Complex Investigation of the Fermi Surface in HfB₂
The electronic structure, Fermi surface, angle dependence of the cyclotron masses, and extremal cross sections of the Fermi surface of the hafnium diboride are investigated from the first principles using the fully relativistic and full-potential linear muffin-tin orbitals’ methods. A good agreement...
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irk-123456789-1120242017-01-17T03:02:54Z Complex Investigation of the Fermi Surface in HfB₂ Sichkar, S.M. Antonov, V.N. Электронные структура и свойства The electronic structure, Fermi surface, angle dependence of the cyclotron masses, and extremal cross sections of the Fermi surface of the hafnium diboride are investigated from the first principles using the fully relativistic and full-potential linear muffin-tin orbitals’ methods. A good agreement with experimental data of cyclotron masses and extremal cross sections of the Fermi surface is achieved. Електронну структуру, поверхню Фермі, кутову залежність циклотронних мас і екстремальні перетини поверхні Фермі дибориду гафнію було досліджено з перших принципів із використанням повністю релятивістського та повнопотенціяльного методів лінійних МТ-орбіталей. Було досягнуто задовільного збігу з експериментальними даними стосовно циклотронних мас і екстремальних перетинів поверхні Фермі. Электронная структура, поверхность Ферми, угловая зависимость циклотронных масс и экстремальные сечения поверхности Ферми диборида гафния были исследованы из первых принципов с использованием полностью релятивистского и полнопотенциального методов линейных МТ-орбиталей. Было достигнуто хорошее согласие с экспериментальными данными циклотронных масс и экстремальных сечений поверхности Ферми. 2015 Article Complex Investigation of the Fermi Surface in HfB₂ / S. M. Sichkar, V. N. Antonov // Металлофизика и новейшие технологии. — 2015. — Т. 37, № 5. — С. 581-591. — Бібліогр.: 21 назв. — англ. 1024-1809 PACS: 63.20.dk, 63.20.kd, 71.15.Mb, 71.15.Rf, 71.18.+y, 71.20.Ps, 74.70.Ad http://dspace.nbuv.gov.ua/handle/123456789/112024 en Металлофизика и новейшие технологии Інститут металофізики ім. Г.В. Курдюмова НАН України |
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Электронные структура и свойства Электронные структура и свойства |
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Электронные структура и свойства Электронные структура и свойства Sichkar, S.M. Antonov, V.N. Complex Investigation of the Fermi Surface in HfB₂ Металлофизика и новейшие технологии |
| description |
The electronic structure, Fermi surface, angle dependence of the cyclotron masses, and extremal cross sections of the Fermi surface of the hafnium diboride are investigated from the first principles using the fully relativistic and full-potential linear muffin-tin orbitals’ methods. A good agreement with experimental data of cyclotron masses and extremal cross sections of the Fermi surface is achieved. |
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Sichkar, S.M. Antonov, V.N. |
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Sichkar, S.M. Antonov, V.N. |
| author_sort |
Sichkar, S.M. |
| title |
Complex Investigation of the Fermi Surface in HfB₂ |
| title_short |
Complex Investigation of the Fermi Surface in HfB₂ |
| title_full |
Complex Investigation of the Fermi Surface in HfB₂ |
| title_fullStr |
Complex Investigation of the Fermi Surface in HfB₂ |
| title_full_unstemmed |
Complex Investigation of the Fermi Surface in HfB₂ |
| title_sort |
complex investigation of the fermi surface in hfb₂ |
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Інститут металофізики ім. Г.В. Курдюмова НАН України |
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2015 |
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Электронные структура и свойства |
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Complex Investigation of the Fermi Surface in HfB₂ / S. M. Sichkar, V. N. Antonov // Металлофизика и новейшие технологии. — 2015. — Т. 37, № 5. — С. 581-591. — Бібліогр.: 21 назв. — англ. |
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Металлофизика и новейшие технологии |
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581
ЭЛЕКТРОННЫЕ СТРУКТУРА И СВОЙСТВА
PACS numbers:63.20.dk, 63.20.kd,71.15.Mb,71.15.Rf,71.18.+y,71.20.Ps, 74.70.Ad
Complex Investigation of the Fermi Surface in HfB2
S. M. Sichkar and V. N. Antonov
G. V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine,
36 Academician Vernadsky Blvd.,
UA-03680 Kyiv, Ukraine
The electronic structure, Fermi surface, angle dependence of the cyclotron
masses, and extremal cross sections of the Fermi surface of the hafnium dibo-
ride are investigated from the first principles using the fully relativistic and
full-potential linear muffin-tin orbitals’ methods. A good agreement with ex-
perimental data of cyclotron masses and extremal cross sections of the Fermi
surface is achieved.
Електронну структуру, поверхню Фермі, кутову залежність циклотронних
мас і екстремальні перетини поверхні Фермі дибориду гафнію було дослі-
джено з перших принципів із використанням повністю релятивістського та
повнопотенціяльного методів лінійних МТ-орбіталей. Було досягнуто задо-
вільного збігу з експериментальними даними стосовно циклотронних мас і
екстремальних перетинів поверхні Фермі.
Электронная структура, поверхность Ферми, угловая зависимость цикло-
тронных масс и экстремальные сечения поверхности Ферми диборида гаф-
ния были исследованы из первых принципов с использованием полностью
релятивистского и полнопотенциального методов линейных МТ-орбита-
лей. Было достигнуто хорошее согласие с экспериментальными данными
циклотронных масс и экстремальных сечений поверхности Ферми.
Key words: electronic structure, Fermi surface, cyclotron masses, LMTO, di-
borides.
(Received March 12, 2015)
1. INTRODUCTION
Ceramics based on transition-metal borides, nitrides and carbides have
extremely high melting points (2500C) and are referred to as ultra-
Металлофиз. новейшие технол. / Metallofiz. Noveishie Tekhnol.
2015, т. 37, № 5, сс. 581—591
Оттиски доступны непосредственно от издателя
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в соответствии с лицензией
2015 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
582 S. M. SICHKAR and V. N. ANTONOV
high temperature ceramics [1, 2]. Among them, diborides such as ZrB2
and HfB2 have a unique combination of mechanical and physical prop-
erties: high melting points (>3000C); high thermal and electrical con-
ductivity; chemical inertness against molten metals; great thermal
shock resistance [1—3]. Thus, although carbides typically have the
highest melting points (3500C); the diborides ZrB2 and HfB2 are
more attractive candidates for high-temperature thermomechanical
structural applications at temperatures 3000C [1, 2].
Potential applications include thermal protective structures for
leading-edge parts on hypersonic re-entry space vehicles [1, 4], propul-
sion systems [1, 4], furnace elements [5], refractory crucibles [5], and
plasma-arc electrodes [5, 6].
The discovery of superconductivity in MgB2 at 39 K by Akimitsu [7]
has led to booming activity in the physics community and activated a
search for superconductivity in other diborides. Natural candidates
for this search are MB2-type light metal diborides (M Li, Be, Al).
However, up to now, superconductivity has not been reported in the
majority of these compounds [8]. Only very recently superconductivity
below 1 K (Tc 0.72 K) has been reported in BeB2.75 [9]. On the other
hand, there is contradictory report about superconductivity up to
Tc 9.5 K in TaB2 [10], which encourages further studies of these dibo-
rides.
By now, the de Haas—van Alphen (dHvA) measurements of the Fer-
mi surface [11], optical ellipsometry measurements [12], magnetic
susceptibility [13, 14], and NMR measurements [15], as well as first-
principles calculations of the electronic structure of diborides includ-
ing HfB2 have been presented [13, 14, 16—19].
Vajeeston et al. [16] investigated the electronic structure of HfB2
using the tight-bonding linear muffin-tin orbitals’ method, they
claimed that metal—metal and metal—boron interactions are less signif-
icant than the p—p covalent interaction of boron atoms.
Zhang et al. [17, 18] investigated theoretically the pressure depend-
ence of elastic constants, bulk modulus and elastic anisotropy of HfB2.
The pressure dependence of structural property shows that the effect
of pressure is little on the structure of HfB2.
They found that high pressure greatly changes the profile of the
density of states (DOS), but it hardly changes the DOS value at Fermi
level. Meanwhile, the Mulliken population analyses are investigated. It
was suggested that as the pressure increases, a number of charge
transfer from Hf to B atoms takes place. Through quasi-harmonic De-
bye model, the variations of the Debye temperature, heat capacity, and
thermal expansion with pressure and temperature were obtained and
discussed.
Fedorchenko and Grechnev with co-authors [13, 14] measured the
temperature dependences of the magnetic susceptibility and its ani-
COMPLEX INVESTIGATION OF THE FERMI SURFACE IN HfB2 583
sotropy for single crystals of transition-metal diborides
MB2 (M Sc, Ti, V, Zr, Hf) in the temperature interval 4.2—300 K. A
transition into the superconducting 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, nonmono-
tonic function of the filling of the hybridized p—d conduction band.
First-principles calculations of the electronic structure of diborides
and the values of the paramagnetic contributions spin and Van Vleck to
their susceptibility show that the behaviour of the magnetic anisotro-
py is caused by the competition between Van Vleck paramagnetism and
orbital diamagnetism of the conduction electrons.
Li et al. [19] studied the thermodynamics of the oxidation of HfB2 at
temperatures of 1000, 1500, 2000, and 2500 K using volatility dia-
grams. They found that HfB2 exhibits oxidation behaviour similar to
ZrB2.
The properties of the Fermi surface of ScB2, ZrB2, and HfB2 single
crystals were studied by Pluzhnikov et al. [11] using the de Haas—van
Alphen effect. The angular dependences of the frequencies of the
dHvA oscillations in the planes (101 0), (112 0) and (0001), and the
values of their effective cyclotron masses were measured. The fre-
quencies of the oscillations lie in the interval (0.96—0.87)102T and the
measured cyclotron masses
*
cm lie in the range (0.09—0.87)m0.
Despite a lot of publications, there are still many open questions re-
lated to the electronic structure and physical properties of HfB2 dibo-
ride. Up to now, there is no theoretical explanation of the real form of
Fermi surface as well as angle dependence of the cyclotron masses and
extremal cross sections of the Fermi surface in HfB2.
The aim of this work is a complex investigation of the electronic
structure, Fermi surface, angle dependence of the cyclotron masses,
and extremal cross sections of the Fermi surface of the hafnium dibo-
ride. The paper is organized as follows. Section 2 presents the details of
the lattice structure in HfB2. Section 3 is devoted to the electronic
structure. Section 4 focuses on angle dependence of the cyclotron
masses and extremal cross sections of the Fermi surface. The results
are compared with available experimental data. Finally, the main re-
sults are summarized in Sec. 5.
2. GEOMETRIC STRUCTURE OF HfB2
It is simply a hexagonal lattice in which closely-packed transition-
metal layers are present alternative with graphite-like B layers (Fig.
1). These diborides cannot be exactly layered compounds because the
interlayer interaction 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 neighbour boron atoms in
584 S. M. SICHKAR and V. N. ANTONOV
each plane. The M atoms lie directly in the centres of each boron hexa-
gon, but midway between adjacent boron layers. Each transition metal
atom has twelve nearest neighbour B atoms and eight nearest neigh-
bour transition metal 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).
3. ENERGY BAND STRUCTURE
Figure 2 presents the energy band structure and total density of states
of HfB2. The partial DOSs for HfB2 are shown in Fig. 3. Our results for
the electronic structure of HfB2 are in agreement with earlier calcula-
tions [14, 16]. A common feature for all transition metal diborides is
the deep DOS minimum (pseudo-gap) at the Fermi energy separating
the valence band and the conduction band. According to Pasturel et al.
[20], a pseudo-gap arises because of a strong chemical interaction. The
M—B covalent bonding is believed to be responsible for this effect. The
Fig. 1. Schematic representation of the HfB2 crystal structure.
Fig. 2. Energy band structure and total DOS [in states/(cell eV)] of HfB2.
COMPLEX INVESTIGATION OF THE FERMI SURFACE IN HfB2 585
Hf4f5/2,7/2 states in HfB2 are situated at the 14.5 eV to 10 eV. The
Hf5d states are the dominant features in the interval from 12.5 eV to
14 eV. These tightly bound states show overlap with B2p and, to a less-
er extent, with B2s states both above and below F, implying considera-
ble covalence. Higher-energy states between 9 eV and 17 eV above F
appear to arise from Hf6p and 6s states hybridized with B2p states.
The crystal field at the Hf site (D6h point symmetry) causes the split-
ting of Hfd orbitals into a singlet a1g (d3z
21) and two doublets e1g (dyz
and dxz) and e2g (dxy and dx
2y
2). The crystal field at the B site (D3h point
symmetry) causes the splitting of Bp orbitals into a singlet a4 (pz) and a
doublet e2 (px and py). Bs states occupy a bottom of valence band be-
tween 14.6 eV and 3.0 eV and hybridize strongly with Bpx and py and
Hfdyz and dxz states located at 12.5 eV to 0.5 eV. Bpx and py occupied
states are located between 12.5 eV and 0.5 eV. Bpz states occupied a
smaller energy interval from 7.5 eV to 0.5 eV with a very strong and
narrow peak structure at around 4 eV.
Fig. 3. Partial DOSs [in states/(atom eV)] of HfB2.
586 S. M. SICHKAR and V. N. ANTONOV
4. FERMI SURFACE
The Fermi surfaces (FS) of ScB2, HfB2, and HfB2 were studied by Plu-
zhnikov et al. [11] using the dHvA effect. Theoretical calculations
show a ring-like electron FS around the K symmetry point (Fig. 4) and
of a wrinkled dumbbell-like hole FS at the A point (Fig. 5) in HfB2. The
electron and hole Fermi surfaces have threefold and sixfold symme-
tries, respectively.
Figure 6 shows the calculated cross section areas in the plane per-
pendicular z direction and crossed A symmetry point for hole FS (upper
panel) and crossed point for electron FS (lower panel) of HfB2.
Fig. 4. The calculated electron sheets of the Fermi surface around K symmetry
point from the 6th energy band of HfB2.
Fig. 5. The calculated hole sheets of the Fermi surface at the A symmetry point
from the 5th energy band of HfB2.
COMPLEX INVESTIGATION OF THE FERMI SURFACE IN HfB2 587
Figure 7 represents angular variations of the experimentally meas-
ured dHvA frequencies [11] for HfB2 in comparison with the first-
principle calculations for field direction in the (101 0), (112 0), and
(0001) planes. The observed frequencies of , , , and oscillations
belong to electron FS around the K point. The , , and orbits belong
to the hole wrinkled dumbbell FS. The frequencies have four branch-
es at the (101 0) plane and three branches at the (112 0) plane. The
lower frequencies have one branch in both planes. The theory reason-
ably well reproduces the frequencies measured experimentally. How-
ever, there are still some discrepancies. The orbits have additional
two branches at higher frequencies on the (112 0) and (0001) planes
not observed experimentally. The experiment for high frequencies de-
tected only orbits in vicinity of the <0001> direction in HfB2. We
found , and orbits similar to the corresponding orbits observed ex-
perimentally in isostructural and isovalent ZrB2 [11].
These orbits have not been detected in the dHvA experiment [11].
Fig. 6. The calculated cross-sections in the plane perpendicular z direction and
crossed A symmetry point (upper panel) and point (lower panel) for HfB2
(full curves).
588 S. M. SICHKAR and V. N. ANTONOV
One of the possible reasons for that is the relatively large cyclotron
mass for these orbits. Figure 8 shows the theoretically calculated angu-
lar dependence of the cyclotron masses (mb) and the experimentally
measured masses (
*
cm ) for high-symmetry directions in HfB2.
The cyclotron effective masses were determined from the tempera-
ture dependences of the amplitudes of the dHvA oscillations. The cy-
clotron masses for , , and orbits in HfB2 are much higher than the
corresponding low-frequency oscillations , , and .
The fact that the masses for electron Fermi surface are significantly
larger than for the hole one may explain a negative experimentally
measured Hall coefficient [21] and confirms electrons as the dominant
charge carriers in HfB2. A Wiedemann—Franz analysis also indicates
the dominance of electronic contributions to heat transport [21].
We note that band cyclotron effective masses mb are renormalized by
the electron—phonon interaction
*
cm mb(1 ), where is the constant
Fig. 7. The calculated (open circles for the electron (lower part) and hole (up-
per part) surfaces) and experimentally measured [11] (black full squares) an-
gular dependence of the dHvA oscillation frequencies in the HfB2 compound.
COMPLEX INVESTIGATION OF THE FERMI SURFACE IN HfB2 589
of the electron—phonon interaction.
By comparing the experimentally measured cyclotron masses with
band masses, we can estimate the . It is strongly varied on the orbit
type and magnetic direction. We estimate the constant of the electron-
phonon interaction to be equal to 0.18—0.23 for the orbits and 0.36
and 0.75 for the and orbits respectively, with H 0001). For the
<101 0> and <112 0> directions, the for the orbits are reduced (to
0.10 and 0.12 values, respectively).
5. SUMMARY
We investigated in details the Fermi surface, angle dependence of the
cyclotron masses, and extremal cross sections of the Fermi surface of
Fig. 8. The calculated angular dependence of the cyclotron masses for the elec-
tron Fermi surface (open circles in upper part) and the hole Fermi surface
(open triangles in lower part) and experimentally measured ones [11] (black
full squares) in the compound HfB2.
590 S. M. SICHKAR and V. N. ANTONOV
HfB2. Theoretical calculations show a ring-like electron FS in HfB2
around the K symmetry point and a wrinkled dumbbell-like hole FS at
the A point. Theory reproduces the experimentally measured dHvA
frequencies in HfB2 reasonably well. We found that masses for low-
frequency oscillations , , , and are less than 0.25m0. Masses for
high-frequency oscillations , , and lie in the range from 0.3 to
0.65m0. The experiment for high frequencies detected only orbits in
vicinity of the <0001> direction in HfB2. We found , , and orbits
similar to the corresponding orbits observed experimentally in
isostructural and isovalent ZrB2 [11]. These orbits have not been de-
tected in the dHvA experiment. One of the possible reasons for that is
the relatively large cyclotron mass for these orbits.
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|