1College of Information and Control Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China 2Department of Mechanical and Electrical Engineering, Hetao College, Bayannur Inner Mongolia 015000, China 3School of Microelectronics, Xidian University, Xi’an 710071, China 4Functional Materials Laboratory, School of Materials Science and Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China
In this work, a novel carbon allotrope tP40 carbon with space group P4/mmm is proposed. The structural stability, mechanical properties, elastic anisotropy, and electronic properties of tP40 carbon are investigated systematically by using density functional theory (DFT). The calculated elastic constants and phonon dispersion spectra indicate that the tP40 phase is a metastable carbon phase with mechanical stability and dynamic stability. The B/G ratio indicates that tP40 carbon is brittle from 0 GPa to 60 GPa, while tP40 carbon is ductile from 70 GPa to 100 GPa. Additionally, the anisotropic factors and the directional dependence of the Poisson’s ratio, shear modulus, and Young’s modulus of tP40 carbon at different pressures are estimated and plotted, suggesting that the tP40 carbon is elastically anisotropic. The calculated hardness values of tP40 carbon are 44.0 GPa and 40.2 GPa obtained by using Lyakhov–Oganov’s model and Chen’s model, respectively, which means that the tP40 carbon can be considered as a superhard material. The electronic band gap within Heyd–Scuseria–Ernzerhof hybrid functional (HSE06) is 4.130 eV, and it is found that the tP40 carbon is an indirect and wider band gap semiconductor material.
* Project supported by the National Natural Science Foundationof China (Grant Nos. 61804120 and 61901162), the China Postdoctoral Science Foundation (Grant Nos. 2019TQ0243 and 2019M663646), the Young Talent Fund of University Association for Science and Technology in Shaanxi Province, China (Grant No. 20190110), the National Key Research and Development Program of China (Grant No. 2018YFB1502902), and the Key Program for International Science and Technolog Cooperation Projects of Shaanxi Province, China (Grant No. 2019KWZ-03).
Cite this article:
Heng Liu(刘恒), Qing-Yang Fan(樊庆扬)†, Fang Yang(杨放), Xin-Hai Yu(于新海), Wei Zhang(张伟), and Si-Ning Yun(云斯宁)‡ tP40 carbon: A novel superhard carbon allotrope 2020 Chin. Phys. B 29 106102
Fig. 1.
Crystal structure of tP40 carbon (a) along the a axis (b) and b axis (c), and crystal structure of P carbon (d).
Crystal
Method
a
b
c
V
ρ
tP40 carbon
GGAa
8.414
4.383
7.756
2.571
LDAa
8.314
4.329
7.482
2.666
C64
7.180b
2.511
6.022
2.562
Pnma-BN
GGAc
4.890
2.589
4.284
13.557
LDAc
4.795
2.557
4.243
13.007
C96
PW91d
9.020
2.700
GGAe
9.004
C72
9.460f
11.760
1.690
Diamond
GGAa
3.566
11.341
3.566g
11.337
LDAa
3.526
10.961
3.525g
10.950
Exp.h
3.567
11.346
Table 1.
Calculated values of lattice constant (in unit Å), cell volume (in unit Å3), and density (in unit g/cm3) for tP40 carbon, C64, Pnma-BN, C96, and diamond.
Fig. 2.
Plots of ratios of (a) a/a0, c/c0 and (b) V/V0versus pressure for tP40 carbon, C96 carbon, P2221 carbon, c-BN carbon, and diamond.
tP40
C64a
Pnma-BNb
C96c
C72d
P carbone
Diamond
Diamondf
C11
542
598
392
623
273
754
1053
1076
C12
174
99
108
139
152
120
125
C13
82
256
56
C22
770
C23
116
C33
575
677
675
979
C44
240
254
299
194
81
401
563
577
C55
272
C66
261
187
285
B
259
264
298
279
183
334
431
442
G
234
217
227
219
75
360
522
E
540
510
543
521
198
795
1116
v
0.152
0.178
0.310
0.104
0.070
B/G
1.106
1.220
2.46
0.928
0.826
Table 2.
Calculated values of elastic constants of Cij (in unit GPa), bulk modulus B (in unit GPa), shear modulus G (in unit GPa), Young’s modulus E (in unit GPa), Poisson’s ratio v, and B/G ratio for each of tP40 carbon, C64, Pnma-BN, C96, C72, and diamond.
Fig. 3.
Phonon spectra for tP40 carbon at (a) 0 GPa and (b) 100 GPa.
vp/(m/s)
vs/(m/s)
vm/(m/s)
ΘD/K
B/G
0
14908
9545
10487
1579
1.106
10
15271
9568
10538
1606
1.229
20
15483
9447
10434
1607
1.352
30
15886
9546
10559
1641
1.435
40
16039
9387
10409
1633
1.586
50
16280
9397
10443
1649
1.668
60
16584
9504
10558
1681
1.712
70
17051
9578
10659
1701
1.847
80
16898
9356
10423
1682
1.929
90
17036
9278
10349
1680
2.039
100
17184
9227
10303
1682
2.134
Table 3.
Estimated shear wave velocity (vs), compressional wave velocity (vp), average sound velocity vm, Debye temperature ΘD, and B/G ratio results for tP40 carbon.
Fig. 4.
Plots of (a) elastic constants and (b) B, G, and E for tP40 carbon versus pressures.
Pressure
A1
A3
Ba
Bc
ABa
ABc
0
1.008
1.421
807.58
721.47
1.00
0.893
10
0.994
1.487
928.01
817.32
1.00
0.881
20
0.981
1.491
1044.87
908.75
1.00
0.870
30
0.940
1.022
1179.82
990.59
1.00
0.840
40
0.932
0.932
1295.93
1081.51
1.00
0.835
50
0.922
0.984
1403.18
1160.81
1.00
0.827
60
0.910
1.159
1508.43
1241.13
1.00
0.823
70
0.765
1.307
1561.37
1619.36
1.00
1.037
80
0.865
1.338
1724.73
1394.34
1.00
0.808
90
0.825
1.440
1838.87
1456.74
1.00
0.792
100
0.813
1.567
1942.09
1529.97
1.00
0.788
Table 4.
Anisotropy factors of tP40 carbon from 0 GPa to 100 GPa.
Fig. 5.
Directional dependence of Young’s modulus at (a) 0 GPa, (b) 50 GPa, and (c) 100 GPa; the shear modulus at (d) 0 GPa, (e) 50 GPa, and (f) 100 GPa; and Passion’s ratio at (g) 0 GPa, (h) 50 GPa, and (i) 100 GPa for tP40 carbon.
Emax
Emin
Ratio
Gmax
Gmin
Ratio
vmax
vmin
Ratio
0
595
480
1.24
261
184
1.42
0.31
0.09
3.44
10
631
490
1.29
272
180
1.51
0.37
0.10
3.70
20
651
500
1.30
283
177
1.60
0.41
0.11
3.73
30
683
607
1.13
282
225
1.25
0.35
0.14
2.50
40
718
584
1.23
286
207
1.38
0.41
0.14
2.93
50
748
619
1.21
293
219
1.34
0.41
0.15
2.73
60
779
635
1.23
314
223
1.41
0.43
0.15
2.87
70
994
626
1.59
368
212
1.74
0.48
0.12
4.00
80
835
621
1.34
335
208
1.61
0.49
0.15
3.27
90
864
615
1.41
348
202
1.72
0.52
0.15
3.47
100
888
604
1.47
358
195
1.84
0.55
0.14
3.92
Table 5.
Mximum values and minimum values of Young’s modulus (in unit GPa), shear modulus (in unit GPa), and Poisson’s ratio and Emax/Emin, Gmax/Gmin, and vmax/vmin for tP40 carbon.
P
(100) (010)
(011) (101)
(001)
(110)
(111)
Emax
Emin
Ratio
Emax
Emin
Ratio
Emax
Emin
Ratio
Emax
Emin
Ratio
Emax
Emin
Ratio
0
555
480
1.16
562
480
1.17
595
480
1.24
595
548
1.09
595
520
1.14
10
604
490
1.23
595
490
1.21
631
490
1.29
631
585
1.08
631
543
1.16
20
645
500
1.29
618
500
1.24
651
500
1.30
651
613
1.06
651
563
1.16
30
683
607
1.13
627
607
1.03
616
607
1.01
683
615
1.11
642
614
1.05
40
718
614
1.17
645
607
1.06
614
584
1.05
718
584
1.23
660
584
1.13
50
748
626
1.19
662
626
1.06
626
619
1.01
748
619
1.21
683
619
1.10
60
779
635
1.23
688
635
1.08
704
635
1.11
779
688
1.17
710
665
1.07
70
994
626
1.59
708
626
1.13
758
626
1.21
994
708
1.40
765
665
1.15
80
835
621
1.34
713
621
1.15
765
621
1.23
835
711
1.74
765
664
1.15
90
864
615
1.40
720
615
1.17
801
615
1.30
864
713
1.21
801
675
1.19
100
888
604
1.47
735
604
1.22
839
604
1.39
888
725
1.22
839
656
1.28
Table 6.
Maximum and minimum Young’s moduli and the values of Emax/Emin in different planes at different pressures for tP40 carbon.
Fig. 6.
Electronic band phases according to (a) DFT, (b) HSE06, and (c) DOS for tP40 carbon at zero pressure.
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