First-principles study of hypothetical boron crystals: Bn(n = 13, 14, 15)

Solid State Sciences 14 (2012) 1636e1642

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First-principles study of hypothetical boron crystals: Bn(n ¼ 13, 14, 15) Sezgin Aydın, Mehmet S¸ims¸ek* Department of Physics, Faculty of Sciences, Gazi University, Teknikokullar, 06500 Ankara, Turkey

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 November 2011 Received in revised form 12 April 2012 Accepted 23 April 2012 Available online 30 April 2012

First-principles simulations within density functional theory are performed to investigate structural, electronic and mechanical properties of hypothetical boron crystals Bn(n ¼ 13, 14, 15). These hypothetical crystals are generated by inserting boron atom(s) to the space in three-dimensional network of a-boron (a-B12). The effects of inserted atom(s) and their site(s) on the lattice parameters, mechanical and electronic properties are discussed. Cohesive energies and formation enthalpies are calculated to discuss energetic stability of purposed compounds, and also the elastic constants are determined to study mechanical stability and mechanical properties such as bulk, shear and Young moduli. To check the phase stability, molecular dynamics simulations and transition state search calculations are performed and to emphasize distinction of the phases energy-volume curves for all phases are presented. From calculated density of states and Mulliken atomic charges/bond overlap populations, it is observed that the charge transfers exist between inserted boron atom(s) located at different sites and icosahedral boron atoms. By mean of the optimized ground state geometry and other first-principles results, the micro-hardnesses of each boron phases are calculated. Ó 2012 Elsevier Masson SAS. All rights reserved.

Keywords: Boron phases Density-functional theory Elastic properties Electronic properties Metastable phases

1. Introduction Most of elemental boron crystals and compounds include building blocks such as B12 icosahedron, B6 octahedron, linear atomic chains and/or atomic clusters in the three-dimensional network [1e5]. Especially, the chains could be formed by a combination of one and two kinds of atoms with different numbers, for example, the chains are formed by two oxygen atoms in boron suboxide (B6O) [6,7], three or two carbon atoms in boron carbide

* Corresponding author. Tel.: þ90 3122021235. E-mail address: [email protected] (M. S¸ims¸ek). 1293-2558/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2012.04.030

(B12C3, B13C2) [8e10], and two boron atoms in recently synthesized orthorhombic boron (g-B28) [11e13], etc. On the other hand, because of its electron deficient character and the mixing of twoand three-center bondings, boron compounds exhibit fascinating geometric morphology [3,4]. Among the boron allotropes, a-rhombohedral boron (a-B12) is the simplest icosahedral boron phase, and in its crystal structure, B12 icosahedron is building block, and each of them has 12 neighbors, 6 of them are nearest neighbor. However, boron atoms are located to two kinds of non-equivalent sites (polar sites and equatorial sites) on the icosahedral units (B12). Boron atoms in B12 icosahedra are linked to the five intra-icosahedral boron atoms, and B(polar)-B(polar) covalent bonds are formed between the polar

S. Aydın, M. S¸ims¸ek / Solid State Sciences 14 (2012) 1636e1642

atoms, which are located at top and bottom of the icosahedra, as the intra-icosahedral three-center bonds. Also, there are intraicosahedral three-center bonds between polar and equatorial boron atoms, which have high electron density [2]. On the other hand, the bonds between the neighboring icosahedra generally tend to form as strong covalent two-electron-two-center, oneelectron-two-center and also two-electron weak three-center bonds, due to the its electron-deficiency character [3]. It would also be noted here that, although it is not stable by itself [5], a-B12 is generally more stable phase than b-boron phase at low temperatures [14]. However, there are controversy results for ground state of crystalline boron, and the debate continues over the more stable phases of pressure- and temperature-dependency [15e17]. The purpose of this paper is to give a systematic analysis of hypothetical stable boron phases, which are produced by inserting different number of boron atom(s) to the different Wyckoff sites in inter-icosahedral network. This inserting mechanism can generate different metastable boron phases. The phases consist of one icosahedron and the inserted atom(s), located at inter-icosahedral space, had been studied previously to calculate enthalpy of formation and crystallographic parameters with together other icosahedron unit and chain systems [9], i.e., the six suggested structures had all been previously constructed in Ref. [9]. In this study, we focused on B13, B14 and B15 phases for detailed investigation of their structural, mechanical and electronic properties and phase stabilities by first-principles calculations. For the sake of comparison, we also calculate the properties of a-B12. 2. Calculation methods In this study, first-principles calculations were performed based on density functional theory, as implemented in CASTEP simulation code [18]. Exchange correlation effects were treated within the generalized gradient approximation (GGA-PW91) [19]. The Vanderbilt ultrasoft pseudopotentials [20] were used. After convergence tests, a kinetic energy cut-off value for plane wave expansions and k-point meshes for Brillouin-zone sampling were chosen as 500 eV and 10  10  10, respectively. In order to obtain relaxed geometry of the phases, the BroydeneFletchereGoldfarbeShannon (BFGS) algorithm was applied, where all cell parameters and atomic coordinates were optimized simultaneously. The ultra-fine setup of the software package was chosen for better convergence and consistent results. When the four criteria: (i) maximum ionic Hellman-Feynman force, (ii) maximum displacement, (iii) maximum energy change and (iv) maximum stress between two sequential cycles were smaller than the individual limit values in a CASTEP simulation, the convergence could be succeed. In ultra-fine setup, the special limit values for the

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four convergence criteria above are 0.01 eV/Å, 5.0  104 Å, 5.0  106 eV/atom, and 0.02 GPa, respectively. Furthermore, the parameters obtained from first-principles calculations such as Mulliken population of the bonds, coordination number of atoms and bond length are imported to Gao’s hardness method (reader can be found the details of the hardness method in Ref.[21e23]), and then, individual bond hardnesses and total Vickers micro-hardness of hypothetical B13, B14 and B15 boron phases are calculated. As a next step, we calculate other mechanical properties for the constituent rhombohedral crystals, there are five independent elastic constants; c11, c33, c44, c12 and c13, and Reuss (R)-Voigt (V)-Hill (H) bulk modulus (B) and shear modulus (G) are given as in Ref. [24]. Thus, the mechanical stability criteria for hexagonal/rhombohedral crystals are given as [24],

c44 >0; c11 >jc12 j; ðc11 þ 2c12 Þc33 >2c213 :

3. Results and discussion Primitive cells of the inspected hypothetical B13, B14 and B15 phases consist of one icosahedron and the inserted atom(s), located at inter-icosahedral space, as one, two and three atoms, respectively (see Fig. 1(a)e(f)). There are two different phases for both B13 and B14: a- and b-phases. The difference of a- and bphases comes from the location of inserted boron atoms: if the inserted atom(s) is far away from the center of cell, we call them aphases and the others b-phases, where neighboring equatorial boron atoms are connected each other. However, the cell of B15 phase contains a three-atom chain (BeBeB), like CeCeC, CeBeC and CeCeB chains in boron carbide (B4C) [9,10]. Here, it is note that a-B13, b-B13 phases indicated as B12-BVaVa and B12-VaBVa, and a-B14 and b-B14 as B12-BVaB and B12-BBVa, also B15 as B12-BBB, respectively (Va stands for vacancy) in a previous study of Saal et al. [9]. The calculated structural parameters, densities and cohesive energies of Bn phases are listed in Table 1, and calculated energyvolume curves of them are shown in Fig. 2. All of primitive cells have rhombohedral crystal system as a-B12, space groups of the phases are R-3m(166), except for a-B13 (R3m). However, contrary to that of a-B12, apex angles of B13, B14 and B15 are greater than ideal case (a ¼ 60 ). It is seen from Table 1 and Fig. 2 that all structures are energetically stable, and considered new boron phases are not more stable than a-B12. Moreover, cohesive energy can be used to discuss stability of the structures. The structure which has higher cohesive energy will become more thermodynamically stable (at low temperatures) [25]. Then, the stability order can be given as aB12 > a-B14 > a-B13 > B15 > b-B14 > b-B13, and the stability order

Fig. 1. Crystal structures of (a) a-B12, (b) a-B13, (c) b-B13, (d) a-B14, (e) b-B14, (f) B15. For an easy view, inserted boron atom(s) and their bonds are colored by red, the icosahedra in front face are colored by violet, and those in back face are colored by blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Table 1 Calculated structural parameters, densities, cohesive energies and formation enthalpies for Bn(n ¼ 13, 14, 15). a-B13

B12 N Space group a (Å)

a ( )

12 13 R-3m (166) R3m (160) 4.974 5.178 4.974 [36] 4.967 [16] 5.057 [27] 58.205 65.935 58.171 [36] 83.452 110.757 2.581 2.107

V (Å3) Density (g/cm3) Ecoh 7.081 (eV/atom) DH (eV) 0

b-B13

a-B14

b-B14

B15

13 R-3m (166) 4.848

14 R-3m (166) 5.182

14 R-3m (166) 5.084

15 R-3m (166) 5.144 [26]

65.517

67.681 [26]

64.595 65.983

88.699 111.113 104.062 111.990 2.631 2.262 2.415 2.404

6.8564

6.624

6.940

6.849

6.8557

2.925

5.947

1.979

3.252

3.384

and structural parameters agree with the previous ab initio results [9,16,26,27]. The phases labeled as a- are more stable than b-phases. b-B13 phase has the smallest lattice parameter and the highest density. On the other hand, formation enthalpies of Bn phases are also listed in Table 1 to give a discussion about probabilities of synthesizing of the phases. It is found that all formation enthalpies are positive, suggesting the metastable or stable nature of the Bnphases [28]. At the same time, this situation indicates that high temperature and/or high pressure are necessary for the experimental synthesis of these hypothetical boron phases [29], i.e., they may be synthesized by using the synthesis techniques in solid-state chemistry as reviewed by Jansen [30] and/or Wentorf [11] as handling of orthorhombic boron B28 [12,13]. Furthermore, to investigate whether the structures are unstable or (meta)stable, inserted atomic coordinates slightly shifted and then the geometry of atoms and cell parameters are optimized simultaneously, so that the if system evolves with continuously decreasing energy to a local or global minimum then the system is at a (meta)stable structure, otherwise the system unstable, but it may be always not true because the system may be at a saddle point [31]. In order to practical control of the phase stability for B13- and B14-structures, different initial structures are generated with respect to locations of the inserted boron atom(s). For B13-phases, when the optimization started with the slightly shifted atomic coordinates of inserted atom, it evolves continuously decreasing

Fig. 2. Energy-volume curves of Bn-phases.

energy to a local minimum of b-B13, but it is surprisingly observed that, when symmetry on the unit cell is broken, the optimizations of b-B13 phase converged to globally minimum of a-B13 phase. For B14-phase, it is showed that when the distance between the inserted boron atoms is to be smaller than 2 Å, all phases converged to a local minimum of b-B14 phase, and for the distance higher than 2 Å, all phases converged to a global minimum of a-B14 phase. It is note that a universal representation of the states of matter, which are including both metastable and stable phases were investigated by Jansen et.al [32e34]. In that general approach for addressing metastability of the matter based on local ergodicity [33], and a metastable state is defined by a local equilibrium, whereas stable state is defined by global equilibrium. To test the minima are local or global, we have performed molecular dynamics simulations (MD). By using NPT ensemble which means constant pressure and constant temperature we check stability of the phases. Temperature and pressure are chosen as 300 K and 0 GPa, respectively. MD simulations are made for 1000 steps, corresponding to 1.0 ps. The calculated MD results for B13 and B14 phases are proved existence of the a- and b-phases for B13 and B14 on the Bn phase space. It is observed that there is a transition between B13phases from b-B13 to the considerably more stable structure of a-B13 aboutw0.28 ps, while there is a transition between B14-phases from b-B14 to the considerably more stable structure of a-B14 aboutw0.26 ps. Finally, we have also performed transition state search calculations by using the Linear Synchronous Transit (LST) optimization [35] of CASTEP package for B13- and B14-phases. It is found that the activation barrier for transition from b-B14 phase to a-B14 phase is 0.79 eV. In case of B13-phase, it is not obtained an activation barrier from LST optimizations, because energy of the phase is decreases continuously from b-phase to a-B13 phase. This situation is compatible with geometry optimization and MD results when symmetry is broken. As a practical approach, we can say that the symmetry constraint can be created by the activation barrier [31]. When symmetry is applied and broken for B13-atom at the same lattice, difference in corresponding energies should be given the activation barrier as 3.60 eV. In the globally stable phase of B13 (a-B13), the bonds can be classified into nine different types: 3  13 ¼ 39 valence electrons have constructed 36 covalent bonds, 6 of them inter- and 30 intraicosahedral bonds, three short bonds are inter-icosahedral bonds (Bc-Be), which are linking inserted boron atom to equatorial boron atom with length of 1.628 Å and Mulliken bond population of 1.01. On the other hand, in the globally stable phase of B14 (a-B14), the bonds can be classified into six different types: 3  14 ¼ 42 valence electrons have constructed 39 covalent bonds, 9 of them inter- and 30 intra-icosahedral bonds. 6 of them are short bonds (intericosahedral kinds, Bc-Be) which are linking inserted boron atoms to equatorial boron atoms in edge of icosahedron with length of 1.652 Å and Mulliken bond population of 1.04. The shortest bond in b-B14 phase occurs between inserted boron atoms (BceBc bond), as a dumbbell (B2) with 1.620 Å bond length and 1.26 Mulliken bond populations. On the other hand, it is note that the electronic density maps (see Fig. 3(e)) show that boron atoms of B2 dumbbell is strongly covalent bonded to each other, as in orthorhombic boron (g-B28) [36]. In case of B15, the bonds can be classified into 6 different types: 3  15 ¼ 45 valence electrons have constructed 41 covalent bonds, 11 of them inter- and 30 intra-icosahedral bonds, and 2 of them are short bonds (intra-chain kinds, BceBc), linking inserted boron atoms each other as 1.590 Å with 1.07 Mulliken bond population. The bonding nature and degree of covalency in the bonds (or Mulliken populations) can be supported by electron density maps (see Fig. 3(a)e(f)). If electron density is located between two atoms, the bond is covalent. And, when electron density on the bond increases, its covalency increases. If the

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Fig. 3. Calculated electron density maps in plane that including inserted atom(s) of (a) a-B12, (b) a-B13, (c) b-B13, (d) a-B14, (e) b-B14, (f) B15. Electron density is high in red-colored regions, and is low in blue-colored regions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

electron density is centered to the atomic locations and has nearly spherical symmetry, the bond between the atoms becomes ionic. Furthermore, the Mulliken overlap population can be used to investigate the covalent or ionic nature of a bond. Positive and negative bond population values indicate bonding and antibonding states, respectively. A high value of the bond population indicates a covalent bond, while a low value indicates an ionic bond, however, zero value indicates an ideal ionic bond [37,38]. Calculated Mulliken overlap populations and bond lengths of different bond types in Bn(n ¼ 13, 14, 15) phases are listed in Table 2. For each structure, inter-icosahedral bonds (BpeBp*, BceBe and BceBc bonds) have higher population than intra-icosahedral bonds of icosahedron (BpeBp, Bp-Be*, BpeBe and Be-Be bonds), and except for b-B14, the bonds of BpeBp* in all structures have the higher population values. Generally, in all structures the outer (inter-icosahedral) bond lengths are smaller than the inner (intraicosahedral) bonds. For BpeBp* bonds, the bond in a-B12 has the highest Mulliken population, and the bond in b-B13 has the smallest bond length. There may be charge transfer in the bonds between inserting atom(s) (positively charged) and icosahedral atoms (negatively charged). It is seen from Fig. 3(a)e(f) that intericosahedral bonds of icosahedron have high electron density, and then high population value. In order to determine possible charge transfers in Bn-phases, we calculated Mulliken atomic charges, and results are listed in Table 3. It is observe that charge transfers exist between boron atoms located at different sites. However, total

Table 2 Calculated bond lengths, d(Å) and Mulliken overlap populations (P) In Bn(n ¼ 13, 14, 15) with those of a-B12. Bond classification is done in accordance with Ref. [23]. Additionally, Bc-Be bond is linking equatorial boron atoms to chain atoms, while BceBc bond is between chain atoms.

BpeBp* BpeBp BpeBe* BpeBe BeeBe BceBe BceBc

d P d P d P d P d P d P d P

a-B12

a-B13

b-B13

a-B14

b-B14

B15

1.648 1.13 1.718 0.43 1.773 0.53 1.764 0.47 1.750 0.48 e e e e

1.698 1.05 1.865 0.44 1.718 0.58 1.798 0.38 1.682 0.58 1.628 1.01 e e

1.601 1.12 1.773 0.49 1.723 0.48 1.756 0.48 1.838 0.46 e e e e

1.705 1.06 1.845 0.49 1.762 0.52 1.788 0.43 1.737 0.49 1.652 1.04 e e

1.671 1.10 1.816 0.51 1.760 0.49 1.757 0.50 1.739 0.54 1.777 0.74 1.620 1.26

1.699 1.10 1.869 0.48 1.756 0.53 1.774 0.47 1.771 0.49 1.673 0.90 1.590 1.07

charges of all phases are very small and also the charges of boron atoms in polar sites are very close to zero. The charges of equatorial boron atoms in all constituent structures, except for a-B13 are same and total charges of icosahedra are negative, while total charges of inserted atom(s) are positive. The mechanical properties of the structures are examined. We calculated the elastic constants by using stress-strain method, in order to check mechanical stability of the structures, and to calculate related physical properties. The physical/mechanical properties such as bulk modulus, shear modulus, Young modulus and Poisson’s ratio can be determined from elastic constants by

Table 3 Calculated Mulliken atomic charges of boron atoms located at different sites in Bn-phases. Qe-I and Qe-II are charges of boron atoms located at equatorial site on icosahedron, Qe-I stands for charge of boron atoms linked to chain/inserted atom(s). Qp-I and Qp-II are charges of boron atoms located at up-polar and down-polar sites on icosahedron, respectively. Qc and Qm are charges of boron atoms located at the edges and center of chain, respectively. Qico, Qsubs and Qnet are total charges of icosahedron, inserted atoms, and the phase. Qe-I

a-B12 0.00 a-B13 b-B13 a-B14 b-B14 B15

0.14 0.01 0.08 0.00 0.02

Qe-II

Qp-I

Qp-II

Qc

Qm

Qico

Qsubs

Qnet

0.00 0.11 0.01 0.08 0.00 0.02

0.00 0.02 0.00 0.00 0.02 0.00

0.00 0.01 0.00 0.00 0.02 0.00

e 0.21 0.04 0.23 0.06 0.17

e e e e e 0.43

0.00 0.18 0.06 0.48 0.12 0.12

0.00 0.21 0.04 0.46 0.12 0.09

0.00 0.03 0.02 0.02 0.00 0.03

Table 4 Calculated elastic constants (cij), bulk (B), shear (G) And Young (E) modulus in GPa unit, Poisson’s ratio (v) And mechanical stability (MS) of Bn-phases.

c11 c33 c44 c12 c13 c66 BV GV BR GR B G E v MS

a-B12

a-B13

b-B13

a-B14

b-B14

B15

473.8 632.2 225.0 108.7 43.5 182.6 219.1 218.8 217.9 213.2 218.5 216.0 487.4 0.13 Yes

210.0 398.6 121.4 24.7 32.1 92.6 110.7 115.7 101.3 110.3 106.0 113.0 250.2 0.11 Yes

29.8 561.9 98.8 218.7 126.4 94.5 173.8 30.6 124.2 571.4 149.0 301.0 539.7 0.10 No

338.4 381.7 107.7 245.8 59.0 46.3 198.5 98.6 194.4 74.3 196.4 86.5 226.2 0.31 Yes

25.5 461.8 19.0 363.2 134.3 168.9 197.4 49.3 185.0 45.8 191.2 47.6 155.6 0.64 No

423.7 363.7 1.5 167.5 55.8 128.1 196.6 88.4 190.6 3.8 193.6 46.1 128.0 0.39 Yes

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Table 5 m Calculated number of bonds (dm in Å), valence electron density (Ne ), bond ionicity m (fi), individual bond hardnesses (Hv in GPa), and total hardness (Hv in GPa) of Bn-phases. m

m

Crystal

Bond

Pm

dm

nm

Ne

fi

Hv (GPa)

Hv (GPa)

a-B13

BpeBp* BceBe BeeBe BpeBe*

1.05 1.01 0.58 0.58 0.48 0.54 0.38 0.49 0.44 1.04 1.06 0.49 0.52 0.43 0.49 0.9 1.10 1.07 0.48 0.53 0.49 0.47

1.698 1.628 1.682 1.718 1.770 1.715 1.798 1.834 1.865 1.652 1.705 1.737 1.762 1.788 1.845 1.673 1.699 1.590 1.869 1.756 1.771 1.774

3 3 6 6 6 3 3 3 3 6 3 6 12 6 6 6 3 2 6 6 6 12

0.351 0.597 0.361 0.339 0.310 0.341 0.296 0.279 0.265 0.626 0.380 0.359 0.344 0.329 0.300 0.532 0.406 1.113 0.305 0.367 0.358 0.357

0.000 0.000 0.111 0.111 0.230 0.053 0.472 0.204 0.330 0.000 0.000 0.204 0.120 0.354 0.204 0.000 0.000 0.000 0.230 0.089 0.204 0.256

46.41 73.37 42.39 38.51 29.25 41.63 20.43 25.71 20.52 73.00 48.39 34.88 36.18 25.61 26.62 63.49 51.01 117.80 25.22 39.52 33.13 30.97

35.09

BpeBe BpeBp a-B14

B15

BceBe BpeBp* BeeBe BpeBe* BpeBe BpeBp BceBe BpeBp* BceBc BpeBp BpeBe* BeeBe BpeBe

37.07

38.68

mean of polycrystalline Hill approach [39]. Calculated elastic constants and connected physical properties are listed in Table 4. It is seen from Table 4, all of phases are mechanically stable, except for b-phases of B13 and B14. For each mechanically stable phase, c33 has the highest value, and c11 follows it. The order of bulk modulus (B) from the highest to the lowest is given as B12 > aB14 > B15 > a-B13. Furthermore, the orders of shear modulus and Young modulus are same as B12 > a-B13 > a-B14 > B15. On the other hand, Young modulus (E) and Poisson’s ratio (v) values give technological properties of material such as stiffness and deformation to volume changing with compression and or decompression by pressure and/or heating. As seen from Table 4 the largest value of Poisson’s ratio is 0.39 of B15, and it corresponds to the larger deformation to the volume changing, and then, it may have highest anisotropy. Calculated individual bond hardnesses and total hardness of mechanically stable hypothetic Bn-phases are presented in Table 5. As seen from Table 5, calculated total hardnesses of Bn (n ¼ 13, 14, 15) phases are 35.09 and 37.07 GPa, and 38.68 GPa, respectively, then the order of hardness is B15 > a-B14 > a-B13. BceBc and Bc-Be bonds are the hardest bonds in the structures, and BpeBp* bonds follow them. As expected, due to lower population value, higher bond lengths and higher ionicity, intra-icosahedral bonds have lower hardness.

Fig. 4. Calculated band structure (a) And PDOS (b) For a-B13.

Fig. 5. Calculated band structure (a) And PDOS (b) For a-B14.

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Fig. 6. Calculated band structure (a) And PDOS (b) For B15.

Fig. 7. Calculated band structure (a) And PDOS (b) For a-B12.

In order to reveal the effects of inserted atoms on the electronic structure, we also calculated band structures and partial density of states (PDOS) of Bn-phases are shown in Figs. 4e6 (a) and (b), respectively. For comparison, those of a-B12 are also shown in Fig. 7. It is seen that calculated band structure and energy gap of 1.66 eV

Fig. 8. Calculated total density of states (TDOS) of Bn-phases.

for a-B12 agree well with calculated value of 1.70 eV in Ref.[40]. However, the calculated energy gap with GGA approach is smaller than experimental value of 2.0 eV [41], it is note that the DFT calculations with GGA approach always lower estimate than the experimental values [41]. Although the base structure a-B12 is semiconductor, all of mechanically stable hypothetic Bn-phases (aB13, a-B14 and B15) exhibit metallic character, because density of states in Fermi level is different from zero. At the same time, one band of a-B13 (see Fig. 4), two bands of a-B14 (see Fig. 5) and two bands of B15 (see Fig. 6) are crossing the Fermi level. From PDOS curves of Bn-phases, boron s-states are dominant at the lower energy range of the valence band. At the higher energy range and the vicinity of the Fermi level, p-states become dominant. However, valence states of a-B13 and a-B14 are similar to those of a-B12, and can be split to three main regions: (i) w(15 eV), (ii) w(10 eV), and (iii) energy range from 10 eV to Fermi level, respectively. On the other hand, valence states of B15 have some difference from those of a-B13 and a-B14: there are two main peak groups instead of three main peak groups. One group is centered about 15 eV, other group is located at energy range from 13 eV to the Fermi level. Furthermore, total DOS (TDOS) curves of all Bn-phases are plotted in Fig. 8, and it can be seen from this figure that, when number of inserted boron atoms increase, density of states at the Fermi level (N(EF)) increases. On the other hand, the anisotropic nature of bonding structure in metallic a-B13 yields strong dispersion on the band of p-electrons along BeU, UeA and also GeZ directions in the

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vicinity of the Fermi level, in which the uppermost valence band crossing the Fermi level. The most anisotropic structure is B15 and the second one is a-B14. In both structure two bands of p-states located at the vicinity of the Fermi level crossing the Fermi level more than that of B13. It is meaning that the high electron conductivity is accessible in a-B14 and B15. The important features of the p-states around the uppermost valence band and the lowest conduction bands can be qualified to the localized directional intericosahedral bonds. 4. Conclusions The structural, mechanical and electronic properties of a-B13, bB13, a-B14, b-B14 and B15 have been investigated by first-principles plane wave density functional calculations. It is shown that all of these recently presented hypothetical boron compounds are energetically stable and also mechanically stable, except for bphases. MD and LST calculations indicate that b-phases of B13 and B14 are metastable states at 0 GPa pressure. Although the host structure (a-boron) is semiconductor, recently designed phases are metallic. However, inserted boron atoms play an important role on mechanical and electronic behavior of the icosahedral structure. Among the energetically and mechanically stable phases, B15 is the most anisotropic material. It is observed that there are occupied and unoccupied nearly flat bands located at the vicinity of the Fermi level. From the hardness analysis, the covalent bonds between inserted atom(s) (BceBc), and between inserted atom(s) and equatorial boron atoms (BceBe) are harder than the others. Total hardnesses of Bn (n ¼ 13, 14, 15) phases are 35.09 and 37.07 GPa, and 38.68 GPa, respectively, and these values are considerably close to superhardness limit of 40 GPa. Acknowledgments This work is supported partly by the State of Planning Organization of Turkey under Grant No. 2011K120290 and Gazi University BAP, under grant no. 05/2010-82. References [1] D. Emin, Phys. Today 40 (1987) 55.

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