Hardness and elastic properties of covalent/ionic solid solutions from firstprinciples theory
Qing-Miao Hu, Krisztina Kádas, Sture Hogmark, Rui Yang, Börje Johansson et al.
Citation: J. Appl. Phys. 103, 083505 (2008); doi: 10.1063/1.2904857
View online: http://dx.doi.org/10.1063/1.2904857
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v103/i8
Published by the American Institute of Physics.
Related Articles
A thermodynamic model of sliding friction
AIP Advances 2, 012179 (2012)
Mechanism of adaptability for the nano-structured TiAlCrSiYN-based hard physical vapor deposition coatings
under extreme frictional conditions
J. Appl. Phys. 111, 064306 (2012)
Structure related hardness and elastic modulus of bulk metallic glass
J. Appl. Phys. 111, 053518 (2012)
Performance of diamond-like carbon-protected rubber under cyclic friction. I. Influence of substrate viscoelasticity
on the depth evolution
J. Appl. Phys. 110, 124906 (2011)
Interchain coupled chain dynamics of poly(ethylene oxide) in blends with poly(methyl methacrylate): Coupling
model analysis
J. Chem. Phys. 135, 194902 (2011)
Additional information on J. Appl. Phys.
Journal Homepage: http://jap.aip.org/
Journal Information: http://jap.aip.org/about/about_the_journal
Top downloads: http://jap.aip.org/features/most_downloaded
Information for Authors: http://jap.aip.org/authors
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
JOURNAL OF APPLIED PHYSICS 103, 083505 共2008兲
Hardness and elastic properties of covalent/ionic solid solutions
from first-principles theory
Qing-Miao Hu,1,2,a兲 Krisztina Kádas,3,4 Sture Hogmark,5 Rui Yang,1 Börje Johansson,2
and Levente Vitos2,6,4
1
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy
of Sciences, 72 Wenhua Road, Shenyang 110016, China
2
Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute
of Technology, Stockholm SE-100 44, Sweden
3
Condensed Matter Theory Group, Physics Department, Uppsala University, Uppsala SE-75121, Sweden
4
Research Institute for Solid State Physics and Optics, Budapest H-1525, P.O. Box 49, Hungary
5
Department of Materials Science, Uppsala University, SE-751 21 Uppsala, Sweden
6
Condensed Matter Theory Group, Physics Department, Uppsala University, Uppsala SE-75121, Sweden;
School of Physics and Optoelectronic Technology and College of Advanced Science and Technology
Dalian University of Technology, Dalian 116024, China
共Received 20 December 2007; accepted 6 February 2008; published online 17 April 2008兲
Most of the engineering materials are alloys 共solid solutions兲 and inevitably contain some impurities
or defects such as vacancies. However, theoretical predictions of the hardness of this kind of
materials have rarely been addressed in literature. In this paper, a hardness formula for
multicomponent covalent solid solution is proposed based on the work of Šimůnek and Vackář
关Phys. Rev. Lett. 96, 085501 共2006兲兴. With this formula, the composition dependence of the
hardness is investigated for titanium nitrogencarbide 共TiN1−xCx兲, off-stoichiometric transition-metal
nitrides 共TiN1−x and VN1−x兲, and B-doped semiconductors. The predicted hardness is in good
agreement with experiments. To investigate the most frequently quoted correlation between
hardness and elastic modulus, the elastic moduli of the systems involved in this paper have also been
calculated. The results show that the elastic moduli cannot be used for rigorous predictions of the
hardness of the solid solutions. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2904857兴
I. INTRODUCTION
Hardness is a measure of the resistance of materials
against permanent deformations. Materials with high hardness are technologically important for cutting and forming
tools, engine components, valves, seals, gears, many types of
wear resistant coatings, etc. The search for harder materials
has a long history and remains one of the most active areas
in materials science. In order to synthesize new hard materials more efficiently, materials scientists have devoted themselves to developing predictive tools for the hardness.
Among these attempts, the empirical correlation between the
hardness and the elastic properties such as bulk modulus and
shear modulus has received considerable attention.1 In metallic materials, the permanent deformation usually occurs by
dislocation glide, and thus their hardness is a measure of the
resistance to plastic deformation. On the other hand, strongly
bonded materials of ionic or covalent type may experience
permanent deformation by microscopic fracture. This fracture occurs by crack initiation and crack propagation, both of
which are associated with the breaking of atomic bonds. In
this way, hardness of brittle materials is directly related to
the bond strength.2–6 Since the bond strength is connected to
the elastic modulus, hardness of brittle materials is usually
associated with their elastic modulus. It was generally considered that higher bulk or shear modulus implies higher
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-8979/2008/103共8兲/083505/9/$23.00
hardness. However, strictly using these moduli for predicting
hardness is greatly limited since many exceptions to this empirical rule exist.1
Based on the connection between the atomic bond and
hardness of covalent and ionic crystals, two different hardness models have been put forward.7,8 The proposed analytic
expressions can be used to determine the hardness of covalent and ionic crystals from first-principles theory. However,
most of such investigations are performed on elemental or
ordered systems. Despite the fact that most of the technically
important materials are actually alloys 共solid solutions兲 and
inevitably contain some defects such as impurities and vacancies, theoretical predictions for the hardness of these materials have rarely been touched upon in literature.
The purpose of the present work is to investigate the
hardness of covalent/inonic solid solutions from firstprinciples theory. The connection between the hardness and
elastic modulus is also discussed. The rest of this paper is
arranged as following: In Sec. II, the electronic structure
based hardness theories proposed in the literature are described, and the hardness theory applied in our work is introduced. In Sec. III, we described the details of our calculations. The hardness and elastic moduli are reported and
discussed in Sec. IV. Finally, we summarize our work in Sec.
V.
II. THEORY
For the completeness of the hardness theory, we first
introduce the hardness formula of Gao et al.7 and Šimůnek
and Vackář.8
103, 083505-1
© 2008 American Institute of Physics
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-2
J. Appl. Phys. 103, 083505 共2008兲
Hu et al.
Gao et al.7 expressed the hardness as
Hij = 350
−1.91f i
N2/3
e e
,
d2.5
共1兲
with Ne being the electron density, d the bond length, and
f i = 1 − E2h / E2g the ionicity of the chemical bond,9 where Eh is
the pure covalent contribution to the average energy gap Eg.
This expression yields hardnesses in excellent agreement
with experiments for typical covalent and polar covalent
crystals. Unfortunately, Eq. 共1兲 is only applicable to the covalent and ionic crystals without any metallic features since
the energy gap explicitly enters into the definition of hardness. He et al.4 have recently defined a new quantity to scale
the ionicity of the atomic bond based on the bond overlap
population. Together with the formula of Gao et al., this
ionicity scale predicts reasonably good hardness for B12
icosahedra and B13C2. However, its validity for the hardness
of other covalent/ionic crystals need to be verified.
Šimůnek and Vackář8 expressed the hardness as
Hij = 共C/⍀兲Sije−␴ f ij ,
with
f ij =
冋 册
ei − e j
ei + e j
Sij = 冑共ei/ni兲共e j/n j兲/dij ,
共2兲
2
.
Rather than a direct geometric average, Šimůnek and
Vackář presented a formula for the hardness of multicomponent compound 关Eq. 共6兲 in Ref. 8兴. Unfortunately, their expression is inconsistent with that for elemental/binary system
共as also noted by Liu et al.10兲 Namely, if one treats the
elemental/binary system as a quasi-multi-component compound, the hardness from their multicomponent formula is
not equivalent to that obtained from the one written for
elementary/binary systems. Furthermore, if the multiplicity
of some of the binary systems involved in the multicomponent crystal are extremely small, Šimůnek and Vackář’s multicomponent hardness expression will generate artificially
low hardness since the overall bond strength is a multiplicity
weighted product of those of the subsystems.
Recently, starting from the theory by Šimůnek and
Vackář,8 we have introduced an alternative model for the
hardness of covalent/ionic solid solutions for the elemental/
binary systems.13 In our approach, we redefine the bond
strength Sij as
共3兲
Here, C and ␴ are constants and ⍀ is the volume of the
system considered. Sij is the so called bond strength defined
as Sij = 冑e je j / 共nijdij兲 where ei and e j are the reference energies of the atoms i and j, respectively, and nij the number of
bonds between atom i and j with bond length of dij. The
reference energy is defined as ei = Zi / Ri with Zi being the
valence-electron number of the atom i and Ri the radius of
atom i within which the atom is electrically neutral. It is
important to point out that besides the purely covalent/ionic
crystals, Eq. 共2兲 works also for the covalent crystals with
some metallic features 共e.g., transition-metal nitride and carbide兲. At the same time, due to the particular definition of Sij,
the above approach becomes somewhat inconvenient if the
coordination numbers for the two atoms forming the bond
are different.10
The hardness formulas by Gao et al.7 and Šimůnek and
Vackář8 introduced above are for perfect elemental or binary
crystals without any defects and solid solute. However, most
of the technologically important materials are actually alloys
共solid solutions兲 and inevitably contain some impurities or
defects. Therefore, we are in need of a hardness formula to
deal with multicomponent systems such as solid solutions.
As proposed in Ref. 7, the hardness of a multicomponent
compound can be calculated as a geometric average of the
hardness of the binary systems involved in the compound.
Due to the geometrical averaging, it is expected that the
hardness of a multicomponent compound should be somewhere in between the lowest and highest hardness of the
binary systems involved. However, this is not the case for
some of the multicomponent compounds. For example, the
hardness of TiN1−xCx has a maximum at x in between 0 and
1 from experiments.2,11,12 We will show this later in this paper.
共4兲
where ni and n j are the coordination numbers for atoms i and
j. This expression captures more accurately the actual
strength of the bond between atoms with different coordination numbers as compared to the original formula by Šimůnek and Vackář.8 In Eq. 共4兲, ei / ni indicates that the reference
energy of atom i is shared by its ni nearest neighbors.
For the multicomponent solid solutions, to avoid the inconsistency between the hardness formulas for multicomponent compound and elemental/binary crystals in Šimůnek
and Vackář’s theory,8 we proposed to express the hardness
for multicomponent covalent/ionic solid solutions as
HM =
where
C
S M e −␴ f M ,
⍀/N
冉兿 冊
m
SM =
共5兲
1/N
SNij ij
.
共6兲
i,j=0
The latter expression indicates that the overall “bond
strength” of the multicomponent compound is a geometric
average of those calculated for the binary subsystems forming the compound. The exponential parameter f M may be
expressed as either the geometric or arithmetic average of
those of the elemental/binary systems, namely,
冉兿 冊
m
f IM
=
f Nij ij
1/N
,
共7兲
i,j=0
or
f IIM
1
=
N
冉兺 冊
m
Nij f ij .
共8兲
i,j=0
Here, N = 兺m
ij=0 Nij represent the number of binary subsystems, with m and Nij being the total number of different
binary subsystems and the multiplicity of each of them, respectively. It should be noted that the hardness of the multicomponent compound with f M of Eq. 共8兲 is actually a geo-
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-3
J. Appl. Phys. 103, 083505 共2008兲
Hu et al.
metric average of the hardness of the binary systems whereas
the hardness with Eq. 共7兲 is not. We will show later that for
some of the system different f M expressions might yield very
different hardness values. In the following, we refer to the
hardness generated using Eqs. 共4兲–共7兲 as HIM , and HIIM to the
hardness generated using Eqs. 共4兲–共6兲 and 共8兲. We note that
our expression for hardness exactly generates the same values as those obtained from Šimůnek and Vackář’s theory8 for
elemental or binary system with equal number of coordinations for the two constituent atoms.
III. CALCULATIONS DETAILS
As is seen from Eqs. 共2兲–共6兲, once the constants C and ␴
are determined, the parameters needed to evaluate the hardness are the atomic radii Ri, the equilibrium bond length and
the volume of the solid solution. These parameters have been
determined from density functional calculations. Here, we
adopted the exact muffin-tin orbitals 共EMTO兲 method in
combination with the generalized-gradient approximation.14
Details about the EMTO method can be found in Refs. 15–19.
In the self-consistent calculations, the one-electron equations
were treated within the scalar relativistic and soft core approximations. The EMTO Green’s function was calculated
for 32 energy points. In the EMTO basis set, s, p, and d
orbitals were included. The irreducible parts of the Brillouin
zones for the ideal and distorted NaCl and diamond structures were sampled using ⬃5000 uniformly distributed k
points. The random distribution of the solute atoms and defects was taken into account using the coherent potential approximation 共CPA兲.20–22 The EMTO-CPA approach has been
successfully applied in the theoretical study of the elastic
constants and phase stability of random Fe-based alloys,23–27
simple and transition-metal alloys,28–31 TiNi–Zr intermetallic
compounds,32 and Hume–Rothery systems,22,33,34 as well as
the crystal structure of complex oxides.35–38 The solid solutions considered in this paper are titanium-nitrocarbide
共TiN1−xCx兲, off-stoichiometric transition-metal 共TM兲 nitride
共TiN1−x and VN1−x兲, and B-doped semiconductors.
Throughout this work, we use the constants C = 1550 and
␴ = 4 parametrized by Šimůnek and Vackář.8 Since C = 1550
and ␴ = 4 work well for a large set of materials,8 it is plausible to assume that these parameters are more or less universal for most covalent/ionic crystals. The basis of the definition of the bond strength in this paper is similar to that of
Šimůnek and Vackář, therefore we expect that the previously
proposed values are naturally applicable in our case as well.
To investigate the often-quoted connection between
hardness and elastic modulus, we have also calculated the
elastic properties of the solid solutions involved in this paper.
For the elastic constant calculations, we first determine the
theoretical equilibrium volume and bulk modulus by Morse
fitting of the total energies versus the volume. To get the
elastic constants C⬘ and C44, we used volume conserving
orthorhombic and monoclinic deformations, i.e.,
TABLE I. Hardness of A3N4 共A = C , Si, Ge兲 共in GPa兲, generated with bond
lengths, reference energies, and volumes listed in Ref. 8.
Compound
HIM = HIIM
Ref. 8
Ref. 40
Expt./ Refs. 41–43
C 3N 4
Si3N4
Ge3N4
66.1
25.5
17.7
70.1
27.4
19.1
56.7
30.9
24.3
21.0⬃ 35.3
冢
1 + ⑀o
0
0
0
1 − ⑀o
0
0
1
0
1 − ⑀2o
冣
,
共9兲
and
冢
⑀m
⑀m
1
0
0
1
0
0
1
2
1 − ⑀m
冣
,
共10兲
respectively. Six strains ⑀ = 0 , 0.01, 0.02, . . . , 0.05 were used
to calculate the total energies E共⑀o兲 and E共⑀m兲. The elastic
constants C⬘ and C44 were obtained by fitting the total energies with respect to ⑀o and ⑀m as E共⑀o兲 = E共0兲 + 2VC⬘⑀2o and
2
, respectively. C11 and C12 are then
E共⑀m兲 = E共0兲 + 2VC44⑀m
evaluated from the bulk modulus B = 31 共C11 + 2C12兲 and the
tetragonal shear constant C⬘ = 21 共C11 − C12兲. The polycrystalline shear modulus G, Young’s modulus E, and Poisson ratio
␯ were calculated using the Hill average of the Voigt and
Reuss bounds.39
IV. RESULTS AND DISCUSSION
A. Assessing the accuracy
Before presenting our results for hardness and elastic
moduli, we address the accuracy of the present approach.
First, we discuss the performance of the hardness formula in
the case of nitride spinel materials, and then we compare the
present calculated elastic properties for TiN, TiC, VN, C共diamond兲, Si, and Ge with former theoretical results and experimental data.
1. Hardness
The present hardness formulas are first applied to the
nitride spinel materials A3N4 共A = C, Si, Ge兲 with two binary
systems as detailed in Ref. 8. To make our results comparable to those in Ref. 8, we use the same reference energies,
bond lengths, and volume of the system as those listed in
Table II of Ref. 8. Table I demonstrates that our hardness
values agree reasonably well with those from Gao et al.,7
Šimůnek and Vackář,8 as well as with the available experimental data.41–43 Note that HIM and HIIM are exactly the same
since the exponential parameters f ij for the two binary systems adopted here as well as in the two references are exactly the same therefore the geometric and arithmetic average yield the same multicomponent f M .
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-4
J. Appl. Phys. 103, 083505 共2008兲
Hu et al.
TABLE II. Elastic constants of TiN, TiC, and VN 共in GPa兲. The present results 共this work兲 are compared to
experimental data 共expt.兲 and former theoretical results obtained within local density approximation 共LDA兲 or
generalized-gradient approximation 共GGA兲.
C⬘
C44
B
G
E
TiN
This work
LDAa
LDAb
GGAb
Expt.c
306
290
321
255
230
239
166
250
168
163
265
326
307
270
318
264
208
276
199
187
595
514
638
479
469
TiC
This work
LDAa
LDAb
GGAb
Expt.d
276
243
250
187
165
212
173
230
167
217
241
286
273
221
199
236
198
238
175
194
534
483
553
415
439
VN
This work
LDAa
Expt.c
276
199
325196
119
133
300
370
267
240
168
156
569
438
393
a
Reference 45.
Reference 46.
Reference 47.
d
Reference 48.
b
c
2. Elastic properties
to study the elastic properties of TiN, TiC, VN, C共diamond兲,
Si, and Ge.
In Table II, we compare the present elastic constants and
polycrystalline elastic moduli of TiN, TiC, and VC with
former theoretical values44,45 and with the available experimental data.46,47 Similar data for C, Si, and Ge in diamond
structure are listed in Table III. We observe that the agreement between different theoretical results and experimental
data is modest. The average deviations between the present
C⬘ and C44 and the corresponding experimental values are
⬃45% and ⬃33% for nitrides and TiC, and ⬃24% and ⬃3%
for semiconductors, respectively. Although, the average deviations are lower in the case of the polycrystalline elastic
moduli, the above errors for nitrides and TiC are larger than
the usual errors obtained in ab initio calculation of elastic
constants.48 On the other hand, comparing different sets of
theoretical results for TiN and TiC obtained within the same
local density approximation 共LDA兲, we find similar large
deviations between them. For example, the LDA values for
C44 for TiN and TiC differ by 51% and 33%, respectively.
The large scatter of different theoretical results indicates the
numerical difficulties associated with such calculations. On
this ground, we conclude that the present method is suitable
B. Solid solutions
1. TiN1−xCx
For TiN1−xCx, our EMTO-CPA calculations were performed for the conventional cell of NaCl structure with Ti
occupying one sublattice and N and C sharing the other one,
in which we make no difference between different Ti atoms
as well as N and C atoms. Therefore, there are two binary
systems 共m = 2兲 contained in this solid solution, Ti–N and
Ti–C. All the atoms are sixfold coordinated 共ni = n j = 6 for
both binary systems兲. The multiplicity, Nij, of each binary
system can be taken as its concentration, i.e., 1 − x and x for
Ti–N and Ti–C, respectively. In this way, the volume of the
multicomponent system, ⍀, should be taken as a quarter of
that of the conventional cell.
Listed in Table IV are the intermediate results for the
hardness calculations. As is seen from the table, with increasing C content, the volume V0 of the solid solution increases
and the overall bond strength S M decreases, both results in a
decrease of the hardness according to Eq. 共5兲. However, with
TABLE III. Elastic constants of C 共diamond兲, Si, and Ge 共in GPa兲. The present results 共this work兲 are compared
to experimental data 共Ref. 58兲 共expt.兲.
C⬘
C44
B
G
E
C 共diamond兲
This work
Expt.
542
477
470
578
442
443
497
535
1085
1145
Si
This work
Expt.
68
51
85
80
96
98
77
67
183
163
Ge
This work
Expt.
51
40
69
67
62
75
61
55
137
132
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-5
J. Appl. Phys. 103, 083505 共2008兲
Hu et al.
TABLE IV. Reference energy ei 共in Å−1兲, equilibrium volume 共in Å3 /bond兲, and bond strength Sij as well as exponential parameter f ij of the binary systems
for TiN1−xCx solid solution.
x
eTi
eN
eC
V0
STiN
STiC
SM
f TiN
f TiC
f IM
f IIM
0.0
0.2
0.4
0.6
0.8
1.0
2.6645
2.6855
2.7045
2.7213
2.7354
2.7456
4.8924
4.8719
4.8515
4.8309
4.8105
¯
¯
3.7715
3.7520
3.7324
3.7116
3.6897
19.048
19.261
19.478
19.708
19.952
20.195
0.2839
0.2833
0.2827
0.2819
0.2808
¯
¯
0.2493
0.2486
0.2478
0.2467
0.2454
0.2839
0.2762
0.2685
0.2609
0.2532
0.2454
0.0869
0.0837
0.0807
0.0780
0.0756
¯
¯
0.0283
0.0263
0.0245
0.0229
0.0215
0.0869
0.0674
0.0516
0.0390
0.0291
0.0215
0.0869
0.0726
0.0590
0.0459
0.0335
0.0215
increasing C content, the reference energy of Ti decreases
whereas those of N and C increases such that the reference
energy difference between Ti and N as well as C decreases,
which yields a decreasing exponential factor f M . A smaller
f M represents a small ionicity, indicating a higher hardness.
The competition between the bond strength/volume and the
ionicity effects controls the trend of the hardness of TiN1−xCx
with respect to x.
Figure 1 compares the theoretical hardness of TiN1−xCx
to that from experiments. From our calculations, the hardness of TiN is lower than TiC. We find that HIM reaches a
maximum at x ⬃ 0.6. This composition dependence is in
good agreement with those measured by different groups11,12
shown in the bottom panel of Fig. 1. The occurrence of the
maximum of hardness has been explained by Jhi et al.2 on
the basis of electronic band structure analysis. They suggested that the high hardness of this system is due to the ␴
bonding states between the nonmetal p and the metal d orbitals that strongly resists shearing strain. These states are
completely filled at a valence-electron concentration of about
8.4, corresponding to x = 0.6. Additional electrons would go
to a higher band and make it less stable against shear deformations.
Also shown in the top panel of Fig. 1 is the hardness HIIM
obtained from Eqs. 共5兲, 共6兲, and 共8兲, which is a simple geometric average of the hardnesses for the constituent binary
systems. Obviously, this scheme fails to reproduce the trend
of the hardness with x correctly: there is no maximum in
between the hardness of TiN and TiC.
Figure 2 displays the composition dependent elastic
properties of TiN1−xCx. As shown in the upper panel of the
figure, both C11 and C⬘ decreases with C content 共x兲. C44 first
goes up with increasing x, reaches a maximum at x ⬃ 0.5, and
then decreases with further increasing of x. Although our C44
is about 50 GPa higher than those calculated by Jhi et al.2
using ab initio pseudopotential method, the calculated composition dependence for C44 in these two studies are in good
agreement with each other. The trends for C44 and hardness
共Fig. 1兲 are also in line, although the maxima are realized for
slightly different x values. However, the most frequently
quoted elastic properties considered as a measure of hardness
are polycrystalline bulk modulus B, shear modulus G, and
Young’s modulus E. The present polycrystalline elastic
moduli are shown in the bottom panel of Fig. 2 along with
the available experimental data for E. We observe that the
theoretical Young’s modulus agrees very well with the ex-
FIG. 1. 共Color online兲 Theoretical hardness of TiN1−xCx 共top panel兲, in
comparison with experimental data 共bottom panel兲, solid circles from Ref.
11 and open ones from Ref. 12.
FIG. 2. 共Color online兲 Elastic constants of TiN1−xCx as a function of carbon
content. The present results are shown by filled symbols, whereas open
triangles refer to the experimental data by Karlsson et al. 共Ref. 12兲.
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-6
J. Appl. Phys. 103, 083505 共2008兲
Hu et al.
TABLE V. Multiplicity of the binary systems in TiN1−x and VN1−x solid
solutions described by supercells.
x
1 / 16
1/8
1/4
Supercell size
TM4N6
TM5N6
TM6N6
2⫻2⫻1
2⫻1⫻1
Conventional cell
1
2
3
4
2
¯
11
4
1
perimental results by Karlsson et al.12 It is seen that the bulk
modulus decreases linearly with increasing x. Both E and G
are smaller for TiC than for TiN. The maxima for E共x兲 and
G共x兲 near x ⬃ 0.4 do not correspond to the maximum hardness 共Fig. 1兲. The comparison between the hardness and elastic moduli seems to support the empirical rule that the correlation between H and E or G is superior compared to that
between H and B. However, none of the elastic moduli can
be considered as a rigorous measure of hardness.
2. TiN1−x and VN1−x
For TiN1−x and VN1−x, the first-principles CPA calculations were performed for the conventional cell of NaCl structure with vacancies randomly distributed on the nitrogen site.
Although the coordination number for TM is not explicitly
considered in the CPA calculations, the presence of nitrogen
vacancies should definitely reduce the coordination for some
of the TM atoms. In order to determine the coordination
numbers for TM, we use several supercells within which
nitrogen vacancies are distributed as homogenously as possible. As an example, for x = 1 / 8 = 12.5%, we choose 2 ⫻ 1
⫻ 1 supercell with one nitrogen vacancy such that we get
two TM atoms fourfold, two TM atoms fivefold, and four
TM atoms sixfold coordinated by nitrogen, and all the nitrogen atoms are sixfold coordinated. Correspondingly, there
are three binary systems 共m = 3兲, TMn=4 − Nn=6, TMn=5
− Nn=6, and TMn=6 − Nn=6, of which the multiplicities 共Nij兲
are 2, 2 and 4, respectively. Details of the binary systems
contained in TMN1−x for different compositions are listed in
Table V.
When looking into the intermediate parameters listed in
Table VI, we note that, again, the hardness of TMN1−x is
controlled by the competition between the overall bond
strength/volume and the ionicity, however, their effects are
FIG. 3. 共Color online兲 Hardness of TiN1−x 共upper panel兲 and VN1−x 共lower
panel兲. The scattered circles and triangles as well as the blue line represent
the hardness from different experiments 共Refs. 49, 50, and 52兲, whereas the
solid squares refer to the present results.
just the opposite to the case of TiN1−xCx. Here, we see that
the volume decreases whereas the bond strength increases
with increasing vacancy concentration, which implies an increases in the hardness. On the other hand, the reference
energy of the TM atoms decrease with increasing vacancy
concentration, but the reference energy for N atom does not
change very much since the number of coordinates of N
atoms remains unchanged. Thus, the difference between the
reference energies of TM and N atoms increases so that the
exponential factor f M increases, which decreases the hardness.
Figure 3 shows our theoretical hardness for TiN1−x and
VN1−x in comparison with experimental data. Since we make
no difference between the TM atoms 共as well as the nitrogen
atoms兲 in our CPA calculations, the reference energies of
these atoms are exactly the same. Thus, the exponential factors f ij for different binary systems are identical. Therefore,
TABLE VI. Reference energy ei 共in Å−1兲, equilibrium volume 共in Å3/bond兲, and bond strength Sij, as well as
exponential parameter f ij of the binary systems for TiN1−x and VN1−x solid solutions.
x
eTM
eN
V0
STM4N6
STM5N6
STM6N6
SM
f ij = f M
0
1 / 16
1/8
1/4
2.6645
2.6288
2.5907
2.5077
4.8924
4.8885
4.8852
4.8819
19.048
18.947
18.850
18.679
TiN1−x
¯
0.3458
0.3438
0.3391
¯
0.3093
0.3075
¯
0.2839
0.2824
0.2807
0.2769
0.2839
0.2926
0.3021
0.3224
0.0869
0.0904
0.0942
0.1032
0
1 / 16
1/8
1/4
3.5153
3.4821
3.4464
3.3661
4.9368
4.9388
4.9422
4.9495
17.396
17.174
16.948
16.487
VN1−x
¯
0.4134
0.4132
0.4124
¯
0.3697
0.3696
¯
0.3376
0.3375
0.3374
0.3368
0.3376
0.3497
0.3631
0.3921
0.0283
0.0299
0.0318
0.0363
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-7
J. Appl. Phys. 103, 083505 共2008兲
Hu et al.
FIG. 4. 共Color online兲 Elastic properties of 共a兲 TiN1−x and 共b兲 VN1−x as a
function of vacancy content. The
present results are shown by filled
symbols, and the open circles in the
left panel refer to the picosecond ultrasonic measurements by Lee et al. 共Ref.
53兲.
f IM = f IIM according to Eqs. 共7兲 and 共8兲. Consequently, HIM
= HIIM . For both TiN1−x and VN1−x, the hardness increases
with increasing vacancy concentration, i.e., there exists vacancy hardening. The hardening rate of TiN1−x is lower than
that of VN1−x. Taking into account the uncertainties of the
experimental data, our theoretical results from Fig. 3 are in
good agreement with observations.49–52
The composition dependent elastic properties of TiN1−x
and VN1−x are shown in Fig. 4. For TiN1−x, the single crystal
elastic constants C11 and C⬘ decrease with increasing vacancy concentration x, whereas C44 remains almost unchanged. All the polycrystalline elastic moduli, B, G, and E,
decrease with increasing x. The elastic properties of TiN1−x
have been investigated both experimentally and theoretically
in the literature. As shown in the top panel of Fig. 4共a兲, our
theoretical C11 is in good agreement with experiments.53 The
theoretical C44 and Young’s modulus E are smaller than the
experimental results 共not shown in the figure兲, however, the
trends of them with respect to x agree with each other. The
bulk modulus of TiN1−x has been calculated by Dridi et al.54
using full-potential linear augmented plane-wave method.
They got a linear dependence of B on x as B = 317− 177x,
which is comparable to B = 266− 158x obtained in our study.
TABLE VII. Reference energy ei 共in Å−1兲, equilibrium volume 共in Å3/bond兲, and bond strength Sij, as well as
exponential parameter f ij of the binary systems for diamond, Si, and Ge doped with B.
SXB
SM
f XB
f IIM
C1−xBx
0.6611
0.6581
0.6549
0.6521
0.6490
0.6461
¯
0.7360
0.7317
0.7278
0.7238
0.7199
0.6611
0.6596
0.6578
0.6564
0.6547
0.6531
¯
0.0124
0.0122
0.0120
0.0118
0.0116
0
0.0002
0.0005
0.0007
0.0009
0.0012
40.581
40.225
39.847
39.497
39.126
38.735
Si1−xBx
0.2853
0.2864
0.2875
0.2886
0.2897
0.2911
¯
0.3372
0.3385
0.3397
0.3410
0.3426
0.2853
0.2873
0.2894
0.2914
0.2935
0.2959
¯
0.0262
0.0262
0.0261
0.0261
0.0261
0
0.0005
0.0010
0.0016
0.0021
0.0026
47.950
47.403
46.910
46.396
45.861
45.330
Ge1−xBx
0.2566
0.2580
0.2592
0.2606
0.2620
0.2634
¯
0.3027
0.3052
0.3068
0.3085
0.3102
0.2566
0.2588
0.2609
0.2631
0.2655
0.2678
¯
0.0262
0.0262
0.0262
0.0262
0.0262
0
0.0005
0.0010
0.0016
0.0021
0.0026
x
eX
eB
V0
0.00
0.02
0.04
0.06
0.08
0.10
4.0808
4.0704
4.0601
4.0502
4.0404
4.0406
¯
5.0916
5.0679
5.0459
5.0246
5.0045
11.318
11.384
11.461
11.528
11.606
11.683
0.00
0.02
0.04
0.06
0.08
0.10
2.6954
2.6976
2.7001
2.7025
2.7051
2.7084
¯
3.7406
3.7428
3.7448
3.7476
3.7515
0.00
0.02
0.04
0.06
0.08
0.10
2.5631
2.5672
2.5705
2.5745
2.5785
2.5826
¯
3.5592
3.5638
3.5694
3.5750
3.5809
SXX
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-8
J. Appl. Phys. 103, 083505 共2008兲
Hu et al.
FIG. 5. 共Color online兲 Hardness of diamond, Si, and Ge doped with B. The
solid symbols represents HIM , whereas the open ones refer to HIIM .
The behavior of the elastic properties of VN1−x, as
shown in Fig. 4共b兲, is different from that of TiN1−x. Similar
to the case of TiN1−x, C11 and B decrease with increasing x.
In contrary to those of TiN1−x, C44, E, and G increase with x.
There is no experimental elastic constants found in literature
for VN1−x. So, our calculations may serve as predictions for
these properties.
Comparing the elastic moduli and the hardness 共see Fig.
3兲, we see that the vacancy concentration x dependence of
the hardness for TiN1−x shows a different behavior compared
to those exhibited by the polycrystalline elastic moduli. For
VN1−x, the trend of hardness with respect to x is in line with
those of Young’s modulus E and shear modulus G, but is in
contrast with that of bulk modulus B. Here, we see again that
the correlation between hardness and elastic moduli is not
unequivocal and monotonic.
3. B doped C, Si, and Ge
The solid solutions considered in this subsection have
the diamond structure with two sublattices: the face centered
cubic 共fcc兲 site and the tetrahedral interstitial sites. All the
atoms in the systems are fourfold coordinated. Since the concentrations of the impurities is relatively low, we assume that
all the impurities replace the fcc site atoms, and are not nearest neighbor of each other, i.e., there is no B–B bond. Therefore, there are two binary systems in each of the solid solutions. Taking C1−xBx as example, Cfcc – Ctet appears with
multiplicity 1 − x and Bfcc – Ctet with multiplicity x. With
these definitions of the multiplicities, the volume for the
solid solution, ⍀, should be set as 1 / 4 of the volume of the
conventional cell.
The reference energy ei, equilibrium volume V0, and
bond strength Sij as well as exponential parameter f ij of the
binary systems for diamond, Si, and Ge doped with B are
listed in Table VII. For all systems, f Iij, as it is defined 关Eq.
共7兲兴, is equal to 0 independent of the B doping since equalatom X – X bonds exist in the systems. The absolute value of
the exponential parameter f IIij is very small, and B doping
does not alter it very much. Therefore, both HIM and HIIM are
determined mainly by V0 and S M , and do not differ from each
other very much. For the B doped diamond, V0 increases
with x whereas S M decrease, both of which result in the
reduction of the hardness 共see Fig. 5兲. This is in agreement
with experimental findings.55,56 For Si and Ge, B doping decreases V0 and strengthens the bond, which enhances the
hardness.57 It should be noted that, for the undoped diamond,
Si, and Ge, we get a hardness of 90.5, 10.9, 8.3 GPa, respectively, in good agreement with the experimental values, 96.0,
11.3, 8.8 GPa accordingly.8
Shown in Fig. 6 are the elastic moduli of diamond, Si,
and Ge as a function of B concentration. For all three systems, the bulk modulus B shows weak x dependence. In accordance with the concentration dependent hardness of B
doping diamond, the shear modulus G and Young’s modulus
E decrease with increasing x. However, there is no clear
correlation between the elastic moduli and hardness for B
doping Si and Ge.
V. CONCLUSION
In summary, we have calculated the hardness of some
solid solutions using a revised Šimůnek and Vackář’s hard-
FIG. 6. 共Color online兲 Elastic constants of B doped 共a兲 diamond, 共b兲 Si, and 共c兲 Ge.
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
083505-9
Hu et al.
ness formula. In general, the obtained theoretical hardness is
in good agreement with experimental data. The connection
between the hardness and elastic moduli for these solid solutions has been reconsidered. We have shown that, in general, the elastic moduli cannot be used as a rigorous measure
of the hardness of covalent/ionic crystal solid solutions.
ACKNOWLEDGMENTS
This work was partly supported by the Chinese MoST
under Grant No. 2006CB605104 and NSFC under Grant No.
50631030. The Swedish Research Council, the Swedish
Foundation for Strategic Research, the Hungarian Scientific
Research Fund 共T046773, T048827, and K-68312兲, and
Böhler Uddeholm AG are also acknowledged for financial
support.
1
V. V. Brazhkin, A. G. Lyapin, and R. J. Hemley, Philos. Mag. A 82, 231
共2002兲, and reference there in.
S. H. Jhi, J. Ihm, S. G. Louie, and M. L. Cohen, Nature 共London兲 399, 132
共1999兲.
3
S. H. Jhi, S. G. Louie, M. L. Cohen, and J. Ihm, Phys. Rev. Lett. 86, 3348
共2001兲.
4
J. He, E. Wu, H. Wang, R. Liu, and Y. Tian, Phys. Rev. Lett. 94, 015504
共2005兲.
5
J. J. Gilman, Mater. Sci. Eng., A 209, 74 共1996兲.
6
D. G. Clerc and H. M. Ledbetter, J. Phys. Chem. Solids 59, 1071 共1998兲.
7
F. Gao, J. He, E. Wu, S. Liu, D. Yu, D. Li, S. Zhang, and Y. Tian, Phys.
Rev. Lett. 91, 015502 共2003兲.
8
A. Šimůnek and J. Vackář, Phys. Rev. Lett. 96, 085501 共2006兲.
9
J. C. Phillips, Rev. Mod. Phys. 42, 317 共1970兲.
10
Z. Y. Liu, X. Guo, J. He, D. Yu, and Y. Tian, Phys. Rev. Lett. 98, 109601
共2007兲.
11
V. Richtet, A. Beger, J. Drobniewski, I. Endle, and E. Wolf, Mater. Sci.
Eng., A 209, 353 共1996兲.
12
L. Karlsson, L. Hultman, M. P. Johansson, J. E. Sundgren, and H.
Ljungcrantz, Surf. Coat. Technol. 126, 1 共2000兲.
13
Q. M. Hu, K. Kadas, S. Hogmark, R. Yang, B. Johansson, and L. Vitos,
Appl. Phys. Lett. 91, 121918 共2007兲.
14
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865
共1996兲.
15
O. K. Andersen, O. Jepsen, and G. Krier, in Lectures on Methods of
Electronic Structure Calculations, edited by V. Kumar, O. K. Andersen,
and A. Mookerjee 共World Scientific, Singapore, 1994兲, p. 63.
16
O. K. Andersen, C. Arcangeli, R. W. Tank, T. Saha-Dasgupta, G. Krier, O.
Jepsen, and I. Dasgupta, Mater. Res. Soc. Symp. Proc. 491, 3 共1998兲.
17
L. Vitos, Phys. Rev. B 64, 014107 共2001兲.
18
L. Vitos, H. L. Skriver, B. Johansson, and J. Kollár, Comput. Mater. Sci.
18, 24 共2000兲.
19
L. Vitos, Computational Quantum Mechanics for Materials Engineers
共Springer-Verlag, London, 2007兲.
20
P. Soven, Phys. Rev. 156, 809 共1967兲.
21
B. L. Györffy, Phys. Rev. B 5, 2382 共1972兲.
22
L. Vitos, I. A. Abrikosov, and B. Johansson, Phys. Rev. Lett. 87, 156401
共2001兲.
23
L. Vitos, P. A. Korzhavyi, and B. Johansson, Phys. Rev. Lett. 88, 155501
共2002兲; Nat. Mater. 2, 25 共2003兲.
24
P. Olsson, I. A. Abrikosov, L. Vitos, and J. Wallenius, J. Nucl. Mater. 321,
84 共2003兲.
25
L. Dubrovinsky, N. Dubrovinskaia, F. Langenhorst, D. Dobson, D. Rubie,
C. Geshmann, I. A. Abrikosov, B. Johansson, V. I. Baykov, L. Vitos, T. Le
2
J. Appl. Phys. 103, 083505 共2008兲
Bihan, W. A. Crichton, V. Dmitriev, and H. P. Weber, Nature 共London兲
422, 58 共2003兲.
26
L. Vitos, P. A. Korzhavyi, and B. Johansson, Phys. Rev. Lett. 96, 117210
共2006兲.
27
N. Dubrovinskaia, L. Dubrovinsky, I. Kantor, W. A. Crichtion, V. Dmitriev, V. Prakapenka, G. Shen, L. Vitos, R. Ahuja, B. Johansson, and I. A.
Abrikosov, Phys. Rev. Lett. 95, 245502 共2005兲.
28
A. Taga, L. Vitos, B. Johansson, and G. Grimvall, Phys. Rev. B 71,
014201 共2005兲.
29
L. Huang, L. Vitos, S. K. Kwon, B. Johansson, and R. Ahuja, Phys. Rev.
B 73, 104203 共2006兲.
30
J. Zander, R. Sandström, and L. Vitos Comput. Mater. Sci. 41, 86 共2007兲.
31
A. E. Kissavos, S. I. Simak, P. Olsson, L. Vitos, and I. A. Abrikosov,
Comput. Mater. Sci. 35, 1 共2006兲.
32
Q. M. Hu, J. M. Lu, R. Yang, L. Wang, B. Hohansson, and L. Vitos, Phys.
Rev. B 76, 224201 共2007兲.
33
B. Magyari-Köpe, G. Grimvall, and L. Vitos, Phys. Rev. B 66, 064210
共2002兲.
34
B. Magyari-Köpe, L. Vitos, and G. Grimvall, Phys. Rev. B 70, 052102
共2004兲.
35
B. Magyari-Köpe, L. Vitos, B. Johansson, and J. Kollár, Acta Crystallogr.,
Sect. B: Struct. Sci. 57, 491 共2001兲.
36
B. Magyari-Köpe, L. Vitos, B. Johansson, and J. Kollár, J. Geophys. Res.
107, 1029 共2002兲.
37
A. Landa, C.-C. Chang, P. N. Kumta, L. Vitos, and I. A. Abrikosov, Solid
State Ionics 149, 209 共2002兲.
38
B. Magyari-Köpe, L. Vitos, G. Grimvall, B. Johansson, and J. Kollár,
Phys. Rev. B 65, 193107 共2002兲.
39
R. Hill, Proc. Phys. Soc. London 65, 349 共1952兲.
40
F. Gao, R. Xu, and K. Liu, Phys. Rev. B 71, 052103 共2005兲.
41
C. M. Sung and M. Sung, Mater. Chem. Phys. 43, 1 共1996兲.
42
J. Z. Jiang, K. Kragh, D. J. Frost, K. Stahl, and H. Lindelov, J. Phys.:
Condens. Matter 13, L515 共2001兲.
43
A. Zerr, M. Schwarz, E. Kroke, M. Goken, and R. Riedel, J. Am. Ceram.
Soc. 85, 86 共2002兲.
44
W. Wolf, R. Podloucky, T. Antretter, and F. D. Fischer, Philos. Mag. B 79,
839 共1999兲.
45
R. Ahuja, O. Eriksson, J. M. Wills, and B. Johansson, Phys. Rev. B 53,
3072 共1996兲.
46
J. O. Kim, J. D. Achenback, P. B. Mirkarimi, M. Shinn, and S. A. Barnett,
J. Appl. Phys. 72, 1805 共1992兲.
47
Y. Kumashiro, H. Tokumoto, E. Sakuma, A. Itoh, Proceedings of the Sixth
International Conference on Internal Friction and Ultrasonic Attenuation
in Solids, edited by R. R. Hasiguti and N. Mikoshiba 共Tokyo University
Press, Tokyo, 1977兲, p. 395.
48
P. Söderlind, O. Eriksson, J. M. Wills, and A. M. Boring, Phys. Rev. B 48,
5844 共1993兲.
49
W. Lengauer, J. Alloys Compd. 186, 293 共1992兲.
50
F. Elstner, A. Ehrlich, H. Giegengack, H. Kupfer, and F. Richter, J. Vac.
Sci. Technol. A 12, 476 共1994兲.
51
H. Holleck, J. Vac. Sci. Technol. A 4, 2661 共1986兲.
52
G. Farges, E. Beauprez, and D. Degout, Surf. Coat. Technol. 54/55, 115
共1992兲.
53
T. Lee, K. Ohmori, C. S. Shin, D. G. Cahill, I. Petrov, and J. E. Greene,
Phys. Rev. B 71, 144106 共2005兲.
54
Z. Dridi, B. Bouhafs, P. Ruterana, and H. Aourag, J. Phys.: Condens.
Matter 14, 10237 共2002兲.
55
H. P. Bovenkerk, U.S. Patent No. 4268276 共19 May 1981兲.
56
R. H. Wentorf and W. A. Rocco, U.S. Patent No. 3745623 共17 July 1973兲.
57
G. Yu, J. Watanabe, K. Izumi, K. Nakashima, T. Jimbo, and M. Umeno,
Jpn. J. Appl. Phys., Part 2 40, L183 共2001兲.
58
W. Martienssen and H. Warlimont, Springer Handbook of Condensed Matter and Materials Data 共Springer, New York, 2005兲.
Downloaded 15 Apr 2012 to 210.72.130.187. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions