Calculations, band theory spin polarization

Earlier it was mentioned that the relativistic theory of electronic states in solids in many respects is identical to that of atoms. Since this is well described elsewhere, this section will only deal with some features of specific implementations of the theory in actual calculation methods used for solids, and the importance of relativistic effects — apart from those already discussed — will be illustrated by examples. Although Section 3 did refer to results of LMTO calculations, we did not describe how these included relativity. This section will deal with these items in the form of an overview, and the basic band structure calculations described relate to the density-functional theory [62,63]. Since magnetism is one of the most important solid state physics fields we shall discuss the simultaneous inclusion of spin-polarization and relativistic effects, in particular the spin-orbit coupling. In that context it appears that several of the materials where such effects are particularly large and interesting are those where electron... 

The question of alternative structure can be answered by electronic-structure theory, and it turns out that a quantitative answer is slightly more complicated because different magnetic properties are calculated for the [NaCl] and [ZnS] types. Nonetheless, non-spin-polarized band-structure calculations are quite sufficient to supply us with a correct qualitative picture. This has been derived using the TB-LMTO-ASA method and the LDA functional, and they give the correct lattice parameters with lowest energies for both structure types [267], just as for the case of CaO. 

The observation of the CT gap should be contrasted with the predictions of band theory. Local-density-approximation (LDA) calculations performed for a number of undoped materials predict these systems to be metals since die Cu 3d and O 2p orbitals form a conduction band which is half-filled (for a review see Pickett 1989). The spin-polarized version of this band theory is not sufficiently accurate to yield an antiferromagnetic state of the insulating compound (Pickett et al. 1992). Thus LDA calculations fail to account for the two principal features of undoped materials the insulating gap and antiferromagnetic ordering. However, despite these serious problems these calculations do yield accurate values for the Fermi surface crossings as observed by angle-resolved photoemission (for a review see Pickett et al. 1992 see also ch. 201 of this Handbook). 

Now let us consider the effect of crystal environment on the magnetic moment of the lanthanides. In Table 10, we show the results of calculations of the magnetic moment of neodymium on several common crystal lattices. A trivalent Nd ion yields a spin moment of 3/lb and an orbital moment of 6/ib- In the final two columns of Table 10, we see that the SIC-LSD theory yields values slightly less than, but very close to, these numbers. This is independent of the crystal structure. The valence electron polarization varies markedly between different crystal structures from 0.34/ib on the fee structure to 0.90/Zb on the simple cubic structure. It is not at all surprising that the valence electron moments can differ so strongly between different crystal structures. The importance of symmetry in electronic structure calculations cannot be overestimated. Eor example, the hep lattice does not have a centre of inversion symmetry and this allows states with different parity to hybridize, so direct f-d hybridization is allowed. However, symmetry considerations forbid f-d hybridization in the cubic structures. Such differences in the way the valence electrons interact with the f-states will undoubtedly lead to strong variations in the valence band moments. 

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