Elastic properties of mono- and polycrystalline
hexagonal AlB2 – like diborides of s, p and d metals
from first-principles calculations
Igor R. Sheina) and Alexander L. Ivanovskii
Institute of Solid State Chemistry, Ural Branch of the Russian Academy of Sciences, 620041
Ekaterinburg, Russia
We have performed accurate ab initio total energy calculations using the fullpotential linearized augmented plane wave (FP-LAPW) method with the
generalized gradient approximation (GGA) for the exchange-correlation potential
to systematically investigate elastic properties of 18 stable, meta-stable and
hypothetical hexagonal (AlB2 – like) metal diborides MB2, where M = Na, Be,
Mg, Ca, Al, Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Ag and Au. For
monocrystalline MB2 the optimized lattice parameters, independent elastic
constants (Cij), bulk modules (B), shear modules (G) are obtained and analyzed in
comparison with the available theoretical and experimental data. For the first time
numerical estimates of a set of elastic parameters of the polycrystalline MB2
ceramics (in the framework of the Voigt-Reuss-Hill approximation), namely bulk
and shear modules, compressibility (β), Young’s modules (Y), Poisson’s ratio (ν),
Lamé’s coefficients (μ, λ), are performed.
Keywords: First principle; Metal diborides; Elastic properties, Polycrystalline ceramic
a)
Electronic mail: shein@ihim.uran.ru
1
I. INTRODUCTION
Transition metal diborides MB2, as well as MB2-based solid solutions and
composites, attract for a long time much attention of physicists and material
scientists due to their unique physical and chemical properties such as hardness,
high melting point, chemical inertness, etc., and belong to the most promising
engineering materials with a wide range of industrial applications. 1-12 Besides bulk
MB2 materials, increasing attention is devoted recently to their nano-sized forms
such as nanopowders, nanowires, nanotubes and nanocomposites. 13-20
An unexpected discovery
21
of superconductivity with the temperature of
critical transition TC ~ 39K in magnesium diboride MgB2 has renewed a new
powerful stimulus to studying this family of materials, see reviews.
Additionally, the interest in metal diborides (such as OsB2, ReB2, RuB2
32-38
22-31
) is
increased now due to search of new ultra-incompressible superhard materials.
Besides the above-mentioned exciting physical properties, metal diborides
also adopt several crystal structures depending on the RM/RB ratio, where RM and
RB are atomic radii of metallic and boron atoms, respectively. The most
representative group of MB2 borides is formed by the layered hexagonal diborides
with AlB2 - like structural type. For these stable AlB2 - like MB2 phases RM/RB
ratio varies in range from 1.14 to 2.06.39
Along with the above MgB2 and AlB2, the most known MB2 phases are
diborides of transition metals of IV-VI groups (Ti, Zr, Hf, Nb etc.). With filling of
the TM d shell, the stability of these phases rapidly decreases, and MoB 2 and WB2
lie on the border of stability. For example, the experimental data on Mo-B phase
equilibria
5,7,40-48
are quite contradictory: in the composition region near to
"molybdenum diboride” the presence of hexagonal MoB2, rhombohedral Mo2B5,
as well as the metal-deficient (Mo1-xB2 or Mo2-xB5) and boron-deficient
compositions (MoB2-x and also Mo2B5-x) is reported depending on the synthetic
route and the thermodynamic parameters of the process. So, the products obtained
47
by mechano-chemical synthesis without external heat (MoB2, Mo2B5 or MoB2 +
2
Mo2B5) are strongly determined by the relation of co-milled boron and Mo powder
mixtures. On the contrary, upon boronizing of Mo in molten salts,
48
only the
Mo2B5 phase was obtained.
In addition, a set of hypothetical AlB 2-like MB2 phases (M = Li, Na, Be, Ca,
Sr, Ag and Au) was discussed recently, and their properties were analyzed by
means of theoretical approaches. 26-28,49-56 So, some theoretical efforts were made to
predict the superconducting transition temperature TC and the stability of these
materials. 26-28,49-56 Let’s note that despite some experimental efforts,
57,58
undoped
samples of these species have not been prepared up to now .
Despite a great number of interesting results on physical properties for AlB2
-like diborides, which were reported in past years, their mechanical behavior is
sufficiently investigated only for several phases: MgB2, TiB2 and ZrB2, whereas
for the majority of other diborides these data, to our knowledge, are rather limited.
Besides, most of the theoretical works devoted to the elastic behavior of MB2 are
performed by means of different approximations and, moreover, deal exclusively
with monocrystals. Meanwhile, the most part of synthesized and experimentally
examined MB2 materials are polycrystalline.
In this paper, in order to get systematic insight into the elastic properties of
these materials, a comparative first-principle study for 18 AlB2-like diborides of IVI groups s, p and d metals (among them there are stable: MgB2, AlB2, ScB2, YB2,
TiB2, ZrB2, HfB2, VB2, NbB2, TaB2 and CrB2, as well as meta-stable: MoB2 and
WB2, and hypothetical phases: NaB2, BeB2, CaB2, AgB2 and AuB2) has been
performed using the FLAPW method within the generalized gradient
approximation (GGA) for the exchange-correlation potential.
As a result, we have evaluated and analyzed a set of physical parameters of
the above-mentioned MB2 monocrystals such as optimized lattice parameters,
density, elastic constants (Cij), bulk modules (B) and shear modules (G).
Additionally, the numerical estimates of mechanical parameters: elastic modules
(B, G), compressibility (β), Young’s modules (Y), Poisson’s ratio (ν), Lamé’s
3
coefficients (μ, λ) for the corresponding polycrystalline MB2 ceramics (in the
Voigt-Reuss-Hill approximation) were obtained and analyzed for the first time.
II. METHOD AND DETAILS OF CALCULATIONS
The AlB2-like metal diborides adopt the layered crystal structure (space group
P6/mmm). It is a simple hexagonal lattice of close-packed metal layers alternating
with graphite-like B layers in sequence ..AHAHAH.. perpendicularly to the c
direction.1-6 The boron atoms are arranged at the corners of a hexagon with three
nearest neighbor B atoms in each plane. The metal atoms are located at the center
of the B hexagons, midway between adjacent boron layers. Thus the M and B
atoms have the [MB12M8] and [BM6B3] coordination polyhedra, respectively. Each
metal center has D6h symmetry, i.e. has 12 borons at the vertices of a hexagonal
prism. In addition, the central metal atom is coordinated also by 8 metals through
the faces of the B12 prisms. The primitive cell contains 1 metal and 2 boron atoms
(Z = 1). There is one B and one metal in nonequivalent atomic positions of M (0,
0, 0) and B (1/3, 2/3, 1/2).
The calculations of the all mentioned diborides (Table I) were carried out by
means of the full-potential method with mixed basis APW+lo (LAPW)
implemented in the WIEN2k suite of programs.59 The generalized gradient
correction (GGA) to exchange-correlation potential of Perdew, Burke and
Ernzerhof
60
was used. The maximum value for partial waves used inside atomic
spheres was l = 12 and the maximum value for partial waves used in the
computation of muffin-tin matrix elements was l = 4. The plane-wave expansion
with RMT × KMAX equal to 7, and k sampling with 10×10×10 k-points mesh in the
Brillouin zone was used. Relativistic effects were taken into account within the
scalar-relativistic approximation. The self-consistent calculations were considered
to converge when the difference in the total energy of the crystal did not exceed
0.001 mRy as calculated at consecutive steps. Other details of the calculations are
described in Ref. 46.
4
III. RESULTS AND DISCUSSION
3.1. Monocrystalline MB2 phases
Firstly, the equilibrium lattice constants (a and c) for all MB2 phases are obtained
from the total energy calculations, followed by the fitting of these results to the
Birch-Murnaghan equation.
61
The results obtained are in reasonable agreement
with the available theoretical and experimental data, Table II. Let’s note that in
Table II the calculated ratio c/a for each MB2 phase is also presented. According
to Pearson’s criterion,
79
for stable AlB2-like phases (see also below) the ratio c/a
should lie in the interval from 0.59 to 1.2. Thus, from the data obtained it may be
concluded that the diborides NaB2, CaB2, AgB2 and AuB2, for which c/a > 1.2,
should be unstable, whereas MgB2 (c/a =1.15) and YB2 (c/a = 1.18) are near to the
upper border of stability.
Let us discuss the mechanical parameters as obtained within the framework
of the FLAPW–GGA calculations for MB2 monocrystals. The values of five
independent elastic constants (Cij, namely C11, C12, C13, C33 and C44; while C66 = ½
(C11 - C12)) for these phases summarized in Table III were found by imposing five
different deformations (monoclinic, triclinic and three - of hexagonal types, see
Ref. 80) to the equilibrium lattice of the hexagonal unit cell and by determining
the dependence of the resulting energy changes on the deformation. There are
some general conclusions we would like to point out from the obtained results.
Firstly, mechanically stable phases should satisfy the well-known Born
criteria: C11 >0, (C11 - C12) >0, C44 >0, and (C11 + C12)C33 - 2C122 > 0. In our case,
these conditions are not met for four phases: BeB2, AgB2 and AuB2, for which C44
< 0, and for NaB2: (C11 - C12) < 0; additionally, for all the four diborides (C11 +
C12)C33 - 2C122 < 0. This means that these hypothetical phases are mechanically
unstable. This result agrees with theoretical estimations of heat of formations for
NaB2, BeB2, AgB2 and AuB2,53 according to which these phases should be
energetically unstable - in comparison with the mechanical mixture of the
5
constituent reagents. Other arguments about instability of these phases, in terms of
electronic density of states and chemical bonding picture, are discussed in Refs.
26,28,51-55. Thus, these phases will not be analyzed below.
Secondly, the estimations of the anisotropy factor A = C33/C11 (Table III)
demonstrate larger elastic anisotropy in the mechanical properties for s and p
metal diborides, as well as for meta-stable MoB2 and WB2 - in comparison with
moderate anisotropy for other d metal diborides. Probably, this tendency may
originate from strong M d – B p bonding between metal and boron sheets in these
compounds - in comparison with weaker p-p and s-p bonding for s and p metal
diborides, whereas for MoB2 and WB2 the reduction of intra-layer interactions (i.e.
the growth of anisotropy) will be the result of filling of anti-bonding d bands.
26,28,51-55
Besides, for anisotropic systems the large C13/C12 and small C33/C11 values
indicate that atomic bonding along the x axis is stronger than that along the z axis.
In our case in the sequence MB2 (where M are s and p metals) → MB2 (where M
are d metals) the values of C13/C12 drastically increase (for example from 0.52
(MgB2) and 0.82 (AlB2) to 1.54 (TiB2) and 1.85 (NbB2)), whereas C33/C11 values
remain comparable enough (for example: 0.65 (MgB2), 0.51 (AlB2), 0.71 (TiB2)
and 0.81 (NbB2)). This fact may be explained as strengthening of metal-metal
bonds for d metal diborides.
The calculated elastic constants allow us to obtain the macroscopic mechanical
parameters of MB2 phases, namely their bulk (B) and shear (G)modules. The
isotropic bulk modules Biso are obtained under the assumption that the c/a ratio
does not change when the lattice is subjected to an isotropic stress. The relevant
expression for Biso 91 coincides with the definition of the bulk modules in Voigt (V)
92
approximation:
BV =
2
1

 C11 + C12 + 2C13 + C33 
9
2

In the same approximation shear modules (GV) are estimated as:
6
GV =
1
(C11 + C12 + 2C33 − 4C13 +12C55 +12C66 )
30
The results obtained are summarized in Table IV in comparison with other
theoretical data. From our results we see that both bulk and shear modules for s
and p metal diborides are as a whole lower than for d metal diborides. It is usually
assumed that hardness of the materials is defined by these elastic modules: the
bulk modulus is a measure of resistance to volume change by applied pressure,
whereas the shear modulus is a measure of resistance to reversible deformations
upon shear stress.
93
Thus, the d metal diborides are the hardest materials among
all MB2 phases.
On the other hand, according to criterion,94 a material is brittle if the B/G ratio is
less than 1.75. In our case, for all the diborides, except WB2 (for which B/G ~ 1.9,
Table IV), these parameters are much less than 1.75, i.e. these materials will behave
in a brittle manner.
Finally, quite interesting conclusions follow from comparison of our data for
MB2 phases and elastic properties of other binary phases, in particular, d metal
monocarbides (MC) and mononitrides (MN), see Fig. 1. Comparison of our bulk
modules for MB2 with the results 95 for MC and MN shows that the general trends in
the dependence of B on the d metal type are very similar to each other. Namely: (i)
the bulk modules of all binary phases for III group metals are minimal; (ii) the bulk
modules of binary phases for III and IV group metals are considerably different,
whereas the bulk modules of phases for V and VI group metals are more close to
each other; (iii) the bulk modules increase for all binary phases of 3d - 5d metals as
going from III to VI group; and (iv) in the same groups the bulk modules decrease
as going from 3d to 4d metals and increase as going from 4d to 5d metals. One
noticeable difference is seen for Ta and W – based phases, where B(TaB2) >
B(WB2), whereas for the corresponding carbides and nitrides B(Ta(C,N)) <
B(W(C,N)). Obviously, this disagreement should be ascribed to the above metastability of the AlB2-like tungsten diboride in the AlB2-like structure because extra
electrons fill the antibonding states.26,53 The mentioned similarities in the change of
the bulk modules of MB2, MC and MN phases in the dependence from the d metal
7
type may seem quite surprising taking into account the fact that MB2 and MC, MN
phases adopt (i) very different structures (layered AlB2 type versus isotropic B1
type) and (ii) different nature of inter-atomic bonding (strong covalent B-B intralayer interactions plus covalent B-M intra-layer interactions for MB2
dominant covalent M-(C,N) bonding for MC and MN
26,28,51-55
versus
95-99
). Therefore we may
suppose that the trends in elastic moduli changes for all these phases are controlled
mainly by their cell volumes (i.e. atomic radii of d metals, and follow the known
semiempirical expression 100-102 B ~ 1/ln, where l is the bond length) and depend to a
much lesser degree on the differences in structure, cohesion energies and the types
of bonds in these phases, see also below.
3.2. Polycrystalline MB2 ceramics
The above elastic parameters are obtained from first-principle calculations of MB2
monocrystals. Meanwhile, the majority of synthesized and experimentally
examined MB2 materials are prepared and investigated as polycrystalline ceramics,
1-7
i.e. in the form of aggregated mixtures of microcrystallites with a random
orientation. Thus, it is useful to estimate the elastic parameters for the
polycrystalline MB2 materials.
For this purpose we utilize the Voigt-Reuss-Hill (VRH) approximation. In
this approach, according to Hill,
103
two main approximations: Voigt
92
and Reuss
(R) 104 are used. In turn, in Reuss’s model the bulk and shear modules (BR and GR)
are estimated as:
(C11 + C12 )C33 − 2C122
BR =
C11 + C12 + 2C33 − 4C13
GR =
[
]
(C11 + C12 )C 33 − 2C122 C 55 C 66
5
2 3BV C55 C 66 + (C11 + C12 )C 33 − 2C122 ( C 55 + C 66 )
[
]
Then, the Voigt and Reuss averages are limits and the actual effective modules for
polycrystals could be estimated in VRH approach by the arithmetic mean of these
8
two bounds. In this way, when the bulk modulus (BVRH) and the shear modulus
(GVRH) are obtained from BV,R and GV,R by the VRH approach in simple forms as:
BVRH = ½ (BV + BR) and GVRH = ½ (GV + GR), averaged compressibility (βVRH =
1/BVRH) and Young’s modules (YVRH) may be calculated by the expression:
YVRH =
9 BVRH GVRH
3BVRH + GVRH
Then, the Poisson’s ratio (ν) and Lame’s constants (μ, λ) were obtained for
polycrystalline MB2 species from BVRH, GVRH and YVRH as:
v=
µ=
3BVRH − 2GVRH
2(3BVRH + GVRH )
YVRH
;
2(1 +ν )
λ=
νYVRH
(1 +ν )(1 − 2ν )
The above-mentioned parameters are summarized in Table V in comparison with
available experimental data for MB2 ceramics. Here we should point out that our
estimations are made in the limit of zero porosity of MB2 ceramics.
It was found for the majority of MB2 ceramics that BVRH > GVRH; this implies
that the parameter limiting the stability of MB2 materials is the shear modulus
GVRH. We have obtained only two deviations from this tendency: ScB2 and TiB2,
for which BVRH ~ GVRH.
The trends of bulk modules for MB 2 ceramics may be discussed as displayed
in Fig. 2. As can be seen, the bulk modules increase as the d metal goes from the
III group to the VI group, or downward on the Periodic Table; the borides of d
metals have larger BVRH than their s, p metal counterparts. As is seen, there is the
above-mentioned distinct correlation between elastic modules and unit cell
volumes of MB2 phases. Generally, these trends are supported by experiment,
Table V.
Thus, diborides of s, p metals, ScB2 and YB2 adopt the minimal BVRH, whereas
the bulk modules for diborides of d metals of IV- VI groups can be varied from
238 to 437 GPa. The maximal bulk modulus (437 GPa) and the minimal
9
compressibility (0.00229 1/GPa) have been obtained for WB2 ceramic. On the
other hand, among IV- VI group d-metal diborides, TiB2, HfB2, VB2 and TaB2
will have the maximal shear modules (GVRH ~ 240 - 270 GPa), i.e. the maximal
bond-restoring energy under elastic shear strain , whereas, according to the GVRH
estimations, CrB2 (GVRH = 174 GPa) will remain a material with the minimal
hardness. The Young’s modulus YVRH has also the maximal and minimal values for
TiB2, HfB2, VB2 and TaB2 (YVRH ~ 560-600 GPa) and CrB2 ceramics (YVRH = 442
GPa), respectively.
The values of the Poisson ratio (ν) for covalent materials are small (ν ~ 0.1),
whereas for metallic materials ν is typically 0.33.105 In our case the values of ν for
MB2 vary from about 0.120 - 0.166 for (Ti,Zr,Hf)B2 to 0.267-0.310 for
(Cr,Mo,W)B2, indicating an increase of metal-metal bonding for the diborides
when d metal goes from the IV group to the VI group. This trend reflects the wellknown situation
26,28,53-55
when the metallicity of MB2 phases increases with the
filling of d orbitals.
IV. CONCLUSIONS
In summary, we have performed FLAPW-GGA calculations to obtain the
systematic trends for elastic properties of 18 hexagonal AlB2-like diborides of s, p
and d metals, among them there are stable (MgB2, AlB2, ScB2, YB2, TiB2, ZrB2,
HfB2, VB2, NbB2, TaB2 and CrB2), meta-stable (MoB2 and WB2) and hypothetical
phases (NaB2, BeB2, CaB2, AgB2 and AuB2). For the first time a set of elastic
parameters of the polycrystalline MB2 ceramics has been estimated.
Our analysis of elastic constants (Cij) shows that NaB2, BeB2, AgB2 and AuB2
belong to mechanically unstable systems. The present study shows also that both
bulk and shear modules for s and p metal diborides are as a whole lower than for d
metal diborides, i.e. d metal diborides should be the hardest materials among all the
examined MB2 phases. On the other hand, all diborides, except WB2, will behave in
a brittle manner.
10
From our studies of MB2 ceramics we may draw the following main
conclusions:
(i)
For the majority of diborides BVRH > GVRH; this implies that a parameter
limiting the stability of MB2 materials is the shear modulus GVRH;
(ii)
The values of the Poisson ratio vary from about 0.120 - 0.166 for
(Ti,Zr,Hf)B2 to 0.267-0.310 for (Cr,Mo,W)B2, indicating an increase of
metal-metal bonding for the diborides when d metal goes from the IV group
to the VI group. This trend can be understood in terms of filling of the
relevant d-like electronic bands.
(iii)
The bulk moduli increase as the d metal goes from the III group to the VI
group, or downward on the Periodic Table; the borides of d metals have
larger BVRH than their s, p metal counterparts. These trends correlate with the
cell volumes of MB2 and generally are supported by experiment.
ACKNOWLEDGEMENTS
The work was supported by RFBR, grant 08-08-00034-а.
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TABLE I. (Color online). Considered hexagonal (AlB2-like) diborides of s, p
and d metals. *
* 1-stable, 2- meta-stable and 3 – hypothetical MB2 phases.
16
TABLE II. Calculated lattice constants (a, c, in Å) for hexagonal (AlB2-like)
diborides of s, p and d metals in comparison with available experimental and
theoretical data.
diboride
NaB2
a
3.002
2.995 a
BeB2
2.898
2.896, a 2.886 b
3.050
MgB2
3.040,a 3.080, b 3.085, c
3.098, d 3.073 e 3.071 f
3.064, g 3.049, q
CaB2
3.191
3.183, a 3.397, b 3.205 h
2.962
AlB2
2.978, a 3.005, c 3.008, l
3.009, m 2.998, n
ScB2
3.114
3.114, a 3.148, c
YB2
3.268
3.253, a 3.290, c
3.006
a
TiB2
3.005, 3.038, c 3.070, b
3.029, e 3.027, i 2.993 j
3.015, k
3.155
a
ZrB2
3.139, 3.130, c 3.127, o
3.183, p 3.169, q
HfB2
3.144
3.111, a 3.141, c
2.970
VB2
2.969, a 2.998, c 3.008, p
3.007, q
NbB2
3.086
a
3.086, 3.086, c 3.111, r
3.181, q
TaB2
3.074
3.072, a 3.088, c 3.164, q
CrB2
2.945
2.970, a 2.973, c
MoB2
3.009
2.990, a 3.039, c 3.012, r
WB2
2.998
2.972, a 3.020, c 3.053, s
AgB2
2.986
2.994, a 3.000, t 3.034 v
AuB2
2.972
2.982, a 2.957, t
a
Reference 53, FP-LAPW + LDA.
b
Reference 54, TB-LMTO.
c
4.297
4.229 a
2.884
2.845, a 3.088 b
3.511
3.448, a 3.532, b 3.523 c
3.520, d 3.527, e 3.528 f
3.493, g 3.446, q
4.051
4.001, a 4.019, b 3.974 h
3.206
3.248, a 3.257, c 3.261, l
3.262, m 3.286, n
3.512
3.465, a 3.517, c
3.849
3.812, a 3.835, c
3.212
a
3.186, 3.239, c 3.262, b
3.220, e 3.240, i 3.147, j
3.222, k
3.542
a
3.499, 3.533, c 3.523,o
3.546, p 3.531, q
3.502
3.409, a 3.470, c
3.029
2.989, a 3.005, c 3.068, p
3.048, q
3.318
a
3.304, 3.306, c 3.309, r
3.357, q
3.209
3.275, a 3.241, c 3.323, q
2.941
2.880, a 3.072, c
3.305
3.342, a 3.055, c 3.330, r
3.334
3.369, a 3.050, c 3.306, s
4.043
4.051, a 3.493, t 4.085, v
4.284
4.197, a 3,785, t
17
c/a
1.43
1.41 a
0.99
0.98, a 1.06 b
1.15
1.13, a 1.15, b 1.14, c
1.14, d 1.15, e 1.15 f
1.14, g 1.13 q
1.27
1.26, a 1.18, b 1.24 h
1.08
1.09, a 1.08, c 1.08, i
1.08, m 1.10 n
1.13
1.11, a 1.12 c
1.18
1.17, a 1.16 c
1.07
a
1.06, 1.07, c 1.06, b
1.06, e 1.07, i 1.05, j
1.07, k
1.12
a
1.12, 1.13, c 1.13, o
1.11, p 1.11, q
1.11
1.10, a 1.11, c
1.02
1.01, a 1.00, c 1.02, p
1.01, q
1.08
a
1.07, 1.07, c 1.06, r
1.06, q
1.05
1.07, a 1.05, c 1.05, q
0.99
0.97, a 1.03, c
1.10
1.12, a 1.05, c 1.11, r
1.11
1.13, a 1.01, c 1.08, s
1.35
1.35, a 1.16, t 1.35, v
1.44
1.41, a 1.28 t
c
Reference 62, experiment.
Reference 63, FP-LAPW + GGA.
e
Reference 64, first-principles pseudopotential method.
f
Reference 66, FLAPW.
g
Reference 66, DFT (CRYSTAL98).
h
Reference 51, FP-LAPW + GGA.
i
Reference 67, Hartree-Fock.
j
Reference 68, NCP-GGA.
k
Reference 69, CASTEP-GGA.
l
Reference 70, plane waves – DFT.
m
Reference 71, FP-LAPW-GGA.
n
Reference 72, VASP-GGA.
o
Reference 73 DFT and GGA.
p
Reference 74, LCAO-DFT.
q
Reference 75, FP-LAPW-GGA.
r
Reference 46, FP-LMTO.
s
Reference 76, CASTEP-LDA.
t
Reference 77, LCAO-DFT.
v
Reference 78, VASP.
d
18
TABLE III. Calculated elastic constants (Cij, in GPa) and anisotropy factors (A
= C33/C11) for hexagonal (AlB2-like) diborides of s, p and d metals in comparison
with available data.
diboride
NaB2
BeB2
C11
C12
C13
C33
158.5
196.2
9.3
151.5
358.1
255.2
114.4
106.1
a
a
a
696
136
88
370a
434.8
62.1
32.8
285.4
b
c
b
c
b
c
MgB2
438, 462,
43, 67,
33, 41,
264, b 254,c
524, d 431, e
58, d 27, e
33, d 31, e
243, d 197, e
f
g
f
g
f
g
446, 365,
68, 98,
39, 65,
284, f 203, g
CaB2
237.5
97.8
34.9
289.8
AlB2
530.0
82.4
67.9
272.3
h
h
h
665
41
17
417 h
ScB2
508.5
41.9
73.5
373.3
YB2
353.9
56.6
85.9
326.7
670.9
64.0
100.9
472.9
656, g 786, i
66, g 127, i
98, g 87, i
461, g 583, i
TiB2
588-711, j
17-410, j
84-320, j
224-440, j
k
l
k
l
k
l
654, 671,
75, 62,
99, 103,
443, k 468, l
626 m
68 m
102 m
444 m
540.5
55.9
111.1
422.2
n
o
n
o
n
o
ZrB2
581, 596,
55, 48,
121, 169,
445, n 482, o
p
p
p
564-606
52-54
118-134
436-477 p
HfB2
592.7
99.6
141.3
481.3
VB2
682.9
110.5
126.5
462.4
699 o
146 o
109 o
552 o
NbB2
608.0
104.1
193.0
493.7
517, q 290, r
95, q 110, r 120, q 190, r
528, q 474, r
TaB2
708.6
129.2
218.6
517.0
CrB2
600.9
165.3
205.8
345.4
MoB2
621.4
116.9
228.4
404.3
WB2
636.1
130.0
263.4
380.9
590-586 s 187-184 s
236-235 s
443-419 s
AgB2
220.9
253.7
82.5
173.5
AuB2
258.8
257.8
88.9
154.3
* mechanically unstable phases, see the text.
a
Reference 81, DFT (CRYSTAL98).
b
Reference 54, TB-LMTO.
c
Reference 63, FP-LAPW + GGA.
d
Reference 82, DFT plane-wave method.
e
Reference 83, DFT.
f
Reference 84, LCAO-DFT.
g
Reference 64, first-principles pseudopotential method (LDA-GGA)
h
Reference 70, plane waves – DFT.
i
Reference 67, Hartree-Fock.
j
Experimenal data cited in Reference 67.
k
Reference 85, FP-LAPW.
l
Reference 86, DFT-CASTEP.
19
C44
41.1
-62.2
119 a
64.4
b
80, 80, c
85, d 85, e
77, f 58, g
96.9
32.7
58 h
189.5
166.3
266.6
259, g 271, i
232-250, j
344,k 269, l
240 m
252.7
m
240, 240, o
256-281 p
262.3
229.0
167 o
220.3
112, q 225, r
236.6
153.6
175.0
137.1
99-94 s
-42.4
-49.4
A
u*
u
0.65
1.22
0.51
0.73
0.92
0.71
0.78
0.81
0.68
0.81
0.73
0.58
0.65
0.60
u
u
m
Reference 69, CASTEP-GGA.
Reference 28, experiment.
o
Reference 16, LCAO-DFT.
p
Reference 15, DFT (LDA-GGA).
q
Reference 87, LCAO-DFT.
r
Reference 88, CASTEP-GGA.
s
Reference 76, CASTEP (LDA-GGA).
n
20
TABLE IV. Calculated bulk modules (B, in GPa) and shear modules (G, in GPa)
for monocrystalline AlB2-like diborides of s, p and d metals in comparison with
available theoretical data.
diboride
MgB2
B
156.7
150, a 163, b 122-161, c 139, d 140,e
CaB2
122.2
134, a
AlB2
196.5
195- 211, f
ScB2
196.5
YB2
156.7
TiB2
260.7
a
d
213, 251-277, 299-306, g 249-304, h
ZrB2
228.8
i
195, 272-275, j 237-260, k
HfB2
270.0
VB2
283.9
175, i 298, j
NbB2
298.9
248, l 298, m
TaB2
340.8
CrB2
300.1
MoB2
310.5
WB2
329.6
322-327, n
a
Reference 54, TB-LMTO.
b
Reference 84, LCAO-DFT.
c
Reference 81, DFT (CRYSTAL98).
d
Reference 64, first-principles pseudopotential method (LDA-GGA)
e
Reference 89, FLAPW-GGA
f
Reference 70, plane waves – DFT.
g
Reference 67, Hartree-Fock.
h
Reference 85, FP-LAPW.
i
Reference 90, TB-LMTO.
j
Reference 74, LCAO-DFT.
k
Reference 73, DFT (LDA-GGA).
l
Reference 87, LCAO-DFT.
m
Reference 88, CASTEP-GGA.
n
Reference 76, CASTEP (LDA-GGA).
21
G
131.5
115, a 90, d
111.5
B/G
1.19
132.1
1.49
202.6
150.0
270.6
261, d 255, h
231.2
0.97
1.05
0.96
239.9
246.5
1.17
1.18
219.8
212, m
243.8
170.7
192.0
171.9
141-144, n
1.22
1.10
0.99
1.36
1.55
1.62
1.92
TABLE V. Calculated values of some elastic parameters for MB2 polycrystalline
ceramics as obtained in the Voigt-Reuss-Hill approximation: bulk modules (BVRH, in
GPa), compressibility (β, in GPa-1), shear modules (GVRH, in GPa), Young’s modules
(YVRH, in GPa), Poisson’s ratio (ν), Lamé’s coefficients (λ, μ, in GPa) and density (ρ,
in kg/m3) in comparison with available experiments.
diboride
MgB2
BVRH
149.9
145, a 120, b
151 c
111.5
184.6
170 d
198.9
β
0.00667
0.00690 a
GVRH
117.0
117.5 a
YVRH
278.6
ν
0.190
λ
71.9
ρ
2695
2650 a
0.00897
0.00541
88.9
98.2
210.7
250.2
0.185
0.274
52.2
119.3
0.00503
201.4
0.121
64.6
YB2
TiB2
171.5
263.2
255, h 276, i
251, j
0.00583
0.00380
0.00392 h
0.00362, i
0.00398 j
150.1
268.9
252, i
237 j
451.8
480 f
348.7
601.9
372-551,f
579 j
71.4
83.9
ZrB2
237.6
218 k
0.00421
230.5
522.6
343-506 e
0.161
0.119
0.110, g
0.108, h
0.151, i
0.141 j
0.134
0.110 e
2869
3314
3190 e
3749
3670 f
5155
4591
4520, e
4510, i
451010
HfB2
278.8
222 k
0.00359
239.5
0.166
0.120 g
119.2
VB2
283.2
0.00353
244.1
327.3
0.00306
221.2
0.165
0.100 e
0.224
120.5
NbB2
TaB2
370.8
0.00270
244.0
0.230
208.2
CrB2
315.7
0.00317
174.2
558.6
500 e
497 k
568.8
268 e
541.7
637 e
600.3
257 e
441.5
221 e
0.267
199.6
MoB2
369.4
0.00271
191.3
498.5
0.279
241.8
WB2
436.8
a
Reference 106.
b
Reference 107.
c
Reference 108.
d
Reference 71.
e
Reference 109.
f
Reference 110.
g
Reference 111.
h
Reference 112.
i
Reference 113.
j
Reference 114..
k
Reference 7.
0.00229
168.8
448.7
0.329
324.3
CaB2
AlB2
ScB2
22
84.0
179.8
6135
6085, e
6100 g
11078
11190 e
5208
5070 e
6952
6970 e
12814
12540 e
5533
5160-520
0e
7533
7780 e
13150
Figures
FIG. 1 (Color online) The calculated bulk modules of monocrystalline AlB2 –like
diborides of 3d - 5d metals in comparison with bulk modules 95 of cubic
nomocarbides and mononitrides of the same metals.
23
FIG. 2. The calculated bulk modules (BVRH, in GPa) of polycrystalline MB2 ceramics
with respect to cell volumes of diborides.
24