CHAPTER -8
CHAPTER: 8 ELECTRONIC STRUCTURE AND ELASTIC PROPERTIES OF CrC AND CrN 8.1 Introduction In this chapter, we have selected CrC and CrN from group VIB transition metal carbides and nitrides for a detail study on electronic, structural and elastic properties. Among transition metal nitrides, CrN has raised many expectations over recent years as it possesses some advantages, such as easy deposition of film, high thermal stability and good corrosion resistance with the prediction of a large bulk modulus. It has also been reported that CrN has a unique antiferromagnetic (AFM) configuration (Corliss et al. 1960). At room temperature, CrN is paramagnetic and has the rocksalt structure. However, upon cooling below 273 286 K, a simultaneous structural and magnetic phase transition takes place. The magnetic ordering is such that pairs of ferromagnetic (FM) planes with alternating spin directions (Corliss et al. 1960; Browne et al. 1970; Ibberson et al. 1992; Eddine et al. 1969) are aligned perpendicular to the (110) direction. This phase transition is accompanied by a (0.56 0.59) % increase in the atomic density and a distortion of the structure which becomes orthorhombic Pmna. It shows hysteresis of 2 3 K and the transition width is extremely sensitive to the N concentration. Corliss et al. (1960) measured a magnetic moment of 2.36μ B per Cr atom. Another experiment by Ibberson and Cywinski (1992) reported a larger value of about 3.17 μ B per Cr atom. In view of the fundamental and technological interest in CrN, a lot of experimental and theoretical work has been done. Although the electrical properties, structural properties and magnetic properties of CrN are thus well established, the same cannot be said about the elastic properties. Earlier results (Browne et al. 1970; Ibberson et al. 1992; Eddine et al. 1969); Tsuchiya et al. 1996) obtained a resistivity as a function of temperature with metallic behavior. The samples used were either powders or polycrystalline films. In these studies CrN showed metallic behavior both in the paramagnetic state and in the antiferromagnetic state below 286 K, although the resistivity dropped by (30 70)% below the transition. Filippetti et al.(1999; 2000) studied CrN using local spin density approximation (LSDA), and Chapter 8: Electronic structure. of CrC and CrN 85
find that assuming cubic symmetry the lowest energy occurs for AFM ordering along (110) plane but with the spins changing every layer instead of every two layers. He and Zin et al. (2011) studied structural, electronic and magnetic properties of CrN under high pressure are investigated by first-principles calculations. They claimed that the antiferromagnetic orthorhombic structure is the preferred ground state structure. CrN undergoes structural and magnetic transitions from an antiferromagnetic rocksalt structure to a non-magnetic phase at 132 GPa. The CrN film is grown on semiconductor substrates, and the compound volume can be very different from its bulk equilibrium value. This is the result of the large strain imposed by the substrate or the host. However, the elastic properties and its derivatives investigation on CrN under pressure are lacking. Zhukov et al. (1989) reported a local-density approximation (LDA)-based electronic structure and same total energy calculation for CrC. They found that the electronic structure of CrC is qualitatively related to that of TiC by a rigid band behavior, and also found that E F of CrC falls near a pronounced peak in the DOS, which is derived from a weakly hybridized Cr d state. They suggested that the additional electron in CrC leads to the occupation of additional antibonding states and this explains the difficulty in synthesizing CrC. However their calculated bulk modulus was significantly higher than that of TiC. Based on this higher bulk modulus and some other considerations, they predict that CrC has the strength in this series. However, they did not examine the lattice stability and other related properties. Liu et al. (1991) reported synthesis of rock salt structure CrC by carbon implantation into Cr films. They have determined that CrC in rock salt phase is unstable above 250 o C, possibly because of the presence of oxygen. These results suggest that rock salt CrC is a metastable phase, possibly with weaker bonding than TiC and VC. However, there are other possibilities. For example, the observed NaCl phase could be stabilized by the presence of impurities or by an off-stiochiometric composition. Singh and Klein (1992) studied the electronic structure, lattice stability and superconductivity of CrC. They have reported that CrC has no sign of instability, and indicated that rock salt structure CrC is a true metastable phase and is a thermodynamically not stable phase. However, they have not calculated the derived properties of elastic constants such as Debye temperature, Poisson s ratio etc. Isaev et al. (2007) studied the phonon related properties of CrC and CrN. The electronic structure and physical properties of the Chromium carbides and nitrides in the B1 structure have been studied by means of the LMTO and FP-LAPW methods, as well as in the framework of ab initio pseudo potential techniques (Guillermet et al. 1992b). Chapter 8: Electronic structure. of CrC and CrN 86
Papaconstantopoulos et al. (1985) have reported energy band structure of CrN in B1 structure. In their calculation, APW method was used. 8.2 Methods of calculation The calculations for CrC and CrN in the rocksalt structures were performed with FP- LAPW within the framework of the density functional theory with generalized gradient approximation Perdew Bruke Ernzerhof 96 (GGA PBE 96) for the exchange correlation potential. Muffin-tin sphere radii (R MT ) of used for the atoms of transition metal, N and C respectively are listed in table 8.1. The valence wave functions inside the MT spheres were expanded into spherical harmonics up to l max = 10 and the R MT K max was taken to be 7.0. 173 number of plane waves were generated for CrC and CrN. In both the cases, 8000 number of k points were initially used which reduces to 165 and 265 inequivalent and irreducible k points for the integration procedure in the Brillouin zone for CrC and CrN respectively. An energy and charge criteria of convergence of 0.0001 Ryd and 0.001e was chosen for the self-consistent cycles and convergence is achieved at 350 number of iteration for both CrC and CrN. The calculations were performed at the equilibrium lattice constants. The plot of the total energy versus volume was fitted using Murnaghan equation of state (Murnaghan, 1944). 8.3Results and discussions 8.3.1Structural properties Fig. 8.1(a) and fig. 8.1(b) shows the total energy versus volume curve for CrC and CrN in rock salt structure. The static equilibrium properties are obtained from these curves. The calculated equilibrium lattice constant for CrC and CrN are listed in table 8.1 along with their respective bulk modulus, pressure derivatives in comparison with available earlier reports. It is seen that the calculated results of the lattice constants and bulk modulus for CrC and CrN agrees well with the reported values in the literature (Filippetti and Hill, 2000; Liu et al. 1991; Singh and Klein, 1992; Isaev et al. 2007; Guillermet et al. 1992; Papaconstantopoulos et al. 1985; Grossman et al. 1999; Toth, 1971; Siegel et al. 2003; Herle et al. 1997; He and Zhi, 2011). There are no experimental data available for elastic constants and their derivatives for comparison. Chapter 8: Electronic structure. of CrC and CrN 87
Fig. 8.1(a) Total energy of CrC as a function of volume Fig. 8.1(b) Total energy of CrN as a function of volume Chapter 8: Electronic structure. of CrC and CrN 88
Table 8.1 Lattice constant a 0, bulk modulus, its pressure derivative, and muffin-tin radii R MT of CrC and CrN System a 0 Bohr) B(GPa) B R MT CrC 7.709 7.615 a, 7.748 b,c, 7.780 d, 7.577 e,f CrN 7.658 7.729 b, 7.796 d, 7.668 g, 7.823) h, 7.838) i, 7.729) c, 7.525 j, 7.759 k 332.722 322 b, 351 c, 333 e,f, a Reference (Liu et al. 1991) experiment 329.254 322 b, 326 g, 361 c, 396 j, 336.2 k, b Reference (Isaev et al. 2007) pseudopotentials+gga c Reference (Grossman et al. 1999) pseudopotentials+lda d Reference (Guillermet et al. 1992) LMTO+LDA e Reference (Toth, 1971) experiment f Reference (Singh and Klein, 1992) LAPW+GGA g Reference (Siegel et al. 2003) VASP+GGA h Reference (Papaconstantopoulos et al. 1985) APW+LDA i Reference (Herle et al. 1997) experiment. j Reference (He and Zhi, 2011) PAW+LDA k Reference (Filippetti and Hill, 2000) pseudopotentials+lda 3.009 R Cr =2.15, R C = 1.92 4.3103, R Cr = 1.90, 4.5 j, 4.5 k R N = 1.80 The lattice constants and bulk modulus of the other compounds of group VIB transition metal carbides and nitrides (i.e., MoC, MoN, WC and WN) have been determined and their values are listed in table 8.2. Table 8.2 Calculated lattice constants and bulk modulus for MoC, MoN, WC and WN Compound Lattice constant in bohr Bulk modulus in GPa MoC 8.277 334.829 MoN 8.212 331.461 WC 8.302 366.057 WN 8.230 365.704 Unlike the previous discussed group IIIB, IVB and VB transition metal carbides and nitrides, the group VIB transition metal carbides are harder than their corresponding nitrides. It is also observed that the hardness increases down the group like group IVB and VB Chapter 8: Electronic structure. of CrC and CrN 89
carbides and nitrides. Tungsten carbide is found to be the hardest compound among the group VI carbides and nitrides. 8.3.2 Elastic properties The elastic constant of solids provides a link between the dynamical and mechanical behavior of crystals, and give important information concerning the nature of the forces operating in solids. In particular, they provide information on the stability and stiffness of materials, and their ab initio calculation requires precise methods. Since the forces and the elastic constants are functions of the first-order and second-order derivatives of the potentials, their calculation will provide a further check on the accuracy of the calculation of forces in solids. They also provide valuable data for developing inter atomic potentials. The calculated elastic constants C ij for CrC and CrN using the volume conserving technique (Brich, 1938) satisfies the necessary condition for mechanical stability for cubic crystal structures (Wallace, 1972) i.e., (C 11 C 12 )>0, (C 11 +2C 12 )>0, C 11 >0, C 44 >0. Thus, both CrC and CrN in rocksalt structures are mechanically stable compounds. The findings are listed in table 8.3. Table 8.3 Calculated elastic constants C 11, C 12 and C 44 of CrC and CrN System C 11 (GPa) C 12 (GPa) C 44 (GPa) CrC 688.391 135.342 80.974 CrN 512.18 217.09 21.88 Zener anisotropy factor A, Poisson s ratio υ, Kleinman parameter ζ, Young s modulus Y, shear modulus C, isotropic shear modulus G, elastic longitudinal v l, elastic transverse v t, elastic wave velocities and the average v m elastic wave velocity and Debye θ D, were find out using the relations given in literatures (Mayer et al. 2003; Harrison, 1989; Johnston et al. 1996; Schreiber et al. 1973; Anderson, 1963). These values are presented in table 8.4 and table 8.5. Table 8.4 Calculated values of density ρ, Zener anisotropy factor A, Poisson s ratio υ, Kleinman parameter ζ, Young s modulus Y, and shear modulus C of CrC and CrN System ρ(kg/m 3 ) A υ ζ Y (GPa) C (GPa) CrC 3466.583 0.293 0.314 0.348 357.4538 276.525 CrC 3251.283 0.148 0.421 0.560 149.669 147.545 Chapter 8: Electronic structure. of CrC and CrN 90
Table 8.5 Calculated values of the isotropic shear modulus G, longitudinal sound velocity v l,, transverse sound velocity v t and average sound velocity v m and Debye temperature θ D. Material G(GPa) v l (m/s) v t (m/s) v m (m/s) θ D (K) CrC 136.054 12022.928 6264.766 7009.831 811.675 CrN 52.666 10891.338 4569.064 4569.064 530.482 CrC and CrN have got small values of Zener anisotropic values of 0.560 and 0.348 respectively. These small values of A imply that these materials have weak anisotropic character. The value of the Poisson s ratio is indicative of the degree of directionality of the covalent bonds. The value of the Poisson s ratio is small (υ = 0.1) for covalent materials, whereas for ionic materials a typical value of υ is 0.25 (Bannikov et al. 2007). The present calculated Poisson s ratios for CrC and CrN are 0.314 and 0.421 respectively. Therefore, the ionic contribution to inter atomic bonding for these compounds is dominant. The values of υ = 0.25 and 0.5 are the lower and upper limits, respectively, for central force solids (Fu et al. 2008). Our υ values are within the range of central force, indicating inter atomic forces are central forces in CrC and CrN. Values of Bulk modulus (B) and shear modulus (G) in table 8.1 and table 8.5 shows that the values of B/G for CrC and CrN are 6.318 and 2.420 respectively. A material is brittle (ductile) if the B/G ratio is less (high) than 1.75 (Shein et al. 2008). Hence preliminary observation can conclude that these materials possess ductile property. 8.3.3 Band structure and density of states. The energy bands of CrC and CrN in rocksalt structure are shown in fig. 8.2(a) and fig. 8.2(b) respectively. The overall features are similar to those of previously published transition-metal carbides and nitrides (Singh, 1992; Herwadkar and Lambrecht, 2009). Values of characteristic band separations are given in table 8.6. For comparison the gap Eg between the non-metal s band and the valence-band complex, the zone-center d-band splitting E d = E(Γ 12 ) E(Γ 25 ), the metal d and nonmetal p 2 3 splitting E d - E p [where Ed E 12 E 25 and Ep E 15 ], and the non-metal p s 5 5 splitting E p E s = E(Γ 15 ) E(Γ 1 ) have been selected. The energy gap E(Γ 15 ) E(L 2 ) is the width of the non-metal 2p band. The difference between the Fermi energy and p band at L point, E F E(L 2 ) is the width of the occupied part of 2p and 3d-bands in case of overlap. The Chapter 8: Electronic structure. of CrC and CrN 91
energy gap between the Fermi level and E(X 3 ) is the width of the occupied part of the 3d bands. It is observed from table 8.6 that the energy gap E g of CrN (0.466 Ry) is greater than CrC (0.191 Ry). The zone centre splitting (ΔEd) for both the CrC (0.132 Ry) is greater than that of CrN (0.070 Ry). The relative position of non-metal p and s bands (E p E s ) in of CrC is less than CrN. The width of 2p band (Γ 15 - L 2 ) of nonmetal in CrC and CrN are 0.467 and 0.512 Ry respectively. The quantity (E d E p ), which is a measure of the relative positions of metal 3d and nonmetal 2p bands shows that the overlapping of these two bands more in CrC than CrN. This indicates the covalency of CrN is less than that of CrC. The total DOS plots of CrC, Cr total and C total in CrC are shown in fig. 8.3 (a). Fig. 8.3(b) and fig. 8.3(c) show the Partial Density of states of Cr (Cr-total, Cr-3d-e g and Cr-3dt 2g ) and C (C-2s and C-2p) in CrC respectively. Similarly the total and partial density of states of CrN are shown in fig. 8.3 (d-f). As observed before, one can see that there are three prominent peaks in the total density of states of CrC. The first peak from the left is dominated by C-s states and it does not contribute to the bonding. The second peak is at around 0.6 Ry and it is dominated by non-metal-2p states and 3d states of Cr. This shows intense hybridization between these states. The third peak is mainly contributed by Cr-5d-t 2g and C- 2p states. In the total density of states of CrN also, we can roughly divide the peaks into three prominent peaks. The first peak from the left is mainly contributed by N-s with negligible contributions from Cr-p and Cr-d states. In CrN, unlike CrC, the second peak is dominated by N-2p states. The third peak is mostly dominated by Cr-d-t2g state. Table 8.6 Characteristic band separations of the band structures as shown in fig. 8.2(a) and fig. 8.2(b) Energy gaps (Ry) CrC CrN E F 0.915 0.916 E g 0.191 0.466 ΔEd 0.132 0.070 E p E s 0.966 1.184 E(Γ 15 ) - E(L 2 ) 0.467 0.512 E d E p 0.115 0.043 E F L 2 0.595 0.622 E F X 3 0.414 0.512 Chapter 8: Electronic structure. of CrC and CrN 92
Fig. 8.2(a) Electronic band structure of CrC Chapter 8: Electronic structure. of CrC and CrN 93
Fig. 8.2(b) Electronic band structure of CrN Chapter 8: Electronic structure. of CrC and CrN 94
Fig. 8.3(a) Density of states of CrC-total, Cr-total, C-total Fig. 8. 3(b) Partial Density of states of Cr (Cr-total, Cr-3d eg and Cr-3d t2g ) in CrC Chapter 8: Electronic structure. of CrC and CrN 95
Fig. 8.3(c) Partial density of states of C (C- total, C-2s and C-2p) in CrC Fig. 8.3(d) Density of states of CrN-total, Cr-total, N-total Chapter 8: Electronic structure. of CrC and CrN 96
Fig. 8.3(e) Partial Density of states of Cr (Cr-total, Cr-3d eg and Cr-3d t2g ) in CrN Fig. 8.3(f) Partial density of states of N (N- total, N-2s and N-2p) in CrN 8.4 Conclusions Full-potential linearized augmented plane wave and generalized gradient approximation based calculation have been performed to elucidate the properties of CrC and CrN. The electronic structure, mechanical stability, elastic constants and their derived Chapter 8: Electronic structure. of CrC and CrN 97
physical properties such as, Zener anisotropy factor, Poisson s ratio, Kleinman parameter, Young s modulus, and shear modulus, Debye temperature is well described. It is also found that the rock salt structure of CrC and CrN can be beaten in to sheets (ductile). The maximum contribution near the Fermi level of CrC is Cr-3d eg. While in CrN, the maximum contribution near the Fermi level of Cr is 3d t2g. Chapter 8: Electronic structure. of CrC and CrN 98