materials

Article Si96: A New Allotrope with Interesting Physical Properties

Qingyang Fan 1,*, Changchun Chai 1, Qun Wei 2, Peikun Zhou 3, Junqin Zhang 1 and Yintang Yang 1

1 Key Laboratory of Ministry of Education for Wide Band-Gap Semiconductor Materials and Devices, School of Microelectronics, Xidian University, Xi’an 710071, China; [email protected] (C.C.); [email protected] (J.Z.); [email protected] (Y.Y.) 2 School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China; [email protected] 3 Faculty of Science, University of Paris-Sud, Paris 91400, France; [email protected] * Correspondence: [email protected]; Tel.: +86-29-8820-2507

Academic Editor: Martin O. Steinhauser Received: 22 January 2016; Accepted: 7 April 2016; Published: 13 April 2016

Abstract: The structural mechanical properties and electronic properties of a new silicon allotrope Si96 are investigated at ambient pressure by using a first-principles calculation method with the ultrasoft pseudopotential scheme in the framework of generalized gradient approximation. The elastic constants and phonon calculations reveal that Si96 is mechanically and dynamically stable at ambient pressure. The conduction band minimum and valence band maximum of Si96 are at the R and G point, which indicates that Si96 is an indirect semiconductor. The anisotropic calculations show that Si96 exhibits a smaller anisotropy than Si in terms of Young’s modulus, the percentage of elastic anisotropy for bulk modulus and shear modulus, and the universal anisotropic index AU. Interestingly, most silicon allotropes exhibit brittle behavior, in contrast to the previously proposed ductile behavior. The void framework, low density, and nanotube structure make Si96 quite attractive for applications such as hydrogen storage and electronic devices that work at extreme conditions, and there are potential applications in Li-battery anode materials.

Keywords: ab initio calculations; structural and anisotropic properties; silicon allotropes

1. Introduction Searching for novel silicon allotropes has been of great interest over the past several decades and has been extensively studied. In other words, it is a hot topic. Experimentally, silicon has been found to have a complicated phase diagram [1]. Many semiconductor silicon structures that have an indirect band gap have been proposed, such as the M phase, Z phase [2], Cmmm phase [3], body-centered-tetragonal Si [4], M4 phase [5], lonsdaleite phase [6], T12 phase [7], ST12 phase [8], C-centered orthorhombic phase [9], C2/m-16, C2/m-20, Amm2, I-4 [10], and P2221 [11], among others. Many semiconductor silicon structures with a direct band gap have been reported, such as P213 phase [12], oF16-Si, tP16-Si, mC12-Si, and tI16-Si [13]. Silicon structures with metallic properties have also been reported, such as the β-Sn phase, R8 phase [14], and Ibam phase [15]. All of the previous studies have opened the possibility of a broader search for new allotropes of silicon that possibly exhibit novel properties. Karttunen et al. [16] have investigated the structural and electronic properties of various clathrate frameworks that are composed of the group 14 element semiconductors, including carbon, silicon, , and tin. Several of the studied clathrate frameworks for silicon possess direct and wide band gaps [16]. Zwijnenburg et al. [17] have studied several new prospective low-energy silicon allotropes that use density functional theory, brute-force random search approaches and hypothetical 4CNs from Treacy

Materials 2016, 9, 284; doi:10.3390/ma9040284 www.mdpi.com/journal/materials Materials 2016, 9, 284 2 of 8 and co-workers. These new low-energy silicon allotropes contain 4-member rings that were previously considered to be incompatible with low-energy silicon structures, and these computational approaches that were employed to explore the energy landscape of silicon were all found to have their own optimal zone of applicability. A systematic search for silicon allotropes was performed by employing a modified ab initio minima hopping structure prediction method by Amsler et al. [18]. They found silicon clathrates that had low density and a stronger overlap of the absorption spectra with the solar spectrum compared to conventional diamond silicon, which are thus promising candidates for use in thin-film photovoltaic applications. Recently, Li et al. [19] found a novel cubic allotrope of carbon, C96-carbon, that has intriguing physical properties. We proposed Si96 (space group: Pm-3m), whose structure is based on C96-carbon [19], with silicon atoms substituting carbon atoms. The physical properties of this new cubic Si allotrope are reported in this paper. The Si96 is composed of six-membered silicon rings that are very similar to graphite-like six-membered carbon rings. Furthermore, the structure of the cubic Si96 phase is porous and has a lower density. Due to its structural porous feature and lower density, Si96 can also be expected to be good hydrogen storage material. The calculation of the Mulliken overlap population ensures the existence of strong covalent bonds in six-membered silicon rings, and thus, the hardness of Si96 is close to diamond Si. Six-membered silicon rings and zigzag six-membered silicon rings cause Si96 to have a lower density among the silicon allotrope materials.

2. Materials and Methods All of the calculations are performed by utilizing the generalized gradient approximation (GGA) functional in the Perdew, Burke and Ernzerrof (PBE) [20] functional form in Cambridge sequential total energy package (CASTEP) [21]. The core-valence interactions were described as Ultra-soft pseudopotentials [22]. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) [23] minimization scheme was used in geometric optimization. The valence electron configurations of Si 3s23p2 are considered. A tested mesh of 4 ˆ 4 ˆ 4 k-point sampling was used for the calculations. For the new Si phase, the ultrasoft pseudopotential was used with the cutoff energy of 300 eV. The self-consistent convergence of the total energy is 5 ˆ 10´6 eV/atom; the maximum force on the atom is 0.01 eV/Å, the maximum ionic displacement is within 5 ˆ 10´4 Å, and the maximum stress is within 0.02 GPa. Both the HSE06 hybrid functional [24] and GGA-PBE methods were used for the calculations of electronic structures. The phonon spectra of Si96 used the linear response approach, also called density functional perturbation theory (DFPT), which is one of the most popular methods of an ab initio calculation of lattice dynamics [25].

3. Results and Discussion

The of Si96 is shown in Figure1; it belongs to the Pm-3m space group of cubic symmetry. The basic building blocks of Si96 are six-membered graphite-like carbon rings, which can be clearly observed in Figure1a. The six-membered rings are normal to the [111] direction in the structure of Si96 (Figure1b). The optimized equilibrium lattice parameter is a = 13.710 Å at ambient pressure, and there are 96 silicon atoms in a conventional cell. The three color spheres represent three non-equivalent atoms. The blue atoms occupy the crystallographic 48n sites in a conventional cell, which is (0.6605, 0.7863, 0.0870). The red atoms occupy the crystallographic 24l1 sites in a conventional cell, which is (0.3792, 0.5, 0.2580). The cyan atoms also occupy the crystallographic 24l2 sites, which have the (0.5, 0.7143, 0.1373) position in a conventional cell, respectively. Furthermore, Si96 has nanotube-like cavities along the crystallographic main axis and the [111] direction, as shown in Figure1a,b. It is well known that the regular arrangement of nanotubes can enhance the efficiency of the hydrogen storage of nanotubes [19]. Thus, the Si96 can be expected to have a good ability for hydrogen storage. Generally, the densities of the materials are closely related to the hydrogen storage ability of materials. The density 3 3 (1.737 g/cm ) of Si96 is smaller than that of diamond Si (calculated value: 2.322 g/cm , experimental value: 2.329 g/cm3). In addition, the densities of I-4 (2.1511 g/cm3), Amm2 (2.1809 g/cm3), C2/m-16 3 3 3 (2.2172 g/cm ), C2/m-20 (2.2251 g/cm )[10] and P2221 (2.227 g/cm )[11] are slightly larger than that Materials 2016, 9, 284 3 of 8

Materials 2016, 9, 284 3 of 8 of Si96. Thus, Si96 can be expected to have a good hydrogen storage ability among the low density are slightly larger than that of Si96. Thus, Si96 can be expected to have a good hydrogen storage ability materialsamong or potentialthe low density applications materials toor potential Li-battery applications anode materials. to Li-battery anode materials.

(a) (b)

Figure 1. Unit cell crystal structures of Si96 (a) and along the [111] direction (b). Figure 1. Unit cell crystal structures of Si96 (a) and along the [111] direction (b).

There are eight bond lengths in Si96, namely, 2.3397, 2.3398, 2.3514, 2.3557, 2.3858, 2.3873, 2.4419 and 2.4540 Å, and each bond length has a difference number; the average bond length is 2.3862 Å, There are eight bond lengths in Si96, namely, 2.3397, 2.3398, 2.3514, 2.3557, 2.3858, 2.3873, 2.4419 and 2.4540which Å, is andslightly each larger bond than length that hasof diamond a difference Si (2.3729 number; Å). The the atoms average occupy bond the length crystallographic is 2.3862 Å, which 24l1 sites, which consist of a four-membered ring. The four bond lengths are all 2.3398 Å. The other is slightly larger than that of diamond Si (2.3729 Å). The atoms occupy the crystallographic 24l sites, atoms occupy the crystallographic 24l1 sites, which consist of a six-membered silicon ring. The 1 whichsix-membered consist of a four-membered ring includes two ring. bondThe lengths, four 2.3514 bond and lengths 2.3398 are Å. allSome 2.3398 of the Å. atoms The otherthat occupy atoms occupy the crystallographicthe crystallographic 24l 124sites,l2 sites which are connected consist to of occupy a six-membered the crystallographic silicon ring.24l1 sites The and six-membered 48n sites, ring includesand two the bondbond lengths lengths, are 2.3514 2.3397 and and 2.33982.3873 Å.Å, re Somespectively. of the Some atoms of thatthe atoms occupy that the occupy crystallographic the crystallographic 24l2 sites connect to themselves, and the bond length is 2.3557 Å. The other 24l2 sites are connected to occupy the crystallographic 24l1 sites and 48n sites, and the bond lengths are six-membered silicon ring consists of the atoms that occupy the crystallographic 48n sites, and it 2.3397 and 2.3873 Å, respectively. Some of the atoms that occupy the crystallographic 24l sites connect includes two bond lengths, 2.4419 and 2.4540 Å. Moreover, there is a zigzag six-membered silicon2 to themselves,ring that consists and the of the bond three length non-equivalent is 2.3557 atoms, Å. The which other includes six-membered three bond lengths, silicon 2.3397, ring consists2.3873 of the atomsand that 2.4419 occupy Å. the crystallographic 48n sites, and it includes two bond lengths, 2.4419 and 2.4540 Å. Moreover,To there understand is a zigzag the six-memberedthermodynamic siliconstability, ring the enthalpies that consists of the of theproposed three structures non-equivalent were atoms, whichcompared includes threewith the bond experimentally lengths, 2.3397, known 2.3873 diamond and Si 2.4419 and β Å.-Sn phase Si and the theoretically Toproposed understand M phase, the Z thermodynamicphase, lonsdaleite phase, stability, P42/ncm the phase, enthalpies C-centered of orthorhombic the proposed (Cco structures) phase, were P42/mnm phase, and P42/mmc phase (tP-16-Si), as depicted in Figure 2. It is obvious that diamond Si compared with the experimentally known diamond Si and β-Sn phase Si and the theoretically proposed remains the most stable phase at ambient pressure. The most unfavorable Si96 is higher in energy than M phase,diamondZ phase, Si by lonsdaleite 0.307 eV/atom phase, at Pambient42/ncm pressure,phase, C-centeredwhile the metastable orthorhombic lonsdaleite (Cco) phase,phase isP 42/mnm phase,0.017 and PeV/atom42/mmc higherphase than (tP diamond16-Si), as Si. depicted In Ref. [13], in the Figure most2 .unfavorable It is obvious tP16-Si that is diamond higher in energy Si remains the most stablethan diamond phase at Si ambient by 0.269 eV/atom pressure. (this The work: most 0.277 unfavorable eV/atom) at Si ambient96 is higher pressure. in energy In Ref. than[12], the diamond Si by 0.307new eV/atom Si20 structure at ambient (space group: pressure, P213) whileis less thestable metastable than Fd-3m lonsdaleite Si by approximately phase is 0.3 0.017 eV/Si eV/atom due to higher than diamondthe distortion Si. Inof the Ref. Si [tetrahedrons.13], the most Generally, unfavorable the distortiontP16-Si of is the higher tetrahedron in energy in these than metastable diamond Si by structures leads to their higher energy. The planar four-membered silicon rings will result in a more 0.269 eV/atom (this work: 0.277 eV/atom) at ambient pressure. In Ref. [12], the new Si structure severe distortion than five-membered silicon rings. The dynamical stability of the structure of 20Si96 (spacewas group: checkedP21 in3) this is less paper. stable The thanphononFd -3dispersionm Si by for approximately Si96 at ambient 0.3pressure eV/Si was due calculated. to the distortion No of the Si tetrahedrons.imaginary frequencies Generally, are observed the distortion throughout of the the whole tetrahedron Brillouin zone, in these which metastable signals dynamically structures the leads to their higherstructural energy. stability The of Si planar96, as shown four-membered in Figure 3. silicon rings will result in a more severe distortion In addition, we calculated the hardness of Si96 and diamond Si using the model of Lyakhov and than five-membered silicon rings. The dynamical stability of the structure of Si96 was checked in this Oganov [26]. The hardness of Si96 is 9.6 GPa, which is slightly smaller than that of diamond Si (13.3 GPa). paper. The phonon dispersion for Si at ambient pressure was calculated. No imaginary frequencies The other results of hardness for diamond96 Si are 8 GPa [27], 9 GPa [28], 12.4 GPa [29], and 2–16 GPa [30]. are observedTo understand throughout the origin the of wholethe hardness, Brillouin it is ne zone,cessary which to understand signals the dynamically Mulliken overlap the structuralpopulation stability of Si96,and as shownbond length. in Figure The average3. bond length of Si96 is 2.3862 Å, a value that is very close to that of Indiamond addition, Si, which we calculated indicates that the the hardness bonds of ofSi96 Si should96 and have diamond as high Sia bond using strength the model as that of ofLyakhov diamond. The average Mulliken overlap population of Si96 (0.66) is very close to that of diamond Si and Oganov [26]. The hardness of Si96 is 9.6 GPa, which is slightly smaller than that of diamond Si (13.3 GPa).(0.73), Thewhich other also resultsconfirms of that hardness the bond for strength diamond of Si96 Si is arevery 8 strong. GPa [ 27], 9 GPa [28], 12.4 GPa [29], and 2–16 GPa [30]. To understand the origin of the hardness, it is necessary to understand the Mulliken overlap population and bond length. The average bond length of Si96 is 2.3862 Å, a value that is very close to that of diamond Si, which indicates that the bonds of Si96 should have as high a bond strength as that of diamond. The average Mulliken overlap population of Si96 (0.66) is very close to that of diamond Si (0.73), which also confirms that the bond strength of Si96 is very strong. Materials 2016, 9, 284 4 of 8

Materials 2016, 9, 284 4 of 8 Materials 2016, 9, 284 4 of 8

Figure 2. Calculated enthalpies of different silicon structures relative to the diamond Si at ambient Figure 2. Calculated enthalpies of different silicon structures relative to the diamond Si at Figurepressure. 2. Calculated enthalpies of different silicon structures relative to the diamond Si at ambient ambientpressure. pressure.

3 3

2 2

1 Frequency (THz) 1 Frequency (THz)

0 0 X R M G R X R M G R Figure 3. Phonon spectrum for Si96. FigureFigure 3. 3.Phonon Phonon spectrum for for Si Si96.96 .

We next investigate the mechanical properties of Si96 and diamond Si for reference. The We next investigate the mechanical properties of Si96 and diamond Si for reference. The Wecalculated next investigate results of Si the96 and mechanical diamond Si properties are listed ofin SiTable96 and 1. The diamond three independent Si for reference. elastic The constants calculated calculated results of Si96 and diamond Si are listed in Table 1. The three independent elastic constants resultsCij of the Si cubicand symmetry, diamond namely Si are C11 listed, C12, and in C Table44, obey1. the The following three generalized independent Born’s elastic mechanical constants Cij of the 96cubic symmetry, namely C11, C12, and C44, obey the following generalized Born’s mechanical stability criteria: C11 > 0, C44 > 0, C11-C12 > 0, and C11 + 2C12 > 0 [31,32]. The elastic modulus is also Cij of the cubic symmetry, namely C11, C12, and C44, obey the following generalized Born’s stabilitycalculated. criteria: Young’s C11 >modulus 0, C44 > 0,E Cand11-C 12Poisson’s > 0, and ratioC11 + v2 Care12 > taken0 [31,32]. as follows:The elastic E =modulus 9BG/(3B is + alsoG), mechanical stability criteria: C > 0, C > 0, C -C > 0, and C + 2C > 0 [31,32]. The elastic calculated.v = (3B − 2G )/[2(3Young’sB + Gmodulus)]. The 11bulk E and modulus Poisson’s44 and shear11ratio12 modulusv are taken are smalleras11 follows: than12 Ethat = of9BG diamond/(3B + G Si.), modulusv = (3B is − also2G)/[2(3 calculated.B + G)]. The bulk Young’s modulus modulus and shearE andmodulus Poisson’s are smaller ratio thanv thatare of taken diamond as follows:Si. The shear modulus and Young’s modulus of Si96 is only approximately one third that of diamond Si. E = 9BGTheIn addition,/(3 shearB + modulusG the), v value= (3andB of´ Young’sPoisson’s2G)/[2(3 modulus ratioB + isG larger)]. ofThe Si96 th is bulkan only that modulus approximately of diamond and Si.shear one Pugh third modulus[33] that proposed of arediamond smallerthe ratio Si. than In addition, the value of Poisson’s ratio is larger than that of diamond Si. Pugh [33] proposed the ratio that ofof diamondbulk to shear Si. modulus The shear (B modulus/G) as an indication and Young’s of ductile modulus versus of brittle Si96 ischaracters. only approximately If B/G > 1.75, onethen third that ofofthe diamondbulk material to shear Si.behaves Inmodulus addition, in a (ductileB/G the) as value way.an indication Otherwis of Poisson’s ofe, ductilethe ratio material versus is larger behaves brittle than characters. in that a brittle of diamond If way. B/G >The 1.75, Si. B Pugh/thenG of [33] the material behaves in a ductile way. Otherwise, the material behaves in a brittle way. The B/G of proposedSi96 is thelarger ratio than of 1.75, bulk which to shear suggests modulus that the (B Si/G96 )allotrope as an indication is prone to of ductile ductile behavior.versus Interestingly,brittle characters. Si96 is larger than 1.75, which suggests that the Si96 allotrope is prone to ductile behavior. Interestingly, If B/Gmost> 1.75, of the then silicon the materialallotropes behaves exhibit brittle in a ductile behavior way. (M Otherwise,-Si: 1.22; Z-Si: the 1.35; material T12-Si: behaves 1.56; I-4: in 1.68; a brittle most of the silicon allotropes exhibit brittle behavior (M-Si: 1.22; Z-Si: 1.35; T12-Si: 1.56; I-4: 1.68; Amm2: 1.54, C2/m-16 Si: 1.60; C2/m-20 Si: 1.50; P2221 Si: 1.54 and diamond Si: 1.40), which contrasts way. The B/G of Si96 is larger than 1.75, which suggests that the Si96 allotrope is prone to ductile Ammwith 2:the 1.54, previously C2/m-16 proposed Si: 1.60; Cductile2/m-20 behavior. Si: 1.50; P222 1 Si: 1.54 and diamond Si: 1.40), which contrasts behavior.with the Interestingly, previously proposed most of ductile the silicon behavior. allotropes exhibit brittle behavior (M-Si: 1.22; Z-Si: 1.35; T12-Si: 1.56;TableI-4: 1. 1.68;The latticeAmm parameters2: 1.54, C 2/(Å),m density-16 Si: (ρ 1.60;: g/cmC3),2/ elasticm-20 constants Si: 1.50; (GPa) P222 and1 Si: elastic 1.54 modulus and diamond Si: Table 1. The lattice parameters (Å), density (ρ: g/cm3), elastic constants (GPa) and elastic modulus 1.40), which(GPa) contrasts of Si96 and with diamond the previouslySi. proposed ductile behavior. (GPa) of Si96 and diamond Si. Materials Work a ρ C11 3 C12 C44 B G B/G E v Table 1. The lattice parameters (Å), density (ρ: g/cm11 ), elastic12 constants44 (GPa) and elastic modulus MaterialsSi96 ThisWork work 13.710a 1.737ρ C89 C33 C26 52B 27G B1.93/G 69E 0.28v (GPa)DiamondSi of96 Si Si96 andThis diamond work Si. 13.7105.436 1.737 2.322 16589 3365 2687 5298 2770 1.931.40 17069 0.280.21 Diamond Si ExperimentalThis work 5.4315.436 1 2.329 2.322 2 166 165 3 6564 8780 10298 70- 1.40- 170- 0.21- 1 2 3 Diamond Si Experimental 1 5.431 2.329 2 166 64 380 102 - - - - Materials Work Ref. a[34] at 300 K;ρ Ref. [35]C11 at 300C 12K; Ref.C44 [36]. BGB/G E v 1 Ref. [34] at 300 K; 2 Ref. [35] at 300 K; 3 Ref. [36]. Si96 This work 13.710 1.737 89 33 26 52 27 1.93 69 0.28 Diamond Si This work 5.436 2.322 165 65 87 98 70 1.40 170 0.21 Diamond Si Experimental 5.431 1 2.329 2 166 3 64 80 102 - - - - 1 Ref. [34] at 300 K; 2 Ref. [35] at 300 K; 3 Ref. [36]. Materials 2016, 9, 284 5 of 8 Materials 2016, 9, 284 5 of 8

It is well known that the electronic structure determines the fundamental physical and chemical It is well known that the electronic structure determines the fundamental physical and chemical properties of materials. The calculated electronic band structures for Si96 utilizing GGA-PBE and properties of materials. The calculated electronic band structures for Si96 utilizing GGA-PBE and HSE06 are presented in Figure 4. The electronic band structure calculation shows that the new silicon HSE06 are presented in Figure4. The electronic band structure calculation shows that the new silicon allotrope Si96 is metallic because the conduction band minimum and valence band maximum both allotrope Si96 is metallic because the conduction band minimum and valence band maximum both overlap the Fermi level. It is known that the calculated band gap with DFT is usually underestimated overlap the Fermi level. It is known that the calculated band gap with DFT is usually underestimated by 30%–50%, and the true band gap must be larger than the calculated results. In consideration of by 30%–50%, and the true band gap must be larger than the calculated results. In consideration of this problem, Heyd et al. proposed a more tractable hybrid functional method, which gave rise to the this problem, Heyd et al. proposed a more tractable hybrid functional method, which gave rise to the Heyd–Scuseria–Ernzerhof (HSE06) functional. The hybrid functional HSE06 is used in the following Heyd–Scuseria–Ernzerhof (HSE06) functional. The hybrid functional HSE06 is used in the following form [37,38]: form [37,38]: HSEHSE HF,HFSR,SR PWPW91,91,SRSR PWPW9191,, LRLR PWPW91 91 Exc E xc“ µExE x pω(q `)  p1(1´ µq)EExx (pω)q `EEx x (pω) q `EcEc (1)(1) where thethe HFHF mixingmixing parameter parameterµ μis is 0.25, 0.25, and and the the screening screening parameter parameter that that provides provides good good accuracy accuracy for ´1 thefor the band band gaps gaps is ω is= ω 0.207 = 0.207 Å Å[−241 [24,38].,38]. The The electronic electronic band band structure structure calculation calculation shows shows that that Si96 Siis96 an is indirectan indirect band band gap gap semiconductor semiconductor with with a banda band gap gap of of 0.474 0.474 eV eV for for the the HSE06 HSE06 hybridhybrid functional.functional. Thus, the band structure calculationscalculations showshow thatthat thethe newnew SiSi9696 isis a a narrow narrow band band gap gap semiconductor semiconductor material. material.

1.0

0.8

0.6

0.4

0.2 HSE06 PBE 0.0

-0.2 Energy (eV) -0.4

-0.6

-0.8

-1.0 X R M G R Si Band structure 96

Figure 4.4. ElectronicElectronic band band structure structure using using Perdew, Perdew, Burke Burke and and Ernzerrof Ernzerrof (PBE) and 96 Heyd–Scuseria–Ernzerhof (HSE06) ofof SiSi96..

The 3D figures of the directional dependences of the reciprocals of Young’s modulus and the The 3D figures of the directional dependences of the reciprocals of Young’s modulus and projections of Young’s modulus at different crystal planes for the new silicon allotrope Si96 and the projections of Young’s modulus at different crystal planes for the new silicon allotrope diamond Si are demonstrated in Figure 5a,c and Figure 5b,d. The three-dimensional (3D) surface Si and diamond Si are demonstrated in Figures5a,c and5b,d. The three-dimensional (3D) construction96 is a valid method for describing the elastic anisotropic behavior of a solid perfectly. surface construction is a valid method for describing the elastic anisotropic behavior of a solid Usually, the anisotropic properties of materials are different due to their various crystal structures [39]. perfectly. Usually, the anisotropic properties of materials are different due to their various The 3D figure appears to be a spherical shape for an isotropic material, while the deviation from the crystal structures [39]. The 3D figure appears to be a spherical shape for an isotropic material, spherical shape exhibits the content of anisotropy [40]. It is clear that the diamond Si exhibits greater while the deviation from the spherical shape exhibits the content of anisotropy [40]. It is clear anisotropy than that of Si96. The maximal and minimal value of diamond Si are 183 and 124 GPa. The that the diamond Si exhibits greater anisotropy than that of Si96. The maximal and minimal maximal and minimal value of Si96 are 70 and 67 GPa, respectively. Emax/Emin(diamond Si) is equal to 1.476 value of diamond Si are 183 and 124 GPa. The maximal and minimal value of Si96 are 70 and and Emax/Emin(Si96) is equal to 1.048, and thus, the diamond Si exhibits greater anisotropy in Young’s 67 GPa, respectively. Emax/Emin(diamond Si) is equal to 1.476 and Emax/Emin(Si96) is equal to 1.048, modulus than that of Si96. In addition, the elastic anisotropy of a crystal can be depicted in many and thus, the diamond Si exhibits greater anisotropy in Young’s modulus than that of Si . different ways. In this work, several anisotropic indices are also calculated, such as the percentage 96of In addition, the elastic anisotropy of a crystal can be depicted in many different ways. In this anisotropy (AB and AG) and the universal anisotropic index (AU). In cubic symmetry, work, several anisotropic indices are also calculated, such as the percentage of anisotropy (AB BV = BR = (C11 + 2C12)/3, GV = (C11 − C12 + 3C44)/5, and GUR = [5(C11 − C12)C44]/(4C44 + 3C11 − 3C12). The equations and AG) and the universal anisotropic index (A ). In cubic symmetry, BV = BR = (C11 + 2C12)/3, used can be expressed as follows: AB = [(BV − BR)/(BV + BR)] × 100%, AG = [(GV − GR)/(GV + GR)] × 100% GV = (C11 ´ C12 + 3C44)/5, and GR = [5(C11 ´ C12)C44]/(4C44 + 3C11 ´ 3C12). The equations used can and AU = 5GV/GR + BV/BR − 6, and there, AU must be greater than or equal to zero. An AU fluctuation be expressed as follows: A = [(B ´ B )/(B + B )] ˆ 100%, A = [(G ´ G )/(G + G )] ˆ 100% B V R V R G V R V U R away Ufrom zero indicates high anisotropic elasticU properties. AB = 0, AG = 3.58%, and AU = 0.336 for and A = 5GV/GR + BV/BR ´ 6, and there, A must be greater than or equal to zero. An A fluctuation diamond Si, and AB = 0, AG = 0.036%, and AU = 0.004 for Si96. In other words, diamond Si exhibits away from zero indicates high anisotropic elastic properties. A = 0, A = 3.58%, and AU = 0.336 for U B G greater anisotropy in Young’s modulus, AB, AUG and A than Si96. diamond Si, and AB = 0, AG = 0.036%, and A = 0.004 for Si96. In other words, diamond Si exhibits To understand the possibility of using Si96 for hydrogen storage and lithium-battery anode greater anisotropy in Young’s modulus, A , A and AU than Si . material, we inset one and two hydrogenB orG lithium atoms into96 diamond Si (8 silicon atoms per conventional cell) and Si96 (96 silicon atoms per conventional cell). This method is consistent with Materials 2016, 9, 284 6 of 8

To understand the possibility of using Si96 for hydrogen storage and lithium-battery anode material, we inset one and two hydrogen or lithium atoms into diamond Si (8 silicon atoms per Materials 2016, 9, 284 6 of 8 conventional cell) and Si96 (96 silicon atoms per conventional cell). This method is consistent with references [[5,41]5,41] toto verifyverify whetherwhether M4 M4 silicon, silicon, diamond diamond Si Si and and other other amorphous amorphous silicon silicon can can be be used used as aslithium lithium battery battery electrode electrode materials. materials. For one For hydrogen one hydrogen (lithium) (lithium) atom insertion, atom insertion, the volume the expansion volume of the Si is ´0.52% (´0.25%), which is much smaller than that of diamond Si (hydrogen 1.95%; expansion96 of the Si96 is −0.52% (−0.25%), which is much smaller than that of diamond Si (hydrogen 1.95%;lithium lithium 2.91%, 2.94%2.91%, [ 52.94%]), Bct-Si [5]), (lithium Bct-Si (lithium 1.41% [5 ])1.41% and M4-Si[5]) and (lithium M4-Si 1.65%(lithium [5]). 1.65% For two [5]). hydrogen For two (lithium) atom insertions, the volume expansion of the Si is ´1.18% (´0.83%), which is much smaller hydrogen (lithium) atom insertions, the volume expansion96 of the Si96 is −1.18% (−0.83%), which is muchthan that smaller of diamond-Si than that of (hydrogen diamond-Si 4.81%; (hydrogen lithium 4.81%; 7.49%, lithium 7.53% [ 57.49%,]). These 7.53% results [5]). showThese that results Si96 showhas a higher capacity as a hydrogen storage and lithium-battery anode material. that Si96 has a higher capacity as a hydrogen storage and lithium-battery anode material.

(a) (b)

(c) (d)

Figure 5. The directional dependence of Young's modulus for Si96 (a) and diamond Si (c); and a 2D Figure 5. The directional dependence of Young’s modulus for Si96 (a) and diamond Si (c); and a 2D representation of Young’s modulus in the xy plane for Si96 (b) and diamond Si (d). representation of Young’s modulus in the xy plane for Si96 (b) and diamond Si (d).

4. Conclusions 4. Conclusions In summary, a new silicon allotrope with space group Pm-3m is predicted, which is mechanically In summary, a new silicon allotrope with space group Pm-3m is predicted, which is mechanically and dynamically stable at ambient pressure. The Si96 is an indirect-gap semiconductor with a band and dynamically stable at ambient pressure. The Si96 is an indirect-gap semiconductor with a band gap of 0.474 eV. The 3D surface contour of Young’s modulus is plotted to verify the elastic anisotropy gap of 0.474 eV. The 3D surface contour of Young’s modulus is plotted to verify the elastic anisotropy of Si96 and diamond Si. At the same time, diamond Si exhibits greater anisotropy than Si96 in Young’s of Si96 and diamond Si. At the same time, diamond Si exhibits greater anisotropy than Si96 in Young’s modulus and a slice of anisotropic indices. The void framework, low density and nanotube structures modulus and a slice of anisotropic indices. The void framework, low density and nanotube structures make Si96 quite attractive for particular applications, such as hydrogen storage and electronic devices make Si96 quite attractive for particular applications, such as hydrogen storage and electronic devices that work at extreme conditions, and for potential applications to Li-battery anode materials. that work at extreme conditions, and for potential applications to Li-battery anode materials.

Acknowledgments: This work was supported by the Natural Sc Scienceience Foundation of China (No. 61474089), 61474089), Open Fund of Key Laboratory of of Complex Complex Electromagnetic Electromagnetic Environment Environment Science Science and and Tec Technology,hnology, China Academy of Engineering Physics (No. 2015-0214. XY.K).

Author Contributions: Qingyang Fan designed the project; Qingyang Fan, Fan, Changchun Changchun Chai Chai and and Qun Qun Wei performed the calculations; Qingyang Fan, Qun Wei, Junqin Zhang, Yintang Yang and Peikun Zhou analyzed the performed the calculations; Qingyang Fan, Qun Wei, Junqin Zhang, Yintang Yang and Peikun Zhou analyzed results; and Qingyang Fan and Changchun Chai wrote the manuscript. the results; and Qingyang Fan and Changchun Chai wrote the manuscript.

Conflicts of Interest: The authors declare no conflict of interest.

Materials 2016, 9, 284 7 of 8

Conflicts of Interest: The authors declare no conflict of interest.

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