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# Point defect formation in M2AlC (M = Zr,Cr) MAX phases and their tendency to disorder and amorphize

## Abstract

First principles calculations are performed on Zr2AlC and Cr2AlC MAX phases to compare their ability to accommodate point defects under irradiation. Interatomic bonding is stronger in Cr2AlC than Zr2AlC but contrary to expectation Zr2AlC exhibits higher vacancy and antisite pair formation energies. However, interstitials and Frenkel defects are generally more difficult to form in Cr2AlC. The results are attributed to the mixed covalent/ionic/metallic nature of the bonding. Detailed comparison of all the energies suggests that the preferred defects in Zr2AlC and Cr2AlC are the VAl+Ali Frenkel and CrAl+AlCr antisite respectively. Thus the potential response of the two phases to irradiation is different and taking account of other competing defects it is suggested that Zr2AlC is less susceptible to amorphization.

## Introduction

A group of nanolaminated hexagonal materials called MAX phases have come under intense scrutiny in recent years due to their unusual physical properties that result from a combination of metallic and ceramic bonding characteristics1. They have the general formula Mn+1AXn where n is 1, 2 or 3, M is an early transition metal, A is an A-group element and X is either C or N. Among the properties that have drawn attention are their relatively high stiffness, good machinability, high thermal and electrical conductivity and good resistance to corrosion, oxidation, creep, fracture and fatigue2. Ti3SiC2 was one of the first MAX phases to be discovered and since then more than 70 others have been synthesized. Various applications for these materials have been suggested or put into practice including surface coatings, heating elements, bearings and armor.

The mixed covalent/ionic/metallic bonding present in MAX phases results in another possible application as an in-core structural material or coating within the hostile environment of a nuclear reactor3, 4. Under irradiation these phases have shown a remarkable ability to accommodate point defects and remain crystalline rather than becoming amorphous even when subjected to high levels of radiation or ion bombardment. However, sustained structural stability depends on the choice of M, A and X elements and some MAX phases are more irradiation tolerant than others. Furthermore, the elements should not have a high neutron cross-section since this will reduce the “permeability” of the material to neutrons and thus lower the reactor’s performance. Also a high neutron cross-section would, in some cases, cause undesirable activation products to form. Ti and Cr have relatively high neutron cross-sections (6.1 and 3.1 barn respectively) while Zr and Al are relatively low (0.184 and 0.233 barn). This suggests that Zrn+1AlCn MAX phases might be candidates for nuclear applications.

Both Zr2AlC5 and Zr3AC2 6, 7 have been recently synthesized although ZrC always appears as a secondary phase and under certain processing conditions Zr2AlC and Zr3AlC2 can exist together. Furthermore related quaternary compositions of the type Zr2(Al, AI)C where AI = Sn, Sb, Pb or Bi have also been reported8,9,10,11. The latter study has suggested that partial substitutions on the A-site help to stabilize the Zr2AlC phase. However, partial substitutions on the M-site have only been successful for (Nbx,Zr1-x)4AlC3 12. The structural stability and irradiation tolerance of Zrn+1AlCn MAX phases can be assessed using density functional theory (DFT) calculations that focus on their formation enthalpy, bond strengths and propensity to form point defects13. If antisite point defect formation is relatively easy then the structure is more likely to remain crystalline than become amorphous since antisite defects retain the coherency of the lattice and act as a recovery mechanism. It is emphasized, however, that DFT calculations of defect energetics can only provide an indicator rather than a complete predictor of the susceptibility of a material to amorphize under irradiation since other process features such as kinetics are not taken into account. In this study we perform DFT calculations on Zr2AlC and compare its properties with Cr2AlC. Chromium is chosen as an alternative M element since it should enhance the high temperature oxidation resistance of the material14 and has a lower neutron cross-section (3.1barn) than Ti. In addition Cr2AlC has been successfully synthesized and the main radiation induced defects have been determined15, 16. Depending on the processing conditions, small amounts of Cr3C2, Cr7C3 and Cr5Al8 can be present in the material17 although generally speaking Cr2AlC has been synthesized (at the moment) with higher purity than Zr2AlC.

Previous computational studies18 using DFT have been performed on the phase stability and bulk properties of (Zr1-x Crx)2AlC including Zr2AlC and Cr2AlC. The calculations show that (Zr1-x Crx)2AlC is unstable with respect to dissociation into the Zr2AlC and Cr2AlC ternary phases and that this is due to the position of the Fermi level which lies at a peak position on the electronic density of states of (Zr0.5Cr0.5)2AlC. Attempts to synthesize such quaternary phases11 failed confirming the instability of these potential solid solutions. Point defect calculations have only been performed on Cr2AlC where it is found that the Cr-Al antisite pair has the lowest formation energy of all possible point defect combinations thus providing a defect mechanism for recovery during irradiation19. This result, as well as the finding that the C Frenkel pair also has relatively low formation energy, is consistent with the recent observations on ion irradiated Cr2AlC15, 16. Although the calculations were based on summing the energies of the individual defects they showed that despite the low magnetic moment of Cr, a non-magnetic treatment of Cr2AlC is sufficient if only trends in defect formation behavior are required. Furthermore, for the nuclear applications of interest here, the materials are synthesized and used at temperatures significantly above the observed Curie temperature (~73 K)20.

## Computational Method

The DFT calculations were performed using the projector augmented-wave (PAW) method21 as implemented in the VASP code22, 23 with the following states treated as valence: Zr (4s4p5s4d), Cr (3p4s3d), Al (3s3p), C (2s2p). Exchange correlation effects were included using the Generalized Gradient Approximation (GGA) as parameterized by Perdew-Burke-Ernzerhof (PBE)24. A Γ-point based Monkhorst-Pack scheme25 was used to sample the Brillouin zone with meshes sizes of 23 × 23 × 7 and 7 × 7 × 7 for perfect unit cells and supercells with defects respectively. A plane wave cutoff energy of 400 eV was used. Since all the MAX phases have metallic character, a Methfessel-Paxton smearing method with a sigma value of 0.2 was applied. In a fully optimized MAX phase structure the forces on each atom were always less than 0.01 eV/Å. Following the conclusions of previous work on Cr-containing systems19, the structures were treated as non-magnetic.

For the point defect calculations the supercell size used was 4 × 4 × 1 in order to minimize intercellular defect interactions. In both Zr2AlC and Cr2AlC, all cationic and anionic vacancies and interstitials were considered together with their associated Frenkel and antisite pairs. Since MAX phases are low-density layered structures, there is ample space between and within the layers for the incorporation of interstitial atoms. We have chosen four such open spaces and labeled them as Ihex, Itet, Ioct and Ipri. The Ihex position is in the Al layer and at the center of two triangular pyramids that connect to adjacent M layers. The Itet position is at the center of a tetrahedron that connects adjacent M layers and crosses a C layer, and the Ioct position is at the center of an octahedron that connects an M layer to an adjacent Al layer. The Ipri position lies at the center of a triangular face of this octahedron within the Al layer. Each of the four symmetry distinct interstitial positions is shown in Fig. 1(c). For Frenkel pairs, the interstitials and vacancies are separated by at least 6 Å in order to avoid spontaneous annihilation of the vacancy-interstitial defect. For antisite pairs, atoms in neighboring layers are interchanged.

The procedure for determining point defect formation energies from supercell calculations is well established, see e.g. ref. 26, and requires knowledge of the chemical potentials of the constituent elements and the application of equilibrium conditions which prevent the formation of competing phases. The method has recently been applied to Ti-based MAX phases13. By calculating the formation enthalpies of the competing phases and applying the equilibrium conditions, the accessible range of chemical potentials required for the synthesis of Zr2AlC and Cr2AlC can be determined (see SI for details) and the general procedure has recently been automated27. The results are shown in Figures S1 and S2. For single point defects (vacancies or interstitials) the choice of chemical potential can significantly affect the formation energies as described below. However, for defect pairs (Frenkels or antisites), which are commonly found in irradiated compounds, the formation energies are independent of the chemical potentials.

To understand how the formation of point defects influences local bonding and hence structural stability we have performed a charge density analysis of bulk Zr2AlC and Cr2AlC with and without the defects. In particular Bader charges (i.e. atomic charges based on charge density) have been calculated together with the charge densities at the bond critical points (bcp) using QTAIMAC (Quantum Theory of Atoms in Molecules and Crystals)28, 29 as implemented in the Critic2 code30. The method is described in more detail in the SI.

## Results and Discussion

### Bulk Properties

The calculated lattice parameters of Zr2AlC and Cr2AlC are shown in Table S1 and compared to experimental values. It is seen that the maximum deviation for Zr2AlC is only 0.16%. The relatively larger deviation for Cr2AlC (−1.12%) is due to the nonmagnetic state considered in this study. The Zr-containing phase has larger lattice parameters than the Cr-containing phase principally because of the larger crystal radius of Zr (13%). Figure 2 shows the calculated Bader charges on different constituent ions and the values of the charge densities at the bond critical points (bcp). Interestingly there is significant charge transfer from Zr to Al and C ions in Zr2AlC whereas charge transfer from Cr and Al to C in Cr2AlC is relatively small. Figure 2(b) clearly suggests that bonding in Cr2AlC is generally stronger than in Zr2AlC.

### Defect Formation

#### Vacancies

We investigated three types of vacancy (i.e.,VZr/VCr, VAl and VC) in both Zr2AlC and Cr2AlC. It is clear from the calculated vacancy formation energies displayed in Fig. 1(a), that vacancies in Zr2AlC are much harder to create than in Cr2AlC and will therefore be less abundant. The values shown were determined using the chemical potentials of the constituent elements and, for Cr2AlC, agree well with the previous work16. The easiest vacancy to form in Cr2AlC is VC whereas in Zr2AlC it is VAl. The Cr and Al vacancy formation energies in Cr2AlC are approximately equal. The larger vacancy formation energies in Zr2AlC are contrary to expectations based on the relative strengths of the bonds suggested by the bcp charge densities (Fig. 2(b)) and are due to the mixed and complex nature of the bonding as discussed in section 3.3.

The relative vacancy formation energies in Zr2AlC are different from Cr2AlC and can be summarized as follows:

$$\begin{array}{c}{{\rm{E}}}_{{\rm{vac}}}({\rm{Zr}}) > {{\rm{E}}}_{{\rm{vac}}}({\rm{C}}) > {{\rm{E}}}_{{\rm{vac}}}({\rm{Al}})\quad \,({{\rm{Zr}}}_{{\rm{2}}}{\rm{AlC}})\\ {{\rm{E}}}_{{\rm{vac}}}({\rm{Al}}) > {{\rm{E}}}_{{\rm{vac}}}({\rm{Cr}}) > {{\rm{E}}}_{{\rm{v}}{\rm{a}}{\rm{c}}}({\rm{C}})\quad ({{\rm{Cr}}}_{{\rm{2}}}{\rm{AlC}})\end{array}$$

Other MAX phases such as Ti3AlC2 and Ti3SiC2 exhibit a trend in vacancy formation similar to Zr2AlC13, 31. However, the results clearly depend on the synthesis conditions, as shown in the SI (Figures S3S6). Figures S3 and S4 present the formation energies at the extreme points on the valid range of chemical potentials whereas Figures S5 and S6 show how these energies vary over the entire valid range. The latter two figures demonstrate that the energies behave uniformly between the extreme points and do not exhibit any local minima or maxima. Taken together several conclusions can be drawn from Figures S3S6. In Zr2AlC, for example, under Zr-rich conditions, the vacancy trend becomes Evac(Zr) > Evac(Al) > Evac(C). It is noteworthy that vacancy formation energies depend strongly on the choice of chemical potentials (i.e., on the synthesis conditions). Under certain synthesis conditions, the vacancy formation energies for some of the vacancies (e.g., VAl, VC) in Zr2AlC become smaller than the corresponding vacancies in Cr2AlC as shown in Figures S3 and S4. The Zr and C vacancies are the least and most affected vacancies in Zr2AlC respectively due to the chemical potentials. The Zr vacancy remains the most difficult vacancy to form in Zr2AlC irrespective of the choice of chemical potential. Nonetheless, VZr becomes relatively easier to form under Al rich conditions. Al vacancies compared to carbon vacancies become relatively harder to form in Zr2AlC under Zr and/or Al rich conditions. The most significant effect of the Zr2AlC synthesis environment is on the formation of C vacancies. VC becomes the most stable vacancy under Zr and/or Al rich conditions. Consequently depending on how Zr2AlC is synthesized, either VAl or VC becomes the most stable vacancy. This is different to the situation in Cr2AlC where the choice of chemical potential has a negligible effect on the vacancy formation energies. For instance, C vacancies remain the most stable vacancies under all the synthesis conditions.

#### Antisite pairs

The antisite pair energies give an indication of how easy it is to create disorder on the different MAX phase sublattices. They also provide a recovery mechanism for the crystal following an irradiation induced displacement cascade. This mechanism depends on the type of the interstitial/vacancy species and the target sublattice. The antisite defect formation energies are shown in Fig. 1(b) for Zr2AlC and Cr2AlC. Unlike single vacancies (or interstitials) the energies do not depend on the chemical environment since the atom species simply interchange and there is no reservoir needed for the deposition or extraction of atoms.

Figure 1(b) shows that antisite pairs are harder to create in Zr2AlC than in Cr2AlC and that of the three antisite combinations, MAl+AlM pairs are the most stable. The results can be understood in simple terms using differences in the crystal radii32 and electronegativities33 of the atoms involved. For example, the percentage difference in crystal radius between M and Al is less than between M and C or Al and C (RZr = 0.86 Å, RCr = 0.76 Å, RAl = 0.53 Å, RC = 0.29 Å). Similarly the percentage difference in electronegativity between M and Al is less than between M and C or Al and C (χZr = 1.33, χCr = 1.66, χAl = 1.61, χC = 2.55). Both differences suggest that MAl+AlM pairs should be the preferred antisite defects and indicate that elastic and electronic effects are playing a role. Comparing ZrAl+AlZr with CrAl+AlCr it is seen that the percentage difference in both quantities is greater in Zr2AlC (e.g. 62% versus 43% for the crystal radii and 21% versus 3% for the electronegativities) explaining the larger formation energy of the ZrAl+AlZr defect. Interchanging the cations (Zr/Cr and Al) with the anions (C) is always relatively difficult. In this case the carbon atom in the cation layer eventually moves into an interstitial position between a Zr/Cr layer and Al layer as shown in Figures S7 and S8. Similarly the cations (Zr/Cr and Al) also do not like to be in a carbon layer and create a vacancy by moving into the next available cation layer.

#### Interstitials

MAX phases are low-density layered structures with ample open space between and within the layers. When these phases are irradiated a large number of atoms are displaced from their equilibrium sites and form interstitials. The energy required to form these interstitials varies depending on the chemistry of MAX phase and the type of interstitial atom and interstice. We have investigated the formation of three different types of interstitial (Zri/Cri, Ali, Ci) in four different interstices (Ihex, Ipri, Ioct and Itet, see Fig. 1(c)) in bulk Zr2AlC and Cr2AlC. The interstitial atoms at the Itet site are found to be either unstable or have relatively large formation energies in both Zr2AlC and Cr2AlC. This is most likely due to the limited space available for Zr/Cr and Al, which are cations with large radii. As a consequence, we now focus only on the salient features of the different interstitials at the sites Ihex, Ipri and Ioct.

The calculated interstitial formation energies are displayed in Fig. 1(d) where it is seen that interstitials are much harder to create in Cr2AlC and will therefore be relatively less abundant. The values shown were determined using the chemical potentials of the constituent elements and, for Cr2AlC, agree well with the previous work16. For Cr2AlC it is also seen that there is little difference in formation energy between the different sites for a given interstitial suggesting that they may have relaxed to similar configurations and that the C interstitial has the lowest energy. For Zr2AlC, it is clear that the Al interstitial has the lowest energy, which is negative, and apparently the same in all sites. The negative formation energy suggests that Zr2AlC can spontaneously become super-stoichiometric in Al and there is some experimental evidence for an excess of Al in some Ti-based MAX phases34. However, as noted in our discussion of vacancy formation energies, the results will depend on the synthesis conditions and Figures S3 and S5 shows that the Al interstitial formation energy becomes almost positive when the thermodynamically valid range of chemical potentials is considered. The trend in behavior, however, does not change (i.e. Ali remains the most favorable interstitial). Interestingly under some synthesis conditions (i.e., values of chemical potential), certain interstitials (e.g., C interstitials) become harder to form in Zr2AlC than in Cr2AlC (see Figures S3/S4 and S5/S6). In Zr2AlC, Zr/Al and C interstitials are the least and most affected by the chemical potentials respectively. Like Al interstitials in Zr2AlC, Zr interstitials remain mostly unaffected by the synthesis conditions except in Al rich conditions that make them relatively harder to form. The most pronounced effect of the chemical potentials is on C interstitials in Zr2AlC. They become harder to form under Zr and/or Al rich conditions. Nevertheless, the Al interstitial remains the most stable interstitial in Zr2AlC irrespective of the chemical environment. On the other hand, the chemical potentials only slightly influence the different interstitial types in Cr2AlC as seen in Figure S6 by the relatively small variation in colors representing the formation energies. The C interstitial remains the most stable interstitial in Cr2AlC under all possible synthesis conditions that stabilize pure Cr2AlC. Cr and Al interstitials become relatively the most difficult to form depending on the synthesis environment. For instance, formation of a particular interstitial (e.g., Cri) becomes harder if any other species (e.g., Al and/or C) is in rich condition.

The almost overlapping energies in Fig. 1(d) suggest that the relaxed configurations should be examined in detail. Figure S9 shows these configurations viewed parallel to the relevant Al layer in the initial structure. We first consider the M(=Zr/Cr) interstitial which is seen to remain in or close to the adjacent Al layer. Interestingly, starting at the Ipri position, these interstitials occupy a lattice site in the Al layer and create an Al interstitial. In the case of Zr2AlC, this lowers the energy considerably, making the process very favorable. The M interstitials in the Ioct position initially are relatively unstable. For Zr2AlC, the Zr atom relaxes into the Al plane whereas for Cr2AlC, the Cr atom stands off from the Al plane. We now consider the Al and C interstitials, which exhibit a particular behavior in Zr2AlC and Cr2AlC, namely, their formation is almost site independent and they always adjust their position in or near the adjacent Al layer. For Zr2AlC, all the Al interstitials spontaneously relax into the Al layer, lowering the energy of the structure considerably as noted above. The same configurations are formed in Cr2AlC but at much higher cost in energy. The difference in behavior can be explained, at least qualitatively, by considering the differences in local structure. Figure S10 compares the Al bond lengths and M-Al layer spacings in Zr2AlC and Cr2AlC with bulk fcc Al. Compared to the bulk metal, Zr2AlC is in tension whereas Cr2AlC is in compression. Thus the formation of an Al interstitial in Zr2AlC is more likely to be favorable than in Cr2AlC.

The C interstitial behaves differently in Zr2AlC and Cr2AlC. For Zr2AlC, the C interstitial in the Al layer remains in the Al layer, as in the case of the Ihex and Ipri sites. The C interstitial starting at the Ioct site comes close to the Al layer but does not settle in this layer. However, for Cr2AlC, the C interstitial moves from the Ihex site to a position in between the Cr and Al layers, as does the C interstitial starting at the Ioct site. On the other hand the C interstitial at Ipri remains in the Al layer. Energetically the most favorable site in Zr2AlC for Zr and Al interstitials is Ipri and for C interstitials is Ioct. For Cr2AlC, Ioct is the most favorable site for the Cr interstitial and Ipri for the C interstitial. The Al interstitials in Cr2AlC are position independent.

#### Frenkel pairs

Frenkel pairs are some of the most commonly found defects in irradiated materials. An atom is displaced off its lattice site to form a vacancy and an interstitial. We have determined Frenkel pair formation energies in Zr2AlC and Cr2AlC in three different ways. Firstly, we have considered isolated Frenkel pairs whose energy is determined simply by summing the formation energies of the corresponding vacancy and interstitial under the same chemical conditions. We note that like antisite defects, the Frenkel formation energies should be independent of the chemical potentials (provided they are the same for the vacancy and interstitial) since no reservoir is needed. This has been the method used in previous defect calculations on MAX phases, e.g. ref. 19. Secondly, we have considered bound Frenkel pairs in which the vacancy and interstitial are relaxed within the same computational cell and initially 5–9 Å apart across M, Al or C layers. In order to keep the number of computations feasible, we have considered only those Frenkel pairs that have the lowest isolated formation energies. Lastly, we have considered a special type of bound Frenkel pair in which vacancy and interstitial are created in the same layer and call them self-Frenkel pairs. Figure 3 compares the formation energies obtained using the three methods.

It is seen that the form and magnitude of the isolated (Fig. 3(a)) and bound (Fig. 3(b)) Frenkel pair formation energies are very similar indicating that the pair binding energy is very small and that bringing the two point defects closer together does not change the preferred configuration, i.e. VAl+Ali in Zr2AlC and VC+Ci in Cr2AlC. It is also seen that M and Al Frenkel pairs are more difficult to form in Cr2AlC than in Zr2AlC whereas C Frenkel pairs are somewhat easier to form in Cr2AlC than in Zr2AlC. This is because M and Al interstitials in Cr2AlC are very difficult to create whereas in Zr2AlC they could be energetically favorable. The relative stability of C and Al Frenkel pairs is also found in other MAX phases. For instance, C Frenkel pairs are found to be stable in Ti3AlC2, Ti2AlN and Cr2GeC and Al Frenkel pairs are found to be stable in Ti3SiC and Ti2AlC33. The M type Frenkel pairs always require a large amount of energy to generate which is a consequence of the relatively strong M-C bonding. The relaxed structures of the bound Frenkel pairs are shown in Figures S11 and S12 for Zr2AlC and Cr2AlC respectively. It is seen that displacements away from the initial configurations are small, which reflects the small binding energies. The Ipri site is the preferred position of the most stable Frenkel pair in Zr2AlC (VAl+Ali) and in Cr2AlC (VC+Ci).

Lastly, we consider the self-Frenkel pair formation energies that are shown in Fig. 3(c). It might be expected that a self-Frenkel pair created in its own layer would recombine on relaxation and thus cost no energy to form. This is what happens with the Al self-Frenkel pair in both Zr2AlC and Cr2AlC but more complex relaxations take place for the M and C self-Frenkel pairs. The displacements they create are shown in Figures S13 and S14. The formation energies of these defects are generally higher than their bound counterparts because secondary defects form. In Zr2AlC, the Zr self-Frenkel pair relaxes to form a ZrAl+AlZr antisite pair and the C self-Frenkel pair stimulates the formation of a Zr Frenkel pair. In Cr2AlC, both the Cr self-Frenkel pair and the C self-Frenkel pair remain but relax to a different orientation.

In order to fully understand the irradiation tolerance of a material at the atomistic level, it is vital to investigate all possible recovery mechanisms that help it to recrystallize and resist amorphization. The above results indicate that the energy to form a vacancy or antisite defect in Zr2AlC is higher than in Cr2AlC. However, interstitials and Frenkel defects are generally more difficult to form in Cr2AlC. Comparison of all the energies suggests that the preferred defects in Zr2AlC and Cr2AlC are the VAl+Ali Frenkel and CrAl+AlCr antisite respectively. However, ZrAl+AlZr antisites and VC+Ci Frenkels are also energetically favorable in these materials and could compete or interact with the preferred defects during irradiation and subsequent cooling. Thus the potential response of the two phases to irradiation is different. Structurally, Zr2AlC tends to form defects that retain the coherency of the lattice (e.g. Figure S11(d) for the VAl+Ali Frenkel and Figure S7(b) for the ZrAl+AlZr antisite) while Cr2AlC tends to form a defect that disturbs coherency (e.g. Figure S12(f) for the VC+Ci Frenkel). Disturbance of the lattice suggests that Cr2AlC is more susceptible to amorphization. The equilibrium concentrations of defects at finite temperature are straightforward to calculate from Boltzmann statistics35. For example at 1100 K, the concentration of CrAl+AlCr antisite pairs in Cr2AlC would exceed the concentration of ZrAl+AlZr antisite pairs in Zr2AlC by about a factor of 100. Similarly, at the same temperature, the concentration of VAl+Ali Frenkel pairs in Zr2AlC would exceed the concentration of Vc+Ci Frenkel pairs in Cr2AlC by about a factor of 2000. However, it is emphasized that these are equilibrium concentrations and likely to be less than those found in situ in a radiation environment.

To further understand how each of these defects affect the strength of the MAX phase we have determined how the charge densities at the bond critical points change using the QTAIMAC method28,29,30. Figure 2(b) clearly shows that all the bonds in Zr2AlC are weaker than the corresponding bonds in Cr2AlC. This is corroborated by another theoretical study36 which also finds that the bulk modulus of Cr2AlC is greater than Zr2AlC. Three types of bonds (M-C, M-Al, Al-Al) exist in bulk M2AlC (M = Zr/Cr) phases where M-C and Al-Al bonds are the strongest and weakest bonds respectively in both systems (Fig. 2(b)). Bonding in Zr2AlC is predominately ionic as indicated by the positive value of the Laplacian of the charge density whereas in Cr2AlC, the Cr-Al bond is found to be covalent (see Table S2). The comparatively weak ionic bonding in Zr2AlC renders it more radiation tolerant than Cr2AlC since defects more easily re-establish themselves with the crystal structure13, 37. It is interesting to note that bonding in several relevant Zr-based compounds (Zr-metal, Zr-carbides, Zr-aluminides) is found to be weaker than the bonding in Cr-based compounds. For instance, Figure S15 compares the strength of M-C, M-Al and Al-Al bonds in Zr2AlC, Cr2AlC and the relevant binaries and metallic systems. Almost all the bonds in Zr-based compounds are relatively weaker than the corresponding bonds in Cr-based compounds. This could imply that Zr-based materials intrinsically have more ability to re-establish defects with the initial crystal structure compared to Cr-based materials. Defects disturb the local bonding by breaking old bonds and creating new bonds. Figure 4 shows the change in bcp charge density (bond strength) due to a defect in Zr2AlC and Cr2AlC. For instance, in the case of cationic antisite MAl+AlM (M = Zr/Cr) defect pairs two new strong bonds, namely M-M and Al-C, are formed in both systems. Most importantly all the bonds including old and newly formed due to defects are stronger in Cr2AlC than in Zr2AlC. Similarly, the most stable defects, namely, Al Frenkel pairs in Zr2AlC and C Frenkel pairs in Cr2AlC also show relatively stronger bonds in Cr2AlC compared to Zr2AlC. The relatively weak and ionic bonding in Zr2AlC makes it comparatively more radiation tolerant than Cr2AlC by ‘recrystallizing’ defects more easily with the initial crystal structure.

## Conclusions

Using DFT calculations combined with a chemical potential and charge density analysis, the relative stability of various point defects in two MAX phases, Zr2AlC and Cr2AlC, has been investigated. The objective has been to determine their relative tendency to disorder and amorphize under irradiation. It is found that interatomic bonding in Cr2AlC is generally stronger than in Zr2AlC but contrary to expectation Zr2AlC exhibits higher vacancy and antisite pair formation energies. Although both materials are metallic they exhibit different degrees of ionicity and covalency, with Zr2AlC being the more ionic. This difference affects the vacancy formation energies, which would otherwise be larger in the stronger metal. Nevertheless, interstitials and Frenkel defects are generally more difficult to form in Cr2AlC. Analysis of the defect formation energies within the accessible range of chemical potentials shows that their relative values do not change significantly compared to pure conditions, especially for Cr2AlC. In Zr2AlC the main effect is to increase the Al interstitial formation energy so that it becomes almost positive. Detailed comparison of all the energies suggests that the preferred defects in Zr2AlC and Cr2AlC are the VAl+Ali Frenkel and CrAl+AlCr antisite respectively. Although DFT calculations are only indicators rather than predictors of a material’s susceptibility to amorphise under irradiation, the current results suggest that that Zr2AlC is less susceptible to amorphization because the defects that form preserve the coherency of the lattice and offer a viable recovery mechanism.

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## Acknowledgements

The calculations were performed at the Cambridge HPCS and the UK National Supercomputing Service, ARCHER. Access to the latter was obtained via the CaFFE consortium and funded by EPSRC under Grant No. EP/M018768/1. All necessary computational data is included in the manuscript, the Supplementary Information or the University of Cambridge Data Repository https://doi.org/10.17863/CAM.8031.

## Author information

### Affiliations

1. #### Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, CB3 0FS, UK

• S. H. Shah
•  & P. D. Bristowe

### Contributions

S.H.S. performed the DFT calculations and analysis under the guidance of P.D.B. Both authors contributed to the discussion of the results and preparation of the manuscript.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding author

Correspondence to P. D. Bristowe.