Mechanical stability and superconductivity of PbO-type phase of thorium monocarbide at high pressure

Computational Materials Science 136 (2017) 238–242

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Mechanical stability and superconductivity of PbO-type phase of thorium monocarbide at high pressure Yan Yan a,b, Fangxu Wang a, Lili Wang a, Rui Chen a, Jian Lv b,⇑ a b

School of Sciences, Changchun University, Changchun 130022, China State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, China

a r t i c l e

i n f o

Article history: Received 28 March 2017 Received in revised form 5 May 2017 Accepted 6 May 2017

Keywords: High pressure Thorium carbides Metallization Superconductor

a b s t r a c t As a potential nuclear fuel for the next generation of nuclear reactors, the structural and physical properties of thorium monocarbide (ThC) under high pressure have attracted a wide range of research interest. Here, the mechanical, electronic, dynamical, and superconducting properties of PbO-type ThC have been systematically investigated through first-principles calculations, which established the thermodynamic and mechanical stability of this phase within the pressure range of 60–140 GPa. Moreover, it is found that pressure significantly affected the electronic states near the Fermi level and superconductivity. The superconducting transition temperature Tc first increases, and then decreases with increasing pressure. The maximal value of Tc reaches 4.64 K at 80 GPa. This phenomenon is a representation of the reconstruction of the Fermi surface and phonon hardening under compression. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Research on actinide carbides began in the 1950s; recently, there has been renewed interest in actinide carbides, such as thorium carbides, for the purpose of potential nuclear fuels for fast nuclear reactors with closed fuel cycles [1–4]. Compared with existing uranium-based nuclear fuels, thorium-based fuels have many excellent physical properties such as, for example, lower thermal expansion coefficients, higher corrosion resistivity, higher melting points, and larger thermal conductivity [5–9]. Owing to the prospective application of thorium carbides as outstanding fuel materials, it is very important to understand their structures and physical properties in order to model the fuel behavior at high pressures/temperatures. Despite the abundant research on actinide compounds, such as Th [10–12], ThN [13], ThC2 [14], and ThO2 [15], however, for ThC, we only know that thorium monocarbide has a B1 structure at ambient conditions. The structural, thermodynamic, electronic, and elastic properties of this structure have been studied by many research groups [16–26]. However, the highpressure properties of ThC, to the authors’ knowledge, are not well known from experimental or theoretical investigations. Meanwhile, due to the discovery of the High-Tc of
203 K of H-S system at high pressure [27], the field of conventional superconductivity

⇑ Corresponding author. E-mail address: [email protected] (J. Lv). http://dx.doi.org/10.1016/j.commatsci.2017.05.008 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.

under pressure get particular attention recently. This significant finding is originally achieved by a theoretical prediction [28]. Motivated by these observations, in this paper, we systematically study the electronic, mechanical, kinetic, and superconductivity properties of a new high-pressure PbO-type phase [29,30] of thorium monocarbide (space group P4/nmm) within the pressure range of 60–140 GPa.

2. Computational methods Ab initio calculations were carried out using the density functional theory (DFT) [31,32] as implemented in the QuantumESPRESSO package [33]. The generalized gradient approximation (GGA) exchange correlation function of Perdew–Burke–Ernzerhof (PBE) was utilized [34]. The thorium 6s26p66d27s2 and carbon 2s22p2 were treated as valence electrons. A cut-off energy of 100 Ry and 10  10  12 Monkhorst–Pack [35] k-point meshes for the electronic Brillouin zone (BZ) sampling were employed, achieving a higher level of accuracy of total energy, within 0.01 mRy/atom. The phonon calculations were based on the density function linear-response method [36–42]. A higher level of accuracy with 26  26  32 k-grid and Gaussians smearing 0.03 Ry was used to achieve phonon modes converged within 0.001 THz. Meanwhile, a 3  3  4 q-points grid in the first BZ was adopted as the interpolation of the force constants about the phonon band calculation. The superconducting calculations employed a dense 15  15  18 MP confirmed k-point grid

Y. Yan et al. / Computational Materials Science 136 (2017) 238–242

239

convergence with Gaussians smearing 0.02 Ry, which essentially achieved the zero-width limit of electron–phonon interacting calculations [43–47]. However, the study on the accurate elastic constants and modulus are investigated within the Voigt–Reuss–Hill approximation [48] as actualized in the Vienna ab initio simulation package (VASP) [49]. 3. Results & discussion The PbO-type phase of of thorium monocarbide is a new highpressure structure and exhibits metallic. Herein, the cell volume and atomic positions for the PbO-type ThC are full optimized at 58.3 GPa. The calculated equilibrium lattice constants a, b and c of this phase are in good agreement with experimental values within 2% (Table 1), supporting the choices of our pseudopotential and functional. The elastic constants show the nature of the stability and mechanical properties of materials by analyzing the behavior of objects under the elastic deformations. Six independent elastic constants, C11, C12, C13, C33, C44, and C66 have been given in a tetragonal structure. In this work, the corresponding elastic constants were acquired by the slopes of the acoustic phonon modes within the long-wave limit, as seen in Fig. 1a. It shows the pressure dependence of the elastic constants of the PbO-type tetragonal structure of ThC. We can see that the independent modes C11, C12, C33, C44, and C66 are monotonously enhanced with increasing pressure. Only the elastic constant C13 first decreases and then increases with increasing pressure. However, all the elastic constants are always positive in our pressure range. This indicates that the structural phase transition does not occur. The bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio v were studied using the Voigt–Reuss–Hill averaging approximation. The physical quantities E and v are related to B and G. Fig. 1b shows the quantities B, G, and E under different pressures, which also increase linearly with pressure. Furthermore, our elastic constants are in accordance with the mechanical stability criteria [50] for the tetragonal symmetry under compression, namely.

C 11 > 0; C 33 > 0C 44 > 0; C 66 > 0; ðC 11  C 12 Þ > 0; ðC 11 þ C 33  2C 13 Þ > 0; ð2C 11 þ C 33 þ 2C 12 þ 4C 13 Þ > 0 This further supports our calculated results. Fig. 2 shows the calculated band structures at 60, 100, and 140 GPa. As seen in the figure, we observed two electronic bands crossing the Fermi level (EF) along the M–C–Z and X–C directions at 60 and 100 GPa. When the pressure increases to 140 GPa, the four electron bands pass along the same line through the Fermi level. Moreover, it is found that electronic band just touching EF at the Z point crosses it with pressure. The electronic bands near the M and X points also pass through EF under compression. Meanwhile, owing to the observable electronic states near the Fermi surface, ThC exhibits obvious metallic behavior. Since the parabolic conduction and valence bands are less dispersed near the Fermi level, it will support superconducting behavior. Furthermore, the electron states of the Fermi surface can play a role in the ordinary transport properties of the metallic system. In the simple condition, the Fermi surface would be a sphere, the radius of which is shown by the Fermi wave vector. Obviously, the Fermi surface of this phase deviates from the spherical shape.

Table 1 Lattice constants of PbO-type ThC at 58.3 GPa obtained from here and experiment. Space group

Method

a (Å)

c (Å)

Ref.

P4/nmm

PWSCF-PBE Expt. data

4.128 4.205

3.063 3.099

This work [30]

Fig. 1. (a) Pressure dependence of the elastic constants in PbO-type ThC. (b) Bulk modulus B, shear modulus G, and Young’s modulus E for ThC as functions of pressure.

In particular, such a case has been encountered in the multivalent states for metals. When depicted in an extended BZ, the Fermi surface associated with the lowest conduction band appears as some changes near the Z, M, and, X points with pressure. Supposing that electrons very close to the Fermi energy level make a main contribution to superconductivity, in many instances, the shape of the Fermi surface with respect to the BZ becomes a guide to studying the electrical properties of the metal. The Fermi surface of new phase of ThC (Fig. 3(a) and (b)) coincide with two energy bands crossing the Fermi level (Fig. 2(a) and (b)). As shown in Fig. 3, the center of all of the Fermi surfaces is located in the C point under different compressions, while the Fermi surface undergoes reconstruction under compression. The Fermi surface of PbO-type ThC (Fig. 3(c)) corresponds with four bands crossing the Fermi level (Fig. 2(c)). In general, if the shape of the Fermi surface changes abruptly, the Van Hove singularity appears in the state, enhanced Tc. However, as will be discussed later, the Tc value of ThC decreases with the increase in pressure (Table 2). Next, we still study the dynamic stability of the PbO-type phase of ThC under compression. No imaginary phonon frequencies are observed in the whole BZ, indicating the dynamic stability of this structure in a wide pressure range. Fig. 4 shows the calculated phonon dispersion relation and the state of the projected phonon density in the tetragonal structure of ThC at 140 GPa. It is indicated that this PbO-type tetragonal phase is an approved structure for thorium monocarbide at high pressures, even to 140 GPa. Owing to the fact that the atomic mass of the thorium atom is much higher than that of the carbon atom, the phonon dispersion relations obviously divide into two portions with a wide gap: one portion is the range between 0 and 8.3 THz, where the contribution of thorium atoms is dominant; the other portion is in the domain of 16.5–25 THz, where the phonon vibration frequencies are mainly from carbon atoms. Subsequently, we further calculate the phonon spectrum, Eliashberg spectral function a2F(x), and the electron–phonon coupling (EPC) constant k for PbO-type ThC to probe its possible superconductivity in Fig. 4. The lack of any imaginary phonon modes indicates lattice dynamical stability of the structure in our pressure range, as shown in Fig. 4(a). A wide gap separates the phonon dispersive curve into two parts. The lower vibration modes are mainly related to the motions of Th atoms; However, the higher frequency modes are related to C atoms. The common contribution (nearly 71.2 and 28.8%, respectively) gives a k value of 0.418 with the pressure up to 140 GPa. Moreover, it is found that there are some

240

Y. Yan et al. / Computational Materials Science 136 (2017) 238–242

Fig. 2. Electronic band structures of PbO-type structure along the main symmetry directions in the first BZ at 60, 100, and 100 GPa. The red solid line represents the Fermi level. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Fermi surface at (a) 60 GPa, (b) 100 GPa, and (c) 200 GPa.

Table 2 Calculated phonon frequency logarithmic average (xlog(K)), electron–phonon coupling parameter (k), critical temperature Tc (l* = 0.1 or 0.13), and electronic density of states at the Fermi level (N(EF) (states/spin/Ry/Unit Cell)) under different pressures. P (GPa)

60 80 100 140

hxlogi

235.591 283.551 309.399 292.454

k

0.549 0.541 0.509 0.418

interesting phonon characteristics under high compressions. For this tetragonal structure, all phonon modes harden up to 140 GPa, and only the transverse acoustic (TA) modes soften near the A and X points with very small frequencies of 1.37 and

Tc (K)

N (EF)

µ* = 0.1

µ* = 0.13

4.048 4.640 4.032 1.564

2.669 3.024 2.481 0.739

10.904 12.208 12.409 10.759

2.15 THz, respectively, resulting in a decreased k, and a lowering value of Tc is also found. The correlation between the superconducting transition temperature Tc and phonon hardening is very clear here. Therefore, we conclude that the high-pressure super-

Y. Yan et al. / Computational Materials Science 136 (2017) 238–242

241

Fig. 4. (a) Comparison of phonon band structure for PbO-type ThC at 80 and 140 GPa, respectively. The individual EPC parameter kqj of each mode (q, j) caused by EPC is illustrated by the size of circle. (b) Eliashberg phonon spectral function a2F(x) and integral EPC parameter k(x) of ThC are compared at different pressures.

conductivity is significantly associated with the vibrations of the low-frequency Th branches, rather than the vibrations from the high-frequency C atoms. In order to further illustrate the contributions related to different phonon modes, all of the calculated EPC parameters kqj (solid circle) of individual phonon modes within the framework of the linear response theory are plotted against the frequency in Fig. 4 (a). As the pressure increases, the phonon frequency increases significantly, and all phonon frequencies provide some contribution to the overall EPC constant (k). However, it is worth noting that the main contribution to EPC comes from the lower phonon frequency. It is shown that the paired electrons of ThC in the tetragonal phase are mediated by the interaction of low-frequency and high-frequency phonons. For the entire BZ along the highsymmetry line, the strong EPC is clear, which shows that the structure of the EPC is almost isotropic; this comes from the typical three-dimensional structure. Therefore, the linear response theory can accurately determine the thermodynamic behavior and the transition temperature [51]. Therefore, it is shown that the contribution from the acoustic phonon modes is greater than optical phonons at all q vectors, implying an important contribution to electron–phonon interaction, particularly from the lowest acoustic phonon modes over the whole BZ. By comparison with the lower phonon modes, the high-frequency optical phonon branches obtain very small k, therefore making a minor contribution to EPC. The conclusions above are also found in the graphic of the Eliashberg spectral function a2F(x). The magnitude of the EPC parameter k reflects all the phonon vibration branches over the entire frequency scope, and decreases with the increase in pressure (Fig. 4(b)). With increasing pressure, either the low- or highfrequency contributions are weakened, owing to k derived from a2F(x)/x. The superconducting transition temperature Tc was calculated using the Allen–Dynes modified McMillan formula [52], and a typical value of the Coulomb pseudopotential l⁄ is generally given as from 0.1 to 0.2. The obtained xlog, EPC parameter k, DOS at the Fermi level (N(EF)), and Tc under different compressions are shown in Table 2. The calculated k is 0.549 with N(EF) 10.904 electroneV1cell1 at 60 GPa, indicating that ThC shows electron– phonon coupling interaction. The estimated xlog is 235.591 K. Using values of l⁄ of 0.1 and 0.13, tetragonal ThC has calculated Tc values of 4.048 and 2.669 K, respectively. Moreover, the change relation of Tc was also investigated with increasing pressure. Under

pressure up to 140 GPa, the EPC parameter k monotonically decreases to 0.418; xlog showed a first increase from 235.591 to 309.399 K, and then decrease to 292.454 K at 140 GPa. Using these results, Tc reaches up to the maximal value of 4.640 and 3.024 K for l⁄ = 0.1 and 0.13, respectively, at 80 GPa. However, with increasing the pressure to 140 GPa, the value of Tc decreases to 1.564 and 0.739 K for l⁄ = 0.1 and 0.13, respectively. The xlog, N(EF), and Tc are first increased, and then decreased with the increase of pressure (Table 2). This is mostly due to phonon modes hardening under compression, while the whole phonon frequencies contributions to EPC are weakened. 4. Conclusion The mechanical, electronic, dynamical, and superconducting properties of PbO-type tetragonal ThC are widely studied by ab initio calculations in our pressure range of 60–140 GPa. The results show that this phase can exist stably up to 140 GPa. Meanwhile, we found that pressure is important to the extent that the effect of pressure on the electronic states near the Fermi level and superconductivity is very large. The value of Tc first increases, and then decreases with increasing pressure, and the maximal value reaches to 4.64 K at 80 GPa. The main contributions to superconductivity come from the low-frequency Th phonon modes and the reduction of Tc is caused by the hardening of phonons. Further experimental study of the superconductivity of high-pressure ThC is necessary to verify our predictions. Acknowledgements This work has been supported by the National Natural Science Foundation of China, No. 11404035 and 11504007, Jilin Provincial Natural Science Foundation of China, 20150101004JC and the Industrial Technology Research Project (2014Y135). References [1] R. Konings, Comprehensive Nuclear Materials, First edition., Elsevier Ltd., Amsterdam, 2012. [2] J.M. Rudy Konings, Thierry Wiss, Christine Guéneau, Nuclear Fuels. In The Chemistry of the Actinide and Transactinide Elements, in: N.M. Edelstein, J. Fuger, L.R. Morss (Eds.), Springer, Berlin, Germany, 2010, vol. 6, pp 3665–3811. [3] K. Maeda, S. Sasaki, M. Kato, Y. Kihara, J. Nucl. Mater. 389 (2009) 78. [4] T. Abram, S. Ion, Energy Policy 36 (2008) 4323.

242

Y. Yan et al. / Computational Materials Science 136 (2017) 238–242

[5] J.-M. Fournier, R. Troc, in: A.J. Freeman, G.H. Lander (Eds.), Handbook on the Physics and Chemistry of the Actinides, vol. 2, p. 29. [6] A.J. Freeman, G.H. Lander (Eds.), Handbook on the Physics and Chemistry of the Actinides, North-Holland, Amsterdam, 1985. [7] R. Benz, A. Naoumidis, Thourium compounds with nitrogen, Gmelin Handbook of Inorganic Chemistry, eighth ed., C3, Springer, Berlin, 1987, Thorium supplement. [8] H. Kleykamp, Thorium Carbides, Gmelin Handbook of Inorganic and Organometallic Chemistry, eighth ed., C6, Springer, Berlin, 1992, Thorium supplement. [9] A.H.M. Evensen, R. Catherall, P. Drumm, P. Van Duppen, O.C. Jonsson, E. Kugler, J. Lettry, O. Tengblad, V. Tikhonov, H.L. Roun, Nucl. Instrum. Meth. Phys. Res. B 126 (1997) 160. [10] N. Richard, S. Bernard, F. Jollet, M. Torrent, Phys. Rev. B 66 (2002) 235112. [11] J. Bouchet, F. Jollet, G. Zérah, Phys. Rev. B 74 (2006) 134304. [12] C.-E. Hu, Z.-Y. Zeng, L. Zhang, X.-R. Chen, L.-C. Cai, Solid State Commun. 150 (2010) 393. [13] Lu. Yong, Da.-Fang. Li, Bao.-Tian. Wang, Rong.-Wu. Li, Ping. Zhang, J. Nucl. Mater. 408 (2011) 136. [14] P. Pogány, A. Kovács, Zoltán Varga, F. Matthias Bickelhaupt, Rudy J.M. Konings, J. Phys. Chem. A 116 (2012) 747. [15] Y. Lu, Y. Yang, P. Zhang, J. Phys.: Condens. Matter 24 (2012) 225801. [16] L. Gerward, J. Staun Olsen, U. Benedict, H. Luo, J. Less-Common Met. 161 (1) (1990) L11. [17] I.R. Shein, K.I. Shein, A.L. Ivanovskii, J. Nucl. Mater. 353 (1) (2006) 19. [18] I.R. Shein, K.I. Shein, A.L. Ivanovskii, Tech. Phys. Lett. 33 (2) (2007) 128. [19] I.R. Shein, A.L. Ivanovski, Phys. Solid State 52 (10) (2010) 2039. [20] I.S. Lim, G.E. Scuseria, Chem. Phys. Lett. 460 (1) (2008) 137. [21] S. Aydin, A. Tatar, Y.O. Ciftci, J. Nucl. Mater. 429 (1) (2012) 55. [22] D. Pérez Daroca, S. Jaroszewicz, A.M. Llois, H.O. Mosca, J. Nucl. Mater. 437 (1–3) (2013) 135. [23] L. Gerward, J. Staun Olsen, U. Benedict, J.-P. Itié, J.C. Spirlet, J. Appl. Crystallogr. 19 (5) (1986) 308. [24] J. Staun Olsen, S. Steenstrup, L. Gerward, U. Benedict, J.-P. Itíe, Physica B+C 139 (140) (1986) 308–310. [25] B.D. Sahoo, K.D. Joshi, S.C. Gupta, AIP Conference Proceedings 1536 (2013) 937. [26] V. Kanchana, G. Vaitheeswaran, A. Svane, S. Heathman, L. Gerward, J. Staun Olsen, Acta Crystallogr. B 70 (3) (2014) 459.

[27] A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontov, S.I. Shylin, Nature 525 (2015) 73. [28] Y.W. Li, J. Hao, H.Y. Liu, Y.L. Li, Y.M. Ma, J. Chem. Phys. 140 (2014) 174712. [29] Y.L. Guo, W.J. Qiu, X.Z. Ke, P. Huai, C. Cheng, H. Han, C.L. Ren, Z.Y. Zhu, Phys. Lett. A 379 (2015) 1607. [30] C. Yu, J. Lin, P. Huai, Y.L. Guo, X.Z. Ke, X.H. Yu, K. Yang, N.N. Li, W.G. Yang, B.X. Sun, R.B. Xie, H.J. Xu, Sci. Rep. 7 (2017) 96. [31] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) 864. [32] W. Kohn, L.J. Sham, Phys. Rev. 140 (1964) A1133. [33] S. Baroni, A. DalCorso, S. deGironcoli, P. Giannozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G. Chiarotti, P. Focher, A. Pasquarello, K. Laasonen, A. Trave, R. Car, N. Marzari, A. Kokalj, J. Phys. Condens. Mat.21 (2009) 395502. http://www. quantum-espresso.org. [34] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [35] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [36] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [37] P. Giannozzi, S. De Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B. 43 (1991) 7231. [38] Y. Li, Y. Ma, T. Cui, Y. Yan, G. Zou, Appl. Phys. Lett. 92 (2008) 101907. [39] Y. Yan, J.Y. Zhang, T. Cui, Y. Li, Y.M. Ma, J. Gong, Z.G. Zong, G.T. Zou, Eur. Phys. J. B 61 (2008) 397. [40] Z.W. Li, Y. Xu, G.Y. Gao, T. Cui, Y.M. Ma, Phys. Rev. B 79 (2009) 193201. [41] L.J. Zhang, Y.C. Wang, X.X. Zhang, Y.M. Ma, Phys. Rev. B 82 (2010) 14108. [42] Y. Ma, M. Eremets, A.R. Oganov, Y. Xie, I. Trojan, S. Medvedev, A.O. Lyakhov, M. Valle, V. Prakapenka, Nature 458 (2009) 182. [43] Y.W. Li, J.X. Yue, X. Liu, Y.M. Ma, T. Cui, G.T. Zou, Phys. Lett. A 374 (2010) 1880. [44] Z.H. Zhu, X.H. Yan, Z.H. Guo, Y.R. Yang, Phys. Lett. A 372 (2008) 1671. [45] S.Y. Li, H.Y. Liu, Ivan I. Naumov, S. Meng, Y.W. Li, John S. Tse, B. Yang, Russell J. Hemley, Phys. Rev. B 93 (2016) 104509. [46] Y. Xie, A.R. Oganov, Y.M. Ma, Phys. Rev. Lett. 104 (2010) 177005. [47] G. Gao, A.R. Oganov, P. Li, Z. Li, H. Wang, T. Cui, Y. Ma, A. Bergara, A.O. Lyakhov, T. Iitaka, Proc. Natl. Acad. Sci. 107 (2008) 1317. [48] R. Hill, Proc. Phys. Soc. London 65 (1952) 350. [49] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [50] J.J. Wang, F.Y. Meng, X.Q. Ma, M.X. Xu, L.Q. Chen, J. Appl. Phys. 108 (2010) 034107. [51] W. Pickett, Nature 418 (2002) 733. [52] P.B. Allen, R.C. Dynes, Phys. Rev. B 12 (1975) 905.