AB-Initio Calculations of the Electronic Structure and Properties of Titanium Carbosulfide
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METHODOLOGY The LKKR method, developed by MacLaren et. al. [3], is a modification of the original KKR technique [4]. Both methods use the muffin-tin approximation to the true crystal potential and use the 'spherical harmonic bases' within the muffin tin and 'plane wave bases' in the interstitial region. In the atomic spheres approximation, one considers overlapping spheres centered at every atom site and there is no interstitial volume in the system. The original KKR method is suitable for bulk systems (systems that possess full three-dimensional periodicity) only. The LKKR method partitions 3-D space into layers and requires Bloch periodicity only within each layer (i.e. the requirement of periodicity is relaxed in the direction perpendicular to the layers) - thus it is well adapted to handle systems that exhibit only 2-D periodicity. This allows for planar defects like surfaces and interfaces to be treated within the same framework as the bulk material.
The calculation employs a Green's function technique adapted to multiple scattering theory - it solves for the one electron Green's function within the semi-relativistic, local spin density and self-consistent field approximations. All desired electronic properties can be obtained once the local energy resolved density of states, p (r,E), for each atom is obtained. The charge density is related to the one electron Green's function as follows: p(r, E)=
Im dk. G(r, k, E)
(1)
where the integration is over the 2-D Brillouin zone. RESULTS & DISCUSSION Variation of Total Energy with Lattice Constant: For the purpose of this calculation, the experimentally reported c/a ratio of 3.5 was held constant and 'a' was allowed to vary between 5.90 Bohr (1 Bohr = 0.5290) and 6.50 Bohr. The variation of the total energy as a function of the lattice constant is depicted in Figure 2. A smooth curve joining the points was obtained by fitting the standard Birch-Murnaghan equation of state to the discrete points. The equilibrium lattice constant (corresponding to the minimum total energy) and the bulk modulus were then obtained from this fit. Initially, the calculation was done sampling 6 k-points in a Brillouin zone wedge and using 37 interstitial plane waves - this predicted an equilibrium lattice constant of 6.25 Bohr (which is about 3% higher than the experimentally reported value) and a bulk modulus of 2.5 Table 1: Predicted Lattice Constant and Bulk Modulus as a function of k-points1 and number of plane waves2
ILattice Constant, a (Bohr) 1 6.25
6.25
I 6.26 I
Bulk Modulus, B (Mbar) 2.51 2.51 2.47 I - number of k-points sampled in one Brillouin zone wedge 2 - number of interstitial plane waves used in the calculation
564
-0.56 -0.58 -
(0
-0.60
CM
-0.62 -0.64
-0.66
5.90 Figure 2
6.00
6.10 6.20 6.30 6.40 Lattice Constant (Bohrs) Total Energy - Lattice Constant Curve
6.50
Mbar. As mentioned earlier, no prior experimental or computational data regarding the bulk
modulus is available to compare our results with; as a measure of internal consistency, we varied both
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