Bulk Fe

 Iron can be considered an obvious choice to start studying magnetic, and hence spin polarised systems. The LDA fails to determine the correct ground state structure of Fe [98]. In nature, Fe is found to be body centred cubic and ferromagnetic. According to the all-electron calculations of reference [98], the LDA gives the face centred cubic non-magnetic structure as the most stable and the PBE GGA predicts the correct structure. The body centred cubic (bcc) and face centered cubic (fcc) structures are shown graphically in Figure 6.7.

Figure 6.7: Fe in the bcc structure (left) and fcc structure (right).

We repeat the LDA and PBE calculations using ultrasoft pseudopotentials constructed with the LDA. We also perform calculations using our WSDA model including the modified cusp condition. To investigate the effect of using consistent pseudopotentials, we repeat the PBE calculations using PBE ultrasoft pseudopotentials. For each model of exchange-correlation, the non-magnetic fcc and bcc structures and the bcc ferromagnetic structures were used. In each of these cases the plane wave kinetic energy cutoff of 500 eV was used and a Monkhorst-Pack grid of dimensions  was used to sample the Brillouin zone. This converges total energy differences to better than 13 meV per atom for the fcc structure and 3 meV per atom for the bcc structure. The total energies obtained are then used to construct an energy volume curve fitted to the Murnaghan equation of state, which allows us to determine the equilibrium lattice parameter and bulk modulus.

Figure 6.8: Total energy against volume for bulk Fe in a number of structures calculated using the LDA.

Figure 6.9: As for Figure 6.8 but for the PBE GGA with LDA pseudopotentials.

Figure 6.10: As for Figure 6.8 but for the PBE GGA with PBE pseudopotentials.

Figure 6.11: As for Figure 6.8 but for our WSDA.

In Table 6.5 we show the equilibrium lattice parameter and bulk modulus for each of the structures and approximations used. For comparison, the experimental lattice parameter is 2.867 Å and the bulk modulus is 172 GPa [99]. The lattice parameter for the ferromagnetic body centred cubic structure is overestimated in all cases and the bulk modulus is underestimated. For the WSDA the predicted lattice parameter is slightly higher than for the other models and the bulk modulus is massively underestimated. This comparison between the WSDA and the other (semi-)local approximations used holds true for the other phases used.

Table 6.5: Equilibrium lattice parameter,  , and bulk modulus,  , of Fe for selected structures and models. All models use LDA pseudopotentials except where marked with *.

	(Å)			(GPa)
Model	bcc(FM)	bcc(NM)	fcc(NM)	bcc(FM)	bcc(NM)	fcc(NM)
LDA	3.05	2.66	3.34	145	340	378
PBE	3.09	2.70	3.37	130	315	342
PBE*	3.14	2.74	3.47	122	296	178
WSDA	3.36	2.90	3.62	37.8	146	162

The energy volume curves obtained are shown in Figures 6.8 - 6.11. For each of the (semi-)local approximations the correct order, as reported in reference [98], is shown. That the correct order is shown by the LDA is possibly a result of fixing the distribution of spin density in our calculations. The differences for the PBE GGA using LDA and PBE pseudopotentials are minor in this case. The exception is the factor of 2 change in the bulk modulus for the face centred cubic non-magnetic phase. Our implementation of the WSDA fails to predict the correct phase order, in fact, the body centred cubic ferromagnetic phase is predicted to be the least stable. This appears to be because of a redistribution of charge between atomic orbitals, specifically to an excess of charge in the  orbital of one of the atoms, leading to an excess spin on one atom and a depletion on the other. This is likely to be a result of errors introduced by the model pair-correlation function taking a unphysical on-top value in regions of high electron density. We will discuss this further in section 6.4.

http://cmt.dur.ac.uk/sjc/thesis_dbj/node77.html