novel method to improve radiation damage tolerance by developing nanoscale-layered structure which can serve as a sink or rapid recovery channel for point defects.


Introduction
Ti 3 SiC 2 belongs to the family of nanolaminated carbides and nitrides, the so-called M +1 AX  (MAX) phases, where M is an early transition metal, A is an IIIA or IVA group element, and X is carbon or nitrogen.In the crystal structure of MAX phase, the nanotwinned M +1 X  layers are intercalated by the A atomic plane, and the structural units are alternatively stacked along the  direction.The M +1 X  layer has the rock-salt-type structure and consists of strong covalent M-X bonds, while the M-A bonds are relatively weaker.Due to the nanolaminated structure, MAX phases exhibit good damage tolerance and machinability, which are the most salient properties compared with typical brittle binary carbides and nitrides [1].Considering the high modulus, high strength, and thermal shock resistance, MAX phases show a great potential applied as high-temperature structure materials.Ti 3 SiC 2 shares the common properties of MAX phase: its fracture toughness ( 1C ) is as high as 7 MPa⋅m 1/2 ; Vicker's hardness is about 5 GPa; Young's modulus is within 322-333 GPa; and flexural strength is about 450 MPa [2].Furthermore, its Weibull modulus is 29.1, ranking the highest value for monolithic ceramics [3].In addition, Ti 3 SiC 2 attracts extensive attentions due to its good high-temperature oxidation resistance, good thermal shock resistance, and good radiation damage tolerance, which highlights the applications in harsh environment, under high temperature, or under nuclear radiation.
Recently, experimental results showed good radiation damage tolerance of Ti 3 SiC 2 [4][5][6][7][8][9].This ternary carbide illustrated structure disorder, instead of amorphization, under heavy ion irradiation up to a damage of ∼25 dpa [6].Ti 3 SiC 2 also demonstrated small swelling of the (0001) atomic planes after irradiation.The swelling is only 2.2% for an average irradiation dose of 4.3 dpa at room temperature [7].At a temperature of above 300 ∘ C, the radiation damage can be easily recovered [8,9].It is interesting to compare the radiation damage tolerance of Ti 3 SiC 2 with the binary carbide SiC.Amorphization appears in SiC at low level of damage, usually <1 dpa; and its swelling is about seven times higher than that of Ti 3 SiC 2 [7].The main origin of good radiation damage tolerance of Ti 3 SiC 2 might be traced back to the nanolaminated crystal structure in which the nanotwinning structure could serve as a sink or recombination channel of point defects.
The behavior of point defect is one of the key factors to control the response of a material to radiation damage.For Ti 3 SiC 2 under radiation, structure disorder instead of amorphization, small swelling, and absence of extended defects, should mainly relate to the formation and recovery of point defects.Therefore, the formation and migration of intrinsic point defects are studied using the first-principles calculations in this work.The results were compared with those of SiC to illustrate the influence of the nanolaminated crystal structure on the point defect behaviors.The aim of this work is twofold: firstly, to explain the unusual radiation damage tolerance by exploring the mechanisms of native point defects in Ti 3 SiC 2 ; secondly, to elucidate the key role of the nanolaminated crystal structure on the generation and recombination of native point defects in Ti 3 SiC 2 .More important, the present study could highlight a novel method to improve radiation damage tolerance by developing nanoscale-layered structure which can serve as a sink or rapid recovery channel for point defects.

Computational Method
This calculation was performed by using the VASP code, based on the density functional theory (DFT) [10].The electron-ion interactions were represented by the projector augmented wave (PAW) method [11].The electronic exchange correlation energy was treated as the generalized gradient approximation (GGA-PBE) [12].The plane wave basis set cut off was 450 eV.Calculation of the defect structure employed a 2 × 2 × 1 supercell, which contains 48 atoms.The special point sampling integration was used over the Brillouin zone by using the Monkhorst-Pack method with 12×12×2 for unit cell and 6 × 6 × 2 for supercell [13].The lattice constants and internal freedom of the unit cell were fully optimized until the total energy difference was smaller than 1 × 10 −6 eV.According to our previous studies on defects in MAX phases, the present calculation method has been proved to be accurate enough to reproduce the defect structure and formation energies [14][15][16].To study the migration of defects, the diffusion energy barrier was calculated by searching the transition state linking the defect configurations before and after the migration process.A generalized synchronous transit method, LST/QST, employed in the CASTEP code was used for locating the transition state [17,18].It combines the linear (LST) or quadratic synchronous transit (QST) methods with conjugate gradient refinements.The LST/optimization and QST/maximization calculations were firstly performed to search the transition state, and then CG minimization was carried out to refine the saddle point geometry.The cycle was The defect formation energy (DFE,   ) is calculated from the following equation: where the  def and  perf are total energies of a defected supercell and a perfect supercell, respectively,   represents the change in the number of atoms of species  ( = Ti, Si, or C) during the process of defect formation, and   denotes the chemical potential of species .The chemical potentials of Ti, Si, and C atoms are assumed to be the bulk hcp Ti, cubic Si, and graphite, respectively.The chemical potentials of these bulk materials are obtained from the total energies of firstprinciples calculation.

Results and Discussion
The crystal structure of Ti 3 SiC 2 is shown in Figure 1.There are two sites (Ti 1 and Ti 2 ) of Ti atoms: one Si site and one C site.According to the crystal structure, the structures of on-lattice defects in Ti 3 SiC 2 can be easily constructed.There are 4 possible on-lattice vacancies (V Ti 1 , V Ti 2 , V Si , and V C ) and 8 kinds of antisites (Si Ti 1 , C Ti 1 , Si Ti 2 , C Ti 2 , Ti Si , C Si , Ti C , and Si C ). Calculation results show that the vacancy formation energy on the Ti 2 site is 1.8 eV lower than that on the Ti 1 site, indicating that the concentration of Ti 2 vacancy is much higher than that of Ti 1 vacancy.Therefore, the vacancy on Ti 2 site is used for further discussion of vacancy on Ti lattice site.Calculated formation energies of on-lattice defects in Ti 3 SiC 2 are listed in Table 1.V Ti has a relatively high formation energy ∼5.5 eV.The values of V Si and V C are both 2.1 eV, only half of the DFE of V Ti in Ti 3 SiC 2 .From Table 1, it can be found that the DFE of antisites between Ti and C atoms is the highest and the DFE of antisites between Ti and Si atoms  is the lowest.The DFE of antisites between C and Si atoms locates between the two end values.The configurations of interstitial defects are complicated compared with the on-lattice defects.For simplification, the unit cell of Ti 3 SiC 2 is divided into two regions, namely, the SiTi 2 and Ti 3 C 2 units.In the SiTi 2 unit, interstitial atoms are initially placed on the  or  position in the Si atomic plane, on the  or  position in the Ti 2 atomic plane, and on the , , or  position between the Si and Ti 2 atomic planes (totally seven configurations as seen in Figure 2(a)).These positions are labeled as I-/-Si, I-/-Ti 2 , and I-//-SiTi 2 , respectively.The Ti 3 C 2 unit in Ti 3 SiC 2 is similar to the rocksalt structure.Since all of the octahedral interstitials are occupied by C atoms in the Ti 3 C 2 unit, it is very hard for the Si atom migrating inside and locating at the interstitial positions.Consequently, only the Ti and C interstitials are considered in Ti 3 C 2 unit.The interstitials in rock-salt carbide have been discussed before [19,20].Two kinds of C  configurations and one kind of Ti  configuration have relatively low formation energies: C  atom is inserted into the center of two neighboring C atoms to form a linear C-C-C trimmer or to occupy the site near the centre of C or Ti tetrahedron; Ti  atom locates at the centre of C or Ti tetrahedron.As a result, four interstitial configurations are adopted in the Ti After the geometry optimization, some configurations are not stable and would change to other configurations.The DFEs of all of the stable interstitial configurations are listed in Table 2.It is found that Ti  is not stable in Ti 3 C 2 unit, and the possible configuration is I--SiTi 2 with the DFE of 3.6 eV.C  will be stable in both the SiTi 2 and the Ti 3 C 2 units.The I--Si and I-CTi 2 configurations are the most possible defects configurations with the DFEs of 0.9 eV and 3.6 eV in SiTi 2 and Ti 3 C 2 units, respectively.The most possible configuration of Si  is the I--Si with the DFE of 2.1 eV.Note that Si  initially located in Ti 2 atomic plane or between the Si and Ti 2 atomic planes converges to the interstitial sites between the C and Ti 2 atomic planes after the geometry optimization, while the corresponding DFE has a very high value of 7.0 eV.The main consequence of displacive radiation damage is the structural disorder caused by the accumulation of point defects.The possibility of accommodating structure disorder is the key factor to prevent amorphization.Sickafus and collaborators did experimental and theoretical investigations on the radiation damage tolerance of A 2 B 2 O 7 pyrochlores [21].In their work, the calculated formation energies of cation antisite pair have very low values and were used to evaluate the resistance to radiation-induced amorphization.The related method has been proved to be a reliable way to predict radiation damage tolerance of a material.Therefore, the structure disorders in Ti 3 SiC 2 are analyzed and compared with those in SiC to illustrate the mechanisms of accommodating radiation damage.
From the calculated point defect formation energies, the formation energies of Frenkel pairs and the antisite pairs in Ti 3 SiC 2 can be obtained accordingly.The following equations are the possible Frenkel and antisite pairs in Ti 3 SiC 2 and their formation energies.The formation energies of point defects in 3C-SiC had been reported by Lucas and Pizzagalli [22].The lowest formation energies of C and Si Frenkel pairs are 6.73 eV and 13.46 eV, respectively.The formation energy of antisite pair of C Si and Si C is 7.5 eV in SiC.So the mechanism of disorder in SiC should be the C Frenkel pair and C Si +Si C antisite pair.It is noted that the formation energies of SiC are obviously much higher than those of the Ti 3 SiC 2 .

Frenkel pairs:
Based on previous analysis, it can be found that the intercalated Si atomic plane, as well as the nanolaminated crystal structure, contributes to the mechanisms of disorders in Ti 3 SiC 2 by two aspects: firstly, to provide a possible process of Si Frenkel pair to accommodate disorder with very low formation energy and, secondly, to present a possible mechanism of Ti Si + Si Ti antisite pair with low formation energy.Therefore, Ti 3 SiC 2 is more resistant to radiation damage than SiC, which had been proved in experiments.
The migrations of point defects dominate the recombination process of defects under radiation damage.Therefore, lower migration energy of a point defect means an easier recovery of Ti 3 SiC 2 from radiation damage.As discussed above, the weakly bonded Si atomic plane provides two kinds of additional disorder mechanisms: Si Frenkel pair and Si-Ti-related antisite pair.As a consequence, the migrations of V Si , Si  , and Ti Si have the close relationships to the recovery of radiation damages.Since Ti Si locates in the Si atomic plane, it is expected that it can migrate via on-lattice V Si site along the Si atomic plane.The calculated migration energy barriers of V Si , Si  , and Ti Si along the Si atomic plane are very low, only 0.9, 0.6, and 0.3 eV, respectively.This indicates that V Si , Si  , and Ti Si defects can easily migrate along the region between the neighbouring nanotwined Ti 3 C 2 blocks.These processes provide quick recovery mechanisms for Ti 3 SiC 2 under radiation damage.The results also illustrate the predominant role of nanolaminated crystal structure on the excellent radiation tolerance of Ti 3 SiC 2 .

Conclusion
In summary, the stable defect structures, defect formation energies, and migration barriers of intrinsic point defects in Ti 3 SiC 2 are studied by using first-principles calculations.The results clearly show that the intercalated Si atomic plane, as well as the nanolaminated crystal structure of Ti 3 SiC 2 , not only provides new disorder mechanisms with low formation energies but also presents the quick migration paths of the related point defects.The reported disorder mechanisms with low formation energies are the Ti Si + Si Ti antisite pair, Si Frenkel pair, and C Frenkel pair.The migration energy barriers of V Si , Si  , and Ti Si along the Si atomic plane are only 0.9, 0.6, and 0.3 eV, respectively.The low formation energies of the defect pairs ensure the high ability of Ti 3 SiC 2 to accommodate structure disorders under displacive radiation damage.At the same time, the low migration energy barriers of point defects in Ti 3 SiC 2 indicate easy recovery of radiation damage.These theoretical results are helpful to explain the good radiation damage tolerance of Ti 3 SiC 2 and can shed a light on the theoretical predication of ceramics with good radiation damage tolerance.This further highlights a novel method to improve the radiation damage tolerance by developing nanoscale-layered structure which can serve as the sink or rapid recovery channel for radiation-induced point defects.

2 Figure 1 :
Figure 1: Crystal structure of Ti 3 SiC 2 .Letters on the left side are the stacking sequence of Ti and Si atoms.

Figure 2 :
Figure 2: Interstitial configurations in SiTi 2 unit (a) and in Ti 3 C 2 unit (b)-(e).Letters above the Si plane in (a) are the stacking positions.Small red balls represent the positions of interstitial atoms.
3 C 2 unit: interstitial atoms locate between C and Ti 1 atomic planes (labeled as I-CTi 1 ), between C and Ti 2 atomic planes (labeled as I-CTi 2 ), inside the C atomic plane (labeled as I-C), and inside the Ti 1 atomic plane (labeled as I-Ti 1 ).The configurations are related to the sites on the center of C or Ti tetrahedron as seen in Figures 2(b) and 2(c) and the C-C-C trimmer as seen in Figures 2(d) and 2(e).C  atoms are initially put on all of the interstitial positions, and Ti  atoms are initially put on the I-CTi 1 and I-CTi 2 positions.

2 ) 3 )
2 eV, C C → V C + C  , 5.7/3.0 eV.(Antisite pairs: Ti Ti + Si Si → Ti Si + Si Ti , 5.0 eV, Ti Ti + C C → Ti C + C Ti , 11.2 eV, Si Si + C C → Si C + C Si , 6.6 eV.(Two formation energies of C Frenkel pairs correspond to the C  in Ti 3 C 2 and SiTi 2 units.If we compare the formation energies, the most possible disorders are the Ti Si + Si Ti antisite pair, Si Frenkel pair, and C Frenkel pair for their low formation energies.

Table 1 :
Defect formation energies (DFEs) (in eV) of on-lattice defects in Ti 3 SiC 2 .Ti  Si  C Si Ti Ti Si C Ti Ti C C Si Si C

Table 2 :
Calculated defect formation energies (in eV) of stable interstitials in Ti 3 SiC 2 .