X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I

The current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 3d compounds is reviewed. Energy band theory based upon the local spin-density approximation (LSDA) describes the XMCD spectra of transition metal compounds reasonably well. Examples which we examine i...

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Дата:	2008
Автори:	Antonov, V.N., Shpar, A.P., Yaresko, A.N.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
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Теми:	Обзоp
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Цитувати:	X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I / V.N. Antonov, A.P. Shpak, A.N. Yaresko // Физика низких температур. — 2008. — Т. 34, № 1. — С. 3-44. — Бібліогр.: 214 назв. — англ.

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spelling	irk-123456789-1166182017-05-12T03:02:33Z X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Antonov, V.N. Shpar, A.P. Yaresko, A.N. Обзоp The current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 3d compounds is reviewed. Energy band theory based upon the local spin-density approximation (LSDA) describes the XMCD spectra of transition metal compounds reasonably well. Examples which we examine in detail are XPt₃ compounds (with X = V, Cr, Mn, Fe, Co, and Ni) in the AuCu₃ structure, Heusler compounds Co₂MnGe, Co₂NbSn, and compounds with noncollinear magnetic structure IrMnAl and Mn₃ZnC. Recently achieved improvements for describing the electronic and magnetic structures of 3d compounds are discussed. 2008 Article X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I / V.N. Antonov, A.P. Shpak, A.N. Yaresko // Физика низких температур. — 2008. — Т. 34, № 1. — С. 3-44. — Бібліогр.: 214 назв. — англ. 0132-6414 PACS: 75.50.Cc; 71.20.Lp; 71.15.Rf http://dspace.nbuv.gov.ua/handle/123456789/116618 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Обзоp Обзоp
spellingShingle	Обзоp Обзоp Antonov, V.N. Shpar, A.P. Yaresko, A.N. X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Физика низких температур
description	The current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 3d compounds is reviewed. Energy band theory based upon the local spin-density approximation (LSDA) describes the XMCD spectra of transition metal compounds reasonably well. Examples which we examine in detail are XPt₃ compounds (with X = V, Cr, Mn, Fe, Co, and Ni) in the AuCu₃ structure, Heusler compounds Co₂MnGe, Co₂NbSn, and compounds with noncollinear magnetic structure IrMnAl and Mn₃ZnC. Recently achieved improvements for describing the electronic and magnetic structures of 3d compounds are discussed.
format	Article
author	Antonov, V.N. Shpar, A.P. Yaresko, A.N.
author_facet	Antonov, V.N. Shpar, A.P. Yaresko, A.N.
author_sort	Antonov, V.N.
title	X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I
title_short	X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I
title_full	X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I
title_fullStr	X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I
title_full_unstemmed	X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I
title_sort	x-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. part i
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2008
topic_facet	Обзоp
url	http://dspace.nbuv.gov.ua/handle/123456789/116618
citation_txt	X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I / V.N. Antonov, A.P. Shpak, A.N. Yaresko // Физика низких температур. — 2008. — Т. 34, № 1. — С. 3-44. — Бібліогр.: 214 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT antonovvn xraymagneticcirculardichroismindandfferromagneticmaterialsrecenttheoreticalprogressparti AT shparap xraymagneticcirculardichroismindandfferromagneticmaterialsrecenttheoreticalprogressparti AT yareskoan xraymagneticcirculardichroismindandfferromagneticmaterialsrecenttheoreticalprogressparti
first_indexed	2025-07-08T10:43:31Z
last_indexed	2025-07-08T10:43:31Z
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fulltext	Fizika Nizkikh Temperatur, 2008, v. 34, No. 1, p. 3–44 X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I (Review Article) V.N. Antonov, A.P. Shpak Institute of Metal Physics, 36 Vernadskii Str., Kiev 03142, Ukraine E-mail: antonov@imp.kiev.ua A.N. Yaresko Max-Planck-Institut for the Physics of Complex Systems, Dresden D-01187, Germany Received April 10, 2007 The current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 3d compounds is reviewed. Energy band theory based upon the local spin-density approximation (LSDA) de- scribes the XMCD spectra of transition metal compounds reasonably well. Examples which we examine in detail are XPt 3 compounds (with X = V, Cr, Mn, Fe, Co, and Ni) in the AuCu3 structure, Heusler compounds Co 2MnGe, Co 2NbSn, and compounds with noncollinear magnetic structure IrMnAl and Mn3ZnC. Recently achieved improvements for describing the electronic and magnetic structures of 3d compounds are dis- cussed. PACS: 75.50.Cc Other ferromagnetic metals and alloys; 71.20.Lp Intermetallic compounds; 71.15.Rf Relativistic effects. Keywords: electronic structure, density of electronic states, x-ray absorption spectra, x-ray magnetic circu- lar dichroism, spin-orbit coupling, orbital magnetic moments. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1. Magneto-optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. LSDA + U method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3. General properties of spin density waves . . . . . . . . . . . . . . . . . . . . 9 3. «Toy» XMCD spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4. 3d metals and compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.1. Cu3Au-type transition metal platinum alloys . . . . . . . . . . . . . . . . . . 11 4.2. Heusler compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.1. Co2MnGe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.2. Co2NbSn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3. Noncollinear magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3.1. IrMnAl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3.2. Mn3ZnC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1. Introduction In 1845 Faraday discovered [1] that the polarization vector of linearly polarized light is rotated upon transmis- sion through a sample that is exposed to a magnetic field parallel to the propagation direction of the light. About 30 years later, Kerr [2] observed that when linearly polarized light is reflected from a magnetic solid, its polarization plane also becomes rotated over a small angle with respect © V.N. Antonov, A.P. Shpak, A.N. Yaresko, 2008 to that of the incident light. This discovery has become known as the magneto-optical (MO) Kerr effect. Since then, many other magneto-optical effects, as for example the Zeeman, Voigt and Cotton–Mouton effects [3], have been discovered. These effects all have in common that they depend on the fact that the interaction of polarized light with a magnetic solid depends on its polarization. The quantum mechanical understanding of the Kerr ef- fect began as early as 1932 when Hulme [4] proposed that the Kerr effect could be attributed to spin-orbit (SO) cou- pling (see, also Kittel [5]). The symmetry between left- and right-hand circularly polarized light is broken due to the SO coupling in a magnetic solid. This leads to differ- ent refractive indices for the two kinds of circularly polar- ized light, so that incident linearly polarized light is re- flected with elliptical polarization, and the major elliptical axis is rotated by the so called Kerr angle from the original axis of linear polarization. The first system- atic study of the frequency dependent Kerr and Faraday effects was developed by Argyres [6] and later Cooper presented a more general theory using some simplifying assumptions [7]. The very powerful linear response tech- niques of Kubo [8] gave general formulas for the conduc- tivity tensor which are being widely used now. A general theory for the frequency dependent conductivity of ferro- magnetic (FM) metals over a wide range of frequencies and temperatures was developed in 1968 by Kondorsky and Vediaev [9]. The first ab initio calculation of MO properties was made by Callaway with co-workers in the middle of the 1970s [10,11]. They calculated the absorp- tion parts of the conductivity tensor elements � xx and� xy for pure Fe and Ni and obtained rather good agreement with experiment. In 1975 the theoretical work of Erskine and Stern showed that the x-ray absorption could be used to deter- mine the x-ray magnetic circular dichroism (XMCD) in transition metals when left- and right-circularly polarized x-ray beams are used [12]. In 1985 Thole et al. [13] pre- dicted a strong magnetic dichroism in the M4,5 x-ray ab- sorption spectra of magnetic rare-earth materials, for which they calculated the temperature and polarization dependence. A year later this MXD effect was confirmed experimentally by van der Laan et al. [14] at the Tb M4,5-absorption edge of terbium iron garnet. The next year Sch�tz et al. [15] performed measurements using x-ray transitions at the K -edge of iron with circularly po- larized x-rays, where the asymmetry in absorption was found to be of the order of 10–4. This was shortly fol- lowed by the observation of magnetic EXAFS [16]. A the- oretical description for the XMCD at the Fe K-absorption edge was given by Ebert et al. [17] using a spin-polarized version of relativistic multiple scattering theory. In 1990 Chen et al. [18] observed a large magnetic dichroism at the L2,3 edge of nickel metal. Full multiplet calculations for 3d transition metal L2,3 edges by Thole and van der Laan [19] were confirmed by several measurements on transition metal oxides. First considered as a rather exotic technique, MXD has now developed as an important mea- surement technique for local magnetic moments. Whereas optical and MO spectra are often swamped by too many transitions between occupied and empty va- lence states, x-ray excitations have the advantage that the core state has a purely local wave function, which offers site, symmetry, and element specificity. Recent progress in devices for circularly polarized synchrotron radiation have now made it possible to explore the polarization de- pendence of magnetic materials on a routine basis. Re- sults of corresponding theoretical investigations can be found e.g. in a review papers [20,21] and a book [22]. We have divided the work into two papers, with this pa- per I, concentrating on the description of the methods and the results for the 3d transition metal compounds including XPt3 compounds (with X = V, Cr, Mn, Fe, Co, and Ni) in the AuCu 3 structure, Heusler compounds Co2MnGe, Co2NbSn, and compounds with noncollinear magnetic structure IrMnAl and Mn3ZnC. Paper II is devoted to the electronic structure and XMCD spectra of 4 f and 5 f com- pounds. As examples of 4 f group we consider GdN com- pound. We also present uranium 5 f compounds. In those compounds where the 5 f electrons are rather delocalized, the LSDA describes the XMCD spectra reasonably well. As example of this group we consider UFe2. Particular dif- ferences occur for the uranium compounds where the 5 f electrons are neither delocalized nor localized, but more or less semilocalized. Typical examples are UXAl (X = Co, Rh, and Pt), and UX (X = S, Se,Te). The semilocalized 5 f 's are, however, not inert, but their interaction with conduc- tion electrons plays an important role. We also consider the electronic structure and XMCD spectra of heavy-fermion compounds UPt3, URu2Si 2 , UPd2Al3, UNi2Al3, and UBe13 where the degree of the 5 f localization is increased in comparison with other uranium compounds. The elec- tronic structure and XMCD spectra of UGe2 which pos- sesses simultaneously ferromagnetism and superconduc- tivity also presented. 2. Theoretical framework 2.1 Magneto-optical effects Magneto-optical effects refer to various changes in the polarization state of light upon interaction with materials possessing a net magnetic moment, including rotation of the plane of linearly polarized light (Faraday, Kerr rota- tion), and the complementary differential absorption of left and right circularly polarized light (circular dichroism). In the near visible spectral range these effects result from ex- citation of electrons in the conduction band. Near x-ray ab- sorption edges, or resonances, magneto-optical effects can 4 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko be enhanced by transitions from well-defined atomic core levels to transition symmetry selected valence states. There are at least two alternative formalisms for describing resonant soft x-ray MO properties. One uses the classical dielectric tensor [20]. Another uses the resonant atomic scattering factor including charge and magnetic contribu- tions [23,24]. The equivalence of these two description (within dipole approximation) is demonstrated in Ref. 25. For the polar Kerr magnetization geometry and a crys- tal of tetragonal symmetry, where both the fourfold axis and the magnetization M are perpendicular to the sample surface and the z-axis is chosen to be parallel to them, the dielectric tensor is composed of the diagonal � xx and � zz , and the off-diagonal � xy component in the form � � � � � � � � � � � � � � xx xy xy xx zz 0 0 0 0 . (1) A complete description of MO effects in this formalism is given by the four nonzero elements of the dielectric ten- sor or, equivalently, by the complex refractive index n( )� n i( ) ( ) ( ) ( )� � � � � �� 1 (2) for several normal modes corresponding to the propaga- tion of pure polarization states along specific directions in the sample. The solution of Maxwell’s equations yields these normal modes [26]. One of these modes is for circu- lar components of opposite ( )� helicity with wave vector h M\| \| having indexes n i ixx xy� � �� 1 � � � . (3) The two other cases are for linear polarization with h M� [25]. One has electric vector E M\| \| and index n \|\| � � � � �1 � �\|\| \|\|i zz . The other has E M� and n� � � � � � �� 1 2 2� � � �i /xx xy xx( ) . At normal light incidence the complex Faraday angle given by [25,27] � � � � �� F F Fi l c n n( ) ( ) ( ) ( )� � � � � � 2 (4) where c is the speed of light, � �F ( ) and � �F ( ) are the Far- aday rotation and the ellipticity. The complex Faraday re- sponse describes the polarization changes to the incident linear polarization on propagation through the film of thickness l. (The incident linearly polarized light is a co- herent superposition of two circularly waves of opposite helicity.) Magnetic circular dichroism is first order in M (or � xy) and is given by � �� or � �� , respectively, the later representing the magneto-optical rotation (MOR) of the plane of polarization (Faraday effect). Magnetic linear dichroism (MLD) n n� � \|\| (also known as the Voigt effect) is quadratic in M. The Voigt effect is present in both ferromagnets and antiferromagnets, while the first order MO effects in the forward scattering beam are absent with the net magnetization in antiferromagnets. The alternative consideration of the MO effects is based on the atomic scattering factor f q( , )� , which pro- vides a microscopic description of the interaction of x-ray photons with magnetic ions. For forward scattering ( )q � 0 f Z f if( ) ( ) ( )� � �� , where Z is the atomic number. �f ( )� and ��f ( )� are the anomalous dispersion corrections related to each other by the Kramers-Kronig transformation. The general equivalence of these two formalisms can be seen by noting the one-to-one corre- spondence of terms describing the same polarization de- pendence for the same normal modes [25]. For a multicomponent sample they relate to � and through: � � � � �( ) ( )� ��2 2 2 c r Z f Ne i i i i (5) � � � �( ) ( )� ��2 2 2 c r f Ne i i i , (6) where the sum is over atomic spheres, each having num- ber density N i , and re is the classical electron radius. The x-ray absorption coefficient � ��( ) of polarization � may be written in terms of the imaginary part of f � �( ) as � � � � �� ( ) ( )� �� 4 r c fe � , (7) where � is the atomic volume. X-ray MCD which is the difference in x-ray absorption for right- and left-circu- larly polarized photons ( )� �� can be presented by ( )�� f f . Faraday rotation � �F ( ) of linear polarization measures MCD in the real part �f � of the resonant mag- netic x-ray-scattering amplitude, i.e. [6], � � � � � � �F el c n n lr f f( ) [ ] [ ( ) ( )]� � � � � �� 2 Re � . (8) Finally, the scattering x-ray intensity from an elemen- tal magnet at the Bragg reflection measured in the reso- nant magnetic x-ray-scattering experiments is just the squared modulus of the total scattering amplitude, which is a linear combination of ( , )� � �� f if f ifz z with the co- efficients fully determined by the experimental geometry [27]. Multiple scattering theory is usually used to cal- culate the resonant magnetic x-ray scattering amplitude ( )� � ��f if [20,27,28]. We should mention that the general equivalence of the dielectric tensor and scattering factor descriptions holds only in the case considering dipole transitions contributing to atomic scattering factor f ( )� . Higher-order multipole terms have different polarization dependence [23]. Using straightforward symmetry considerations it can be shown that all magneto-optical phenomena (XMCD, X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 5 MO Kerr and Faraday effects) are caused by symmetry reduction, in comparison to the paramagnetic state, caused by magnetic ordering [29]. XMCD properties are only manifest when SO coupling is considered in addition. The theoretical description of magnetic dichroism can be cast into four categories. On the one hand, there are one-parti- cle (ground-state) and many-body (excited-state) theories; on the other hand, there are theories for single atoms and those which take into account the solid state. To name a few from each category, for atomic one-particle theories we refer to Refs. 30 and 31, for atomic many-particle multiplet theory to Refs. 32–35, for solid many-particle theories to Ref. 36, and for solid one-particle theories (photoelectron diffraction) to Refs. 37–40. A multiple- scattering approach to XMCD, a solid-state one-particle theory, has been proposed by Ebert et al. [41–43] and Tamura et al. [44]. To calculate the XMCD properties one has to account for magnetism and SO coupling at the same time when dealing with the electronic structure of the material considered. Per- forming corresponding band structure calculations, it is nor- mally sufficient to treat SO coupling in a perturbative way. A more rigorous scheme, however, is obtained by starting from the Dirac equation set up in the framework of relativis- tic spin density functional theory [45]: [ ]c mc V Vsp z n n n� � �� p I k k k 2 � � � � (9) with Vsp ( )r being the spin-polarized part of the ex- change-correlation potential corresponding to the z quan- tization axis, � nk is the four-component Bloch electron wave function. All other parts of the potential are con- tained inV ( )r . The 4�4 matrices � �, , and I are defined by � � � �� 0 1 0 0 1 1 0 0 1� � � � � � � � � � � � � � � � , , I , (10) where � are the standard Pauli matrices, and I is the 2�2 unit matrix. There are quite a few band structure methods available now that are based on the above Dirac equation [45]. In one of the schemes the basis functions are derived from the proper solution of the Dirac equation for the spin depend- ent single-site potentials. In another one, the basis func- tions are obtained initially by solving the Dirac equation without the spin-dependent term and then this term is ac- counted for in the variational step. In spite of this approxi- mation, the latter scheme gives results in a close agreement with the former, while being simpler to implement. Within the one-particle approximation, the absorption coefficient � for incident x-ray of polarization � and pho- ton energy �� can be determined as the probability of electron transition from an initial core state (with wave function � j and energy E j ) to a final unoccupied state (with wave function � nk and energy Enk ) � � � � �� j n n j n j n FE E E E( ) \| \| \| \| ( ) ( ).� � � � � �� k k k k ! 2 � (11) The!� is the dipole electron-photon interaction operator !� �� a , (12) where � are the Dirac matrices, a� is the � polarization unit vector of the photon potential vector [a� � � � �1 2 1 0 0 0 1/ i a z( , , ), ( , , )]. (Here +/– denotes, respec- tively, left- and right-circular photon polarizations with respect to the magnetization direction in the solid.) More detailed expressions of the matrix elements for the spin- polarized fully relativistic LMTO method may be found in Refs. 43,46. While XMCD is calculated using equation (11), the main features can be understood already from a simpli- fied expression for paramagnetic solids. With restriction to electric dipole transitions, keeping the integration only inside the atomic spheres (due to the highly localized core sates) and averaging with respect to polarization of the light one obtains the following expression for the absorp- tion coefficient of the core level with (l j, ) quantum num- bers [47]: � � � � � � lj l j l l j j l lj j 0 1 1 12 1 4 1 ( ) , , , , � � � � � � � � � � � � � � � � �� j j j , � � 1 � � � � � � � � � � �� l l j j l j l j l j j j j N E C E , , , , , ( )( ) ( ) ( ) 1 1 2 1 (13) where N El j� �, ( ) is the partial density of empty states and the C E l j l j , , ( ) � � is radial matrix elements [47]. Equation (13) allows only transitions with "l � �1, "j � �0 1, (dipole selection rules) which means that the ab- sorption coefficient can be interpreted as a direct measure for the sum of ( , )l j -resolved DOS curves weighed by the square of the corresponding radial matrix element (which usually is a smooth function of energy). This simple inter- pretation is also valid for the spin-polarized case [20]. To compare the theoretically calculated and experi- mental XMCD spectra one has take into account the back- ground intensity which affects the high energy part of the experimentally measured spectra due to different kind of inelastic scattering of the electron promoted to the con- duction band above Fermi level (scattering on potentials of surrounding atoms, defects, phonons etc.). To calculate the background spectra can be used the model proposed by Richtmyer et al. [48]. The absorption coefficient for the background intensity is � � � � ( ) ( ) ( ) � � � # $ C dE / E c cf c cfEcf % %2 2 2 2 0 � , (14) 6 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko where E E E Ecf c f c� � , and %c are the energy and the lifetimes broadening of the core hole, Ef is the energy of empty continuum level, Ef0 is the energy of the lowest un- occupied continuum level, and C is a normalization con- stant which in this paper has been used as an adjustable parameter. Concurrent with the development of the x-ray mag- netic circular dichroism experiment, some important magneto-optical sum rules have been derived [49–52]. For the L2 3, edges the l z sum rule can be written as [22] � � � � � � � � � � � $ $ l n d d z h L L L L 4 3 3 2 3 2 � � � � � � ( ) ( ) (15) where nh is the number of holes in the d band n nh d� �10 , � �l z is the average of the magnetic quantum number of the orbital angular momentum. The integration is taken over the whole 2 p absorption region. The s z sum rule can be written as � � � � � �s tz z 7 2 � � � � � $ $ $ � � � � � � � n d d d h L L L L 3 2 3 2 2� � � � � � � � � ( ) ( ) ( ) (16) where t z is the z component of the magnetic dipole opera- tor t s r r r� � �3 2( ) \| \|s / which accounts for the asphericity of the spin moment. The integration $L3 ( )$L2 is taken only over the 2 p /3 2 (2 p /1 2) absorption region. For the K edges the l z sum rule was proposed in Ref. 53 and can be written as � � � � � $ $ � � � � l n d d z h K K 2 3 � � � � � � ( ) ( ) (17) where � �l z represents the expectation value of orbital an- gular momentum, nh is the number of empty states per atom in the 4 p conduction bands. The integration is taken over the whole 1s absorption region. 2.2. LSDA + U method 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 [54]. The main idea is the same as in the Anderson impurity model [55]: the separate treatment of localized f -electrons for which the Coulomb f f� interaction is taken into ac- count by a Hubbard-type term in the Hamiltonian ( )1 2/ U n n i j i j� & (ni are f -orbital occupancies), and delo- calized s p d, , electrons for which the local density approxi- mation for the Coulomb interaction is regarded as sufficient. Hubbard [56,57] was one of the first to point out the importance, in the solid state, of Coulomb correlations which occur inside atoms. The many-body crystal wave function has to reduce to many-body atomic wave func- tions as lattice spacing is increased. This limiting behav- ior is missed in the LDA/DFT. The spectrum of excita- tions for the shell of an f -electron system is a set of many-body levels describing processes of removing and adding electrons. In the simplified case, when every f electron has roughly the same kinetic energy � f and Cou- lomb repulsion energyU , the total energy of the shell with n electrons is given by E n Un n /n f� � �� ( )1 2 and the ex- citation spectrum is given by � �n n n fE E Un� � � ��1 . Let us consider f ion as an open system with a fluctu- ating number of f electrons. The correct formula for the Coulomb energy of f – f interactions as a function of the number of f electrons N given by the LDA should be E UN N /� �( )1 2 [58]. If we subtract this expression from the LDA total energy functional and add a Hubbard-like term (neglecting for a while exchange and non-spheric- ity) we will have the following functional: E E UN N / U n ni i j j� � � � & �LDA ( ) .1 2 1 2 (18) The orbital energies � i are derivatives of (18): � �i i i E n U n� ' ' � � �� LDA 1 2 . (19) This simple formula gives the shift of the LDA orbital energy �U/2 for occupied orbitals ( )ni �1 and �U/2 for un- occupied orbitals ( )ni � 0 . A similar formula is found for the orbital dependent potential V E/ ni i( ) (r r)� � � where variation is taken not on the total charge density(( )r but on the charge density of a particular ith orbita ni ( )r : V V U ni i( ) ( )r r� � �� LDA 1 2 . (20) Expression (20) restores the discontinuous behavior of the one-electron potential of the exact density-functional theory. The functional (18) neglects exchange and non-sphe- ricity of the Coulomb interaction. In the most general rotationally invariant form the LDA + U functional is de- fined as [59,60] E n E E n E nU ULDA L(S)DA dc� � � �[ ( ), �] [ ( )] ( �) ( �)( (r r , (21) X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 7 where E L(S)DA [ ( )]( r is the LSDA (or LDA as in Ref. 58) functional of the total electron spin densities, E nU ( �) is the electron–electron interaction energy of the localized elec- trons, and E ndc ( �) is the so-called «double counting» term which cancels approximately the part of an electron-elec- tron energy which is already included in E LDA . The last two terms are functions of the occupation matrix �n de- fined using the local orbitals { }�lm� . The matrix � \| \| \| \|,n n m m� � �� generally consists of both spin-diagonal and spin-non-diagonal terms. The latter can appear due to the spin-orbit interaction or a non-col- linear magnetic order. Then, the second term in Eq. (21) can be written as [59–61]: E n U nU m m m m m m m m m� � � � ��1 2 1 2 1 2 3 4 3 4 ( , ,{ } , , � � � � � � � � �n U nm m m m m m m m� � � �1 2 1 4 3 2 3 4, , ) (22) where U m m m m1 2 3 4 are the matrix elements of the on-site Coulomb interaction which are given by U a Fm m m m m m m m k k l k 1 2 3 4 1 2 3 4 0 2 � � � , (23) with F k being screened Slater integrals for a given l and a k lm Y lm lm Y lm m m m m k kq q k k kq 1 2 3 4 4 2 1 1 2 3 4� � � � � � �� \| \| \| \|* . (24) The � �lm Y lmkq1 2\| \| angular integrals of a product of three spherical harmonics Ylm can be expressed in terms of Clebsch-Gordan coefficients and Eq. (24) becomes a C m m m m k m m m m k l l 1 2 3 4 1 2 3 4 0 0 0 2� �� , ,( ) � � � C C km m lm lm km m lm lm 1 2 2 1 1 2 3 4 , , . (25) The matrix elements U mmm m' ' and U mm m m' ' which enter those terms in the sum in Eq. (22) which contain a product of the diagonal elements of the occupation matrix can be identified as have the usual meaning of pair Coulomb and exchange integrals U U U Jmmm m mm mm m m mm� � � � � �� , . (26) The averaging of the matrices U mm� and U Jmm mm� �� over all possible pairs of m m, ' defines the averaged Cou- lomb U and exchange J integrals which enter the expres- sion for E dc . Using the properties of Clebsch–Gordan co- efficients one can show that U l U Fmm mm � � �� 1 2 1 2 0 ( ) , (27) U J l l U Jmm mm mm � � � � �� 1 2 2 1( ) ( ) ' � � � �F l C Fn l l k l k0 0 0 0 2 2 21 2 ( ), , (28) where the primed sum is over � &m m. Equations (27) and (28) allow us to establish the following relation between the average exchange integral J and Slater integrals: J l C Fn l l k l k� � �1 2 0 0 0 2 2 2( ), , (29) or explicitly J F F l� � � 1 14 22 4( ) for , (30) J F F F l� � � � 1 6435 286 195 250 32 4 6( ) for . (31) The meaning of U has been carefully discussed by Herring [62]. In, e.g., a f -electron system with nf -elec- trons per atom, U is defined as the energy cost for the re- action 2 1 1( )f f fn n n) �� , (32) i.e., the energy cost for moving a f -electron between two atoms which both initially had nf -electrons. It should be emphasized that U is a renormalized quantity which con- tains the effects of screening by fast s and p electrons. The number of these delocalized electrons on an atom with ( )n �1 f -electrons decreases whereas their number on an atom with ( )n �1 f -electrons increases. The screening re- duces the energy cost for the reaction given by Eq. (32). It is worth noting that because of the screening the value of U in L(S)DA +U calculations is significantly smaller then the bare U used in the Hubbard model [56,57]. In principle, the screened Coulomb U and exchange J integrals can be determined from supercell LSDA calcula- tions using Slater’s transition state technique [63] or from constrained LSDA calculations [64–66]. Then, the LDA +U method becomes parameter free. However, in some cases, as for instance for bcc iron [53], the value of U obtained from such calculations appears to be overestimated. Alternati- vely, the value of U estimated from the photo-emission spectroscopy (PES) and x-ray Bremsstrahlung isochromat spectroscopy (BIS) experiments can be used. Because of the difficulties with unambiguous determination of U it can be considered as a parameter of the model. Then its value can be adjusted so to achieve the best agreement of the results of LDA +U calculations with PES or optical spectra. While the use of an adjustable parameter is generally considered an anathema among first principles practitioners, the LDA + U approach does offer a plausible and practical method to 8 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko approximately treat strongly correlated orbitals in solids. It has been fond that many properties evaluated with the LDA + U method are not sensitive to small variations of the value of U around some optimal value. Indeed, the optimal value of U determined empirically is often very close to the value obtained from supercell or constrained density func- tional calculations. 2.3. General properties of spin density waves The magnetic configuration of an incommensurate spin spiral shows the magnetic moments of certain atomic planes varying in direction. The variation has a well-de- fined period determined by a wave vector q. When the magnetic moment is confined to the lattice sites the mag- netization M varies as [67] M r qr qr( ) cos ( ) sin( ) sin ( ) sin( ) cos ( n n n n n n n n n m� �� ) * + , , , - . / / / , (33) where the polar coordinates are used and mn is the mag- netic moment of atom n with a phase �n at the position rn . Here we consider only planar spirals, that is, � � 0n /� which also give the minimum of the total energy. The magnetization of Eq. (33) is not translationally invariant but transforms as M r R qR M r( ) ( ) ( )� �D , (34) where R is a lattice translation and D is a rotation around the z axis. A spin spiral with a magnetization in a general point r in space can be defined as a magnetic configura- tion which transforms according to Eq. (34). Since the spin spiral describes a spatially rotating magnetization, it can be correlated with a frozen magnon. Because the spin spiral breaks translational symmetry, the Bloch theorem is no longer valid. Computationally, one should use large super-cells to obtain total-energy of the spin spirals. However, when the spin-orbit interaction is neglected spins are decoupled from the lattice and only the relative orientation of the magnetic moments is impor- tant. Then, one can define generalized translations which contain translations in the real space and rotations in the spin space [68]. These generalized translations leave the magnetic structure invariant and lead to a generalized Bloch theorem. Therefore the Bloch spinors can still be characterized by a k vector in the Brillouin zone, and can be written as � k k k i i / u i / d ( ) exp( ) exp( ) ( ) exp( ) ( ) r kr qr r qr r � � � � � �� 2 2 � (35) The functions u k ( )r and d k ( )r are invariant with re- spect to lattice translations having the same role as for nor- mal Bloch functions. Due to this generalized Bloch theo- rem the spin spirals can be studied within the chemical unit cell and no large super-cells are needed. Although the chemical unit cell can be used, the presence of the spin spiral lowers the symmetry of the system. Only the spa- ce-group operations that leave invariant the wave vector of the spiral remain. When considering the general spin space groups, i.e., taking the spin rotations into account, the space-group operations which reverse the spiral vector to- gether with a spin rotation of � around the x axis are sym- metry operations [68]. Although the original formulation of the local-spin-den- sity approximation of density-functional theory allowed noncollinear magnetic order, first-principles calculations for this aspect have begun only recently (for a review, see Ref. 69). One application has been the study of noncollinear ground states, for example, in 1-Fe (Refs. 70–72) or in frustrated antiferromagnets [73,74]. In addition, the non- collinear formulation enables studies of finite-temperature properties of magnetic materials. Since the dominant mag- netic excitations at low temperatures are spin waves which are noncollinear by nature, it is possible to determine the magnon spectra and ultimately the Curie temperature from first principles [75–79]. 3. «Toy» XMCD spectra In order to illustrate the effect of the symmetry break- ing, caused by the simultaneous presence of the spin orbit coupling and spin polarization, and the matrix elements described in the previous section we present in Fig. 1 model K-edge XMCD spectra. These calculations are per- formed assuming that the absorption is determined by transitions between occupied «core» s states and com- pletely empty «valence» p states. The spin splitting of the s states is neglected. All radial matrix elements are as- sumed to be equal. The spectra are obtained by Gaussian broadening of individual contributions from the transi- tions between the discrete atomic-like states. The length of vertical bars in Fig. 1 is proportional to the difference of the squared angular matrix elements for left- and right- circularly polarized photons. When either the spin orbit coupling strength 2 or the effective magnetic field B is equal to zero, as in the lower panels of Fig. 1, p states with the quantum numbers \| \|� and �\| \|� are degenerate. The transitions with "� � �1 oc- cur at the same energy and the absorption spectra for left- and right-circularly polarized x-rays are the same that leads to the absence of the dichroic signal. If both the SO coupling and magnetic field are switched on the degener- acy of the � \| \|� states is lifted. Then, the transitions with "� �1 and "� � �1 have different energies and no longer compensate each other which is reflected in the increas- ing magnitude of the XMCD spectra. If the p states were partially occupied the transition to the occupied states would be prohibited and the low frequency part of the X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 9 spectra up to the energy corresponding to the transition to the first unoccupied p state would become zero. The dependence of the model L 2 3, XMCD spectra on 2 and B is illustrated by Fig. 2. In this example «core» p states are split by strong SO interaction into p /1 2 and p /3 2 sub- shells whereas their exchange splitting is set to zero. As a re- sult a non-zero XMCD signal appears even if the spin orbit coupling of «valence» d states is neglected and only their spin-polarization is taken into account (lower panel in Fig. 2,a). This situation is typical for magnetic 3d metal compounds. When the SO coupling of d states is strong they are also split into well separated d /3 2 and d /5 2 sub-shells (Fig. 2,b). One can note that because of the difference of the corre- sponding angular matrix elements L3 spectrum is mainly formed by p d /3 2 5 2/ ) transitions; the p d /3 2 3 2/ ) con- tribution being much weaker. Similar to the case of the model K spectra switching on the magnetic field B lifts the degeneracy of d3 �, and d3 �,� states which, in turn, leads to the appearance of non-zero XMCD. 4. 3d metals and compounds At the core level edge XMCD is not only element-spe- cific but also orbital specific. For 3d transition metals, the electronic states can be probed by the K , L2 3, and M 2 3, x-ray absorption and emission spectra. The experimental investigations at the K edge of 3d transition metals are rela- tively unproblematic to perform. Because dipole allowed transitions dominate the absorption spectrum for unpo- larized radiation, the absorption coefficient �K E( ) reflects primarily the DOS of unoccupied 4 p -like states of 3d tran- sition metals. The corresponding matrix elements for the 1 41 2 1 2 3 2s p/ / /) , transitions increase monotonically by nearly a factor 3 over the displayed range of energy in pure Fe and Ni [20]. Therefore there is no strict one-to-one cor- respondence �K E( ) and N Ep ( ). Because of the extremely small exchange splitting of the initial 1s state the exchange and spin-orbit splitting of the final 4 p-states that is respon- sible for the observed dichroism at the K edge. For this rea- son the dichroism in the terms of the difference in absorp- tion "� � �K K K� �� for left and right circularly polarized radiation R /K K K� "� �( )2 is found to be only of order of 1%, being highest directly at the absorption edge [20]. Apart from the studies of XMCD spectra at the K edge on the pure elements Fe [15,80–85], Co [80,85], and Ni [83,85], many different systems have been investigated so far. K edge studies supply valuable additional infor- mation, for example, on the exchange coupling in com- pounds (Mn in PtMnSb [80], Fe in Fe3Pt, FePt and FePt3 10 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko 18 19 20 21 22 Energy, eV –0.05 0 0.05 –0.05 0 0.05 In te n si ty , ar b .u n it s a –0.05 0 0.05 –0.10 –0.05 0 0.05 0.10 –0.05 0 0.05 0.10 In te n si ty , ar b .u n it s –0.05 0 0.05 2 = 0.5 2 = 0.2 2 = 0 18 19 20 21 22 Energy, eV bB = 0.5 B = 0.2 B = 0 1+–1– 1+ 1+ 1+ –1– 1+ 1– 1+ –1– 1+ 1– Fig. 1. Model K-edge XMCD spectra calculated for the cases of strong spin-polarization (B = 1) and increasing SO coupling (a) and strong SO coupling (2 = 1) and increasing spin polarization (b) of the «valence» p electrons [21]. [86], Fe2R with R = Ce, Sm, Gd, Tb, Tm, Lu [87], Fe2Gd [88], FenGdm [87], Nd 2Fe 14B [83], Ce 2Fe 17H x [89], al- loys (Fe and Co in Fe xCo 1�x [80,90], Fe and Ni in FexNi1–x) [91], Fe in Fe xPt1–x [80,92] and multilayer sys- tems (Fe in Fe/Co [90], Fe/Cu [93,94], Fe/La [81], Fe/Ce [81], Fe/Nd [95,96], Fe/Gd [97], Co in Co/Cu [93,94,98], Cu in Co/Cu [93,94,98], Fe/Cu [93,94]. In contrast to the K edge, the dichroism spectra at the L2 and L3 edges are also influenced by the spin-orbit cou- pling of the initial 2 p core states. In general, it gives rise to a very pronounced dichroism even if the local spin-po- larization of the absorbing atom is rather small [20]. This was first demonstrated for the L2 3, spectra of 5d transition metals dissolved substantially in Fe [20]. In these systems a relative difference in absorption for left- and right-cir- cularly polarized radiation RL2 3, of up to 20% has been found. The first experimental investigation of XMCD at the L2 3, edge of pure Ni have been done by Chen et al. [18]. Many different systems have been investigated ex- perimentally so far, such as various diluted and concen- trated alloy systems (Fe in FexPd1–x [99], Co in CoxPd1–x [99], Ni in NixPd1–x [99], Cr, Mn, Fe and Co in Ni [100]), compounds and multilayer systems (Fe and V in Fe/V [101,102], Co in Co/Ni, Co/Pd, Co/Pt [103], Co and Cu in Co/Cu [104]). Ab initio calculations of the XMCD spectra of Fe, Co, and Ni at the K and L2 3, edges have been carried out by a number of researches [20,43,105–111]. In the following four sections the results of theoretical investigations on the XMCD in some representative tran- sition metal systems will be presented in details. They are transition-metal alloys consisting of a ferromagnetic 3d element and Pt atom, Heusler alloys and noncollinear IrMnAl and Mn3ZnC compounds. 4.1. Cu3Au-type transition metal platinum alloys Transition-metal alloys consisting of a ferromagnetic 3d element and Pt have drawn attention over the last years due to their wide variety of magnetic properties. In these intermetallic compounds, the Pt sites have induced mag- netic moments due to the hybridization with the transition metal spin-polarized 3d states. The contribution of the or- bital moment to the total moment on the Pt sites is expected to be important because the Pt atom is so heavy (atomic number Z = 78) that the spin-orbit coupling of the Pt 5d electrons is fairly large (coupling constant 4 5d = 0.5 eV). The 3d-Pt system also was found to have a good mag- neto-optical properties (see, e.g., Ref. 112–114). Espe- cially multilayers of Co and Pt or Pd are at present inten- sively studied because of their potential application as optical storage material in MO storage devices [115,116]. X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 11 L3 26 28 30 32 34 Energy, eV –0.05 0 0.05 0.10 –0.05 0 0.05 In te n si ty , ar b . u n it s a –0.05 0 0.05 B = 0 26 28 30 32 34 Energy, eV –0.1 0 0.1 B = 0.2 –0.10 –0.05 0 0.05 0.10 In te n si ty , ar b . u n it s B = 0.5 b –0.05 0 0.05 2 = 0 2 = 0.2 2 = 0.5 L2 L2L3 1+ –1– 1+ 1– 1+ –1– 1+ 1– Fig. 2. Model L2 3, XMCD spectra calculated for the cases of strong spin-polarization (B = 1) and increasing SO coupling (a) and strong SO coupling (2 = 1) and increasing spin polarization (b) of the d electrons [21]. Besides being of interest in applications, transition-metal platinum alloys XPt3 and X3Pt are interesting because the 3d electronic states in the former compounds have less di- rect overlap and are expected to be more localized relative to the 3d states in the X3Pt materials. Also of interest is how the filling of the 3d band across the series affects the hybridization with the Pt 5d states and how this in turn af- fects the Pt spin and orbital moments. The electronic structure, magnetic and MO properties of XPt3 compounds have been investigated intensively both theoretically and experimentally. We briefly review the rather extensive number of studies below. Several band structure calculations within the local spin density approx- imation have been performed. K�bler calculated the spin magnetic moment of VPt3 using the augmented spherical wave (ASW) method [117]. The electronic structure of several ordered phases of the Fe–Pt system, namely, Fe3Pt, FePt3, Fe2Pt2, and Fe3Pt5 have been investigated by Podgorny using the LMTO method [118]. Spin-polarized band structures of MnPt3 and FePt3 were reported by Hasegawa [119]. Tohyama et al. studied the systematic trends in the electronic structure and magnetic moments of XPt3 (X = Ti, V, Cr, Mn, Fe, and Co) based on their tight-binding calculations [120]. Suda et al. carried out APW band calculations for FePt3 in the nonmagnetic an antiferromagnetic phases [121]. Their calculated local magnetic moment at the Fe site agrees well with experi- ment. Shirai et al. performed spin-polarized band structure calculations for XPt 3 (X = V, Cr, Mn, Fe, Co) using the scalar-relativistic version of the LAPW method [122]. They obtained good agreement between their calculated and the experimental spin magnetic moment at the transi- tion metal site. On the other hand, the calculated spin mo- ment of Pt atom differs significantly from the observed magnetic moment due to neglect of the spin-orbit interac- tion. The self-consistent band structure calculations of Fe3Pt were performed using a spin-polarized, scalar rela- tivistic full-potential LAPW method in Ref. 123 to calcu- late the magnetic Compton profile. The orbital and spin magnetic moments of the Cu 3Au- type transition metal intermetallics XPt3 (X = V, Cr, Mn, Fe, and Co) have been evaluated using the full-potential LAPW method with the spin-orbit interaction as a perturbation [124]. The calcula- tion reproduced the experimental trend of the orbital and spin moments well. The orbital moment of Pt in CoPt 3 is, however, underestimated by a factor of about two. Energy band structure calculations for XPt 3 (X = V, Cr, Mn, Fe, Co) were performed in Refs. 125, 126 to investigate their MO properties by means of relativistic ASW and LMTO methods using the local spin density approximation. The electronic structure, optical conductivity tensor and MO spectra of CoPt3 and Co3Pt along with some other multi- layered Co/Pt structures have been investigated experi- mentally and theoretically in Ref. 127. Recently, Maruyama et al. measured magnetic circular x-ray dichroism for CrPt3, MnPt3, CoPt3, and ferromag- netic Fe3Pt and obtained an interesting variation in the or- bital and spin magnetic moments on the Pt sites [128,129]. The spin magnetic moment of Pt in CrPt 3 almost vanishes and the orbital moment is about �0.1 �B , antiferromag- netically coupled with the Cr magnetic moment. On the other hand, the orbital moment of Pt in MnPt3 vanishes al- most completely and a small but positive spin component (of the order of 0.1 �B parallel to the transition metal mo- ment) contributes to the Pt moment. In CoPt 3 and Fe 3Pt, the Pt orbital and spin components are positive and rela- tively larger (in the range 0.1 to 0.3 �B ). Magnetic circular dichroism in the x-ray absorption spectrum in the 2 p–3d excitation region of the transition-metal element was also measured for ferromagnetic Cu 3Au-type alloys, MnPt3, Fe3Pt and CoPt3 in Ref. 130. The x-ray magnetic circular dichroism and Faraday effect were studied in ordered and disordered Fe3Pt at K - and L3-edges in Ref. 131. Angle de- pendent XMCD experiments have been performed at both Co and Pt L2 3, edges in two epitaxial (111) CoPt 3 thin films grown at 690 and 800 K [132]. The analysis of the angular variations of the 3d orbital magnetic moment shows a strong perpendicular magneto-crystalline aniso- tropy (PMA) for the film grown at 690 K. It was related to the existence of anisotropic chemical local order yielding the formation of microscopic Co-rich and Co-poor planar local regions. In contrast, films grown above 800 K pos- sess no PMA. These changes were correlated to their iso- tropic fcc structure of L12 type. The MCD spectrum of fer- romagnetic CoPt3 was observed in the photon energy range of 50 to 80 eV in Ref. 133. MCD spectrum in the Pt N 6 7, ( )4 5f d) region shows very unusual features. This MCD spectral shape deviates significantly from the one expected by considering the conventional selection rule of the dipole transition. It was shown in the frame of the Anderson im- purity model that the unusual line shape of the MCD is caused by a strong interference (Fano effect for resonant photoemission). X-ray Faraday rotation — the dispersive analogue of XMCD — has been measured near the plati- num L edge of ferromagnetic disordered Fe3Pt [134]. The electronic structure, spin and orbital magnetic mo- ments and XMCD spectra of the series XPt3, X = V, Cr, Mn, Fe, Co, Ni and X3Pt (X = Fe, Co, Ni) reported in Ref. 135. The XAS and XMCD spectra were calculated at K , L2 3, and M 2 3, edges for transition metals and L2 3, , M 2 3, , M 4 5, , N 2 3, , N 4 5, , N 6 7, and O2 3, edges at Pt sites. 1. Magnetic moments. Orbital and spin magnetic mo- ments are determined by the interplay of hybridization, ex- change, and Coulomb interactions, crystal-field and spin-or- bit coupling. Table 1 presents the calculated spin Ms and orbital M l magnetic moments in XPt3 and X3Pt com- pounds. Our calculated results are in good agreement 12 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko Table 1. Calculated spin M s and orbital M l magnetic moments (in �B) of XPt3 and X3Pt compounds [135]. Xatom Pt atom X Ms Ml Ms Ml XPt3 V 1.3751 0.0141 –0.0509 –0.0276 Cr 2.6799 0.1789 –0.0300 –0.0558 Mn 3.6985 0.0239 0.1244 –0.0014 Fe 3.1370 0.1004 0.3045 0.0534 Co 1.7079 0.0401 0.2384 0.0486 Ni 0.4379 –0.0770 0.0780 0.0305 X3Pt Fe 2.5062 0.0870 0.2889 0.0522 Co 1.6379 0.0786 0.3153 0.0687 Ni 05178. 00198. 02343. 00866. with those from the FLAPW calculations by Iwashita et al. [124]. The variation in the spin and orbital magnetic moments calculated inside the 3d transition metal and Pt atomic spheres in the XPt3 intermetallics is compared with the experimentally measured moments in Fig. 3. The magnetic moments of the XPt 3 alloys were studied exper- imentally through neutron-scattering experiments already years ago [136,137]. These experiments showed that CoPt3 and MnPt3 are ferromagnets, while CrPt3 and VPt3 are ferrimagnets, and FePt3 is an antiferromagnet. The calculated spin magnetic moments of CoPt3 and MnPt3 at the 3d site are in good agreement with the experiment, and for CrPt3 and VPt3 the ferrimagnetic structure is in accordance with experiment, but the values of the mo- ments are less accurately reproduced. Due to band filling, linear and symmetric behavior centered at MnPt3 is clearly seen in the spin magnetic moments on the 3d tran- sition metal sites (Fig. 3,a.) The characteristic feature of the electronic structure of XPt3 compounds is the strong hybridization of transition metal 3d and Pt 5d states, the later being much more delocalized. Figure 4 shows the spin- and site-projected densities of the electronic states (DOS) for the transition metal site and the Pt site in XPt3 compounds. Strong spin-orbit interaction in the Pt atomic sphere results in splitting of d /3 2 and d /5 2 states with the energy differ- ence between their centers being ~1.5 eV. Inside the X atomic spheres the effect of the spin-orbit coupling is much weaker than the effect of the exchange field. The centers of both Pt d /3 2 and d /5 2 states lie at lower ener- gies than the centers of the corresponding X d states. As a result of the X d – Pt d hybridization, the electronic states at the bottom of the valence band are formed mainly by Pt states while the states in the vicinity of the Fermi level EF have predominantly transition metal d character with an admixture of Pt d states. The hybridization with the ex- change split X d states leads to a strong polarization of Pt d states near EF . The resulting difference in occupation numbers for Pt states with the opposite spin projections gives rise to the appearance of a comparatively large spin magnetic moment at the Pt site. A large energy splitting between the spin-up and spin-down bands is found only for states with predominant 3d character. The minor- ity-spin 3d states form rather narrow bands located near the top of the Pt d band in XPt3 compounds. It is definitely seen in Fig. 4 that, as one proceeds from the lighter to heavier 3d elements, the spin-up 3d band is first filled up to MnPt 3 and then electrons start to occupy the spin-down 3d band. This explains the linear and symmet- ric behaviors in the spin moments of the 3d atoms men- tioned above. The Pt spin moment reflects hybridization of the Pt 5d states with the 3d bands. At the beginning of 3d series (VPt3 and CrPt 3) and at the end (NiPt3) the spin magnetic moment at Pt sites is very small. The spin-up 3d bands in MnPt3, FePt3, and CoPt3 are nearly filled, therefore the spin-down hole in the Pt d states mixes with the empty 3d bands. Due to X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 13 a V Cr Mn Fe Co Ni –1 0 1 2 3 b V Cr Mn Fe Co Ni 0.4 0.2 0 – 0.2 – 0.4 O rb it al m ag n et ic m o m en t ( B ) � S p in m ag n et ic m o m en t ( B ) � X atom Pt atom Fig. 3. Theoretically calculated spin (a) and orbital (b) magnetic moments on the Pt sites (open circles) and X sites (open squares) [135] in comparison with the experimental data for XPt3 and Fe3Pt compounds. The experimental Pt orbital moments (solid circles) for X = Cr, Mn, and Fe are from Ref. 129, Co orbital (solid square) and Pt orbital for CoPt3 are from Ref. 132, the ex- perimental spin magnetic moments on X sites (solid squares) are from Ref. 136 and on Pt sites (solid circles) from Ref. 138. stronger hybridization between the Pt d states and the empty 3d bands in FePt3 and CoPt3 (Fig. 4) the Pt spin moments are larger than in MnPt 3 where the largest 3d moment was found. The agreement between calculated spin magnetic moments at Pt sites and the experimentally derived moments is very good (Fig. 1,a.) As can be seen from Fig. 1,b the variation in the orbital magnetic moments on the Pt sites observed by Maruyama et al. in XMCD experiments [129] is well reproduced by our calculations. However, the interpretation of the or- bital moments is more complicated than in the case of the spin moments. To illustrate the influence of the spin-orbit interaction on the initial and final states involved in the transitions let us introduce a site-dependent function dm Etl ( ) given by [139] dm E l E Etl n tl n z tl n n( ) \| � \| ( )� � � �� k k k k � (36) where �l z is z-projection of the angular momentum opera- tor, Enk and tl nk are the energy of the nth band and the part of the corresponding LMTO wave function formed by the states with the angular momentum l inside the atomic sphere centered at the site t, respectively. In anal- ogy to the l-projected density of states, dm Etl ( ) can be considered as the site- and l-projected density of the ex- pectation value of �l z . This quantity has purely relativistic origins and when the SO interaction is equal to zero dm Etl ( ) � 0. As van Vleck [140] showed for a free ion, the absence of orbital degeneracy is a sufficient condition for the quenching of the orbital moment, which means tha t the f i r s t -order contr ibut ion should vanish : � � � k k\| � \| .l z 0 Thus, dm Etl ( ) can be considered as a measure of the SO interaction of the electronic states. Furthermore, just as the number of states is defined as the integral of DOS, we can define the integral of dm Etl ( ) m E dm E dEtl tl E E b ( ) ( )� $ , (37) where Eb is the bottom of the valence band. Then, the or- bital moment M l at the site t is given by: M m El tl F� ( ) (38) (here and henceforth we will drop the index t for simplicity). Both dm El ( ) and m El ( ) are defined in the local coordi- nate system chosen in such a way that z axis is directed along the magnetization. It is worth noting that the only nonzero matrix elements of the �l z operator calculated be- tween real harmonics with l � 2 are \| \| � \| \|� � ��d l dx y z xy2 2 2 and \| \| � \| \|� � �d l dxz z yz 1. Hence, the largest contribution to m El ( ) can be expected from the d x y2 2� and d xy orbitals. The m El ( ) function is proportional to the strength of the SO interaction and the value of the spin magnetic moment and it depends on the local symmetry. In the particular case of XPt3 compounds the local symmetry for X atoms is O h and for Pt atom it is D4h. For the Oh group, basis functions are E d dg z x y ( , )2 2 2� , and T 2g (d xy , d yz , d xz ) while for the D 4h group they are A g1 (d z2 ), B 1g ( )d x y2 2� , B 2g (d xy) and E g (d yz , d xz ). So the largest contribution to m El ( ) can be expected in the case when we have simul- taneously (at the same energy) a large contribution from the B 1g and B 2g states at Pt sites and the Eg and T 2g sta- tes at X sites in XPt 3 compounds. 14 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko XPt3 compounds P ar ti al d -d en si ty o f st at es , el ec tr o n s/ at o m eV� V spin up spin down 0 2 4 Cr 0 2 4 Mn 0 2 4 6 Fe 0 1 2 3 4 Co 0 1 2 3 Ni –8 –6 –4 –2 0 2 Energy, eV 0 1 2 Pt 0 0.5 1.0 Pt 0 0.5 1.0 Pt 0 0.5 1.0 Pt 0 0.5 1.0 Pt 0 0.5 1.0 Pt –8 –6 –4 –2 0 2 Energy, eV 0 0.5 1.0 Fig. 4. Self-consistent fully relativistic, spin-polarized partial d-density of states of XPt3 compounds (in electrons/atom· eV) [135]. Figure 5 shows the functions dm El ( ) and m El ( ) calcu- lated for Co and Pt sites together with the partial d density of states in CoPt 3 as an example. Here and in the rest of the paper we will only consider the contribution coming from d orbitals to the ml related functions. Both the dm El ( ) and m El ( ) functions show strong energy depend- ence. Although the variations of the functions at Co site are significantly larger in comparison with those at Pt sites, Pt and Co d orbital moments M l are almost equal (see Fig. 3,b and Table 1). When considering the m El ( ) (Fig. 6) as well as dm El ( ) (not shown) functions for XPt 3 compounds, one recog- nizes that especially those of CrPt 3, MnPt 3, FePt 3 and CoPt 3 are very similar for both the X and Pt sites, respec- tively. In going from CrPt 3 to CoPt 3 the Fermi level is simply shifted upwards by filling the bands with elec- trons. In CrPt 3 the Fermi level crosses m El ( ) function at maximum and minimum at Cr and Pt sites, respectively producing rather large Cr and Pt orbital moments with op- posite sign (Fig. 3,b and Table 1). In MnPt 3 due to filling the bands with one more electron, the Fermi level is situ- ated at the local minimum in m El Mn ( ) and at zero crossing in m El Pt ( ). As a result MnPt 3 has a very small orbital magnetic moment at the Mn site and almost a zero Pt or- bital moment. Further shifting of the Fermi level in FePt3 places the Fermi level at a local maximum at both the Fe and Pt sites, providing rather large Fe and Pt orbital mo- ments with the same sign. We should also mention that ab- solute magnitude deviation of the m El ( ) function de- creases at the beginning and end of the 3d row reflecting the decreasing of the spin magnetic moments in the com- pounds. It is interesting to compare the electronic structure and orbital magnetic moments in the XPt 3 and X 3Pt com- pounds. Figure 7 shows d partial density of states in CoPt3 and Co3Pt compounds at both the Co and Pt sites. In the CoPt 3 compound the transition metal site is sur- rounded by 12 Pt sites. On the other hand, in Co 3Pt, 4 of the nearest neighbors are Pt sites and the other 8 are Co sites. This difference in Co–Co coordination has a dra- matic effect on the width of the 3d spin down DOS near the Fermi level. The 3d spin up states are centered more in the middle of the energy range of the Pt 5d states and therefore the Co orbitals hybridize rather effectively with the 5d orbitals in both the CoPt 3 and Co 3Pt compounds. On the other hand, 3d spin down states are centered in vicinity of the Fermi level where the Pt 5d DOS is rather X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 15 Co Pt –0.2 0 0.2 m (E ) l m (E ) l –0.4 0 0.4 d m (E ) l d m (E ) l eg t2g –8 –6 –4 –2 0 2 Energy eV, –2 –1 0 1 N d (E ) N d (E ) –0.1 0 0.1 –0.2 0 0.2 spin up spin down –8 –6 –4 –2 0 2 –0.4 0 0.4 Energy, eV Fig. 5. The dm El( ), m El( ) functions and partial densities of sta- tes (in electrons/atom· eV� spin) for l = 2 in CoPt3 [135]. m (E ) l m (E ) l V –0.2 0 0.2 Cr –0.2 0 0.2 Mn –0.2 0 0.2 Fe –0.4 –0.2 0 0.2 0.4 Co –0.2 0.0 0.2 Ni –6 –4 –2 0 2 –0.2 0 0.2 Pt –0.05 0 0.05 Pt –0.05 0 0.05 Pt –0.05 0 0.05 Pt –0.05 0 0.05 Pt –0.05 0 0.05 Pt –6 –4 –2 0 2 –0.05 0 0.05 Energy, eVEnergy, eV Fig. 6. The m El( ) functions for l = 2 in XPt3 compounds [135]. small. As a result they have smaller 3d–5d hybridization. Thus the 3d spin down states are more localized in CoPt 3 and more itinerant in Co 3Pt. 2. XMCD spectra. At the core level edge XMCD is not only element-specific but also orbital specific. For 3d transition metals, the electronic states can be probed by the K , L2 3, and M 2 3, x-ray absorption and emission spec- tra whereas in 5d transition metals one can use the K , L2 3, , M 2 3, , M 4 5, , N 2 3, , N 4 5, , N 6 7, , and O2 3, spectra. As pointed out above, Eq. (13) for unpolarized absorption spectra � �0( ) allows only transitions with " "l j� � � �1 0 1, , (di- pole selection rules). Therefore only electronic states with an appropriate symmetry contribute to the absorp- tion and emission spectra under consideration (Table 2). We should mention that in some cases quadrupole transi- tions may play an important role, as it occurs, for exam- ple, in rare earth materials (2 4p f) transitions) [141]. 3. K edge of 3d transition metal elements. Figure 8 shows the theoretically calculated K XMCD in XPt 3 (X = Mn, Fe, and Co) compounds. Because dipole allowed transitions dominate the absorption spectrum for unpolarized radiation, the absorption coefficient �5 K E( ) (not shown) reflects pri- marily the DOS of unoccupied 4p like states N Ep ( ) of X above the Fermi level. Due to the energy dependent radial matrix element for the 1 4s p) there is no strict one-to-one correspondence between �K E( ) and N Ep ( ). The exchange splitting of the initial 1s core state is extremely small [142] therefore only the exchange and spin-orbit splitting of the fi- nal 4p states is responsible for the observed dichroism at the K edge. For this reason the dichroism is found to be very small (Fig. 8). It was first pointed out by Gotsis and Strange [107] as well as Brooks and Johansson [143] that XMCD K spectrum reflects the orbital polarization in differential form d l z� �/dE of the p states. As Fig. 8 demonstrates, where K XMCD spectra is shown together with dm El ( ) functions (Eq. (36)), both quantities are indeed closely related to one another giving a rather simple and straightforward interpretation of the XMCD spectra at the K edge. As in the case of absorption, K emission XMCD spec- trum (not shown) is in close relationship with site- and l projected density of the expectation value of �l z . Using the sum rule derived for K spectra in Ref. 110 we obtain a Mn 5 p orbital magnetic moment of around �0.0019 �B in a good agreement with LSDA calculations (�0.0016 �B). 4. L2 3, and M 2 3, edges of 3d transition metal elements. Be- cause of the dipole selection rules, apart from the 4 1 2s / states (which have a small contribution to the XAS due to relatively small 2 4p s) matrix elements [20]) only 3 3 2d / states occur as final states for L2 XAS for unpolarized radiation, whereas for theL3 XAS 3 5 2d / states also contribute (Table 2. Although the 2 33 2 3 2p d/ /) radial matrix elements are only slightly smaller than for the 2 33 2 5 2p d/ /) transitions the angular matrix elements strongly suppress the 2 33 2 3 2p d/ /) contri- bution (see Eq. (13)). Therefore in neglecting the energy 16 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko P ar ti al d -D O S , el ec tr o n s/ at o m eV� Pt spin up spin down –2 –1 0 1 2 Co CoPt3 Co3Pt –8 –6 –4 –2 0 2 Energy, eV –2 0 2 Fig. 7. Self-consistent fully relativistic, spin-polarized partial d-density of states of XPt3 compounds (in electrons/atom· eV) [135]. Table 2. Angular momentum symmetry levels indicating the di- pole allowed transitions from core states to the unoccupied valence states with the indicated partial density of states character. Spectra K L2 L3 M2 M2 M4 M5 N6 N7 N2 N3 N4 N5 O2 O3 Core level 1s1/2 2 p1/2 2 p3/2 3 p1/2 3 p3/2 3d3/2 3d5/2 4 f5/2 4 f7/2 4 p1/2 4 p3/2 4d3/2 4d5/2 5 p1/2 5 p3/2 Valence states p1/2 p3/2 s1/2 s1/2 p1/2 p3/2 d3/2 d /3 2 d3/2 d3/2 p3/2 f5/2 d5/2 g7/2 d5/2 f5/2 f7/2 g7/2 g9/2 dependence of the radial matrix elements, the L2 and L3 spec- tra can be viewed as a direct mapping of the DOS curve for 3 3 2d / and 3 5 2d / character, respectively. In contrast to the K edge, the dichroism at the L2 and L3 edges is also influenced by the spin-orbit coupling of the initial 2 p core states. This gives rise to a very pro- nounced dichroism in comparison with the dichroism at the K edge. Figure 9 shows the theoretically calculated 3d transition metal L2 3, XMCD spectra in XPt 3 alloys in comparison with the experimental data [130]. One finds that the theoretical XMCD spectra for the late transition metals to be in good agreement with experiment. For MnPt 3 the calculated magnetic dichroism is somewhat too high at the L2 edge. As one can see, the XMCD spectra for Fe 3Pt, CoPt 3 and NiPt 3 are very similar with strong decreasing of the dichroism in NiPt 3 reflecting first of all the decrease of the spin magnetic moment in the later compound at the transition metal site (Table 1). The XMCD spectra of ferrimagnetic VPt 3 and CrPt 3 differ significantly at the L3 edge from the spectra of the ferro- magnetic compounds. The former spectra have an addi- tional positive peak in the low energy region. The XMCD spectra at the L2 3, edges are mostly deter- mined by the strength of the SO coupling of the initial 2 p core states and spin-polarization of the final empty 3 3 2 5 2d / , / states while the exchange splitting of the 2 p core states as well as the SO coupling of the 3d valence states X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 17 X M C D ab so rp ti o n sp ec tr a K -e d g e, ar b . u n it s CrPt 3 dml XMCD 0 0.005 MnPt 3 0 0.005 FePt 3 –0.002 0 0.002 0.004 CoPt 3 0 2 4 6 8 Energy, eV –0.002 –0.001 0 0.001 0.002 Fig. 8. Theoretically calculated 3d transition metal K XMCD absorption spectra in comparison with dm El( ) function in XPt3 alloys [135]. L edge2,3 M edge2,3 X -r ay m ag n et ic ci rc u la r d ic h ro is m , ar b . u n it s VPt3 –2 0 2 CrPt3 –2 0 2 L3 L2 MnPt3 theory exper. –2 0 2 Fe Pt3 –2 0 2 CoPt3 –2 0 2 NiPt3 –10 0 10 20 –0.3 0 0.3 –0.4 –0.2 0 0.2 –0.4 –0.2 0 0.2 M3 M2 –0.4 –0.2 0 0.2 –0.4 –0.2 0 0.2 –0.4 –0.2 0 0.2 –10 0 10 20 Energy, eVEnergy, eV –0.1 0 0.1 Fig. 9. Theoretically calculated 3d transition metal L3 (full line), L2 (dotted line) and M 3 (full line), M 2 (dotted line) XMCD spectra in XPt3 alloys [135] in comparison with exper- imental data (circles) [130]. are of minor importance for the XMCD at the L2 3, edge of transition metals [20]. To investigate the influence of the initial state on the resulting XMCD spectra we calculated also the XAS and XMCD spectra of XPt3 compounds at the M 2 3, edge. The spin-orbit splitting of the 3 p core level is of one order of magnitude smaller (from about 0.73 eV in V to 2.2 eV in Ni) than for the 2 p level (from 7.7 eV in V to 17.3 eV in Ni) at the X-site in the XPt 3 compounds. As a result the magnetic dichroism at the M 2 3, edge is much smaller than at the L2 3, edge (Fig. 9). Besides the M 2 and the M 3 spec- tra are strongly overlapped and the M 3 spectrum contrib- utes to some extent to the structure of the total M 2 3, spec- trum in the region of the M 2 edge. To decompose a corresponding experimental M 2 3, spectrum into its M 2 and M 3 parts will therefore be quite difficult in general. It worth to mentioning that the shape of L3 and M 3 XMCD spectra are very similar. 5. Pt L2 3, edges. As mentioned above, XMCD investi- gations supply information on magnetic properties in a component resolved way. This seems especially interest- ing if there is a magnetic moment induced at a normally nonmagnetic element by neighboring magnetic atoms. The underlying mechanism of the magnetic and mag- neto-optical properties of the systems considered here is the well known ability of transition metals to induce large spin polarization of Pt via strong 3d d�5 hybridization and exchange interaction. A very extreme example for this situation occurs for Pt in the XPt 3 compounds. Results of the theoretical calculations for the circular dichroism at the L2 3, edge of Pt are shown in Fig. 10 in comparison with the experimental data [128,129]. As one can see, a rather pronounced XMCD is found. In ferro- magnetic compounds MnPt 3, Fe 3Pt, and CoPt 3, the XMCD spectrum is negative at the L3 and positive at the L2 edge as has been seen for the XMCD spectra at L2 3, edges of the 3d transition metals (Fig. 9). The XMCD in MnPt 3 at the L3 and L2 edges are of nearly equal magni- tude, which suggest that an orbital magnetic moment al- most vanishes in the Pt 5d states in this compound [129]. In ferrimagnetically ordered VPt 3 and CrPt 3 the XMCD spectra at the L3 edge are positive with a double peak structure in a good agreement with the experimental mea- surements [128]. The experimental XMCD spectrum of CrPt3 at the Pt L2 edge shows a positive sign although the theoretically calculated spectrum has additional negative components at both the low and high energy sides of the main peak (Fig. 10). 6. Pt M, N and O edges. To investigate the influence of the initial state on the resulting Pt XMCD spectra we calcu- lated also the XAS and XMCD spectra of XPt 3 compounds at the M 2 3, , M 4 5, , N 2 3, , N 4 5, , N 6 7, , and O2 3, edges. We found a systematic decreasing of the XMCD spectra in terms of R /� "� �( )2 0 in the row L M N2 3 2 3 2 3, , ,� � edges. Although the magnetic dichroism at the O2 3, and N 6 7, edges became almost as large as at the L2 3, edge. Besides, the lifetime widths of the core O2 3, and N 6 7, levels are much smaller than L2 3, ones [144]. Therefore the spectros- copy of Pt atoms in the ultra-soft x-ray energy range at the O2 3, and N 6 7, edges may be a very useful tool for investi- gating the electronic structure of magnetic materials. Pt M 4 5, and N 4 5, spectra to some extent can be con- sidered as an analog of the K spectrum. As it was men- tioned above, K absorption spectrum at both sites re- flects the energy distribution of empty p /1 2 and p /3 2 energy states (Table 2). The M 4 (N 4) absorption spectra due to the dipole selection rules occur for the transition 18 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko Pt L3 Pt L2 X -r ay m ag n et ic ci rc u la r d ic h ro is m , ar b . u n it s VPt3 VPt3 –0.2 0 0.2 –0.2 0 0.2 CrPt3 CrPt3 theory exper. –0.2 0 0.2 –0.2 0 0.2 MnPt3 MnPt3 theory exper. –0.2 0 0.2 –0.2 0 0.2 Fe Pt3 Fe Pt3 –0.4 0 0.4 –0.4 0 0.4 CoPt3 CoPt3 theory exper. –0.4 –0.2 0 0.2 –0.4 –0.2 0 0.2 NiPt3 NiPt3 –20 0 20 –0.2 0 0.2 –20 0 20 –0.2 0 0.2 Energy, eVEnergy, eV Fig. 10. Theoretically calculated Pt L2 3, XMCD spectra in XPt3 alloys (full line) [135] in comparison with the experiment (cir- cles), the experimental data for CrPt3 are from Ref. 128, MnPt3 from Ref. 129, and CoPt3 from Ref 132. from the 3 3 2d / (4 3 2d / ) core states to the p /1 2, p /3 2, and f 5 2/ valence states above Fermi level, whereas for the M 5 (N 5) XAS the p /3 2, f 5 2/ , and f 7 2/ states contributes. Results of the theoretical calculations of the circular dichroism in absorption at the N 4 5, edge of Pt in the XPt3 (X = Cr, Mn and Fe) are shown in Fig. 11. Comparing this spectra with the corresponding XMCD spectra of transition metals at the K edge (Fig. 8) one can see an ob- vious resemblance between these two quantities (the magnetic dichroism at the N 4 edge has an opposite sign to the XMCD at the K and N 5 edge). Such a resemblance reflects the similarity of the energy distribution of unoc- cupied p local partial densities of states N Ep ( ) just above the Fermi level at X and Pt sites (not shown). It oc- curs due to a strong X p–Pt p hybridization effect. The major difference is seen at 0 to 2 eV. It can be attributed to an additional contribution of the f / /5 2 7 2, energy states to the N 4 5, spectra and to the difference in the radial ma- t r i x e l e m e n t s (1 1 2 3 2s p / /) , i n K s p e c t r a a n d 4 3 2 5 2 1 2 3 2d p/ / / /, ,) in N 4 5, spectra). Although the later plays a minor role due to the fact that radial matrix ele- ments are smooth functions of energy. It is interesting to compare Pt XAS and XMCD spectra at the L2 3, , O2 3, and N 6 7, edges. Due to the dipole selec- tion rules, for unpolarized radiation (apart from the s /1 2 states which have a small contribution to the XAS) only 3 3 2d / states occur as final states for L2 as well as for O2 spectra (Table 2). The L3 and O3 spectra reflect the energy distribution of both the 3 3 2d / and 3 5 2d / empty states. On the other hand, the N 7 absorption spectrum reflects only the 3 5 2d / states (the density of the g / /7 2 9 2, states is really very small) whereas for the N 6 XAS both the 3 3 2d / and 3 5 2d / states contribute. Therefore we have an inverse sit- uation: N 6 absorption spectra correspond to the L3 and O3 spectra, whereas the N 7 is the analog of the L2 and O2 ones. This situation is clearly seen in Fig. 12 where the theoretically calculated XMCD spectra of XPt 3 com- pounds at the O2 3, and N 6 7, edges is presented. The XMCD spectra at O3 edges are almost identical to the spectra at the N 6 edges. The XMCD spectra at O2 edges are also very similar to the spectra at the N 7 edges (but not identical because the energy distribution of Pt 3 3 2d / and 3 3 2d / states is not exactly the same due to SO inter- action). The magnetic dichroism (e.g. in CoPt 3) is nega- tive at the O3 edge and positive at the O2 edge (as it was at L2 3, edges, see Fig. 10), but the XMCD is positive at the N 7 edge and negative at the N 6 one. However, we empha- size that O3 (O2) and N 6 (N 7) XMCD spectra are not identical to the L3 (L2) ones. One can argue that at least for Pt the L2 3, spectra predominantly reflect atomic as- pects of the valence band while for the O2 3, and N 6 7, edge the itinerant aspects are more important. This is espe- cially pronounced in ferrimagnetic VPt 3, CrPt 3 and fer- romagnetic MnPt 3 with more itinerant character of the valence states than in CoPt 3 and NiPt 3 with relatively more localized 3d states. Because of the relatively small spin-orbit splitting of the 4 f states of Pt (~3.3 eV), the N 6 and the N 7 spectra have an appreciable overlap. For this reason the N 7 spec- trum contributes to some extent to the structure of the to- tal N 6 7, spectrum in the region of the N 6 edge, as can be seen from the Fig. 12. To decompose a corresponding ex- perimental N 6 7, spectrum into its N 6 and N 7 parts will therefore be quite difficult in general. Finally, we explored the anisotropy of the XMCD spectra with respect to the magnetization direction in these compounds. The influence of the direction of the magnetization on the XMCD spectra was found to be very small in XPt 3 compounds. The comparatively small de- pendence of the XMCD spectra on the magnetization di- rection is related to the high degree of isotropy inherent to the Cu 3Au structure. 7. XMCD sum rules. It is interesting to compare the spin and orbital magnetic moments obtained from the the- oretically calculated XAS and XMCD spectra through the sum rules (Eqs. (15), (16)) with directly calculated LSDA values. In this case we at least avoid all the experimental problems. Figure 13 shows direct and sum rule derived spin and orbital magnetic moments from the theoretical XMCD L2 3, spectra. The general trend of the sum rule results is in a good agreement with the LSDA calculated spin and or- bital magnetic moments for both the X and Pt sites. The X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 19 X -r ay ab so rp ti o n sp ec tr a N 4 ,5 -e d g e, ar b . u n it s N5 N4 CrPt3 –0.02 0 0.02 MnPt3 –0.02 0 0.02 FePt3 0 5 10 15 20 25 –0.02 0 0.02 Energy, eV Fig. 11. Theoretically calculated Pt N4 5, XMCD spectra in XPt3 alloys [135]. orbital magnetic moments at X and Pt sites agree well with the direct calculations, but the spin magnetic mo- ments deduced from the theoretical XMCD spectra are underestimated for VPt 3, CrPt 3, and MnPt 3 for both the transition metal and Pt sites. The disagreement at X sites reaches the 40% in VPt 3 and reduces to 19% in CrPt 3, 15% in MnPt 3 and becomes less than 10% in Fe 3Pt, CoPt3 and NiPt 3 compounds. Such behavior arises be- cause the sum rules ignore the p transitions which play an essential role in the formation of the spin magnetic mo- ments in early transition metals. In Fe, Co and Ni the rela- tive contribution of the s states is reduced and the effect plays a minor role. Thus, first principles determinations of both the XMCD spectra and ground state properties ( M l and M s) are probably required for quantitative inter- pretation of the experimental results. 4.2. Heusler compounds 4.2.1. Co 2MnGe. Electronic devices exploiting the spin of an electron have attracted great scientific interest [145]. The basic element is a ferromagnetic electrode pro- viding a spin-polarized electric current. Materials with a complete spin-polarization at the Fermi level would be most desirable. The rapid development of magneto-elec- tronics intensified the interest in such materials. Adding the spin degree of freedom to conventional electronic de- vices has several advantages such as the nonvolatility, the increased data processing speed, the decreased electric power consumption and the increased integration densities [145,146]. The current advances in new materials are pro- mising for engineering new spintronic devices in the near future [146]. A metal for spin-up and a semiconductor for spin-down electrons is called half-metallic ferromagnet [147] (HMF) and Heusler compounds have been consid- ered potential candidates to show this property [147]. The full Heusler alloys are defined as well-ordered ternary intermetallic compounds, at the stoichiometric composi- tion X YZ2 , which have the cubic L21 structure. These compounds involve two different transition metal atoms X and Y and a third element Z which is a nonmagnetic 20 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko O edge2,3- N edge6,7- O3 O2 –0.5 0 0.5 –0.5 0 0.5 –0.5 0 0.5 –0.5 0 0.5 –0.5 0 0.5 0 10 20 Energy, eV Energy, eV –0.5 0 0.5 N7 N6 VPt3 –0.5 0 0.5 CrPt3 –0.5 0 0.5 MnPt3 –0.5 0 0.5 FePt3 –0.5 0 0.5 CoPt3 –0.5 0 0.5 NiPt3 0 10 20 –0.5 0 0.5 X -r ay m ag n et ic ci rc u la r d ic h ro is m , ar b . u n it s Fig. 12. Theoretically calculated Pt O2 3, and N6 7, XMCD spec- tra in XPt3 alloys [135]. X atom S p in m ag n et ic m o m en t, B � O rb it al m ag n et ic m o m en t, B � a V Cr Mn Fe Co Ni –1 0 1 2 3 b V Cr Mn Fe Co Ni –0.2 0 0.2 d V Cr Mn Fe Co Ni –0.2 0 0.2 Pt atom c V Cr Mn Fe Co Ni LSDA sum rules –0.5 0 0.5 1.0 Fig. 13. Theoretically LSDA calculated spin (a), (c) and orbital (b), (d) magnetic moments at the X and Pt sites (open circles) in comparison with estimated data using the sum rules (solid cir- cles) for XPt3 and Fe3Pt compounds [135]. metal or nonmetallic element. Currently the Heusler al- loys are at the focus of a large scientific interest due to their potential for applications in magnetic field sensors and spintronics devices [145]. Numerous theoretical and experimental studies of Heusler alloys have been carried out, and it has been shown that composition and heat treatment are important parameters determining their magnetic properties. Co 2MnGe has a very high Curie temperature of about 905 K and a huge magnetization (M B~ 5� ). The half-me- talicity in the Co 2MnGe bulk compound was predicted for the first time in the pioneering Korringa–Kohn–Rostoker (KKR) calculations performed by Fujii et al. [148]. More recently, Ishida et al. [149] showed that the half-metallic character of Co 2MnGe strongly depends on the type of sur- face termination (Mn and Ge-terminated [001] surfaces are generally half metallic, Co-terminated ones are not), growth direction, and thickness of the film. Within this same frame- work, density-functional layer KKR simulations using the coherent potential approximation by Orgassa et al. [150,151] for the half-Heusler compound NiMnSb showed that atomic disorder (i.e., interchange of Ni and Mn or va- cancies) leads to the disappearance of the half-metallic char- acter. The Co antisite defect in Co 2MnGe were investigated in Ref. 152. It was found that half-metalicity could be lost because of defect-induced minority gap states. However, these states are efficiently screened out, as shown by the fast decay of the minority-spin charge density around EF as a function of the distance from the defective site. The same authors extend their analysis by the investigation of the ef- fects of several kind of defects such as swaps and antisites in both Co 2MnSi and Co 2MnGe hosts, in terms of formation energy and defect-induced electronic and magnetic proper- ties [153]. They show that Mn antisites have the lowest for- mation energy and a retain half-metallic character. The ef- fect of atomic disorder on the half-metalicity of similar Heusler alloys Co2CrAl and Co 2FeAl has been investigated by Miura et al. [154,155]. They have shown that disorder between Cr and Al does not significantly reduce the spin po- larization of Co 2CrAl, while disorder between Co and Cr causes a considerable reduction of the spin polarization. Block et al. [156] studied the effect of uniform strain and tetragonal distortion on the electronic and magnetic proper- ties of the Heusler compounds Co 2CrAl and NiMnSb. They showed that the half-metallic character of the Heusler phases is lost with only a few percent uniform stress or tetragonal distortion of the lattice. The orbital magnetism in some half-metallic Heusler alloys including Co 2MnGe have been studied recently by Galanakis [157] using the fully relativistic screened KKR method. It was found that the calculated orbital magnetic moments are negligible com- pared with the spin moments. X-ray magnetic circular dichroism in the ferromag- netic Co 2MnGe alloy has been measured at the Co and Mn L2 3, edges [158,159]. Using the magneto-optical sum rules the orbital moments of Co and Mn have been de- duced to be 0.07 and 0.03 �B , respectively. 1. Crystal structure. The Heusler-type X YZ2 com- pound crystallizes in the cubic L21 structure with Fm m3 space group (No. 225). It is formed by four interpenetrat- ing fcc sublattices as shown in Fig. 14. The X ions oc- cupy the 8c Wyckoff positions (x /�1 4, y �1 4/ , z /�1 4). TheY ions occupy the 4a positions (x = 0, y = 0, z = 0), and the Z ions are placed at the 4b sites (x /�1 2, y /�1 2, z /�1 2). All atoms have eight nearest neighbors at the same distance. The Y and Z atoms have eight X atoms as nearest neighbors, while for X there are four Y and four Z atoms. 2. Energy band structure. The total and partial DOS’s of Co 2MnGe are presented in Fig. 15. The results agree wel l wi th prev ious band s t ruc ture ca lcu la t ions [148,149,152,153]. The occupied part of the valence band can be subdivided into several regions. Ge 2s states appear between –12.0 and –9.9 eV. The states in the energy range –7.0 to 4.0 eV are formed by Co and Mn d states and Ge p states. Our calculations show that Co2MnGe has a local magnetic moments of 0.956 �B on Co, 3.121 �B on Mn and –0.068 �B on Ge. The orbital moments are equal to 0.032 �B and 0.024 �B on the Co and Mn sites, respec- tively. The results are in good agreement with recent rela- tivistic KKR calculations [157]. The interaction between the transition metals is ferromagnetic, leading to a total calculated moment of 4.964 �B . The crystal field at the Co 8c site (Td point symmetry) causes the splitting of d orbitals into a doublet e (3 12z � and x y2 2� ) and a triplet t 2 (xy, yz, and xz). The crystal field at the Mn 4a site (Oh point symmetry) splits Mn d states into eg (3 12z � and x y2 2� ) and t g2 (xy, yz, and xz) states. The hybridization between Mn and Co d states plays an impor- tant role in the formation of the band structure of Co2MnGe. X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 21 X Y Z Fig. 14. Schematic representation of the L21 structure. The cu- bic cell contains four primitive cells. It leads to the splitting of the d states into the bonding states which have Co and Mn d character and antibonding states with stronger contribution of Mn d states. Without the spin-orbit coupling the strong hybridiza- tion between the minority spin Mn d and Co d states leads to the opening of a gap of 0.1 eV. Thus, according to the spin-polarized calculations Co 2MnGe is a half-metallic ferromagnet. Although spin-orbit splitting of the d energy bands for both the Co and Mn atoms is much smaller than their spin and crystal-field splittings this interaction closes the energy gap for the minority spin states. The Fermi level falls in a region of very low but finite minor- ity-spin DOS. The spin polarization of the electron states at the Fermi energy is equal to 95%. Similar results for some other Heusler alloys were obtained in Refs. 161,162 where it was shown that the spin-orbit interaction can re- sult in a nonvanishing density of states in the minor- ity-spin gap of the half metals around the Fermi level, which reduces the spin polarization at EF . 3. XMCD spectra. Recently x-ray magnetic circular dichroism in the ferromagnetic Co 2MnGe alloy has been measured at the Co and Mn L2 3, edges [158,159]. The ex- perimentally measured dichroic lines have different signs at the L3 and L2 edges of Co and Mn [159]. Figure 16 shows the XAS and XMCD spectra at the L2 3, edges of Co calculated in the LSDA approach to- gether with the experimental data [159]. The correspond- ing spectra for Mn are presented in Fig. 17. The contribu- tion from the background scattering is shown by dotted line in the upper panel of Figs. 16 and 17. Because of the dipole selection rules, apart from the 4 1 2s / states (which have a small contribution to the XAS due to relatively small 2 4p s) matrix elements) only3 3 2d / states occur as final states for L2 XAS for unpolarized radia- tion, whereas for the L3 XAS the 3 5 2d / states also contrib- ute [22]. Although the 2 33 2 3 2p d/ /) radial matrix ele- ments are only slightly smaller than for the 2 33 2 5 2p d/ /) transitions the angular matrix elements strongly suppress the 2 33 2 3 2p d/ /) contribution [22]. Therefore neglecting the energy dependence of the radial matrix elements, the L2 and the L3 spectrum can be viewed as a direct mapping of the DOS curve for 3 3 2d / and 3 5 2d / character, respectively. 22 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko T o ta l D O S P ar ti al d en si ty o f st at es spin-up spin-down–5 0 5 Ge s p –1 0 1 Mn d eg t2g –2 0 2 Co d e t2 –10 –5 0 5 10 Energy, eV –2 0 2 Fig. 15. The total (in states/(cell�eV)) and partial (in stat- es/(atom�eV)) density of states of Co2MnGe [160]. The Fermi energy is at zero. X M C D ar b . u n it s , X A S ar b . u n it s , L3 L2 Co 0 5 770 780 790 800 810 Energy eV, – 1.0 – 0.5 0 0.5 Fig. 16. Upper panel: calculated (thin full line) and experimen- tal (circles) x-ray absorption spectra for the right polarized x-rays �� of Co2MnGe at the Co L2 3, edges. Experimental spec- tra (see Ref. 159) were measured by mean of total photoelectron yield with external magnetic field (1.4 T). Dotted lines show the theoretically calculated background spectra, full thick lines are sum of the theoretical XAS and background spectra. Low panel: theoretically calculated (full lines) and experimental (circles) XMCD spectra of Co2MnGe at the Co L2 3, edges [160]. The experimental Co XAS has a pronounced shoulder at the L3 peak at around 782 eV shifted by about 3 eV with re- spect to the maximum to higher photon energy. This struc- ture is less pronounced at the L2 edge. This result can be as- cribed to the lifetime broadening effect because the lifetime of the 2 1 2p / core hole is shorter than the 2 3 2p / core hole due to the L L V2 3 Coster-Kronig decay. This feature is partly due to the interband transitions from 2 p core level to 3d empty states at around 4 eV. Actually as can be seen from Fig. 18 Co d partial DOS’s have two pronounced peaks at 4 eV and 6.5 eV above the Fermi level. Both the features are reflected in the theoretically calculated XAS at the Co L3 edge around 782 and 784.5 eV, respectively (Fig. 16). Al- though the second peak is less pronounced in the experi- mental spectrum. Figure 17 presents the calculated XAS as well as XMCD spectra of the Co2MnGe compound at the Mn L2 3, edges compared with the experimental data [159]. The first peak in the empty Mn d partial DOS’s at 4 eV above the Fermi level is less intensive in comparison with the corresponding peak in the Co d PDOS (see Fig. 18), therefore the theoretically calculated XAS has only one pronounced feature at the high energy part of the XAS at around 645 eV (Fig. 17). It may be seen in the upper part of Fig. 16 that the ex- perimentally measured Co L3 XAS has some additional intensity around 782 eV which is not completely repro- duced by the theoretical calculations. A similar situation also appear in the Mn L3 XAS (Fig. 17) where the theoret- ical one-particle calculations does not reproduce all the intensity at around 643 eV. This might indicate that addi- tional satellite structures may appear due to many-body effects at the high energy tails of both the Co and Mn L2 3, XAS’s. This question needs additional theoretical investi- gation using an appropriate many-body treatment. The XMCD spectra at the L2 3, edges are mostly deter- mined by the strength of the SO coupling of the initial 2 p core states and spin-polarization of the final empty 3 3 2 5 2d / /, states while the exchange splitting of the 2 p core states as well as the SO coupling of the 3d valence states are of minor importance for the XMCD at the L2 3, edge of 3d transition metals [22]. 4. Magnetic moments. In magnets, the atomic spin M s and orbital M l magnetic moments are basic quantities and their separate determination is therefore important. Meth- ods of their experimental determination include tradi- tional gyromagnetic ratio measurements [163], magnetic form factor measurements using neutron scattering [164], and magnetic x-ray scattering [165]. In addition to these, the recently developed x-ray magnetic circular dichroism combined with several sum rules [49–52] has attracted much attention as a method of site- and symmetry-selec- tive determination of M s and M l . Because of the significant implications of the sum rules, numerous experimental and theoretical studies aimed at in- vestigating their validity for itinerant magnetic systems have been reported, but with widely different conclusions. The claimed adequacy of the sum rules varies from very good (within 5% agreement) to very poor (up to 50% discrepancy) [22]. This lack of a consensus may have several origins. For example, on the theoretical side, it has been demonstrated by circularly polarized 2 p resonant photoemission measure- X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 23 X M C D ar b . u n it s , X A S ar b . u n it s , Mn 0 5 10 15 630 640 650 660 670 Energy eV, – 2 0 L3 L2 Fig. 17. Upper panel: calculated (thin full line) and experimen- tal (circles) x-ray absorption spectra for the right polarized x-rays �� of Co2MnGe at the Mn L2 3, edges. Experimental spec- tra (see Ref. 159) were measured by mean of total photoelectron yield with external magnetic field (1.4 T). Dotted lines show the theoretically calculated background spectra, full thick lines are sum of the theoretical XAS and background spectra. Low panel: theoretically calculated (full lines) and experimental (circles) XMCD spectra of Co2MnGe at the Mn L2 3, edges [160]. d -p ar ti al D O S Co Mn 3 4 5 6 7 8 Energy, eV 0 0.2 0.4 Fig. 18. The Co and Mn d-partial density of unoccupied states (in states/(atom�eV)) of Co2MnGe [160]. Energies are relative to the Fermi energy. ments of Ni that both the band structure effects and elec- tron-electron correlations are needed to satisfactorily account for the observed MCD spectra [166]. However, it is ex- tremely difficult to include both of them in a single theoretical framework. Besides, the XAS as well as XMCD spectra can be strongly affected (especially for the early transition met- als) by the interaction of the excited electron with the created core hole [167,168]. On the experimental side, the indirect x-ray absorption techniques, i.e., the total electron and fluorescence yield methods, are known to suffer from saturation and self-ab- sorption effects that are very difficult to correct for [100]. The total electron yield method can be sensitive to the varying applied magnetic field, changing the electron de- tecting efficiency, or, equivalently, the sample photo- current. The fluorescence yield method is insensitive to the applied field, but the yield is intrinsically not propor- tional to the absorption cross section, because the radia- tive to non-radiative relative core-hole decay probability depends strongly on the symmetry and spin polarization of the XAS final states [169]. The Mn L3 and the L2 spectra in Co 2MnGe are strongly overlapped therefore the decomposition of a cor- responding experimental L2 3, spectrum into its L3 and L2 parts is quite difficult and can lead to a significant error in the estimation of the magnetic moments using the sum rules (the integration L3 $ and L2 $ in Eq. (16) must be taken over the 2 3 2p / and 2 1 2p / absorption regions sepa- rately). Besides, the experimentally measured Co and Mn L2 3, x-ray absorption spectra have background scattering intensity and the integration of the corresponding XASs may lead to an additional error in the estimation of the magnetic moments using the sum rules. It is interesting to compare the spin and orbital mo- ments obtained from the theoretically calculated XAS and XMCD spectra through sum rules [Eqs. (15), (16)] with directly calculated LSDA values in order to avoid addi- tional experimental problems. The number of the transi- tion metal 3d electrons is calculated by integrating the oc- cupied d partial density of states inside the corresponding atomic sphere which gives the values averaged for the nonequivalent sites nCo = 7.786 and nMn = 5.682. Sum rules reproduce the spin magnetic moments within 29%, and 23% and the orbital moments within 41% and 54% for Co and Mn, respectively (Table 3). XMCD sum rules are derived within an ionic model using a number of ap- proximations. For L2 3, , they are [20]: (1) ignoring the ex- change splitting of the core levels; (2) replacing the inter- action operator � �a l in Eq. (11) by � �a�; (3) ignoring the asphericity of the core states; (4) ignoring the difference of d /3 2 and d /5 2 radial wave functions; (5) ignoring p s) transitions; (6) ignoring the energy dependence of the radial matrix elements. To investigate the influence of the last point we applied the sum rules to the XMCD spec- tra neglecting the energy dependence of the radial matrix elements. As can be seen from Table 3 using the energy independent radial matrix elements reduces the disagree- ment in spin magnetic moments to 3% and 2% and in the orbital moment to 3% and 4% for Co and Mn, respec- tively. Additionally the omitting of the p s) transitions leads to almost perfect agreement between LSDA and sum rule results within 1% for the spin moments and 0% for the orbital moments. These results show that the en- ergy dependence of the matrix elements and the presence of p s) transitions affect strongly the values of both the spin and the orbital magnetic moments derived from the sum rules. Table 3. The experimental and calculated spin M s and orbital M l magnetic moments (in �B ) of Co2MnGe. Method Atom Ms Ml LSDA Ge –0.068 0.001 Co 0.956 0.032 Mn 3.121 0.024 Sum rules Co 0.681 0.019 Mn 2.412 0.011 Sum rules a Co 0.925 0.031 Mn 3.060 0.023 Sum rules b Co 0.944 0.032 Mn 3.101 0.024 LSDA + U Co 0.989 0.041 Mn 3.049 0.031 Expt. c Co 1.04 0.07 Mn 3.17 0.03 Comment: a Sum rules applied for the XMCD spectra calculated ignoring the energy dependence of the radial matrix elements. b Sum rules applied for the XMCD spectra calculated ignoring the energy dependence of the radial matrix elements and ignor- ing p s) transitions. c Reference 159. The value of the orbital magnetic moments derived from the experimental XMCD spectra is considerably higher in comparison with our band structure calculations (Table 3). It is a well-known fact, however, that LSDA calculations are inaccurate in describing orbital magne- tism [20,22]. In the LSDA, the Kohn-Sham equation is described by a local potential which depends on the elec- tron spin density. The orbital current, which is responsi- ble for M l , is, however, not included in the equations. This means, that although M s is self-consistently deter- mined in the LSDA, there is no framework to determine simultaneously M l self-consistently. Numerous attempts have been made to better estimate M l in solids. They can be roughly classified into two categories. One is based on 24 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko the so-called current density functional theory [170–172] which is intended to extend density functional theory to include the orbital current as an extra degree of freedom, which describes M l . Unfortunately an explicit form of the current density functional is at present unknown. The other category includes orbital polarization [173–176], self-interaction correction, [177] and LSDA + U [59,60] approaches. To calculate M l beyond the LSDA scheme we used the rotationally invariant LSDA + U method [60] using a dou- ble-counting correction term in the fully-localized limit approximation [178.179]. We used U J� �1 0. eV for the transition metal sites. In this case U U Jeff � � � 0 and the effect of the LSDA + U comes from non-spherical terms which are determined by F 2 and F 4 Slater integrals. This approach is similar to the orbital polarization corrections mentioned above [22]. The LSDA + U calculations produce orbital magnetic moments equal to 0.041 �B and 0.031 �B for Co and Mn sites, respectively. The Co LSDA + U orbital moment is somewhat smaller than the experimental estimates, however, the value for Mn is in perfect agreement with the experimental data. We should mention that the shape of the XMCD spectra in Co2MnGe is less sensitive to the orbital polarization correction in comparison with the evaluated orbital magnetic moments. The XAS and XMCD spectra calculated in LSDA and LSDA + U approximations have almost identical shape. 4.2.2. Co2NbSn. Ferromagnetic shape-memory alloys displaying large magnetic-field-induced strain have re- cently emerged as a new class of active materials, very promising for actuator and sensor applications. The inter- est in ferromagnetic shape memory compounds stems from the possibility of controlling the phase transition by application of a magnetic field [180]. The response of the system to a field is potentially faster than that obtained by changing temperature or applying stress, thus substan- tially increasing the range of applications. Ni 2MnGa is the only known ferromagnetic material exhibiting a martensitic transition from a high-temperature Heusler structure to a related tetragonal form [18] with a 6.6% c axis contraction at Ts � 200 K. Associated with this phase transition the material exhibits shape memory properties enabling the system to reverse large deformations in the martensitic phase by heating into the cubic phase. Co2NbSn is also reported to belong to the group of fer- romagnetic shape memory Heusler alloys. However, un- like Ni 2MnGa, for which the martensitic phase transition (Ts = 200 K) occurs within the ferromagnetically ordered phase (Tc = 376 K), for Co2NbSn the martensitic phase transition temperature Ts = 233 K is well above the ferro- magnetic ordering temperature of Tc = 116 K [182]. Con- sequently the shape memory behavior in Co 2NbSn can- not be controlled by a magnetic field. A striking feature of Co 2NbSn is the very strong sam- ple dependence of its physical properties. Sample-to- sample variations of transition temperatures by 20 K (Tc) and 50 K (Ts) are observed [183–185]. Moreover, a resis- tivity ratio ( (300 4 2 1K K/ . � and a negative temperature derivative d /dT( over a wide temperature range are ob- served. This is in conflict with a fairly large density of states at the Fermi level, as derived from the electronic specific heat coefficient [184,185]. Further, at the first- order transition, Ts , the accompanying specific heat anomaly is smeared out over a wide temperature range of 60 K. Such behavior points to the presence of crystallo- graphic disorder, which has been identified recently by means of a detailed microscopic study of the local crystal- lographic site symmetry in Ref. 182. Co 2NbSn is disor- dered on an atomic scale, i.e., there are random displace- ments of Co, Nb and Sn ions from their nominal positions in both the Fm m3 and Pmma lattices. The band structure calculations of Ni 2MnGa and Co 2NbSn both in the cubic and tetragonal phases were car- ried out in early work of Fujii et al. [186] using the KKR method. The structural transition in these compounds has been proposed as being driven by a band Jahn–Teller ef- fect, i.e., an energy gain is provided by the lifting of the de- generacy of electron bands. One should mention that al- though the energy band structure of the low temperature phase of Co 2NbSn was evaluated in Ref. 186 as tetragonal, recent structural investigations [182] shows that the low temperature phase of this compound has orthorhombic Pmma symmetry. The study of the magnetic and structural phase transitions of the pseudo-ternary Heusler series (Co1�xNi x) 2NbSn (x = = 0, 0.18, 0.56, 0.81, and 1) are presented in Ref. 187. Upon alloying Co2NbSn with Ni, both the structural and the mag- netic phase transition temperatures are suppressed to zero temperature with Ni content at x = 0.32 (Tc) and 0.47 (Ts). This might indicate a very strong interlocking of the two phe- nomena. Moreover, if such a coincidence of a first- and sec- ond-order phase transition occurs, it might be interesting in the context of quantum phase transitions. For instance, for MnSi a second-order phase transition is replaced by a first-or- der transition upon approaching the quantum critical point [188]. Authors of Ref. 187 have also presented arguments for the disorder to control the unusual magnetic domain be- havior, i.e., the formation of very narrow domain walls occur- ring in an itinerant magnet. Recently x-ray magnetic circular dichroism in the fer- romagnet Co 2NbSn has been measured at the Co L2 3, edges [189]. Using the magneto-optic sum rules the spin moment and the orbital moment of Co have been deduced to be 0.38, and 0.09 �B, , respectively. 1. Crystal structure. The Heusler-type Co 2NbSn com- pound crystallized at high temperatures in the cubic L21 structure with Fm m3 symmetry (No. 225) is formed by X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 25 four interpenetrating fcc sublattices. The Co ions occupy the 4b Wyckoff positions (x /�1 4, y /�1 4, z /�1 4). The Nb ions occupy the 4a positions (x = 0, y = 0, z = 0) and the Sn ions are placed at the 4b sites (x /�1 2, y /�1 2, z /�1 2). Both the Co and Nb atoms have eight nearest neighbors at the same distance. Nb has eight Co atoms as nearest neighbors, while for Co there are four Nb and four Sn atoms (Table 4). Co 2NbSn undergoes a structural transition at Ts 233 K from the cubic Heusler Fm m3 high-temperature phase into a simple orthorhombic low-temperature lattice of Pmma symmetry (No. 51) [182]. Two inequivalent Co ions oc- cupy the 4h (x = 0, y = 0.253, z /�1 2) and 4k (x /�1 4, y � 0.746, z = 0.051) Wyckoff positions. The Nb ions oc- cupy also two inequivalent Wyckoff positions 2a (x = 0, y = 0, z = 0) and 2 f (x /�1 4, y /�1 2, z = 0.527). Sn ions are placed at the 2e (x /�1 4, y = 0, z = 0.537) and 2b (x = 0, y /�1 2, z = 0) [182] sites. (Fig. 19). In the fol- lowing we will refer to Co 1, Co 2, Nb 1, Nb 2, Sn1, and Sn 2 as situated at the 4h, 4k, 2a, 2 f , 2e, and 2b sites, respec- tively. The structural transition gives rise to a large varia- tion of interatomic distances up to 4%, compared to the cu- bic lattice (Table 4). In contrast, the volume of the unit cell remains almost constant over the whole temperature range. While on a macro- and mesoscopic scale Co 2NbSn ap- pears to be fully ordered, from microscopic M�ssbauer and NMR studies it was found that on an atomic scale site disorder is present in form of random displacements of Co, Nb and Sn ions from their nominal positions in both the Fm m3 and Pmma lattices [182]. The energy band structure and total DOS of Co 2NbSn for the high temperature Fm m3 structure obtained from fully relativistic LDA calculations are presented in Fig. 20. The results agree well with previous band struc- ture calculations [186]. The occupied part of the valence band can be subdivided into several regions. The tin 2s bands appear between �11.4 and �9.0 eV. The next three energy bands in the energy region �6.9 to �3.1 eV are the tin 5 p bands. The Co and Nb d energy bands are located above and below E F at about �3.1 to 2.0 eV. The corre- sponding spin-polarized partial densities of states are shown in Fig. 21. Because of strong Sn p– p hybridization Sn p states are split into bonding and antibonding states. The former are located between approximately �7 and �2 eV, while 26 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko x y z Co1 Co2 Nb1 Nb2 Sn1 Sn2 Fig. 19. The orthorhombic low-temperature lattice structure of Co2NbSn with space group Pmma (No. 51). Table 4. Number and distance of the nearest neighbors for diffe- rent type of atoms in the high (Fm m3 ) and low (Pmma) phases of Co2NbSn (in arb. unuts). Fm m3 Pmma Atom Neighbors Distance Atom Neighbors Distance Co 4� Sn 5.034 Co1 2 2� Sn 5.026 2 1� Sn 5.039 Co2 1 1� Sn 4.973 2 2� Sn 5.004 1 1� Sn 5.167 4� Nb 5.034 Co1 2 2� Nb 4.997 2 1� Nb 5.063 Co2 1 21� Nb 4.853 2 12� Nb 5.054 1 21� Nb 5.189 6� Co 5.813 Co1 1 1� Co 5.533 2 1� Co 5.610 1 1� Co 5.667 2 1� Co 6.217 Co2 1 2� Co 5.511 1 2� Co 5.690 Nb 8� Co 5.034 Nb1 4 2� Co 5.052 4 1� Co 5.063 Nb2 2 2� Co 4.853 4 1� Co 4.997 2 2� Co 5.189 Co NbSn2 Fm3m DOS – 10 – 5 0 5 E n er g y, eV X W K L W UX 0 5 10 15 Fig. 20. The fully relativistic LDA energy band structure and total DOS (in states/(cell�eV)) of Co2NbSn for the high tem- perature Fm m3 structure [190]. the latter are spread over a broad energy range above –1 eV. The center of Nb d states, defined as the energy at which the corresponding logarithmic derivative is equal to � �l 1, lies at �6 � 0 28. eV just above the bottom of the antibonding Sn p states. The crystal field at the Nb 4a site (Oh point symmetry) splits Nb d states into Eg (3 12z � and x y2 2� ) and T g2 (xy, yz, and xz) ones. The Eg states, which form � bonds with Sn p states, are strongly hybrid- ized with the latter and give a significant contribution to the bonding states below �2 eV. The T g2 states form weaker Ni d – Sn p � bonds but they hybridize strongly with d states of eight Co nearest neighbors. The center of Co d states (�6 � �1 2. eV) is found in a gap between the bonding and antibonding Sn p states. The crystal field at the Co 4b site (Td point symmetry) causes the splitting of d orbitals into a doublet E (3 12z � and x y2 2� ) and a triplet T2 (xy, yz, and xz). The hybrid- ization between Co T2 and Nb T g2 states, causes the split- ting of the Nb T g2 states into two peaks, the bonding one located at ~2.5 eV below the Fermi level EF and the unoc- cupied antibonding peak centered at 1.5 eV. The Co E orbitals are weakly hybridized with Sn and Nb states, however, they are split into well separated bonding and antibonding peaks by strong Co d –Co d � hopping. It is important to note, that in the Fm m3 structure the anti- bonding Co E states form a peak in the DOS at the Fermi level. The presence of the peak of antibonding states at EF is a possible source of structure instability and can be responsible for the site disorder on an atomic scale in the Fm m3 structure of Co 2NbSn [182]. In addition to the crystal field splitting, the d levels of the Co and Nb atoms are split due to the exchange interaction. The exchange splitting between the spin-up and down d electrons on the Co atom is about 0.5 eV. The corresponding splitting on the Nb atom is much smaller. Spin-orbit splitting of the d energy bands for both the Co and Nb atoms is much smaller than their spin and crystal-field splittings. The spin-polarized calculations show that Co2NbSn in the high-temperature phase is not a half-metallic ferromagnet. The Fermi level crosses both the majority and minority spin energy bands. The local symmetry in the low-temperature orthorhombic phase of Co 2NbSn reduces to C 2 and Cs at the 4h and 4k Co sites and to C h2 and C 2v at the 2a and 2 f Nb sites, X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 27 spin-up spin-down Sn p – 0.5 0 0.5 Co d e t2– 2 0 2 Nb d eg t2g – 6 – 4 – 2 0 2 4 6 8 Energy, eV – 2 0 2 P ar ti al d en si ty o f st at es , el ec tr o n s/ (a to m ·e V ) Fig. 21. The symmetry separated partial density of states of Co2NbSn for the high temperature Fm m3 structure [190]. spin-up spin-down Fm3m – 2 0 2 Pmma Co1 – 2 0 2 Pmma Co2 – 3 – 2 – 1 0 1 2 Energy, eV – 2 0 2P ar ti al d en si ty o f st at es , el ec tr o n s/ (a to m ·e V ) Fig. 22. The partial Co 3d density of states of Co2NbSn for high, Fm m3 , and low, Pmma, temperature structures [190]. respectively. The crystal field causes the d orbitals to split into five singlets at each site. An important feature of the electronic structure of Co2NbSn in the high temperature Fm m3 structure is a high DOS of the minority-spin electrons at the Fermi energy, EF . The Fermi level cuts the major peak of the density of antibonding Co d states of E symmetry at 10 meV from its maximum (Fig. 21). A large DOS at EF signals an instabil- ity with respect to a structural phase transition. In the low temperature Pmma phase the Fermi level is situated in a lo- cal minimum of both the Co1 and Co 2 d DOS’s (Fig. 22). The value of the total DOS is decreased by more than a fac- tor of two in going from the high to low temperature phase. These changes of the density of states in the vicinity of the Fermi level are closely related to the changes of Co– Co interatomic distances (see Table 4). In the low-temperature phase Co1–Co1 and Co 2–Co 2 distances along the y direc- tion shrink. The energy of Co orbitals forming � bonds be- tween these atoms (E orbitals in the Fm m3 phase) increases and the corresponding peak of DOS is pushed above EF . On the contrary, two of the four Co nearest neighbors in the xz plane move away from a Co site. The energy of the Co orbitals, that participate in � bonds between these more distant atoms, lowers and the orbitals become occupied. This picture supports the suggestion made by Fujii et al. [186] that the driving force for the structural phase transi- tion in Co 2NbSn is the band Jahn-Teller effect. The experimentally measured dichroic lines have dif- ferent signs at the L3 and L2 edges [189]. In order to com- pare relative amplitudes of the L3 and L2 XMCD spectra we first normalize the corresponding isotropic x-ray ab- sorption spectra (XAS) to the experimental ones taking into account the background scattering intensity as de- scribed in Sec. 2. Figure 23 shows the calculated isotropic x-ray absorption and XMCD spectra of Co at the L2 3, edges in the LSDA approach together with the experimen- tal data [189]. The contribution from the background scattering is shown by dotted line in the upper panel of Fig. 23. The two nonequivalent Co ions produce similar XAS’s with slightly different energy positions and differ- ent intensities, it leads to the peak structure with a high energy shoulder in L3 and L2 XAS’s. This structure is less pronounced at the L2 edge. This result can be ascribed to the lifetime broadening effect because the lifetime of the 2 1 2p / core hole is shorter than the 2 p /3 2 core hole due to the Coster-Kronig decay [22]. Table 5 presents the comparison between calculated and experimental magnetic moments in Co2NbSn. The spin magnetic moment at the tin site is very small. The spin moment at Nb sites is also small and for the low tem- perature phase has an opposite direction to the spin mo- ment at Co sites. The LSDA overestimates the spin moment on the Co site (Table 5). It might be connected with the presence of crystallographic disorder in this compound [182]. To in- vestigate the crystallographic disorder we reduce the num- ber of the symmetry operations to make all the atoms in the unit cell of Pmma lattice nonequivalent. Then we modeled the disorder by shifting some of Co, Nb, and Sn atoms from their nominal positions. We found that the displacement of only Nb and Sn atoms by 2% (with Co atoms in their nomi- nal positions) reduces the average Co spin magnetic mo- ment from 0.451 �B to 0.446 �B. However, the displace- ment of Co atoms by 2% (keeping Nb and Sn atoms in their nominal positions) leads to the reduction of the average Co spin magnetic moment to 0.325 �B. It is interesting to note that the displacements of Co atoms more than 4% from their nominal positions lead to a nonmagnetic ground state. We also investigated the effect of the possible substitutional disorder in the compound by interchanging the Co ions with Sn or Nb ones. We found that the sub- stitutional disorder greatly influences the spin magnetic moment of Co. The average Co spin magnetic moment is equal to 0.345 �B and 0.02 �B when 12.5% or 25% Co ions are interchanged with Sn or Nb ions, respectively. It is interesting to compare the spin and orbital mo- ments obtained from the theoretically calculated XAS and XMCD spectra through sum rules [Eqs. (15), (16)] with directly calculated LSDA values. In this case we at least avoid all the experimental problems. Sum rules reproduce the spin magnetic moments within 22% and 16% and the 28 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko X M C D , ar b . u n it s In te n si ty , ar b . u n it s L3 L2 0 5 10 15 0 10 20 Energy, eV – 2 – 1 0 1 L2 L3 Fig. 23. Theoretically calculated (thick full line) and experimen- tal (circles) isotropic absorption and XMCD spectra of Co2NbSn at the Co L2 3, edges. Experimental spectra [189] were measured at 50 K. The upper panel also shows the background spectra (dot- ted line) due to the transitions from inner 2 p / /1 2 3 2, levels to the continuum of unoccupied levels [48]. Full and dashed lines pres- ent the spectra for Co1 and Co2, respectively [190]. orbital moments within 22% and 29% for Co 1 and Co 2, respectively (Table 5). XMCD sum rules are derived within an ionic model using a series of approximations. We applied the sum rules to the XMCD spectra calculated neglecting the energy dependence of the radial matrix ele- ments and the p s) transitions. As can be seen from Ta- ble 5 using the energy independent radial matrix elements reduces the disagreement in spin magnetic moments to 5% and 3% and in the orbital moment to 6% and 7% for Co 1 and Co 2, respectively. An additional omitting of the p s) transitions leads to almost perfect agreement be- tween LSDA and sum rule results within 1–2% for the spin moments and 3% for the orbital moments. These re- sults show that the energy dependence of the matrix ele- ments and the presence of p s) transitions affect strongly the values of both the spin and the orbital mag- netic moments derived using the sum rules. The value of the orbital magnetic moment derived from the experimental XMCD spectra (M l exp = 0.09 �B [189]) is considerably higher in comparison with our band structure calculations. It is a well-known fact, how- ever, that LSDA calculations be inaccurate in describing orbital magnetism [20,22]. In the LSDA, the Kohn-Sham equation is described by a local potential which depends on the electron spin density. The orbital current, which is responsible for M l , is, however, not included in the equa- tions. This means, that although M s is self-consistently determined in the LSDA, there is no framework to deter- mine simultaneously M l self-consistently. To calculate M l beyond the LSDA scheme we used the rotationally in- variant LSDA + U method [60]. We used U J� �1 0. eV. The LSDA + U calculations produce the orbital magnetic moments equal to 0.046 �B for Co sites. This value is in better agreement with the experimental data but still smaller than the experimental estimations. We should mention that in order to obtain the spin and orbital magnetic moments the authors of Ref. 189 first es- timate the ratio � � � �l / sz z without � �t z term. To estimate the absolute values of M s and M l they used the value of the total magnetic moment at the Co site obtained by the magnetization measurements under high magnetic fields, neglecting the magnetic moments of other atoms. It is well known that the magnetic dipole operator t z is quite small for cubic systems [22]. In our particular case the term ( )7 2/ t z� � in Eq. (16) is equal to 0.001 �B at Co site in the high-temperature cubic Fm m3 structure. On the other hand, the experimental measurements in Ref. 189 have been done below Curie temperature, but for the low temperature orthorhombic Pmma structure the term ( )7 2/ t z� � is not small and is equal to 0.061 and 0.046 �B for Co 1 and Co 2, respectively. These two approximations can lead to additional errors in the estimation of the spin and orbital magnetic moments provided in Ref. 189. The magnetic and structural phase transitions of the pseudo-ternary Heusler series (Co 1�xNi x) 2NbSn (x = 0, 0.18, 0.56, 0.81, and 1) are experimentally studied in Ref. 187. The authors found that upon alloying Co 2NbSn with Ni, both the structural and the magnetic phase transition temperatures are suppressed to zero temperature with Ni content at x = 0.32 (Tc) and 0.47 (Ts). We have calculated the electronic and magnetic struc- tures as well as the XMCD spectra of the pseudo-ternary Heusler series (Co 1�xNi x) 2NbSn for x = 0, 0.125, 0.25, 0.375, 0.5, 0.675, 0.75 and 1.0. The simplest way to construct a pseudo-ternary Heusler alloy (Co 1�xNi x) 2NbSn for x = 0.5 is to substi- tute one of the two inequivalent Co atoms by Ni in Pmma Co 2NbSn. The Ni atom can substitute Co atoms both at the 4h and 4k sites. The total energy calculations show that the more preferable site for Ni is the 4k site. Figu- re 24 presents the DOS’s of CoNiNbSn compound in the Pmma crystal structure with Ni ions at the 4k site. To cal- culate the electronic structure of (Co 1�xNi x)2NbSn sys- tem for x = 0.125, 0.25, 0.375, 0.675 and 0.75 we reduce the number of the symmetry operations and make all the atoms in the unit cell of the Pmma lattice nonequivalent. X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 29 Table 5 The experimental and calculated spin M s and orbital M l magnetic moments (in �B) of Co2NbSn. Method Atom Ms Ml Fm3m Co 0.454 0.044 Nb 0.0 0.012 Sn 0.004 –0.001 Pmma Co1 0.491 0.031 Co2 0.410 0.028 Nb1 –0.013 0.007 Nb2 –0.015 0.010 Sn1 0.002 0.0 Sn2 0.005 0.0 sum rules Co1 0.380 0.024 Co2 0.343 0.020 sum rules a Co1 0.468 0.029 Co2 0.398 0.026 sum rules b Co1 0.480 0.030 Co2 0.416 0.027 expt. [189] Co 0.38 0.09 Comment: a sum rules applied for the XMCD spectra calculated with ignore the energy dependence of the radial matrix element b sum rules footnote{sum rules applied for the XMCD spectra calculated with ignore the energy dependence of the radial ma- trix elements and ignore p s) transitions. In contrast to the Pmma Co 2NbSn compound where the Fermi level is situated in a local minimum of the DOS, CoNiNbSn is characterized by quite high DOS at the Fermi energy. In the later case the Fermi level cuts the majority peak of Co spin-down d DOS at 20 meV from its maximum with the Ni atom at the 4k site (Fig. 24). When we place the Ni atom in the 4h site, EF is situated exactly at the peak of Co spin-down density of states (not shown). The value of the total DOS in the (Co1�xNi x) 2NbSn system is increased by more than 30% going from x = 0.0 to x = 0.5. It should be mentioned, however, that this behavior is not monotonic and the lowest DOS was found for the alloy with x = 0.25. In this alloy the DOS is even smaller than in the stable Co 2NbSn compound. A large DOS at EF for x = 0.5 alloy signals an instability of the crystal structure and can be considered as theoretical support of the experimen- tal observation that upon alloying Co 2NbSn with Ni the structural phase transition temperatures are suppressed to zero temperature with Ni content at x = 0.47. Above this concentration the compound remains in the cubic Fm m3 structure in the whole temperature range [187]. Our theoretical calculations give the ferromagnetic so- lution for the cubic Fm m3 (Co 1�xNi x) 2NbSn compounds only for for x 7 0 125. . For larger Ni concentrations LSDA produces nonmagnetic solutions. Figure 25 presents par- tial DOS’s of cubic Fm m3 (Co 1�xNi x) 2NbSn compound for x � 0 5. . The DOS at the Fermi level in this case is considerably smaller in comparison with the DOS for the same compound in the Pmma crystal structure. Table 6 presents the calculated magnetic moments for (Co1�xNi x) 2NbSn systems in the Pmma structure. The value of the magnetic moments in CoNiNbSn depends strongly on which site is occupied by the Ni atom. The band structure calculations give ferromagnetic solutions for the (Co1�xNi x) 2NbSn systems for x up to x = 0.5. Only alloys with x 8 0.625 have no ferromagnetic solution. On the other hand, the experimental study of the magnetic phase transitions of the (Co1�xNi x) 2NbSn systems [187] reveals that the magnetic phase transition is suppressed al- ready for x = 0.32. The LSDA overestimation of the 30 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko spin-up spin-down Co – 2 0 2 Ni – 2 0 2 4 Nb – 4 – 2 0 2 Energy, eV – 2 – 1 0 1 2 P ar ti al d en si ty o f st at es , el ec tr o n s/ (a to m ·e V ) Fig. 24. The partial d density of states of (Co1�xNix)2NbSn for x � 0 5. system for Pmma structure with Ni atoms at the 4k site [190]. Co E T2 – 5 0 5 Ni – 2 0 2 Nb – 3 – 2 – 1 0 1 2 Energy, eV –2 0 2P ar ti al d en si ty o f st at es , el ec tr o n s/ (a to m ·e V ) Fig. 25. The symmetry separated partial density of states of (Co1�xNix)2NbSn for the x � 0 5. system for the cubic Fm m3 structure [190]. tendency toward the magnetism in the (Co1�xNi x) 2NbSn systems is probably connected with the crystallographic disorder in these alloys [187]. The effect of Ni substitution for Co on the XMCD spec- tra of the pseudo-ternary Heusler series (Co1�xNi x) 2NbSn (x = 0.5) results simply in the disappearing of one of the components (dashed line at low panel in Fig. 23) because Ni L2 3, spectra are situated in other energy range. As a re- sult Co L2 3, XMCD spectra are reduced in absolute value and slightly shifted to lower energy. 4.3. Noncollinear magnetism 4.3.1. IrMnAl. IrMnAl is classified as a weakly ferro- magnetic compound with a Curie temperature of about 400 K [191]. IrMnAl crystallizes in the fluorite (C1) struc- ture [191]. A Heusler structure differs from the C1 struc- ture in having additional atoms at the edge centers and at the body center. The characteristic feature of the crystal structure of the intermetallic compound IrMnAl is the chemical disorder of Ir and Al atoms which randomly oc- cupy the 8c positions in the C1structure. Atomic disorder in the Heusler compounds is not a rare phenomenon and is well studied theoretically [150–156] and experimentally [192–195]. Recently x-ray magnetic circular dichroism in the ferromagnet IrMnAl has been measured at Ir L2 3, edges [196]. The observation shows that Ir has a 5d magnetic moment in IrMnAl. Using the magneto-optic sum rules the spin moment and the orbital moment of Ir at 30 K have been deduced to be 0.018, and -0.0031 �B , respectively. The magnetic moment in IrMnAl comes mainly from Mn atoms and has been reported to be 0.4 �B on the Mn site [191]. An even smaller magnetic moment of 0.123 �B /atom for IrMnAl was estimated from the magnetiza- tion measurements [196]. This magnetic moment is much smaller than the total moment of 2�5 �B found in ferro- magnetic Mn compounds with either L21 or C b1 crystal structure. For example, Mn has a moment of 4 �B in the Heusler alloy PtMnSb [197]. It is still not clear why Mn has a very small magnetic moment in IrMnAl although the Curie temperature is comparable to other Mn based ferromagnets. The small value of the net magnetic mo- ment and the large Curie temperature of IrMnAl can be reconciled with either a small and partly delocalized Mn moment in an itinerant electron picture or with a large mo- ment of Mn associated with a noncollinear magnetic structure. Here we address the possibility of noncollinear magnetic configurations in the C1structure. We study the noncollinear magnetism by calculating the total energies for different spin spirals. 1. Crystal structure. The intermetallic compound IrMnAl crystallizes in the C1 structure [196] (space group: F m43 , No. 225) in which Mn atoms occupy the 4a Wyckoff positions (x y z� � � 0) and Ir as well as Al at- oms randomly occupy the 8c positions (x y z /� � �1 4), the vacancies are situated at the 4b sites (x y z /� � �1 2). The lattice constant a is found to be 5.992 � [196]. To simplify the band structure calculations we assumed an additional ordering of Ir and Al ions in 8c sublattice (see Fig. 26) using the fcc structure with space group F m43 (No. 216). We place Ir and Al ions in 4c ( , , )1 4 1 4 1 4/ / / and 4d ( , , )3 4 3 4 3 4/ / / sites, respectively, leaving vacancies at the 4b ( , , )1 2 1 2 1 2/ / / sites. One should mention that in their early work. Matsumoto and. Watanabe [191] proposed another atomic occupation in the C1 structure of IrMnAl, namely, Mn atoms at the 4a sites, but Ir and Al randomly occupy 4b and 4d sites (with vacancies at the 4c sites). In this case 4b and 4d sites are not equivalent. Although the first configuration proposed in Ref. 196 seems to be more correct in our band structure cal- culations we used both the configurations to study possible intersite disorder in IrMnAl. Table 7 presents the number and distance of the near- est neighbors for different type of atoms in the real struc- ture (group 225) and model phases (group 216 and group 129) of IrMnAl investigated in Ref. 198. 2. Band structure. The total and partial DOS's of ferro- magnetic IrMnAl obtained from the LSDA calculation are X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 31 Table 6. The calculated spin M s and orbital M l magnetic mo- ments (in �B) of (Co1�xNix)2NbSn systems in the Pmma structure for x = 0.125, 0.5 and 0.375. (We present averaged magnetic mo- ments for different inequivalent sites.) Method Atom Ms Ml x = 0.5 Co(4h) 0.506 0.032 Ni in 4k Ni(4k) 0.006 0.001 Nb(2a) –0.019 0.0 Nb(2 f) –0.026 0.010 x = 0.5 Co(4k) 0.351 0.031 Ni in 4h Ni(4h) 0.004 0.001 Nb(2a) –0.009 0.009 Nb(2 f) –0.021 0.002 x = 0.125 Co 0.435 0.032 Ni 0.040 0.006 x = 0.25 Co 0.380 0.027 Ni 0.025 0.001 x = 0.375 Co 0.431 0.030 Ni 0.012 0.002 x = 0.625 Co 0.0 0.0 Ni 0.0 0.0 presented in Fig. 27. The occupied part of the valence band can be subdivided into several regions. The Al 3s bands appear between –8.4 and –5.5 eV. Al p states are spread over a broad energy range above –4.8 eV. The Ir and Mn d energy states are strongly hybridized and lo- cated above and below EF at about –4.5 to 3.4 eV. The d levels of the Mn and Ir atoms are split due to the crystal field. At both the Ir and Mn sites in the F43m structure the crystal field causes the d orbitals to split into a doublet e z(3 12 � and x y2 2� ) and a triplet t xy yz2( , , and xz ). The spin-polarized calculations show that IrMnAl is not a half-metallic ferromagnet. The Fermi le- vel crosses both the majority and minority spin energy bands. Spin-orbit splitting of the d energy bands for the Mn atoms is smaller than their spin and crystal-field splittings. The characteristic feature of the electronic structure of IrMnAl alloy is the strong hybridization of Mn 3d and Ir 5d states the later being more delocalized. Due to the hybridization the nonmagnetic Ir atom in IrMnAl becomes magnetic. The value of the Ir magnetic 32 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko Mn Ir Al Fig. 26. The fcc structure of IrMnAl with space group F m43 (No. 216) with ordered Mn and Ir ions. T o ta l D O S P ar ti al d en si ty o f st at es spin-up spin-down – 5 0 5 Al s Al p – 1 0 1 Ir d e t2– 1 0 1 Mn d e t2 – 5 0 5 10 15 Energy, eV – 1 0 1 Fig. 27. The total (in states/(cell�eV)) and partial (in sta- tes/(atom�eV)) density of states of ferromagnetic IrMnAl in F m43 structure [198]. The Fermi energy is at zero. Table 7. The number and distance of the nearest neighbors for different type of atoms in the real structure (group 225) and model phases (group 216 and group 129) of IrMnAl (in arb. units.). Atom Neighbors Distance Atom Neighbors Distance group 216 Ir in 4c group 216 Ir in 4b Mn 4�Ir 4.894 Mn 4�Al 4.894 4�Al 4.894 6�Ir 5.651 12�Mn 7.992 12�Mn 7.992 Ir 4�Mn 4.894 Ir 4�Al 4.894 6�Al 5.651 6�Mn 5.651 12�Ir 7.992 12�Ir 7.992 group 129 group 225 Mn 4�Ir 4.894 Mn 4�Ir 4.894 4�Al 4.894 4�Al 4.894 12�Mn 7.992 12�Mn 7.992 Ir 4�Mn 4.894 Ir 4�Mn 4.894 4�Ir 5.651 6�Ir/Al 5.651 2�Al 5.651 12�Ir/Al 7.992 4�Ir 7.992 8�Al 7.992 moment depends on the number of the nearest neighbor Mn atoms and the Mn–Ir distances. The spin and orbital magnetic moments at the Ir site are equal to –0.063 �B and –0.094 �B , respectively. Due to strong Mn 3d and Ir 5d hybridization the spin magnetic moment at the Mn site is reduced to 1.105 �B , the orbital moment was found to be 0.114 �B . 3. Noncolinear magnetism. First, we have studied the possibility of noncollinear ordering by studying the energetics of spiral configurations. The total energy is calculated as a function of the spiral wave vector q, and the wave vector is varied along the high-symmetry direc- tions [111], [100], and [110]. Vector q is given in units of 2�/a where a is the lattice constant of the C1structure. The total energies for the spiral wave vector q varying along the high-symmetry direction [111] are shown in Fig. 28. The variation of the total energy in [111] direc- tion has the lowest energy for the antiferromagnetic con- figuration at q = (0.5, 0.5, 0.5). The corresponding spin magnetic structure is shown in Fig. 29 where we present only Mn and Ir spin moments in the positions according to the Fig. 26. In order to obtain a deeper understanding of the energy dispersion we look into the behavior of magnetization. The Mn and Ir magnetic moments projected onto the local spin axis defined by the polar angles � � 90 � and � � qrn where rn is the position of Mn or Ir site are shown in Fig. 28. The atomic magnetizations show that within the Mn spheres the magnetization is substantially increases going from the collinear ferromagnetic configuration q = (0, 0, 0) to the antiferromagnetic order q = (0.5, 0.5, 0.5). The magnetic moment in Ir changes a sign from negative to positive go- ing from the collinear ferromagnetic configuration to the antiferromagnetic one. The total energies for the spiral wave vector q varying along the high-symmetry directions [100] and [110] are shown in Fig. 30 together with the Mn and Ir spin magnetic moments. The variation of the total energy in [100] direc- tion has the same tendency as it was observed in the [111] direction, namely the total energy monotonously lowers when going from the collinear ferromagnetic configuration q = (0, 0, 0) to the antiferromagnetic one q = (1, 0, 0). Ho- wever, the total energy is slightly lower for the antifer- romagnetic ordering with q vector along [100] direction in comparison with the [111] direction. The magnetization in Mn atomic sphere increases monotonically in the [100] di- rection going from the ferromagnetic to antiferromagnetic configuration. The lowest total energy was found to be near the Brillouin-zone boundary at q = (0.75, 0.75, 0) correspond- ing to the spiral order which implies that the exchange cou- pling between nearest and next nearest Mn atoms is anti- ferromagnetic. In the [110] direction, the magnetization in Mn atomic sphere increases more rapidly reaching its max- imum value of 2.603 �B at q = (0.6, 0.6, 0) before decreas- ing again to 2.45 �B in the antiferromagnetic state at q = = (1, 1, 0). The behavior of the Ir moment in the [110] sym- metry direction shows stronger variation in comparison with the [100] direction. The spin magnetic moment at the Ir site reaches its maximum positive value at around q = (0.4, 0.4, 0) and vanishes for the antiferromagnetic configuration at q = (1, 1, 0). Figure 31 presents the spin magnetic structure for Mn and Ir sites corresponding to the spiral order for q = (0.75, 0.75, 0) possessing the lowest total energy. The electronic partial density of states for this spiral structure is presented in Fig. 32. Comparing Figs. 27 and 32 we can conclude that the electronic states are strongly redistributed going from the collinear ferromagnetic configuration to the magnetic spiral order. These calculations show that the small value of the net magnetic moment in IrMnAl discovered experimentally [191] might be associated with noncollinear magnetic structure. A similar conclusion was drawn also by authors X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 33 –2 –1 0 Ir –0.05 0 0.05 Mn 0 0.1 0.2 0.3 0.4 0.5 [qqq] 1.0 1.5 2.0 2.5 M , s B � M , s B � E , m R y to t Fig. 28. Total energy and spin magnetic moments as a function of the spiral vector q varying along the [111] direction [198]. of Ref. 196. Because we consider here only planar spirals, that is, � �n /� 2, the Mn and Ir spin magnetic moments are situated in the xy plane (Figs. 29 and 31) therefore the magnetization along the z axes is equal to zero. From our band structure calculations IrMnAl has tendency for the antiferromagnetic ordering, the exchange coupling be- tween nearest and next nearest Mn atoms is antiferromagnetic. On the other hand, the experimental magnetization measure- ments classified IrMnAl as a weakly ferromagnetic material with small net magnetization of 0.123�B /atom [196]. The dis- agreements might be explained by the possible Mn — vacancy intersite disorder. To investigate such effect we carried out sev- eral super-cell band structure calculations in which we placed Mn atoms at variable vacant sites. We found that if some amount of the Mn atoms (from 12.5% to 25%) occupy vacant places the compound becomes ferrimagnetically ordered with the net magnetization of 0.068 –0.169 �B /atom depending on the number and the type of crystallographic positions occupied by the Mn atoms. It is interesting to note that a full-Heusler compound Ir2MnAl in L21 crystal structure without vacant places at all has an antiferromagnetic order [191]. 4. XMCD spectra. At the core level edge XMCD is not only element-specific but also orbital specific. Recently, x-ray magnetic circular dichroism in the ferromagnet IrMnAl has been measured at Ir L2 3, edges [196]. It was found that the experimentally measured dichroic lines have different signs at the L3 and L2 edges [196]. Figure 33,a shows the isotropic x-ray absorption and XMCD spectra of Ir at the L2 3, edges calculated in the LSDA approach to- gether with the experimental data [196]. The experimental measurements were carried out in external magnetic field of 1.15 T, and although the magnetization does not saturate at the highest applied field of 5.5 T we compare the measured XMCD spectra with calculated ones for the ferromag- netically ordered IrMnAl. It would be interesting to repeat the experiment at much higher magnetic fields to see how much the magnetic moments and the XMCD spectra change. The experimental Ir L3 XAS has a pronounced high en- ergy shoulder. This structure is less pronounced at the L2 edge. This result can be ascribed to the lifetime broadening effect because the lifetime of the 2 1 2p / core hole is shorter than the 2 3 2p / core hole due to the L L V2 3 Coster–Kronig decay. These features are due to the interband transitions from 2 p core level to 3d empty states at around 8–20 eV above the Fermi level (Fig. 27) and quite well reproduced by the theoretical calculations (Fig. 33). The theoretically calculated Ir L2 3, XMCD spectra are in good agreement with the experiment, although small 34 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko x y z Fig. 29. Spin magnetic structure for spiral direction [111] and the spiral vector q = 0.5 (thick vector shows the spin direction of Mn atoms and thin one Ir atoms) [198]. <qq0> <q00> –3 –2 –1 0 <qq0> <q00> <q00> Ir –0.05 0 0.05 <qq0> Mn 0.0 0.2 0.4 0.6 0.8 1.0 q 1.0 1.5 2.0 2.5 M , s B m M , s B m E , m R y to t Fig. 30. Total energy and spin magnetic moments as a func- tion of the spiral vector q varying along the [110] and [100] di- rections [198]. shoulders at the high energy part of both the L3 and L2 spectra are not reproduced by our calculations. However, if we place Ir atoms at the 4b sites with Al atoms and va- cancies at the 4d and 4c sites, respectively (as proposed by Matsumoto and Watanabe [191]) we have been able to reproduce these structures, but with much larger dichroic signal at the L3 edge in comparison with the experimental data (not shown). We can conclude that probably some amount of Ir and Al atoms as well as vacant places ran- domly distributed between all the sites 4c, 4d, and 4b in the real IrMnAl crystal structure. 4.3.2. Mn 3ZnC. Ternary manganese compounds with a formula Mn 3MX (M = Ga, Sn, Zn and X = C and N) and the cubic crystal structure of a perovskite type have attracted much interest due to their large variety of mag- netic orderings and structural transformations [199=207]. Mn 3GaC becomes a ferromagnet below Tc = 255 K. Furthermore, as the temperature decreases from Tc the first-order transition from a ferromagnet to an antifer- romagnet, which involves a change of crystal structure from a cubic lattice to a rhombohedral one, has been ob- served at Tt = 165 K [208]. The antiferromagnetic order- ing below Tt is of the second kind, with an easy axis along the [111] direction. In the case of Mn 3SnC the crystal structure is cubic in the whole temperature range and be- low Tc = 265 K this compound becomes magnetic. The magnetic ordering was reported to be complex [208]. In Mn 3ZnC, a magnetic phase transition occurs at T t = = 233 K, which has been classified as a second-order transi- tion from ferromagnetic phase (Tc = 380 K) to ferrimagnetic one with a non-collinear magnetic structure [206,207,209]. The transition is accompanied by a structural change from cubic to tetragonal. In the low-temperature phase, the tetragonality is monotonically decreased with decreasing temperature [210]. The magnetization also gradually de- creases with temperature and eventually shows a cusp at T t . In addition, Mn 3ZnC is magnetically hardened at low tem- peratures; the hysteresis loop becomes larger with decreas- ing temperature and the coercive force is linearly increased. In these three compounds the results of neutron dif- fraction show that the magnetic moments of Mn atoms are X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 35 T o ta l D O S P ar ti al d en si ty o f st at es spin up– spin down– –5 0 5 Al s Al p –1 0 1 Ir d –2 0 2 Mn d –5 0 5 10 15 Energy eV, –2 0 2 Fig. 32. The partial density of states of IrMnAl in F m43 struc- ture for spiral direction [110] and the spiral vector q = 0.75 [198]. The Fermi energy is at zero. y z x Fig. 31. Spin magnetic structure for spiral direction [110] and the spiral vector q = 0.75 (thick vector shows the spin direc- tion of Mn atom and thin one Ir atom) [198]. much smaller than the 4 5� �B observed in other ordered manganese alloys [22]. This result indicates a strong itin- erant character of 3d electrons of Mn atoms in Mn 3MX. The observed paramagnetic susceptibility obeys the Cu- rie–Weiss law with � = 1.9 �B for Mn 3ZnC and � = = 2.41 �B for Mn 3GaC [210]. In the case of Mn 3SnC the observed Curie-Weiss susceptibility has a slight upward convexity [207]. The electronic structure of these compounds plays the key role in determining their magnetic and structural properties. The energy band structure of the Mn 3MX sys- t e m h a s b e e n c a l c u l a t e d b y v a r i o u s m e t h o d s [208,211–213]. Jardin and Labbe [211] performed a band calculation of cubic perovskite compounds Mn3MX by applying a simple tight-binding approximation to the d electrons of Mn atoms and p electrons of X atoms. They found a sharp singularity in the electronic density of states at the Fermi level and pointed out that the singular- ity of the density of states could explain the nature of the magnetic and structural phase transitions. In Ref. 208 the electronic bands of Mn 3ZnC, Mn 3GaC, Mn 3InC and Mn3SnC were calculated in the non-magnetic state of the cubic perovskite structure, by a selfconsistent augmented plane wave (APW) method. The ferromagnetic bands of Mn 3GaC has also been calculated in cubic structure. The energy band structure of Mn3GaC in nonmagnetic, ferro- magnetic (FM) and antiferromagnetic (AFM) states were calculated using a self-consistent linearized aug- mented-plane-wave (LAPW) method in Ref. 212. It was found that the conduction bands around the Fermi level consist mainly of the Mn 3d orbitals which are not bond- ing with C 2 p orbitals. Total energies for both the FM and AFM states were calculated as a function of the unit cell volume. The 4 p states in transition metals usually attract only minor interest because they are not the states of constitut- ing magnetic or orbital orders. Recently, however, under- standing 4 p states has become important since XMCD spectroscopy using K edges of transition metals became popular, in which the 1s core electrons are excited to the 4 p states through the dipolar transition. The K edge XMCD is sensitive to electronic structures at neighboring sites, because of the delocalized nature of the 4 p states. Recently in order to study the magnetic phase transition in Mn 3ZnC at 233 K, x-ray magnetic circular dichroism has been measured at the Mn K-edge as a function of tem- perature and magnetic field [210]. When the transition appears, the spectral intensity of the prominent peak shows an abrupt drastic increase with decreasing temper- ature and a linear reduction with increasing magnetic field. 1. Crystal structure..Mn3ZnC at room temperature crys- tallize in the cubic perovskite-type structure with Pm m3 space group (No. 221). Mn atoms being at the face centers, Zn atoms at the corners, and C atoms at the body center (see Fig. 34). The Mn atoms have two carbon nearest neighbors at the 1.965 � distance. The second coordination consists of 8 Mn atoms and 4 Zn atoms at the 2.779 �. A ferromagnetic to ferrimagnetic phase transition in Mn 3ZnC at T t = 233 K is accompanied by a structural change from cubic to tetragonal. Magnetic unit-cell of the low-temperature ferrimagnetic phase (P mmm4 space group, No. 123) is composed of 4 formula units, and Mn ions form 4 layers and occupy 3 different sites [206]: Mn 1 possesses magnetic moments of 2.7 �B canting � 65� from the c �axis to [111] direction, the Mn2 and Mn3 have magnetic moments of 1.67 �B/Mn ferromagnetically aligned to the c �axis (Fig. 35). Therefore, the Mn mo- ments constitute a non-collinear magnetic structure. In the low-temperature tetragonal structure Mn�C inter- atomic distances are slightly decreased in comparison with the high temperature cubic phase up to the 1.958 �. Mn1 atoms are surrounded by 2Mn 2 and 2Mn 3 atoms at the 2.762 � distance and 4Mn1 atoms at the 2.769 � distance. Therefore, Mn�Mn interatomic distances are decreased through the cubic�tetragonal structure transition. 2. Energy band structure. The total and partial density of states of cubic ferromagnetic Mn3ZnC are presented in Fig. 36. The results agree well with previous band struc- ture calculations [212]. The occupied part of the valence band can be subdivided into several regions. C 2s states ap- 36 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko X M C D , ar b . u n it s In te n si ty , ar b . u n it s aIr 0 0.5 1.0 1.5 b L2 L3 Ir theory exper. 0 50 100 150 Energy, eV –0.002 0 0.002 Fig. 33. Theoretically calculated [198] (dotted line) and experi- mental [196] (circles) isotropic absorption spectra of IrMnAl at the Ir L2 3, edges. Experimental spectra were measured with external magnetic field (1.15 T) at 30 K. Dashed lines show the theoreti- cally calculated background spectra, full thick lines are sum of the theoretical XAS and background spectra (a); Theoretically calcu- lated (full lines) and experimental [196] (circles) XMCD spectra of IrMnAl at the Ir L2 3, edges. (b). pear between –13.1 and –10.6 eV. Zn 3d states are fully oc- cupied and cross the bottom of C 2 p bands in a very narrow energy interval from –7.4 to �6.7 eV. C 2 p states extended from –7 eV up to 15 eV. The states in the energy range –3.2 to 3.5 eV are formed by Mn d states. The crystal field at the Mn site (D h4 point symmetry) causes the splitting of d orbitals into three singlets a g1 and b g1 (3 12z � and x y2 2� ), b xzg2 ( ) and a doublet eg (xy, yz). The a bg g1 � and b eg g2 � splittings are negligible in comparison with its width in LSDA calculations, therefore we present in Fig. 36 DOS of eg orbitals as a sum of the a g1 and b g1 ones and t g2 orbitals as a sum of the b g2 and eg ones. One should mention that there is quite a small C 2 p–Mn d hy- bridization in the valence bands below the Fermi level. Mn3ZnC in cubic perovskite type crystal structure has a local magnetic moments of 2.362 �B on Mn, –0.062 �B on Zn and –0.196 �B on C. The orbital moments are equal to 0.028 �B , 0.001 �B and 0.0001 �B on the Mn, Zn and C sites, respectively. Our calculations produce larger magnetic moments at the Mn site in comparison with the Takahashi and Igarashi re- sults [213]. The interaction between the transition metals is ferrimagnetic, leading to a total calculated moment of 6.935 �B . Mn3ZnC partial DOS’s for the low-temperature tetragonal structure are presented in Fig. 37. For this crystal structure a spin magnetic moments are of 2.269 �B on noncollinear Mn1 atom sites, 1.956 �B on collinear Mn 2 3, ones, –0.041�B on Zn and –0.114�B on C sites. The orbital moments are equal to –0.022 �B , 0.009 �B , –0.002 �B and �0.002 �B on the Mn1, Mn 2 3, , Zn and C sites, respectively. Our calculations produce smaller spin magnetic moments at the Mn1 sites and larger at the Mn 2 3, sites in comparison with the experimental data [206]. 3. XMCD spectra. Figure 38,b shows the theoretically calculated Mn K -edge x-ray absorption spectra as well as XMCD spectra in Mn 3ZnC in comparison with the corre- sponding experimental data [210]. In order to compare relative amplitudes of the K XMCD spectra we first normalize the theoretically calculated x-ray absorption spectra to the experimental ones taking into account the background scattering intensity [48] (Fig. 38,a). There are no large differences in the shape of XAS for low- and high-temperature phases of Mn 3ZnC in agreement with the experimental measurements which show no signifi- cant temperature variation of XAS [210]. The experimen- tal x-ray absorption spectra have three humps around 1.5, 5 and 10 eV above the Fermi level, which are well repro- duced by the theoretical calculations. Figure 38,b shows the experimental XMCD spectrum [210] measured at 300 K and the theoretically calculated one using the LSDA approximation for cubic high tem- perature phase. The theory is in good agreement with pre- vious calculations [213] and the experimental measure- ments, although the calculated magnetic dichroism is somewhat too high at the 5 eV (peak C) and in the 9 to 13 eV energy range. The 4 3p d� hybridization and the spin-orbit interaction (SOI) in the 4 p states play a crucial role for the K edge dichroism. We calculated the site-dependent function dm Etl ( ) (Eq. (36)) [139]. As can be seen from Fig. 38,b the K XMCD spectrum and dm Etl ( ) function are indeed closely related to one another giving a rather simple and straightforward interpretation of the XMCD spectra at the K edge. The K XMCD spectra come from the orbital polariza- tion in the empty p states, which may be induced by (1) the spin polarization in the p states through the SOI, and (2) the orbital polarization at neighboring sites through hybridization. We calculated the XMCD spectra at Mn sites in the ferromagnetic cubic phase of Mn 3ZnC with turning the SOI off separately on the Mn 4 p states and the Mn 3d states, respectively. We found that when the SOI on X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 37 Mn Zn C Fig.34. Cubic perovskite-type crystal structure of Mn3ZnC at room temperature. Mn1 Mn2,3 Zn1,2 Zn3,4 C Fig. 35. Low-temperature magnetic structure of Mn3ZnC. the Mn 3d states is turned off the spectrum above 3.5 eV does not change, while the negative peak B is reduced and the prominent peak near the K edge (peak A) is largely di- minished. On the other hand, when the SOI on the Mn 4 p orbital is turned off, peaks A and B keep a similar shape, while peak C is reduced intensity and the minimum D al- most vanishes. We can conclude that the 3d orbital polar- ization at neighboring Mn sites induces the p orbital polarization near the edge through the 4 3p d� hybridiza- tion. The spectrum for 3.5 eV above the Fermi level origi- nates from the spin polarization in the 4 p symmetric states through the SOI. Similar results have been obtained 38 Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 V.N. Antonov, A.P. Shpak, and A.N. Yaresko T o ta l D O S P ar ti al d en si ty o f st at es spin up– spin down––10 0 10 C s C p –0.5 0 0.5 Zn d –10 0 10 Mn d eg t2g –10 –5 0 5 10 Energy, eV –1 0 1 Fig. 36. The total (in states/(cell�eV)) and partial (in states/(atom�eV)) ferromagnetic density of states of Mn3ZnC in cubic perovskite-type structure [214]. The Fermi energy is at zero. P ar ti al d en si ty o f st at es Cp spin up– spin down– –0.5 0 0.5 Zn d –10 0 10 Mn d2 –2 0 2 Mn d1 –5 0 5 Energy, eV –2 0 2 C p Fig. 37. Partial density of states (in states/(atom�eV)) of Mn3ZnC in low temperature non-collinear tetragonal structure [214]. The Fermi energy is at zero. by Takahashi and Igarashi in the ferromagnetic cubic phases of Mn3GaC [213]. The experimental Mn K XMCD spectrum shows a noteworthy variation with temperature [210]. As the tem- perature is decreased the dichroic intensity of peak A is rapidly increased, with other peaks being almost unaf- fected. When the positive peak A overcomes the negative contribution of peaks B, C and D the value of 4 p orbital magnetic moment M L p indicates a reversal of sign from positive to negative around 185 K. The change in orbital moment is obviously associated with the appearance of the noncollinear magnetic structure. Figure 38,c presents the calculated XMCD spectra for low-temperature tetragonal Mn3ZnC compound at the Mn K edges compared with the experimental data [210]. Al- though the number of Mn1 atoms which are canting �65� from the c axis to [111] direction is two times larger than the number of ferromagnetically aligned Mn 2 and Mn 3 at- oms (Fig. 35) the main contribution to the prominent peak A situated near the Fermi-edge comes from the Mn2,3 at- oms (Fig. 38,c). We found that dm El�1( ) function is three times larger at the Mn 2 3, sites than at Mn1 one around the Fermi edge. The orbital magnetic moment in p projected DOS is equal to –0.00010 �B at Mn1 and –0.00102 �B at the Mn 2 3, sites. Therefore the effect of SO coupling in 4 p states is different for particular Mn sites in Mn 3ZnC. To estimate the Mn orbital moment in the p �projected states the authors of Ref. 210 integrated the experimental XMCD spectrum over a range of –5 eV to 13 eV. The inte- grated XMCD intensity has a positive value in the tetragonal ferrimagnetic phase, which signifies a negative orbital mag- netic moment in the p-projected bands by the relation M lL z� � � � �B . On the assumption of nh = 6, the magni- tude of M L p was estimated to be around –0.00075 �B /Mn from the sum rules (Eq. (17)). We apply sum rules to our theoretically calculated XMCD spectra and obtained M L p = �0.00012 �B and –0.00136�B for the Mn1 and the Mn 2 3, sites, respectively, which gives the value of the M L p = –0.00054/Mn�B in rea- sonable agreement with the experimental estimations and band structure calculations. We found that the dm El�1( ) function is 1.5 times larger at the Mn 2 3, sites than at the Mn site in cubic hight tem- perature phase around the Fermi edge. Besides, due to lowering crystal symmetry from cubic to tetragonal through structural phase transition and decreasing the Mn �Mn interatomic distances the 4 3p d� hybridization is in- creased. It affects mostly the states near the Fermi-edge because the Mn d DOS are rather small at 8 3.5 eV above the Fermi level (Fig. 37), therefore the main changes of the intensity of dichroic signal due to this effect one would expect at the prominent peak A (Fig. 38,c). We can conclude that the increase of the 4 3p d� hy- bridization and change of the effect of SO coupling in the 4 p states leads to a strong increase of the spectral inten- sity of the prominent peak A through structural and mag- netic phase transitions at Tt � 233 K. The authors of Ref. 210 measured magnetic field vari- ation of the dichroic spectrum and found surprising re- sults: the peaks B and D are sensitive and increase with negative intensity, whereas the peak A is rather insensi- tive to the applied magnetic field (just slightly increases with increasing magnetic field from 0.5 to 10 T). There- X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Fizika Nizkikh Temperatur, 2008, v. 34, No. 1 39 X A S , ar b .u n it s X M C D , ar b .u n it s X M C D , ar b .u n it s a collinear noncollinear exper. 300 K 0 5 10 15 b A B C D collinear xmcd exper. 300 K 0 0.2 noncollinear c Mn1 Mn + Mn2 3 sum exper. 30 K -5 0 5 10 15 Energy, eV 0 0.2 dml Fig. 38. Theoretically calculated [214] isotropic absorption spec- tra of Mn3ZnC at the Mn K edge for cubic high temperature phase (dashed line) and low temperature tetragonal structure (full line) in comparison with the experimental spectrum [210] (cir- cles) measured at 300 K. The dotted line shows the theoretically calculated background spectrum (a); The experimental XMCD spectrum [210] measured at 300 K (circles) and theoretically cal- culated XMCD spectra for cubic high temperature phase (full line), the dotted line presents the dm El�1( ) function (see the ex- planation in the text) (b); Theoretically calculated XMCD spec- trum for low temperature tetragonal structure in comparison with the experimental measurements at 30 K [210] (circles); dashed and dotted lines show the theoretically calculated contributions from Mn1 and Mn2 + Mn3 sites, respectively, the thick full line is the total spectrum (c). fore the dichroic spectral peak makes a different response to the external parameters; roughly, the peak A is sensi- tive to temperature, while the other peaks to magnetic field. The authors speculate that this behavior indicates a process in which the canted Mn orbital moments are forcedly aligned to the direction of applied magnetic field. We investigated the influence of the Mn 1 canting an- gle on the XMCD spectra and 4 p orbital magnetic mo- ments at Mn sites. Figure 39 presents the theoretically calculated XMCD spectra at the Mn 1 and Mn 2 3, sites as a function of the Mn 1 canting angle for the low temperature tetragonal structure. We found that the prominent peak A is rather insensitive to the Mn1 canting in the 65� to 35� angle interval. On the other hand, the peak B strongly in- creased in the same angle range. It can be explained by different behavior of the these peaks at the Mn 1 and Mn2,3 sites. In decreasing of the canted angle the intensity of peak A is increased at the Mn 1 site and decreased at the Mn1 sites, almost compensating each other. However, the peak B increased at both nonequivalent sites, with an es- pecially strong increase at the Mn 1 site. We found a simi- lar behavior also for the function dm El�1( ) (not shown). Figure 40 presents variation of Mn 4 p orbital moments at the Mn1 and Mn 2 3, sites with the canting Mn1 angle. Both the orbital moments are negative for most of the angle interval. The Mn 2 3, 4 p orbital moments are slightly de- creased when canting angle changes from 65� to 40 � , and more quickly decrease with further decreasing of the cant- ing angle. On the other hand, the Mn1 4 p orbital moment is first increased from 65� to 40 � and then decreased. As the canting Mn1 angle is decreased further, the value of M L p indicated a reversal of sign from negative to positive around 15� and 7� for the Mn1 and Mn2,3sites, respec- tively, which is responsible for the trend that the negative peaks B and D overcome the positive contribution of peaks A and C. 1. M. Faraday, Philos. Trans. R. Soc. 136, 1 (1846). 2. J. Kerr, Philos. Mag. 3, 321 (1877). 3. M.J. Freiser, IEEE Transactions on Magnetics 4, 1 (1968). 4. H.R. Hulme, Proc. Roy. Soc. (London) A135, 237 (1932). 5. C. Kittel, Phys. 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