Materials Transactions, Vol. 51, No. 3 (2010) pp. 574 to 577
#2010 The Japan Institute of Metals
RAPID PUBLICATION
First-Principles Calculations of the Specific Heats of Cubic Carbides and Nitrides
Satoshi Iikubo1 , Hiroshi Ohtani2 and Mitsuhiro Hasebe2
1
2
Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, Kitakyushu 808-0196, Japan
Department of Materials Science and Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan
Calculated specific heats of several carbides and nitrides with a B1 structure have been compared with those of experimental data up to
3000 K. The specific heats at constant pressure are calculated with the quasiharmonic approximation, using phonon dispersions calculated from
direct method, and pseudopotential plane-wave method. The calculated results of HfC, TaC, TiC, ZrC, and ZrN are in excellent agreement with
the experimental data up to 3000 K. For NbC, TiN, and VC, the calculated results are also in excellent agreement with the experimental data up
to 2000 K, while they show an excessive rise over the experimental data above 2000 K. The deviation of the calculation from the experiment at
high temperature is caused by instability of the B1 structure or anharmonic effect. [doi:10.2320/matertrans.MBW200913]
(Received October 14, 2009; Accepted November 19, 2009; Published January 7, 2010)
Keywords: specific heat, first-principles calculations, quasiharmonic approximation, carbide and nitride with B1 structure
1.
Introduction
The CALPHAD (CALculation of PHAse Diagrams)
approach has been successfully used in a wide area of
applications, from alloys to surfaces, interfaces, and clusters.1) In the current approach, the thermodynamic parameters necessary for calculating phase diagrams are estimated
from the experimental data such as phase boundaries,
activities and so on. It is difficult to estimate these parameters
for metastable phases for which the physical and chemical
properties are not experimentally accessible. One effective
way to obtain information on the metastable phase is an
electronic structure calculation based on the first-principles
method.2) The application of the first-principles calculations
to the study of thermodynamic properties at finite temperature remains challenging, because this calculation provides
the thermodynamic properties at the ground state, i.e., zero
temperature. Therefore, a method is required to calculate
the specific heats for extending the ground state structure
properties to those in a finite temperature region.
The specific heat, which is closely related to the free
energy, has been examined theoretically, both semi-empirically and with the first-principles. To calculate the specific
heat at constant pressure, the anharmonic effect of the lattice
vibration must be considered. The isopiestic specific heats at
finite temperatures below 300 K have been discussed on the
basis of the Debye and Grüneisen models to derive their
thermal expansion.3) Using this model, Lu et al. show the
thermodynamic properties of carbides and nitrides4,5) at
temperatures up to 3000 K. An alternative approach for
determining the lattice contribution to the free energy is the
quasiharmonic approximation. Calculation based on this
method suggests that the quasiharmonic approximation
provides a reasonable description of the thermodynamic
properties of simple metals,6,7) oxides,8–10) and nitrides11,12)
below their melting points. Comparing these two methods,
the former uses a fitting parameter to reproduce the
experimental results in the extended temperature region,
while the latter does not need any experimental data. Thus,
the quasiharmonic approach matches our needs, and we
attempted to calculate the specific heat, thermal expansion,
and bulk modulus for HfC, NbC, TaC, TiC, TiN, VC, ZrC,
and ZrN with the B1 structure, employing the direct method
for obtaining phonon spectra. The essential results are
recorded in this paper.
2.
Theory and Calculation Procedure
Then, the specific heat at constant volume is described by,
@E
;
ð1Þ
Cv ¼
@T v
where E denotes the internal energy. The experimental
specific heat of a crystalline phase should be constructed
from the electronic and lattice contributions. The internal
energy due to the electron excitations Eel and phonon
excitations Eph can be expressed as
Z1
Z F
el
el
D ðÞ f ð; TÞd Del ðÞd; ð2Þ
E ðTÞ ¼
1
1
Z1
Eph ðTÞ ¼
Dph ð!Þgð!; TÞh !d!;
ð3Þ
0
where , F , Del ðÞ, f ð; TÞ, !, Dph ð!Þ, gð!; TÞ are electron
energy, the Fermi energy, electronic density of state, Fermi
distribution function, phonon frequency, the phonon density
of states, and Bose distribution function, respectively.
Then, the specific heat at constant pressure Cpph is given by,
Cpph ¼ CVph þ v ðTÞ2 B0 ðTÞVðTÞT;
ð4Þ
where B0 ðTÞ is the bulk modulus and VðTÞ is the volume at
temperature T. The coefficient of volume thermal expansion
v ðTÞ is expressed as
1 @V
:
ð5Þ
v ¼
V @T p
To estimate the anharmonic lattice contribution included
in Cpph , we used the quasiharmonic approximation. The
vibrational free energy of the lattice ions F ph ðV; TÞ is
XX h !j ðq; VÞ
ln 2 sinh
;
F ph ðV; TÞ ¼ EðVÞ þ kB T
2kB T
q
j
ð6Þ
First-Principles Calculations of the Specific Heats of Cubic Carbides and Nitrides
where EðVÞ is the energy of the static lattice at a given
volume V, and !j ðq; VÞ is the frequency of the jth phonon
band at point q in the Brillouin zone. A quasiharmonic
approximation, assuming that phonon frequencies depend
only on the cell parameters, was employed here. This enabled
us to take thermal expansion into account. To relate the
vibrational free energy F ph ðV; TÞ and thermal properties, the
third-order Birch–Murnaghan equation of state13) is used.
#3
( " 2
9V0 B0
V0 3
EðVÞ ¼ E0 þ
1 B00
16
V
" 2
#2 "
2 #)
V0 3
V0 3
þ
1
64
;
ð7Þ
V
V
575
(a) TaC
(b) NbC
-200
Free energy, F
ph
/ (kJ/mol/F.U.)
-100
-300
-400
3.
Results and Discussion
We first show the Helmholtz free energy calculated using
the quasiharmonic approximation, and note a problem in
estimating the anharmonic effect at higher temperature.
Figure 1 shows a plot of the vibrational free energy F ph for
TaC and NbC against a lattice volume at temperatures from 0
to 3000 K. The lattice volumes with minimum free energy
were determined from fitting curves of the third-order Birch–
Murnaghan equation of state at each temperature, given by
the solid lines in Fig. 1. As shown in this figure, the lattice
volume for both the compounds increases with increasing
temperature. For TaC, the lattice volume of minimum free
energy is 21.72 Å3 at 0 K, and substantially increase to
23.66 Å3 at 3000 K, whereas, for NbC, the figure shows that
the lattice volume of the minimum free energy is 22.2 Å3 at
0 K and leaves the range of the figure above 2700 K, as shown
in Fig. 1(b). It is difficult to estimate the minimum free
energy at high temperature for NbC, because the system
shows the anomalous behavior of the phonon dispersion.
Figure 2 shows the phonon spectrum for NbC at (a)
V0 ¼ 22:03 Å3 and (b) V0 ¼ 25:14 Å3 . Three acoustic and
optical branches, separated by a frequency gap, are shown
20
21
22
23
24
21
3
22
23
24
3
Volume, V/Å
Volume, V/Å
Fig. 1 Temperature dependence of the free energy F on crystal volume
is denoted by crosses for (a) TaC and (b) NbC at temperatures
0 < T < 3000 K at 100 K intervals. The dashed lines are results of fittings
using the third-order Birch–Murnaghan equation of state. The solid line
connects the F minima for different temperatures.
25
(b) NbC, V0 = 25.14 Å3
(a) NbC, V0 = 22.03 Å3
20
Frequency, ω/THz
where V0 is the volume for the minimum energy, and B00 ¼
@B0 =@P is the pressure derivative of the bulk modulus.
First-principles calculations were carried out using the
Vienna Ab Initio Simulation Package (VASP),14) which is a
calculation code based on the density functional theory for
systems with periodic boundary conditions. To obtain the
thermodynamic data at 0 K, the local density approximation
with non spin polarization was applied and 520 eV was
chosen for the plane-wave cutoff. The PHONON code,15)
which calculates the phonon properties of a crystal by a direct
method, was used to calculate the vibrational properties. In
this method, the crystal supercell is first optimized by a full
quantum mechanical electronic structure calculation. In such
calculations, one can make a series of small displacements of
one atom at a time and calculate the Hellmann–Feynman
forces exerted on all atoms. The force constants are then
derived from the ab initio Hellmann–Feynman forces. The
first-principles and phonon calculations were carried out with
64 atoms for a perfect crystal of (TMX)4 (TM = transition
metal, X ¼ C, N), which corresponds to a 2 2 2 unit
cell of a fluorite structure.
15
10
5
0
W
L
Γ
X
W K
W
L
Γ
X
W K
Fig. 2 The phonon spectrum for NbC calculated at (a) V0 ¼ 22:03 Å3 and
(b) V0 ¼ 25:14 Å3 along high symmetry directions in the Brillouin zone.
in the figures. A softening of the acoustic modes near the
L point can be observed in Fig. 2(b), consistent with the
previous report.16) By introducing volume expansion to the
system, the frequency of L point becomes an imaginary
number, which is incompatible with the dynamic stability of
NbC with the B1 structure. The same behavior is observed for
VC, and TiN above 2500 K. Plausible causes of this problem
might be the anharmonic effect of the crystal lattice, or
a possible high-temperature structural transition, i.e., the
order-disorder transition. For example, Isaev et al. reported
structural instability of the B1 structure for a particular
valence electron number, e.g., NbN.16)
The calculated specific heats presented in Figs. 3(a)–3(h)
are compared with the available experimental data,17,18)
as shown by the circles. We note an excellent agreement
between the theoretical and experimental results, especially
for HfC, TaC, TiC, ZrC, and ZrN in a wide range of
temperatures up to 3000 K. It can also be seen that below
1000 K, the difference between Cvph and Cpph is very small.
At high temperature, Cvph approaches the classical constant
value, while Cpph increases monotonously with the temper-
ph
Cp
ph
Cp
+
experiment
0
1000
2000
Temperature, T/K
80
(c) TiC
60
ph
Cv
40
ph
Cp
ph
Cp
20
0
Specific Heat, C/(J/mol/K)
0
1000
2000
Temperature, T/K
(e) ZrN
60
ph
Cv
40
ph
Cp
ph
20
el
Cp + Cv
experiment
0
1000
2000
Temperature, T/K
(g) TiN
60
ph
Cv
ph
Cp
ph
20
el
Cp + Cv
experiment
0
0
1000
2000
Temperature, T/K
0
3000
1000
2000
Temperature, T/K
3000
80
(d) ZrC
60
ph
Cv
40
ph
Cp
ph
Cp
20
el
+ Cv
experiment
0
0
1000
2000
Temperature, T/K
3000
80
(f) NbC
TiC
ZrN
80
300
HfC
TaC
250
TaC
60
HfC
TiC
200
ZrC
ZrC
40
150
ZrN
VC
NbC
NbC
100
20
TiN
TiN
50
60
ph
VC
Cv
40
ph
0
Cp
ph
Cp
20
0
0
el
Cv
+
experiment
2500
Tempelature, T/K
0
2500
Temperature, T/K
0
3000
80
40
0
3000
80
0
Specific Heat, C/(J/mol/K)
el
+ Cv
experiment
Cv
ph
Cp
ph
el
Cp + Cv
experiment
20
3000
Specific Heat, C/(J/mol/K)
Specific Heat, C/(J/mol/K)
0
Specific Heat, C/(J/mol/K)
20
el
Cv
ph
40
(b)
(a)
Bulk modulus, B0 /GPa
ph
Cv
40
350
(b) TaC
60
-1
60
80
-6
(a) HfC
Linear thermal expansion coefficient, α l x10 /K
80
Specific Heat, C/(J/mol/K)
S. Iikubo, H. Ohtani and M. Hasebe
0
Specific Heat, C/(J/mol/K)
Specific Heat, C/(J/mol/K)
576
1000
2000
Temperature, T/K
3000
Fig. 4 Temperature dependence of (a) linear thermal expansion coefficient
and (b) bulk modulus B0 ðTÞ for HfC, TaC, TiC, ZrC, ZrN, NbC, TiN,
and VC, respectively. The curves are the calculated results and the
symbols for represent experimental data from Ref. 19).
80
(h) VC
60
ph
Cv
40
ph
Cp
ph
20
el
Cp + Cv
experiment
0
0
1000
2000
Temperature, T/K
3000
Fig. 3 Calculated temperature dependence of heat capacity of (a) HfC,
(b) TaC, (c) TiC, (d) ZrC, (e) ZrN, (f) NbC, (g) TiN, and (h) VC at constant
pressure (Cp ) and constant volume (Cv ). The experimental data for Cp
(Ref. 13) and 14)) are denoted by circles.
ature. The difference between the solid and dashed lines in
Fig. 3 denotes the electronic contribution to the specific
heats. It can readily be understood that the contribution is
quite small, but not negligible. On the other hand, we see a
marked difference in the high temperature region for NbC,
TiN, and VC, which show the soft mode in the phonon
calculation for a large volume. The calculated specific heats
are much larger than the experimental ones above approximately 2000 K. This large discrepancy in high temperature is
attributed to the anharmonic contribution or high-temperature structural instability.
Figures 4(a) and (b) show the temperature dependence of
the linear thermal expansion coefficient l and bulk modulus
B0 obtained from the F–V curve fitting. In the regular system,
as the temperature increase, the l and V are expected to
increase, while B0 decreases. Available experimental results
for linear thermal expansion coefficient l 19) are also
indicated in this figure. Our calculations are in good
agreement with the experimental data of l for HfC, TaC,
TiC, ZrC, and ZrN. Thermal volume expansion can be
indirectly reproduced by the quasiharmonic approximation.
For TaC, we note a finite deviation between the calculation
and experiment for l , while the specific heat shows good
agreement with the experiment. A large deviation in l values
for NbC, TiN, and VC is observed at high temperature. The
difference between the calculated specific heats and exper-
imental ones are clearly attributable to the over estimation of
l and V, because the anharmonic contribution to isopiestic
specific heat is described by a multiplication of these
properties. These compounds also show anomalous softening
of B0 with increasing temperature. The deviation in the l
values and anomalous softening of B0 are obviously related
to the anharmonic effect or high-temperature structural
instability.
4.
Conclusions
The isopiestic specific heats for HfC, NbC, TaC, TiC, TiN,
VC, ZrC, and ZrN with the B1 structure were calculated
by taking the electron and lattice contributions into account
at constant volume as well as an additional anharmonic
component of the lattice vibration. The anharmonic part
was described by the quasiharmonic approximation. The
comparison between the experimental data and calculated
results showed an excellent agreement even at high temperature. The present results show a possible development for
the CALPHAD method that includes metastable structures
that are not experimentally accessible.
Acknowledgment
The authors are grateful to Dr. S. Minamoto at CTC
Technology Co. and Prof. I. Tanaka at Kyoto University for
useful discussions. This work was partly supported by the
Grant-in-Aid for Young Scientists (B) (21740239) from the
Japanese Ministry of Education, Culture, Sports, Science and
Technology. One author (H. O.) also acknowledges financial
support by the Grant-in-Aid for Scientific Research on
Priority Area, ‘‘Atomic Scale Modification’’ (Area No. 424)
from the Japanese Ministry of Education, Culture, Sports,
Science and Technology.
First-Principles Calculations of the Specific Heats of Cubic Carbides and Nitrides
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