Fully relativistic spin-polarized LMTO calculations of the magneto-optical Kerr effect of d and f ferromagnetic materials. II. Neodymium chalcogenides

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1 PHYSICAL REVIEW B VOLUME 59, NUMBER 22 1 JUNE 1999-II Fully relativistic spin-polarized LMTO calculations of the magneto-optical Kerr effect of d and f ferromagnetic materials. II. Neodymium chalcogenides V. N. Antonov* and B. N. Harmon Ames Laboratory, Iowa State University, Ames, Iowa A. Ya. Perlov and A. N. Yaresko Institute of Metal Physics, 36 Vernadsky Street, Kiev, Ukraine Received 21 October 1998 The optical and magneto-optical spectra of neodymium NaCl-type compounds NdX (X S, Se, and Te and Nd 3 S 4 with Th 3 P 4 crystal structure are investigated theoretically using an energy-band approach in combination with the linear-response formalism. The energy band structure is obtained within the local spin-density approximation LSDA, and within its extension which explicitly takes into account the onsite 4 f Coulomb interaction U (LSDA U). Although the LSDA predicts a large Kerr rotation in the compounds under consideration, we find that the LSDA U gives a better description of the Kerr spectra of NdS and Nd 3 S 4. The origin of the Kerr rotation realized in the compounds is examined. S I. INTRODUCTION Rare-earth compounds and alloys exhibit a great variety of unusual properties. Among them one finds heavy-fermion systems, intermediate valence compounds, Kondo metals, and Kondo insulators. To correctly understand the physical properties of these materials, it is necessary to investigate in detail their electronic structure. In this light, optical and MO spectroscopy has proven to be an extremely useful tool for the study of the f states in rare-earth and actinide compounds. 1 Over the past decade the MO properties of rare earth compounds have attracted large interest, 1 which increased after the discovery of the maximal observable rotation of 90 for the Kerr angle in CeSb by Pittini et al. 2 The majority of MO investigations deal with compounds and alloys of the light rare earth ions Ce 3 and Nd 3, the half filled shell ion Eu 2, and the heavy rare earths Tm 2 and Yb 3. 1 As with most lanthanides, cerium and neodymium form face centered cubic FCC rock-salt structure binary chalcogenides with the VIA elements of the periodic table of the elements S, Se, Te and the same FCC structure with the pnictides, the VA elements N, P, As, Sb, Bi. 3 With the exception of the nitrides, the majority of cerium and neodymium chalcogenides and pnictides order antiferromagnetically in zero magnetic field. CeN has no order down to 1.5 K, while NdN orders ferromagnetically. 1 The MO properties of cerium chalcogenides are well investigated experimentally, 1,2,4 and also theoretically. 5 8 The MO properties of neodymium chalcogenides are less studied. Experimental measurements of MO spectra have been performed only for NdS and Nd 3 S 4 compounds. 1,9,10 There are no previous theoretical MO calculations available for these neodymium compounds that we are aware of. In the present work we report a detailed theoretical investigation of the optical and MO Kerr spectra of NdX (X S, Se, Te and Nd 3 S 4 compounds, as the second in a series of three papers. The previous paper 11 is devoted to theoretical investigation of MO spectra of chromium spinel chalcogenides and the third one to MO properties of uranium chalcogenides and pnictides. 12 The paper is organized as follows. The computational details are explained in Sec. II. Section III presents the electronic structure, optical and MO spectra of NdS, NdSe, NdTe, and Nd 3 S 4 compounds calculated in the LSDA and LSDA U approximations. The results are compared to the experimental data. Finally, a summary and conclusions are given in Sec. IV. II. COMPUTATIONAL APPROACH The application of standard LSDA methods to f-shell systems meets with problems in most cases, because of the correlated nature of the f electrons. To account better for the on-site f-electron correlations, we have adopted as a suitable model Hamiltonian that of the LSDA U approach. 13 The main idea is the same as in the Anderson impurity model: 14 the separate treatment of localized f electrons for which the Coulomb f -f interaction is taken into account by a Hubbardtype term in the Hamiltonian 1 2 U i j n i n j (n i are f orbital occupancies, and delocalized s, p,d electrons for which the local density approximation is regarded as sufficient. Let us consider the f ion as an open system with a fluctuating number of f electrons. The formula for the Coulomb energy of f -f interactions as a function of the number of f electrons N given by the LSDA is E UN(N 1)/2. If we subtract this expression from the LSDA total energy functional, add a Hubbard-like term and take into account the exchange interaction we obtain the following functional: 13 where E E LSDA 1 2 U n m n m m,m, 1 2 U J m m,m, n m n m d.c., /99/59 22 / /$15.00 PRB The American Physical Society

2 ANTONOV, HARMON, PERLOV, AND YARESKO PRB 59 FIG. 1. Self-consistent fully relativistic, spin-polarized energy band structure and total DOS in states/ unit cell ev calculated for NdS treating the 4 f states as 1 fully localized (4 f in core ; 2 itinerant LSDA ; and 3 partly localized (LSDA U). d.c. U N N 1 2 JN N 1 2 JN N 1, 2 N is the total number of localized f electrons, N and N are the number of f electrons with spin-up and spin-down, respectively, U is the screened Coulomb parameter, and J is the exchange parameter. The orbital energies i are derivatives of Eq. 1 with respect to orbital occupations n i : i E E n LSDA i U J 1 2 i n E LSDA U eff 1 2 n i. This simple formula gives the shift of the LSDA orbital energy U eff /2 for occupied f orbitals (n i 1) and U eff /2 for unoccupied f orbitals (n i 0). A similar formula is found for the orbital dependent potential V i (r ) E/ n i (r ) where 2 FIG. 2. LSDA total in states/ unit cell ev and partial in states/ atom ev densities of states calculated for NdS. variation is taken not on the total charge density (r ) but on the charge density of a particular ith orbital n i (r ): V i r VLSDA r Ueff 1 2 n i. 3

3 PRB 59 FULLY RELATIVISTIC SPIN-.... II TABLE I. The experimental and LSDA U calculated spin, orbital, and total magnetic moments in B )ofndx (X S, Se, and Te and Nd 3 S 4 compounds. The experimental data are from Ref. 1. Compound Atom M s M l M total Expt. NdS Nd S NdSe Nd Se NdTe Nd Te Nd 3 S 4 Nd S The advantage of the LSDA U method is the ability to treat simultaneously delocalized conduction band electrons and localized 4 f electrons in the same computational scheme. With regard to these electronic structure calculations, we mention that the present approach is still essentially a single particle description, even though intra-atomic 4 f Coulomb correlations are explicitly taken into account. The LSDA U method has proven to be a very efficient and reliable tool in calculating the electronic structure of systems containing localized orbitals where the Coulomb interaction is much larger than the bandwidth. It works not only for 4 f orbitals of rare-earth ions, but also for such systems as transition metal oxides, where localized 3d orbitals hybridize quite strongly with oxygen 2p orbitals see review article Ref. 15. The LSDA U method was recently applied to heavy-fermion compounds YbPtBi 16 and Yb 4 As 3 Ref. 17 and it has also been used to explain the nature of the huge polar Kerr rotation in CeSb Refs. 5 and 7. The computational details of the LSDA band structure and optical calculations have been presented in a previous paper. 11 We note that for large Kerr rotation, it is not possible to use the approximate expression for the polar Kerr rotation K ( ) and Kerr ellipticity K ( ) spectra, namely 1 xy K i K. 1 xx xx xx and xy are the diagonal and off-diagonal components of the dielectric tensor, which are related to the optical conductivity tensor through ( ) (4 i/ ) ( ). Equation 4 is valid only for small K, K, and xy xx. Instead one has to use the exact expression 1 tan K e 2i K 1 n 1 n 5 1 tan K 1 n 1 n with n ( xx i xy ) 1/2, the complex indices of refraction. From Eq. 5 it can be seen that the maximal observable K is 90. Self-consistent energy band-structure calculations of NdX (X S, Se, Te and Nd 3 S 4 were performed by means 4 FIG. 3. Calculated absorptive diagonal part of the optical conductivity 1xx in s 1 ) and the optical reflectivity R of NdS compared with experimental data Ref. 9. of the fully relativistic, spin-polarized linear-muffin-tinorbital SPR-LMTO method in the atomic sphere approximation with combined corrections included The LSDA part of energy band structure calculations was based on the spin-density-functional theory with von Barth Hedin parametrization 22 of the exchange-correlation potential. Core charge densities were recalculated at every iteration of the self-consistency loop. The basis consisted of the Nd s, p, d, f, and g; S,SeandTes,p, and d, LMTO s. The energy expansion parameters E Rl were chosen at the centers of gravity of the occupied parts of the partial state densities, this gives high accuracy for the charge density. The lattice parameters are 5.679, 5.911, and Å for NdS, NdSe, and NdTe, respectively. 3 The neodymium chalcogenide Nd 3 S 4 crystallizes in the cubic Th 3 P 4 crystal structure with two formula units per primitive cell. The space group is I4 3d No. 220 with Nd at the 12a positions and S at the 16c positions. The lattice constant is equal to Å. 3 To more accurately represent the potential and charge density we include additional empty spheres in the 12b positions. The k integrated functions charge density, DOS, and l-projected DOS s were calculated by the improved tetrahedron method 23 on a grid of 1330 k points in the irreducible part of Brillouin zone BZ for rock-salt type structure of NdX and 1891 k points in the Th 3 P 4 crystal structure of Nd 3 S 4.

4 ANTONOV, HARMON, PERLOV, AND YARESKO PRB 59 FIG. 4. Calculated and experimental Kerr rotation ( K ) and Kerr ellipticity ( K ) spectra in deg. of the NdS compounds with the 4 f electrons treated as itinerant electrons LSDA, fully localized (4 f in the core and partly localized (LSDA U). The experimental data are those of Ref. 9. III. RESULTS AND DISCUSSION A. NdS, NdSe, and NdTe In our optical and MO calculations we have performed three independent fully relativistic spin-polarized band structure calculations. We consider the 4 f electrons as 1 itinerant electrons, using the LSDA; 2 fully localized, putting them in the core; and 3 partly localized, using the LSDA U approximation. We note that a difference with respect to treating the 4 f electrons as core electrons is that in the LSDA U calculation all optical transitions from the 4 f states are taken into account. Figure 1 shows the energy band structure of NdS for all the three approximations. Figure 2 presents the LSDA total and partial DOS s. The energy band structure of NdS with the 4 f electrons in the core can be subdivided into three regions separated by energy gaps. The bands in the lowest region around 14 ev not shown in Fig. 1 have mostly S s character with some amount of Nd sp character mixed in Fig. 2. The next six energy bands are S p bands separated from the s bands by an energy gap of about 0.65 ev. The highest region can be characterized as Nd spin-split Nd d bands. There is a strong hybridization between S p and Nd d states. It is important that there is an energy gap between S p and Nd d states at 3.1 to 2.7 ev. It means that all the interband transitions in the energy interval of 0 to 3.1 ev take part inside of Nd d bands see below. The sharp peaks in the DOS at the Fermi energy and at around 2 ev above are due to 4 f 5/2 and 4 f 7/2 states, respectively see Figs. 1 and 2. There are three 4 f energy bands crossing the Fermi level. In our LSDA U band structure calculations we started from a 4 f 3 configuration for the Nd 3 ion with three on-site 4 f energies shifted downward by U eff /2 and 11 levels shifted upwards by this amount. The energies of occupied and unoccupied f levels are separated by approximately U eff.we emphasize, however, that the 4 f states are not completely localized, but may hybridize, and together with all other states their energy positions relax to self consistency. It is still not clear how to choose the projections of the orbital momentum onto the spin direction m l if we have more than one occupied state. The value of the magnetic moment and MO spectra strongly depends on the m l s and it may be better to regard the values of m l s as parameters and try to specify them from comparison of the calculated physical properties with experiments. We performed calculations for every possible combination of the m l s and found that the best agreement between calculated and measured MO spectra can be achieved with m l 0, 3. These values also give total magnetic moments equal to 3.09, 3.11, and 3.14 B for NdS, NdSe, and NdTe, respectively Table I. The experimental ordered magnetic moments are equal to 2.24, 2.34, and 2.58 B for NdS, NdSe, and NdTe respectively. 1 The experimental data are from magnetization measurements in fields of 200 koe with probably not complete saturation. Usually the Hubbard-like U eff is evaluated by comparison of theoretically calculated energy positions of f bands with XPS and UPS measurements. From photoemission measurements U eff is found to be in the range of 5 to 7 ev for Nd compounds. 24 It can be also calculated from atomic Dirac-

5 PRB 59 FULLY RELATIVISTIC SPIN-.... II FIG. 5. Decomposition of the Kerr rotation spectrum of NdS in separate contributions. Top panel: calculated real and imaginary part of the diagonal dielectric function, (1) xx, and (2) xx. Third panel from the top: The imaginary part of D 1 which results from (1) xx and (2) xx. Bottom panel: The Kerr rotation in deg. which results as a product of Im D 1 and (2) xy second panel from the top in s 2 ). The experimental Kerr angle spectrum is from Ref. 9. Hartree-Fock DHF approximation, 24 Green-function impurity calculations, 25 and from band structure calculations in the super-cell approximation. 26 The DHF calculation gives U eff 6.7 ev and our calculations of U eff in the super-cell approximation give U eff 7.2 ev. The calculated value of U eff strongly depends on theoretical approximations and it may be better to regard the value of U eff as a parameter and try to specify it from comparison of the calculated physical properties with experiments. We found that the best agreement between calculated and measured MO spectra can be achieved with U eff 7 ev. The LSDA U energy bands and total DOS of NdS for U eff 7 ev are shown in Fig. 1. For Nd 3 ions three 4 f bands are fully occupied and hybridize with S p states but the other eleven 4 f bands are well above the Fermi level and are separated from them by the correlation energy U eff. Figure 3 shows the calculated absorptive diagonal part of the optical conductivity 1xx and the optical reflectivity compared with experimental data. 9 We mention, furthermore, that we have convoluted the calculated spectra with a Lorentzian whose width is 0.4 ev to approximate a lifetime broadening. There is a small peak at 0.2 ev and a large broad structure centered around 8 ev for the calculations treating 4 f electrons as core states. The first peak originates from interband transitions between eigenvalue surfaces that are quasiparallel around the symmetry point. These quasiparallel bands are visible along -L and -K symmetry directions see Fig. 1. The very broad structure centered around 8 ev is due to transitions from bands with predominantly S 3p character to unoccupied bands which have predominantly Nd 5d character. The calculation of the optical conductivity where the 4 f s are treated as band states reproduces two additional structures in the 0.0 to 3.1 ev energy interval, namely, the peaks at 0.06 and 1.55 ev Fig. 3. Both the peaks involve 4 f electron interband transitions. The second peak is responsible for the deep minimum in the optical reflectivity at 1.3 ev which is not observed in the experiment. 9 The best agreement between theory and the experiment for the optical reflectivity was found to be when we used the LSDA U approximation Fig. 3. The Coulomb repulsion U eff strongly influences the electronic structure and the optical spectra of NdS. Due to removing the 4 f states from the Fermi level, the diagonal part of the optical conductivity

6 ANTONOV, HARMON, PERLOV, AND YARESKO PRB 59 FIG. 6. Self-consistent LSDA U energy band structure and total DOS in states/ unit cell ev of NdSe and NdTe (U eff 7 ev. FIG. 8. LSDA and LSDA U energy band structure and total DOS in states/ unit cell ev calculated for Nd 3 S 4. FIG. 7. Same as Fig. 5, but for NdSe and NdTe using the LSDA U approximation. 1xx in the energy interval of 0 to 3.1 ev has only two small peaks see Fig. 3. The first peak has the same nature as a corresponding peak in the calculations with the 4 f electrons in the core. The second peak at around 1.4 ev consists of Nd 5d 4 f transitions originating from interband transitions around the symmetry point and along -L and -K symmetry directions see Fig. 1. After consideration of the band structure and optical properties we turn to the magneto-optical spectra. In Fig. 4 we show the experimental 9 K ( ) and K ( ) MO Kerr spectra of NdS, as well as the spectra calculated with LSDA, LSDA U and with the 4 f electrons in the core. This picture clearly demonstrates that the better description is achieved in the LSDA U approach. The most prominent discrepancy in the LSDA spectra is the extra peak below 1 ev which is caused by extra structure present in the interband dielectric tensor. Responsible are interband transitions involving the hybridized 4 f states, which in the LSDA approach exhibit a maximum resonance near E F. In the LSDA U approach, the 4 f state energies are shifted due to the on-site Coulomb interaction U eff. As a result, the transitions involving the 4 f states do not take place at small photon energies any more, and the erroneous peak structure around 1 ev disappears from Kerr spectra. The calculations in which the 4 f electrons are treated as quasicore are able to reproduce a very similar structure as the LSDA U calculations, but, due to the lack of corresponding 5d 4 f interband transitions, the offdiagonal part of the optical conductivity 2xy is nearly zero, so that a very small Kerr rotation is obtained. The situation is clearly seen in Fig. 5 where we show the separate contributions of both the numerator, i.e., xy ( ) and the denominator, D( ) xx 1 (4 i/ ) xx 1/2, which factor together to give the Kerr angle of NdS see the explanations in the previous paper 11. Due to the wrong position of the 4 f bands

7 PRB 59 FULLY RELATIVISTIC SPIN-.... II FIG. 9. Theoretically calculated optical spectra of Nd 3 S 4 and La 3 S 4 and the experimental spectra for Nd 3 S 4 from Ref. 28. in the LSDA calculations the imaginary part of the inverse denominator times the photon frequency, Im D 1, displays a double resonance structure at about 1.0 and 2.4 ev which leads to a disagreement with the experimental Kerr spectra. The LSDA U calculations as well as the calculations which treated the 4 f electrons as core states produce a single resonance maximum of the denominator at about 3 ev. This resonance is even larger in the former calculations, but the imaginary part of xy, i.e., 2xy, displays a very small value at 3 ev. Therefore the peak in the Kerr rotation at 3.1 ev results as a combination of a deep resonance structure of the denominator and interband Nd 5d 4 f transitions contributing into 2xy. Let us consider briefly the electronic structure and MO properties of NdSe and NdTe. The LSDA U energy band structure and total DOS of NdSe and NdTe are shown in Fig. 6. The main trend in the electronic structure of the sequence of NdX compounds (X S, Se, Te results from the characteristic trend in the chalcogenide p wave functions and from the systematic change of the lattice parameters. The counteraction of screening by inner atomic shells and of relativistic effects leads to the characteristic trend in the position of the atomic p state and hence of the center of gravity of the chalcogenide p band monotonically increasing from S to Te. On the other hand, the center of gravity for the Nd f bands is monotonically decreasing from NdS to NdTe. The SO splitting at the center of gravity is equal to 0.45 and 0.19 ev for Nd f and d, respectively, for all three compounds, but SO splitting for chalcogenide p states shows one order of magnitude increase from S 0.09 ev to Te 0.96 ev. The 6p bandwidth is monotonically increasing from 3.4 ev in NdS to 4.3 ev in NdTe due to the increasing extension of the

8 ANTONOV, HARMON, PERLOV, AND YARESKO PRB 59 FIG. 10. Calculated and experimental Kerr rotation ( K ) and Kerr ellipticity ( K ) spectra in deg. of Nd 3 S 4 with the 4 f electrons treated as itinerant electrons LSDA, fully localized (4 f in the core and partially localized (LSDA U approach. The experimental data are those of Ref. 28. atomic wave function. With the increasing chalcogenide p band widths the indirect energy gap between the top of the chalcogenide p and the bottom of the Nd 5d bands decreased in NdSe and closed in NdTe see Fig. 6. Although the SO interaction increases from S to Te, the MO Kerr spectrum is decreasing from NdS to NdTe. The explanation of this phenomena can be found in comparison of Figs. 5 and 7. In NdS the energy position of the resonance structure of the inverse denominator times the photon frequency Im D 1, coincides exactly with the maximum in 2xy. However, if one moves from NdS to NdTe through the series, the real diagonal part of complex dielectric func- FIG. 11. Same as Fig. 5, but for Nd 3 S 4 with the LSDA U approximation.

9 PRB 59 FULLY RELATIVISTIC SPIN-.... II tion 1xx crosses the energy axis at 2.91, 2.67, and 2.19 ev in NdS, NdSe, and NdTe respectively and, hence, the position of the resonance structure of the inverse denominator shifts to smaller energies. As a consequence, in NdSe and, in particular NdTe, the well pronounced resonance structure coming from the inverse denominator is situated in the energy region with a small 2xy value. B. Nd 3 S 4 Among the many neodymium-sulphur compounds NdS, Nd 2 S 3, Nd 2 S 4, Nd 4 S 7, NdS 2 ) only Nd 3 S 4 orders ferromagnetically 27 at 47 K. La 3 S 4 is isostructural but diamagnetic and serves as a reference material. Near normal incidence reflectivity was experimentally determined for Nd 3 S 4 and La 3 S 4 from 0.03 to 12 ev in Ref. 28. Other optical functions have been derived from the reflectivity data using a Kramers-Kronig transformation. The polar Kerr rotation and ellipticity was also measured. 28 It was found that the optical spectra of Nd 3 S 4 and La 3 S 4 are very similar, which was interpreted 28 as implying there are no transitions related to the occupied 4 f states in the Nd 3 S 4 compound. The LSDA and LSDA U energy band structures and total DOS s of Nd 3 S 4 are shown in Fig. 8. The LSDA energy band structure of Nd 3 S 4 can be subdivided into several regions separated by energy gaps. The bands in the lowest region at 13 ev not shown in Fig. 8 have a mostly S s character with some amount of Nd s character mixed in. The next group of bands is formed by S p states with some admixture of Nd p and d states. The large narrow peak situated at the Fermi energy is formed by 4 f bands of Nd. Unoccupied 5d bands of Nd are separated from 4 f 5/2 bands by an energy gap. Thus, applied to Nd 3 S 4, the LSDA places f states just at the Fermi energy. The main difference between LSDA NdS and Nd 3 S 4 energy band structures is in the relative position of the Nd 4 f and 5d states. Nd 4 f energy bands lay inside of wide Nd 5d bands in NdS and strongly hybridize with them. But in the case of Nd 3 S 4,4f 5/2 bands are situated below the Nd 5d bands and separated from them by an energy gap. 4 f energy bands in La 3 S 4 are empty and situated well above the Fermi energy. In our LSDA U band structure calculations of Nd 3 S 4 six on-site 4 f states for2nd atoms per unit cell were shifted downward by U eff /2 and 22 levels shifted upwards, by this amount. The energies of occupied and unoccupied 4 f levels are separated by approximately U eff Fig. 8 for which we take the same value as in the case of NdS, namely U eff 7 ev. Now the Nd 5d band is partly occupied, hence the LSDA U electronic structure of Nd 3 S 4 is similar to the LSDA La 3 S 4 one not shown in the vicinity of the Fermi energy. In Fig. 9 we show the experimental 28 diagonal parts of the complex dielectric function and optical conductivity together with calculated spectra within LSDA, LSDA U and the 4 f as core electrons approximations. The corresponding LSDA calculations for La 3 S 4 are also shown. Calculated plasma frequencies p are equal to 1.89 and 0.31 ev in NdS and Nd 3 S 4, respectively. As a consequence a deep minimum in the optical reflectivity in Nd 3 S 4 shifts towards smaller energies in comparison to the NdS compound see Figs. 3 and 9. Above 6 ev the calculated optical reflectivity of Nd 3 S 4 is higher than the experimental one, possibly due to nonperfect sample polishing or oxidation of the surface. As a result, the calculated imaginary part of complex dielectric function 2xx is higher than the experimental one above 6 ev. Except for this, there is good agreement between experiment and theory for all the three approximations. Moreover, the optical properties of La 3 S 4 are very similar to those of Nd 3 S 4, in good correspondence to experimental measurements. 28 Although the optical spectra of Nd 3 S 4 are not sensitive to the position of the 4 f bands, it is not the case for the magneto-optics. In Fig. 10 we show the experimental 28 and theoretically calculated K ( ) and K ( ) MO Kerr spectra of Nd 3 S 4. In the case of Nd 3 S 4, as well as NdS, the better description is given by the LSDA U approach. The calculation with fully localized, corelike 4 f electrons gives completely inappropriate results and is not discussed further. The most prominent discrepancy in the LSDA spectra is the extra peak about 2 ev which is caused by extra structures present in the interband dielectric tensor. Responsible are 4 f 5d interband transitions involving the occupied 4 f states, which in the LSDA approach are situated at E F. In the LSDA U approach, the 4 f states are shifted due to the on-site Coulomb interaction U eff. As a result, the 4 f 5d transitions do not take place at such photon energies, and the erroneous peak structure around 2 ev disappears from the Kerr spectra. The LSDA calculation also is not able to produce the right value of the main peak in the Kerr rotation at about 0.9 ev. This peak results from a combination of a deep resonance structure of the inverse denominator Im D 1 and interband transitions between occupied Nd 5d states near the Fermi energy and unoccupied Nd 4 f 5/2 states contributing to 2xy see Fig. 11. Although the LSDA calculations display an even larger resonance structure in the inverse denominator than for the LSDA U calculations, the imaginary part of xy, i.e., 2xy displays a very small value at 0.9 ev due to wrong position of the LSDA 4 f bands. IV. SUMMARY In conclusion we have shown that the MO spectra of NdX (X S, Se, Te and Nd 3 S 4 are very sensitive tools for drawing conclusions about the appropriate model description. Based on the calculated MO spectra we conclude that NdS and Nd 3 S 4 MO spectra are best described using the LSDA U approach. It was found that the main peak in the Kerr rotation of NdS at 3.1 ev and at 0.9 ev in Nd 3 S 4 results from a combination of a deep resonance structure of the denominator and interband Nd 5d 4 f transitions contributing into 2xy. In our LSDA U calculations we step aside from ab initio band structure calculations introducing additional parameters, namely, the value of the screened Coulomb parameter U eff 7 ev and the projections of the orbital momentum onto the spin direction m l 0, 3. The evaluation of U eff and m l from first principles requires further investigation. ACKNOWLEDGMENTS This work was carried out at the Ames Laboratory, which is operated for the U.S. Department of Energy by Iowa State University under Contract No. W This work was supported by the Director for Energy Research, Office of Basis Energy Sciences of the U.S. Department of Energy.

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