Magnetic moment collapse induced axial alternative compressibility of Cr2TiAlC2 at 420 GPa from first principle

The electronic structure and thermodynamical properties of Cr2TiAlC2 are studied by first principles under pressure. The obtained results observed that the ferromagnetic order is the most stable ground state and the magnetic moment will collapse at about 50 GPa. As a result, the lattice a axis becomes stiffer above about 420 GPa, ultimately presenting the same axial compressibility trends with those of nonmagnetic compounds Mo2TiAlC2 and hypothetical Cr2TiAlC2. The elastic constants and phonon dispersion curves demonstrate the structural stability during the disappearance of magnetic moment and occurrence of axial alternative compressibility. The density of states and energy band calculations confirmed the existence of magnetic moment of Cr2TiAlC2 at 0 GPa and disappearance at high pressures above 50 GPa. Evolutions of magnetic moment collapse with pressure are confirmed by a variety of properties. The obtained grüneisen parameter and thermal expansion coefficients show the maximum value among the known MAX phases, to date and to the author’s knowledge.


Results and Discussion
Structural properties. Previous investigations for Cr 2 AlC, Cr 2 GaC, and Cr 2 GeC 27 demonstrated these Cr-containing MAX compounds are weak correlated materials, if applicable it should be U < 1 eV and therefore it is unnecessary to use LDA + U method (localized density approximation), our detailed test calculations for its usability are shown in Table S1. The atomic arrangement and Brillouin zone (BZ) orientation of Cr 2 TiAlC 2 are same 9 with that of Mo 2 TiAlC 2 . Therefore, several collinear magnetic configurations of Cr 2 TiAlC 2 are considered to search for its ground state, including non-magnetic(NM), ferromagnetic (FM), various antiferromagnetic (AFM), and two in-plane AFM, in which no correlation effect of the Cr d electrons is applied. These magnetic orders are same 13 with that of Mo 2 GaC, we here ignore the schematic illustrations for simplicity. Relative to the minimum energy of the NM unit cell, the comparison for all the considered magnetic states of Cr 2 TiAlC 2 is presented in Figure S1. With the results of Table S2 and Figure S1, it is reasonable to consider FM configuration as the ground phase, consisting with the theoretical calculations 1 .
Usually, the contraction of c axis under pressure is faster than that of a axis, such as NM Mo 2 TiAlC 2 9 , whereas our simulations observe that the compressibility along a axis is faster than that of c axis within 0~420 GPa in FM Cr 2 TiAlC 2 . Interestingly, an alternative case occurs with pressure continuous increasing, as is shown in Fig. 1. For comparison purpose we also simulate the structural evolution for the hypothetical NM Cr 2 TiAlC 2 and find that the two axial compressibilities are almost identical below about 100 GPa, henceforth the stiffer a axis occurs. The detailed structural evolutions for FM/NM Cr 2 TiAlC 2 , and NM Mo 2 TiAlC 2 are shown in Table S3. The rapidest shift of Cr atom in FM Cr 2 TiAlC 2 among the three compounds might resist the c-axis contraction. The NM Cr 2 TiAlC 2 present almost identical c-axis contractions with that of NM Mo 2 TiAlC 2 , the slightly larger a-axis compressibility of NM Cr 2 TiAlC 2 might originates from its smaller atomic radials. The magnetic moment definitely decelerates the c-axis but accelerates the a-axis contractions in FM Cr 2 TiAlC 2 in comparison with those of NM Cr 2 TiAlC 2 . In addition, the volumetric compressibility of NM Mo 2 TiAlC 2 is evidently smaller than that of NM Cr 2 TiAlC 2 due probably to the larger Mo radius, the largest volumetric compressibility of FM Cr 2 TiAlC 2 due mainly to the fact of its easily compressed a axis.
Our calculated elastic constants are listed in Figs 2 and S2. Using the stability criteria 28 , i.e. It is found that FM Cr 2 TiAlC 2 is mechanical stable up to 500 GPa. To confirm this conclusion, we calculate the phonon spectra at 0, 420, and 500 GPa by the finite displacement method using a cutoff radium of 5 Å with a supercell volume of 16 times larger than the unit cell under the same precision settings with that of geometrical relaxation, as is shown in Figure S3. The results indicate that FM Cr 2 TiAlC 2 is dynamic stable within 0~500 GPa. The calculated elastic constants of FM/NM Cr 2 TiAlC 2 and NM Mo 2 TiAlC 2 9 are shown in Fig. 2, in which the discrepancy between the FM/NM Cr 2 TiAlC 2 tends to zero with pressure increasing to about 40 GPa, suggesting that the magnetic moment are totally collapsed.
Previous calculations for Cr 2 GeC 29 observed a magnetic moment collapse phenomenon at about 25 GPa, which is far smaller than the present 50 GPa, indicating that the Ti atom probably resists the collapse process and therefore could stabilize the lattice 9 , as is shown in Fig. 3. In addition, it is about 40 GPa in Cr 2 AlC from our simulations. The difficult collapse evolution possibly means the existence of more complex intermediate transition  magnetic configurations. This magnetic transition (from FM to NM) of Cr 2 TiAlC 2 should be the internal driven force of axial alternative compressibility at about 420 GPa.
Previous measurements 30,31 confirmed that Cr 2 GeC and Cr 2 AlC exist net magnetic moments and the magnitudes nearly vanish at about 100 K, whereas the crude estimation of the magnetic moment of per Cr atom is 0.05 30 /0.02 31 μ B in Cr 2 AlC and 0.02 30 μ B in Cr 2 GeC, far smaller than available calculations such as FM Cr 2 AlC 32 (0.9 μ B at U = 1.95 and 2.5 μ B at U = 2.95), AFM Cr 2 AlC 33 (0.7 μ B ), AFM Cr 2 GeC 34 (1.4 μ B ). However other measurement 35 for AFM Cr 2 AlC obtained a value of 0.64 μ B . Our computed sum of the absolute magnetic moment of spin-up and spin-down directions is 5.3749 μ B in AFM Cr 2 AlC in one unit cell (four Cr atoms, 0.6719 μ B per Cr atom), consisting well with previous calculations 27  The difference between the present FM Cr 2 TiAlC 2 and AFM Cr 2 AlC is about 0.13 μ B , whereas they are nearly identical in a recent calculation 38 , with values of 0.99 μ B in FM Cr 2 TiAlC 2 and 1.0 μ B in AFM Cr 2 AlC, respectively. The slab of Ti-C between the two nearest Cr 2 AlC stack blocks in unit cell of Cr 2 TiAlC 2 possibly affect the atomic moment as the Ti atom could strongly stabilize the unavailable Cr 3 AlC 2 39 .
The calculated bond length compressibility and bond population of FM Cr 2 TiAlC 2 and NM Mo 2 TiAlC 2 are shown in Figures S4 and 5, through which a clear correlation between the bond population of Al-Cr and a-axis stiffening above 420 GPa is undoubtedly seen. The abnormal increase of bond population in Al-Cr bond above 420 GPa means that the antibonding interaction becomes stronger again after it approaches the first local minimum. To deep understand the axial compressibility of FM Cr 2 TiAlC 2 , we systematically studied the bond rotation angle variations under pressure and the obtained results are shown in Table S4. Apart from the contributions of the bond nature to the axial compressibility, the number of bonds rotating towards orientation with increasingly larger projection angle to the ab plane is also more than the cases of opposite rotations, i.e., three (C-Cr, C-Ti, and Cr-Ti) versus one (Al-Cr). The three rotations contribute larger to c axis than to a axis. Owing to the rapid immigration of Cr atom along c axis, all of the four bonds rotate far larger angles in FM Cr 2 TiAlC 2 than those of NM Mo 2 TiAlC 2 , particularly for the low pressure range below 50 GPa. These behaviors might be one reasonable explanation to the stiffer c axis of FM Cr 2 TiAlC 2 at low pressure.
For understanding the chemical bonding of FM Cr 2 TiAlC 2 , we calculate the band structure and density of states (DOS) at 0 and 50 GPa, respectively, comprising of spin-up (alpha) and spin-down (beta) components, as is seen in Figs 4 and 5. These spin-up/down levels shift their profiles towards higher/lower-energy sides as a whole. However, as far as the orbitals with energies crossing the Fermi level are concerned, the average covered energy ranges (the broadened width of each orbital) of all the spin-down orbitals decrease with pressure below about 50 GPa firstly and increase with pressure above 50 GPa subsequently, as is shown in Figure S6, which also provides correlation to the magnetic moment collapse behavior at about 50 GPa. Still, the minimum range of average covered beta orbital approximately equals to the energy threshold of alpha orbital at 0 GPa. In addition, the variation slopes of the two different curves present synchrotron responses to the external compression below 50 GPa, explaining the existence of magnetic moment below 50 GPa. The two values ultimately become equal to each other at 500 GPa, demonstrating the fact that driving the spin orbital overlap is extremely difficult. This is also the reason why the axial alternative compressibility happens at extreme high pressure (about 420 GPa) other than at about 50 GPa.
To explore the shift trend of the whole beta orbitals under pressure, we further test all of the orbitals crossing the Fermi level, as is shown in Figure S7. As expected, the Cr 3d dominant orbitals present substantial discrepancies between alpha and beta spin orbitals at 0 GPa, whereas such discrepancies are rapidly reduced under pressure. Their contrary shift tendencies illustrated the energy evolution shown in Figure S6.
Both the t 2g and e g states are half-filled, whereas the low-spin states provide six more unoccupied orbitals and high-spin states fill in five more valence orbitals and therefore they provide the net magnetic to the system. Analysis to all the orbitals observed that the low-lying energy levels are contributed mainly by Cr 3d (t 2g ) states in the vicinity of Fermi level, whereas the high-lying energy levels are composed mainly by Cr 3d (e g ). Energy levels crossing Fermi level present highly hybridized characters. Energy levels sited in the conduction band are contributed mainly by Cr 3d (e g ) and Ti 3d (e g ) states. These distributions consist with the bond population features shown in Figure S5 such as the antibonding Cr-Ti population.
The values (per unit cell) of the spin-up DOS at Fermi level are 4.74, 4.27, 3.51, 1.08 states/eV for 0, 20, 50, and 500 GPa, respectively. The correspondent values of spin-down counterparts are − 2.41, − 2.73, − 3.39, and − 1.08 states/eV. The total DOS value 9 is 5.58 states/eV at 0 GPa in NM Mo 2 TiAlC 2 , far smaller than the current 7.15 states/eV of FM Cr 2 TiAlC 2 , denoting the significant influence of magnetic moment. In Figure S8, the energy band of FM Cr 2 TiAlC 2 , particularly for its spin-up components, present similar metallic and energy-level features with that of NM Mo 2 TiAlC 2 at 0 GPa 9 , indicating the possibility of whole disappearance of the magnetic influence under pressure.
Electron density difference (EDD) shown in Fig. 6 correlates well with that of Mulliken charge variations. C atom gains charges (− 0.59 e) mainly from Ti (0.81 e) and Al (0.32 e) atoms, respectively. Moreover, Cr atom loses its minor charges (0.03 e) at ambient conditions in FM Cr 2 TiAlC 2 . However the number of charges in C atoms keeps almost unchanged even under higher pressure, which is important to sustain the extensive stability of C-Cr and C-Ti bond populations under pressure. The Al atoms lose its charge firstly and gain charge from others subsequently under pressure, which is just contrary with that of Cr atom, whereas Ti atoms monotonically lose its charges with pressure. These charge transfer direction forms significantly larger bond populations along Al-Cr orientation and relatively small values along C-related bonds including C-Cr and C-Ti, which probably means the strong charge saturation of C atoms.
Electron localization function (ELF) generally reflects the general and total orbital bonding features and the charge transfer trends as well as the atomic charge distributions, as is shown in Fig. 7. The C atom attracts substantial charges around it and builds polarized bonding along C-Cr direction with ionic dominance and partial covalent participation character. At pressures below 50 GPa, the charge distributions of Cr d states present strong anisotropy feature, whereas such feature rapidly decreased under pressure, indicating the strong hybridization of Cr d states. Moreover, the partial-filling feature of Cr d states is undoubtedly shown, such as the prominent t 2g (d xy ) at low pressure and e g (d z 2) at high pressure. However, less Mo 3d orbital feature is discernible 9 in NM  Under pressure, the giant reduction of ELF value in NM Mo 2 TiAlC 2 9 means the weakening of bonding at 20 GPa. With pressure continuous increasing, the inter-atomic bonding of NM Mo 2 TiAlC 2 behaves similar variations with those of FM Cr 2 TiAlC 2 . The large electron localization could be seen in the region between adjacent atoms in NM Mo 2 TiAlC 2 , indicative of nearly completely filled and slightly stronger covalent bonding, consisting well with its relatively larger mechanical quantities. Both FM Cr 2 TiAlC 2 and NM Mo 2 TiAlC 2 show similar polarized bonding features with directional orientations around the C atoms at high pressure, which means the anisotropic bonding characters and different chemical bonding styles. The small ELF value between C and Cr and the nearly spherically symmetry distribution of Cr sites in FM Cr 2 TiAlC 2 demonstrate the mixture bonding character. A directional bonding between Cr and Al is polarized towards the Cr sites judged from an arc shape. There is a maximum value between C and Ti, indicative of covalent bonding. The nearly square distribution around C means its partially ionic constitutions. In addition, a predominantly antibonding orbital along Cr-Ti is built.
The calculated Fermi surfaces (FS) of orbitals crossing Fermi level are shown in Figure S9. The FS evolution provides a direct evidence for the magnetic moment collapse under pressure. Alpha orbitals with higher energies locate at nearer distance from the center of the Brillouin zone (BZ), whereas beta orbitals with higher energies locate at farer distance from the center of BZ. In particular, the spatial orientations of the alpha and beta orbitals are extremely different. This is the typical anisotropy of the energy level distribution features, meaning the existence of the residual net magnetic moment.
The whole FS profiles of orbitals 51 and 52 are shifted to first BZ at 20 GPa. Meanwhile, the higher energy of the orbital exists, the nearer distance of its FS to the center of the first BZ will be, which is just contrary to the case of beta spin at 0 GPa. Orbitals 49 and 50, with beta spin belonging to higher-energy orbitals in comparison with the other orbitals crossing Fermi level, site farer from the center of the first BZ relative to the other ones at 0 GPa, whereas they transform to the lower-energy orbitals with same orbital sites at 20 GPa, meaning the huge shift of the beta orbital towards lower-energy side under 20 GPa. Such orbital energy shift is about 0.6 eV, far larger than the case of alpha orbital which shows a global shift of about 0.2 eV towards higher-energy side. The just opposite shift tendencies will cause more overlap between them and ultimately induce them are totally overlapped at 500 GPa. Despite the magnetic moment has already been disappeared at 50 GPa, the orbital energies and the FS profiles are not totally merged, still existing an energy discrepancy of about 0.1 eV between the different spin orbitals.
Elastic properties. The small c 33 means that the c-axis direction is relatively soft, which is partially inconsistent with the axial compressibility. An evident dip of c 33 corresponds to the c-axis softening above 420 GPa in FM Cr 2 TiAlC 2 , as is shown in Figure S2. Anisotropic parameter A = c 33 /c 11 , A = 1, denoting isotropic crystal, any ratio higher or lower than 1 means an elastic anisotropy. The obtained A are 1.1671 at 0 GPa and 0.9978 at 500 GPa, respectively. The   For covalent and ionic compounds, the relationships between bulk (B) and shear (G) moduli are G ≈ 1.1B and G ≈ 0.6B, respectively. For FM Cr 2 TiAlC 2 the simulated values of G/B are 0.8643 at 0 GPa, 0.4576 at 100 GPa, and 0.2977 at 500 GPa, respectively, indicating that the mixed bonding is suitable for FM Cr 2 TiAlC 2 at 0 GPa and the degree of ionicity increases with pressure. All of these values are larger than their counterparts in NM Mo 2 TiAlC 2 , corresponding to 0.715 at 0 GPa and 0.403 at 100 GPa, respectively 9 . However, the comparable values of 0.5146 in NM Mo 2 TiAlC 2 9 and 0.5187 in FM Cr 2 TiAlC 2 at 50 GPa mean that the pressure could effectively tune the chemical bonds of magnetic materials.
Pugh et al. use the B/G ratio 41 to estimate crystal ductility (< 0.57) or brittleness (< 1.75). The FM Cr 2 TiAlC 2 is brittle (G/B = 0.8643) at 0 GPa, and the degree of brittleness increases with pressure, such as the values are 0.4576 at 100 GPa and 0.2977 at 500 GPa, respectively. However those values are 0.715 at 0 GPa and 0.4031 at 100 GPa in NM Mo 2 TiAlC 2 9 , demonstrating that the FM Cr 2 TiAlC 2 is more brittleness owing to the appearance of the magnetic moment. Moreover, the current B/C 44 is 1.2068, which is also slightly smaller than that of 1.44 9 of NM Mo 2 TiAlC 2 but just within the range (1.2-1.7) of M n+1 AX n phases.
The Poisson's ratios are 0.1634 (0 GPa), 0.3015 (100 GPa), and 0.3646 (500 GPa) in FM Cr 2 TiAlC 2 , respectively, far smaller 9 than those of 0.2667 (0 GPa) and 0.3423 (100 GPa) in NM Mo 2 TiAlC 2 , meaning that the FM Cr 2 TiAlC 2 is covalent and ionic materials at 0 GPa, and a mixed bonding with partial metallic and certain ionic participation combination could be assumed at higher pressure. The Poisson's ratios of NM/FM Cr 2 TiAlC 2 present similar variations with the only exception of lower counterparts within 10~40 GPa as there is an obvious reduction in FM one, which could be attributed to the influence of magnetic moment. Thermodynamical properties. Several thermodynamical properties are studied for FM Cr 2 TiAlC 2 by the quasi-harmonic Debye model, the calculation details 42 could be obtained elsewhere. Grüneisen parameter γ characterizes the anharmonicity of lattice, the calculated γ are shown in Figures S11 and 12, in which the γ of FM Cr 2 TiAlC 2 is far larger than that of NM 9 Mo 2 TiAlC 2 , particularly at low pressure, such difference is obviously decreased with pressure. In Figure S11, the inserted small figure means the values of FM Cr 2 TiAlC 2 and NM 9 Mo 2 TiAlC 2 at 0 GPa within 0~1500 K, the significantly larger value of Cr-containing compound represents the stronger phonon-phonon interaction originating probably from the effect of magnetism at low temperature, whereas the value of γ approaches its maximum limit at about 600~700 K and then decreases gradually with temperature in FM Cr 2 TiAlC 2 , such inverse variation indicates probably the opposite strength response of the phonon-phonon interaction with the volume change. Previous calculations for Ti 2 SC 43 also detected such decreasing trend at 0 GPa within 0-2000 K. The general variation trends of γ in FM Cr 2 TiAlC 2 are similar with those 9 of NM Mo 2 TiAlC 2 and thus we here neglect the detailed discussions. Generally, the values of γ are within 1.5~2 44 , previous calculations 44 for several MAX phases found that Cr 2 GeC presents largest γ with a value of 2.38, which is still smaller than the present FM Cr 2 TiAlC 2 which is 2.63 at 0 GPa and 0 K, meaning that the present compound behaves the largest γ among all known MAX phases till now, to the author's knowledge.
The variation trend of the thermal expansion coefficient α with temperature and pressure of FM Cr 2 TiAlC 2 is generally the same with that 9 of NM Mo 2 TiAlC 2 , as are shown in Figures S13 and 14. The α approaches its upper limit at about 1200 K in FM Cr 2 TiAlC 2 , with a value of 3.5343 × 10 −5 × K −1 , which is far larger than those of counterparts 9 of NM Mo 2 TiAlC 2 , with a maximum value of 2.064 × 10 −5 × K −1 at 1500 K. Such larger α can also be found in other Cr-containing MAX compounds, such as Cr 2 GaC and Cr 2 AlC 45 , with respective values are 2.9885 × 10 −5 × K −1 and 2.4312 × 10 −5 × K −1 at 1500 K. Moreover, previous calculations for Ti 2 SC 43 still found a peak in such evolution curve, with a maximum value of 1.8909 × 10 −5 × K −1 K at 900 K, and it decreases to 1.8007 × 10 −5 × K −1 at 1500 K. The detailed summary of α and other coefficients could be found in a recent review 46 . The present α probably is the largest value among all the MAX phases.
The isothermal (B T ) and adiabatic (B S ) bulk moduli present similar variation trends with those 9 of NM Mo 2 TiAlC 2 with the only exception of 0 GPa, as is shown in Figure S15 and the inserted figure. Usually, B T and B S exist small difference owing to the small α and γ, B S = B T (1 + αγT), where T means temperature. Variations of the discrepancy in (Mo 2 TiAlC 2 -Cr 2 TiAlC 2 ) for B T and B S present just opposite variation trends between 0 GPa and higher pressures because the novel variation trends of FM Cr 2 TiAlC 2 (inserted the small figure), which approaches the minimum value 130.63 GPa at about 1400 K in B T and 144.08 GPa at 800 K in B S . Such nonlinear variation is closely related to the variations of α and γ. Previous calculations for Ti 2 SC 43 also observed such phenomenon in B T and B S .

Conclusion
The magnetic moment collapse induced axial alternative compressibility in FM Cr 2 TiAlC 2 at 420 GPa is observed for the first time in this family of compounds. The correctness of this conclusion could be evidenced by the electronic and mechanical properties. The strong influence of the magnetic moment caused many excellent thermodynamic properties. The implication of the current investigation is that both the spin transition and the charge rearrangement could be adjusted by high pressure. The ferromagnetic moment collapse has crossed a series of antiferromagnetic order state and stabilized ultimately at a nonmagnetic order state.