Optical conductivity and direct interband transitions in Gd

Structure found in the optical conductivity for radiation polarized parallel and perpendicular to the c-axis of a single-crystal of Gd below the Curie temperature has been correlated with direct interband transitions as derived from the spin-polarized conduction bands with the inclusion of the spin-orbit interaction as a perturbation. Spin-flip transitions not allowed for in the eleclric dipole transitions are shown to be only weakly allowed. There have been many attempts [l-61 recently to associate the experimentally observed structure in the optical conductivity of Gd to the onset of direct interband transitions and the possibility of transitions due to the creation of new energy gaps in the conduction bands by the s-f exchange interaction [7, 81. The prediction of the creation of new energy gaps in the conduction bands by the s-f exchange interaction is based on the application of the nearly free electron theory (NFE) to rare earth metals with more than half-filled 4f shells and exhibiting a spin-screw type magnetic ordering which is strongly dependent upon temperature. Gd has been found to exhibit a simple ferromagnetic alignment through out the temperature range below the Curie point. Also Dimmock et al. [9] have shown from energy band calculations based on the augmentedplane wave (APW) method that the conduction bands of Gd differ considerably from those bands calculated from the NFE method and instead resemble more closely the conduction bands of the transition metals. In addition, Dimmock et al. have shown that the energy gaps in the conduction bands created by the s-f exchange interaction are associated with superzone boundaries due to the onset of antiferromagnetic ordering and are absent in the case of a ferromagnet such as Gd. Recently, Harmon and Freeman [lo] have extended the APW method to the calculation of the spin-polarized energy-band structure of Gd emphazing the polarization of the conduction electrons due to the indirect exchange interaction with the 4f core-type electrons. They found that the conduction bands split into spin-up and spin-down bands very similar in shape to the paramagnetic bands [9] and that the exchange splitting increases as the energy eigenvalue increases, being proportional to the amount of d-character in the bands. In this study we take the spin-split band structure of Harmon and Freeman and evaluate the band excitation energies for ferromagnetic Gd with the spinorbit interaction as a perturbation on the spin-split bands. Based on a detailed consideration of the paramagnetic band structure [9] we have found that, at the four points of high symmetry, i.e., F, K, H and A in the Brillouin zone where optical transitions are allowed in the presence of spin-orbit interaction, the energy bands at the point K warrant the most attention when considering the structure in the optical conductivity as measured by polarized radiation on a single crystal of Gd [ll]. The spin-up and spin-down energy bands of interest at K are K,, K,, K, and K,. The selection rules for optical transitions with and without spin-orbit coupling at the point K have been discussed in detail by Dimmock et al. [9] and therefore only the pertinent possible transitions will be given here. These transitions are shown schematically in figure 1. The paramagnetic bands are deduced from Harmon and Freeman by simply taking the mean of the respective spin-up and spin-down bands. The exchange-split bands are taken directly as published. The magnitudes of the exchanges splitting of the four bands are Ex = 0.374 eV for K,, Ex = 0.418 eV for K,, Ex = 0.513 eV fpr K, and Ex = 0.691 eV for K,. Note that Ex.= 6.69 eV as cited in the literature [9] is in agreement only with the K, band which is an empty band lying completely above the Ferrni surface. Applying the spin-orbit interaction as a perturbation on the energy bands after exchange splitting gives the following new levels. K, f J and K, f J which are singlets do not split due to the spin-orbit interaction and simply go to K, f J and K, f J respectively. The K, f J bands are doubly degenerate and split into Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979529 OPTICAL CONDUCTIVITY AND DIRECT INTERBAND TRANSITIONS IN Gd C5-79 1 ' a (b) ( c ) : '~X29--