materials
Article
A Reinvestigation of a Superhard Tetragonal sp3
Carbon Allotrope
Mengjiang Xing *, Binhua Li, Zhengtao Yu and Qi Chen
Faculty of Information Engineering and Automation, Kunming University of Science and Technology,
Kunming 650051, China; [email protected] (B.L.); [email protected] (Z.Y.); [email protected] (Q.C.)
* Correspondence: [email protected]; Tel.: +86-871-6742-8866
Academic Editor: Martin O. Steinhauser
Received: 5 May 2016; Accepted: 13 June 2016; Published: 17 June 2016
Abstract: I4–carbon was first proposed by Zhang et al., this paper will report regarding this phase of
carbon. The present paper reports the structural and elastic properties of the three-dimensional carbon
allotrope I4–carbon using first-principles density functional theory. The related enthalpy, elastic
constants, and phonon spectra confirm that the newly-predicted I4–carbon is thermodynamically,
mechanically, and dynamically stable. The calculated mechanical properties indicate that I4–carbon
has a larger bulk modulus (393 GPa), shear modulus (421 GPa), Young’s modulus (931 GPa),
and hardness (55.5 GPa), all of which are all slightly larger than those of c-BN. The present results
indicate that I4–carbon is a superhard material and an indirect-band-gap semiconductor. Moreover,
I4–carbon shows a smaller elastic anisotropy in its linear bulk modulus, shear anisotropic factors,
universal anisotropic index, and Young’s modulus.
Keywords: elastic properties; anisotropic properties; electronic structure; carbon allotrope
PACS: 61.50.-f; 62.20.-x; 63.20.dk; 71.20.-b
1. Introduction
The group 14 elements, such as carbon, silicon, and germanium, have attracted much interest and
have been extensively studied [1–20]. Carbon is found on the Earth mainly in the form of graphite
and diamond. The quest for carbon materials with desired properties is of great interest in both
fundamental science and advanced technology. One of the most famous carbon materials is graphene.
Recently, some scholars of great research found several carbon allotropes with low-energy metastable
structures, such as monoclinic M-carbon [1,2], F-carbon [3], orthorhombic W-carbon [4], Z-carbon [5],
H-carbon and S-carbon [6], C-carbon [7], Imma-carbon [8,9], M585-carbon [10], T12-carbon [11],
C2/m-16 carbon [12], P2221 -carbon [13], and Cco-carbon [14]. We easily found that these carbon
allotropes, with low-energy metastable structures with sp3 hybridization, could possibly explain the
superhard property of materials. Hardness is an important property that determines many of the
technological applications of materials. The mechanical properties of some carbon allotropes (such as
T-carbon [21] and Y-carbon [22]) are not excellent, as discussed above, but can be improved through
modulation [23,24]. At the moment, there are a multitude of metastable carbon allotropes that have
been predicted using the quantum-chemical methods of calculation, e.g., evolutionary metadynamics
technique [25–28], and Universal Structure Prediction: Evolutionary Xtallography (USPEX) [29–32],
or post-graphite superhard phase synthesized by cold-compressing graphite [33–39]. Compressed
graphite appears as an initial material for transition to different superhard phases. An ocean of possible
carbon modifications can be obtained by combining four-, five-, six-, seven-, even eight-membered
carbon rings. The discussion above of carbon allotropes basically adopts this structure. Wang et al. [40]
found a dynamically stable and energetically favourable carbon allotrope, i.e., J-carbon. It has a wider
Materials 2016, 9, 484; doi:10.3390/ma9060484
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Materials 2016, 9, 484
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band gap (5.89 eV within LDA-HSE06) than that of diamond and a larger bulk modulus (395 GPa).
Currently, Z-carbon [33,34] is the most stable and hardest material predicted by the theoretical methods
compared with other theoretical structures, but it has not exceeded diamond’s bulk modulus and
hardness. The inclusion in the Z-carbon of additional diamond blocks generates another superhard
carbon allotrope family, investigated in [41]. Li et al. [42] predicted a new cubic carbon allotrope,
namely, C96 carbon. Unfortunately, its mechanical properties are not excellent and it cannot be used
as a potential superhard material. Recently, another sp3 carbon allotrope, i.e., diamond nanothread,
was synthesized experimentally; Fitzgibbons et al. [43] and Zhan et al. [44] then reported regarding
its mechanical properties. These nanothreads show extraordinary properties, such as strength and
stiffness, higher than those of sp2 carbon nanotubes.
Moreover, I4–carbon was first predicted in [45].
However, the physical properties,
i.e., the structural, mechanical, and electronic properties, of I4–carbon were not studied in [45],
nor in other studies. The physical properties of a novel superhard carbon allotrope with tetragonal
I4 symmetry (16 atoms/cell) will be detailed in this paper using first-principles calculations.
2. Materials and Methods
The calculations were performed using the density functional theory (DFT) [46,47] by using the
Cambridge Serial Total Energy Package (CASTEP) code [48]. The electron–ionic core interaction
was represented using the ultrasoft pseudopotentials [49]. The structural optimizations were
conducted using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization [50]. The calculations
were performed using the local density approximation (LDA) [51,52] and generalized gradient
approximation (GGA) in the form of the Perdew–Burke–Ernzerhof (PBE) [53] exchange correlation
potential. The total energy convergence tests showed that convergence to within 1 m¨eV/atom was
achieved using the above calculation parameters. The electron and core interactions were included by
using the ultrasoft pseudopotentials method, with a plane-wave energy cutoff energy of 400 eV,
where 2s2 2p2 are treated as valence electrons for C. Highly dense k-point [54] sampling with a
grid spacing of less than 2π ˆ 0.025 Å´1 (7 ˆ 7 ˆ 7 for I4–carbon) in the Brillouin zone was used.
The self-consistent convergence of the total energy was 5 ˆ 10´6 eV/atom; the maximum force on the
atom was 0.01 eV/Å; the maximum ionic displacement was within 5 ˆ 10´4 Å; the maximum stress
was within 0.02 GPa.
3. Results and Discussion
3.1. Structural Properties
There is a new tetragonal carbon phase, namely, I4–carbon, which belongs to the I4 space group.
The crystal structure of I4–carbon is shown in Figure 1. Different atomic positions are denoted
by different colours of spheres. There are five inequivalent carbon atoms in its conventional cell,
located at C1: 8g (´0.1219, ´0.2709, ´0.6225), C2: 8g (0.4044, ´0.1809, 0.0492), C3: 8g (0.4812, 0.2514,
´0.3772), C4: 2a (0.5000, 0.5000, 0.50000) and C5: 2d (0.0000, 0.5000, 0.7500). The optimized lattice
parameters within the GGA and LDA level of I4–carbon at ambient pressure are listed in Table 1.
As listed, a = 5.5628 (5.5017) Å and c = 5.5082 (5.4471) Å are within the GGA (LDA) level and c is
slightly smaller than a. These lattice parameters are in excellent agreement with the theoretical values
predicted by Zhang et al. [45]. The results within GGA are closer to the theoretical values predicted
by Zhang et al. [45]; thus, the following discussions use the results of the GGA level. I4–carbon has
seven bond lengths, which are also listed in Table 1. Between C1 and C2, C1 and C3, there are two
different bond lengths, i.e., 1.5586 and 1.5224 Å, 1.5555 and 1.5207 Å, respectively, within the GGA
level. The average bond length of I4–carbon is 1.5510 Å, while it is 1.567 Å and 1.551 Å for C2/m–16
carbon and M carbon, respectively. They are all slightly greater than that of diamond (1.535 Å).
The hardness of I4–carbon, C2/m–16 carbon, M carbon, c-BN and diamond is calculated by using
Lyakhov and Oganov’s model [55]. The hardness of I4–carbon is 55.5 GPa, which is slightly smaller
Materials 2016, 9, 484
Materials 2016, 9, 484
3 of 15
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than
of C2/m–16
C2/m–16 carbon
than that
that of
carbon(59.5
(59.5 GPa)
GPa) [12]
[12] and
and M
M carbon
carbon (66.6
(66.6 GPa),
GPa), approximately
approximately half
half of
of that
that of
of
diamond
diamond (89.7
(89.7 GPa),
GPa), and
and slightly
slightly larger
larger than
than that
that of
of c-BN
c-BN (49.9
(49.9 GPa);
GPa); the
the hardness
hardness of
of diamond
diamond is
is very
very
4
close to
the
Lyakhov
and
Oganov’s
model.
TheThe
hardness
of Iof
to the
theresult
resultof
ofRef
Ref[55]
[55](91.0
(91.0GPa)
GPa)using
using
the
Lyakhov
and
Oganov’s
model.
hardness
–carbon
is is
83.0
I4–carbon
83.0GPa
GPaininRef
Ref[45],
[45],the
themain
mainreason
reasonfor
forthis
thissituation
situationisisthat
that the
the empirical
empirical formula
formula may
over or underestimate the value of the material’s
material’s hardness.
hardness. Most
Most researchers
researchers agree
agree that
that “superhard”
“superhard”
materials are those with Hv exceeding 40 GPa. Although
Although there
there are slight differences between the
4040
GPa,
indicating
that
I4–carbon
is a
results of the
the empirical
empiricalmodels
modelsabove,
above,all
allofofthem
themgreatly
greatlyexceed
exceed
GPa,
indicating
that
I 4 –carbon
superhard
material.
is
a superhard
material.
Figure
I4 –carbon.
–carbon.
Figure 1.
1. The
The crystal
crystal structure
structure of
of I4
Table
I 4–carbon.
–carbon.
Table 1.
1. The
Thecalculated
calculatedlattice
latticeparameters
parameters (in
(in Å)
Å) and
and bond
bond lengths
lengths (in
(in Å)
Å) for
for I4
Methods
a
Methods
GGA
5.5628
GGA
LDA
5.5017
LDA
dC1-C2
dC1-C2
1.5586
5.5628
5.5082
1.5586
1.5224
1.5224
5.4471 5.44711.5435
5.5017
1.5435
1.5036
1.5036
c
5.5082
a
c
dC1-C3
dC1-C3
1.5555
1.5555
1.5207
1.5207
1.5408
1.5408
1.5039
1.5039
dC2-C3
dC2-C5
dC2-C5
dC3-C4
1.5694
1.5871
dC2-C3
1.5694
1.5871
1.5430
1.5531
1.5531
1.5683
1.5683
1.5253
dC3-C4
1.5430
1.5253
3.2.
3.2. Mechanical
Mechanical Properties
Properties
I 4 –carbon,
seven
independent
elastic
constants
Cij were
determined
within
For the tetragonal
tetragonal I4–carbon,
seven
independent
elastic
constants
Cij were
determined
within
the
the
GGA
level
from
the
stress
of
the
strained
structure
with
a
finite
strain.
The
elastic
constants
of
GGA level from the stress of the strained structure with a finite strain. The elastic constants of I4–carbon
under
different
pressures
are pressures
listed in Table
2. From
one
can find
that
stability
I 4 –carbon
under
different
are listed
inTable
Table2, 2.
From
Table
2, the
onemechanical
can find that
the
of
I4–carbon
satisfies
Born’s
criterion
for
a
tetragonal
crystal
[56]:
C
>
0,
i
=
1,
3,
4,
6;
C
–C
>
0;
ii
11 C12
mechanical stability of I 4 –carbon satisfies Born’s criterion for a tetragonal
crystal [56]:
ii > 0,
C
C33
C 2C
) 13
+ >C33
+ 4C2(C
is mechanically
stable
13 >110,
0; C2(C
11 +11
C+
33 − 12
0 and
+ indicating
C12) + C33 +that
4C13I4–carbon
> 0, indicating
that I 4 –carbon
i =111,+ 3,
4, ´
6; 2C
C1113–C>120>and
at
0–100
GPa.
Moreover,
almost
all
elastic
constants
increase
with
increasing
pressure.
C
increases
is mechanically stable at 0–100 GPa. Moreover, almost all elastic constants increase with11increasing
the fastestC(dC
followed
by C=334.57),
(dC33followed
/dP = 4.24);
is 33the
= 0.63).
11 /dP = 4.57),
pressure.
11 increases
the fastest
(dC11/dP
by CC
3344
(dC
/dPslowest
= 4.24); (dC44/dP
C44 is the slowest
C
with
pressure
(dC16/dP
= ´0.35).
Furthermore,
it is important
explore
16 decreases
(dC44/dP
= 0.63).
C16 increasing
decreases with
increasing
pressure
(dC16/dP
= −0.35). Furthermore,
it is to
important
the
dynamic
stability
and
thermodynamic
stability
for
further
experimental
synthesis.
The
phonon
to explore the dynamic stability and thermodynamic stability for further experimental synthesis. The
spectra and
the enthalpies
as a function
pressure
shownare
in Figure
Clearly,
noClearly,
imaginary
phonon
spectra
and the enthalpies
as a of
function
of are
pressure
shown 2a–d.
in Figure
2a–d.
no
mode is shown in the phonon spectra. This strongly confirms that the I4–carbon with the I4 structure
imaginary mode is shown in the phonon spectra. This strongly confirms that the I 4 –carbon with the
is dynamically stable; thus, it is a new metastable phase in the carbon family. The enthalpy of
I 4 structure is dynamically stable; thus, it is a new metastable phase in the carbon family. The
different carbon allotropes is quantified in terms of the following formation enthalpies formula: ∆H
enthalpy of different carbon allotropes is quantified in terms of the following formation enthalpies
= Hcarbon allotropes/n ´ Hdiamond/n2 , where n denotes the number of atoms in a conventional
formula: ΔH = Hcarbon1 allotropes/n1 − Hdiamond/n
2, where n denotes the number of atoms in a
cell for carbon allotropes or graphite. As shown in Figure 2c,d, the calculated formation enthalpies of
conventional cell for carbon allotropes or graphite. As shown in Figure 2c,d, the calculated formation
I4–carbon are slightly smaller than those of M-carbon below 32.25 GPa. Bulk modulus B and shear
enthalpies of I 4 –carbon are slightly smaller than those of M-carbon below 32.25 GPa. Bulk modulus
B and shear modulus G are calculated by using the Voigt–Reuss–Hill approximation. It is known that
Materials 2016, 9, 484
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Materials 2016, 9, 484
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the Voigt bound is obtained by using the average polycrystalline modulus, based on an assumption
of uniform strain throughout a polycrystalline, and is the upper limit of the actual effective modulus,
G are
calculated
by using
Voigt–Reuss–Hill
approximation.
It is lower
knownlimit
that the
Voigt
whilemodulus
the Reuss
bound
is obtained
bythe
assuming
a uniform
stress and is the
of the
actual
bound
is
obtained
by
using
the
average
polycrystalline
modulus,
based
on
an
assumption
of
uniform
effective modulus. The arithmetic average of Voigt and Reuss bounds is referred to as the Voigt–
strain throughout
a polycrystalline,
is the upper
limitPoisson’s
of the actual
effective
while
the the
Reuss–Hill
approximations.
Young’sand
modulus
E and
ratio
v aremodulus,
obtained
using
Reuss bound is obtained by assuming a uniform stress and is the lower limit of the actual effective
following equations: E = 9BG/(3B + G), v = (3B − 2G)/(6B + 2G), respectively. The calculated results of
modulus. The arithmetic average of Voigt and Reuss bounds is referred to as the Voigt–Reuss–Hill
elastic
modulus forYoung’s
I 4 –carbon
areE also
shown in
Table
The bulk
shear
modulus,
approximations.
modulus
and Poisson’s
ratio
v are 2.
obtained
usingmodulus,
the following
equations:
Poisson’s
ratio,
and
Young’s
modulus
all
increase
with
increasing
pressure.
The
bulk
modulus
E = 9BG/(3B + G), v = (3B ´ 2G)/(6B + 2G), respectively. The calculated results of elastic modulus
for
i.e.
,
dB
/
dP
=
3.30
and
dG
/
dP
= 0.86,
increases
the
fastest,
while
shear
modulus
increases
the
slowest,
I4–carbon are also shown in Table 2. The bulk modulus, shear modulus, Poisson’s ratio, and Young’s
) and
Lewandowski
respectively.
[57] proposed
the ratio
of bulk
shear
modulus
(B/Gthe
modulus Pugh
all increase
with increasing
pressure.
The to
bulk
modulus
increases
fastest,
while shear [58]
modulus
increases
the
slowest,
i.e.,
dB/dP
=
3.30
and
dG/dP
=
0.86,
respectively.
Pugh
[57]
and v > 0.26
proposed Poisson’s ratio as indications of a ductile versus brittle characteristic. B/G > 1.75proposed
ratiomaterial
of bulk torepresents
shear modulus
(B/G) and
Lewandowski
proposed
as indications
B/G < 1.75[58]
and
v < 0.26Poisson’s
usuallyratio
indicate
brittleness.
for athe
solid
ductility,
while
characteristic. B/G > 1.75 and v > 0.26 for a solid material represents ductility,
Fromof0a ductile
to 100 versus
GPa,brittle
I 4 –carbon
is brittle; namely, the brittleness of I 4 –carbon decreases with
while B/G < 1.75 and v < 0.26 usually indicate brittleness. From 0 to 100 GPa, I4–carbon is brittle;
increasing pressure. Compared with other carbon allotropes (P2221–carbon: 0.872 [13]; C2/m–16
namely, the brittleness of I4–carbon decreases with increasing pressure. Compared with other carbon
carbon:
0.867 [12];
0.858
[12]; diamond:
0.831
[12]),
I 4 –carbon
(0.933)
the least
allotropes
(P2221Imma–carbon:
–carbon: 0.872 [13];
C2/m–16
carbon: 0.867
[12];
Imma–carbon:
0.858
[12]; has
diamond:
brittleness
at ambient
pressure.
0.831 [12]),
I4–carbon
(0.933) has the least brittleness at ambient pressure.
40
(a)
50
(b)
35
40
Frequency (THz)
30
25
30
20
20
15
10
10
5
0
Z
A
M
G Z
R
X
G
0
Z
A
0 GPa
ΔH (eV/atom)
0.25
M
G
Z
R
X G
100 GPa
(c)
(d)
4.0
C2/m-16
M
I-4
P2221
C96
Imma
TY
T
Y
0.20
3.5
3.0
2.5
2.0
0.15
1.5
1.0
0.10
0
20
40
60
80 100 0 20
Pressure (GPa)
40
60
80 100
Figure
2. Phonon
spectra
forfor
I4I4
–carbon
100GPa
GPa(b);
(b);enthalpies
enthalpies
I4 –carbon
Figure
2. Phonon
spectra
–carbonatat00GPa
GPa (a)
(a) and
and 100
of of
I4 –carbon
and and
otherother
carbon
allotropes
relative
totographite
ofpressure
pressure
(c,d).
carbon
allotropes
relative
graphiteas
asaa function
function of
(c,d).
Incompressibility can reflect the size of the bulk modulus from the other side. The V/V0 of
I 4 –carbon, C2/m–16 carbon, M–carbon, P2221–carbon, c–BN, and diamond as a function of pressure
are shown in Figure 3a. From Figure 3a, it is clear that c–BN shows a weaker incompressibility than
that of the other carbon allotropes, while I 4 –carbon exhibits the weakest incompressibility among
the carbon allotropes. The order of incompressibility for carbon allotropes and c-BN is diamond >
P2221–carbon > M–carbon > C2/m–16 carbon > I 4 –carbon > c–BN. The bulk moduli of carbon
allotropes and c-BN are also in this order: diamond (439 GPa) > P2221–carbon (409 GPa) > M–carbon
Materials 2016, 9, 484
5 of 15
Table 2. The calculated elastic constants (in GPa) and elastic modulus (in GPa) of I4 –carbon at
different pressures.
Materials 2016, 9, P
484
C11
C12
C13
C16
C33
C44
C66
B
G
5 of 15
0
926
133
101
´28
1017
398
472
393
421
10
1015
142 > I´35
1028
408 > c–BN
493 (370
441GPa).434
(399 GPa) > C2/m–16
carbon175
(398 GPa)
4 –carbon
(393 GPa)
Regarding the
20
1034
189
153
´40
1081
422
511
460
446
incompressibility
of 1079
I 4 –carbon
investigations,
the lattice
a455
/a0 and c/c0 as
30
220 in detailed
182
´43
1132
428
528 parameters
495
40
1128
248
208
´46
1181
436
544
529
466
functions of pressure are illustrated in Figure 3b. Regarding the incompressibility of lattice
235close.´49
1229
442
559lattice563
475 c is slightly
a0 and 1174
c/c0, they274
are very
The incompressibility
of
parameter
parameters a/50
60
1220
304
261
´53
1275
449
573
596
484
fitting332
the calculated
data
the polynomial
fitting629
methods,
larger than that
70 of a. By
1263
287
´56 using
1320
454
586
491the following
relationships 80
at 0 GPa
and 0 K
are obtained:
1306
360
303
´59
1364
458
598
661
499
90
1347
387
340
´62
1403
462
608
692
504
−6P2 − 7.85386 × 10−4P + 0.99970
a/419
a0 = 1.96545
(1)
100
1383
366 × 10´63
1441
461
614
723
507
c/c0 = 1.93918 × 10−6P2 − 7.79164 × 10−4P + 0.99965
(2)
Incompressibility can reflect the size of the bulk modulus from the other side. The V/V 0 of
where
the unit
of pressure
is GPa.
By fitting
using
the polynomial,
is clear that
incompressibility
I4–carbon,
C2/m–16
carbon,
M–carbon,
P222
c–BN, anditdiamond
as athe
function
of pressure
1 –carbon,
is slightly
better3a,
than
that
of athat
. c–BN shows a weaker incompressibility than
of
parameter
arelattice
shown
in Figurec3a.
From Figure
it is
clear
that of the other carbon allotropes, while I4–carbon exhibits the weakest incompressibility among
Table 2.allotropes.
The calculated
(in GPa) and for
elastic
modulus
(in GPa)
of c-BN
I4 –carbon
at
the carbon
Theelastic
order constants
of incompressibility
carbon
allotropes
and
is diamond
different
pressures.
> P2221 –carbon > M–carbon > C2/m–16 carbon > I4–carbon > c–BN. The bulk moduli of carbon
allotropes and c-BN are also in this order: diamond (439 GPa) > P2221 –carbon (409 GPa) > M–carbon
P
C11
C12 C13 C16
C33
C44 C66
B
G
(399 GPa) > C2/m–16 carbon (398 GPa) > I4–carbon (393 GPa) > c–BN (370 GPa). Regarding the
0
926 133 101 −28 1017 398 472 393 421
incompressibility of I4–carbon in detailed investigations, the lattice parameters a/a0 and c/c0 as
10 1015 175 142 −35 1028 408 493 441 434
functions of pressure are illustrated in Figure 3b. Regarding the incompressibility of lattice parameters
20 1034 189 153 −40 1081 422 511 460 446
a/a0 and c/c0 , they are very close. The incompressibility of lattice parameter c is slightly larger than that
30 1079
220 182
−43 1132fitting
428 methods,
528 495
455
of a. By fitting the calculated
data using
the polynomial
the following
relationships at
40
1128
248
208
−46
1181
436
544
529
466
0 GPa and 0 K are obtained:
50 1174 274 235 −49 1229 442 559 563 475
4
60a{a01220
304 ˆ261
1275 ˆ
449
596 484
“ 1.96545
10´6−53
P2 ´ 7.85386
10´573
P + 0.99970
(1)
70 1263 332 287 −56 1320 454 586 629 491
“ 1.93918
10´6−59
P2 ´ 7.79164
10´4598
P + 0.99965
(2)
80c{c01306
360 ˆ303
1364 ˆ
458
661 499
90 is
1347
340 using
−62 the
1403
462 608
504 the incompressibility
where the unit of pressure
GPa. 387
By fitting
polynomial,
it is 692
clear that
1383 better
419 than
366 that
−63
of lattice parameter c100
is slightly
of a.1441 461 614 723 507
(b)
(a)
1.00
1.00
0.98
0.99
0.96
0.94
V/V0
0.92
0.97
0.90
I-4
C2/m-16
M
P2221
0.88
0.86
0.82
0.96
0
20
40
0.95
I-4 a/a0
c-BN
Diamond
0.84
X/X0
0.98
I-4 c/c0
60
20
80 100 0
Pressure (GPa)
40
60
80
100
0.94
Figure
3. The
volume V/V
V/V0 0asasaafunction
I4 –carbon,
–carbon, P222
P22211-carbon,
Figure 3.
The primitive
primitive cell
cell volume
functionof
of pressure
pressure for
for I4
-carbon,
C2/m-16
carbon,
M-carbon,
c-BN
and
diamond
(a);
lattice
constants
a/a
0
,
c/c
0
compression
C2/m-16 carbon, M-carbon, c-BN and diamond (a); lattice constants a/a0 , c/c0 compression of
of
I4 –carbon as a function of pressure (b).
I4
The defined Debye temperature parameter, which can be interpreted as the temperature at
which the highest-frequency mode (and hence every mode) is excited, is of great importance,
allowing us to predict the heat capacity at any temperature in an atomic solid. The formula for
evaluating the Debye temperature is given by [59]:
Materials 2016, 9, 484
6 of 15
1
h  3n  N A   3
D 
  vm
 
k B  4  M 
Materials 2016, 9, 484
6 of 15
(3)
The defined Debye temperature parameter, which can be interpreted as the temperature at which
where h denotes Planck’s constant, kB denotes Boltzmann’s constant, n denotes the number of atoms
the highest-frequency mode (and hence every mode) is excited, is of great importance, allowing us to
A denotes Avogadro’s number, M denotes molar mass, and ρ denotes density. The
per formula
unit,
predict
theN
heat
capacity at any temperature in an atomic solid. The formula for evaluating the Debye
vm by
is described
by the following formula:
average sound
velocity
temperature
is given
[59]:
„
ˆ
˙ 1
3
h 3n NA ρ
1
ΘD “
vm
k B  4π
M  3

(3)
1 2 1 
vm   Boltzmann’s
 3 constant, n denotes the number of atoms
where h denotes Planck’s constant, kB denotes
3
 3  vt vl 
(4)
per formula unit, NA denotes Avogadro’s number, M denotes molar mass, and ρ denotes density.
The average sound velocity vm is described by the following formula:
The transverse and longitudinal elastic wave velocity, vt and vl, can be obtained using Navier’s
« ˜
¸ff´ 1
equation, as follows [60]:
3
1 2
1
vm “
3
v3
`
(4)
v3
t
l
3B  4G
G
vt 
and vl 
3 vl , can be obtained using Navier’s
ρ wave velocity, vt and
The transverse and longitudinal elastic
(5)
equation, as follows [60]:
d
d
where B and G are the bulk modulus and shear
modulus,
Based on Equation (3), the
G
3B `respectively.
4G
vt “
and vl “
(5)
Debye temperature of I 4 –carbon ΘD = 2024ρK. The Debye3ρtemperature of I 4 –carbon as a function
of pressure
is plotted
inthe
Figure
4a. It isand
found
the Debye
temperature
increases
with
increasing
where
B and G are
bulk modulus
shearthat
modulus,
respectively.
Based on Equation
(3), the
Debye
of I4–carbon
ΘD = 2024
The Debye
temperature
of I4–carbon
as a function
of pressure
pressure.temperature
Furthermore,
the greater
the K.
pressure,
the
more slowly
the Debye
temperature
increases.
is
plotted
in
Figure
4a.
It
is
found
that
the
Debye
temperature
increases
with
increasing
pressure.
The calculated results of the density, Debye temperature, transverse and longitudinal elastic wave
Furthermore, the greater the pressure, the more slowly the Debye temperature increases. The calculated
velocity vt and vl, and the mean sound velocity vm at different pressures are listed in Table 3. The
results of the density, Debye temperature, transverse and longitudinal elastic wave velocity vt and
vl the
increase
withvelocity
increasing
pressure.
With are
thelisted
increase
of3.pressure,
density and
vl , and
mean sound
vm at different
pressures
in Table
The densitythe
andtransverse
vl
v
t
shows
no
monotonic
increase
or
decrease.
However,
the
longitudinal
elastic
wave
velocity
increase with increasing pressure. With the increase of pressure, the transverse longitudinal elastic mean
wave velocity
vt showswith
no monotonic
increase
or decrease.
theand
mean
sound velocity
vm
vm increases
increasing
pressure
from 0 However,
to 80 GPa
decreases
with increasing
sound velocity
increases
with
increasing
pressure
from
0
to
80
GPa
and
decreases
with
increasing
pressure
from
pressure from 80 to 100 GPa. The main reason is that sometimes vt increases with increasing80pressure
to 100 GPa. The main reason is that sometimes vt increases with increasing pressure and sometimes
and sometimes decreases, i.e., there is no law.
decreases, i.e., there is no law.
2180
(a)
5.6
(b)
2160
Debye temperature (K)
2120
5.4
2100
2080
Band gap (eV)
5.5
2140
5.3
2060
2040
2020
5.2
0
20
40
60
80 100
0
20
40
60
80 100
Pressure (GPa)
4. Debye
temperature
(a) and
bandgap
gap(b)
(b) of
as aas
function
of pressure.
Figure 4.Figure
Debye
temperature
(a) and
band
ofI4I4–carbon
–carbon
a function
of pressure.
Table 3. The calculated density (ρ, in g/cm3), the longitudinal, transverse, and mean elastic wave
velocity (vs, vp, vm, in m/s), and the Debye temperature (ΘD, in K) for I 4 –carbon.
P
0
10
ρ
3.2764
3.3564
vl
17,069
17,434
vt
11,336
11,373
vm
12,398
12,467
ΘD
2024
2051
Materials 2016, 9, 484
7 of 15
Table 3. The calculated density (ρ, in g/cm3 ), the longitudinal, transverse, and mean elastic wave
velocity (vs , vp , vm , in m/s), and the Debye temperature (ΘD , in K) for I4 –carbon.
P
ρ
vl
vt
vm
ΘD
0
10
20
30
40
50
60
70
80
90
100
3.2764
3.3564
3.4332
3.5059
3.5750
3.6412
3.7045
3.7654
3.8243
3.8814
3.9365
17,069
17,434
17,529
17,730
17,939
18124
18303
18465
18617
18751
18852
11,336
11,373
11,402
11,395
11,415
11417
11427
11424
11417
11400
11345
12,398
12,467
12,503
12,513
12,549
12566
12589
12598
12602
12594
12547
2024
2051
2073
2089
2109
2124
2141
2154
2166
2176
2178
3.3. Anisotropic Properties
It is well known that the anisotropy of elasticity is an important implication in engineering science
and crystal physics [61], hence, it is worthwhile to investigate the elastic anisotropy of materials.
The shear anisotropic factors provide a measure of Ba , Bb , and Bc , i.e., the bulk modulus along the a, b
and c axes, respectively, which can be calculated by using the following equations:
dP
Λ
“
,
da
1`α`β
(6)
Bb “ b
dP
Ba
“
,
db
α
(7)
Bc “ c
Ba
dP
“
,
dc
β
(8)
Ba “ a
Λ “ C11 ` 2C12 α ` C22 α2 ` C33 β2 ` 2C13 βp1 ` αq,
pC11 ´ C12 q
,
pC22 ´ C12 q
(10)
pC22 ´ C12 qpC11 ´ C13 q ´ pC11 ´ C12 qpC13 ´ C12 q
pC22 ´ C12 qpC33 ´ C13 q
(11)
α“
β“
(9)
The calculated Ba , Bb , and Bc at different pressures are shown in Figure 5a. Due to the lattice
constant a being equal to b, the bulk modulus along the a-axis is equal to that along the b axis, in other
words, Ba = Bb . It is clear that Ba , Bb , and Bc increase with increasing pressure and that Ba , Bb (87.69%)
increase more than Bc (76.50%). The calculated directional bulk modulus suggests that it is the largest
along the a-axis and the smallest along the c-axis, indicating that the compressibility along the c-axis is
the smallest, while along the a-axis, is the largest. This is in accordance with the relationships between
the ratios a/a0 , c/c0 and pressure, as shown in Figure 3b. Therefore, the anisotropy of the linear bulk
modulus should also be considered. The anisotropy of the bulk modulus along the a-axis and c-axis
with respect to the b-axis can be estimated by:
A Ba “
Ba
Bc
, and A Bc “
Bb
Bb
(12)
The anisotropy factors of the bulk modulus along the a-axis and c-axis for I4–carbon at T = 0 K as a
function of pressure are shown in Figure 5b. Note that a value of 1.0 indicates elastic isotropy and any
departure from 1.0 represents elastic anisotropy. The anisotropy of the bulk modulus along the a-axis
and c-axis with respect to the b-axis shows that ABa is elastic isotropy and ABc is elastic anisotropy.
Materials 2016, 9, 484
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A1 
Materials 2016, 9, 484
(a)
(13)8 of 15
(b)
1.08
Anisotropy of the bulk modulus
2200
4C44
,
C11  C33  2C13
Bulk modulus (GPa)
2000
1.06
1800
1.04
1600
ABa
ABc
1400
Ba=Bb
1200
1.00
Bc
0
1.30
1.02
20 40 60 80 100 0
20 40 60 80 100
Pressure (GPa)
(c)
0.09
(d)
1.25
1.15
0.07
U
1.10
0.06
1.05
1.00
A1=A2
0.95
A3
0.05
U
A
0.90
0
20
40
60
80
A
Shear anisotropic factor
0.08
1.20
100
0
20
Pressure (GPa)
40
60
80
0.04
100
Figure 5. Linear bulk modulus Ba, Bb, and Bc of I4–carbon at 0 K as a function of pressure (a);
Figure 5. Linear bulk modulus Ba , Bb , and Bc of I4 –carbon at 0 K as a function of pressure (a); anisotropy
anisotropy factors of I4–carbon at 0 K as a function of pressure, (b) ABb and ABc; (c) A1, A2 and A3; (d)
factors of I4 –carbon at 0 K as a function of pressure, (b) ABb and ABc ; (c) A1 , A2 and A3 ; (d) AU .
AU.
AfterFor
discussing
anisotropy
of the[1bulk
now discuss
the anisotropy of the shear
the (0 1 0)the
shear
planes between
0 1] modulus,
and [0 0 1] we
directions,
it is [60]:
modulus. The shear anisotropic factors provide a measure
of the degree of anisotropy in the bonding
4C55
A2  anisotropic
,
(14)
between atoms in different planes. The shear
factor
for the (1 0 0) shear planes between
C22  C33  2C23
[0 1 1] and [0 1 0] directions and the (0 1 0) shear planes between [1 0 1] and [0 0 1] directions is [62]:
For the (0 0 1) shear planes between [1 1 0] and [0 1 0] directions, it is [60]:
4C
4C44
A1 “
A3 C
 ` C 66´ 2C ,
11
C  C33  2C 13
11
22
12
(15)
(13)
For the
0) shearcrystal,
planesthe
between
[11,0A1]
andA[0
0 1]1.0,
directions,
is [60]:
2, and
3 are
while any itvalue
smaller or larger than
For (0
an1isotropic
factors A
1.0 is a measure of the elastic anisotropy possessed by the materials. The anisotropy factors of I 4 –
4C55
carbon at T = 0 K as a function of pressure
A2 “are shown in Figure 5c.
, Due to the symmetry of the crystal (14)
C
`
2C23 Figure 5c, A1 first increases and then
22 A1C=33
A´
2. From
structure, C11 = C22, C44 = C55 and C13 = C23; thus,
decreases with increasing pressure, while A3 increases monotonically with increasing pressure.
For the (0 0 1) shear planes between [1 1 0] and [0 1 0] directions, it is [60]:
I 4 –carbon shows elastic anisotropy in the bulk modulus and shear modulus. The universal
A3 “
4C66
C11 ` C22 ´ 2C12
(15)
For an isotropic crystal, the factors A1 , A2 , and A3 are 1.0, while any value smaller or larger
than 1.0 is a measure of the elastic anisotropy possessed by the materials. The anisotropy factors of
I4–carbon at T = 0 K as a function of pressure are shown in Figure 5c. Due to the symmetry of the crystal
structure, C11 = C22 , C44 = C55 and C13 = C23 ; thus, A1 = A2 . From Figure 5c, A1 first increases and
Materials 2016, 9, 484
9 of 15
then decreases with increasing pressure, while A3 increases monotonically with increasing pressure.
I4–carbon shows elastic anisotropy in the bulk modulus and shear modulus. The universal anisotropic
index (AU = 5GV /GR + BV /BR ´ 6) combines the shear modulus and bulk modulus to exhibit the
anisotropy of the material. The AU of I4–carbon increases with pressure (see Figure 5d), indicating that
the anisotropy of I4–carbon increases with increasing pressure. The AU of I4–carbon is 0.0369, which is
smaller than that of C2/m–carbon (0.0766) and P2221 –carbon (0.0526).
Young’s modulus is not always the same in all orientations of a material; it will change depending
on the direction of the force vector. Engineers can use this directional phenomenon to their advantage
in creating structures. The directional dependence of the anisotropy in Young’s modulus is calculated
by using the Elastic Anisotropy Measures (ELAM) [63,64] code. The 2D representation of Young’s
modulus for I4–carbon in the (001), (010), (100), (111), (001), and (101) planes is shown in Figure 6a–f,
respectively. The black, red, and blue solid curves represent Young’s modulus for I4–carbon in the
(001), (010), (100), (111), (001), and (101) planes at 0, 50 and 100 GPa, respectively. It is obvious that the
anisotropy of Young’s modulus increases with increasing pressure in the (001), (010), (100), (111), (001),
and (101) planes. To investigate the anisotropy of Young’s modulus in detail, we calculate the maximal
and minimal values of Young’s modulus for I4–carbon in the (001), (010), (100), (111), (001), and (101)
planes and for all possible directions, together with the ratio Emax /Emin , which are listed in Table 4.
The conclusion from Table 4 is consistent with the conclusion we obtained from Figure 6, i.e., that the
anisotropy of Young’s modulus increases with increasing pressure in the (001), (010), (100), (111), (001),
and (101) planes. Regarding the distribution of Young's modulus in the (001) plane and (001) plane,
the maximal and minimal values are the same, while in the (010) plane and (100) plane, they are also the
same. The (001) plane and (001) plane exhibit the largest anisotropy of Young’s modulus for I4–carbon,
while the (101) plane exhibits the smallest anisotropy of Young’s modulus. This situation also appears
under high pressure. All of the degrees of special plane anisotropy are smaller than the overall
performance of the material anisotropy because the minimal value of the entire Young’s modulus
was not obtained. The anisotropy of Young’s modulus of I4–carbon (Emax /Emin = 1.133) is slightly
smaller than that of C2/m-16 carbon (Emax /Emin = 1.275) and P2221-carbon (Emax /Emin = 1.173). Thus,
I4–carbon shows a smaller elastic anisotropy in its linear bulk modulus, shear anisotropic factors,
universal anisotropic index, and Young’s modulus.
Table 4. The maximum and minimum values of Young’s modulus (in GPa) in different planes and for
pressures for I4 –carbon.
Plane
P
Maximum
Minimum
Emax /Emin
(001)
0
50
100
0
50
100
0
50
100
0
50
100
0
50
100
0
50
100
0
50
100
999
1250
1427
998
1153
1292
998
1153
1292
992
1233
1402
999
1250
1427
949
1160
1298
999
1250
1427
887
1054
1168
889
1053
1150
889
1053
1150
888
1045
1139
887
1054
1168
885
1050
1147
882
1042
1138
1.126
1.186
1.222
1.123
1.095
1.123
1.123
1.095
1.123
1.117
1.180
1.231
1.126
1.186
1.222
1.072
1.105
1.132
1.133
1.200
1.254
(010)
(100)
(111)
(001)
(101)
All
Materials 2016, 9, 484
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100
0
50
100
All
Materials 2016, 9, 484
1500
1298
999
1250
1427
1147
882
1042
1138
1500
(a)
750
750
0
0
-750
-750
-1500
-1500
1500
-750
0
001 plane
750
1500
750
750
0
0
-750
-750
-1500
-1500
-750
0
750
1500
-750
(e)
-1500
-1500
1500
750
750
0
0
-750
-750
-1500
-1500 -750 - 0
750
001 plane
0
750
1500
0
750
1500
0
750
101 plane
1500
010 plane
(d)
-750
100 plane
1500
10 of 15
(b)
-1500
-1500
1500
(c)
1.132
1.133
1.200
1.254
111 plane
(f)
-1500
1500 -1500
-750
Figure
2Drepresentation
representationof
ofYoung’s
Young’smodulus
modulus in the
(101)
planes
1 ), and
Figure
6.6.2D
the (001),
(001),(010),
(010),(100),
(100),(111),
(111),( 00
(001
), and
(101)
planes
forfor
I4 –carbon,
I4–carbon,shown
shownin
in(a–f),
(a–f),respectively.
respectively.
3.4.
ElectronicProperties
Properties
3.4.
Electronic
I 4 –carbonare
Theband
bandstructures
structures of
of I4–carbon
are shown
shown in Figure
GPa,
respectively.
The
Figure 7a,b
7a,batat0 0and
and100
100
GPa,
respectively.
Semiconductor
materials
have
direct
or indirect
In direct
band
gap materials,
the
Semiconductor
materials
have
direct
or indirect
bandband
gaps.gaps.
In direct
band gap
materials,
the maximum
of band
the valence
and minimum
of the conduction
bandoccur
(CBM)atoccur
at thevalue
same of
of maximum
the valence
(VBM)band
and(VBM)
minimum
of the conduction
band (CBM)
the same
of vector,
the wave
vector,
while in indirect
band
gap materials,
they occur
for different
values
of the
thevalue
wave
while
in indirect
band gap
materials,
they occur
for different
values
of the
wave
4 –carbon is an and
indirect-band-gap
and wide-band-gap
waveIt vector.
It that
is obvious
vector.
is obvious
I4–carbonthat
is an Iindirect-band-gap
wide-band-gap semiconductor
material.
semiconductor
material.
The VBM
is located
at the
M point
(0.500,
0.500, is
0.000),
while
The
VBM of I4–carbon
is located
atof
theI 4M–carbon
point (0.500,
0.500,
0.000),
while
the CBM
located
from
the (0.000,
CBM is0.000,
located
fromthrough
point (0.000,
0.000,
0.3667)
through
(0.000,(0.000,
0.000, 0.000,
0.000)0.500),
to the whether
Z point at
point
0.3667)
G (0.000,
0.000,
0.000)
to the G
Z point
(0.000, pressure
0.000, 0.500),
whether
ambient pressure
or under
pressure.
From
Figuregap
7a,b,
it is
ambient
or under
highatpressure.
From Figure
7a,b, high
it is shown
that
the band
increases
shown
that
the
band
gap
increases
with
increasing
pressure;
more
details
are
shown
in
Figure
4b;
with increasing pressure; more details are shown in Figure 4b; the band gap of I4–carbon increasesthe
from
4 –carbon
increases
from 5.19 the
eV to
5.59
The gap
greater
the pressure,
the less the
band
of IeV.
5.19
eV gap
to 5.59
The greater
the pressure,
less
theeV.
band
increases.
To determine
theband
reason
gapwe
increases.
Tothe
determine
the reason
why,
weand
calculate
energy
valuespressures.
of the CBM
and
VBM
why,
calculate
energy values
of the
CBM
VBM the
under
different
The
calculated
under
different
pressures.
The
calculated
results
are
illustrated
in
Figure
7c.
The
energy
of
VBM
for eV
results are illustrated in Figure 7c. The energy of VBM for I4–carbon is 7.96 eV at 0 GPa and 10.58
I 4 –carbon
isVBM
7.96=eV
at eV.
0 GPa
10.58ofeVCBM
at 100
ΔEVBM is
= 2.62
of CBM
at 100
GPa; ∆E
2.62
Theand
energy
forGPa;
I4–carbon
13.15eV.
eVThe
at 0energy
GPa and
16.17 for
eV at
–carbon
is 13.15
at ∆E
0 GPa
16.17
at 100
ΔEis
CBM
= 3.02
CBM
− ΔE
VBM gap
= 0.40
eV,
100I 4GPa;
∆ECBM
= 3.02eVeV.
´ ∆E
= 0.40
eV,GPa;
which
the
sameeV.
as ΔE
with
the
band
increase.
CBMand
VBMeV
In other words, as pressure increases, the conduction band and valence band energy will increase,
but the energy of the conduction band will increase more than that of the valence band. The partial
density of states (PDOS) of five inequivalent carbon atoms in I4–carbon at 0 and 100 GPa are shown
in Figure 8. Figure 8a,c,e,g,i show the partial density of states of five inequivalent carbon atoms at
0 GPa; Figure 8b,d,f,h,j show the partial density of states of five inequivalent carbon atoms at 100 GPa.
mainly by the C-2s orbitals, while in the other energy part, is contributed mainly by the C-2p orbitals.
Materials 2016, 9, 484
11 of 15
10
10
which is the same as with the band gap increase. In other words, as pressure increases, the conduction
8 band energy will increase, but the8 energy of the conduction band will increase
band and valence
Materials
2016,
9,
484
11 of 15
more than that 6of the valence band. The partial density
6 of states (PDOS) of five inequivalent carbon
Energy (eV)
atoms in I 4 –carbon at 0 and 100 GPa are shown in Figure 8. Figure 8a,c,e,g,i show the partial density
4
of states of five 4inequivalent carbon atoms at 0 GPa; Figure
8b,d,f,h,j show the partial density of states
No matter the position of the carbon atoms, the partial density
of states in the conduction band part
of five inequivalent carbon atoms at 100 GPa. No matter the position of the carbon atoms, the partial
2 and the valence band deviates from the Fermi
is, with increasing2pressure intensity, near low energy,
density of states in the conduction band part is, with increasing pressure intensity, near low energy,
level. The PDOS in the lower energy part is contributed mainly by the C-2s orbitals, while in the other
and the valence0band deviates from the Fermi level. The
0 PDOS in the lower energy part is contributed
energy
part,
contributed
mainly
theother
C-2p energy
orbitals.
whileby
in the
part, is contributed mainly by the C-2p orbitals.
mainly
byisthe
C-2s orbitals,
-2
-2
10
-4
8
-6
8
-66
6
4
Energy (eV)
-8
-10
10
-4
Z
A
4
-8
2
2
0
M
G Z
R
-100
X G
(a) 0 GPa
-2
0-4 GPa
G
Z
A
CBM
M
X
G
100 GPa
16.17 eV
-8
CBM
Z
R
(b) 100 GPa
-6
-8
G Z
R
X G
-10
Z
A M
G
Z
R
X
G
(a) 0 GPa
100 GPa
ΔE
=3.02 eV100 (b)GPa
0 GPaCBM
Eg=5.59
eV
16.17 eV
13.15 eVCBM
ΔECBM=3.02 eV
Eg=5.19 eV
CBM
VBM
A M
-4
-6
-10
Z
-2
VBM
13.15 eV
Eg=5.19 eV
VBM
Eg=5.59 eV
10.58
ΔEVBM=2.62 eV
VBM
7.96 eV
ΔEVBM=2.62 eV
7.96 eV
(c)
eV
10.58 eV
(c)
I4the
Figure 7. Electronic
band
structure
for the
–carbon
atambient
ambient
pressure
(a)
and
100
GPa (b); the
I4I4
Figure
Electronic
band structure
structure
for
–carbon
at
pressure
(a) and(a)100
GPa
(b);GPa
the (b);
Figure
7. 7.
Electronic
band
forthe
–carbon
at ambient
pressure
and
100
calculated
energy
values
of
CBM
and
VBM
under
different
pressures
(c).
calculated energy
valuesenergy
of CBM
and
VBM
different
pressures
(c).
the calculated
values
of CBM
andunder
VBM under
different
pressures (c).
0.8
0.8
0.6
0.6
(a)
0.0
0.6
0.2
C1-s
C1-p
(c)
C2-s
C2-p
(d)
C2-s
C2-p
0.2
(b)
0.0
0.6
0.4
0.2
C1-s
C1-p
C1-s
C1-p
0.2
0.0
0.6
(c)
0.4
0.2
0.4
0.0
0.2
0.0
0.6
(b)
0.4
0.4
0.0
0.6
C1-s
C1-p
0.2
0.4
0.0
0.6
(a)
0.4
-20
-15
-10
-5
0
5
C2-s
C2-p
10
(d)
C2-s
C2-p
0.4
0.2
0.0
-20
-15
-10
-5
Figure 8. Cont.
0
5
10
Materials 2016, 9, 484
12 of 15
Partial density of states (electrons/eV)
Materials 2016, 9, 484
0.8
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
(e)
C3-s
C3-p
(f)
C3-s
C3-p
(g)
C4-s
C4-p
(h)
C4-s
C4-p
0.8
(i)
0.6
0.4
0.2
0.0
(j)
0.6
0.4
0.2
0.0
-20
-15
-10
-5
0
5
12 of 15
10
C5-s
C5-p
C5-s
C5-p
-20
-15
-10
-5
Energy (eV)
0
5
10
Figure 8. Partial density of state for I4 –carbon; (a,c,e,g,i) are at ambient pressure, while (b,d,f,h,j) are
Figure 8. Partial
density of state for I4–carbon; (a,c,e,g,i) are at ambient pressure, while (b,d,f,h,j) are
at 100 GPa.
at 100 GPa.
4. Conclusions
4. Conclusions In
conclusion, the low-energy metastable structures of I4–carbon have been systematically
investigated based on the density functional theory. The related enthalpy, elastic constants and
I4 –carbon havemechanically,
In conclusion,
the confirm
low-energy
structures
been systematically
phonon spectra
that themetastable
newly-predicted
I4–carbon of
is thermodynamically,
dynamically
stable.
As a potential
superhard
material for
engineering
mechanical
investigatedand
based
on the
density
functional
theory.
The
related applications,
enthalpy,theelastic
constants and
properties of I4–carbon indicate that it is a superhard material. I4–carbon shows a smaller anisotropy
phonon spectra confirm that the newly-predicted I4–carbon is thermodynamically, mechanically,
in its linear bulk modulus, shear anisotropic factors, universal anisotropic index, and Young’s modulus.
and dynamically
As athat
potential
material
for engineering
applications, the
The band stable.
structure shows
I4–carbon issuperhard
an indirect-band-gap
and wide-band-gap
semiconductor
Finally,
its mechanical
reveal that I4–carbon
possesses
a high
I4 –carbon
–carbon of
indicate
that properties
it is a superhard
material.
shows a
mechanical material.
properties
oftheI4calculations
bulk and shear modulus as well as a low Poisson’s ratio and B/G ratio (<1.75). Moreover, I4–carbon
smaller anisotropy
in its linear bulk modulus, shear anisotropic factors, universal anisotropic index,
has a larger Debye temperature (ΘD = 2024 K). Due to their wide-band-gap and higher bulk moduli,
hardness,
they The
are attractive
for semiconductor
superhard
material with and wideI4–carbon and
is an
indirect-band-gap
and Young’s
modulus.
band structure
showsdevice
that applications
potential technological and industrial applications.
band-gap semiconductor material. Finally, the calculations of its mechanical properties reveal that I4
Acknowledgments:
This work
supported
by theas
Fund
for as
Talents
of Yunnan
Province,
–carbon possesses
a high bulk
andwas
shear
modulus
well
a low
Poisson’s
ratioChina
and B/G ratio
(Grant No. KKSY201403006) and the National Natural Science Foundation of China (No. 61564005).
I4–carbonMengjiang
has a larger
Debye
(ΘD =Mengjiang
2024 K).
Due
to their
(<1.75). Moreover,
Author Contributions:
Xing and
Binhua temperature
Li designed the project;
Xing,
Binhua
Li andwide-bandZhengtao Yu performed the calculations; Mengjiang Xing, Zhengtao Yu and Qi Chen determined the results;
gap and higher
bulk moduli, hardness, they are attractive for semiconductor device applications and
Mengjiang Xing and Binhua Li wrote the manuscript.
superhard material
potential
and
industrial applications.
Conflicts of with
Interest:
The authorstechnological
declare no conflict of
interest.
Acknowledgments: This work was supported by the Fund for Talents of Yunnan Province, China (Grant No.
KKSY201403006) and the National Natural Science Foundation of China (No. 61564005).
Author Contributions: Mengjiang Xing and Binhua Li designed the project; Mengjiang Xing, Binhua Li and
Zhengtao Yu performed the calculations; Mengjiang Xing, Zhengtao Yu and Qi Chen determined the results;
Mengjiang Xing and Binhua Li wrote the manuscript.
Materials 2016, 9, 484
13 of 15
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