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Theoretical investigations on the structural, lattice dynamical and thermodynamic properties of XC (X=Si, Ge, Sn)
Solid State Communications 151 (2011) 1545–1549
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Theoretical investigations on the structural, lattice dynamical and thermodynamic properties of XC (X = Si, Ge, Sn) Xudong Zhang ∗ , Shanyu Quan, Caihong Ying, Zhijie Li School of Science, Shenyang University of Technology, Shenyang 110870, China
article
info
Article history: Received 5 April 2011 Received in revised form 12 July 2011 Accepted 22 July 2011 by J. R. Chelikowsky Available online 12 August 2011 Keywords: A. Semiconductors D. Lattice dynamics D. Thermodynamics properties
abstract First-principles calculations, which is based on the plane-wave pseudopotential approach to the density functional perturbation theory within the local density approximation, have been performed to investigate the structural, lattice dynamical, and thermodynamic properties of SiC, GeC, and SnC. The results of ground state parameters, phase transition pressure and phonon dispersion are compared and agree well with the experimental and theoretical data in the previous literature. The obtained phonon frequencies at the zone-center are analyzed. We also used the phonon density of states and quasiharmonic approximation to calculate and predict some thermodynamic properties such as entropy, heat capacity, internal energy, and phonon free energy of SiC, GeC, and SnC in B3 phase. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, IV–IV ground compounds such as SiC, GeC, and SnC have attracted both scientific and technological interest. Due to their good chemical stability, wide band gap, high hardness, high stiffness, high melting point and high thermal conductivity, they are considered to be a promising material for electronic and optical device. It is necessary to study their fundamental properties: lattice constants, bulk modulus, structural stability, electronic structure, and lattice dynamics. Yoshida et al. [1] used X-ray diffraction measurements in static diamond-anvil cell (DAC) and investigated that the pressureinduced phase transition of SiC from B3 to B1 structure occurred at 100 GPa or higher. Some theoretical group also studied the phase transition of SiC using different methods [2–8]. These theoretical studies presented the critical pressure of SiC from B3 to B1 structure to be around 60 GPa. Hao et al. [8] also investigated the critical pressure of GeC and SnC to be around 89 and 32 GPa, respectively, and they also calculated the electronic, elastic and optical properties under high pressure using first-principles calculations based on the density functional theory with the plane wave. Pandey et al. [9] studied the structural stability, electronic, and optical properties of GeC and SnC at ambient pressure using a linear combination of atomic orbital approach. They presented the band gap from indirect to direct from SiC to GeC to SnC. Khenata
∗
Corresponding author. Tel.: +86 24 25496502; fax: +86 24 25496502. E-mail address: [email protected] (X. Zhang).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.07.047
et al. [10,11] investigated the structural and electronic, and optical properties of SiC, GeC, and SnC using the full potential linear augmented plane wave (FP-LAPW) method. Sekkal and Zaoui [12] studied the structural and thermodynamic properties of GeC in B3 structure using an MD method. They predicted Debye temperature, thermal expansion, heat capacity, and Grüneisen parameter of GeC at zero pressure. Feldman et al. [13] experimentally investigated the dynamics property of SiC using Raman scattering measurements. They presented the phonon dispersion curves of SiC. Karch et al. [14,15] calculated phonon frequencies along the stacking direction of SiC using the self-consistent study, they found only small differences to Raman scattering measurements. Wang et al. [16] calculated the pressure dependence of Born effective charges, dielectric constant, and lattice dynamics in SiC using the linearized augmented plane wave (LAPW) method. A lot of studies have been carried out to calculate the structure and electronic properties of the B3 XC (X = Si, Ge, Sn), but many of their dynamical and thermodynamic properties are still not well established. To the best of our knowledge, there are no theoretical or experimental works exploring lattice dynamics properties of GeC, SnC and phonon contribution to the phonon free energy 1F , the internal energy 1E and the entropy S. Therefore, the aim of this work is to clarify the dynamical and thermodynamic properties of the XC (X = Si, Ge, Sn) in B3 phase. The paper is organized as follows. In Section 1, we introduced the research progress briefly. In Section 2, we describe the computational method used in this work. We present our results and discussions in Section 3. Finally, a summary of the work is given in Section 4.
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X. Zhang et al. / Solid State Communications 151 (2011) 1545–1549
Table 1 Calculated lattice constants (a in Å), bulk modulus (B0 in Mbar) and its first pressure derivative (B′0 ) are compared to other theoretical calculations and experiments. Lattice constant (Å)
B0 (Mbar)
B′0
SiC ZB This work 4.35 Experiment 4.36 [29] Other works, 4.30 [7], 4.363 [8], 4.40 [9], 4.34 [10], 4.40 [11], 4.30 [12], 4.28 [26], 4.33 [27],4.32 [31], 4.35 [32]
2.31 2.25 [30] 2.27 [7], 2.116 [8], 2.06 [9], 2.28 [10], 2.15 [11], 2.25 [12], 2.45 [26], 2.29 [27], 2.25 [31], 2.22 [32]
3.967 – 3.79 [8], 5.3 [9], 4.02 [10], 3.70 [11], 3.90 [12], 3.94 [26], 4.11 [31], 3.8 [32]
RS This work 4.04 Other works 4.03 [2], 3.97 [7], 4.04 [8], 4.03 [10], 4.13 [12]
2.56 2.666 [7], 2.435 [8], 2.30 [10], 1.62 [12]
4.39 4.64 [7], 5.5 [10], 4.36 [12]
GeC ZB This work 4.59 Other works 4.589 [8], 4.61 [9], 4.54 [10], 4.62 [11], 4.43 [12], 4.49 [26], 4.53 [27]
2.01 1.75 [8], 1.81 [9], 2.03 [10], 2.00 [11], 1.88 [12], 2.18 [26], 1.88 [27]
4.17 4.2 [9], 3.73 [10], 4.15 [11], 3.45 [12],
RS This work 4.32 Other works 4.359 [8], 4.33 [10], 4.25 [12]
1.81 1.74 [8], 2.00 [10], 1.33 [12]
4.52 3.29 [10], 4.42 [12]
SnC ZB This work 5.13 Other works 5.113 [8], 5.17 [9], 5 [10], 5.10 [11], 4.84 [27]
1.17 1.198 [8], 1.19 [9], 1.47 [10], 1.25 [11], 1.33 [28]
4.24 4.3 [9], 4.51 [10], 4.31 [11]
RS This work 4.72 Other works 4.829 [8], 4.70 [10]
1.59 1.346 [8], 1.66 [10]
4.39 4.7 [10]
2. Computational details The ABINIT code [17–19], which is based on the plane-wave pseudopotential approach in the framework of the functional theory (DFT) and the density functional perturbation theory (DFPT), has been applied to calculate the structural, dynamical and thermodynamic properties of XC (X = Si, Ge, Sn) in B3 phase. The exchange-correlation term has been considered within the local density approximation (LDA). The norm-conserving local density approximation pseudopotentials of C, Si, Ge, and Sn are generated in the scheme of Troullier and Martins [20], which come from the ABINIT web site. Troullier and Martins pseudopotentials describe the interaction between the nuclei and core electrons and valence electrons. Per primitive unit cell contains two atoms in the ab initio calculations. To obtain the ground-state parameters, the B3 structures of SiC, GeC and SnC are first optimized by using a Broyden–Fletcher–Goldfarb–Shanno (BFGS) procedure [21]. An 8 × 8 × 8 Monkhorst–Pack [22] grid is chosen for k-grid samplings in the Brillouin zone. The plane-wave kinetic energy cutoff of 40 hartree is set to guarantee the total energy errors within 0.0005 hartree in all the calculations. Convergence tests prove that the Brillouin zone sampling and the kinetic energy cutoff are reliable to guarantee excellent convergence. Phonon frequencies are calculated using the DFPT linear-response method [23,24]. The thermodynamic properties are calculated using the phonon density of states and quasiharmonic approximation. 3. Results and discussions 3.1. Structural properties The equilibrium lattice parameters a, bulk modulus B0 , and their pressure derivatives of bulk modulus B′0 were calculated by means of fitting Murnaghan’s [25] equation of state (EOS) to total energies versus volume. We summarized our results and the experimental
Table 2 Calculated phase transition pressure from B3 to B1 phase. Material
Reference
Pt (GPa)
SiC
Present (LDA) Experimental [1] Theory (LDA) [2] Theory (LDA) [3] Theory (GGA) [4] Theory (LDA) [5] Theory (LDA) [6] Theory (GGA) [7] Theory (LDA) [8]
62 100 66.5 65.3 63 92 63 75.4 65.1
GeC
Present (LDA) Theory (LDA) [8]
87 89.4
SnC
Present (LDA) Theory (LDA) [8]
33 32.5
and other theoretical values in the B1 and B3 phase in Table 1. It can be clearly seen that the calculated structural parameters are in excellent agreement with experimental values and other theoretical ones. The slight underestimation in the equilibrium lattice parameter is a common feature with LDA calculations. The calculated bulk modulus decreases from SiC to SnC, suggesting that the compressibility increases from SiC to SnC. We note that the anion atoms are the same in the three compounds, the cation atoms size are different. The different size of the cation atoms could be the responsible for the lattice constant increasing from SiC to SnC. We calculated the phase transition pressure at T = 0 K using the usual method of equal enthalpies, i.e., the enthalpy as the function of pressure, H = E + PV . The enthalpy as a function of pressure for SiC, GeC, and SnC in B3 and B1 phases is illustrated in Fig. 1. Our results of phase transition pressures from B3 to B1 structure of SiC, GeC, and SnC are tabulated in Table 2. The calculated phase transition pressure of SiC is similar to
X. Zhang et al. / Solid State Communications 151 (2011) 1545–1549
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Fig. 2. Calculated phonon dispersion and phonon density of states (PDOS) of SiC.
Fig. 3. Calculated phonon dispersion and phonon density of states (PDOS) of GeC.
3.2. Dynamical properties The calculated phonon dispersion spectra and the total density of phonon states (PDOS) along the principal symmetry direction of the Brillouin zone (BZ) are displayed in Figs. 2–4. The calculated phonon frequencies at the high-symmetry points Γ , X , and L for SiC, GeC, and SnC in B3 phase are listed in Table 3 and are compared to experimental data and previous theoretical results. Our calculated LO and TO frequencies of SiC at the high-symmetry points Γ , X , and L agree well with the experimental data of Feldman et al. [13] as well as with other ab initio values of Karch et al. [15] and Wanget al. [16]. This illustrated that our computational method is correct. We can reasonably use this method to predict the dynamic properties of GeC and SnC, though no experimental results are available up until now. The prominent characteristics of the phonon dispersion of SiC, GeC, and SnC are the following. Fig. 1. Calculated enthalpy as a function of pressure for SiC, GeC, and SnC in B3 and B1 phases.
the previous theoretical results. The predicted transition pressure differs considerably from the experimental data. This may be due to the fact that the experimental transition pressure was obtained for the forward transition, where an excess pressure beyond the equilibrium value appears and is included in the measurements [26]. We found that the phase transition pressure of GeC is higher than that of SiC. The reason may be that the existing of the d-core states in Ge (nonexistent in Si) results in an extra repulsion, which leads to a higher transition pressure [8].
(a) The form of the phonon dispersion curves of SiC, GeC, and SnC are similar to each other. (b) The transverse optical (TO) phonon modes of XC become flat along the X → Γ → L directions, and a strong peak in the PDOS. The longitudinal optical (LO) phonon modes of XC along the L → X → W symmetry direction, the LO branch shows flat dispersion for SiC, GeC and SnC, respectively, which induces a weak peak in the PDOS. (c) The top of the bands are located at the zone center Γ point for three compounds. The LO and TO branches are clearly separated in three compounds. The LO–TO splitting at the
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Table 3 Calculated phonon frequencies (in cm−1 ) of SiC, GeC, and SnC, compared to experimental and theoretical results.
ΓTO
ΓLO
XTA
XLA
XTO
XLO
LTA
LLA
LTO
LLO
SiC This work Experiment [13] Karch et al. [15] Wang et al. [16]
783 796 783 774
953 972 956 945
364 373 366 361
623 640 629 622
749 761 755 741
811 829 829 807
260 266 261 257
608 610 610 601
755 766 766 747
832 838 838 817
GeC This work
626
748
214
348
617
697
162
331
612
705
SnC This work
456
558
134
216
450
503
109
199
440
516
Fig. 4. Calculated phonon dispersion and phonon density of states (PDOS) of SnC.
Fig. 5. Calculated internal energy 1E of SiC, GeC, and SnC.
zone-center Γ point is 170, 122, and 102 cm−1 for SiC, GeC, and SnC, respectively. These splitting are relatively bigger than other semiconductors. It can be seen from Figs. 2–4 that the mass difference between anion C and cations Si, Ge, Sn significantly affects the shapes of the dispersion curves and the corresponding PDOS. So it is clear that the TO branch and LO branch have relatively flat dispersion. You can see that a clear gap is formed between acoustic and optical branches for three compounds. No imaginary phonon frequency is observed in the whole BZ as suggested by the phonon dispersion curves and PDOS in Figs. 2–4, indicating the dynamical stability of the B3 phase for these compounds. 3.3. Thermodynamic properties After obtaining the phonon spectrum over the entire BZ, we calculated and obtained the phonon free energy 1F , the internal energy 1E, the entropy S, and the constant-volume specific heat Cv, at zero pressure of XC using the phonon density of states and quasiharmonic approximation [33]. Expressions used for phonon free energy 1F , the internal energy 1E, the entropy S, and the constant-volume specific heat Cv are given as follows [34]:
1F = 3nNkB T
ωmax
∫
ln 2 sinh 0
1E = 3nN
h¯
ωmax
∫
2
ω coth
0
ωmax [
∫ S = 3nNkB
h¯ ω 2kB T
0
h¯ ω
2kB T
h¯ ω
2kB T
coth
h¯ ω 2kB T
g (ω) dω
(1)
g (ω) dω
− ln 2 sinh
(2) h¯ ω 2kB T
× g (ω) dω 2 ∫ ωmax h¯ ω h¯ ω 2 CV = 3nNkB csc h g (ω) dω 0
2kB T
2kB T
] (3) (4)
Fig. 6. Calculated phonon free energy 1F of SiC, GeC, and SnC.
where kB is the Boltzmann constant, n is the number of atoms per unit cell, N is the number of unit cells, ωmax is the largest phonon frequency, ω is the phonon frequency, ω and g (ω) is the normalized phonon density of states with 0 max g (ω) dω = 1. Figs. 5–8 display the calculated 1E, 1F , S, and Cv within the scope of temperature from 0 to 2000 K at zero pressure, respectively. In Fig. 5, it can be seen that when the temperature increases, the internal energies approach each other gradually and increase almost linearly with temperature. At the same temperature, the value of internal energies decreases with increasing atom mass (Si → Ge → Sn). Fig. 6 shows the variations of phonon free energy with temperature for the XC compounds. The phonon free energy curves are similar and move down with increasing temperature. At the same temperature, the value of phonon free energy also increases with decreasing atom mass (Sn → Ge → Si). Fig. 7
X. Zhang et al. / Solid State Communications 151 (2011) 1545–1549
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4. Conclusions In this work, we have calculated the ground state parameters of XC in the B3 and B1 structures and phase transition pressure from B3 to B1 structures by using the density functional theory within LDA. The ground state parameters are in good agreement with available theoretical and experimental studies. We have calculated the phonon frequency using the DFPT linear-response method and described the main features of the phonon dispersion curves and compared with the other literature. We have also calculated and presented the thermodynamic quantities such as the internal energy, free energy, specific heat and entropy. Acknowledgment
Fig. 7. Calculated constant-volume specific heat Cv of SiC, GeC, and SnC.
This work has been supported by Education Department Foundation of Liaoning Province of China (201064145 and 2010397). References
Fig. 8. Calculated entropy S of SiC, GeC, and SnC.
displays the temperature dependence of the heat capacities of the XC compounds. We note that the heat capacities follow the Debye model and approach the Petit and Dulong limit (C v ∼ 50 J/mol c K−1 ) at high temperatures for SiC, GeC, and SnC. At low temperature, the curvature of Cv increases with increasing atom mass (Si → Ge → Sn). The shape of the calculated Cv curve seems to be typical for III–V previous results [15]. This trend supports our predictive results. Fig. 8 shows the variations of entropy with temperature for SiC, GeC, and SnC. Obviously, the entropy exhibits a similar tendency for all XC compounds. We can see that the entropy increases when the size of the atom mass increases (Si → Ge → Sn). In an ensemble where the sample volume and temperature are independent variables, the relevant potential is the free energy, F = E − TS. So, the entropy is present as a T ∗ S product to allow comparison with the internal energy and free energy. Above 400 K, the value of T ∗ S increases faster than the internal energy as the temperature increases, resulting in the value of the free energy for each compound, changed from positive to negative in different temperatures. From Figs. 5–8, one can see the variations of E, F , S, and Cv for the XC compounds as a function of temperature and the atomic mass of X. Because of the mass difference between Si, Ge, and Sn, the bond length of Si–C, Ge–C and Sn–C are different. As we know, the bond length strongly affects some physical properties, such as thermodynamics, elastic constant, hardness and so on [35]. So the trends of E, F , S, and Cv are normal.
[1] M. Yoshida, A. Onodera, M. Ueno, K. Takemura, O. Shimomura, Phys. Rev. B 48 (1993) 10587. [2] K.J. Chang, M.L. Cohen, Phys. Rev. B 35 (1987) 8196. [3] K. Karch, F. Bechstedt, P. Pavone, D. Strauch, Phys. Rev. B 53 (1996) 13400. [4] M.S. Miao, M. Prikhodko, W.R.L. Lambrecht, Phys. Rev. B 66 (2002) 064107. [5] M. Catti, Phys. Rev. Lett. 87 (2001) 035504. [6] M.S Miao, W.R.L. Lambrecht, Phys. Rev. B 68 (2003) 092103. [7] Y.P. Lu, H.D. Wei, J. Zhu, X.D. Yang, Physica B 403 (2008) 3543. [8] A.M. Hao, X.C Yang, X.M Wang, Y Zhu, X Liu, R.P. Liu, J. Appl. Phys. 108 (2010) 063531. [9] R. Pandey, M. Rérat, C. Darrigan, M. Causà, J. Appl. Phys. 88 (2000) 6462. [10] R. Khenata, H. Baltache, M. Sahnoun, M. Driz, M. Rérat, B. Abbar, Physica B 336 (2003) 321. [11] M. Sahnoun, R. Khenata, H. Baltache, M. Rérat, M. Driz, B. Bouhafs, B. Abbar, Physica B 355 (2005) 392. [12] W. Sekkal, A. Zaoui, New J. Phys. 4 (2002) 9. [13] D.W. Feldman, J.H. Parker Jr., W.J. Choyke, L. Patrick, Phys. Rev. 173 (1968) 787. [14] K. Karch, F. Bechstedt, P. Pavone, D. Strauch, J. Phys.: Condens. Matter 8 (1996) 2945. [15] K. Karch, P. Pavone, W. Windl, O. Schütt, D. Strauch, Phys. Rev. B 50 (1994) 17054. [16] C.Z. Wang, R.C. Yu, Henry Krakauer, Phys. Rev. B 53 (1996) 5430. [17] X. Gonze, J.M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.Y. Raty, D.C. Allan, Comput. Mater. Sci. 25 (3) (2002) 478. [18] X. Gonze, G.M. Rignanese, M. Verstraete, J.M. Beuken, Y. Pouillon, R. Caracas, F. Jollet, M. Torrent, G. Zerah, M. Mikami, Ph. Ghosez, M. Veithen, J.Y. Raty, V. Olevano, F. Bruneval, L. Reining, R. Godby, G. Onida, D.R. Hamann, D.C. Allan, Z. Kristallogr, 220 (5) 2005, 558. [19] The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated, and other contributors (URL http://www.abinit.org) [20] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [21] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipesin Fortran 77: The Art of Scientific Computing, second ed. Cambridge University Press, Cambridge, 1992, 418. [22] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [23] X. Gonze, C. Lee, Phys. Rev. B 55 (1997) 10355. [24] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [25] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1947) 244. [26] A.F. Goncharov, E.V. Yokovenko, S.M. Stishov, JEPT Lett. 52 (1990) 491. [27] H. Rûcker, M. Methfessel, Phys. Rev. B 52 (1995) 11059. [28] O.F. Sankey, A.A. Demkov, W.T. Petuskey, P.F. McMillan, Model. Simul. Mater. Sci. Eng. 1 (1993) 741. [29] P.E. Van Camp, V.E. Doren, J.T. Devreese, Phys. Rev.B 34 (1986) 1314. [30] W.R.L. Lambrecht, B. Segall, M. Metfessel, M. Van Schilfgaade, Phys. Rev. B 44 (1991) 3685. [31] T. Meijie, S. Yip, Phys. Rev. B 52 (1995) 15150. [32] K. Karch, P. Pavone, W. Windl, D. Srauch, F. Bechstedt, Int. J. Quantum Chem. 56 (1995) 801. [33] The QHA code has been written and developed by E. I. Isaev, https://qeforge.org [34] C. Lee, X. Gonze, Phys. Rev. B 51 (1995) 8610. [35] X.H. Zhang, X.G. Luo, J.C. Han, J.P. Li, W.B. Han, Comput. Mater. Sci. 44 (2008) 411.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Theoretical investigations on the structural, lattice dynamical and thermodynamic properties of XC (X = Si, Ge, Sn) Xudong Zhang ∗ , Shanyu Quan, Caihong Ying, Zhijie Li School of Science, Shenyang University of Technology, Shenyang 110870, China
article
info
Article history: Received 5 April 2011 Received in revised form 12 July 2011 Accepted 22 July 2011 by J. R. Chelikowsky Available online 12 August 2011 Keywords: A. Semiconductors D. Lattice dynamics D. Thermodynamics properties
abstract First-principles calculations, which is based on the plane-wave pseudopotential approach to the density functional perturbation theory within the local density approximation, have been performed to investigate the structural, lattice dynamical, and thermodynamic properties of SiC, GeC, and SnC. The results of ground state parameters, phase transition pressure and phonon dispersion are compared and agree well with the experimental and theoretical data in the previous literature. The obtained phonon frequencies at the zone-center are analyzed. We also used the phonon density of states and quasiharmonic approximation to calculate and predict some thermodynamic properties such as entropy, heat capacity, internal energy, and phonon free energy of SiC, GeC, and SnC in B3 phase. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, IV–IV ground compounds such as SiC, GeC, and SnC have attracted both scientific and technological interest. Due to their good chemical stability, wide band gap, high hardness, high stiffness, high melting point and high thermal conductivity, they are considered to be a promising material for electronic and optical device. It is necessary to study their fundamental properties: lattice constants, bulk modulus, structural stability, electronic structure, and lattice dynamics. Yoshida et al. [1] used X-ray diffraction measurements in static diamond-anvil cell (DAC) and investigated that the pressureinduced phase transition of SiC from B3 to B1 structure occurred at 100 GPa or higher. Some theoretical group also studied the phase transition of SiC using different methods [2–8]. These theoretical studies presented the critical pressure of SiC from B3 to B1 structure to be around 60 GPa. Hao et al. [8] also investigated the critical pressure of GeC and SnC to be around 89 and 32 GPa, respectively, and they also calculated the electronic, elastic and optical properties under high pressure using first-principles calculations based on the density functional theory with the plane wave. Pandey et al. [9] studied the structural stability, electronic, and optical properties of GeC and SnC at ambient pressure using a linear combination of atomic orbital approach. They presented the band gap from indirect to direct from SiC to GeC to SnC. Khenata
∗
Corresponding author. Tel.: +86 24 25496502; fax: +86 24 25496502. E-mail address: [email protected] (X. Zhang).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.07.047
et al. [10,11] investigated the structural and electronic, and optical properties of SiC, GeC, and SnC using the full potential linear augmented plane wave (FP-LAPW) method. Sekkal and Zaoui [12] studied the structural and thermodynamic properties of GeC in B3 structure using an MD method. They predicted Debye temperature, thermal expansion, heat capacity, and Grüneisen parameter of GeC at zero pressure. Feldman et al. [13] experimentally investigated the dynamics property of SiC using Raman scattering measurements. They presented the phonon dispersion curves of SiC. Karch et al. [14,15] calculated phonon frequencies along the stacking direction of SiC using the self-consistent study, they found only small differences to Raman scattering measurements. Wang et al. [16] calculated the pressure dependence of Born effective charges, dielectric constant, and lattice dynamics in SiC using the linearized augmented plane wave (LAPW) method. A lot of studies have been carried out to calculate the structure and electronic properties of the B3 XC (X = Si, Ge, Sn), but many of their dynamical and thermodynamic properties are still not well established. To the best of our knowledge, there are no theoretical or experimental works exploring lattice dynamics properties of GeC, SnC and phonon contribution to the phonon free energy 1F , the internal energy 1E and the entropy S. Therefore, the aim of this work is to clarify the dynamical and thermodynamic properties of the XC (X = Si, Ge, Sn) in B3 phase. The paper is organized as follows. In Section 1, we introduced the research progress briefly. In Section 2, we describe the computational method used in this work. We present our results and discussions in Section 3. Finally, a summary of the work is given in Section 4.
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X. Zhang et al. / Solid State Communications 151 (2011) 1545–1549
Table 1 Calculated lattice constants (a in Å), bulk modulus (B0 in Mbar) and its first pressure derivative (B′0 ) are compared to other theoretical calculations and experiments. Lattice constant (Å)
B0 (Mbar)
B′0
SiC ZB This work 4.35 Experiment 4.36 [29] Other works, 4.30 [7], 4.363 [8], 4.40 [9], 4.34 [10], 4.40 [11], 4.30 [12], 4.28 [26], 4.33 [27],4.32 [31], 4.35 [32]
2.31 2.25 [30] 2.27 [7], 2.116 [8], 2.06 [9], 2.28 [10], 2.15 [11], 2.25 [12], 2.45 [26], 2.29 [27], 2.25 [31], 2.22 [32]
3.967 – 3.79 [8], 5.3 [9], 4.02 [10], 3.70 [11], 3.90 [12], 3.94 [26], 4.11 [31], 3.8 [32]
RS This work 4.04 Other works 4.03 [2], 3.97 [7], 4.04 [8], 4.03 [10], 4.13 [12]
2.56 2.666 [7], 2.435 [8], 2.30 [10], 1.62 [12]
4.39 4.64 [7], 5.5 [10], 4.36 [12]
GeC ZB This work 4.59 Other works 4.589 [8], 4.61 [9], 4.54 [10], 4.62 [11], 4.43 [12], 4.49 [26], 4.53 [27]
2.01 1.75 [8], 1.81 [9], 2.03 [10], 2.00 [11], 1.88 [12], 2.18 [26], 1.88 [27]
4.17 4.2 [9], 3.73 [10], 4.15 [11], 3.45 [12],
RS This work 4.32 Other works 4.359 [8], 4.33 [10], 4.25 [12]
1.81 1.74 [8], 2.00 [10], 1.33 [12]
4.52 3.29 [10], 4.42 [12]
SnC ZB This work 5.13 Other works 5.113 [8], 5.17 [9], 5 [10], 5.10 [11], 4.84 [27]
1.17 1.198 [8], 1.19 [9], 1.47 [10], 1.25 [11], 1.33 [28]
4.24 4.3 [9], 4.51 [10], 4.31 [11]
RS This work 4.72 Other works 4.829 [8], 4.70 [10]
1.59 1.346 [8], 1.66 [10]
4.39 4.7 [10]
2. Computational details The ABINIT code [17–19], which is based on the plane-wave pseudopotential approach in the framework of the functional theory (DFT) and the density functional perturbation theory (DFPT), has been applied to calculate the structural, dynamical and thermodynamic properties of XC (X = Si, Ge, Sn) in B3 phase. The exchange-correlation term has been considered within the local density approximation (LDA). The norm-conserving local density approximation pseudopotentials of C, Si, Ge, and Sn are generated in the scheme of Troullier and Martins [20], which come from the ABINIT web site. Troullier and Martins pseudopotentials describe the interaction between the nuclei and core electrons and valence electrons. Per primitive unit cell contains two atoms in the ab initio calculations. To obtain the ground-state parameters, the B3 structures of SiC, GeC and SnC are first optimized by using a Broyden–Fletcher–Goldfarb–Shanno (BFGS) procedure [21]. An 8 × 8 × 8 Monkhorst–Pack [22] grid is chosen for k-grid samplings in the Brillouin zone. The plane-wave kinetic energy cutoff of 40 hartree is set to guarantee the total energy errors within 0.0005 hartree in all the calculations. Convergence tests prove that the Brillouin zone sampling and the kinetic energy cutoff are reliable to guarantee excellent convergence. Phonon frequencies are calculated using the DFPT linear-response method [23,24]. The thermodynamic properties are calculated using the phonon density of states and quasiharmonic approximation. 3. Results and discussions 3.1. Structural properties The equilibrium lattice parameters a, bulk modulus B0 , and their pressure derivatives of bulk modulus B′0 were calculated by means of fitting Murnaghan’s [25] equation of state (EOS) to total energies versus volume. We summarized our results and the experimental
Table 2 Calculated phase transition pressure from B3 to B1 phase. Material
Reference
Pt (GPa)
SiC
Present (LDA) Experimental [1] Theory (LDA) [2] Theory (LDA) [3] Theory (GGA) [4] Theory (LDA) [5] Theory (LDA) [6] Theory (GGA) [7] Theory (LDA) [8]
62 100 66.5 65.3 63 92 63 75.4 65.1
GeC
Present (LDA) Theory (LDA) [8]
87 89.4
SnC
Present (LDA) Theory (LDA) [8]
33 32.5
and other theoretical values in the B1 and B3 phase in Table 1. It can be clearly seen that the calculated structural parameters are in excellent agreement with experimental values and other theoretical ones. The slight underestimation in the equilibrium lattice parameter is a common feature with LDA calculations. The calculated bulk modulus decreases from SiC to SnC, suggesting that the compressibility increases from SiC to SnC. We note that the anion atoms are the same in the three compounds, the cation atoms size are different. The different size of the cation atoms could be the responsible for the lattice constant increasing from SiC to SnC. We calculated the phase transition pressure at T = 0 K using the usual method of equal enthalpies, i.e., the enthalpy as the function of pressure, H = E + PV . The enthalpy as a function of pressure for SiC, GeC, and SnC in B3 and B1 phases is illustrated in Fig. 1. Our results of phase transition pressures from B3 to B1 structure of SiC, GeC, and SnC are tabulated in Table 2. The calculated phase transition pressure of SiC is similar to
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Fig. 2. Calculated phonon dispersion and phonon density of states (PDOS) of SiC.
Fig. 3. Calculated phonon dispersion and phonon density of states (PDOS) of GeC.
3.2. Dynamical properties The calculated phonon dispersion spectra and the total density of phonon states (PDOS) along the principal symmetry direction of the Brillouin zone (BZ) are displayed in Figs. 2–4. The calculated phonon frequencies at the high-symmetry points Γ , X , and L for SiC, GeC, and SnC in B3 phase are listed in Table 3 and are compared to experimental data and previous theoretical results. Our calculated LO and TO frequencies of SiC at the high-symmetry points Γ , X , and L agree well with the experimental data of Feldman et al. [13] as well as with other ab initio values of Karch et al. [15] and Wanget al. [16]. This illustrated that our computational method is correct. We can reasonably use this method to predict the dynamic properties of GeC and SnC, though no experimental results are available up until now. The prominent characteristics of the phonon dispersion of SiC, GeC, and SnC are the following. Fig. 1. Calculated enthalpy as a function of pressure for SiC, GeC, and SnC in B3 and B1 phases.
the previous theoretical results. The predicted transition pressure differs considerably from the experimental data. This may be due to the fact that the experimental transition pressure was obtained for the forward transition, where an excess pressure beyond the equilibrium value appears and is included in the measurements [26]. We found that the phase transition pressure of GeC is higher than that of SiC. The reason may be that the existing of the d-core states in Ge (nonexistent in Si) results in an extra repulsion, which leads to a higher transition pressure [8].
(a) The form of the phonon dispersion curves of SiC, GeC, and SnC are similar to each other. (b) The transverse optical (TO) phonon modes of XC become flat along the X → Γ → L directions, and a strong peak in the PDOS. The longitudinal optical (LO) phonon modes of XC along the L → X → W symmetry direction, the LO branch shows flat dispersion for SiC, GeC and SnC, respectively, which induces a weak peak in the PDOS. (c) The top of the bands are located at the zone center Γ point for three compounds. The LO and TO branches are clearly separated in three compounds. The LO–TO splitting at the
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Table 3 Calculated phonon frequencies (in cm−1 ) of SiC, GeC, and SnC, compared to experimental and theoretical results.
ΓTO
ΓLO
XTA
XLA
XTO
XLO
LTA
LLA
LTO
LLO
SiC This work Experiment [13] Karch et al. [15] Wang et al. [16]
783 796 783 774
953 972 956 945
364 373 366 361
623 640 629 622
749 761 755 741
811 829 829 807
260 266 261 257
608 610 610 601
755 766 766 747
832 838 838 817
GeC This work
626
748
214
348
617
697
162
331
612
705
SnC This work
456
558
134
216
450
503
109
199
440
516
Fig. 4. Calculated phonon dispersion and phonon density of states (PDOS) of SnC.
Fig. 5. Calculated internal energy 1E of SiC, GeC, and SnC.
zone-center Γ point is 170, 122, and 102 cm−1 for SiC, GeC, and SnC, respectively. These splitting are relatively bigger than other semiconductors. It can be seen from Figs. 2–4 that the mass difference between anion C and cations Si, Ge, Sn significantly affects the shapes of the dispersion curves and the corresponding PDOS. So it is clear that the TO branch and LO branch have relatively flat dispersion. You can see that a clear gap is formed between acoustic and optical branches for three compounds. No imaginary phonon frequency is observed in the whole BZ as suggested by the phonon dispersion curves and PDOS in Figs. 2–4, indicating the dynamical stability of the B3 phase for these compounds. 3.3. Thermodynamic properties After obtaining the phonon spectrum over the entire BZ, we calculated and obtained the phonon free energy 1F , the internal energy 1E, the entropy S, and the constant-volume specific heat Cv, at zero pressure of XC using the phonon density of states and quasiharmonic approximation [33]. Expressions used for phonon free energy 1F , the internal energy 1E, the entropy S, and the constant-volume specific heat Cv are given as follows [34]:
1F = 3nNkB T
ωmax
∫
ln 2 sinh 0
1E = 3nN
h¯
ωmax
∫
2
ω coth
0
ωmax [
∫ S = 3nNkB
h¯ ω 2kB T
0
h¯ ω
2kB T
h¯ ω
2kB T
coth
h¯ ω 2kB T
g (ω) dω
(1)
g (ω) dω
− ln 2 sinh
(2) h¯ ω 2kB T
× g (ω) dω 2 ∫ ωmax h¯ ω h¯ ω 2 CV = 3nNkB csc h g (ω) dω 0
2kB T
2kB T
] (3) (4)
Fig. 6. Calculated phonon free energy 1F of SiC, GeC, and SnC.
where kB is the Boltzmann constant, n is the number of atoms per unit cell, N is the number of unit cells, ωmax is the largest phonon frequency, ω is the phonon frequency, ω and g (ω) is the normalized phonon density of states with 0 max g (ω) dω = 1. Figs. 5–8 display the calculated 1E, 1F , S, and Cv within the scope of temperature from 0 to 2000 K at zero pressure, respectively. In Fig. 5, it can be seen that when the temperature increases, the internal energies approach each other gradually and increase almost linearly with temperature. At the same temperature, the value of internal energies decreases with increasing atom mass (Si → Ge → Sn). Fig. 6 shows the variations of phonon free energy with temperature for the XC compounds. The phonon free energy curves are similar and move down with increasing temperature. At the same temperature, the value of phonon free energy also increases with decreasing atom mass (Sn → Ge → Si). Fig. 7
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4. Conclusions In this work, we have calculated the ground state parameters of XC in the B3 and B1 structures and phase transition pressure from B3 to B1 structures by using the density functional theory within LDA. The ground state parameters are in good agreement with available theoretical and experimental studies. We have calculated the phonon frequency using the DFPT linear-response method and described the main features of the phonon dispersion curves and compared with the other literature. We have also calculated and presented the thermodynamic quantities such as the internal energy, free energy, specific heat and entropy. Acknowledgment
Fig. 7. Calculated constant-volume specific heat Cv of SiC, GeC, and SnC.
This work has been supported by Education Department Foundation of Liaoning Province of China (201064145 and 2010397). References
Fig. 8. Calculated entropy S of SiC, GeC, and SnC.
displays the temperature dependence of the heat capacities of the XC compounds. We note that the heat capacities follow the Debye model and approach the Petit and Dulong limit (C v ∼ 50 J/mol c K−1 ) at high temperatures for SiC, GeC, and SnC. At low temperature, the curvature of Cv increases with increasing atom mass (Si → Ge → Sn). The shape of the calculated Cv curve seems to be typical for III–V previous results [15]. This trend supports our predictive results. Fig. 8 shows the variations of entropy with temperature for SiC, GeC, and SnC. Obviously, the entropy exhibits a similar tendency for all XC compounds. We can see that the entropy increases when the size of the atom mass increases (Si → Ge → Sn). In an ensemble where the sample volume and temperature are independent variables, the relevant potential is the free energy, F = E − TS. So, the entropy is present as a T ∗ S product to allow comparison with the internal energy and free energy. Above 400 K, the value of T ∗ S increases faster than the internal energy as the temperature increases, resulting in the value of the free energy for each compound, changed from positive to negative in different temperatures. From Figs. 5–8, one can see the variations of E, F , S, and Cv for the XC compounds as a function of temperature and the atomic mass of X. Because of the mass difference between Si, Ge, and Sn, the bond length of Si–C, Ge–C and Sn–C are different. As we know, the bond length strongly affects some physical properties, such as thermodynamics, elastic constant, hardness and so on [35]. So the trends of E, F , S, and Cv are normal.
[1] M. Yoshida, A. Onodera, M. Ueno, K. Takemura, O. Shimomura, Phys. Rev. B 48 (1993) 10587. [2] K.J. Chang, M.L. Cohen, Phys. Rev. B 35 (1987) 8196. [3] K. Karch, F. Bechstedt, P. Pavone, D. Strauch, Phys. Rev. B 53 (1996) 13400. [4] M.S. Miao, M. Prikhodko, W.R.L. Lambrecht, Phys. Rev. B 66 (2002) 064107. [5] M. Catti, Phys. Rev. Lett. 87 (2001) 035504. [6] M.S Miao, W.R.L. Lambrecht, Phys. Rev. B 68 (2003) 092103. [7] Y.P. Lu, H.D. Wei, J. Zhu, X.D. Yang, Physica B 403 (2008) 3543. [8] A.M. Hao, X.C Yang, X.M Wang, Y Zhu, X Liu, R.P. Liu, J. Appl. Phys. 108 (2010) 063531. [9] R. Pandey, M. Rérat, C. Darrigan, M. Causà, J. Appl. Phys. 88 (2000) 6462. [10] R. Khenata, H. Baltache, M. Sahnoun, M. Driz, M. Rérat, B. Abbar, Physica B 336 (2003) 321. [11] M. Sahnoun, R. Khenata, H. Baltache, M. Rérat, M. Driz, B. Bouhafs, B. Abbar, Physica B 355 (2005) 392. [12] W. Sekkal, A. Zaoui, New J. Phys. 4 (2002) 9. [13] D.W. Feldman, J.H. Parker Jr., W.J. Choyke, L. Patrick, Phys. Rev. 173 (1968) 787. [14] K. Karch, F. Bechstedt, P. Pavone, D. Strauch, J. Phys.: Condens. Matter 8 (1996) 2945. [15] K. Karch, P. Pavone, W. Windl, O. Schütt, D. Strauch, Phys. Rev. B 50 (1994) 17054. [16] C.Z. Wang, R.C. Yu, Henry Krakauer, Phys. Rev. B 53 (1996) 5430. [17] X. Gonze, J.M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.Y. Raty, D.C. Allan, Comput. Mater. Sci. 25 (3) (2002) 478. [18] X. Gonze, G.M. Rignanese, M. Verstraete, J.M. Beuken, Y. Pouillon, R. Caracas, F. Jollet, M. Torrent, G. Zerah, M. Mikami, Ph. Ghosez, M. Veithen, J.Y. Raty, V. Olevano, F. Bruneval, L. Reining, R. Godby, G. Onida, D.R. Hamann, D.C. Allan, Z. Kristallogr, 220 (5) 2005, 558. [19] The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated, and other contributors (URL http://www.abinit.org) [20] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [21] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipesin Fortran 77: The Art of Scientific Computing, second ed. Cambridge University Press, Cambridge, 1992, 418. [22] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [23] X. Gonze, C. Lee, Phys. Rev. B 55 (1997) 10355. [24] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [25] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1947) 244. [26] A.F. Goncharov, E.V. Yokovenko, S.M. Stishov, JEPT Lett. 52 (1990) 491. [27] H. Rûcker, M. Methfessel, Phys. Rev. B 52 (1995) 11059. [28] O.F. Sankey, A.A. Demkov, W.T. Petuskey, P.F. McMillan, Model. Simul. Mater. Sci. Eng. 1 (1993) 741. [29] P.E. Van Camp, V.E. Doren, J.T. Devreese, Phys. Rev.B 34 (1986) 1314. [30] W.R.L. Lambrecht, B. Segall, M. Metfessel, M. Van Schilfgaade, Phys. Rev. B 44 (1991) 3685. [31] T. Meijie, S. Yip, Phys. Rev. B 52 (1995) 15150. [32] K. Karch, P. Pavone, W. Windl, D. Srauch, F. Bechstedt, Int. J. Quantum Chem. 56 (1995) 801. [33] The QHA code has been written and developed by E. I. Isaev, https://qeforge.org [34] C. Lee, X. Gonze, Phys. Rev. B 51 (1995) 8610. [35] X.H. Zhang, X.G. Luo, J.C. Han, J.P. Li, W.B. Han, Comput. Mater. Sci. 44 (2008) 411.