Novel Super hard Orthorhombic and Hexagonal C12 allotropes: Crystal structure rationale and DFT investigations

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Matar This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7127640/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Aug, 2025 Read the published version in Structural Chemistry → Version 1 posted 7 You are reading this latest preprint version Abstract Novel orthorhombic and hexagonal C 12 allotropes with respective topologies unj and umy were devised from crystal chemistry supported by subsequent density functional theory DFT-based calculations of ground state structures and energy derived physical properties. Both allotropes were found with high density magnitude of 3.24 g/cm 3 below diamond (3.55 g/cm 3 ) and mechanically stable with large Vickers hardness magnitudes: H V ( unj C 12 ) = 72 GPa and H V ( umy C 12 ) = 45 GPa, letting consider them as ultrahard and superhard respectively. Being dynamically stable with positive phonon frequencies, the two allotropes show relationship trends with diamond experimental C V =f(T) heat capacity discreet values. The electronic band structures show insulation properties with large band gap like diamond for unj C 12 and a smaller gap for semi-conducting umy -C 12 . A holistic interrelationship: “crystal structure ↔ mechanical ↔ dynamic ↔ electronic properties” is proposed. Carbon allotropes topology DFT hardness phonons specific heat Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction The field of carbon research has a particular position among scientists, with a focus on allotropes related to diamond’s physical properties, especially the mechanical and the electronic ones. Many carbon allotropes are identified artificially thanks to programs such as CALYPSO based on evolutionary crystallography [1]. Nevertheless, approaches based on crystal chemistry rationale also help find original allotropes [2]. A library of such original carbon allotropes was conceived to store the devised structures, namely SACADA database [3,4]. Therein, C allotropes are described in different topologies identified using TopCryst program [5]. For instance, Diamond is labeled “ dia” , and its rare hexagonal form lonsdaleite , is labeled “ lon ”. This exceptional material is known as the hardest material with Vickers hardness amounting to 95 GPa is characterized by a high density of ρ ~ 3.55 g/cm 3 arising from the perfectly covalent character of the C-C short connections (1.45 Å) within C(sp 3 ) -like tetrahedral C4 . In this work, starting from crystal engineering of tetrahedral stacking with corner and edge sharing C4 tetrahedra, we identify original superhard allotropes, namely orthorhombic and hexagonal C 12 . Subsequent quantum mechanics calculations based on the Density Functional Theory (DFT) [6,7], the crystal structures were geometrically optimized to the ground state energy and their physical properties were then derived and discussed establishing potential relationship with diamond where it applies. The C 12 allotropes were then identified with unj and umy topologies. On one hand, carbon allotrope with unj topology was found by Conesa back in 2002 in the hexagonal system with C 6 stoichiometry in a work reporting on computer modeled allotropes of Si and Ge [8]; so, presently devised based centered orthorhombic C 12 is proposed as a novel allotrope. On the other hand, C 12 with umy topology is original, not documented in databases to the best of author’s knowledge. It needs to be highlighted that such systems like other ones (cf. [9] and therein references) are metastable carbon allotropes -from the cohesive energy point of view- versus diamond, despite their stability. After this Introduction, the paper is organized as follows: the Theoretical framework and the Computational methodology are given in Section 1; the Crystal Structure characteristics are presented in Section 2; the Mechanical properties from the elastic constants are addressed in Section 3; The Dynamic properties from the Phonons are detailed in Section 4; the pertaining Temperature dependence of the heat capacity in comparison with Diamond experimental data is given in Section 5. Section 6 presents the Electronic band structures. The paper is ended with a Conclusion. 1 Theoretical framework and Computational methodology To determine the ground state structures corresponding to the energy minimum and to derive the mechanical and dynamic properties as well as the electronic band structures, quantum mechanics computations were carried out based on the widely accepted framework of the density functional theory DFT [6,7]. Within DFT, the calculations were performed using the Vienna Ab initio Simulation Package (VASP) code [10,11] and the Projector Augmented Wave (PAW) method [11,12] for the atomic potentials. DFT exchange-correlation (XC) effects were considered using the generalized gradient approximation (GGA) [13]. Relaxation of the atoms onto the ground state structures was performed with the conjugate gradient algorithm according to Press et al . [14]. The Bloechl tetrahedron method [15] with corrections according to the scheme of Methfessel and Paxton [16] was used for geometry optimization and energy calculations. Brillouin-zone (BZ) integrals were approximated by a special k -point sampling according to Monkhorst and Pack [17]. Structural parameters were optimized until atomic forces were below 0.02 eV/Å and all stress components < 0.003 eV/Å 3 . The calculations were converged at an energy cutoff of 400 eV for the plane-wave basis set in terms of the automatic high precision k - point integration in the reciprocal space to obtain a final convergence and relaxation to zero strains for the original stoichiometries presented in this work. In the post-processing of the ground state electronic structures, the charge density projections were operated on the lattice sites. The mechanical stability was inferred from the calculation of the elastic constants Cij. Their processing was operated thanks to the ELATE online program [18]. The outcome provides the bulk (B) and shear (G) modules along different averaging methods; the Voigt method [19] was used here for B V and G V . Two methods of microscopic theory of hardness by Tian et al. [20] and Chen et al. [21] were used to estimate the Vickers hardness (H V ) from the bulk and shear modules B V and G V ( vide infra ). For the assessment of the dynamic stabilities phonons band structures were calculated based on a high resolution of the respective Brillouin zone according to Togo et al . [22]. Experimental specific heat C V data of diamond needed to assess the calculated results of the two allotropes versus experiment were extracted from Victor works [23]. The electronic band structures were obtained using the all-electron DFT-based ASW method [24] and the GGA XC functional [13]. The VESTA (Visualization for Electronic and Structural Analysis) program [25] was used to visualize the crystal structures. 2 Crystal structures analyses. From the fact that the stability of diamond from the perfectly covalent character of the C-C short connections within regular C(sp 3 )-like tetrahedral carbon, crystal chemistry protocols for the search of such carbon allotropes were adopted and carried out. This is somehow a more rational research endeavor than using structure predication techniques. Orthorhombic and hexagonal C 12 were then sketched and submitted to unconstrained geometry optimization, with following topology analyses. The C 12 orthorhombic system characterized in No. 20 C 222 1 space group was identified in unj topology that was firstly referred to in the work of Conesa [8] for hexagonal C 6 and other group IV A elements: Si and Ge. This allotrope is catalogued in SACADA database under N°29 with a calculated density of 3.24 g/cm 3 . The bulk B V and shear G V modules (V referring to Voigt averaging) were given as follows: B V = 402 GPa and G V = 420 GPa. The crystal structure is shown in Fig. 1 a in ball-and stick and polyhedral representations. It can be noted that the tetrahedra are corner sharing throughout the structure; this feature characterizes diamond. The second C 12 allotrope was identified in the hexagonal system with P 6 1 22 space group (No.178) and identified in umy topology considered as original being not catalogued in SACADA database. The structure is shown in Fig. 1 b. The polyhedral representation indicates that the tetrahedra are corner and edge sharing, opposite to orthorhombic C 12 . This feature should be consequent regarding the mechanical and electronic properties, especially the hardness magnitude and the band structure. From Table 1 , the building block tetrahedra are distorted with angles below and above the ideal value characterizing the perfect tetrahedron of 109.47° of C(sp 3 ) like hybridization. The relevant angles are ∠C-C-C = 107° in unj -C 12 and ∠C-C-C = 111–116° in umy -C 12 . In both allotropes the density is ρ = 3.24 g/cm 3 , close to the magnitude of hexagonal C 6 by Conesa [8]. Regarding the ground state energies, the atom averaged values obtained after deducting C atomic energy in a large box (-6.6 eV) from total energy, show a clear trend to decrease of (E coh /at.) from unj to umy C 12 . Both remain smaller than diamond value (-2.47 eV/atom) letting consider them as metastable. Nevertheless, both allotropes will be shown to possess relationship of physical properties with diamond, i.e., the dynamic, thermal, and electronic properties. Table 1 Crystal structural properties and topologies of two C 12 allotropes in base centered orthorhombic and hexagonal space groups Allotrope topology unj C 12 C 222 1 No.20 umy C 12 P 6 1 22 No.178 a, Å 3.5639 4.1449 b, Å 6.1671 - c, Å 3.3673 4.9685 Shortest dist. Å 1.56 1.60 Angle ∠C-C-C ° 107 111/116 Volume, Å 3 74.01 73.93 V/at., Å 3 6.17 6.16 Density ρ (g/cm 3 ) 3.24 3.23 Atomic positions C1 (4 b ) 0; 0.2324, ¼ C2 (8 c ) 0.1514, 0.3838, 0.9167 C1 (12 c ) 0.6350, 0.7186, 0.9519 E total , eV E tot ./at., eV E coh /at., eV -107.8 -8.98 -2.38 -99.12 -8.26 -1.66 E(C) BOX = -6 6 eV 3 Mechanical properties from the elastic constants The investigation of the mechanical properties was based on the calculations of the elastic properties determined by performing finite distortions of the lattice and deriving the elastic constants from the strain-stress relationship. The calculated sets of elastic constants C ij (i and j indicating directions) are given in Table 2 . All C ij values are positive signaling stability of the two allotropes while obeying the stability rules regarding the orthorhombic and hexagonal systems. Using ELATE program introduced above [18], the bulk and the shear modules obtained by averaging the elastic constants using Voigt's [19] method are also given, namely B V and G V . Table 2 Elastic Constants and Voigt-average properties of bulk B V and shear G V as well as hardness H V along two models (all values are in GPa units). C ij C 11 C 22 C 12 C 13 C 33 C 44 C 55 C 66 B V G V G V /B V H V 1 H V 2 unj C 12 932 930 180 126 871 374 489 496 400 425 1.06 72 72 umy C 12 800 800 124 129 613 338 256 256 331 292 0.88 44 45 The largest bulk modulus B V is observed for unj C 12 with B V =400 GPa and G V = 425 GPa. These values are close to hexagonal unj C 6 . Nevertheless, they remain lower than diamond’s: B V =444 GPa and G V = 534 GPa [26]. The corresponding Pugh ratios G V /B V [27] relevant to “ductile-to-brittle” criteria were then calculated. For G V /B V 1 a brittle behavior is deducted. Indeed, for diamond, known for its brittleness, G V /B V = 1.20. The Pugh ratio for the unj C 12 allotrope is close to unity (1.06) below diamond ratio. This shows a trend to brittleness versus much lower magnitude of 0.88 for the other C 12 allotrope assigned with a trend to ductile behavior. Much lower hardness is then expected since the Pugh ratio intervenes in the calculation of the Vickers hardness as shown here below in two models of microscopic theory of hardness: Hv 1 = 0.92(G V /B V ) 1.137 G V 0.708 (Tian et al.) [20] H V 2 = 2(G V 3 /B V 2 ) 0.585 −3 (Chen et al.) [21] The corresponding Vickers hardness (H V ) magnitudes obtained along the two methods are given in the last two columns of Table 2 . The two models lead to close magnitudes with the largest value found for unj C 12 of 72 GPa versus a reduced magnitude for umy C 12 The largest Vickers hardness magnitude remains smaller than diamond’s which amounts to H V =96 GPa [26]. 4 Dynamic and thermodynamic properties To verify the dynamic stability of the carbon allotropes, an analysis of their phonon properties was performed. The phonon band structures are shown in Fig. 2 a and 2 b. They were obtained from a high resolution of the orthorhombic and hexagonal Brillouin zones BZ in accordance with the method of Togo et al . [22]. The bands (red lines) develop along the main directions of the orthorhombic ( unj C 12 ) and hexagonal ( umy C 12 ) BZ (horizontal x -axis), separated by vertical lines for better visualization, while the vertical direction ( y -axis) represents the frequencies ω, given in terahertz (THz). For each crystal system the phonons band structures include 3N bands (N number of atoms) describing three acoustic modes starting from zero energy (ω = 0) at the Γ point (the center of the Brillouin zone) and reaching up to a few terahertz, and 3N-3 optical modes at higher energies. The low-frequency acoustic modes are relevant to the rigid translation modes (two transverse and one longitudinal) of the crystal lattice. The phonon frequencies are all positive, indicating that the allotropes are dynamically stable. The highest bands are observed around ~ 40 THz. Such magnitude is close to the value observed for diamond by Raman spectroscopy [28], letting expect relationship with diamond for the two carbon allotropes. Support for such an assumption needs a study of the thermal properties of both allotropes in comparison to diamond. 5 Temperature dependence of the heat capacity The thermodynamic properties were calculated from the phonon frequencies using the statistical thermodynamic approach [29] on a high-precision sampling mesh in the respective BZ’s. The temperature dependencies of the entropy S and the heat capacity C V are presented in Fig. 3 . Available experimental C V discrete data for diamond [23] are also reported in red closed circles. In both C 12 stoichiometries, S increases with temperature as expected from increased disorder with T. The calculated C V = f (T) curves follow the experimental discrete experimental points of diamond with a better fit observed for unj C 12 whereas umy C 12 shows less agreement. The good agreement of the former with diamond likely arises, besides the corner sharing tetrahedra structure and the high cohesiveness, from the differences of the electronic band structures. 6 Electronic band structures Using the crystal parameters in Table 1 , the electronic band structures were obtained for the two carbon allotropes using the all-electrons DFT-based augmented spherical wave method (ASW) [24] and GGA XC approximation [13]. The band structures are displayed in Fig. 4 . The bands develop along the main directions of the respective Brillouin zones. Along the vertical direction the two panels exhibit an energy gap signaling semi-conducting to insulating behavior. The zero energy is then considered at E V , i.e. at the top of the valence band (VB) separated by an energy gap from the higher energy conduction band (CB). In Fig. 4 a representing unj C 12 the band gap is close to 5 eV, a magnitude observed for diamond, whereas half this magnitude is observed in Fig. 4 b for umy C 12 . Both gaps are of indirect nature, like in diamond. Such electronic band structures let establish trends with the thermal properties (Fig. 3 ): The closest allotrope to diamond thermally is unj C 12 , opposite to umy C 12 . Conclusion Based on quantum mechanics calculations of ground state crystal structures and pertaining physical properties, two novel carbon allotropes with C 12 stoichiometry in orthorhombic and hexagonal crystal systems respectively were proposed with properties related to diamond. Specifically, the structures were found with distorted C4 tetrahedra that are corner sharing for orthorhombic C 12 and corner and edge sharing for hexagonal C 12 . Such specific tetrahedra led to different hardness with the largest magnitude Vickers hardness found for the corner sharing tetrahedra orthorhombic allotrope with H V ( unj C 12 ) = 72 GPa versus H V ( umy C 12 ) = 45 GPa, letting consider them as ultrahard and superhard respectively. Dynamically both allotropes were found stable with positive frequencies revealed from their phonons band structures. Pertaining thermodynamic properties showed specific heat C V curves with different levels of agreement with diamond’s literature experimental values. The closest fit was found for the most cohesive and hardest allotrope unj C 12 qualified with the largest electronic band gap. From the present study, a holistic interrelationship: “crystal structure ↔ mechanical ↔ dynamic ↔ electronic properties” is deducted. Declarations Declaration of Competing Interest The author declares that he has no known competing interests or personal relationships that could have appeared to influence the work reported in this paper. Ethical Approval Not applicable Funding No funding to declare Availability of data and materials Data will be made available on request References S. Zhang, J. He, Z. Zhao, D. Yu, Y. Tian, Discovery of superhard materials via CALYPSO methodology. Chinese Phys. B 28 (2019) 106104. S.F. Matar. From layered 2D carbon to 3D tetrahedral allotropes C 12 and C 18 with physical properties related to diamond: Crystal chemistry and DFT investigations. Progress in Solid State Chemistry, 76, 100492, 2024. R. Hoffmann, A. Kabanov, A. A. Golov, D. M. Proserpio. Homo Citans and Carbon Allotropes: For an Ethics of Citation. 55 (2016) 10962-10976. SACADA (Samara Carbon Allotrope Database). www.sacada.info V. A. Blatov, A.P. Shevchenko; D. M. Proserpio. 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Generalization of intrinsic ductile-to-brittle criteria by Pugh and Pettifor for materials with a cubic crystal structure. Scientific Reports | (2021) 11:4531 R.S. Krishnan. Raman spectrum of diamond. Nature 1945; 155:171. M.T. Dove. Introduction to lattice dynamics. New York, USA: Cambridge University Press; 1993. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Aug, 2025 Read the published version in Structural Chemistry → Version 1 posted Editorial decision: Revision requested 17 Jul, 2025 Reviews received at journal 16 Jul, 2025 Reviewers agreed at journal 16 Jul, 2025 Reviewers invited by journal 15 Jul, 2025 Editor assigned by journal 15 Jul, 2025 Submission checks completed at journal 15 Jul, 2025 First submitted to journal 15 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7127640","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":486639306,"identity":"67d81860-179d-44e5-86e7-033719e4b094","order_by":0,"name":"Samir F. Matar","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0klEQVRIiWNgGAWjYBADOT4wxcbAwCdBjPoDDAzGbDAtbMRqSWwjWot8A3fi5w8199Lb+NcYMHwoO8zAJt2AX4vBAd7NEgeOFee2SbwxYJxxDqhF5gABLQy8GyQOsCUAtZwxYOZtA2qRSCDkMN7NPw78S0hnA2n5S4wWhgO82yQOtiUksPH3GDAzEqPF4DDvNouzfQmGbRJsBQd7zqXzEPSLfHvv5hsV3xLk+fkPb3zwo8xajp9QiDEwwxhA94CM5yGgHhnwE3DPKBgFo2AUjFwAABN2PiXUUWr6AAAAAElFTkSuQmCC","orcid":"","institution":"Lebanese German University (LGU)","correspondingAuthor":true,"prefix":"","firstName":"Samir","middleName":"F.","lastName":"Matar","suffix":""}],"badges":[],"createdAt":"2025-07-15 07:38:35","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7127640/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7127640/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11224-025-02576-6","type":"published","date":"2025-08-20T16:29:35+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":87062696,"identity":"b60f980b-0e9a-425b-8998-ecc336bf8eb1","added_by":"auto","created_at":"2025-07-18 17:30:41","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":183275,"visible":true,"origin":"","legend":"\u003cp\u003eCrystal structures in ball and stick and polyhedral representations (white spheres correspond to a different carbon crystal sites cf. Table 1): a) \u003cstrong\u003eunj \u003c/strong\u003eC\u003csub\u003e12\u003c/sub\u003e, and b) \u003cstrong\u003eumy \u003c/strong\u003eC\u003csub\u003e12\u003c/sub\u003e. The spheres with different colors correspond to the two carbon crystal sites (cf. Table 1).\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7127640/v1/f3178bf980f8879b43ae1856.png"},{"id":87062209,"identity":"e4345ea1-04d6-4259-8f39-cd434be29a33","added_by":"auto","created_at":"2025-07-18 17:22:41","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":76677,"visible":true,"origin":"","legend":"\u003cp\u003ePhonon band structures along the major lines of the respective Brillouin zones. a) \u003cstrong\u003eunj\u003c/strong\u003e C\u003csub\u003e12\u003c/sub\u003e , b) \u003cstrong\u003eumy\u003c/strong\u003e C\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7127640/v1/ecda6d86af657b2cb658a7fc.png"},{"id":87062208,"identity":"34520fe6-8624-4378-9f52-9beb962b050b","added_by":"auto","created_at":"2025-07-18 17:22:41","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":15205,"visible":true,"origin":"","legend":"\u003cp\u003eTemperature dependence of the Entropy (S) and the Specific Heat C\u003csub\u003eV\u003c/sub\u003e: a) Orthorhombic C\u003csub\u003e12 \u003c/sub\u003e\u003cstrong\u003eunj\u003c/strong\u003e , b) Hexagonal \u003cstrong\u003eumy \u003c/strong\u003eC\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7127640/v1/b6401bc061117e354dbed3b6.png"},{"id":87063228,"identity":"ae53e3e5-f82a-4108-b169-dd05cc96663a","added_by":"auto","created_at":"2025-07-18 17:38:41","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":91286,"visible":true,"origin":"","legend":"\u003cp\u003eElectronic band structures of a) Orthorhombic C\u003csub\u003e12 \u003c/sub\u003e\u003cstrong\u003eunj\u003c/strong\u003e , b) Hexagonal \u003cstrong\u003eumy \u003c/strong\u003eC\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7127640/v1/ca25af9f708c96829f983fb2.png"},{"id":89847201,"identity":"93fe3971-5285-4217-8382-3e4ead49ac38","added_by":"auto","created_at":"2025-08-25 16:41:59","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":974287,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7127640/v1/de01825f-7ccd-4fbb-9e2c-15c49c0b2c49.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eNovel Super hard Orthorhombic and Hexagonal C\u003csub\u003e12\u003c/sub\u003e allotropes: Crystal structure rationale and DFT investigations\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe field of carbon research has a particular position among scientists, with a focus on allotropes related to diamond\u0026rsquo;s physical properties, especially the mechanical and the electronic ones. Many carbon allotropes are identified artificially thanks to programs such as CALYPSO based on evolutionary crystallography [1]. Nevertheless, approaches based on crystal chemistry rationale also help find original allotropes [2]. A library of such original carbon allotropes was conceived to store the devised structures, namely SACADA database [3,4]. Therein, C allotropes are described in different topologies identified using TopCryst program [5]. For instance, Diamond is labeled \u0026ldquo;\u003cb\u003edia\u0026rdquo;\u003c/b\u003e, and its rare hexagonal form \u003cem\u003elonsdaleite\u003c/em\u003e, is labeled \u0026ldquo;\u003cb\u003elon\u003c/b\u003e\u0026rdquo;. This exceptional material is known as the hardest material with Vickers hardness amounting to 95 GPa is characterized by a high density of ρ\u0026thinsp;~\u0026thinsp;3.55 g/cm\u003csup\u003e3\u003c/sup\u003e arising from the perfectly covalent character of the C-C short connections (1.45 \u0026Aring;) within C(sp\u003csup\u003e3\u003c/sup\u003e) -like tetrahedral \u003cem\u003eC4\u003c/em\u003e. In this work, starting from crystal engineering of tetrahedral stacking with corner and edge sharing \u003cem\u003eC4\u003c/em\u003e tetrahedra, we identify original superhard allotropes, namely orthorhombic and hexagonal C\u003csub\u003e12\u003c/sub\u003e. Subsequent quantum mechanics calculations based on the Density Functional Theory (DFT) [6,7], the crystal structures were geometrically optimized to the ground state energy and their physical properties were then derived and discussed establishing potential relationship with diamond where it applies. The C\u003csub\u003e12\u003c/sub\u003e allotropes were then identified with \u003cb\u003eunj\u003c/b\u003e and \u003cb\u003eumy\u003c/b\u003e topologies. On one hand, carbon allotrope with \u003cb\u003eunj\u003c/b\u003e topology was found by Conesa back in 2002 in the hexagonal system with C\u003csub\u003e6\u003c/sub\u003e stoichiometry in a work reporting on computer modeled allotropes of Si and Ge [8]; so, presently devised based centered orthorhombic C\u003csub\u003e12\u003c/sub\u003e is proposed as a novel allotrope. On the other hand, C\u003csub\u003e12\u003c/sub\u003e with \u003cb\u003eumy\u003c/b\u003e topology is original, not documented in databases to the best of author\u0026rsquo;s knowledge. It needs to be highlighted that such systems like other ones (cf. [9] and therein references) are metastable carbon allotropes -from the cohesive energy point of view- versus diamond, despite their stability.\u003c/p\u003e\u003cp\u003eAfter this Introduction, the paper is organized as follows: the Theoretical framework and the Computational methodology are given in Section 1; the Crystal Structure characteristics are presented in Section 2; the Mechanical properties from the elastic constants are addressed in Section 3; The Dynamic properties from the Phonons are detailed in Section 4; the pertaining Temperature dependence of the heat capacity in comparison with Diamond experimental data is given in Section 5. Section 6 presents the Electronic band structures. The paper is ended with a Conclusion.\u003c/p\u003e"},{"header":"1 Theoretical framework and Computational methodology","content":"\u003cp\u003eTo determine the ground state structures corresponding to the energy minimum and to derive the mechanical and dynamic properties as well as the electronic band structures, quantum mechanics computations were carried out based on the widely accepted framework of the density functional theory DFT [6,7]. Within DFT, the calculations were performed using the Vienna Ab initio Simulation Package (VASP) code [10,11] and the Projector Augmented Wave (PAW) method [11,12] for the atomic potentials. DFT exchange-correlation (XC) effects were considered using the generalized gradient approximation (GGA) [13]. Relaxation of the atoms onto the ground state structures was performed with the conjugate gradient algorithm according to Press \u003cem\u003eet al\u003c/em\u003e. [14]. The Bloechl tetrahedron method [15] with corrections according to the scheme of Methfessel and Paxton [16] was used for geometry optimization and energy calculations. Brillouin-zone (BZ) integrals were approximated by a special \u003cb\u003ek\u003c/b\u003e-point sampling according to Monkhorst and Pack [17]. Structural parameters were optimized until atomic forces were below 0.02 eV/\u0026Aring; and all stress components\u0026thinsp;\u0026lt;\u0026thinsp;0.003 eV/\u0026Aring;\u003csup\u003e3\u003c/sup\u003e. The calculations were converged at an energy cutoff of 400 eV for the plane-wave basis set in terms of the automatic high precision \u003cb\u003ek\u003c/b\u003e\u003cem\u003e-\u003c/em\u003epoint integration in the reciprocal space to obtain a final convergence and relaxation to zero strains for the original stoichiometries presented in this work. In the post-processing of the ground state electronic structures, the charge density projections were operated on the lattice sites.\u003c/p\u003e\u003cp\u003eThe mechanical stability was inferred from the calculation of the elastic constants Cij. Their processing was operated thanks to the ELATE online program [18]. The outcome provides the bulk (B) and shear (G) modules along different averaging methods; the Voigt method [19] was used here for B\u003csub\u003eV\u003c/sub\u003e and G\u003csub\u003eV\u003c/sub\u003e. Two methods of microscopic theory of hardness by Tian et al. [20] and Chen et al. [21] were used to estimate the Vickers hardness (H\u003csub\u003eV\u003c/sub\u003e) from the bulk and shear modules B\u003csub\u003eV\u003c/sub\u003e and G\u003csub\u003eV\u003c/sub\u003e (\u003cem\u003evide infra\u003c/em\u003e).\u003c/p\u003e\u003cp\u003eFor the assessment of the dynamic stabilities phonons band structures were calculated based on a high resolution of the respective Brillouin zone according to Togo \u003cem\u003eet al\u003c/em\u003e. [22].\u003c/p\u003e\u003cp\u003eExperimental specific heat C\u003csub\u003eV\u003c/sub\u003e data of diamond needed to assess the calculated results of the two allotropes versus experiment were extracted from Victor works [23]. The electronic band structures were obtained using the all-electron DFT-based ASW method [24] and the GGA XC functional [13]. The VESTA (Visualization for Electronic and Structural Analysis) program [25] was used to visualize the crystal structures.\u003c/p\u003e"},{"header":"2 Crystal structures analyses.","content":"\u003cp\u003eFrom the fact that the stability of diamond from the perfectly covalent character of the C-C short connections within regular C(sp\u003csup\u003e3\u003c/sup\u003e)-like tetrahedral carbon, crystal chemistry protocols for the search of such carbon allotropes were adopted and carried out. This is somehow a more rational research endeavor than using structure predication techniques. Orthorhombic and hexagonal C\u003csub\u003e12\u003c/sub\u003e were then sketched and submitted to unconstrained geometry optimization, with following topology analyses.\u003c/p\u003e\u003cp\u003eThe C\u003csub\u003e12\u003c/sub\u003e orthorhombic system characterized in No. 20 \u003cem\u003eC\u003c/em\u003e222\u003csub\u003e1\u003c/sub\u003e space group was identified in \u003cb\u003eunj\u003c/b\u003e topology that was firstly referred to in the work of Conesa [8] for hexagonal C\u003csub\u003e6\u003c/sub\u003e and other group \u003cem\u003eIV\u003c/em\u003e\u003csup\u003eA\u003c/sup\u003e elements: Si and Ge. This allotrope is catalogued in SACADA database under N\u0026deg;29 with a calculated density of 3.24 g/cm\u003csup\u003e3\u003c/sup\u003e. The bulk B\u003csub\u003eV\u003c/sub\u003e and shear G\u003csub\u003eV\u003c/sub\u003e modules (V referring to Voigt averaging) were given as follows: B\u003csub\u003eV\u003c/sub\u003e= 402 GPa and G\u003csub\u003eV\u003c/sub\u003e= 420 GPa. The crystal structure is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea in ball-and stick and polyhedral representations. It can be noted that the tetrahedra are corner sharing throughout the structure; this feature characterizes diamond. The second C\u003csub\u003e12\u003c/sub\u003e allotrope was identified in the hexagonal system with \u003cem\u003eP\u003c/em\u003e6\u003csub\u003e1\u003c/sub\u003e22 space group (No.178) and identified in \u003cb\u003eumy\u003c/b\u003e topology considered as original being not catalogued in SACADA database. The structure is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. The polyhedral representation indicates that the tetrahedra are corner and edge sharing, opposite to orthorhombic C\u003csub\u003e12\u003c/sub\u003e. This feature should be consequent regarding the mechanical and electronic properties, especially the hardness magnitude and the band structure.\u003c/p\u003e\u003cp\u003eFrom Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the building block tetrahedra are distorted with angles below and above the ideal value characterizing the perfect tetrahedron of 109.47\u0026deg; of C(sp\u003csup\u003e3\u003c/sup\u003e) like hybridization. The relevant angles are \u0026ang;C-C-C\u0026thinsp;=\u0026thinsp;107\u0026deg; in \u003cb\u003eunj\u003c/b\u003e-C\u003csub\u003e12\u003c/sub\u003e and \u0026ang;C-C-C\u0026thinsp;=\u0026thinsp;111\u0026ndash;116\u0026deg; in \u003cb\u003eumy\u003c/b\u003e-C\u003csub\u003e12\u003c/sub\u003e. In both allotropes the density is ρ\u0026thinsp;=\u0026thinsp;3.24 g/cm\u003csup\u003e3\u003c/sup\u003e, close to the magnitude of hexagonal C\u003csub\u003e6\u003c/sub\u003e by Conesa [8]. Regarding the ground state energies, the atom averaged values obtained after deducting C atomic energy in a large box (-6.6 eV) from total energy, show a clear trend to decrease of (E\u003csub\u003ecoh\u003c/sub\u003e/at.) from \u003cb\u003eunj\u003c/b\u003e to \u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e. Both remain smaller than diamond value (-2.47\u0026nbsp;eV/atom) letting consider them as metastable. Nevertheless, both allotropes will be shown to possess relationship of physical properties with diamond, i.e., the dynamic, thermal, and electronic properties.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eCrystal structural properties and topologies of two C\u003csub\u003e12\u003c/sub\u003e allotropes in base centered orthorhombic and hexagonal space groups\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAllotrope topology\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eunj C\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e \u003cp\u003e\u003cem\u003eC\u003c/em\u003e222\u003csub\u003e1\u003c/sub\u003e No.20\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eumy C\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e \u003cp\u003e\u003cem\u003eP\u003c/em\u003e6\u003csub\u003e1\u003c/sub\u003e22 No.178\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ea, \u0026Aring;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e3.5639\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e4.1449\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eb, \u0026Aring;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e6.1671\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ec, \u0026Aring;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e3.3673\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e4.9685\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eShortest dist. \u0026Aring;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1.56\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1.60\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAngle \u0026ang;C-C-C \u0026deg;\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e107\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e111/116\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eVolume, \u0026Aring;\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e74.01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e73.93\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eV/at., \u0026Aring;\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e6.17\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e6.16\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eDensity ρ (g/cm\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e3.24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e3.23\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAtomic positions\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eC1 (4\u003cem\u003eb\u003c/em\u003e) 0; 0.2324, \u0026frac14;\u003c/p\u003e\u003cp\u003eC2 (8\u003cem\u003ec\u003c/em\u003e) 0.1514, 0.3838, 0.9167\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eC1 (12\u003cem\u003ec\u003c/em\u003e) 0.6350, 0.7186, 0.9519\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eE\u003csub\u003etotal\u003c/sub\u003e, eV\u003c/p\u003e\u003cp\u003eE\u003csub\u003etot\u003c/sub\u003e./at., eV\u003c/p\u003e\u003cp\u003eE\u003csub\u003ecoh\u003c/sub\u003e/at., eV\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e-107.8\u003c/p\u003e\u003cp\u003e-8.98\u003c/p\u003e\u003cp\u003e-2.38\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e-99.12\u003c/p\u003e\u003cp\u003e-8.26\u003c/p\u003e\u003cp\u003e-1.66\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003ctfoot\u003e\u003ctr\u003e\u003ctd colspan=\"3\"\u003eE(C)\u003csub\u003eBOX\u003c/sub\u003e= -6 6 eV\u003c/td\u003e\u003c/tr\u003e\u003c/tfoot\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"3 Mechanical properties from the elastic constants","content":"\u003cp\u003eThe investigation of the mechanical properties was based on the calculations of the elastic properties determined by performing finite distortions of the lattice and deriving the elastic constants from the strain-stress relationship. The calculated sets of elastic constants C\u003csub\u003eij\u003c/sub\u003e (i and j indicating directions) are given in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. All C\u003csub\u003eij\u003c/sub\u003e values are positive signaling stability of the two allotropes while obeying the stability rules regarding the orthorhombic and hexagonal systems. Using ELATE program introduced above [18], the bulk and the shear modules obtained by averaging the elastic constants using Voigt's [19] method are also given, namely B\u003csub\u003eV\u003c/sub\u003e and G\u003csub\u003eV\u003c/sub\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eElastic Constants and Voigt-average properties of bulk B\u003csub\u003eV\u003c/sub\u003e and shear G\u003csub\u003eV\u003c/sub\u003e as well as hardness H\u003csub\u003eV\u003c/sub\u003e along two models (all values are in GPa units).\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"14\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c11\" colnum=\"11\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c12\" colnum=\"12\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c13\" colnum=\"13\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c14\" colnum=\"14\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eC\u003csub\u003eij\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eC\u003csub\u003e11\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eC\u003csub\u003e22\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eC\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eC\u003csub\u003e13\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003eC\u003csub\u003e33\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003eC\u003csub\u003e44\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e\u003cp\u003eC\u003csub\u003e55\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c9\"\u003e\u003cp\u003eC\u003csub\u003e66\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c10\"\u003e\u003cp\u003eB\u003csub\u003eV\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c11\"\u003e\u003cp\u003eG\u003csub\u003eV\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c12\"\u003e\u003cp\u003eG\u003csub\u003eV\u003c/sub\u003e/B\u003csub\u003eV\u003c/sub\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c13\"\u003e\u003cp\u003eH\u003csub\u003eV\u003c/sub\u003e\u003csup\u003e1\u003c/sup\u003e\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c14\"\u003e\u003cp\u003eH\u003csub\u003eV\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e932\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e930\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e180\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e126\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e871\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e374\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e489\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e496\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e400\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e\u003cp\u003e425\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e\u003cp\u003e1.06\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e\u003cp\u003e72\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e\u003cp\u003e72\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e800\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e800\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e\u003cp\u003e124\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e129\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e613\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e338\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e256\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e256\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e331\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c11\"\u003e\u003cp\u003e292\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c12\"\u003e\u003cp\u003e0.88\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c13\"\u003e\u003cp\u003e44\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c14\"\u003e\u003cp\u003e45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThe largest bulk modulus B\u003csub\u003eV\u003c/sub\u003e is observed for \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e with \u003cem\u003eB\u003c/em\u003e\u003csub\u003eV\u003c/sub\u003e =400 GPa and \u003cem\u003eG\u003c/em\u003e\u003csub\u003eV\u003c/sub\u003e= 425 GPa. These values are close to hexagonal \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e6\u003c/sub\u003e. Nevertheless, they remain lower than diamond\u0026rsquo;s: \u003cem\u003eB\u003c/em\u003e\u003csub\u003eV\u003c/sub\u003e =444 GPa and \u003cem\u003eG\u003c/em\u003e\u003csub\u003eV\u003c/sub\u003e= 534 GPa [26]. The corresponding Pugh ratios G\u003csub\u003eV\u003c/sub\u003e/B\u003csub\u003eV\u003c/sub\u003e [27] relevant to \u0026ldquo;ductile-to-brittle\u0026rdquo; criteria were then calculated. For G\u003csub\u003eV\u003c/sub\u003e/B\u003csub\u003eV\u003c/sub\u003e \u0026lt; 1 a trend to ductile behavior is expected whereas for G\u003csub\u003eV\u003c/sub\u003e/B\u003csub\u003eV\u003c/sub\u003e \u0026gt;1 a brittle behavior is deducted. Indeed, for diamond, known for its brittleness, G\u003csub\u003eV\u003c/sub\u003e/B\u003csub\u003eV\u003c/sub\u003e = 1.20. The Pugh ratio for the \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e allotrope is close to unity (1.06) below diamond ratio. This shows a trend to brittleness versus much lower magnitude of 0.88 for the other C\u003csub\u003e12\u003c/sub\u003e allotrope assigned with a trend to ductile behavior. Much lower hardness is then expected since the Pugh ratio intervenes in the calculation of the Vickers hardness as shown here below in two models of microscopic theory of hardness:\u003c/p\u003e\u003cp\u003eHv\u003csup\u003e1\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.92(G\u003csub\u003eV\u003c/sub\u003e/B\u003csub\u003eV\u003c/sub\u003e)\u003csup\u003e1.137\u003c/sup\u003e G\u003csub\u003eV\u003c/sub\u003e\u003csup\u003e0.708\u003c/sup\u003e (Tian et al.) [20]\u003c/p\u003e\u003cp\u003eH\u003csub\u003eV\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e= 2(G\u003csub\u003eV\u003c/sub\u003e\u003csup\u003e3\u003c/sup\u003e/B\u003csub\u003eV\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e)\u003csup\u003e0.585\u003c/sup\u003e\u0026minus;3 (Chen et al.) [21]\u003c/p\u003e\u003cp\u003eThe corresponding Vickers hardness (H\u003csub\u003eV\u003c/sub\u003e) magnitudes obtained along the two methods are given in the last two columns of Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The two models lead to close magnitudes with the largest value found for \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e of 72 GPa versus a reduced magnitude for \u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e The largest Vickers hardness magnitude remains smaller than diamond\u0026rsquo;s which amounts to H\u003csub\u003eV\u003c/sub\u003e=96 GPa [26].\u003c/p\u003e"},{"header":"4 Dynamic and thermodynamic properties","content":"\u003cp\u003eTo verify the dynamic stability of the carbon allotropes, an analysis of their phonon properties was performed. The phonon band structures are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea and \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb. They were obtained from a high resolution of the orthorhombic and hexagonal Brillouin zones BZ in accordance with the method of Togo \u003cem\u003eet al\u003c/em\u003e. [22]. The bands (red lines) develop along the main directions of the orthorhombic (\u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e) and hexagonal (\u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e) BZ (horizontal \u003cem\u003ex\u003c/em\u003e-axis), separated by vertical lines for better visualization, while the vertical direction (\u003cem\u003ey\u003c/em\u003e-axis) represents the frequencies ω, given in terahertz (THz).\u003c/p\u003e\u003cp\u003eFor each crystal system the phonons band structures include 3N bands (N number of atoms) describing three acoustic modes starting from zero energy (ω\u0026thinsp;=\u0026thinsp;0) at the Γ point (the center of the Brillouin zone) and reaching up to a few terahertz, and 3N-3 optical modes at higher energies. The low-frequency acoustic modes are relevant to the rigid translation modes (two transverse and one longitudinal) of the crystal lattice. The phonon frequencies are all positive, indicating that the allotropes are dynamically stable. The highest bands are observed around ~\u0026thinsp;40 THz. Such magnitude is close to the value observed for diamond by Raman spectroscopy [28], letting expect relationship with diamond for the two carbon allotropes. Support for such an assumption needs a study of the thermal properties of both allotropes in comparison to diamond.\u003c/p\u003e"},{"header":"5 Temperature dependence of the heat capacity","content":"\u003cp\u003eThe thermodynamic properties were calculated from the phonon frequencies using the statistical thermodynamic approach [29] on a high-precision sampling mesh in the respective BZ\u0026rsquo;s. The temperature dependencies of the entropy S and the heat capacity C\u003csub\u003eV\u003c/sub\u003e are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Available experimental C\u003csub\u003eV\u003c/sub\u003e discrete data for diamond [23] are also reported in red closed circles. In both C\u003csub\u003e12\u003c/sub\u003e stoichiometries, S increases with temperature as expected from increased disorder with T. The calculated C\u003csub\u003eV\u003c/sub\u003e = \u003cem\u003ef\u003c/em\u003e(T) curves follow the experimental discrete experimental points of diamond with a better fit observed for \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e whereas \u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e shows less agreement. The good agreement of the former with diamond likely arises, besides the corner sharing tetrahedra structure and the high cohesiveness, from the differences of the electronic band structures.\u003c/p\u003e"},{"header":"6 Electronic band structures","content":"\u003cp\u003eUsing the crystal parameters in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the electronic band structures were obtained for the two carbon allotropes using the all-electrons DFT-based augmented spherical wave method (ASW) [24] and GGA XC approximation [13]. The band structures are displayed in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The bands develop along the main directions of the respective Brillouin zones. Along the vertical direction the two panels exhibit an energy gap signaling semi-conducting to insulating behavior. The zero energy is then considered at E\u003csub\u003eV\u003c/sub\u003e, i.e. at the top of the valence band (VB) separated by an energy gap from the higher energy conduction band (CB). In Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea representing \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e the band gap is close to 5 eV, a magnitude observed for diamond, whereas half this magnitude is observed in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb for \u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e. Both gaps are of indirect nature, like in diamond. Such electronic band structures let establish trends with the thermal properties (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e): The closest allotrope to diamond thermally is \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e, opposite to \u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eBased on quantum mechanics calculations of ground state crystal structures and pertaining physical properties, two novel carbon allotropes with C\u003csub\u003e12\u003c/sub\u003e stoichiometry in orthorhombic and hexagonal crystal systems respectively were proposed with properties related to diamond.\u003c/p\u003e\u003cp\u003eSpecifically, the structures were found with distorted \u003cem\u003eC4\u003c/em\u003e tetrahedra that are corner sharing for orthorhombic C\u003csub\u003e12\u003c/sub\u003e and corner and edge sharing for hexagonal C\u003csub\u003e12\u003c/sub\u003e. Such specific tetrahedra led to different hardness with the largest magnitude Vickers hardness found for the corner sharing tetrahedra orthorhombic allotrope with H\u003csub\u003eV\u003c/sub\u003e(\u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;72 GPa versus H\u003csub\u003eV\u003c/sub\u003e(\u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;45 GPa, letting consider them as ultrahard and superhard respectively. Dynamically both allotropes were found stable with positive frequencies revealed from their phonons band structures. Pertaining thermodynamic properties showed specific heat C\u003csub\u003eV\u003c/sub\u003e curves with different levels of agreement with diamond\u0026rsquo;s literature experimental values. The closest fit was found for the most cohesive and hardest allotrope \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e qualified with the largest electronic band gap.\u003c/p\u003e\u003cp\u003eFrom the present study, a holistic interrelationship: \u0026ldquo;crystal structure \u0026harr; mechanical \u0026harr; dynamic \u0026harr; electronic properties\u0026rdquo; is deducted.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eDeclaration of Competing Interest\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe author declares that he has no known competing interests or personal relationships that could have appeared to influence the work reported in this paper.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eEthical Approval\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eNot applicable\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFunding\u003c/p\u003e\n\u003cp\u003eNo funding to declare\u003c/p\u003e\n\u003cp\u003eAvailability of data and materials\u003c/p\u003e\n\u003cp\u003eData will be made available on request\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eS. Zhang, J. He, Z. Zhao, D. Yu, Y. Tian, Discovery of superhard materials via CALYPSO methodology. Chinese Phys. B 28 (2019) 106104.\u003c/li\u003e\n\u003cli\u003eS.F. Matar. From layered 2D carbon to 3D tetrahedral allotropes C\u003csub\u003e12\u003c/sub\u003e and C\u003csub\u003e18\u003c/sub\u003e with physical properties related to diamond: Crystal chemistry and DFT investigations. Progress in Solid State Chemistry, 76, 100492, 2024.\u003c/li\u003e\n\u003cli\u003eR. Hoffmann, A. Kabanov, A. A. Golov, D. M. Proserpio. Homo Citans and Carbon Allotropes: For an Ethics of Citation. 55 (2016) 10962-10976. \u003c/li\u003e\n\u003cli\u003eSACADA (Samara Carbon Allotrope Database). www.sacada.info \u003c/li\u003e\n\u003cli\u003eV. A. Blatov, A.P. Shevchenko; D. M. Proserpio. Applied topological analysis of crystal structures with the program package ToposPro, Cryst. Growth Des., , 14 (2014), 3576\u0026ndash;3586.\u003c/li\u003e\n\u003cli\u003eP. Hohenberg, W. Kohn. Inhomogeneous electron gas. 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New York, USA: Cambridge University Press; 1993.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"structural-chemistry","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stuc","sideBox":"Learn more about [Structural Chemistry](https://www.springer.com/journal/11224)","snPcode":"11224","submissionUrl":"https://submission.nature.com/new-submission/11224/3","title":"Structural Chemistry","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Carbon allotropes, topology, DFT, hardness, phonons, specific heat","lastPublishedDoi":"10.21203/rs.3.rs-7127640/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7127640/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eNovel orthorhombic and hexagonal C\u003csub\u003e12\u003c/sub\u003e allotropes with respective topologies \u003cb\u003eunj\u003c/b\u003e and \u003cb\u003eumy\u003c/b\u003e were devised from crystal chemistry supported by subsequent density functional theory DFT-based calculations of ground state structures and energy derived physical properties. Both allotropes were found with high density magnitude of 3.24 g/cm\u003csup\u003e3\u003c/sup\u003e below diamond (3.55 g/cm\u003csup\u003e3\u003c/sup\u003e) and mechanically stable with large Vickers hardness magnitudes: H\u003csub\u003eV\u003c/sub\u003e(\u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;72 GPa and H\u003csub\u003eV\u003c/sub\u003e(\u003cb\u003eumy\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;45 GPa, letting consider them as ultrahard and superhard respectively. Being dynamically stable with positive phonon frequencies, the two allotropes show relationship trends with diamond experimental C\u003csub\u003eV\u003c/sub\u003e=f(T) heat capacity discreet values. The electronic band structures show insulation properties with large band gap like diamond for \u003cb\u003eunj\u003c/b\u003e C\u003csub\u003e12\u003c/sub\u003e and a smaller gap for semi-conducting \u003cb\u003eumy\u003c/b\u003e-C\u003csub\u003e12\u003c/sub\u003e. A holistic interrelationship: \u0026ldquo;crystal structure \u0026harr; mechanical \u0026harr; dynamic \u0026harr; electronic properties\u0026rdquo; is proposed.\u003c/p\u003e","manuscriptTitle":"Novel Super hard Orthorhombic and Hexagonal C12 allotropes: Crystal structure rationale and DFT investigations","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-18 17:22:36","doi":"10.21203/rs.3.rs-7127640/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-07-17T04:35:19+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-16T15:33:34+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"182734483670752382150809874561106952984","date":"2025-07-16T05:38:08+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-07-15T18:59:43+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-07-15T18:54:59+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-07-15T16:43:50+00:00","index":"","fulltext":""},{"type":"submitted","content":"Structural Chemistry","date":"2025-07-15T07:36:48+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"structural-chemistry","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stuc","sideBox":"Learn more about [Structural Chemistry](https://www.springer.com/journal/11224)","snPcode":"11224","submissionUrl":"https://submission.nature.com/new-submission/11224/3","title":"Structural Chemistry","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"a2b8acb6-899d-4bb1-8bf3-b98d75ff589f","owner":[],"postedDate":"July 18th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-08-25T16:36:01+00:00","versionOfRecord":{"articleIdentity":"rs-7127640","link":"https://doi.org/10.1007/s11224-025-02576-6","journal":{"identity":"structural-chemistry","isVorOnly":false,"title":"Structural Chemistry"},"publishedOn":"2025-08-20 16:29:35","publishedOnDateReadable":"August 20th, 2025"},"versionCreatedAt":"2025-07-18 17:22:36","video":"","vorDoi":"10.1007/s11224-025-02576-6","vorDoiUrl":"https://doi.org/10.1007/s11224-025-02576-6","workflowStages":[]},"version":"v1","identity":"rs-7127640","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7127640","identity":"rs-7127640","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}