The structural, elastic and electronic properties of ...

International Journal of Modern Physics C Vol. 26, No. 1 (2015) 1550003 (9 pages) # .c World Scienti¯c Publishing Company DOI: 10.1142/S0129183115500035

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The structural, elastic and electronic properties of A2 C2 (A¼Li, Na, K, Rb and Cs): First-principles calculations

Chun-Lin Tang, Guang-Lin Sun and Yan-Ling Li* School of Physics and Electronic Engineering Jiangsu Normal University Xuzhou, 221116, P. R. China *[email protected] Received 7 January 2014 Accepted 25 April 2014 Published 28 May 2014 The structural, elastic and electronic properties of A2 C2 (A¼Li, Na, K, Rb and Cs) at zero temperature were investigated by ¯rst-principles total energy calculations. The optimized equilibrium structural parameters agree well with available experimental values. Elastic constants, bulk modulus, Young's modulus and Poissons ratio were given. All the structures studied are stable mechanically and all stable A2 C2 studied has strong compressibility, which originates from weak Coulomb repulsion between metal atoms and carbon atoms. The electronic structure calculations show that binary alkali metal carbides studied here are insulators. Keywords: Alkali-metal carbide; elastic constant; electronic property; density functional theory. PACS Nos.: 61.50.f, 62.20.x, 71.15.Mb, 71.20.b, 77.74.Bw, 78.20.Ga.

1. Introduction Binary alkali metal carbides are of intense interest for researchers for a long time. Not only do they have remarkable structural and physical properties, but they have important technical applications in industry. For example, manufacturing graphite intercalation compounds (GICs) can be used as high-performance electrode materials or hydrogen-stored materials.1–3 Most recently, pressure-induced peculiar structural evolutional behavior and superconducting property were noticed for Li2C2 and CaC2 via ¯rst-principles calculations.4,5 At ambient pressure, orthorhombic Li2C2 (space group Immm, Z ¼ 2) represents salt-like acetylides consisting of C 2 dumbbell 2 anions. The dumbbell units in Li2C2 are expected to polymerize into zigzag chains of carbon atoms at pressures around 5 GPa,6 and the system develops into semimetals (P -3m1-Li2C2) and metals (Cmcm-Li2C2) with polymeric anions (chains, layers, strands) by increasing pressure up to 20 GPa, and semimetallic P -3m1-Li2C2 displays an electronic structure close to that of graphene.4,7 By ¯rst-principles total 1550003-1


C.-L. Tang, G.-L. Sun & Y.-L. Li

energy calculations, it is shown that in the athermal limit the orthorhombic polymorph of Cs2C2 is more stable by about 7 kJ/mol with respect to the hexagonal modi¯cation, while for Rb2C2 the orthorhombic polymorph is more stable by around 4 kJ/mol.8 Prior to those theoretical researches, many experiments have been done to address their crystal structure.9–12 X-ray powder di®raction has been used to con¯rm structure of pure polycrystalline Li2C2 (Immm, No. 71, Z ¼ 2), shown that the polycrystalline Li2C2 is isotropic to Rb2O2 and Cs2O2, where Li ions occupy the 4j sites (x ¼ 0, y ¼ 0:5, z ¼ 0:2360) and C atoms reside on the 4g sites (x ¼ 0, y ¼ 0:1269, z ¼ 0).13 A reversible phase transition to a cubic modi¯cation (Fm-3m, No. 225, Z ¼ 4) has been observed at about 500 K, and this high temperature modi¯cation can be described as an anti°uorite-structure with disordered C 2 2 dumbbells.9 The crystal structure of Na2C2 (and K2C2) can be understood as another distorted variant of the anti-CaF2 structure, which can also be obtained from the cubic CaF2 structure by group–subgroup transitions.11 A phase transition from the room temperature structure to the undistorted cubic anti-°uorite structure (Fm-3m, Z ¼ 4) was observed at about 570 K (Na2C2) and 423 K (K2C2), respectively.14 As temperature increases, the motion of the C 2 2 dumbbells for low-temperature K2C2 are consistent with a motion restricted to a double cone with the cone angle, while the C 2 2 dumbbells in high-temperature are less restricted and undergo a fast reorientation.12 Although some experimental and theoretical studies have been reported, few works were given with regard to their elastic behavior. Here, the structural, elastic, and electronic properties of binary alkali metal carbides (A2 C2, A¼Li, Na, K, Rb, and Cs) were investigated by ¯rst-principles calculations based on DFT.15 First, crystal structural parameters were optimized by total energy calculations. Second, the elastic constants are given by ¯nite strain technique under the framework of linear response theory. Finally, electronic properties are calculated to discuss microscopic mechanism of mechanic behavior. 2. Computational Details The ground-state and electronic structures of the A2 C2 series were investigated by using the Vienna ab initio simulations package (VASP) based on DFT.16 The projected-augmented wave (PAW)17 potentials provided with the VASP package were employed to represent electron-ion interactions, and exchange and correlation terms were described with the generalized gradient approximation (GGA). The PAW potentials explicitly treat three valence electrons for Li (1s22s1), nine valence electrons for Na (2s22p63s1), K (3s23p64s1), Rb (4s24p65s1) and Cs (5s25p66s1), and four valence electrons for C (2s22p2). A k-point grid of spacing 2 0:03Å1 was applied to carry out integral in Brillouin zone. The atoms are steadily relaxed toward equilibrium until the Hellmann–Feynman force is less than 0.002 eV/Å and the tolerance in the self-consistent ¯eld (SCF) calculation is 1:0 106 eV/atom. A relaxation of internal degrees of freedom is allowed at each unit cell compression or 1550003-2

The structural, elastic and electronic properties of A2 C2 (A ¼ Li, Na, K, Rb and Cs)

expansion. From the full elastic constant tensor, one can determine bulk modulus B and shear modulus G according to the Voigt–Reuss–Hill (VRH) approximation.18 9B ¼ ðc11 þ c22 þ c33 Þ þ 2ðc12 þ c23 þ c31 Þ;

ð1Þ

15G ¼ ðc11 þ c22 þ c33 Þ ðc12 þ c23 þ c31 Þ þ 3ðc44 þ c55 þ c66 Þ:

ð2Þ

The Young's modulus, E, Poissons ratio can be calculated via formulae


E¼

9BG ; 3B þ G

¼

3B 2G : 2ð3B þ GÞ

ð3Þ

3. Results and Discussions 3.1. Structural properties Seven reported structure types, including Li2C2 (Orthorhombic Immm),9 Na2C2 (Orthorhombic Immm and Tetragonal I41 =acd),19,20 K2C2 (Tetragonal I41 =acd),14 Rb2C2(Orthorhombic Pnma)21 and Cs2C2 (Orthorhombic Pnma and hexagonal P -62m)21 were considered in our theoretical calculations. The optimized equilibrium structural parameters, density, C–C bond length and atomic fractional coordinates of A2 C2 carbides were listed in Table 1. For comparison, we also present available Table 1. Equilibrium lattice parameters: V0 (Å3), space group, a/Å, b/Å, c/Å, density (g/cm3), bond length dCC /Åof C2 dumbbell and atomic fractional coordinates. V0 is the volume of conventional cell. A2 C2

space group

Li2C2

Immm

Na2C2

K2C2 Rb2C2

Cs2C2

b

c

V0

dCC

Reference

Atomic fractional coordinates

4.8239 4.8312 4.833 6.7735 6.7432 6.7560 5.4239 5.3862 7.6355 7.580 4.7837 4.7322 4.724

5.3517 5.4344 5.440 12.6059 12.6743 12.6880 6.3662 6.4007 14.5921 14.690 9.8455 9.7146 10.093

93.28 95.88 96.10 578.35 576.31 579.13 144.00 144.43 850.29 844.03 444.95 425.87

1.350 1.313 1.309 1.608 1.614 1.606 1.614 1.610 1.597 1.609 2.910 3.040

1.258 1.226 1.199 1.262 1.200 1.204 1.262 1.200 1.266 1.351 1.268 1.273 1.254

this 9 22 this 19 23 this 20 this 14 this 21 8

Li 4j (0.5000, 0.0000, 0.2618) C 4g (0.0000, 0.1304, 0.0000)

9.6795 5.0512 10.5413 515.39 3.735 9.5453 5.0011 10.3743 495.20 3.888 9.703 5.140 10.482

1.270 1.391 1.258

this 21 8

1.270, 1.271

this

Na 4i (0.0000, 0.0000, 0.2449) C 4g (0.0000, 0.3838, 0.0000) K 16e (0.2500, 0.3081, 0.1250) C 16f (0.0587, 0.0587, 0.2500) Rb1 4c(0.3393, 0.2500, 0.5720) Rb2 4c(0.0131, 0.2500, 0.6685) C1 4c(0.2895, 0.2500, 0.9090) C2 4c(0.1688, 0.2500, 0.8526) Cs1 4c(0.3418, 0.2500, 0.5715) Cs2 4c(0.4879, 0.2500, 0.1675) C1 4c(0.2852, 0.2500, 0.9127) C2 4c(0.1755, 0.2500, 0.8466) Cs1 3f(0. 3991, 0.0000, 0.0000)

21 8

Cs2 3g(0.7316, 0.0000, 0.5000) C1 2e(0.0000, 0.0000, 0.1069)

a

3.6131 3.6520 3.655 I41 =acd 6.7735 6.7432 6.7560 Immm 4.1668 4.1893 I41 =acd 7.6355 7.580 Pnma 9.4473 9.2638 9.373 Pnma

P -62m 8.7232 8.7232 8.6372 8.6372 8.741 8.741

5.9402 391.45 3.688 5.6743 366.61 5.909

1.259, 1.260

Na 16e (0.2500, 0.3115, 0.1250) C 16f (0.0659, 0.0659, 0.2500)

C2 4h(0.6667, 0.3333, 0.3930)

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Fig. 1. (Color online) Conventional unit cells of Immm, I41 =acd, Pnma and P -62m. Carbon and metal atoms are represented as red and gray balls, respectively.

theoretical and experimental values. It can be seen that the optimized lattice constants of A2 C2 carbides agree well with available experimental values with the deviations less than 2.5% except hexagonal Cs2C2, indicating that the selected exchange correlation function, cuto® energy and other convergence criteria are reasonable. The reason for the disagreement between theoretical and experimental values is that the hexagonal Cs2C2 is a high temperature-induced phase, the experimental structure parameters are determined at about 470 K while the theoretical values are calculated at zero temperature. The crystal structures were plotted in Fig. 1. In Immm structure of Li2C2 or Na2C2, C 2 2 dumbbells are aligned parallel to a axis of the orthorhombic unit cell. Each Li or Na atom is surrounded by six carbon atoms, while the C 2 2 dumbbells are placed in an eight-fold co-ordination polyhedron. For I41 =acd structure of Na2C2 or K2C2, Aþ ions and C 2 2 groups are arranged in a distorted anti-CaF2 type packing with C 2 groups replacing calcium ions. From 2 2 Fig. 1, one can see that the C 2 dumbbells hold the (0 0 1) plane perpendicular to the tetragonal c-axis. Like in Immm structure, each Na or K atom is surrounded by six carbon atoms from four C2 units and the C2 units are coordinated by eight Na or K atoms. Both Rb2C2 and Cs2C2 can crystallize in an orthorhombic modi¯cation (Pnma, Z ¼ 4), and the Cs2C2 can also crystallize in a hexagonal Na2O2-type structure (P -62m, Z ¼ 3). Hexagonal Cs2C2 have two crystallographically distinct sites for the metal atoms and carbon atoms, respectively. Cs1 is coordinated by six 1550003-4



carbon atoms, while Cs2 is coordinated by eight carbon atoms. Two in-equivalent carbon atoms result in two distinct C 2 2 dumbbells with di®erent bond length (see Table 1). For Pnma structure of Rb2C2 or Cs2C2 it is related to the anti-PbCl2 structure type with the center of gravity of the C2 units lying in the Pb position. There are also two distinct Wycko® positions for the alkali metal and carbon atoms, respectively. Rb1 (Cs1) is coordinated by seven carbon atoms, while Rb2 (Cs2) is surrounded by ¯ve carbon atoms. Di®ering from the hexagonal modi¯cation of Rb2C2 and Cs2C2, the two in-equivalent carbon atoms only bring about one kind of C2 dumbbell with a nine-fold coordination polyhedron of the alkali metals. C–C distance in C2 unit increases slightly with the increase of atomic number from Li to Cs. C–C distance varying from 1.258 Å to 1.271 Å obtained in A2 C2 solids signals its C–C triple bond behavior, which is insigni¯cantly di®erent from 1.191 Å in CaC2. The loose packing among C2 dumbbells in A2 C2 is also manifested in an unusually low crystal density varying from 1.31 g/cm3 to 3.74 g/cm3 (see Table 1). In addition, ¯rst-principles total energy calculations show that it is di±cult to determine relative stability of I41 =acd-Na2C2 and Pnma-Na2C2 due to very close enthalpy between them at zero pressure. For Cs2C2 system, the orthorhombic Cs2C2 is more stable by about 35 meV per molecular than the hexagonal Cs2C2, which agrees well with previous theoretical calculation.8 3.2. Elastic properties Elastic properties of a solid are related to various physical properties, such as interatomic potentials, phonon spectra and equation of states.24 The elastic behavior of a completely asymmetric material is speci¯ed by 21 independent elastic constants, while for an isotropic material, the number is two. There are nine independent elastic constants for orthorhombic crystals, six independent elastic constants for tetragonal crystals and ¯ve independent elastic constants for hexagonal crystals. For a stable orthorhombic structure, the nine independent elastic constants cij (c11 , c22 , c33 , c44 , c55 , c66 , c12 , c13 , c23 ) should satisfy the well-known Born–Huang criteria25,26: c11 þ c33 2c13 > 0; c11 > 0; c33 > 0; c44 > 0; c55 > 0; c66 > 0; c11 þ c22 þ c33 þ 2c12 þ 2c13 þ 2c23 > 0;

ð4aÞ ð4bÞ

c11 þ c22 2c12 > 0;

ð4cÞ

c22 þ c23 2c23 > 0:

For a tetragonal structure, the six independent elastic constants cij (c11 , c33 , c44 , c66 , c12 , c13 ) follow the inequalities27–29: c11 c12 > 0; c44 > 0;

c11 þ c33 2c13 > 0;

c66 > 0;

c11 > 0;

c33 > 0;

2c11 þ c33 þ 2c12 þ 4c13 > 0:

ð5aÞ ð5bÞ

Five independent elastic constants cij (c11 , c33 , c44 , c12 , c13 ) for a stable hexagonal structure satisfy formulae30: c12 > 0;

c33 > 0;

c11 c12 > 0;

c44 > 0;

1550003-5

ðc11 þ c12 Þc33 2c 213 > 0:

ð6Þ

C.-L. Tang, G.-L. Sun & Y.-L. Li Table 2. Elastic constants cij (GPa), the isotropic bulk modulus B (GPa), shear modulus G (GPa), Young's modulus E (GPa), elastic anisotropy constants B=G and Poissons ratio . A2 C2

Space group

c11

c22

c33

c44

c55

c66

c12

c13

c23

B

G

B=G

E

Li2C2 Na2C2

Immm I41 =acd Immm I41 =acd Pnma Pnma P -62m

84 51 51 33 27 20 25

172 51 62 33 29 24 25

94 58 84 38 28 22 9

23 14 9 7 9 8 6

29 14 13 7 9 8 6

19 25 16 11 7 6 6

11 27 16 16 11 11 14

18 17 14 12 17 17 9

16 17 17 12 13 11 9

31 20 49 32 18 16 19

57 9 35 18 8 6 6

0.54 2.22 1.40 1.78 2.25 2.67 3.17

107 24 84 45 21 16 15

0.346 0.087 0.150 0.110 0.082 0.072 0.044


K2C2 Rb2C2 Cs2C2

In order to get more accurate elastic constants of A2 C2, all considered structures are ¯rstly optimized by using VASP code. The calculated elastic constants are presented in Table 2. It is obvious that the elastic constants cij in Table 2 for orthorhombic Li2C2, Na2C2, Rb2C2 and Cs2C2, tetragonal Na2C2 and K2C2 and hexagonal Cs2C2 satisfy the Born–Huang stability criteria, indicating their stability mechanically. From the elastic constants calculated above, bulk modulus B, shear modulus G, and Young's modulus E were obtained using VRH formulas, and they are important to understand the elastic properties of A2 C2. From Table 2, one can see that all bulk moduli B are small suggesting that all stable structures have strong compressibility. Strong compressibility attributes to weak Coulomb repulsion between alkali metal atoms and carbon atoms.28 The factor that measures stability of a crystal against shear is the Poissons ratio. The smaller the value is, the structure tends to go against the shear. This ability against shear for orthorhombic Li2C2 is apparently weak for its high Poissons ratio. Young's modulus is used to provide a measure of the sti®ness of the material. The calculated E values of Li2C2 and orthorhombic Na2C2 are 107 GPa and 84 GPa respectively, indicating their stronger sti®ness than other systems discussed. It is well-known that the elastic anisotropy of crystals plays an important role in engineering applications, since it is highly correlated with both hardness and microcracks in materials.24 Therefore, it is fundamental to calculate elastic anisotropy in order to understand the characteristics and improve the durability of materials. Considering that the shear modulus G represents the resistance to plastic deformation and the bulk modulus B represents the resistance to fracture, the quotient of bulk modulus to shear modulus of polycrystalline phases has been introduced to estimate the elastic anisotropy properties by Pugh.31 The critical value of B=G, 1.75, separates ductile and brittle materials.24 A high B=G value indicates the strong ductility. From the B=G values for all structures listed in Table 2, it is obvious that Li2C2 has a great brittleness, while the hexagonal Cs2C2 has strong ductility, meaning that hexagonal Cs2C2 might be the potential ductile material. 1550003-6



Fig. 2. (Color online) Total and projected density of states (PDOS) for Li2C2-Immm and Rb2C2-Pnma. The PDOS for other ¯ve structures were not listed here in view of their similarity as Li2C2-Immm or Rb2C2-Pnma.

3.3. Electronic properties To understand the correlation between electronic properties and mechanical behavior, the PDOS (see Fig. 2) and energy band are calculated. Band structure calculations show that the orthorhombic Li2C2, tetragonal Na2C2, tetragonal K2C2 and orthorhombic Rb2C2 are insulators with indirect band gap of 3.349, 3.636, 3.261, and 3.557 eV, respectively. While orthorhombic Na2C2 and Cs2C2 as well as hexagonal Cs2C2 are also insulators but with direct band gap of 3.342, 3.113, and 2.7 eV, respectively. Considering that present calculations underestimate band gaps due to well-known reason, A2 C2 solids should have more wider band gaps than those predicted here. The electrons from the C-p states dominate near the top of valence band

Fig. 3. (Color online) The three-dimensional charge density distribution of Li2C2-Immm at an isosurface value of 0.1 eletrons/Å3. Carbon and Li atoms are represented as red and gray balls, respectively.

1550003-7


for all systems studied here. Valence electrons from alkali-metal atoms have minor contribution to the top of valence band. That is, near the top of valence band, the weak hybridization exists between electronic states from carbon and those of A-electronic states, which is consistent with the little charge accumulation of electrons between carbon C2 unit and alkali-metal atoms (see Fig. 3), indicating weak Coulomb repulsion and thus high compressibility in A2 C2 solids.


4. Conclusion The structural, elastic and electronic properties of A2 C2 were investigated using ¯rst-principles total energy calculations. The calculated structural parameters are in good agreement with experimental values. The elastic constants, bulk modulus, shear modulus, Young's modulus and Poissons ratio were obtained. Lower bulk modulus shows that all systems have strong compressibility. The electronic properties calculations suggest that insulating behavior was elucidated for seven stable binary carbides, in which electrons from the C-p states dominate the top of the valence band. Near the top of valence band, weak hybridization between carbon and alkali-metal leads to weak Coulomb repulsion and thus brings about smaller bulk moduli. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11347007) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). Part of the calculations was performed in Center for Computational Science of CASHIPS. References 1. M. Inagaki, J. Mater. Res. 4, 1560 (1989). 2. R. Matsumotoa, M. Arakawa, H. Yoshida and N. Akuzawa, Synth. Met. 162, 2149 (2012). 3. M. S. Dresselhaus and G. Dresselhaus, Adv. Phys. 51, 1 (2002). 4. D. Benson, Y. L. Li, W. Luo, R. Ahuja, G. Svensson and U. Häussermann, Inorg. Chem. 52, 6402 (2013). 5. Y. L. Li, W. Luo, Z. Zeng, H. Q. Lin, H. K. Mao and R. Ahuja, Proc. Natl. Acad. Sci. USA 110, 9289 (2013). 6. X. Q. Chen, C. L. Fu and C. Franchini, J. Phys. Condens. Matter 22, 292201 (2010). 7. J. Nylen, S. Konar, P. Lazor, D. Benson and U. Häussermann, J. Chem. Phys. 137, 4507 (2012). 8. B. Winkler and V. Milman, Solid State Commun. 121, 155 (2002). 9. U. Ruschewitz and R. P€ottgen, Z. Anorg. Allg. Chem. 625, 1599 (1999). 10. D. R. Secrist and L. G. Wisnyi, Acta. Cryst. 15, 1042 (1962). 11. U. Ruschewitz, Coord. Chem. Rev. 244, 115 (2003). 12. B. Zibrowius, C. Bähtz, M. Knappw and U. Ruschewitz, Phys. Chem. Chem. Phys. 6, 5237 (2004). 1550003-8



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