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Mechanical properties, lattice thermal conductivity, infrared and Raman spectrum of the fullerite C24
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Mechanical properties, lattice thermal conductivity, infrared and Raman spectrum of the fullerite C24 Shuangyu Kang, Zhihao Xiang, Huijing Mu, Yingxiang Cai ∗ Department of Physics, Nanchang University, Jiangxi, Nanchang 330031, PR China
a r t i c l e
i n f o
Article history: Received 10 July 2019 Received in revised form 31 August 2019 Accepted 30 September 2019 Available online xxxx Communicated by R. Wu Keywords: Fullerite C24 Carbon polymorphs Mode Grüneisen parameters Raman and infrared spectra
a b s t r a c t Lightweight carbon materials with excellent thermal and mechanical properties have important applications in aerospace industry. In this study, the stability, mechanical properties, lattice thermal conductivity, electronic structure, infrared and Raman spectrum of sp3 hybridized low-density fullerite C24 were investigated according to density functional theory (DFT) calculations. It was found that the fullerite C24 was both thermodynamic and dynamic stable. Quasi-harmonic approximation and Grüneisen parameter calculations clarified why the fullerite C24 had a positive thermal expansion coefficient at low temperature. The fullerite C24 also exhibited excellent mechanical properties. Interestingly, the Vickers hardness of carbon allotropes was found to almost be linear proportional to the density of a carbon material. HSE06 electronic structure calculations showed that it was a semiconductor with direct bandgap of 2.56 eV. Anharmonic lattice dynamic calculations showed that its thermal conductivity was higher than semiconductor silicon. Besides, Raman and infrared active modes as well as the corresponding spectra were presented. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Due to the flexibility of bond hybridization, carbon can form many polymorphs by means of linear (sp), trigonal (sp2 ), and tetrahedral (sp3 ) coordination bonds or combining them together [1–4]. Among carbon polymorphs, all sp3 hybridized structures usually exhibit excellent mechanical properties, thermodynamic and chemical stabilities. For instance, wide-bandgap diamond [5], body-centred tetragonal C4 (bct-C4 ) [6], chiral C6 [7], C-centred orthorhombic C8 [8], M-carbon [9] and W-carbon [10] are superhard materials. These carbon allotropes could be used in many fields, such as electrodes [11] and cutting tool [12]. To obtain all sp3 hybridized three-dimensional (3D) carbon crystals, low-dimensional carbon allotropes have been used as fundamental building blocks. For example, carbon polymorphs (α -Carbon, β -Carbon and γ -Carbon) can be achieved by transversal compressing one-dimensional carbon nanotubes (CNTs) [13,14]. Twodimensional graphite under pressure can transform W carbon, Mcarbon, bct-C4 , hex-diamond, and cub-diamond under high pressure [10,15,16]. In addition, zero-dimensional (0D) fullerenes have also been utilized to prepare 3D carbon materials [17,18]. However, most 0D fullerenes could not generate ideal all sp3 hybridized structures due to the mismatch of point group symmetry.
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Corresponding author. E-mail address: [email protected] (Y. Cai).
https://doi.org/10.1016/j.physleta.2019.126035 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Fullerene molecules are clusters of carbon Cn (20 ≤ n and n is even number, except n = 22). C20 is the smallest fullerene but the least stable structure [19,20]. C24 fullerenes with 4-6 and 5-6 rings are more stable with respect to C20 and its isomers [21]. The 5-6 ring C24 fullerenes with D6 symmetry are lower in total energy than 4-6 ring fullerenes [19,22,21,23], but they can not polymerize into all sp3 hybridized 3D crystals. In contrast, the 4-6 ring fullerenes consist of six tetragons and eight hexagons with three equivalent C4 axes and four equivalent C3 axes, which can form two all sp3 hybridized three-dimensional crystals. One is bcc-C6 [24] also named as 3d-C24 [25], truncated octahedral [26], C6 [27], CA6 [28], clathrate VII [29], sodalite [30] and KI [31]. The other is the simple cubic fullerite C24 [32,33], also named as SCF-C24 [34], cubic C3 [35], CA4 [36] and cubic C24 [28] in previous studies. Recently study shows that the superhard 3d-C24 fullerene crystal has extremely high tensile strength and superior thermal conductivity [25]. However, less study has been performed to investigate the physical properties of fullerite C24 including phonon dispersion, thermal conductivity, Vickers hardness, Raman and infrared (IR) spectra. In this study, the thermodynamic and kinetic stability of fullerite C24 was firstly evaluated. Secondly, its thermal expansion coefficient, macroscopic and microscopic Grüneisen parameters were calculated. Thirdly, electronic band structure, density of states (DOS) and the charge density of valence and conduction bands near Fermi level were presented. Then, its elastic modulus, ten-
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sile properties and Vickers hardness were determined. Finally, its phonon lifetimes, phonon group velocity, lattice thermal conductivity, Raman and IR spectra were investigated. 2. Theoretical methods Our calculations were carried out using generalized gradient approximation (GGA) [37,38] density functional theory (DFT) as implemented in VASP code [39,40]. Projector augmented wave (PAW) method [41,42] was used to describe the interactions between the nucleus and valence electrons of carbon atoms. A planewave basis with a cutoff energy of 400 eV was used to expand the wave functions for all carbon allotropes investigated in this study. The geometries of carbon polymorphs were fully relaxed including the atomic positions and lattice parameters until the residual forces on each atom was less than 0.0001 eV· Å−1 . The Brillouin zone sample meshes were dense enough and the k spacing was less than 0.3 Å−1 for all carbon allotropes. The phonon band structure was determined by the direct supercell method as implemented in the Phonopy code [43]. Bulk and shear modulus were evaluated using the Voigt-Reuss-Hill average scheme implement in the ELATE code [36]. The ideal strength was determined using the method described in previous studies [44–46]. The lattice vectors were incrementally deformed in the direction of the applied strains. Equilibrium volumes at any temperature were calculated by direct minimization of the Helmholtz free energy with respect to its independent lattice constant using the quasi-harmonic approximation. Mode Grüneisen parameter γi and macroscopic Grüneisen parameter γ were calculated from Eq. (1) and (2), respectively.
γi (qν ) = −
V
ωi (qν )
∂ ωi (qν ) ∂V
(1)
where the V was the volume, the quency.
γ = i
ωi (qν ) was the phonon fre-
γi c V ,i i
(2)
c V ,i
where the c V ,i was the partial vibrational mode contributions to the heat capacity, such that C v = ρ1V i c V ,i . The lattice thermal conductivity of fullerite C24 was calculated by solving the linearized phonon Boltzmann equation within the single-mode relaxation time approximation (RTA) and implemented by the Phono3py code [47]. For fullerite C24 , a 7 × 7 × 7 q mesh was utilized to calculate the thermal conductivity according to the Eq. (3).
κ=
1 NV0
C λ νλ
νλ τλ
(3)
λ
where the V 0 was the volume of a unit cell, and the v λ and τλ were the group velocity and phonon mode λ, respectively. The group velocity can be obtained directly from the eigenvalue equation,
υα (λ) ≡ =
∂ ωλ ∂ qα 1 2ωλ
κκ β γ
W β (κ , λ)
∂ D β γ κκ , q W γ κ , λ ∂ qα
(4)
where the ωλ was harmonic frequency, q was wave vector, Wβ (κ , λ) was polarization vector and Dβ γ κκ , q was a dynamical matrix. IR and Raman spectra were simulated using Spectroscopy tools [48], which interfaces with the Phonopy code.
Fig. 1. (a) A C24 fullerene molecule. (b) A C24 cluster formed by chemical binding a core C24 molecule with other six C24 molecules. (c) The top view of a 2 × 2 × 2 fullerite C24 supercell.
3. Results and discussion Fig. 1(a) shows a C24 fullerene molecule. It belongs to the O h point group and consists of six tetragons and eight hexagons. By means of tetragon face-to-face bonding, each C24 fullerene molecule can chemically bind with six other identical molecules to form a fullerene cluster as shown in Fig. 1(b). While more C24 fullerene molecules join the cluster, a 3D fullerite C24 crystal (see Fig. 1(c)) is finally achieved. From C24 molecules to the fullerite C24 crystal, the cohesive energy is about 0.255 eV/carbon. Fullerite C24 ¯ space is an all sp3 -hybridized structure and belongs to the Pm3m group. Its equilibrium lattice parameter, a, is 5.902 Å and three non-equivalent bonds have the length of 1.592 Å for four-six bond, 1.568 Å for four-eight bond and 1.473 Å for six-eight bond. Carbon atoms occupy the Wyckoff positions of 24k (0.633, 0, 0.809). The density of fullerite C24 is only 2.327 g/cm3 , which is slightly higher than that (2.27 g/cm3 ) of graphite [49] or (2.30 g/cm3 ) of h-carbon [50], but far less than that (3.527 g/cm3 ) of diamond [51]. Both thermodynamic and kinetic stabilities are evaluated for the fullerite C24 as shown in Fig. 2. In view of that different exchange correlation functions almost have no effect on the relative stability of carbon allotropes [52], only PAW-PBE is used to investigate the relative stability of fullerite C24 with respective to a few typical carbon allotropes including M-carbon [9], Z-carbon [53], diamond, graphdiyne, T-C8 [54], α -carbon [13], bct C4 [6], T-carbon [55] and h-carbon [50] (see Fig. 2(a)). Except for T-C8 , graphdiyne and h-carbon, other carbon polymorphs are sp3 -hybridized structures. It can be found that the less the volume per carbon atom is, the lower the total energy is for sp3 -hybridized carbon polymorphs. In other words, the higher the density is, the more stable a sp3 carbon allotrope is. The total energy of fullerite C24 is far less than that of T-carbon [55] previous reported, and thus it is thermodynamically more stable. Secondly, we calculate the phonon dispersion of fullerite C24 along the high symmetry direction in the first Brillouin zone. Since no imaginary frequencies (or soft phonon modes) are observed (see Fig. 2(b)), the fullerite C24 is kinetically stable. At point, the highest phonon frequency is 41.88 THz. Since both thermodynamic and kinetic stabilities of fullerite C24 have been confirmed, it should be stable at normal conditions. Three independent elastic constants of fullerite C24 are calculated to be c 11 = 551.1, c 12 = 141.8 and c 44 = 226.3 GPa, respectively. Since they satisfy the Born-Huang mechanical stability criterion (c 11 > 0, c 44 > 0, c 11 > |c12 | and c 11 + 2c12 > 0) [56], the fullerite C24 is also mechanically stable. Based on quasi-harmonic approximation, the free energy of fullerite C24 is calculated from 0 to 900 K with 100 K interval as shown in Fig. 3(a). Red line connects minimum energy points at each temperature. Thermal expansion can be indirectly reproduced by the quasi-harmonic approximation without taking anharmonicity into account. Fig. 3(b) and Fig. 3(c) show the volume and thermal expansion coefficient as functions of temperature. It can be
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Fig. 2. (a) Total energy as a function of volume for fullerite C24 and other typical carbon materials. (b) Phonon dispersion of fullerite C24 . (For interpretation of the colours in the figures, the reader is referred to the web version of this article.)
Fig. 3. (a) Free energy of fullerite C24 as a function of volume at different temperatures. (b) and (c) show the changes of volume and thermal expansion with respective to temperature, respectively. (d) Mode Grüneisen parameters γi . (e) Weighted Grüneisen parameter γ as a function of temperature.
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Fig. 4. (a) Band structures and the density of state (DOS) of fullerite C24 . , X, R, A, Z and M denote the high symmetry points of (0, 0, 0), (0.5, 0, 0), (0.5, 0, 0.5), (0.5, 0.5, 0.5), (0, 0, 0.5) and (0.5, 0.5, 0), respectively. (b) Charge density of valence band maximum (VBM). (c) Charge density of conduction band minimum (CBM).
seen that the thermal expansion coefficient of fullerite C24 is only 3.14 × 10−6 /K at room temperature (300 K). The mode Grüneisen parameter (γi ) of fullerite C24 is shown in Fig. 3(d). According to the equation (1), the γi is related to phonon frequency and volume. The phonon mode balance between the positive and negative γi determines whether the solid expands or contracts during heating. Since most γi are positive and negative γi occur only around 10 THz, the fullerite C24 will expand with the temperature increasing. Fig. 3(e) shows the weighted average γ of mode Grüneisen parameters calculated according to the equation (2). It can be seen that the γ is always positive, which also explains the thermal expansion of fullerite C24 at low temperature. Fig. 4(a) shows the electronic band structure and density of states (DOS) of fullerite C24 . Both valence band maximum (VBM) and the conduction band minimum (CBM) are emphasized with different colours. It can be seen that fullerite C24 is a semiconductor with a direct bandgap of 1.94 eV occurring at the A point in the first Brillouin zone. In contrast, the other C24 fullerene crystal, 3d-C24 [25], is an indirect bandgap semiconductor. In view of that the PBE functional usually underestimates the bandgap, HSE06 functional is also utilized to evaluate the bandgap of fullerite C24 . At the HSE06 level, the bandgap is up to 2.56 eV. Fig. 4(b) and 4(c) show the charge densities of the VBM and CBM, respectively. Both VBM and CBM charge are quite localized, which also discloses the semiconducting nature of fullerite C24 . The VBM charge mainly concentrates on the whole fullerene cage but the CBM charge mainly distributes on the tetragons.
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Fig. 5. (a), (b) and (c) show the spatial dependence of Young’s modulus, shear modulus and Poisson’s ratio. (d) Ideal tensile strength along [100], [110] and [111] directions. (e) Density and Vickers hardness of diamond, Z-carbon, M-carbon, bct C4 , α -Carbon, h-carbon, T-C8 , T-carbon and our materials.
Fig. 5(a)-(c) show the Young’s modulus, shear modulus and Poisson’s ratio in xy plane, respectively. As can be seen that the fullerite C24 behaves almost isotropic. The maximum (minimum) value of Young’s modulus (E), shear modulus (G) and Poisson’s ratio of fullerite C24 are 534.1 (493.0) GPa, 226.3 (204.63) GPa and 0.22 (0.16), respectively. Its bulk modulus (K) is 278.2 GPa. Fig. 5(d) shows the stress-strain curve. It can be found that the ideal tensile strength (σ ) is 63.8, 46.3 and 59.1 GPa along the [100], [110] and [111] directions, respectively. The σ along [100] direction is about one-third of diamond (203.9 GPa) [57] but is close to that of MCarbon (75.1 GPa) [58] and T-C8 (72.9 GPa) [54]. According to the formula H v = 2(G 3 / K 2 )0.585 − 3 [59], the Vickers hardness of fullerite C24 is estimated to be 31.9 GPa. Although the H v of fullerite C24 is not competitive with the superhard carbons, such as Mcarbon (79.2 GPa) [57] and Z-carbon (83 GPa) [57], it is close to the h-carbon (35.52 GPa) [50] and even higher than T-C8 (20.76 GPa) [54]. To disclose the main factors affecting the hardness of carbon allotropes, the H v of diamond, Z-carbon, M-carbon, h-carbon, bct C4 , α -Carbon, T-carbon, T-C8 are also investigated as shown in Fig. 5(e). Interestingly, we find that the H v is almost linear proportional to the density of carbon material. It should be noted that the H v of T-carbon is only 5.56 GPa, which is far less than 61.1 GPa [55] predicted by Sheng et al. In view of its lower density, the H v of T-carbon might be seriously overestimated, which have been proposed in recent study performed by Chen et al. [60]. Fig. 6(a) shows the phonon group velocity. Most of the phonons have the group velocity in the range of 0-5 km/s. Only a few lowfrequency phonons have the speed over 10 km/s. Fig. 6(b) is the phonon lifetime (τ ) of fullerite C24 at room temperature (300 K). The τ is around 2.5 ps for the high frequency (∼ 40 THz) phonons. For lower frequency phonons, the τ can reach 20 THz, but the phonon density is quite low. In contrast, the phonon density at 30 THz is the highest and the τ is about 5 km/s. Fig. 6(c) shows lattice thermal conductivity (κ L ) of fullerite C24 . At 300 K, the κ L is only 228.5 W/m·k, which is far less than that (879.2 W/m·k) of 3d-C24 [25] but is still higher than some typical semiconductors, such as Si (153 W/m·k) [61] and Ge (58 W/m·k) [62]. With the increasing of temperature, the κ L abrupt decreases due to enhancing phonon dispersion. The IR and Raman vibration modes of fullerite C24 at ( ) point are also investigated as shown in Fig. 6(d). The space group theoretical analysis indicates that fullerite C24 has five IR active modes (5T 1u ) and ten Raman active modes (2 A 1g + 4E g + 4T 2g ). For Raman spectrum, eight shift peaks
Fig. 6. (a) Phonon group velocities of fullerite C24 . (b) Phonon lifetime of fullerite C24 at 300 K. (c) Calculated lattice thermal conductivity of fullerite C24 as a function of temperature. (d) Simulated Raman and IR spectra for fullerite C24 , respectively.
occur at the frequency of 24.3 THz (810 cm−1 ), 25 THz (833.3 cm−1 ), 27.9 THz (930 cm−1 ), 29.3 THz (976.7 cm−1 ), 32.7 THz (1090 cm−1 ), 37.4 THz (1246.7 cm−1 ), 40.7 THz (1356.7 cm−1 ) and 41.9 THz (1396.7 cm−1 ), respectively. For IR spectrum, four visible peaks occur at 20.1 THz (670 cm−1 ), 26.4 THz (880 cm−1 ), 30.4 THz (1013 cm−1 ) and 38.7 THz (1290 cm−1 ), respectively. Its unique Raman and IR spectrum could be used as fingers to distinguish fullerite C24 from other carbon allotropes. 4. Conclusions In conclusion, our DFT calculations confirmed that sp3 hybridized fullerite C24 was both thermodynamics and dynamic stability. Its thermal properties including, thermal expansion coefficient and Grüneisen parameters were determined by quasiharmonic approximation. Electron structure calculations indicated that fullerite C24 was a semiconductor with a direct band gap of 1.94 eV at the PBE level and 2.56 eV at the HSE06 level. Lightweight fullerite C24 also exhibited excellent mechanical properties. Its maximum (minimum) Young’s modulus (E) and shear modulus (G) were 534.1 (493.0) and 226.3 (204.63) GPa, respectively. Its bulk modulus (K) was 278.2 GPa. Along the [100], [110] and [111] directions, its ideal tensile strength (σ ) was 63.8, 46.3 and 59.1 GPa, respectively. Our study also showed that the Vickers hardness was almost linear proportional to the densities of carbon allotropes. The Vickers hardness of fullerite C24 is 31.9 GPa. Anharmonic lattice dynamics calculations showed that the thermal conductivity of fullerite C24 was 228.5 W/m·k at room temperature due to low phonon group velocities and short phonon lifetimes for most phonons. In addition, the IR and Raman spectra of fullerite C24 were determined, which could be used to distinguish fullerite C24 from other carbon allotropes. Acknowledgements This work has been supported financially by the National Natural Science Foundation of China (Grant No. 11464028), Natural Science Foundation of Jiangxi Province, China (Grant No. 20171ACB21007) and Department of Education of Jiangxi Province, China (Grant No. GJJ150025).
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Mechanical properties, lattice thermal conductivity, infrared and Raman spectrum of the fullerite C24 Shuangyu Kang, Zhihao Xiang, Huijing Mu, Yingxiang Cai ∗ Department of Physics, Nanchang University, Jiangxi, Nanchang 330031, PR China
a r t i c l e
i n f o
Article history: Received 10 July 2019 Received in revised form 31 August 2019 Accepted 30 September 2019 Available online xxxx Communicated by R. Wu Keywords: Fullerite C24 Carbon polymorphs Mode Grüneisen parameters Raman and infrared spectra
a b s t r a c t Lightweight carbon materials with excellent thermal and mechanical properties have important applications in aerospace industry. In this study, the stability, mechanical properties, lattice thermal conductivity, electronic structure, infrared and Raman spectrum of sp3 hybridized low-density fullerite C24 were investigated according to density functional theory (DFT) calculations. It was found that the fullerite C24 was both thermodynamic and dynamic stable. Quasi-harmonic approximation and Grüneisen parameter calculations clarified why the fullerite C24 had a positive thermal expansion coefficient at low temperature. The fullerite C24 also exhibited excellent mechanical properties. Interestingly, the Vickers hardness of carbon allotropes was found to almost be linear proportional to the density of a carbon material. HSE06 electronic structure calculations showed that it was a semiconductor with direct bandgap of 2.56 eV. Anharmonic lattice dynamic calculations showed that its thermal conductivity was higher than semiconductor silicon. Besides, Raman and infrared active modes as well as the corresponding spectra were presented. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Due to the flexibility of bond hybridization, carbon can form many polymorphs by means of linear (sp), trigonal (sp2 ), and tetrahedral (sp3 ) coordination bonds or combining them together [1–4]. Among carbon polymorphs, all sp3 hybridized structures usually exhibit excellent mechanical properties, thermodynamic and chemical stabilities. For instance, wide-bandgap diamond [5], body-centred tetragonal C4 (bct-C4 ) [6], chiral C6 [7], C-centred orthorhombic C8 [8], M-carbon [9] and W-carbon [10] are superhard materials. These carbon allotropes could be used in many fields, such as electrodes [11] and cutting tool [12]. To obtain all sp3 hybridized three-dimensional (3D) carbon crystals, low-dimensional carbon allotropes have been used as fundamental building blocks. For example, carbon polymorphs (α -Carbon, β -Carbon and γ -Carbon) can be achieved by transversal compressing one-dimensional carbon nanotubes (CNTs) [13,14]. Twodimensional graphite under pressure can transform W carbon, Mcarbon, bct-C4 , hex-diamond, and cub-diamond under high pressure [10,15,16]. In addition, zero-dimensional (0D) fullerenes have also been utilized to prepare 3D carbon materials [17,18]. However, most 0D fullerenes could not generate ideal all sp3 hybridized structures due to the mismatch of point group symmetry.
*
Corresponding author. E-mail address: [email protected] (Y. Cai).
https://doi.org/10.1016/j.physleta.2019.126035 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Fullerene molecules are clusters of carbon Cn (20 ≤ n and n is even number, except n = 22). C20 is the smallest fullerene but the least stable structure [19,20]. C24 fullerenes with 4-6 and 5-6 rings are more stable with respect to C20 and its isomers [21]. The 5-6 ring C24 fullerenes with D6 symmetry are lower in total energy than 4-6 ring fullerenes [19,22,21,23], but they can not polymerize into all sp3 hybridized 3D crystals. In contrast, the 4-6 ring fullerenes consist of six tetragons and eight hexagons with three equivalent C4 axes and four equivalent C3 axes, which can form two all sp3 hybridized three-dimensional crystals. One is bcc-C6 [24] also named as 3d-C24 [25], truncated octahedral [26], C6 [27], CA6 [28], clathrate VII [29], sodalite [30] and KI [31]. The other is the simple cubic fullerite C24 [32,33], also named as SCF-C24 [34], cubic C3 [35], CA4 [36] and cubic C24 [28] in previous studies. Recently study shows that the superhard 3d-C24 fullerene crystal has extremely high tensile strength and superior thermal conductivity [25]. However, less study has been performed to investigate the physical properties of fullerite C24 including phonon dispersion, thermal conductivity, Vickers hardness, Raman and infrared (IR) spectra. In this study, the thermodynamic and kinetic stability of fullerite C24 was firstly evaluated. Secondly, its thermal expansion coefficient, macroscopic and microscopic Grüneisen parameters were calculated. Thirdly, electronic band structure, density of states (DOS) and the charge density of valence and conduction bands near Fermi level were presented. Then, its elastic modulus, ten-
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sile properties and Vickers hardness were determined. Finally, its phonon lifetimes, phonon group velocity, lattice thermal conductivity, Raman and IR spectra were investigated. 2. Theoretical methods Our calculations were carried out using generalized gradient approximation (GGA) [37,38] density functional theory (DFT) as implemented in VASP code [39,40]. Projector augmented wave (PAW) method [41,42] was used to describe the interactions between the nucleus and valence electrons of carbon atoms. A planewave basis with a cutoff energy of 400 eV was used to expand the wave functions for all carbon allotropes investigated in this study. The geometries of carbon polymorphs were fully relaxed including the atomic positions and lattice parameters until the residual forces on each atom was less than 0.0001 eV· Å−1 . The Brillouin zone sample meshes were dense enough and the k spacing was less than 0.3 Å−1 for all carbon allotropes. The phonon band structure was determined by the direct supercell method as implemented in the Phonopy code [43]. Bulk and shear modulus were evaluated using the Voigt-Reuss-Hill average scheme implement in the ELATE code [36]. The ideal strength was determined using the method described in previous studies [44–46]. The lattice vectors were incrementally deformed in the direction of the applied strains. Equilibrium volumes at any temperature were calculated by direct minimization of the Helmholtz free energy with respect to its independent lattice constant using the quasi-harmonic approximation. Mode Grüneisen parameter γi and macroscopic Grüneisen parameter γ were calculated from Eq. (1) and (2), respectively.
γi (qν ) = −
V
ωi (qν )
∂ ωi (qν ) ∂V
(1)
where the V was the volume, the quency.
γ = i
ωi (qν ) was the phonon fre-
γi c V ,i i
(2)
c V ,i
where the c V ,i was the partial vibrational mode contributions to the heat capacity, such that C v = ρ1V i c V ,i . The lattice thermal conductivity of fullerite C24 was calculated by solving the linearized phonon Boltzmann equation within the single-mode relaxation time approximation (RTA) and implemented by the Phono3py code [47]. For fullerite C24 , a 7 × 7 × 7 q mesh was utilized to calculate the thermal conductivity according to the Eq. (3).
κ=
1 NV0
C λ νλ
νλ τλ
(3)
λ
where the V 0 was the volume of a unit cell, and the v λ and τλ were the group velocity and phonon mode λ, respectively. The group velocity can be obtained directly from the eigenvalue equation,
υα (λ) ≡ =
∂ ωλ ∂ qα 1 2ωλ
κκ β γ
W β (κ , λ)
∂ D β γ κκ , q W γ κ , λ ∂ qα
(4)
where the ωλ was harmonic frequency, q was wave vector, Wβ (κ , λ) was polarization vector and Dβ γ κκ , q was a dynamical matrix. IR and Raman spectra were simulated using Spectroscopy tools [48], which interfaces with the Phonopy code.
Fig. 1. (a) A C24 fullerene molecule. (b) A C24 cluster formed by chemical binding a core C24 molecule with other six C24 molecules. (c) The top view of a 2 × 2 × 2 fullerite C24 supercell.
3. Results and discussion Fig. 1(a) shows a C24 fullerene molecule. It belongs to the O h point group and consists of six tetragons and eight hexagons. By means of tetragon face-to-face bonding, each C24 fullerene molecule can chemically bind with six other identical molecules to form a fullerene cluster as shown in Fig. 1(b). While more C24 fullerene molecules join the cluster, a 3D fullerite C24 crystal (see Fig. 1(c)) is finally achieved. From C24 molecules to the fullerite C24 crystal, the cohesive energy is about 0.255 eV/carbon. Fullerite C24 ¯ space is an all sp3 -hybridized structure and belongs to the Pm3m group. Its equilibrium lattice parameter, a, is 5.902 Å and three non-equivalent bonds have the length of 1.592 Å for four-six bond, 1.568 Å for four-eight bond and 1.473 Å for six-eight bond. Carbon atoms occupy the Wyckoff positions of 24k (0.633, 0, 0.809). The density of fullerite C24 is only 2.327 g/cm3 , which is slightly higher than that (2.27 g/cm3 ) of graphite [49] or (2.30 g/cm3 ) of h-carbon [50], but far less than that (3.527 g/cm3 ) of diamond [51]. Both thermodynamic and kinetic stabilities are evaluated for the fullerite C24 as shown in Fig. 2. In view of that different exchange correlation functions almost have no effect on the relative stability of carbon allotropes [52], only PAW-PBE is used to investigate the relative stability of fullerite C24 with respective to a few typical carbon allotropes including M-carbon [9], Z-carbon [53], diamond, graphdiyne, T-C8 [54], α -carbon [13], bct C4 [6], T-carbon [55] and h-carbon [50] (see Fig. 2(a)). Except for T-C8 , graphdiyne and h-carbon, other carbon polymorphs are sp3 -hybridized structures. It can be found that the less the volume per carbon atom is, the lower the total energy is for sp3 -hybridized carbon polymorphs. In other words, the higher the density is, the more stable a sp3 carbon allotrope is. The total energy of fullerite C24 is far less than that of T-carbon [55] previous reported, and thus it is thermodynamically more stable. Secondly, we calculate the phonon dispersion of fullerite C24 along the high symmetry direction in the first Brillouin zone. Since no imaginary frequencies (or soft phonon modes) are observed (see Fig. 2(b)), the fullerite C24 is kinetically stable. At point, the highest phonon frequency is 41.88 THz. Since both thermodynamic and kinetic stabilities of fullerite C24 have been confirmed, it should be stable at normal conditions. Three independent elastic constants of fullerite C24 are calculated to be c 11 = 551.1, c 12 = 141.8 and c 44 = 226.3 GPa, respectively. Since they satisfy the Born-Huang mechanical stability criterion (c 11 > 0, c 44 > 0, c 11 > |c12 | and c 11 + 2c12 > 0) [56], the fullerite C24 is also mechanically stable. Based on quasi-harmonic approximation, the free energy of fullerite C24 is calculated from 0 to 900 K with 100 K interval as shown in Fig. 3(a). Red line connects minimum energy points at each temperature. Thermal expansion can be indirectly reproduced by the quasi-harmonic approximation without taking anharmonicity into account. Fig. 3(b) and Fig. 3(c) show the volume and thermal expansion coefficient as functions of temperature. It can be
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Fig. 2. (a) Total energy as a function of volume for fullerite C24 and other typical carbon materials. (b) Phonon dispersion of fullerite C24 . (For interpretation of the colours in the figures, the reader is referred to the web version of this article.)
Fig. 3. (a) Free energy of fullerite C24 as a function of volume at different temperatures. (b) and (c) show the changes of volume and thermal expansion with respective to temperature, respectively. (d) Mode Grüneisen parameters γi . (e) Weighted Grüneisen parameter γ as a function of temperature.
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Fig. 4. (a) Band structures and the density of state (DOS) of fullerite C24 . , X, R, A, Z and M denote the high symmetry points of (0, 0, 0), (0.5, 0, 0), (0.5, 0, 0.5), (0.5, 0.5, 0.5), (0, 0, 0.5) and (0.5, 0.5, 0), respectively. (b) Charge density of valence band maximum (VBM). (c) Charge density of conduction band minimum (CBM).
seen that the thermal expansion coefficient of fullerite C24 is only 3.14 × 10−6 /K at room temperature (300 K). The mode Grüneisen parameter (γi ) of fullerite C24 is shown in Fig. 3(d). According to the equation (1), the γi is related to phonon frequency and volume. The phonon mode balance between the positive and negative γi determines whether the solid expands or contracts during heating. Since most γi are positive and negative γi occur only around 10 THz, the fullerite C24 will expand with the temperature increasing. Fig. 3(e) shows the weighted average γ of mode Grüneisen parameters calculated according to the equation (2). It can be seen that the γ is always positive, which also explains the thermal expansion of fullerite C24 at low temperature. Fig. 4(a) shows the electronic band structure and density of states (DOS) of fullerite C24 . Both valence band maximum (VBM) and the conduction band minimum (CBM) are emphasized with different colours. It can be seen that fullerite C24 is a semiconductor with a direct bandgap of 1.94 eV occurring at the A point in the first Brillouin zone. In contrast, the other C24 fullerene crystal, 3d-C24 [25], is an indirect bandgap semiconductor. In view of that the PBE functional usually underestimates the bandgap, HSE06 functional is also utilized to evaluate the bandgap of fullerite C24 . At the HSE06 level, the bandgap is up to 2.56 eV. Fig. 4(b) and 4(c) show the charge densities of the VBM and CBM, respectively. Both VBM and CBM charge are quite localized, which also discloses the semiconducting nature of fullerite C24 . The VBM charge mainly concentrates on the whole fullerene cage but the CBM charge mainly distributes on the tetragons.
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Fig. 5. (a), (b) and (c) show the spatial dependence of Young’s modulus, shear modulus and Poisson’s ratio. (d) Ideal tensile strength along [100], [110] and [111] directions. (e) Density and Vickers hardness of diamond, Z-carbon, M-carbon, bct C4 , α -Carbon, h-carbon, T-C8 , T-carbon and our materials.
Fig. 5(a)-(c) show the Young’s modulus, shear modulus and Poisson’s ratio in xy plane, respectively. As can be seen that the fullerite C24 behaves almost isotropic. The maximum (minimum) value of Young’s modulus (E), shear modulus (G) and Poisson’s ratio of fullerite C24 are 534.1 (493.0) GPa, 226.3 (204.63) GPa and 0.22 (0.16), respectively. Its bulk modulus (K) is 278.2 GPa. Fig. 5(d) shows the stress-strain curve. It can be found that the ideal tensile strength (σ ) is 63.8, 46.3 and 59.1 GPa along the [100], [110] and [111] directions, respectively. The σ along [100] direction is about one-third of diamond (203.9 GPa) [57] but is close to that of MCarbon (75.1 GPa) [58] and T-C8 (72.9 GPa) [54]. According to the formula H v = 2(G 3 / K 2 )0.585 − 3 [59], the Vickers hardness of fullerite C24 is estimated to be 31.9 GPa. Although the H v of fullerite C24 is not competitive with the superhard carbons, such as Mcarbon (79.2 GPa) [57] and Z-carbon (83 GPa) [57], it is close to the h-carbon (35.52 GPa) [50] and even higher than T-C8 (20.76 GPa) [54]. To disclose the main factors affecting the hardness of carbon allotropes, the H v of diamond, Z-carbon, M-carbon, h-carbon, bct C4 , α -Carbon, T-carbon, T-C8 are also investigated as shown in Fig. 5(e). Interestingly, we find that the H v is almost linear proportional to the density of carbon material. It should be noted that the H v of T-carbon is only 5.56 GPa, which is far less than 61.1 GPa [55] predicted by Sheng et al. In view of its lower density, the H v of T-carbon might be seriously overestimated, which have been proposed in recent study performed by Chen et al. [60]. Fig. 6(a) shows the phonon group velocity. Most of the phonons have the group velocity in the range of 0-5 km/s. Only a few lowfrequency phonons have the speed over 10 km/s. Fig. 6(b) is the phonon lifetime (τ ) of fullerite C24 at room temperature (300 K). The τ is around 2.5 ps for the high frequency (∼ 40 THz) phonons. For lower frequency phonons, the τ can reach 20 THz, but the phonon density is quite low. In contrast, the phonon density at 30 THz is the highest and the τ is about 5 km/s. Fig. 6(c) shows lattice thermal conductivity (κ L ) of fullerite C24 . At 300 K, the κ L is only 228.5 W/m·k, which is far less than that (879.2 W/m·k) of 3d-C24 [25] but is still higher than some typical semiconductors, such as Si (153 W/m·k) [61] and Ge (58 W/m·k) [62]. With the increasing of temperature, the κ L abrupt decreases due to enhancing phonon dispersion. The IR and Raman vibration modes of fullerite C24 at ( ) point are also investigated as shown in Fig. 6(d). The space group theoretical analysis indicates that fullerite C24 has five IR active modes (5T 1u ) and ten Raman active modes (2 A 1g + 4E g + 4T 2g ). For Raman spectrum, eight shift peaks
Fig. 6. (a) Phonon group velocities of fullerite C24 . (b) Phonon lifetime of fullerite C24 at 300 K. (c) Calculated lattice thermal conductivity of fullerite C24 as a function of temperature. (d) Simulated Raman and IR spectra for fullerite C24 , respectively.
occur at the frequency of 24.3 THz (810 cm−1 ), 25 THz (833.3 cm−1 ), 27.9 THz (930 cm−1 ), 29.3 THz (976.7 cm−1 ), 32.7 THz (1090 cm−1 ), 37.4 THz (1246.7 cm−1 ), 40.7 THz (1356.7 cm−1 ) and 41.9 THz (1396.7 cm−1 ), respectively. For IR spectrum, four visible peaks occur at 20.1 THz (670 cm−1 ), 26.4 THz (880 cm−1 ), 30.4 THz (1013 cm−1 ) and 38.7 THz (1290 cm−1 ), respectively. Its unique Raman and IR spectrum could be used as fingers to distinguish fullerite C24 from other carbon allotropes. 4. Conclusions In conclusion, our DFT calculations confirmed that sp3 hybridized fullerite C24 was both thermodynamics and dynamic stability. Its thermal properties including, thermal expansion coefficient and Grüneisen parameters were determined by quasiharmonic approximation. Electron structure calculations indicated that fullerite C24 was a semiconductor with a direct band gap of 1.94 eV at the PBE level and 2.56 eV at the HSE06 level. Lightweight fullerite C24 also exhibited excellent mechanical properties. Its maximum (minimum) Young’s modulus (E) and shear modulus (G) were 534.1 (493.0) and 226.3 (204.63) GPa, respectively. Its bulk modulus (K) was 278.2 GPa. Along the [100], [110] and [111] directions, its ideal tensile strength (σ ) was 63.8, 46.3 and 59.1 GPa, respectively. Our study also showed that the Vickers hardness was almost linear proportional to the densities of carbon allotropes. The Vickers hardness of fullerite C24 is 31.9 GPa. Anharmonic lattice dynamics calculations showed that the thermal conductivity of fullerite C24 was 228.5 W/m·k at room temperature due to low phonon group velocities and short phonon lifetimes for most phonons. In addition, the IR and Raman spectra of fullerite C24 were determined, which could be used to distinguish fullerite C24 from other carbon allotropes. Acknowledgements This work has been supported financially by the National Natural Science Foundation of China (Grant No. 11464028), Natural Science Foundation of Jiangxi Province, China (Grant No. 20171ACB21007) and Department of Education of Jiangxi Province, China (Grant No. GJJ150025).
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