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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Computationally Designed Crystal Structures of the Supertetrahedral AlX (X=B, C, Al, Si) Solids 4

Iliya V. Getmanskii, Vitaliy V. Koval, Alexander I. Boldyrev, Ruslan M. Minyaev, and Vladimir I. Minkin J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b10248 • Publication Date (Web): 14 Dec 2018 Downloaded from http://pubs.acs.org on December 18, 2018

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The Journal of Physical Chemistry

Computationally Designed Crystal Structures of the Supertetrahedral Al4X (X=B,C,Al,Si) Solids Iliya V. Getmanskii1, Vitaliy V. Koval1, Alexander I. Boldyrev1,2,* Ruslan M. Minyaev1,* and Vladimir I. Minkin1* 1Institute of Physical and Organic Chemistry, Southern Federal University, 194/2 Stachka Ave., 344090, Rostov-on-Don, Russian Federation 2Department of Chemistry and Biochemistry, Utah State University, Logan, Utah 84322, United States of America *e-mail:[email protected] *e-mail:[email protected] *e-mail:[email protected]

ABSTRACT New metastable crystalline forms of the supertetrahedral Al4X (X=B, C, Al, Si) solids have been computationally designed using density functional theory calculations with imposing of periodic boundary conditions. The geometric and electronic structures of the predicted new systems were calculated on the basis of the diamond lattice in which all carbon atoms are replaced by the Al4X structural units, where X is boron, carbon, aluminum and silicon atoms. The calculations showed that the dynamic stability of the Al4X crystal structures critically depends on the nature of the bridging atom X: supertetrahedral Al4C and Al4Si solids are dynamically stable, whereas Al4B and Al4Al ones are unstable. INTRODUCTION Acquisition of new materials with the non-standard structure and unusual properties is a challengeable task of the contemporary micro- and nanoelectronics. An important role in the search for such type structures is played by the computational design and theoretical appraisal of new compounds and structural blocks. This kind of research is well represented by the vastly developed area of the molecular modeling of various modifications of crystalline forms of carbon1,2 started with the seminal work of Burdett and Lee,3 who were the first to note that a tetrahedral carbon atom in the crystal lattice of diamond can be replaced by a C4 tetrahedron without loss of symmetry and periodicity of the crystal form with a new allotropic supertetrahedral form of carbon. In a similar way, another new allotropic form of carbon was built up by replacing carbon atoms by C8 cubes to give the "supercubane" which, according to the EHMO calculations,4 is denser and harder than diamond. The later DFT calculations did not, however, confirm this conclusion and the calculations performed on the supertetrahedral diamond (T-carbon) showed5 that this allotropic form of carbon would have lower hardness than ACS Paragon Plus Environment

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diamond. Neither of the subsequently, proposed various modifications of the diamond-like structures based on bridging carbon atoms with acetylene and bis-acetylene linkers was found to excel diamond in density and hardness.1,2 We have previously shown that the idea to use the diamond lattice as the template for the construction of stable crystal forms can be successfully exploited in the computational design of novel allotropic modifications of other than carbon main-group elements, superterahedral boron6 and supertetrahedral aluminum.7 In the present paper, this approach is extended to the study of geometric and electronic structures of solids constructed on the basis of a diamond lattice in which pairs of the carbon atoms are replaced by Al4–X fragments, where atoms X (X = B, C, Al, Si) serve as the linkers between tetrahedral Al4 units. METHODS The calculations were carried out using Vienna Ab initio Simulations Package(VASP)8-11 with PAW pseudopotentials12,13 and the PBEsol density functional.14 The plane-wave cutoff energy 750 eV of the associated pseudopotentials was used. The Brillouin zone has been sampled by Monkhorst-Pack method15 with automatic generated grid 15×15×15. The calculations of phonon spectra were performed using the PHONOPY package.16 The cohesive energy (𝐸c) per primitive unit cell for the Al4X crystal is calculated by using the expression 𝐸c = 𝐸bulk ―4𝐸Al ― 𝐸X, where 𝐸bulk is the total energy per primitive unit cell, including 5 atoms; 𝐸Al and 𝐸X are the total energies of isolated Al and X atoms, respectively. The hardness of the crystals is estimated using methods proposed by Šimůnek17 and Tian18. The elastic constants of the crystals are calculated on the basis of the stress tensor at different distortions of the lattice using the VASP package. The bulk modulus, shear modulus, Young's modulus and Poisson's ratio of polycrystals are obtained in the framework of Voigt-Reuss-Hill (VRH) approximation19. The frequency dependent dielectric function was calculated in the independent particle approximation (IPA), as implemented in the VASP code. The calculated geometric characteristics of the studied systems have been visualized by using VESTA software.20 RESULTS AND DISCUSSION A schematic view of the crystalline mixed forms of supertetrahedral structures and their supercells is shown in Fig.1.

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Figure 1. Schematic depiction of the conventional cubic unit cell of the crystal structure cFAl4X (X=B, C, Al, Si). Atoms Al and X are designated by blue and brown colors, respectively. The calculated phonon spectra of cF-Al4B and cF-Al4Al depicted in Fig.S1-S2 contain low-frequency branches entering the imaginary region indicative of the dynamic instability of the consideredcrystal structures. The calculated phonon spectra for cF-Al4C and cF-Al4Si shown in Fig.2-3 have no lowfrequency branches entering the imaginary region. Therefore, the calculations predict that these crystal structures are dynamically stable.



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Figure 2. Calculated phonon dispersion curves along high-symmetry lines in the first Brillouin zone (left panel) and phonon density of states (right panel) for cF-Al4C.

Figure 3. Calculated phonon dispersion curves along high-symmetry lines in the first Brillouin zone (left panel) and phonon density of states (right panel) for cF-Al4Si.

The cF-Al4C structure has a face-centered cubic lattice (space group 𝐹43𝑚, number 216) with four aluminum atoms and one carbon atom per primitive unit cell. The lattice constant of cF-Al4C is 8.160 Å (see Table 1). Carbon atoms occupy Wyckoff positions 4a, which have coordinates (0, 0, 0). Aluminum atoms occupy Wyckoff positions 16e, which have coordinates (0.13710, 0.13710, 0.13710). The covalent Al–C bond length is equal to 1.938 Å. Four aluminum atoms of the same tetrahedron are bound together by four three-center two-electron (3c-2e) bonds with the distance between two Al atoms of the same tetrahedron equals to 2.606 Å.


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Figure 4. NBO orbitals in (Al4H3)4C compound: left – corresponding to 2c-2e inter-tetrahedral C–Al bond; right – corresponding to 3c-2e bonds in the faces of one tetrahedron. The NBO analysis21,22 performed on a molecular model X-(Al4H3)4 (X=C) (Fig. 4) pertinent to the considered solids reveals two main types of bonds responsible for their stability. Since it simulates all the main orbital interactions between the atoms of aluminum and carbon, as well as between the atoms of aluminum inside tetrahedrons. In this model, the lengths of the Al–C and Al–Al bonds obtained using DFT calculations are in good agreement with the bond lengths obtained in solid-state calculations with the imposition of periodic conditions. The bonds between central carbon atom and the nearby aluminum atoms are the covalent two-electron twocenter (2c-2e) bonds, whereas the Al–Al bonds forming the tetrahedron faces are the 3c-2e bonds. According to results of the AdNDP analysis,23,24 the mentioned 2c-2e bonds have 1.998 |e| occupation numbers (ONs). Also these nearby aluminum atoms connected with 2 aluminum atoms lying on one of faces of the corresponding tetrahedron with 1.908 |e| ONs. The calculated density of the cF-Al4C crystal solid (1.47 g/cm3) is more than twice higher than that predicted for the supertetrahedral aluminum cF-Al8. Vickers hardness decreases in the row of cF-Al4C, cF-Al4Si, cF-Al8. The same trend is observed for the calculated elastic constants, bulk modulus, shear modulus, and Young's modulus (Table 2). The elastic properties of cF-Al4C are similar to those of samarium and bismuth.25 The Poisson's ratio value (0.369) found for cFAl4C is the lowest among studied systems. The calculated electronic band structure of cF-Al4C shown in Fig. 5 allows assignment of this crystal to an indirect-band-gap semiconductor with a band gap of 0.94 eV. The valence band ACS Paragon Plus Environment


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maximum is at the X point and the conduction band minimum is located on the symmetry line Δ, occuring 0.65 of the way from Γ to X at an energy value of 0.04 eV below the Γ-point minimum.

Figure 5. Calculated electronic band structure along high-symmetry lines in the first Brillouin zone (left panel) and electronic density of states (right panel) for cF-Al4C. Table 1. Calculated lattice constants (𝑎, in Å), cohesive energies per primitive unit cell (𝐸𝑐, in eV), Wyckoff position coordinates for Al atom (x, x, x), Al–X bond length or inter-tetrahedral Al–Al bond lengths (𝑑1, in Å), intra-tetrahedral Al–Al bond lengths (𝑑2, in Å), densities (𝜌, in g/cm3) and Vickers hardness (𝐻, in GPa) for cF-Al4X (X = C, Si) and cF-Al8 structures. Structure cF-Al4C cF-Al4Si cF-Al8

𝑎 8.160 9.287 13.322

𝐸𝑐 -20.45 -16.20 -21.20

𝑥 0.13710 0.15085 0.06924

𝑑1 1.94 2.43 2.57

𝑑2 2.61 2.60 2.60

𝜌 1.47 1.13 0.61

𝐻Simunek17 𝐻Tian18 5.5 1.4 3.8 0.2 1.7 0.4

Table 2.The calculated elastic constants (𝑐𝑖𝑗, in GPa), bulk modulus (𝐾, in GPa), shear modulus (𝐺, in GPa), Young's modulus (𝐸, in GPa) and Poisson's ratio (𝜈) for cF-Al4X (X = C, Si) and cF-Al8 structures. Structure cF-Al4C cF-Al4Si cF-Al8

𝑐11 60.21 34.35 13.82

𝑐12 42.19 24.52 10.53

𝑐44 18.60 2.69 4.24

𝐾 48.20 27.79 11.63


𝐺 13.90 3.44 2.90

𝐸 38.04 9.90 8.04

𝜈 0.369 0.441 0.385

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The calculated plots of the real and imaginary parts of the complex dielectric constant versus the photon energy are shown in Fig.6. The absorption peak is expected to be observed at 431 nm in the violet region of the spectrum.

Figure 6. Frequency dependence of the real (red curve) and imaginary (blue curve) parts of complex dielectric permittivity for cF-Al4C. The cF-Al4Si structure has a face-centered cubic lattice (space group 𝐹43𝑚, number 216) with four aluminum atoms and one silicon atom per primitive unit cell. The lattice constant of cF-Al4Si is 9.287 Å. Silicon atoms occupy Wyckoff positions 4a, which have coordinates (0, 0, 0). Aluminum atoms occupy Wyckoff positions 16e, which have coordinates (0.15085, 0.15085, 0.15085). The covalent Al–Si bond length is 2.427 Å. Four aluminum atoms of the same tetrahedron are bound together by four three-center two-electron (3c-2e) bonds. The distance between two Al atoms of the same tetrahedron is equal to2.604 Å. Density of the substance is 1.13 g/cm3. The bulk, shear, and Young's modules of polycrystalline cF-Al4Si are 27.79, 3.44 and 9.90 GPa, respectively, and its Poisson's ratio is equal to 0.4406. The elastic properties of cF-Al4Si are similar to those of thallium and lead.25 The calculated electronic band structure of cF-Al4Si is shown in Fig. 7. The crystal is an indirect-band-gap semiconductor with a band gap of 0.77 eV. The valence band maximum is at the X point and the conduction band minimum is located on the symmetry line Δ, occuring 0.61 of the way from Γ to X.



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Figure 7. Calculated electronic band structure along high-symmetry lines in the first Brillouin zone (left panel) and electronic density of states (right panel) for cF-Al4Si. The calculated plots of the real and imaginary parts of the complex dielectric constant versus the photon energy are shown in Fig. 8. The absorption peaks are observed at 611 nm (2.03 eV), 517 nm (2.40 eV) and 393 nm (3.15 eV) in the orange, green and violet regions, respectively.

Figure 8. Frequency dependence of the real (red curve) and imaginary (blue curve) parts of complex dielectric permittivity for cF-Al4Si. CONCLUSIONS ACS Paragon Plus Environment

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To sum up, we have computationally assessed a new mixed-atom cF-Al4X (X=B, C, Al, Si) form of the supertetrahedral aluminum on the basis of a diamond lattice in which carbon atoms are replaced by the Al4X units. The DFT PBEsol/750 eV quantum chemical calculations confirmed that the three-dimensional crystal structures cF-Al4C and cF-Al4Si conform to the metastable phases of these solids with the properties of a semimetal exhibiting high plasticity and substantial hardness. Supertetrahedral cF-Al4B and cF-Al4Al solids are dynamically unstable (see Figures S1 and S2 in Supporting Information) and therefore here are not discussed. ASSOCIATED CONTENT Supporting Information. File (docx) containing: translation vectors in Cartesian coordinates, Cartesian and direct coordinates of atoms, and free energies for cF-Al4C, cF-Al4Si, cF-Al4B and cF-Al4Al structures. Calculated phonon dispersion curves along high-symmetry lines in the first Brillouin zone and phonon density of states for cF-Al4B (Figure S1) and cF-Al4Al (Figure S1). AUTHOR INFORMATION Corresponding Authors *(A.I.B.) E-mail: [email protected]. *(R.M.M.) E-mail: [email protected] *(V.I.M.) E-mail: [email protected]

ORCID Iliya V. Getmanskii: 0000-0001-8042-2665 Vitaliy V. Koval: 0000-0002-9724-513X Alexander I. Boldyrev: 0000-0002-8277-3669 Ruslan M. Minyaev: 0000-0001-9563-736X Vladimir I. Minkin: 0000-0001-6096-503X Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS



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The work was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement No. 14.Y26.31.0016) and by the USA National Science Foundation (grant CHEM-1664379) to AIB. REFERENCES 1. Georgakilas, V.; Perman, J. A.; Tucek,J.; Zboril,R. Broad Family of Carbon Nanoallotropes: Classification, Chemistry, and Applications of Fullerenes, Carbon Dots, Nanotubes, Graphene, Nanodiamonds, and Combined Superstructures.Chem. Rev. 2015, 115, 4744–4822. 2. Heimann, R. B.; Evsvukov, S. E.; Koga, Y. Carbon Allotropes: A Suggested Classification Scheme Based on Valence Orbital Hybridization. Carbon 1997, 35, 1654–1658. 3. Burdett, J. K.; Lee, S. Moments and the Energies of Solids. J. Am. Chem. Soc. 1985, 107, 3063–3082. 4. Johnston, R. L.; Hoffmann, R. Superdense Carbon, C8: Supercubane or Analog of silicon?J. Am. Chem. Soc. 1989, 111, 810–819. 5. Sheng, X. L.; Yan, Q. B.; Ye, F.; Zheng, Q. R.;Su, G.T-Carbon: A Novel Carbon Allotrope. Phys. Rev. Lett. 2011, 106, 155703. 6. Getmanskii, I. V.; Minyaev, R. M.; Steglenko, D. V.; Koval, V. V.; Zaitsev, S. A.; Minkin, V. I.From Two- to Three-Dimensional Structures of a Supertetrahedral Boron Using Density Functional Calculations. Angew. Chem., Int. Ed. 2017, 56, 10118–10122. 7. Minyaev, R. M.; Getmanskii, I. V.; Minkin, V. I. Supertetrahedral Aluminum and Silicon Structures and Their Hybrid Analogues. Russ. J. Inorg. Chem.2014, 59, 332– 336 (Zh. Neorg. Khim. 2014, 59, 487–491 (in Russian)). 8. Kresse, G.; Hafner, J. Ab Initio Molecular Dynamics for Liquid Metals. Phys. Rev. B 1993, 47, 558–561. 9. Kresse, G.; Hafner, J. Ab Initio Molecular-Dynamics Simulation of the Liquid-Metal– Amorphous-Semiconductor Transition in Germanium. Phys. Rev. B 1994, 49, 14251– 14269. 10. Kresse, G.; Furthmüller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169–11186.


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25. Handbook of the Physicochemical Properties of the Elements; Samsonov, G. V., Ed.; IFI/Plenum: New York, 1968.

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