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Prediction of structural, lattice dynamical, and mechanical properties of CaB2
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Published on 25 September 2012 on http://pubs.rsc.org | doi:10.1039/C2RA21807K
P. Vajeeston,*ab P. Ravindranac and H. Fjellvåga
Received 16th June 2012, Accepted 20th September 2012
DOI: 10.1039/c2ra21807k
The structural phase stability of CaB2 at ground state and high pressures was investigated using stateof-the-art ab initio calculations based on density functional theory. The calculations predicted that at
equilibrium conditions CaB2 crystallizes in an orthorhombic structure (a-phase; space group Cmmm)
and at high pressure it transforms into hexagonal structure (b-phase; space group P6/mmm). From
lattice dynamics simulation and studies on mechanical properties we have found that the predicted
phases are dynamically as well as mechanically stable. The character of chemical bonding in CaB2 is
discussed on the basis of electronic structures, charge density, charge transfer, and bond overlap
population analyses. The band structures and density of states reveal that both a- and b-CaB2
polymorphs have metallic behaviour.
Introduction
The discovery of unexpected superconductivity with high
transition temperature (TC) in MgB21 has initiated many
experimental2–10 as well as theoretical11–18 investigations on
metallic diborides. The transition temperature for MgB2 (TC #
39 K) exceeds by almost twice the record values of TC for
conventional B1- and A15-type intermetallic superconductors
(SC).19 Compared to high-temperature SC, MgB2 has an
exclusively simple composition and crystal structure (it belongs
to a large group of p-, d-, and f-metal diborides with the AlB2type structure; space group P6/mmm and Z = 1).20 This finding
triggered many investigations of related materials including
various binary and ternary phases. One of the interesting
candidates expected to have higher TC is the hypothetical simple
hexagonal CaB2,21–25 with the same crystal symmetry as MgB2,
with all Mg atoms being replaced by Ca atoms. First-principles
calculations predicted that the simple hexagonal CaB2 should
have a greater unit cell volume (both a and c lattice constants are
longer) and a much larger value for the density of states at the
Fermi energy (EF) than MgB2.21,22,25,26 Both these features are
favorable for a higher TC, since MgB2 is found to have a positive
dependence of TC on the unit-cell volume,27 and more generally
a larger density of states at EF can result in a greater electron–
phonon coupling constant. Although CaB2 is an attractive
candidate for superconductivity, it has not yet been synthesized
and hence its equilibrium geometry is not yet known.
Furthermore, it has not been extensively studied theoretically
a
Center of Theoretical and Computational Chemistry, Department of
Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo,
Norway
b
Center for Materials Sciences and Nanotechnology, Department of
Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo,
Norway. E-mail: ponniahv@kjemi.uio.no
c
Department of Physics, Central University of Tamil Nadu, Thiruvarur 610
004, Tamil Nadu, India
This journal is ß The Royal Society of Chemistry 2012
in the perspective of structural stability and superconductivity. A
recent theoretical study by Choi et al.24 demonstrated that CaB2
in the AlB2-type structure is dynamically stable. In the present
study we have investigated the structural phase stability,
electronic, mechanical, and lattice dynamical properties of the
hitherto experimentally unexplored CaB2 phase.
Computational details
The first-principles calculations were performed based on density
functional theory and the pseudo-potential method implemented
in the CASTEP code.28 Ultrasoft pseudo-potentials were
employed to describe the electron–ion interactions with the
plane-wave cutoff energy of 500 eV. The exchange and
correlation terms were described with generalized gradient
approximations in the scheme of Perdew–Burke–Ernzerhof.29
The geometric optimization of the unit cell was carried out with
the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization
algorithm provided in this code. For each structure the lattice
parameters and atomic positions were fully optimized using the
force and stress minimization. The k-points were generated using
the Monkhorst–Pack method with a grid size of 8 6 8 6 6 for
structural optimization of the AlB2 structure. A similar density
of k-points and energy cut-off were used to estimate total energy
as a function of volume for all the structures considered in the
present study. Iterative relaxation of atomic positions was
stopped when the change in total energy between successive
steps was less than 1 meV/cell. With this criterion, the
forces generally acting on the atoms were found to be less than
1023 eV Å21.
Density functional perturbation theory (DFPT)30 was used for
phonon calculations. For the phonon calculation we have used
norm-conserving pseudo-potentials with 850 eV energy cut-off
for all atoms together with a 16 6 16 6 12 mesh of k points,
with the energy conversion threshold of 0.01 meV/atom,
RSC Adv., 2012, 2, 11687–11694 | 11687
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maximum displacement of 0.001 Å and maximum force of 0.03
eV Å21, yielding a high accuracy for the energy and atomic
displacements. For B and Ca atoms the valence states were
modelled using the 2s2, 2p1 and 4s2 electrons, respectively.
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Structure models considered
In general, at ambient conditions most of the di-borides (AB2
family) crystalize in AlB2-type (P6/mmm; space group 191)
structure. In addition to that, a few of them also crystalize in
RuB2-type (PmmnZ; space group 59), CaSi2-type (example:
MoB2; R3-mH; space group 166), and ReB2-type (P63/mmc;
space group 194) structures. Experimentally one can generally
find the crystal structure of the system from X-ray diffraction,
powder neutron diffraction, and Raman spectra studies. On the
other hand no unique method is available to identify the ground
state structure on a theoretical basis. The equilibrium crystal
structures predicted based on first principles calculations using
the structural inputs from inorganic crystal structure data base
(ICSD)31 mostly agree well with experimental structures. In our
long experience on predicting structural properties of hydrides
and oxides we found that the ICSD approach is more reliable
when more existing structural information (within similar
chemical formula; e.g., for the present case AB2; A and B are
elements in the periodic table) is used as a starting point. The
reliability of the calculation depends upon the number of input
structures considered in the calculations. It is a tedious process to
select input structures from the 3035 entries for the AX2
composition in the ICSD database, which also involves
tremendous computations. Several compounds/phases have the
same structure type and some cases have only a small variation
in the positional parameters (for certain atoms). Even though we
used different positional parameters, these structures mostly
converted to a similar type of structural arrangement during the
full geometry optimization and hence these possibilities are
omitted. In this particular composition almost 30 structure types
have unique structural arrangements. The involved structure
types are BeH2 (IBAM), XeF2 (I4/mmm), a-SnF2 (C2/c), PdF2
(Pa3̄), HfH2 (I4/m), TiO2 (I41/AMDS), TiO2 (Pbca), TiO2 (P42/
MNM), a-PbO2 (Pbcn), PbF2 (Pnma), InNi2 (P63/mmc), CaF2
Fig. 1 Calculated unit cell volume vs. total energy for a- and
b-variations of CaB2. For more clarity we have presented only the total
energy vs. volume curve for a- and b-CaB2 variations.
11688 | RSC Adv., 2012, 2, 11687–11694
(Fm-3m), AuSn2 (Pbca), GeS2 (P1c1), SiO2 (C1c1), VO2 (C12/
m1), FeS2 (P1), BeF2 (P6222), CdI2 (P3-m1), SiS2 (I4-2d),
a-TeO2 (P41212), Ag2S (IM3-m), ZrOS (P213), CaCl2 (Pnnm),
BS2 (P121/c1), WB2 (P63/mmc), CaSi2 (R3-mH), AlB2 (P6/
mmm), SiO2 (P3221) and RuB2 (PmmnZ).31
Among the considered structures for our structural optimization, the calculated total energy at the equilibrium volume for the
RuB2-derived atomic arrangements occur at the lowest total
energy (see Fig. 1), and in particular, the starting structure is
transformed into another structure. This type of situation
sometimes arises when we carry out the full geometry optimization. Hence, we performed the symmetry analysis for the
optimized structural data for all the phases with low total
energies. Our symmetry analysis shows that during the structural
relaxation process the low symmetry RuB2-type (orthorhombic;
PmmnZ) phase of CaB2 transforms into a somewhat highsymmetry proto-type (orthorhombic; Cmmm; Table 1) phase
(hereafter this phase is referred as a-CaB2). It is well known that,
instead of relaxing to the local minimum, the system sometimes
relaxes to the global minima as is the case here. It should be
noted that the recent work of Ozisik et al.32 demonstrated that at
ambient conditions CaB2 stabilizes in the OsB2 structure. On the
other hand in the present study we found that during the
structural relaxation process the low symmetric RuB2-type
(orthorhombic; PmmnZ) phase of CaB2 transforms into a
somewhat high-symmetry proto-type (orthorhombic; Cmmm)
phase. It is important to note that CaB2 is the first compound in
the AX2 series that is stabilized in Cmmm space group, while
both RuB2 and OsB2 compounds crystalize in PmmnZ space
group (space group number 59). Our calculated lattice parameters for the CaB2 phase are in good agreement with those
reported in ref. 33, thus indicating that during the structural
relaxation process Ozisik et al. may also end up with the Cmmm
space group. However, this information was not given in ref. 32.
The a-CaB2 structure consists of linear chains of B atoms
along the y-axis (see Fig. 2a). This is one of the distinct features
of the a-CaB2 phase compared with the other di-borides known
in the literature to date. In general, in the diboride family (AB2),
the B atoms are formed as a hexagonal lattice in which closepacked alternate A layers are present with graphite-like B layers.
In the a-phase, at the equilibrium volume the calculated B–B
distance along the linear chain is 1.588 Å and the average Ca–B
distance is 2.891 Å. The next energetically favorable phase is an
AlB2-type derived structure model, which occurs at a reduced
volume i.e. at high pressures. It is interesting to note that during
the structural optimizations many of the initially assumed trial
structures (for example: CaCl2, CdI2, PbF2, InNi2, AuSn2)
relaxed towards the AlB2-type structure (AlB2-type; named
b-CaB2). This finding strongly emphasizes that this particular
atomic arrangement is one of the more favorable structures for
the CaB2 phase at high pressures. The crystal structure of AlB2type transition-metal diborides is designated as C32 with the
space group symmetry P6/mmm. It is simply a hexagonal lattice
in which close-packed Ca layers are arranged alternately to the
graphite-like B layers (see Fig. 2b). However, these diborides
cannot be seen exactly as layered compounds because the
interlayer interaction is strong (see the chemical bonding
section). The boron atoms are placed at the corners of hexagons
with three nearest neighbor boron atoms in each plane. The Ca
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Table 1 The calculated equilibrium structural parameters (a, b, and c are in Å) and heat of formation (DH; in kJ mol21) for the CaB2 phases
Compound space group
Cell parameters (Å)
Atom
Site
Position
(DH)
a-CaB2
Cmmm (65)
a = 7.410
b = 3.135
c = 3.716
a = 3.264
c = 4.137
Ca
B
2b
4h
0, K, 0
20.2662, K, K
249.6
Ca
B
1a
2d
0,0,0
1/3, 2/3, K
238.8
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b-CaB2
P6/mmm (191)
AlB2-type
Fig. 2 Theoretically predicted crystal structures for CaB2: (a) orthorhombic a-phase at ambient conditions and (b) hexagonal b-phase at high
pressure. The legends for the different kinds of atoms are given in the
illustration.
atoms are present directly in the centers of each boron hexagon,
but midway between adjacent boron layers; each Ca atom has 12
nearest-neighbor B atoms and six nearest-neighbor in-plane Ca
atoms. There is one formula unit per primitive cell and the
crystal has simple hexagonal symmetry (D6h).
The calculated transition pressure for the a- to b-CaB2
conversion is 8.2 GPa and the involved energy difference
between these two variations is 0.11 eV/f.u. at their equilibrium
volumes. At the a-to-b phase transition point for CaB2 the
estimated difference in cell volume is y6.70 Å3/f.u. The pressure
induced a-to-b transition involves reconstructive rearrangements
of the Ca and B lattices with breaking and reconstruction of
bonds. Usually, application of pressure reduces the covalency in
solids and makes the valence electrons more diffuse than at
Fig. 3 Calculated a/b ratio as a function of unit cell volume/f.u. in
a-CaB2.
This journal is ß The Royal Society of Chemistry 2012
ambient conditions. Due to the reduction in the covalency, this
lower symmetry a-phase of CaB2 transforms into the highly
symmetric b-phase. It is interesting to note that when we
calculate the total energy as a function of cell volume without
any constraint to the geometry and atom positions, the a-phase
automatically transformed into the b-phase at the volume
y38 Å3/f.u., (i.e. the volume versus total energy curve for the
a-phase has two minima). From Fig. 3 it is evident that the a/b
ratio is reduced when we reduce the cell volume. The a/b ratio
becomes 1 close to the volume of y38 Å3/f.u. and this is the
region where the system transforms into the b-phase.
In order to understand the dynamical stability of the predicted
a- and b-phases, we have also calculated the phonon density of
states (DOS) at the equilibrium volume for these two variations,
which are shown in Fig. 4. From this figure it is clear that both aand b-phases have no imaginary frequency, which indicates that
the predicted crystal structures for these compositions are either
ground-state and stable within the thermodynamical conditions,
or at least dynamically stable phases.33 As the predicted phases
are found to be dynamically stable, experimental verification is
needed to confirm our results. Owing to the differences in the
atomic arrangement of the crystal structure of these two phases
(Fig. 2), the a- and b-phases have quite different phonon DOS.
The calculated phonon DOS for the a-phase is very narrow
owing to the one dimensional B chains along y-direction. Since
the mass of B atom is much smaller than that of Ca atom, the
phonon frequencies originating from B atoms have a higher
value. Hence, for the a-and b-phases, the high frequency modes
above 10 THz are dominated by B atoms. In both phases the low
frequency modes below 10 THz are mainly dominated by Ca
atoms.
Fig. 4 Calculated total phonon density of states for a- and b-phases of
CaB2. The variations are noted in the corresponding panel.
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In general, the prepared samples from various methods often
contain more than one phase (mixed phases) if the enthalpy
difference between the phases is small. In such situations it is
very difficult to characterize these samples experimentally where
we can use the theoretically-simulated phonon spectra. The
calculated frequencies of Raman and IR spectra can be used to
distinguish different phases within a sample. So, the theoretical
studies on spectroscopic properties of materials will be complementary to experimental studies to characterize contaminated
or multiphase samples. In connection with this we have
calculated the Raman and IR active modes for the predicted aand b-variations of CaB2. The obtained Raman and IR
frequencies for a- and b-CaB2 are given along with the available
literature values for b-variation in Table 2. Due to the symmetry
changes both a- and b-phases have considerably different
Raman and IR modes. The lower symmetry a-phase has several
Raman and IR modes and the corresponding phonon frequencies for all these peaks are listed in Table 2. However, the high
symmetry b-phase has only one Raman (E2g) and two IR (Eu and
A2u) active modes due to very simple crystal structure. In order
to understand the stability we have calculated the formation
energy (DH) using the following equation: DHCaB2 = ECaB2 2
(ECa + 2EB) where, ECaB2, ECa, and EB are the optimized total
energy (with respect to stress and strain) of a-/b-CaB2, Ca (cubic;
Fm-3m), and B (hexagonal; R-3m) respectively. The calculated
negative value of the formation energies (Table 1) for the a- and
b-phases indicate that these phases are stable and might exist in
certain conditions. To verify our prediction, more careful
experimental efforts are needed.
The total DOS of both a- and b-CaB2 phases given in Fig. 5
show that these two phases have metallic character with finite
electrons at the Fermi level (EF). The calculated site projected
electronic DOS for a- and b-CaB2 are given in Fig. 6. The
energies in this figure are shifted so that the Fermi energies are
aligned with zero (shown by a dotted line). The electronic
structures of CaB2 polymorphs exhibit well-localized peaks
compared to superconducting MgB2 (which has a broad valence
band) due to the smaller lattice constants of MgB2. The
predicted a- and b-CaB2 phases have a larger total electronic
DOS value at the Fermi energy than MgB2. The more localized
electronic band feature of CaB2 is associated with a large
interatomic distance between the constituents, which reduces the
overlap interactions between electronic states leading to narrow
bands. The expanded volume as well as large DOS at the EF are
favorable for a higher superconducting transition temperature,
since the isoelectronic MgB2 is found to have a positive
dependence of TC on the unit-cell volume. In general, within
Fig. 5 Comparison of the calculated total electronic density of states for
MgB2, a- and b-variations of CaB2.
the Bardeen-Cooper-Schrieffer (BCS) mechanism for superconductivity, a larger electronic DOS at EF can result in a
higher value of electron–phonon coupling constant. In this
aspect both a- and b-CaB2 have a larger electronic DOS at EF
and one can expect higher TC. This finding is consistent with
previous findings (only for the b-CaB2) from theoretical
studies.21,22,25,26 However, our calculated electronic structure of
both phases (shown in Fig. 7) do not exhibit the distinct doublydegenerate flat band feature in the vicinity of EF which is
believed to bring higher TC in superconducting MgB2.25 Hence,
even if superconductivity is observed in the polymorphs of CaB2,
the superconducting TC may not be as high as that in MgB2.
In order to understand the origin of metallic behavior of the
CaB2 polymorphs, we have conducted an orbital-decomposed
electronic structure analysis of both the phases. As shown in
Fig. 6 we found that for the a-phase the metallic nature mainly
originates from the B py and pz states due to the one-dimensional
character of the crystal structure. However, for the b-phase the
metallic nature mainly originates from the px, py and pz states of
B atoms (see Fig. 6). The broad band feature of the p-states
suggests that p–p s bonding is present between the B atoms. The
stronger covalent interaction between B–B within the linear
chain and the large separation between chains localize the px
states in the a-phase. In both cases the Ca s-electrons contribute
very little to the valence band and are mainly reflected in the
unoccupied state. Hence one can conclude that the Ca atoms
donate their valence electrons to the boron chains/layers. The B
2s electrons are well localized and hence their contribution at EF
Table 2 The calculated Raman and IR frequency (in cm21) for the modes at the C point of the Brillouin zone for CaB2 phases. The Mullikenpopulation charge density analysis for CaB2 polymorphs are given in terms of bond overlap population (denoted as BOP) and Mulliken-effective
charges (MEC) (given in e)
Compound
Raman active modes
IR active modes
Atom
MEC
BOP
a-CaB2
Ag: 165, 502, 529
B1g: 1202
B2g: 255, 533, 561
B3g: 96, 1204
E2g: 588 (664)a
B1u: 281, 314
B2u: 123
B3u: 231, 289
Ca
B
20.79
+1.59
2.83 (B–B)
0.05 (Ca–B)
A2u: 305 (290)a
E1u: 263
Ca
B
20.71
+1.42
0.89 (B–B)
0.16 (Ca–B)
b-CaB2
a
From ref. 24.
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Fig. 6 Calculated total site projected electronic DOS for a- and b-variations of CaB2. The Fermi level is set at zero energy and marked by the vertical
dotted line.
Fig. 7 Electronic band structure of (a) a- and (b) b-CaB2. The Fermi level is set to zero.
is found to be minimum. The B px in a-phase and pz states in the
b-phase are present in a wide energy range and dominate the
bottom of the valence band. In contrast, the B p states are
dominant in the valence band and are mainly responsible for the
metallic behaviour.
Single crystal elastic constants and mechanical stability
In order to understand the mechanical stability of the predicted
phases, we have computed single crystal elastic constants from
linear response DFPT calculations.34 The elastic constants of a
material describe its response to an applied stress or, conversely,
the stress required to maintain a given deformation. Both stress
and strain have three tensile and three shear components, giving
six components in total. The linear elastic constants in a crystal
form a 6 6 6 symmetric matrix, having 27 different components
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and 21 of which are independent. However, any symmetry
presented in the structure may reduce the number of these
components. For an orthorhombic crystal, the independent
elastic stiffness tensor reduces to nine components C11, C22, C33,
C44, C55, C66, C12, C13 and C23 in the Voigt notation.35 The
well-known Born stability criteria36 for an orthorhombic system
are:
B1 = C11 + C22 + C33 + 2(C12 + C13 + C23) . 0,
(1)
B2 = C11 + C22 2 2C12 . 0
(2)
B3 = C11 + C33 2 2C13 . 0, and
(3)
B4 = C22 + C33 2 2C23 . 0.
(4)
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Table 3 The calculated single crystal elastic constants Cij (in GPa), bulk
modulus (in GPa), Youngs modulus (in GPa), and compressibility
(GPa21) for CaB2 polymorphs
Properties
a-CaB2
b-CaB2
Elastic constants (Cij)
C11 = 119
C12 = 225
C13 = 35
C22 = 490
C23 = 230
C33 = 125
C44 = 1
C55 = 38
C66 = 19
61
109 (along x)
479 (along y)
114 (along z)
0.017
C11
C12
C13
C33
C44
Bulk modulus
Young’s modulus
Compressibility
=
=
=
=
=
269
65
38
269
93
121
250(along x and y)
260 (along z)
0.0083
All the nine calculated independent single crystalline elastic
stiffness constants for a-CaB2 are given in Table 3. The
computed B1, B2, B3, and B4 values for a-CaB2 are 714, 659,
174, and 675 GPa, respectively. All the four conditions for
mechanical stability given in eqn (1)–(4) are simultaneously
satisfied and this clearly indicates that the predicted a-phase is a
mechanically stable phase.
For a stable hexagonal structure, Cij should satisfy the well
known Born stability criteria:36
C12 . 0 , C33 . 0 , C44 . 0,
B5 = (C112C12)/2 . 0, and
(5)
B6 = (C11+C12)C3322C132 . 0
(6)
The computed B5 and B6 values for b-CaB2 in its equilibrium
volume are 102, and 86 958 GPa, respectively. All the conditions
given in eqn (5)–(6) are simultaneously satisfied and this clearly
indicates that, similar to the a-phase, the predicted b-phase is
also mechanically stable. Like the elastic constant tensor, the
bulk and shear moduli contain information regarding the
hardness of a material with respect to various types of
deformation. Properties such as bulk moduli, shear moduli,
Young’s moduli and Poisson’s ratio can be computed from the
values of elastic constants and the calculated values are tabulated
in Table 3. There is strong anisotropy in the calculated Young’s
modulus (more than four times larger along the y direction
compared with the x and z directions) in the a-phase and this is
associated with the presence of one-dimensional borohydride
chains in this phase. In general, most of the borides are
reasonably hard materials and also having higher bulk modulus
value. This is partially due to the strong bonding interaction
between B–B. On the other hand the present a-CaB2 phase has a
much lower bulk modulus and higher compressibility than the
b-CaB2 and other di-borides. The possible reason for such lower
hardness can be explained by the formation of one dimensional
B chains along the (010) direction along with the relatively higher
equilibrium volume.
11692 | RSC Adv., 2012, 2, 11687–11694
Chemical bonding
The a- and b-CaB2 exhibit similar bonding characteristics, but,
the magnitude of the interaction is different. Fig. 8 shows the
charge-density distribution at the Ca and B sites, from which it is
evident that the highest charge density resides in the immediate
vicinity of the nuclei. Furthermore, the spherical charge
distribution shows that the bonding interactions between Ca–B
have predominantly ionic character. On the other hand, the nonspherical distribution of charge at the B sites and the presence of
charge density between B atoms indicate that the interactions
between B–B have mainly a directional character (see Fig. 8).
The substantial difference in the electro-negativity between Ca
and B suggests the presence of strong ionic character (i.e., the Ca
valence electrons are transferred to the B sites). From chargetransfer plots in Fig. 8 we found that finite numbers of electrons
are present between B–B, which means that there is a significant
degree of covalent interaction between B and B in the chain/
layers. Furthermore, the charge-transfer plot shows that charges
are depleted from Ca (not shown in Fig. 8) and the depletion is
spherically symmetric at the Ca site, which implies that the
bonding interaction between Ca and B can be regarded as ionic.
In order to quantify the bonding and estimate the amount of
electrons on and between the participating atoms, we have
conducted a Mulliken-population analysis.37 Although there is
no unique definition to identify how many electrons are
associated with an atom in a molecule or an atomic grouping
in a solid, it has nevertheless proved useful in many cases to
perform population analyses. Due to its simplicity, the Mullikenpopulation scheme has become the most popular approach.
However, the method is more qualitative than quantitative,
providing results that are sensitive to the atomic basis. The
calculated Mulliken charges are reported in Table 2 for CaB2
polymorphs. The Mulliken effective charge (MEC) value for Ca
varies from +1.42 to +1.59e (minimum in b-phase and maximum
in a-phase). This finding is consistent with the charge density
analyses and the magnitude of the MEC shows that Ca does not
completely donate its two valence electrons to the B atoms,
which is much smaller than in a pure ionic picture.
To understand the bonding interaction between the constituents the bond overlap population (BOP) values are also
calculated on the basis of Mulliken population analysis. The
BOP values can provide useful information about the bonding
property between atoms. A high BOP value indicates a strong
covalent bond, while a low BOP value indicates an ionic/nonbonding interaction. The calculated BOP values for the CaB2
polymorphs are listed in Table 2. The BOP values for the Ca–B
bonds in the calculated two structures vary between 0.05 and
0.16. Similarly, the calculated BOP values for the B–B bonds
vary between 0.89 and 2.83. Therefore, the Ca–B bonds in these
borides can be regarded as having dominant ionic character,
whereas the B–B interactions have a strong covalent character.
When we compare the BOP values for the different bonding
interactions between atoms in a- and b-phases, overall the
bonding interactions are much stronger in a-CaB2 and relatively
weaker in b-CaB2. This finding is in contrast to the bulk modulus
calculation where the calculated bulk modulus is much higher in
b-phase and at a minimum in the a-phase. The possible reason is
that, in the a-phase, even though the B–B interactions are
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Fig. 8 The calculated valence electron charge density distribution (a, c, d, and f) and charge difference plot (b and e) for the a- (top panel) and b-CaB2
(bottom panel) phases.
relatively stronger, the basic structure is one dimensional. This
results in weaker interaction between the chains and brings the
equilibrium volume higher than that in the b-phase. On the other
hand, in the b-phase, the B atoms are arranged in a stronger 6
ring configuration. Moreover, the equilibrium volume for the
b-phase is much smaller than that of a-CaB2 and this could
explain the higher value of the bulk modulus in the b-phase.
Conclusions
We have performed first-principles DFT calculations to predict
the phase stability of CaB2 and the potential ground state crystal
structure was identified. The elastic, mechanical, and lattice
dynamical properties of CaB2 in two different polymorphs were
studied using state-of-the-art density functional calculations. The
ground-state crystal structures have been identified from
structural optimization of a number of structures using force
as well as stress minimizations. At ambient conditions CaB2
crystallizes in an orthorhombic phase, which has a fascinating
one dimensional B chain along the y-axis. This is the first
compound in the AX2 series (including, borides, oxides,
fluorides, etc) to stabilize in the Cmmm space group. At 8.2
GPa pressure the ground state orthorhombic structure transforms into an AlB2-type hexagonal structure. Furthermore, the
predicted polymorphs of CaB2 have been found to be
mechanically and dynamically stable. The chemical bonding
character of these phases was analysed using DOS, charge
density, charge transfer, and Mulliken population and found
that the interaction between B–B is highly covalent and that
between B–Ca is dominantly ionic.
Acknowledgements
The authors gratefully acknowledge the Research Council of
Norway 460829 and European Union seventh frame work
This journal is ß The Royal Society of Chemistry 2012
program under the ‘‘NanoHy’’ (Grant agreement no.: 210092)
project for financial support. PV acknowledges the Research
Council of Norway for providing computer time at the
Norwegian supercomputer (titan.uio.no).
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