X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I
The curreThe current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 4 f and 5 f compounds is reviewed. Energy band theory based upon the local spin-density approximation (LSDA) describes the XMCD spectra of transition metal compounds with high accuracy. Howe...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1168102017-05-17T03:02:37Z X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part I Antonov, V.N. Shpak, A.P. Yaresko, A.N. Обзоp The curreThe current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 4 f and 5 f compounds is reviewed. Energy band theory based upon the local spin-density approximation (LSDA) describes the XMCD spectra of transition metal compounds with high accuracy. However, the LSDA does not suffice for lanthanide compounds which have a correlated 4 f shell. A satisfactory description of the XMCD spectra could be obtained by using a generalization of the LSDA, in which explicitly f electron Coulomb correlation are taken into account (LSDA + U approach). As examples of this group we consider GdN compound. We also consider 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 differences 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 conduction electrons plays an important role. We also consider the electronic structure and XMCD spectra of heavy-fermion compounds UPt₃, URu₂Si₂, UPd₂Al3₃, UNi₂Al₃, and UBe₁₃ where the degree of the 5 f localization is increased in comparison with other uranium compounds. The electronic structure and XMCD spectra of UGe₂ which possesses simultaneously ferromagnetism and superconductivity also presented. Recently achieved improvements for describing 5 f 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, № 2. — С. 107–147. — Бібліогр.: 198 назв. — англ. 0132-6414 PACS: 75.50.Cc; 71.20.Lp; 71.15.Rf http://dspace.nbuv.gov.ua/handle/123456789/116810 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Обзоp Обзоp |
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Обзоp Обзоp Antonov, V.N. Shpak, 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 curreThe current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 4 f
and 5 f compounds is reviewed. Energy band theory based upon the local spin-density approximation
(LSDA) describes the XMCD spectra of transition metal compounds with high accuracy. However, the
LSDA does not suffice for lanthanide compounds which have a correlated 4 f shell. A satisfactory description
of the XMCD spectra could be obtained by using a generalization of the LSDA, in which explicitly f
electron Coulomb correlation are taken into account (LSDA + U approach). As examples of this group we
consider GdN compound. We also consider 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 differences 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
conduction electrons plays an important role. We also consider the electronic structure and XMCD spectra
of heavy-fermion compounds UPt₃, URu₂Si₂, UPd₂Al3₃, UNi₂Al₃, and UBe₁₃ where the degree of the 5 f localization
is increased in comparison with other uranium compounds. The electronic structure and XMCD
spectra of UGe₂ which possesses simultaneously ferromagnetism and superconductivity also presented. Recently
achieved improvements for describing 5 f compounds are discussed. |
| format |
Article |
| author |
Antonov, V.N. Shpak, A.P. Yaresko, A.N. |
| author_facet |
Antonov, V.N. Shpak, 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/123456789/116810 |
| 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, № 2. — С. 107–147. — Бібліогр.: 198 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT antonovvn xraymagneticcirculardichroismindandfferromagneticmaterialsrecenttheoreticalprogressparti AT shpakap xraymagneticcirculardichroismindandfferromagneticmaterialsrecenttheoreticalprogressparti AT yareskoan xraymagneticcirculardichroismindandfferromagneticmaterialsrecenttheoreticalprogressparti |
| first_indexed |
2025-07-08T11:04:39Z |
| last_indexed |
2025-07-08T11:04:39Z |
| _version_ |
1837076504805113856 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2, p. 107–147
X-ray magnetic circular dichroism in d and f ferromagnetic
materials: recent theoretical progress. Part II
(Review Article)
V.N. Antonov and A.P. Shpak
Institute of Metal Physics, 36 Vernadskii Str., 03142 Kiev, Ukraine
E-mail: antonov@imp.kiev.ua
A.N. Yaresko
Max-Planck-Institut for the Physics of Complex Systems, D-01187 Dresden, Germany
Received April 10, 2007
The current status of theoretical understanding of the x-ray magnetic circular dichroism (XMCD) of 4 f
and 5 f compounds is reviewed. Energy band theory based upon the local spin-density approximation
(LSDA) describes the XMCD spectra of transition metal compounds with high accuracy. However, the
LSDA does not suffice for lanthanide compounds which have a correlated 4 f shell. A satisfactory descrip-
tion of the XMCD spectra could be obtained by using a generalization of the LSDA, in which explicitly f
electron Coulomb correlation are taken into account (LSDA + U approach). As examples of this group we
consider GdN compound. We also consider uranium 5 f compounds. In those compounds where the 5 f elec-
trons are rather delocalized, the LSDA describes the XMCD spectra reasonably well. As example of this
group we consider UFe 2. Particular differences 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
conduction electrons plays an important role. We also consider the electronic structure and XMCD spectra
of heavy-fermion compounds UPt 3, URu 2Si 2, UPd 2Al 3, UNi 2Al 3, and UBe13 where the degree of the 5 f lo-
calization is increased in comparison with other uranium compounds. The electronic structure and XMCD
spectra of UGe 2 which possesses simultaneously ferromagnetism and superconductivity also presented. Re-
cently achieved improvements for describing 5 f compounds are discussed.
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. Rare-earth compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
1.1. GdN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2. Uranium compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.1. Intermetallic compounds . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.1.1. UFe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.1.2. UXAl (X = Co, Rh, and Pt) . . . . . . . . . . . . . . . . . . . . . . . 119
2.2. Uranium monochalcogenides . . . . . . . . . . . . . . . . . . . . . . . . 123
2.3. Heavy-fermion compounds . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.3.1. UPt3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.3.2. URu2Si2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2.3.3. UPd2Al3 and UNi2Al3. . . . . . . . . . . . . . . . . . . . . . . . . . 132
2.3.4. UBe13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
2.4. UGe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
© V.N. Antonov, A.P. Shpak, and A.N. Yaresko, 2008
1. Rare-earth compounds
The astonishing variety of magnetic behaviors in
rare-earth-3d transition metal (R-TM) intermetallics re-
flects the complexity of the exchange mechanism involv-
ing direct and indirect interaction mediated by the band
states. These mechanisms are still not completely under-
stood. X-ray magnetic circular dichroism (XMCD), being
element and orbital selective, offers the opportunity to
probe the TM 4 p and 3d band states by scanning through
their K and L2 3, edges, respectively. Information on the
rare-earth ground state is usually obtained by performing
XMCD measurements at the rare-earth M 4 5, edges since
these edges involve the 3 4d f� transitions, i.e., they probe
the electronic states of the 4 f shell. On the other hand, L 2 3,
edges of rare-earth ion provide the information on the R 5d
band states through the 2 5p d� transitions. Such studies of
XCMD have shown to be very useful and give new insite on
both the magnetic properties of the R-TM compounds and
the interpretation of the XMCD spectra.
Recently systematic studies have been performed on
several series of R-TM intermetallic crystals, amorphous
materials and insulating ferromagnetic oxides in order to
extract the relevant physical effects which govern the
XCMD of K , L2 3, and M 4 5, edges. Giorgetti et al. [1] pres-
ent the XMCD studies at the L2 3, edges of Ce and K edge
of transition metals in CeFe2, Ce(Fe0.8Co0.2) 2, CeCo5,
Ce2Co17, CeRu2Ge2, Ce3Al11, CePd, CoFe2H3.8 com-
pounds. Suga and Imada [2] studied a dense Kondo mate-
rial, Sm4As3. They performed the M 4 5, and N 4 5, XMCD
at the Sm edges. The authors also measure XMCD spectra
in the permanent magnet Nd2Fe14B at the M 4 5, and L 2 3,
edges of Nd and Fe, respectively. The shape of the spectra
agree with atomic calculations.
XMCD at Er M 4 5, [3] was used to follow the H T, mag-
netic phase diagram of an amorphous Er12Fe73 alloy. In
these samples, macroscopic measurements of the mag-
netic moment show a strong evolution of the compensa-
tion temperature with the applied magnetic field. The
variation of XMCD at Er M 4 5, is consistent with a mag-
netic structure of both the Er and Fe atoms. It shows the
existence of temperature-induced, as well as field-in-
duced, flip of the Er sublattice with respect to the direc-
tion of the magnetic field, evidenced by the change of
sign of the dichroism. The authors of Ref. 4 present a
XMCD study of a CeCuSi compound and a Ce/Fe multi-
layer performed at the Ce M 4 5, absorption edge. In the
Ce/Fe multilayered structure (MLS), Ce atoms are in the
highly hybridized � phase, characterized by a strong mix-
ing between the 4 f electrons with the valence band, and
carry an ordered moment. XMCD experiments show the
part of this moment is due to 4 f electrons. The difference
in the shape of the XMCD signals of a typical �-like com-
pound CeCuSi and of the Ce/Fe multilayer demonstrate
that the XMCD spectra reflect the hybridization in the
ground state of the cerium atoms in the multilayer. Ce M 4 5,
XMCD spectra in strongly correlated ferromagnetic sys-
tems CeCuSi, CeRh3B2, and CeFe2 measured in Ref. 5.
By applying sum rules, it was shown that these experi-
ments are able to yield both the magnitude and the direc-
tion of the 4 f magnetic moments on Ce.
A systematic XMCD study at the Fe K edge on RFe14B
series (R = rare earth and Y) performed in Ref. 6. The
study identifies the influence of the rare-earth magnetic
state into the K edge XMCD signals in RFe14B inter-
metallic compounds. This signal results from the addition
of two components, each one being due to the magnetic
contribution of both the iron and the rare-earth sublat-
tices. The contribution of the R sublattice to the XMCD
signal has been extracted yielding a direct correlation to
the R magnetic moment. XMCD spectra has been mea-
sured in R–Co compounds (R = La, Tb, and Dy) at the Co
K edge [7]. The experimental results have been inter-
preted within the multiple-scattering framework includ-
ing the spin-orbit coupling. In the three systems, the
XMCD spectra in the near edge region are well repro-
duced. Co K edge XMCD spectra in crystalline and amor-
phous Gd–Co alloys measured in Ref. 8. The results anal-
yses using a semirelativistic full multiple scattering
approach. It was shown that the spin polarization on Co
atoms in GdCo5 alloys is smaller than that in Co metal.
XMCD experiments have been performed in Ref. 9 at
the R L2 3, (R = rare earth) and Ni K edges on single crys-
tals of GdNi5 and TbNi5. The spectra present huge and
well-structured dichroic signals at both the R L2 3, and N K
edges. In TbNi5 the negative XMCD structure, observed
3 eV below the edge at the Tb L3 edge, was interpreted as
the quadrupolar (2 4p f� ) transitions. A systematic
study of XMCD, x-ray resonance magnetic scattering
(XRMS), and resonance inelastic x-ray scattering (RIXS)
at the L2 3, edges of Nd on Nd2Fe14B presented in Ref. 10
allowes to assign a dipole (E1) or quadrupole (E2) origin
to different features appearing in the experimental spectra
and to study the thermal dependence of the Nd moment
orientation below the spin reorientation transition which
take place at TSRT = 135 K. A single crystal of Tb as a pro-
totype system for a one-element magnet was used to in-
vestigate XMCD at the L2 3, edge [11]. The high resolution
of the experimental data allows for a clear identification
of the E1 and E2 transitions. On the basis of ab initio cal-
culations a simple procedure for extracting of the E2 part
from the experimental XMCD data was developed.
Fe L2 3, XMCD spectra on a single crystal of Fe17Dy2
measured by Castro et al. in Ref. 12. XMCD study at the K
edge in the R6Fe23 series (R = Ho and Y) presented
in Ref. 13. This study identifies the influence of the
rare-earth magnetic state on the K edge XMCD signals.
The results demonstrate that the contribution of both Fe
and R to the K edge XMCD spectra can be easily isolated
108 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
following its temperature-dependent behavior through
compensation temperature and that they can be directly
correlated to the Fe and R magnetic moments.
The distinguishing feature of the rare-earth elements
in solids is the atomic character of the 4 f levels, which lie
relatively deep in the ion while having very small binding
energies. It is this feature that account for the chemical
similarity and magnetic diversity found in the series. A
well-known consequence of this localized behavior is that
a number of solid-state and spectroscopies involving the
4 f electrons can be explained with the multiplet structure
found from atomic calculations, with only small correc-
tions being necessary to incorporate solid-state effects.
A 3d absorption process in rare-earth ions involves the
electronic excitations to the 4 f or the valence band (VB)
levels [14], i.e.,
3 4 3 410 9 1d f d fn x n x( ) ( )VB VB� � (1)
or
3 4 3 410 9 1d f d fn x n x( ) ( )VB VB� � . (2)
The final-state configuration contains two open shells
and due to strong 3d–4f overlap gives rise to large Cou-
lomb and exchange correlation energies and produces a
wide spread over the multiplet levels. The 3d � VB exci-
tations (2) have much weaker strength in comparison with
the first ones (1) not only due to low 3 6d p� cross sec-
tion but also because near threshold the empty valence
band states have mostly 5d character with a little hybrid-
ized 6 p states. The total 3 49 1d f n � multiplet structure is
very complex and the total numbers of levels runs into
thousands for many elements in the middle of the
rare-earth series. There have been several calculations of
the 3 4 49 1d d f n( ) � multiplet structure in individual ele-
ments [15–26].
The theoretical analysis of magnetic circular dich-
roism of 4 f photoemission spectra in Gd and Tb ions re-
ported in Refs. 24–26. Imada and Jo calculate the M 4 5,
and N 4 5, x-ray absorption spectra (XAS) [23] for left and
right circularly polarized light in trivalent rare-earth ions.
Thole et al. [20] measured and calculated in intermediate
coupling the M 4 5, XAS for all the rare-earth metals.
The energy band approximation was used in Ref. 27 to
calculated XMCD spectra of Gd 5(Si 2Ge 2) compound.
To treat the correlation effects at a simple level the
LSDA + U method was used.
1.1. GdN
The Gd pnictides form an interesting family of materi-
als, because of the great variety of their magnetic and
electrical properties, despite their common simple crystal
structure, the face-centered cubic of sodium chloride.
While most Gd pnictides have been found to be antiferro-
magnetic, stoichiometric GdN after a controversial dis-
cussion over three decades [28,29] seems to be recog-
nized now as a ferromagnet. It has a Curie temperature TC
around 60 K and a magnetic saturation moment near
7 � B /Gd ion consistent with the 8S 7 2/ half filled 4 f shell
configuration of Gd 3 � with zero orbital angular momen-
tum [30].
An appealing property of GdN is that it is ferromag-
netic with a large gap at the Fermi energy in the minority
spin states, according to the electronic structure calcula-
tions based on the local density approximation [31–33].
At the same time, GdN is semimetallic in majority spin
states with electron and hole pockets at the Fermi surface
[32]. This latter property has led to some interest in GdN
as a possible candidate for spin-dependent transport
devices [34], exploiting the spin filter, giant magneto-
resistance, or tunneling magnetoresistance effects.
X-ray absorption spectra and XMCD at the gadolinium
M 4 5, and N K edges have been measured in GdN
by Leuenberger at al. [35]. The ordered 4 f moment ex-
tracted from the M 4 5, XMCD spectra was consistent with
the 8S 7 2/ configuration of Gd3+. The exchange field gen-
erated by the Gd 4 f electrons in the ferromagnetic phase
of GdN induces a magnetic polarization of the N p band
states, as can be concluded from the observation of strong
magnetic circular dichroism at the K edge of nitrogen.
However, a comparison of the spectra with the theoretical
partial density of vacant N p states shows considerable
disparities that are not well understood.
Figure 1 shows the fully relativistic spin-polarized en-
ergy band structure of GdN. In these calculations the 4 f
states have been considered as: (1) itinerant using the lo-
cal spin-density approximation, (2) fully localized, treat-
ing them as core states, and (3) partly localized using the
LSDA + U approximation.
The energy band structure of GdN with the 4 f elec-
trons in core can be subdivided into three regions sepa-
rated by energy gaps. The bands in the lowest region
around �12.9 to �11.1 eV have mostly N s character with
some amount of Gd sp character mixed in. The next six
energy bands are primarily N p bands separated from the s
bands by an energy gap of about 6.2 eV. The width of the
N p band is about 4.5 eV and is influenced by hybridiza-
tion with Gd 5d states. The spin splitting of the N p bands
is small (about 0.2 eV at the X symmetry point (Fig. 1)).
The highest region can be characterized as Gd crystal
field and spin-split d bands.
An important issue is the energy position of the occu-
pied 4 f states in the electron band structure of GdN. The
LSDA calculations place the empty 4 f states of Gd in
GdN at 1 to 2 eV above the Fermi level with the occupied
majority-spin 4 f states situated at around �4 to �3.2 eV
below Fermi level, EF . It is well known that LSDA usu-
ally gives a wrong energy position for the 4 f states in
rare-earth compounds. For nonzero 4 f occupation it
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 109
places the 4 f states right at the Fermi level [37,38] in con-
tradiction with various experimental observations. In the
case of Gd compounds the LSDA places the empty 4 f
states too close to the Fermi energy. For example, the
LSDA calculations produce the empty 4 f states in pure
Gd metal at 2.7 eV above the Fermi level [39], although
according to the x-ray bremsstrahlung isochromat spec-
troscopy (BIS) measurements they are situated around
5.5 eV above the Fermi level [40,41]. The XPS spectrum
measured by Leuenberger et al. [35] in the valence band
region of GdN shows the Gd 3 � 4 f 6 final state multiplet
located at around 8 eV below the Fermi level.
Figure 1 also presents the energy band structure of
GdN calculated in the LSDA + U approximation. In such
an approximation the Gd 4 f empty states are situated
around 5 eV above the Fermi level, well hybridized with
Gd 5d and N 2 p minority states. The majority-spin 4 f
states form a narrow band well below the Fermi energy
and occupy a �7 to �8 eV energy interval in good agree-
ment with the XPS measurements [35].
The partial density of states (DOS) of cubic ferromag-
netic GdN are presented in Fig. 2 for the LSDA +U calcu-
lations. The majority 4 f electrons create an exchange
field that leads to spin splitting of the N p band. Further-
more, there is a visible Gd d–N sp as well as Gd 4 f –N p
hybridization in occupied part of GdN valence band. One
of the consequences is that the N anion should carry a
magnetic moment. The Gd f states above the Fermi level
hybridize with the Gd d t g2
states only in the minority
channel (Fig. 2). The Gd d eg
states shift to higher energy
due to the crystal-filed splitting and almost don’t hybrid-
ize with the Gd 4 f states. The orbital moments are equal
to 0.057 � B and �0.0007� B on the Gd and N sites, respec-
tively. Exchange and hybridization induce spin splitting
of the conduction band states. As a result, the itinerant Gd
5d and N 2 p derived band electrons carry small spin mag-
netic moments of 0.107 � B and �0.098 � B , respectively,
that are of opposite each other and nearly cancel. The Gd
5d and N 2 p orbital moments are equal to �0.0066 � B and
�0.0007 � B , respectively.
110 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
GdN 4f in core DOS
–10
–5
0
5
E
n
er
g
y,
eV
X W K L W U X 0 5 10
LSDA
–10
–5
0
5
E
n
er
g
y,
eV
X W K L WU X 0 5 10 15 20
LSDA+U
–10
–5
0
5
E
n
er
g
y,
eV
X W K L WU X 0 5 10 15 20
Fig. 1. Self-consistent fully relativistic spin-polarized energy
band structure and total DOS (in states/(unit cell�eV)) calculated
for GdN treating the 4 f states as: (1) fully localized (4 f in core);
(2) itinerant (LSDA); and (3) partly localized (LSDA + U ) [36].
LSDA+U
N
spin up
spin down
s
p
–2
–1
0
1
Gd d
eg
t2g–0.5
0
0.5
Gd f
–10 –5 0 5 10
Energy, eV
–10
0
10
20
Fig. 2. Partial density of states of GdN [36]. The Fermi energy
is at zero.
One should mention that although Gd 3 � free ion con-
sistent with the 8S 7 2/ half filled 4 f shell configuration
possesses a zero orbital angular momentum, in solids Gd
has small but nonzero orbital moment of around 0.063 � B
due to hybridization with other states and also because in
solids spin-up states are the linear combination of the
4 5 2f / and 4 f 7 2/ states and ml for each state can be
noninteger.
The study of the 4 f electron shell in rare-earth com-
pounds is usually performed by tuning the energy of the
x-ray close to the M 4 5, edges of rare-earth where elec-
tronic transitions between 3d 3 2 5 2/ , / and 4 f 5 2 7 2/ , / states
are involved. Figure 3 shows the calculated XAS and
XMCD spectra in the LSDA + U approximation for GdN
at the M 4 5, edges together with the corresponding experi-
mental data [35]. The experimentally measured dichroism
is large, as is common for Gd-based systems at the 3d
threshold; it amounts to more than 20 %.
The theoretically calculated XAS spectra have a rather
simple line shape composed of two white line peaks at the
M 5 and M 4 edges, however the experimentally measured
spectra have well pronounced fine structures at high-en-
ergy part of the M 5 and M 4 XAS’s. This fine structure are
believed to be due to the multiplet structures which have
not been included in the band structure calculations.
Figure 3,b shows the calculated XMCD spectra in the
LSDA + U approximation for GdN together with the cor-
responding experimental data [35]. The dichroism is
mostly negative at the Gd M 5 edge and positive at the M 4
one. The calculations describe correctly the deep negative
minimum at the Gd M 5 edge and the low-energy positive
peak at the M 4 edge, however they don’t produce the
high-energy fine structures at both the edges, which are
probably caused by the multiplet structure as described
above. The XMCD at the M 5 edge also possesses an addi-
tional small positive lobe at the low-energy side which is
not in the theoretical calculations. The LSDA + U theory
underestimates the intensity for the XMCD spectrum at
M 5 edge and overestimates it at the M 4 edge in compari-
son with the experiment.
We investigate also the effect of the core-hole effect in
the final state using the supercell approximation. When
the 3d core electron is photoexcited to the unoccupied 4 f
states, the distribution of the charge changes to account
for the created hole. The final-state interaction improves
the agreement between theory and the experiment at the
M 5 edge in the intensity of the prominent negative peak
and by producing correctly a positive lobe at the low-en-
ergy side.
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 111
X
A
S
,
ar
b
.
u
n
it
s
X
M
C
D
,
ar
b
.
u
n
it
s
M
4
2
b
M5 a
LSDA+U
LSDA+U with hole
exper.
1180 1200 1220
Energy, eV
–2
–1
0
1
Fig. 3. (a) Theoretically calculated [36] (dotted lines) and
experimental [35] (circles) isotropic absorption spectra of GdN
at the Gd M 4 5, edges. Experimental spectra were measured with
external magnetic field (0.1 T) at 15 K. (b) Experimental [35]
(circles) XMCD spectra of GdN at the Gd M 4 5, edges in com-
parison with theoretically calculated ones using the LSDA + U
approximation without (dotted lines) and with (full lines) tak-
ing into account core-hole effect.
X
A
S
,
ar
b
.
u
n
it
s
X
M
C
D
,
ar
b
.
u
n
it
s
L3 L2
a
0
0.5
1.0
1.5
b
theory
exper.
7200 7250 7300 7350 7400 7450
Energy, eV
–0.05
0
0.05
Fig. 4. (a) Theoretically calculated [36] (dashed line) and
experimental [42] (circles) isotropic absorption spectra of GdN
at the Gd L2 3, edges (L2 spectra are shifted towards smaller
energy by 550 eV). Experimental spectra were measured at a
bulklike GdN layer deposited on a Si substrate at 450 °C in to-
tal fluorescence yield mode. Dotted lines show the theoretically
calculated background spectra, full thick lines are sum of the
theoretical XAS and background spectra. (b) Experimental [42]
(circles) XMCD spectra of GdN at the Gd L2 3, edges in compar-
ison with theoretically calculated ones using the LSDA + U
(full lines) approximation.
Figure 4 shows the calculated XAS and XMCD spectra
in the LSDA + U approximation at the L2 3, edges together
with the corresponding experimental data measured at
bulklike layers of GdN [42].
Our calculations of Gd L2 3, XAS produce two addi-
tional peaks at the high-energy side of the prominent
peak, the position of the second high-energy peak is in
good agreement with the experiment, however the first
one is less pronounced in the experimental spectra.
The dichroism at the L2 3, edges has two lobes, a posi-
tive and a negative one. The positive lobe is larger in com-
parison with the negative one for L3 spectrum and vice
versa for the L2 edge. Our LSDA +U calculations overes-
timate the smaller lobe and underestimate the larger one
at both the L3 and L2 edges.
We found minor influence of the final-state interaction
on the shape of the Gd L2 3, XMCD spectra in the whole
energy interval. A small core-hole effect might come from
the fact that the Gd 5d states are less localized in compari-
son with the 4 f states and have smaller amplitude inside
the MT sphere and thus are less subject to the core hole
potential.
A qualitative explanation of the XMCD spectra shape
is provided by the analysis of the corresponding selection
rules, orbital character and occupation numbers of indi-
vidual 5d orbitals. Because of the electric dipole selection
rules (�l
1; �j
0 1, ) the major contribution to the
absorption at the L2 edge stems from the transitions
2 51 2 3 2p d/ /� and that at the L3 edge originates primarily
from 2 p3 2/ � 5d 5 2/ transitions, with a weaker contribu-
tion from 2 p3 2/ � 5d 3 2/ transitions. For the later case the
corresponding 2 p3 2/ � 5d 3 2/ radial matrix elements are
only slightly smaller than for the 2 p3 2/ � 5d 5 2/ transi-
tions. The angular matrix elements, however, strongly
suppress the 2 p3 2/ � 5d 3 2/ contribution. Therefore the
contribution to XMCD spectrum at the L3 edge from the
transitions with �j 0 is one order of magnitude smaller
than the transitions with �j 1[43].
The selection rules for the magnetic quantum number
m j (m j is restricted to � �j j, ... ) are �m j � 1for � � 1
and �m j �1 for � �1. Table 1 presents the dipole al-
lowed transitions for x-ray absorption spectra at the L3
and L2 edges for left (� � 1) and right (� �1) polarized
x-rays.
To go further, we need to discuss the character of the
3d empty DOS. Since l and s prefer to couple antiparallel
for less than half-filled shells, the j l s / � 3 2 has a
lower energy than the j l s / � 5 2 level. Due to the
intra-atomic exchange interaction the lowest sublevel of
the j / 3 2 will be m /3 2 3 2/ � , however, for the j / 5 2
the lowest sublevel will be m /5 2 5 2/ � . This reversal in
the energy sequence arises from the gain in energy due to
alignment of the spin with the exchange field.
Table 1. The dipole allowed transitions from core 2 p1 2 3 2/ , / levels
to the unoccupied 5d3 2 5 2/ , / valence states for left ( )� �1 and
right (� �1) polarized x-rays
Edge � = +1 � = –1
L3
–3/2 � –1/2 –3/2 � –5/2
–1/2 � +1/2 –1/2 � –3/2
+1/2 � +3/2 +1/2 � –1/2
+3/2 � +5/2 +3/2 � +1/2
L2
–1/2 � +1/2 –1/2 � –3/2
+1/2 � +3/2 +1/2 � –1/2
The contribution to the L3 absorption spectrum from
the first two transitions (Table 1) for � � 1 cancels to a
large extent with the contribution of opposite sign from
the last two transitions for � �1 having the same final
states. Thus the XMCD spectrum of Gd at the L3 edge
( )I ��� � � can be approximated by the following sum
of m j -projected partial densities of states: (N �5 2
5 2
/
/ +
� ��N N3 2
5 2
3 2
5 2
/
/
/
/) ( + N 5 2
5 2
/
/ ). Here we use the notation N
m
j
j
for the density of states with the total momentum j and its
projection m j . From this expression one would expect the
L3 XMCD spectrum with two peaks of opposite signs
with almost the same intensity. The corresponding L2
XMCD spectrum can be approximated by the following
partial DOS’s: (N N N� �� �1 2
3 2
3 2
3 2
1 2
3 2
/
/
/
/
/
/) ( + N 3 2
3 2
/
/ ). From
this expression one would also expect two peak structure
of L2 XMCD spectrum with an opposite signs. Besides,
due to the reversal energy sequences for the j 3 2/ and
j 5 2/ sublevels the energy positions of the positive and
negative peaks are opposite to each other for the L3 and L2
XMCD spectra.
We should note, however, that the explanation of the
XMCD line shape in terms of m j -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 ma-
trix elements; second, in our consideration we neglect
cross terms in the transition matrix elements. Besides, we
have used here the jj-coupling scheme, however, the
combination of the hybridization, Coulomb, exchange
and crystal-field energies may be so large relative to the
5d spin-orbit energy that the jj-coupling is no longer an
adequate approximation.
The XMCD spectra at the Gd L2 3, edges are mostly de-
termined by the strength of the spin-orbit coupling of the
initial Gd 2 p core states and spin-polarization of the final
empty 5d 3 2 5 2/ , / states while the exchange splitting of the
Gd 2 p core states as well as the SO coupling of the 5d va-
lence states are of minor importance for the XMCD at the
Gd L2 3, edges of GdN.
112 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
The XAS and XMCD spectra in metals at the K edge in
which the 1s core electrons are excited to the p states
through the dipolar transition usually attract only minor
interest because p states are not the states of influencing
magnetic or orbital order. Recently, however, understand-
ing p states has become important since XMCD spectro-
scopy using K edges of transition metals became popular.
The K edge XMCD is sensitive to electronic structures at
neighboring sites, because of the delocalized nature of the
p states.
It is documented that sizable XMCD signals can be de-
tected at the K edge of nonmagnetic atoms, like sulfur and
oxygen in ferromagnetic EuS [44] and EuO [45], respec-
tively. The experimental K edge photoabsorption and
XMCD spectra of nitrogen in GdN were investigated by
Leuenberger et al. [35]. It was found that the dichroic
peak amplitude amounts to 4% of the edge jump of the
isotropic XA spectrum at 401 eV (Fig. 5), which is a re-
markably large value for K edge XMCD. The N K edge
dichroic signal in GdN is about three times larger than at
the K edge of oxygen in EuO and exceeds that at the K
edge of sulfur in EuS by an order of magnitude; it sur-
passes even that at the onsite Fe K edge of iron metal
where it is on the order of 0.3% [46].
A comparison of the XMCD spectra with the theoreti-
cal partial density of empty N p states calculated by Aerts
et al. [47] shows considerable disparities that were not
well understood [35]. Clearly, to reproduce the XMCD
spectra one has to include the transition matrix elements.
Figure 5 shows the theoretically calculated x-ray ab-
sorption spectra at the N K edge as well as XMCD spectra
in GdN in comparison with the corresponding experimen-
tal data [35]. The experimentally measured XA spectrum
has a three peak structure. The first maximum in the spec-
trum is at around 400 eV which has a low energy shoulder
not reproduced in the theoretical LSDA or LSDA + U cal-
culations. The energy position of the theoretical second
peak at around 402 eV is in good agreement with the
experimental measurements. The position of the third
high-energy peak is shifted to higher energy in the theory.
Figure 5,b shows the experimental XMCD spectrum
[35] and theoretically calculated ones using the LSDA ap-
proximation and with 4 f electrons placed in the core. The
experimental spectrum is very complicated and consists
of three positive (A, B, C) and two negative (D, F) peaks.
The LSDA calculations as well the calculations with 4 f
electrons in the core give a completely inadequate de-
scription of the shape of N K XMCD spectrum. The most
prominent discrepancy in the LSDA XMCD spectrum is
the resonance structure with negative and positive peaks
at around 396 to 398 eV which is caused by the strong hy-
bridization of unoccupied Gd N p states with the 4 f states
situated too close to the Fermi level in the LSDA calcula-
tions. This structure disappears when we put 4 f electrons
in core.
The N 2 p–Gd (4 f , 5d) hybridization and the spin-orbit
interaction in the 2 p states play crucial roles for the N K
edge dichroism. The K XMCD spectra come from the or-
bital polarization in the empty p states, which may be in-
duced by (1) the spin polarization in the p states through
the spin-orbit interaction, and (2) the orbital polarization
at neighboring sites through hybridization. We calculated
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 113
X
A
S
,
ar
b
.
u
n
it
s
N
K
X
M
C
D
,
ar
b
.
u
n
it
s
LSDA
LSDA+U
with hole
exper.
0
5
10
A
D
F
LSDA
4f in core
exper.
–0.04
0
0.04
LSDA+U
with hole
exper.
395 400 405 410
Energy, eV
–0.04
0
0.04
a
b
c
B C
Fig. 5. (a) The experimental [35] (circles) isotropic absorption
spectrum of GdN at the N K edge in comparison with the cal-
culated ones [36] using the LSDA (full line) and LSDA + U ap-
proximations without (dashed line) and with (dotted line) tak-
ing into account the core hole-effect. Experimental spectra
were measured with external magnetic field (0.1 T) at 15 K.
Dashed-dotted line shows the theoretically calculated back-
ground spectrum. (b) Experimental [35] (circles) XMCD spec-
trum of GdN at the Gd K edge in comparison with theoreti-
cally calculated ones using the LSDA (full line) and putting
the 4 f states in core (dashed line) approximations; (c) experi-
mental (circles) XMCD Gd K spectrum in comparison with
theoretically calculated using the LSDA + U approximation
with (full line) and without (dashed line) taking into account
the core-hole effect.
the K XMCD spectrum at N site with turning the SOI off
separately on the N 2 p states and at the Gd site (at both the
4 f and 5d states), respectively. We found that the K
XMCD spectrum is slightly changed when the SOI on the
N site is turned off, while the spectrum almost disappears
(reduced its intensity almost two order of magnitude) when
the SOI on the Gd site is turned off. This indicates that the
SOI on Gd site is influencing the orbital mixture of N 2 p
states through the N (2 p)–Gd (d f, ) hybridization.
The LSDA +U approach (Fig. 5,c) improves the agree-
ment between theory and the experiment, especially in
describing the peak B. However, LSDA +U theory fails to
produce the peak A, besides the peaks B and D are shifted
toward lower energy at around 0.6 eV in comparison with
the experimental measurements. Also for the energies
higher than peak C theory gives some additional oscillat-
ing structures, while the experimental spectrum is a
smooth positive function of energy.
We investigate also the core-hole effect in the final
state using the supercell approximation. In our calcula-
tions we used a supercell containing eight conventional
GdN cells. At one of the eight N atoms we create a hole at
the 1s level for the self-consistent LSDA + U calculations
of the K spectrum. We found that the core-hole interactions
significantly improve the agreement between theoretically
calculated and experimentally measured N K XMCD spec-
tra (Fig. 5,c). The oscillation behavior of the high-energy
part of the theoretical spectrum above 405 eV could possi-
bly be damped by the quasiparticle life-time effect, which
is not taken into account in our calculations. The core-hole
effect improves also the agreement in the energy position
of the third high-energy peak in the XAS (Fig. 5,a).
However, all the calculations were not able to produce
the first maximum of the N K XAS above the edge at
around 400 eV. One of the possible reasons for such dis-
agreements might be the surface effect. The N K edge oc-
curs at a relatively small energy and one would expect
larger surface affects at the N K edge than, for example, at
the Gd M 4 5, or L2 3, edges. To model the surface effects
we carried out band structure calculations using a tetra-
gonal supercell containing 4 unit cells of GdN along the z
direction in which 3 GdN layers are replaced by 3 layers
of empty spheres. We calculated the XAS and XMCD
spectra at N K edge for such a 5 layer slab separated by
3 layers of empty spheres (5/3 multilayered structure
(MLS)) using the LSDA + U approximation. We also car-
ried out the band structure calculations for a 9 layer slab
separated by 3 layers of empty spheres (9/3 MLS). We
found that the K XMCD spectrum for N in the middle of
the 9/3 MLS (5th layer) is identical to the corresponding
bulk LSDA +U spectrum (not shown). The corresponding
spectrum for the middle layer in the 5/3 MLS (3th layer) is
still slightly different from the bulk spectrum, therefore
the convergence was achieved only in the 9/3 MLS. Fi-
gure 6 shows the N p empty partial DOS’s for the surface
layer in the 9/3 MLS and the bulk structure in comparison
with the experimental XA spectrum at the N K edge. It can
be seen that the partial DOS strongly increases at the first
maximum above the edge for the surface layer.
Actually the importance of the surface effect has some
experimental evidence. The authors of Ref. 35 mention
that the spectral feature at 400 eV was not contained in a
preliminary N K edge XA spectrum recorded on a Cr-co-
vered 30 � GdN layer using the total fluorescence yield
detection mode due to the larger probing depth of this
method compared to the measurements with the total elec-
tron yield (TEY) detection in Ref. 35. This indicates that
the first maximum above the edge in the XA spectrum at
400 eV is likely related to the GdN surface or interface
where the TEY detection is sensitive. The peak is a signa-
ture of the surface GdN XA behavior of the sample. This
also applies for the slowly rising part of the XMCD signal
below 400 eV. This result supports our conclusion that the
first maximum above the edge in the XA spectrum might
be related to the GdN surface or interface.
It is also important to note that the energy position of
the first XA maximum above the edge at around 400 eV
coincides with the position of the Gd 4 f DOS and any
kind of change in the N 2 p–Gd 4 f hybridization (which we
discussed in previous paragraph) might influence the inten-
sity of the XAS at that energy. The possible existence of in-
terstitial N atoms may also influence the low-energy part of
the spectrum via stronger direct Gd 4 f –N 2 p hybridization.
Due to the delocalized nature of the p states and wide
spread of p wave functions K XMCD spectra are very
sensitive to the surrounding neighborhood and, hence, the
K XMCD spectroscopy can be used as an effective probe
which can detect details of magnetic interatomic interac-
tions in rare-earth compounds.
114 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
N
p
-p
ar
ti
al
D
O
S
bulk
surf. layer
XAS exper.
0 5 10
Energy, eV
0
5
10
Fig. 6. N p empty partial DOS’s (in arbitrary units) for the sur-
face layer in the 9/3 MLS (full line) and the bulk structure
(dashed line) [36] in comparison with the experimental XA
spectrum at the N K edge [35] (circles).
2. Uranium compounds
Uranium compounds exhibit rich variety of properties
to large extent because of the complex behavior of 5 f
electrons which is intermediate between the itinerant be-
havior of 3d electrons in transition metals and the local-
ized one of 4 f electrons in rare-earth compounds. The
dual character of 5 f electrons alongside with the pres-
ence of strong spin-orbit coupling make the determina-
tion of the electronic structure of U compounds a chal-
lenging task because in many of them the width of 5 f
bands, their spin-orbit splitting, and the on-site Coulomb
repulsion in the partially filled 5 f shell are of the same or-
der of magnitude and should be taken into account on the
same footing. An interest to uranium compounds has re-
cently been renewed, especially after the discovery of
such unusual effects as heavy fermion superconductivity
and coexistence of superconductivity and magnetism.
Because of the great number of papers which have
been produced in recent years on actinide intermetallics,
and in particular on heavy-fermion compounds, these
would deserve one or more specialized review articles.
Various aspects of these systems and of general
heavy-fermion systems have already been reviewed in the
past [48–60]. For this reason we will merely outline some
general concepts relevant to uranium intermetallics,
rather than doing a systematic review of all their physical
properties.
For heavy-fermion compounds the attribute «heavy»
is connected to the observation of a characteristic energy
much smaller than in ordinary metals that reflects a ther-
mal effective mass m* of the conduction electrons orders
of magnitude larger than the bare electron mass. These
heavy masses manifest themselves, for example, by
a large electronic coefficient � of the specific heat C
( for )� �C/T T 0 , an enhanced Pauli susceptibility, a
huge T 2 term in the electrical resistivity, and highly tem-
perature-dependent de Haas–van Alphen oscillation
amplitudes at very low temperatures. The large m* value
is usually believed to derive from the strong correlation
electrons. While at high temperature the 5 f electrons and
conduction electrons interact weakly, at low temperature
these two subsets of electrons become strongly coupled,
resulting in the formation of a narrow resonance in the
density of states near the Fermi energy. Thus, at a suffi-
ciently low temperature, the heavy-fermion compounds
behave like a system of heavy itinerant electrons, the
properties of which can be described in the framework of
a Landau Fermi-liquid formalism.
Among uranium heavy-fermion compounds supercon-
ductivity is observed in UBe13, UPt3, URu2Si2, U2PtC2,
UPd2Al3, and UNi2Al3. Superconductivity usually in
these compounds coexists with AF order and this has led
to the suggestion that the effective attractive interaction
between the superconducting electrons may be mediated
by spin fluctuations, rather than by the electron-phonon
interaction. This is supported by the fact that the observed
superconducting states are highly anisotropic, with nodes
in the gap function not explainable by a s-wave theory.
A fascinating aspect of this class of compounds is the
observation that, within the heavy-fermion 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
heavy-fermion properties at low temperature is still lack-
ing. The heavy-fermion ground-state properties are
highly sensitive to impurities, chemical composition, and
slight changes of external parameters. This sensitivity in-
dicates that a subtle interplay between different interac-
tions produces a richness of experimental phenomena. It
is widely believed that the competition between the
Kondo effect (reflecting the interaction between the lo-
calized 5 f moments and the conduction electrons) and
the magnetic correlations between the periodically ar-
ranged 5 f moments constitutes the key factor for as far as
the magnetic properties of heavy-fermion compounds are
concerned [48].
The x-ray magnetic circular dichroism technique de-
veloped in recent years has evolved into a powerful mag-
netometry tool to separate orbital and spin contributions
to element specific magnetic moments. Study of the 5 f
electron shell in uranium compounds is usually per-
formed by tuning the energy of the x-ray close to the M 4 5,
edges of uranium (located at 3552 and 3728 eV, respec-
tively) where electronic transitions between 3d 3 2 5 2/ , / and
5 f 5 2 7 2/ , / states occur. Recently XMCD measurements have
been successfully performed for uranium compounds such
as US [61,62], USb0.5Te0.5 [63], U xLa1�xS [64], UBe13 and
UPt3 [65], UFe2 [66,67], UNi2Al3 [68], UPd2Al3 and
URu2Si2 [69], URhAl [70], UCoAl and UPtAl [71].
There are some features in common for all the uranium
compounds investigated up to now. First, the dichroism at
the M 4 edge is much larger, sometimes of one order of
magnitude, than at the M 5 one. Second, the dichroism at
the M 4 edge has a single negative lobe that has no distinct
structure, on the other hand, two lobes, a positive and a
negative one, are observed at the M 5 edge. Concerning
the line shape of the XMCD signal, the investigated me-
tallic uranium compounds fall into two types according to
a relative intensity of the positive and negative lobes ob-
served at the M 5 edge. The two lobes have almost equal
intensity for UP3, UPd2Al3, UPtAl, and UBe13. On the
other hand, the positive lobe is smaller in comparison with
the negative one for US, USb0.5Te0.5, UFe2, URu2Si2,
UCoAl, and URhAl.
With the aim of undertaking a systematic investigation
of the trends in uranium compounds we present the theo-
retically calculated electronic structure and XMCD spec-
tra at M 4 5, edges for the following uranium compounds:
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 115
UPt3, URu2Si2, UPd2Al3, UNi2Al3, UBe13, UFe2, UPd3,
UXAl (X = Co, Rh, and Pt), and UX (X = S, Se, and Te).
The first five compounds belong to heavy-fermion super-
conductors, UFe2 is widely believed to be an example of
compound with completely itinerant 5 f electrons, while
UPd 3 is the only known compound with completely local-
ized 5 f electrons. The electronic structure and XMCD
spectra of UGe2 which possesses simultaneously ferro-
magnetism and superconductivity also presented.
2.1. Intermetallic compounds
2.1.1. UFe 2
Figure 7 shows the calculated fully relativistic spin-po-
larized partial 5 f density of states of ferromagnetic UFe2
[72]. Because of the strong spin-orbit interaction of 5 f
electrons, j / 5 2 and j / 7 2 states are well separated in
energy and the occupied states are composed mostly of
5 5 2f / states whereas 5 f 7 2/ states are almost empty. One
can note, however, that an indirect hybridization between
j / 5 2 and j / 7 2 states via Fe 3d states is rather strong.
In order to compare relative amplitudes of M 4 and M 5
XMCD spectra we first normalize the corresponding iso-
tropic x-ray absorption spectra (XAS) to the experimental
ones taking into account the background scattering inten-
sity. Figure 8 shows the calculated isotropic x-ray absorp-
tion and XMCD spectra in the LSDA and LSDA + U (OP)
approximations together with the experimental data [66].
The contribution from the background scattering is
shown by dashed lines in the upper panel of Fig. 8.
The experimentally measured dichroic M 4 line con-
sists of a simple nearly symmetric negative peak that has
no distinct structure. Such a peak is characteristic of the
M 4 edge of all uranium systems. The dichroic line at the
M 5 edge has an asymmetric s shape with two peaks — a
stronger negative peak and a weaker positive peak. The
dichroism at the M 4 edge is more than two times larger
than at the M 5 one.
Because of the electric dipole selection rules (�l
1;
�j
0 1, ) the major contribution to the absorption at the
M 4 edge stems from the transitions 3 53 2 5 2d f/ /�
and that at the M 5 edge originates primarily from
3 55 2 7 2d f/ /� transitions, with a weaker contribution
from 3 55 2 5 2d f/ /� transitions. For the later case the cor-
responding 3 55 2 7 2d f/ /� radial matrix elements are only
slightly smaller than for the 3 55 2 7 2d f/ /� transitions.
The angular matrix elements, however, strongly suppress
the 3 55 2 5 2d f/ /� contribution. Therefore the contribu-
tion to XMCD spectrum at M 5 edge from the transitions
with �j 0 is about 15 times smaller than the transitions
with �j 1.
The selection rules for the magnetic quantum number
m j (m j is restricted to � �j j, ... ) are �m j � 1 for � = +1
and �m j �1 for � �1. Table 2 presents the dipole al-
lowed transitions for x-ray absorption spectra at M 5 and
M 4 edges for left (� � 1) and right (� �1) polarized
x-rays.
To go further, we needs to discuss the characteristic of
the 5 f empty DOS. Since l and s prefer to couple antiparallel
for less than half-filled shells, the j l s / � 5 2 has a lower
energy than the j l s / � 7 2 level. Due to the intra-ato-
mic exchange interaction the lowest sublevel of the
j / 5 2 will be m /5 2 5 2/ � , however, for the j / 7 2 the
lowest sublevel will be m /7 2 7 2/ � . This reversal in the
116 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
5f
5/2
5f7/2
–3 –2 –1 0 1 2 3 4
Energy, eV
0
2
4
6
8
Fig. 7. The LSDA partial 5 f 5 2/ and 5 f 7 2/ density of states in
UFe2 [72].
A
b
so
rp
ti
o
n
,
ar
b
.
u
n
it
s
X
M
C
D
,
ar
b
.
u
n
it
s
UFe2M5
M4
0
50
100
LSDA
LSDA+U(OP)
exper.
–20 0 20 40 60 80 100 120
Energy, eV
–3
–2
–1
0
1
Fig. 8. Isotropic absorption and XMCD spectra of UFe2 at the
uranium M4 5, edges calculated in the LSDA (solid lines) and
LSDA + U (OP) (dashed lines) approximations [72]. Experi-
mental spectra [66] (circles) were measured at 20 K and at
magnetic field 2 T (the U M 4 spectrum is shifted by �95 eV to
include it in the figure). Upper panel also shows the back-
ground spectra (dashed line) due to the transitions from inner
3 3 2 5 2d / , / levels to the continuum of unoccupied levels.
energy sequence arises from the gain in energy due to
alignment of the spin with the exchange field [65].
Table 2. The dipole allowed transitions from core 3 3 2 5 2d / , / levels
to the unoccupied 5 f 5 2 7 2/ , / valence states for left (� �1) and
right (� �1) polarized x-rays
Edge � = +1 � = –1
M5
–5/2 � –3/2 –5/2 � –7/2
–3/2 � –1/2 –3/2 � –5/2
–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
M4
–3/2 � –1/2 –3/2 � –5/2
–1/2 � +1/2 –1/2 � –3/2
+1/2 � +3/2 +1/2 � –1/2
+3/2 � +5/2 +3/2 � +1/2
The 5 f 7 2/ states are almost completely empty in all the
uranium compounds. Therefore all the transitions listed
in Table 2 are active in the M 5 absorption spectrum. The
contribution from the first four transitions for � � 1can-
cels to a large extent with the contribution of the opposite
sign from the last four transitions for � �1 having the
same final states. Thus the XMCD spectrum of U at the
M 5 edge (I ��� � � ) can be roughly approximated by
the following sum of m j -projected partial densities of
states [72]: (N �7 2
7 2
/
/ + N N� �5 2
7 2
7 2
7 2
/
/
/
/) ( + N 5 2
7 2
/
/ ). Here we use
the notation N
m
j
j
for the density of states with the total
momentum j and its projection m j . As a result, the shape
of the M 5 XMCD spectrum contains of two peaks of op-
posite signs — a negative peak at lower energy and a posi-
tive 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 m mj j
| |
increases with the increase of the magnetization at the U
site, the amplitude of the M 5 spectrum should be propor-
tional to the U magnetic moment.
A rather different situation occurs in the case of the
M 4 x-ray absorption spectrum. Usually in uranium com-
pounds the U atom is in 5 f 3 (U 3� ) or 5 f 2 (U 4� ) confi-
gurations and has partly occupied 5 f 5 2/ states. In the first
case the 5 f 5 2/ states with m /j �5 2, �3 2/ , and �1 2/ are
usually occupied. The dipole allowed transitions for � � 1
are �1 2/ � � 1 2/ , � 1 2/ � � 3 2/ and � 3 2/ � � 5 2/ and
those for � �1 are � � �3 2 1 2/ / . The transitions with
the same final states m j = +1/2 mostly cancel each other
and the XMCD spectrum of U at the M 4 edge can be
roughly represented by the sum [72] � �( )/
/
/
/N N3 2
5 2
5 2
5 2 . The
corresponding analysis for the 5 f 2 (U 4� ) configuration
with occupied f 5 2 5 2/ , /� and f 5 2 3 2/ , /� states shows that the
dipole allowed transitions for � � 1 are � � �3 2 1 2/ / ,
� � �1 2 1 2/ / , � � �1 2 3 2/ / , and � � �3 2 5 2/ / and for
� �1: � � �1 2 1 2/ / and � � �3 2 1 2/ / . Again , the
XMCD spectrum of U at the M 4 edge can be approxi-
mated by � �( )/
/
/
/N N3 2
5 2
5 2
5 2 [72]. This explains why the
dichroic M 4 line in uranium compounds consists of a sin-
gle nearly symmetric negative peak.
We should note, however, that the explanation of the
XMCD line shape in terms of m j -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 ma-
trix elements; second, in our consideration we neglect
cross terms in the transition matrix elements; third, there
is no pure 5 f 3 or 5 f 2 configurations in uranium com-
pounds. It is always difficult to estimate an appropriate
atomic 5 f occupation number in band structure calcula-
tions. Such a determination is usually obtained by the in-
tegration of the 5 f electron charge density inside of the
corresponding atomic sphere. In the particular UFe2 case,
the occupation number of U 5 f states is around 2.9 in the
LSDA calculations. We, however, should keep in mind
that some amount of the 5 f states are derived from the
so-called «tails» of Fe 3d states arising as a result of the
decomposition of the wave function centered at Fe atoms.
The careful analysis in the case of UPd3 presented in
Ref. 73 shows that the occupation number of the «tails» of
Pd 4d states sum up to give the 5 f occupation of 0.9 elec-
trons in the U atomic sphere. We should also note that due
to the strong hybridization between U 5 f and Fe 3d
states, the U 5 f 7 2/ states in UFe2 are not completely
empty, some of them are occupied, also some amount of U
5 f 5 2/ states, which we have been considering as fully oc-
cupied, are partially empty.
The overall shapes of the calculated and experimental
uranium M 4 5, XMCD spectra correspond well to each
other (Fig. 8). The major discrepancy between the calcu-
lated and experimental XMCD spectra is the size of the
M 4 XMCD peak. The LSDA underestimates the integral
intensity of the XMCD at the M 4 edge. As the integrated
XMCD signal is proportional to the orbital moment [74]
this discrepancy may be related to an underestimation of
the orbital moment by LSDA-based computational me-
thods. On the other hand, the LSDA + U (OP) approxima-
tion gives larger intensity for the M 4 XMCD spectrum in
comparison with the experimentally measured one. It re-
flects the overestimation of the orbital moment at U site in
the LSDA + U (OP) calculations. In the case of the M 5
XMCD spectrum, the LSDA reproduces the amplitude of
the positive peak and overestimates the amplitude of the
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 117
negative peak. The LSDA + U (OP) approximation, in
contrast, gives good agreement in the amplitude of the
negative peak but overestimates that of the positive peak.
To investigate the influence of the initial state on the
resulting U XMCD spectra we calculated also the XAS
and XMCD spectra of UFe2 compound at the N 4 5, and
O4 5, edges (not shown). We found a substantial decrease
of the XMCD in terms of R / �� �( )2 0 at N 4 5, edges in
comparison with the M 4 5, ones. The theoretically calcu-
lated dichroic N 4 line consists of a simple nearly symmet-
ric negative peak that has no distinct structure as was
observed at the M 4 XMCD spectrum. The LSDA calcula-
tions give much smaller dichroic signal at the N 4 edge in
comparison with the LSDA + U ( )OP calculations. The
dichroic line at the N 5 edge is reminiscent of the corre-
sponding M 5 spectrum and has an asymmetric s shape
with two peaks — a stronger negative peak and much
weaker positive peak. In contrast to the dichroism at the
M 4 5, edges where XMCD at M 4 edge is more than two
times larger than at the M 5 one, the dichroism at the N 4
edge has almost the same intensity as at the N 5 edge.
Due to MO selection rules the O4 XMCD spectrum re-
sembles the M 4 spectrum, whereas the O5 spectrum is
similar to the M 5 one. Because of the relatively small
spin-orbit splitting of the 5d states of U ( � 11 eV), the O4
and O5 spectra almost overlap each other. The magnetic
dichroism at quasi-core O4 5, edges is of one order of mag-
nitude larger than the dichroism at the N 4 5, edges and be-
come almost as large as that at the M 4 5, edge. Besides, the
lifetime broadening of the core O4 5, levels is much
smaller than the broadening of the M 4 5, ones [75]. There-
fore the spectroscopy of U atoms in the ultra-soft x-ray
energy range at the O4 5, edges may be a very useful tool
for investigation of the 5 f electronic states in magnetic U
materials.
The XAS at the M 4 5, , N 4 5, , and O4 5, edges involve
electronic transitions between nd 3 2 5 2/ , / (n = 3, 4, and 5)
and 5 f 5 2 7 2/ , / states and therefore are used to study of the
5 f empty electronic states in uranium compounds. To
investigate the 6d states of U one should tune the energy of
the x-ray close to the M 2 3, , N 2 3, , O2 3, , or N 6 7, edges
of uranium. The first three doublets are due to the
np d1 2 3 2 3 2 5 26/ , / / , /� (n = 3, 4, and 5) interband transitions.
Figure 9 presents the theoretically calculated XMCD
spectra of U M 2 3, , N 2 3, , and O2 3, edges. The XMCD sig-
nals at these edges are two order of magnitude less than
the corresponding signals at the M 4 5, edges.
Because of the dipole selection rules, apart from the
ns1 2/ states (which have a small contribution to the XAS’s
due to relatively small np s� 7 matrix elements only
6 3 2d / states occur as final states for the M 2, N 2, and O2
XAS’s for unpolarized radiation, whereas for the M 3, N 3,
and O3 XAS’s the 6d 5 2/ states also contribute. Although
the np d3 2 3 26/ /� radial matrix elements are only slightly
smaller than for the np d3 2 5 26/ /� transitions the angular
matrix elements strongly suppress the np d3 2 3 26/ /� con-
tribution. Therefore, neglecting the energy dependence of
the radial matrix elements, the M 2, N 2, and O2 absorp-
tion spectra can be viewed as a direct mapping of the DOS
curve for 6d 3 2/ , and the M 3, N 3, and O3 XAS’s reflect the
DOS curve for 6d 5 2/ states. The shape of X 3 (X M N , ,
or O) XMCD spectra consists of two peaks of opposite
sign — a negative peak at lower energy and a positive
peak at higher energy. The shape of X 2 (X M N , , or O)
XMCD spectra also have two peaks of an opposite sign,
118 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
M3 M2
–0.2
–0.1
0
0.1
N3
N2
–0.1
0
O3
O2
–0.1
0
N7
N6
0 20 40 60 80 100
Energy, eV
–0.05
0
0.05
U
ra
n
iu
m
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. 9. XMCD spectra of UFe2 at the uranium M2 3, , N2 3, , O2 3,
and N6 7, edges calculated in the LSDA approximation [72]. All
the XMCD spectra are multiplied by a factor 102 (the M 2 and
N2 spectra are shifted by –800 and –150 eV, respectively, to
include them in the figure).
but the negative peaks situated at higher energy and the
positive peak at lower energy (Fig. 9).
Figure 9 also presents the theoretically calculated
XMCD spectra at U N 6 7, edges. Because of the electric
dipole selection rules the major contribution to the ab-
sorption at the N 7 edge stems from the transitions
4 67 2 5 2f d/ /� and that at the N 6 edge originates primar-
ily from 4 65 2 3 2f d/ /� transitions (the contribution from
4 65 2 5 2f d/ /� transitions are strongly suppressed by the
angular matrix elements). The XMCD signals at these
edges are even smaller than the corresponding signals at
the X 2 3, (X M N , , or O) edges. Because of the relatively
small spin-orbit splitting of the 4 f states of U, the N 6 and
N 7 spectra have an appreciable overlap. Besides, in the
case of N 6 7, XAS one would expect a strong electrostatic
interaction between the created 4 f core hole and the va-
lence states. It can lead to an additional multiplet struc-
ture in the XAS and XMCD spectra at the N 6 7, edges. We
have not considered multiplet structure in our XMCD cal-
culations. This structure can be captured using full atomic
multiplet structure calculations.
We also calculated the x-ray magnetic circular
dichroism at the Fe K , L2 3, , and M 2 3, edges, with the re-
sults being presented in Fig. 10. For comparison we also
show the XMCD spectra in bcc Fe. Although the XMCD
signal at the Fe K edge has almost the same amplitude
both in bcc Fe and UFe 2, their shapes are quite different
(Fig. 10).
The dichroism at Fe L2 and L3 edges is influenced by
the spin-orbit coupling of the initial 2 p core states. This
gives rise to a very pronounced dichroism in comparison
with the dichroism at the K edge. Figure 10 shows the the-
oretically calculated Fe L2 3, XMCD spectra in UFe 2 and
bcc Fe. The dichroism at the L3 edge has a negative sign
and at the L2 edge a positive one. The XMCD dichroic
signals at the Fe L2 3, and M 2 3, edges are three times
smaller in UFe 2 than the corresponding XMCD in bcc Fe
due to strongly reduced magnetic moment at the Fe site in
UFe 2 in comparison with pure Fe. Besides, the shape of
the spectra is more asymmetrical in UFe 2.
The magnetic dichroism at the Fe M 2 3, edges is much
smaller than at the L2 3, edges (Fig. 10). Besides the M 2
and the M 3 spectra are strongly overlapped and the M 3
spectrum contributes to some extent to the structure of the
total M 2 3, spectrum in the region of the M 2 edge. To de-
compose a corresponding experimental M 2 3, spectrum
into its M 2 and M 3 parts will therefore be quite difficult
in general. It worth mentioning that the shape of Fe L3 and
M 3 XMCD spectra are very similar.
2.1.2. UXAl (X = Co, Rh, and Pt)
The group of ternary uranium compounds with compo-
sition UTX, where T is a transition metal (Fe–Ni and 4d,
5d equivalents) and X a p element (Al, Ga, Ge, Sn), has
recently attracted attention [76]. These compounds pro-
vide wide possibilities for study via the variation of atom
types. The compounds forming with atoms to the left of
the transition metal series (Fe, Co, and Ru) are paramag-
netic — although UCoAl is metamagnetic — while
URhAl, UIrAl and UPtAl are ferromagnetic and UNiAl is
antiferromagnetic.
One of the key questions to be addressed when dis-
cussing actinide compounds is the degree of localization
of the 5 f electrons, which may range from nearly local-
ized to practically itinerant, depending on the specific
compound. Since the 5 f electrons are simultaneously in-
volved in the chemical bonding and magnetism, a broad
variety of physical properties may emerge from the degree
of 5 f localization. UTAl (T = Co, Rh, and Pt) compounds
have been also considered in this respect [71,77–80].
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 119
F
e
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
K
–0.1
0
0.1
0.2
L
3
L
2
Fe
Fe in UFe2
–2
–1
0
1
M
3
M
2
0 5 10 15 20
Energy, eV
–0.4
0
0.4
Fig. 10. XMCD spectra of UFe2 at the Fe K, L2 3, , and M2 3,
edges in bcc Fe and Fe in UFe2 calculated in the LSDA ap-
proximation [72]. The XMCD spectrum at the K edge has been
multiplied by a factor 102.
UCoAl shows no magnetic ordering down to the low-
est temperatures, but in a relatively low magnetic field, of
about 0.7 T, applied along the c axis a metamagnetic tran-
sition to a ferromagnetic state is observed at low tempera-
tures. The metamagnetic transition in UCoAl is attributed
to band metamagnetism [71]. The metamagnetism is in-
duced only when the magnetic field is applied along the c
axis, whereas in fields in a perpendicular direction UCoAl
behaves like a Pauli paramagnet and no metamagnetic
transition is observed in magnetic fields up to 42 T [77].
The strong uniaxial magnetic anisotropy is preserved in
UCoAl, at least up to room temperature. It is of interest to
note a rather low ordered magnetic moment of UCoAl
which amounts to 0.30 � B /f.u. at 4.2 K, above the meta-
magnetic transition. The moment steadily increases with
magnetic field, showing no saturation tendency up to 35 T
where it reaches the value of 0.6 � B /f.u. [77,78].
The UPtAl compound is an appropriate reference system
for the same structure and with composition and bonding
similar to that of UCoAl. It orders ferromagnetically below a
TC of 43 K with a saturated magnetization of 1.38 � B /f.u.
at 2 K in fields applied along the c axis [81]. The strong
uniaxial anisotropy is manifested by the fact that the mag-
netization measured along the a axis is much smaller and
has no spontaneous component. In fact, it resembles the
magnetic response of a paramagnet exhibiting 0.28
� B /f.u. at 40 T.
As for the URhAl compound, a sizable induced mo-
ment of 0.28 � B on the Rh atom within the basal uranium
plane was detected in a polarized neutron study, whereas,
interestingly, only a very small induced moment of 0 03. � B
was detected on the equally close Rh site out of the plane
[79]. The large anisotropy in the induced Rh moments
clearly reflects the anisotropy of the U(5 f )–Rh(4d) hy-
bridization: a strong hybridization occurs between the va-
lence orbitals of the U and Rh atoms within the basal
plane, but the hybridization between the valence orbitals
of the U atom and those of the equally close Rh atom in
the adjacent plane is much smaller.
Later, inelastic neutron-scattering experiments found
a peak at 380 meV, which was interpreted as the signature
of an intermultiplet transition [80], thus promoting the lo-
calized picture. The 380 meV peak occurred at the same
energy where a uranium intermultiplet transition was ob-
served [82] in UPd 3, which is one of the uranium com-
pounds where the 5 f electrons are undoubtedly localized.
Five electronic band structure calculations for URhAl
were carried out recently [71,83–86]. These indicated,
first, that the bonding and magnetism are governed by the
U (5 f )–Rh (4d) hybridization [84] and, second, that the
calculated magneto-optical Kerr spectrum [83] — based
on the assumption of delocalized 5 f ’s — compares rea-
sonably well to the experimental Kerr spectrum. Besides,
the authors of Ref. 85 were able to describe satisfactory
the equilibrium volume, bulk modulus, and magneto-
crystalline anisotropy in URhAl using the LSDA-based
full potential relativistic LAPW method. Somewhat less
well explained were the uranium orbital moment and the
XMCD spectra.
Experimental and theoretical x-ray magnetic circular
dichroism studies of the intermetallic compounds UCoAl
and UPtAl at the uranium M 4 and M 5 edges are reported
in Ref. 71. The results show that the orbital-to-spin mo-
ment ratio is of comparable value, M /Ml s � �2, for both
compounds. The reduction of the M /Ml s ratio compared
to the U 3 � (5 f 3) free ion value of �2 57. , and the sizable
decrease of orbital and spin moments, especially for
UCoAl, indicate a significant delocalization of the 5 f elec-
tron states in these compounds.
1. Band structure. UTAl (T = Co, Rh, or Pt) crystallize
in the hexagonal ZrNiAl structure (Fe2P type), which
contains three formula units per unit cell. The ZrNiAl
structure has a layered structure, consisting of planes of
uranium atoms admixted with one-third of the T atoms,
that are stacked along the c axis, while two adjacent ura-
nium planes are separated from one another by a layer
consisting of the remaining T atoms and the Al atoms.
The uranium atoms have transition metal nearest neigh-
bors and vice versa, so both uranium and T atoms are well
separated from atoms of the same type. The uranium
interlayer exchange coupling is relatively weak and de-
pends sensitively on the specific T elements, which gives
rise to a variety of magnetic behaviors observed in the
UTX compounds [76].
The fully relativistic spin-polarized LSDA energy
band structure and total density of states of the ferromag-
netic UTAl (T = Co, Rh, and Pt) compounds are shown in
Fig. 11 [86]. The bands in the lowest region of UPtAl, be-
tween �9.2 and �6.0 eV, have mostly Al s character with a
small amount of U spd and Al p character mixed in. The
energy bands between �6.0 and �3.0 eV are predomi-
nantly Pt 5d states. Due to increasing of the spatial expan-
sion of valence transition metal d states in going from Co
to Pt the corresponding d energy widths are increased and
shifted downwards. Co 3d energy bands are occupied in
the �1.2 to �2.8 eV energy interval in UCoAl, the 4d
bands of Rh in URhAl are situated in the �2.0 to �4.5 eV
energy range, and Pt 5d bands are in the �3.0 to �6.0 eV
interval. Therefore the valence d energy band widths are
equal to 1.6, 2.5, and 3.0 eV in UCoAl, URhAl, and
UPtAl, respectively (Fig. 11). The U 5 f energy bands oc-
cupy the same energy interval above and below E F in all
the compounds under consideration, namely, about �1.0
to 2.0 eV. There is a strong hybridization between the
U 6d, transition metal d, and Al p states.
The itinerant character of electron states usually im-
plies a strong reduction of the orbital magnetic moment
with respect to the free-atom expectation value. Never-
120 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
theless, in contrast to 3d electrons in transition metals,
sizable orbital magnetic moments are observed in U
intermetallic compounds with apparently strongly
delocalized 5 f electrons. It is the very strong spin-orbit
coupling present in actinides that enhances an orbital mo-
ment in the case of itinerant 5 f electron states. Analyzing
spin and orbital magnetic moments in various actinide
compounds, Lander et al. suggested that the ratio of the
orbital to the spin moments provides information on the
strength of 5 f ligand hybridization, and consequently the
delocalization of the 5 f electrons [87]. The individual
values of orbital and spin components, however, contain
essential information, and therefore relevant experiments
and first-principles electronic structure calculations
which independently evaluate orbital and spin moments
become an important issue for 5 f electron compounds.
The recently developed x-ray magnetic circular dich-
roism experimental method combined with several sum
rules [74,88] has attracted much attention as a site- and
symmetry-selective way to determine M s and M l . It
should be mention, however, that the reported quantita-
tive results inferred from the XMCD spectra are based on
a sum rule analysis of the spin-orbit split spectra of the
core levels of uranium. The sum rules enable one to esti-
mate the spin and orbital components of the uranium ions,
however, the values of magnetic moments rely on theoret-
ical inputs such as the number of holes in the 5 f subshell
and a value of the dipolar term. In particular, the spin mo-
ment is retrieved with a higher relative error. Comparing
the XMCD-derived moments with the results of polarized
neutron diffraction and first-principles calculations, one
usually obtains smaller moments from the XMCD sum
rules for uranium compounds [71,85]. A more reliable
quantity that can be extracted from the sum rule analysis
is the ratio between orbital and spin moments and their
relative orientation.
Table 3 lists the calculated spin M s , orbital M l , and to-
tal M t magnetic moments (in � B ) of UTAl (T = Co, Rh,
and Pt) as well as the ratio M l /M s [86]. Our LSDA results
are in good agreement with previous LSDA-based calcu-
lations [71,85]. All the LSDA calculations strongly un-
derestimate the orbital moment in the compounds. The
inclusion of the orbital polarization (OP) correction in
Ref. 84 brings the calculated total U moment in URhAl to
0.60 � B , in better agreement with experiment (0.94 � B
according to Ref. 79) in comparison with the LSDA cal-
culations (Table 3).
Table 3. The experimental and calculated spin Ms, orbital Ml, and
total Mt magnetic moments at the uranium site (in �B) of UCoAl,
URhAl, and UPtAl
Compound Method Ms Ml Mt –Ml/Ms
LSDA –0.92 1.09 0.17 1.18
LSDA +U(OP) –1.14 2.29 1.15 2.01
UCoAl LSDA +U –1.50 3.47 1.97 2.31
LSDA [71] –1.01 1.19 1.18 1.18
exper. [71] — — — 1.95
LSDA –1.23 1.72 0.49 1.40
LSDA +U(OP) –1.40 2.94 1.54 2.10
LSDA +U –1.66 3.83 2.17 2.31
URhAl LSDA [71] –1.22 1.59 0.37 1.30
LSDA [85] –1.24 1.63 0.39 1.31
LSDA + OP [84] –1.01 1.61 0.60 1.59
exper. [79] –1.16 2.10 0.94 1.81
LSDA –1.63 2.08 0.45 1.28
LSDA +U(OP) –1.60 3.32 1.72 2.08
UPtAl LSDA +U –1.85 4.26 2.41 2.30
LSDA [71] –1.63 2.06 0.43 1.26
exper. [71] — — — 2.10
As mentioned, we also carried out energy band struc-
ture calculations for the UTAl compounds using a gene-
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 121
UCoAl LSDA
–5
0
5
E
n
er
g
y,
eV
M K A L H A 0 10 20 30
URhAl
–5
0
5
E
n
er
g
y,
eV
M K A L H A 0 10 20 30
UPtAl
–5
0
5
E
n
er
g
y,
eV
M K A L H A 0 10 20 30
DOS
Fig. 11. The LSDA self-consistent fully relativistic, spin-polar-
ized energy band structure and total DOS (in states/(unit cell�eV))
of UCoAl, URhAl, and UPtAl [86].
ralization of the LSDA + U method [73]. In these calcula-
tions we used U J 0 5. eV, which gives U eff = 0 (the
LSDA + U (OP) approximation) as well as U 2 0. eV and
J 0 5. eV. Figure 12 shows the 5 f 5 2/ partial density of
states in UPtAl calculated in the LSDA, LSDA + U (OP),
and LSDA + U approximations. As can be seen from
Fig. 12 the LSDA + U (OP) approximation, which takes
into account the correlations between spin and orbital
magnetic moment directions, strongly affects the relative
energy positions of m j projected 5 f density of states and
substantially improves their orbital magnetic moments
(Table 3). The ratio M l /M s in the LSDA +U (OP) calcula-
tions is equal to 2.01, 2.10, and 2.08 for UCoAl, URhAl,
and UPtAl, respectively. The correspondent experimental
data are 1.95, 1.81, and 2.10 estimated from the XMCD
measurements [71].
The orbital magnetic moments calculated in the
LSDA + U approximation with U 2 0. eV and J 0 5. eV
are larger than those calculated using U J 0 5. eV,
which leads to slightly overestimated ratio M l /M s in
comparison with the experimental data (Table 3).
2. XMCD spectra. Figure 13 shows the calculated
x-ray isotropic absorption and XMCD spectra in the
LSDA, LSDA + U (OP), and LSDA + U approximations
for UPtAl [86] together with the experimental data [71].
To calculate the x-ray isotropic absorption M 4 5, spectra
we take into account the background intensity which ap- pears due to the transitions from inner levels to the
continuum of unoccupied levels [89].
Due to underestimation of the orbital magnetic moment
the theory produces much smaller intensity of the XMCD
spectrum at the M 4 edge in comparison with the experi-
ment in the LSDA calculations and simultaneously gives a
larger dichroic signal at the M 5 edge of UPtAl (Fig. 13).
On the other hand, the LSDA + U (OP) approximation pro-
duces an excellent agreement not only for the value of the
magnetic moments but also in the shape and intensity of
XMCD spectra both at the M 4 and M 5 edges. The LSDA +
U approximation with U 2 0. eV and J 0 5. eV overesti-
mates the negative signal at the M 4 edge due to the overes-
timation of the U orbital magnetic moment. This approxi-
mation also underestimates the positive peak and strongly
overestimates the negative one at the M 5 edge (Fig. 13).
In the case of URhAl the LSDA + U (OP) approxima-
tion also produces an XMCD spectrum at the M 4 edge in
excellent agreement with experiment, but slightly overes-
timates the value of the positive shoulder at the M 5 edge
(Fig. 14). The LSDA + U approximation with U 2 0. eV
and J 0 5. eV overestimates the negative signal at the M 4
edge, although, slightly improves the agreement with the
experimental spectrum at U M 5 edge (Fig. 14).
The LSDA + U (OP) approximation overestimates and
the LSDA + U one strongly overestimates the intensity of
XMCD signal at the M 4 edge in UCoAl, probably due to
the fact that the measured spontaneous magnetic moment
122 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
UPtAl
LSDA 5f
5/2
mj = –5/2
mj = –3/2
mj = –1/2
other
0
1
2
3
LSDA+U(OP)
0
1
2
3
LSDA+U
–2 –1 0 1 2 3
Energy, eV
0
1
2
Fig. 12. The partial 5 f 5 2/ density of states in UPtAl calculated
in the LSDA and LSDA + U (OP) approximations [86].
A
b
so
rp
ti
o
n
,
ar
b
.
u
n
it
s
X
M
C
D
,
ar
b
.
u
n
it
s
UPtAlM5
M
4
0
2
4
6
8
10
LSDA
LSDA+U(OP)
LSDA+U
exper.
–20 0 20 40 60 80 100 120
Energy, eV
–3
–2
–1
0
Fig. 13. Isotropic absorption and XMCD spectra of UPtAl at
the uranium M4 5, edges calculated in the LSDA (dotted lines)
and LSDA + U (OP) (solid lines) approximations [86]. Experi-
mental spectra [71] (circles) were measured at 10 K and at
magnetic field 2 T (the U M 4 spectrum is shifted by �95 eV to
include it in the figure).
of UCoAl is far from the saturation in the experimentally
applied external magnetic field of 7 T [71]. One would ex-
pect, therefore, that in a higher magnetic field UCoAl will
have larger orbital magnetic moment and, hence, larger
dichroism at the M 4 edge. As was the case for URhAl, the
LSDA + U calculations with U 2 0. eV and J 0 5. eV
give a better description of the positive peak at the M 5
edge in UCoAl (Fig. 14).
The 5 f 7 2/ states are almost completely empty in all the
uranium compounds, therefore the XMCD spectrum of U
at the M 5 edge can be roughly represented by the follow-
ing m j projected partial density of states [72]: [N �7 2
7 2
/
/ +
� ��N N5 2
7 2
7 2
7 2
/
/
/
/] [ + N 5 2
7 2
/
/ ]. Thus the shape of M 5 XMCD
spectrum consists of two peaks of opposite sign: a nega-
tive peak at lower energy and a positive peak at higher en-
ergy. The XMCD spectrum of U at the M 4 edge can be
represented by the �[ /
/N 3 2
5 2 + N 5 2
5 2
/
/ ] DOS’s, [72] thus it
consists of a single negative peak.
In UCoAl (above the metamagnetic transition) the
dichroic line at the M 5 edge has an asymmetric s shape
with two peaks: a stronger negative peak and a weaker
positive peak. The shape of the M 5 XMCD spectrum
strongly depends on the value of the external magnetic
field, the positive peak is increased relative the negative
one upon increasing the external magnetic field from
0.9 to 7 T (see Fig. 2 in Ref. 71). From the qualitative
description of the M 5 XMCD spectra in terms of partial
density of states we can conclude that the shape of the M 5
XMCD spectrum depends on the relative energy positions
of the [N 7 2
7 2
/
/ + N 5 2
7 2
/
/ ] and [N �7 2
7 2
/
/ + N �5 2
7 2
/
/ ] partial DOS’s
which depend on the value of crystal field and Zeeman
splittings of the 5 f 7 2/ electronic states [72]. Upon increas-
ing of the external magnetic field the Zeeman splitting is in-
creased, leading to larger separations between the m j pro-
jected partial DOS’s. Figure 15 shows uranium M 5 XMCD
spectrum of UCoAl calculated in the LSDA + U (OP) ap-
proximation and the spectra calculated with [N 7 2
7 2
/
/ +
� N 5 2
7 2
/
/ ] and [N �7 2
7 2
/
/ + N �5 2
7 2
/
/ ] DOS’s artificially shifted by
10 and 20 meV. It is clearly seen that model calculations
correctly reproduce the experimental tendency in the
shape of UCoAl M 5 XMCD spectrum in the external
magnetic field.
In conclusion, the LSDA + U approximation with
U 2 0. eV and J 0 5. eV overestimates the negative sig-
nal at the M 4 edge for all the compounds under the
consideration due to the overestimation of the U orbital
magnetic moment. This approximation provides poor de-
scription of the XMCD spectrum at the M 5 edge in
UPtAl, but gives rather good agreement with the experi-
ment in the case of URhAl and UCoAl. One can conclude
that the U 5 f states in UPtAl have more itinerant charac-
ter than those in URhAl and UCoAl.
2.2. Uranium monochalcogenides
The uranium compounds US, USe, and UTe belong to
the class of uranium monochalcogenides that crystallize
in the NaCl structure and order ferromagnetically (on the
uranium sublattice) at Curie temperatures of 178, 160,
and 102 K, respectively (see, e.g., the review [49]). These
uranium compounds exhibit several unusual physical
phenomena, which are the reason for a continuing on-go-
ing interest in these compounds. Despite their relatively
simple and highly symmetrical NaCl structure, it has been
found that the magnetic ordering on the uranium atoms is
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 123
U
M
4
,5
X
M
C
D
,
ar
b
.
u
n
it
s
UCoAlM
5
M4
–1.0
–0.5
0
URhAl
LSDA
LSDA+U(OP)
LSDA+U
exper.
0 20 40 60 80 100
Energy, eV
–2
–1
0
1
Fig. 14. The XMCD spectra of UCoAl and URhAl at the ura-
nium M4 5, edges calculated in the LSDA (dashed lines) and
LSDA + U (OP) (solid lines) approximations [86]. Experimental
spectra for UCoAl [71] (circles) were measured at magnetic field
7 T. Experimental data for URhAl is from Ref. 70 (the U M 4
spectra are shifted by –95 eV to include them in the figure).
U
M
5
X
M
C
D
,
ar
b
.
u
n
it
s
–20 –10 0 10 20
Energy, eV
– 0.2
– 0.1
0
0.1
Fig. 15. Uranium M5 XMCD spectrum of UCoAl calculated in
the LSDA + U OP) approximation (full line) and spectra calcu-
lated with [N7 2/ + N5 2/ ] and [N�7 2/ + N�5 2/ ] DOS’s artificially
shifted by 10 meV (dashed line) and 20 meV (dotted line) [86].
strongly anisotropic [90,91], with the uranium moment
favoring a [111] alignment. The magnetic anisotropy in
US, e.g., is one of the largest measured in a cubic mate-
rial, with a magnetic anisotropy constant K 1 of more than
2�108 erg/cm 3 [92]. Also the magnetic moment itself is
unusual, consisting of an orbital moment that is about
twice as large as the spin moment, and of opposite sign
[93–95]. A bulk magnetization measurements [91] yields
an ordered moment of 1.55 � B per unit formula and neu-
tron scattering measurements [96] show a slightly larger
value of 1.70 � B , which is assigned to the 5 f magnetic
moment. These values are far smaller than that expected
for the free ion, indicating that some sort of «solid state
effect» takes place with the 5 f states. From several exper-
imental results (for instance, photoemission [97], electri-
cal resistivity [98], pressure dependence of Curie temper-
ature [99], and specific heat measurements [100,101]),
the 5 f electrons of US are considered to be itinerant.
It has been suggested that uranium monochalco-
genides are mixed valence systems [102]. Low-tempera-
ture ultrasonic studies on USe and UTe were performed in
the context of questioning the possibility of the coexis-
tence of magnetism and intermediate valence behavior
[103]. They found a monotonic trend of the Poisson’s ra-
tio, which decreases with increasing chalcogenide mass,
and is positive in US, negative in USe and UTe. This indi-
cates the possibility of intermediate valence in the last
two compounds. Indeed, a negative Poisson’s ratio, i.e., a
negative C12 elastic constant, is quite common for inter-
mediate valence systems, and its occurrence seems to be
due to an anomalously low value of the bulk modulus. A
negative C12 means that it costs more energy to distort the
crystal from cubic to tetragonal structure, than to modify
the volume. Thus, when uniaxially compressed along a
[100] direction, the material will contract in the [010] and
[001] directions, trying to maintain a cubic structure. An
explanation for a negative C12 may be given through
a breathing deformability of the actinide ion due to a
valence instability [104].
The dependence of the Curie temperatures TC of US,
USe and UTe on hydrostatic pressure up to 13 GPa has
been determined in Ref. 105. For USe and UTe, TC ini-
tially increases with applied pressure, passing through
maxima at pressure of about 6 and 7 GPa, respectively.
For US, TC decreases monotonically with pressure, which
is compatible with pressure-dependent itinerant electron
magnetism. Pressure increases the bandwidth and corre-
spondingly decreases the density of states at the Fermi
level, which leads to a decrease of TC . The behavior of
USe and UTe is suggestive of localized interacting 5 f
moments undergoing Kondo-type fluctuations, which be-
gin to exceed the magnetic interaction when TC passes
through maximum. A theoretical analysis of these experi-
ments is given in Ref. 106. On the basis of band structure
calculations it is argued that the nonmonotonic behavior of
TC under pressure is solely the result of pressure-driven in-
creased 5 f itineracy.
It must be remarked that the behavior of uranium
monochalcogenides cannot be explained entirely by a
simple trend of increasing localization with increasing
chalcogen mass [48]. Whereas such a trend is evident in
the dynamic magnetic response, in the pressure-depend-
ence of the Curie temperatures and in the value of the or-
dered moment, the behavior of Poisson’s ratio and of the
Curie temperature is the opposite from what one would
naively expect.
There are several band structure calculations of ura-
nium monochalcogenides in literature [95,107–116].
Kraft et al. [110] have performed the LSDA calculation
with the spin-orbit interaction in a second variational treat-
ment for ferromagnetic uranium monochalcogenides (US,
USe, and UTe) using the ASW method, and have shown
that the magnitude of the calculated orbital magnetic mo-
ment M l is larger than that of spin moment M s and they
couple in an antiparallel way to each other. However, the
magnitude of the total magnetic moment (M s + M l ) is too
small compared to the experimental data, indicating that
the calculated M l is not large enough.
The optical and MO spectra of uranium monochalco-
genides have been investigated theoretically in Refs. 107,
108, 110, 112. These theoretical spectra are all computed
from first principles, using Kubo linear-response theory,
but it appears that there are large differences among them.
Cooper and co-worker [109] find good agreement with
experiment for the real part of the diagonal conductivity
( )( )
xx
1 of UTe, but the much more complicated off-diago-
nal conductivity (
xy
( )2 ) of US and UTe is about 4 times
larger than experiment and also the shape of their spec-
trum is different from the experimental one. Halilov and
Kulatov [107] also find an off-diagonal conductivity
which is much larger than the experimental one, but they
additionally obtain a diagonal conductivity
xx
( )1 that dif-
fers substantially from experiment. Gasche [108] find a
Kerr rotation spectrum of US that is quite different from
experiment, and subsequently consider the effect of an or-
bital polarization term to improve the ab initio Kerr spec-
tra. Kraft et al. [110] obtained for US, USe, and UTe rea-
sonable agreement with experiment for the absolute value
of the Kerr spectra. However, the shape of the Kerr spectra
is not reproduced by LSDA theory, since the theoretical
spectra exhibit a double-peak structure, but experimental
spectra have only a one-peak structure. The LSDA +U cal-
culations presented in Ref. 112 take into account the
strong Coulomb correlations among the 5 f orbitals and
are greatly improve the agreement between theory and ex-
periment for all three materials. This finding appears to
be consistent with the quasilocalized nature of the 5 f
electrons in these compounds.
124 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
3. Band structure. All the three chalcogenides,
namely, US, USe, and UTe crystallize in the NaCl type
structure (B1) with space group symmetry Fm m3 . The ura-
nium atom is positioned at (0,0,0) and chalcogen at
(1/2,1/2,1/2).
The LSDA energy band structure of US (Fig. 16) can
be subdivided into three regions separated by energy
gaps. The bands in the lowest region around �15 eV have
mostly S s character with a small amount of U sp character
mixed in. The next six energy bands are S p bands sepa-
rated from the s bands by an energy gap of about 6 eV. The
width of the S p band is about 4 eV. U 6d bands are broad
and extend between �2.5 and 10 eV. The sharp peaks in
the DOS just below and above the Fermi energy are due to
5 f 5 2/ and 5 f 7 2/ states, respectively. Figure 16 also shows
the energy bands and total density of states of US in the
LSDA + U approximation [116]. The Coulomb repulsion
splits partially occupied U 5 f 5 2/ states and the LSDA + U
calculations give a solution with three localized 5 f elec-
trons in US. U 5 f states just above the Fermi level are
formed by the remaining 5 f 5 2/ states whereas the peak of
5 f 7 2/ states is pushed about 1 eV upward from its LSDA
position.
Table 4 presents the comparison between calculated
and experimental magnetic moments in uranium
monochalcogenides. For comparison, we list also the re-
sults of previous band structure calculations. Our LSDA
results obtained by fully relativistic spin-polarized
LMTO method are in good agreement with the ASW
Kraft et al. results [110]. The LSDA calculations for fer-
romagnetic uranium monochalcogenides (US, USe, and
UTe) give the magnitude of the total magnetic moment
M t too small compared to the experimental data, indicat-
ing that the calculated M l is not large enough.
It is a well-known fact, however, that the LSDA calcu-
lations fail to produce the correct value of the orbital
moment of uranium compounds [95,117,119–121]. In
LSDA, the Kohn–Sham equation is described by a local
potential including the spin-dependent electron density.
The electric current, which describes M l , is, however, not
included. This means, that although M s is self-consis-
tently determined in LSDA, there is no framework to si-
multaneously determine M l self-consistently.
Using the LSDA + OP method Brooks [95] obtained
larger magnitude of M l and improvement in M t . How-
ever, they have stated that the individual magnitudes of
M s and M l are considered to be too large from the analy-
sis of the magnetic form factor and the ratio M l /M s is still
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 125
US LSDA
–15
–10
–5
0
5
E
n
er
g
y,
eV
X W K L WUX 0 4 8
LSDA+U
–15
–10
–5
0
5
E
n
er
g
y,
eV
X W K L W UX 0 4 8
DOS
Fig. 16. Self-consistent fully relativistic energy band structure
and total DOS (in states/(unit cell�eV)) of US calculated within
the LSDA and LSDA + U approximations with U = 2 eV and
J = 0.5 eV [116].
Table 4. The experimental and calculated spin Ms, orbital Ml,
and total Mt magnetic moments at uranium site (in �B) of US, USe,
and UTe [116]
Compound Method Ms Ml Mt –Ml/Ms
US
LSDA –1.53 2.14 0.60 1.41
LSDA +U(OP) –1.48 3.21 1.72 2.17
LSDA +U –1.35 3.42 2.07 2.53
LSDA [110] –1.6 2.5 0.9 1.6
LSDA + OP [95] –2.1 3.2 1.1 1.5
OP scaled HF [117] –1.51 3.12 1.61 2.07
HF(TB) [114] – 1.49 3.19 1.70 2.14
exper. [96] –1.3 3.0 1.7 2.3
exper. [91] — — 1.55 —
USe
LSDA –1.75 2.54 0.79 1.45
LSDA +U(OP) –1.65 3.65 2.00 2.21
LSDA +U –1.96 4.61 2.65 2.35
LSDA [110] –1.8 2.8 1.0 1.5
LSDA + OP [95] –2.4 3.4 1.0 1.4
exper. [96] — — 2.0 —
exper. [91] — — 1.8 —
UTe
LSDA –2.12 3.12 1.00 1.47
LSDA +U(OP) –1.91 4.09 2.17 2.14
LSDA +U –2.13 4.95 2.81 2.32
LSDA [110] –2.2 3.4 1.2 1.5
LSDA + OP [95] –2.6 3.4 0.8 1.3
exper. [118] –1.57 3.48 1.91 2.21
exper. [96] — — 2.2 —
exper. [91] — — 1.9 —
far from the experimental value for all the three uranium
monochalcogenides (Table 4).
Table 4 presents the calculated magnetic moments in
uranium monochalcogenides using a generalization of the
LSDA + U method [73,122]. In this calculations we used
U 2 0. eV and J 0 5. eV. Table 4 presents also the LSDA +
U calculated magnetic moments with U J 0 5. eV (the
LSDA + U (OP) approximation).
Figure 17 shows 5 f 5 2/ partial density of states in US cal-
culated within the LSDA, LSDA + U (OP) and LSDA + U
approximations [116]. The LSDA + U (OP) approxima-
tion strongly affects the relative energy positions of m j
projected 5 f density of states and substantially improve
their orbital magnetic moments (Table 4). For example,
the ratio M l /M s in the LSDA + U (OP) calculations is
equal to �2.17 and �2.14 for US and UTe, respectively.
The corresponding experimental value are �2.3 for US
from the neutron measurements [96] and �2.21 for UTe
from the magnetic Compton profile measurements [118].
The 5 f spin M s and orbital M l magnetic moments in
US have been also calculated in Ref. 114 on the basis of
the HF approximation for an extended Hubbard model.
The tight-binding model includes the intra-atomic 5 f –5 f
multipole interaction and the SOI in the 5 f state. The pa-
rameters involved in the model were determined by fitting
with the energy of Bloch electrons in the paramagnetic
state obtained in the LDA band structure calculation. The
calculated ratio of the moments M l /M s of �2.14 and M l
of �3.19 � B are in good agreement with available experi-
mental results (Table 4).
We should mention that the results of the LSDA +U (OP)
calculations are in close agreement with the results ob-
tained using the HF approximation for an extended Hub-
bard model [114] (Table 4). Both the approximations take
into account the SOI and the intra-atomic 5 f –5 f Cou-
lomb interaction in Hubbard model. The small differences
in magnetic moments are due to slightly different values
of U eff . In our calculations we used U J 0 5. eV, which
gives U eff = 0. Authors of Ref. 114 used U 0 76. eV and
J 0 5. eV, which gives U eff = 0.26 eV. Besides, there are
some small differences in F 2, F 4 and F 6 Slater integrals
in two the calculations.
Figure 17 also shows the m j projected 5 f 5 2/ density of
states in US calculated in the LSDA + U approximation
with U 2 0. eV and J 0 5. eV [116]. The corresponding
partial DOS’s for USe and UTe are presented in Fig. 18.
The degree of localization of occupied 5 f 5 2/ states is in-
creasing going from US to UTe. In US the 5 f 5 2/ states with
m /j �5 2 is strongly hybridized with other occupied
states, while the hybridization in USe and particularly in
UTe almost vanishes. The 5 f 5 2/ states with m /j �5 2 are
responsible for the narrow single peak in UTe (Fig. 18).
The orbital magnetic moments calculated in the LSDA +
U approximation are larger than calculated in the LSDA +
U (OP) approximation, which leads to slightly overesti-
mated ratio M l /M s in comparison with the experimental
data for the LSDA + U calculations (Table 4).
4. XMCD spectra. Figure 19 shows the XMCD spectra
of US, USe, and UTe at the uranium M 4 5, edges calcu-
lated within the LSDA and LSDA + U approximations
[116]. It is clearly seen that the LSDA calculations give
126 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
US
LSDA 5f
5/2
mj = –5/2
mj = –3/2
mj = –1/2
mj = +1/2
other
0
1
2
3
LSDA+U(OP)
0
1
2
3
LSDA+U
–1 0 1 2
Energy, eV
1
2
3
Fig. 17. The partial 5 f 5 2/ and 5 f 7 2/ density of states within
US, USe, and UTe calculated in the LSDA and LSDA + U ap-
proximations [116].
P
ar
ti
al
D
O
S
,
st
at
es
/(
at
o
m
eV�
�
USe 5f5/2
other
0
1
2
3
UTe
–2 –1 0 1 2
Energy, eV
1
2
3
4
m = –3/2j
m = –5/2j
m = – /2j 1
m = + /2j 1
Fig. 18. The partial 5 f 5 2/ density of states in USe and UTe cal-
culated within the LSDA + U approximation [116].
inappropriate results. The major discrepancy between the
LSDA calculated and experimental XMCD spectra is the
size of the M 4 XMCD peak. The LSDA underestimates
the integral intensity of the XMCD at M 4 edge. As the in-
tegrated XMCD signal is proportional to the orbital
moment [74] this discrepancy could be related to an un-
derestimation of the orbital moment by LSDA-based
computational methods (Table 4). On the other hand, the
LSDA + U approximation produces good agreement with
the experimentally measured intensity for the M 4 XMCD
spectrum. In the case of the M 5 XMCD spectrum, the
LSDA strongly overestimates the value of the positive
peak. The LSDA + U (OP) approximation gives a good
agreement in the shape and intensity of the XMCD spec-
trum at the M 5 edge.
The behavior of the 5 f electrons ranges from nearly
delocalized to almost localized: US is considered to be
nearly itinerant [123], while UTe is considered to be
quasilocalized [124]. So the failure of LSDA description
of XMCD spectra in US comes as a surprise, because, if
the 5 f electrons are itinerant, one would expect the
delocalized LSDA approach to be applicable. However,
as the integrated XMCD signal is proportional to the or-
bital moment [74] this discrepancy could be related to an
underestimation of the orbital moment by LSDA-based
computational methods.
It is interesting to note, that the LSDA + U (OP) and
LSDA + U calculations give similar results for XMCD
spectrum at the M 5 edge in the case of US and became rel-
atively more different going through USe and UTe, proba-
bly, reflecting the increase of degree of localization of 5 f
electrons. Besides, the relative intensity of the M 5 and
M 4 XMCD spectra is strongly increased going from US
to UTe. The experimental measurements of the XMCD
spectra in USe and UTe are highly desired.
2.3. Heavy-fermion compounds
2.3.1. UPt3
UPt3 is a well known heavy-fermion system [125,126].
The Sommerfeld coefficient of the linear low-tempe-
rature specific heat is strongly enhanced, i.e., � =
= 420 mJ/(mol U�K 2). Strong electron-electron correla-
tions are also manifest in a T T3 log term in the low-tem-
perature specific heat, which is believed to be due to spin
fluctuations. At low temperature UPt3 is a superconductor,
with a TC of 0.54 K [59]. UPt3 is the archetype of a
heavy-fermion system. It has the qualitative properties of a
Fermi liquid, but the magnitude of the effective masses, re-
flected in the specific heat and magnetic susceptibility, is
very much larger than the free-electron value. The heavi-
ness of the electrons is generally attributed to electron cor-
relations which come from the strong Coulomb interac-
tions among the localized 5 f electrons on the U sites.
UPt3 has attracted a great deal of interest from
band-structure theorists [127–131], particularly when it
became clear that reliable experimental information on
the Fermi surface could be obtained by measurements of
the de Haas–van Alphen (dHvA) effect [132,134]. These
experiments unambiguously confirm that UPt3 has to be
regarded as a strongly correlated Fermi liquid. Although a
detailed picture of the low-temperature phase of UPt3 has
emerged, a comprehensive theoretical picture of the
heavy quasiparticles is still missing.
It has been considered a success of the LSDA that the
dHvA frequencies could be related to extremal orbits on
the Fermi surface obtained by band-structure calculations
which treat the U 5 f states as itinerant. There are good
reasons that standard band-structure calculations repro-
duce well the complex topology of the Fermi surface in
UPt3. In great contrast, however, no such agreement is
found for the measured cyclotron masses. The calculated
energy bands are too broad for explaining the effective
masses: dHvA masses are by a factor of order 20 bigger
than the band masses mb obtained from the LSDA calcu-
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 127
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
US
LSDA
LSDA+U(OP)
LSDA+U
exper.–2
–1
0
1
USe
–3
–2
–1
0
1
2
UTe
0 20 40 60 80 100
Energy, eV
–3
–2
–1
0
1
M
4
,5
U
Fig. 19. The XMCD spectra of US, USe, and UTe at the ura-
nium M4 5, edges calculated within the LSDA (dashed lines),
LSDA + U (OP) (dotted lines), and LSDA + U (solid lines) ap-
proximations [116]. Experimental spectra of US [71] (circles)
were measured at magnetic field 2 T (the U M 4 spectra are
shifted by �95 eV to include them in the figure).
lations [129–131]. This is of course the defining charac-
teristic of a heavy-fermion compound and is due to the
strong electron-electron correlations not included in the
band-structure calculations. It is interesting that even in
the presence of such strong correlations, there is no evi-
dence of any breakdown of Fermi-liquid theory. The stan-
dard Lifshitz–Kosevich formula for the field and tempera-
ture dependence of the amplitude of quantum oscillations
is perfectly verified down to 10 mK and up to 18 T [59].
UPt3 shows a static antiferromagnetic order below
about TN = 5 K with a very small staggered moment of or-
der 0.01 � B /U atom. This ordering was first noticed in
muon spin relaxation measurements by Heffner et al.
[135] and was soon confirmed by neutron scattering
[136]. The magnetic order is collinear and commensurate
with the crystal lattice, with a moment aligned in the basal
plane. It corresponds to antiferromagnetic coupling
within planes and ferromagnetic coupling between
planes. All aspects of this ordering were reproduced by
later neutron studies on a different crystal [137,138] and
by magnetic x-ray scattering [139]. The moment at lower
temperatures grows to a maximum magnitude of
0.02–0.03 � B /U atom.
1. Band structure. UPt3 crystallizes in the MgCd3-type
structure. The uranium atoms form a closed-packed hex-
agonal structure with the platinum atoms bisecting the
planar bonds. There are two formula units per unit cell.
The compound belongs to the space group P /mmc6 3 and
the point group D h6 . The lattice parameters are a =
= 5.753 � and c/a = 4.898. The nearest U–U distance is
between atoms in adjacent layers, equal to 4.132 �, and
the conductivity is greatest along the c axis.
The fully relativistic spin-polarized LSDA energy
band structure and total DOS of the ferromagnetic UPt 3
compound is shown in Fig. 20 [140]. The occupied part of
the valence band is formed predominantly by Pt 5d states.
The characteristic feature of the LSDA band structure is a
narrow peak of U 5 f 5 2/ states situated just at the Fermi
level (EF ) 1.0 eV above the top of Pt 5d states. U 5 f 7 2/
states are split off by strong SO coupling and form an-
other narrow peak 1 eV above EF .
Figure 20 also shows the band structure of UPt 3 calcu-
lated in the LSDA +U approximation withU = 2.0 eV and
J = 0.5 eV [140]. The Coulomb repulsion splits partially
occupied U 5 f 5 2/ states and the LSDA + U calculations
give a solution with two localized 5 f electrons. These lo-
calized 5 f states are situated above the top of Pt 5d and
form a rather narrow peak at 0.2 eV below EF . The posi-
tion of the peak agrees well with the results of recent reso-
nant photo-emission spectroscopy (PES) [141] and angu-
lar resolved PES (ARPES) [142] measurements. U 5 f
states just above the Fermi level are formed by the re-
maining 5 f 5 2/ states whereas the peak from the 5f7/2
states is pushed from its LSDA position at 1 eV above EF
to 2.3 eV.
An orbital resolved DOS corresponding to the orbitals
with the largest occupation numbers is shown in Fig. 21
for UPt3 and for UPd3 as a reference material. Two peaks
at –1.0 to –0.5 eV in UPd3 are formed by 5 f 5 2/ states with
m /j �5 2 and m /j �3 2. Their occupation numbers are
n5 2/ = 0.988 and n3 2/ = 0.982, which corresponds to an
f 2 configuration of the U ion [73]. The corresponding
128 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
UPt3 LSDA
–10
–5
0
5
E
n
er
g
y,
eV
M K A L H A 0 10 20 30
LSDA+U
–10
–5
0
5
E
n
er
g
y,
eV
MK A L H A 0 10 20 30
DOS
Fig. 20. The self-consistent fully relativistic, spin-polarized energy
band structure and total DOS (in states/(unit cell�eV)) of UPt3 cal-
culated in the LSDA and LSDA +U approximations [140].
UPt3
other
0
2
4
UPd3
–3 –2 –1 0 1 2 3
Energy, eV
0
4
5f5/2
m = –5/2j
m = – /2j 3
P
ar
ti
al
D
O
S
,
st
at
es
/(
at
o
m
·e
V
)
Fig. 21. The partial 5 f 5 2/ density of states in UPt3 and UPd3
calculated in the LSDA + U approximation.
states in UPt3 are situated in –0.5 to 0.2 eV energy range,
very close to the Fermi level and partially occupied. Such
a different energy position of occupied 5 f 5 2/ states in
UPd3 and UPt3 can be explained by the larger spatial ex-
tent of Pt 5d wave functions as compared to the Pd 4d
states which causes a proportional increase of the part of
f electron density at U site provided by the «tails» of d
states. The screening of the localized U 5 f states by this
delocalized density becomes stronger in UPt3 and their
occupied 5 f 5 2/ states shift to higher energy [73].
The above-mentioned self-consistent LSDA + U solu-
tions for UPd3 and UPt3 are magnetic with a rather large
U magnetic moment. This is contrary to the experimental
data which show that the ordered magnetic moment is
only 0.01 � B and 0.02–0.03 � B per U atom in UPd3 and
UPt3, respectively [137–139,143]. This extremely small
U magnetic moment is explained by the fact that accord-
ing to the crystalline electric field (CEF) level scheme de-
rived from neutron scattering experiments, the lowest
CEF level of U4+ ion in both compounds is a singlet
[139,144] which leads to a nonmagnetic ground state for
these compounds. The LSDA + U is still a one electron
approximation and can not fully account for the subtle
many-body effects responsible for the small value of the
U magnetic moment in the UPd3 and UPt3. It tries to obey
the Hund’s rules in the only way it is allowed to, i.e., by
producing a magnetic solution. A possible way to over-
come this discrepancy between the calculations and the
experiment is to force a nonmagnetic ground state in the
LSDA + U calculations as it was done by H. Harima et al.
in Refs. 143, 145. We have verified, however, that this
leads to an increase of the total energy as compared to
magnetic states obtained in the calculations.
It should be mentioned that depending on the starting
conditions another self-consistent LSDA + U solution
very close in total energy can obtained for UPd3 as well as
for UPt3. This solution also results in two localized U 5f
electrons but in this case the occupied states are | ,5 2 5 2/ /� �
and | ,5 2 1 2/ /� � (here we used the notation | ,j m j � for the
state with the total momentum j and its projection m j )
[140]. The existence of two almost degenerate solutions
can be understood if one compares the matrix elements of
Coulomb interaction U m mj j, � calculated between 5 f 5 2/
states with different m j [73]. The matrix elements
U / /5 2 3 2, andU 5 2 1 2/ , / are equal and the energy difference is
caused not by the on-site Coulomb interaction but instead
by a difference in the hybridization between U 5 f 5 2/ and
conduction electrons. Also, the lowest unoccupied 5 f
state, which is either | ,5 2 1 2/ /� � or | ,5 2 3 2/ /� �, feels the
same Coulomb repulsion of the localized electrons. Total
energy calculations, however, show that lower energy so-
lution is associated with | ,5 2 3 2/ /� � occupied states.
2. XMCD spectra. As we mentioned above, for the 5 2f
configuration in UPt 3 we have two solutions with close to-
tal energies, in the first case the 5 f 5 2/ states with m /j �5 2
and �3 2/ are occupied, in the second case the occupied
states are m /j �5 2 and �1 2/ . In the first case the dipole al-
lowed transitions for left circularly polarized light, � � 1
a r e � � �3 2 1 2/ / , � � �1 2 1 2/ / , � � �1 2 3 2/ / , a n d
� � �3 2 5 2/ / and for right circularly polarization � �1:
� � �1 2 1 2/ / and � � �3 2 1 2/ / . The transitions with
equal final states m /j �1 2 and m /j � 1 2 mostly cancel
each other and the XMCD spectrum of U at the M 4 edge
( )I ��� � � can be roughly represented by � �[ /
/
/
/N N3 2
5 2
5 2
5 2]
partial density of states [72]. In the second case, however,
the dipole allowed transitions for � � 1are � � �1 2 1 2/ / ,
� � �1 2 3 2/ / , a n d � � �3 2 5 2/ / a n d f o r � �1:
� � �1 2 3 2/ / and � � �3 2 1 2/ / . Therefore U M 4 XMCD
spectrum can be roughly represented by N N N1 2
5 2
3 2
5 2
5 2
5 2
/
/
/
/
/
/[� � ]
partial density of states. One would expect therefore
smaller intensity of dichroic signal at the M 4 edge for the
second case in comparison with the first one due to the com-
pensation between N 1 2/ and [ ]/ /N N3 2 5 2� partial density
of states in the second case.
The 5 f 7 2/ states are almost completely empty in all the
uranium compounds. Therefore the XMCD spectrum of U
at the M 5 edge can be roughly represented by the m j pro-
jected partial density of states [72]: [N N� �� �7 2
7 2
5 2
7 2
/
/
/
/ ]
� �[ /
/
/
/N N7 2
7 2
5 2
7 2]. As a result, the shape of the M 5 XMCD
spectrum consists of two peaks of an opposite sign: a ne-
gative 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.
Although we neglect cross terms in the transition ma-
trix elements and there is no full compensation between
transitions with equal final states due to difference in the
angular matrix elements, such a simple representation
qualitatively reproduces all the peculiarities of the experi-
mentally measured XMCD spectra in UPt 3. It gives a
simple, slightly asymmetric negative peak at the M 4 edge
and an s shaped two-peak structure at the M 5 edge
(Fig. 22). It also correctly gives the dichroism at the M 4
edge of approximately one order of magnitude larger than
at the M 5 one. The spectrum at the M 4 edge is very sensi-
tive to the character of the occupied 5 f 5 2/ states and has
larger intensity for the solution with occupied | ,5 2 3 2/ /� �
states.
Figure 23 shows the calculated XMCD spectra in the
LSDA and LSDA + U approximations for UPt 3 [140] to-
gether with the experimental data [65]. The intrinsic
broadening mechanisms have been accounted for by fold-
ing the XMCD spectra with a Lorentzian of 3.2 and 3.4 eV
for M 5 and M 4 spectra, respectively. The overall shapes
of the calculated and experimental uranium M 4 5, XMCD
spectra correspond well to each other. The major discrep-
ancy between the calculated and experimental XMCD
spectra is the size of the M 4 XMCD peak. The LSDA the-
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 129
ory produces a much smaller intensity for the XMCD
spectrum at M 4 edge in comparison with the experiment
and simultaneously gives a larger dichroic signal at M 5
edge. On the other hand, the LSDA + U approximation
produces excellent agreement in the shape and intensity
of XMCD spectra both at the M 4 and M 5 edges for the so-
lution with the | ,5 2 3 2/ /� � state occupation. The solution
with | ,5 2 1 2/ /� � occupation produces a smaller intensity
for the XMCD spectrum at the M 4 edge in comparison
with the experiment. This observation is consistent with
the total energy calculations which show that the lowest
energy state has the solution with | ,5 2 3 2/ /� � states
occupied.
The LSDA + U (OP) approximation, which describes
the correlations between spin and orbital magnetic mo-
ment directions (U eff = 0) gives a correct value of the
XMCD spectrum at U M 4 edge, but slightly overesti-
mates the positive peak and underestimates the negative
one at the M 5 edge (not shown).
Figure 23 shows also the XMCD spectra in UPd 3 cal-
culated using the LSDA + U approximation for the solu-
tion with occupied | ,5 2 3 2/ /� � states [140]. The XMCD
spectra of UPd3 and UPt3 are very similar, except, the
positive peak at the M 5 edge is slightly less pronounced
in UPd3 than in UPt3. Experimental measurements of
XMCD spectra in UPd3 are highly desired.
2.3.2. URu2Si2
The heavy-fermion superconductor URu2Si2 has at-
tracted continuous attention in the last decade for its un-
usual ground-state properties. URu2Si2 crystallizes in the
body-centered tetragonal ThCr2Si2 structure with lattice
constant a = 4.126 � and c/a = 2.319. At TN = 17.5 K the
system undergoes an antiferromagnetic phase transition
which is accompanied by a sharp peak in the specific heat
[146,147] and thermal expansion [148]. A second transi-
tion occurs at TC = 1.2 K and indicates the onset of super-
conductivity which coexists with the antiferromagnetic
order. Neutron-scattering measurements [149,150] re-
vealed a simple antiferromagnetic structure with a tiny or-
dered moment of (0.04
0.01) � B /U atom, oriented
along the c axis of the tetragonal crystal structure. The
formation of an energy gap in the magnetic excitation
spectrum is reflected by an exponential temperature de-
pendence of the specific heat [146,147], the thermal ex-
pansion [148] and the NMR and nuclear quadruple-reso-
nance NQR relaxation rates [151] in the ordered state.
Electrical resistivity [152] and point-contact spectros-
copy measurements [153] show a similar energy gap, in-
dicating a strong scattering of the conduction electrons by
130 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
P
ar
ti
al
d
en
si
ty
o
f
st
at
es
,
ar
b
.
u
n
it
s
a
–0.3
0.3
b
0 20
Energy, eV
–3
–2
–1
0
1
–20
0
Fig. 22. The model representation of the M5 (a) and M4 (b)
XMCD of UPt3 for two solutions with | ,5 2 3 2/ /� � occupied states
(full lines) and | ,5 2 1 2/ /� � ones (dashed lines): (a) presents the
partial densities of states [N N N N� �� � �7 2
7 2
5 2
7 2
7 2
7 2
5 2
7 2
/
/
/
/
/
/
/
/] [ ];
( ) [ /
/
/
/b N N� �3 2
5 2
5 2
5 2] (full line) and N N N1 2
5 2
3 2
5 2
5 2
5 2
/
/
/
/
/
/[� � ] (dashed
lines) [140] (see the explanation in the text).
U
M
X
M
C
D
,
a
rb
.
u
n
it
s
4
,5
UPt3
LSDA
LSDA+U(m = 1/2)j
LSDA+U(m = 3/2)j
exper.
–1
0
1
UPd3
LSDA+U
0 20 40 60 80 100
Energy, eV
–1
0
1
Fig. 23. The XMCD spectra of UPt3 and UPd3 at the uranium
M4 5, edges calculated in the LSDA, LSDA + U (OP), and
LSDA + U approximations [140]. Experimental spectra for
UPt3 [65] (circles) were measured in a magnetic field of 5 T at
20 K (the U M 4 spectra are shifted by �95 eV to include them
in the figure).
the magnetic excitations. Magnetization measurements in
high magnetic fields [154,155] show a suppression of the
heavy-fermion state in three consecutive steps at 35.8,
37.3, and 39.4 T for fields along the easy axis (B c| | ). These
transitions have been confirmed in high-field measure-
ments of the magnetoresistance and Hall coefficient [156].
There are several LSDA band structure calculations of
URu2Si2 in the literature [157–160]. A self-consistent
calculation of electronic band structure for antiferro-
magnetically ordered URu2Si2 was performed using an
all-electron fully relativistic spin-polarized LAPW
method by Yamagami and Hamada [160]. They obtained a
magnetic moment at the uranium site with a tiny value of
0.09 � B due to cancellation between the spin and the or-
bital moments. The theoretically calculated frequencies
as functions of the direction of applied magnetic field are
in reasonable agreement with the dHvA frequencies mea-
sured by Ohkuni et al. [161].
The electronic band structure and the Fermi surface of
paramagnetic URu2Si2 have been studied also with
high-resolution angle-resolved photoemission spectros-
copy in Ref. 162. It was found that Ru 4d bands form the
main body of the valence band and exhibit a remarkable
energy dispersion in qualitatively good agreement with
the band structure calculations. In addition to the
dispersive Ru 4d bands, a less dispersive band was found
near the Fermi level, which can be assigned to the U
5 f –Ru 4d hybridized band.
1. Band structure. Self-consistent LSDA calculations
produce an antiferromagnetic ground state in URu2Si2
[140] in agreement with the experimental observation
[148]. The spin moment at the U site is obtained as
�0 04. � B , the orbital moment is 0.09 � B . The total mag-
netic moment is, therefore, 0.05 � B . This is in a good
agreement with the magnetic moment of 0.04 � B ob-
served by neutron-scattering measurements [149,150].
The fully relativistic spin-polarized LSDA energy band
structure and total DOS of the antiferromagnetic URu2Si2
is shown in Fig. 24. Figure 25 shows the LSDA partial
density of states of URu2Si2 [140]. Si 3s states are located
mostly at the bottom of the valence band in the –11 to
–8 eV energy interval. Si 2 p states hybridize strongly
with Ru 4d, U 6d and U 5 f valence states and occupy a
wide energy range from –6.5 to 11 eV. There is an energy
gap of around 0.5 eV between Si 3s and 3 p states. Ru 4d
states are situated below and above Fermi level in the –6.5
to 3.5 eV range. The Fermi level falls in the local mini-
mum of Ru 4d states (Fig. 25). U 6d states are strongly hy-
bridized with Ru 4d as well as Si 3 p and even Si 3s states.
A narrow peak of U 5 f 5 2/ states situated just at the Fermi
level EF . U 5 f 7 2/ states are split off by strong SO cou-
pling and form another narrow peak 1.2 eV above EF . Be-
cause U 5 f states are situated at the local minimum of Ru
4d states there is rather week U 5 f –Ru 4d hybridization.
Figure 24 also shows the band structure of URu2Si2 cal-
culated in the LSDA + U approximation with U = 2.0 eV
and J = 0.5 eV [140]. The Coulomb repulsion U eff
strongly influences the electronic structure of URu2Si2.
The occupied on-site 5 f energies are shifted downward
by U eff /2 and the unoccupied levels are shifted upwards
by this amount. As a result both the occupied and empty U
5 f states move to a position with large Ru 4d DOS and the
degree of U 5 f –Ru 4d hybridization increases going from
the LSDA to the LSDA + U solution. In the Hartree–Fock
like LSDA + U solution with nonspherical correction to
Coulomb matrix elements, three particular 5 f 5 2/ states
(m /j �5 2, �3 2/ , and �1 2/ ) are occupied which leads to
large spin (–2.01 � B ) and orbital (4.78 � B ) magnetic mo-
ments for the U atom. U 5 f states just above the Fermi
level are formed by the remaining 5 5 2f / states whereas the
peak of 5 f 7 2/ states is pushed from its LSDA position
above EF by 2.8 eV.
2. XMCD spectra. Figure 26 shows the calculated
x-ray isotropic absorption and XMCD spectra in the
LSDA and LSDA +U approximations for URu 2Si 2 [140]
together with the experimental data [69]. To calculate the
x-ray isotropic absorption M 4 5, spectra we take into ac-
count the background intensity which appears due to tran-
sitions from occupied levels to the continuum of unoccu-
pied levels [89].
The theory [140] produces a much smaller intensity of
the XMCD spectrum at the M 4 edge in comparison with
the experiment in the LSDA calculations. It also gives a
larger positive peak and a two times smaller negative
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 131
URu Si2 2 LSDA
–10
–5
0
5
E
n
er
g
y,
eV
X M A Z M 0 10 20 30
LSDA+U
–10
–5
0
5
10
E
n
er
g
y,
eV
X M A Z M 0 10 20 30
DOS
Fig. 24. The self-consistent fully relativistic, spin-polarized
energy band structure and total DOS (in states/(unit cell�eV))
of URu2Si2 calculated in the LSDA and LSDA + U approxima-
tions [140].
peak at the M 5 edge (Fig. 26). The LSDA +U approxima-
tion with J = 2.0 and J = 0.5 eV and nonspherical correc-
tions to Coulomb matrix elements [69] produces excellent
agreement in shape and intensity for the XMCD spectra
both at the M 4 and M 5 edges. This can be considered as
evidence in favor of a picture of partly localized U 5 f
states in URu2Si2.
One should mention that the LSDA + U (OP) calcula-
tions (U eff = 0) underestimate the negative XMCD peak
and overestimate the positive one at the M 5 edge (not
shown). This approximation also slightly underestimates
the XMCD signal at the M 4 edge.
2.3.3. UPd2Al3 and UNi2Al3
The most recently discovered heavy-fermion super-
conductors UPd2Al3 and UNi2Al3 [163,164] exhibit
coexistence between superconductivity and a magnetic
state with relatively large ordered magnetic moments.
UPd2Al3 was found to exhibit a simple antiferromagnetic
structure [wave vector q = (0,0,1/2)] below TN � 14.5 K
and static magnetic moments of U lying in the basal plane
[165]. The neutron-scattering data are consistent with an
ordered magnetic moment M t � 0.85 � B , reduced
compared to the effective moment obtained from the
high-temperature susceptibility, but exceeding by up to
two orders of magnitude the small moments found, for ex-
ample, in UPt3. Hence, in contrast to UPt3, a picture of lo-
cal-moment magnetism seems to describe the magnetic
state in UPd2Al3. Surprisingly, this large-moment magne-
tism was found to coexist with heavy-fermion supercon-
ductivity exhibiting the highest TC reported to date for
this class of materials.
The electronic structure and Fermi surface of the
antiferromagnetic UPd2Al3 were calculated using the
LSDA approximation in Refs. 166–168. The calculated
magnetic moment was in good agreement with experi-
ment as was the calculated magnetocrystalline aniso-
tropy. The calculations reveal the importance of hybrid-
ization of the U 5 f states with the valence states of Pd and
Al even though this hybridization appears to be rather
weak and to influence only a restricted energy interval in
the U 5 f bands. The calculated dHvA frequencies are
found to be in good agreement with the experimental data.
However, the observed heavy masses cannot be obtained
within the LSDA [168].
The measured (in Ref. 169) x-ray photoemission and
bremsstrahlung isochromat spectra of UPd2Al3 are well
reproduced by the LSDA calculated U 5 f density of
states. On the other hand, the resonance photoemission
spectra of UPd2Al3 does not match the calculated U 5f
DOS in shape or position, while the calculated Pd 4d DOS
matches very well with the off-resonance spectrum [170].
132 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
Sis
p
0
1
Rud
1
2
Uf
d
–10 –5 0 5 10
Energy, eV
0
5
10
15
0
Fig. 25. The partial density of states in URu2Si2 calculated in
the LSDA approximation [140] (the 6d partial DOS has been
multiplied by factor 3 for clarity).
A
b
so
rp
ti
o
n
,
ar
b
.
u
n
it
s
X
M
C
D
,
ar
b
.
u
n
it
s
URu2Si2
M5
M4
0
20
40
60
LSDA
LSDA+U
exper.
0 20 40 60 80 100
Energy, eV
–2
–1
0
Fig. 26. Isotopic absorption and XMCD spectra of URu2Si2 at
the uranium M4 5, edges calculated in the LSDA (dashed lines)
and LSDA + U (full lines) approximations [140]. Experimental
spectra [69] (circles) were measured at 50 K and in a magnetic
field of 5 T (the U M 4 spectra are shifted by �95 eV to include
them in the figure).
The superconducting and magnetic properties of
UNi2Al3 are not so well documented compared to those
of UPd2Al3 owing to the difficulties of preparing good
single crystals [68]. UNi2Al3 undergoes transitions to
antiferromagnetism at TN � 4.6 K and to superconduc-
tivity at TC � 1.2 K [164]. Muon spin rotation (�SR) ex-
periments [171] on polycrystalline UNi2Al3 showed evi-
dence for antiferromagnetism with an ordered moment of
the order of 0.1 � B . Elastic neutron scattering from a sin-
gle-crystal sample of UNi2Al3 has revealed the onset of
long-range magnetic order below TN = 4.6 K [172].
The order is characterized by wave vector of the form
( , , )1 2 0 1 2/ /
� , with � = 0.110
0.0003, indicating an in-
commensurate magnetic structure within the basal plane,
which is simply stacked antiferromagnetically along c to
form the full three-dimensional magnetic structure. The
maximum amplitude of the ordered moment is estimated
to be (0.21
0 10. ) � B .
1. Band structure. UPd2Al3 and UPd2Al3 crystallize in
a rather simple hexagonal structure P /mmm6 (D h6
1 ,
PrNi3Al3-type structure) with lattice constant a = 5.365 �
and c/a = 4.186 for UPd2Al3 and a = 5.207 � and c/a =
= 4.018 for UNi2Al3.
The fully relativistic spin-polarized LSDA energy
band structures and total DOS’s of the antiferromagnetic
UPd2Al3 and UNi2Al3 are shown in Fig. 27 [140]. The re-
sults of our band structure calculations of UPd2Al3 are in
good agreement with previous calculations of Sandratskii
et al. [167]. Al 3s states are located mostly at the bottom
of the valence band in the �9.7 to �5 eV energy interval.
Al 3 p states occupy wide energy range from �6 to 11 eV
hybridized strongly with Pd 4d, U 6d and U 5 f valence
states. Pd 4d states are almost fully occupied and situated
below Fermi level in the �5 to �2.5 eV range. The mag-
netic moment at the Pd site, therefore, is extremely small.
U 6d states are strongly hybridized with Pd 4d as well as
Al 3 p states. The characteristic feature of the LSDA band
structure is a narrow peak of U 5 f 5 2/ states situated just at
the Fermi level EF . U 5 f 7 2/ states are split off by strong
spin-orbit coupling and form another narrow peak 1.2 eV
above EF . Because Pd 4d states are located far below the
Fermi level, there is rather week U 5 f –Pd 4d hybridiza-
tion. We should mention, however, that this hybridization
is of primary importance and influences greatly the form
and width of the 5 f peaks (the analysis of the hybridiza-
tion effects in UPd2Al3 are presented in Ref. 167).
In agreement with experiment [165] we found the
basal plane of the hexagonal structure to be the plane of
easy magnetization in UPd2Al3. The magnetic structures
with magnetic moments lying in the xy plane possess
lower energy than those with atomic moments along the z
axis. A rotation of the magnetic moment within the xy
plane does not noticeably change the energy of the con-
figuration as well as the value of the spin and orbital mag-
netic moments.
Our calculations, unfortunately, yield for the total en-
ergy of the in-plane ferromagnetic structure a slightly
lower value than for the energy of the corresponding
antiferromagnetic structure, although the difference of
the total energy of the ferromagnetic and antiferro-
magnetic in-plane solutions is very small, about 9 meV
per formula unit, and is close to the accuracy limit of our
LMTO–LSDA calculations. This disagrees with experi-
ment which shows the ground-state magnetic structure to
be antiferromagnetic [165]. The same results were ob-
tained by Sandratskii et al. in Ref. 167.
The energy band structure of UNi2Al3 and UPd2Al3
are very similar (Fig. 27) [140]. The major difference is in
the energy location and width of the transition metal
bands. Due to less spatial expansion of Ni 3d wave func-
tions compared to Pd 4d wave functions the Ni 3d energy
band is 1.5 times narrower than the corresponding 4d
band in UPd2Al3. The Ni 3d energy band is situated in the
�3 to �1.2 eV energy interval. Due to a shift of the Ni 3d
band toward the Fermi level, the U 5 f –Ni 3d hybridiza-
tion in UNi2Al3 is increased in comparison with the
U 5 f –Pd 4d hybridization in UPd2Al3. A stronger inter-
action between 5 f and conduction electrons when replac-
ing Pd by Ni is manifested in a shift toward higher tem-
peratures of the maxima of both the resistivity and the
susceptibility together with the decrease of the magnetic
ordering temperature TN , the superconductivity tempera-
ture TC , the antiferromagnetic moment and the smaller
entropy change at TN [68].
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 133
UPd Al2 3 LSDA
–10
–5
0
5
E
n
er
g
y,
eV
M K A L H A
0 10 20 30 40
UNi2Al3
–10
–5
0
5
E
n
er
g
y,
eV
M K A L H A
0 10 20 30 40
DOS
Fig. 27. The self-consistent fully relativistic, spin-polarized
energy band structure and total DOS (in states/(unit cell�eV))
of UPd2Al3 and UNi2Al3 calculated in the LSDA approxima-
tion [140].
Figure 28 shows m j projected 5 f 5 2/ density of states in
UPd 2Al 3 calculated in the LSDA and LSDA + U approxi-
mations [140]. We performed two LSDA +U band structure
calculations. In the first calculation we usedU J 0 5. eV,
which gives U eff = 0 (the so-called LSDA + U (OP) ap-
proximation). In the second oneU 2 0. eV and J 0 5. eV.
The LSDA approximation places the 5 f 5 2/ density of
states in close vicinity of the Fermi level at �0.5 to 0.5 eV
with strong hybridization between states with different
m j . The Coulomb repulsion U eff strongly influences the
electronic structure of UPd2Al3 and UNi2Al3. In the
Hartree–Fock-like LSDA + U solution with nonspherical
corrections to Coulomb matrix elements, three particular
5 5 2f / states (m /j �5 2, �3 2/ , and �1 2/ ) are almost com-
pletely occupied producing the 5 f 3 configuration for U
in UPd2Al3 and UNi2Al3.
Table 5 lists the calculated spin M s , orbital M l , and to-
tal M t magnetic moments at uranium site (in � B ) as well
as the ratio M l /M s for UPd2Al3 and UNi2Al3 [140]. Our
LSDA results are in good agreement with previous LSDA
calculations [167]. Surprisingly, LSDA calculations pro-
duce the total magnetic moments in UPd2Al3 and
UNi2Al3 in good agreement with the experimental data.
On the other hand, the LSDA calculations strongly under-
estimate the ratio M l /M s (especially in UNi2Al3) due
to the underestimation of the orbital moment by
LSDA-based computational methods. The ratio M l /M s in
the LSDA + U (OP) calculations is in reasonable agree-
ment with the experimental data for both the compounds.
2. XMCD spectra. Figure 29 shows the calculated XMCD
spectra in the LSDA, LSDA + U (OP) and LSDA + U ap-
proximations for UPd2Al3 [140] together with the corre-
sponding experimental data [69]. The overall shapes of
the calculated and experimental uranium M 4 5, XMCD
spectra correspond well to each other. The major discrep-
ancy between the calculated and experimental XMCD
spectra is the size of the M 4 XMCD peak. The LSDA the-
ory produces much smaller intensity for the XMCD spec-
trum at the M 4 edge in comparison with experiment and
simultaneously strongly overestimates the negative peak
at the M 5 edge. On the other hand, the LSDA +U (OP) ap-
proximation produces an excellent agreement in the shape
and intensity of the XMCD spectra both at the M 4 and
M 5 edges. The LSDA + U calculations with U = 2.0 eV
slightly overestimate the intensity of the dichroic signal at
the M 4 edge and produce a larger negative peak and
smaller positive one at the M 5 edge.
Figure 29 shows also the XMCD spectra for UNi2Al3
[140]. The experimental data exist only for the M 4 edge
in this compound [68]. For the LSDA calculations the the-
ory produces a smaller intensity of the XMCD spectrum
at the M 4 edge in comparison with the experiment. On the
other hand, the intensity of the experimentally measured
M 4 XMCD spectrum is in between the results obtained by
LSDA + U (OP) and LSDA + U approximations.
2.3.4. UBe13
The system UBe13 was the first U-based heavy-fer-
mion superconductor discovered [173] and, similar to
UPt3, it shows peculiar properties, pointing to an uncon-
ventional superconducting order parameter. UBe13 is cer-
134 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
UPd2Al3
LSDA 5f5/2
mj = 5/2–
mj = 3/2–
mj = 1/2–
other
0
1
2
3
4
LSDA+U(OP)
0
1
2
3
LSDA+U
–1 0 1 2 3
Energy, eV
0
1
2
3
Fig. 28. The partial 5 f 5 2/ density of states in UPd2Al3 [140].
Table 5. The experimental and calculated spin Ms, orbital Ml, and
total Mt magnetic moments at uranium site (in �B) of UPd2Al3 and
UNi2Al3. The magnetic moments calculated for easy magnetic axes,
namely, hexagonal plane in UPd2Al3 and c axis in UNi2Al3 [140]
Compound Method Ms Ml Mt –Ml/Ms
LSDA –1.38 2.22 0.84 1.61
LSDA [167] –1.62 2.49 0.87 1.54
LSDA +U(OP) –1.59 3.73 2.14 2.34
UPd2Al3 LSDA + U –1.92 4.61 2.69 2.40
exper. [165] — — 0.85 —
exper. [68] — — — 2.01
exper. [69] — — — 1.91
LSDA –0.47 0.54 0.07 1.15
LSDA +U(OP) –1.22 2.90 1.68 2.38
UNi2Al3 LSDA +U –1.74 4.46 2.72 2.56
exper. [165] — — 0.2 —
exper. [68] — — — 2.49
tainly the most anomalous of the heavy-fermion super-
conductors.
The specific heat in UBe13 is very weakly dependent
upon magnetic field and highly sensitive to pressure
[174]. The low-temperature value of the electronic spe-
cific heat coefficient, � is of order 1000 mJ/(mol�K2),
corresponding to an effective mass of several hundred
free-electron masses. The magnetic susceptibility is weakly
pressure dependent in comparison with the specific heat and
under pressure has a completely different temperature de-
pendence [175]. Doping on the U sublattice which drives
away the specific heat anomaly leaves the low-temperature
susceptibility essentially unchanged. The magnetization is
linear in fields up to 20 T [174].
The dynamic magnetic susceptibility reveals no signif-
icant structure on the scale of 1 meV as is evidenced in
C/T and instead shows a broad «quasielastic» response on
the scale of 15 meV as evidenced in both neutron scatter-
ing and Raman spectra. Concomitant with the peak in � ''
is a Schottky anomaly in the specific heat, suggesting that
the 15 meV peak represents highly damped crystal-field
levels for which further evidence appears in the nuclear
magnetic relaxation of the 9Be sites. This dynamic sus-
ceptibility peak integrates to give 80% of the static sus-
ceptibility up to the experimental cut-off. This places a
stringent bound on any hypothetical moment-carrying
state in the low-frequency region; given a 10 K Kondo
scale, to explain the residual susceptibility the effective
squared moment must be less than 0.25 � B , which would
appear to rule out an interpretation in terms of a 5 3f �6
ground state [174].
There are several different interpretations of these ex-
perimental data in literature. Miranda and coworkers sug-
gested the non-Fermi-liquid (NFL) behavior of UBe13
could be driven by disorder [176]. Cox proposed, based
on symmetry grounds, the NFL behavior can be explained
by the two-channel Kondo model description [177]. More
recently, Anders et al. tackled the problem for the corre-
sponding lattice model [178]. They also performed a cal-
culation of the optical properties within such a two-chan-
nel Anderson lattice model for which the suppression of
the low-frequency Drude component and the develop-
ment of a mid-infrared absorption in the excitation spec-
trum at low temperatures have been suggested [178].
One framework for describing the low-temperature
properties of UBe13 characterizes the material’s behavior
in terms of its energy scales. Whereas common metals
may be characterized by a single energy scale (the Fermi
energy), UBe13 appears to require several. One may con-
sidered four energy scales [174]: a crystal field splitting
of 150–189 K, a Kondo temperature of about 25 K, a
spin-fluctuation temperature of about 2 K, and the super-
conducting transition temperature of about 0.8 K.
The energy band structure and Fermi surface of UBe13
have been investigated in Refs. 179–182 in a frame of the
LSDA approximation. It was shown [182] that the hybrid-
ization between the U 5 f states and the Be 2 p states oc-
curs in the vicinity of the Fermi level. The sheets of the
Fermi surface are all small in size and closed in topology.
The cyclotron effective mass calculated for the dHvA
branches in the three symmetry directions varies from
1.08 m0 to 4.18 m0. The theoretical electronic spe-
cific-heat coefficient � band
LDA is 13.0 mJ/(K 2�mol) [182].
The theoretical results for the electronic specific-heat co-
efficient are much less than the experimental ones, sug-
gesting a large enhancement due to many-body effects.
This disagreement between theory and experiment might
be ascribed to the enhancements due to the electron corre-
lations and/or the electron-phonon interaction which the
LDA fails to take into account.
1. Band structure. UBe13 crystallizes in the NaZn13-type
fcc structure with the space group Oh
6–Fm c3 (No 226) and
contains 28 atoms per unit cell. There are two distinct Be
sites, Be1 and Be2, with the 24 Be2 sites having a very low
site symmetry (only a mirror plane). The U atoms are sur-
rounded by cages of 24 Be2 atoms (Fig. 30) at the distance
of 3.02 �. Eight Be1 atoms are separated from the U atom
by 4.443 �. This ensures that the U atoms widely sepa-
rated. The U atoms form a simple cubic sublattice with a
large U–U nearest-neighbor distance of a/2 = 5.13 �,
which guarantees that the f–f overlap is negligible. There-
fore, all broadening of the U 5 f states into bands results
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 135
U
M
4
,5
X
M
C
D
,
ar
b
.
u
n
it
s
M5
x 5 LSDA
LSDA+U(OP)
LSDA+U
exper.
–2
0
2
x 5
0 20 40 60 80 100
Energy, eV
–2
–1
0
UPd Al2 3
UNi2Al3
M4
Fig. 29. The XMCD spectra of UPd2Al3 and UNi2Al3 at the
uranium M4 5, edges calculated in LSDA and LSDA + U ap-
proximations [140]. Experimental spectra for UPd2Al3 [69]
were measured in a magnetic field of 5 T and 35 K. The expe-
rimental data for the U M 4 XMCD spectrum of UNi2Al3 is
from Ref. 68 (the U M 4 spectra are shifted by �95 eV to in-
clude them in the figure).
entirely from hybridization with the conduction bands,
rather than partially from direct f–f overlap, as occurs in
many U compounds.
Self-consistent LSDA calculations produce a nonmag-
netic ground state in UBe13 [140]. To calculate the elec-
tronic structure and XMCD spectra of UBe13 in the LSDA
approximation, the term 2 � B B s� which couples the spin
of an electron to the external magnetic field was added to
the Hamiltonian at the variational step. The fully relativ-
istic spin-polarized LSDA energy band structure and total
DOS of UBe 13 is shown in Fig. 31 calculated in an exter-
nal magnetic field of 20 T [140]. The occupied part of the
valence band is formed predominantly by Be 2s and 2 p
states. U 5 f 5 2/ states are situated just at the Fermi level
1.0 eV above the top of Be 2 p states. U 5 f 7 2/ states are
split off by strong SO coupling and form another narrow
peak 1 eV above EF . Be 2s states are located mostly at the
bottom of the valence band. Be 2 p states are strongly hy-
bridized with U 6d states in the �6 to �1 eV energy inter-
val. On the other hand, there is quite large U 5 f –Be 2 p
hybrization in vicinity of the Fermi level in the �0.6 to 1.4
eV energy range. Although every individual Be atom pro-
duces a quite small 2 p partial density of states, due to the
large number of Be atoms they sum up to a 2 p DOS com-
parable in intensity with the U 5 f DOS (Fig. 31).
Figure 31 also shows the band structure of UBe13 cal-
culated in the LSDA + U approximation with U = 2.0 eV
and J = 0.5 eV. Partially occupied U 5 f 5 2/ states split due
to the Coulomb repulsion and the LSDA + U calculations
give a solution with three localized 5 f electrons. These
localized 5 f states form a rather narrow peak at 0.6 eV
below EF . U 5 f states just above the Fermi level are
formed by the remaining 5 f 5 2/ states whereas the peak of
5 7 2f / states is pushed from its LSDA position at 1.2 eV
above EF to 2.2 eV.
Figure 32 shows m j projected 5 f 5 2/ and total 5 f 7 2/ den-
sity of states in UBe13 calculated in the LSDA and LSDA + U
approximations [140]. We performed two LSDA + U band
structure calculations both withU = 2.0 eV and J = 0.5 eV.
In the first calculation we used the LSDA + U method
with nonspherical corrections to the Coulomb matrix ele-
ments [73]. The effect of a less asymmetric density of lo-
calized 5 f electrons can be simulated by replacing the
matrix elements U mmm m' ' and J mm m m' ' by averaged Cou-
lomb U and exchange J integrals, respectively, and set-
ting all other matrix elements to zero [73]. In the
nonrelativistic limit this would correspond, except for the
approximation to the double counting term, to the original
version of the LSDA +U method proposed in Ref. 183. In
this case all unoccupied U 5 f electrons independently of
their angular momentum experience the same Coulomb
repulsion as the localized ones. In the Hartree–Fock-like
LSDA + U solution with nonspherical corrections to the
Coulomb matrix elements three particular 5 f 5 2/ states
(m /j �5 2, �3 2/ , and �1 2/ ) are occupied which leads to
(i) large spin (–1.95 � B ) and orbital (4.47 � B ) magnetic
moments of U atom and (ii) strongly anisotropic Coulomb
interaction of the remaining 5 f electrons with the occu-
pied ones. In the calculations using the LSDA +U method
with spherically averaged U and J an unoccupied U 5 f
electrons feel a much more isotropic repulsive potential
and is situated closer to the Fermi energy. This gives
smaller magnetic moments (spin moment is equal to �1.82
� B and orbital moment 4.08 � B ) in comparison with the
nonspherial solution. The 5 f 5 2/ states with m j �1 2/ be-
came partly empty for the calculations with spherically
averaged U and J and the main peak of N �1 2/ DOS is situ-
ated just above the Fermi level (Fig. 32).
136 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
U
Be 1
Be 2
Fig. 30. Crystal structure of UBe13.
UBe13 LSDA
–10
–5
0
5
E
n
er
g
y,
eV
X W L K X 0 10 20
LSDA+U
–10
–5
0
5
E
n
er
g
y,
eV
X W L K X 0 10 20 30
DOS
30
Fig. 31. The energy band structure and total density of states
(in states/(unite cell�eV) in UBe13 calculated in the LSDA and
LSDA + U approximations [140].
The three calculations presented in Fig. 32 produce ra-
ther different energy locations for the empty 5 f states [140].
The principal question of the energy position of the empty
5 f states is usually answered by bremsstrahlung isochromat
spectroscopy (BIS) measurements. Figure 33 shows the ex-
perimental BIS spectrum of UBe13 [184] compared with the
calculated energy distribution for the unoccupied partial U
5 f density of states in the LSDA and LSDA + U approxima-
tions. The LSDA places empty 5 f states too close to the
Fermi level (Fig. 33). The LSDA + U calculations with
nonspherical solution place the maximum of empty 5 f
states more than 1 eV higher than the experiment. The
LSDA +U calculations with spherically averagedU and J
give the correct position of empty 5 f states within the ex-
perimental resolution (Fig. 33). The main peak in the BIS
spectrum is derived from the U 5 f 7 2/ states, while the low
energy shoulder split off from the main peak is from the
5 5 2f / states.
2. XMCD spectra. Figure 34 shows the UBe13 x-ray
isotropic absorption and XMCD spectra calculated in the
LSDA and LSDA +U approximations [140] together with
the experimental data [65]. The LSDA calculations pro-
duce much smaller intensity of the XMCD spectrum at the
M 4 edge in comparison with the experiment and simulta-
neously give larger dichroic signal for the negative peak
and do not produce the positive shoulder at the M 5 edge
(Fig. 34). On the other hand, the LSDA + U calculations
improve the agreement between the theory and the experi-
ment in the shape and intensity of XMCD spectra both at
the M 4 and M 5 edges. The LSDA + U method with
nonspherical corrections to the Coulomb matrix elements
slightly overestimates the dichroic signal at the M 4 edge,
underestimates the intensity of the positive peak and
strongly overestimates the negative peak at the M 5 edge.
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 137
UBe13
LSDA
0
2
4
6
8
10
LSDA+U(spher.)
0
2
4
6
8
LSDA+U
–1 0 1 2 3 4 5
0
2
4
6
5f5/2
5f7/2
mj = 5/2–
mj = 3/2–
mj = 1/2–
other
Energy, eV
Fig. 32. The mj projected 5 f 5 2/ and total 5 f 7 2/ density of
states in UBe13 calculated in the LSDA and LSDA + U appro-
ximations [140].
In
te
n
si
ty
,
ar
b
.
u
n
it
s
BIS LSDA
LSDA+U
LSDA+U(spher.)
exper.
–2 0 2 4 6
Energy, eV
Fig. 33. Comparison of the calculated U partial 5 f DOS in the
LSDA (dotted line), LSDA + U approximations with the exper-
imental BIS spectrum (circles) of UBe13 [184]. Dashed line
presents DOS calculated with nonspherical correction to Cou-
lomb matrix elements whereas full line are calculated with
averaged U and J [140].
UBe13
M5
M4
0
5
10
LSDA
LSDA+U
LSDA+U (spher.)
exper.
0 20 40 60 80 100
Energy, eV
–4
–2
0
2
U
M
4
,5
X
M
C
D
,
ar
b
.
u
n
it
s
Fig. 34. Isotropic absorption and XMCD spectra of UBe13 at
the uranium M4 5, edges calculated in the LSDA (dotted lines),
and LSDA + U approximations. The dashed line presents
XMCD spectra calculated with nonspherical corrections to Cou-
lomb matrix elements whereas the full line results are calculated
with averaged U and J [140]. Experimental spectra [65] (cir-
cles) were measured at 12 K and in a magnetic field of 5 T (the
U M 4 spectrum is shifted by –95 eV to include it in the figure).
The LSDA + U calculations with averaged U and J give a
correct value of the positive peak at the M 5 edge and the
negative peak at the M 4 one but still overestimate the in-
tensity of the negative peak at the M 5 edge.
UBe13 is unlike the other heavy-fermion compounds
in that the better description of its XMCD and BIS spectra
requires spherically averaged U and J values. The physi-
cal reason for that is not clear, however there are some in-
dications from the calculations. Compare the orbital re-
solved 5 f 5 2/ DOS’s shown in Fig. 32 one can see that in
the LSDA + U solution with nonspherical corrections to
the Coulomb matrix elements, three particular 5 f 5 2/
states (m /j �5 2, �3 2/ , and �1 2/ ) are fully occupied
which leads to a pure 5 f 3 configuration. The calculations
using the spherically averaged U and J values give a solu-
tion with partly empty m /j �1 2 states with the main
peak of the N �1 2/ DOS very close to the Fermi level
(Fig. 32). This is the typical situation for a system with
mixed valence [38,185]. One should mention that the
LSDA + U method which combines LSDA with a basi-
cally static, i.e., Hartree–Fock-like, mean-field approxi-
mation for a multi-band Anderson lattice model does not
contain true many-body physics and cannot treat a sys-
tems with mixed valence properly. The evaluation of the
electronic structure of UBe13 needs further theoretical
investigations.
2.4. UGe2
The coexistence of ferromagnetism (FM) and super-
conductivity (SC) has been at the forefront of condensed
matter research since a pioneering paper by Ginzburg
[186]. 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 in-
ternal 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 understanding of how magnetic order and super-
conductivity can coexist. It seems to be generally ac-
cepted that antiferromagnetism with local moments com-
ing from rare-earth elements readily coexists with type-II
superconductivity [187]. This is because superconductiv-
ity and magnetism are carried by different types of
electrons; magnetism is connected with deeply seated 4 f
electrons, while superconductivity is fundamentally related
to the outermost electrons such as s, p, and d electrons. In
the case of a ferromagnetic superconductor the situation is
more complex because internal fields are not canceled out in
the range of a superconducting coherence length in contrast
with an antiferromagnetic superconductor.
Recently, UGe2 has attracted considerable attention
because the coexistence of SC and FM was found under
high pressure [188,189]. It is particularly interesting to
note that both of ferromagnetism and superconductivity
may be carried by itinerant 5 f electrons, which can be ho-
mogeneously spread in the real space, although it is still a
matter of debate and remains to be resolved.
UGe2 crystallizes in the orthorhombic ZrGa2 structure
(space group Cmmm). At ambient pressure, UGe2 orders
ferromagnetically below the Curie temperature TC = 52 K
with the ordered moment of 1.4 � B . The magnetic proper-
ties are strongly anisotropic, and the easy magnetization
axis is the crystallographic a axis of the ZrGa2 structure.
Superconductivity is found in the pressure range of 1.0 to
1.6 GPa. The highest superconducting critical tempera-
ture TSC = 0.8 K at the pressure PC = 1.2 GPa, while TC =
= 35 K at that pressure. As the applied pressure increases,
the superconductivity disappears where the ferromagne-
tism disappears at around 1.7 GPa. Therefore, the super-
conductivity and ferromagnetism in UGe2 seem to be
closely related, although the mechanism of superconductiv-
ity has not been understood yet, and it is very important to
characterize the magnetic properties of UGe2. The XMCD
technique developed in recent years has evolved into a pow-
erful magnetometry tool to separate orbital and spin contri-
butions to element specific magnetic moments. XMCD ex-
periments measure the absorption of x-rays with opposite
(left and right) states of circular polarization.
In a recent publication [190] we reported on the x-ray
absorption and magnetic circular dichroism measure-
ments performed at the M 4 5, edges of uranium in the fer-
romagnetic superconductor UGe2. The spectra are well
described with the LSDA +U electronic structure compu-
tation method. Combined with the analysis of the pub-
lished (i) x-ray photoemission spectrum, (ii) two-dimen-
sional electron positron momentum density, and (iii)
angular dependence of the de Haas–van Alphen frequen-
cies, we infer for the Coulomb repulsion energy within
the 5 f electron shell U = 2 eV.
The present work is an extension of the previous study.
Recently, Okane et al. [191] measured x-ray absorption
magnetic circular dichroism at the U N 4 5, and N 2 3, edges
as well as at the Ge L2 3, ones for the ferromagnetic super-
conductor UGe2 in the normal state. The orbital and spin
magnetic moments deduced from the sum rule analysis of
the XMCD data indicate that the U atom in UGe2 is con-
sidered to be closer to the trivalent state rather than the
tetravalent state. The XMCD measurement at the U N 2 3,
indicates that the U 6d electrons have negligibly small
magnetic contributions.
Inada et al. [192] also performed XMCD experiments
at the Ge K edge in UGe2. The Ge K edge XMCD spec-
trum shows a main negative peak near the edge and a
small positive one at 7 eV above the edge. The amplitude
of this spectrum is unusually very large in spite of being at
ligand sites.
138 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
1. U N 4 5, XMCD spectra. Figure 35 shows the calcu-
lated XAS and XMCD spectra in the LSDA and LSDA +U
approximations for UGe2 at the N 4 5, edges together with
the corresponding experimental data [191]. The experi-
mentally measured XAS spectra have a rather simple line
shape composed of two white line peaks at the N 5 and N 4
edges and no distinct fine structures due to multiplet split-
ting were observed. This justifies the description of the ab-
sorption of the incident x-rays in terms of a one-particle ap-
proximation. Hence, valuable information on the nature of
the 5 f electrons can be obtained from comparison of experi-
mental data to results of band-structure calculations.
The XMCD signals at the N 5 and N 4 edges have the
same sign, and the XMCD signals at the N 4 edge have a
much higher intensity than those at the N 5 edge. These
behaviors were commonly observed in the XMCD spectra
at the U M 4 5, edges of the ferromagnetic uranium com-
pounds [43] from which one can conclude that the orbital
and the spin magnetic moments are directed in the oppo-
site direction to each other.
A qualitative explanation of the XMCD spectra shape
is provided by the analysis of the orbital character, occu-
pation numbers of individual 5 f orbitals and correspond-
ing selection rules. Because of the electric dipole selec-
tion rules (�l
1; �j
0 1, ) the major contribution to
the absorption at the N 4 edge stems from the transitions
4 43 2 5 2d f/ /� and that at the N 5 edge originates primar-
ily from 4 55 2 7 2d f/ /� transitions, with a weaker contri-
bution from 4 55 2 5 2d f/ /� transitions. The selection
rules for the magnetic quantum number m j (m j is re-
stricted to � �j j, ... ) are �m j = +1 for � � and
�m j �1for � �.
In our previous paper [72] we show that qualitatively
the XMCD spectrum of U at the M 5 edge (I � �� �— )
can be roughly represented by the following m j projected
partial density of states: [N �7 2
7 2
/
/ + N N� �5 2
7 2
7 2
7 2
/
/
/
/] [ +
+ N 5 2
7 2
/
/ ]. Here we used the notation N
m
j
j
with the total mo-
mentum j and its projection m j . As a result, the shape of
M 5 XMCD spectrum usually results in two peaks of op-
posite sign: a negative peak at lower energy and a positive
peak at higher energy. Relative intensity of the negative
and positive lobes depends on the value of crystal field
and Zeeman splitting of the 5 f 7 2/ electronic states [116].
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. Similar con-
sideration is valid also for the N 5 edge.
It can be shown (see [72]) that the XMCD spectrum of
U at the M 4 and N 4 edges can be fairly well represented
by considering m j projected partial density of states:
� �[ /
/N 3 2
5 2 N 5 2
5 2
/
/ ]. It explains why the dichroic M 4 as well
as N 4 lines in uranium compounds consist of a single
nearly symmetric negative peak.
We should note, however, that the explanation of the
XMCD line shape in the terms of partial DOS’s presented
above should be considered only qualitatively. First, there
is no full compensation between transitions with equal fi-
nal states due to difference in the angular matrix ele-
ments; second, in our consideration we neglect cross
terms in the transition matrix elements. Besides, we have
used here the jj-coupling scheme where the total momen-
tum j is written as j l s � . However, the combination of
the hybridization, Coulomb, exchange and crystal-field
energies may be so large relative to the 5 f spin-orbit en-
ergy that the jj-coupling is no longer an adequate
approximation.
Figure 35,b shows the calculated XMCD spectra in the
LSDA and LSDA + U approximations for UGe2 together
with the corresponding experimental data [191]. The
overall shapes of the calculated and experimental ura-
nium N 4 5, XMCD spectra correspond well to each other.
The major discrepancy between the calculated and expe-
rimental XMCD spectra is the size of the N 4 XMCD
peak. The LSDA theory produces much smaller intensity
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 139
X
A
S
,
ar
b
.
u
n
it
s
X
M
C
D
,
ar
b
.
u
n
it
s
a
N5
N4
0
4
8
b
LSDA
LSDA+U
exper.
720 740 760 780 800
Energy, eV
–0.4
0
Fig. 35. (a) Theoretically calculated [193] (dashed line) and ex-
perimental [191] (circles) isotropic absorption spectra of UGe2 at
the U N4 5, edges. Experimental spectra were measured with ex-
ternal magnetic field (2 T) at 25 K. Dotted lines show the theo-
retically calculated background spectra, full thick lines are sum
of the theoretical XAS and background spectra. (b) Experimen-
tal [191] (circles) XMCD spectra of UGe2 at the U N4 5, edges
in comparison with theoretically calculated ones using the
LSDA (dotted lines) and LSDA + U (full lines) approximations.
for the XMCD spectrum at the N 4 edge in comparison
with the experiment. It also can’t produce the correct
shape of the N 5 XMCD spectrum. On the other hand, the
LSDA + U approximation with U = 2 eV produces excel-
lent agreement in the shape and intensity of XMCD spec-
tra at the N 4 5, edges.
Now we focus on values of moments of the 5 f shell.
The orbital magnetic moment can be estimated from the
XMCD sum rules [74,194]. By integrating the experimen-
tally measured XAS and XMCD spectra at the M 4 5, edges
we obtained � L = 1.91 and 1.75 � B for the hypothetical
f 2 and f 3 configurations, respectively [190]. A similar
procedure has been used by Okane et al. at the N 4 5, edges
[191], they obtained � L = 1.89 and 2.35 � B for the f 2 and
f 3 configurations, respectively. Although the values for
the f 2 configuration are very close, the values for the f 3
configuration differ more than 30 %. One of the possible
reasons for such disagreement might be connected with
the fact that the application of the sum rule is valid only
when the spin orbit splitting of the core level is suffi-
ciently large compared with other interactions including
the core-valence Coulomb and exchange interaction. The
condition may not be so clear at the U N 4 5, edges because
the spin-orbit splitting is considerably smaller than that at
the U M 4 5, edges [191]. One should mention also that
XMCD sum rules are derived within an ionic model using
a number of approximations [43,195]. The largest mis-
take comes from the ignorance of the energy dependence
of the radial matrix elements in sum rules, sometimes it
can produce an error up to 100 % [196].
From our LSDA + U band structure calculations with
U 2 eV we obtain a larger 5 f orbital magnetic moment:
M l = 3.46 � B , which may indicate that the LSDA + U is
producing too much localization for the 5 f orbitals [73].
The analysis of the orbital projected DOS provided in
our previous paper shows that for U 2 eV the two most
populated 5 f orbitals become almost completely occu-
pied and corresponding peaks of orbital resolved DOS are
found below the Fermi energy, EF (see Fig. 3 in [190]).
The third most occupied orbital remains only partially oc-
cupied. Whereas the main peak of DOS projected onto
this orbital is situated below EF , an additional narrow
peak can be seen just above the Fermi level. Even for
U 4 eV the third peak remains partially occupied. We
can conclude that the U atom in UGe2 possesses a valency
somewhat in between U 4 � ( f 2) and U 3 � ( f 3).
One should mention that the ratio R /L S �� � of the
orbital to spin moment is not in disagreement with the ex-
periment: our LSDA + U calculations produce R 2.25,
while the experimental estimations give 2.24 and 2.51 for
f 3 configurations by integrating the spectra at the M 4 5,
and N 4 5, edges, respectively [190,191].
2. U N 2 3, and Ge L2 3, XMCD spectra. In order to in-
vestigate the contribution of the U 6d electrons to the
magnetization, Okane et al. [191] have measured XMCD
at the U N 2 3, edges, too. Figure 36 shows the calculated
XAS and XMCD spectra in the LSDA + U approxima-
tions for UGe2 at the N 3 edge together with the corre-
sponding experimental data [191]. The experimentally
measured XAS spectrum has quite large background in-
tensity. One can see that no appreciable XMCD signals
are observed at the U N 3 edge.
The theoretical LSDA + U calculations also produce a
XMCD spectrum of very small intensity Fig. 36,b. It
might be connected with quite a small U 6d spin and or-
bital magnetic moments equal to 0.075 and �0.041 � B ,
respectively.
Okane et al. also measured XAS and XMCD spectra in
the region of the Ge L2 3, absorption edges [191]. The
spectra have quite complicated line shapes and it is hard
to separate the Ge L2 3, signal from U N 2 and Gd M 4
XMCD signals. The later arises from the sample holder.
Figure 37 presents the calculated XMCD spectra of the
UGe2 at the Ge L2 3, edges compared with the experimen-
tal data [191]. The authors of [191] consider a positive
peak B at 1215 eV, a negative peak C at 1228 and another
140 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
X
A
S
,
ar
b
.u
n
it
s
X
M
C
D
,
ar
b
.
u
n
it
s
a
N3
0
0.5
1.0
b
theory
exper.
1020 1040 1060 1080
Energy, eV
–0.02
0
0.02
Fig. 36. (a) Theoretically calculated [193] (dashed line) and ex-
perimental [191] (circles) isotropic absorption spectra of UGe2
at the U N3 edge. Dotted lines show the theoretically calculated
background spectrum, full thick line is sum of the theoretical
XAS and background spectrum. (b) Experimental [191] (cir-
cles) XMCD spectrum of UGe2 at the U N3 edge in compari-
son with theoretically calculated ones using the LSDA + U ap-
proximations (full line).
negative peak D at 1255 eV as the XMCD spectra of the
Ge L2 3, edges, since the energy separation between those
structures is close to the spin orbit splitting of the Ge L2 3,
core level 30 eV. A strong negative peak at around 1183 eV
(peak A) apparently comes from the Gd M 5 spectrum of
the sample holder. A positive XMCD peak at 1215 eV may
include probably not only the Ge L3 contribution but also a
contribution from Gd M 4 , and the broad hump at around
1270 eV may arise from U N 2 contributions [191].
Our band structure calculations perfectly describe the
peaks B and C as the L3 XMCD spectrum, while the L2
XMCD spectrum well reproduces the fine structure D.
Due to larger U 4 1 2p / electron energy binding in compari-
son with the Ge 2 1 2p / one, U N 2 XMCD spectrum is situ-
ated at the higher energy side of the Ge L2 spectrum
(peak E). The values of Gd 5d orbital (spin) magnetic mo-
ments are equal to 0.018 (0.019), 0.022 (0.013) and 0.010
(0.011) � B at the Ge1, Ge2, and Ge3 sites, respectively.
The main contribution to the intensity of XMCD L2,3
spectra come from Ge1 and Ge2 sites because they have
larger magnitude for their spin and orbital polarizations
(Fig. 37,a).
Through turning the SOI off separately on the Ge 4d
and the U 5 f states we found that the negative peak C
originates from the spin polarization in the Ge 4d sym-
metric states through the SOI while the Ge 4d and U 5 f
hybridization is responsible for large positive XMCD at
around 1215 eV (peak B).
One should mention that XMCD spectra at the U N 2 3,
and Ge L2 3, edges are mostly determined by the strength
of the SO coupling of the initial U 4 p and Ge 2 p core
states and spin-polarization of the final empty d 3 2 5 2/ , /
states while the exchange splitting of the U 4 p and Ge 2 p
core states as well as the SO coupling of the d valence
states are of minor importance for the XMCD at the U
N 2 3, and Ge L2 3, edges of UGe2.
3. Ge K XMCD spectrum. The 4 p states in transition
metals usually attract only minor interest because they are
not the states responsible for magnetic or orbital orders.
Recently, however, understanding 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
states at neighboring sites, because of delocalized nature of
the 4 p states. It is expected that the ligand site XMCD is a
candidate for one of the effective probes which can detect
the mixing between p and f states in uranium compounds.
Figure 38,b shows the calculated XMCD spectra in the
LSDA + U approximations for UGe2 at the K edge to-
gether with the corresponding experimental data [192].
The experimental XMCD spectrum shows a main nega-
tive peak near 11100 eV and a small positive peak at about
7 eV higher. One might expect only tiny signals of XMCD
from the 4 p band, because it does not possess a large mag-
netic moment. However, the intensity of the negative
peak of UGe2 K XMCD spectrum reaches about 3% of the
intensity of the fluorescence (or absorption) from K edge
[192]. This value is large. Even the iron K edge XMCD is
only on the order of 0.3% [197].
The K XMCD spectra come from the orbital polariza-
tion in the empty p states, which may be induced by (i) the
spin polarization in the p states through the spin-orbit in-
teraction (SOI), and (ii) the orbital polarization at neigh-
boring sites through hybridization.
We calculated the XMCD spectra at Ge site with turn-
ing the SOI off separately on the Ge 4 p and the U 5 f
states, respectively. We found that the prominent negative
peak is reduced in intensity more than one order of magni-
tude when the SOI on the U 5 f states is turned off, while
the small positive lobe almost does not change. When the
SOI on the Ge 4 p orbital is turned off the negative promi-
nent peak is slightly changed and the positive lobe is di-
minished. We can conclude that the positive lobe origi-
nates from the spin polarization in the Ge 4 p symmetric
states through the SOI. The Ge 4 p and U 5 f hybridization
is responsible for large negative XMCD near the Ge K
edge. This indicates that the Ge 4 p orbital polarization
originates mainly from the large 5 f orbital polarizations
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 141
X
M
C
D
,
ar
b
.
u
n
it
s
G
e
L
2
,3
a
Ge L3
Ge L2
U N2
Ge1
Ge2
Ge3
–0.001
0
0.001
0.002
b
A
B
C
D
E
theory
exper.
1200 1250
Energy, eV
–0.008
–0.004
0
0.004
Fig. 37. (a) Theoretically calculated [193] XMCD spectra of
UGe2 at the U N2 and Ge L2 3, edges at different Ge sites;
(b) experimental [191] (circles) XMCD spectra of UGe2 at the
Ge L2 3, and U N2 edges in comparison with theoretically cal-
culated ones using the LSDA + U approximations (full line).
at neighboring U atoms through Ge 4 p–U 5 f hybridiza-
tion. This mechanism seems different from the XMCD in
transition metal compounds in which the 4 p orbital polar-
ization is induced mostly by the 4 p spin polarization at
the atom itself through the SOI [43].
Similar results have been obtained by Usuda et al.
[198] for the magnetic resonant x-ray scattering (MRXS)
spectra at Ga sites in the antiferromagnetic cubic phase of
UGa 3: the MRXS intensity largely decreased when the
SOI on the U 5 f states is turned off, while it was only
slightly reduced when the SOI on the Ga 4 p orbital is
turned off.
From our LSDA + U band structure calculations the
value of the orbital magnetic moment in the p projected
bands are equal to �0.025, �0.031, and �0.006 � B at Ge1,
Ge2, and Ge3 sites, respectively. The contributions to the
intensity of XMCD K spectrum from different Ge sites are
related to the magnitude of their orbital polarizations
(Fig. 38,a).
3. Summary
Recent progress in first-principles calculations of the
x-ray magnetic dichroism illustrates that the XMCD spectra
are developing into a powerful tool for tracing the electronic
and magnetic structure of solids. The density-functional the-
ory in the local-density approximation gives a fully satisfac-
tory explanation of the XMCD spectra of transition metal
compounds and alloys in most cases. Morover, theory can
help to understand the nature of XMCD spectra and gives
some recommendations how to create compounds with ap-
propriate magnetic properties.
We demonstrated that XMCD K spectrum reflects the
orbital polarization in differential form of the p states.
Due to small exchange splitting of the initial1s core states
only the exchange and spin-orbit splitting of the final 4 p
states is responsible for the observed dichroism at the K
edge. The XMCD spectra of transition metals for the L2 3,
edge are mostly determined by the strength of the SO cou-
pling of the initial 2 p core states and spin-polarization of
the final empty 3d 3 2 5 2/ , / states while the exchange split-
ting of the 2 p core states as well as SO coupling of the 3d
valence states are of minor importance.
The recently derived sum rules for the orbital and spin
magnetic moments were tested for several compounds.
XMCD sum rules are derived within an ionic model using
a number of approximations. For L2 3, , they are: (1) ignor-
ing the exchange splitting of the core levels; (2) replacing
the interaction operator � �a � 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 ra-
dial matrix elements. The last point is the most important.
We 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
magnetic moments derived from the sum rules.
In most of the 4 f systems, the f electrons are localized
and form a Hund’s rule ground state. The application of
plain LDA calculations to 4 f electron systems encounters
problems in most cases, because of the correlated nature
of electrons in the f shell. To better account for strong
on-site electron correlations the LSDA + U approach
should be used, in which a model Hamiltonian explicitly
including the on-site Coulomb interaction,U , for localized
states is combined with the standard band structure calcu-
lation Hamiltonian for extended states. The LSDA + U
method provides a rather good description of the elec-
tronic structure and the XMCD properties of some lan-
thanide compounds.
Actinide compounds occupy an intermediate position
between itinerant 3d and localized 4 f systems, and one of
the fundamental questions concerning the actinide mate-
rials is whether their f states are localized or itinerant.
This question is most frequently answered by comparison
142 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
G
e
K
X
M
C
D
,
ar
b
.
u
n
it
s
a
–0.02
0
b
theory
exper.
11090 11100 11110 111
Energy, eV
–0.08
–0.04
0
Ge1
Ge2
Ge3
Fig. 38. (a) Theoretically calculated [193] XMCD spectra of UGe2
at the K edge at different Ge sites; (b) theoretically calculated
XMCD spectrum of UGe2 at the Ge K edge using the LSDA + U
approximations (full line) in comparison with the experimental
one [192] (circles). Experimental spectrum was measured at 3 K
with external magnetic field (0.5 T) applied along a axis.
between experimental spectroscopies and the different
theoretical descriptions. X-ray absorption spectroscopy,
photoelectrons spectroscopy and bremsstrahlung isochro-
mat spectroscopy supply direct information about the en-
ergy states (both occupied and unoccupied) around the
Fermi energy, and can provide a means of discrimination
between the two theoretical limits. The dual character of
5 f electrons alongside with the presence of strong SO
coupling make the determination of the electronic struc-
ture of uranium compounds a challenging task because in
many of them the width of 5 f bands, their spin-orbit split-
ting, and the on-site Coulomb repulsion in the partially
filled 5 f shell are of the same order of magnitude and
should be taken into account on the same footing.
There are some features in common for all the uranium
compounds investigated up to now. First, the dichroism at
the M 4 edge is much larger, sometimes of one order of
magnitude, than at the M 5 one. Second, the dichroism at
the M 4 edge has a single negative peak that has no dis-
tinct structure, on the other hand, two peaks, a positive
and a negative one, are observed at the M 5 edge. The pe-
culiarities of the XMCD spectra can be understood quali-
tatively considering the partial density of states and the
electric dipole selection rules.
The overall shapes of the calculated and experimental
uranium M 4 5, XMCD spectra correspond well to each
other. The major discrepancy between the calculated and
experimental XMCD spectra is the size of the M 4 XMCD
peak. The LSDA theory produces usually much smaller
intensity for the XMCD spectrum at the M 4 edge in com-
parison with the experiment and simultaneously gives in-
appropriate dichroic signal strength at the M 5 edge. It
fails to produce a correct intensity of dichroic signal at the
M 4 edge even in UFe2 which is widely believed to have
itinerant 5 f electrons. As the integrated XMCD signal is
proportional to the orbital moment this discrepancy could
be related rather to an underestimation of the orbital mo-
ment by LSDA-based computational methods rather than
to a failure in the description of the energy band structure
of the itinerant 5 f systems. The LSDA +U approximation
gives much better agreement in the shape and intensity of
the XMCD spectra both at the M 4 and M 5 edges in ura-
nium compounds.
Concerning the best description of line shape and
intensity of the XMCD spectra, the investigated metallic
uranium compounds fall into two groups according to the
type of LSDA + U method used. The LSDA + U (OP)
approximation (U eff = 0) better describes the XMCD spec-
tra in UFe2, UXAl (X = Co, Rh, and Pt), UPd2Al3, and
UNi2Al3 compounds. But the XMCD spectra of UPt3,
URu2Si2, and UBe13 are better described by the LSDA + U
method with U = 2.0 eV and J = 0.5 eV. It might be con-
cluded to some extent that the last three compounds have
a larger degree of localization than the compounds from
the first group.
1. C. Giorgetti, S. Pizzini, E. Dartige, A. Fontaine, F.
Baudelet, C. Brouder, P. Bauer, G. Krill, S. Miraglia, D.
Fruchart, and J.P. Kappler, Phys. Rev. B48, 12732 (1993).
2. S. Suga and S. Imada, J. Electron. Spectrosc. Relat.
Phenom. 78, 231 (1996).
3. L.M. Garcia, S. Pizzini, J.P. Rueff, J. Galera, A. Fontaine,
J.P. Kappler, G. Krill, and J. Goedkoop, J. Appl. Phys. 79,
6497 (1996).
4. M. Finazzi, F.M.F. de Groot, A.M. Dias, B. Kierren, F.
Bertran, P. Sainctavit, J.P. Kappler, O. Schulte, W. Felsch,
and G. Krill, Phys. Rev. Lett. 75, 4654 (1995).
5. J.P. Schille, F. Bertran, M. Finazzi, C. Brouder, J.P. Kappler,
and G. Krill, Phys. Rev. B50, 2985 (1994).
6. J. Chaboy, H. Maruama, L.M. Garcia, J. Bartolome, K.
Kobayashi, N. Kawamura, A. Marcelli, and L. Bozukov,
Phys. Rev. B54, R15637 (1996).
7. J.P. Raeff, R.M. Galera, C. Giorgetti, E. Dartige, C. Brouder,
and M. Alouani, Phys. Rev. B58, 12271 (1998).
8. I. Yamamoto, S.N.T. Nakamura, T. Fujikawa, and S. Nanao,
J. Electron. Spectrosc. Relat. Phenom. 125, 89 (2002).
9. R.M. Galera and A. Rogalev, J. Appl. Phys. 85, 4889 (1999).
10. F. Bartolome, J.M. Tonnerre, L. Seve, D. Raoux, J.E. Lorenzo,
J. Chaboy, L.M. Garcia, J. Bartolome, M. Krisch, A. Rogalev,
R. Serimaa, G.C. Kao, G. Cibin, and A. Marcelli, J. Appl.
Phys. 83, 7091 (1998).
11. H. Wende, Z. Li, A. Scherz, G. Ceballos, K. Babeerschke,
A. Ankudinov, J.J. Rehr, F. Wilhelm, A. Rogalev, D.L.
Schlagel, and T.A. Lograsso, J. Appl. Phys. 91, 7361 (2002).
12. A.R.B. de Castro, G.B. Fraguas, P.T. Fonseca, R.N. Suave, S.
Gama, A.A. Coelho, and L.A. Santos, J. Electron. Spectrosc.
Relat. Phenom. 101–103, 725 (1999).
13. J. Chaboy, L.M. Garcia, F. Bartolome, H. Maruyama, S.
Uemura, N. Kawamura, and A.S. Markosayan, J. Appl. Phys.
88, 336 (2000).
14. W. Kossel, Z. Physik 1, 119 (1920).
15. C. Bonnelle, R.C. Karnatak, and J. Sugar, Phys. Rev. A9,
1920 (1967).
16. V.F. Demekhin, Fiz. Tverd. Tela (Leningrad) 16, 1020
(1974) [Sov. Phys.-Solid State 16, 659 (1974)].
17. J. Sugar, Phys. Rev. A6, 1764 (1972).
18. J. Sugar, Phys. Rev. B5, 1785 (1972).
19. C. Bonnelle, R.C. Karnatak, and N. Spector, J. Phys. B10,
795 (1977).
20. B.T. Thole, G. van der Laan, J.C. Fuggle, G.A. Sawatzky,
R.C. Karnatak, and J.-M. Esteva, Phys. Rev. B32, 5107
(1985).
21. J.B. Goedkoop, B.T. Thole, G. van der Laan, G.A. Sawatzky,
F.M.F. de Groot, and J.C. Fuggle, Phys. Rev. B37, 2086
(1988).
22. G. van der Laan and B.T. Thole, Phys. Rev. B42, 6670
(1990).
23. S. Imada and T. Jo, J. Phys. Soc. Jpn. 59, 3358 (1990).
24. Z. Hu, K. Starke, G. van der Laan, E. Navas, A. Bauer, E.
Weschke, C. Sch�ssler-Langeheime, E. Arenholz, A. M�hlig,
G. Kaindl, J.B. Goodkoop, and N.B. Brookes, Phys. Rev.
B59, 9737 (1999).
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 143
25. G. van der Laan, E. Arenholz, Z. Hu, E. Weschke, C.
Sch�ssler-Langeheime, E. Navas, A. M�hlig, G. Kaindl,
J.B. Goodkoop, and N.B. Brookes, Phys. Rev. B59, 8835
(1999).
26. K. Starke, E. Navas, E. Arenholz, Z. Hu, L. Baumgarten,
G. van der Laan, C.T. Chen, and G. Kaindl, Phys. Rev.
B55, 2672 (1997).
27. B.N. Harmon and V.N. Antonov, J. Appl. Phys. 93, 4678
(2003).
28. P. Wachter and E. Kaldis, Solid State Commun. 34, 241
(1980).
29. P. Wachter, in: Handbook of the Physics and Chemistry of
Rare Earths, K.A. Gschneidner, L. Eyring, and S. H�fner
(eds.), North-Holland, Amsterdam (1994), Vol. 19, p. 177.
30. D.X. Li, Y. Haga, H. Shida, T. Suzuki, Y.S. Kwon, and G.
Kido, J. Phys.: Condens. Matter 9, 10777 (1997).
31. A. Hasegawa and A. Yanase, J. Phys. Soc. Jpn. 42, 492
(1977).
32. A.G. Petukhov, W.R.L. Lambrecht, and B. Segall, Phys.
Rev. B53, 4324 (1996).
33. W.R.L. Lambrecht, Phys. Rev. B62, 13538 (2000).
34. S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton,
S. von Molnar, M.L. Roukes, A.Y. Chtchelkanova, and D.M.
Treger, Science 294, 1488 (2001).
35. F. Leuenberger, A. Parge, W. Felsch, K. Fauth, and M.
Hessler, Phys. Rev. B72, 014427 (2005).
36. V. Antonov, B. Harmon, A. Yaresko, and A. Shpak, Phys.
Rev. B76,184422 (2007).
37. V.N. Antonov, B.N. Harmon, and A.N. Yaresko, Phys.
Rev. B63, 205112 (2001).
38. V.N. Antonov, B.N. Harmon, and A.N. Yaresko, Phys.
Rev. B66, 165208 (2002).
39. W.E. Picket, A.J. Freeman, and D.D. Koelling, Phys. Rev.
B22, 2695 (1980).
40. J. Lang, Y. Baer, and P. Cox, J. Phys. F11, 121 (1981).
41. Y. Baer and W.D. Schneider, in: Handbook of the Physics
and Chemistry of Rare Earths, K.A. Gschneidner, L.
Eyring, and S. H�fner (eds.), North-Holland, Amsterdam
(1987), Vol. 10, p. 1.
42. F. Leuenberger, A. Parge, W. Felsch, F. Baudelet, C.
Giorgetti, E. Dartyge, and F. Wilhelm, Phys. Rev. B73,
214430 (2006).
43. V. Antonov, B. Harmon, and A. Yaresko, Electronic struc-
ture and magneto-optical properties of solids, Kluwer Aca-
demic Publishers, Dordrecht, Boston, London (2004).
44. A. Rogalev, J. Goulon, and C. Brouder, J. Phys. Condens.
Matter 11, 1115 (1999).
45. P.G. Steeneken, Ph.D. thesis, University of Groningen,
Groningen (2002).
46. G. Sch�tz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller,
R. Frahm, and G. Materlik, Phys. Rev. Lett. 58, 737 (1987).
47. C.M. Aerts, P. Strange, M. Horne, W.M. Temmerman, Z.
Szotek, and A. Svane, Phys. Rev. B69, 045115 (2004).
48. P. Santini, R. Lemanski, and P. Erdoes, Adv. Phys. 48, 537
(1999).
49. J.-M. Fournier and R. Troch, in: Handbook on the Physics
and Chemistry of the Actinides, A.J. Freeman and G.H.
Lander (eds.), North-Holland, Amsterdam (1985), Vol. 2,
p. 29.
50. H.R. Ott and Z. Fisk, in: Handbook on the Physics and
Chemistry of the Actinides, A.J. Freeman and G.H. Lander
(eds.), North-Holland, Amsterdam (1987), Vol. 5.
51. V. Sechovsky and L. Havela, in: Intermetallic Compounds
of Actinides. Ferromagnetic Materials, E.P. Wohlfarth and
K.H.J. Buschow (eds.), Elsvier Amsterdam (1998), Vol. 4.
52. T. Endstra, S.A.M. Mentink, G.J. Nieuwenhuys, and J.A.
Mydosh, in: Magnetic Properties of Uranium Based 1-2-2
Intermetallics. Selected Topics in Magnetism, L.C. Gupta
and M.S. Multani (eds.), World Scienti, Singapore (1992).
53. M.R. Norman and D.D. Koelling, in: Electronic Structure,
Fermi Surfaces and Superconductivity in f Electron Met-
als. Handbook on the Physics and Chemistry of Rare
Earths, G.H.L.K.A. Gschneidner Jr., L. Eyring, and G.R.
Choppin (eds.), Elsevier, Amsterdam (1993), Vol. 17.
54. J.M. Fournier and E. Gratz, in: Transport Properties of Rare
Earth and Actinide Intermetallics. Handbook on the Physics
and Chemistry of Rare Earths, G.H.L.K.A. Gschneidner Jr.,
L. Eyring, and G.R. Choppin (eds.), Elsevier, Amsterdam
(1993), Vol. 17.
55. W. Potzel, G.M. Kalvius, and J. Gal, in: Moessbauer Stud-
ies on Electronic Structure of Intermetallic Compounds.
Handbook on the Physics and Chemistry of Rare Earths,
G.H.L.K.A. Gschneidner Jr., L. Eyring and G.R. Choppin
(eds.), Elsevier, Amsterdam (1993), Vol. 17.
56. P. Fulde, J. Keller, and G. Zwicknagl, in: Solid State Phys-
ics, H. Ehrenreich and D. Turnbull (eds.), Academic Press,
San Diego, CA (1988), Vol. 17.
57. A. Amato, Rev. Mod. Phys. 69, 1119 (1997).
58. M.B. Maple, J. Alloys Comp. 303–304, 1 (2000).
59. R. Joynt and L. Taillefer, Rev. Mod. Phys. 74, 235 (2002).
60. M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).
61. S.P. Collins, D. Laundy, C.C. Tang, and G. van der Laan,
J. Phys.: Condens. Matter 7, 9325 (1995).
62. N. Kernavanois, P.D. de Reotier, A. Yaouanc, J.P.
Sanchez, V. Honkim�ki, T. Tschentscher, J. McCarthy, and
O. Vogt, J. Phys.: Condens. Matter 13, 9677 (2001).
63. P.D. de Reotier, J.P. Sanchez, A. Yaouanc, M. Finazzi, P.
Sainctavit, G. Krill, J.P. Kappler, J. Goedkoop, J. Goulon,
C. Goulon-Ginet, A. Rogalev, and O. Vogt, J. Phys.:
Condens. Matter 9, 3291 (1997).
64. A. Bombardi, N. Kernovanois, P.D. de Reotier, G.H. Lander,
J.P. Sanchez, A. Yaouanc, P. Burlet, E. Lelievre-Berna, A.
Rogalev, O. Vogt, and K. Mattenberger, Eur. Phys. J. B21,
547 (2001).
65. P.D. de Reotier, A. Yaouanc, G. van der Laan, N. Ker-
navanois, J.P. Sanchez, J.L. Smith, A. Hiess, A. Huxley, and
A. Rogalev, Phys. Rev. B60, 10606 (1999).
66. M. Finazzi, P. Sainctavit, A.M. Dias, J.P. Kappler, G. Krill,
J.P. Sanchez, P.D. de Reotier, A. Yaouanc, A. Rogalev, and
J. Goulon, Phys. Rev. B55, 3010 (1997).
67. P.D. de Reotier, J.P. Sanches, and A. Yaouanc, J. Alloys
Comp. 271–273, 414 (1998).
68. N. Kernavanois, J.X. Boucherle, P.D. de Reotier, F. Givord,
E. Lelievre-Berna, E. Ressouche, A. Rogalev, J.P. Sanchez,
N. Sato, and A. Yaouanc, J. Phys.: Condens. Matter 12,
7857 (2000).
144 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
69. A. Yaouanc, P.D. de Reotier, G. van der Laan, A. Hiess, J.
Goulon, C. Neumann, P. Lejay, and N. Sato, Phys. Rev.
B58, 8793 (1998).
70. W. Grange, M. Finazzi, J.P. Kappler, A. Dellobe, G. Krill,
P. Sainctavit, J.P. Sanchez, A. Rogalev, and J. Goulon, J.
Alloys Comp. 275–277, 583 (1998).
71. M. Kucera, J. Kunes, A. Kolomiets, M. Divis, A.V. Andreev,
V. Sechovsky, J.P. Kappler, and A. Rogalev, Phys. Rev. B66,
144405 (2002).
72. V.N. Antonov, B.N. Harmon, and A.N. Yaresko, Phys.
Rev. B68, 214425 (2003).
73. A.N. Yaresko, V.N. Antonov, and P. Fulde, Phys. Rev.
B67, 155103 (2003).
74. B.T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys.
Rev. Lett. 68, 1943 (1992).
75. J.C. Fuggle and J.E. Inglesfield, Unoccupied Electronic
States. Topics in Applied Physics, Springer, New York
(1992), Vol. 69.
76. V. Sechovsky and L. Havela, in: Handbook of Magnetic
Materials, K.H.J. Buschow (eds.), Elsevier, Amsterdam
(1998), Vol. 11, p. 1.
77. N.V. Mushnikov, T. Goto, K. Kamishima, H. Yamada,
A.V. Andreev, Y. Shiokawa, A. Iwao, and V. Sechovsky,
Phys. Rev. B59, 6877 (1999).
78. M. Wulff, J.M. Fournier, A. Delapalme, B. Gillon, V.
Sechovsky, L. Havela, and A.V. Andreev, Physica B163,
331 (1990).
79. J.A. Paix�o, G.H. Lander, P.J. Brown, H. Nakotte, F.R. de
Boer, and E. Br�ck, J. Phys.: Condens. Matter 4, 829 (1992).
80. A. Hiess, L. Havela, K. Prokevs, R.S. Eccleston, and G.H.
Lander, Physica B230–232, 89 (1997).
81. A.V. Andreev, Y. Shiokawa, M. Tomida, Y. Homma, V.
Sechovsky, and N.V. Mushnikov, J. Phys. Soc. Jpn. 68,
1999 (1999).
82. K.A. McEwen, U. Steigenberger, and J.L. Martinez, Physica
B186–188, 670 (1993).
83. P.M. Oppeneer, A.Y. Perlov, V.N. Antonov, A.N. Yaresko,
T. Kraft, and M.S. Brooks, J. Alloys Comp. 271–273, 831
(1998).
84. T. Gascher, M.S.S. Brooks, and B. Johansson, J. Phys.:
Condens. Matter 7, 9511 (1995).
85. J. Kunes, P. Novak, M. Divis, and P.M. Oppeneer, Phys.
Rev. B63, 205111 (2001).
86. V.N. Antonov, B.N. Harmon, O. Andryushchenko, L.
Bekenev, and A.N. Yaresko, Phys. Rev. B68, 214425 (2003).
87. G.H. Lande, M.S.S. Brooks, and B. Johansson, Phys. Rev.
B43, 13672 (1991).
88. P. Carra, B.T. Thole, M. Altarelli, and X. Wang, Phys.
Rev. Lett. 70, 694 (1993).
89. F.K. Richtmyer, S.W. Barnes, and E. Ramberg, Phys. Rev.
46, 843 (1934).
90. D.L. Tillwick and P. de V. du Plessis, J. Magn. Magn. Ma-
ter. 3, 329 (1976).
91. G. Busch, O. Vogt, A. Delpalme, and G.H. Lander, J.
Phys. C12, 1391 (1979).
92. G.H. Lander, M.S.S. Brooks, B. Lebech, P.J. Brown, O.
Vogt, and K. Mattenberger, J. Appl. Phys. 69, 4803 (1991).
93. M.S.S. Brooks and P.J. Kelly, Phys. Rev. Lett. 51, 1708
(1983).
94. G.H. Lander, Physica B186–188, 664 (1993).
95. M.S.S. Brooks, Physica B130, 6 (1985).
96. F.A. Wedgwood, J. Phys. C5, 2427 (1972).
97. B. Reihl, J. Less-Common Met. 128, 331 (1987).
98. J. Schoenes, B. Frick, and O. Vogt, Phys. Rev. B30, 6578
(1984).
99. J. Schoenes, B. Frick, and O. Vogt, Phys. Rev. B30, 6578
(1984).
100. C.Y. Huang, R.J. Laskowski, C.E. Olsen, and J.L. Smith,
J. Phys. C40, 26 (1979).
101. E.F. Westrum, R.R. Walters, H.E. Flotow, and D.W.
Osborne, J. Chem. Phys. 48, 155 (1968).
102. P. Erdoes and J. Robinson, The Physics of Actinide Com-
pounds, Plenum Press, New York, London (1983).
103. J. Neuenschwander, O. Vogt, E. Vogt, and P. Wachter,
Physica B144, 66 (1986).
104. H. Bilz, G. Guentherodt, W. Kleppmann, and W. Kress,
Phys. Rev. Lett. 43, 1998 (1979).
105. A.L. Cornelius, J.S. Schilling, O. Vogt, K. Mattenberger,
and U. Benedict, J. Magn. Magn. Mater. 161, 169 (1996).
106. O.G. Sheng and B.R. Cooper, J. Magn. Magn. Mater.
164, 335 (1996).
107. S.V. Halilov and E.T. Kulatov, J. Phys.: Condens. Matter
3, 6363 (1991).
108. T. Gasche, Ph.D. thesis, Uppsala (1993).
109. B.R. Cooper, Q.G. Sheng, S.P. Lim, C. Sanchez-Castro,
N. Kioussis, and J.M. Wills, J. Magn. Magn. Mater. 108,
10 (1992).
110. T. Kraft, P.M. Oppeneer, V.N. Antonov, and H. Eschrig,
Phys. Rev. B52, 3561 (1995).
111. J. Trygg, J.M. Wills, and M.S.S. Brooks, Phys. Rev. B52,
2496 (1995).
112. P.M. Oppeneer, V.N. Antonov, A.Y. Perlov, A.N.
Yaresko, T. Kraft, and H. Eschrig, Physica B230–232,
544 (1997).
113. H. Yamagami, J. Phys. Soc. Jpn. 67, 3176 (1999).
114. T. Shishidou, T. Oguchi, and T. Jo, Phys. Rev. B59, 6813
(1999).
115. T. Shishidou and T. Oguchi, Phys. Rev. B62, 11747 (2000).
116. V.N. Antonov, B.N. Harmon, O. Andryushchenko, L.
Bekenev, and A.N. Yaresko, Fiz. Nizk. Temp. 30, 411
(2004) [Low Temp. Phys. 30, 305 (2004)].
117. L. Severin, M.S.S. Brooks, and B. Johansson, Phys. Rev.
Lett. 71, 3214 (1993).
118. H. Sakurai, H. Hashimoto, A. Ochiai, T. Suzuki, M. Ito,
and F. Itoh, J. Phys.: Condens. Matter 7, L599 (1995).
119. M.S.S. Brooks and B. Johansson, in: Handbook of Mag-
netic Materials, K.H.J. Buschow (eds.), North-Holland,
Amsterdam (1993), Vol. 7, p. 139.
120. O. Eriksson, M.S.S. Brooks, and B. Johansson, Phys. Rev.
B41, 7311 (1990).
121. A. Mavromaras, L. Sandratskii, and J. K�bler, Solid State
Commun. 106, 115 (1998).
122. I.V. Solovyev, A.I. Liechtenstein, and K. Terakura, Phys.
Rev. Lett. 80, 5758 (1998).
123. M.S.S. Brooks, T. Gasche, and B. Johansson, J. Phys.
Chem. Solids 56, 1491 (1995).
124. B. Reihl, N. Martensson, and O. Vogt, J. Appl. Phys. 53,
2008 (1982).
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 145
125. G.R. Stewart, Z. Fisk, J.O. Willis, and J.L. Smith, Phys.
Rev. Lett. 52, 679 (1984).
126. G. Stewart, Rev. Mod. Phys. 56, 755 (1984).
127. R. Albers, A.M. Boring, and N.E. Christensen, Phys. Rev.
B33, 8116 (1986).
128. M.R. Norman, R.C. Albers, A.M. Boring, and N.E.
Christensen, Solid State Commun. 68, 245 (1988).
129. N. Kimura, R. Settai, Y. Onuki, H. Toshima, E. Yamamoto,
K. Maezawa, H. Aoki, and H. Harima, J. Phys. Soc. Jpn.
64, 3883 (1995).
130. S.R. Julian, G.J. Teunissen, N. Dorion-Leyraund, A.D.
Huxley, M.P. Ray, M.R. Norman, J. Floquet, and G.G.
Lonzarich, The Fermi Surface of UPt3, University of
Cambridge, Cambridge (2000).
131. G. Zwicknagl, A.N. Yaresko, and P. Fulde, Phys. Rev.
B65, R081103 (2002).
132. L. Taillefer and G.G. Lonzarich, Phys. Rev. Lett. 60, 1570
(1988).
133. L. Taillefer, R. Newbury, G.G. Lonzarich, Z. Fisk, and
J.L. Smith, J. Magn. Magn. Mater. 63–64, 372 (1987).
134. N. Kimura, T. Komatsubara, D. Aoki, Y. Onuki, Y. Haga,
E. Yamamoto, H. Aoki, and H. Harima, J. Phys. Soc. Jpn.
67, 2185 (1998).
135. R.H. Heffner, D.W. Cooke, A.L. Giorgi, R.L. Hutson,
M.E. Schillaci, H.D. Rempp, J.L. Smith, J.O. Willis, D.E.
MacLaughlin, C. Boekema, R.L. Lichti, J. Oostens, and
A.B. Denison, Phys. Rev. B39, 11345 (1989).
136. G. Aeppli, E. Bucher, A.I. Goldman, G. Shirane, C. Bro-
holm, and J.K. Kjems, J. Magn. Magn. Mater. 76–77, 385
(1988).
137. S.M. Hayden, L. Taillefer, C. Vettier, and J. Flouquet,
Phys. Rev. B46, 8675 (1992).
138. B. Lussier, B. Ellman, and L. Taillefer, Phys. Rev. B53,
5145 (1996).
139. E.D. Isaacs, P. Zschack, C.L. Broholm, C. Burns, G.
Aeppli, A.P. Ramirez, T.T.M. Palstra, R.W. Erwin, N.
Stucheli1, and E. Bucher, Phys. Rev. Lett. 75, 1178 (1995).
140. A.N. Yaresko, V.N. Antonov, and B.N. Harmon, Phys.
Rev. B68, 214426 (2003).
141. J.W. Allen, J.D. Denlinger, Y.X. Zhang, G.-H. Gweon, S.-H.
Yang, S.-J. Oh, E.-J. Cho, W.P. Ellis, D.A. Gajewski, R.
Chau, and M.B. Maple, Physica B281–282, 725 (2000).
142. T. Ito, H. Kumigashira, S. Souma, T. Takahashi, Y. Tokiwa,
Y. Haga, and Y. Onuki, Physica B312–313, 653 (2002).
143. Y. Tokiwa, K. Sugiyama, T. Takeuchi, M. Nakashima, R.
Settai, Y. Inada, Y. Haga, E. Yamamoto, K. Kindo, H.
Harima, and Y. Onuki, J. Phys. Soc. Jpn. 70, 1731 (2001).
144. W.J.L. Buyers, A.F. Murray, T.M. Holden, and E.C. Svens-
son, Physica B+C102, 291 (1980).
145. H. Harima, J. Magn. Magn. Mater. 226–230, 83 (2001).
146. T.T.M. Palstra, A.A. Menovsky, J. van der Berg, J. Dirk-
maat, P.H. Kes, G.J. Nieuwenhuys, and J.A. Mydosh,
Phys. Rev. Lett. 55, 2727 (1985).
147. W. Schlabitz, J. Baumann, B. Pollit, U. Rauchschwalbe,
H.M. Mayer, U. Ahlheim, and C.D. Bredl, Z. Physik 62,
171 (1986).
148. A. de Visser, F.E. Kayzel, A.A. Menovsky, J.J.M. Franse,
J. van der Berg, and G.J. Nieuwenhuys, Phys. Rev. B34,
8168 (1986).
149. C. Broholm, J.K. Kjems, W.J.L. Buyers, P. Matthews,
T.T.M. Palstra, A.A. Menovsky, and J.A. Mydosh, Phys.
Rev. Lett. 58, 1467 (1987).
150. C Broholm, H. Lin, P.T. Matthews, T.E. Mason, W.J.L.
Buyers, M.F. Collins, A.A. Menovsky, J.A. Mydosh, and
J.K. Kjems, Phys. Rev. B43, 12809 (1991).
151. Y. Kohori, K. Matsuda, and T. Kohara, J. Phys. Soc. Jpn.
65, 1083 (1996).
152. S.A.M. Mentink, T.E. Mason, S. Su�llow, G.J. Nieu-
wenhuys, A.A. Menovsky, and J.A.A.J. Peerenboom,
Phys. Rev. B53, R6014 (1996).
153. K. Hasselbach, J.R. Kirtley, and P. Lejay, Phys. Rev.
B46, 5826 (1992).
154. A. de Visser, F.R. de Boer, A.A. Menovsky, and J.J.M.
Franse, Solid State Commun. 64, 527 (1987).
155. K. Sugiyama, H. Fuke, K. Kindo, K. Shimohata, A.A.
Menovsky, J.A. Mydosh, and M. Date, J. Phys. Soc. Jpn.
59, 3331 (1990).
156. K. Bakker, A. de Visser, A.A. Menovsky, and J.J.M. Franse,
Physica B186–188, 720 (1993).
157. M.R. Norman, T. Oguchi, and A.J. Freeman, Phys. Rev.
B38, 11193 (1988).
158. L.M. Sandratskii and J. K�bler, Phys. Rev. B50, 9258
(1994).
159. A. Continenza and P. Johansson, J. Magn. Magn. Mater.
140–144, 1401 (1995).
160. H. Yamagami and N. Hamada, Physica B284–288, 1295
(2000).
161. H. Ohkuni, T. Ishida, Y. Inada, Y. Haga, E. Yamamoto,
Y. Onuki, and S. Takahashi, J. Phys. Soc. Jpn. 66, 945
(1997).
162. T. Ito, H. Kumigashira, T. Takahashi, Y. Haga, E. Yama-
moto, T. Honma, H. Ohkuni, and Y. Onuki, Physica
B281–282, 727 (2000).
163. C. Geibel, C. Schank, S. Thies, H. Kitazawa, C.D. Bredl,
A. Booehm, M. Rau, A. Grauel, R. Caspary, R. Helfrich,
U. Ahlheim, G. Weber, and F. Steglich, Z. Physik 84, 1
(1991).
164. C. Geibel, S. Thies, D. Kaczorowski, A. Mehner, A. Grauel,
B. Seidel, U. Ahlheim, R. Helfrich, K. Petersen, C.D.
Bredl, and F. Steglich, Z. Physik 83, 305 (1991).
165. A. Krimmel, P. Fischer, B. Roessli, H. Maletta, C. Geibel,
C. Schank, A. Grauel, A. Loidl, and F. Steglich, Z. Physik
86, 161 (1992).
166. J. Sticht and J. K�bler, Z. Physik B87, 299 (1992).
167. L.M. Sandratskii, J. K�bler, P. Zahn, and I. Mertig, Phys.
Rev. B50, 15834 (1994).
168. K. Kn�pfle, A. Mavromaras, L.M. Sandratskii, and J. K�bler,
J. Phys.: Condens. Matter 8, 901 (1996).
169. S. Fujimori, Y. Saito, M. Seki, K. Tamura, M. Mizuta, K.
Yamaki, K. Sato, T. Okane, A. Tanaka, N. Sato, T.
Komatsubara, Y. Tezuka, S. Shin, S. Suzuki, and S. Sato, J.
Electron. Spectrosc. Relat. Phenom. 101–103, 439 (1999).
170. T. Rjima, S.Sato, S. Suzuki, S. Fujimori, M. Yamada, N.
Sato, Y. Onuki, T. Komatsubara, Y. Tezuka, Shin, and T.
Ishii, J. Electron. Spectrosc. Relat. Phenom. 78, 147 (1996).
171. A. Amato, C. Geibel, F.N. Gygax, R.H. Hefner, E. Knetsch,
D.E. MacLaughlin, C. Schank, A. Schenck, F. Steglich,
and M. Weber, Z. Physik 86, 159 (1992).
146 Fizika Nizkikh Temperatur, 2008, v. 34, No. 2
V.N. Antonov, A.P. Shpak, and A.N. Yaresko
172. J.G. Lussier, M. Mao, A.A. Schr�der, J. Garret, B.D.
Gaulin, S.M. Shapiro, and W.J.L. Buyers, Phys. Rev. B56,
11749 (1997).
173. H.R. Ott, H. Rudigier, Z. Fisk, and J.L. Smith, Phys. Rev.
Lett. 50, 1595 (1983).
174. D.L. Cox and A. Zawadowski, Adv. Phys. 47, 599 (1998).
175. M.W. McElfresh, M.B. Maple, J.O. Willis, D. Schiferl,
J.L. Smith, Z. Fisk, and D.L. Cox, Phys. Rev. B48, 10395
(1993).
176. E. Miranda, V. Dobrosavljevic, and G. Kottliar, Phys.
Rev. Lett. 78, 290 (1997).
177. D.L. Cox, Phys. Rev. Lett. 59, 1240 (1987).
178. F.B. Anders, M. Jarell, and D.L. Cox, Phys. Rev. Lett. 78,
2000 (1997).
179. K. Takegahara, H. Harima, and T. Kasuya, J. Phys. F16,
1691 (1986).
180. M.R. Norman, W.E. Picket, H. Krakauer, and C.S. Wang,
Phys. Rev. B36, 4058 (1987).
181. K. Takegahara and H. Harima, Physica B281–282, 764
(2000).
182. T. Maehira, A. Higashia, M. Higuchi, H. Yasuhara, and
A. Hasegawa, Physica B312–313, 103 (2002).
183. V.I. Anisimov, J. Zaanen, and O.K. Andersen, Phys. Rev.
B44, 943 (1991).
184. E. Wuilloud, Y. Baer, H.R. Ott, Z. Fisk, and J.L. Smith,
Phys. Rev. B29, 5228 (1984).
185. V.N. Antonov, B.N. Harmon, and A.N. Yaresko, Phys.
Rev. B66, 165209 (2002).
186. V.L. Ginzburg, Sov. Phys. JETP 4, 153 (1957).
187. N. Aso, G. Motoyama, Y. Uwatoko, S. Ban, S. Nakamura,
T. Nishioka, Y. Homma, Y. Shiokawa, K. Hirota, and
N.K. Sato, Phys. Rev. B73, 054512 (2006).
188. S. Saxena, P. Agarwal, K. Ahilan, F. Grosche, R. Hasel-
wimmer, M. Steiner, E. Pugh, I. Walker, S. Julian, P.
Monthoux, G. Lonzarich, A. Huxley, I. Sheikin, D.
Braithwaite, and J. Flouquet, Nature 406, 587 (2000).
189. A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois, D.
Braithwaite, R. Calemczuk, and J. Flouquet, Phys. Rev.
B63, 144519 (2001).
190. A.N. Yaresko, P.D. de R�otier, A. Yaouanc, N. Kernavanois,
J.-P. Sanchez, A.A. Menovsky, and V.N. Antonov, J. Phys.:
Condens. Matter 17, 2443 (2005).
191. T. Okane, J. Okamoto, K. Mamiya, S. Fujimori, Y. Takeda,
Y. Saitoh, Y. Muramatsu, A. Fujimori, Y. Haga, E. Yama-
moto, A. Tanaka, T. Honda, Y. Inada, and Y. Nuki, J.
Phys. Soc. Jpn. 75, 024704 (2006).
192. Y. Inada, T. Honma, N. Kawamura, M. Suzuki, H.
Miyagawa, E. Yamamoto, Y. Haga, T. Okane, S.
Fujimori, and Y. Onuki, Physica B359–361, 1054 (2005).
193. V. Antonov, B. Harmon, and A. Yaresko, J. Phys.:
Condens. Matter 19, 186222 (2007).
194. G. van der Laan and B.T. Thole, Phys. Rev. B53, 14458
(1996).
195. H. Ebert, Rep. Prog. Phys. 59, 1665 (1996).
196. V.N. Antonov, O. Jepsen, A.N. Yaresko, and A.P. Shpak,
J. Appl. Phys. 100, 043711 (2006).
197. H. Maruyama, M. Suzuki, N. Kawamura, M. Ito, E.
Arakawa, J. Kokubun, K. Hirano, K. Horie, S. Uemura, K.
Hagiwara, M. Mizumaki, S. Goto, H. Kitamura, K.
Namikawa, and T. Ishikawa, J. Synchrotron Rad. 6, 1133
(1999).
198. M. Usuda, J. Igarashi, and A. Kodama, Phys. Rev. B69,
224402 (2004).
X-ray magnetic circular dichroism in d and f ferromagnetic materials: recent theoretical progress. Part II
Fizika Nizkikh Temperatur, 2008, v. 34, No. 2 147
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