Optical conductivity and direct interband transitions in Gd

HAL Id: jpa-00218946

https://hal.archives-ouvertes.fr/jpa-00218946

Submitted on 1 Jan 1979

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Optical conductivity and direct interband transitions in Gd

P. Ferguson

To cite this version:

P. Ferguson. Optical conductivity and direct interband transitions in Gd. Journal de Physique

Colloques, 1979, 40 (C5), pp.C5-78-C5-80. �10.1051/jphyscol:1979529�. �jpa-00218946�

JOURNAL DE PHYSIQUE

Colloque C5, supplkment au no 5 , Tome 40, Mai 1979, page C5-78

Optical conductivity and direct interband transitions in Gd

P. Ferguson

Centre d'Etudes Nucltaires de Grenoble, Dtpartement de Recherche Fondamentale,

Grenoble

France

R6sumB.

- Les structures trouvkes dans les courbes de conductivitk optique pour des radiations polarisees parall6lement et perpendiculairement A l'axe

d'un monocristal de Gd au-dessous de la tempkrature de Curie ont Ctk comparCes avec les transitions directes interbandes. Ces transitions sont dkduites des bandes de conduction

spin polarisk, en considkrant I'interaction spin-orbite comme une perturbation. Les transitions spin-flip inter- dites dans les transitions dipolaires klectriques sont faiblement permises.

Abstract. -

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 tran- sitions 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

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 conduc- tion 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 anti- ferromagnetic 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 struc- ture of Gd emphazing the polarization of the conduc- tion 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 exci- tation energies for ferromagnetic Gd with the spin- orbit 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 cou- pling 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 transi- tions are shown schematically in figure 1. The para- magnetic 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,

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 ^'

^{: '~X29--}

^-

-1.5

Fig. 1.

-

K,

J. and K, f 1. Similarly, the K,

1 bands are doubly degenerate and split into K,

1 and K, fl.

The magnitude of the spin-orbit energy was taken as Es-,

0.17 eV [12]. This spin-orbit energy was calculated from the Dirac wave equation in the Hartree-Fock scheme without imposing any approxi- mations on the exchange interaction. A previous calculation employing the Schroedinger wave equa- tion with the spin-orbit interaction included as a perturbation and using the Slater exchange approxi- mation yielded an inordinately larger value for the spin-orbit energy, i.e., Es-,

0.388 eV [13].

The possible direct interband transitions with exchange and spin-orbit splitting are listed in table I for the optical wave polarized with the electric field perpendicular to the c-axis (E I c) and in table I1 with the electric field parallel to the c-axis (E

c). All transitions as listed are allowed by electric dipole interaction. As discussed extensively in the literature [4, 9, 141, spin-flip transitions are not allowed in the electric dipole limit. However, the spin-orbit inter- action can mix the band functions and consequently spin-flip transitions are allowed. The degree of mixing of the band functions and the strength of the transitions are strongly dependent upon the magnitude of the exchange splitting. We note that E../Es-,

3.02 for K, and 4.06 for K,. That is, the spin-orbit energy is not comparable to the exchange splitting and therefore these transitions are only weakly allowed.

The optical absorption of a single crystal of Gd was measured it 4.2 K by Weaver and Lynch [Ill.

Table I.

Possible direct interband transitions a t the symmetry point K in the presence of exchange and spin-orbit interaction for E I c-axis.

Transitions Energy (eV)

- -

Kg f

K7 f ^1.832

2.000

K 8

1

1 0.969

1.139 2.149 2.319

Table 11. - Possible direct interband transitions a t the symmetry point K in the presence of exchange and spin-orbit interaction for E

c-axis.

Transitions Energy (eV)

- -

f

K7 f 1.832

2.000

Ks T

^K7 f ^{0.83 1}

1.008 1.171

The optical conductivities as a function of photon energy were measured for E I c and E

c.

For E l c there is pronounced structure at E

0.7 eV, 1.2 eV and a very broad structure between E

1.8-2.5 eV. For E

c there is a broad peak at E

0.7 eV and a more pronounced peak at E

1.75 eV. Hodgson and Cleyet [2] found similar structure in measurements of optical conductivity at 105 K for polycrystalline films of Gd. The structures at 0.7 eV and 1.2 eV were not found in measurements recorded above the Curie temperature. Myers [5]

also showed that the structures at 0.7 eV and 1.2 eV were not present above the Curie temperature in polycrystalline films but were present below the Curie temperature. Petrakian [6] found similar results also in polycrystalline films.

The transitions listed in tables I and I1 can be correlated with the structure observed by Weaver and Lynch. The structure at 0.7 eV for both orienta- tions can be assigned tentatively to the K, f

K7 t

at 0.831 eV. The structure at 1.2 eV for E I c can be

assigned to either the transition of K7 1

K, 1 at

1.234 eV or the transition of K,

K, f at

E

1.171 eV. These transitions are in agreement with

C5-80 P. FERGUSON

the large peak observed in the joint density of states calculated from the relativistic energy bands [15]

by Knyazev and Noskov [16]. There are, as listed in table 11, similar transitions for E

c but no structure has been found in this photon energy range. The structure at 1.75 eV for E

c can be assigned to the transition for K,

K,

at 1.832 eV with some broadening due to the transition a t 2.00 eV. The very flat broad structure for E I c extending from 1.7 eV to 2.5 eV can be due to a multiplicity of transitions

such as K, f

K, f at 2.00 eV and K, 1

K, 1 ^at

2.149 and 2.319 eV. These assignments can only be considered as tentative due to the rudimentary manner of applying the spin-orbit splitting to the spin-pola- rized conduction bands. It remains then to apply the spin-orbit interaction as a perturbation to the rela- tivistic ferromagnetic band functions in order to evaluate the various optical matrix elements and consequently the optical conductivity. This work is in progress.

References

[I] FERGUSON, P. E. and ROMAGNOLI, R. J., J. Appl. Phys. 40 (1969) 1236.

[2] HODGSON, J. N. and CLEYET, B., J. Phys. C 2 (1969) 97.

[3] MILLER, R. F. et al., J . Phys. F : Metal Phys. 4 (1974) 2338.

[4] ERSKINE, J. L. and STERN, E. A., Phys. Rev. B 8 (1973) 1239.

[5] MYERS, H. P., J. Phys. P : Metal Phys. 6 (1976).

[6] PETRAKIAN, J. P. et al., Solid State Commun. 24 (1977) 397.

[7] MIWA, H., Prog. Theor. Phys. 29 (1963) 477.

[8] MACINTOSH, A. R., Phys. Rev. Lett. 9 (1962) 90.

[9] DIMMOCK, J. 0. et al., Proc. Intl. Coll. Metals,Paris (John Wiley and sons. N.Y.) 1965.

[ l l ] WEAVER, J. H. and LYNCH, D. H., Phys. Rev. Lett. 34 (1975) 1324.

[I21 DESCLAUX, J. P., Atomic Data and Nuclear Data Tables 12 (1973) 312.

[13] HERMANN, F. and SKILLMAN, S., Atomic Structure Calculation (Prentice-Hall, Englewood Cliffs, N.J.) 1965.

[14] FREEMAN, A. J., Magnetic Properties of Rare Earth Metals, R. J. Elliot Ed. (Plenum Press) 1972.

[15] KEETON, S. C. and Loucm, T. L., Phys) Rev. 169 (1968) 672.

[16] KNUYAZEV, Y. V. and N o s ~ o v , M. M., Phys. Met. Metall. 30 (1970) 230.

[lo] HARMON, B.

k.