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Advances in Condensed Matter Physics
Volume 2013 (2013), Article ID 136274, 7 pages
Structural, Elastic, and Electronic Properties of Antiperovskite Chromium-Based Carbides ACCr3 (A = Al and Ga)
1Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China
2High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China
Received 29 October 2012; Accepted 24 December 2012
Academic Editor: Laifeng Li
Copyright © 2013 D. F. Shao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , 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 . And it was found there is spin density wave antiferromagnetism that coexists with superconductivity in , , and (, 0.03, 0.06, and 0.10) . For chromium-based antiperovskites, Wiendlocha et al.  discussed the possibility of superconductivity in GaNCr3 and RhNCr3, and recent phonon and electron-phonon coupling calculation of RhNCr3  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  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  of . The self-consistent calculations were considered to be converged when the total energy of the system was stable within 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  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 : where the coefficients and 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 (, , and ). Therefore, here we introduced 3 Lagrangian strain tensors labeled as in terms of . and the corresponding coefficients 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 , bulk modulus and shear modulus of cubic crystal can be defined as where the subscripts and denote the Voigt, and Reuss average, respectively. In (3), Thus 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 can be determined as where , , 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. The mean sound velocity is defined as where and are the longitudinal and transverse sound velocities, which can be obtained from bulk modulus and shear modulus :
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).
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 , which is defined as the difference between the total energies of initial reagents (elements) and final compounds: Here, , , and are the total energies of the initial reagents.
According to the definition of , 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 , the compounds are unlikely to be synthesized under normal conditions.
The calculated 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 . For AlCCr3 and GaCCr3, the values of 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, is varied with a step of 0.0025 in every case for . 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, , , and , which indicates that the compounds are mechanically stable.
Cracks in crystals are directly related to the anisotropy of thermal and elastic properties . The 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 . The elastic constant represents the shear resistance in a  direction, which is related to the C–Cr bonding. The values of for the two compounds are almost the same, which implies the similar C–Cr bonding nature of the two compounds. turns out to be the stiffness associated with a shear in a  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 are listed in Table 3. The different A–Cr bondings of the two compounds lead to different anisotropy. For AlCCr3, indicates that AlCCr3 is almost isotropic. However, GaCCr3 seems like more anisotropic ().
Young’s modulus defines the ratio between the linear stress and strain. The 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. The values of for AlCCr3 and GaCCr3 are very close to each other, suggesting the similar covalent nature of the two compounds.
The Cauchy’s pressure is defined as the difference between the two particular elastic constants . 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 . As shown in Table 3, the values of for AlCCr3 and GaCCr3 are positive, which is a clear indication for the compounds to be ductile. Another index of ductility is the ratio . According to Pugh , the ductile/brittle properties of materials could be related empirically to the ratio . If , the material would be ductile; otherwise the material behaves in a brittle manner. For ACCr3 (A = Al and Ga), all the calculated 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 of ACCr3 (A = Al and Ga), listed in Table 3, are very close to the typical value of 0.33 for ductile metallic materials . All the parameters clearly show the ductility of ACCr3 (A = Al and Ga).
Using bulk modulus , shear modulus , and Young’s modulus , the Debye temperature can be calculated. As a matter of fact, a higher would imply a higher thermal conductivity associated with the material . For the present calculation, the are estimated to be 532.43 K and 497.12 K for AlCCr3 and GaCCr3, respectively. The similar values of 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 -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 with values of 4.79 states/eV for AlCCr3 and 5.62 states/eV for 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 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 and M points, which forms a cubic cage-like electron pocket surrounding 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.
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 discussed before.
The electronic properties of AlCCr3 and GaCCr3 are very similar to those of the superconducting antiperovskite MgCNi3 , 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 , the electron-phonon coupling constant can be calculated from the following expression , where is the atomic mass and is the averaged phonon frequency and can be approximated as . For superconducting MgCNi3, there is a peak (a van Hove singularity) in DOS just below Fermi level, which leads to a large . The is not large enough to lead magnetic instability, but it can lead to a large electron-phonon coupling constant . 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.  pointed out that in chromium systems, chromium atoms have very large values of the McMillan-Hopfield parameters , which may lead to a very strong electron-phonon coupling. The recently reported work about RhNCr3  supports Wiendlocha et al.’s prediction. The phonon and electron-phonon coupling calculations show that RhNCr3, which has a large , 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.
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 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.
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.
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