Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi
The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the Heusler alloy Co₂FeSi were investigated theoretically from first principles, using the fully relativistic Dirac linear MT-orbital (LMTO) band structure method. Densities of valence states, orbital and spin magneti...
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| Цитувати: | Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi / V.N. Antonov, D.A. Kukusta, A.P. Shpak, A.N. Yaresko // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 627-639. — Бібліогр.: 59 назв. — англ. |
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irk-123456789-1195752017-06-08T03:03:19Z Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi Antonov, V.N. Kukusta, D.A. Shpak, A.P. Yaresko, A.N. The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the Heusler alloy Co₂FeSi were investigated theoretically from first principles, using the fully relativistic Dirac linear MT-orbital (LMTO) band structure method. Densities of valence states, orbital and spin magnetic moments as well as polarization of the electronic states at the Fermi level are analyzed and discussed. The origin of the XMCD spectra in the Co₂FeSi compound is examined. The calculated results are compared with available experimental data. Електронна структура та спектри рентгенiвского магнiтного циркулярного дихроїзму (РМЦД) хойслерiвського сплаву Co₂FeSi були дослiдженi теоретично з перших принципiв за допомогою повнiстю релятивiстського Дiракiвського лiнiйного методу ЛМТО. Проаналiзованi та обговоренi щiльностi валентних електронiв, орбiтальнi та магнiтнi спiновi моменти, а також поляризацiя електронiв на рiвнi Фермi.Вивчається походження спектрiв РМЦД у Co₂FeSi. Результати обчислень порiвнюються з еспериментальними даними. 2008 Article Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi / V.N. Antonov, D.A. Kukusta, A.P. Shpak, A.N. Yaresko // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 627-639. — Бібліогр.: 59 назв. — англ. 1607-324X PACS: 71.28.+d, 71.25.Pi, 75.30.Mb DOI:10.5488/CMP.11.4.627 http://dspace.nbuv.gov.ua/handle/123456789/119575 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the Heusler alloy Co₂FeSi
were investigated theoretically from first principles, using the fully relativistic Dirac linear MT-orbital (LMTO)
band structure method. Densities of valence states, orbital and spin magnetic moments as well as polarization
of the electronic states at the Fermi level are analyzed and discussed. The origin of the XMCD spectra in the
Co₂FeSi compound is examined. The calculated results are compared with available experimental data. |
| format |
Article |
| author |
Antonov, V.N. Kukusta, D.A. Shpak, A.P. Yaresko, A.N. |
| spellingShingle |
Antonov, V.N. Kukusta, D.A. Shpak, A.P. Yaresko, A.N. Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi Condensed Matter Physics |
| author_facet |
Antonov, V.N. Kukusta, D.A. Shpak, A.P. Yaresko, A.N. |
| author_sort |
Antonov, V.N. |
| title |
Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi |
| title_short |
Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi |
| title_full |
Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi |
| title_fullStr |
Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi |
| title_full_unstemmed |
Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi |
| title_sort |
electronic structure and x-ray magnetic circular dichroism in the heusler alloy co₂fesi |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119575 |
| citation_txt |
Electronic structure and x-ray magnetic circular dichroism in the Heusler alloy Co₂FeSi / V.N. Antonov, D.A. Kukusta, A.P. Shpak, A.N. Yaresko // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 627-639. — Бібліогр.: 59 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT antonovvn electronicstructureandxraymagneticcirculardichroismintheheusleralloyco2fesi AT kukustada electronicstructureandxraymagneticcirculardichroismintheheusleralloyco2fesi AT shpakap electronicstructureandxraymagneticcirculardichroismintheheusleralloyco2fesi AT yareskoan electronicstructureandxraymagneticcirculardichroismintheheusleralloyco2fesi |
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2025-07-08T16:12:06Z |
| last_indexed |
2025-07-08T16:12:06Z |
| _version_ |
1837095850216521728 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 4(56), pp. 627–639
Electronic structure and x-ray magnetic circular
dichroism in the Heusler alloy Co2FeSi
V.N.Antonov1, D.A.Kukusta1, A.P.Shpak1, A.N.Yaresko2
1 Institute for Metal Physics of the National Academy of Sciences of Ukraine, 36 Vernadsky Str., 03142 Kiev
2 Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D–70569 Stuttgart, Germany
Received December 11, 2007, in final form June 18, 2008
The electronic structure and x-ray magnetic circular dichroism (XMCD) spectra of the Heusler alloy Co2FeSi
were investigated theoretically from first principles, using the fully relativistic Dirac linear MT-orbital (LMTO)
band structure method. Densities of valence states, orbital and spin magnetic moments as well as polarization
of the electronic states at the Fermi level are analyzed and discussed. The origin of the XMCD spectra in the
Co2FeSi compound is examined. The calculated results are compared with available experimental data.
Key words: strongly correlated systems, electronic structure, x-ray magnetic circular dichroism
PACS: 71.28.+d, 71.25.Pi, 75.30.Mb
1. Introduction
Electronic devices exploiting the spin of an electron have attracted great scientific interest [1].
Their basic element is a ferromagnetic electrode providing a spin-polarized electric current. In
this context, the most interesting materials are the ones having a complete spin-polarization at the
Fermi level. The rapid development of magneto-electronics intensified the interest to such materials.
The application of the spin degree of freedom to conventional electronic devices has got several
advantages such as non-volatility, an increased data processing speed, a decreased electric power
consumption and increased integration densities [1,2]. The current advances in new materials are
promising for engineering new spintronic devices in the near future [2].
A metal for spin-up and a semiconductor for spin-down electrons is called half-metallic ferro-
magnet (HMF) [3], and Heusler compounds have been considered potential candidates to show this
property. de Groot et al. first predicted half-metallic ferromagnetism for the Heusler-like compound
NiMnSb [3]. The full Heusler alloys are defined as well-ordered ternary intermetallic compounds,
at the stoichiometric composition X2Y Z, which have the cubic L21 structure. These compounds
involve two different transition metal atoms X , and Y , and a third element Z which is a non-
magnetic metal or nonmetallic element. Currently the Heusler alloys are at the focus of a large
scientific interest due to their potential for applications in magnetic field sensors and spintronic
devices [1]. Numerous theoretical and experimental studies of Heusler alloys have been carried out,
and it has been shown that composition and heat treatment are important parameters determining
their magnetic properties.
In this paper we investigate the Heusler alloy Co2FeSi. This compound is one of the most promis-
ing candidates for spintronic applications. Recently, Balke et al. [4] have conclusively showed that
Co2FeSi crystallizes in the ordered L21 structure. Balke et al. [5] revealed that Co2FeSi undergoes
the L21 ↔ B2 phase transition at the temperature ≈ 1031 K. Wurmehl et al. [6] have found that
L21-ordered Co2FeSi is the Heusler compound as well as the half-metallic ferromagnet with the
highest Curie temperature (1100 ± 20 K) in the classes of Heusler compounds and half-metallic
ferromagnets. The measured magnetic moment in saturation was found to be 5.97 ± 0.05 µB at
5 K which corresponds to 1.49 µB per atom. An extrapolation to 0 K gives an integer magnetic
moment of 6µB per unit cell within the experimental uncertainty, as expected for a half-metallic
c© V.N.Antonov, D.A.Kukusta, A.P.Shpak, A.N.Yaresko 627
V.N.Antonov et al.
ferromagnet. Mössbauer spectroscopy was performed at magnetostructural investigations, the ob-
served pattern of the spectrum being typical of a magnetically ordered system. Moreover, XMCD
spectra taken at the L2,3 absorption edges of Fe and Co at photoabsorption were used in order
to investigate the site specific magnetic properties. Schneider et al. [7] have grown thin films of
Co2FeSi and provided their full experimental characterization including the measurements of the
x-ray absorption spectra (XAS) as well as the XMCD spectra. They estimated the element-specific
spin and orbital magnetic moments using sum rules. Furthermore, investigations of integral macro-
scopic magnetic transport properties, such as the magnetization, resistivity, magnetoresistance, and
spin polarization of Co2FeSi films were carried out. The authors found out that the saturation mag-
netization of the films does not depend on the substrate chosen and exhibits a T 3/2 behavior over
the 0–300 K interval. It can be extrapolated to 5.0 µB/f.u. at 0 K. This value is much smaller than
the value of saturation magnetization observed on the bulk sample [6]. That might be caused by
an incomplete atomic order in thin films with respect to the L21 structure. Schneider et al. [7] also
estimated the surface spin polarization using the spin-resolved photoemission spectroscopy. They
found quite small spin polarization at EF in Co2FeSi thin films (≈6%). This value is drastically re-
duced with respect to the expected 100% in case of half-metallic behavior as predicted in [6]. It has
been shown that the observed reduction of spin polarization at EF is not confined to the outermost
surface layer and originates from disorder effects which introduce minority states at EF. Resis-
tivity measured over all 0–300 K temperature interval revealed metallic behavior of Co2FeSi thin
films. Anisotropic magnetoresistance effect with respect to the current direction was also observed
which implies a significant contribution of spin-flip scattering process to the magnetoresistance.
X-ray magnetic circular dichroism photoemission electron microscopy has been used for a direct
observation of the domain structure of single- and polycrystalline samples of Co2FeSi in [8]. Spin
polarized photoemission from a single domain of single crystal shows a spin polarization of 16% at
the Fermi energy and up to 35% in the d-bands, at room temperature.
Despite a failure to reach 100% polarization at the Fermi level and relatively small saturation
magnetization, thin films of Co2FeSi have been successfully used for fabrication of magnetic tunnel
junctions [9]. The tunnel magneto-resistance (TMR) ratios of 60% at low temperature and 41% at
room temperature suggest that an improvement in the materials is still necessary for a successful
use in the devices [8].
Theoretically, the electronic structure and other properties of Co2FeSi were studied in [5,6,10,
11]. Self-consistent band-structure calculations were carried out by Kandpal et al. using the full-
potential linear augmented-plane-wave (FLAPW) method [10]. The main focus in [10] was made
on the magnetic moment and on its very strong discrepancy while comparing the experimental
and the calculated values for Co2FeSi. Wurmehl et al. [6] systematically investigate the effect of
different approximations to the exchange-correlation potential as well as of different band structure
methods on the magnetic moments in Co2FeSi. They used LMTO method, the FLAPW method
and the Korringa-Kohn-Rostocker (KKR) method. The latter method was allowed to calculate the
band structure in the muffin-tin (MT) and the atomic sphere (ASA) approximations. Moreover,
it provides the coherent potential approximation (CPA) to be used for disordered systems. The
calculations were carried out with the most common parameterizations of the exchange-correlation
functional as given by Moruzzi, Janak, and Williams [12], von Barth and Hedin [13], Vosco, Wilk,
and Nussair [14], and Engel and Vosko [15]. The generalized gradient approximation (GGA) was
used in the form given by Perdew et al. [16,17]. The calculated total magnetic moments range from
4.9µB to 5.7µB in the LSDA approximation. Thus, they are throughout too low compared to the
experiment. Therefore, the LSDA band structure calculations are not sufficient to reproduce half-
metalicity in Co2FeSi. However, the LSDA+U scheme satisfactorily reproduces the experimental
observations. Values of Ueff from 2.5 eV to 5.0 eV for Co and 2.4 eV to 4.8 eV for Fe result in
a magnetic moment of 6 µB and a gap in the minority states. Finally, it was shown for Co2FeSi
that correlation could also contribute to the destruction of the half-metallic behavior if the corre-
lation becomes as strong as 5 eV. The substitutional series of the quaternary Heusler compound
Co2Mn1−xFexSi (x=0, 0.5, 1) was synthesized and investigated both experimentally and theoreti-
cally by Balke et al. [5]. The structural and magnetic properties of the Co2Mn1−xFexSi Heusler
628
Electronic structure and x-ray magnetic circular dichroism in the Co2FeSi
alloys were investigated by means of x-ray diffraction, high- and low-temperature magnetome-
try, Mössbauer spectroscopy, and differential scanning calorimetry. The electronic structure was
explored by means of high energy photoemission spectroscopy at about 8 keV photon energy. To
calculate the electronic and magnetic structures they used the FLAPW method. All the compounds
under consideration were found to be half-metallic in the LDA+U approach. Sargolzaei et al. [18]
have carried out density functional calculations of Co2Y Z (Y = Mn,Fe; Z = Al, Si,Ga,Ge) us-
ing the relativistic version of the full-potential local-orbital (FPLO) minimum-basis band-structure
method and the Perdew-Wang parameterization of the exchange-correlation potential in the LSDA
approach. The authors optimized the equilibrium lattice parameters using LSDA total-energy cal-
culations. The calculated lattice constants a turned out to be 2–3% smaller than the experimental
values. It was found that Co2FeSi shows considerably different magnetic moments at the LSDA and
experimental lattice constants. It originates from the fact that at the theoretical lattice constant,
the Fermi level is situated in a steep slope at the band edge of an almost empty minority spin 3d
band, and flat minority spin energy bands cross the Fermi level in Co2FeSi. The latter gives rise
to the mentioned steep band edge and thus to the sensitivity of the magnetic moment of Co2FeSi
with respect to the lattice spacing. It was found that spin-orbit coupling reduces the degree of spin
polarization of the density of states at Fermi level by a few percent. The authors present an al-
ternative explanation of the measured integer total moment of Co2FeSi, contrasting the suggested
correlation-induced half-metalicity. While the LDA+U approach yields a half-metallic state with
an integer total moment at the experimental lattice constant, the LSDA+OP approach yields the
experimental total moment at a 2.0% expanded lattice constant. The authors propose that the
measured moment of Co2FeSi need not be caused by a correlated half-metallic state. It could, e.g.,
arise from a small but influential disorder of the sample.
The effect of different types of disorder on properties of Co2FeSi was investigated by Nakatani
et al. [11]. It turned out that in Co2FeSi EF lies close to the upper edge of the gap, near the
minimum energy of the conduction band. In such a case, the half-metalicity can be spoiled by the
increase of temperature and structural disorder because some types of disorder (B2- and A2-type)
form additional states in the minority gap. A perfect L21 structure assumed for all the above
calculations is not present in real samples; disorders are usually expected. This important fact
was considered in calculations [11]. The investigations of the disordering effects on the electronic
structure of Co2FeSi alloy have shown that moderate disordering between Fe, Co, and Si atoms
forms additional electronic states near the gap edge, which immediately destroys the half-metalicity
and reduces spin-polarization at EF level. This indicates that a higher spin polarization can be
achieved with a higher degree of order.
In the present study, we focus our attention on the x-ray magnetic circular dichroism in the
Heusler alloy Co2FeSi. The XMCD technique developed in recent years has evolved into a powerful
magnetometry tool to separate orbital and spin contributions into the element specific magnetic
moments. XMCD experiments measure the absorption of x-rays with opposite (left and right) states
of circular polarization. Recently x-ray magnetic circular dichroism in the ferromagnetic Co2FeSi
alloy has been measured at the Co and Fe L2,3 edges for the bulk [6] and thin film samples [7].
These two measurements gave almost similar results in the shape of the XAS and XMCD spectra.
This paper is organized as follows. Section 2 presents a description of the Heusler alloys Co2FeSi
crystal structure and the computational details. Section 3 is devoted to the electronic structure
and XMCD spectra of the Co2FeSi compound calculated in the fully relativistic Dirac LMTO
band structure method. The calculated results are compared with the available experimental data.
Finally, the results are summarized in section 4.
2. Crystal structure and computational details
The Heusler-type X2Y Z compound crystallizes in the cubic L21 structure with Fm3m space
group (No. 225). It is formed by four interpenetrating fcc sublattices as shown in figure 1. The X
ions occupy the 8c Wyckoff positions (x = 1
4
, y = 1
4
, z = 1
4
). The Y ions occupy the 4a positions
(x=0, y=0, z=0), and the Z ions are placed at the 4b sites (x = 1
2
, y = 1
2
, z = 1
2
). All atoms have
629
V.N.Antonov et al.
eight nearest neighbors at the same distance. The Y and Z atoms have eight X atoms as nearest
neighbors, while for X there are four Y and four Z atoms.
Figure 1. (Color online) Schematic representation of the L21 structure. The cubic cell contains
four primitive cells.
Using straightforward symmetry considerations it can be shown that all magneto-optical phe-
nomena (XMCD, MO Kerr and Faraday effects) are caused by the symmetry reduction, in com-
parison to the paramagnetic state, caused by magnetic ordering [19]. This symmetry lowering
occurs only when spin-orbit (SO) coupling is included. Therefore, in order to calculate the XMCD
properties one has to account for both magnetism and SO coupling at the same time.
Within the one-particle approximation, the absorption coefficient µ for incident x-rays is de-
termined by the probability of electron transitions from an initial core state (with wave function
ψj and energy Ej) to final unoccupied states (with wave functions ψnk and energies Enk) as
µλ
j (ω) =
∑
nk
|〈Ψnk|Jλ|Ψj〉|2δ(Enk −Ej − ~ω)θ(Enk −EF) , (1)
where ~ω is the photon energy, λ is its polarization and Jλ = −eαaλ being the dipole electron-
photon interaction operator, α are Dirac matrices, aλ is the λ polarization unit vector of the
photon vector potential [a± = 1/
√
2(1,±i, 0), az = (0, 0, 1)]. (Here +/− denotes, respectively, left
and right circular photon polarizations with respect to the magnetization direction in the solid).
In order to simplify the comparison of the theoretical x-ray isotropic absorption L2,3 spectra
of Co2FeSi to the experimental ones we take into account the background intensity which affects
the high energy part of the spectra and is caused by different kind of inelastic scattering of the
electron promoted to the conduction band above Fermi level due to x-ray absorption (scattering
on potentials of surrounding atoms, defects, phonons etc.). To calculate the background spectra we
used the model proposed by Richtmyer et al. [20]. The absorption coefficient for the background
intensity is
µ(ω) =
CΓc
2π
∫ ∞
Ecf0
dEcf
(Γc/2)2 + (~ω −Ecf)2
, (2)
where Ecf = Ec−Ef , Ec, and Γc are the energy and the lifetimes broadening of the core hole, Ef is
the energy of empty continuum level, Ef0
is the energy of the lowest unoccupied continuum level,
and C is a normalization constant which in this paper has been used as an adjustable parameter.
We have adopted the LSDA+U method [21] as a different level of approximation to treat the
electron-electron correlations. We used the rotationally invariant LSDA+U method. This method is
described in detail in previous paper [22]. The effective on-site Coulomb repulsion U was considered
as an adjustable parameter. We used U = 4.0 eV. For this value, we found good agreement between
the calculated and the extrapolated value of total spin magnetic moment of Co2FeSi [6]. For the
exchange integral J , the value of 0.89 eV estimated from constrained LSDA calculations was used.
630
Electronic structure and x-ray magnetic circular dichroism in the Co2FeSi
Concurrent with the development of the x-ray magnetic circular dichroism experiment, some
important magneto-optical sum rules have been derived [23–26].
For the L2,3 edges the lz sum rule can be written as [27]
〈lz〉 = nh
4
3
∫
L3+L2
dω(µ+ − µ−)
∫
L3+L2
dω(µ+ + µ−)
, (3)
where nh is the number of holes in the d band nh = 10 − nd, 〈lz〉 is the average of the magnetic
quantum number of the orbital angular momentum. The integration is taken over the whole 2p
absorption region. The sz sum rule can be written as
〈sz〉 +
7
2
〈tz〉 = nh
∫
L3
dω(µ+ − µ−) − 2
∫
L2
dω(µ+ − µ−)
∫
L3+L2
dω(µ+ + µ−)
, (4)
where tz is the z component of the magnetic dipole operator t = s− 3 r (r · s)/|r|2 which accounts
for the asphericity of the spin moment. The integration
∫
L3
(
∫
L2
)
is taken only over the 2p3/2
(
2p1/2
)
absorption region.
The details of the computational method are described in the previous papers [28–31], and here
we only mention some aspects specific to the present calculations. The calculations were performed
for the experimentally observed lattice constant a = 5.658 Å[32] as well as for the one theoretically
calculated in total energy optimization using the spin-polarized fully relativistic linear-muffin-tin-
orbital (SPR LMTO) method [33,34] with the combined correction term taken into account. The
LSDA part of the calculations was based on the spin-density functional with the Perdew-Wang
[35] of the exchange-correlation potential. Brillouin zone (BZ) integrations were performed using
the improved tetrahedron method [36] and charge self-consistency was obtained on a grid of 349 k
points in the irreducible part of the BZ. The basis consisted of transition metal s, p, d and Si s, p
and d LMTO’s.
The intrinsic broadening mechanisms have been accounted for by folding the XMCD spectra
with a Lorentzian. For the finite lifetime of the core hole, a constant width Γc, in general from [37],
has been used. The finite apparative resolution of the spectrometer has been accounted for by a
Gaussian of 1.0 eV.
3. Results and discussion
3.1. Energy band structure
The total and partial DOS’s of Co2FeSi calculated in the LSDA approximation are presented
in figure 2 for the experimentally measured lattice constant. The occupied part of the valence band
can be subdivided into several regions. Si 2s states appear between −12.0 eV and −9.0 eV. The
states in the energy range from −7.0 eV to 2.0 eV are formed by Co and Fe d states and Si p
states. Our calculations show that Co2FeSi has local spin magnetic moments of 1.224µB on Co,
2.709µB on Fe and −0.033µB on Si. The orbital moments are equal to 0.049µB, 0.080µB, and
0.002µB on the Co, Fe, and Si sites, respectively. The interaction between the transition metals is
ferromagnetic which leads to a total calculated moment of 5.125µB. This value is in good agreement
with previous calculations [7,10] and differs from the experimental value of 6.00±0.05 µB [6].
The crystal field at the Co 8c site (Td point symmetry) causes the splitting of d orbitals into
a doublet e (3z2 − 1 and x2 − y2) and a triplet t2 (xy, yz, and xz). The crystal field at the Fe 4a
site (Oh point symmetry) splits Fe d states into eg (3z2 − 1 and x2 − y2) and t2g (xy, yz, and xz)
states. The hybridization between Fe and Co d states plays an important role in the formation of
the band structure of Co2FeSi. It leads to the splitting of the d states into the bonding states which
are of Co and Fe d character and antibonding states with stronger contribution of Fe d states.
631
V.N.Antonov et al.
Figure 2. (Color online) The total
(in states/(cell eV)) and partial (in
states/(atom eV)) density of states of
Co2FeSi. The Fermi energy is at zero.
Figure 3 shows the band structure and density of
states calculated using the LSDA+U method. The
effective Coulomb-exchange parameters Ueff were set
to 3.11 eV at the Co and Fe sites (U= 4 eV and J=
0.89 eV). The minority DOS figure 3 exhibits a clear
gap around EF. Thus, according to the spin-polarized
LSDA+U calculations Co2FeSi is a half-metallic fer-
romagnet if the spin-orbit coupling is neglected. The
high density below EF is dominated by d states being
located at Co and Fe sites. Inspecting the majority
DOS, one finds a small density of states near EF.
This density is mainly derived from the states located
at Co and Si sites. Although spin-orbit splitting of
the d energy bands for both the Co and Fe atoms is
much smaller than their spin and crystal-field splitti-
ngs, this interaction destroys the energy gap for the
minority spin states. The Fermi level falls within a
region of very small but finite minority-spin DOS.
In this case the spin polarization of electron states
at the Fermi energy is equal to 94.5%. Similar re-
sults were obtained in [38,39] for some other Heusler
alloys. The authors show that the spin-orbit interac-
tion can cause a nonvanishing density of states within
the minority-spin band gap of half-metals around the
Fermi level and reduce the spin polarization at the
EF. Due to closing, the gap total spin magnetic mo-
ment became equal to 5.99µB, slightly less than in-
teger value predicted in [6].
Figure 3. Band structure of Co2FeSi calculated in LSDA+U approximation. The Fermi energy
is at zero.
632
Electronic structure and x-ray magnetic circular dichroism in the Co2FeSi
3.2. XMCD spectra
At the core level edge XMCD is not only element-specific but also orbital specific. For 3d
transition metals, the electronic states can be probed by the K, L2,3 and M2,3 x-ray absorption
and emission spectra. Recently x-ray magnetic circular dichroism in the ferromagnetic Co2FeSi
alloy has been measured at the Co and Fe L2,3 edges [7]. The experimentally measured dichroic
lines have different signs at the L3 and L2 edges of Co and Fe [7].
In order to compare relative amplitudes of the L3 and L2 XMCD spectra we first normalize the
corresponding x-ray absorption spectra for the right polarized x-rays µ− to the experimental ones
taking into account the background scattering intensity as described in section 2. Figure 4 shows
the XAS and XMCD spectra at the L2,3 edges of Co calculated in the LSDA+U approach together
with the experimental data [7]. The corresponding spectra for Fe are presented in figure 5. The
contribution from the background scattering is shown by dotted line in the upper panel in figures 4
and 5.
X
M
C
D
(a
rb
.u
ni
ts
)
X
A
S
(a
rb
.u
ni
ts
)
L3
L2
Co
0
5
10
Theory aexper
Theory aLSDA
Exper.
770 780 790 800 810
Energy (eV)
-0.4
-0.2
0.0
0.2
Figure 4. (Color online) Upper panel: calculated in LSDA+U approximation for the experimen-
tal aexpr (blue full line) and theoretical atheory (red dashed lines) values of lattice parameter,
as well as experimental (circles) x-ray absorption spectra for the right polarized x-rays µ− of
Co2FeSi at the Co L2,3 edges. Experimental spectra [7] were measured by means of total photo-
electron yield with external magnetic field (1.6 T). Dotted line shows the theoretically calculated
background spectra. Low panel: theoretically calculated for the experimental aexpr (blue thin
full line) and theoretical atheory (red dashed lines) values of lattice parameter and experimental
[7] (circles) XMCD spectra of Co2FeSi at the Co L2,3 edges.
Due to the dipole selection rules, apart from the 4s1/2 states (which have a small contribution
to the XAS due to relatively small 2p→ 4s matrix elements) only 3d3/2 states occur as final states
for L2 XAS for unpolarized radiation, whereas for the L3 XAS, the 3d5/2 states also contribute
[27]. Although the 2p3/2 → 3d3/2 radial matrix elements are only slightly smaller than for the 2p3/2
→ 3d5/2 transitions, the angular matrix elements strongly suppress the 2p3/2 → 3d3/2 contribution
[27]. Therefore, neglecting the energy dependence of the radial matrix elements, the L2 and the
L3 spectrum can be viewed as a direct mapping of the DOS curve for 3d3/2 and 3d5/2 character,
respectively.
The experimental Co XAS has a pronounced shoulder at the L3 peak at around 785 eV shifted
by about 3 eV with respect to the maximum to higher photon energy. This structure is less
pronounced at the L2 edge. This result can be ascribed to the lifetime broadening effect because
633
V.N.Antonov et al.
L3
L2
Fe
0.0
0.4
0.8
In
te
ns
ity
(a
rb
.u
ni
ts
)
Theory aexper
Theory aLSDA
Exper.
710 720 730
Energy (eV)
-0.6
-0.3
0.0
X
M
C
D
(a
rb
.u
ni
ts
)
Figure 5. (Color online) Upper panel: calculated in LSDA+U approximation for experimental
aexpr (blue full line) and theoretical atheory values of lattice parameter (red dashed line), as well
as experimental (circles) x-ray absorption spectra for the right polarized x-rays µ− of Co2FeSi at
the Fe L2,3 edges. Experimental spectra [7] were measured by means of total photoelectron yield
with external magnetic field (1.6 T). Dotted line shows the theoretically calculated background
spectra. Low panel: theoretically calculated for experimental aexpr (blue full line) and theoretical
atheory values of lattice parameter (red dashed line) and experimental [7] (circles) XMCD spectra
of Co2FeSi at the Fe L2,3 edges.
the lifetime of the 2p1/2 core hole is shorter than the 2p3/2 core hole due to the L2L3V Coster-
Kronig decay. This feature is partly due to the interband transitions from 2p core level to Co 3d
empty states at around 4 eV above the Fermi level. Actually, as can be seen in figure 6, Co d partial
DOS’s have two pronounced peaks at 4 eV and approximately 6 eV above the Fermi level. Both
the features are reflected in the theoretically calculated XAS at the Co L3 edge around 785 and
789 eV, respectively (figure 4), although the second peak is less pronounced in the experimental
spectrum.
Figure 6. (Color online) The Co and Fe d-partial density of unoccupied states (in states/(atom
eV)) of Co2FeSi. Energies are relative to the Fermi energy.
Figure 5 presents the calculated XAS as well as XMCD spectra of the Co2FeSi compound at
the Fe L2,3 edges compared with the experimental data [7]. The peak in the empty Fe d partial
634
Electronic structure and x-ray magnetic circular dichroism in the Co2FeSi
DOS at 5.8 eV above the Fermi level is less intensive in comparison with the corresponding peak
in the Co d partial DOS (see figure 6). Therefore, the theoretically calculated XAS has only one
pronounced feature at the high energy part of the XAS at 717 eV (figure 5).
It may be seen in the upper part of figure 4 that the experimentally measured Co L3 XAS has
some additional intensity around 785 eV which is not completely reproduced by the theoretical
calculations. A similar situation also appears in the Fe L3 XAS (figure 5) where the theoretical
one-particle calculations do not reproduce all the intensity at around 717 eV. This might indicate
that additional satellite structures may appear due to many-body effects at the high energy tails
of both the Co and Fe L2,3 XAS’s. This issue needs additional theoretical investigation using an
appropriate many-body treatment.
The XMCD spectra at the L2,3 edges are mostly determined by the strength of the SO coupling
of the initial 2p core states and spin-polarization of the final empty 3d3/2 and 3d5/2 states while
the exchange splitting of the 2p core states as well as the SO coupling of the 3d valence states are
of minor importance for the XMCD at the L2,3 edge of 3d transition metals [27]. The calculated Co
and Fe L2,3 XMCD spectra are in good agreement with the experiment [7], although the calculated
magnetic dichroism is somewhat too high at the L2 edge. One of the reasons for this discrepancy
might be the core-hole effect. When the 2p core electron is photo-excited to the unoccupied d
states, the distribution of the charge changes to account for the created hole. This effect is not
taken into account in the method we used for the electronic structure calculations, and it is likely
to lead to the observed discrepancy [40].
To investigate the effect of lattice constant on the magnetic moments and XMCD spectra we
optimized the equilibrium lattice parameter a using LSDA total-energy calculations. The calculated
lattice constant a=5.545 Å is 2% smaller than the experimental value a=5.658 Å. Similar results
were obtained by Sargolzaei et al. [18]. For the theoretically calculated LSDA lattice constant, spin
magnetic moments are reduced by approximately 7% and 4% for Co and Fe, respectively. However,
LSDA+U approach still produces a half-metallic solution for the theoretically calculated lattice
constant. We found that the XMCD spectra are almost identical for the LSDA and experimentally
measured lattice constants (figures 4 and 5).
3.3. Magnetic moments
In magnets, the atomic spin Ms and orbital Ml magnetic moments are basic quantities. There-
fore, their separate determination is important. Methods of their experimental determination in-
clude traditional gyromagnetic ratio measurements [41], magnetic form factor measurements using
neutron scattering [42], and magnetic x-ray scattering [43]. In addition to these, the recently de-
veloped x-ray magnetic circular dichroism combined with several sum rules [23–26] has attracted
much attention as a method of site- and symmetry-selective determination of Ms and Ml.
Due to significant implications of the sum rules, numerous experimental and theoretical studies
have been reported aimed at investigating their validity for itinerant magnetic systems, but with
widely different conclusions. The claimed adequacy of the sum rules varies from very good (within
5% agreement) to very poor (up to 50% discrepancy) [27]. This lack of a consensus may have several
origins. For example, on the theoretical side, it has been demonstrated by circularly polarized 2p
resonant photoemission measurements of Ni that both the band structure effects and electron-
electron correlations are needed to satisfactorily account for the observed XMCD spectra [44].
However, it is extremely difficult to include both of them in a single theoretical framework. Besides,
the XAS as well as XMCD spectra can be strongly affected (especially for the early transition
metals) by the interaction of the excited electron with the created core hole [40,45].
On the experimental side, the indirect x-ray absorption techniques, i.e., the total electron
and fluorescence yield methods, are known to suffer from saturation and self-absorption effects
that are very difficult to correct for [46]. The total electron yield method can be sensitive to
the varying applied magnetic field, changing the electron detecting efficiency, or, equivalently,
the sample photocurrent. The fluorescence yield method is insensitive to the applied field, but
the yield is intrinsically not proportional to the absorption cross section, because the radiative
to non-radiative relative core-hole decay probability strongly depends on the symmetry and spin
635
V.N.Antonov et al.
Table 1. The experimental measured (using transmition mode (TM) and total electron yield
(TEY) methods) and calculated (for the experimentally measured lattice constant) spin Ms,
orbital Ml total Ms+Ml magnetic moments (in µB) of Co2FeSi (sum rules1 applied for the
XMCD spectra calculated neglecting the energy dependence of the radial matrix elements, sum
rules2 applied for the XMCD spectra calculated neglecting the energy dependence of the radial
matrix elements and neglecting p→ s transitions).
method atom Ms Ml Ms+Ml
Si –0.033 0.002 –0.031
LSDA Co 1.224 0.049 1.273
Fe 2.709 0.080 2.789
Si –0.025 0.002 –0.023
LSDA+OP Co 1.293 0.059 1.352
Fe 2.717 0.123 2.840
Si –0.036 0.008 –0.028
LSDA+U Co 1.622 0.111 1.733
Fe 2.663 0.145 2.808
Sum rules Co 1.122 0.072 1.194
Fe 1.861 0.096 1.957
Sum rules1 Co 1.526 0.110 1.636
Fe 2.363 0.141 2.504
Sum rules2 Co 1.554 0.112 1.666
Fe 2.394 0.144 2.538
Expt. (TM)[6] Co 1.2 ± 0.1
bulk sample Fe 2.6 ± 0.1
Expt. (TM) [7] Co 1.25 ÷ 1.28 0.11 ÷ 0.13 1.36 ÷ 1.41
thin films Fe 2.43 ÷ 2.46 0.07 ÷ 0.12 2.50 ÷ 2.58
Expt. (TEY)[7] Co 1.07 ÷ 1.13 0.04 ÷ 0.14 1.11 ÷ 1.27
thin films Fe 2.46 ÷ 2.47 0.05 ÷ 0.10 2.51 ÷ 2.57
polarization of the XAS final states [47].
The Fe L3 and L2 spectra in Co2FeSi are strongly overlapped. Therefore, the decomposition of a
corresponding experimental L2,3 spectrum into its L3 and L2 parts is quite difficult and can lead to
a significant error in the estimation of the magnetic moments using the sum rules (the integration
∫
L3
and
∫
L2
in equation (4) should be taken over the 2p3/2 and 2p1/2 absorption regions separately).
Besides, the experimentally measured Co and Fe L2,3 x-ray absorption spectra have background
scattering intensity and the integration of the corresponding XASs may lead to an additional error
in the estimation of the magnetic moments using the sum rules.
The value of the orbital magnetic moments derived from the experimental XMCD spectra is
considerably higher in comparison with our LSDA band structure calculations (Table 1). It is a
well-known fact, however, that LSDA calculations are inaccurate in describing orbital magnetism
[48,27]. In the LSDA, the Kohn-Sham equation is described by a local potential which depends on
the electron spin density. The orbital current, which is responsible for Ml, is, however, not included
in the equations. This means that although Ms is self-consistently determined in the LSDA, there
is no framework to determine simultaneously Ml self-consistently. Numerous attempts have been
made to better estimate Ml in solids. They can be roughly classified into two categories. One is
based on the so-called current density functional theory [49–51] which is intended to extend density
functional theory to include the orbital current as an extra degree of freedom, which describes Ml.
Unfortunately an explicit form of the current density functional is at present unknown. The other
category includes orbital polarization [52–55], self-interaction correction [56], and LSDA+U [57,22]
approaches.
To calculate Ml beyond the LSDA scheme we used the rotationally invariant LSDA+U method
636
Electronic structure and x-ray magnetic circular dichroism in the Co2FeSi
[22] using a double-counting correction term in the fully-localized limit approximation [58,59].
We used U = J = −0.89 eV for the transition metal sites. In this case Ueff = U − J = 0 and
the effect of the LSDA+U comes from non-spherical terms which are determined by F2 and F4
Slater integrals. This approach is similar to the orbital polarization (OP) correction [27]. The
LSDA+OP calculations (LSDA+U with Ueff = 0) produce orbital magnetic moments equal to
0.059 µB and 0.123 µB for Co and Fe sites, respectively. The Co LSDA+OP orbital moment is
somewhat smaller than the experimental estimates. However, the value for Fe is in good agreement
with the experimental data. A full LSDA+U method produces larger orbital magnetic moments
(0.111 µB and 0.145 µB for Co and Fe sites, respectively). We should mention that the shape of the
XMCD spectra in Co2FeSi is less sensitive to the orbital polarization correction in comparison with
the evaluated orbital magnetic moments. The XAS and XMCD spectra calculated in the LSDA,
LSDA+OP and LSDA+U approximations are almost of identical shape.
It is interesting to compare the spin and orbital moments obtained from the theoretically
calculated XAS and XMCD spectra through sum rules [equation (3),(4)] with directly calculated
LSDA+U values in order to avoid exra experimental problems. The number of the transition
metal 3d electrons is calculated by integrating the occupied d partial density of states inside the
corresponding atomic sphere which gives the values averaged for the nonequivalent sites nCo=7.742
and nFe=6.542. Sum rules reproduce the spin magnetic moments within 31%, and 30% and the
orbital moments within 35% and 34% for Co and Fe, respectively (Table 1). XMCD sum rules
are derived within an ionic model using a number of approximations. For L2,3, they are [48]: (1)
neglecting the exchange splitting of the core levels; (2) replacing the interaction operator α · aλ in
equation (1) by ∇ ·aλ; (3) neglecting the asphericity of the core states; (4) neglecting the difference
of d3/2 and d5/2 radial wave functions; (5) neglecting p → s transitions; (6) neglecting the energy
dependence of the radial matrix elements. To effect the effect of the last point we applied the sum
rules to the XMCD spectra neglecting the energy dependence of the radial matrix elements. As can
be seen in Table 1 using the energy independent radial matrix elements, the disagreement in spin
magnetic moments is reduced to 6% and 11% and in the orbital moment to 1% and 3% for Co and
Fe, respectively. Moreover, the omission of the p → s transitions leads to a quite good agreement
between the LSDA+U and sum rule results (within 4% and 10% for the spin moments at the Co
and Fe sites, respectively, and less than 1% for the orbital moments for both sites). These results
show that the energy dependence of the matrix elements and the presence of p → s transitions
strongly affect the values of both the spin and the orbital magnetic moments derived from the sum
rules.
4. Summary
We have studied the electronic structure and x-ray magnetic circular dichroism spectra of the
Heusler alloy Co2FeSi by means of an ab initio fully-relativistic spin-polarized Dirac linear muffin-
tin orbital method.
The spin-polarized LSDA+U calculations without the SO coupling show that Co2FeSi is a half-
metallic ferromagnet. The calculated total DOS shows a gap of 1.0 eV for minority spin electrons.
The gap in the minority bands is due to the Co–Fe interactions as the strongest bonding interacti-
ons. The spin-orbit coupling destroys the energy gap in the minority-spin states. Nevertheless, the
spin polarization of electron states at the Fermi energy in the Heusler alloys Co2FeSi is still high
(about 95%).
The band structure calculations in the LSDA+U approximation reproduce quite well the shape
of the XAS and XMCD spectra at the Co and Fe L2,3 edges. The energy dependence of the matrix
elements as well as the presence of p → s transitions strongly affect the values of both the spin
and the orbital magnetic moments derived from the calculated XAS and XMCD spectra using the
sum rules.
637
V.N.Antonov et al.
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Електронна структура та рентгенiвський магнiтний
циркулярний дихроїзм хойслерiвського сплаву Co2FeSi
В.Н.Антонов1, Д.А.Кукуста1, А.П.Шпак1, А.Н.Яресько2
1 Iнститут металофiзики Нацiональної Академiї Наук України, бульвар Вернадського, 36, Київ, 03142
2 Iнститут Макса Планка фiзики твердого тiла, Гейзенбергштрассе 1, 70569 Штуттгарт, Нiмеччина
Отримано 11 грудня 2007 р., в остаточному виглядi – 18 червня 2008 р.
Електронна структура та спектри рентгенiвского магнiтного циркулярного дихроїзму (РМЦД) хой-
слерiвського сплаву Co2FeSi були дослiдженi теоретично з перших принципiв за допомогою повнiс-
тю релятивiстського Дiракiвського лiнiйного методу ЛМТО. Проаналiзованi та обговоренi щiльностi
валентних електронiв, орбiтальнi та магнiтнi спiновi моменти, а також поляризацiя електронiв на рiв-
нi Фермi.Вивчається походження спектрiв РМЦД у Co2FeSi. Результати обчислень порiвнюються з
еспериментальними даними.
Ключовi слова: сильно скорельованi системи, електронна структура, рентгенiвский магнiтний
циркулярний дихроїзм
PACS: 71.28.+d, 71.25.Pi, 75.30.Mb
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