hypothetical planar and nanotubular crystalline structures with five interatomic bonds of kepler nets type

Hypothetical planar and nanotubular crystalline structures with five interatomic bonds of Kepler nets type Aleksey I... Kochaev Department of Physics, Ulyanovsk State Technical Universit

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

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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 2 February 2017)

The possibility of metastable existence of planar and non-chiral nanotubular crys-talline lattices in the form of Kepler nets of 34324, 3342, and 346 types (the

nota-tions are given in Schl¨afly symbols), using ab initio calculanota-tions, 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 inter-atomic bonds corresponding to the steadiest states for such were determined We found that among these new composed structures crystals encountered strong elas-tic properties Besides, some of them can possess pyroelectric and piezoelectric properties Our results can be used for nanoelectronics and

nanoelectromechani-cal devices designing © 2017 Author(s) All article content, except where

other-wise 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 well-known book “Harmonices Mundi”1proved 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.1and are denoted

by Schl¨afly symbols as it is accepted now.2The 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 482means 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 63net (first picture on second line in Fig.1) The possibility

of existence of more complex, then graphene, planar carbon nets from sp2hybridized carbon atoms

is theoretically proved.8

There are 482, 46 12, and 3 122nets 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.9It is shown

on second line (second picture) in Fig.1

In,10–12supplementary, sp3 nanoallotropes of carbon and other atoms from the 14th group of periodic table of elements crystallization in the form of 3 122and 46 12 Kepler nets are mathematically simulated By using a random searching C Pickard and R J Needs13meet 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,10tubular structures were called supracrystalline nanotubes.17It means more difficult over crystalline structure

of such atomic lattices (from latin supra – over) In14–19we 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 1 Kepler nets The notations are given in Schl¨afly 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.20Phosphorene is an example of 2D material consisting of atoms from 15thgroup 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,

geo-metrical 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 34324, 3342, and 346 Kepler nets as it is shown in Fig.2

II ABILITY TO FORM THE COVALENT BONDS

A Elements of 15 th group

In ground state the electronic configuration of valence electrons for such atoms is ns2np3where

n = 2 (N), 3 (P), 4 (As), 5 (Sb), 6 (Bi) Nitrogen atom can’t have intra-shell exited states because

there aren’t free orbitals in its valence shell It can form only 3 covalent bonds and 1 donor-acceptor

bond Even, if to allow mixing of ns and all three np orbitals (sp3hybridization), 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 ns1np3nd1 The conditions of 5 equivalent hybrid orbitals in the sheet plane arise, and the possibility to form 5 covalent bonds with neighbouring

atoms appears This situation corresponds to sp3d hybridization of atomic orbitals It should be noted

that in scientific literature sp3d hybridization usually is understood as hexahedral hybridization when

3 bonds in the sheet plane and 2 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 3343, 346, and 34324 Kepler nets can be metastable

FIG 2 Atomic models of planar and tubular Kepler crystals (KC): (a) KC in form 34324 Kepler net; (b) non-chiral nanotube (0,3) in form 34324 Kepler net; (c) KC in form 3342Kepler net; (d) non-chiral nanotubes (6,6) left and (0,4) right in form

34324 Kepler net; (e) KC in form 346 Kepler net.

B Elements of 16 th group

For this category of atoms, the ground state of valence electrons is ns2np4 where n = 2 (O),

3 (S), 4 (Se), 5 (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: ns2np3nd1and ns1np3nd2 In the first

cause sp3d hybridization takes place Thus two different situations are possible hypothetically: 4

equivalent sp orbitals are located in sheet plane and 2 free electrons are in d orbital perpendicular

to the sheet; 5 equivalent spd orbitals are in the sheet plane, and 1 free electron is in d orbital The last situation is energetically more favourable In the second exited state, not described before, sp3d2

hybridization appears when 6 equivalent spd orbitals in the sheet plane can exist.

In the first exited state atoms can crystallize in the form of 3343, 346, 34324 Kepler nets, and in the second exited state in the form of 36one 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.2one can see also that planar KC of 34324 type belong to the class of point symmetry

mmm and other two types belong to the class 1 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.25–27In all cases ion cores were treated using Vanderbilt pseudopotentials.28Electron 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 mesh30of 28 × 28 × 1 for planar and

3 × 3 × 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 bwas calculated as follows:

E b=E t−NE

Here E is the total energy of an isolated atom, N is the number of atoms in the translating cluster (unit cell), and E tis 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 < 0 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.31–33The 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 c ijklis defined as

c ijlm=∂x∂2F

ij ∂x lm

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 4 and

6 independent components of elastic tensor for planar KC belong to the mmm and class 1 point

symmetry group, respectively For the planar KC shown in Fig.2, they are as follows:

mmm class 1

* ,

c11c12 0

c12c22 0

0 0 c33

+ /

-* ,

c11c12c13

c12c22c23

c13c23c33

+ / -

Here, we used the matrix representations of the tensor c ijlmusing 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

The two-dimensional Young’s modulus Y 2Dfor the tension (compression) deformation in

arbi-trary direction and Poisson’s ratio σ as an evaluate of the lateral compression along h accompanied

by tension along k are given by17

Y 2D= 1

a 1i a 1j a 1l a 1m s ijlm, σ=s hk

Here, s ijlmare the components of the elastic compliances tensor in the crystallographic coordinate

system, (a 1n) 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 2Dand the Poisson’s

ratio σ for mmm class of point symmetry group in 2 mutually perpendicular directions are given by

Y 2Dh10i=c11c22−c

2 12

c22 , Y 2Dh01i=c11c22−c

2 12

σ = σh10i= σh01i= −c12

and for class 1 to the same directions are given by

Yh10i

2D =c11c22c33−c11c

2

23−c212c33+ 2c12c13c23−c213c22

c22c33−c2

23

,

Y 2Dh01i=c11c22c33−c11c

2

23−c2

12c33+ 2c12c13c23−c2

13c22

c11c33−c2

13

,

(6)

σ = σh10i= σh01i= −c12c33−c13c23

c11c33−c2

13

TABLE I The bond length b (Å), binding energy per atom E b (eV/atom), values of the Young’s modulus Y2D (N/m) for different directions and Poisson’s ratio σ for different types of planar crystals made from elements of 15 th and 16 th groups of the periodic table.

Kepler nets Atoms b, Å E b, eV/atom Y 2Dh01i, N/m Y 2Dh10i, N/m σ

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 <10> 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

<01>.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 TableI We see that the binding energy in all cases doesn’t

FIG 3 Binding energy from the bond length for 343 2 4 type of planar crystals: (a) elements of 15 th group, (b) elements of

16thgroup.

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 C20is calculated to be 4.01 eV,34for tetrahedrane

C4H43.90 eV,35and for cubane 4.42 eV.36This circumstance allows to hope that crystalline structures offered by us can exist in metastable state at room temperature For the elements of 15thgroup 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 TableI 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 Y2D (N/m) for longitudinal direction and Poisson’s ratio σ for different types of nanotubes (0,n) and (n,n) made from elements of 15 th and

16 th groups of the periodic table.

Kepler nets Atoms Type nanotubes b, Å E b, eV/atom Y2D , N/m σ

P

S

P

S

025202-8 Aleksey I Kochaev AIP Advances 7, 025202 (2017)

TABLE III The values of the Young’s modulus Y2D (N/m) for longitudinal direction and Poisson’s ratio σ for known types

of 2D crystals and nanotubes.

3342) slightly large then corresponding for graphene and octagraphene.9Obviously due to the atoms located at the vertices of triangles and squares (Fig.2c)

Fig.3show the binding energy per atom E b as a function of bond length b for 34324 type of considered crystalline structures both formed from the elements of 15thand 15thgroups 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 34324 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 34324 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 34324 lattice two different

spin’s orientation can take place In first case pzelectrons on the neighbour atoms located in the squares

having opposite directions of the spins And in second case the directions of all pzelectrons 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 TableII

The two-dimensional Young’s modulus Y2D 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 3342 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 – 42for comparison values of some parameters for known types of 2D crystals and nanotubes are given in TableIII

V CONCLUSIONS

As it appears from an electronic configuration of the exited atoms of 15th group of period-ical table, in KC 34334, 3342, 346 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

elec-trons which are located in pz sub-shells perpendicular to plane of a crystal Such crystals can be

metals or semimetals Moreover, KC of 3442 and 346 types aren’t centrosymmetric and, there-fore, can find piezoelectric and pyroelectric properties If they are synthesized, it will be the first

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 are synthesized

ACKNOWLEDGMENTS

The reported study was funded by RFBR, according to the research project No 16-32-60041 mol a dk

1J Kepler, Weltharmonik II Buch der Weltharmonik (R Oldenbourg Verlag, M¨unchen – Berlin, 1939), p 63.

2H S M Coxeter, Regular polytopes, Third Edition (Dover Publications, New York, 1973), pp 14, 69, 149.

3 K S Novoselov, A K Geim, S V Morozov, D Jiang, Y Zheng, S V Dubonos, I V Grigorieva, and A A Firsov, Science

306, 666 (2004).

4 B Aufray, A Kara, S Vizzini, H Oughaddon, Ch Leandri, B Ealet, and G Le Lay, Appl Phys Lett.96, 183102 (2010).

5 M E Davila, L Xian, S Cahangirov, O Rubio, and G Le Lay, New J Phys.16, 095002 (2014).

6C R Dean, A F Young, and I Eric, Nanotech 5, 722 (2010).

7 Y Xu, B Yan, H Zhang, Y Wang, G Xu, P Tang, W Duan, and S Zhang, Phys Rev Lett.111, 136804 (2014).

8 A T Balaban, Comput Mat Appl.17, 397 (1989).

9 X L Sheng, H J Cui, F Ye, O B Yan, Q R Zhang, and G Su, J Appl Phys.112, 074315 (2012).

10 A Karenin, J Phys.: Conf Ser.315, 012025 (2012).

11 A N Enyashkin and A L Ivanovskii, Phys Status Solidi B8, 1879 (2011).

12 E A Belenkov and V A Greshnyakov, Phys Solid State55, 1754 (2013).

13 C Pickard and R J Needs, Phys Rev B81, 014106 (2010).

14 R A Brazhe, A A Karenin, A I Kochaev, and R M Meftakhutdinov, Phys Solid State53, 1481 (2011).

15 A I Kochaev, A A Karenin, R M Meftakhutdinov, and R A Brazhe, J Phys.: Conf Ser.345, 012007 (2012).

16 R A Brazhe and A I Kochaev, Phys Solid State54, 1612 (2012).

17 R A Brazhe, A I Kochaev, and V S Nefedov, Phys Solid State54, 1430 (2012).

18 P A Brazhe and V S Nefedov, Phys Solid State54, 1528 (2012).

19 R A Brazhe and V S Nefedov, Phys Solid State56, 626 (2014).

20 A Castellanos-Gomez, L Vicarelli, E Prada, K L Narasimha-Acharaya, S I Blanter, D I Groenendijk, M Buscema,

G A Steele, J V Alvarez, Z W Zandbergen, J J Palacios, and H S J Van der Zant, 2D Materials1, 025001 (2014).

21 S Chandratre and P Sharma, Appl Phys Lett.100, 023144 (2012).

22 M T Ong and E T Reed, ASC Nano6, 1387 (2012).

23 R A Brazhe, A I Kochaev, and A A Sovetkin, Phys Solid State55, 1925 (2013).

24 R A Brazhe, A I Kochaev, and A A Sovetkin, Phys Solid State55, 2094 (2013).

25 G Kresse and J Furtm¨uller, Phys Rev B54, 11169 (1996).

26 G Kresse and J Furtm¨uller, Comput Mat Sci.6, 15 (1996).

27G Kresse and J Furtm¨uller, VASP the Gyide (University of Vienna, 2009)http://cms.mpi.univei.ac.at/vasp/

28 D Vanderbilt, Phys Rev B41, 7892 (1990).

29 J P Perdew, K Burke, and M Ernserhof, Phys Rev Lett.77, 3865 (1996).

30 H T Monkhorst and J D Pack, Phys Rev B13, 5188 (1976).

31 P Koskinen, S Malola, and H Hakkinen, Phys Rev Lett.101, 115502 (2008).

32 B I Dunlapy and J C Boetger, J Phys B: At Mol Opt Phys.29, 4907 (1996).

33 V V Ivanovskaya, A Zobelly, D Teilet-Billy, N Rougeau, V Didis, and P R Briddon, Eur Phys J B76, 481 (2010).

34 J C Grossman, L Mitas, and K Paghachari, Appl Phys Lett.75, 3870 (1995).

35 M M Maslov and K R Katin, Chem Phys.386, 66 (2011).

36 M M Maslov, D A Lobanov, A I Podlivaev, and L A Openov, Phys Solid State51, 645 (2009).

37 J Zhao, H Liu, Z Yu, R Quhe, S Zhou, Y Wang, C C Liu, H Zhong, N Han, J Lu, Y Yao, and K Wu, Prog Mat Sci.

83, 24 (2016).

38 H Tang and S Ismail-Beigi, Phys Rev Let.99, 115501 (2007).

39 A Rubio, J L Corkil, and M L Cohen, Phys Rev B49, 5081 (1994).

40 K Sakaushi and M Antonietti, Bull Chem Soc Jpn88, 386 (2015).

41 S Ida, Bull Chem Soc Jpn88, 1619 (2015).

42 D Deng, K S Novoselov, Q Fu, N Zheng, Z Tian, and X Bao, Nat Nano11, 218 (2016).