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First-principles study of a novel superhard sp3 carbon allotrope
Physics Letters A 378 (2014) 3326–3330
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
First-principles study of a novel superhard sp 3 carbon allotrope Yangming Liu a , Mingchun Lu a , Miao Zhang b,c,∗ a b c
Department of Aeronautical Engineering Vocational Technology, Jilin University of Chemical Technology, Jilin 132102, China Department of Physics, Beihua University, Jilin 132013, 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 3 May 2014 Received in revised form 21 June 2014 Accepted 22 June 2014 Available online 3 October 2014 Communicated by R. Wu Keywords: Carbon allotrope Crystal structure Superhard materials Hardness
a b s t r a c t We systematically study the mechanical and electronic properties of a novel superhard sp 3 carbon allotrope (Imma-carbon). This polymorph was identified using a developed methodology on the theoretical design of superhard material materials based on the CALYPSO algorithm, and predicted to be more stable than graphite for pressures above 12.9 GPa. The structural, electronic, and mechanical properties of Imma-carbon were investigated by means of density functional theory (DFT). It is dynamically stable and a semiconductor with a direct band gap of ∼2.6 eV. The calculated elastic constants for Imma-carbon satisfy the stability condition. Calculations of bulk modulus and hardness indicate that this Imma-carbon is an ultra-incompressible and superhard material. Meanwhile, we extensively investigated stress–strain relations of Imma-carbon under various tensile and shear loading directions. The present results provide insight for understanding its mechanical properties at the atomic level. Our study provides valuable guidance in designing new strong covalent superhard materials. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Superhard materials are of particular importance in industrial applications due to their superior properties of high compressional strength, chemical inertness, and high hardness. As it is well known, diamond is the hardest material in the world. However, diamond is neither stable in presence of oxygen even at moderate temperature, nor a suitable abrasive for machining ferrous alloys. In the meanwhile, the difficult synthesized condition limits the extensive application of the second hardest superhard material c-BN. Therefore, these limitations encourage us to explore new novel superhard materials with good mechanical property. Since carbon element can easily form strong covalent sp 3 hybridized bonds, its allotropes have always stimulated considerable scientific interest are the key for designing new novel superhard materials. Recently, many theoretical carbon allotrope structures are proposed [1–22], such as M-carbon [1], W -carbon [2], oC32-carbon [3], and Cco-C8 [4], et al. However, the hardness of these carbon allotropes are far lower than diamond and c-BN. In order to design new superhard materials compared to diamond and c-BN, we put our focus on exploring the new carbon allotropes with high hardness values.
*
Corresponding author at: State Key Laboratory of Superhard Materials, Jilin Univ., 2699, Qianjin Str., Changchun 130012, PR China. Tel.: +86 431 85167557. E-mail address: [email protected] (M. Zhang). http://dx.doi.org/10.1016/j.physleta.2014.06.050 0375-9601/© 2014 Elsevier B.V. All rights reserved.
In this study, a novel superhard sp 3 hybridized crystalline carbon allotrope with an orthorhombic phase is reported. This Imma-carbon is a semiconductor with a direct band gap of approximately 2.6 eV. It has a high bulk modulus value of 444.7 GPa and a Vickers hardness value of 83.5 GPa, which are larger than that of c-BN (66.3 GPa and 403.0 GPa). Furthermore, we extensively investigated stress–strain relations of Imma-carbon under various tensile and shear loading directions. These results provide insights into understanding its mechanical properties at strains. Our study provides an important key step toward designing new strong covalent superhard materials. 2. Computational method A developed methodology to design superhard materials for given chemical systems under external conditions, was employed [31]. The first-principles calculations were fully carried out using the density functional theory within the local density approximation (LDA) [23,24] as implemented in the Vienna Ab initio Simulation Package code [25]. The all-electron projector augmented wave method [26] was adopted with 2s2 2p 2 for C treated as valence electrons. The cutoff energy of 800 eV for the expansion of the wave function into plane waves and Monkhorst and Pack [27] k-points were chosen to 3 × 6 × 6 in the Brillouin zone for Imma-carbon to ensure that all the calculations are well converged. The phonon frequencies for Imma-carbon were calculated using the direct supercell method, which uses the forces obtained by the
Y. Liu et al. / Physics Letters A 378 (2014) 3326–3330
Table 1 Crystallographic data for the Imma-carbon at zero pressure. Crystal
Space group
a (Å)
b (Å)
c (Å)
Imma-carbon Atom C1 C2
Imma Site 16j 4h
7.5147 x 0.16060 0.5
4.2363 y 0.06700 −0.06450
4.3173 z 0.33830 0.33490
Fig. 1. (Color online.) (a), (b) Polyhedral views of the crystal structure of Immacarbon along two different directions, respectively. (c) Enthalpy differences of various carbon allotropes relative to graphite.
Hellmann–Feynman theorem [28,29] calculated from the optimized supercell (216 atoms). Elastic constants were simulated by the strain-stress method, and bulk modulus and shear modulus were thus derived from the Voigt–Reuss–Hill averaging scheme [30]. 3. Results and discussion The structural predictions were carried out at ambient pressure with simulation cells containing 1–24 atoms. Our structure
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searches can successfully reproduce previously predicted diamond, Lonsdaleite (hexagonal diamond), M-carbon, Z -carbon, W -carbon etc., thereby demonstrating the reliability of our computational scheme. Besides, our structure simulations predict a new sp 3 carbon allotrope with space group of Imma and 24 carbon atoms in unit cell. The structural parameters of Imma-carbon at 0 GPa are a = 7.5147 Å, b = 4.2363 Å, and c = 4.3173 Å, with two inequivalent carbon atoms occupying 16j (0.1606, 0.067, 0.3383) and 4h (0.5, −0.0645, 0.3349) Wyckoff positions, as listed in Table 1. The structure consists of exclusively three-dimensional sp 3 hybridized covalent bonds. Fig. 1(a) and (b) shows the 2D projection of Imma-carbon along the two different directions, indicating the 4 + 6 + 8 even membered patterns. It can be seen that Imma-carbon is energetically stable than some of the previously proposed carbon allotrope structures and surpass graphite above 12.9 GPa, indicating it is a metastable structure, as shown in Fig. 1 (c). To explore the underlying origin of stability and electronic properties of Imma-carbon, we have thus computed its phonon dispersion curves, electronic band structures and density of states. As displayed in Fig. 2(a), no imaginary phonon frequency was detected in the whole Brillouin zone, thereby indicating the dynamically stability of Imma-carbon at ambient pressures. Fig. 2(b) shows the band structure near the Fermi energy (left panel) and electronic densities (right panel) of Imma-carbon at 0 GPa. It is clearly seen that this structure is a semiconductor with a direct band gap of approximately 2.6 eV. The valence band maximum and conduction band minimum are located at the Γ point. On the other hand, the partial density of states shows the main peak of valence density of states originates mostly from the C-p state. This is due to sp 3 hybridization leading to charge in s orbital transfer into p orbital. These electronic properties indicate this Imma-carbon phase is a strong sp 3 hybridization system and should be a potential superhard material. Moreover, the direct band gap feature suggests it will be applied for the solar cell or photography industry, and can be potentially applied in light-emitting. The calculated elastic constants of Imma-carbon are listed in Table 2. It clearly shows that the calculated results satisfy the mechanical stability criteria [32], indicating that it is elastically stable. The large values of C 11 , C 22 , and C 33 for Imma-carbon indicate that it is extremely difficult to be compressed along the a-axis, b-axis, and c-axis, respectively. As shown in Table 2, the bulk modulus, shear modulus, Young’s modulus, and Vickers hardness of diamond, c-BN, and Imma-carbon. It can be seen that all of the bulk modulus, shear modulus, and Young’s modulus of Imma-carbon are larger than that of c-BN, indicating its strong resistance to shape changes at a constant volume. It is well known that the bulk and shear modulus has been considered as very im-
Fig. 2. (Color online.) (a) Calculated phonon dispersion curves of Imma-carbon at zero pressure. (b) Calculated electronic band structure (left) and electronic densities (right) of oC32 carbon at zero pressure. The zero energy refers to the top of valence bands.
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Y. Liu et al. / Physics Letters A 378 (2014) 3326–3330
Table 2 Calculated elastic constants C i j (in GPa), bulk modulus B 0 (GPa), shear modulus G (GPa), Young’s modulus Y (GPa), and Vickers hardness H v (GPa) of diamond, c-BN, and Imma-carbon.
Diamond c-BN Imma-carbon
C 11
C 12
1104 820 1150
149 194 72
C 13
113
C 22
1168
C 23
51
C 33
C 44
1213
599 477 456
C 55
543
C 66
B0
G
Y
Hv
394
467.4 403.0 444.7
550.1 411.5 498.3
1185.3 921.0 1088.4
94.0 66.3 83.5
Fig. 3. (Color online.) The calculated stress–strain relations of Imma-carbon in various tension deformation directions.
portant parameters, governing the indentation hardness. This suggests Imma-carbon should be a good candidate of superhard material. To further analyze the hardness of Imma-carbon, we adopt the recently introduced empirical scheme [33] to evaluate the Vickers hardness (H v ) determined by the bulk modulus (B 0 ) and shear modulus (G), where: H v = 2(G 3 / B 20 )0.585 − 3. The values of Vickers hardness for diamond, c-BN, and Imma-carbon are 94.0 GPa, 66.3 GPa, and 83.5 GPa, respectively. Strikingly, the hardness value of Imma-carbon is higher than that of c-BN, whereas it is slightly lower than that of diamond. Therefore, Imma-carbon would be promised to become an excellent potential candidate of superhard material. We now turn to a detailed analysis of the atomistic structural deformation modes in Imma-carbon to examine the microscopic mechanism of its reduced ideal strength and the fracture behavior under the tensile and shear loading conditions based on firstprinciples level. Such calculations have been extensively applied under a variety of loading conditions [34–42]. To explore ideal strength of the Imma-carbon, we have systematically examined the stress–strain relations of Imma-carbon under tensile loading. The calculated results (Fig. 3) show that Imma-carbon has strong stress responses in the 001, 010, 011, 100, 101, and 110 directions with the peak tensile stresses between 70 and 130 GPa. It can be clearly seen that the weakest peak tensile stress occurs in the 010 direction, with the strength of 73.3 GPa. Fig. 4 shows the snapshots of the strained structures for Imma-carbon, together with the 3D electron localization function (ELF) isosurfaces at ELF = 0.85. At equilibrium (T0), the lengths of the atomic bonds C1–C2 and C3–C4 are 1.553 Å and 1.570 Å, respectively. With increasing strain and at ε = 0.116 (T1), the bond lengths of C1–C2 and C3–C4 increase to 1.687 Å and 1.897 Å, respectively. As the tensile deformation increases in the 010 direction, the C3–C4 bond breaks. However, the C1–C2 bond still remains with the length of 1.735 Å (see T3), because there is electron location between C1–C2. We further investigate the stress–strain relations of Immacarbon under shear deformation, as shown in Fig. 5. The calculated stress–strain relations along various shear paths indicate that the
Fig. 4. (Color online.) The calculated stress–strain relations of Imma-carbon under the 010 tensile direction (top) and the snapshots of the strained structures, corresponding to the filled symbols in the stress–strain plot (bottom).
¯ ] direction with a highest shear strength appears in the (011)[011 peak shear stress of 125.8 GPa. The lowest peak stress of 79.7 GPa ¯ ] direction. To further analyze the bondoccurs in the (110)[110 breaking mechanism of Imma-carbon in its weakest pure shear ¯ ], we examine the strain energy deformation direction (110)[110 and bond length variations compared to the shear-strain relations ¯ ] direction, as shown in Fig. 6 (left along the easy-slip (110)[110 panel). The snapshots of the strained structures, corresponding to the filled symbols in the stress–strain plot are also plotted in Fig. 6 (right panel). It shows that the C–C bonds remain very strong up to the bond-breaking point at the critical shear strain of ε = 0.231 (S1) where the sp 3 bonding character is clearly seen. The bond lengths of C1–C2 and C3–C4 are 1.693 Å and 1.881 Å, respectively. Above the critical shear strain, a transformation into the graphite structure occurs at ε = 0.254 (S2). The lengths of C1–C2 and C3–C4 increase to 2.852 Å. 4. Conclusion In summary, we study the mechanical properties of a novel superhard sp 3 carbon allotrope, Imma-carbon, using a developed methodology on the theoretical design of superhard material materials based on the CALYPSO algorithm. We have performed systematic first-principles simulations to examine the structural stability, electronic properties, Vickers hardness, tensile and shear ideal strength of Imma-carbon. This Imma-carbon is a semiconductor with a direct band gap of ∼2.6 eV. Furthermore, it has a high bulk modulus of 444.7 GPa and a Vickers hardness of 83.5 GPa, which are larger than that of c-BN (66.3 GPa and 403.0 GPa). Our study not only provides an important key step toward designing new strong covalent superhard materials, but also expands the list
Y. Liu et al. / Physics Letters A 378 (2014) 3326–3330
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Fig. 5. (Color online.) The calculated stress–strain relations of Imma-carbon in various shear deformation directions.
Fig. 6. (Color online.) Left panel: the calculated stress–strain relation of Imma-carbon (top), the corresponding strain energy (middle), and the bond lengths (bottom) of Imma-carbon under (110) [−110] shear direction. Right panel: The snapshots of the strained structures, corresponding to the filled symbols in the stress–strain plot.
of high-pressure carbon allotropes with good mechanical properties. Acknowledgements This work is supported by the Natural Science Foundation of China under grants 11304167, 51374132, 11304141, 11304140, and 51202084. The authors acknowledge the High Performance Computing Center of Jilin University for supercomputer time. References [1] Q. Li, Y. Ma, A.R. Oganov, H. Wang, Y. Xu, T. Cui, H.K. Mao, G. Zou, Phys. Rev. Lett. 102 (2009) 175506. [2] J.T. Wang, C. Chen, Y. Kawazoe, Phys. Rev. Lett. 106 (2011) 075501.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
First-principles study of a novel superhard sp 3 carbon allotrope Yangming Liu a , Mingchun Lu a , Miao Zhang b,c,∗ a b c
Department of Aeronautical Engineering Vocational Technology, Jilin University of Chemical Technology, Jilin 132102, China Department of Physics, Beihua University, Jilin 132013, 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 3 May 2014 Received in revised form 21 June 2014 Accepted 22 June 2014 Available online 3 October 2014 Communicated by R. Wu Keywords: Carbon allotrope Crystal structure Superhard materials Hardness
a b s t r a c t We systematically study the mechanical and electronic properties of a novel superhard sp 3 carbon allotrope (Imma-carbon). This polymorph was identified using a developed methodology on the theoretical design of superhard material materials based on the CALYPSO algorithm, and predicted to be more stable than graphite for pressures above 12.9 GPa. The structural, electronic, and mechanical properties of Imma-carbon were investigated by means of density functional theory (DFT). It is dynamically stable and a semiconductor with a direct band gap of ∼2.6 eV. The calculated elastic constants for Imma-carbon satisfy the stability condition. Calculations of bulk modulus and hardness indicate that this Imma-carbon is an ultra-incompressible and superhard material. Meanwhile, we extensively investigated stress–strain relations of Imma-carbon under various tensile and shear loading directions. The present results provide insight for understanding its mechanical properties at the atomic level. Our study provides valuable guidance in designing new strong covalent superhard materials. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Superhard materials are of particular importance in industrial applications due to their superior properties of high compressional strength, chemical inertness, and high hardness. As it is well known, diamond is the hardest material in the world. However, diamond is neither stable in presence of oxygen even at moderate temperature, nor a suitable abrasive for machining ferrous alloys. In the meanwhile, the difficult synthesized condition limits the extensive application of the second hardest superhard material c-BN. Therefore, these limitations encourage us to explore new novel superhard materials with good mechanical property. Since carbon element can easily form strong covalent sp 3 hybridized bonds, its allotropes have always stimulated considerable scientific interest are the key for designing new novel superhard materials. Recently, many theoretical carbon allotrope structures are proposed [1–22], such as M-carbon [1], W -carbon [2], oC32-carbon [3], and Cco-C8 [4], et al. However, the hardness of these carbon allotropes are far lower than diamond and c-BN. In order to design new superhard materials compared to diamond and c-BN, we put our focus on exploring the new carbon allotropes with high hardness values.
*
Corresponding author at: State Key Laboratory of Superhard Materials, Jilin Univ., 2699, Qianjin Str., Changchun 130012, PR China. Tel.: +86 431 85167557. E-mail address: [email protected] (M. Zhang). http://dx.doi.org/10.1016/j.physleta.2014.06.050 0375-9601/© 2014 Elsevier B.V. All rights reserved.
In this study, a novel superhard sp 3 hybridized crystalline carbon allotrope with an orthorhombic phase is reported. This Imma-carbon is a semiconductor with a direct band gap of approximately 2.6 eV. It has a high bulk modulus value of 444.7 GPa and a Vickers hardness value of 83.5 GPa, which are larger than that of c-BN (66.3 GPa and 403.0 GPa). Furthermore, we extensively investigated stress–strain relations of Imma-carbon under various tensile and shear loading directions. These results provide insights into understanding its mechanical properties at strains. Our study provides an important key step toward designing new strong covalent superhard materials. 2. Computational method A developed methodology to design superhard materials for given chemical systems under external conditions, was employed [31]. The first-principles calculations were fully carried out using the density functional theory within the local density approximation (LDA) [23,24] as implemented in the Vienna Ab initio Simulation Package code [25]. The all-electron projector augmented wave method [26] was adopted with 2s2 2p 2 for C treated as valence electrons. The cutoff energy of 800 eV for the expansion of the wave function into plane waves and Monkhorst and Pack [27] k-points were chosen to 3 × 6 × 6 in the Brillouin zone for Imma-carbon to ensure that all the calculations are well converged. The phonon frequencies for Imma-carbon were calculated using the direct supercell method, which uses the forces obtained by the
Y. Liu et al. / Physics Letters A 378 (2014) 3326–3330
Table 1 Crystallographic data for the Imma-carbon at zero pressure. Crystal
Space group
a (Å)
b (Å)
c (Å)
Imma-carbon Atom C1 C2
Imma Site 16j 4h
7.5147 x 0.16060 0.5
4.2363 y 0.06700 −0.06450
4.3173 z 0.33830 0.33490
Fig. 1. (Color online.) (a), (b) Polyhedral views of the crystal structure of Immacarbon along two different directions, respectively. (c) Enthalpy differences of various carbon allotropes relative to graphite.
Hellmann–Feynman theorem [28,29] calculated from the optimized supercell (216 atoms). Elastic constants were simulated by the strain-stress method, and bulk modulus and shear modulus were thus derived from the Voigt–Reuss–Hill averaging scheme [30]. 3. Results and discussion The structural predictions were carried out at ambient pressure with simulation cells containing 1–24 atoms. Our structure
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searches can successfully reproduce previously predicted diamond, Lonsdaleite (hexagonal diamond), M-carbon, Z -carbon, W -carbon etc., thereby demonstrating the reliability of our computational scheme. Besides, our structure simulations predict a new sp 3 carbon allotrope with space group of Imma and 24 carbon atoms in unit cell. The structural parameters of Imma-carbon at 0 GPa are a = 7.5147 Å, b = 4.2363 Å, and c = 4.3173 Å, with two inequivalent carbon atoms occupying 16j (0.1606, 0.067, 0.3383) and 4h (0.5, −0.0645, 0.3349) Wyckoff positions, as listed in Table 1. The structure consists of exclusively three-dimensional sp 3 hybridized covalent bonds. Fig. 1(a) and (b) shows the 2D projection of Imma-carbon along the two different directions, indicating the 4 + 6 + 8 even membered patterns. It can be seen that Imma-carbon is energetically stable than some of the previously proposed carbon allotrope structures and surpass graphite above 12.9 GPa, indicating it is a metastable structure, as shown in Fig. 1 (c). To explore the underlying origin of stability and electronic properties of Imma-carbon, we have thus computed its phonon dispersion curves, electronic band structures and density of states. As displayed in Fig. 2(a), no imaginary phonon frequency was detected in the whole Brillouin zone, thereby indicating the dynamically stability of Imma-carbon at ambient pressures. Fig. 2(b) shows the band structure near the Fermi energy (left panel) and electronic densities (right panel) of Imma-carbon at 0 GPa. It is clearly seen that this structure is a semiconductor with a direct band gap of approximately 2.6 eV. The valence band maximum and conduction band minimum are located at the Γ point. On the other hand, the partial density of states shows the main peak of valence density of states originates mostly from the C-p state. This is due to sp 3 hybridization leading to charge in s orbital transfer into p orbital. These electronic properties indicate this Imma-carbon phase is a strong sp 3 hybridization system and should be a potential superhard material. Moreover, the direct band gap feature suggests it will be applied for the solar cell or photography industry, and can be potentially applied in light-emitting. The calculated elastic constants of Imma-carbon are listed in Table 2. It clearly shows that the calculated results satisfy the mechanical stability criteria [32], indicating that it is elastically stable. The large values of C 11 , C 22 , and C 33 for Imma-carbon indicate that it is extremely difficult to be compressed along the a-axis, b-axis, and c-axis, respectively. As shown in Table 2, the bulk modulus, shear modulus, Young’s modulus, and Vickers hardness of diamond, c-BN, and Imma-carbon. It can be seen that all of the bulk modulus, shear modulus, and Young’s modulus of Imma-carbon are larger than that of c-BN, indicating its strong resistance to shape changes at a constant volume. It is well known that the bulk and shear modulus has been considered as very im-
Fig. 2. (Color online.) (a) Calculated phonon dispersion curves of Imma-carbon at zero pressure. (b) Calculated electronic band structure (left) and electronic densities (right) of oC32 carbon at zero pressure. The zero energy refers to the top of valence bands.
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Y. Liu et al. / Physics Letters A 378 (2014) 3326–3330
Table 2 Calculated elastic constants C i j (in GPa), bulk modulus B 0 (GPa), shear modulus G (GPa), Young’s modulus Y (GPa), and Vickers hardness H v (GPa) of diamond, c-BN, and Imma-carbon.
Diamond c-BN Imma-carbon
C 11
C 12
1104 820 1150
149 194 72
C 13
113
C 22
1168
C 23
51
C 33
C 44
1213
599 477 456
C 55
543
C 66
B0
G
Y
Hv
394
467.4 403.0 444.7
550.1 411.5 498.3
1185.3 921.0 1088.4
94.0 66.3 83.5
Fig. 3. (Color online.) The calculated stress–strain relations of Imma-carbon in various tension deformation directions.
portant parameters, governing the indentation hardness. This suggests Imma-carbon should be a good candidate of superhard material. To further analyze the hardness of Imma-carbon, we adopt the recently introduced empirical scheme [33] to evaluate the Vickers hardness (H v ) determined by the bulk modulus (B 0 ) and shear modulus (G), where: H v = 2(G 3 / B 20 )0.585 − 3. The values of Vickers hardness for diamond, c-BN, and Imma-carbon are 94.0 GPa, 66.3 GPa, and 83.5 GPa, respectively. Strikingly, the hardness value of Imma-carbon is higher than that of c-BN, whereas it is slightly lower than that of diamond. Therefore, Imma-carbon would be promised to become an excellent potential candidate of superhard material. We now turn to a detailed analysis of the atomistic structural deformation modes in Imma-carbon to examine the microscopic mechanism of its reduced ideal strength and the fracture behavior under the tensile and shear loading conditions based on firstprinciples level. Such calculations have been extensively applied under a variety of loading conditions [34–42]. To explore ideal strength of the Imma-carbon, we have systematically examined the stress–strain relations of Imma-carbon under tensile loading. The calculated results (Fig. 3) show that Imma-carbon has strong stress responses in the 001, 010, 011, 100, 101, and 110 directions with the peak tensile stresses between 70 and 130 GPa. It can be clearly seen that the weakest peak tensile stress occurs in the 010 direction, with the strength of 73.3 GPa. Fig. 4 shows the snapshots of the strained structures for Imma-carbon, together with the 3D electron localization function (ELF) isosurfaces at ELF = 0.85. At equilibrium (T0), the lengths of the atomic bonds C1–C2 and C3–C4 are 1.553 Å and 1.570 Å, respectively. With increasing strain and at ε = 0.116 (T1), the bond lengths of C1–C2 and C3–C4 increase to 1.687 Å and 1.897 Å, respectively. As the tensile deformation increases in the 010 direction, the C3–C4 bond breaks. However, the C1–C2 bond still remains with the length of 1.735 Å (see T3), because there is electron location between C1–C2. We further investigate the stress–strain relations of Immacarbon under shear deformation, as shown in Fig. 5. The calculated stress–strain relations along various shear paths indicate that the
Fig. 4. (Color online.) The calculated stress–strain relations of Imma-carbon under the 010 tensile direction (top) and the snapshots of the strained structures, corresponding to the filled symbols in the stress–strain plot (bottom).
¯ ] direction with a highest shear strength appears in the (011)[011 peak shear stress of 125.8 GPa. The lowest peak stress of 79.7 GPa ¯ ] direction. To further analyze the bondoccurs in the (110)[110 breaking mechanism of Imma-carbon in its weakest pure shear ¯ ], we examine the strain energy deformation direction (110)[110 and bond length variations compared to the shear-strain relations ¯ ] direction, as shown in Fig. 6 (left along the easy-slip (110)[110 panel). The snapshots of the strained structures, corresponding to the filled symbols in the stress–strain plot are also plotted in Fig. 6 (right panel). It shows that the C–C bonds remain very strong up to the bond-breaking point at the critical shear strain of ε = 0.231 (S1) where the sp 3 bonding character is clearly seen. The bond lengths of C1–C2 and C3–C4 are 1.693 Å and 1.881 Å, respectively. Above the critical shear strain, a transformation into the graphite structure occurs at ε = 0.254 (S2). The lengths of C1–C2 and C3–C4 increase to 2.852 Å. 4. Conclusion In summary, we study the mechanical properties of a novel superhard sp 3 carbon allotrope, Imma-carbon, using a developed methodology on the theoretical design of superhard material materials based on the CALYPSO algorithm. We have performed systematic first-principles simulations to examine the structural stability, electronic properties, Vickers hardness, tensile and shear ideal strength of Imma-carbon. This Imma-carbon is a semiconductor with a direct band gap of ∼2.6 eV. Furthermore, it has a high bulk modulus of 444.7 GPa and a Vickers hardness of 83.5 GPa, which are larger than that of c-BN (66.3 GPa and 403.0 GPa). Our study not only provides an important key step toward designing new strong covalent superhard materials, but also expands the list
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Fig. 5. (Color online.) The calculated stress–strain relations of Imma-carbon in various shear deformation directions.
Fig. 6. (Color online.) Left panel: the calculated stress–strain relation of Imma-carbon (top), the corresponding strain energy (middle), and the bond lengths (bottom) of Imma-carbon under (110) [−110] shear direction. Right panel: The snapshots of the strained structures, corresponding to the filled symbols in the stress–strain plot.
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