Electronic and magnetic properties of the ferromagnetic superconductor UCoGe

Electronic and magnetic properties of the ferromagnetic superconductor UCoGe

The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the ferromagnetic superconductor UCoGe at the U N₄,₅, Ge and Co K and Co L₂,₃ edges were investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital band structure meth...

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Автор: Antonov, V.N.
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Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2017
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К 100-летию со дня рождения И.М. Лифшица
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Цитувати:Electronic and magnetic properties of the ferromagnetic superconductor UCoGe / V.N. Antonov // Физика низких температур. — 2016. — Т. 43, № 1. — С. 68-80. — Бібліогр.: 64 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-129354
record_format dspace
spelling Antonov, V.N.
2018-01-19T13:56:33Z
2018-01-19T13:56:33Z
2017
Electronic and magnetic properties of the ferromagnetic superconductor UCoGe / V.N. Antonov // Физика низких температур. — 2016. — Т. 43, № 1. — С. 68-80. — Бібліогр.: 64 назв. — англ.
0132-6414
PACS: 71.28.+d, 75.30.Mb
https://nasplib.isofts.kiev.ua/handle/123456789/129354
The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the ferromagnetic superconductor UCoGe at the U N₄,₅, Ge and Co K and Co L₂,₃ edges were investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital band structure method. The electronic structure is obtained with the local spin-density approximation (LSDA), as well as with a generalization of the LSDA+ U method which takes into account the non-diagonal occupation matrix (in spin indexes) of localized electrons. A stable ferromagnetic ground state was found. The uranium total magnetic moment is quite small (about −0.171μB) in the LSDA approximation as a result of almost complete cancellation between the spin magnetic moment of 0.657μB and the opposite orbital magnetic moment of −0.828μB, resulting from strong spin-orbit coupling at the uranium site. Valency of U ion in UCoGe is close to 3+. The ratio orbital and spin magnetic moments M l/M s ranged from 1.163 in the GGA approach up to 2.456 for the LSDA+ U calculations is smaller than the corresponding ratio for the free ion U³⁺ value (2.60), it can indicate a significant delocalization of the 5 f-electron states due to the hybridization of the U 5f electrons with the conduction band and Co 3d electrons. The line shape of the dichroic spectra at the U M₅ and M₄ edges predicted by considering the magneto-optical selection rules as well as the occupation and the energy sequence of the mj-projected partial densities of states. The theoretically calculated XMCD spectra at the U M₄,₅, Ge and Co K and Co L₂,₃ edges are in good agreement with the experimentally measured spectra.
This work was supported by Science and Technology Center in Ukraine STCU, Project No. 6255.
en
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
Физика низких температур
К 100-летию со дня рождения И.М. Лифшица
Electronic and magnetic properties of the ferromagnetic superconductor UCoGe
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Electronic and magnetic properties of the ferromagnetic superconductor UCoGe
spellingShingle Electronic and magnetic properties of the ferromagnetic superconductor UCoGe
Antonov, V.N.
К 100-летию со дня рождения И.М. Лифшица
title_short Electronic and magnetic properties of the ferromagnetic superconductor UCoGe
title_full Electronic and magnetic properties of the ferromagnetic superconductor UCoGe
title_fullStr Electronic and magnetic properties of the ferromagnetic superconductor UCoGe
title_full_unstemmed Electronic and magnetic properties of the ferromagnetic superconductor UCoGe
title_sort electronic and magnetic properties of the ferromagnetic superconductor ucoge
author Antonov, V.N.
author_facet Antonov, V.N.
topic К 100-летию со дня рождения И.М. Лифшица
topic_facet К 100-летию со дня рождения И.М. Лифшица
publishDate 2017
language English
container_title Физика низких температур
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format Article
description The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the ferromagnetic superconductor UCoGe at the U N₄,₅, Ge and Co K and Co L₂,₃ edges were investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital band structure method. The electronic structure is obtained with the local spin-density approximation (LSDA), as well as with a generalization of the LSDA+ U method which takes into account the non-diagonal occupation matrix (in spin indexes) of localized electrons. A stable ferromagnetic ground state was found. The uranium total magnetic moment is quite small (about −0.171μB) in the LSDA approximation as a result of almost complete cancellation between the spin magnetic moment of 0.657μB and the opposite orbital magnetic moment of −0.828μB, resulting from strong spin-orbit coupling at the uranium site. Valency of U ion in UCoGe is close to 3+. The ratio orbital and spin magnetic moments M l/M s ranged from 1.163 in the GGA approach up to 2.456 for the LSDA+ U calculations is smaller than the corresponding ratio for the free ion U³⁺ value (2.60), it can indicate a significant delocalization of the 5 f-electron states due to the hybridization of the U 5f electrons with the conduction band and Co 3d electrons. The line shape of the dichroic spectra at the U M₅ and M₄ edges predicted by considering the magneto-optical selection rules as well as the occupation and the energy sequence of the mj-projected partial densities of states. The theoretically calculated XMCD spectra at the U M₄,₅, Ge and Co K and Co L₂,₃ edges are in good agreement with the experimentally measured spectra.
issn 0132-6414
url https://nasplib.isofts.kiev.ua/handle/123456789/129354
citation_txt Electronic and magnetic properties of the ferromagnetic superconductor UCoGe / V.N. Antonov // Физика низких температур. — 2016. — Т. 43, № 1. — С. 68-80. — Бібліогр.: 64 назв. — англ.
work_keys_str_mv AT antonovvn electronicandmagneticpropertiesoftheferromagneticsuperconductorucoge
first_indexed 2025-11-25T22:46:37Z
last_indexed 2025-11-25T22:46:37Z
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fulltext Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1, pp. 68–80 Electronic and magnetic properties of the ferromagnetic superconductor UCoGe V.N. Antonov Institute for Metal Physics, 36 Vernadsky Str., 03142 Kiev, Ukraine E-mail: antonov@imp.kiev.ua Received June 7, 2016, published online November 25, 2016 The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the ferromagnetic su- perconductor UCoGe at the U N4,5, Ge and Co K and Co L2,3 edges were investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital band structure method. The electronic structure is obtained with the local spin-density approximation (LSDA), as well as with a generalization of the LSDA+U method which takes into account the non-diagonal occupation matrix (in spin indexes) of loca- lized electrons. A stable ferromagnetic ground state was found. The uranium total magnetic moment is quite small (about –0.171µB) in the LSDA approximation as a result of almost complete cancellation between the spin magnetic moment of 0.657 µB and the opposite orbital magnetic moment of –0.828 µB, resulting from strong spin-orbit coupling at the uranium site. Valency of U ion in UCoGe is close to 3+. The ratio orbital and spin magnetic moments Ml/Ms ranged from 1.163 in the GGA approach up to 2.456 for the LSDA+U calcula- tions is smaller than the corresponding ratio for the free ion U3+ value (2.60), it can indicate a significant de- localization of the 5f-electron states due to the hybridization of the U 5f electrons with the conduction band and Co 3d electrons. The line shape of the dichroic spectra at the U M5 and M4 edges predicted by conside- ring the magneto-optical selection rules as well as the occupation and the energy sequence of the mj-projected partial densities of states. The theoretically calculated XMCD spectra at the U M4,5, Ge and Co K and Co L2,3 edges are in good agreement with the experimentally measured spectra. PACS: 71.28.+d Narrow-band systems; intermediate-valence solids; 75.30.Mb Valence fluctuation, Kondo lattice, and heavy-fermion phenomena. Keywords: heavy fermion, x-ray magnetic circular dichnoism spectra. 1. Introduction Uranium compounds exhibit a rich variety of properties to a large extent because of the complex behavior of their 5f electrons. The 5f states in U are intermediate between the itinerant 3d electrons in transition metals and the local- ized f electrons in rare-earth compounds. The determina- tion of the electronic structure of U compounds is a chal- lenging task because in many of them the width of the 5f bands, their spin-orbit splitting, and the on-site Coulomb repulsion in the partially filled 5f shell are of the same or- der of magnitude and should be taken into account on the same footing. Interest in uranium compounds has recently been renewed, especially after the discovery of such unu- sual effects as heavy-fermion superconductivity and the coexistence of superconductivity and magnetism. The coexistence of ferromagnetism (FM) and super- conductivity (SC) has been at the forefront of condensed matter research since a pioneering paper by Ginzburg [1]. The interplay between two long-range orderings FM and SC is a fascinating aspect in strongly correlated electron systems because generally SC does not favorably coexist with FM since the FM moment gives rise to an internal magnetic field, which breaks the pairing state. During the last three decades, however, the discovery of a number of magnetic superconductors has allowed for a better under- standing of how magnetic order and superconductivity can coexist. It seems to be generally accepted that antifer- romagnetism with local moments coming from rare-earth (RE) elements readily coexists with type-II superconduc- tivity [2]. This is because superconductivity and mag- netism are carried by different types of electrons; mag- netism is connected with deeply seated 4f electrons, while superconductivity is fundamentally related to the outer- most electrons such as s, p, and d electrons. In the case of a ferromagnetic superconductor the situation is more com- plex because internal fields are not canceled out in the © V.N. Antonov, 2017 mailto:antonov@imp.kiev.ua Electronic and magnetic properties of the ferromagnetic superconductor UCoGe range of a superconducting coherence length in contrast with an antiferromagnetic superconductor. A fascinating aspect of this class of compounds is the observation that, within superconducting regime, a wealth of ground states can occur. Although a myriad of experiments have been devoted to the characterization of these ground states, a comprehensive understanding of the ferromagnetic super- conductor properties at low temperature is still lacking. The ferromagnetic superconductor ground-state properties are highly sensitive to impurities, chemical composition, and slight changes of external parameters. This sensitivity indicates that a subtle interplay between different interac- tions produces a richness of experimental phenomena. The coexistence of FM and SC was first discovered in UGe2 [3] under pressure in 2000, almost two decades after the discovery of SC in CeCu2Si2. Soon afterward, the SC was found in the weak ferromagnet URhGe for the first time at ambient pressure [4]. Recently UCoGe with identi- cal crystal structure of URhGe was found to be a ferro- magnetic superconductor, as well [5]. In all of these com- pounds, scT is lower than TC, indicating that SC phase exists in the FM phase, which is contrary to the case such as ErRh4B4 [6]. UCoGe belongs to the family of intermetallic UTX com- pounds (where T is a transition metal and X is Si or Ge) [7], crystallizing in the orthorhombic TiNiSi-type structure. Orig- inally, it was reported to have a paramagnetic ground state [7]. Magnetization measurements show that UCoGe is a weak ferromagnet with a Curie temperature TC = 3 K and a tiny ordered moment µ = 0.03µB. Superconductivity is ob- served with a resistance transition temperature scT = 0.8 K. Additional thermal-expansion and specific-heat measure- ments provide clear evidence of bulk magnetism and super- conductivity [5]. The proximity to a ferromagnetic instability suggests the electrons forming triplet Cooper pairs and su- perconductivity mediated by ferromagnetic fluctuations [5]. Nonlinear field response of the Shubnikov–de Haas frequen- cy in UCoGe was observed above 20 T [8] and a possible field-induced topological Fermi surface transition, also known as a Lifshitz transition, supported by thermopower [9], and magnetoresistivity measurements [8,10]. Besides URhGe [4], UCoGe is the second known example of a fer- romagnetic superconductor in the TiNiSi-type family of compounds. There has been considerable impetus to understand the electronic structure and magnetism in 5f materials, includ- ing this series of superconducting ferromagnets, owing to the wide variety of ground-state properties exhibited. The- oretical models are required to explain the properties of interactions and fluctuations, and a consequence of this is the need of direct knowledge of the spin and orbital mo- ments. The x-ray magnetic circular dichroism technique developed in recent years has evolved into a powerful magnetometry tool to separate orbital and spin contribu- tions to element specific magnetic moments. X-ray mag- netic circular dichroism experiments consist of measuring the absorption of x rays with opposite (left and right) states of circular polarization. A unique situation can be formed where the spin orbit coupling is typically of a similar mag- nitude to the crystal field. The delicate balance between these can lead to different ground states in apparently simi- lar compounds, depending on the degree of localization of the 5f electrons. For U, Hund's rules, which describe a lo- cal moment system, can be used to obtain the ratio of the orbital moment (Ul) and the spin moment (Us). In a free ion the ratio is given by Ul/Us = –3.29 for U4+ and Ul/Us = = –2.60 for U3+, and values below these are then used to characterize the itinerancy of the 5f electrons [11]. The strong interplay between magnetism and supercon- ductivity is a common feature of the ferromagnetic super- conductors. While the magnetism of UGe2 and URhGe is well established and understood, this is not the case for UCoGe where the respective contribution of U and Co is still under debate. There is an urgent need for a detailed knowledge of the magnetism of UCoGe. Band structure calculations [12] and neutron experiments [11] have en- deavored to explore the orbital and spin part of the ordered moment, but contradictory results were published. On one hand, theoretical calculations in the LSDA approach [12] predict a small uranium moment (~0.1 µB) due to an al- most cancellation of substantial orbital and spin moments, and unexpectedly a large cobalt moment (0.2–0.5 µB) ei- ther parallel or antiparallel to the U moment. On the other hand, comparison of NMR data of UCoGe led to conclude that the ferromagnetism in UCoGe originates predominant- ly from U 5f electrons at least at low field [13]. Surprising- ly, polarized neutron diffraction experiments [14] show that, in an applied field of 3 T, the small ordered moment is essentially carried by the U atoms (~0.1 µB), while at 12 T a substantial moment (~0.2 µB) antiparallel to the U moment is induced at the Co site and a parallel magneti- zation is observed in the interstitial regions (~0.3 µB). This paper is concentrated on the theoretical investigation of x-ray absorption spectra (XAS) and x-ray magnetic circu- lar dichroism (XMCD) in UCoGe. Study of the 5f electron shell in uranium compounds is usually performed by tuning the energy of the x ray close to the 4,5M edges of uranium (located at 3552 and 3728 eV, respectively) where electronic transitions between 3/2,5/23d and 5/2,7/25 f states occur. Recently XMCD measurements have been successfully per- formed for UCoGe compound [15,16]. Taupin et al. [15] measured XAS and XMCD spectra at the U 4,5M and Co/Ga K edges. The orbital (–0.70 µB) and spin (0.30 µB) moments of U at 2.1 K and 17 T have been determined. The XMCD at the Co/Ge K edges reveal the presence of small Co 4p and Ge 4p orbital moments parallel to the macroscopic magnetization. In addition, the Co 3d moment is estimated to be at most of the order of –0.1 µB at 17 T. Butchers et al. [16] measured the XAS and XMCD signal of UCoGe at 1.5 K and 6 T at Co L2,3 edges. They estimated the same value of Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 69 V.N. Antonov the spin magnetic moment at U site equal to ~0.30 µB, how- ever, U orbital moment was found to be equal –0.40 µB al- most as twice smaller than data from Taupin et al. [15]. Both the publications obtain rather small ratio Ul/Us (–2.3 in Ref. 15 and –1.3 in Ref. 16) suggesting a significant delocali- zation of the 5f-electron states in UCoGe. There are some features in common for all the uranium compounds investigated up to now including UCoGe. First, the dichroism at the 4M edge is much larger, some- times one order of magnitude larger, than at the 5M one. Second, the dichroism at the 4M edge has a single nega- tive lobe that has no distinct structure, on the other hand, two lobes, a positive and a negative one, are observed at the 5M edge. With the above as background, we have performed calculations to evaluate the XMCD properties for UCoGe. The comparison between experiment and theo- ry provides insight into the nature of the 5f electrons and offers some evaluation of the suitability of several elec- tronic structure methods for treating 5f electrons. This paper is organized as follows. Section 2 presents a description of the crystal structure of UCoGe and the compu- tational details. Section 3 is devoted to the electronic struc- ture and XMCD properties of UCoGe calculated in the LSDA, GGA and LSDA+U approximations. The magneto- optical (MO) and XMCD theoretical calculations are com- pared to the experimental measurements. Finally, the results are summarized in Sec. 4. 2. Crystal structure and computational details 2.1. Crystal structure The crystal structure of UCoGe is orthorhombic TiNiSi- type (Pnma, space group No 62). The crystal structure is shown in Fig. 1, with the cell parameters taken from Huy et al. [5], and the internal atomic positions from Canepa et al. [17]. These parameters are shown in Table 1. The U atom forms the zig-zag chain along a axis with the distance of U Ud − = 3.477 Å, which is close to the so-called Hill limit associated with the direct overlap of 5f wave functions [18]. Of the U atoms substructure each U atom has four U neighbors, two along the chain, and the other two in contig- uous chains. The four neighbors form an unsymmetrical tetrahedron around the central U atom. This situation is similar to C in diamond, that is, the U atoms form a distorted diamond structure. U has six nearest Co neighbors: one at the 2.8420 Å distance, two at the 2.9330 Å distance, two at the 3.0715 Å distance and one at the 3.1050 Å distance. Two Ge atoms are situated at the 2.9378 Å distance from U, one Ge is at the 3.0238 Å, another at the 3.0246 Å, and last two Ge atoms are at the 3.0358 Å distance. 2.2. X-ray magnetic circular dichroism 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 excitation of electrons in the conduction band. Near x-ray absorption edges, or resonances, magneto-optical effects can be enhanced by transitions from well-defined atomic core levels to transition symmetry selected valence states. Within the one-particle approximation, the absorption coefficient ( )j λµ ω for incident x ray of polarization λ and photon energy ω can be determined as the probability of electronic transitions from initial core states with the total angular momentum j to final unoccupied Bloch states 2( ) = | | | | ( )j n jm n jmj j m nj E Eλλµ ω 〈Ψ Π Ψ 〉 δ − − ω ×∑∑ k k k  ( ),n FE E× θ −k (1) where jm j Ψ and jm j E are the wave function and the energy of a core state with the projection of the total angu- lar momentum ;jm nΨ k and nE k are the wave function and the energy of a valence state in the nth band with the wave vector k; FE is the Fermi energy. Table 1. Wyckoff positions (x, y, z) for UCoGe [17]. Lattice constants a = 6.845 Å, b = 4.206 Å and c = 7.222 Å [5] Atom Site x y z U 4c 0.0101 0.25 0.7075 Co 4c 0.2887 0.25 0.4172 Ge 4c 0.1967 0.25 0.0870 Fig. 1. (Color online) The crystal unit cell of UCoGe of the TiNiSi-type (Pnma, space group No 62) structure. 70 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 Electronic and magnetic properties of the ferromagnetic superconductor UCoGe λΠ is the electron-photon interaction operator in the dipole approximation = ,eλ λΠ − aα (2) where α are the Dirac matrices, λa is the λ polarization unit vector of the photon vector potential, with ||= 1/ 2(1, ,0), = (0, 0, 1).a i a± ± Here, + and – denotes, respectively, left and right circular photon polarizations with respect to the magnetization direction in the solid. Then, x-ray magnetic circular and linear dichroism are gi- ven by + −µ − µ and || ( )/2,+ −µ − µ + µ respectively. More detailed expressions of the matrix elements in electric di- pole approximation may be found in Refs. 19–22. Matrix elements due to magnetic dipole and electric quadrupole corrections presents in Ref. 22. Concurrent with the development of the x-ray magnetic circular dichroism experiment, some important magneto- optical sum rules have been derived [23–26]. For the 2,3L edges the zl sum rule can be written as [21] 3 2 3 2 ( ) 4= , 3 ( ) L L z h L L d l n d + − + + − + ω µ − µ 〈 〉 ω µ + µ ∫ ∫ (3) where hn is the number of holes in the d band = 10h dn n− , zl〈 〉 is the average of the magnetic quantum number of the orbital angular momentum. The integration is taken over the whole 2p absorption region. The zs sum rule can be written as 3 2 3 2 ( ) 2 ( ) 7 = , 2 ( ) L L z z h L L d d s t n d + − + − + − + ω µ − µ − ω µ − µ 〈 〉 + 〈 〉 ω µ + µ ∫ ∫ ∫ (4) where zt is the z component of the magnetic dipole oper- ator 2= 3 ( )/ | |− ⋅t s r r s r which accounts for the asphericity of the spin moment. The integration 3L∫ 2L     ∫ is taken only over the 3/2 1/22 (2 )p p absorption region. In order to simplify the comparison of the theoretical x-ray isotropic absorption M4,5 spectra of UCoGe to the experimental ones we take into account the background intensity which affects the high-energy part of the spec- tra. The shape of x-ray absorption caused by the transi- tions from inner levels to the continuum of unoccupied levels was first discussed by Richtmyer et al. in the early thirties [27]. The absorption coefficient with the assump- tion of equally distributed empty continuum levels is 2 2 0 ( ) = 2 ( / 2) ( ) cfc c cfEcf dEC E ∞Γ µ ω π Γ + ω −∫  , (5) where = ,cf c fE E E− cE and cΓ are the energy and the width of a core level, fE is the energy of empty continu- um level, 0f E is the energy of the lowest continuum level, and C is a normalization constant which has been used as an adjustable parameter. 2.3. Calculation details The details of the computational method are described in our previous papers [28,29], and here we only mention some aspects specific to the present calculations. The calculations presented in this work were performed using the spin- polarized fully relativistic linear-muffin-tin-orbital (LMTO) method [30–32] for the experimentally observed lattice con- stants [5]. The LSDA part of the calculations was based on the spin-density functional with the Perdew–Wang [33] of the exchange-correlation potential. The exchange-correlation functional of a GGA-type was also used in the version of Perdew, Burke and Ernzerhof [34,35]. The basis consisted of the s, p, and d LMTO's for Co and Ge sites and the s, p, d, and f LMTO's for U site. The k-space integrations were performed with the improved tetrahedron method [36] and the self-consistent charge density was obtained with 175 irreducible k-points in UCoGe. The x-ray absorption and dichroism spectra were calcu- lated taking into account the exchange splitting of core levels. The finite lifetime of a core hole was accounted for by folding the spectra with a Lorentzian. The widths of core level spectra for U, Co, and Ge was taken from Ref. 37. The finite apparative resolution of the spectrome- ter was accounted for by a Gaussian of width 0.6 eV. It is a well-known fact that the LSDA calculations fail to produce the correct value of the orbital moment of ura- nium compounds [38–42]. 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 re- sponsible for Ml, is, however, not included into the equa- tions. This means, that although Ms is self-consistently determined in the LSDA, there is no framework to deter- mine simultaneously Ml self-consistently. Numerous attempts have been made to better estimate Ml in solids. They can be roughly classified into two cate- gories. One is based on the so-called current density func- tional theory [43–45] which is intended to extend density functional theory to include the orbital current as an extra degree of freedom, which describes Ml. Unfortunately an explicit form of the current density functional is at present unknown. The other category includes orbital polarization (OP) [39–42], self-interaction correction (SIC) [46], and LSDA+U [47,48] approaches, which provide a means to calculate Ml beyond the LSDA scheme. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 71 V.N. Antonov Solovyev et al. [47] argued that the key parameter re- sponsible for the exchange-correlation enhancement of the orbital magnetic moments in solids is the “Hubbard U” rather than the intra-atomic Hund's second rule coupling, being consistent with a more general concept of the orbital polarization. This leads to a unified rotationally invariant LSDA+U prescription for the orbital magnetism. We have adopted the LSDA+U method [48,49] as a dif- ferent level of approximation to treat the electron-electron correlation. We used a generalization of the LSDA+U meth- od which takes into account spin-orbit coupling so that the occupation matrix of localized electrons becomes non- diagonal in spin indexes. This method is described in detail in our previous paper [48] including the procedure to calculate the screened Coulomb U and exchange J integrals, as well as the Slater integrals F2, F4, and F6. We used J = 0.5 eV, U = 2.0 eV and U = 0.5 eV for the U site. In the last case of = = 0.5U J eV, eff = = 0U U J− and the effect of the LSDA+U comes from non-spherical terms which are deter- mined by F2 and F4 Slater integrals. This approach is similar to the orbital polarization corrections mentioned above [21]. For the last LSDA+U approach with = = 0.5U J eV, eff = = 0U U J− we will use the notation LSDA+OP through the paper. 3. Results and discussion 3.1. Band structure For electronic structure calculations where there are heavy elements it is important to include spin-orbit interac- tions, and the magnetization is now dependent of the crys- tallographic direction. The results show that the most sta- ble magnetization direction in UCoGe is in the c axis, by 1.5 and 2.8 meV/f.u. with respect to the a and b direc- tions, respectively. The fully relativistic spin-polarized LSDA partial DOSs of the ferromagnetic UCoGe compound are shown in Fig. 2. The results are in good agreement with previous band struc- ture calculations [12,50–53]. The occupied part of the DOS can be decomposed into two regions. The first region, from –10.7 to –7.9 eV, consists mainly of the Ge 4s states, which are split off from the main valence-band group (see Fig. 2). There is a band gap from –7.9 to –5.6 eV. The bottom of the main valence-band group is situated at –5.6 eV. This band group represents mainly the Co 3d states with the major part located below the Fermi level (EF) between –5.6 and –0.07 eV. There is a pronounced hybridization between the Co 3d and the U 5f states between –5.7 and 2 eV. Typically for other UTM compounds, the spin-orbit cou- pling leads to two main U 5f peaks, split into the 5f5/2 and 5f7/2 states (in energy range between –0.4 and 2.2 eV), shifted from each other by about 1 eV. The bands crossing EF are dominated by the U 5f5/2 states (see Fig. 3), being partly hybridized with the Co 3d states, and these together create a metallic bond. The Co 3d states are clearly present in the whole range of valence- band energies with some tail at EF. Some U 6d and Ge 4p band tails occur at EF as well, also contributing to a metal- lic bond. There is also a wide contribution from the U 6d and Ge 4p states in the same energy range as that of the U 5f states, hybridizing with the latter. Figure 4 shows ARPES spectra of UCoGe measured at 20 K with the photon energies equal to 500 eV for T–Z–T high-symmetry line [53]. The experimental ARPES spectra are rather featureless. The spectra basically consist of three high-intensity parts located at around EB = EF, 0.4 eV, and 0.8 eV. The former two are the contributions mainly from U 5f states, while the last one is mainly from Co 3d states although they are strongly hybridized (see Fig. 3). Further- more, there are clear gaps in the intensity at around EB = = 0.2 eV and between 0.5–0.7 eV. Figure 4 also shows the calculated band structures of UCoGe (full black curves) using the LSDA, LSDA+OP, and LSDA+U approximations. Although, many dispersive bands exist in the calculation, and its comparison with the experimental spectra is not Fig. 2. (Color online) The LSDA self-consistent fully relativistic, spin-polarized partial DOSs (in states/(atom·eV)) of UCoGe. 72 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 Electronic and magnetic properties of the ferromagnetic superconductor UCoGe straightforward the correspondences between the ARPES spectra and the calculations are clearly recognized. Mean- while, the three approximations produce rather different structures of the energy bands in that energy interval. The LSDA approximation well describes the dispersion of U 5f energy bands in close vicinity of the Fermi level, however, fails to deliver correct energy position of the bands around 0.4 and 0.8 eV. The LSDA+OP approximation, on the other hand, significantly improves the agreement between theoret- ically calculated and experimentally measured band posi- tions in these two-energy intervals. Both the approximations well describe the gap in the intensity at around EB = 0.2 eV but not around 0.5–0.7 eV. For the LSDA+U approach (with Hubbard U = 2 eV) the agreement between theory and ex- periment becomes worse especially in states near EF and gap at 0.2 eV. We can conclude that the LSDA+OP approach produces the energy bands in closer agreement with the ARPES measurement in general comparing with other two approximations. 3.2. XMCD spectra 3.2.1. U M4,5 spectra. In order to compare relative ampli- tudes of the 4M and 5M XMCD spectra we first normalize the corresponding isotropic x-ray absorption spectra (XAS) to the experimental ones taking into account the background scattering intensity as described in Sec. 2. Figure 5 shows the calculated isotropic x-ray absorption and XMCD spectra in the LSDA, LSDA+OP, and LSDA+U approximations together with the experimental data [15]. The contribution from the background scattering is shown by dashed lines in the upper panel of Fig. 5. The experimentally measured dichroic 4M line consists of a simple nearly symmetric negative peak that has no distinct structure. Such a peak is characteristic of the 4M edge of all uranium systems. The dichroic line at the 5M edge has an asymmetric s shape with two peaks — a stronger negative peak and a weaker positive peak. The dichroism at the 4M edge is almost three times larger than at the 5M one. We recall that the M4 (M5) edge corresponds to 3/2 5/23 (3 ) 5d d f→ transitions. Because of the electric dipole selection rules ( = 1l∆ ± ; = 0, 1j∆ ± ) the major con- tribution to the absorption at the 4M edge stems from the transitions 3/2 5/23 5d f→ and that at the 5M edge origi- nates primarily from 5/2 7/23 5d f→ transitions, with a weaker contribution from 5/2 5/23 5d f→ transitions. For Fig. 3. (Color online) The LSDA self-consistent fully relativistic, spin-polarized energy band structure in “fat band” representation and U 5f (red curve) and Co 3d partial DOSs (in states/(atom·eV)) of UCoGe in the vicinity of EF. The amounts of the U 5f (open red circles) and Co 3d (full blue circles) states characters are marked by the thickness of the bands. Fig. 4. (Color online) Blowup of the experimental ARPES spec- tra of UCoGe [53] and the band-structure calculations (full black curves) for the LSDA, LSDA+OP, and LSDA+U approximations along the T–Z–T high-symmetry lines. Fig. 5. (Color online) The experimental XAS and XMCD spectra [15] (open circles) at the U M4,5 edges in UCoGe compared with the theoretically calculated ones using the LSDA (dashed green curves), LSDA+OP (full blue curves), and LSDA+U (dotted red curves) approximations. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 73 V.N. Antonov the later case the corresponding 5/2 5/23 5d f→ radial ma- trix elements are only slightly smaller than for the 5/2 7/23 5d f→ transitions. The angular matrix elements, however, strongly suppress the 5/2 5/23 5d f→ contribu- tion. Therefore the contribution to XMCD spectrum at the M5 edge from the transitions with = 0j∆ is about 15 times smaller than the transitions with = 1j∆ . The selection rules for the magnetic quantum number jm ( jm is restricted to ,... )j j− + are = 1jm∆ + for = 1λ + and = 1jm∆ − for = 1.λ − Table 2 presents the dipole allowed transitions for x-ray absorption spectra at the M5 and M4 edges for left ( = 1)λ + and right ( = 1)λ − polarized x rays. To go further, we needs to discuss the characteristic of the 5f empty DOS. Since l and s prefer to couple antiparal- lel for less than half-filled shells, the = = 5/2j l s− has a lower energy than the = = 7/2j l s+ level. Due to the in- tra-atomic exchange interaction the lowest sublevel of the = 5/2j will be 5/2 = 5/2,m − however, for the = 7/2j the lowest sublevel will be 7/2 = 7/2.m + This reversal in the energy sequence arises from the gain in energy due to alignment of the spin with the exchange field [54]. The 7/25 f states are almost completely empty in all the uranium compounds. Therefore all the transitions listed in Table 2 are active in the 5M absorption spec- trum. The contribution from the first four transitions for = 1λ + cancels to a large extent with the contribution of the opposite sign from the last four transitions for = 1λ − having the same final states. Thus the XMCD spectrum of U at the 5M edge ( = )I − +µ − µ can be roughly ap- proximated by the following sum of mj-projected partial densities of states: 7/2 7/2(N− + 7/2 7/2 5/2 7/2) (N N− − + 7/2 5/2 ).N Here we use the notation j m j N for the density of states with the total momentum j and its projection mj. As a result, the shape of the 5M XMCD spectrum contains of two peaks of opposite signs — a negative peak at lower energy and a positive peak at higher energy. As the separation of the peaks is smaller than the typical lifetime broadening, the peaks cancel each other to a large extent, thus leading to a rather small signal. Since the splitting of states with = | |j jm m± increases with the increase of the magneti- zation at the U site, the amplitude of the 5M spectrum should be proportional to the U magnetic moment. A rather different situation occurs in the case of the 4M x-ray absorption spectrum. Usually in uranium compounds the U atom is in 3 35 ( )f U + or 2 45 ( )f U + configurations and has partly occupied 5f5/2 states. In the first case the 5f5/2 states with = 5/2,jm − 3/2,− and 1/2− are usually occu- pied. The dipole allowed transitions for = 1λ + are 1/2 1/2,− → + 1/2 3/2,+ → + and 3/2 5/2+ → + and those for = 1λ − are 3/2 1/2.+ → + The transitions with the same final states mj = +1/2 mostly cancel each other and the XMCD spectrum of U at the 4M edge can be roughly rep- resented by the sum 5/2 5/2 3/2 5/2( ).N N− + The corresponding analysis for the 2 45 ( )f U + configuration with occupied 5/2, 5/2f − and 5/2, 3/2f − states shows that the dipole allowed transitions for = 1λ + are 3/2 1/2,− → − 1/2 1/2,− → + 1/2 3/2,+ → + and 3/2 5/2+ → + and for = 1λ − : 1/2 1/2+ → − and 3/2 1/2.+ → + Again, the XMCD spec- trum of U at the 4M edge can be approximated by 5/2 5/2 3/2 5/2( ).N N− + This explains why the dichroic 4M line in uranium compounds consists of a single nearly symmetric negative peak. We should note, however, that the explanation of the XMCD line shape in terms of mj-projected DOS's pre- sented above should be considered as only qualitative. First, there is no full compensation between transitions with equal final states due to difference in the angular matrix elements; second, in our consideration we neglect cross terms in the transition matrix elements; third, there is no pure 35 f or 25 f configurations in uranium com- pounds. It is always difficult to estimate an appropriate atomic 5f occupation number in band structure calcula- tions. Such a determination is usually obtained by the integration of the 5f electron charge density inside of the corresponding atomic sphere. In the particular UCoGe case, the occupation number of U 5f states is around 2.92 in the LSDA calculations. We, however, should keep in mind that some amount of the 5f states are derived from the so-called “tails” of Co 3d and Ge 4p states arising as a result of the decomposition of the wave function cen- tered at Co and Ge atoms. The careful analysis in the case of UPd3 presented in Ref. 48 shows that the occupation number of the “tails” of Pd 4d states sum up to give the 5f occupation of 0.9 electrons in the U atomic sphere. We should also note that due to the strong hybridization be- tween U 5f and Co 3d states, the U 7/25 f states in UCoGe are not completely empty, some of them are oc- cupied, also some amount of U 5/25 f states, which we have been considering as fully occupied, are partially empty. Table 2. The dipole allowed transitions from core 3d3/2,5/2 levels to the unoccupied 5f5/2,7/2 valence states for left (λ = +1) and right (λ = –1) polarized x rays Edge λ = +1 λ = –1 –5/2 ® –3/2 –5/2 ® –7/2 –3/2 ® –1/2 –3/2 ® –5/2 M5 –1/2 ® +1/2 –1/2 ® –3/2 +1/2 ® +3/2 +1/2 ® –1/2 +3/2 ® +5/2 +3/2 ®+1/2 +5/2 ® +7/2 +5/2 ® +3/2 –3/2 ® –1/2 –3/2 ® –5/2 M4 –1/2 ® +1/2 –1/2 ® –3/2 –5/2 ® –3/2 +1/2 ® –1/2 +3/2 ® +5/2 +3/2 ® +1/2 74 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 Electronic and magnetic properties of the ferromagnetic superconductor UCoGe The overall shapes of the calculated and experimental uranium 4,5M XMCD spectra correspond well to each other (Fig. 5). The major discrepancy between the calcu- lated and experimental XMCD spectra is the size of the M4 XMCD peak. The LSDA underestimates the integral inten- sity of the XMCD at the M4 edge. As the integrated XMCD signal is proportional to the orbital moment [24] this discrepancy may be related to an underestimation of the orbital moment by LSDA-based computational meth- ods (see Table 4). On the other hand, the LSDA+U approx- imation gives larger intensity for the M4 XMCD spectrum in comparison with the experimentally measured one. It reflects the overestimation of the orbital moment at U site in the LSDA+U calculations (Table 4). The LSDA+OP approximation, in contrast, gives good agreement in the amplitude of the negative peak at the M4 edge. The LSDA+OP approximation also slightly better reproduces the shape of the M5 XMCD spectrum in comparison with the LSDA and LSDA+U approaches. 3.2.2. Co L2,3 spectra. Figure 6 shows the calculated XAS and XMCD spectra at the Co 2,3L edges in UCoGe together with the experimental spectra [16]. Because of the dipole selection rules, apart from the 4s1/2 states (which have a small contribution to the XAS due to relatively small 2p ® 4s matrix elements) only 3/23d states occur as final states for 2L XAS for unpolarized radiation, whereas for the 3L XAS the 5/23d states also contribute [21]. Although the 2p3/2 ® 3d3/2 radial matrix elements are only slightly small- er than for the 2p3/2 ® 3d5/2 transitions the angular matrix elements strongly suppress the 2p3/2 ® 3d3/2 contribution [21]. Therefore neglecting the energy dependence of the radial matrix elements, the 2L and the 3L spectra can be viewed as a direct mapping of the DOS curve for 3/23d and 5/23d character, respectively. The Co 3L XAS spectrum has a pronounced shoulder at the 3L peak shifted by about 3 eV with respect to the maximum to higher photon energy. The theory represents this shoulder but with smaller intensi- ty in comparison with the experiment. Similar situation is observed for the Co 2L spectrum where theory also under- estimates the intensity of high energy shoulder. Figure 6 (lower panel) shows the theoretically calculated Co 2,3L XMCD spectra in UCoGe using the LSDA, LSDA+OP, and LSDA+U approximations in comparison with the experimentally measured spectra [16]. A qualita- tive explanation of the XMCD spectra shape is provided by the analysis of the corresponding selection rules, or- bital character and occupation numbers of individual 3d orbitals. Table 3 presents the dipole allowed transitions for x-ray absorption spectra at the 3L and 2L edges for left and right polarized x rays. The dichroism at the 3L edge has a major negative peak with positive high energy shoulder. The LSDA calcu- lations underestimate the intensity both the major negative peak and high energy shoulder. The LSDA+OP and LSDA+U produce similar results, however, LSDA+OP still better describes the relative intensities of the two fine structures. 3.2.3. Co and Ge K spectra. Figure 7 shows the calcu- lated XAS and XMCD spectra at the Co K edge using the LSDA, LSDA+OP, and LSDA+U approximations in UCoGe together with the experimental spectra [15]. Be- cause dipole allowed transitions dominate the absorption spectrum for unpolarized radiation, the absorption coeffi- cient 0 ( )K Eµ (not shown) reflects primarily the DOS of unoccupied 4p-like states ( )pN E of Co above the Fermi level. Due to the energy-dependent radial matrix element for the 1 4s p→ there is no strict one-to-one correspond- Table 3. The dipole allowed transitions from core 2p1/2,3/2 levels to the unoccupied 3d3/2,5/2 valence states for left (λ = +1) and right (λ = –1) polarized x rays Edge λ = +1 λ = –1 –3/2 ® –1/2 –3/2 ® –5/2 L3 –1/2 ® +1/2 –1/2 ® –3/2 2p3/2 ® 3d5/2 +1/2 ® +3/2 +1/2 ® –1/2 +3/2 ® +5/2 +3/2 ® +1/2 –3/2 ® –1/2 –1/2 ®–3/2 L3 –1/2 ® +1/2 +1/2 ® –1/2 2p3/2 ® 3d3/2 +1/2 ® +3/2 +3/2 ® +1/2 L2 –1/2 ® +1/2 –1/2 ® –3/2 2p1/2 ® 3d3/2 +1/2 ® +3/2 +1/2 ® –1/2 Fig. 6. (Color online) The experimental XAS and XMCD spectra [16] (open circles) at the Co L2,3 edges in UCoGe compared with the theoretically calculated ones using the LSDA (dashed green curves), LSDA+OP (full blue curves), and LSDA+U (dotted red curves) approximations. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 75 V.N. Antonov ence between ( )K Eµ and ( )pN E . The exchange splitting of the initial 1s-core state is extremely small [55] therefore only the exchange and spin-orbit splitting of the final 4p states is responsible for the observed dichroism at the K edge. For this reason the dichroism is found to be very small (Fig. 7). It was first pointed out by Gotsis and Strange [56] as well as Brooks and Johansson [57] that XMCD K-spectrum reflects the orbital polarization in dif- ferential form /zd l dE〈 〉 of the p states. It gives a rather simple and straightforward interpretation of the Co XMCD spectrum at the K edge [58]. The x-ray absorption spectrum at the Co K edge con- sists of the major peak at around 17 eV above the edge and low energy shoulder at 2 eV. Theory well describe the en- ergy position of these fine structures. All the three approx- imations produce rather similar K XAS spectra. However, the deviations in the different approximations are well rec- ognized for the XMCD spectra (lower panel of Fig. 7). The LSDA produces well the low-energy positive peak at 1 eV but completely fails to describe the second negative peak at 4 eV above the edge. The LSDA+U calculations overesti- mate the intensity of this peak. The LSDA+OP gives the best agreement with experimental spectrum. Figure 8 shows the calculated XAS and XMCD spectra at the Ge K edge using the LSDA, LSDA+OP, and LSDA+U approximations in UCoGe together with the ex- perimental spectra [15]. Theory well reproduces the major fine structures of Ge K XAS spectrum: the main peak at 9 eV, low energy shoulder at 2 eV and high energy shoul- der at 13 eV. However, theory overestimates the intensity of last shoulder. Again, all the three approximations give identical XAS spectra. In contrast, these approaches give very different dichroism spectra. The LSDA approach underestimates the dichroism at the Ge K edge (lower panel of Fig. 8), the LSDA+U overestimates it and only the LSDA+OP give reasonable agreement with experimental measurements. It is important to note that the Co and Ge K XMCD spectra (Figs. 7 and 8) which reflect energy distribution of the 4p valence state are much more sensitive to the type of the approximation for the U 5f states in comparison with the corresponding Co 2,3L XMCD spectra (Fig. 6). This fact indicates the stronger U 5f – Co(Ge) 4p hybridization in comparison with U 5f – Co 3d one due to more expand- ed character of the 4p wave functions. 3.3. Magnetic moments In magnets, the atomic spin sM and orbital lM magne- tic moments are basic quantities and their separate determi- nation is therefore important. Methods of their experimental determination include traditional gyromagnetic ratio meas- Fig. 7. (Color online) The experimental XAS and XMCD spectra [15] (open circles) at the Co K edge in UCoGe compared with the theoretically calculated ones using the LSDA (dashed green curves), LSDA+OP (full blue curves), and LSDA+U (dotted red curves) approximations. Fig. 8. (Color online) The experimental XAS and XMCD spectra [15] (open circles) at the Ge K edge in UCoGe compared with the theoretically calculated ones using the LSDA (dashed green curves), LSDA+OP (full blue curves) and LSDA+U (dotted red curves) approximations. 76 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 Electronic and magnetic properties of the ferromagnetic superconductor UCoGe urements [59], magnetic form factor measurements using the neutron scattering [60], and magnetic x ray scattering [61]. In addition to these, the recently developed x ray magnetic circular dichroism combined with several sum rules [24,25] has attracted much attention as a method of site- and sym- metry-selective determination of sM and lM . The experimentally observed net magnetization in UCoGe are highly controversial. On one hand, Huy et al. [5] estimated the total magnetic moment as quite small (Mt = 0.03µB per f.u.), on the other hand, the effective par- amagnetic moment found by Troc and Tran is much larger Mt = 1.7µB [62]. The XMCD measurements provide total magnetic moments in between of these extreme values: 0.16µB [16] and 0.44µB [15]. Our band structure calcula- tions produce the total magnetic moment equal to 0.298, 0.411, 1.345, and 2.007µB for the LSDA, GGA, LSDA+OP, and LSDA+U approximations, respectively. Table 4 presents the comparison between calculated and experimental magnetic moments in UCoGe. It is clear that in the case of LSDA as well as the GGA calculations, the val- ues of antiparallel spin and orbital moments on a uranium atom almost compensate each other. The spin and orbital moments on the Co atom are rather small, but parallel, and thus enhance each other. The total moment per U atom alig- ned along the c, b, and a axes is as small as –0.171,–0.123, and –0.164µB, respectively, in the LSDA approach. At the same time, the Ge atom delivers a negligibly negative total magnetic moment. Finally, UCoGe has totM (per f.u.) equal to about –0.298, –0.246, and –0.333µB, for alignments along the above three axes, respectively. The situation is changed when OP correction is included in the U 5f states in the cal- culations. This leads to a marked increase of especially the orbital moment lM on the U site (–2.093 µB) and, hence, the value of the total U magnetic moment, which becomes as large as –1.073 µB for the c axis. A similar situation also oc- curs for the remaining axes. The LSDA+U produces even larger U orbital moment (–3.077µB) and the total magnetic moment became equal to –1.824 µB at the U site. To investigate a possible error of the sum rules we compare the spin and orbital moments obtained from the theoretically calculated XAS and XMCD spectra through the sum rules with directly calculated LSDA and LSDA+OP values in order to avoid additional experi- mental problems. The sum rules reproduce the spin mag- netic moments within 7%, 15%, and 22% and the orbital moments within 5%, 21%, and 0% for the U, Co, and Ge sites, respectively, for the LSDA approach (Table 4). Taupin et al. [15] experimentally determined the branching ratio B for the 3/2,5/23 5d f→ transition of U as 5/2 5/2 3/2= /( )B A A A+ , where 5/2A and 3/2A are the integrated areas of the isotropic white lines at the 5,4M edges, respectively. B was found to be 0.701. Thus the experimentally determined B is in between those calculat- ed for the 25 f (U4+) and 35 f (U3+) electronic configura- tions (0.686 and 0.729, respectively) [15]. Thus the 5f elec- tron count is 52 < < 3.f en This is in agreement with our band structure calculations, 5 f en is estimated to be 2.92 (in the LSDA calculations) and 2.96 (for the GGA approach), as well as with core level photoelectron spectroscopy [64] which shows that 5 f en is less than but close to 3. These results invalidate the occurrence of U4+ ions as suggested from neutron form factor analysis [14]. From the ratio Ml/Ms the degree of localization of 5f states can be estimated. If this ratio fall below the free ion U3+ value (2.60), it can indicate a significant delocalization of the 5f electron states due to the hybridization of the U 5f electrons with the conduction band and Co 3d electrons [11]. This ratio is range from 1.163 in the GGA approach up to 2.456 for the LSDA+U calculations. The XMCD measurements estimate this ratio equal to 1.3 [16] and 2.34 [15]. Therefore, one can conclude that the on-site Coulomb repulsion on the U site is suppressed, most likely because of the strong hybridization between U 5f and Co 3d/4p and Ge 4p states. As a consequence, U 5f electrons in UCoGe demonstrate almost purely itinerant behavior. Table 4. The experimental and calculated spin Ms, orbital Ml, and total Mt magnetic moments (in µB) of UCoGe at U, Co and Ge sites with magnetization along c direction Method U Co Ge Ms Ml Mt –Ml/Ms Ms Ml Ms Ml LSDA 0.657 –0.828 –0.171 1.453 –0.100 –0.019 –0.009 –0.001 GGA 0.955 –1.105 –0.150 1.163 –0.201 –0.041 –0.017 –0.002 LSDA+OP 1.020 –2.093 –1.073 2.052 –0.191 –0.066 –0.017 –0.002 LSDA+U 1.253 –3.077 –1.824 2.456 –0.135 –0.032 –0.007 –0.009 LSDA [12] 1.076 –1.215 –0.139 1.129 –0.231 –0.057 –0.036 –0.002 LSDA+OP [12] 1.254 –2.666 –1.412 2.126 –0.300 –0.183 –0.045 –0.010 LSDA [63] 1.083 –1.181 –0.098 1.090 –0.472 –0.063 –0.026 0 Exper. [15] 0.297 –0.695 –0.398 2.34 – – – – Exper. [16] 0.30 –0.40 –0.16 1.3 –0.16 – – – Sum rules (LSDA) 0.609 –0.868 –0.259 1.425 –0.115 –0.015 –0.007 –0.001 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 77 V.N. Antonov 4. Summary We have studied by means of an ab initio fully- relativistic spin-polarized Dirac linear muffin-tin orbital (LMTO) method the electronic structure and the x-ray magnetic circular dichroism in UCoGe at the U M4,5, Co L2,3, Co K, and Ge K edges. The bands crossing FE are dominated by the U 5/25 f states, being partly hybridized with the Co 3d states, and these together create a metallic bond. We found that the LSDA+OP approach produces the energy bands in closer agreement with the ARPES measurement in general com- paring with the LSDA and LSDA+U approximations. In the case of LSDA as well as the GGA calculations, the values of antiparallel spin and orbital moments on a uranium atom almost compensate each other. The spin and orbital mo- ments on the Co atom are rather small, but parallel, and thus enhance each other. The total moment per U atom aligned along the c, b, and a axes is as small as –0.171, –0.123, and –0.164 µB, respectively. The situation is changed when OP correction is included in the U 5f states in the calculations. This leads to a marked increase of especially the orbital moment lM on the U site (–2.093 µB) and, hence, the total U magnetic moment, which becomes as large as –1.073 µB for the c axis. A similar situation also occurs for the re- maining axes. The experimentally measured dichroic 4M line consists of a simple nearly symmetric negative peak that has no distinct structure. The dichroic line at the 5M edge has an asymmetric s shape with two peaks — a stronger negative peak and a weaker positive peak. The overall shapes of the calculated and experimental uranium 4,5M XMCD spectra correspond well to each other. The major discrepancy be- tween the calculated and experimental XMCD spectra is in the amplitude of the 4M XMCD peak. The LSDA under- estimates the integral intensity of the XMCD at 4M edge, the LSDA+U approach overestimates the dichroism at this edge. The LSDA+OP approximation, in contrast, gives the best agreement with the experimental spectrum. Similar situation was observed in the case of the 5M XMCD spec- trum where the LSDA+OP gives the best agreement with the experimental spectrum. The line shape of the dichroic spectra can be qualitatively understood considering the MO selection rules as well as the occupation and the energy sequence of the mj-projected par- tial densities of states. The 5 7/2f states are almost com- pletely empty in all the uranium compounds and the XMCD spectrum of U at the 5M edge can be roughly approximated by the following sum of partial densities of 7/25 f states: 7/2 7/2 7/2 7/2 7/2 5/2 7/2 5/2( ) ( ).N N N N− −+ − + As a result, the shape of the 5M XMCD spectrum stems from two peaks of opposite signs — a negative peak at lower energy and a positive peak at higher energy. As the separation of the peaks is smaller than the typical lifetime broadening, the peaks cancel each other to a large extent, thus leading to a rather small signal. A rather different situation occurs in the case of the M4 x-ray absorption spectrum. Uranium compounds have partially occupied 5/25 f states and the XMCD spectrum of U at the 4M edge can be approximated by 5/2 5/2 3/2 5/2( ).N N− + This explains why the dichroic 4M line in uranium compounds consists of a single nearly symmetric negative peak. The XMCD signals at U 2,3M , 2,3N , 2,3O and 6,7N edges are two orders of magnitude weaker than the corresponding signals at the 4,5M edges. Due to small exchange splitting of the initial 1s-core states only the exchange and spin-orbit splitting of the final 4p states are responsible for the observed dichroism at Co and Ge K edges. The XMCD spectra of Co for the 2,3L edge are mostly determined by the strength of the SO coupling of the initial 2p-core states and spin- polarization of the final empty 3/2,5/23d states while the exchange splitting of the 2p-core states as well as the SO coupling of the 3d valence states are of minor im- portance. The 5f electron count is 52 < < 3f en from the XMCD measurement and close to 3 from core level photoelectron spectroscopy. This is in agreement with our band structure calculations, 5 f en is estimated to be 2.92 (in the LSDA calculations) and 2.96 (for the GGA approach). Therefore valency of U ion in UCoGe is close to 3+. 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Phys.: Condens. Matter 20, 285221 (2008). 64. S. Fujimori, T. Ohkochi, I. Kawasaki, A. Yasui, Y. Takeda, T. Okane, Y. Saitoh, A. Fujimori, H. Yamagami, Y. Haga, E. Yamamoto, Y. Tokiwa, S. Ikeda, T. Sugai, H. Ohkuni, N. Kimura, and Y. Ōnuki, J. Phys. Soc. Jpn. 81, 014703 (2012). 80 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 http://journals.jps.jp/author/Yamamoto%2C+Etsuji http://journals.jps.jp/author/Yamamoto%2C+Etsuji http://journals.jps.jp/author/Tokiwa%2C+Yoshifumi http://journals.jps.jp/author/Ikeda%2C+Shugo http://journals.jps.jp/author/Sugai%2C+Takashi http://journals.jps.jp/author/Ohkuni%2C+Hitoshi http://journals.jps.jp/author/Kimura%2C+Noriaki http://journals.jps.jp/author/Kimura%2C+Noriaki http://journals.jps.jp/author/%C5%8Cnuki%2C+Yoshichika 1. Introduction 2. Crystal structure and computational details 2.1. Crystal structure 2.2. X-ray magnetic circular dichroism 2.3. Calculation details 3. Results and discussion 3.1. Band structure 3.2. XMCD spectra 3.3. Magnetic moments 4. Summary Acknowledgments

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