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To extend the theoretical work beyond the RHF free atom approach, electronic structure calculations have been performed using two methods: the FP-LAPW DFT method employed using the ELK code and a wavefunction based method using GAMESS.

7.4.1 Calculation Details

The electronic structure of CeB6 has been modelled using the ELK FP-LAPW code.

For all calculations performed, spin-orbit coupling was turned on. In order to converge the calculations, rgkmaxwas set to6.0.

The electronic structure work of Suvasini et al.167 was used as an initial suggestion for the placement of the Ce and B core/valence electron states. This initial calculation placed the Ce 4f1, 5d1 and 6s2 states into the valence. However, these calculations

were found to converge to a non-magnetic groundstate when replicated. Instead, the valence states of the Ce were described as 4f1, 5s2, 5p6, 5d1 and 6s2 and the valence

states of the B were described as 2s2 and 2p1. All other electron states were treated as

core states. This calculation converged to a spin-polarised ground-state and it is this calculation which will be analysed in the proceeding sections.

7.4.2 Lattice Optimisation

An immediate difficulty in modelling CeB6 using DFT relates to its very narrow 4f

band which lies close toEF. This means that very subtle changes in the band structure

can have significant effects on the macroscopic properties (in particular, the magnetic moment). As a result of this, optimising the lattice parameter is essential to calculating the true groundstate properties of the system. The lattice optimisation was performed by varying the lattice parameter within a range of±10 %. Figure 7.6 presents the BM plot for CeB6 calculated with the GGA exchange correlation functional using 1000 k

points within the IBZ. The BM 3rd order equation of state fitting to the data found the minimum lattice parameter to have a value a0 = 4.135 ˚A, in excellent agreement

with the literature.143,167,168 All subsequent ELK work used this lattice parameter as a

3.4 3.6 3.8 4.0 4.2 4.4 4.6 -9014.65 -9014.60 -9014.55 -9014.50 -9014.45 -9014.40 -9014.35 -9014.30 -9014.25 -9014.20 Birch-Murnaghan 3 rd Order EoS CeB 6 GGA 1000 k points E n e r g y [ H a r t . ] Lattice Parameter [Å]

Figure 7.6: The lattice optimisation of CeB6 as performed using the ELK code. The

lattice parameter was varied by as much as ±10 % of the lattice parameter quoted by Suvasini et al. as 4.119 ˚A.167 and fit using the BM 3rd order equation of state. The minimum lattice parameter was found to be 4.135 ˚A, in excellent agreement with the quoted value.

7.4.3 Band Structure Study

Figure 7.7 plots the DOS of CeB6. Immediately apparent is the very narrow 4f band

which sits near EF in both spin channels. The DOS at EF is dominated by the Ce f

contribution. The contributions at E−EF <2 eV are comprised mainly of B p states

which are hybridised with the B s states. There are also minimal contributions from Ce s, p and d states. Above EF is where the DOS is comprised predominantly from

unoccupied dstates. The magnetic moment seems to originate from the asymmetry in the DOS generated by the majority and minority Ce 4f peaks. The sharpness of these peaks makes it crucial that the band structure from the calculation is satisfactory as very small deviations in this band structure will create significant changes to the magnetic properties of the material.

-14 -12 -10 -8 -6 -4 -2 0 2 4 -15 -10 -5 0 5 10 15 D e n s i t y o f S t a t e s [ s t a t e s / e V ] E-E F [eV] Total DOS Ce 4 f

Figure 7.7: Spin polarised DOS of CeB6. The arrows denote their respective spin

channels. EF is dominated by Ce 4f states.

Figure 7.8 plots the calculated non-magnetic band structure of CeB6 where the Ce 4f

electron is treated in the valence. The band structure is in very reasonable agreement with the calculation of Suvasini et al. Deviations from their calculations are likely due to differences in calculation method used. The ELK code is afull potential code whereas theFully Relativistic Spin-Polarised Linear Muffin Tin Orbital(SPR-LMTO) method169 used by Suvasiniet al. is a fully relativistic calculation. In the ELK calculation, an ad- ditional term is present in ˆHto account for SOC. This does not allow the ELK code to qualify as a fully-relativistic calculation however. Both theoretical paradigms for cal- culating the properties of CeB6 therefore have their inherent strengths and weaknesses.

As a result, small deviations in the calculated band structure are to be expected. In particular, there is quite a deviation in which bands crossEFand this is reflected in the

shapes of the Fermi surfaces (see§7.4.4).

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 M X Z R A M E - E F [ e V ]

Figure 7.8: Non-magnetic band structure plot of CeB6 with the Ce 4f electron treated

as valence.

amining the DOS between the two different schemes showed no discernible difference between the two schemes. Table 7.2 compares several quantities of interest between the two functionals. Comparing the LDA and GGA exchange correlation functionals, the GGA calculates a lower energy groundstate than that of the LDA. Examining the calcu- lated total, spin and orbital moments finds both functionals give very similar results for the material’s magnetic properties. The difference between the DOS atEF is also very

EXC Total Energy [Hart.] DOS at EF Total Moment [µB] Spin Moment [µB] Orbital Moment [µB] LDA -9006.69896 192.60108 0.2197 -0.2684 0.4881 GGA -9014.59817 181.46936 0.2405 -0.2895 0.5300

Table 7.2: Comparison of the LDA and GGA ELK calculations for CeB6. Changing

the exchange correlation functional appears to make minimal changes to the various properties of the material. Note the magnetic moment values will be slightly lower than expected since they are calculated only over the muffin-tins.

small. In conclusion, the GGA makes a very small change over the LDA. Both function- als give the same character to the material. Examining the differences in the calculated EMDs will give a better and more quantitative comparison of the two functionals as the differences will be easily identifiable in the calculated MCPs and 2D-EMD plots.

7.4.4 Fermi Surface Calculations

As with all metallic materials, CeB6 has a Fermi surface. Various investigations have

been performed using either the de Haas-van Alphen effect or positron annihilation radia- tion (2D-ACAR) to probe the Fermi surface of CeB6and sister compound LaB6.167,170,171

Attempting to replicate their work can reinforce confidence in the electronic structure calculation we have chosen to calculate the EMD from. The resulting Fermi surface calculated is heavily dependent on the treatment of the Ce 4f electron as being part of the core or a valence state. As such, both types of calculation were performed. An 80×80×80 mesh of k points was used in reciprocal space to accurately calculate the non-magnetic Fermi surface. The Fermi surfaces calculated from treating the single Ce 4f electron in the core and as valence have been plotted in figure 7.9. The core cal- culation is characterised by the development of a single band crossing EF, generating

4f1 _{Core 4f}1 _Valence

Figure 7.9: The calculated Fermi Surfaces for CeB6 where the single Ce 4f1 electron

has been treated as either a core or valence state. The projection is along the principle axis.

a set of periodic oval shapes with small holes along the larger radii. This calculation is in excellent agreement with the Fermi surface measurements of LaB6 studied in the

available literature.170,171 This is reasonable since La lacks an outer 4f electron. Setting

Ce’s 4f electron to be a core state forces the electron to be described with a purely sym- metric, localised wavefunction, effectively making it inert. Fermiologically, this creates a calculation which is not dissimilar to that of a LaB6, and this is reflected in the stellar

agreement between the calculated core-4f Fermi surface and the measured LaB6 Fermi

surface.

The second Fermi surface was calculated with the Ce 4f electron being treated as a valence state. Now the Ce 4f electron’s wavefunction is described as a planewave. The Fermi surface from this calculation is characterised by the development of three bands, crossing the Fermi level. This results in a significantly more complicated Fermi Surface when compared to the core-4f calculation. The widest band (coloured in blue) contributes the greatest to the DOS and is characterised by a set ofcones which point alongb in the BZ. The second and third bands contribute significantly less to the DOS and originate in the corners of the unit cell. This calculation disagrees quite significantly with Suvasini et al.’s work primarily due to the high sensitivity the Fermi surface’s topology has to changes in the band structure. This makes sense when returning to the calculated band structure and DOS plots (section) which while giving very reasonable agreement with the Suvasini calculations, the band structure plot in particular displays subtle variations in band structure around EF which cause the deviations in Fermi

surface agreement.