A first-principles study on the defective properties of MAX phase Cr₂AlC: the magnetic ordering and strong correlation effect

Abstract

Our first-principles calculations indicate the defective properties of Cr₂AlC are very different from the commonly studied Ti-based MAX phases. Al vacancies are predicted to have high formation energies while Cr vacancies have low formation energies in Cr₂AlC. Compared with previously reported results of MAX phases, the formation energy of the anion antisite defect pair in Cr₂AlC is the lowest (∼1.9 eV), predicting a good irradiation-resistance property. The reduction in the formation energy of the Cr–Al antisite defect is attributed to the magnetic ordering and strong correlation effect of Cr atoms. Analysis suggests that the majority spin state of the d_z2 orbital of Cr is lowered in energy while the minority spin states become empty, which makes the Cr more ionic in character.

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Introduction

Recently, MAX phase materials have attracted increased attention due to their unique combination of ceramic and metallic properties. The MAX phases form a large family of ternary carbides with the general formula M_n+1AX_n, where n varies from 1–3, M is an early transition metal, A is an A-group element, and X is C or N. Cr₂AlC is a MAX phase material that exhibits many advantageous properties. For example, it has a high thermal (17.5–22.5 W mK⁻¹) and electrical conductivity (1.4 to 2.3 × 10⁶ Ω⁻¹ m⁻¹),^1–3 and a large tolerance to damage (retaining ∼70% of its flexural strength when subjected to a 100 N indentation load).⁴ Cr₂AlC also displays resistance against oxidation and corrosion, high elastic stiffness and maintains high strength at high temperature.^5,6 All of these properties make Cr₂AlC a suitable candidate to be adopted in applications where materials are subject to extreme environments, such as nuclear reactors.

To evaluate the performance of Cr₂AlC as a structural material for special applications, a microscopic understanding of the electronic and structural properties of intrinsic defects in the material is required. Defects can be unintentionally introduced into Cr₂AlC during growth and also when subject to an extreme environment of high-irradiation. Indeed, several recent experimental studies⁷ have investigated the properties of Cr₂AlC when subject to irradiation with 7 MeV Xe²⁶⁺ ions and 500 keV He²⁺ ions. Although prior theoretical studies have investigated the properties of many other MAX phase materials^8–13 and demonstrated good agreement with experiment,^14–17 there is no theoretical investigation of the defective properties of Cr₂AlC up to now.

The main problem in further investigation on the defective properties of Cr₂AlC by first-principles study should come from the discrepancies between theoretical and experimental results in bulk properties. It is reported that the existing theoretical results are not consistent with experimental reports of the density of states,¹⁸ bulk modulus and Young's modulus.^18–20 Until recent years, several works explained why this has been the case and how to overcome it. Therefore, the discrepancies have been minimized and explained.^17,21

Two reasons for the possible discrepancies between the existing theoretical results on Cr₂AlC and experimental studies are strong correlation effect^21,22 introduced by the presence of localized d-orbitals and the magnetic ordering of unpaired d-electrons.^23,24 Failing to account for these effects self-consistently can lead to quantitatively and qualitatively incorrect predictions of the structural, electronic and optical properties of Cr₂AlC. By considering the Cr 3d on-site Coulomb energy, Du et al.²¹ obtained reasonable unit cell volumes compared with experimental data. However, Cr₂AlC was treated as a ferromagnetic material in their work, which is inconsistent with available experimental data. Prior experiments have reported a low magnetic moment of Cr (∼0.002 μB per Cr atom), and a low transition temperature (∼73 K), which was interpreted as very weak ferromagnetic (FM) or canted antiferromagnetic (AFM) spin configuration.^25,26 Recent investigations of the structural, optical and elastic properties of Cr₂AlC within the GGA+U framework¹⁷ demonstrated better agreement with experiment when taking the antiferromagnetic ordering of the Cr atoms into account. Based on these previous studies for bulk Cr₂AlC, the defective properties of Cr₂AlC are firstly studied in this work by using the calculation settings previously used.¹⁷

Our calculations demonstrate the importance of taking into account the anti-ferromagnetic ordering of the Cr atoms when calculating the formation energies of point defects in Cr₂AlC. Using the AFM+U calculations, we show the Cr–Al antisites defects have much lower formation energies compared to calculations at the GGA level of theory. The lower formation energies for these defects are consistent with experimental observations of large anti-site formation found in Cr₂AlC.⁷

Calculation details

Our calculations were performed under the framework of density functional theory as implemented in the Vienna ab initio simulation package (VASP).^27,28 The projected augmented wave method (PAW)²⁹ and the generalized gradient approximation (GGA)³⁰ were used. The exchange and correlation energies were calculated using the Perdew–Burke–Ernzerhof (PBE) functional.³¹ The wave functions were expanded in a plane-wave basis set with an energy cutoff of 400 eV. Spin polarized calculation was included for Cr contained systems to correctly account for the magnetic properties. The strong on-site Coulomb repulsion was modified by the Hubbard U correction.³² Based on previous results,¹⁷ the in-AFM1 magnetic ordering configuration was found to be the ground state ordering which we adopt for our calculations using a U value of 1 eV. The previous work has reported that the usage of the U value equal to 1 eV is suitable, since the structural parameters are close to the ones by experimental measurement and by calculations with hybrid exchange–correlation functional, HSE06.¹⁷ The lattice constants and internal freedom of the unit cell were fully optimized until the Hellmann–Feynman forces on the atoms were less than 0.01 eV Å⁻¹. The effective charge for each atom (charge difference after bonding) was calculated using Bader charge analysis.³³

Calculation of the defect structure employed a 4 × 4 × 1 supercell, which contains 128 atoms. According to our previous studies of defects in the 211-MAX phases,³⁴ a supercell of this size has been proven to lead to formation energies that are sufficiently converged. The special k-point sampling integration was used over the Brillouin zone by using the Γ-centered 5 × 5 × 5 for this supercell.³⁵ Defect formation energies were calculated using bulk bcc-Cr, fcc-Al and graphite as the chemical potentials of the limiting phase. Calculations of bulk Cr and Al were performed by using a 30 × 30 × 30 k-point grid. The computed equilibrium lattice constants of bulk bcc-Cr and fcc-Al are 2.84 Å and 4.04 Å, respectively, which agree well with the experimental values of 2.88 Å and 4.05 Å.³⁶ All these calculation were checked using larger energy cutoffs and k-meshes; the results of total energy and Hellmann–Feynman forces are converged within 0.01 eV and 0.01 eV Å⁻¹, respectively.

Results and discussion

The details of the calculation methods and parameters are illustrated in the ESI.† As a member of the 211 phase, Cr₂AlC has a hexagonal structure with a space group P6₃/mmc, as shown in Fig. 1. Its unit cell contains two formula units, the carbon atoms are positioned at the (2a), Cr at the (4f) and Al at the (2c) Wyckoff positions. The structure of Cr₂AlC can be regarded as an alternating stacking of a two dimensional sheet of edge-sharing Cr₆C octahedral and a planar close-packed Al layer. Based on the results of Dahlqvist et al.,¹⁷ the ground state in-AFM1 magnetic ordering is considered. The configuration of in-AFM1 spin ordering is shown in Fig. 1(a), within the Cr plane, the Cr atoms has parallel spins along the [1̄, 2, 1̄, 0] direction and anti-parallel spins along the [2, 1̄, 1̄, 0] direction. When viewed along the stacking direction, the Al layers act as mirror planes to separate the Cr with the same spin. With the magnetic ordering and strong correlation effect considered, the obtained lattice constants and the mechanical properties are listed in Table S1.†

Fig. 1 The crystal structure and defect configuration of Cr₂AlC. The sideview (a) and topview (b) of the original Cr₂AlC structure and magnetic ordering configuration. Arrows on Cr atoms illustrate the spin directions. (c) The different defect configurations considered in this paper. The white, green and gray balls represent Cr, Al and C atoms, respectively. The small red ball in (c) represents the positions of interstitial atoms.

As can be seen from Table S1,† the difference between the experimentally measured and calculated structural parameters can vary significantly. According to these experimental results, we can find the non-spin polarized PBE calculations underestimate the lattice constants, and overestimate the bulk modulus B, shear modulus G, and Young's modulus E. In particular, the bulk modulus is greatly overestimated. The experimental measured values of bulk modulus are obtained using different methods. By using diamond anvil cell method, the measured bulk modulus is 165 GPa, which is 19% lower than the non-spin polarized calculation result for Cr₂AlC. By using sound velocity method, the value is ∼139 GPa, which is 40% lower than the non-spin polarized calculation result. The value of the bulk modulus can be decreased by taking into account certain magnetic ordering for Cr₂AlC without using any +U method as first shown in M. Dahlqvist et al.³⁷ With both of the in-AFM1 magnetic ordering considered and U parameter set, the values of lattice constants are increased, while the moduli B, G and E are decreased, which brings the calculated structural properties closer to the experimental reports. Therefore, the discrepancies have been minimized and explained, which is also pointed out by ref. 17 and 21. According to these results, the defective properties of Cr₂AlC are firstly studied by first-principles in this work.

According to the crystal structure, the configurations of native point defects in Cr₂AlC are illustrated in Fig. 1(c). There are 3 types of vacancies (V_Cr, V_Al, and V_C), 3 possible interstitial defects (Cr_i, Al_i and C_i), and 6 types of antisites (Cr_Al, Cr_C, Al_C, Al_Cr, C_Al, and C_Cr). Theoretically, there are two possible interstitial sites for the 211 MAX phases. However, prior studies have demonstrated interstitials are not stable between the M–X layers in Ti-contained MAX phases,³⁸ which is consistent with our own calculations of interstitials in Cr₂AlC. The interstitials are much more stable between Cr–Al layers than between the Cr–C layers, as shown in Table S2.† Therefore, in this work, only the interstitial site between the Cr–Al layers is considered as shown in Fig. 1(c). Considering the strong correlation effect and magnetic ordering, the calculated formation energies of point defects in Cr₂AlC are shown in Fig. 2 (details are listed in Table S3†), compared with the non-magnetic calculation results of Cr₂AlC and Ti₃SiC₂.³⁸

Fig. 2 Defect formation energies of on-lattice defects in Cr₂AlC. The results of E_def for the most studied Ti₃SiC₂ calculated by previous work³⁸ are also plotted for comparison. All the defect formation energies were calculated with the chemical potentials of atoms assumed to be the bulk crystals of elementary substance.

Prior studies of defects in Ti-based MAX phase materials^9–13,38,39 identified a consistent trend in the formation of vacancies; V_Al < V_C < V_Ti for Ti₂AlC, Ti₃AlC₂, and V_Si < V_C < V_Ti for Ti₃SiC₂. This suggests the group-A element (Al, Si) vacancies are more easily formed. These calculated results are consistent with experimental demonstrations of the facile ability to electrochemically etch the A-group elements to form 2D layers of the MAX phases, called “MXene”.⁴⁰ In contrast, the defect properties of Cr₂AlC could be very different from the Ti-based MAX phase materials according to our results. The calculated E_def follow the sequence of V_C < V_Cr < V_Al for Cr₂AlC, suggesting that the Al vacancy is not easily formed in Cr₂AlC, while the Cr vacancy becomes the most likely defect. A wide range of formation energies have been reported for the transition metal vacancies in the MAX phase materials. It has been reported the vacancy of Ti is unlikely to occur given its large formation energy.^38,39 With the various chemical potentials considered, Liao et al.¹⁰ concluded the E_def of V_Ti should be above 4 eV for Ti₂AlC. And with the atomic energy chosen as chemical potential, the V_Ti is 5.5 eV for Ti₃SiC₂. In contrast, the E_def of V_Cr is only ∼1 eV for Cr₂AlC, indicating a high concentration of Cr vacancies may occur under certain conditions.

Next, we consider the role of interstitial defects. The relaxed interstitial configurations are demonstrated in Fig. 3. It can be seen interstitial Cr and C are located at the octahedral canter formed by three Al and three Cr atoms. In contrast, the configuration of Al_i is very different, in which the interstitial Al incorporates into the Al atomic plane from its initial setting position after relaxation. The formation of Cr_i and Al_i both result in large relaxations of the lattice structures. For the chromium interstitial, the bond lengths between interstitial Cr and the nearest Al and Cr (2.26 and 2.22 Å) are much larger than the ones in interstitial carbon (1.99 and 1.98 Å). Interstitial Al incorporates into the Al atomic plane, 2.25 Å from its nearest neighbour Al atoms. Therefore, the Cr_i and Al_i have higher formation energies compared with C_i, as shown in Fig. 2 and Table S2.†

Fig. 3 Calculated atomic configurations for interstitial Cr (a), C (b), and Al (c) defects in Cr₂AlC. The topview of interstitial Al is shown in (d) for clarity. The bond lengths obtained by AFM+U (in-AFM1) calculations are indicated in angstroms, with the results by NM calculations indicated in brackets. The white, green and gray balls represent Cr, Al and C atoms, respectively.

For the antisite defects, we find E_def is greatly reduced compared with previous reported Ti-contained MAX phases. It can be noticed that some antisite defects even have negative formation energies. The negative values mean the defects are very easily formed, which may even be created spontaneously under certain experimental conditions.⁴¹ A negative value also occurs if a large chemical potential of the atomic species is chosen. To compare with previous results of other MAX phases and for simplicity, the chemical potentials of elementary substance are used in this work. To exclude the influence of the chemical potentials, the formation energies of defect pairs are calculated to verify our conclusion.

From the calculated point defect formation energies, the formation energies of Frenkel, Schottky and antisite pairs in Cr₂AlC can be obtained accordingly. Since the formation energies of defect pairs do not depend on the choice of the reference chemical potentials, it is more convenient to make comparison between different MAX phases. As shown in Table 1, it is clear that E_def of Cr–Al antisite pair is significantly lower than the previously reported values of other MAX phases. It is known that the formation of cation antisite pairs plays an important role in the irradiation tolerant properties of MAX phases. The main consequence of displacive radiation damage is the structural disorder caused by the accumulation of point defects. The possibility of accommodating structural disorder is a key factor to prevent amorphization. The calculated formation energies of cation antisite pair can be used to evaluate the resistance to radiation-induced amorphization.⁴² For MAX phases, the cation antisite pairs are related to the M–A antisite pairs, whose formation energy has been proved to be a reliable quantity to evaluate the radiation damage tolerance of MAX phases.^38,39 For example, Zhao et al.³⁹ calculated the formation energy of M–A antisite pairs in Ti₃AlC₂ and Ti₃SiC₂. They found Ti₃AlC₂ has a lower E_def of Ti–Al (3.18 eV) antisite pair than Ti₃SiC₂ (3.58 eV). Therefore, they conclude Ti₃AlC₂ should have a higher irradiation tolerance compared to Ti₃SiC₂, which is consistent with the experimental observations. In this work, we show the E_def of Cr–Al antisite pair in Cr₂AlC is only 1.9 eV, which is lower than Ti-contained system by 1.3–1.7 eV. The low formation energy of antisite Cr–Al pair suggests good irradiation tolerance. This conclusion is in agreement with our previous experimental work observed a saturation of irradiation effects, which are related to the generation of a high concentration of Cr–Al antisite pairs.⁷

Table 1 The formation energies (in eV) of the possible Frenkel, Schottky and antisite pairs in Cr₂AlC. The NM and AFM+U represent the results obtained by non-magnetic PBE calculation and PBE+U calculation with antiferromagnetic spin-ordering (in-AFM1), respectively

Defect	Equation	Denotation	NM	AFM+U
Frenkel	CrCr → VCr + Cri	CrFP	6.294	6.576
	AlAl → VAl + Ali	AlFP	6.524	6.638
	CC → VC + Ci	CFP	3.144	3.049
Antisite	CrCr + AlAl → CrAl + AlCr	CrAlAP	2.265	1.898
	CrCr + CC → CrC + CCr	CrCAP	9.480	9.395
	AlAl + CC → AlC + CAl	AlCAP	20.661	20.312
Schottky	2ICr + IAl + IC	Isch	15.348	17.230
	2VCr + VAl + VC	Vsch	6.906	5.610

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Next, we discuss the impact of the magnetic ordering and strong correlation effect on the defective properties of Cr₂AlC. Although the changes in the crystal structure induced by the AFM+U (in-AFM1) are negligible, the AFM+U approach alters the defect formation energy. Most importantly, it greatly reduces the formation energy of the Cr–Al antisite pairs, which is a key parameter to evaluate the radiation tolerance of Cr₂AlC. In order to reveal the origin of AFM+U effects on defect formation energy, the projected density of states (PDOS) and charge density analysis are carried out.

The deformation charge density is shown in Fig. 4(a), which is defined as the difference between the crystal charge and the atomic charge distribution. It is clear Cr and Al lose their electrons to C. There is a strong polarized covalent bond between Cr and C. With the magnetic ordering and strong correlation effect considered (AFM+U calculation), the distribution of the charge density changes. Fig. 4(b) shows the corresponding charge density redistribution, defined as Δρ = ρ^AFM − ρ^NM. Here ρ^AFM represents the total charge density of Cr₂AlC using the AFM+U calculation with in-AFM1 magnetic ordering. The red isosurface corresponds to an increase in electron accumulation, while the blue regions are a depletion of electron density. The blue distribution around the Cr atom has clear d_z² orbital character, while the red isosurface has d_xz + d_yz character. The anti-ferromagnetic ordering and the coulomb repulsion term, U, results in a strong charge transfer between the Cr-d_z² and C-p_z orbitals. Interestingly, a back-donation effect⁴³ is observed from the electron increase in Cr-d_xz + d_yz orbitals. The charge transfer from the Cr-d_z² states to the C-p_z states results in the effective ionic radius of Cr being lowered and becoming comparable to the ionic radius of Al. As a result, the inclusion of in-AFM1 ordering and an on-site Coulomb repulsion term lowers the formation energy of the Cr–Al antisite pairs.

Fig. 4 Deformation charge density (difference between the crystal charge and the atomic charge distribution ρ^diff = ρ^NM − ρ^atomic) (a) and the charge redistribution (ρ^diff = ρ^AFM − ρ^NM) by AFM+U calculation with in-AFM1 magnetic ordering (b) of perfect Cr₂AlC in (2, 1̄, 1̄, 0) plane. Contours are added with the interval 0.005 and 0.002 electrons per Bohr³ for (a) and (b), respectively.

The site and orbital resolved densities of state (DOS) are computed to illustrate the atomic interactions in Cr₂AlC as shown in Fig. 5. The DOS of Cr₂AlC at the Fermi level is dominated by the 3d states of Cr. The hybridized C-2p and Cr-3d states are located between −5 eV and −3.5 eV, which indicates strong p–d covalent bonding between Cr–C. The hybridization of the Cr-3d states and the Al-3p states occurs approximately 2 eV below the Fermi level, which suggests a relatively weaker bond compared to Cr–C. To further reveal the origin of the effect induced by AFM+U (in-AFM1) method. The density of states are projected to each orbital with the structure of Cr₂AlC is rotated to make the C–Cr bonding direction parallel to the direction of d_z² orbital. It is clearly shown in Fig. 6 that, different from other d-orbitals of Cr, there is a strong splitting in Cr-d_z² orbitals after AFM+U method introduced. The majority spin states of d_z² orbital of Cr are lowered in energy while the minority spin states are pushed above the Fermi lever, which makes the Cr more ionic in character. As mentioned above, the lower in the effective ionic radius of Cr should benefit the reduction in the formation energy of the Cr–Al antisite defect.

Fig. 5 The projected density of states of Cr₂AlC. The PDOSs of Cr-3d, Al-3p and C-2p orbitals are shown by blue, green, and red lines, respectively. The solid and dashed lines represent the NM and AFM+U (in-AFM1) calculations, respectively. The scale of Al-3p PDOS is magnified by three times for clarity. The Fermi-level is fixed at 0 eV.

Fig. 6 The projected density of states (PDOS) of Cr₂AlC for Cr-d and C-p orbitals. The structure of Cr₂AlC is rotated to make the C–Cr bonding direction parallel to the direction of d_z² orbital. The solid and dashed lines represent the PDOS by NM and AFM+U (in-AFM1) calculations, respectively. It is clear that a strong spin-splitting of d_z² is generated by AFM+U method indicated by green arrows, which does not occur for other d-orbitals.

The above analysis gives a qualitative understanding to the reduction in the formation energy of the Cr–Al antisite defect by AFM+U calculation. To further support our conclusion, the charge density redistribution of the Cr_Al and Al_Cr configuration is calculated using the same approach. As shown in Fig. 7, the electrons on the bonds of Al_Cr and the nearest Al or C are increased with AFM+U calculation, which indicates stronger interaction between them. In contrast, the interaction between the Cr_Al and surrounding Al atoms is not obviously enhanced. These results are consistent with the previous results in Table S3† that the decrease in CrAl^AP formation energy is mainly corresponding to the decrease in formation energy of Al_Cr defect.

Fig. 7 The charge density differences between results by NM and AFM+U calculation of (a) Al_Cr defects in (1̄, 1, 0, 1) plane and (b) Cr_Al defect in (0,0,0,1) plane. The Cr, Al and C atoms in these planes are plotted out. The red and blue isosurfaces (±5 × 10⁻⁴ electrons per Bohr³) correspond to electron increase and depletion zone, respectively. Contours are added with the interval 1 × 10⁻⁴ electrons per Bohr³. In order to make the charge differences, the structures are fixed to be the relaxed structures by AFM+U (in-AFM1) calculation. The difference is negligible when we use the structures by NM calculation.

At last, since magnetism has been proved to be important in simulation of Cr₂AlC,¹⁷ more magnetic structures are calculated to study their influence on the defect formation energies. Here, we use the same magnetic structures as in ref. 17. As shown in Fig. 8, they are FM, AFM[0001]^A₂, AFM[0001]^X₂, AFM[0001]₁, in-AFM1 and in-AFM2, which we noted as (a)–(f) for clarity.

Fig. 8 Six magnetic structures considered for Cr₂AlC in this work according to the previous work by Dahlqvist et al.¹⁷ The white, green and gray balls represent Cr, Al and C atoms, respectively. Arrows on Cr atoms illustrate the spin directions.

Following the same approach, the formation energies of different point defects (Table S4†) and defect pairs (Table 2) can be obtained. Here, in order to study the influence of magnetic ordering on the defect formation energy, the calculation is carried out without Hubbard U added. As shown in Table 2, it is obvious that the formation energy varies greatly from one calculation to the next depending on the magnetic structure. For example, the formation energy of Cr Frenkel pairs defect is 6.46 eV for (e), while 5.75 eV for (f). Although the difference is large between different magnetic structures, it can be found a consistent trend in the formation energies, which is CrAl_AP < C_FP < Cr_FP < Al_FP < V_sch < CrC_AP < I_sch < AlC_AP. Therefore, the choice of magnetic structures does not change the conclusion that the Cr–Al antisite pairs and C Frenkel pairs should be the most easily formed in Cr₂AlC, which is different from the Ti-based MAX phases. To summarise, a NM calculation is likely to be sufficient to obtain the trend in the defect formation energy. However, a calculation with magnetic structure and strong correlation effect considered is recommended for a high precision calculation to obtain a reliable value of defect formation energy for Cr-based MAX phases.

Table 2 The formation energies (in eV, U = 0 eV) of the possible Frenkel, Schottky and antisite pairs in Cr₂AlC with different magnetic structures illustrated in Fig. 7

Denotation	NM	(a)	(b)	(c)	(d)	(e)	(f)
CrFP	6.29	5.96	6.32	6.30	6.46	6.37	5.75
AlFP	6.52	6.01	6.75	6.68	7.38	6.65	6.17
CFP	3.14	2.68	3.47	3.44	3.62	3.35	3.08
CrAlAP	2.27	1.58	2.04	2.20	2.36	2.11	1.58
CrCAP	9.48	9.16	7.95	8.01	8.15	9.59	7.14
AlCAP	20.7	20.2	20.5	20.8	20.4	20.7	19.5
Isch	15.3	14.7	15.1	15.2	15.8	15.2	14.2
Vsch	6.91	5.95	7.77	7.51	8.08	7.56	6.59

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Conclusions

In summary, we have systematically studied the properties of intrinsic point defects in Cr₂AlC and demonstrated that they are different from the conventional Ti-based MAX phases. We find Al vacancies have high formation energies and hence are unlikely to form in Cr₂AlC. The lowest formation energy occurs for the cation antisite defect. The formation energy of the Cr–Al antisite defect pairs is only 1.90 eV, providing a substitution mechanism to accommodate the defects create by irradiation. Therefore, it should be reasonable to predict that Cr₂AlC has a higher irradiation resistance than previously reported MAX phases. The magnetic ordering and strong correlation effect of Cr atoms are considered by AFM+U calculation. The formation energies of the antisite defects of Cr₂AlC are decreased by AFM+U calculation. The charge density redistribution shows that by using AFM+U calculation, the interactions between the antisite anions and their surrounding atoms are enhanced. The mechanism is explained by the minority spin of the Cr-d_z² orbital being pushed above the Fermi level which lowers the effective ionic radius of Cr. The calculation with different magnetic structures indicates that the magnetic ordering is important for a high precision calculation for defect formation energies of Cr₂AlC. These results may help in understanding the magnetic ordering and strong correlation effects present in Cr₂AlC and other Cr containing MAX phase materials.

Acknowledgements

This work was supported by the Program of International S&T Cooperation (Grant No. 2014DFG60230), National Natural Science Foundation of China (No. 91326105, U1404111, 11504089, 21501189, 21571185, 21306220, 11605273), the Shanghai Municipal Science and Technology Commission (16ZR1443100), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA02040104). We also thank the Supercomputing Center of Chinese Academy of Sciences (SCCAS) and the Shanghai Supercomputing Center for computer resources.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available: The calculated structural and mechanical parameters of Cr₂AlC compared with experimental data; the defect formation energies of interstitials with various positions; the detailed data for defect formation energies in Cr₂AlC; the projected density of states of Cr₂AlC for Cr-3d and C-2p orbitals; the detailed data for defect formation energies of Cr₂AlC with different magnetic structures. See DOI: 10.1039/c6ra15366f

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