Hypothetical planar and nanotubular crystalline structures with five interatomic bonds of Kepler nets type Aleksey I Kochaev Citation: AIP Advances 7, 025202 (2017); doi: 10.1063/1.4975707 View online: http://dx.doi.org/10.1063/1.4975707 View Table of Contents: http://aip.scitation.org/toc/adv/7/2 Published by the American Institute of Physics Articles you may be interested in Absorption enhancement and total absorption in a graphene-waveguide hybrid structure AIP Advances 7, 025101025101 (2017); 10.1063/1.4975706 Proposed manipulation of spin currents in GaAs crystal using the Pockels effect AIP Advances 7, 025201025201 (2017); 10.1063/1.4975221 AIP ADVANCES 7, 025202 (2017) Hypothetical planar and nanotubular crystalline structures with five interatomic bonds of Kepler nets type Aleksey I Kochaev Department of Physics, Ulyanovsk State Technical University, 32 Severny Venets St., 432027, Russia (Received 10 December 2016; accepted 25 January 2017; published online February 2017) The possibility of metastable existence of planar and non-chiral nanotubular crystalline lattices in the form of Kepler nets of 3432 4, 33 42 , and 34 types (the notations are given in Schlăafly symbols), using ab initio calculations, has researched Atoms of P, As, Sb, Bi from 15th group and atoms of S, Se, Te from 16th group of the periodic table were taken into consideration The lengths of interatomic bonds corresponding to the steadiest states for such were determined We found that among these new composed structures crystals encountered strong elastic properties Besides, some of them can possess pyroelectric and piezoelectric properties Our results can be used for nanoelectronics and nanoelectromechanical devices designing © 2017 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4975707] I INTRODUCTION In the seventeenth century the famous German mathematician and astronomer Johannes Kepler who attempted to explain the proportions of the natural world for geometrical reasons in his wellknown book “Harmonices Mundi”1 proved that only 13 nets can be constructed from regular polygons and the identical vertices to which these polygons adjoin Two of these nets are enantiomorphic and only 11 Kepler nets are usually taken into consideration There are shown in Fig and are denoted by Schlăafly symbols as it is accepted now.2 The numerals indicate the type and the number of regular polygons meeting in one node, and the superscripts correspond to the number of identical polygons For example, the notation 482 means that one 4-gon and two 8-gons meet in each net’s node Times passed, and in the beginning of twenty century it was unexpectedly turned out that Kepler nets can be realized in planar and tube-like nanostructures So, graphene obtained by A Geim and K Novoselov in 2004,3 silicene,4 germanene,5 two-dimensional boron nitride,6 and stanene7 are crystallizing in the form of honeycomb 63 net (first picture on second line in Fig 1) The possibility of existence of more complex, then graphene, planar carbon nets from sp2 hybridized carbon atoms is theoretically proved.8 There are 482 , 46 12, and 122 nets forming the structures consists of 4- and 8-gons; 4-, 6-, and 12-gons; 3- and 12-gons One of them, 482 , is realized in the structure of octagraphene.9 It is shown on second line (second picture) in Fig In,10–12 supplementary, sp3 nanoallotropes of carbon and other atoms from the 14th group of periodic table of elements crystallization in the form of 122 and 46 12 Kepler nets are mathematically simulated By using a random searching C Pickard and R J Needs13 meet hypothetical material like 3464 form of Kepler sets (first structure on thirst row in Fig 1) Two-dimensional (2D) atomic structures in the nodes of which crystalline lattice not individual atoms but their symmetrically organized complexes are displaced were named supracrystals,10 tubular structures were called supracrystalline nanotubes.17 It means more difficult over crystalline structure of such atomic lattices (from latin supra – over) In14–19 we represented some their physical properties received by computer simulation and numerical calculation The term “supracrystals” for designation of similar structures wasn’t widely adopted, moreover, it allows ambiguous interpretation Because 2158-3226/2017/7(2)/025202/9 7, 025202-1 © Author(s) 2017 025202-2 Aleksey I Kochaev AIP Advances 7, 025202 (2017) FIG Kepler nets The notations are given in Schlăafly symbols all these atomic lattices are representatives of the Kepler nets, perhaps, they can be called as Kepler crystals (KC) Recently a message appears on obtaining of two-dimensional crystalline phosphorus called the phosphorene.20 Phosphorene is an example of 2D material consisting of atoms from 15th group of the periodic table Unlike graphene and other (2D) crystals discussed above each phosphorene’s atom is connected with the nearest atoms by means of covalent bonds In this paper we present the results of ab initio simulation for thermodynamic stability, geometrical characteristics and elastic properties of other planar and tubular nanoallotropes composed from chemical elements of 15th and 16th groups of the periodic table The atomic models of these new crystalline structures can be described within 3432 4, 33 42 , and 34 Kepler nets as it is shown in Fig II ABILITY TO FORM THE COVALENT BONDS A Elements of 15th group In ground state the electronic configuration of valence electrons for such atoms is ns2 np3 where n = (N), (P), (As), (Sb), (Bi) Nitrogen atom can’t have intra-shell exited states because there aren’t free orbitals in its valence shell It can form only covalent bonds and donor-acceptor bond Even, if to allow mixing of ns and all three np orbitals (sp3 hybridization), the fifth electron from external shell, “running across” all hybrid orbitals, doesn’t allow to form covalent structure So, the existence of nitrogen crystals is impossible All other elements from this group can be in exited state ns1 np3 nd The conditions of equivalent hybrid orbitals in the sheet plane arise, and the possibility to form covalent bonds with neighbouring atoms appears This situation corresponds to sp3 d hybridization of atomic orbitals It should be noted that in scientific literature sp3 d hybridization usually is understood as hexahedral hybridization when bonds in the sheet plane and bonds perpendicular to a sheet take place It is probable that flat version of such hybridization is also possible, but it can show only experiment or correct mathematical modelling Of cause, the atoms excitation requires energy, but its expenses pay off formation of additional bonds between the atoms So, the considered regular structures in the form of 33 43 , 34 6, and 3432 Kepler nets can be metastable 025202-3 Aleksey I Kochaev AIP Advances 7, 025202 (2017) FIG Atomic models of planar and tubular Kepler crystals (KC): (a) KC in form 3432 Kepler net; (b) non-chiral nanotube (0,3) in form 3432 Kepler net; (c) KC in form 33 42 Kepler net; (d) non-chiral nanotubes (6,6) left and (0,4) right in form 3432 Kepler net; (e) KC in form 34 Kepler net B Elements of 16th group For this category of atoms, the ground state of valence electrons is ns2 np4 where n = (O), (S), (Se), (Te) As well as for nitrogen, the intra-shell excitation of oxygen atom is impossible owing to lack of the corresponding free orbitals Therefore, planar crystallization of oxygen is also impossible Other elements of 16th group can be in two exited states: ns2 np3 nd and ns1 np3 nd In the first cause sp3 d hybridization takes place Thus two different situations are possible hypothetically: equivalent sp orbitals are located in sheet plane and free electrons are in d orbital perpendicular to the sheet; equivalent spd orbitals are in the sheet plane, and free electron is in d orbital The last situation is energetically more favourable In the second exited state, not described before, sp3 d hybridization appears when equivalent spd orbitals in the sheet plane can exist In the first exited state atoms can crystallize in the form of 33 43 , 34 6, 3432 Kepler nets, and in the second exited state in the form of 36 one However, such exited state stability apparently is very 025202-4 Aleksey I Kochaev AIP Advances 7, 025202 (2017) small as the electron should “jump” from ns sub-shell at once in nd sub-shell passing np one For this reason, such structures aren’t considered From Fig one can see also that planar KC of 3432 type belong to the class of point symmetry mmm and other two types belong to the class These two structures have not a symmetry centre in unit cell and, therefore, they have to find piezoelectric and pyroelectric properties It is very important and promising result because to turn well-known graphene-like planar and tubular crystals into piezoelectric materials they needed to be “spoiled” by a perforation breaking the central symmetry or by alien atoms doping.21–24 III USED METHODS Our calculations were performed with the density functional theory (DFT) implemented within the Vienna Ab initio Simulation Package (VASP) of 4.6 version.25–27 In all cases ion cores were treated using Vanderbilt pseudopotentials.28 Electron exchange and correlation effects were described using the spin-polarized generalized-gradient corrected Perdew☞Burke☞Ernzerhof approximation.29 The electronic wave function was expanded in a plane wave basis set with an energy cutoff of 550 eV Brillouin zone sampling was done using a Monkhorst☞Pack mesh30 of 28 × 28 × for planar and × × 28 for tubular structure cases and a Gaussian smearing of 0.01 eV was used for the electronic occupation All calculations were done using periodic boundary conditions To avoid the interaction between adjacent atoms in the direction perpendicular to the sheet a lattice cell parameter much greater than possible bond length was used (6 Å) The same value exactly was chosen for free space to allocate on each side nanotube The binding energy per one atom E b was calculated as follows: Et − NE (1) N Here E is the total energy of an isolated atom, N is the number of atoms in the translating cluster (unit cell), and E t is the total energy of the cluster The assumption we made was that zero level of energy corresponds to disintegration of the system, i.e E < 0, E t < Because |E t | > |NE|, the binding energy turns out negative Such method led to the binding energy for graphene equal – 7.83 eV that is attractive fit to the results received by other authors.31–33 The fit supports our approach After we made sure the equilibrium metastable forms of researched structures are possible we can evaluate their elastic properties Performing ab initio calculation for the total energy of a structure we can extract the elastic data For an arbitrary anisotropic bulk media there are 81 components of fourth-rank elastic tensor cijkl is defined as Eb = cijlm = ∂2F , ∂xij ∂xlm (2) where F – the density of the potential energy of elastic-deformed body determined with the total energy, x ij , x lm – the components of strain tensor In our case the elastic tensor contains and independent components of elastic tensor for planar KC belong to the mmm and class point symmetry group, respectively For the planar KC shown in Fig 2, they are as follows: mmm c11 c12 class c11 c12 c13 c12 c22 c12 c22 c23 0 c33 c13 c23 c33 Here, we used the matrix representations of the tensor cijlm using convolution in pairs of symmetric indices such as: 11↔1, 22↔2, 12↔3, 21↔3 For planar structures components of above matrix are main because it allows to determine all the important elastic parameters of any material including Young’s modulus Y and Poisson’s ratio σ For tubular structures, these either characteristics are the major 025202-5 Aleksey I Kochaev AIP Advances 7, 025202 (2017) The two-dimensional Young’s modulus Y 2D for the tension (compression) deformation in arbitrary direction and Poisson’s ratio σ as an evaluate of the lateral compression along h accompanied by tension along k are given by17 Y2D = shk , σ= a1i a1j a1l a1m sijlm skk (3) Here, sijlm are the components of the elastic compliances tensor in the crystallographic coordinate system, (a1n ) is the matrix of the direction cosines of a moving reference system respect to the crystallographic axes.17 The expressions derived from (3) for the two-dimensional Young’s modulus Y 2D and the Poisson’s ratio σ for mmm class of point symmetry group in mutually perpendicular directions are given by 10 Y2D = c11 c22 − c12 , c22 σ=σ 10 01 Y2D = =σ 01 =− c11 c22 − c12 , c11 (4) c12 c11 (5) and for class to the same directions are given by 10 Y2D = 01 Y2D = − c2 c + 2c c c − c2 c c11 c22 c33 − c11 c23 12 13 23 12 33 13 22 c22 c33 − c23 − c2 c + 2c c c − c2 c c11 c22 c33 − c11 c23 12 13 23 12 33 13 22 c11 c33 − c13 σ=σ 10 =σ 01 =− , (6) , c12 c33 − c13 c23 c11 c33 − c13 (7) TABLE I The bond length b (Å), binding energy per atom E b (eV/atom), values of the Young’s modulus Y 2D (N/m) for different directions and Poisson’s ratio σ for different types of planar crystals made from elements of 15th and 16th groups of the periodic table Atoms b, Å E b , eV/atom 01 Y2D , N/m 10 Y2D , N/m σ 3432 P As Sb Bi S Se Te 2.50 2.65 3.06 3.20 2.34 2.80 3.24 –2.26 –2.20 –0.80 –0.39 –2.29 –1.17 –0.62 204 110 64.2 11.3 176 141 51.5 203 109 60.9 9.27 178 134 49.6 0.53 0.47 0.44 0.41 0.49 0.47 0.46 33 42 P As Sb Bi S Se Te 2.36 2.68 3.08 3.15 2.44 2.76 3.10 –2.20 –2.20 –0.76 –0.38 –2.30 –1.20 –0.58 375 229 98.5 68.3 289 194 87.8 370 232 99.4 61.0 287 198 84.6 0.59 0.57 0.50 0.47 0.55 0.54 0.51 34 P As Sb Bi S Se Te 2.40 2.70 3.09 3.17 2.48 2.80 3.10 –2.22 –2.19 –0.78 –0.27 –2.16 –1.08 –0.45 120 92.4 41.3 10.6 94.4 42.6 27.7 116 84.9 35.3 5.48 75.3 41.0 13.4 0.42 0.40 0.38 0.37 0.41 0.41 0.37 Kepler nets 025202-6 Aleksey I Kochaev AIP Advances 7, 025202 (2017) The superscripts indicate how to set the crystallographical and crystallophysical axes between themselves If we consider the both elastic characteristic coefficients along a horizontal direction (Fig 2), then they will be labelled In other case, we consider the elastic properties in the vertical direction which is perpendicular to the previous one, then this direction will be defined as .17 IV RESULTS The equilibrium values of bond lengths, binding energy, Young’s modulus and Poisson’s ratio for all types of considered KC are given in Table I We see that the binding energy in all cases doesn’t FIG Binding energy from the bond length for 3432 type of planar crystals: (a) elements of 15th group, (b) elements of 16th group 025202-7 Aleksey I Kochaev AIP Advances 7, 025202 (2017) exceed (on modulus) 2.30 eV, thus bond length is in the range of 2.2☞2.5 Å It is more than three times less, then binding energy of graphene, though it is commensurable with the binding energy for some fullerenes and fullerenes For example, the binding energy for fullerene C20 is calculated to be 4.01 eV,34 for tetrahedrane C4 H4 3.90 eV,35 and for cubane 4.42 eV.36 This circumstance allows to hope that crystalline structures offered by us can exist in metastable state at room temperature For the elements of 15th group planar crystals composed of phosphorus atoms are the steadiest and for the elements of 16th group sulfur structures seem to be steadier As it shown in Table I within each periodical group while increasing of atomic number the value of Young’s modulus decrease The maximum magnitude (375 N/m for phosphorus KC in the form TABLE II The bond length b (Å), binding energy per atom E b (eV/atom), values of the Young’s modulus Y 2D (N/m) for longitudinal direction and Poisson’s ratio σ for different types of nanotubes (0,n) and (n,n) made from elements of 15th and 16th groups of the periodic table Kepler nets 3432 Atoms Type nanotubes b, Å E b , eV/atom Y 2D , N/m σ P (0,4) (0,8) (0,15) (0,20) 2.49 2.49 2.49 2.50 –2.16 –2.14 –2.20 –2.21 235 264 250 210 0.26 0.31 0.44 0.53 As (0,4) (0,8) (0,15) (0,20) 2.63 2.62 2.65 2.65 –2.13 –2.15 –2.18 –2.19 167 177 168 125 0.26 0.31 0.38 0.39 S (0,4) (0,5) (0,15) (0,20) 2.33 2.34 2.32 2.33 –2.21 –2.23 –2.26 –2.26 228 245 213 199 0.28 0.35 0.44 0.48 (0,4) (0,8) (0,15) (0,20) 2.35 2.35 2.36 2.36 –2.17 –2.15 –2.18 –2.29 402 426 415 380 0.44 0.45 0.46 0.49 (4,4) (8,8) (15,15) (20,20) 2.35 2.35 2.35 2.36 –2.17 –2.15 –2.16 –2.29 400 414 406 379 0.44 0.44 0.46 0.49 (0,4) (0,8) (0,15) (0,20) 2.66 2.65 2.66 2.66 –2.16 –2.17 –2.19 –2.19 255 317 261 244 0.29 0.46 0.46 0.52 (4,4) (8,8) (15,15) (20,20) 2.64 2.65 2.65 2.65 –2.17 –2.15 –2.16 –2.29 265 320 265 257 0.29 0.45 0.46 0.52 (0,4) (0,8) (0,15) (0,20) 2.42 2.42 2.43 2.42 –2.23 –2.24 –2.26 –2.27 255 317 261 244 0.34 0.40 0.52 0.54 (4,4) (8,8) (15,15) (20,20) 2.42 2.42 2.43 2.42 –2.24 –2.25 –2.28 –2.28 339 356 321 304 0.34 0.44 0.52 0.54 P 33 42 As S 025202-8 Aleksey I Kochaev AIP Advances 7, 025202 (2017) TABLE III The values of the Young’s modulus Y 2D (N/m) for longitudinal direction and Poisson’s ratio σ for known types of 2D crystals and nanotubes Structure b, Å E b , eV/atom 01 Y2D , N/m σ Graphene3,9 Octagraphene9 Graphyne9 Graphdiyne9 Silicene4,37 Borophene38,39 Carbon nanotubes (0,10)3 BN nanotubes (0,10)6 1.42 1.44 1.31 1.33 2.30 1.65 1.42 1.52 –7.92 –7.39 –7.26 –7.14 –5.51 –6.33 –8.30 –7.31 350 306 245 174 63.8 ··· 412 403 0.21 0.13 0.38 0.41 0.33 ··· 0.53 0.57 33 42 ) slightly large then corresponding for graphene and octagraphene.9 Obviously due to the atoms located at the vertices of triangles and squares (Fig 2c) Fig show the binding energy per atom E b as a function of bond length b for 3432 type of considered crystalline structures both formed from the elements of 15th and 15th groups of the periodic table Minima of the curves correspond to equilibrium states of each structure, and the numerical values are equal to binding energy The distinctive feature of 3432 crystals is the possibility of their existence in two metastable forms differing each other by the value of interatomic bond length (see Fig 3a) We believe that it is caused by existence of additional type of normal oscillations in 3432 crystalline lattice Really, besides the periodic two-dimensional bulk tension-compression deformations, the deformation can take place here at which the bases of triangles (for example, located horizontally in Fig 2) increase, and the bases of other triangles (located vertically) decrease, then on the contrary Respectively, the squares periodically rotate at first in one, then in other direction To include the rotational degrees of freedom an additional energy is required For this reason, the corresponding local minima of binding energy settle above, and the average distance between atoms increases When we have the elements of 16th group (Fig 3b) in the nodes of 3432 lattice two different spin’s orientation can take place In first case pz electrons on the neighbour atoms located in the squares having opposite directions of the spins And in second case the directions of all pz electrons spins are identical The first opportunity is energetically more favourable Therefore, the corresponding binding energy minimum is deeper and even can become dominating The equilibrium values of bond lengths, binding energy, Young’s modulus and Poisson’s ratio for most stabile types of considered nanotubes are given in Table II The two-dimensional Young’s modulus Y 2D for wide nanotubes come near the same for the planar form KC as well as the binding energy per atom Despite the lower stability of phosphorus nanotubes formed from Kepler net 33 42 the elastic constants are similar to carbon nanotubes which confirms that these nanotubes are also strong Poisson’s ratio takes in the range values from 0.26 to 0.54 The cited9,37–42 for comparison values of some parameters for known types of 2D crystals and nanotubes are given in Table III V CONCLUSIONS As it appears from an electronic configuration of the exited atoms of 15th group of periodical table, in KC 3433 4, 33 42 , 34 types formed by them there are no free electrons Therefore, depending on width of the forbidden gap, they should to be dielectrics or semiconductors On the contrary, in crystals of the same types created from the elements of 16th group, there are free electrons which are located in pz sub-shells perpendicular to plane of a crystal Such crystals can be metals or semimetals Moreover, KC of 34 42 and 34 types aren’t centrosymmetric and, therefore, can find piezoelectric and pyroelectric properties If they are synthesized, it will be the first 025202-9 Aleksey I Kochaev AIP Advances 7, 025202 (2017) natural piezo- and pyroelectrics We hope that described here new 2D crystals of Kepler nets type will receive a practical application for nanoelectronic and nanoectromechanical devices designing if 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There are 482 , 46 12, and 122 nets forming the structures consists of 4- and 8-gons; 4-, 6-, and 12-gons; 3- and 12-gons One of them, 482 , is realized in the structure of octagraphene.9 It is