Introduction

Iron (Fe) is known to have a bcc crystal structure at temperatures lower than 1100 K [1]. Its optimal lattice parameters for this structure are derived using Density-Functional Theory (DFT) methods. The Cambridge Serial Total Energy Package (CASTEP) software package, which uses plane-wave basis sets in order to analyze crystal structures, is implemented to analyze the bcc lattices of Fe to determine the optimal parameters. Two functionals in the generalized gradient approximations, PBE and PW91, are used to derive these parameters. The PBE and PW91 functionals are expected to produce virtually identical results [2]. A comparison between the results obtained from these functionals is made along with the computational costs associated with both of them.  

Methodology 

We use the plane wave DFT calculations to evaluate the optimal lattice parameter for Fe bcc crystal structure. All the calculations were performed using the CASTEP Simulation Package in Material Studio. The exchange and correlation energies were calculated using the PBE and the PW 91 functional described within the generalized gradient approximation (GGA). Electronic convergence tolerance of 2E-06 was used for all the calculations. The core electrons were treated using “on the fly” generated (OTFG) ultra-soft pseudo-potential with a core radius of 2.4 Bohrs (1.27 Angstrom) generated with a panel of 8 valence electrons ( 3d6 4s2). All the calculations are performed with the spin-polarized option ON as Fe is a magnetic element. 

Cut-off Energy and K-Point Optimization 

The essential step before performing a plane-wave basis set is to optimize the k-points and cut-off energy. This process is performed with both the functionals. 

K-point  

The experimentally determined optimal lattice parameter for Fe bcc crystal structure is 2.856 Angstrom [1]. However, it is more appropriate to perform k-point optimization on the lowest lattice dimension being investigated. As the smallest lattice parameter in real space gives the longest parameter in k-space. Hence the K-point optimization is performed at the lattice parameter of 2.3 Angstrom. A very high cut-off energy value is selected here 500 eV to avoid its effects on k-point optimization.

Table 1: K-point grid vs the number of K-points used

K-point grid	K-points used
3x3x3	4
4x4x4	6
5x5x5	10
6x6x6	14
7x7x7	20
8x8x8	26
9x9x9	35
10x10x10	44
12x12x12	68
15x15x15	120

Figs.1 and 2 show the absolute change in the consecutive total energy (|ΔE|) with respect to the number of irreducible k-points. The corresponding K-point grid is shown in Table 1. As a convergence criterion of |ΔE| less than 0.04 eV is used for both the functionals. This corresponds to the optimal grid of 9x9x9. 

Cut-off energy 

In Figs. 3 and 4, the absolute change in the consecutive total energy (|ΔE|) is plotted as a function of the cut-off energy. As a convergence criterion, it is assumed that the total energy is converged if |ΔE| is less than 0.03 eV. From Figs. 3 and 4, it is clear that the cut-off energy of 400 eV sufficient considering both the functionals. A lattice parameter of 2.856 Angstrom and K-point grid of 9x9x9 is used for this convergence. 

Fe lattice optimization 

To optimize the Fe bcc lattice parameter, the crystal structure energy is calculated for the bcc geometry while changing the lattice parameters from 2.3 to 3.5 angstrom with an interval of 0.1 Angstrom. Based on the K-point and cut off energy convergence study, cut-off energy of 400 eV and a K-point grid of 9x9x9 is employed in these calculations.  

Results

 

Fig. 5 shows the lattice parameter optimization. A curve is fitted to the DFT data to get an estimate of the energy minima with respect to the lattice parameter. A polynomial fit of order 4 is used for this purpose for data ranging from lattice parameter 2.6 to 3. Table 2 compares the optimum lattice parameters calculated using both the functionals. Both the functionals give optimum lattice parameter values close to the experimental data. There is hardly any difference between the values determined by both functionals. Table 2 shows the time required for calculations using both the functionals, which is not much different. 

Table 2: Results of calculations

Property	Experimental optimum lattice parameter	DFT optimum lattice parameter (PBE) in Angstrom	DFT optimum lattice parameter (PW91) in Angstrom	Total time for calculations (PBE) in s	Total time for calculations (PW91) in s
Value	2.856	2.811	2.799	61.82	59.42

Conclusions 

Density Functional Theory-based calculations are used to ascertain the lattice parameters of the Fe bcc crystal. It is determined by varying the lattice parameters and calculating their respective energies. The optimized lattice parameters obtained are reasonably coherent with the experimental results. A comparative study of the PBE and the PW91 functionals is also performed. It shows that both the functionals give very similar results while using similar computational time although PBE is a simplification of PW91 functional [2]. 

References

1. Greenwood, Norman Neill, and Alan Earnshaw. Chemistry of the Elements. Elsevier, 2012.
2. Mattsson, Ann E., et al. “Nonequivalence of the generalized gradient approximations PBE and PW91.” Physical Review B 73.19 (2006): 195123.