Four sp2sp3 hybrid carbon allotropes are proposed on the basis of first principles calculations. These four carbon allotropes are energetically more favorable than graphite under suitable pressure conditions. They can be assembled from graphite through intralayer wrinkling and interlayer buckling, which is similar to the formation of diamond from graphite. For one of the sp2sp3 hybrid carbon allotropes, mC24, the electron diffraction patterns match these of i-carbon, which is synthesized from shock-compressed graphite (H. Hirai and K. Kondo, Science, 1991, 253, 772). The allotropes exhibit tunable electronic characteristics from metallic to semiconductive with band gaps comparable to those of silicon allotropes. They are all superhard materials with Vickers hardness values comparable to that of cubic BN. The sp2sp3 hybrid carbon allotroes are promising materials for photovoltaic electronic devices, and abrasive and grinding tools.

Carbon adopts a wide range of allotropes, such as graphite, diamond, fullerene, nanotubes, graphdiyne, and amorphous carbon, because of its ability to form sp-, sp2-, and sp3-hybridized bonds. Studies have investigated the nature of the conversion mechanisms between different allotropes under pressure among various carbon configurations with different hybridizations, such as from sp2 to sp3. An sp2-hybridized fullerene C60 instantaneously transforms into sp3-hybridized diamond by cold compressing at high non-hydrostatic pressures (20 ± 5 GPa) or at hydrostatic pressures of up to 15 GPa and temperatures of up to 1200 °C.1,2 C60 is assembled into sp2sp3 hybrid one-, two-, or three-dimensional polymers at low pressures or temperatures.3–6 Carbon nanotubes with only sp2-hybridized atoms can be assembled into sp2sp3 hybrid polymers7 before these nanotubes transform into diamond.2 A previous study systematically investigated the polymerization of carbon nanotubes under pressure and proposed a series of sp2sp3 hybrid nanotube polymers.8 

sp2-hybridized graphite readily transforms into sp3-hybridized diamond under high pressure and temperature. Moreover, experimental studies have demonstrated numerous novel carbon allotropes that can be generated from compressed graphite. One of the least well understood of these carbon allotropes is i-carbon, which is obtained alongside diamond and n-diamond from shock-compressed graphite.9,10 Although i-carbon was first prepared via ion-beam technique decades ago,11 its crystal structure still remains inscrutable. The accompanying n-diamond has frequently been detected in compressed graphite,9,10,12 and several structural models, such as modified diamond13 and CHx,14 have been theoretically proposed to interpret its crystal structure. Another little understood carbon allotrope is a transparent superhard carbon polymorph. It is observable under a high pressure range of 10–25 GPa,15–21 but only quenchable at low temperatures (<100 K) and at ambient pressure.15 This post-graphite phase has been extensively explored to identify candidate crystal structures, including sp2sp3 hybrid (3,0)/(4,0),22 and sp3-hybridized M-carbon, bct-C4, F-carbon, X-carbon, Y-carbon, W-carbon, Z-carbon, and P-carbon.23–30 

Considering that i-carbon is usually accompanied by diamond, n-diamond, or amorphous carbon, theoretical simulations are a necessary and powerful tool to elucidate its ambiguous crystal structure. In this study, four sp2sp3 hybrid carbon allotropes are proposed on the basis of first principles calculations. The four allotropes can be assembled by compressing graphite along the c-axis with partially sp2-hybridized atoms bulked into diamond-like sp3-hybridized atoms to construct sp2sp3 hybrid carbon allotropes, which are similar to C60 and nanotube polymers. The four sp2sp3 hybrid carbon allotropes are dynamically stable and energetically more favorable than graphite under appropriate pressures. One of the allotropes readily explains the ambiguous i-carbon based on consistent electron diffraction patterns. The transition mechanism from graphite to a sp2sp3 hybrid allotrope involves the intralayer wrinkling and interlayer buckling of the graphite sheets; this is similar to the formation of diamond from graphite and would explain the coproduction of i-carbon and diamond. The sp2sp3 hybrid carbon allotropes exhibit tunable electronic properties from metallic to semiconductive. Mechanically, sp2sp3 hybrid carbon allotropes show superhardness, and high Young’s, bulk, and shear moduli.

Basic structure searching simulations were performed through use of the swarm-intelligence-based CALYPSO code,31 which has successfully proposed several novel carbon allotropes,32–34 with system sizes containing up to 24 carbon atoms. The calculations of the structural relaxation and electronic and mechanical properties were performed using the DFT framework with the CASTEP code.35 The Vanderbilt ultrasoft pseudopotential was adopted at an energy cutoff of 310 eV for plane-wave expansion. The exchange correlation terms were described on the basis of the local density approximation (LDA) developed by Ceperley and Alder and parameterized by Perdew and Zunger (CA-PZ).36,37 Monkhorst–Pack k point meshes with a separation of 2π × 0.04 Å−1 were chosen for Brillouin zone sampling.38 The finite displacement method was employed to calculate the phonon frequencies based on CASTEP code. The variable-cell nudged elastic band (VC-NEB) method, which was generalized to determine the minimum energy route and the activation pathways between the specified initial and end structures for a phase transition, was applied to estimate the transition path at ambient pressure.39 

In addition to diamond and theoretical post-graphite candidates, such as M-carbon, bct-C4, F-carbon, X-carbon, Y-carbon, O-carbon, or P-carbon, our structural search through the CALYPSO code yielded four unexpected novel sp2sp3 hybrid carbon allotropes, namely mC12, mC24, oC16, and oC24, as depicted in Fig. 1. mC12 and mC24 are monoclinic crystals with 12 and 24 carbon atoms in a unit cell, respectively, while oC16 and oC24 possess orthorhombic symmetry with 16 and 24 carbon atoms in a unit cell, respectively. mC12 and oC24 are characterized by sandwich-like layered structures with a …sp2sp3sp2sp3… stacking sequence along one direction. In oC16 and mC24, sp3 atoms establish diamond-like building blocks and sp2 atoms act as linking nodes. In Fig. 1 the atomic blocks highlighted by yellow and purple rectangles are fold graphite layers; therefore, we proposed the new sp2sp3 hybrid carbon allotropes can be constructed from graphite through interlayer buckling (see discussion later). The crystal symmetries, lattice parameters, and atomic Wyckoff positions of the four carbon allotropes are summarized in Table I. The atomic densities of these sp2sp3 hybrid carbon allotropes are 0.162–0.170 atoms/Å3; i.e., the values are between those of graphite (0.116 atoms/Å3) and diamond (0.182 atoms/Å3).

FIG. 1.

Crystal structures of (a) mC12, (b) mC24, (c) oC16, and (d) oC24. The light gray spheres represent sp2-hybridized carbon atoms, and the dark gray spheres represent sp3-hybridized carbon atoms. The yellow and purple rectangles mark the neighboring wrinkled graphite layers.

FIG. 1.

Crystal structures of (a) mC12, (b) mC24, (c) oC16, and (d) oC24. The light gray spheres represent sp2-hybridized carbon atoms, and the dark gray spheres represent sp3-hybridized carbon atoms. The yellow and purple rectangles mark the neighboring wrinkled graphite layers.

Close modal
TABLE I.

Space group (S.G.), lattice parameters (L.P., Å), and atomic Wyckoff positions of sp2sp3 hybrid carbon allotropes.

Structure S.G. L.P. Atomic positions
(3,0)/(4,0)  Cmmm (65)  a = 11.175  8q (-0.120, 0.315, -1/2) 
b = 4.128  8p (0.186, 0.814, 0) 
c = 2.516  4j (0, 0.164, -1/2) 
(4,0)ab  Cmca (64)  a = 4.131  16g (-1.315, -0.828, 0.092)
8d (-1.338, -1, -1/2) 
b = 7.398 
c = 4.767 
(4,0)  Immm (71)  a = 6.860  8n (0.306, -0.316, -1/2)
4g (0, -0.664, -2) 
b = 4.125 
c = 2.552 
mC12  C2/m (12)  a = 5.035  8j (0.276, -0.181, 0.382)
4g (0, -0.342, 0) 
b = 4.262 
c = 5.144 
β = 138.018° 
oC16  Cmmm (65)  a =14.781
b =2.550
c =2.491 
4h (0.455, 1/2, 1.500) 
4h (0.220, 1/2, 1.500) 
4g (0.600, 1/2, 2) 
4g (0.339, 1, 2) 
mC24  C2/m (12)  a =12.306
b =2.473
c =4.914
β = 79.196° 
4i (0.392, -1/2, 1.012) 
4i (0.280, -1/2, 0.498) 
4i (0.278, -1/2, 0.978) 
4i (0.602, -1/2, 0.648) 
4i (0.531, 0, 0.625) 
4i (0.544, 0, 1.087) 
oC24  Cmcm (63)  a = 4.257  16h (0.319, 0.366, 0.559)
8g (-0.158, 0.378, 3/4) 
b = 5.036 
c = 6.886 
Structure S.G. L.P. Atomic positions
(3,0)/(4,0)  Cmmm (65)  a = 11.175  8q (-0.120, 0.315, -1/2) 
b = 4.128  8p (0.186, 0.814, 0) 
c = 2.516  4j (0, 0.164, -1/2) 
(4,0)ab  Cmca (64)  a = 4.131  16g (-1.315, -0.828, 0.092)
8d (-1.338, -1, -1/2) 
b = 7.398 
c = 4.767 
(4,0)  Immm (71)  a = 6.860  8n (0.306, -0.316, -1/2)
4g (0, -0.664, -2) 
b = 4.125 
c = 2.552 
mC12  C2/m (12)  a = 5.035  8j (0.276, -0.181, 0.382)
4g (0, -0.342, 0) 
b = 4.262 
c = 5.144 
β = 138.018° 
oC16  Cmmm (65)  a =14.781
b =2.550
c =2.491 
4h (0.455, 1/2, 1.500) 
4h (0.220, 1/2, 1.500) 
4g (0.600, 1/2, 2) 
4g (0.339, 1, 2) 
mC24  C2/m (12)  a =12.306
b =2.473
c =4.914
β = 79.196° 
4i (0.392, -1/2, 1.012) 
4i (0.280, -1/2, 0.498) 
4i (0.278, -1/2, 0.978) 
4i (0.602, -1/2, 0.648) 
4i (0.531, 0, 0.625) 
4i (0.544, 0, 1.087) 
oC24  Cmcm (63)  a = 4.257  16h (0.319, 0.366, 0.559)
8g (-0.158, 0.378, 3/4) 
b = 5.036 
c = 6.886 

The phonon spectra of mC12, oC16, mC24, and oC24 at ambient pressure were calculated to examine their structural stabilities. As shown in Fig. 2, no imaginary frequency is found throughout the entire Brillouin zone of the four polymorphs, confirming their dynamic stabilities. In mC12, oC16, and oC24, the C–C bonds are single and double bonds, which show distinct phonon vibrational frequencies; the top-most frequency mode is therefore isolated from the other modes. While there are benzene ring structures (except for the single and double bonds) in mC24, continuous phonon frequencies are found. Fig. 3 presents the enthalpies of the novel carbon allotropes as well as those of (3,0)/(4,0), (4,0), (4,0)ab (Table I),22 M-carbon, and bct-C4 relative to graphite under a pressure range of 0–60 GPa. The sp2sp3 hybrid (3,0)/(4,0), (4,0), and (4,0)ab are honeycomb structures with each individual honeycombs that can be viewed as narrow carbon nanotubes. The sp3-hybridized M-carbon and bct-C4 are assembled from wrinkled and buckled graphene layers. At ambient pressure, sp2sp3 hybrid carbon allotropes are energetically more stable than C60. oC16 is more favorable than (4,0) and (4,0)ab, and is comparable to (3,0)/(4,0), which is a potential candidate for the experimental post-graphite phase. These sp2sp3 hybrid carbons are pressure-induced phases, and their relative enthalpies decrease with decreasing pressure. oC16 and (3,0)/(4,0) become more stable than graphite at 16.7 and 16.6 GPa, respectively, and mC12 and oC24 show higher stabilities than graphite above 53.2 and 52.8 GPa, respectively. Interestingly, mC24 is compressed to a new sp3-hybridized carbon allotrope (Fig. 3(b)) under compression at 10 GPa with neighboring layered sp2 atoms buckled into sp3 atoms.

FIG. 2.

Phonon spectra of mC12, oC16, mC24, and oC24 at ambient pressure.

FIG. 2.

Phonon spectra of mC12, oC16, mC24, and oC24 at ambient pressure.

Close modal
FIG. 3.

(a) Enthalpy of bct-C4, M-carbon, (3,0)/(4,0), (4,0)ab, (4,0), mC12, oC16, oC24, and mC24 as a function of pressure relative to graphite. Insert shows the enthalpy of C60 relative to graphite. (b) The new sp3-hybridized carbon allotrope from compressed mC24 at 10 GPa. The yellow sticks represent the formed C–C bonds.

FIG. 3.

(a) Enthalpy of bct-C4, M-carbon, (3,0)/(4,0), (4,0)ab, (4,0), mC12, oC16, oC24, and mC24 as a function of pressure relative to graphite. Insert shows the enthalpy of C60 relative to graphite. (b) The new sp3-hybridized carbon allotrope from compressed mC24 at 10 GPa. The yellow sticks represent the formed C–C bonds.

Close modal

To further define the structural models, we simulate the electron diffraction patterns to compare them with the experimental data. In the recovered samples of shock compressed graphite, electron diffraction patterns using a transmission electron microscope (TEM) demonstrated a novel carbon allotrope, named as i-carbon, together with n-diamond and diamond.9 The observed d-spacing points to a mysterious carbon allotrope that has remained a puzzle since its first synthesis in 1981.11 The simulated d-spacings of mC24, mC12, oC16, oC24, (3,0)/(4,0), and M-carbon are listed in Table II and compared with the experimental data. An excellent d-spacing match suggests that i-carbon possesses a mC24 crystal structure, while other carbon allotropes are at bad match for i-carbon’s d-spacing pattern. This finding further demonstrates that sp2sp3 hybrid carbon allotropes can be synthesized from compressed graphite.

TABLE II.

Experimental d-spacing (dexp, Å) of i-carbon compared with the calculated d-spacing (dcal, Å) of mC24, mC12, oC16, oC24, (3,0)/(4,0), and M-carbon.

dexp dcal
Ref. 9  Ref. 11  mC24 mC12 oC16 oC24 (3,0)/(4,0) M-carbon
3.04  3.01  3.02  –  –  2.94  –  – 
2.42  2.46  2.40, 2.41  2.37, 2.64  2.46, 2.49  –  –  2.40 
2.08  2.12  2.09, 2.11  2.11, 2.13  2.06  2.03, 2.13  2.07, 2.11  2.08, 2.10 
1.70  1.74  1.71, 1.73  1.71, 1.72  1.68, 1.75  1.70, 1.72  –  1.69, 1.78 
1.49  1.50  1.50, 1.52  1.48, 1.54  1.48, 1.53  1.47, 1.52  1.49, 1.50  1.47, 1.50 
1.26  1.28  1.24, 1.27  1.23, 1.26  1.26, 1.27  1.26, 1.27  1.26, 1.28  1.26, 1.28 
1.19    1.18  1.19  1.19  1.18  1.19  1.19 
dexp dcal
Ref. 9  Ref. 11  mC24 mC12 oC16 oC24 (3,0)/(4,0) M-carbon
3.04  3.01  3.02  –  –  2.94  –  – 
2.42  2.46  2.40, 2.41  2.37, 2.64  2.46, 2.49  –  –  2.40 
2.08  2.12  2.09, 2.11  2.11, 2.13  2.06  2.03, 2.13  2.07, 2.11  2.08, 2.10 
1.70  1.74  1.71, 1.73  1.71, 1.72  1.68, 1.75  1.70, 1.72  –  1.69, 1.78 
1.49  1.50  1.50, 1.52  1.48, 1.54  1.48, 1.53  1.47, 1.52  1.49, 1.50  1.47, 1.50 
1.26  1.28  1.24, 1.27  1.23, 1.26  1.26, 1.27  1.26, 1.27  1.26, 1.28  1.26, 1.28 
1.19    1.18  1.19  1.19  1.18  1.19  1.19 

The VC-NEB method, which has successfully yielded the formation mechanism of diamond40 and the post-graphite phase bct-C4, M-carbon, and F-carbon,25,26 is employed to reveal the formation mechanism of sp2sp3 hybrid carbon allotropes from graphite. As featured in Fig. 4, mC12, oC16, mC24, and oC24 are assembled through intralayer wrinkling and interlayer buckling of graphite sheets, which is similar to the construction of diamond from graphite. Accordingly, sp2sp3 hybrid carbon allotropes are expected to form in compressed graphite, which strengthens the validity of a mC24 structure for i-carbon.

FIG. 4.

Calculated energy barriers of mC12, oC16, mC24, and oC24 from graphite at ambient pressure. The light gray and dark spheres represent carbon atoms located in neighboring graphite layers.

FIG. 4.

Calculated energy barriers of mC12, oC16, mC24, and oC24 from graphite at ambient pressure. The light gray and dark spheres represent carbon atoms located in neighboring graphite layers.

Close modal

Considering thermodynamic stability, we found that kinetic processes play a critical role in the recovery of a metastable crystal; this can be assessed through studying the transition energy barrier. The formation energy barrier determines the specific requirements (such as pressure and temperature) to activate a phase transition. By using the state-of-the-art VC-NEB method, we investigated the formation energy barriers from graphite to mC12, oC16, mC24, and oC24 at ambient pressure (Fig. 4). The formation energy barrier from graphite to diamond is 0.326 eV/atom using the VC-NEB method under ambient pressure.40 Therefore, extreme pressures and temperatures are necessary to produce diamond from graphite. The activation barriers of mC12, oC16, mC24, and oC24 are 0.443, 0.454, 0.378, and 0.466 eV/atom; these are all higher than for diamond. Consequently, higher pressures or temperatures are needed to obtain these allotropes from graphite. However, it is should be noted that the formation energy barriers of mC24 and diamond are comparable, indicating that the two carbon allotropes may coexist during formation. This may explain the mixture of i-carbon and diamond in shock compressed graphite.9 

Owing to their distinct sp2 and sp3 hybridization states, graphite and diamond are conductive and semiconductive, respectively; in particular, diamond has a wide electronic band gap. Tunable electronic properties (Fig. 5) are expected for carbon allotropes that integrate the two bonding types. Our calculations reveal that (3,0)/(4,0) and (4,0) are metallic, while (4,0)ab is semiconductive with a band gap of 2.12 eV, which is consistent with the previously reported value of 2.3 eV.22oC16 is metallic with a valence band and a conduction band that cross the Fermi level, which corresponds to electron and hole conductivities. Conversely, mC24 is semiconductive with a narrow direct band gap of 0.06 eV, which is less than that of diamond-structured silicon (0.66 eV)41 as calculated via DFT. The indirect band gaps of mC12 and oC24 carbons are 1.07 eV and 1.29 eV, respectively, and therefore narrower than those of guest-free silicon clathrates (1.85–1.9 eV).42,43 Therefore, the new sp2sp3 hybrid carbon allotropes are promising for photovoltaic device applications.

FIG. 5.

Electronic band structures of oC16, mC24, mC12, and oC24 at ambient pressure.

FIG. 5.

Electronic band structures of oC16, mC24, mC12, and oC24 at ambient pressure.

Close modal

Because of their unique atomic arrangements and molecular hybridization states, graphite and diamond have very different material properties: graphite is soft and ductile, whereas diamond is ultrahard and brittle. Considering that sp2sp3 hybrid carbon allotropes show a combination of the structural characteristics of graphite and diamond, we propose that these allotropes likely exhibit superb and multifunctional mechanical properties (Table III). On the basis of Chen’s model,44 we found six sp2sp3 hybrid carbon allotropes that are superhard, with Vickers hardnesses comparable to that of cubic BN (65.2 GPa).44 Bulk modulus (B) represents the resistance of materials to fracture, and shear modulus (G) represents the resistance of materials to plastic deformation. Hence, the B/G ratio is defined as a quantitative index to evaluate the brittleness or ductility of a crystal. A high B/G value is associated with ductility, whereas a low B/G value corresponds to brittleness.45 The bulk moduli and shear moduli of sp2sp3 hybrid carbon allotropes are weaker than for diamond because of their lower atomic density, but their B/G values are higher and comparable to cubic BN (0.99).44 This finding demonstrates that the sp2sp3 hybrid carbon allotropes are more ductile than diamond. The in-plane isotropic Young’s modulus of a graphene sheet and the axial Young’s modulus of a nanotube can both reach 1 TPa;46,47 such high values originate from the high stiffness of sp2 buckling. In Table III, the Young’s moduli of (4,0)ab, (4,0), oC16, mC24, and oC24 in one direction all exceed 1.1 TPa and are 100–200 GPa higher than for diamond. These sp2sp3 hybrid carbon allotropes exhibit superhardness, high Young’s moduli, and good ductility; they therefore have potential for use as materials for abrasive and grinding tools.

TABLE III.

Vickers hardness Hv (GPa), bulk modulus B (GPa), shear modulus G (GPa), B/G ratio, and Young’s moduli along x-, y-, and z-axes (Yx/Yy/Yz, GPa) of carbon allotropes at ambient pressure.

Structure Hv B G B/G Yx/Yy/Yz
diamond  95.9  454.5  545.7  0.83  1062.4/1062.4/1062.4 
(4,0)ab  65.6  376.0  390.4  0.96  1266.2/943.4/580.1 
361a 
(4,0)  60.6  373.0  372.0  1.00  1203.0/1280.9/636.6 
356a 
mC12  51.5  371.4  339.8  1.09  557.8/1038.4/665.6 
oC16  42.9  391.3  318.8  1.23  1024.7/602.7/1183.9 
mC24  65.3  329.0  356.3  0.92  1114.3/1211.8/443.7 
oC24  51.7  371.4  340.4  1.09  1040.0/663.4/1106.7 
Structure Hv B G B/G Yx/Yy/Yz
diamond  95.9  454.5  545.7  0.83  1062.4/1062.4/1062.4 
(4,0)ab  65.6  376.0  390.4  0.96  1266.2/943.4/580.1 
361a 
(4,0)  60.6  373.0  372.0  1.00  1203.0/1280.9/636.6 
356a 
mC12  51.5  371.4  339.8  1.09  557.8/1038.4/665.6 
oC16  42.9  391.3  318.8  1.23  1024.7/602.7/1183.9 
mC24  65.3  329.0  356.3  0.92  1114.3/1211.8/443.7 
oC24  51.7  371.4  340.4  1.09  1040.0/663.4/1106.7 
a

Ref. 22.

Four sp2sp3 hybrid carbon allotropes, namely mC12, oC16, mC24, and oC24, are reported based on first principles calculations. The four allotropes are energetically more favorable than C60 at ambient pressure and are more stable than graphite under compression. By comparing the electron diffraction d-spacing and simulating the formation mechanism of mC24 from graphite with experimental data on i-carbon, which is obtained from compressed graphite, it can be shown that mC24 readily explains i-carbon. Transition pathway investigations indicate that sp2sp3 hybrid carbon allotropes are expected to be synthesized from compressed graphite through intralayer wrinkling and interlayer buckling. oC16 is metallic, whereas mC12, mC24, and oC24 are semiconductive with narrow band gaps comparable to those of silicon allotropes. The sp2sp3 hybrid carbon allotropes show a combination of excellent mechanical characteristics such as superhardness, high Young’s modulus, and good ductility. Therefore, these carbon allotropes could potentially be used as materials in photovoltaic devices or abrasive and grinding tools.

This work was supported by the National Science Foundation of China (Grants Nos. 51421091, 51332005, and 51272227), the Natural Science Foundation for Distinguished Young Scholars of Hebei Province of China (Grant No. E2014203150), and Postgraduate Innovation Project of Hebei Province of China (Grant No. 00302-6370007).

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