Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations
The electronic structure, magnetism as well as the excited-state properties such as the optical and x-ray magnetic circular dichroism (XMCD) spectra of the Heusler alloys Co₂TiSn and Co₂ZrSn were investigated theoretically from first principles using the fully relativistic Dirac LMTO band structu...
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Інститут фізики конденсованих систем НАН України
2005
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| Cite this: | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations / L.V. Bekenov, V.N. Antonov, A.P. Shpak, A.N. Yaresko // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 565–577. — Бібліогр.: 38 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859469255559348224 |
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| author | Bekenov, L.V. Antonov, V.N. Shpak, A.P. Yaresko, A.N. |
| author_facet | Bekenov, L.V. Antonov, V.N. Shpak, A.P. Yaresko, A.N. |
| citation_txt | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations / L.V. Bekenov, V.N. Antonov, A.P. Shpak, A.N. Yaresko // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 565–577. — Бібліогр.: 38 назв. — англ. |
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| description | The electronic structure, magnetism as well as the excited-state properties
such as the optical and x-ray magnetic circular dichroism (XMCD) spectra
of the Heusler alloys Co₂TiSn and Co₂ZrSn were investigated theoretically
from first principles using the fully relativistic Dirac LMTO band structure
method. The origin of the XMCD spectra at the Co L₂,₃ edges in the compounds
is examined. Densities of valence states, orbital and spin magnetic
moments as well as optical spectra are analyzed and discussed. The calculated
results are compared with the available experimental data.
Використовуючи повністю релятивістський діраківський ЛМТО метод зонних розрахунків було теоретично досліджено електронну
структуру, магнітні властивості та властивості збудженого стану, такі
як оптичні спектри та спектри рентгенівського магнітного циркулярного дихроїзму, сплавів Гейслера Co₂TiSn та Co₂ZrSn. з’ясовано
походження L₂,₃-спектрів рентгенівського циркулярного магнітного
дихроїзму на атомах кобальту. Проаналізовано та обговорено щільності валентних станів, величини орбітального та спінового магнітних моментів та оптичні спектри. Результати розрахунків порівняно з експериментальними даними.
|
| first_indexed | 2025-11-24T06:15:37Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 3(43), pp. 565–577
Electronic structure and excited-state
properties of Co2TiSn and Co2ZrSn
from ab initio calculations
L.V.Bekenov 1 , V.N.Antonov 1 , A.P.Shpak 1 , A.N.Yaresko 2
1 Institute of Metal Physics,
36 Vernadsky Street, 252142 Kiev, Ukraine
2 Max Planck Institute for Physics of Complex Systems,
D–01187 Dresden, Germany
Received June 24, 2005
The electronic structure, magnetism as well as the excited-state properties
such as the optical and x-ray magnetic circular dichroism (XMCD) spectra
of the Heusler alloys Co2TiSn and Co2ZrSn were investigated theoretically
from first principles using the fully relativistic Dirac LMTO band structure
method. The origin of the XMCD spectra at the Co L2,3 edges in the com-
pounds is examined. Densities of valence states, orbital and spin magnetic
moments as well as optical spectra are analyzed and discussed. The cal-
culated results are compared with the available experimental data.
Key words: electronic structure, Heusler alloys, x-ray magnetic circular
dichroism, optics
PACS: 71.28.+d, 71.25.Pi, 75.30.Mb
1. Introduction
Heusler intermetallic alloys have attracted great interest during the last century
due to the possibility to study, in the same family of alloys, a series of interesting
diverse magnetic phenomena like itinerant and localized magnetism, antiferromag-
netism, helimagnetism, Pauli paramagnetism, or heavy-fermionic behaviour [1–3].
Recently the rapid development of spintronics [4–7] intensified the interest in the
Heusler alloys [8]. Most magnetoelectronic devices rely on an imbalance in the num-
ber of majority and minority spin carriers, with the ideal material exhibiting a com-
plete (100%) spin-polarization at the Fermi surface (i.e., a half-metallic ferromagnet)
[9], and some of the Heusler compounds have been predicted from first-principles
calculations to be half-metallic [9–11].
Theoretical and experimental investigations of the Co2TiSn and Co2ZrSn Heusler
alloys have been performed for a few decades [12–19]. Co2TiSn and Co2ZrSn have
c© L.V.Bekenov, V.N.Antonov, A.P.Shpak, A.N.Yaresko 565
L.V.Bekenov et al.
been found to be ferromagnets with Curie temperatures (TC) around 371 and 448 K,
respectively [12,15]. The saturation magnetic moments estimated from magnetiza-
tion measurements at 4.2 K in a field of 18 kOe for Co2TiSn and Co2ZrSn are
reported to be 1.96 µB and 1.46 µB per formula unit (f.u.), respectively [15]. Very
similar results were obtained by the authors of [18], whose high-field magnetization
measurements up to 150 kOe at 4.2 K gave 1.92 µB/f.u. for Co2TiSn and 1.64 µB/f.u.
for Co2ZrSn. The ferromagnetic behaviour of the considered alloys results from the
presence of a large amount of the magnetic Co atoms. It is interesting to note that
the corresponding Ni compounds are Pauli paramagnets, and by varying the Ni/Co
ratio, it is possible to change their paramagnetic behaviour to ferromagnetic one;
for Ni2−xCoxTiSn such a change occurs at x = 1.2 [16].
The electronic structure calculations of the Co2TiSn alloy using the symmetrized
augmented plane-wave (SAPW) method were carried out in [14]. It was found that
the Co d bands are characterized by the hump near the Fermi level of the DOS curves
and are mainly occupied for both spin states, while the major peaks of the Ti d states
lie in the high energy region and are mainly unoccupied in both spin states. The
reported evaluated total magnetic moment values strongly depend on the exchange
parameters which were used to form the crystal potential from a superposition of
the atomic potentials. The authors of [17] presented the outcome of high-resolution
x-ray photoemission spectroscopy measurements of the Co2ZrSn valence bands in
comparison with the total density of states distribution curves obtained from their
band structure calculations of Co2ZrSn, which were fulfilled by means of the spin-
polarized tight-binding linear muffin-tin orbital (TBLMTO) method, and found a
very good agreement between the experimental and theoretical results. The total
magnetic moment was obtained to be 1.74 µB/f.u..
The x-ray magnetic circular dichroism technique developed in recent years has
evolved into a powerful magnetometry tool to separate orbital and spin contribu-
tions to element specific magnetic moments. X-ray magnetic circular dichroism ex-
periments measure the absorption of x-rays with opposite (left and right) states of
circular polarization. Recently x-ray magnetic circular dichroism in Co2TiSn and
Co2ZrSn has been measured at the Co L2,3 edges [18]. Using the magneto-optic sum
rules the spin moment and the orbital moment of Co for Co2TiSn have been de-
duced to be 0.87 and 0.09 µB, respectively, and for Co2ZrSn 0.7 and 0.12 µB. The
authors also calculated the Co 3d partial density of states using the LMTO-ASA
band structure method to interpret the XMCD spectra. However, they did not in-
clude the spin-orbit (SO) interaction in their calculations and, hence, were not able
to obtain the orbital moment.
Optical properties investigations of Co2TiSn and Co2ZrSn exhibit abnormal be-
haviour of the real part of the dielectric function ε1(ω), which becomes negative
only at the wave lengths greater than 4 µm for Co2TiSn and 13 µm for Co2ZrSn and
which module values |ε1| are comparatively low, and the real part of the optical con-
ductivity σ1(ω), which does not show the Drude-like behaviour at low energies [19].
The aim of this work is the theoretical investigation of the electronic structure of
Co2TiSn and Co2ZrSn Heusler alloys using the fully relativistic Dirac LMTO band
566
Electronic structure and excited-state properties . . .
structure method. We calculated the x-ray absorption as well as the x-ray magnetic
circular dichroism spectra in these compounds and the energy dependences of some
of the optical characteristics.
This paper is organized as follows. Section 2 presents a description of the Co2TiSn
and Co2ZrSn Heusler alloys crystal structure and the computational details. Secti-
on 3 is devoted to the electronic structure, XMCD spectra, magnetic and optical
properties of the alloys calculated in the fully relativistic Dirac LMTO band struc-
ture method. Finally, the results are summarized in section 4.
2. Crystal structure and computational details
The Heusler-type Co2TiSn and Co2ZrSn have a cubic L21 structure (space group
Fm3m), which can be thought of as a simple cubic lattice for Co atoms, with the Sn
and Ti or Zr atoms arranged at alternate body centered positions. The Co, Ti and
Zr atoms have eight nearest neighbors at the same distance. Ti and Zr has eight Co
atoms as nearest neighbors, while for Co there are four Ti or Zr and four Sn atoms.
The electronic structures of the alloys were calculated for the experimentally ob-
served lattice constants [13], which are a = 11.4517 a.u. for Co2TiSn and a = 11.8051
a.u. for Co2ZrSn, using the spin-polarized fully relativistic linear-muffin-tin-orbital
(SPR LMTO) method [20,21] in the atomic sphere approximation (ASA) with the
combined correction term taken into account. The LSDA part of the calculations
was based on the spin-density functional with the von Barth-Hedin parametrizati-
on [22] of the exchange-correlation potential. Brillouin zone (BZ) integrations were
performed using the improved tetrahedron method [23] and charge self-consistency
was obtained on a grid of 641 k points in the irreducible wedge of the BZ. The basis
consisted of s, p and d LMTO’s.
Once energies εkn and functions |kn〉 for the n bands are obtained self-consistently,
the interband contribution to the imaginary part of the dielectric tensor ε2(ω) can
be calculated by summing transitions from occupied to unoccupied states (with fi-
xed k vector) over the BZ, weighted with the appropriate matrix element for the
probability of the transition. To be specific, the components of ε2(ω) are given by
εij
2 (ω) =
V e2
2πh̄m2ω2
∫
d3k
∑
nn′
〈kn|pi|kn′〉 〈kn′|pj|kn〉
× fkn(1 − fkn′)δ(εkn′ − εkn − h̄ω). (1)
Here (px, py, pz) = p is the momentum operator and fkn is the Fermi distribution.
Further details about the evaluation of matrix elements are given elsewhere [24,25].
The real part of the components of the dielectric tensor ε1(ω) is then calculated us-
ing the Kramer-Kroning transformation. Knowledge of both the real and imaginary
parts of the dielectric tensor permits the calculation of important optical constants.
The calculations yield unbroadened functions. To reproduce the experimental condi-
tions more correctly, it is necessary to broaden the calculated spectra, although the
exact form of the broadening function is unknown. Also the instrumental resolution
567
L.V.Bekenov et al.
smears out many fine features. To simulate these effects the lifetime broadening was
simulated by convoluting the calculated optical constants with a Lorentzian, and
the experimental resolution was simulated by broadening the final spectra with a
Gaussian.
In order to calculate the XMCD properties one has to take into account both
magnetism and SO coupling when dealing with the electronic structure of the mate-
rial considered, since the symmetry reduction in comparison with the paramagnetic
state, which causes XMCD effect and is due to magnetic ordering, has consequences
only when SO coupling is considered in addition [26].
Within the one-particle approximation, the absorption coefficient µ for incident
x-rays of polarization λ and photon energy h̄ω can be determined as the probability
of electron transitions from an initial core state (with wave function ψj and energy
Ej) to the final unoccupied states (with wave functions ψnk and energies Enk)
µλ
j (ω) =
∑
nk
|〈Ψnk|Jλ|Ψj〉|2δ(Enk − Ej − h̄ω)θ(Enk − EF) , (2)
with Jλ = −eαaλ being the dipole electron-photon interaction operator, where 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 absorpti-
on L2,3 spectra of Co2TiSn and Co2ZrSn with the experimental ones we take into
account the background intensity which affects the high energy part of the spectra.
The shape of x-ray absorption caused by the transitions from inner levels to the
continuum of unoccupied levels was first discussed by Richtmyer et al. in the early
thirties [27]. The absorption coefficient with the assumption of equally distributed
empty continuum levels is
µ(ω) =
CΓc
2π
∫
∞
Ecf0
dEcf
(Γc/2)2 + (h̄ω − Ecf )2
, (3)
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.
Finally, the intrinsic broadening mechanisms have been considered by folding
XMCD spectra with a Lorentzian. For the finite lifetime of the core hole a constant
width Γc, in general from [28], has been used. The finite apparative resolution of the
spectrometer has been considered by means of a Gaussian.
Concurrent with the x-ray magnetic circular dichroism experimental develop-
ments, some important magneto-optical sum rules have been derived in recent
years [29].
568
Electronic structure and excited-state properties . . .
For the L2,3 edges the lz sum rule can be written as
〈lz〉 = nh
4
∫
L3+L2
dω(µ+ − µ−)
3
∫
L3+L2
dω(µ+ + µ−)
, (4)
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 is written as
〈sz〉 +
7
2
〈tz〉 = nh
∫
L3
dω(µ+ − µ−) − 2
∫
L2
dω(µ+ − µ−)∫
L3+L2
dω(µ+ + µ−)
, (5)
where tz is the z component of the magnetic dipole operator t = s − 3r(r · s)/|r|2
which takes into account the asphericity of the spin moment. The integration
∫
L3
(
∫
L2
) is taken only over the 2p3/2 (2p1/2) absorption region.
3. Results and discussion
3.1. Electronic structure
The fully relativistic spin-projected energy band structures and total DOS of
Co2TiSn and Co2ZrSn obtained from our LSDA calculations are presented in figure 1.
As one can see, the band structures of both alloys are quite similar, though it is
evident that there are some differences in the DOS of these Heusler compounds due
to both the presence of different atoms and different volumes. The occupied part of
spin-down
-10
-5
0
5
E
ne
rg
y
(e
V
)
Γ X W K Γ L W U X 0 5 10
Co2TiSn
spin-up
DOS
-10
-5
0
5
E
ne
rg
y
(e
V
)
Γ X W K Γ L W U X 0 5 10
spin-down
-10
-5
0
5
E
ne
rg
y
(e
V
)
Γ X W K Γ L W U X 0 5 10
Co2ZrSn
spin-up
DOS
-10
-5
0
5
E
ne
rg
y
(e
V
)
Γ X W K Γ L W U X 0 5 10
Figure 1. Self-consistent fully relativistic spin-polarized LSDA energy band struc-
tures and total DOS (in states/(cell eV)) of Co2TiSn and Co2ZrSn.
569
L.V.Bekenov et al.
Co2TiSn
P
ar
tia
l
de
ns
ity
of
st
at
es
(s
ta
te
s/
(a
to
m
eV
))
spin-up
spin-down
Sn s
Sn p
-1
0
1
Co d
e
t2-2
0
2
Ti d
eg
t2g
-9 -6 -3 0 3 6
Energy (eV)
-2
0
2
Co2ZrSn
spin-up
spin-down
Sn s
Sn p
-1
0
1
Co d
e
t2-2
0
2
Zr d
eg
t2g
-9 -6 -3 0 3 6
Energy (eV)
-2
0
2
Figure 2. Symmetry separated partial density of states of Co2TiSn and Co2ZrSn.
the valence bands can be subdivided into several regions. The lowest valence band
appeared in both the majority and minority spin states between −11.7 and −9 eV
for Co2TiSn and between −11.4 and −9 eV for Co2ZrSn is entirely due to the tin 5s
electrons and is separated with respect to the other hybridized bands, being basically
unaffected by the Co and Ti or Zr exchange interaction. The next three energy bands
in the energy region between approximately −7 and −3 eV are the tin 5p bands.
The upper dispersed bands, which are located above and below EF from about −3
to 5 eV, are due to the strong hybridization of Co and Ti or Zr d energy bands. The
corresponding spin-projected partial densities of states are shown in figure 2.
Due to the strong Sn p–p hybridization, Sn p states are split into bonding and
antibonding states. The former are located between approximately −7 and −1 eV,
while the latter are spread over a broad energy range above −1 eV. The centers of
Ti and Zr d states, defined as the energy at which the corresponding logarithmic
derivative is equal to −l− 1, lie at εν = 0.76 eV and εν = 1.76 eV, respectively. The
crystal field at the Ti and Zr 4a site (Oh point symmetry) splits their d states into
eg and t2g ones. The eg states, which form σ bonds with Sn p states, are strongly
hybridized with the latter and give a significant contribution to the bonding states
below −3 eV. The t2g states form weaker Ti d – Sn p and Zr d – Sn p π bonds but
they strongly hybridize with d states of eight Co nearest neighbors.
570
Electronic structure and excited-state properties . . .
The center of Co d states (εν = −1.19 eV for Co2TiSn and εν = −1.08 eV
for Co2ZrSn) is found in a gap between the bonding and antibonding Sn p states.
The crystal field at the Co 4b site (Td point symmetry) causes the splitting of d
orbitals into a doublet e (d3z2−1 and dx2−y2) and a triplet t2 (dxy, dyz, and dxz). The
hybridization between Co t2 and Ti t2g states in Co2TiSn and Co t2 and Zr t2g states
in Co2ZrSn causes the splitting of the Ti and Zr t2g states into two peaks, the bonding
ones located at ∼2 eV below the Fermi level and the unoccupied antibonding peaks
centered at ∼1.5 eV for Co2TiSn and at ∼2.5 eV for Co2ZrSn. The Co e orbitals
are split into well separated bonding and antibonding peaks. The bonding states are
rather strongly hybridized with Zr, Ti and Sn states, while the antibonding states
provide a peak in the minority spin band which is located very close to the Fermi
level and is quite similar between Co2TiSn and Co2ZrSn.
In addition to the crystal field splitting, the d levels of the Co, Ti and Zr atoms
are split due to the exchange interaction. The exchange splitting between the spin-up
and -down d electrons on the Co atom is about 0.5 eV. The corresponding splitting
on the Ti and Zr atom is much smaller. Spin-orbit splitting of the d energy bands for
the Co, Ti and Zr atoms is much smaller than their spin and crystal-field splittings.
In the vicinity of the Fermi energy the minority spin bands of both alloys show a
gap (at E = −0.18 eV for Co2TiSn and at E = −0.14 eV for Co2ZrSn). However,
the Fermi level crosses both the majority and minority spin energy bands, so neither
Co2TiSn nor Co2ZrSn are half-metallic ferromagnets.
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 whereas in 4d transition metals one can use
the K, L2,3, M2,3 and M4,5 spectra.
The experimentally measured dichroic lines have different signs at the L3 and L2
edges [18]. In order to compare relative amplitudes of the L3 and L2 XMCD spectra
we first normalize the corresponding isotropic x-ray absorption spectra (XAS) to
the experimental ones taking into account the background scattering intensity as
described in section 2. Figure 3 shows the calculated isotropic x-ray absorption and
XMCD spectra of Co at the L2,3 edges for both alloys in the LSDA approach together
with the experimental data [18]. The contribution from the background scattering
is shown by dotted line in the XAS panels of figure 3.
Because of the dipole selection rules, apart from the 4s1/2 states (which have a
small contribution to the XAS due to relatively small 2p → 4s matrix elements) only
3d3/2 states occur as final states for L2 XAS for unpolarized radiation, whereas for the
L3 XAS 3d5/2 states also contribute [29]. 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 [29]. Therefore in
neglecting the energy dependence of the radial matrix elements, the L2 and the L3
spectra can be viewed as a direct mapping of the DOS curve for 3d3/2 and 3d5/2
character, respectively. The XMCD spectra at the L2,3 edges are mostly determined
571
L.V.Bekenov et al.
Co2TiSn
X
M
C
D
(a
rb
.u
ni
ts
)
X
A
S
(a
rb
.u
ni
ts
)
L3
L2
0
50
100
L3
L2
theory
exper.
0 10 20
Energy (eV)
-50
0
Co2ZrSn
L3
L2
0
50
100
L3
L2
theory
exper.
0 10 20
Energy (eV)
-50
0
Figure 3. Theoretically calculated (full line) and experimental (triangles) isotrop-
ic absorption and XMCD spectra of Co2TiSn and Co2ZrSn at the Co L2,3 edges.
Experimental spectra [18] were measured at 50 K. The XAS panel also shows the
background spectra (dotted line) due to the transitions from inner 2p1/2,3/2 levels
to the continuum of unoccupied levels [27].
by the strength of the SO coupling of the initial 2p core states and spin-polarization
of the final empty 3d3/2,5/2 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 [29]. The theoretically calculated
Co L2,3 XMCD spectra are in good agreement with the experiment, although the
calculated magnetic circular dichroism is somewhat too high at the L2 edge. The
main reason for this discrepancy is 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 hole created. This effect is not taken into account by the electronic
structure calculations and leads to the observed discrepancy [30].
3.3. Magnetic properties
In magnets, the atomic spin Ms and orbital Ml magnetic moments are basic
quantities and their separate determination is therefore important. Methods of their
experimental determination include traditional gyromagnetic ratio measurements
[31], magnetic form factor measurements using neutron scattering [32], and mag-
netic x-ray scattering [33]. In addition to these, the recently developed x-ray mag-
netic circular dichroism combined with several sum rules [34,35] has attracted much
attention as a method of site- and symmetry-selective determination of Ms and Ml.
Table 1 presents the comparison between calculated and experimental magnetic mo-
572
Electronic structure and excited-state properties . . .
Table 1. The experimental and calculated spin Ms and orbital Ml magnetic mo-
ments (in µB) of Co2TiSn and Co2ZrSn.
atom Ms Ml atom Ms Ml
Co2TiSn Co 0.741 0.037 Co2ZrSn Co 0.864 0.052
Ti -0.101 0.010 Zr -0.107 0.009
Sn 0.021 0.001 Sn 0.029 0.001
sum rules Co 0.525 0.024 sum rules Co 0.604 0.034
sum rulesa Co 0.692 0.036 sum rulesa Co 0.795 0.050
sum rulesb Co 0.707 0.037 sum rulesb Co 0.811 0.051
experimentc Co 0.87 0.09 experimentc Co 0.70 0.12
asum rules applied for the XMCD spectra calculated with ignoring the energy dependence of
the radial matrix elements.
bsum rules applied for the XMCD spectra calculated with ignoring the energy dependence of
the radial matrix elements and ignoring p → s transitions.
cReference [18].
ments in Co2TiSn and Co2ZrSn. The spin magnetic moment at the tin site is very
small. The spin moment at the Ti and Zr sites is also small and has an opposite
direction to the spin moment at the Co sites. In Co2ZrSn the spin moment on the Co
atom is larger than in Co2TiSn; this can be ascribed to the larger lattice constant
of the Co sublattice in Co2ZrSn, which results in a smaller Co 3d’s direct hybridi-
zation and a consequently larger exchange interaction. It is generally believed in
transition-metal compounds that the orbital contribution becomes larger when the
3d states are more localized [36]. Therefore, our results for the orbital moment also
suggest that the Co 3d states are more localized in Co2ZrSn than in Co2TiSn. The
calculated total magnetic moment for Co2TiSn is 1.402 µB/f.u. and for Co2ZrSn it
is 1.650 µB/f.u. while the experimental values [18] for these alloys are 1.92 µB/f.u.
and 1.64 µB/f.u., respectively.
It is interesting to compare the spin and orbital moments obtained from the the-
oretically calculated XAS and XMCD spectra through sum rules [Equations (4),(5)]
with directly calculated LSDA values. In this case we at least avoid all the experi-
mental problems. The number of the Co 3d electrons is calculated by integrating the
occupied d partial density of states inside the corresponding atomic sphere which
gives the values nCo=7.871 for Co2TiSn and nCo=7.892 for Co2ZrSn. Sum rules re-
produce the spin magnetic moments within 30% and the orbital moments within
35% for both Co2TiSn and Co2ZrSn (table 1). XMCD sum rules for L2,3 are derived
within an ionic model using a series of approximations, particularly disregarding
the energy dependence of the radial matrix elements and p → s transitions [37]. To
investigate the effect of these two factors we applied the sum rules to the XMCD
spectra calculated neglecting the energy dependence of the radial matrix elements
and the p → s transitions. As can be seen from table 1 using the energy independent
radial matrix elements reduces the disagreement in spin magnetic moments to 7%
and 8% and in the orbital moment to 3% and 4% for Co2TiSn and Co2ZrSn, re-
573
L.V.Bekenov et al.
spectively. An additional omission of the p → s transitions reduces the discrepancy
between LSDA and the sum rule results up to 5% and 6% for the spin moments and
∼%0 and 3% for the orbital moments for Co2TiSn and Co2ZrSn, respectively. These
results show that the energy dependence of the matrix elements and the presence of
s → p transitions strongly affect the values of both the spin and the orbital magnetic
moments derived using the sum rules.
The values of the orbital magnetic moment derived from the experimental XMCD
spectra (M exp
l =0.09 µB for Co2TiSn and M exp
l =0.12 µB for Co2ZrSn [18]) are consid-
erably higher in comparison with our band structure calculations. It is a well-known
fact, however, that LSDA calculations be inaccurate in describing orbital magnetism
[29,37]. 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 simulta-
neously Ml self-consistently. To calculate Ml beyond the LSDA scheme we used the
rotationally invariant LSDA+U method [38]. We used Ueff = 0 (U = J = 1.0 eV). In
this case the effect of the LSDA+U comes from non-spherical terms and the approach
is similar to the orbital polarization corrections [29]. The LSDA+U calculations pro-
duced the orbital magnetic moments for Co2TiSn and Co2ZrSn equal to 0.057 µB
and 0.074 µB per Co atom, respectively. These values are in better agreement with
the experimental data but still smaller than the experimental estimations.
Co2TiSn
theory
exper.
0.6
0.9
R
ef
le
ct
iv
ity
0 1 2 3 4 5
Energy (eV)
30
60
σ1
xx
,1
01
4
s-
1
Co2ZrSn
theory
exper.
0.3
0.6
0.9
0 1 2 3 4 5
Energy (eV)
30
60
Figure 4. Theoretically calculated (full line) and experimentally measured [19]
(triangles) reflectivity R(ω) and diagonal σ1
xx absorptive part of optical conduc-
tivity tensor for Co2TiSn and Co2ZrSn.
574
Electronic structure and excited-state properties . . .
3.4. Optical properties
The optical properties of materials originate from interband transitions from
occupied to unoccupied bands, involving not only the occupied and unoccupied
parts of the electronic structure but also the character of the bands.
Figure 4 shows the results of our calculations of the reflectivity R(ω) and the
diagonal σ1
xx(ω) absorptive part of optical conductivity tensor in comparison with
the experimental energy dependences of these constants [19]. The theoretical reflec-
tivity spectra are in good agreement with the experimental data. Both theory and
experiment give a rather sharp slope of reflectivity at low energies and then a slow
decreasing behaviour. Theory overestimates the slope region within about 0.5 eV
for both alloys and the reflectivity values in that region.
The positions of the σ1
xx maxima for both alloys are about 1 eV overestimated
by theory, and their theoretical values are larger than the experimental ones. The
theoretical spectra of σ1
xx at low energies have a pronounced Drude-like behaviour,
while the experimental ones exhibit a decrease of the Drude contribution to σ1
xx. The
main reason of this discrepancy is an arrangement disorder of the Sn, Ti and Zr atoms
in the experimentally investigated alloys [19], which results in a sharp increase of
the conduction electron scattering. The authors of [19] also believe that the decrease
of the Drude contribution to σ1
xx observed in the experimentally considered alloys
is partly caused by their half-metallic nature. However, as we have pointed out, our
fully relativistic calculations predict that Co2TiSn and Co2ZrSn are not half-metallic
ferromagnets.
4. Summary
We have studied by means of a fully relativistic spin-polarized Dirac linear muffin-
tin orbital method the electronic and magnetic structures as well as optical prop-
erties and x-ray magnetic circular dichroism spectra of Co2TiSn and Co2ZrSn. The
most characteristic features of the Co d states are an antibonding peak in the mi-
nority spin band which is located very close to the Fermi level and is quite similar
between Co2TiSn and Co2ZrSn, and an energy gap at E = −0.18 eV for Co2TiSn
and at E = −0.14 eV for Co2ZrSn in the minority spin band. The spin moment
in both alloys has a significant value only for the Co atoms, and for Co2ZrSn it
is greater than for Co2TiSn; its values for the other atoms are close to zero. The
total magnetic moment of Co2ZrSn is very close to its experimental value and in
contrast to the experiment is larger than that of Co2TiSn, while for the latter it is
27% smaller than the experimental result. The calculated reflectivity spectra are in
good agreement with the experimental data. The theoretical spectra of σ1
xx at low
energies in contrast to the experiment have a pronounced Drude-like behaviour. The
x-ray absorption and XMCD spectra at the Co L2,3 edges are reproduced quite well
by our LSDA band structure calculations, although the calculated magnetic circular
dichroism is somewhat too high at the L2 edge due to the core-hole effect.
575
L.V.Bekenov et al.
Acknowledgements
This work was supported by the CRDF program, project No. 14589.
References
1. Webster P.J., Ziebeck K.R.A., Alloys and Compounds of d-Elements with Main Group
Elements. Part 2, edited by Wijn H.R.J., vol. 19/C of Landolt-Börnstein, New Series,
Group III, 75–184, Springer-Verlag, Berlin, 1988.
2. Pierre J., Skolozdra R.V., Tobola J., Kaprzyk S., Hordequin C., Kouacou M.A.,
Karla I., Currat R., Leliévre-Berna E., J. Alloys Comp., 1997, 262–263, 101.
3. Tobola J., Pierre J., J. Alloys Comp., 2000, 296, 243.
4. Prinz G.A., Science, 1998, 282, 1660.
5. Ohno Y., Young D.K., Beschoten B., Matsukura F., Ohno H., Awschalom D.D.,
Nature, 1999, 402, 790.
6. Dietl T., Ohno H., Matsukura F., Cibert J., Ferrand D., Science, 2000, 287, 1019.
7. Park J.H., Vescovo E., Kim H.J., Kwon C., Ramesh R., Venkatesan T., Nature, 1998,
392, 794.
8. Deb A., Itou M., Sakurai Y., Hiraoka N., Sakai N., Phys. Rev. B, 2001, 63, 064409.
9. de Groot R.A., Mueller F.M., van Engen P., Buschow K.H.J., Phys. Rev. Lett., 1983,
50, 2024.
10. Pickett W., Phys. World, 1998, 11, 22.
11. Galanakis I., Ostanin S., Alouani M., Dreyssé H., Wills J.M., Phys. Rev. B, 2000, 61,
4093.
12. Ziebeck K.R.A., Webster P.J., J. Phys. Chem. Solids, 1974, 35, 1.
13. Gorlich E.A., Kmiec R., Latka K., Matlak T., Ruebenbauer K., Szytula A., Tomala K.,
Phys. Status Solidi A, 1975, 30, 765.
14. Ishida S., Akazawa S., Kubo Y., Ishida J., J. Phys. F: Met. Phys., 1982, 12, 1111.
15. van Engen P.G., Buschow K.H.J., Erman M., J. Magn. Magn. Mater., 1983, 30, 374.
16. Pierre J., Skolozdra R.V., Stadnyk Y.V., J. Magn. Magn. Mater., 1993, 128, 93.
17. Ślebarski A., Jezierski A., Lütkehoff S., Neumann M., Phys. Rev. B, 1998, 57, 6408.
18. Yamasaki A., Imada S., Arai R., Utsunomiya H., Suga S., Muro T., Saitoh Y.,
Kanomata T., Ishida S., Phys. Rev. B, 2002, 65, 104410.
19. Shreder E.I., Kirillova M.M., Dyakina V.P., Phys. Met. Metallogr., 2000, 90, No. 4,
48 (in Russian).
20. Andersen O.K., Phys. Rev. B, 1975, 12, 3060.
21. Nemoshkalenko V.V., Krasovskii A.E., Antonov V.N., Antonov V.N., Fleck U.,
Wonn H., Ziesche P., Phys. Status Solidi B, 1983, 120, 283.
22. von Barth U., Hedin L., J. Phys. C, 1972, 5, 1629.
23. Blöchl P.E., Jepsen O., Andersen O.K., Phys. Rev. B, 1994, 49, 16.
24. Alouani M., Wills J.M., Phys. Rev. B, 1996, 54, 2480.
25. Ahuja R., Auluck S., Wills J.M., Alouani M., Johansson B., Eriksson O., Phys. Rev. B,
1997, 55, 4999.
26. Kleiner W.H., Phys. Rev., 1966, 142, 318.
27. Richtmyer F.K., Barnes S.W., Ramberg E., Phys. Rev., 1934, 46, 843.
28. Fuggle J.C., Inglesfield J.E., Unoccupied Electronic States. Topics in Applied Physics,
vol. 69. Springer, New York, 1992.
576
Electronic structure and excited-state properties . . .
29. Antonov V., Harmon B., Yaresko A., Electronic structure and magneto-optical prop-
erties of solids. Kluwer Academic Publishers, Dordrecht, Boston, London, 2004.
30. Schwitalla J., Ebert H., Phys. Rev. Lett., 1998, 80, 4586.
31. Scott G.G., J. Phys. Soc. Jpn., 1962, 17, 372.
32. Marshall W., Lovsey S.W., Theory of Thermal Neutron Scattering. Oxford University
Press, Oxford, 1971.
33. Blume M., J. Appl. Phys., 1985, 57, 3615.
34. Thole B.T., Carra P., Sette F., van der Laan G., Phys. Rev. Lett., 1992, 68, 1943.
35. Carra P., Thole B.T., Altarelli M., Wang X., Phys. Rev. Lett., 1993, 70, 694.
36. Okutani M., Jo T., J. Phys. Soc. Jpn., 2000, 69, 598.
37. Ebert H., Rep. Prog. Phys., 1996, 59, 1665.
38. Yaresko A.N., Antonov V. N., Fulde P., Phys. Rev. B, 2003, 67, 155103.
Електронна структура та властивості збудженого
стану Co2TiSn та Co2ZrSn, отримані з ab initio
розрахунків
Л.В.Бекенов 1 , В.М.Антонов 1 , А.П.Шпак 1 ,
О.М.Яресько 2
1 Інститут металофізики ім. Г.В.Курдюмова НАН України,
бульв. Вернадського 36, 03142 Київ
2 Інститут Макса Планка Комплексних систем,
D–01187 Дрезден, Німеччина
Отримано 24 червня 2005 р.
Використовуючи повністю релятивістський діраківський ЛМТО ме-
тод зонних розрахунків було теоретично досліджено електронну
структуру, магнітні властивості та властивості збудженого стану, такі
як оптичні спектри та спектри рентгенівського магнітного цирку-
лярного дихроїзму, сплавів Гейслера Co2TiSn та Co2ZrSn. з’ясовано
походження L2,3-спектрів рентгенівського циркулярного магнітного
дихроїзму на атомах кобальту. Проаналізовано та обговорено щіль-
ності валентних станів, величини орбітального та спінового магніт-
них моментів та оптичні спектри. Результати розрахунків порівняно
з експериментальними даними.
Ключові слова: електронна структура, сплави Гейслера,
рентгенівський магнітний циркулярний дихроїзм, оптичні
властивості
PACS: 71.28.+d, 71.25.Pi, 75.30.Mb
577
578
|
| id | nasplib_isofts_kiev_ua-123456789-119751 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-11-24T06:15:37Z |
| publishDate | 2005 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Bekenov, L.V. Antonov, V.N. Shpak, A.P. Yaresko, A.N. 2017-06-08T11:11:51Z 2017-06-08T11:11:51Z 2005 Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations / L.V. Bekenov, V.N. Antonov, A.P. Shpak, A.N. Yaresko // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 565–577. — Бібліогр.: 38 назв. — англ. 1607-324X PACS: 71.28.+d, 71.25.Pi, 75.30.Mb DOI:10.5488/CMP.8.3.565 https://nasplib.isofts.kiev.ua/handle/123456789/119751 The electronic structure, magnetism as well as the excited-state properties such as the optical and x-ray magnetic circular dichroism (XMCD) spectra of the Heusler alloys Co₂TiSn and Co₂ZrSn were investigated theoretically from first principles using the fully relativistic Dirac LMTO band structure method. The origin of the XMCD spectra at the Co L₂,₃ edges in the compounds is examined. Densities of valence states, orbital and spin magnetic moments as well as optical spectra are analyzed and discussed. The calculated results are compared with the available experimental data. Використовуючи повністю релятивістський діраківський ЛМТО метод зонних розрахунків було теоретично досліджено електронну структуру, магнітні властивості та властивості збудженого стану, такі як оптичні спектри та спектри рентгенівського магнітного циркулярного дихроїзму, сплавів Гейслера Co₂TiSn та Co₂ZrSn. з’ясовано походження L₂,₃-спектрів рентгенівського циркулярного магнітного дихроїзму на атомах кобальту. Проаналізовано та обговорено щільності валентних станів, величини орбітального та спінового магнітних моментів та оптичні спектри. Результати розрахунків порівняно з експериментальними даними. This work was supported by the CRDF program, project No. 14589. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations Електронна структура та властивості збудженого стану Co₂TiSn та Co₂ZrSn, отримані з ab initio розрахунків Article published earlier |
| spellingShingle | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations Bekenov, L.V. Antonov, V.N. Shpak, A.P. Yaresko, A.N. |
| title | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations |
| title_alt | Електронна структура та властивості збудженого стану Co₂TiSn та Co₂ZrSn, отримані з ab initio розрахунків |
| title_full | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations |
| title_fullStr | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations |
| title_full_unstemmed | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations |
| title_short | Electronic structure and excited-state properties of Co₂TiSn and Co₂ZrSn from ab initio calculations |
| title_sort | electronic structure and excited-state properties of co₂tisn and co₂zrsn from ab initio calculations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/119751 |
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