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Electronic structure and elastic properties of scandium carbide and yttrium carbide: A first principles study
Physica B 406 (2011) 4041–4045
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Electronic structure and elastic properties of scandium carbide and yttrium carbide: A first principles study Jameson Maibam a, B. Indrajit Sharma a,n, Ramendu Bhattacharjee a, R.K. Thapa b, R.K. Brojen Singh c a
Department of Physics, Assam University, Silchar, Assam 788011, India Department of Physics, Mizoram University, Tanhril, Aizawl 796009, Mizoram, India c Centre for Interdisciplinary Research in Basic Sciences, Jamia Millia Islamia, New Delhi 110025,India b
a r t i c l e i n f o
abstract
Article history: Received 1 April 2011 Received in revised form 19 July 2011 Accepted 20 July 2011 Available online 3 August 2011
We have studied the electronic, structural, and elastic properties of scandium carbide and yttrium carbide by means of accurate first principles total energy calculations using the full-potential linearized plane wave method (FP-LAPW). We have used the generalized gradient approximation (GGA) for the exchange and correlation potential. Volume optimization, energy band structure, and density of states (DOS) of the systems are presented. The second order elastic constants have been calculated and other related quantities such as the Zener anisotropy factor, Poisson’s ratio, Young’s modulus, Kleinman parameter, Debye temperature, and sound velocities have been determined. The band gap calculation shows that YC is relatively more ionic than ScC. & 2011 Elsevier B.V. All rights reserved.
Keywords: DFT Electronic structure DOS Elastic constants
1. Introduction Transition metalmono carbides are known as refractory compounds. They have, relatively, high brittleness, hardness, melting point. In addition, they also have interesting optical, electronic, catalytic, and magnetic properties [1–4]. Due to these interesting properties, they are involved in wide technological application areas such as making tools for machine buildings, chemical, and nuclear industries. The structural, elastic, and mechanical properties of transition metal monocarbides have been extensively studied before theoretically using various methods [5–13], whereas the literature of electronic, structural, and elastic properties of ScC and YC in rock salt structure is still scarce. These compounds show three types of bonding characteristics: ionic, covalent, and metallic [6]. The occurrence of ionic like structure in combination with covalent like hardness is very interesting. In addition, they show metallic conductivities comparable with those of pure transition metals. Zhang et al. [11] studied the chemical bonding of ScC. Recently Korir et al. calculated the e bulk properties of YC [12]. Isaev et al. calculated the phonon related properties of both ScC and YC in rock salt structure using first principles ultrasoft potential [14]. ¨ Haglund et al. [15] have studied scandium carbide using self-consistent linear muffin-tin orbital method. Experimental
n
Corresponding author. Tel.: þ91 03842 270843. E-mail address: indraoffi[email protected] (B. Indrajit Sharma).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.07.036
work on ScC and YC was done by Villars and Calvert [16]. Ferna´ndez Guillermet et al. [17] studied the cohesive energy of YC using the self-consistent linear muffin–tin orbital method. Kalemos et al. [18] studied the diatomic molecule of ScC. However, some of the physical properties of these two compounds have received less attention. To our knowledge the elastic properties, which are the important bulk properties of solids, have not been considered either experimentally or theoretically for ScC and YC. The primary purpose of this work is to provide some additional information to the existing data on the physical properties of ScC and YC by using ab initio total energy calculations. We try to focus on the band structure, density of states, elastic, structural, and mechanical properties will be discussed in brief.
2. Computational methods The calculations for ScC and YC in the rock salt structures were performed with FP-LAPW within the framework of the density functional theory with generalized gradient approximation — Perdew–Bruke–Ernzerhof 96 (GGA—PBE 96) for the exchange correlation potential [19]. We have employed the full-potential (linearized) augmented plane waves plus local orbital (FP-LAPWþ lo) method as implemented in the WEIN2k code. [20]. This method has been extensively tested and is among the most accurate methods for performing electronic structure calculations of crystals. In this method, the unit cell is divided into non-overlapping atomic spheres whose center is at atomic
4042
J. Maibam et al. / Physica B 406 (2011) 4041–4045
Table 1 Lattice constant a0, bulk modulus B, its pressure derivative B0 , and muffintin radii RMT. System
a0 (Bohr)
B (GPa)
B0
RMT (Bohr)
ScC
8.86 8.84a, 8.92b, 8.89c 9.63 9.60a, 9.65b, 9.817d, 9.47(LDA)e, 9.61(GGA)e
155.722 153a 126.213 128a, 141.8(LDA)e, 124.3(GGA)e
3.009
RSc ¼ 2.15, RC ¼ 1.92
4.451
RY ¼2.35, RC ¼ 2.3
YC
a
Ref. [14]. Ref. [16] (experiment). Ref. [11]. d Ref. [17]. e Ref. [12]. b c
-1604.672 -1604.674 -1604.676 Energy (Ryd.)
position and interstitial region. Inside the muffin-tin region, the potential is a product of radial function and spherical harmonics. For the interstitial region, i.e., outside the muffin–tin sphere, the potential is expanded in plane waves. Muffin-tin spheres for metal atoms, carbon atoms in ScC, and YC are listed in Table 1. For every case the wave functions inside the MT spheres, which are expanded into spherical harmonics, are up to l ¼9 and RKmax ¼10. The number of k points used for the integration procedure is 7000, which reduces to 222 irreducible k points inside the Brillion zone including five high symmetry points W, L, G, X, and K. The calculations were performed with the equilibrium lattice constants that are determined from the plot of total energy against the unit cell volume by fitting to the Murnaghan equation of state [21]. For calculation of elastic constants, we have used the procedure of Brich [22].
-1604.678 -1604.680 -1604.682 -1604.684 -1604.686 155
160
165
175
180
185
190
195
volume (a.u.3)
3. Results and discussion 3.1. Structural properties
-6847.618 -6847.620 Energy (Ryd.)
In Fig. 1(a) and (b), we show the total energy curve as a function of unit cell volume for ScC and YC in rock salt structure. For the determination of static equilibrium properties, we use the Murnaghan equation of states [21]. In Table 1, we present our calculated values of the equilibrium lattice constants, its pressure derivatives, and muffin-tin radii, as well as the experimental and theoretical values. It is found that the equilibrium lattice constants of ScC and YC are slightly smaller about 0.67% and 0.21% than the experimental value [16]. The comparison between theory and experiment is complicated by the fact that the stiochiometry, i.e. the composition, is not perfect in the real crystal, for example, presence of defects, which is ignored in the present calculation, which is based on the ideal rock salt structure. For YC the earlier reported value for lattice constant is made by extrapolation by assuming a linear variation of a0 of Y with atomic fraction of C. The bulk modulus is a fundamental physical property of solids and it can be used a measure of the average bond strengths of atoms of crystals. All physical properties are related to the total energy. For example, the lattice constant that minimizes the total energy is the equilibrium lattice constant of a crystal. Any physical property related to the total energy can be determined if the total energy is calculated. Unfortunately, for the present crystals ScC and YC there are no theoretical or experimental results to compare with the present calculated value for elastic constants and their derivatives.
170
-6847.622 -6847.624 -6847.626 -6847.628 -6847.630 200
210
220
230
240
250
volume (a.u.3) Fig. 1. (a) Total energy as a function of lattice constant of ScC. (b) Total energy as a function of lattice constant of YC.
Table 2 Calculated elastic constants C11, C12, and C44. System
C11 (GPa)
C12 (GPa)
C44 (GPa)
ScC YC
312.650 293.928
77.258 69.365
63.502 52.161
3.2. Elastic properties In this study, to compute the elastic constants Cij, we use the ‘volume conserving’ technique [22]. For cubic crystal structures
the necessary condition for mechanical stability [23] is given by (C11–C12)40, (C11 þ2C12) 40, C11 40, C44 40. The findings are listed in Table 2.
J. Maibam et al. / Physica B 406 (2011) 4041–4045
One can observe that C11, C12, C44 values for ScC and YC satisfy (C11–C12)40, (C11 þ2C12)40, C11 40, C44 40. Thus, ScC and YC both in rock salt structures are mechanically stable carbides. The Zener anisotropy factor A, Poisson’s ratio u, shear modulus C0 , and Young’s modulus Y are calculated using the relations given by Mayer et al.[24]: A¼
2C44 C11 C12
u¼
1 Bð2=3ÞG 2 B þ ð1=3ÞG
Y¼
9GB G þ 3B
is the isotropic shear modulus, GV is Voigt’s shear modulus corresponding to the upper bound of G values, and GR is Reuss’s shear modulus corresponding to the lower bound of G values. They can be expressed as GV ¼
ð5Þ
5 4 3 þ ¼ GR ðC11 C12 Þ C44
ð2Þ
The Kleinman parameter z describes the relative positions of the cation and anion sub-lattices under volume conserving strain distortions, for which positions are fixed by symmetry. We use the following relations:
ð3Þ
z¼
GV þ GR 2
C11 C12 þ3C44 5
ð1Þ
where G¼
4043
ð4Þ
C11 þ8C12 7C11 þ 2C12
System
r (kg/m3)
A
u
Z
Y (GPa)
C0 (GPa)
ScC YC
1834.871 2532.628
0.540 0.612
0.277 0.284
0.397 0.437
208.201 163.323
117.696 85.286
Table 4 Calculated values of the isotropic shear modulus G, longitudinal sound velocity vl, transverse sound velocity vt, and average sound velocity vm, and Debye temperature yD. Material
G (GPa)
vl (m/s)
vt (m/s)
vm (m/s)
yD (K)
ScC YC
81.509 63.583
12004.07 9127.373
6665.001 5010.540
7423.517 5585.699
748.114 517.956
ð7Þ
for the Kleinman parameter and C0 ¼
Table 3 Calculated values of density r, Zener anisotropy factor A, Poisson’s ratio u, Kleinman parameter z, Young’s modulus Y, and shear modulus C0 of ScC and YC.
ð6Þ
C11 C12 2
ð8Þ
for the shear modulus [25]. The calculated values of density r, Zener anisotropy factor A, Poisson’s ratio u, Kleinman parameter z, Young’s modulus Y, and shear modulus C0 of ScC and YC are listed in Table 3. The Debye temperature yD is calculated from the elastic constants data using the average sound velocity vm, by the following common relation given in Ref. [26]: h 3n NA r 1=3 yD ¼ um ð9Þ k 4p M where h is Plank’s constant, k is the Boltzmanns constant, NA is Avogadro’s number , n is the number of atoms per formula unit, M is the molecular mass per formula unit, r (¼ M/V) is the density, and vm is given [27] as " !#ð1=3Þ 1 2 1 um ¼ þ ð10Þ 3 u3t u3l
Fig. 2. (a) Electronic energy band structure of ScC. (b) Electronic energy band structure of YC.
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J. Maibam et al. / Physica B 406 (2011) 4041–4045
Table 5 Characteristic band Fig. 2(a) and (b).
separations
of
the
band
structures
as
shown
in
Energy gaps (Ry)
ScC
YC
Eg
0.1931 0.0786 0.9131 0.7260
0.2632 0.1377 0.8161 0.6284
DEd Ep–Es E (G15)–E(L1)
where vl and vt are the longitudinal and the transverse elastic wave velocities, respectively, which are obtained from Navier’s equations [28] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3B þ 4G ul ¼ ð11Þ 3r
ut ¼
sffiffiffiffi G
ð12Þ
r
90 Density of states (states/Ryd.)
Density of states (states/Ryd.)
ScC (total DOS) Sc (total) C (total)
100 80 60 40 20 0
Sc (p) Sc (d-eg) Sc (d-t2g)
75 60 45 30 15 0
0.0
0.5 Energy (Ryd.)
1.0
1.5
0.0
0.5 Energy (Ryd.)
1.0
1.5
30
C (s) C (p)
Density of States (states/Ryd.)
Density of states (states/Ryd.)
35
25 20 15 10 5
60
40
20
0
0 0.0
0.5 Energy (Ryd.)
1.0
0.0
1.5
20
0.5 Energy (Ryd.)
1.0
1.5
50
Y (p) Y (d-eg) Y (d-t2g)
Density of States (states/Ryd.)
Density of States (states/Ryd.)
YC (total) Y (total) C (total)
80
15
10
5
C (s) C (p)
40 30 20 10 0
0 0.0
0.5 Energy (Ryd.)
1.0
1.5
0.0
0.5 1.0 Energy (Ryd.)
1.5
Fig. 3. (a) Density of states of ScC-total, Sc-total, and C-total. (b) Partial density of states of Sc (Sc-3p, Sc-3d(eg) and Sc-3d(t2g)) in ScC. (c) Partial density of states of C (C-2s and C-2p) in ScC. (d) Density of states of YC-total, Y-total, and C-total. (e). Partial density of states of Y (Y-3p, Y-3d(eg) and Y-3d(t2g)) in YC. (f) Partial density of states of C (C-2s and C-2p) in YC.
J. Maibam et al. / Physica B 406 (2011) 4041–4045
4045
YC. The highest contribution towards the total density of states near the Fermi level of both ScC and YC is C-2p.
The calculated values of the longitudinal, transverse, and average sound velocities along with the Debye temperature are given in Table 4. The Zener anisotropy factor (A) is a measure of the degree of anisotropy in the solid structures. The present calculated values of A for rock salt structure of ScC and YC are found to be 0.540 and 0.612, respectively. The small value of A implies that these materials have weak anisotropic character. It can also be concluded that YC is, relatively, more anisotropic than ScC. Poisson’s ratio u is the contraction relative to stretching. For rubber, gold, and titanium the values are 0.5, 0.42, and 0.34, respectively. It can be seen that the Poisson’s ratio for ScC and YC are 0.276 and 0.284, respectively. The Poisson’s ratio for ionic material is about u¼ 0.25 [29]. Therefore, one can say that the atomic bondings in these compounds are ionic in nature. The Kleinman parameter z of ScC and YC are 0.397 and 0.437, respectively. It describes the relative positions of the cation and anion sub-lattices under volume conserving strain distortions, for which positions are fixed by symmetry. The hardness of a material can also be predicted by isotropic shear modulus better than the bulk modulus. The calculated isotropic shear moduli G for ScC and YC in rock salt structure are 81.509 GPa and 63.583 GPa, respectively. In comparison with density, Young’s modulus and hardness of these materials, it can be seen that ScC is lighter, stiffer, and harder than YC. The effective cutoff frequency of materials, Debye temperature yD, for ScC and YC is 748.114 K and 517.956 K, respectively.
In this paper, we present a complete analysis of elastic, structural, and electronic properties of ScC and YC, using FP-LAPW. Both of the two compounds are found to be metallic.The elastic constant analysis shows that ScC is lighter, stiffer, and harder as compared with YC but YC is relatively more anisotropic and also both the compounds possess ionic character. The band structure of these two compounds reveals that YC is more ionic than ScC. The experimental data for rock salt structure of ScC and YC are quite limited to compare with. In our opinion, the reason for narrowing of C-2p band is due to the involvement of interaction with Y 4p state. Near the Fermi level, the highest contribution towards the total density of states for both ScC and YC is C-2p. We are not aware of any published data for elastic constants and its derived parameters of ScC and YC, so our theoretical calculation can be used to cover this lack of data.
3.3. Band structure and density of states
References
The electronic band structures of ScC and YC are shown in Fig. 2(a) and (b). It can be seen from both the band structures that three bands cross the Fermi level and therefore both structures show a metallic character. We have selected for comparison the band gap Eg between the C-s band and the valence band complex at L points. Characteristic band separations such as DEd ¼E(G12)– E(G250 ) give the zone-center metal d band, Ep–Es ¼E (G15)–E(G1) gives the Cp–Cs energy gap and E (G15)–E(L1) gives the width of C-p states, and are listed in Table 5. The calculated d band (DEd) for ScC is found to be smaller as compared to YC. The Eg value at X points for ScC is 0.1931 Ry. and that of YC is 0.2632 Ry. This indicates that the ionicity of YC is more than that of ScC. The width of C-2p band, E (G15)–E(L1), in the case of ScC is larger wider than that of YC . It is due to the interaction between Y-4p in addition to Y-4d states with C-p states (as shown in total and partial DOS of YC). The total DOS plots of ScC, Sc-total, and C total in ScC are shown in Fig. 3(a) Fig. 3(b) and (c) shows the partial density of states of Sc (Sc-3p, Sc-3d(eg), and Sc-3d(t2g)) and C (C-2s and C-2p) in ScC, respectively. Similarly the total and partial densities states of YC are shown in Fig. 3(d)(f). In ScC DOS plots, we can see that the peaks may be divided into three prominent peaks. The first peak from left is dominated by C-s states with a significant contribution from Sc-d(eg) state. The second peak is dominated by Sc-d(t2g) states with small contributions from Sc-d(eg) and C-p states. The Sc-p states have negligible contribution on the total density of states. But in the case of YC, the first peak is mainly contributed by C-s with significant contributions from Y-p and Y-d states. This is much different from the ScC case, where Sc-p contribution is absent. This may be the cause of pulling up of the lower band structure in the band structure of YC in comparison with ScC as shown in Fig. 2(a) and (b). We can also see that metal p state’s contribution is much large in the second peak of DOS in
[1] L. Toth, Transition Metal Carbides and Nitrides, Academic Press, New York, 1971. [2] E.K. Storms, The Refractory Carbides, Academic Press, New York, 1967. [3] R. Freer (Ed.), The Physics and Chemistry of Carbides, Nitrides and Borides, in: NATO ASI Series. E, vol. 185, Kluwer, Dordrecht, 1990. [4] S.T. Oyama (Ed.), The Chemistry of Transition Metal Carbides and Nitrides, Blacklie Academic, Professional, London, 1996. [5] C. Padauni, J. Phys.: Condens. Matter 20 (2008) 225014. [6] K. Schwarz, J. Phys. C 10 (1997) 195. [7] Xiaoju Guo, Bo Xu, Julong He, Dongli Yu, Zhongyuan Liu, Yongjun Tian, Appl. Phys. Lett. 93 (2008) 041904. [8] D.J. Chadi, Marvin L. Cohen, Phys. Rev. B 10 (1974) 496. [9] P. Blaha, J. Redinger, K. Schwarz, Phys. Rev. B. 31 (1985) 2316. [10] Jing Chen, L.L. Boyer, H. Krakauer, M.J. Mehl, Phy. Rev. B 37 (1988) 3295. [11] Y. Zhang, J. Li, L. Zhou, S. Xiang, Solid State Commun. 121 (2002) 411. [12] K.K. Korir, G.O. Amolo, N.W. Makau, D.P. Joubert, Diamond Relat. Materials 20 (2011) 157. [13] L. Ivanovskii, Russ. Chemi. Rev. 78 (4) (2009) 303. [14] E.I. Isaev, S.I. Simak, I.A. Abrikosov, R. Ahuja, Yu.Kh. Vekilov, M.I. Katsnelson, A.I. Lichtenstein, B. Johansson, J. Appl. Phys. 101 (2007) 123519. [15] J. H¨aglund, G. Grimvall, T. Jarlborg, A.F. Guillermet, Phys. Rev. B 43 (1991) 14400. [16] P.V. Villars, L.D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, American Society for Metals, Metal Park, OH, 1985. [17] A. Ferna´ndez Guillermet, J. H¨aglund, G. Grimvall, Phys. Rev. B 45 (1992) 11557. [18] A. Kalemos, A. Mavridis, J.F. Harrison, J. Phys. Chem. A 105 (2001) 755–759. [19] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Let. 77 (1996) 3865. [20] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k: An Augmented Plane Wave þLocal Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz, Techn. Universitat Wien, Austria, 2001. [21] F.D. Murnaghan, Proc. USA Natl. Acad. Sci 30 (1944) 244. [22] F. Brich, J. Appl. Phys. 9 (1938) 279–288. [23] D.C. Wallace, Thermodynamics of crystals, Wiley, new York, 1972 Chapter. 1. [24] Mayer, H. Anton, E. Bott, M. Methfessel, J. Sticht, P.C. Schmidt, Intermetallics 11 (2003) 23. [25] W.A. Harisson, Electronic Structures and Properties of solids, Dover, New York, 1989. [26] Johnston, G. Keeler, R. Rollins, S. Spicklemeire, Solid State Physics Simulations, The Consortium for Upper-Level Physics Software, Wiley, New York, 1996. [27] O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909. [28] E. Schreiber, O.L. Anderson, N. Soga, Elastic Constants and Their Measurements, McGraw-Hill, New York, 1973. [29] V.V. Bannikov, I.R. Shein, A.L. Ivanovskii, Phys. Status Solidi Rapid Res. Lett 3 (2007) 89.
4. Conclusions
Acknowledgements Financial support under the Fast Track Project (SR/FTP/PS-33/ 2006 dt.06.05.2008) from Department of Science and Technology (DST), Government of India, is gratefully acknowledged.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Electronic structure and elastic properties of scandium carbide and yttrium carbide: A first principles study Jameson Maibam a, B. Indrajit Sharma a,n, Ramendu Bhattacharjee a, R.K. Thapa b, R.K. Brojen Singh c a
Department of Physics, Assam University, Silchar, Assam 788011, India Department of Physics, Mizoram University, Tanhril, Aizawl 796009, Mizoram, India c Centre for Interdisciplinary Research in Basic Sciences, Jamia Millia Islamia, New Delhi 110025,India b
a r t i c l e i n f o
abstract
Article history: Received 1 April 2011 Received in revised form 19 July 2011 Accepted 20 July 2011 Available online 3 August 2011
We have studied the electronic, structural, and elastic properties of scandium carbide and yttrium carbide by means of accurate first principles total energy calculations using the full-potential linearized plane wave method (FP-LAPW). We have used the generalized gradient approximation (GGA) for the exchange and correlation potential. Volume optimization, energy band structure, and density of states (DOS) of the systems are presented. The second order elastic constants have been calculated and other related quantities such as the Zener anisotropy factor, Poisson’s ratio, Young’s modulus, Kleinman parameter, Debye temperature, and sound velocities have been determined. The band gap calculation shows that YC is relatively more ionic than ScC. & 2011 Elsevier B.V. All rights reserved.
Keywords: DFT Electronic structure DOS Elastic constants
1. Introduction Transition metalmono carbides are known as refractory compounds. They have, relatively, high brittleness, hardness, melting point. In addition, they also have interesting optical, electronic, catalytic, and magnetic properties [1–4]. Due to these interesting properties, they are involved in wide technological application areas such as making tools for machine buildings, chemical, and nuclear industries. The structural, elastic, and mechanical properties of transition metal monocarbides have been extensively studied before theoretically using various methods [5–13], whereas the literature of electronic, structural, and elastic properties of ScC and YC in rock salt structure is still scarce. These compounds show three types of bonding characteristics: ionic, covalent, and metallic [6]. The occurrence of ionic like structure in combination with covalent like hardness is very interesting. In addition, they show metallic conductivities comparable with those of pure transition metals. Zhang et al. [11] studied the chemical bonding of ScC. Recently Korir et al. calculated the e bulk properties of YC [12]. Isaev et al. calculated the phonon related properties of both ScC and YC in rock salt structure using first principles ultrasoft potential [14]. ¨ Haglund et al. [15] have studied scandium carbide using self-consistent linear muffin-tin orbital method. Experimental
n
Corresponding author. Tel.: þ91 03842 270843. E-mail address: indraoffi[email protected] (B. Indrajit Sharma).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.07.036
work on ScC and YC was done by Villars and Calvert [16]. Ferna´ndez Guillermet et al. [17] studied the cohesive energy of YC using the self-consistent linear muffin–tin orbital method. Kalemos et al. [18] studied the diatomic molecule of ScC. However, some of the physical properties of these two compounds have received less attention. To our knowledge the elastic properties, which are the important bulk properties of solids, have not been considered either experimentally or theoretically for ScC and YC. The primary purpose of this work is to provide some additional information to the existing data on the physical properties of ScC and YC by using ab initio total energy calculations. We try to focus on the band structure, density of states, elastic, structural, and mechanical properties will be discussed in brief.
2. Computational methods The calculations for ScC and YC in the rock salt structures were performed with FP-LAPW within the framework of the density functional theory with generalized gradient approximation — Perdew–Bruke–Ernzerhof 96 (GGA—PBE 96) for the exchange correlation potential [19]. We have employed the full-potential (linearized) augmented plane waves plus local orbital (FP-LAPWþ lo) method as implemented in the WEIN2k code. [20]. This method has been extensively tested and is among the most accurate methods for performing electronic structure calculations of crystals. In this method, the unit cell is divided into non-overlapping atomic spheres whose center is at atomic
4042
J. Maibam et al. / Physica B 406 (2011) 4041–4045
Table 1 Lattice constant a0, bulk modulus B, its pressure derivative B0 , and muffintin radii RMT. System
a0 (Bohr)
B (GPa)
B0
RMT (Bohr)
ScC
8.86 8.84a, 8.92b, 8.89c 9.63 9.60a, 9.65b, 9.817d, 9.47(LDA)e, 9.61(GGA)e
155.722 153a 126.213 128a, 141.8(LDA)e, 124.3(GGA)e
3.009
RSc ¼ 2.15, RC ¼ 1.92
4.451
RY ¼2.35, RC ¼ 2.3
YC
a
Ref. [14]. Ref. [16] (experiment). Ref. [11]. d Ref. [17]. e Ref. [12]. b c
-1604.672 -1604.674 -1604.676 Energy (Ryd.)
position and interstitial region. Inside the muffin-tin region, the potential is a product of radial function and spherical harmonics. For the interstitial region, i.e., outside the muffin–tin sphere, the potential is expanded in plane waves. Muffin-tin spheres for metal atoms, carbon atoms in ScC, and YC are listed in Table 1. For every case the wave functions inside the MT spheres, which are expanded into spherical harmonics, are up to l ¼9 and RKmax ¼10. The number of k points used for the integration procedure is 7000, which reduces to 222 irreducible k points inside the Brillion zone including five high symmetry points W, L, G, X, and K. The calculations were performed with the equilibrium lattice constants that are determined from the plot of total energy against the unit cell volume by fitting to the Murnaghan equation of state [21]. For calculation of elastic constants, we have used the procedure of Brich [22].
-1604.678 -1604.680 -1604.682 -1604.684 -1604.686 155
160
165
175
180
185
190
195
volume (a.u.3)
3. Results and discussion 3.1. Structural properties
-6847.618 -6847.620 Energy (Ryd.)
In Fig. 1(a) and (b), we show the total energy curve as a function of unit cell volume for ScC and YC in rock salt structure. For the determination of static equilibrium properties, we use the Murnaghan equation of states [21]. In Table 1, we present our calculated values of the equilibrium lattice constants, its pressure derivatives, and muffin-tin radii, as well as the experimental and theoretical values. It is found that the equilibrium lattice constants of ScC and YC are slightly smaller about 0.67% and 0.21% than the experimental value [16]. The comparison between theory and experiment is complicated by the fact that the stiochiometry, i.e. the composition, is not perfect in the real crystal, for example, presence of defects, which is ignored in the present calculation, which is based on the ideal rock salt structure. For YC the earlier reported value for lattice constant is made by extrapolation by assuming a linear variation of a0 of Y with atomic fraction of C. The bulk modulus is a fundamental physical property of solids and it can be used a measure of the average bond strengths of atoms of crystals. All physical properties are related to the total energy. For example, the lattice constant that minimizes the total energy is the equilibrium lattice constant of a crystal. Any physical property related to the total energy can be determined if the total energy is calculated. Unfortunately, for the present crystals ScC and YC there are no theoretical or experimental results to compare with the present calculated value for elastic constants and their derivatives.
170
-6847.622 -6847.624 -6847.626 -6847.628 -6847.630 200
210
220
230
240
250
volume (a.u.3) Fig. 1. (a) Total energy as a function of lattice constant of ScC. (b) Total energy as a function of lattice constant of YC.
Table 2 Calculated elastic constants C11, C12, and C44. System
C11 (GPa)
C12 (GPa)
C44 (GPa)
ScC YC
312.650 293.928
77.258 69.365
63.502 52.161
3.2. Elastic properties In this study, to compute the elastic constants Cij, we use the ‘volume conserving’ technique [22]. For cubic crystal structures
the necessary condition for mechanical stability [23] is given by (C11–C12)40, (C11 þ2C12) 40, C11 40, C44 40. The findings are listed in Table 2.
J. Maibam et al. / Physica B 406 (2011) 4041–4045
One can observe that C11, C12, C44 values for ScC and YC satisfy (C11–C12)40, (C11 þ2C12)40, C11 40, C44 40. Thus, ScC and YC both in rock salt structures are mechanically stable carbides. The Zener anisotropy factor A, Poisson’s ratio u, shear modulus C0 , and Young’s modulus Y are calculated using the relations given by Mayer et al.[24]: A¼
2C44 C11 C12
u¼
1 Bð2=3ÞG 2 B þ ð1=3ÞG
Y¼
9GB G þ 3B
is the isotropic shear modulus, GV is Voigt’s shear modulus corresponding to the upper bound of G values, and GR is Reuss’s shear modulus corresponding to the lower bound of G values. They can be expressed as GV ¼
ð5Þ
5 4 3 þ ¼ GR ðC11 C12 Þ C44
ð2Þ
The Kleinman parameter z describes the relative positions of the cation and anion sub-lattices under volume conserving strain distortions, for which positions are fixed by symmetry. We use the following relations:
ð3Þ
z¼
GV þ GR 2
C11 C12 þ3C44 5
ð1Þ
where G¼
4043
ð4Þ
C11 þ8C12 7C11 þ 2C12
System
r (kg/m3)
A
u
Z
Y (GPa)
C0 (GPa)
ScC YC
1834.871 2532.628
0.540 0.612
0.277 0.284
0.397 0.437
208.201 163.323
117.696 85.286
Table 4 Calculated values of the isotropic shear modulus G, longitudinal sound velocity vl, transverse sound velocity vt, and average sound velocity vm, and Debye temperature yD. Material
G (GPa)
vl (m/s)
vt (m/s)
vm (m/s)
yD (K)
ScC YC
81.509 63.583
12004.07 9127.373
6665.001 5010.540
7423.517 5585.699
748.114 517.956
ð7Þ
for the Kleinman parameter and C0 ¼
Table 3 Calculated values of density r, Zener anisotropy factor A, Poisson’s ratio u, Kleinman parameter z, Young’s modulus Y, and shear modulus C0 of ScC and YC.
ð6Þ
C11 C12 2
ð8Þ
for the shear modulus [25]. The calculated values of density r, Zener anisotropy factor A, Poisson’s ratio u, Kleinman parameter z, Young’s modulus Y, and shear modulus C0 of ScC and YC are listed in Table 3. The Debye temperature yD is calculated from the elastic constants data using the average sound velocity vm, by the following common relation given in Ref. [26]: h 3n NA r 1=3 yD ¼ um ð9Þ k 4p M where h is Plank’s constant, k is the Boltzmanns constant, NA is Avogadro’s number , n is the number of atoms per formula unit, M is the molecular mass per formula unit, r (¼ M/V) is the density, and vm is given [27] as " !#ð1=3Þ 1 2 1 um ¼ þ ð10Þ 3 u3t u3l
Fig. 2. (a) Electronic energy band structure of ScC. (b) Electronic energy band structure of YC.
4044
J. Maibam et al. / Physica B 406 (2011) 4041–4045
Table 5 Characteristic band Fig. 2(a) and (b).
separations
of
the
band
structures
as
shown
in
Energy gaps (Ry)
ScC
YC
Eg
0.1931 0.0786 0.9131 0.7260
0.2632 0.1377 0.8161 0.6284
DEd Ep–Es E (G15)–E(L1)
where vl and vt are the longitudinal and the transverse elastic wave velocities, respectively, which are obtained from Navier’s equations [28] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3B þ 4G ul ¼ ð11Þ 3r
ut ¼
sffiffiffiffi G
ð12Þ
r
90 Density of states (states/Ryd.)
Density of states (states/Ryd.)
ScC (total DOS) Sc (total) C (total)
100 80 60 40 20 0
Sc (p) Sc (d-eg) Sc (d-t2g)
75 60 45 30 15 0
0.0
0.5 Energy (Ryd.)
1.0
1.5
0.0
0.5 Energy (Ryd.)
1.0
1.5
30
C (s) C (p)
Density of States (states/Ryd.)
Density of states (states/Ryd.)
35
25 20 15 10 5
60
40
20
0
0 0.0
0.5 Energy (Ryd.)
1.0
0.0
1.5
20
0.5 Energy (Ryd.)
1.0
1.5
50
Y (p) Y (d-eg) Y (d-t2g)
Density of States (states/Ryd.)
Density of States (states/Ryd.)
YC (total) Y (total) C (total)
80
15
10
5
C (s) C (p)
40 30 20 10 0
0 0.0
0.5 Energy (Ryd.)
1.0
1.5
0.0
0.5 1.0 Energy (Ryd.)
1.5
Fig. 3. (a) Density of states of ScC-total, Sc-total, and C-total. (b) Partial density of states of Sc (Sc-3p, Sc-3d(eg) and Sc-3d(t2g)) in ScC. (c) Partial density of states of C (C-2s and C-2p) in ScC. (d) Density of states of YC-total, Y-total, and C-total. (e). Partial density of states of Y (Y-3p, Y-3d(eg) and Y-3d(t2g)) in YC. (f) Partial density of states of C (C-2s and C-2p) in YC.
J. Maibam et al. / Physica B 406 (2011) 4041–4045
4045
YC. The highest contribution towards the total density of states near the Fermi level of both ScC and YC is C-2p.
The calculated values of the longitudinal, transverse, and average sound velocities along with the Debye temperature are given in Table 4. The Zener anisotropy factor (A) is a measure of the degree of anisotropy in the solid structures. The present calculated values of A for rock salt structure of ScC and YC are found to be 0.540 and 0.612, respectively. The small value of A implies that these materials have weak anisotropic character. It can also be concluded that YC is, relatively, more anisotropic than ScC. Poisson’s ratio u is the contraction relative to stretching. For rubber, gold, and titanium the values are 0.5, 0.42, and 0.34, respectively. It can be seen that the Poisson’s ratio for ScC and YC are 0.276 and 0.284, respectively. The Poisson’s ratio for ionic material is about u¼ 0.25 [29]. Therefore, one can say that the atomic bondings in these compounds are ionic in nature. The Kleinman parameter z of ScC and YC are 0.397 and 0.437, respectively. It describes the relative positions of the cation and anion sub-lattices under volume conserving strain distortions, for which positions are fixed by symmetry. The hardness of a material can also be predicted by isotropic shear modulus better than the bulk modulus. The calculated isotropic shear moduli G for ScC and YC in rock salt structure are 81.509 GPa and 63.583 GPa, respectively. In comparison with density, Young’s modulus and hardness of these materials, it can be seen that ScC is lighter, stiffer, and harder than YC. The effective cutoff frequency of materials, Debye temperature yD, for ScC and YC is 748.114 K and 517.956 K, respectively.
In this paper, we present a complete analysis of elastic, structural, and electronic properties of ScC and YC, using FP-LAPW. Both of the two compounds are found to be metallic.The elastic constant analysis shows that ScC is lighter, stiffer, and harder as compared with YC but YC is relatively more anisotropic and also both the compounds possess ionic character. The band structure of these two compounds reveals that YC is more ionic than ScC. The experimental data for rock salt structure of ScC and YC are quite limited to compare with. In our opinion, the reason for narrowing of C-2p band is due to the involvement of interaction with Y 4p state. Near the Fermi level, the highest contribution towards the total density of states for both ScC and YC is C-2p. We are not aware of any published data for elastic constants and its derived parameters of ScC and YC, so our theoretical calculation can be used to cover this lack of data.
3.3. Band structure and density of states
References
The electronic band structures of ScC and YC are shown in Fig. 2(a) and (b). It can be seen from both the band structures that three bands cross the Fermi level and therefore both structures show a metallic character. We have selected for comparison the band gap Eg between the C-s band and the valence band complex at L points. Characteristic band separations such as DEd ¼E(G12)– E(G250 ) give the zone-center metal d band, Ep–Es ¼E (G15)–E(G1) gives the Cp–Cs energy gap and E (G15)–E(L1) gives the width of C-p states, and are listed in Table 5. The calculated d band (DEd) for ScC is found to be smaller as compared to YC. The Eg value at X points for ScC is 0.1931 Ry. and that of YC is 0.2632 Ry. This indicates that the ionicity of YC is more than that of ScC. The width of C-2p band, E (G15)–E(L1), in the case of ScC is larger wider than that of YC . It is due to the interaction between Y-4p in addition to Y-4d states with C-p states (as shown in total and partial DOS of YC). The total DOS plots of ScC, Sc-total, and C total in ScC are shown in Fig. 3(a) Fig. 3(b) and (c) shows the partial density of states of Sc (Sc-3p, Sc-3d(eg), and Sc-3d(t2g)) and C (C-2s and C-2p) in ScC, respectively. Similarly the total and partial densities states of YC are shown in Fig. 3(d)(f). In ScC DOS plots, we can see that the peaks may be divided into three prominent peaks. The first peak from left is dominated by C-s states with a significant contribution from Sc-d(eg) state. The second peak is dominated by Sc-d(t2g) states with small contributions from Sc-d(eg) and C-p states. The Sc-p states have negligible contribution on the total density of states. But in the case of YC, the first peak is mainly contributed by C-s with significant contributions from Y-p and Y-d states. This is much different from the ScC case, where Sc-p contribution is absent. This may be the cause of pulling up of the lower band structure in the band structure of YC in comparison with ScC as shown in Fig. 2(a) and (b). We can also see that metal p state’s contribution is much large in the second peak of DOS in
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4. Conclusions
Acknowledgements Financial support under the Fast Track Project (SR/FTP/PS-33/ 2006 dt.06.05.2008) from Department of Science and Technology (DST), Government of India, is gratefully acknowledged.