Structural, Elastic, and Electronic Properties of Antiperovskite Chromium-Based Carbides ACCr₃ (A = Al and Ga)

Abstract

We theoretically investigated antiperovskite chromium-based carbides ACCr₃ through the first-principles calculation based on density functional theory (DFT). The structure optimization shows that the lattice parameter of ACCr₃ is basically proportional to the radius of A-site elements. The 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 and may 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. The Fermi level locates at the vicinity of the density of states (DOSs) peak, which leads to a large DOS at Fermi level dominated by Cr-3d electrons. The band structures of AlCCr₃ and GaCCr₃ are very similar to those of the superconducting antiperovskite MgCNi₃. The similarity may make AlCCr₃ and GaCCr₃ behave superconductively, which needs to be further investigated in theoretical and experimental studies.

1. Introduction

Recently, antiperovskite compounds AXM₃ (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, nickel-based and manganese-based antiperovskites were extensively studied. Abundant physical properties were shown in the two kinds of compounds, such as superconductivity [1–3], giant magnetoresistance (MR) [4, 5], large negative magnetocaloric effect (MCE) [6, 7], giant negative thermal expansion [8, 9], magnetostriction [10], and nearly zero temperature coefficient of resistivity [11, 12]. But there are only a few reports about other 3d-metal-based antiperovskites so far. The difficulty restricting researchers is the exploration of new materials that can be experimentally synthesized. Therefore, theoretical investigations on these potential 3d-metal-based antiperovskites are useful to find 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−𝑥Re_𝑥(𝑥>0.17) [14], Cr_1−𝑥Ru_𝑥(𝑥>0.17) [15], and (Cr_1−𝑥Mo_𝑥)₇₅Ru₇₅ (𝑥=0, 0.03, 0.06, and 0.10) [16]. For chromium-based antiperovskites, Wiendlocha et al. [17] discussed the possibility of superconductivity in GaNCr₃ and RhNCr₃, and recent phonon and electron-phonon coupling calculation of RhNCr₃ [18] supports Wiendlocha et al.’s prediction. In the present work, we theoretically investigated antiperovskite chromium-based carbides ACCr₃ through the first-principles calculation based on density functional theory (DFT). The optimized lattice parameter of ACCr₃ is basically proportional to the radius of A-site elements. From the analysis of formation energies, we predict that only AlCCr₃ and GaCCr₃ can be synthesized at ambient pressure, and they are stable with nonmagnetic ground states. The elastic and electronic properties of the two compounds are specifically discussed. AlCCr₃ and GaCCr₃ show ductile natures and may have similar thermal properties. The electron and hole bands cross the Fermi level, implying the multiple-band nature of AlCCr₃ and GaCCr₃. The 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. The bands properties of AlCCr₃ and GaCCr₃ are very similar to those of superconducting antiperovskite MgCNi₃. The similarity may make AlCCr₃ and GaCCr₃ show superconductivity.

2. Computational Details

The calculations were performed by projected augmented-wave (PAW) [19, 20] method using the ABINIT code [21–23]. The PAW method can lead to very accurate results comparable to other all-electron methods. For the exchange-correlation 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. The self-consistent calculations were considered to be converged when the total energy of the system was stable within 10⁻⁶ 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.

The elastic constants are evaluated according to the finite-strain continuum elasticity theory [26, 27]. The strain energy per unit mass can be expressed as a polynomial of the strain parameter 𝜉 [28]: 𝜙(𝜉)=12𝐴₂𝜉²+16𝐴₃𝜉³+𝑂𝜉⁴, (1) where the coefficients 𝐴₂ and 𝐴₃ are combinations of second- and third-order elastic constants of the crystal, respectively. For a cubic structure, there are three independent second-order elastic constants (𝐶₁₁,  𝐶₁₂, and 𝐶₄₄). Therefore, here we introduced 3 Lagrangian strain tensors labeled as 𝜂_𝛼 in terms of 𝜉. 𝜂_𝛼 and the corresponding coefficients 𝐴₂ are listed in Table 1.

Table 1 

The coefficient 𝐴₂ in (1) as combinations for second order elastic constant for cubic crystal. 𝜂_𝛼 denotes the Lagrangian strain tensors in terms of 𝜉.

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 𝐵 and shear modulus 𝐺 of cubic crystal can be defined as 𝐶𝐵=¹¹+2𝐶₁₂3,𝐺𝐺=^𝑉+𝐺_𝑅2, (2)(3) where the subscripts 𝑉 and 𝑅 denote the Voigt, and Reuss average, respectively. In (3), 𝐺_𝑉=𝐶₁₁−𝐶₁₂+3𝐶₄₄5,𝐺_𝑅=5𝐶₁₁−𝐶₁₂𝐶₄₄4𝐶₄₄+3𝐶₁₁−𝐶₁₂. (4) Thus Young’s modulus 𝐸 and Poisson’s ratio 𝜎 can be obtained using the relations 𝐸=9𝐵𝐺,3𝐵+𝐺𝜎=3𝐵−2𝐺2().3𝐵+𝐺 (5)

Using the calculated bulk modulus 𝐵, shear modulus 𝐺, and Young’s modulus 𝐸, the Debye temperature Θ_𝐷 can be determined as Θ_𝐷=ℎ𝑘_𝐵3𝑛𝜌𝑁_𝐴4𝜋𝑀^1/3𝑣_𝑚, (6) where ℎ, 𝑘_𝐵, and 𝑁_𝐴 are Planck’s, Boltzmann’s constants, and Avogadro’s number, respectively. 𝜌 is the mass density, 𝑀 is the molecular weight, and 𝑛 is the number of atoms in the unit cell. The mean sound velocity is defined as 𝑣_𝑚=132𝑣³_𝑡+1𝑣³_𝑙^1/3, (7) where 𝑣_𝑙 and 𝑣_𝑡 are the longitudinal and transverse sound velocities, which can be obtained from bulk modulus 𝐵 and shear modulus 𝐺: 𝑣_𝑙=3𝐵+4𝐺3𝜌^1/2,𝑣_𝑡=𝐺𝜌^1/2. (8)

3. Results and Discussion

3.1. Ground State and Formation Energy

The structures were fully optimized with respect to lattice parameter and atomic positions. The optimized lattice parameters of ACCr₃ for different A-site elements are listed in Table 2. The results show that the lattice parameter of ACCr₃ is basically proportional to the radius of A-site elements (see Figure 1).

Table 2 

Lattice parameter 𝑎, magnetic moments per Cr atom 𝑀, and the formation energies Δ𝐸 of ACCr₃.

Figure 1 

Relation between the lattice parameter 𝑎 and the radius of A-site elements 𝑟A.

The common technique for producing antiperovskite carbides is the solid-state synthesis from the parent materials. Therefore we assumed that the antiperovskite chromium-based carbides ACCr₃ could be synthesized in the same routine. We analyze the stability of compounds by calculating the formation energies Δ𝐸, which is defined as the difference between the total energies of initial reagents (elements) and final compounds: Δ𝐸=𝐸^ACCr3_tot−𝐸^A_tot+𝐸^C_tot+3𝐸^Cr_tot. (9) Here, 𝐸^A_tot, 𝐸^C_tot, and 𝐸^Cr_tot are the total energies of the initial reagents.

According to the definition of Δ𝐸, 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 Δ𝐸>0, the compounds are unlikely to be synthesized under normal conditions.

The calculated Δ𝐸 for the series of A-site elements with different magnetic configurations are listed in Table 2. Except for AlCCr₃ and GaCCr₃, all other compounds show positive Δ𝐸. For AlCCr₃ and GaCCr₃, the values of Δ𝐸 for different magnetic configurations 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 AlCCr₃ and GaCCr₃. From the analysis of formation energies, it seems that ACCr₃ can be synthesized only when A = Al and Ga. The elastic and electronic properties of the two compounds will be discussed later.

3.2. Elastic and Mechanical Properties

In order to obtain accurate elastic constants, 𝜉 is varied with a step of 0.0025 in every case for 𝜂_𝛼. As an example, the strain energies for AlCCr₃ and the fitted polynomials are shown in Figure 2. The 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 find that they satisfy the mechanical stability criteria for a cubic crystal, that is, 𝐶₁₁>𝐶₁₂, 𝐶₄₄>0, and 𝐶₁₁+2𝐶₁₂>0 [30], which indicates that the compounds are mechanically stable.

Table 3 

Elastic constants and derived quantities for ACCr₃ (A = Al and Ga).

Figure 2 

The strain-energy relation of AlCCr3.

Cracks in crystals are directly related to the anisotropy of thermal and elastic properties [31]. The elastic anisotropy factor 𝐴 of cubic crystal can be described as 𝐴=2𝐶₄₄𝐶₁₁−𝐶₁₂. (10) For a complete isotropic system, the anisotropy factor 𝐴 takes the value of unity, and the deviation from unity measures the degree of elastic anisotropy [32]. The elastic constant 𝐶₄₄ represents the shear resistance in a [100] direction, which is related to the C–Cr bonding. The values of 𝐶₄₄ for the two compounds are almost the same, which implies the similar C–Cr bonding nature of the two compounds. 𝐶₁₁−𝐶₁₂ 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 ACM₃, it still can influence the anisotropy of the material. The anisotropic factors 𝐴 are listed in Table 3. The different A–Cr bondings of the two compounds lead to different anisotropy. For AlCCr₃, 𝐴=0.91 indicates that AlCCr₃ is almost isotropic. However, GaCCr₃ seems like more anisotropic (𝐴=0.78).

Young’s modulus 𝐸 defines the ratio between the linear stress and strain. The larger the value of 𝐸, 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. The values of 𝐸 for AlCCr₃ and GaCCr₃ are very close to each other, suggesting the similar covalent nature of the two compounds.

The Cauchy’s pressure 𝐶_𝑝 is defined as the difference between the two particular elastic constants 𝐶_𝑝=𝐶₁₂−𝐶₄₄. It is considered to serve as an indication of ductility: 𝐶_𝑝>0 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 𝐶_𝑝 for AlCCr₃ and GaCCr₃ are positive, which is a clear indication for the compounds to be ductile. Another index of ductility is the ratio 𝐵/𝐺. According to Pugh [34], the ductile/brittle properties of materials could be related empirically to the ratio 𝐵/𝐺. If 𝐵/𝐺>1.75, the material would be ductile; otherwise the material behaves in a brittle manner. For ACCr₃ (A = Al and Ga), all the calculated 𝐵/𝐺 ratios are much larger than 1.75, which clearly highlights the ductile nature of ACCr₃ (A = Al and Ga). Poisson’s ratio generally quantifies the stability of the crystal against shear and takes the value between −1 and 0.5, which are the lower and the upper bounds. The lower bound is where the material does not change its shape, and the upper bound is where the volume remains unchanged. The calculated 𝜎 of ACCr₃ (A = Al and Ga), listed in Table 3, are very close to the typical 𝜎 value of 0.33 for ductile metallic materials [35]. All the parameters clearly show the ductility of ACCr₃ (A = Al and Ga).

Using bulk modulus 𝐵, shear modulus 𝐺, and Young’s modulus 𝐸, the Debye temperature Θ_𝐷 can be calculated. As a matter of fact, a higher Θ_𝐷 would imply a higher thermal conductivity associated with the material [36]. For the present calculation, the Θ_𝐷 are estimated to be 532.43 K and 497.12 K for AlCCr₃ and GaCCr₃, respectively. The similar values of Θ_𝐷 suggest the similar thermal characteristics of the two compounds.

3.3. Electronic Properties

The calculated electronic band structures along the high symmetry directions in the Brillouin zones together with the total and site-projected 𝑙-decomposed DOS at equilibrium lattice parameters for ACCr₃ (A = Al and Ga) are shown in Figures 3 and 4. The 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 𝑁(𝐸_𝐹) with values of 4.79 states/eV for AlCCr₃ and 5.62 states/eV for GaCCr₃.

(a)

(b)

Figure 3 

The band structures for (a) AlCCr3 and (b) GaCCr3. The colored lines denote the bands crossing the Fermi level.

(a)

(b)

Figure 4 

The density of states of (a) AlCCr3 and (b) GaCCr3.

The Fermi surfaces of the two compounds are similar (see Figure 5). There 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). The 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 AlCCr₃, which makes the electron pockets surround the corners of the Brillouin zones connected to each other at M point. However, for GaCCr₃, 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. The presence of electron and hole bands crossing the Fermi level indicates the multiple-band natures for AlCCr₃ and GaCCr₃.

(a)

(b)

(c)

Figure 5 

(a) Brillouin zone of ACCr3. Fermi surfaces of (b) AlCCr3 and (c) GaCCr3.

In order to understand the bonding nature among the atoms in AlCCr₃ and GaCCr₃, we plot the contour maps of the charge density of the two compounds in Figure 6. The C–Cr bonds are very strong, which coincides with the strong hybridization between C-2p and Cr-3d electrons shown in DOS figures. The similarity of the bonding nature for the two compounds coincides with the similar Young’s modulus 𝐸 discussed before.

(a)

(b)

Figure 6 

The contour map of electron charge density of (a) AlCCr3 and (b) GaCCr3.

The electronic properties of AlCCr₃ and GaCCr₃ are very similar to those of the superconducting antiperovskite MgCNi₃ [37], for which the Fermi level locates at the vicinity of the DOS peak as well, and there is multiple-band nature present. The similarity possibly makes AlCCr₃ and GaCCr₃ potential superconductors. According to McMillan’s coupling theory [38], the electron-phonon coupling constant 𝜆 can be calculated from the following expression 𝜆=∑_𝛼(𝜂_𝛼/𝑀_𝛼⟨𝜔²_𝛼⟩), where 𝑀_𝛼 is the atomic mass and ⟨𝜔²_𝛼⟩ is the averaged phonon frequency and can be approximated as ⟨𝜔²_𝛼⟩=Θ²_𝐷/2. For superconducting MgCNi₃, there is a peak (a van Hove singularity) in DOS just below Fermi level, which leads to a large 𝑁(𝐸_𝐹). The 𝑁(𝐸_𝐹) 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-Hopfield parameters [38], which may lead to a very strong electron-phonon coupling. The recently reported work about RhNCr₃ [18] supports Wiendlocha et al.’s prediction. The phonon and electron-phonon coupling calculations show that RhNCr₃, which has a large 𝑁(𝐸_𝐹), is a strong coupling superconductor with Tc above 16 K. Therefore, we consider that there is possibility for superconductivity appearing in AlCCr₃ and GaCCr₃. Phonon and electron-phonon coupling calculations will be carried out in the future to confirm the possibility.

4. Conclusion

In conclusion, we theoretically investigated the antiperovskite chromium-based carbides ACCr₃ through the first-principles calculation based on density functional theory. The optimized lattice parameter of ACCr₃ is basically proportional to the radius of A-site elements. Only AlCCr₃ and GaCCr₃ have negative formation energies, implying that the two compounds can be synthesized at ambient pressure and are stable with nonmagnetic ground states. AlCCr₃ and GaCCr₃ show ductile natures, and they may have similar thermal properties. Similar to superconducting antiperovskite MgCNi₃, there are electron and hole bands that cross the Fermi level for AlCCr₃ and GaCCr₃, indicating multiple-band natures. The Fermi level locates at the vicinity of the DOS peak, which leads to a large 𝑁(𝐸_𝐹) dominated by Cr-3d electrons. These similarities possibly make AlCCr₃ and GaCCr₃ show superconductivity, which needs to be further investigated from theoretical and experimental studies.

Acknowledgments

This work was supported by the National Key Basic Research under Contract no. 2011CBA00111 and the National Nature Science Foundation of China under Contract nos. 91222109, 11274311, 51171177, 11174295, U1232139, and 50701042. The calculations were partially performed at the Center for Computational Science, CASHIPS.