materials Article
Mechanical and Electronic Properties of XC6 and XC12 Qun Wei 1, *, Quan Zhang 2 and Meiguang Zhang 3, * 1 2 3
*
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China School of Microelectronics, Xidian University, Xi’an 710071, China;
[email protected] College of Physics and Optoelectronic Technology, Baoji University of Arts and Sciences, Baoji 721016, China Correspondence:
[email protected] (Q.W.);
[email protected] (M.Z.); Tel.: +86-29-8846-9165 (Q.W.); +86-917-336-4258 (M.Z.)
Academic Editor: Martin O. Steinhauser Received: 14 July 2016; Accepted: 22 August 2016; Published: 25 August 2016
Abstract: A series of carbon-based superconductors XC6 with high Tc were reported recently. In this paper, based on the first-principles calculations, we studied the mechanical properties of these structures, and further explored the XC12 phases, where the X atoms are from elemental hydrogen to calcium, except noble gas atoms. The mechanically- and dynamically-stable structures include HC6 , NC6 , and SC6 in XC6 phases, and BC12 , CC12 , PC12 , SC12 , ClC12 , and KC12 in XC12 phases. The doping leads to a weakening in mechanical properties and an increase in the elastic anisotropy. C6 has the lowest elastic anisotropy, and the anisotropy increases with the atomic number of doping atoms for both XC6 and XC12 . Furthermore, the acoustic velocities, Debye temperatures, and the electronic properties are also studied. Keywords: first-principles calculations; carbides; mechanical properties; elastic anisotropy
1. Introduction Elemental carbon exhibits a rich diversity of structures and properties, due to its flexible bond hybridization. A large number of stable or metastable phases of the pure carbon, including the most commonly known, graphite and diamond, and other various carbon allotropes [1–4] (such as lonsdaleite, fullerene, and graphene, etc.), and diversified carbides [5–11], have been studied in experiments and theoretical calculations. Graphite, which is the most stable phase at low pressure, has a sp2 -hybridized framework and is ultrasoft semimetallic, whereas diamond, stable at high pressure, is superhard, insulating with a sp3 network. Recently, a novel one-dimensional metastable allotrope of carbon with a finite length was first synthesized by Pan et al. [1], called Carbyne. It has a sp-hybridized network and shows a strong purple-blue fluorescence. The successful synthesis of Carbyne is a great promotion for the further analysis on properties and applications. The 2D material MXenes as a promising electrode material, which is early transition metal carbides and carbon nitrides, is reported [11], owing to its metallic conductivity and hydrophilic nature. These properties of different carbides are appealing. To find superhard superconductors, researches designed some carbide superconductors, such as boron carbides and XC6 structure with cubic symmetry. The diamond-like Bx Cy system, which is superhard and superconductive, has also attracted much interest [5–10]. The best simulated structure of the synthesized d-BC3 (Pmma-b phase) has a Vickers hardness of 64.8 GPa, showing a superhard nature, and its Tc reaches 4.9–8.8 K [5]. The P-4m2 polymorph of d-BC7 with a low energy also has a high Vickers hardness of 75.2 GPa [8]. Furthermore, Wang et al. [9] explored more potential superhard structures of boron carbide, uncovering the stability is mainly contributed by the elemental boron at low pressure, and by the carbon at high pressure. The novel metastable carbon structure C6 bcc is predicted with a cubic symmetry [12]. It is an indirect band gap semiconductor with 2.5 eV, calculated by the local density approximation. Recently, doped with simple metals, Lu et al. [13]
Materials 2016, 9, 726; doi:10.3390/ma9090726
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simple metals, Lu et al. [13] studied a series of sodalite-based carbon structures, similar to the boronstudied a seriesAlthough of sodalite-based carbon structures, similar the boron-doped diamond. Although doped diamond. they found these structures are alltometastable, some of these structures they found these structures are all metastable, some of these structures show a superconductivity, show a superconductivity, e.g., the critical temperature of NaC6 is 116 K. In this paper, we mainly e.g., the the critical temperature of NaC K. InXC this6 phases paper, we mainly properties 6 is 116 study mechanical properties of these eleven (HC 6, LiC6study , NC6,the OCmechanical 6, FC6, NaC 6, AlC6, of these eleven XC phases (HC , LiC , NC , OC , FC , NaC , AlC , SiC , PC , SC , and ClC 6 6) which is of6dynamical 6 6 6 and, 6 for comparison, 6 6 6 calculated. 6 SiC6, PC6, SC6, and ClC stability C66 is also In 6 ) which is of dynamical stability and, for comparison, C is also calculated. In addition, the XC addition, the XC12 structures are systematically explored, in6which the X atom is from H to Ca, except 12 structures are The systematically explored, in which the constant, X atom ismodulus, from H tothe Ca, except He, and Ar. He, Ne, and Ar. doping-induced changes in elastic anisotropy ofNe, elasticity The doping-induced changes in elastic constant, modulus, the anisotropy of elasticity and acoustic and acoustic velocity, Debye temperature, and the electronic structures are also studied. velocity, Debye temperature, and the electronic structures are also studied. 2. Results and Discussion 2. Results and Discussion As shown in Figure 1a, the structure of XC6 is obtained by doping the X atom into the C6 bcc As shown in Figure 1a, the structure of XC6 is obtained by doping the X atom into the C6 bcc structure at (0, 0, 0). It is of Im-3m symmetry (No. 229), consisting of two formula units (f.u.) per unit structure at (0, 0, 0). It is of Im-3m symmetry (No. 229), consisting of two formula units (f.u.) per unit cell. Each C atom has four nearest neighbors with the bond angle of 90° or 120°. The XC6 structure cell. Each C atom has four nearest neighbors with the bond angle of 90◦ or 120◦ . The XC structure has four C4 rings and eight C6 rings. In Table 1, the calculated lattice parameter a of C6 has6 a good has four C4 rings and eight C6 rings. In Table 1, the calculated lattice parameter a of C6 has a good agreement with the available result [12], and is smaller than that of the XC6 structures. By removing agreement with the available result [12], and is smaller than that of the XC6 structures. By removing the corner atoms and only leaving the center X atom, the XC12 structure is obtained (Figure 1b). All the corner atoms and only leaving the center X atom, the XC12 structure is obtained (Figure 1b). All of of the XC12 phases are smaller than the corresponding XC6 phases, but larger than the C6 phase in the the XC12 phases are smaller than the corresponding XC6 phases, but larger than the C6 phase in the lattice parameter. lattice parameter.
Figure 1. Unit cell of XC6 (a) and XC12 (b). The black and blue spheres represent C and X atoms, respectively. Figure 1. Unit cell of XC6 (a) and XC12 (b). The black and blue spheres represent C and X atoms, respectively. Table 1. Calculated lattice parameter a, elastic constants Cij (GPa), mechanical stability, bulk modulus B (GPa), shear modulus (GPa), Young’s modulus E (GPa), Poisson’s ratio ν, and B/G ratio. Table 1. Calculated lattice G parameter a, elastic constants Cij (GPa), mechanical stability, bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Poisson’s ratio ν, and B/G ratio. Mechanical Materials a C11 C12 C44 Mechanical B G E ν B/G Stability
Materials
a
C11
Diamond 3.566 a 1053 a a C6 4.375 803 a Diamond 3.566 1053 b 4.34 C6 4.375 803 HC6 4.390 b 607 4.34 LiC6 4.491 634 HC 6 4.390 607 NC6 4.446 414 OC66 4.434 196 LiC 4.491 634 FC6 6 4.427 269 NC 4.446 414 NaC6 4.566 659 OC 6 4.434 196 AlC6 4.618 497 FC 6 4.427 269 SiC6 4.614 527 PC6 6 4.605 542 NaC 4.566 659 SC6 4.608 683 AlC6 4.618 497 ClC6 4.613 92
SiC6 PC6
4.614 4.605
527 542
C12
C44
120 a 95 a 120
563 a a 307 563
Stability
95
307
stable
215 118 215 295 407 118 370 295 91 407 162 370 165 179 91 115 162 374
165 179
344 −78 344 162 216 −78 335 162 −548 216 −59 335 −66 −132 −548 90 −59 104
−66 −132
stable stable unstable stable stable unstable unstable unstable stable unstable unstable unstable unstable unstable unstable unstable stable unstable unstable
unstable unstable
E
ν
B/G
325
735
0.13
1.018
352 346 335
275 108
652 293 0.186 0.354 1.258 3.102
335
108
293
B
431 a 331 431 a b 352 331 346 b
305
G
522 a 325a 522 275
146
1116 a 0.07 a a a 735 0.070.13 1116 652
378
0.186
0.354
0.294
1.018 1.258
3.102
2.089
Materials 2016, 9, 726 Materials 2016, 9, 726
SC6 ClC6 HC12 LiC12 BeC 12 Materials BC12 HC CC 12 12 LiC12 NC12 BeC12 OC 12 BC 12 FC CC 12 12 NC 12 12 NaC OC12 MgC12 FC12 AlC 12 NaC 12 MgC SiC 12 12 AlC 12 12 PC SiC12 12 PCSC 12 ClC SC12 12 ClC 12 12 KC KC12 CaC12 CaC12
4.608 4.613 4.383 4.444 4.451 a 4.439 4.383 4.376 4.444 4.415 4.451 4.404 4.439 4.401 4.376 4.415 4.476 4.404 4.508 4.401 4.513 4.476 4.508 4.511 4.513 4.504 4.511 4.502 4.504 4.503 4.502 4.503 4.512 4.512 4.543 4.543
3 of 12
683 92 103 695 743 C11 684 103689 695 275 743 −661 684 689−33 275741 −661 667 −33 741645 667559 645645 559 645397 397349 349779 779 734 734
115 374 461 108 98 C12 136 461 146 108 361 98 830 136 529 146 361 77 830 108 529 123 77 108 170 123 141 170 273 141 297 273 297 53 53 58
90 stable 104 unstable 336 Tableunstable 1. Cont. 32 stable Mechanical 289 stable C44 Stability 233 stable 336 unstable 214 stable 32 stable 278 unstable 289 stable 526 unstable 233 stable 476 unstable 214 stable 278 unstable −9 unstable 526 unstable 31 stable 476 unstable 569 stable − unstable 31 stable −25 unstable 56 stable 144 stable −25 unstable 251 stable 144 stable 295 stable 251 stable 295 stable 18 stable 18 stable −2166 unstable
305
146
378
0.294
3 of 12 2.089
304 313 B 319 327
93 302 G 248 235
253 686 E 591 569
0.361 0.135 ν 0.191 0.21
3.269 1.036 B/G 1.286 1.391
93 302 248 235
253 686 591 569
294 297
89 110
240 294
0.363 0.335
294 297309
89 110 181
240 294 454
0.363 3.303 0.335 2.700 0.255 1.707
309314 314314 314295 295
144 181 123 144 123 93 93
375 454 326 375 326 252
0.301 0.255 2.181 1.707 0.326 0.301 2.553 2.181 0.326 3.172 2.553 0.357
304 313 319 327
252
0.361 0.135 0.191 0.21
0.357
3.269 1.036 1.286 1.391
3.303 2.700
3.172
−2166 unstable a Ref [14]; b Ref [12]. a b
58
Ref [14]; Ref [12].
The formation enthalpies of XC6 in [13] and XC12 structures are calculated reference to diamond The formation enthalpies of XC6 in [13] and XC12 structures are calculated reference to and the most stable X phase at ambient pressure. The equations are given by diamond and the most stable X phase at ambient pressure. The equations are given by H XC6 =((HHXC6 −HHX −66H H C ))/7, 7 ,and and H XC12 ( H XC12 H X 12 H C ) 13 , and the calculated ∆HXC ∆HXC12 = ( HXC12 − HX − 12HC )/13, and the calculated results X XC6 C 6 results are shown in Figure The positive these phases are metastable. The two are shown in Figure 2. The 2. positive values values indicateindicate these phases are metastable. The two curves curves the formation enthalpy a similar where the F-doped carbides the lowest of the of formation enthalpy followfollow a similar trend,trend, where the F-doped carbides havehave the lowest ∆H, ΔH, and the PC 6 and CC 12 have the largest ΔH in XC 6 and XC 12 , respectively. Compared to other and the PC6 and CC12 have the largest ∆H in XC6 and XC12 , respectively. Compared to other doped doped elements of the second theperiods third periods in 6the XCXC 6 and XC12, fluorine (F) possesses the elements of the second and theand third in the XC and (F) possesses the largest 12 , fluorine largest electronegativity difference to C, leading to a stronger interaction between C electronegativity difference relativerelative to C, leading to a stronger interaction between F and F C and atoms; atoms; thus, FC 6 and FC 12 phases are more stable. thus, FC6 and FC12 phases are more stable. 1.8
Formation enthalpy (eV/atom)
SiC6 PC6
NC6
1.5 1.2
OC6
HC6
0.9
0.3 1.2
FC6 CC12
1.0
SiC12
BC12
BeC12
0.8
NC12 MgC12 OC12
0.6
HC12
LiC12
2
4
AlC12
PC12 SC12 ClC 12
CaC12 KC12
NaC12
FC12
0.4 0
ClC6
NaC6
LiC6
0.6
SC6
AlC6
6
8
10
12
14
16
18
20
Atomic number
Figure 2. 2. Formation Formation enthalpy enthalpy of of XC XC66and andXC XC1212. . Figure
The 1. The The calculated calculated elastic elastic constants constants and and moduli moduli are are listed listed in in Table Table 1. The generalized generalized Born’s Born’s mechanicalstability stabilitycriteria criteriaof of cubic cubic phase phase are are given given by by [15]: [15]: C > 0, C > 0, C > C | | , and C C , mechanical C11 0, C 0, 44 11 11 44 11 1212 and C11 + 2C12 ) > 0. In Table 1, the C6 and HC6 , NC6 , and SC6 have the mechanical stability, and they ((C 2C12 ) 0. In Table 1, the C6 and HC6, NC6, and SC6 have the mechanical stability, and they are11also dynamically stable [13]. The XC12 has ten mechanically stable phases, but only six of these are also have dynamically stable [13]. The(BC XC12 ,has ten mechanically stable phases, but only six of these phases the dynamical stability 12 CC12 , PC12 , SC12 , ClC12 , and KC12 ) due to the absence of phases have the dynamical stability (BC12, CC12, PC12, SC12, ClC12, and KC12) due to the absence of the
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imaginary frequency in the whole Brillouin zone (see Figures 3 and 4). The S is the only element that imaginary frequency the whole Brillouin (see(see Figures 3 and 4). The S is Sthe onlyonly element that is capable to make notinonly XCwhole 6, but also XCzone 12, zone stable. the imaginary frequency in the Brillouin Figures 3 and 4). The is the element is capable to make not not onlyonly XC6XC , but alsoalso XC12XC , stable. that is capable to make 6 , but 12 , stable. (a)
40
40
(a)
20
10 10
BC12
X
R
M
BC12R
R
M
R
0
X
30
20 20
10 10
CC12
0
X
R
M
CC12
0
X
R
(d)
(e)
(d)
(e)
30
30
M
20
10 10
SC12
0
X
R
M
SC12 R
X
R
M
R
0
10 10
R
PC12
M
0
R
PC12 X
X
R
M
R R
(f) (f)
30
20 20
10 10
ClC12
0 0
20 20
0
30
20
30
R
Frequency (THz) Frequency (THz)
30
Frequency (THz) Frequency (THz)
20
0
Frequency (THz) Frequency (THz)
(c)
30
30
30
Frequency (THz)(THz) Frequency
30
(c)
(b)
(b)
40
Frequency (THz) Frequency (THz)
Frequency Frequency (THz)(THz)
40
X
R
X
R
ClC12
M M
30
20 20
10 10
KC12
0
R R
X
0
X
R R
M M
KC12
R
R
Figure3.3.Phonon Phononspectra spectraofofdynamically dynamically stable phases 12; (b) CC12; (c) PC12; (d) SC12; (e) ClC12; Figure stable phases (a) (a) BCBC 12 ; (b) CC12 ; (c) PC12 ; (d) SC12 ; (e) ClC12 ; Figure 3. Phonon spectra of dynamically stable phases (a) BC 12; (b) CC12; (c) PC12; (d) SC12; (e) ClC12; and(f) (f)KC KC12.. and and (f) KC1212. (b)
(a) (a)
40 (b) 40
Frequency (THz) Frequency (THz)
Frequency (THz) Frequency (THz)
30 30 20 20 10 10
LiC LiC1212
0 0
X X
R R
M M
30 30 20 20 10 10
0
-10 -10 X
R R
X
(c) (c)
R R
M M
R
R
(d) (d) 30 30
Frequency Frequency(THz) (THz)
30 30
Frequency(THz) (THz) Frequency
BeC BeC1212
0
20 20
10 10
20 20
10 10
0
0 0
AlC12 AlC 12
0
MgC12 MgC 12 X X
R R
M M
RR
XX
RR
MM
RR
Figure 4. Phonon spectra of dynamically unstable phases (a) LiC12 ; (b) BeC12 ; (c) MgC12 ; and (d) AlC12 . Figure dynamically unstable unstablephases phases(a) (a)LiC LiC1212; ;(b) (b)BeC BeC1212; ;(c) (c)MgC MgC ; and(d) (d)AlC AlC Figure 4. 4. Phonon spectra of dynamically 1212 ; and 12.12.
By Voigt-Reuss-Hill approximations [16–18], the bulk modulus B and shear modulus G can be By approximations [16–18], [16–18], the thebulk bulkmodulus modulusBBand andshear shearmodulus modulusGGcan canbebe By Voigt-Reuss-Hill Voigt-Reuss-Hill approximations obtained, and the Young’s modulus E and Poisson’s ratio ν are defined as [19,20] E = 9BG/(3B + G ) obtained, and the Young’s modulus E and Poisson’s ratio ν are defined as [19,20] obtained, modulus E and Poisson’s ratio ν are defined as [19,20] and ν = (3B − 2G )/[2(3B + G )]. HC6 has the largest bulk modulus of 346 GPa, showing the best EE 99BG / (3B G) and (3B B 22G G))//[2(3 [2(3BBGG)]. )]. HC HC6 6has hasthe thelargest largestbulk bulkmodulus modulusofof346 346 BG ability to resist the compression. The shear modulus is often used to qualitatively predict the hardness, GPa, showing the best toasresist resist the between compression. The shear shear to modulus often used GPa,Young’s showing to the compression. modulus often used toto and modulus E isability defined the ratio stressThe and strain measureisis the stiffness of qualitatively hardness, Young’s modulus EEisisdefined asasthe ratio stress and predict the 1, hardness, and Young’s modulus defined the ratiobetween between stress and aqualitatively solid material. In Table C6 is theand largest in shear modulus and Young’s modulus, which means to measure measure the stiffness of aa solid strain to solid material. material. In InTable Table1,1,CC6 6isisthe thelargest largestininshear shearmodulus modulusand and strain Young’s modulus, modulus, which means Young’s means that that doping doping leads leads to to aa weakening weakeningininmechanical mechanicalproperties. properties.The The
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that doping leads to a weakening in mechanical properties. The Poisson’s ratio exhibits the plasticity; usually, the larger the value, the better the plasticity. According to Pugh [21], C6 , HC6 , BC12 , CC12 , and PC12 are brittle materials (B/G < 1.75), while NC6 , SC6 , SC12 , ClC12 , and KC12 are ductile materials (B/G > 1.75). This conforms the calculated results of Poisson’s ratio. The elastic anisotropy is important for the analysis on the mechanical property and, thus, the universal elastic anisotropy index (AU ), Zener anisotropy index (A), and the percentage anisotropy in compressibility and shear are calculated. For the cubic phase, the universal elastic anisotropy index [22] is defined as: AU = 5GV /GR + BV /BR − 6, the nonzero value suggests an anisotropy characteristic. Furthermore, it is known that C44 represents the resistance to deformation with respect to a shear stress applied across the (100) plane in the [010] direction, and (C11 − C12 )/2 represents the resistance to shear deformation by a shear stress applied across the (110) plane in the [110] direction. For an isotropic crystal, the two shear resistances turn to identical. Therefore, Zener [23] introduced A = 2C44 /(C11 − C12 ) to quantify the extension of anisotropy. The value of 1.0 represents the isotropy, and any deviation from 1.0 indicates the degree of the shear anisotropy. The percentage anisotropy in compressibility and shear are given by: A B = ( BV − BR )/( BV + BR ) and AG = ( GV − GR )/( GV + GR ) [24]. The AB is always 0.0 for a cubic phase. As shown in Table 2, C6 has the lowest anisotropy. The universal elastic anisotropy index and the percentage anisotropy in shear is increasing with the atomic number of doped element for both XC6 and XC12 , and the anisotropy which obtains from the shear anisotropic factor is also increasing, except SC6 and KC12 . Furthermore, owing to the percentage anisotropy in shear of C6 , BC12 , and CC12 being slight, they are almost isotropic. Table 2. Universal elastic anisotropy index (AU ), Zener anisotropy index (A), and percentage anisotropy in shear (AG ). Parameter C6 AU A AG (%)
0.024 0.8672 0.243
HC6
NC6
SC6
BC12
CC12
PC12
SC12
ClC12
KC12
0.398 1.755 3.752
1.30 2.723 11.567
1.77 0.317 15.016
0.032 0.851 0.315
0.068 0.788 0.678
0.3814 0.572 3.714
2.752 4.048 21.596
11.084 11.346 53.098
21.252 0.0496 68.612
The elastic anisotropies are calculated with the elastics anisotropy measures (ElAM) code [25,26] which makes the representations of non-isotropic materials easy and visual. For the cubic phase, the representation in xy, xz, and yz planes are identical, as a result, only the xy plane is presented. The 2D figures of the differences in each direction of Poisson’s ratio are shown in Figure 5. The maximum value curves and minimum positive value curves of C6 and XC6 stable phases are illustrated in Figure 5a,b, and those of XC12 stable phases are shown in Figure 5c,d. Particularly, the SC12 and ClC12 have the negative minimum Poisson’s ratio. It is seen that all of the structures are anisotropic and C6 has the lowest anisotropy, suggesting the doping increase the elastic anisotropy. The largest value of maximum curve is in the same direction of the lowest value of minimum positive value curve for each structure. Furthermore, for XC12 phases, the anisotropy of Poisson’s ratio is increasing with the atomic number. The negative minimum Poisson’s ratio of SC12 and ClC12 indicate these two phases have auxeticity [27], and ClC12 is more prominent than SC12 . The directional dependence of the Young’s modulus [28] are demonstrated in Figures 6 and 7. The distance from the origin of system of coordinate to the surface equals the Young’s modulus in this direction, and thus any departure from the sphere indicates the anisotropy. As shown, all of the phases are anisotropic, and the anisotropy of Young’s modulus is increasing with the doping atomic number. For the S-doped phases, which have stable XC6 and XC12 structures, the maximum (minimum) values of SC6 and SC12 are 650 (291) and 371 (175) GPa, respectively. The Emax /Emin ratio of SC6 (2.23) is slightly larger than that of SC12 (2.12), indicating the SC6 is more anisotropic.
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6 of 12 6 of 12 6 of 12 (a) max max
0.6 (a) 0.6
C6 C6
HC6 HC6
NC6 NC6
SC6 SC6
(b) minp (b) minp
0.4 0.4
C6 C6
HC6 HC6
NC6 NC6
SC6 SC6
0.4 0.4 0.2 0.2
0.2 0.2 0.0 0.0
0.0 0.0
-0.2 -0.2
-0.2 -0.2
-0.4 -0.4 -0.6 -0.6 -0.6 -0.6
-0.4 -0.4 -0.4 -0.4
(c) max (c) max
-0.2 -0.2
BC12 BC12 ClC12 ClC12
0.0 0.0
CC12 CC12 PC12 PC12
0.2 0.2
SC12 SC12 KC12 KC12
0.4 0.4
0.9 0.9
0.6 0.6
-0.4 -0.4
-0.2 -0.2
0.0 0.0
(d) minp BC12 CC12 (d) minp BC12 CC12 PC12 KC12 minn PC12 KC12 minn
0.2 0.2
0.4 0.4
SC12 SC12 SC12 SC12
ClC12 ClC12 ClC12 ClC12
0.2 0.2
0.4 0.4
0.4 0.4
0.6 0.6
0.2 0.2
0.3 0.3
0.0 0.0
0.0 0.0 -0.3 -0.3
-0.2 -0.2
-0.6 -0.6
-0.4 -0.4
-0.9 -0.9 -0.9 -0.9
-0.6 -0.6
-0.3 -0.3
0.0 0.0
0.3 0.3
0.6 0.6
0.9 0.9
-0.4 -0.4
-0.2 -0.2
0.0 0.0
Figure representations of Poisson’s ratio.ratio. (a) Maximum of C6 and stable phases; (b)phases; minimum Figure5.5.2D 2D representations of Poisson’s (a) Maximum of XC C6 6and XC 6 stable (b) Figure 5. 2D representations of Poisson’s ratio. (a) Maximum of C6 and XC6 stable phases; (b) positive of C phases; (c)phases; maximum of XC12 stable and (d) minimum positive minimum positive of 6Cstable 6 and XC 6 stable (c) maximum of XCphases; 12 stable phases; and (d) minimum 6 and XC minimum positive of C6 and XC6 stable phases; (c) maximum of XC12 stable phases; and (d) minimum and minimum negative of XC12 stable only SC12 only and ClC have the positive and minimum negative of XCphases; 12 stableparticularly, phases; particularly, SC1212and ClC 12 negative have the positive and minimum negative of XC12 stable phases; particularly, only SC12 and ClC12 have the minimum Poisson’s ratio, the solid and dash lines represent the minimum positive and minimal negative minimum Poisson’s ratio, the solid and dash lines represent the minimum positive and negative minimum Poisson’s ratio, the solid and dash lines represent the minimum positive and negative, respectively. minimal negative, respectively. minimal negative, respectively.
Figure 6.Directional Directional dependenceofofthe the Young’smodulus modulus ofCC6(a); (a); HC6 (b); NC6 (c); (c); and and SC SC6 (d). (d). Figure Figure 6. 6. Directional dependence dependence of the Young’s Young’s modulusof of C66 (a); HC HC66 (b); NC66 (c); and SC66 (d).
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Figure 7. (d); Figure 7. Directional dependence of the the Young’s Young’s modulus modulus of ofBC BC1212 (a); (a); CC CC12 12 (b); (b); PC PC12 12 (c); SC12 12 (d); ClC (e); and KC (f). 12(f). 12 (e); and KC12 ClC12
The acousticvelocity velocity is a fundamental parameter tothe measure chemical bonding The acoustic is a fundamental parameter to measure chemicalthe bonding characteristics, characteristics, and it is determined by the symmetry of the crystal and propagation direction. and it is determined by the symmetry of the crystal and propagation direction. Brugger [29] provided Brugger [29] providedtoancalculate efficientthe procedure to calculate the transverse phase velocities of pure transverse and an efficient procedure phase velocities of pure and longitudinal modes from longitudinal modes from the single crystal elastic constants. The cubic structure only has three the single crystal elastic constants. The cubic structure only has three directions [001], [110], and [111] directions [001], [110], and for the pure transverse anddirections longitudinal other directions for the pure transverse and[111] longitudinal modes and other are modes for the and qusi-transverse and are for the qusi-transverse and qusi-longitudinal acoustic of a cubic are phase qusi-longitudinal waves. The acoustic velocities ofwaves. a cubicThe phase in the velocities principal directions [30]:in the principal directions are [30]: √ √ for [100], vl = C11 /ρ, [010]vt1 = [001]vt2 = C44 /ρ, p , √ for [110], [100], vvl=p(CC11 [010] vt1 )/2ρ, [001] vt 2 ]v =C44 +, C12 + 2C [110 (C11 − C12 )/2ρ, [001]vt2 = C44 /ρ, for 44 t1 l p 11 p for [111], + 2C + 4C ) )/3ρ, 112]vvt1t1 = C44 )v/3ρ. [110], vvl l= ((C C11 2 , [[110] vt2(C=11 (CC1211) −2C 12 , + [001] C44 , 11 C1212 2C 4444 t2 for [111], (C11 of 2Cthe 4C44 ) 3v,l is[112] vt1 vt 2 (acoustic C11 C12velocity, C44 ) and 3 . vt1 and vt2 refer where ρ isvthe structure, the longitudinal l density 12 the first transverse mode and the second transverse mode, respectively. It should be noted that there where ρ is thefor density of theofstructure, vl is the longitudinal acoustic velocity,isand vt1 and vt2 refer the is a misprint equation [110]vt1 in [30]. Here, the correct expression given. Based on the first transverse mode and the second transverse mode, respectively. It should be noted that there is a elastic constants, the anisotropic properties of acoustic velocities indicate the elastic anisotropy in these vt1 in [30]. Here, the correlates correct expression isphysical given. Based on the misprintAs foraequation of [110] crystals. fundamental physical parameter which with many properties of elastic solids, h i1/3 N ρ h 3n A constants, the anisotropic properties acoustic velocities indicate theΘelastic anisotropy in these the Debye temperature can be obtained of from the average acoustic velocity: vm , D = k B 4π M crystals. As a fundamental physical parameter which correlates with many physical properties where h and kB are the Planck and Boltzmann constants, respectively; NA is Avogadro’s number; n of is solids, Debye temperature can unit; be M obtained from the average acoustic the total the number of atoms in the formula is the mean molecular weight, and ρ is thevelocity: density. p −1/3 1/3 The average is vm = (2/v3tm + 1/v3lm )/3 , where vlm = ( B + 4G/3)/ρ is the h 3acoustic n N A velocity D longitudinal where h and kB are √ the Planck and Boltzmann constants, respectively; vm , velocity, average and vtm = G/ρ is the average transverse acoustic velocity. k B 4 M acoustic All of the calculated acoustic velocities and Debye temperatures of diamond and stable XC6 and NA is Avogadro’s number; n is the total number of atoms in the formula unit; M is the mean molecular XC12 phases are shown in Table 3. Diamond is larger than C6 and doped structures in anisotropic and 1 3 3 3 1/ vlm ) /are vm acoustic 3decreasing isvelocity. weight, and the density. The average acoustic velocity , where average acoustic The densities are increasing and the is average (2 / vtm velocities with the atomic number, except NC6 , which has a much smaller shear modulus. Compared to C6 , the vlm results ( B 4G / 3) / is the average longitudinal acoustic velocity, and vtm ForGthe / element is the doping in a decrease in the average acoustic velocity and Debye temperature. average acoustic velocity. S, whichtransverse makes both XC6 and XC12 phases stable, the average acoustic velocity of SC6 decreases by All of the calculated acoustic velocities and Debye temperatures of diamond and stable XC6 and XC12 phases are shown in Table 3. Diamond is larger than C6 and doped structures in anisotropic and
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average acoustic velocity. The densities are increasing and the average acoustic velocities are decreasing with the atomic number, except NC6, which has a much smaller shear modulus. Materials 2016, 9, 726 8 of 12 Compared to C6, the doping results in a decrease in the average acoustic velocity and Debye temperature. For the element S, which makes both XC6 and XC12 phases stable, the average acoustic 38.65% than , and that by of SC 35.96%. Furthermore, be found that the Debye temperature velocity of SCC6 6decreases 38.65% C6, and that of SCit12 can by 35.96%. Furthermore, it can be found 12 by than is decreasing the atomic except . The Θ characterizes the strength of the covalent that the Debyewith temperature is number, decreasing withSC the atomic number, except SC 6 . The Θ D characterizes 6 D bond in solids, so the strength of the covalent bond is lower phase bond whichishas the for larger the strength of the covalent bond in solids, so the strength offor thethe covalent lower the atomic phase number ofthe doping atom. which has larger atomic number of doping atom. Table anisotropic acoustic acoustic velocities (m/s) (m/s)and andaverage averageacoustic acousticvelocity velocity(m/s). (m/s). Table3.3.Density Density(g/cm (g/cm33),), anisotropic Parameters Diamond C6 Parameters Diamond C6 ρ 3.517 2.857 ρ vl 3.517 2.857 [100] 17,303 16,765 17,303 16,765 [100] [010]vvt1l 12,652 10,366 [010]v 12,652 10,366 t2 t1 12,652 10,366 [001]v [001]v 12,652 10,366 16,267 [110] vl t2 18,079
HC6 2.869 2.869 14,546 14,546 10,950 10,950 10,950 10,950 16,222
NC6 3.252 3.252 11,283 11,283 7058 7058 7058 7058 12,603
SC6 3.535 3.535 13,900 13,900 5046 5046 5046 5046 11,762
BC12 2.941 2.941 15,251 15,251 8901 8901 8901 8901 14,786
CC12 2.992 2.992 15,175 15,175 8457 8457 8457 8457 14,528
PC12 3.182 3.182 14,237 14,237 6727 6727 6727 6727 12,991
SC12
SC12 3.206 3.206 11,128 11,128 8848 8848 8848 8848 13,520
ClC12
KC12
[110] [110] vvt1l [110]v [001]vt2 t1 [001]vt2 [111] vl [111] vl vt1,2 [112] ]vt1,2 [112 vl vl vt vt vmvm D ΘΘ D
16,222 8265 12,603 4277 8265 4277 10,950 7058 10,950 7058 16,744 13,013 16,744 13,013 9247 5367 9247 5367 15,761 12,136 12,136 15,761 9791 5763 9791 5763 10,792 6483 6483 10,792 1766 1047 1766 1047
11,762 8963 8963 5046 5046 10,956 10,956 7877 7877 11,889 11,889 6427 6427 7173 7173 1118 1118
14,786 9652 9652 8901 8901 14,628 14,628 9409 9409 14,851 14,851 9183 9183 10,128 10,128 1598 1598
14,528 9526 9526 8457 8457 14,306 14,306 9184 9184 14,629 14,629 8863 8863 9795 9795 1551 1551
12,991 8899 8899 6727 6727 12,548 12,548 8239 8239 13,151 13,151 7542 7542 8378 8378 1303 1303
13,520 4398 4398 8848 8848 14,228 14,228 6244 6244 12,563 12,563 6702 6702 7487 7487 1165 1165
13,758 2822 2822 9505 9505 14,722 14,722 5952 5952 12,100 12,100 6138 6138 6880 6880 1069 1069
11,446 10,467 10,467 2331 2331 9813 9813 8652 8652 11,246 11,246 5298 5298 5963 5963 926 926
18,079 11,517
11,517 12,652 12,652 18,330 18,330 11,907 11,907 17,901 17,901 12,183 12,183 13,282 13,282 2219 2219
16,267 11,131 11,131 10,366 10,366 16,098 16,098 10,882 10,882 16,356 16,356 10,666 10,666 11,692 11,692 1823 1823
HC6
NC6
SC6
BC12
CC12
PC12
ClC12 3.265 3.265 10,339 10,339 9505 9505 9505 9505 13,758
KC12 3.313 3.313 2331 2331 2331 2331 2331 2331 11,446
Figure 8 shows the electronic band structure and density of state (DOS) of XC12 stable phases. Figure 8 shows the electronic band structure and density of state (DOS) of XC12 stable phases. The dash line represents the Fermi level (EF). The electronic properties of XC6 have been studied in The dash line represents the Fermi level (EF ). The electronic properties of XC6 have been studied [13]. For XC12, all of the band structures cross the Fermi level in the Brillouin zone, showing the in [13]. For XC , all of the band structures cross the Fermi level in the Brillouin zone, showing the metallic nature.12 The conduction band and valence band are mainly characterized by the contributions metallic nature. The conduction band and valence band are mainly characterized by the contributions of C-p states, whereas the DOS near the Fermi level originated from the p orbital electrons of the of C-p states, whereas the DOS near the Fermi level originated from the p orbital electrons of the doped doped element, except the ClC12 and KC12. element, except the ClC12 and KC12 .
Figure 8. Cont.
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Figure 8. Cont.
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Figure 8. 8. Electronic Electronic band of of state of BC CC12 (b); PC PC(c); SC SC(d); Figure band structure structureand anddensity density state of 12 BC(a); 12 (a); CC 12 (b); 12 12 (c); 12 12 ClC (e); and KC (f). 12 12 (d); ClC12 (e); and KC12 (f).
ComputationalMethods Methods 3.3.Computational The calculations calculations are are performed The structural structural The performed with with the the first-principles first-principles calculations. calculations. The optimizations are using the density functional theory (DFT) [31,32] with the generalized gradient optimizations are using the density functional theory (DFT) [31,32] with the generalized gradient approximation (GGA), which is parameterized by Perdew, Burke, and Ernzerrof (PBE) [33]. approximation (GGA), which is parameterized by Perdew, Burke, and Ernzerrof (PBE) [33]. The The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme was used thegeometry geometry Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme [34][34] was used ininthe optimization, and the total energy convergence tests are within 1 meV/atom. When the total optimization, and the total energy convergence tests are within 1 meV/atom. When the totalenergy energyis − 6 −6 eV/atom,the themaximum maximumionic ionicHellmann-Feynman Hellmann-Feynmanforce forceisis0.01 0.01eV/Å, eV/Å,the themaximum maximumstress stress is5.0 5.0××10 10 eV/atom, 4 Å, the structural relaxation will stop. 0.02GPa GPaand and the the maximum maximum ionic ionic displacement displacement is is 5.0 5.0 ××10 10−4−Å, isis0.02 the structural relaxation will stop. −1 in the Brillouin zone. Theenergy energycutoff cutoffisis400 400eV, eV, and and the the K-points K-points separation separation is is 0.02 0.02 Å Å−1 The in the Brillouin zone. Conclusions 4.4.Conclusions Byusing using the the first-principles first-principles calculations, properties of XC the 6 and By calculations,the theanalyses analyseson onthe themechanical mechanical properties of XC 6 and further exploration of XC structures are given. The formation enthalpies of dynamically stable XC the further exploration of 12 XC12 structures are given. The formation enthalpies of dynamically stable6 XC6 phases and all of the XC12 structures, and the elastic constants, are calculated. There are ten
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phases and all of the XC12 structures, and the elastic constants, are calculated. There are ten structures which have the mechanical and dynamical stability (C6 , HC6 , NC6 , SC6 , BC12 , CC12 , PC12 , SC12 , ClC12 , and KC12 ). The elastic modulus and anisotropy of the ten structures are studied and, in these structures, C6 has the lowest elastic anisotropy and the anisotropy increases with the atomic number. The doping leads to the weakening in mechanical properties and the increase in the elastic anisotropy. In addition, Debye temperatures and the anisotropy of acoustic velocities are also studied. The electronic properties studies show the metallic characteristic for XC6 and XC12 phases. Acknowledgments: This work was financially supported by the Natural Science Foundation of China (No. 11204007), Natural Science Basic Research plan in Shaanxi Province of China (grant No.: 2016JM1026, 20161016), and Education Committee Natural Science Foundation in Shaanxi Province of China (grant No.: 16JK1049). Author Contributions: Qun Wei and Meiguang Zhang designed the project; Quan Zhang and Qun Wei performed the calculations, Qun Wei and Quan Zhang prepared the manuscript, Meiguang Zhang revised the paper, all authors discussed the results and commented on the manuscript. Conflicts of Interest: The authors declare no conflict of interest.
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