Electronic structure, phonon spectra and electron–phonon interaction in ScB₂
The electronic structure, Fermi surface, angle dependence of the cyclotron masses and extremal cross sections of the Fermi surface, phonon spectra, electron–phonon Eliashberg and transport spectral functions, temperature dependence of electrical resistivity of the ScB₂ diboride were investigated fro...
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irk-123456789-1186692017-05-31T03:03:12Z Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ Sichkar, S.M. Antonov, V.N. Электронные свойства проводящих систем The electronic structure, Fermi surface, angle dependence of the cyclotron masses and extremal cross sections of the Fermi surface, phonon spectra, electron–phonon Eliashberg and transport spectral functions, temperature dependence of electrical resistivity of the ScB₂ diboride were investigated from first principles using the fully re-lativistic and full potential linear muffin-tin orbital methods. The calculations of the dynamic matrix were carried out within the framework of the linear response theory. A good agreement with experimental data of electron–phonon spectral functions, electrical resistivity, cyclotron masses and extremal cross sections of the Fermi sur-face was achieved. 2013 Article Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ / S.M. Sichkar, V.N. Antonov // Физика низких температур. — 2013. — Т. 39, № 7. — С. 771–779. — Бібліогр.: 40 назв. — англ. 0132-6414 PACS: 75.50.Cc, 71.20.Lp, 71.15.Rf http://dspace.nbuv.gov.ua/handle/123456789/118669 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электронные свойства проводящих систем Электронные свойства проводящих систем |
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Электронные свойства проводящих систем Электронные свойства проводящих систем Sichkar, S.M. Antonov, V.N. Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ Физика низких температур |
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The electronic structure, Fermi surface, angle dependence of the cyclotron masses and extremal cross sections of the Fermi surface, phonon spectra, electron–phonon Eliashberg and transport spectral functions, temperature dependence of electrical resistivity of the ScB₂ diboride were investigated from first principles using the fully re-lativistic and full potential linear muffin-tin orbital methods. The calculations of the dynamic matrix were carried out within the framework of the linear response theory. A good agreement with experimental data of electron–phonon spectral functions, electrical resistivity, cyclotron masses and extremal cross sections of the Fermi sur-face was achieved. |
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Sichkar, S.M. Antonov, V.N. |
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Sichkar, S.M. Antonov, V.N. |
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Sichkar, S.M. |
| title |
Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ |
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Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ |
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Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ |
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Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ |
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Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ |
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electronic structure, phonon spectra and electron–phonon interaction in scb₂ |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электронные свойства проводящих систем |
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Electronic structure, phonon spectra and electron–phonon interaction in ScB₂ / S.M. Sichkar, V.N. Antonov // Физика низких температур. — 2013. — Т. 39, № 7. — С. 771–779. — Бібліогр.: 40 назв. — англ. |
| series |
Физика низких температур |
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AT sichkarsm electronicstructurephononspectraandelectronphononinteractioninscb2 AT antonovvn electronicstructurephononspectraandelectronphononinteractioninscb2 |
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2025-07-08T14:25:19Z |
| last_indexed |
2025-07-08T14:25:19Z |
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1837089133839777792 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7, pp. 771–779
Electronic structure, phonon spectra and electron–phonon
interaction in ScB2
S.M. Sichkar1 and V.N. Antonov1,2
1Institute of Metal Physics, 36 Vernadsky Str., 03680 Kiev-142, Ukraine
2Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011, USA
E-mail: antonov@imp.kiev.ua
Received February 11, 2013, revised March 1, 2013
The electronic structure, Fermi surface, angle dependence of the cyclotron masses and extremal cross sections
of the Fermi surface, phonon spectra, electron–phonon Eliashberg and transport spectral functions, temperature
dependence of electrical resistivity of the ScB2 diboride were investigated from first principles using the fully re-
lativistic and full potential linear muffin-tin orbital methods. The calculations of the dynamic matrix were carried
out within the framework of the linear response theory. A good agreement with experimental data of electron–
phonon spectral functions, electrical resistivity, cyclotron masses and extremal cross sections of the Fermi sur-
face was achieved.
PACS: 75.50.Cc Other ferromagnetic metals and alloys;
71.20.Lp Intermetallic compounds;
71.15.Rf Relativistic effects.
Keywords: diborides, superconductivity, Fermi surface.
1. Introduction
The discovery of superconductivity in MgB2 at 39 K
by Akimitsu [1] has lead to booming activity in the phys-
ics community and activated a search for superconductivi-
ty in other diborides. Natural candidates for this search are
AB2-type light metal diborides (A = Li, Be, Al). However,
up to now superconductivity has not been reported in the
majority of these compounds [2]. Only very recently su-
perconductivity below 1 K (Tc = 0.72 K) has been re-
ported in BeB2.75 [3]. According to Ref. 4 no supercon-
ducting transition down to 0.42 K has been observed in
powders of diborides of transition metals (M = Ti, Zr, Hf,
V, Cr, Mo, U). NbB2 is expected to superconduct with a
rather low transition temperature (< 1 K), and contradicto-
ry reports about superconductivity up to Tc = 9.5 K in
TaB2 can be found in Ref. 4. For ScB2 only low-
temperature superconductivity was found with Tc ~ 1.5 K
[5]. Finally, the reported Tc = 7 K in ZrB2 encourages
further studies of these diborides [2].
Quite a number of theoretical studies of the electronic
properties of the diborides are known to date [6–23]. Iva-
novskii et al. [9] performed full potential linear muffin-tin
orbital (FP-LMTO) calculations of all hexagonal diborides
of 3d (Sc, Ti, ..., Fe), 4d (Y, Zr, ..., Ru), and 5d (La, Hf, ...,
Os) metals and analyzed the variations in their chemical
stability and some other properties (e.g., melting tempera-
tures, enthalpies of formation). They found that the evolu-
tion of their band structures can be described within a ri-
gid-band model. For M = Ti, Zr, Hf energy Fermi ( )Fε
falls near the density of state (DOS) minimum (pseudogap)
between the fully occupied bonding bands and unoccupied
antibonding bands. Sc, Y, La diborides have partially un-
occupied bonding bands. Vajeeston et al. [11] investigated
the electronic structure and ground state properties of AlB2
type transition metal diborides MB2 (M = Sc, Ti, V, Cr,
Mn, Fe, Y, Zr, Nb, Mo, Hf, Ta) using the self consistent
tight-binding linear muffin-tin orbital method. The equili-
brium volume, bulk moduli, pressure derivative of bulk
moduli, cohesive energy, heat of formation, and electronic
specific heat coefficient were calculated for these systems
and compared with the available experimental and other
theoretical results. The bonding nature of these diborides
was analyzed via the density of states histogram as well as
the charge density plots, and the chemical stability was
analyzed using the band filling principle. The variation in
the calculated cohesive properties of these materials was
correlated with the band filling effect. The existence of a
pseudogap in the total density of states was found to be a
common feature for all these compounds. The reason for
the creation of the pseudogap was found to be due to the
strong covalent interaction between boron p states. Fedor-
© S.M. Sichkar and V.N. Antonov, 2013
S.M. Sichkar and V.N. Antonov
chenko and Grechnev with coauthors [13,14] measured the
temperature dependences of the magnetic susceptibility χ
and its anisotropy = ⊥∆χ χ −χ
for single crystals of tran-
sition-metal diborides MB2 (M = Sc, Ti, V, Zr, Hf) in the
temperature interval 4.2–300 K. A transition into the su-
perconducting state was not found in any of the diborides
studied, right down to liquid-helium temperature. It
was found that the anisotropy is weakly temperature-
dependent, a nonmonotonic function of the filling of the
hybridized p–d conduction band. First-principles calcula-
tions of the electronic structure of diborides and the values
of the paramagnetic contributions spin and Van Vleck to
their susceptibility show that the behavior of the magnetic
anisotropy is due to the competition between Van Vleck
paramagnetism and orbital diamagnetism of the conduction
electrons. Duan et al. [23] calculated elastic constants of 24
compounds of the AlB2-type diborides including ScB2 by
first-principles with the generalized gradient approximation
using the Voigt–Reuss–Hill averaging scheme. Values of all
independent elastic constants as well as bulk modulus in a
and c directions were predicted. It was founded that AlB2 is
more ductile while ScB2 is more brittle, and AlB2 has a
highest elastic anisotropy in the 24 AlB2-type compounds.
Deligoz et al. [15] investigated the structural and lattice dy-
namical properties of MB2 (M = Sc, V, Ti) using first-
principles total energy calculations. Specifically, the lattice
parameters (a, c) of the stable phase, the bond lengths of
M–B and B–B atoms, phonon dispersion curves and the
corresponding density of states, and some thermodynami-
cal quantities such as internal energy, entropy, heat capaci-
ty, and their temperature-dependent behaviors, were pre-
sented. The obtained results for structural parameters are in
a good agreement with the available experimental data.
Zhang et al. [19] presented extensive structure searches to
uncover the high-pressure structures of MB2 (M = Sc, Ti,
Y, and Zr) up to 300 GPa using the ab initio evolutionary
algorithm [24]. They show that ZrB2 persists up to
300 GPa within an ambient-pressure AlB2-type structure,
while pressure-induced transitions into monoclinic phases
(C2/m, Z = 4) occur for ScB2 at 208 GPa and YB2 at
163 GPa. A tetragonal R-ThSi2 structure (I41/amd, Z = 4)
for TiB2 at 215 GPa was predicted. The phase transforma-
tion mechanism has been discussed.
The properties of the Fermi surface of ScB2, ZrB2, and
HfB2 single crystals were studied by Pluzhnikov et al. [25]
using the de Haas–van Alphen effect. The angular depen-
dences of the frequencies of the dHvA oscillations in the
planes (1010), (1120), and (0001) and the values of their
effective cyclotron masses were measured. The frequen-
cies of the oscillations in ScB2 found lie in the interval
(2.09–23.6)⋅102 T and the measured cyclotron masses lie
in the range (0.26–0.87)m0.
Despite a lot of publications, there are still many open
questions related to the electronic structure and physical
properties of ScB2 diboride. The theoretical efforts were
devoted mostly to the lattice and mechanical properties of
ScB2. There is no theoretical explanation of the Fermi sur-
face as well as angle dependence of the cyclotron masses
and extremal cross sections of the Fermi surface, electron–
phonon interaction and electrical resistivity in ScB2. The
aim of this work is a complex investigation of the electron-
ic structure, Fermi surface, angle dependence of the cyclo-
tron masses and extremal cross sections of the Fermi sur-
face, phonon spectra, electron–phonon Eliashberg and
transport spectral functions, and temperature dependence
of electrical resistivity of the ScB2 diboride.
The paper is organized as follows. Section 2 presents
the details of the calculations. Section 3 is devoted to the
electronic structure as well as the Fermi surface, angle de-
pendence of the cyclotron masses and extremal cross sec-
tions of the Fermi surface, phonon spectra, electron–
phonon interaction and electrical resistivity using the fully
relativistic and full potential LMTO band structure me-
thods. The results are compared with available experimen-
tal data. Finally, the results are summarized in Sec. 4.
2. Computational details
Most known transition-metal (M) diborides MB2 are
formed by group III–VI transition elements (Sc, Ti, Zr, Hf,
V, Nb, and others) and have a layered hexagonal C32
structure of the AlB2-type with the space group symmetry
6/P mmm (number 191). It is simply a hexagonal lattice in
which closely-packed transition metal layers alternating
with graphite-like B layers (Fig. 1). These diborides cannot
be exactly layered compounds because the interlayer inte-
raction is strong even though the M layers alternate with
the B layers in their crystal structure. The boron atoms lie
on the corners of hexagons with the three nearest neighbor
boron atoms in each plane. The M atoms lie directly in the
centers of each boron hexagon, but midway between adja-
cent boron layers. Each transition metal atom has 12 nearest
neighbor B atoms and eight nearest neighbor transition met-
al atoms (six are on the metal plane and two out of the metal
plane). There is one formula unit per primitive cell and the
crystal has simple hexagonal symmetry (D6h). By choosing
Fig. 1. (Color online) Schematic representation of the ScB2 crys-
tal structure.
772 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7
Electronic structure, phonon spectra and electron–phonon interaction in ScB2
appropriate primitive lattice vectors, the atoms are posi-
tioned at Sc (0,0,0), B (1/3,1/6,1/2), and B (2/3,1/3,1/2) in
the unit cell. The distance between Sc–Sc is equal to c. This
structure is quite close packed, and can be coped with effi-
ciently and accurately by the atomic sphere approximation
method. However, for precise calculation of the phonon
spectra and electron–phonon interaction, a full potential ap-
proximation should be used.
The Eliashberg function (the spectral function of the
electron–phonon interaction) expressed in terms of the
phonon linewidths νγq has the form [26]
2 1( ) = ( ).
2 ( )F
F
N
ν
ν
νν
γ
α ω δ ω−ω
π ε ω∑ q
q
qq
(1)
The line-widths characterize the partial contribution of
each phonon:
2
,= 2 | | ( ) ( ).j F j Fj j
jj
g ν
′ν ν +′+
′
γ πω δ ε − ε δ ε − ε∑ q
q q k k qk q k
k
(2)
The electron–phonon interaction constant is defined as
2
ph
0
= 2 ( ).e
d F
∞
−
ω
λ α ω
ω∫ (3)
It can also be expressed in terms of the phonons line-
widths:
ph 2= ,
( )
e
FN
ν
−
ν ν
γ
λ
π ε ω
∑ q
q q
(4)
were ( )FN ε is the electron density of states per atom and
per spin on the Fermi level and q
qj jg ν
′+k k is the electron–
phonon interaction matrix element. The double summation
over Fermi surface in Eq. (2) was carried out on dense mesh
(793 point in the irreducible part of the Brilloin zone (BZ)).
Calculations of the electronic structure and physical
properties of the ScB2 diborides were performed using fully
relativistic LMTO method [27] with the experimentally ob-
served lattice constants: a = 3.117 Å and c = 3.407 Å [15].
For the calculation of the phonon spectra and electron–
phonon interaction a scalar relativistic FP-LMTO method
[28] was used. In our calculations we used the Perdew–
Wang [29] parameterization of the exchange-correlation
potential in general gradient approximation. BZ integra-
tions were performed using the improved tetrahedron me-
thod [30]. Phonon spectra and electron–phonon matrix
elements were calculated for 50 points in the irreducible
part of the BZ using the linear response scheme developed
by Savrasov [28]. The 3s and 3p semi-core states of ScB2
were treated as valence states in separate energy windows.
Variations in charge density and potential were expanded
in spherical harmonics inside the MT sphere as well as
2894 plane waves in the interstitial area with 88.57 Ry
cut-off energy. As for the area inside the MT spheres, we
used 3k–spd LMTO basis set energy (–0.1, –1, –2.5 Ry)
with one-center expansions inside the MT-spheres per-
formed up to lmax = 6.
3. Results and discussion
3.1. Energy band structure
Figure 2 presents the energy band structure and total
density of states of ScB2. The partial DOSs ScB2 are
shown in Fig. 3. Our results for the electronic structure of
ScB2 are in a good agreement with earlier calculations
[9,11,13,14,23]. The Sc 3d states are the dominant features
in the interval from − 10.0 to 12 eV. These tightly bound
states show overlap with B 2p and, to a lesser extent, with
B 2s states both above and below ,Fε implying consider-
Fig. 2. (Color online) Energy band structure and total DOS [in
states/(cell⋅eV)] of ScB2.
Fig. 3. (Color online) Partial density of states of ScB2.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7 773
S.M. Sichkar and V.N. Antonov
Fig. 4. (Color online) The calculated hole sheet of the Fermi sur-
face around A symmetry point from the 3rd energy band (a) and
electron sheet from the 5th energy band (b) in ScB2.
able covalency. The crystal field at the Sc site (D6h point
symmetry) causes the splitting of Sc d orbitals into a sing-
let 1 23 1
( )g z
a d
−
and two doublets 1ge ( yzd and )xzd and
2ge ( xyd and 2 2 ).
x y
d
−
The crystal field at the B site
(D3h point symmetry) causes the splitting of B p orbitals
into a singlet 4 ( )za p and a doublet 2e ( xp and ).yp B s
states occupy a bottom of valence band between − 11.1 eV
and − 1.0 eV and hybridize strongly with B xp and yp
and Sc xzd and yzd states. B xp and yp occupied states
are located between − 11.0 eV and .Fε B zp states occu-
pied a smaller energy interval from − 5.5 eV to Fε with a
very strong and narrow peak structure at around − 2 eV.
ScB2 diboride has partially unoccupied bonding bands.
The Sc d band along MΓ − is below Fε (Fig. 2) and the
large contribution to ( )FN ε is due to Sc d states. There is
a small hole concentration of B ,2 x yp states at A symme-
try point. Thus, one can expect for ScB2 only low-
temperature superconductivity.
3.2. Fermi surface
The Fermi surface (FS) of ScB2 consists of three sheets:
two small almost identical closed hole ellipsoids around
A symmetry point (Fig. 4(a)) and open large electron
sheet derived from the crossing of the Fermi energy by the
5th energy band (Fig. 4(b)). Figure 5 shows the calculated
Fermi surface cross section areas of ScB2 in the plane per-
pendicular to the z direction and crossed A symmetry
point, crossed Γ point and in the plane at half distance
between A and Γ points.
The Fermi surfaces of ScB2, ZrB2, and HfB2 were stu-
died by Pluzhnikov et al. [25] using the dHvA effect. Fig-
ure 6 represents angular variations of the experimentally
measured dHvA frequencies [25] for ScB2 in comparison
with the first-principle calculations for field direction in
the (1010), (1120), and (0001) planes. The observed de-
pendences ,τ ϕ, and Ψ show that these fragments of the
FS are strongly anisotropic, and the dependences of π and
σ are close to an ellipsoidal frequencies. The theoretical
calculations reveal that π and σ oscillations indeed be-
long to hole ellipsoids around the A symmetry point. The
ϕ, ,τ and Ψ orbits belong to large open electron FS. Two
Fig. 5. (Color online) The calculated Fermi surface cross section areas of ScB2 in the plane perpendicular to the z direction and cross A
symmetry point (a), cross Γ point (c) and in the plane at half distance between A and Γ points (b). The cross sections of hole ellipsoids
present by full blue curve and red dashed curve for the 3rd and 4th energy bands, respectively. The magenta curves at all three panels show
the cross sections of the Fermi surface for the electron sheet corresponding to the 5th energy band.
774 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7
Electronic structure, phonon spectra and electron–phonon interaction in ScB2
hole ellipsoids around A point have almost identical form
and slightly different size, as a result they have similar
Fermi surface cross section areas (π and σ orbits for the
3rd and 4th energy bands, respectively) appeared at all the
three planes (Fig. 6). The theory reasonably well reproduc-
es the experimentally measured frequencies for the ,σ ϕ,
,τ and Ψ orbits. However, at the (1120) plane we were
not able to detect low frequency β orbit observed experi-
mentally. The experiment for high frequencies detected
only Ψ orbits in vicinity of the < 0001 > direction in
ScB2. We found an additional ,α β and γ orbits situated
at the electron FS. These orbits have not been detected in
the dHvA experiment [25]. One of the possible reasons for
that is the relatively large cyclotron masses for these orbits.
Figure 7 shows the theoretically calculated angular depen-
dence of the cyclotron masses ( )bm and the experimental-
ly measured masses *( )cm in ScB2. The cyclotron effective
masses were determined from the temperature depen-
dences of the amplitudes of the dHvA oscillations. The
cyclotron masses for the α , β and γ orbits are much larg-
er than for the corresponding low-frequency oscillations
,π ,σ ,τ and .Ψ
We note that band cyclotron effective masses bm are
renormalized by the electron–phonon interaction * =cm
(1 ),bm= + λ where λ is the constant of the electron–
phonon interaction. By comparing the experimentally
measured cyclotron masses with band masses we can esti-
mate the .λ It is strongly varied on the orbit type and
magnetic direction. We estimate the constant of the elec-
tron–phonon interaction to be equal to 0.16 and 0.41 for the
π and σ hole orbits, respectively, with H || <0001>. The
constant λ for the electron orbits is significantly larger and
equal to 0.9 and 1.1 for the Ψ and τ orbits, respectively.
3.3. Phonon spectra
The unit cell of ScB2 contains three atoms, which gives
in general case a nine phonon branches. Figure 8 shows
theoretically calculated phonon dispersion curves along Γ–A
symmetry direction in ScB2. Figure 9 shows theoretically
calculated phonon density of state for ScB2 (full blue
curve). The DOS for ScB2 can be separated into three dis-
tinct regions. Based on our analysis of relative directions
of eigenvectors for each atom in unit cell, we found that
the first region (with a peak in phonon DOS at 26.9 meV)
is dominated by the motion of Sc. This region belongs
Fig. 6. (Color online) The calculated (open red, blue and magenta
circles) and experimentally measured [25] (black full squares)
angular dependence of the dHvA oscillation frequencies in the
compound ScB2.
Fig. 7. (Color online) The calculated angular dependence of the
cyclotron masses in ScB2 (open red, blue and magenta circles)
and experimentally measured ones [25] (black full squares).
Fig. 8. (Color online) Theoretically calculated phonon dispersion
curves along Γ–A symmetry direction in ScB2.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7 775
S.M. Sichkar and V.N. Antonov
to the acoustic phonon modes. The second wide region
(54–83 meV) results from the coupled motion of Sc and
the two B atoms in the unit cell. The 1 ,uE 2 ,gA 1gB pho-
non modes lie in this area (see Fig. 8 and Table 1). The pho-
non DOS in the third region extends from 83 to 100 meV.
This is due to the movement of boron atoms and is ex-
pected since boron is lighter than Sc. The covalent charac-
ter of the B–B bonding is also crucial for the high frequen-
cy of phonons. The in-plane 2gE mode belongs to this
region. The second and third regions represent optical pho-
non modes in crystals. The most significant feature in the
phonon DOS is a gap around 33–54 meV. This gap is a
consequence of the large mass difference between B and
Sc, which leads to decoupling of the transition metal and
boron vibrations.
Currently, there are no data concerning the experimen-
tally measured phonon DOS in ScB2. So we compare our
results with theoretically calculated phonon DOS by Deli-
goz et al. [15] (Fig. 9). Calculations of these authors were
based on the density functional formalism and generalized
gradient approximation. They used the Perdew–Burke–
Ernzerhof functional [31] for the exchange-correlation
energy as it is implemented in the SIESTA code [32]. This
code calculates the total energies and atomic Hellmann–
Feynman forces using a linear combination of atomic or-
bitals as the basis set. The basis set consists of finite range
pseudoatomic orbitals of the Sankey–Niklewsky type [33]
was generalized to include multiplezeta decays. The inte-
ractions between electrons and core ions were simulated
with the separable Troullier–Martins [34] normconserving
pseudopotentials. In other words, they used the so-called
“frozen phonon” technique and built an optimized rhombo-
hedral supercell with 36 atoms. This method is inconvenient
for calculating phonon spectra for small q points as well as
for compounds with large number of atoms per unit cell.
There is a good agreement between our calculations and
the results of Deligoz et al. [15] in a shape and energy posi-
tion of the first peak in the phonon DOS. The second and
third regions consist of two peak each, in both the regions
high-energy peaks have smaller intensity in our calculations
in comparison with results of Deligoz et al. [15] (see also
Table 1). There is also small high-energy shift of the third
peak in Deligoz et al. [15] in comparison with our results.
3.4. Electron–phonon interaction
Figure 10(a) shows theoretically calculated Eliashberg
functions for ScB2. There are two main differences in
comparison with phonon DOS curve. Low-energy peak is
significantly reduced and shifted downwards by 1.6 meV.
Wide 62–83 meV shoulder transform into two peaks on
68.3 and 72.8 meV. Analysis of electron–phonon prefactor
2 ( )α ω 2 2( ( ) ( ) ( )/ ( ))F Fα ω ≡ α ω ω ω shows that 2gB pho-
non mode has strong coupling with electronic subsystem and
significantly expands middle-energy region on Eliashberg
curve. As consequence, averaged electron–phonon interac-
tion constant phe−λ is reasonably large and equal to 0.47.
A unique feature of electron–phonon coupling in MgB2
(Tc = 39 K) is the down-shift of the in-plane 2gE mode
well below the out-of-plane 1gB mode [35]. Such an in-
version of the usual sequence of mode frequencies give
one wide continuous peak formed from medium- and high-
energy regions. ScB2 to some extend is situated in the
middle between MB2 (M = Ti, Zr, and Hf) and MgB2 dibo-
rides. Superconductivity in MgB2 is also related to the ex-
istence of B ,2 x yp band hole along the Γ–A direction.
Moreover, according to Ref. 36, the existence of degene-
rate ,x yp states above Fε in the Brillouin zone is crucial
Fig. 9. (Color online) Theoretically calculated phonon density of
states (full blue line) for ScB2. The open circles presents the cal-
culated phonon DOS of ScB2 by Deligoz et al. [15].
Table 1. Theoretically calculated phonon frequencies (in
meV) in the Γ symmetry point for ScB2 and calculated ones by
Deligoz et al. [15]
Reference E1u A2g B1g E2g
Our results 55.247 56.85 79.113 99.29
Ref. 15 55.72 56.11 77.08 99.72
Fig. 10. (Color online) Theoretically calculated Eliashberg function
2 ( )Fα ω of ScB2 (a) and electron–phonon prefactor 2( )α ω (b).
776 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7
Electronic structure, phonon spectra and electron–phonon interaction in ScB2
for the supercondactivity in diborides. The ,2 –2 x yD p
bands in ScB2 are partially filled with small hole concen-
tration near A point (see Figs. 2 and 3). On the other hand,
the Fermi level for MB2 (M = Ti, Zr, and Hf) diborides
falls in the pseudogap and the B ,2 x yp bands are com-
pletely filled [14,35,37]. A systematic experimental search
in the past for superconductivity in the d diborides showed
that Tc for MB2 (M = Ti, Zr, Hf, V, Nb, Cr, Mo, U) is below
0.4 K [2].
To calculate the Tc for ScB2, we used in our calcula-
tions McMillan formula modified by Allen–Dynes [38]:
log
*
1.04(1 )= ,
1.2 (1 0.62 )
cT
ω + λ
− λ −µ + λ
(5)
where logω is the effective logarithmically averaged pho-
non frequency, *µ is the screening Coulomb pseudopoten-
tial. We obtained Tc =1.62 K *(µ = 0.14) in good agree-
ment with the experimental value 1.5 K [5]. Relatively low
value can be explained by smallest logω among all transi-
tion metal diborides (see Table 2). This quantity represents
the effective average frequency of the coupling modes and
sets the energy scale for the pairing interaction. Its small
value indicates that the pairing interaction is mainly me-
diated by the d atom vibrations and not by the boron modes.
3.5. Electrical resistivity
In the pure metals (excluding low-temperature region),
the electron–phonon interaction is the dominant factor go-
verning electrical conductivity of the substance. Using
lowest-order variational approximation, the solution for the
Boltzmann equation gives the following formula for the
temperature dependence of ( ):I Tρ
2
2cell
tr2 2
0
( ) = ( ),
( ) sinh
B
I
F I
k T dT F
N v
∞πΩ ω ξ
ρ α ω
ωε 〈 〉 ξ∫ (6)
where the subscript I specifies the direction of the elec-
trical current. In our work, we investigate two direction:
[0001] (c axis or z direction) and [1010] (a axis or x direc-
tion). 2
Iv〈 〉 is the average square of the I component of
the Fermi velocity, = /2 .Bk Tξ ω
Mathematically, the transport function tr ( )Fα ω dif-
fers from ( )Fα ω only by an additional factor
2[1 ( ) ( )/ ],I I Iv v v′− 〈 〉k k which preferentially weights the
backscattering processes.
Formula (6) remains valid in the range
tr tr/5 < < 2TΘ Θ [28], where
2 1/2
tr tr ,Θ ≡ 〈ω 〉 (7)
2 2
tr tr
tr 0
2= ( )F d
∞
〈ω 〉 ωα ω ω
λ ∫ , (8)
2
tr tr
0
= 2 ( ) dF
∞ ω
λ α ω
ω∫ . (9)
The low-temperature electrical resistivity is the result of
electron–electron interaction, size effects, scattering on
impurities, etc., however, for high temperatures it is neces-
sarily to take into account the effects of anharmonicity and
the temperature smearing of the Fermi surface. In our cal-
culations trΘ = 606.69 K for c axis, and 500.7 for a axis
for ScB2.
Figure 11 represents the theoretically calculated tem-
perature dependence of electrical resistivity in ScB2. We
obtained anisotropy ratio of electrical resistivity at T =
= 300 K: /x zρ ρ = 1.33. Among previously studied dibo-
rides (HfB2, ZrB2 and TiB2) [39,40] ScB2 has the highest
degree of anisotropy.
4. Summary
We have studied the electronic structure and physical
properties of ScB2 using fully relativistic and full potential
linear muffin-tin orbital methods. We study the electron
and phonon subsystems as well as the electron–phonon
interaction in this compound.
We investigated the Fermi surface, angle dependence of
the cyclotron masses, and extremal cross sections of the
Fermi surface of ScB2 in details. Theoretical calculations
show that the Fermi surface of ScB2 consists of three
sheets: two small almost identical closed hole ellipsoids
Fig. 11. (Color online) Temperature dependence of electrical
resistivity in ScB2. Theoretically calculated for the <0001> di-
rection (dashed blue curve) and the basal <1010> direction (full
red curve).
Table 2. The values of phe−λ and logω for transition metal
diborides
Diborides ScB2 ZrB2 [39] TiB2 [39] HfB2 [40]
phe−λ 0.47 0.14 0.15 0.17
logω 372.23 520.37 582.13 459.25
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7 777
S.M. Sichkar and V.N. Antonov
around A symmetry point and open large electron sheet
derived from the crossing of the Fermi energy by the 5th
energy band. The angle variation of the ,τ ϕ, and Ψ show
that these fragments of the FS are strongly anisotropic, and
the dependencies of π and σ are close to an ellipsoidal
frequencies. The theoretical calculations reveal that π and
σ oscillations belong to hole ellipsoids around the A
symmetry point. The ϕ, ,τ and Ψ orbits belong to large
open electron FS. Two hole ellipsoids around A point
have almost identical form and slightly different size, as a
result they have similar Fermi surface cross section areas.
The theory reproduces the experimentally measured fre-
quencies for the ,σ ϕ, ,τ and Ψ orbits reasonably well.
However, at the (1120) plane we were not able to detect
low frequency β orbit observed experimentally. The expe-
riment for high frequencies detected only Ψ orbits in vi-
cinity of the <0001> direction in ScB2. We found an addi-
tional ,α β and γ orbits situated at the electron FS.
These orbits have not been detected in the dHvA experi-
ment [25]. One of the possible reasons for that is the rela-
tively large cyclotron masses for these orbits. We found
that the cyclotron masses for the ,α β and γ orbits are
much larger than for the corresponding low-frequency os-
cillations ,π ,σ ,τ and .Ψ
Calculated phonon spectra and phonon DOSs for ScB2
is in good agreement with previous calculations. The aver-
aged electron–phonon interaction constant was found to be
rather large phe−λ = 0.47 for ScB2. We calculated the
temperature dependence of the electrical resistivity for
ScB2 in the lowest-order variational approximation of the
Boltzmann equation. We obtained rather large anisotropy
ratio of electrical resistivity at T = 300 K: /x zρ ρ = 1.33.
Acknowledgments
This work was carried out at the Ames Laboratory,
which is operated for the U.S. Department of Energy by
Iowa State University under Contract No. DE-AC02-
07CH11358. This work was supported by the Director for
Energy Research, Office of Basic Energy Sciences of the
U.S. Department of Energy. V.N.A. gratefully acknow-
ledges the hospitality during his stay at Ames Laboratory.
This work was also supported by the National Academy
of Sciences of Ukraine in the framework of the State Tar-
get Scientific and Technology Program “Nanotechnology
and Nanomaterials” for 2010–2014 (No. 0277092303) and
Implementation and Application of Grid Technologies for
2009-2013 (No. 0274092303).
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