Structural , Elastic , and Electronic Properties of Antiperovskite Chromium-Based Carbides ACCr 3 ( A = Al and Ga )

We theoretically investigated antiperovskite chromium-based carbides ACCr3 through the �rst-principles calculation based on density functional theory (DFT). e structure optimization shows that the lattice parameter of ACCr3 is basically proportional to the radius of A-site elements. e calculated formation energies show that AlCCr3 and GaCCr3 can be synthesized at ambient pressure and are stable with nonmagnetic ground states. Based on the calculation of elastic constants, some elastic, mechanical, and thermal parameters are derived and discussed. AlCCr3 and GaCCr3 show ductile natures andmay have similar thermal properties. From the analysis of the electronic structures, it was found that there are electron and hole bands that cross the Fermi level for AlCCr3 and GaCCr3, indicating multiple-band natures. e Fermi level locates at the vicinity of the density of states (DOSs) peak, which leads to a largeDOS at Fermi level dominated byCr-3d electrons.e band structures of AlCCr3 andGaCCr3 are very similar to those of the superconducting antiperovskite MgCNi3. e similarity may make AlCCr3 and GaCCr3 behave superconductively, which needs to be further investigated in theoretical and experimental studies.


Introduction
Recently, antiperovskite compounds AXM 3 (A, main group elements; X, carbon, boron, or nitrogen; M, transition metal) have attracted considerable attention.Due to the high concentration of transition metals in a cell, it can be deduced that interesting properties will be found in the family of compounds.In the antiperovskites family, nickelbased and manganese-based antiperovskites were extensively studied.Abundant physical properties were shown in the two kinds of compounds, such as superconductivity [1][2][3], giant magnetoresistance (MR) [4,5], large negative magnetocaloric effect (MCE) [6,7], giant negative thermal expansion [8,9], magnetostriction [10], and nearly zero temperature coef-�cient of resistivity [11,12].But there are only a few reports about other 3d-metal-based antiperovskites so far.e difficulty restricting researchers is the exploration of new materials that can be experimentally synthesized.erefore, theoretical investigations on these potential 3d-metal-based antiperovskites are useful to �nd the easily prepared stable materials and explore the new physical properties.
In the earlier years, researchers have found that in chromium compounds there are varieties of interesting physical properties.Many of chromium alloys such as Cr-Ru, Cr-Rh, and Cr-Ir alloys show superconductivity [13].And it was found there is spin density wave antiferromagnetism that coexists with superconductivity in Cr 1− Re  (  1 [14], Cr 1− Ru  (  1 [15], and (Cr 1− Mo   5 Ru 5 (  , 0.03, 0.06, and 0.10) [16].For chromium-based antiperovskites, Wiendlocha et al. [17] discussed the possibility of superconductivity in GaNCr 3 and RhNCr 3 , and recent phonon and electron-phonon coupling calculation of RhNCr 3 [18] supports Wiendlocha et al. 's prediction.In the present work, we theoretically investigated antiperovskite chromium-based carbides ACCr 3 through the �rst-principles calculation based on density functional theory (DFT).e optimized lattice parameter of ACCr 3 is basically proportional to the radius of A-site elements.From the analysis of formation energies, we predict that only AlCCr 3 and GaCCr 3 can be synthesized at ambient pressure, and they are stable with nonmagnetic ground states.e elastic and electronic properties of the two compounds are speci�cally discussed.

Computational Details
e calculations were performed by projected augmentedwave (PAW) [19,20] method using the ABINIT code [21][22][23].e PAW method can lead to very accurate results comparable to other all-electron methods.For the exchangecorrelation functional, the generalized gradient approximation (GGA) according to the Perdew-Burke-Ernzerhof [24] parametrization was used.Electronic wavefunctions are expanded with plane waves up to an energy cutoff of 1200 eV.Brillouin zone sampling is performed on the Monkhorst-Pack (MP) mesh [25] of 16 × 16 × 16. e self-consistent calculations were considered to be converged when the total energy of the system was stable within 1 −6 Ha.Nonmagnetic, ferromagnetic (FM), and antiferromagnetic (AFM) states were tested in the study.For AFM states, we only considered the simplest case: the spins of Cr atom in [111] layers are parallel with each other in the same layer and are antiparallel with spins in the neighboring layers.e elastic constants are evaluated according to the �nitestrain continuum elasticity theory [26,27].e strain energy per unit mass can be expressed as a polynomial of the strain parameter  [28]: where the coefficients  2 and  3 are combinations of secondand third-order elastic constants of the crystal, respectively.For a cubic structure, there are three independent secondorder elastic constants ( 11 ,  12 , and  44 ).erefore, here we introduced 3 Lagrangian strain tensors labeled as   in terms of .  and the corresponding coefficients  2 are listed in Table 1.
Once the single-crystal elastic constants are calculated, the related elastic and mechanic parameters may also be evaluated.Using the Voigt-Reuss-Hill approximation [29], bulk modulus  and shear modulus  of cubic crystal can be de�ned as where the subscripts  and  denote the Voigt, and Reuss average, respectively.In ( 3 us Young's modulus  and Poisson's ratio  can be obtained using the relations Using the calculated bulk modulus , shear modulus , and Young's modulus , the Debye temperature Θ  can be determined as where ℎ,   , and   are Planck's, Boltzmann's constants, and Avogadro's number, respectively. is the mass density,  is the molecular weight, and  is the number of atoms in the unit cell.e mean sound velocity is de�ned as where   and   are the longitudinal and transverse sound velocities, which can be obtained from bulk modulus  and shear modulus :

Ground State and Formation
Energy.e structures were fully optimized with respect to lattice parameter and atomic positions.e optimized lattice parameters of ACCr 3 for different A-site elements are listed in Table 2. e results show that the lattice parameter of ACCr 3 is basically proportional to the radius of A-site elements (see Figure 1).e common technique for producing antiperovskite carbides is the solid-state synthesis from the parent materials.erefore we assumed that the antiperovskite chromiumbased carbides ACCr 3 could be synthesized in the same routine.We analyze the stability of compounds by calculating the formation energies Δ, which is de�ned as the difference between the total energies of initial reagents (elements) and �nal compounds: Here,  A tot ,  C tot , and  Cr tot are the total energies of the initial reagents.
According to the de�nition of Δ, negative values indicate that the formation of compounds from the initial components is energetically favorable and the compounds are stable with respect to a mixture of the initial elements.Conversely, if Δ  , the compounds are unlikely to be synthesized under normal conditions.e calculated Δ for the series of A-site elements with different magnetic con�gurations are listed in Table 2. Except for AlCCr 3 and GaCCr 3 , all other compounds show positive Δ.For AlCCr 3 and GaCCr 3 , the values of Δ for different magnetic con�gurations are very close, and the magnetic moments per Cr atom for the FM and AFM states are very small, which implies the nonmagnetic ground states of AlCCr 3 and GaCCr 3 .From the analysis of formation energies, it seems that ACCr 3 can be synthesized only when A = Al and Ga.e elastic and electronic properties of the two compounds will be discussed later.

Elastic and Mechanical Properties.
In order to obtain accurate elastic constants,  is varied with a step of 0.0025 in every case for   .As an example, the strain energies for AlCCr 3 and the �tted polynomials are shown in Figure 2. e calculated elastic constants at ambient pressure and other derived mechanical parameters are listed in Table 3. From the calculated values of the elastic constants, one can �nd that they satisfy the mechanical stability criteria for a cubic crystal, that is,  11   12 ,  44  , and  11 + 2 12   [30], which indicates that the compounds are mechanically stable.Cracks in crystals are directly related to the anisotropy of thermal and elastic properties [31].e elastic anisotropy factor  of cubic crystal can be described as For a complete isotropic system, the anisotropy factor  takes the value of unity, and the deviation from unity measures the degree of elastic anisotropy [32].e elastic constant  44 represents the shear resistance in a [100] direction, which is related to the C-Cr bonding.e values of  44 for the two compounds are almost the same, which implies the similar C-Cr bonding nature of the two compounds. 11 −  12 turns out to be the stiffness associated with a shear in a [110] direction, which can be connected with the bonding between A-Cr bonding.Although A-M bonding is usually very weak in antiperovskite ACM 3 , it still can in�uence the anisotropy of the material.e anisotropic factors  are listed in Table 3. e different A-Cr bondings of the two compounds lead to different anisotropy.For AlCCr 3 ,   .1 indicates that AlCCr 3 is almost isotropic.However, GaCCr 3 seems like more anisotropic (  .).
Young's modulus  de�nes the ratio between the linear stress and strain.e larger the value of , the stiffer is the material.In general, as Young's modulus increases, the covalent nature of the compound also increases, which further has an impact on the ductility of the compounds.e values of  for AlCCr 3 and GaCCr 3 are very close to each other, suggesting the similar covalent nature of the two compounds.
e Cauchy's pressure   is de�ned as the difference between the two particular elastic constants     12 −  44 .It is considered to serve as an indication of ductility:   >  suggests that the material is expected to be ductile; otherwise, the material is expected to be brittle [33].As shown in Table 3, the values of   for AlCCr 3 and GaCCr 3 are positive, which is a clear indication for the compounds to be ductile.Another index of ductility is the ratio .According to Pugh [34], the ductile/brittle properties of materials could be related empirically to the ratio .If  > 1., the material would be ductile; otherwise the material behaves in a brittle manner.For ACCr 3 (A = Al and Ga), all the calculated  ratios are much larger than 1.75, which clearly highlights the ductile nature of ACCr 3 (A = Al and Ga).Poisson's ratio generally quanti�es the stability of the crystal against shear and takes the value between −1 and 0.5, which are the lower and the upper bounds.e lower bound is where the material does not change its shape, and the upper bound is where the volume remains unchanged.e calculated  of ACCr 3 (A = Al and Ga), listed in Table 3, are very close to the typical  value of 0.33 for ductile metallic materials [35].All the parameters clearly show the ductility of ACCr 3 (A = Al and Ga).
Using bulk modulus , shear modulus , and Young's modulus , the Debye temperature Θ  can be calculated.As a matter of fact, a higher Θ  would imply a higher thermal conductivity associated with the material [36].For the present calculation, the Θ  are estimated to be 532.43K and 497.12K for AlCCr 3 and GaCCr 3 , respectively.e similar values of Θ  suggest the similar thermal characteristics of the two compounds.

Electronic Properties. e calculated electronic band structures along the high symmetry directions in the
Brillouin zones together with the total and site-projected -decomposed DOS at equilibrium lattice parameters for ACCr 3 (A = Al and Ga) are shown in Figures 3 and 4. e band structures of the two compounds are very similar.Bands from −13 eV to −10 eV are mainly from the C-2s and Cr-3d states with strong hybridizations characteristic.From −8 eV to 2 eV, there are hybridizations between C-2p and Cr-3d states, and the Cr-3d states contribute dominantly to the bands, which suggests the itinerant nature of Cr-3d electrons.From Figure 4, it can be seen the Fermi level locates at the vicinity of the DOS peak, which leads large DOS at the Fermi level   ) with values of 4.79 states/eV for AlCCr 3 and 5.62 states/eV for GaCCr 3 .
e Fermi surfaces of the two compounds are similar (see Figure 5).ere are four bands that cross the Fermi level.Hole pockets surrounding Γ point come from the lower two bands (denoted with red and green colors in Figure 3).Electron pockets surrounding R point come from the upper two bands (denoted with blue and purple colors in Figure 3).e differences of the Fermi surface of the two compounds come from the band denoted with blue color in Figure 3.It crosses the Fermi level at the vicinity of M point for AlCCr 3 , which makes the electron pockets surround the corners of the Brillouin zones connected to each other at M point.However, for GaCCr 3 , it crosses the Fermi level in the place between Γ and M points, which forms a cubic cage-like electron pocket surrounding Γ point and connecting with the corner-centered electron pockets.e presence of electron and hole bands crossing the Fermi level indicates the multiple-band natures for AlCCr 3 and GaCCr 3 .In order to understand the bonding nature among the atoms in AlCCr 3 and GaCCr 3 , we plot the contour maps of the charge density of the two compounds in Figure 6.e C-Cr bonds are very strong, which coincides with the strong hybridization between C-2p and Cr-3d electrons shown in DOS �gures.e similarity of the bonding nature for the two compounds coincides with the similar Young's modulus  discussed before.
e electronic properties of AlCCr 3 and GaCCr 3 are very similar to those of the superconducting antiperovskite MgCNi 3 [37], for which the Fermi level locates at the vicinity of the DOS peak as well, and there is multiple-band nature present.e similarity possibly makes AlCCr 3 and GaCCr 3 potential superconductors.According to McMillan's coupling theory [38], the electron-phonon coupling constant  can be calculated from the following expression   ∑  (  /  ⟨ 2  ⟩), where   is the atomic mass and ⟨ 2  ⟩ is the averaged phonon frequency and can be approximated as ⟨ 2   ⟩  Θ 2  /2.For superconducting MgCNi 3 , there is a peak (a van Hove singularity) in DOS just below Fermi level, which leads to a large (  ).e (  ) is not large enough to lead magnetic instability, but it can lead to a large electron-phonon coupling constant [39].To the best of our knowledge, except for the superconducting nickel antiperovskites, there are no other antiperovskites that have such a nature.Wiendlocha et al. [17] pointed out that in chromium systems, chromium atoms have very large values of the McMillan-Hop�eld parameters [38], which may lead to a very strong electronphonon coupling.e recently reported work about RhNCr 3 [18] supports Wiendlocha et al. 's prediction.e phonon and electron-phonon coupling calculations show that RhNCr 3 , which has a large   ), is a strong coupling superconductor with Tc above 16 K.erefore, we consider that there is possibility for superconductivity appearing in AlCCr 3 and GaCCr 3 .Phonon and electron-phonon coupling calculations will be carried out in the future to con�rm the possibility.

Conclusion
In conclusion, we theoretically investigated the antiperovskite chromium-based carbides ACCr 3 through the �rst-principles calculation based on density functional theory.e optimized lattice parameter of ACCr 3 is basically proportional to the radius of A-site elements.Only AlCCr 3 and GaCCr 3 have negative formation energies, implying that the two compounds can be synthesized at ambient pressure and are stable with nonmagnetic ground states.AlCCr 3 and GaCCr 3 show ductile natures, and they may have similar thermal properties.Similar to superconducting antiperovskite MgCNi 3 , there are electron and hole bands that cross the Fermi level for AlCCr 3 and GaCCr 3 , indicating multiple-band natures.e Fermi level locates at the vicinity of the DOS peak, which leads to a large   ) dominated by Cr-3d electrons.ese similarities possibly make AlCCr 3 and GaCCr 3 show superconductivity, which needs to be further investigated from theoretical and experimental studies.

F 3 :
e band structures for (a) AlCCr 3 and (b) GaCCr 3 .e colored lines denote the bands crossing the Fermi level.
T 1: e coefficient  2 in (1) as combinations for second order elastic constant for cubic crystal.denotes the Lagrangian strain tensors in terms of .AlCCr 3 and GaCCr 3 show ductile natures and may have similar thermal properties.e electron and hole bands cross the Fermi level, implying the multiple-band nature of AlCCr 3 and GaCCr 3 .e Fermi level locates at the vicinity of the density of states (DOS) peak, which leads to a large DOS at Fermi level dominated by Cr-3d electrons.e bands properties of AlCCr 3 and GaCCr 3 are very similar to those of superconducting antiperovskite MgCNi 3 .e similarity may make AlCCr 3 and GaCCr 3 show superconductivity. ), T 2: Lattice parameter , magnetic moments per Cr atom , and the formation energies Δ of ACCr 3 .Elastic constants and derived quantities for ACCr 3 (A = Al and Ga).