Electronic structure and magnetic properties of transition metal diborides

Article abstract of DOI:10.1016/j.jallcom.2009.03.123

The temperature dependencies of the magnetic susceptibility χ and its anisotropy Δ χ = χ_{{norm of matrix}} - χ_⊥ were measured for the hexagonal single crystalline TB₂ compounds (T = Sc, Ti, Zr, Hf, V and Cr) in the temperatu

Full text of DOI:10.1016/j.jallcom.2009.03.123

Journal of Alloys and Compounds 481 (2009) 75–80
Contents lists available at ScienceDirect
Journal of Alloys and Compounds
journal homepage: www.elsevier.com/locate/jallcom
Electronic structure and magnetic properties of transition metal diborides
G.E. Grechnev , A.V. Fedorchenko , A.V. Logosha , A.S. Panfilova,∗, I.V. Svechkareva,
a
a
a
b
b
b
V.B. Filippov , A.B. Lyashchenko , A.V. Evdokimova
B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61103 Kharkov, Ukraine
I. Frantsevich Institute for Problems of Material Science, 3 Krzhyzhanovsky Street, 03680 Kiev, Ukraine
a
b
a r t i c l e i n f o
a b s t r a c t
Article history:
The temperature dependencies of the magnetic susceptibility ꢀ and its anisotropy ꢁꢀ = ꢀ − ꢀ were
ꢀ
⊥
Received 28 January 2009
Received in revised form 19 March 2009
Accepted 21 March 2009
measured for the hexagonal single crystalline TB2 compounds (T = Sc, Ti, Zr, Hf, V and Cr) in the temper-
ature range 4.2–300 K. It was found that ꢁꢀ varies strongly and nonmonotonously with the filling of the
hybridized d-band, being almost temperature independent and the largest for the diborides of group 4
metals. Ab initio calculations of the electronic structure and susceptibility of the diborides provide evi-
dence that magnetic anisotropy originates from competing Van Vleck paramagnetism and the orbital
diamagnetism of conduction electrons.
Available online 31 March 2009
PACS:
7
1.20.−b
©
2009 Elsevier B.V. All rights reserved.
72.80.Ga
75.10.Lp
75.30.Gw
Keywords:
Diborides
Electronic structure
Magnetic susceptibility
Magnetic anisotropy
1. Introduction
properties of TB are still limited and ambiguous. Also these data
2
were mostly obtained for the polycrystalline samples of different
quality and origin. The recent technological progress in growing
the single crystal samples has made possible more detailed and
extended experimental investigations of the physical properties of
diborides, including studies of the Fermi surface features and mani-
festations of the lattice anisotropy by using the de Haas–van Alphen
effect [6].
Transition metal (T) diborides TB2 are of great scientific and
technological interest due to their unique physical and chemical
properties such as high melting point, hardness, thermal conduc-
tivity, and chemical inertness [1]. The most known TB compounds
2
are diborides of transition metals of groups 3–6 (Sc, Ti, Zr, Hf, V, Nb,
etc.) which have the layered hexagonal C32 structure of AlB2 type.
These compounds retain the C32 crystal structure in a wide range
of conduction band occupation numbers in respective 3d-, 4d- and
In this paper we present results of experimental and theoreti-
cal investigations of magnetic properties of the single crystalline
5
d-series, providing a variety of structural, electronic and magnetic
diborides TB in relation to their electronic structure. The main
2
properties [1–7].
objective of this work is to study the magnetic susceptibility and
its anisotropy in the representative series of the C32 diborides.
These diborides appear to be an appropriate set of the isostruc-
tural compounds, allowing to analyse variations of the magnetic
properties with substantial changes in lattice parameters and the
filling of strongly hybridized p–d states.
The discovery of superconductivity in MgB at T = 39 K has
2
initiated detailed studies of superconducting and other physical
properties of AlB -type borides, and there are controversial and
2
contradictory reports on the superconductivity and value of Tc for
TB2 (see e.g. [5] and references therein). Also, there is only scarce
and incomplete information on magnetic properties of diborides
The detailed experimental studies of magnetic susceptibility for
TB2 compounds are supplemented by electronic structure calcula-
tions within LSDA approximation of the density-functional theory
by using a full-potential LMTO method. Within the unified ab initio
approach, the electronic structure, magnetic and elastic proper-
ties of ScB , TiB , VB , CrB , ZrB , and HfB were evaluated and
[
1–3]. Despite a great number of experimental studies in the past
years, the available data on superconducting, magnetic and elastic
∗ Corresponding author. Fax: +380 573 403 370.
E-mail address: panfilov@ilt.kharkov.ua (A.S. Panfilov).
2
2
2
2
2
2
compared with experimental data.
0925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.jallcom.2009.03.123

76
G.E. Grechnev et al. / Journal of Alloys and Compounds 481 (2009) 75–80
Fig. 1. Temperature dependence of the magnetic susceptibility for the diborides of
group 4 metals.
Fig. 2. Temperature dependence of the magnetic susceptibility for ScB2, VB2 and
CrB2 compounds.
3. Computational details and results
2.
Experimental details and results
The diborides single crystals were obtained by high-frequency crucibleless zone
3.1. Band structure calculations
melting of the initial diborides powders prepared by reduction of the oxides of the
corresponding metals of 99.95% purity with amorphous boron of 99.9% purity (for
details see Refs. [6,8]). The de Haas–van Alphen effect has been recently reported
The investigated diborides possess the hexagonal C32 crystal
structure with c/a ratio close to unity [1]. This crystal lattice is
composed of transition metal layers alternating with graphite-like
boron layers stacked perpendicularly to the [0 0 1]axis. Ab initio cal-
culations of the electronic structure of diborides were carried out
by employing a modified FP-LMTO method [14,15]. The exchange-
correlation potential was treated in the LDA [16] and GGA [17]
approximations of the density functional theory. In the present cal-
culations for TB2 the FP-LMTO basis set included the 2s, 2p, and 3d
orbitals of boron, as well as the np, (n + 1)s, (n + 1)p, and nd orbitals
of T within a single, fully hybridizing energy panel, where n is the
principal quantum number for d valence states of the transition
metal.
[
6] for the same set of samples that supports their high quality. The measure-
ments of the magnetic susceptibility and its anisotropy were carried out by a
Faraday method in the temperature range 4.2–300 K and in magnetic field around
H ꢁ 1 T.
The magnetic susceptibility of diborides versus temperature ꢀ(T) was measured
for the magnetic field applied both along and perpendicular to the six-fold c-axis of
the hexagonal structure, ꢀꢀ and ꢀ⊥, respectively. As seen in Figs. 1 and 2, in most
cases the ꢀ(T) dependence is rather weak, except for ScB2 and CrB2, and the mag-
netic anisotropy ꢁꢀ = ꢀꢀ − ꢀ⊥ appears to be nearly temperature independent. Also,
in the series of TB2ꢁꢀ reveals a strong and nonmonotonous dependence on the con-
duction band filling, and the largest anisotropy is found for the diborides of group
4
metals.
CrB2 is the first magnetic compound in the series, which orders antiferromag-
netically (AFM) with the Néel temperature TN = 85–88 K [2,9,10]. According to the
neutron-diffraction measurements [9], CrB2possesses a helical magnetic structure
of cycloidal type, which magnetic moment (of about 0.6 ꢂB per Cr atom at T = 0)
turns in the a–c plane. As it follows from the available data on magnetic properties of
the single crystalline CrB2 sample [10], in the paramagnetic (PM) phase the suscep-
tibility of CrB2is an order of magnitude higher than that of other diborides, and there
is no noticeable anisotropy of ꢀ in this compound. Our data in Fig. 2 corresponds to
a polycrystalline sample of CrB2, and appears to be in agreement with the results of
Ref. [10]. The distinctive feature of the observed ꢀ(T) is a sharp cusp at TN ꢁ 87 K,
where the AFM–PM transition takes place.
The integration over the Brillouin zone was performed using the
tetrahedron method [18]. The results reported here were obtained
by using up to 4000 k-points in the irreducible wedge of the
Brillouin zone for the field-induced spin-polarized calculations
described below. The corresponding calculated total energies were
−6
well converged (∼10 Ry) with respect to all parameters involved,
such as k-space sampling and basis set truncation.
The room temperature values of the magnetic susceptibility and its anisotropy
for the studied diborides are summarized in Table 1 together with the available
experimental data. As seen from the table, for some systems, especially the diborides
of group 4 metals, there is a noticeable discrepancy between our results and the
literature data obtained for polycrystalline samples. This suggests that magnetic
susceptibility of the diborides is strongly dependent on a sample quality and its
stoichiometry.
The band structure calculations were performed for a number
of lattice parameters close to the experimental ones, and the c/a
ratio was fixed at its experimental value for each TB2 compound.
The equilibrium lattice spacings and corresponding theoretical bulk
moduli BLDA were determined from calculated (within LDA) volume
dependencies of the total energy E(V) by employing the well known

G.E. Grechnev et al. / Journal of Alloys and Compounds 481 (2009) 75–80
77
Table 1
Magnetic susceptibility of TB2 at T = 293 K.
TB2
ꢀꢀ (×10− cm /mol)
4
3
ꢀ⊥ (×10−4 cm /mol)
3
ꢀꢀ − ꢀ⊥ (×10−4 cm /mol)
3
ꢀ¯ = (ꢀꢀ + 2ꢀ⊥)/3 (×10−4 cm /mol)
3
a
ScB2
1.045
0.066
0.094
−0.303
0.255
–
0.989
−0.740
−0.671
−0.936
0.271
–
0.056
0.806
0.765
1.008
−0.471
−0.416
−0.725
0.265
5.14
∼0.8
b
−0.40 ; 0.31 d
c
TiB2
ZrB2
HfB2
VB2
CrB2
NbB2
TaB2
d
−0.68
d
0.633
−0.04
c
−0.016
0.21 ; 0.34 d
5.4 e; 3.9 d
0.08 d
e
∼ 0
–
–
–
–
–
–
–
–
d
−0.65
a
Ref. [11].
b
c
The sample was kindly presented by F.A. Sidorenko.
Ref. [12].
Ref. [13].
Ref. [10].
d
e
Table 2
Experimental lattice parameters (from Ref. [1]), calculated bulk moduli B and contributions to magnetic susceptibility of diborides (see text for details).
TB2
a (Å)
c (Å)
B (GPa)
ꢀston
ꢀ¯ spin
(×10 cm /mol)
ꢀ¯ orb
ꢁꢀorb
(×10 cm /mol)
ꢀdia
ꢀsuma
×10− cm /mol)
4
3
−4
3
(×10 cm /mol)
−4
3
−4
3
(×10 cm /mol)
−4
3
(×10−4 cm /mol)
3
(
ScB2
TiB2
VB2
CrB2
YB2
ZrB2
NbB2
MoB2
HfB2
TaB2
3.148
3.028
2.998
2.969
3.298
3.165
3.115
3.040
3.140
3.097
3.516
3.228
3.057
3.066
3.843
3.547
3.264
3.120
3.470
3.225
240
290
270
230
230
280
260
250
310
290
0.40
0.10
0.68
4.0
0.39
0.08
0.38
0.58
0.07
0.35
0.57
0.15
0.90
7.03
0.43
0.09
0.40
0.62
0.09
0.37
0.39
0.99
0.83
0.60
0.23
0.50
0.56
0.55
0.41
0.47
0.043
0.131
0.080
0.010
0.025
0.053
0.068
0.008
0.032
0.055
−0.10
−0.12
−0.11
−0.10
−0.25
−0.23
−0.22
−0.20
−0.36
−0.35
0.86
1.02
1.62
7.53
0.41
0.36
0.74
0.97
0.14
0.49
a
ꢀsum = ꢀ¯ spin + ꢀ¯ orb + ꢀdia.
Murnaghan equation [14]. This equation is based on the assumption
chemical stability of these borides. Within the series ScB –MnB ,
2 2
ꢂ
that the pressure derivative B of the bulk modulus B is constant. By
and also YB –MoB , the Fermi level passes through a deep mini-
2 2
ꢂ
using the values of B , evaluated from the Murnaghan equation, we
mum in DOS N(E), related to the pseudo-gap in electronic spectrum,
and falls into the area of antibonding states and large values of N(EF).
Consequently, this leads to instability of the C32 phase of diborides
from the middle of the transition metals row.
have estimated bulk moduli B, corresponding to the experimental
lattice parameters, and these estimations are given in Table 2.
According to our previous calculations [15,19], such corrections
for BLDA provided the theoretical estimations of B close to experi-
mental values of the bulk moduli. The obtained high values of bulk
moduli of diborides (Table 2) are in a qualitative agreement with
the theoretical B of Ref. [7], calculated within DFT-GGA approxi-
mation. Though there are scarce and contradictory experimental
data on elastic properties of diborides (see Refs. [7,1] and refer-
ences therein), the results of present calculations, as well as ones of
Ref. [7], allow to reveal the trends of bulk moduli behavior among
diborides and to supplement deficient experimental data. This is
of particular importance in connection with the recently found
very high bulk moduli in the ultraincompressible diborides RuB2,
OsB and ReB [20], having somewhat different crystal structure [1].
3.2. Calculated magnetic properties
The FP-LMTO calculations of the field-induced spin and orbital
(Van Vleck) magnetic moments were carried out for the diborides
2
2
The present results also indicate, that previously reported in Ref. [4]
theoretical bulk moduli of the hexagonal C32 diborides were sub-
stantially underestimated, presumably due to the employed atomic
sphere approximation for the total energy calculations.
As seen in Figs. 3 and 4, the calculated densities of electronic
states (DOS) N(E) of the diborides are similar, but vary in details
and positions of the Fermi level EF. The vertical lines in Figs. 3 and 4
indicate the conduction band filling for the corresponding 3d- and
4
d-diborides. According to the calculations, the partial DOSs in
Figs. 3 and 4 exhibit a strong hybridization of d-states of transition
metal with p-states of boron. Also, in addition to the p–d hybridiza-
tion, the filling of conduction band is expected to be substantially
responsible for properties of transition metal diborides.
Fig. 3. Density of states for VB2. The total DOS and partial contributions of vanadium
d-states and boron p-states are indicated by solid line, dashed line, and dashed-
dotted line, respectively. The vertical lines mark the conduction band filling for the
corresponding 3d-diborides.
For the diborides of metals from group 4 (TiB , ZrB , HfB ) and 5
2
2
2
(
VB , NbB , TaB ) the bonding states are filled, whereas antibonding
2 2 2
states are almost unoccupied. This can explain hardness and high

7
8
G.E. Grechnev et al. / Journal of Alloys and Compounds 481 (2009) 75–80
where ꢄ (r) are the partial wave functions, and g(ꢃ(r)) is a function
l
of the charge density [16]. The values of the calculated enhanced
Pauli susceptibility ꢀston are also presented in Table 2.
4. Discussion
The calculation of the magnetic susceptibility of metallic sys-
tems is a rather difficult problem (see [15,22] and references
therein). The total susceptibility in the absence of spontaneous
magnetic moment can be expressed as the sum:
ꢀtot = ꢀspin + ꢀorb + ꢀdia + ꢀL,
(5)
where these terms correspond to the Pauli spin susceptibility,
a generalization of the Van Vleck orbital paramagnetism, the
Langevin diamagnetism of closed shells, and generalization of Lan-
dau conduction electrons diamagnetism, respectively. The Langevin
contributions ꢀdia were estimated based on results of Refs. [22–24]
and appeared to be between free-atom and free-ionic diamagnetic
susceptibilities (see Table 2). Obviously, these ꢀdia terms do not
reveal any anisotropy.
In order to analyse the experimental data on ꢀ and ꢁꢀ we
use the calculated contributions to ꢀ in Table 2. As can be seen
from the table, in the non-magnetic transition metal diborides the
orbital Van Vleck term ꢀorb provides substantial contribution to the
total paramagnetic susceptibility. In some diborides ꢀorb even over-
comes ꢀspin, particularly this shows up in the diborides of group
Fig. 4. Density of states for NbB2. The total DOS and partial contributions of niobium
d-states and boron p-states are indicated by solid line, dashed line, and dashed-
dotted line, respectively. The vertical lines mark the conduction band filling for the
corresponding 4d-diborides.
in the external field H of 10 T. The effect of the external magnetic
field was taken into account self-consistently, within LSDA in line
with the method described in Refs. [15,19], by means of the Zeeman
operator:
4
metals (TiB , ZrB , HfB ). Also, it can be seen in Table 2 that
2 2 2
ˆ
ˆ
ˆ
HZ = H · (2s + l),
(1)
the Stoner model provides smaller values of the spin susceptibil-
ity ꢀston, comparatively to the field-induced calculated ꢀspin. This
is in agreement with the recent observation [19], that for the param-
agnetic metallic systems the Stoner approach underestimates spin
susceptibility, whereas the LSDA field-induced calculations take
into account non-uniform induced magnetization density in the
which was incorporated in the FP-LMTO Hamiltonian. Here sˆ is
the spin operator and l the orbital angular momentum opera-
tor. When the field-induced spin and orbital magnetic moments
are calculated, the corresponding volume magnetization can be
evaluated, and the ratio between the magnetization and the field
strength provides the related contributions to susceptibility, ꢀspin
ˆ
unit cell and provide more accurate values of ꢀspin
.
To calculate the Landau diamagnetic contribution ꢀL is a con-
siderably more difficult problem (see [22,25–27] and Refs. therein).
The free-electron Landau limit is often used for estimations, giving
and ꢀorb
.
For the hexagonal C32 crystal structure, the components of these
contributions, ꢀ and ꢀ , are derived from the magnetic moments
obtained in an external field, applied parallel and perpendicular
iꢀ
i⊥
0
a ꢀL that equals −(1/3) of the Pauli spin susceptibility, though for
many systems this crude approximation was found not to provide
even the correct order of magnitude of the diamagnetic suscepti-
bility [25,27–29].
to the c-axis, respectively. The averaged values of the calculated
ꢀspin and ꢀorb components ꢀ¯ i = (ꢀ + 2ꢀ )/3 and the evaluated
iꢀ
i⊥
orb
anisotropy of the orbital contribution ꢁꢀ
= ꢀ
− ꢀ
are
orbꢀ
orb⊥
According to the experimental data in Table 1 and Figs. 1 and 2,
the observed anisotropy of susceptibility appears to be the largest
and almost temperature independent for the diborides of group 4
metals. As can be seen in Table 2, the calculated anisotropy of the
Van Vleck paramagnetic contribution ꢀorb ranges only from 5 to
listed in Table 2. The employed number of 4000 k-points provides
relative precision better than 10% for the calculated ꢁꢀ, or about
ꢂB for corresponding field-induced moments, using the exter-
nal field of 10 T.
For comparison, the Pauli spin contribution to the magnetic sus-
ceptibility was calculated within the Stoner model:
−
5
10
15% of the experimental ꢁꢀ for these diborides. In addition, there
is a conspicuous discrepancy between the calculated contributions
2
B
−1
to the magnetic susceptibility ꢀsum = ꢀ¯
+ ꢀ¯ + ꢀ in Table 2
ꢀ
ston = SꢀP ≡ ꢂ N(EF)[1 − IN(EF)]
,
(2)
spin
orb
dia
and the experimental values, ꢀexp = ꢀ¯ (Table 1), which can be
2
B
where ꢀP = ꢂ N(EF), S is the Stoner enhancement factor, and
attributed to the Landau diamagnetic contribution ꢀ in Eq. (5). For
L
ꢂB the Bohr magneton. The Stoner integral I, describing the
exchange-correlation interaction of the conduction electrons can
be expressed in terms of the calculated parameters of the electronic
structure (see Ref. [21] and references therein):
the diborides of group 4 metals we can evaluate ꢀ ꢁ ꢀ
sum
− ꢀ to
L
exp
−
4
3
be about −10 cm /mol. The relatively large magnitude of this esti-
0
L
mate, which far exceeds the free-electron Landau value ꢀ , requires
further examination.
ꢀ
It has been established, that small groups of quasi-degenerate
electronic states with light effective masses, situated in the close
vicinity of the Fermi level EF (about 0.1 eV or closer), yield pro-
nounced and anisotropic orbital diamagnetic contributions to
susceptibility in non-transition metals Cd and Zn [25], graphite
[26], Be [28], Bi–Sb alloys [27] and even in systems with sub-
1
I =
N (EF )J
ꢂ
Nql
ꢂ
(EF ).
(3)
ql
qll
N(EF)2
ꢂ
qll
Here N(EF) is the total density of electronic states at the Fermi level
EF, N (EF) is the partial density of states for atom q in the unit cell,
ql
Jqll
ꢂ
are the local exchange integrals:
stantial admixture of d- and f-states at EF (YbPb , YbSn3 and
3
ꢁ
La(In,Sn)3 alloys [29]). Such ꢀL contributions can be many times
2
2
0
Jll
ꢂ
=
g(ꢃ(r))ꢄ (r) ꢄ
ꢂ
(r) dr,
(4)
larger than the free-electron Landau estimation ꢀ and this anoma-
l
l
L
lous diamagnetism is determined by a delicate interaction between

G.E. Grechnev et al. / Journal of Alloys and Compounds 481 (2009) 75–80
79
Fig. 6. The energy bands of VB2, NbB2, TaB2, and CrB2 along the symmetry directions.
The Fermi level is marked by a horizontal dashed line.
Fig. 5. Energy bands of TiB2, ZrB2, HfB2, and ScB2along the symmetry directions.
The Fermi level is marked by a horizontal dashed line.
sis of ꢀL of the diborides in a future. As the first step in this direction,
we identified here appropriate electronic states near EF as possible
sources of the large diamagnetism in diborides.
effective masses, small spin–orbit splittings, and the relative posi-
tion of EF. In this connection we should note that the position of EF
with respect to the degeneracy points of E(k) spectrum may be not
quite correctly predicted by ab initio calculations.
As can be seen in Fig. 5, the present band structure calculations
point to the presence of quasi-degenerate hybridized electronic
Two other diborides studied in the present work, ScB and CrB ,
2
2
appear to be paramagnetic (see Table 1) with expected moderate
diamagnetic contributions to ꢀ. As can be seen in Table 2, the field-
induced calculated spin susceptibility ꢀspin is noticeably higher
than the spin contribution ꢀston calculated within the Stoner model,
and the orbital Van Vleck contribution ꢀorb is comparable to ꢀspin
states close to EF in TiB . The bands crossing in the basal plane
2
around the symmetry point K is of particular importance in con-
nection with a manifestation of the anomalously large ꢀL, which
was found to originate from similar degeneracy points [26–28].
Also, the analogous quasi-degenerate states with small effective
and necessary to describe the experimental data for ScB . Moreover,
2
the calculated anisotropy of ꢀorb is consistent with the experimen-
tally observed ꢁꢀ in ScB2 and CrB2.
We note, that for ScB2 the sum of calculated paramagnetic con-
tributions appears to be lower than the experimental ꢀ. This should
be considered as rather expected underestimation for the LSDA
masses exist at EF in the diborides of other group 4 metals, ZrB and
2
HfB , and appear to be slightly affected by the spin–orbital coupling
2
(
see directions K → ꢅ in Fig. 5). On the other hand, as also seen in
Fig. 5, for ScB , the diboride of the group 3 metal, the corresponding
2
ground state of ScB . Due to the well known overbonding tendency
2
bands crossing lies substantially above EF. It should be noted, that in
the recent de Haas–van Alphen studies [6] the very light cyclotron
masses (about 0.1 of the free-electron mass) have been observed
of LSDA [14] the theoretical equilibrium volume is smaller than the
experimental one, and this resulted in slightly suppressed values
of ꢀspin and ꢀorb, which were obtained within the field-induced
self-consistent FP-LMTO calculations.
in ZrB2 and HfB . This band structure similarity together with the
2
observed close values of ꢀ¯ and ꢁꢀ in TiB , ZrB2 and HfB2 (see
2
In CrB2 the Stoner criterion is nearly fulfilled, IN(EF) ꢁ 1, with
the Fermi level located at the steep slope of N(E) peak where DOS
rapidly grows with energy, and the main contribution to N(EF)
comes mostly from d-states of Cr (see Fig. 3). The calculated sus-
ceptibility enhancement factor S appears to be about 8, which
is comparable with earlier estimations (S ꢁ 10, [3,4]). In the PM
Table 1) indicate that the corresponding quasi-degenerate states
might determine the diamagnetism of the diborides of group 4
metals.
Available experimental data on susceptibility of the group 5
diborides, (VB , NbB2 and TaB , see Table 1), together with the cal-
2
2
culated paramagnetic terms ꢀspin and ꢀorb from Table 2, also imply
the presence of a substantial diamagnetic contribution to ꢀ. Even
phase of CrB the magnetic susceptibility rises with decreasing
2
−
4
3
temperature and becomes ꢀexp ꢁ 6.5 × 10
cm /mol at T = 90
though VB appeared to be paramagnetic, the large ꢀ
and ꢀorb
2
spin
K. The extrapolated PM susceptibility, ꢀexp(T → 0), provides the
contributions have to be almost compensated by ꢀL. Also the sign
of ꢁꢀorb in Table 2 is not consistent with the experimental ꢁꢀ for
−
4
3
estimation ꢀexp(0) ꢁ 7.5 × 10 cm /mol, which is in agreement
with the calculated paramagnetic contributions ꢀ
and ꢀorb from
spin
VB . This suggests that only the larger anisotropy of ꢀL may compete
2
Table 2.
with ꢁꢀorb and outperform it.
The band structure calculations for the low temperature AFM
The calculated band structure of the group 5 diborides is pre-
sented in Fig. 6, where one can see the presence of quasi-degenerate
states with small effective masses at EF, namely the bands cross-
ing in the ꢅ → Adirection. We note that the ꢅ → A bands crossing
helical magnetic structure of CrB are extremely difficult, and in
2
the present work the electronic structure calculation has been per-
formed for the FM phase of CrB , which provided the magnetic
2
moment of 0.8 ꢂB, in a fair agreement with the experiment [9,2].
distinctly approaches EF in the series VB , NbB and TaB . One can
2
2
2
point out, that for CrB diboride of group 6 the corresponding bands
2
crossing is also not far from EF. However in this case, as shown
below, the spin paramagnetic susceptibility is definitely dominant
due to substantial exchange enhancement.
5. Conclusions
The magnetic susceptibility and its anisotropy was studied for
the first time on single crystals of the diborides ScB , TiB , VB ,
It should be emphasized that rigorous theoretical analysis of ꢀL
is rather cumbersome procedure, which includes a derivation of
accurate multi-band k · p models for groups of quasi-degenerate
states at EF, and then exploring a variety of possible options to
handle analytical and numerical problems. A choice of scenario for
solution can depend on dimension of suitable k · p models and a
relative position of the Fermi level. This task goes beyond the aims
of the present work, and we hope to be able to perform such analy-
2
2
2
ZrB , and HfB . It was found that the value of anisotropy is strongly
2
2
dependent on the filling of p–d hybridized conduction band, and
appeared to be the largest for Ti-group diborides. The ab initio
electronic structure calculations in an external magnetic field have
allowed to evaluate the paramagnetic spin and orbital Van Vleck
contributions to magnetic susceptibility of the diborides and their
anisotropy. It has been demonstrated that LSDA provides an ade-

8
0
G.E. Grechnev et al. / Journal of Alloys and Compounds 481 (2009) 75–80
quate description of the magnetic properties for ScB2 and strongly
enhanced CrB2 compounds.
[8] G. Levchenko, A. Lyashchenko, V. Baumer, A. Evdokimova, V. Filippov, Yu.
Paderno, N. Shitsevalova, J. Solid State Chem. 179 (2006) 2949.
[
9] S. Funahashi, Y. Hamaguchi, T. Nanaka, E. Bannai, Solid State Commun. 23 (1977)
59.
For the diborides of groups 4 and 5, the comparison of the
experimental and calculated susceptibilities has revealed a large
8
c
[10] G. Balakrishnan, S. Majumdar, M.R. Lees, D.M K. Paul, J. Cryst. Growth 274 (2005)
294.
−
4
3
and anisotropic diamagnetic contribution about −10 cm /mol,
which can originate from the generalized Landau diamagnetism
of conduction electrons, ꢀL. It is anticipated that the large magni-
tude of ꢀL in the diborides of groups 4 and 5 has its origin in the
quasi-degenerate electronic states close to the Fermi energy.
[
[
11] P. Peshev, J. Etourneau, R. Naslain, Mater. Res. Bull. 5 (1970) 319.
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Structure and Physical Properties of Solids: The Uses of the LMTO Method,
Springer, Berlin, 2000, p. 148.
[
Acknowledgments
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[
15] G.E. Grechnev, R. Ahuja, O. Eriksson, Phys. Rev. B 68 (2003) 64414.
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The authors thank F.A. Sidorenko for giving the single crystalline
sample of TiB2 compound and A. Grechnev for fruitful scientific
discussions.
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