# T-carbon: experiments, properties, potential applications and derivatives

Xin-Wei Yi  These authors make equal contribution to this work. School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Zhen Zhang  These authors make equal contribution to this work. School
of Physical Sciences, University of Chinese Academy of Sciences, Beijing
100049, China  Zheng-Wei Liao  These authors make equal contribution to this
work. School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Xue-Juan Dong  School of Physical Sciences, University
of Chinese Academy of Sciences, Beijing 100049, China  Jing-Yang You
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, National
University of Singapore, 117551, Singapore  Gang Su<EMAIL_ADDRESS>Kavli
Institute for Theoretical Sciences, and CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China  School of Physical Sciences, University of Chinese
Academy of Sciences, Beijing 100049, China

###### Abstract

T-carbon is a novel carbon allotrope with many appealing properties. Since the
proposal of T-carbon, there has been a lot of intensive studies devoted to its
physical, chemical, optical, magnetic, thermoelectrical and topological
properties and possible applications in diverse areas in recent years. In this
review, we provide a comprehensive review on the advances of the experiments,
intriguing properties and various potential applications of T-carbon in energy
storage, optoelectronics, thermoelectrics, topological states, etc. As
intercalation and doping are effective methods to modify the electronic
structure of materials or even to convert into sparkly different new
structures or phases, we also discuss different atom doped T-carbon, which
exhibit more intriguing properties and lead to promising potential
applications in solar cells, photocatalysis, magnetism, superconductivity and
so on. In addition, it is interesting to mention that the hydrogenated
T-carbon molecules were found to be possibly related to the physical origin of
the UV extinction feature in interstellar medium that has been the half-
century long unsolved puzzle. A number of novel derivative structures either
derived or inspired from T-carbon are included as well. Finally, we give
prospects and outlook for future directions of study on T-carbon and related
structures.

###### Contents

  1. I Carbon allotropes
    1. I.1 Diamond
    2. I.2 Graphite and Graphene
    3. I.3 Graphdiyne
    4. I.4 Carbon nanotubes
    5. I.5 Carbyne
    6. I.6 Fullerenes
    7. I.7 Amorphous carbon
    8. I.8 Carbon dots
    9. I.9 T-carbon
  2. II Experiments on T-carbon
    1. II.1 Synthesis of T-carbon nanowires by picosecond laser irradiation
    2. II.2 Preparation of T-carbon by plasma enhanced chemical vapor deposition
  3. III Properties of T-carbon
    1. III.1 Chemical bonding
    2. III.2 Mechanical properties
    3. III.3 Thermoelectric properties
    4. III.4 Excitonic effect
    5. III.5 Optical properties
    6. III.6 Topological phonons
  4. IV Potential applications of T-carbon
    1. IV.1 Hydrogen storage
    2. IV.2 Li-ion batteries
    3. IV.3 Solar cells
    4. IV.4 Photocatalysis
    5. IV.5 Magnetism in Ni-doped T-carbon
    6. IV.6 Superconductivity in Na-doped T-carbon
    7. IV.7 Hydrogenated T-carbon molecules in interstellar medium
  5. V Derivative structures from T-carbon
    1. V.1 TY-carbon
    2. V.2 T-II carbon
    3. V.3 Bct-C16
    4. V.4 sp2-diamond
    5. V.5 L-carbon, C–c–C4, H–c–C4, H–2c–C4
    6. V.6 Carboneyane
    7. V.7 Cycligraphyne and Cyclicgraphdiyne
  6. VI Prospects and outlook

Figure 1: Carbon allotropes: (a) amorphous carbon, (b) graphite, (c) diamond,
(d) T-carbon, (e) graphdiyne, (f) carbon nanotube, (g) carbyne, and (f) C60
fullerene.

##  I Carbon allotropes

Carbon is an extremely versatile element with highest mass content in earth’s
crust. Each carbon atom has four valence electrons with 2$s^{2}p^{2}$. As
carbon atom can form $sp$, $sp^{2}$ and $sp^{3}$ hybrid chemical bonds, it has
strong ability to bind itself with other elements to form a variety of
allotropes or compounds with different chemical bonds, leading to the colorful
world. It is thought that 10$\%$ to 25$\%$ of interstellar carbon in the form
of polycyclic aromatic hydrocarbons (PAHs) that may explain the origin of
life. Very recently, one group has discovered the PAHs in the interstellar
cloud via spectral matched filtering for the first time [1], where the
abundance of PAHs is found to far exceed the expected target. Studying these
PAHs and other similar molecules can help human better understand how life
began in space. Besides the best known and studied graphite and diamond, since
1980s, several new carbon allotropes, including fullerenes [2, 3], carbon
nanotubes [4, 5], and graphene [6], etc., have been synthesized, which
inspired numerous attempts to find new carbon structures in the past decades.
Due to the abundant physical and chemical properties, the research of carbon
materials has always been a hot topic. According to the statistics of SACADA
(Samara Carbon Allotrope Database), there are currently more than 500 carbon
allotropes that have been proposed [7]. In the following, we will briefly
discuss several typical carbon allotropes that have been extensively studied,
as shown in Fig. 1.

###  I.1 Diamond

Diamond has two different structures: cubic and hexagonal with all $sp^{3}$
hybrid chemical bond. Diamond is an insulator with a band gap of 5.5 eV and
has high thermal conductivity. It is the hardest material that occurs
naturally, and thus is widely used in industry, such as arts and crafts,
cutting tools, and so on. It is also a precious gem.

Nanodiamond has many new mechanical and optical properties, such as high
specific surface area and adjustable surface structure, non-toxicity and
biocompatibility, which are very suitable for the preparation of specific
drugs, drug delivery, electrochemistry, and new materials [8]. As a green
catalyst, metal-free nanodiamond also shows great potential in the remediation
of organic pollutants. Some scientists have designed nanodiamond-based
catalysts with specific active sites and clear oxidation mechanism, and
established a very promising catalytic oxidation system for the green
remediation of actual polluted water [9].

Diamond can be produced naturally or synthetically. Common synthetic
techniques include CVD, high temperature and high pressure technology,
chemical synthesis, and so on. Recently, an experiment realized the chemical
induced transformation from AB-stacked bilayer graphene grown by CVD to
fluorinated single-layer diamond [10].

Conductive diamond for electrochemical research can be obtained by doping
diamond, and doped diamond is usually synthesized by various CVD technologies.
Among the doped diamond, boron doped diamond (BDD) is the most popular one,
which can achieve high conductivity through high concentration doping.
Conductive diamond has wide electrochemical potential window, low and stable
background current, fast electron transfer rate, long-term response, chemical
inertness, biocompatibility, and rich surface reactivity [11]. With these
properties, conductive diamond films have been applied in many different
fields, such as chemical and biochemical sensing, anode and cathode
electrosynthesis of different compounds, electrocatalytic reduction of carbon
dioxide, and manufacture of batteries and supercapacitors [11]. In addition,
it is worth noting that BBD has been widely studied as an anode for
electrochemical oxidation of organic wastewater [12].

Nitrogen vacancy (NV) is a kind of point defect that fixes a nitrogen atom
near the vacancy of diamond. Electron spin-based NV in diamond can be used as
magnetometer to explore the static and dynamic magnetic structures and
electronic transport phenomena [13]. In recent years, the analysis of related
theories and experiments has been in forefront of the field of magnetometry
[13]. Besides, the NV and group-IV color centers in diamond are used as main
quantum emitters in the field of quantum information [14, 15, 16].

The high thermal conductivity, low expansion coefficient, and low anisotropy
of diamond make it a promising heat-dissipating material and thermal
accelerator for thermal management composites. Recently, 3D porous diamond
foam with 3D connected diamond network was obtained by the hot filament
assisted CVD technology [17]. Adding diamond foam into latent heat storage
devices can achieve high thermal response and good structural stability.

It is well known that diamond has brittle fracture, which hinders its
application in certain areas. Recently, the existence of ultra-large,
reversible elastic deformation in nanoscale single-crystalline and
polycrystalline diamond needles has been demonstrated [18]. This surprising
high elastic deformation is attributed to the lack of defects in the nanoscale
needle-like structure and relatively smooth surface [18].

Table 1: The equilibrium density ($\rho$), Hardness, bulk modulus ($B$), cohesive energy (Ecoh), energy gap ($E_{g}$), electrical conductivity ($\sigma$) and thermal conductivity ($\kappa$) at zero pressure of several typical carbon allotropes. |  $\rho$ (g/ m3) |  Hardness (GPa) |  $B$ (102 GPa) |  Ecoh (eV/atom) |  $E_{g}$ (eV) |  $\sigma$ ($\Omega^{-1}$cm-1) |  $\kappa$ (Wm-1K-1) |  Reference
---|---|---|---|---|---|---|---|---
c-diamond |  3.52 |  96$\pm$5 |  4.43 |  7.37 |  5.45 |  10-18 |  200 |  [19]
h-diamond |  3.52 |  92.8 |  4.30 |  7.732 |  3.13 |  - |  - |  [20]
Graphite |  2.27 |  - |  2.86-3.19 |  7.374 |  -0.04 |  2.5$\times$104 |  2000 |  [21]
Amorphous carbon |  1.4-1.8 |  0.2 |  - |  - |  0.4-0.7 |  10-2-10-3 |  2.2 |  [22]
Carbon nanotubes |  0.01 |  - |  1-8 |  - |  - |  105-106 |  3500 |  [23, 24, 25, 26]
Fullerene (C60) |  1.65 |  30 |  7.0-9.0 |  - |  1.8 |  10-14 |  0.4 |  [27, 28, 29, 30, 31, 32]
M-carbon |  3.45 |  83.1 |  4.31 |  7.636 |  3.56 |  - |  - |  [33]
Bct-C4 |  3.35 |  92.2 |  4.09 |  7.533 |  2.47 |  - |  - |  [34]
T-carbon |  1.50 |  $<$10 |  1.69 |  6.573 |  2.25 |  - |  - |  [20, 35]

###  I.2 Graphite and Graphene

Graphite is a three-dimensional (3D) van der Waals bulk carbon material with
$sp^{2}$ hybridization, which is stacked in terms of ABAB… [Fig 1 (b)]. The
adjacent graphene sheets in graphite can slide easily with each other, which
makes graphite soft and lubricating. The special electronic distribution in
graphite makes it thermally and electrically conductive. Graphite is the most
stable carbon allotrope under ambient temperature and pressure and is inert
with almost all materials. Owing to the special structure and excellent
properties, graphite is widely used in industry and science.

Due to the large interspace between graphene layers, graphite can host many
atoms and molecules. Based on the specific structure of graphite, researchers
have devoted to the modification of graphite. There are three main forms of
modification, including graphite oxide (GrO), graphite intercalation compounds
(GICs) and expanded graphite [36]. GrO is oxidized functionalized graphite,
which is insulating and hydrophilic. The distance between neighboring layers
of GrO is larger than that of graphite, making it easier for ions with a large
radius such as Na+ to pass through, which increases the possibility to use it
as the anode of the battery. The reduced GrO was reported to have potential
applications in the anode of lithium- (LIB) and sodium-ion batteries (NIB) due
to the excellent electrochemical performance and large specific capacity,
which can reach 917 mAh$\cdot$g-1 in LIB [37]. GrO can also be used for the
cheap production of graphene [38]. There are many ways to prepare GrO,
including Brodie, Staudenmaier, Hummers, modified Hummers, and improved
Hummers methods [39]. Among them, the modified Hummers method has realized the
massive production of GrO with low-cost expanded graphite [40].

GICs, obtained by intercalating metal oxides, metal chlorides, bromides,
fluorides, oxyhalides, acidic oxides, Lewis acids, etc. into the interlayer of
graphite, exhibit excellent physical and chemical properties. For instance,
several metal-intercalated compounds of graphite, such as KC8, CaC6 and YbC6,
are reported to be superconductors [41]. Several GICs exhibit better physical
and chemical properties than the pristine graphite and can be used as
electrodes [42]. GICs has been used as anode materials of LIB in
commercialization for many years. Additionally, graphite as anodes for NIB
[43] and K-ion batteries (KIB) [44] has high reversible charge storage
capacity. The different host-guest interactions between different alkali metal
elements and graphite have been discussed in detail [45]. It was reported that
the limited cycle time and inferior stability of KIB can be solved by
constructing an organic-rich passivation layer on the graphite anodes [46].
High-performance NIB can be obtained by adjusting the relative stability of
ternary graphite intercalation compounds and the solvent activity in
electrolytes [47]. Graphite can be used not only as the anode material by
inserting cations, but also as the cathode material by inserted anions. The
structure, characterization methods, mechanical and electrochemical properties
of graphite cathode have been widely investigated [48]. The study on graphite
cathodes mainly focus on Al-ion and dual-ion batteries (also called “dual-
graphite batteries”), where GICs are used as both anode and cathode. The
advantages of dual-ion batteries are high energy density, environment-
fridenly, low cost and switchable polarity [42, 48]. The further research and
commercialization of the related directions is very promising.

In 2004, single-layer graphite was obtained by mechanical exfoliation, named
graphene [6]. Since then, there has been an upsurge in the study of two-
dimensional (2D) materials dominated by graphene. In graphene, carbon atoms
are hybridized with $sp^{2}$ bond. Graphene is a Dirac semimetal with the
electron mobility of 2.5$\times$105 cm2$\cdot$V-1$\cdot$s-1 [49] and the
thermal conductivity of 3000-5000 W/mK [50]. Since the energy dispersion of
graphene near the Fermi energy is linear, the carriers exhibit the
characteristics of the Dirac fermion. Owing to the intriguing optical,
electrical and mechanical properties, graphene has important applications in
materials science, micro nano processing, energy, biomedicine and drug
delivery, and is considered a revolutionary material.

Graphene with high electron mobility is gradually used in transistors, which
can work at low voltage and room temperature [51]. Due to the large surface-
to-volume ratio and high electrical conductivity etc., the application of
graphene sensors has been widely studied in recent years, including
biosensors, diagnostics, DNA sensors and gas sensors [51, 52]. Graphene can
improve the photovoltaic and electrical properties of perovskite solar cells,
which has already been used as conductive electrode, carrier transporting
material and stabilizer material for perovskite solar cells [53]. Spin
transport in graphene was confirmed in 2007 [54]. Since then, the emergence of
graphene based heterostructures has been discussed [55]. It seems that low-
power all-spin-logic devices based on 2D van der Waals heterostructures will
be seen in a few years [55]. Both graphene oxide and reduced graphene oxide
contain active oxygen groups, which brings new properties to the materials.
They are widely applied as the main fillers of nanocomposites, and can be used
in electrochemical biosensors, solar cells, supercapacitors [56].

To realize the industrialization of graphene requires its rapid and economic
mass production. This is also the main research hotspot of graphene synthesis
technology in recent years. The synthesis methods of graphene are mainly
divided into two categories, namely constructive and destructive methods. The
former is mainly to delaminate graphite into graphene, including mechanical
exfoliation, arc discharge, liquid-phase exfoliation (LPE), oxidative
exfoliation-reduction, etc., while the latter uses atomic level precursors to
construct graphene, such as chemical vapor deposition (CVD), epitaxial growth,
substrate-free gas-phase synthesis, etc. [57]. Among them, oxidative
exfoliation-reduction, CVD and LPE have been shown to possess high industrial
application in the future [57].

Gao $et$ $al.$ used scanning tunneling microscope (STM) to manipulate
graphene, and realized that graphene can be folded and unfolded repeatedly in
any direction, with the maximum twist angle of 60∘ [58]. The folded single-
crystal graphene nanoislands can form a special chiral tube edge, which has
one-dimensional electronic characteristics similar to carbon nanotubes. The
atomic-level manipulation of graphene provides a new possibility to construct
quantum devices with carbon nanostructures.

Two recent experiments provide evidence for 2D electron fluid in graphene [59,
60], revealing the flow behavior of quantum fluid that cannot be observed in
water. On the basis of relevant research, we can design new thermoelectric
materials, which can convert electric current into thermal current (and vice
versa).

Simple single-layer graphene (SLG) has made many remarkable achievements. Due
to the interlayer interaction, multilayer graphene (MLG) stacks usually show
other novel properties different from SLG [61]. If the graphene layer in MLG
is not stacked in the form of “AB” in graphite, but with an angular twist, it
is called twisted graphene (tBLG). Due to the overlap of two Dirac cones, DOS
of tBLG shows the van Hove singularity [62]. When the rotation angle is small,
the electronic properties [63] and Raman spectra [64] of tBLG are different
from those of SLG, whereas when the rotation angle is large, they are
basically the same as SLG. In addition, the optical properties [65] and moiré
patterns [66] of superlattice structure are also dramatically dependent on the
rotation angle.

In 2018, unconventional superconductivity was discovered in magic-angle $\sim
1.1^{\circ}$ twisted bilayer graphene (TBG) superlattices [67]. This amazing
property was realized by appropriately charging the carrier density using an
external gate. The superconducting temperatures in the two devices were
measured to be 0.5 and 1.7 K, respectively. Since then, breakthrough research
of magic-angle TBG has emerged in an endless stream. There have been many
studies to explain the novel superconducting properties [68, 69, 70, 71, 72,
73, 74]. Recently, two different groups [75, 76] have independently
established Josephson junctions with magic-angle TBG, thus forming a
controllable superconducting valve. It is reported that Moiré superlattices
based on magic-angle TBG can be controlled by unconventional ferroelectricity
[77] and magnetism [78], which will bring new directions for the innovation of
new generation of quantum materials and electronic devices. Additionally, the
superconductivity of TBG was extended to a trilayer graphene structure [79].
The superconductivity of the sandwich magic-angle TBG is believed to be from
the strong electron couplings.

###  I.3 Graphdiyne

Graphyne is a 2D planar network structure formed by inserting one alkyne bond
between the benzene rings of graphene, which was first proposed in 1987 [80].
Another structure, named graphdiyne, can be obtained by inserting a
diacetylene bond (two alkyne bonds) between the benzene rings of graphene
[81], which was first synthesized experimentally in 2010 [82]. Graphdiyne has
the hexagonal structure with space group of P6/mmm, and its lattice constant
is 9.48 Å [83]. It is a narrow band gap semiconductor with a band gap of 0.46
eV [83]. At present, the main synthesis techniques of graphdiyne include dry
chemistry [84, 85, 86, 87] and wet chemistry [82, 88, 89, 90, 91]. As
graphdiyne is a 2D porous structure, it has a broad application in separation
of mixed gases. In 2011, Smith $et$ $al.$ theoretically studied the ability of
graphdiyne to separate H2 from CH4 and CO [92], and found that the near center
adsorption diffusion process of H2 has the lowest energy, which therefore
exhibits the best permeability, 104 times faster than that of porous graphene.
One year later, Buehler $et$ $al.$ studied this issue in depth through
molecular dynamics simulations [93], and found that these gases need a certain
pressure to release through graphdiyne, and at atmospheric pressure, only H2
can pass through.

At room temperature, the intrinsic electron mobility of graphdiyne is 105
cm${}^{2}\cdot$V${}^{-1}\cdot$s-1, while its hole mobility is an order of
magnitude lower than the electron mobility. The high carrier mobility,
suitable band gap and special structure make it have great potential in
photocatalysis. Wang $et$ $al.$ synthesized TiO2/graphdiyne composites by
hydrothermal method, and found that compared with TiO2/graphene composites,
TiO2/graphdiyne can better catalyze the degradation of methylene blue [94].
Then, by first-principles calculations, they showed that the TiO2/graphdiyne
composite has more positive charges than TiO2/graphene, easier to capture
electrons, and has stronger photocatalytic effect [95]. In addition, Venugopal
$et$ $al.$ designed and synthesized a ZnO/graphdiyne composite, using ZnO
instead of TiO2, with similar photocatalytic performance [96]. Similarly, the
nice performance of graphdiyne has also promoted the development of
photoelectrochemistry and electrolytic water decomposition catalysts [97, 98,
99, 100, 101, 102, 103].

Because graphdiyne combines the characteristics of 2D materials and porous
materials, it exhibits good electronic transport properties and can be used in
energy storage devices. Theoretical calculations show that the atomic density
of graphdiyne is reduced due to the existence of diacetylene bonds, and
therefore it can form LiC3 by the surface adsorption of lithium. The
theoretical specific capacity of graphdiyne is about 744 mAh$\cdot$g-1 [104,
105], which is twice that of graphite [106, 107]. Graphdiyne [108, 109] and
its derivatives, such as N-graphdiyne [110], Cl-graphdiyne [111], H-graphdiyne
[112], B-graphdiyne [113] participated in the preparation of lithium battery,
which has the characteristics of large capacity, good cycle stability and high
Coulomb efficiency. At the same time, graphdiyne can also be used as a good
material to assemble solar cells [114, 115, 116], which can greatly improve
the efficiency of solar cells. In addition, graphdiyne has great potential in
hydrophobic materials [117, 118] and temperature sensors [119].

###  I.4 Carbon nanotubes

Carbon nanotubes (CNTs) are 1D systems with carbon atoms bonding each other by
entirely $sp^{2}$ hybridization. As early as 1952, Radushkevich and
Lukianovich published a clear image showing carbon nanotubes with a diameter
of 50 nm [120, 121]. The rediscovery of CNTs in 1991 by lijima [122] set off
an upsurge in the study of CNTs [123, 124]. The CNTs include single-walled
(SWCNTs) and multi-walled carbon nanotubes (MWCNTs). MWCNTs contain double-
walled (DWCNTs) and triple-walled carbon nanotubes (TWCNTs) [125] and so on.
The longest CNTs are known to be more than 0.5 m [126], and the shortest are
cycloparaphenylene [127]. The structure of CNTs is determined by the diameter
and chiral angle (helicity), which are usually described by two Hamada indexes
(n, m). The synthesis methods of CNTs include arc discharge [128], laser
ablation [129], microwave heating [130], hydrocarbons CVD [131] and so on.

The diversity of diameters and chiral angles leads to a wide variety of
structures, accompanied by a wealth of properties, such as electronic, thermal
and mechanical properties [132]. The density of CNTs was observed to be 1.6
g$\cdot$cm-3 [133]. The Young’s modulus of SWCNTs is 1-5 TPa, while that of
MWCNTs is 0.2-0.95 TPa. CNTs can behave as both metal and semiconductor, which
is determined by Hamada indexe (n, m). For SWCNTs, if n-m can be divided by 3,
then CNTs are considered as a metal, otherwise, CNTs act as semiconductors.
Metallic SWCNTs can be used to study the single-electron charging, Luttinger
liquid, quantum interference, etc [134, 135], whereas semiconducting CNTs have
been studied in transistors, sensory devices, etc. Superconductivity was
observed in SWCNTs with diameters of 1.4 and 0.4 nm at about 0.55 K [136] and
15 K [137], respectively. Moreover, the phonon propagation in CNTs is
anisotropic, and the thermal conductivity of a single MWCNT along the tube
direction is about 3000 W/mK at room temperature, which is much larger than
that of diamond and graphite ($\sim$2000 W/mK). Therefore, CNTs are expected
to be a good thermal conductor along the tube direction, while they are
thermal insulating transverse to the tube axis. The thermal stability of CNTs
is maintained at 2800 ∘C in vacuum and 750 ∘C in air.

CNTs have a wide range of applications. MWCNTs as electrically conductive
components in polymer composites are the first to realize commercial
application. CNTs can also be used as supercapacitor electrodes, because the
capacitance of SWCNT and MWCNT electrodes are 180 and 102 F/g, respectively
[138]. In addition, CNTs can also be used for hydrogen storage, field emission
devices [139, 140], sensors and probes [141], etc. N-doped CNTs have good
stability and can be implemented as efficient metal-free catalysts for
nitrobenzene hydrogenation [142, 143].

###  I.5 Carbyne

Carbon chain is a 1D carbon allotrope formed by multiple carbon atoms arranged
in a straight line, which has been studied since 1885 [144]. According to the
chemical bonds connected, it can be divided into two types: 1) carbon atoms
are alternately connected by single and triple bonds, i.e. polyyne [145]; 2)
carbon atoms are continuously arranged by double bond, i.e. cumulene [146]. In
general, with the increase of the length of these carbon chains, the stability
will become worse, and it is easy to transform into other structures. An ideal
1D carbon chain is infinitely long, and its properties do not change with the
length, so it is called carbyne. Carbyne was first detected in 1961 [147], and
was reproduced in 1978 [148, 149]. In 2016, Shi $et$ $al.$ synthesized a
stable carbon chain composed of more than 6000 carbon atoms by using the
carbon nanotube template method [150]. In 2018, Heeg $et$ $al.$ studied this
ultra-long carbon chain by near-field Raman spectroscopy, and confirmed that
the Raman spectrum peak is no longer affected by the length when the length
exceeds 30 nm, thus experimentally confirming the existence of carbyne [151].
At present, how to control the synthesis of carbon chain with different
length, as well as the synthesis of stable carbyne, is still a concern.

Theoretical calculations show that the mechanical strength of carbyne is
higher than that of diamond [152], which is expected to be used in mechanical
reinforcement materials. Meanwhile, carbon fiber compounds composed of carbyne
may be stronger than existing carbon fiber compounds, and are expected to make
progress in aerospace and many fields [153]. In addition, the band gap of
carbyne can be adjusted with length and external strain, and the minimum band
gap is 1.8 eV. This means that carbyne is a typical semiconductor, which has
great potential in the synthesis of semiconductor materials with a single atom
width [154]. The nonlinear optical properties of carbyne also contribute to
the fabrication of frequency conversion devices [155]. It also has potential
in energy storage [156, 157] and hydrogen storage [158].

###  I.6 Fullerenes

Fullerene is another kind of carbon allotrope, whose molecules are connected
by carbon atoms through double or single bonds to form a closed curved
surface. It contains a series of closed cage molecules, such as C60, C70, C76,
C78, C80, C84 and so on. A common form of fullerene molecule C60, composed of
60 carbon atoms in a football shape, is known as C60 or buckminsterfullerene,
which was first discovered in 1985 by Harold Kroto, Richard E. Smalley and
Robert F. Curl [3]. Highly symmetrical cage C60 is a truncated icosahedron
with 60 carbon atoms forming 20 hexagons and 12 pentagons. There are 6
fivefold axes, 10 threefold axes and 60 twofold axes in C60, where two ring
bonds with a bond length of 6:6 between two hexagons and 6:5 between a hexagon
and a pentagon coexist. The van der Waals diameter of C60 molecule is about
1.1 nm [159]. Fullerenes can be obtained based on the thermal decomposition of
graphite firstly, and then extracted from the thermal decomposition products
by solvents and sorbents.

Fullerenes exhibit unique physical and chemical properties. Crystalline
fullerenes are semiconductors with a band gap of 1.2$\sim$1.9 eV and good
photoconductivity. When they are exposed to the visible light, the electrical
resistance decreases, which can be used to make photodetectors and
optoelectronic devices. However, C60 crystal can be tuned to a metal by doping
alkali metal atoms. The C60 compounds doped with potassium or Rubidium alkali
metal atoms (K3C60, Rb3C60) are superconductors with the transition
temperature of 18-40 K [160]. The ferromagnetic materials can be obtained by
adding platinum group to C60. Fullerenes possess high electronegativity and
act as strong oxidants in chemical process. Fullerenes have poor solubility,
and they are soluble in some organic solvents (such as carbon disulfide,
toluene, benzene, tetrachloromethane, decane, hexane, pentane) and insoluble
in polar solvents. Fullerenes can absorb, bind or react with hydrogen [161],
making them useful in hydrogen storage [162], oxygen production [163],
polymerization [164], free radical reaction [165] and so on.

The application of fullerenes is mainly focused on medicine and pharmacology
[166], and the idea of using the combination of fullerenes’ endohedral
compounds and radioisotopes to develop anticancer drugs has attracted
extensive attention. Fullerenes are also predicted as the basis of the
battery, because compared with lithium battery, they have the advantage of
high efficiency, light weight and environmental protection. Fullerenes also
show nonlinear optical properties due to their solubility in nonpolar
solvents, which provide an opportunity for fullerenes to be used as optical
closures-limiters of laser radiation intensity. Because the optical absorption
spectrum of fullerenes film is between 180 and 680 nm, the application of
fullerenes film in solar cells is also possible.

###  I.7 Amorphous carbon

In contrast to the crystalline diamond and graphite, amorphous carbon contains
a certain degree of disorder, in which the carbon atoms does not form the
crystalline structure and the bonds between carbon atoms are irregular.
$sp^{3}$, $sp^{2}$ and even $sp$ hybrid bonds can exist in amorphous carbon,
where the dangling $\pi$-bond could be hydrogen-passivated to stabilize it. In
addition, amorphous carbon could be turned to other forms of carbon under
certain conditions. For examples, microdiamonds can be grown by irradiating
the amorphous film with nanosecond lasers at room temperature [167]; high
quality graphene can be obtained by thermal annealing of amorphous carbon
[168]; and amorphous carbon powders could be converted to the crystalline
graphite by microwave heating [169].

Among the amorphous carbon, diamond-like carbon (DLC) is a metastable form,
where most of the carbon atoms in DLC are connected with $sp^{3}$ hybrid bond,
and a small amount of $sp^{2}$ bond. DLC is a metastable anisotropic
semiconductor. There are also cubic and hexagonal DLC, which inherit the
hardness of diamond and are easy to deform. DLC contains both carbon and
hydrogen atoms, which can be prepared by physical vapor deposition, CVD and
other methods [170]. DLC films with different smoothness, $sp^{3}$ bonding
ratio, hydrogen content and deposition amount can be obtained by different
methods.

At present, the study of DLC films mainly focuses on the wear resistance,
antifriction, antibacterial and corrosion resistance of the coating, as well
as improving the electrochemical performance without changing the overall
performance of the substrate. They are environment-friendly materials and can
be used in mechanical wear-resistant parts, clocks, aerospace and medical
materials [171, 172, 173, 174].

There are four main forms of DLC: hydrogen-free amorphous carbon film (a-C),
tetrahedral hydrogen-free amorphous carbon film (ta-C), hydrogenated amorphous
carbon film (a-C:H) and tetrahedral hydrogenated amorphous carbon film
(ta-C:H). In the ternary diagram of $sp^{3}$ covalent carbon, $sp^{2}$
covalent carbon and hydrogen, they belong to different regions [175]. Among
them, ta-C has a large number of carbon atoms bonded into tetrahedron by
$sp^{3}$ bond, which is similar to diamond in properties and can be used in
electronic devices [176, 177, 178]. Recently, Ohtake $et$ $al.$ redefined the
DLC region on ternary diagram by analyzing the $sp^{3}$ ratio, hydrogen
content, and other properties of 74 kinds of amorphous carbon films [179].

Undoped DLC coating has some challenges in thickness, adhesion and stability,
which can be solved by preparing DLC containing metal nanoparticles [180].
DLC-based metal nanocomposites can also be applied to novel biosensors that
combine the resonant response of waveguide structures [180]. Recently, in view
of the well-known bactericidal effect of silver in medicine, Ag-doped DLC
coating was synthesized by hot electron vacuum arc deposition, which has good
corrosion resistance, electrochemical stability and antibacterial effect
against S. aureus [181].

The $sp^{3}$ bond in DLC is easy to break at high temperature, which leads to
graphitization. Graphitization is usually accompanied by thermal degradation
of DLC. Doping silicon, boron, nitrogen, etc. can significantly improve the
high-temperature tribological properties of DLC films [182]. A recent report
showed that in silicon doped DLC, Si atoms often replace $sp^{2}$ hybrid
carbon atoms, which leads to a significant increase in the proportion of
$sp^{3}$ chemical bond, thus inhibiting graphitization [183]. Si doping can
improve the nano-wear resistance, hardness and bonding strength. Another
report pointed out that the Si-C bond is longer than the C-C bond, which
enables the Si-doped DLC film to accommodate high disorder and reduces the
frequency of occurrence of highly strained C-C bond [184].

Figure 2: Overview of exotic properties and potential applications of
T-carbon. Parts of the figure are reprinted (adapted) with permission from ref
[185], Copyright (2019) Royal Society of Chemistry; ref [186], Copyright
(2020) American Physical Society; ref [187], Copyright (2014) Elsevier; ref
[188], Copyright (2018) Elsevier; ref [189], Copyright (2020) Elsevier;
https://www.sigmaaldrich.com/technical-documents/articles/technology-
spotlights; NASA&ESA.

###  I.8 Carbon dots

Carbon Dots (CDs), as a well-known 0D carbon nano allotrope that is mostly
smaller than 10 nm, have attracted wide attention in recent years. It consists
of nanoscale carbon cores and passivated chemical functional groups shell,
which is similar to a soft corona-like shell [190]. According to the core
structure, CDs can be roughly divided into three main categories: carbon
nanodots (CNDs), carbonized polymer dots (CPDs), graphene quantum dots (GQDs).
In GQDs, the carbon core possesses one or few graphene sheets with diameters
of 2-20 nm [191]. CNDs usually have amorphous inner structures, and carbon
quantum dots (CQDs) mainly composed of inner graphitic crystalline structures
are also regarded as a special case of CNDs [192]. CPDs exhibit a
polymer/carbon hybrid structure, including rich short polymer chains and
functional groups on the surface [193]. Although CDs have been found for only
15 years [194], many different approaches have been implemented in the
synthesis of CDs [190, 195]. For all methods, the advantages of nanoscale
carbon particles and sufficient passivation of the surface should be ensured.
The carbon cores and surface passivated functional groups make CDs have
versatile properties.

The most attractive feature of CDs is its bright and colorful fluorescence
emission ranging from ultraviolet (UV) to near-infrared. The main
photoluminescence mechanisms include: (1) quantum confinement effect (QCE);
(2) surface and defect states; (3) molecular fluorophores (MFs); (4)
crosslink-enhanced emission (CEE) effect; (5) thermally activated delayed
fluorescence (TADF) and (6) phosphorescence [192, 196]. Any CDs system
contains more than one photoluminescence mechanism, which makes the
photoluminescence spectrum and quantum yield of CDs very complex. For GQDs
with a non-zero tunable band gap, its photoluminescence mainly depends on the
QCE and $\pi$-plasmons in the core states. In addition, the effective
intersystem crossing (ISC) in the core states allows GQDs to display TADF and
phosphorescence. For CNDs, the main contribution for photoluminescence is
surface states due to doping effect and MFs. The heteroatoms in CNDs can be
controlled by experiments so that the photoluminescence characteristics of
functional groups change dramatically. On the other hand, the CEE effect plays
a leading role in the photoluminescence of CPDs. It should be noted that
molecular photoluminescence is determined only by the fluorescent molecules in
CNDs without spatial limitation. However, the potential photoluminescence
centers in CPDs are spatially limited [192]. Another interesting feature of
CDs is its unique photoinduced redox characteristics as both excellent
electron donors and acceptors. This can be verified by the experimental high-
efficient fluorescence quenching of electron donor and acceptor molecules as
quenchers [190].

In addition to the above photoluminescence and redox characteristics, CDs also
have the characteristics of chemical inertia, high specific surface area,
nontoxicity, low cost and biocompatibility. Using these advantages, CDs are
widely used in the fields of photocatalysis, sensors, sterilization,
optoelectronics, food and safety. For example, a sensitive and effective
wireless bacteria detection and ablation sensing system based on fluorescent
CDs (FCD) and CsWO3 nanohybrid has recently been designed. The surface-
coatable CsWO3–FCD nanohybrid shows excellent solubility, photothermal effect,
and sensing ability [197]. Another example is that a new type of CDs was
synthesized last year, which integrates FeCo2O4 nanoflowers supported on Ni
foam. Such new metal oxide-integrated CDs show excellent performance in
electrochemical decomposition of water [198].

###  I.9 T-carbon

T-carbon was theoretically predicted in 2011 [20] and first fabricated in
experiment in 2017 [199]. T-carbon is a 3D carbon allotrope, which was
obtained by substituting each carbon atom in diamond with a regular carbon
tetrahedron composed of four carbon atoms with $sp^{3}$ hybridization.
T-carbon has a porous structure and is a semiconductor with a direct band gap
of about 5 eV. Due to its exotic properties, T-carbon has great applications
such as hydrogen storage, photocatalysis, superconductivity, etc. [200, 185,
186] (Fig. 2).

Figure 3: The structure of T-carbon. (a) The cubic crystalline structure of
T-carbon obtained by replacing each carbon atom in diamond with a carbon
tetrahedron. (b)–(d) Views from [100], [110] and [111] directions of T-carbon,
respectively. Copyright (2011) American Physical Society. Reprinted (adapted)
with permission from ref [20].
Figure 4: Electronic band structures and corresponding DOS for T-carbon.
Copyright (2011) American Physical Society. Reprinted (adapted) with
permission from ref [20].

T-carbon has the same space group of $Fd\bar{3}m$ as diamond, as shown in Fig.
3, and its primitive cell contains two tetrahedral units with eight carbon
atoms. The three lattice vectors are $a_{1}=(1,1,0)l/2$, $a_{2}=(0,1,1)l/2$
and $a_{3}=(1,0,1)l/2$ with $l=7.52$ Å, and the carbon atoms occupy the
Wyckoff position $32e(0.0706,0.0706,0)$. The comparison of T-carbon with
several typical carbon allotropes are listed in Table 1. It is noted that
T-carbon has much lower equilibrium density (1.5 g/cm3) and bulk modulus (169
GPa) among 3D carbon materials. The electronic band structure and density of
states (DOS) of T-carbon are plotted in Fig. 4. One observes that T-carbon is
a semiconductor with a direct band gap by different calculation methods [20,
200, 201]. The projected DOS shows that 2$p_{x}$, 2$p_{y}$ and 2$p_{z}$
orbitals make more contribution to the valence band than 2$s$ orbitals,
indicating the anisotropic $sp^{3}$ bonds in T-carbon.

In this review, we will first discuss recent experiments on T-carbon. Then, we
will present the intriguing properties of T-carbon, including the mechanical,
electronic, optical, topological and superconducting properties, and its
potential applications in solar cells, photocatalysis, interstellar medium,
etc. Finally, we will outline future prospects and outlook on T-carbon.

##  II Experiments on T-carbon

Recently, T-carbon has been successfully synthesized in two independent
experiments by picosecond laser irradiation [199] and plasma enhanced chemical
vapor deposition [202], respectively.

###  II.1 Synthesis of T-carbon nanowires by picosecond laser irradiation

In 2017, T-carbon nanowires (NWs) have been successfully synthesized by
irradiating multi-walled carbon nanotubes (MWCNTs) dissolved in methanol
solution with picosecond laser at a specific frequency [199]. The first
synthesis experiment of T-carbon lays a basis for further investigation and
application of T-carbon.

Figure 5: The experimental layout of T-carbon nanowires from a pseudo-
topotactic conversion of multi-walled carbon nanotubes under picosecond laser
irradiation. Reproduction from http://gr.xjtu.edu.cn/web/jinying-zhang
publications.
Figure 6: HRTEM images of (a) shortened MWCNTs with FFT pattern and the
structure of pristine MWCNTs inserted in the upper right corner (scale bar =
20 nm) and (b) carbon nanowire produced from picosecond laser irradiation with
the low magnification in the inset (scale bar = 5 nm). FFT patterns of (c) the
nanowire corresponding to (b) (scale bar = 5 nm-1), and (d) another nanowire
(scale bar = 2 nm-1 ). Reprinted (adapted) with permission from ref [199].

The MWCNTs used in the experiment were obtained by CVD. 0.5 mg of MWCNTs were
dispersed in absolute methanol (40 ml), and then the suspension was irradiated
for an hour by the Q-switched laser with the wavelength of 532 nm, pulse
duration of 10 ps, repetition rate of 1000 Hz and pulse power of 75 mW. After
irradiation, the suspension became transparent and gas was produced during the
reaction. The experimental layout of T-carbon nanowires is presented in Fig.
5. Images of the transparent suspension after reaction obtained by the high-
resolution transmission electron microscopy (HRTEM) show that there were new
NWs structures appearing as presented in Figs. 6(a) and (b). Besides, the
suspension also contains unreacted MWCNTs and some amorphous phases. A fast
Fourier transform (FFT) pattern calculated from HRTEM images shows that the
new NWs had a cubic lattice as shown in Figs. 6(c) and (d). The results of
electronic energy loss spectroscopy (EELS) show that the ratio of $sp^{3}$
components in the suspension is dramatically increased after reaction compared
with the suspension before laser irradiation, and only a small amount of
carbon atoms with $sp^{2}$ hybrid bonds (16$\pm$4%) left, which correspond to
amorphous structures. To verify that there were only $sp^{3}$ hybrid bonds in
the NWs, a comparative experiment was conducted, where the methanol solvent
without MWCNTs was irradiated under the same conditions, and the EELS results
show that the irradiated methanol solution also had a lot of the same
amorphous structures. Moreover, the FFT patterns of NWs with different tilting
angles were consistent with the diffraction patterns of different crystal
planes of T-carbon, and the measured intra-planar and inter-planar angles of
NWs were consistent with the calculated results of T-carbon, indicating that
the new NWs were T-carbon. The obtained T-carbon NWs have the lattice constant
of 7.80 Å, which is in good agreement with the calculated value of 7.52 Å. The
observed ultraviolet (UV) absorption spectra and photoluminescence spectra of
the suspension after irradiation exhibited an obvious absorption peak, which
meets well with the calculated UV and photoluminescence spectra of T-carbon
(see Fig. 18). These results revealed that the T-carbon nanowires were really
generated [199].

The average diameter of T-carbon NWs was 11.8$\pm$2.8 nm, which was consistent
with the statistical average of MWCNTs (11.7$\pm$2.2 nm), indicating that
T-carbon NWs came from the pseudo-topotactic conversion of MWCNTs, which
belongs to the first-order phase transition [199].

###  II.2 Preparation of T-carbon by plasma enhanced chemical vapor deposition

Figure 7: The schematic diagrams of T-carbon synthesis by plasma enhanced CVD.
Copyright (2020) Elsevier. Reprinted (adapted) with permission from ref [202].

Plasma enhanced CVD (PECVD) is a mature technology that is widely used in
material growth. Kai Xu $et$ $al$. used PECVD to grow T-carbon samples on the
substrate of polycrystalline cubic diamond and single-crystalline cubic
diamond under high temperature and high pressure, and obtained T-carbon with
two different morphologies as shown in Fig. 7 [202]. For the substrate of the
polycrystalline cubic diamond, the plasma feed gas was the mixture of
Ar/H2/CH4 with 3 standard liters per minute (slm) of Ar, 3 slm of H2 and 30
standard cubic centimeters per minute (sccm) of CH4, and the plasma was
lighted and sustained by direct current arcjet (DC arc). The pressure in the
reaction chamber was kept at 8 kPa, and the substrate was heated to around
1000 ℃ for 4 h. For the substrate of the single-crystalline cubic diamond, the
plasma feed gas was the mixture of Ar/H2/CH4 with 4 slm of Ar, 1.2 slm of H2
and 60 sccm of CH4 and the plasma was lighted and sustained by the dual radio
frequency inductively coupled plasma (RF ICP), where the pressure was also
kept at 8 kPa, and the substrate temperature was maintained at 890 ℃ for 200
h. The obtained product in the former experiment looks like the rods extended
from the cauliflower-like diamond and connected with each other, forming kink
joints, and the extended length of the sticking carbon rods can reach several
microns. The obtained product in the latter experiment is similar to the rod
bunch connected together. The thinnest stick-like sample thinned by the focus
ion beam method was about 80 nm.

As the sample produced by the above two experiments was small, the XRD spectra
were measured based on the newly generated phases together with their
surrounding substrate. By comparing the measured XRD pattern of T-carbon with
the simulated result by DFT calculations, it was found that both two samples
have some peaks which meet well with the calculated result. For example, two
peaks of $2\theta=39.41^{\circ}$ and 61.34∘ in the XRD spectrum for the sample
grown on the single crystalline diamond substrate correspond to the (222) and
(115) planes of T-carbon, respectively. Among all known carbon allotropes only
T-carbon has such XRD peaks, indicating that the new rod-like phase was
T-carbon. According to Bragg’s law, the lattice constant was then deduced to
be 7.89 Å and 7.85 Å, respectively, which are in good agreement with the
calculated value of 7.52 Å [20] and the value (7.8 Å) in the previous
experiment [199] for T-carbon. The weight percentage of T-carbon took up 1.46%
in the total mixed sample in terms of the relative ratio of XRD diffraction
peaks. The results of selected area diffraction of TEM also show that there
was T-carbon in the mixed sample. Moreover, the EELS spectrum showed that
there were only $sp^{3}$ bonds in the sample. Although the ratio of T-carbon
in the sample was very small, many peaks of the Raman spectrum corresponding
to T-carbon were still obtained by comparing with the calculated results. In
addition, the measured and simulated Fourier transform infrared spectroscopy
(FT-IR) of the sample also show obvious consistent peaks (see Fig. 18).

In the above experiments, the temperature of the plasma was much higher than
that of the substrate, so the hotter carbon atoms moved upward and the colder
moved downward. When colder atoms moved downward to deposit onto the
substrate, they would be subject to the upward impact from the hotter atoms,
which is equivalent to experiencing a negative pressure environment. The
negative pressure environment was shown to be more suitable for the synthesis
of T-carbon [203]. In addition, carbon atoms are prone to form T-carbon phase
on the diamond substrate because T-carbon has a similar lattice structure and
chemical bond with diamond. The experimental preparation of T-carbon with
PECVD shows that the massive production of T-carbon is feasible.

The realization of T-carbon in experiments not only added a new member to the
carbon family, but also offered great opportunities for the further study on
T-carbon. In the near future, the massive production and practical
applications of T-carbon are highly expected.

##  III Properties of T-carbon

###  III.1 Chemical bonding

Figure 8: Schematic plots of (a) $sp$, $sp^{2}$ and $sp^{3}$ hybridization and
(b) $\sigma$ and $\pi$ bonds.

As we mentioned above, carbon atoms can form $sp$, $sp^{2}$, and $sp^{3}$
hybrid chemical bonds, which consist of $\sigma$ and $\pi$ bonds that differ
in the overlapping of atomic orbitals as shown in Fig. 8. Covalent bonds are
formed by the overlapping of atomic orbitals. $\sigma$ bonds are a result of
the head-to-head overlapping of atomic orbitals whereas $\pi$ bonds are formed
by the lateral overlapping of two atomic orbitals. The $\sigma$ bonds are
generally stronger than the $\pi$ bonds, owing to the significantly lower
degree of overlapping of $\pi$ bonds. In general, single bonds ($sp^{3}$
hybridization) consist of $\sigma$ bonds, whereas a double bond ($sp^{2}$
hybridization) is made up one $\sigma$ and one $\pi$ bonds, and a triple bond
($sp$ hybridization) is composed of two $\pi$ and one $\sigma$ bonds. The $s$
character, which means the percentage of the $s$ orbital initially involved in
the hybridization process, is related to the bond length and strength, because
the $s$ orbital is smaller than the $p$ orbitals, and it has stronger
interaction between the electrons and nuclei. Therefore, C-C bond in $sp$
hybridization is shortest and strongest, and that in $sp^{3}$ is the longest
and weakest. Different bonding methods or ratios will lead to completely
different properties, resulting in the rich diversity of carbon materials.

Figure 9: The structural motif (top), local DOS (middle) and MDOS for
$sp^{3}$-mixed levels (bottom) of diamond and T-carbon. Copyright (2017)
Elsevier. Reprinted (adapted) with permission from ref [204].
Figure 10: Projected COHPs of diamond and T-carbon. From left to right: C-C
bond in diamond, C-C bond in the tetrahedron of T-carbon, and C-C bond between
the tetrahedrons of T-carbon. Copyright (2017) Elsevier. Reprinted (adapted)
with permission from ref [204].

T-carbon possesses $T_{d}$ symmetry around the center of the tetrahedral units
consisting of four carbon atoms (C4 units). The calculated total energies of
graphite, diamond and T-carbon show that the energy of T-carbon is 125.6
kJ/mol higher than that of graphite, and 113.2 kJ/mol higher than that of
diamond as a result of different chemical bonds [204]. However, due to the
existence of dynamic barriers, T-carbon is still a stable structure and in the
metastable state. For diamond, the C-C bond has a bond length of 1.547 Å with
the bond angle of 109.5°, which is formed by $sp^{3}$ hybridization of the $s$
and $p$ orbitals. For T-carbon, the bond length and bond angle in the C4 units
are 1.501 Å and 60°, respectively, while between two adjacent C4 units they
are 1.416 Å and 144.74°, respectively. To further investigate the
hybridization of orbitals in T-carbon, the weighted mixing of local density of
states (LDOS) and mixing density of states (MDOS) were performed with the
$sp^{3}$-orbital mixing technique [205] as shown in Fig. 9. Compared to
diamond, the orbital mixing of $s$\- and $p$-level of T-carbon is much lower.
The projected Crystal Orbital Hamilton Population (pCOHP) uses the Hamiltonian
matrix element between two atoms to effectively weight DOS by the bond energy,
and the pCOHP results for diamond and T-carbon are plotted in Fig. 10.
Generally, the energy of the bonding state is less than zero (energy
lowering), while that of antibonding state is above zero compared to the non-
bonding state of zero energy. The shorter bond between two C4 units in
T-carbon is 49% stronger than the longer one within C4 units, and has largely
$s$-character in the main bonding peak range from -17 to -15 eV (see Fig. 10),
while the higher $p$ orbitals do not contribute much to stabilization in
contrast there is a leftward anti-bonding region area -5 to -2 eV. This anti-
bonding characteristic is due to the obvious deviation of the bond angle from
its parent allotrope, resulting in the systematic strain and the hybrid
efficiency reduction of the $s$ and $p$ orbitals. Overall, T-carbon is
different from traditional carbon allotropes in the way of bonding. On the one
hand, the unique bonding brings a variety of good properties for T-carbon, on
the other hand, it makes people have a deeper understanding of the law of
carbon bonding, which plays a great instructive role in the search for new
carbon allotropes.

###  III.2 Mechanical properties

The unique lattice structure and bond nature of T-carbon imply that it has
excellent mechanical properties, which have been studied in several works.

Figure 11: (a) The average atomic potential as a function of strain in [100],
[110] and [111] directions, and (b) corresponding evolution of the ratio
($f_{sp^{2}}$) of $sp^{2}$ hybrid bond for T-carbon NWs. Stress-strain curves
for T-carbon NWs along (c) [110] and (d) [100] directions. Copyright (2018)
Elsevier. Reprinted (adapted) with permission from ref [188].

Bai $et$ $al.$ studied the structures and mechanical properties of T-carbon
NWs under tensile strains in three different directions [100], [110] and [111]
at the temperature of 300 K by molecular dynamics simulation [188]. The
results show that T-carbon exhibits excellent ductility with high failure
strain and mechanical anisotropy in Fig. 11. It is noted that the ratio of
$sp^{2}$ hybrid bond to all chemical bonds ($f_{sp^{2}}$) grows rapidly after
a certain strain and can reach up to 80$\%$ with the increase of strain in
[111] and [110] directions as shown in Fig. 11(a), indicating the appearance
of a global graphitization, while $f_{sp^{2}}$ slightly increases until the
failure of T-carbon NWs with the increase of strain in [100] direction. The
variation of average atomic potential energy ($\phi$) under tensile strains
further reflects the evolution of global graphitization [Fig. 11(b)]. The
smaller $\phi$ is, the more stable the system is. With the increase of strain
in [110] and [111] directions, the $\phi$ decreases first due to the global
graphitization, indicating that the carbon networks become stable after the
global graphitization, and then dramatically increases, corresponding to the
deformation of the $sp^{2}$ network. Figure 11(c) shows that the slope of
stress-strain ($\sigma-\varepsilon$) in the [110] direction is large before
and after graphitization (stage I and stage III), while during the
graphitization, the stress increases slowly and fluctuates (stage II). The
fluctuation of stress during the graphitization enables T-carbon NWs to
withstand a larger strain. However, stress rapidly increases with the increase
of strain in the [100] direction, indicating that no global graphitization
occurs in this case.

Figure 12: (a) The tensile stress-strain curves for nanocrystalline T-carbon
with an average grain size of 5.4 nm under different strain rates. (b) The
stress-strain curves of nanocrystalline T-carbon with different grain sizes at
a strain rate of 1$\times$10${}^{8}/s$. Three-dimensional contour plots of (c)
Young’s modulus and (d) Poisson’s ratio for T-carbon. Copyright (2019)
Elsevier. Reprinted (adapted) with permission from ref [206]. Copyright (2019)
Elsevier. Reprinted (a) and (b) with permission from ref [206]. Reprinted (c)
and (d) with permission from ref [207].

The process of global graphitization can be also viewed from the
microstructural evolution of T-carbon. With the increase of strain in [110]
and [111] directions, $sp^{2}$ clusters appear at the boundary and then
propagate to the entire NWs, leading to the the evolution of NWs from $sp^{3}$
to $sp^{2}$ network. The change in the number of chemical bonds during the
graphitization can be implemented to analyze the fracture of chemical bonds.
It was found that when applying strain to the above three directions, the
number of chemical bonds between the C4 units was remained, while the total
number of chemical bonds in NWs is slightly increased, which shows that the
internal chemical bonds within the C4 units were broken during the global
graphitization. This graphitization process in T-carbon is different from
previously reported carbon materials, because the $sp^{2}$ clusters in other
carbon materials like diamond [208], diamond-like carbon [209] tend to be
localized and do not spread to the entire network, resulting in brittle
fracture. Moreover, for diamond and diamond-like carbon, their failure stress
is larger than that of T-carbon NWs, because the $sp^{2}$ dominated network
cannot withstand much high stress and with a few strains the $sp^{2}$ clusters
of T-carbon NWs will be cracked and fractured. The unique global
graphitization process for T-carbon makes it the excellent ductility, whereas
the failure stress of T-carbon is much smaller than that of diamond and
diamond-like carbon, because T-carbon has a lower density and does not have
the dense structure to withstand great stress.

The above study was based on the single-crystal T-carbon NWs. The
nanocrystalline T-carbon samples composed of a large number of single-crystal
grains with random directions, sizes and shapes were also investigated [206],
and the relation between stress and strain under different strain rates is
shown in Fig. 12(a), from which the Young’s modulus, failure stress and
failure strain can be obtained. The authors found that with the increase of
strain rate, the failure stress, failure strain and Young’s modulus of
nanocrystalline T-carbon are increased. Compared with nanocrystalline diamond,
the failure strain of nanocrystalline T-carbon was much higher, while the
failure stress and Young’s modulus were lower than that of diamond. These
results are similar to that of single-crystalline T-carbon NWs we mentioned
above [188]. Compared to nanocrystalline materials including the carbon
materials and metal materials, the strain rate sensitivity (defined as
$\partial\sigma/\partial\dot{\varepsilon}$) of T-carbon NWs are much higher.

The stress-strain relation for different grain sizes at a fixed strain rate
was also investigated as shown in Fig. 12(b). It was shown that with the
increase of grain size, the failure strain is decreased, while the failure
stress and Young’s modulus have no obvious changes.

The atoms inside the grains mainly contain $sp^{3}$ bonds, while the boundary
of grains consists of the atoms with $s$, $sp$ and $sp^{2}$ bonding states,
corresponding to the carbon atoms connected to one, two and three adjacent
atoms within the 0.2 nm radius, respectively. With the increase of tensile
strain, the number of $s$, $sp$, $sp^{2}$ bonds increases, corresponding to
the increase of the boundary thickness. Because of more disordered bonds at
the grain boundaries and the relatively concentrated stress at their junction
interface, the cracks always initiate and propagate from the grain boundaries,
resulting in the brittle fracture of the nanocrystalline T-carbon.

The mechanical properties including Young’s modulus, shear modulus and
Poisson’s ratio were also investigated in Ref. [207]. 3D diagrams of Young’s
modulus and Poisson’s ratio for T-carbon are shown in Figs. 12(c) and (d). The
Young’s modulus and Poisson’s ratio show the elastic anisotropy of T-carbon,
which is higher than that of diamond, but lower than that of Y-carbon and TY-
carbon, because of their different arrangements of atoms or types of chemical
bonds. The results of the ratio of the maximum and minimum of Young’s modulus
reveal that the elastic anisotropy of the above carbon allotropes in each
plane is arranged in the following order: (111) $<$ (001) = (010) = (100) $<$
(011) = (110) = (101) plane, because they have similar cubic symmetry and
their respective three lattice constants are the same.

T-carbon was calculated to exhibit a Vickers hardness of 61GPa by Sheng $et$
$al.$ [20]. However, one other study [35] reported that the hardness of
T-carbon is less than 10 GPa, and the reasons are listed in the following: (1)
The empirical formula for calculating the hardness of T-carbon used by Sheng
$et$ $al.$ proposed by Gao $et$ $al.$ [210] as well as the commonly used
Simunek and Vackar’s (SV) method [211] is suitable for the materials with
uniform bond distribution. As we mentioned above that the $sp^{3}$ bond in
T-carbon is anisotropic, the Vickers hardness obtained by the above two
methods may be not accurate. (2) Considering the high density of C-C covalent
bonds in carbon tetrahedrons of T-carbon, the C4 unit can be regarded as a
whole, called as an artificial super-atom. For an artificial super-atom, the
four $sp^{3}$ chemical bonds are uniform. By applying this concept to the
above two methods, the hardness of T-carbon was estimated to be 8.2 and 7.7
GPa, respectively; (3) The formula of ideal elastic hardness given by Sneddon
[212] was used to estimate the approximate range of hardness, which is defined
as $H_{id}=\dfrac{E_{cot\phi}}{2(1-\nu^{2})}$, where $E$ is Young’s modulus,
$\nu$ is Poisson’s ratio, and cot$\phi\approx$ 0.5 for pyramid indentation and
the real hardness would be (0.01-0.2)$H_{id}$ at high strains. By substituting
the Young’s modulus value $E$ = 185 GPa provided by Sheng $et$ $al.$ into
Sneddon’s formula, the hardness value of T-carbon was estimated in the range
of 0.5 to 10 GPa; (4) J. S. Tse showed that the shear stress of 3 GPa can be
used as the upper limit of hardness of T-carbon [213]; (5) When the Pugh
modulus ratio is greater than 0.571, the material is generally considered to
be brittle and hard [214]. The Pugh modulus ratio of T-carbon is only 0.414,
indicating that it has strong ductility and low hardness; (6) The hardness of
T-carbon was calculated to be 5.6 GPa by Chen $et$ $al.$ with their own model
[215]. This model was also used to estimate the hardness of super-hard
materials, such as diamond, and the obtained results were consistent with
those obtained by empirical formula and SV model. As the calculations on
hardness of a material depend strongly on the designed model, T-carbon
provides a great opportunity to test various models on hardness because of its
exotic chemical bondings. Therefore, how to get a reasonable hardness of
T-carbon is an intriguing issue that needs more explorations.

###  III.3 Thermoelectric properties

The thermoelectric effect is a phenomenon that can directly convert a
temperature difference to an electric voltage or vice versa. A thermoelectric
device creates a voltage when a different temperature is distributed on each
side, whereas when a voltage is applied to it, a temperature difference will
be created on the two ends. The high thermoelectric figure of merit (ZT) means
the low-cost conversion, which plays an essential role in the reuse of waste
heat.

Figure 13:  (a) Temperature dependent lattice thermal conductivity for several
carbon allotropes: T-carbon, BCO-C16 ($\vec{a}$, $\vec{b}$, $\vec{c}$),
graphene and diamond. (b) The power factor (PF) of calcium (Ca) or magnesium
(Mg) doped T-carbon, T-carbon with compressive strain, and T-carbon cut into
two-dimensional structures along the (111) direction. (c) Seebeck coefficient
(thermopower) and (d) the figure of merit $ZT$ in contour plots in the plane
of chemical potential ($\mu$) and temperature. Copyright (2017) American
Physical Society. Reprinted (a) with permission from ref [216]. Copyright
(2019) Royal Society of Chemistry. Reprinted (b), (c) and (d) with permission
from ref [185].

Yue $et$ $al.$ studied the heat transport properties of T-carbon [216] and the
lattice thermal conductivity $\kappa_{p}$ as a function of temperature was
plotted in Fig. 13(a). At room temperature (300 K), the lattice thermal
conductivity of T-carbon is 33.06 W/mK, which is much smaller than that of
graphite, diamond, and BCO-C16. The low thermal conductivity of T-carbon comes
from the strong anharmonicity of phonons in T-carbon.

Thermoelectric efficiency (ZT) defined as $ZT=S^{2}\sigma T/\kappa$, where
$S$, $\sigma$, $\kappa$, T represent Seebeck coefficient, electrical
conductivity, total thermal conductivity, and absolute temperature,
respectively, is inversely proportional to $\kappa$. The low lattice thermal
conductivity of T-carbon shows that it is likely to be a good thermoelectric
material. Figure 13(c) shows that the Seebeck coefficient of T-carbon can be
as high as 2000 $\mu$V/K [185], which is higher than that of many superior
thermoelectric materials, such as SnSe (550 $\mu$eV/K) [217]. The maximum ZT
value of T-carbon is around 0.43 as shown in Fig. 13(d), which is much smaller
than that of SnSe with the maximum ZT of 2.6, because the former possesses
lower electrical conductivity and higher thermal conductivity.

The conductivity of T-carbon can be improved by applying strain [218], doping
elements [219], or cutting it into low-dimensional structures [220]. Figure
13(b) shows the power factor defined as PF= $S^{2}\sigma$ as a function of
temperature for T-carbon doped with Ca and Mg elements, applied with
compressive strain, or cut into two-dimensional (2D) structures along the
(111) direction. On one hand, doping elements can effectively improve the PF
of T-carbon; on the other hand, after doping the thermal conductivity of
T-carbon will decrease, leading to the significant enhancement of ZT of
T-carbon. The PF of T-carbon can also be improved by applying a few
compressive strain. In addition, by cutting T-carbon into a 2D structure along
the (111) direction, PF can be greatly improved at low temperatures, which
means that T-carbon can be transformed from high-temperature to low-
temperature thermoelectric materials. In general, low thermal conductivity and
high Seebeck coefficient make T-carbon a potential thermoelectric material. In
addition, its thermoelectric properties can be further improved by improving
the conductivity through various ways. Thermoelectric devices based on
T-carbon may enable solid-state cooling to replace vapor-compression cycle
technologies and convert energy from waste heat sources under ambient
conditions.

###  III.4 Excitonic effect

Electron-phonon (e-ph) interactions play an important role in carriers
lifetime and transport properties of materials [221, 222]. The absorption and
recombination of excitons directly influence the light absorption and
luminescence of semiconductors. The excitonic effect and the relative energy
position of the bright and dark excitons have an important effect on the
photo-physical properties for semiconductors [223].

Figure 14: Mode-resolved scattering rates and total relaxation time of (a)
holes and (b) electrons, where the relative contribution from the TO branch is
inserted. Reprinted (adapted) with permission from ref [224].

The scattering rate and relaxation time (RT) of different phonon modes,
including lateral acoustics (TA), longitudinal acoustics (LA), lateral optics
(TO), and longitudinal optics (LO) phonon modes, are shown in Fig. 14(a)
[224]. In general, the contribution to scattering from optical phonons (mainly
TO mode) is much larger than that from acoustic phonons in a wide energy
range. It can be found that the acoustic phonons dominate the scattering at
the CBM, while the optical phonons dominate the scattering at the VBM.
However, the electron RT ($\sim$ 0.5 ps) from optical phonons scattering at
the VBM is higher than that ($\sim$ 0.25 ps) from acoustic phonons scattering
at the CBM, indicating that TO mode phonons make the most contributions to
electroacoustic scattering. Figure 14(b) shows an energy band valley of about
0.22 eV at point L (EL) above the CBM. When the energy is higher than the
valley, the inter-valley scattering occurs between $\Gamma$ valley at CBM and
L valley, and the scattering rate is greatly enhanced. The mean free path
(MFP) of hot carriers is directly related to the RT, because MFP is the
product of group velocity and RT. Since the group velocity of holes is greater
than that of electrons, and the RT of holes is about twice than that of
electron, the free path of hot holes (up to 80 nm) is much larger than that of
hot electrons (only 15 nm).

Figure 15: The optical absorption of T-carbon with inclusion of excitonic
effect based on GW plus independent-particle (IP) approximation and Bethe-
Salpeter equation (BSE), respectively. The first dark and the first bright
exciton are denoted as D1 and A, respectively, and their corresponding
absorption process was inserted. Reprinted (adapted) with permission from ref
[224].

The electron-phonon scattering mainly affects the lifetime of hot carriers,
and the electron-hole interactions mainly influence the lifetime of excitons.
The optical absorption including the excitonic effect is shown in Fig. 15. The
position marked by letter A corresponds to the lowest photon energy of the
bright exciton ($\sim$ 3.39 eV), indicating a transfer from the first valence
band to the first conduction band near $\Gamma$ point. There are four more
dark excitons with the first one below the bright exciton locating at 3.23 eV
denoted by D1, which originates from the transfer between VBM/VBM-1 and CBM.
The radiative lifetimes of the first bright exciton and the first dark exciton
were calculated to be 2.8 ns and 3.4 ns, respectively. The longer exciton
lifetime of dark excitons will trap more excitons population, leading to the
low photoluminescence quantum yield of T-carbon. It is an important research
topic and highly expected to improve the photoluminescence quantum yield of
T-carbon.

###  III.5 Optical properties

By studying optical absorption characteristics of materials, various
information such as the electronic band structure, impurity defects, etc. can
be directly obtained. The optical properties of T-carbon were extensively
studied both in theories and experiments [199, 201, 202, 207].

Figure 16: Calculated optical properties for several carbon allotropes with
similar structures: (a) Conductivity, (b) dielectric function, (c) absorption
spectra, and (d) EELS. Reprinted (adapted) with permission from ref [207].

The optical properties of several carbon allotropes are shown in Fig. 16. For
T-carbon, the real part of the conductivity becomes positive when the photon
frequency is higher than 1.780 1/fs, and T-carbon has the widest photon energy
range among these carbon materials as shown in Fig. 16(a). Fig. 16(b) shows
the static dielectric constant of T-carbon is about 4.590. The results of
absorption coefficient in Fig. 16(c) show that only T-carbon, Cubane-yane and
TY carbon can absorb visible light, and T-carbon can absorb the light with
photo energy above 1.8 eV. The absorption coefficients of T-carbon in the
visible light range are comparable to that of GaAs with the highest absorption
coefficient currently used, exhibiting the potential applications of T-carbon
in optoelectronic materials. The electron energy loss spectrum (EELS)
describes the energy loss of electrons in the process of rapidly passing
through the material, which becomes larger near 20 eV as shown in Fig. 16(d).

Figure 17: (a) Joint density of states (JDOS) with partial enlarged views at
specific energies in the inset, (b) real part and (c) imaginaty part of the
dielectric function, (d) absorption and (e) reflectivity spectra, and (f) EELS
of T-carbon in $z$ polarization direction under different pressure. Copyright
(2019) Elsevier. Reprinted (adapted) with permission from ref [201].

The optical properties of T-carbon under different hydrostatic pressures have
also been studied [201]. When the T-carbon is subjected to the hydrostatic
pressure of 0, 3, 6, and 9 GPa, the change of the joint density of states
(JDOS), the complex dielectric function, light absorption spectra, reflection
spectra, and ELSS in the incident light energy at the range of 0-30 eV in $z$
polarization direction are presented in Fig. 17. One may observe that with the
increase of static pressure, the spectra are slightly shifted to the region
with higher intensity of the incident light (blue shift). In addition, the
spectra in the $x$-axis and $z$-axis polarization direction are consistent due
to cubic crystal structure of T-carbon with high symmetries. From the JDOS in
the inset of Fig. 17(a) the band gap of T-carbon can be obtained, and as the
pressure increases, the band gap of T-carbon gradually decreases as shown in
the inset of Fig. 17(a).

Figure 18: (a) UV–visiable absorption of T-carbon NWs with the simulated
absorption spectrum inserted. (b) The photoluminescence spectrum. (a) and (b)
Reprinted (adapted) with permission from ref [199]. (c) FT-IR spectra of the
sample grown on the substrate of single crystalline diamond. (d) The simulated
FT-IR spectra of T-carbon. Copyright (2020) Elsevier. (c) and (d) Reprinted
(adapted) with permission from ref [202].

Figure 18(a) shows the experimental data of UV absorption spectrum for
T-carbon NWs. There are three obvious absorption peaks at 225, 250, and 276
nm, which are comparable with the calculated values of 226, 240, and 263 nm
for T-carbon as shown in the inset of Fig. 18(a). Figure 18(b) shows the
photoluminescence spectrum of T-carbon NWs. The quantum yield defined as the
ratio of the number of emitted photons to the absorbed photons at the
wavelength of 436 nm is 5.4$\pm$0.2$\%$. Figure 18(c) shows the experimental
data of FT-IR spectra for the sample synthesized by PECVD on a single crystal
diamond substrate, which shows a consistent peak with the calculated result at
1085 cm-1 as shown in Fig. 18(d).

###  III.6 Topological phonons

So far, the topological band structure is mainly discussed in electronic
systems [225, 226, 227, 228, 229, 230, 231, 232]. Recently, topological
phonons have also attracted much attention [233, 234, 235, 236, 237, 238],
because of their potential implications in electron-phonon coupling, dynamic
instability [239], and phonon diodes [235]. The spin-orbit coupling (SOC) of
carbon atom is very small, and carbon materials are well known as the
candidates of topological semimetal (metal) [240], which may have a family of
ideal candidates of topological phonons. However, it is not easy to find a
realistic carbon material featuring topological phonons, because it requires
the topological surface states of phonons that should be well separated from
the bulk phonon spectrum.

Figure 19: (a) The phonon spectra of T-carbon along high-symmetry paths, where
the BZ of the primitive cell of T-carbon with high symmetry points and paths
is inserted. (b) The enlarged views of phonon spectra of T-carbon in the range
of 13 to 17 THz. Copyright (2021) American Physical Society. Reprinted
(adapted) with permission from ref [189].
Figure 20: (a) The distribution of Weyl points at the boundaries of BZ in
T-carbon, i.e, in $k_{x,y,z}=2\pi/a$ and $(\pm 1,\pm 1,\pm 1)$ planes, where
the red- and blue-colored signs represent the Weyl point at the square and
hexagonal surfaces, respectively, and the ”+” and ”-” indicate the chirality
of Weyl points. (b) Schematic depiction of the gimbal. (c) Shape of the
hourglass Weyl loop (black-colored loop) in $k_{z}=0$ plane obtained from the
DFT calculations. The color map corresponds to the frequency difference
between the two crossing phonon bands. (d) Distribution of type-II Weyl
phonons at about 14.5 THz in (111) plane obtained from the DFT calculations.
Copyright (2021) American Physical Society. Reprinted (adapted) with
permission from ref [189].

T-carbon as a realistic carbon material has a high-symmetry crystal structure
with the same space group as diamond, which exhibits exotic topological
phonons [189]. The phonon spectrum of T-carbon and its enlarge view in the
range of 13 to 17 THz were shown in Fig. 19. It is clear that there are two
flatbands above a large phonon band gap of about 10 THz between 22 to 32 THz,
which are related to the specific geometry of T-carbon [241]. Since high
frequency phonons make little contribution to the electron-phonon coupling and
thermal transport [186], we only study the relative low-frequency phonons
below 20 THz. From Fig. 19(b), we find that two phonon bands from optical
branch cross linearly at about 15 THz on $\Gamma-X$, $K-\Gamma$ and $U-X$
paths, and at about 14.5 THz on $L-U$ path. After a careful study within
$k_{z}=0$ plane, we find that the crossing points on $\Gamma-X$ and $\Gamma-K$
paths are not isolated, but form a nodal loop in $k_{z}=0$ plane. Besides
these band crossing points, there is double degeneracy appearing on $X-W$
path, which is protected by the combination of glide mirrors $\tilde{M_{x}}$
and $\tilde{M_{z}}$ because of their anticommutating relation on $X-W$ path.
Considering the crystal symmetry of T-carbon, we can obtain three intersecting
nodal loops (named as nodal gimbal phonons) around $\Gamma$ point and 6 pairs
of type-I Weyl phonons around the center of squares at the boundary of BZ at
about 15 THz, and 12 pairs of type-II Weyl phonons around the center of
hexagons at the boundary of BZ at about 14.5 THz as shown in Fig. 20. By
$k\cdot p$ perturbative analysis, the above crossing points (loops) were
demonstrated to be protected by the symmetries of T-carbon [189].

Figure 21: (a) Local density of states (LDOS) and (b) a constant energy slice
at 15 THz projected on the semi-infinite (001) surface of T-carbon. (c) LDOS
and (d) a constant energy slice at 15 THz projected on the semi-infinite (111)
surface. (e) Distribution of Weyl phonons in $k_{x}=2\pi/a$ plane of T-carbon
obtained from the DFT calculations, where the Weyl phonons with opposite
chirality are marked as ”+” and ”-”, respectively, and the redefined Cartesian
coordinate system $(q_{x},q_{y},q_{z})$ in $k_{x}=2\pi/a$ plane is also
indicated. (f) The phonon surface arcs projected on the semi-infinite (100)
surface of T-carbon at a constant frequency slice 15.2 THz. Copyright (2021)
American Physical Society. Reprinted (adapted) with permission from ref [189].

The surface of a nodal line semimetal features the drumhead like states. From
Figs. 21(a) and 21(c), where we show the phonon surface states of T-carbon on
(001) and (111) surfaces, respectively, one may observe the drumhead surface
bands that emanate from the bulk nodal points, which connects the two nodal
lines through the surface BZ boundary. In Figs. 21(b) and 21(d), we plot the
constant frequency slice at 15 THz, which cuts through the drumhead, forming a
few arcs or circles, because the drumhead is not completely flat in frequency.
The surface of a Weyl semimetal features the Fermi arcs connected two Weyl
points with opposite chirality. From Figs. 21(e) and 21(f), it is clear to see
two Fermi arcs connected with two opposite Weyl points at the ends, which is
the fingerprint of nontrivial Weyl phonons.

##  IV Potential applications of T-carbon

Figure 22: Hydrogen storage in T-carbon.

Although T-carbon itself possesses many intriguing properties, as we discussed
above, the wide energy gap of T-carbon might make it less practical
application. Thus, it is important to tune the band gap of T-carbon. The most
commonly used method in tuning the band gaps is to introduce electronic or
hole states by doping that may optimize the performance of materials. Owing to
the large interspaces between carbon atoms along certain directions, many
kinds of atoms can be doped into T-carbon to form stable compounds, which
makes T-carbon have important applications in hydrogen storage [20], ion
battery [185], solar cells [200], photocatalysis [242, 243], magnetism and
superconductivity [186]. These advances will be discussed in subsequent
sections. Moreover, B and Al doped T-carbon are predicted to be a kind of
stable structure with impurity bands and zero band gap, which promotes the
conductivity of T-carbon. Meanwhile, the absorption, reflection and loss peaks
of Al doped T-carbon in UV light are decreased, indicating its potential
application in optoelectronic and microelectronic devices [244]. In addition,
by substituting one C atom with one N atom in each C4 unit in T-carbon, a new
type of stable carbonitride C3N can be obtained, which is a multi-layered
structure, and its single-layer film has great potential application in water
nanofiltration, providing a new solution for wastewater treatment and seawater
desalination [245].

###  IV.1 Hydrogen storage

Since the 1970s, hydrogen has been regarded as one of the most promising
alternatives to fossil fuels, because the only by-product of its combustion is
water, which has great advantages in reducing greenhouse gases and harmful
pollutants. On the other hand, hydrogen is light in weight and can provide
higher energy density, making that hydrogen powered engine is more efficient
than internal combustion engines. The main reason that hinders the
popularization of hydrogen energy is that it is challenging to obtain and
store a large amount of hydrogen safely, densely, and quickly. For this
reason, discovering the next generation of new materials for hydrogen storage
is essential. As carbon materials have the characteristics of high specific
surface area and light weight, searching for new porous carbon materials for
hydrogen storage has always been a hot topic [246, 247, 248, 249].

Due to the fluffy structure, T-carbon has potential application in hydrogen
storage [20]. T-carbon has a low density, about two-third that of graphite and
half that of diamond. According to the maximum number of hydrogen molecules
adsorbed, the hydrogen storage value can be estimated. The hydrogen adsorption
energy for T-carbon can be defined as
$E_{adsorption}=(E_{T-carbon}+nE_{H_{2}}-E_{compound})/n$, where
$E_{adsorption}$, $E_{H_{2}}$ and $E_{compound}$ are the total energy of
T-carbon, one hydrogen molecule and T-carbon absorbing $n$ hydrogen molecules,
respectively. For T-carbon absorbing 8 hydrogen molecules, the adsorption
energy is about 0.173 eV. By increasing the number of absorbed hydrogen
molecules until the number of hydrogen molecules reaches 16, the absorption
energy becomes negative, indicating that the highest absorption capacity of
hydrogen molecules is reached. Due to the strong C-C bond in T-carbon and the
weak repulsion between the adsorbed hydrogen molecules H2, T-carbon is still
stable after absorbing 16 hydrogen molecules in a unit cell. According to the
maximum number of adsorbed hydrogen molecules as shown in Fig. 22, the
hydrogen storage capacity of T-carbon is calculated to be about 7.7 wt$\%$
[20], which is quite competitive compared with some high-capacity hydrogen
storage materials [185].

###  IV.2 Li-ion batteries

Figure 23: The potential application of T-carbon in ion batteries: (a) the
migration of M (Li, Na, K, Mg) atoms in T-carbon, (b) the energy change of M
atoms diffusing along the minimum migration path as indicated in (a).
Copyright (2019) Royal Society of Chemistry. Reprinted (adapted) with
permission from ref [185].

Rechargeable energy storage devices, such as lithium-ion batteries and
portable power sources, play an important role in electronic devices,
biomedicine, aerospace and electric vehicles, etc [250]. Carbon-based
materials are widely used in lithium-ion batteries, among which graphite is
the most commonly used negative electrode material, because graphite has the
high specific surface area and large interlayer space to accommodate lithium
atoms, resulting in a high energy storage capacity of 372 mA$\cdot$h$\cdot$g-1
[251].

As a new member of the carbon family, T-carbon has good application potential
in lithium-ion batteries and other rechargeable energy storage devices due to
its fluffy structure [185]. The specific volume of metal atoms is defined as
$C=nF/M_{C_{x}}$ [250], where $n$, $F$ and $M_{C_{x}}$ represent the number of
electrons per metal atom involved in the electrochemical process (for Li, Na
and K, $n$ = 1; for Mg, $n$ = 2), the Faraday constant with the value of
26.801 A$\cdot$h$\cdot$g-1, and the mass of $C_{x}$ ($x$ is the number of
carbon atoms, and for T-carbon, $x$ is 8), respectively. The specific volumes
of Li (Na or K) and Mg are 588 and 1176 mA$\cdot$h$\cdot$g-1, both are much
higher than that of graphite.

The most stable adsorption position for metal ions M (Li, Na, K, Mg) is the
center T1 of the cage in T-carbon. The migration of M atoms in T-carbon is
shown in Fig. 23(a). The migration barriers of M atoms in T-carbon are 0.075,
0.233, 0.528 and 0.688 eV, respectively, as shown in Fig. 23(b). The Li-ion
migration barrier in T-carbon is much smaller than that in graphite with the
battier of 0.327 eV [251]. According to Arrhenius law, the diffusion constant
of lithium ion in T-carbon, which is defined as $D\sim exp(-E/k_{B}T)$, where
$E$ is the energy barrier, was calculated to be 1.7$\times$104 times that in
graphene [252]. Therefore, T-carbon is a good lithium ion diffusion material
and would have great application potential in ultra-fast charging and
discharging of rechargeable energy storage devices.

Recently, theoretical calculations show that T-carbon has a maximum lithium
capacity of 1116 mA$\cdot$h$\cdot$g-1 and tends to form an energy favorable
mixture of C64Li, C64Li16 and C64Li32 [253]. Since too high and too low
lithium concentrations are not conducive to the adsorption and release of
lithium by T-carbon anode respectively, only half of the maximum lithium
capacity ($\sim$ 487 mA$\cdot$h$\cdot$g-1) in T-carbon can be used for anode
discharge, which may provide guidance for the future research of lithium,
sodium or potassium ion battery electrodes in new energy resources.

###  IV.3 Solar cells

Table 2: The effective masses of electron ($m_{e}/m_{0}$) and hole ($m_{h}/m_{0}$), electron ($\nu_{e}$) and hole ($\nu_{h}$) mobility, band gaps ($E_{g}$) and energy level of CBM ($E_{\rm CBM}$) for T-carbon and several common used solar cell materials. |  $m_{e}/m_{0}$ |  $\nu_{e}$ (${\rm cm^{2}\cdot s^{-1}\cdot V^{-1}}$) |  $m_{h}/m_{0}$ |  $\nu_{h}$ (${\rm cm^{2}\cdot s^{-1}\cdot V^{-1}}$) |  $E_{g}$ (eV) |  $E_{\rm CBM}$ (eV) |  Reference
---|---|---|---|---|---|---|---
T-carbon |  0.22 |  2360 |  0.66 |  270 |  2.27 |  -4.4 |  [200]
MAPbI3 |  0.14-0.26 |  5-10 |  0.2-0.44 |  1-5 |  1.5 |  -3.9 |  [254, 255, 256]
P |  0.17-1.12 |  1000 |  0.15-6.35 |  10000 |  1.51 |  - |  [257]
TlO2 |  0.096-0.424 |  3300 |  0.144-4.092 |  4300 |  1.56 |  - |  [258]
Si |  1.08 |  1500 |  0.59 |  450 |  1.12 |  - |  [259]
TiO2 |  5-10 |  18 |  0.6-1.0 |  0.16 |  3.2 |  -4.1 |  [260, 261]
MoS2 |  0.46-0.48 |  60-72 |  0.57-0.60 |  152-200 |  1.6 |  - |  [262]
SnO2 |  0.26,0.21 |  166,229 |  1.27,1.6 |  15.7,11 |  3.66 |  -4.42 |  [263, 264]
ZnO |  0.29 |  205 |  0.78 |  50 |  3.4 |  -4.4 |  [265, 263, 260]

As the most widely available renewable energy in nature, solar energy is of
great significance in the development of next-generation green energy. In
recent years, organic-inorganic hybrid perovskite solar cells, as a new member
of the solar cell family, have attracted great attention for the next
generation of solar cell technology due to their excellent properties [266,
267, 268]. Since 2010, the energy conversion efficiency of perovskite solar
cells has increased from 3.8$\%$ to 22.7$\%$ [269, 270]. At present, the most
commonly used electron transport material in perovskite batteries is TiO2,
which is applied to collect and transport electrons from the perovskite
absorber layer [271, 272]. Although electrons can enter TiO2 from the
perovskite absorption layer quickly, on one hand, the electron recombination
rate is high because of the low electron mobility in TiO2, and on the other
hand, the required sintering temperature is high for the preparation of TiO2
conductive crystals, limiting the application of TiO2 in perovskite solar
cells [273]. Therefore, it is necessary to design new and efficient electron
transport materials, which requires a high electron mobility of the materials
to meet the rapid electron transport and avoid interface electronic
recombination, and the materials to have energy levels matched with perovskite
in order to facilitate electron injection and hole blocking [274].

Due to the cubic structure, T-carbon exhibits isotropic transport properties.
The effective mass $m^{\\*}/m_{0}$ of the electron and hole of T-carbon was
calculated to be 0.22 and 0.66, respectively, which are comparable with that
(0.2-1.3) of perovskite MAPbI3 and MASnI3 [254]. The electron mobility of
T-carbon was calculated to be 2360 cm${}^{2}\cdot$s${}^{-1}\cdot$V-1, which is
about 9 times of the hole mobility (270 cm${}^{2}\cdot$s${}^{-1}\cdot$V-1),
which indicates that the electron transport capacity of T-carbon is better
than the hole transport capacity [200]. The electron mobility of T-carbon is
comparable with or even higher than that of some commonly used materials, such
as black phosphorus (1000 cm${}^{2}\cdot$s${}^{-1}\cdot$V-1) [257], TlO2 (3300
cm${}^{2}\cdot$s${}^{-1}\cdot$V-1) [258], Si (1500
cm${}^{2}\cdot$s${}^{-1}\cdot$V-1) [259] and TiO2 (0.0524
cm${}^{2}\cdot$s${}^{-1}\cdot$V-1) [261], and the hole mobility of T-carbon is
much larger than that of MoS2 (200 cm${}^{2}\cdot$s${}^{-1}\cdot$V-1) [262],
perovskite MAPbI3 (1-5 cm${}^{2}\cdot$s${}^{-1}\cdot$V-1) [255, 256] and TiO2
(0.16 cm${}^{2}\cdot$s${}^{-1}\cdot$V-1) [261] as listed in Table 2. The high
electron mobility of T-carbon can further improve the short-circuit current
and efficiency [275]. Therefore, T-carbon may be a viable alternative to
traditional electron transport material TiO2.

In perovskite solar cells, the bottom of CBM for the electron transport
materials should be lower than the perovskite to achieve rapid electron
injection [276, 277, 278, 279]. T-carbon has a direct energy gap of about
2.273 eV with the energy level of the bottom of CBM located at 4.4 eV [200],
which is close to the energy levels of SnO2 and ZnO that have faster electron
injection efficiency than TiO2 [274, 280, 281, 282], indicating that T-carbon
may have a higher electron injection rate than TiO2.

###  IV.4 Photocatalysis

To solve the problem of environmental pollution caused by the overuse of
fossil fuels, producing hydrogen from water using sunlight is an essential and
natural choice. Therefore, searching for stable and effective photo-catalysts
is really meaningful. The typical photo-catalysis materials include TiO2, CdS,
C3N4 and so on [283, 284, 285, 286, 287].

Figure 24: (a) The crystal structure of Pb doped T-carbon with one C4 unit
substituted by one Pb atom in primitive cell of T-carbon. (b) The band
structure of Pb doped T-carbon with HSE06 method, where the red line is the
impurity state of Pb atom. (c) The total and partial DOS of Pb doped T-carbon.
(d) Absorption spectra of Pb (red solid line) and Sn (blue dash line) doped
T-carbon, respectively. (e) The crystal structure of B-N codoped T-carbon and
(f) its correspdong band structure. Copyright (2019) Elsevier. (a), (b), (c)
and (d) Reprinted (adapted) with permission from ref [242].

Appropriate band gap is an important criterion for photocatalytic materials.
To realize it, doping is usually an effective method to manipulate the
electronic band structures. In 2018, Ren $et$ $al.$ adopted various doping
methods to tune the band gap of T-carbon, and predicted a stable structure
obtained by replacing a C4 unit in T-carbon with a single atom in IVA group
[242]. Moreover, in 2020, Tian $et$ $al.$ proposed another stable structure by
$p-n$ codoping with one boron and one nitrogen atoms substituting two carbon
atoms in each C4 unit of T-carbon to form an intra-tetrahedron B-N bond [243].
The corresponding two structures are shown in Figs. 24(a) and (e). According
to the calculation of phonon spectra and cohesive energies, the two doping
structures are demonstrated to be kinetically stable and slightly
thermodynamically less stable than pristine T-carbon. Unlike T-carbon itself,
which has a direct band gap, the single-atom M (M=Si, Ge, Sn, Pb) in IVA group
doped T-carbon has an indirect band gap, and the impurity states were
successfully introduced into the energy gap of T-carbon as shown in Fig.
24(b). Among these doped structures, Si and Ge doped T-carbon have a larger
energy gap than T-carbon, while Sn and Pb doped T-carbon have a smaller energy
gap. The change of energy gap is mainly due to the shift of the bottom of CBM
relative to the Fermi level, which is attributed to the doped atomic states,
indicating the strong localization of electronic state near the impurity. For
B-N codoped T-carbon, it remains a direct band gap semiconductor with a
smaller energy gap than T-carbon. It is noted that the energy gap of Pb doped
T-carbon and B-N codoped T-carbon is 1.62 and 2.55 eV, respectively. In
addition, Pb doped T-carbon performs the good optical absorption properties in
the visible light region with a broad peak at 500-550 nm and a weak peak at
600-650 nm, corresponding to green and red light radiation, respectively.
Because of the suitable band gap, localized electronic states and good optical
absorption, Pb doped T-carbon is expected to be a high performance
photocatalyst material [288].

###  IV.5 Magnetism in Ni-doped T-carbon

Figure 25: (a) The cubic crystalline structure of Ni-doped T-carbon NiC8 with Ni atoms (orange balls) occupying the Wyckoff position $4a$. The green arrows indicate the magnetization direction of the Ni atom. (b) The phonon spectra, (c) electron band structure, and (d) partial density of states (PDOS) of the Ni-doped T-carbon. Table 3: The lattice constants $l$ (Å), magnetic moments $\langle S\rangle$ ($\mu_{B}$), energy differences per magnetic atom between ferromagnetic and antiferromagnetic configurations $\Delta E$ (meV/Ni), and Curie temperatures $T_{C}$ (K) for Ni-doped T-carbon and face centered cubic (fcc) Ni, respectively. |  $l$ (Å) |  $\langle S\rangle$ ($\mu_{B}$) |  $\Delta E$ (meV/Ni) |  $T_{C}$ (K)
---|---|---|---|---
Ni-doped T-carbon | 7.52 | 1.2 | -5.5 | 33
Fcc Ni | 3.52 | 0.6 | -106 | 627 [289]

The observation of magnetism in carbon materials has attracted much attention,
such as a charge transfer complex of C60 and tetrakis ethylene [290], carbon
nanofoams, carbon nanodiamonds [291, 292], untreated graphite with intrinsic
point defects [293, 294], highly oriented pyrolytic graphite irradiated with
high-energy protons [295] and so on. As we discussed above, T-carbon has many
potential applications, and the exploration of possible magnetic properties
based on T-carbon would greatly enrich the practical application of T-carbon.

A natural idea to introduce magnetism into T-carbon is to dope magnetic atoms,
where transition metals are good choice. When a transition metal atom is
intercalated into the primitive cell of T-carbon to form C8M (M = transition
metal atom), we find that the most energetically preferred structure with the
transition metal atoms forms a fcc structure, as shown in Fig. 25(a). After
checking phonon spectra of all doped structures, only Ni-doped T-carbon is
dynamically stable with no imaginary frequency modes in the whole Brillouin
zone. The optimized lattice constant of Ni-doped T-carbon is about 7.52 Å.
Moreover, Ni-doped T-carbon is demonstrated to possess a ferromagnetic ground
state with the magnetization along [111] direction [see Fig. 25(a)]. The band
structure and PDOS of Ni-doped T-carbon is depicted in Figs. 25(c) and (d).
Unlike T-carbon itself, Ni-doped T-carbon behaves as a half-metal with only
one spin species (spin down) passing through the Fermi level, and the
magnetism is introduced due to the different occupations of spin up and down
electrons. The DOS near Fermi level is mainly dominated by the $e_{g}$
orbitals of Ni atoms. In Ni-doped T-carbon, T-carbon maintains the direct band
gap semiconductor, but the band gap slightly decreases. To estimate the Curie
temperature of Ni-doped T-carbon, we calculate the energy differences ($\Delta
E$) between ferromagnetic and antiferromagnetic configurations of Ni-doped
T-carbon with fcc Ni using the same method [GGA+U (U=4 eV)], and $\Delta E$ of
the former is about 20 times smaller than that of the latter as listed in
Table 3. It is known that the Curie temperature of fcc Ni is 627 K [289], and
the Curie temperature of Ni-doped T-carbon is estimated to be about 33 K. The
magnetism in T-carbon and the half-metal property make Ni-doped T-carbon might
have potential applications in spintronics, such as spin valve, spin
transistor, magneto-optical devices and so on.

###  IV.6 Superconductivity in Na-doped T-carbon

The observation of superconductivity has long been a popular topic in
condensed matter physics and materials science, which has received wide
interest in recent years [296, 297]. Except for the pure carbon
superconductors, such as single-walled carbon nanotubes and twisted bilayer
graphene with low superconducting transition temperatures [137, 67, 298],
several carbon intercalation compounds were reported to be superconductors,
including graphite intercalation compounds [299], alkali-metal doped C60 [300,
301, 302, 303, 304, 305] and diamond intercalation compounds [306].

Figure 26: (a) The cubic crystalline structure of Na-doped T-carbon NaC8 with
Na atoms (green balls) occupying the Wyckoff position $4a$. (b) The electron
band structure and partial density of states (PDOS) of the Na-doped T-carbon
with inclusion of the SOC. (c) The phonon spectra and (d) phonon density of
states (PhDOS) (divided by 40), Eliashberg spectral function
$\alpha^{2}F(\omega)$, and cumulative frequency-dependent of EPC
$\lambda(\omega)$ of Na-doped T-carbon at ambient pressure. Copyright (2020)
American Physical Society. Reprinted (adapted) with permission from ref [186].

Due to the large interspaces between carbon atoms, many different atoms can be
intercalated into T-carbon. The crystal structure of Na-doped T-carbon was
shown in Fig. 26(a), which was a compensate metal with a dispersive band
contributed by the $s$ and $p$ orbitals of C atoms and $s$ orbital of a Na
atom passing through the Fermi level [Fig. 26(b)] [186]. There is no imaginary
frequency mode in Fig. 26(c), indicating the dynamical stability of Na-doped
T-carbon. From the PhDOS in Fig. 26(d), one can observe a peak from Na atoms
at the low frequency because the relative larger atomic mass of Na atom than
that of C atom. As the Na-doped T-carbon is a metal with an electronic Fermi
surface, superconductivity may be induced by phonon-mediated electron pairing.
The electron-phonon coupling strength $\lambda(\omega)$ was calculated to be
0.63. Utilizing the calculated Eliashberg spectral function
$\alpha^{2}F(\omega)$ and $\lambda(\omega)$ together with a typical value of
the effective screened Coulomb repulsion constant $\mu^{\\*}$ = 0.1, the
superconducting transition temperature Tc of Na-doped T-carbon was calculated
to be about 11 K at ambient pressure with the McMillan-Allen-Dynes (MAD)
approach [307, 308].

Table 4: The pressure dependent superconductive parameters of N(EF) (in unit of states/spin/eV/cell), volume (Å3), $\omega_{log}$ (in K), $\lambda$, and Tc (in K) for Na-doped T-carbon. P(GPa) | N(EF) | V(Å3) |  $\omega_{log}$(K) | $\lambda$ | Tc(K)
---|---|---|---|---|---
0 | 0.85 | 108.85 | 413.2 | 0.63 | 10.9
3 | 0.83 | 106.81 | 389.8 | 0.67 | 12.1
5 | 0.82 | 105.54 | 369.8 | 0.70 | 12.9
7 | 0.82 | 104.35 | 348.0 | 0.74 | 13.8
10 | 0.80 | 102.67 | 297.2 | 0.86 | 16.0
12 | 0.79 | 101.61 | 254.4 | 0.98 | 17.2
14 | 0.79 | 100.60 | 181.0 | 1.36 | 18.7

Figure 27: (a) Temperature dependence of electronic specific heat C in the
normal state of Na-doped T-carbon superconductor. The upper and lower insets
are the enlarged parts of the specific heat at temperature T $<$ 15 K and 15 K
$<$ T $<$ 50 K, respectively. The dash lines are fitting curves at different
temperature regions. (b) Temperature dependent Lorenz number $L$
[$L=\kappa$/($\sigma$T)]. The inset exhibits the temperature dependent
electrical ($\sigma$) and thermal ($\kappa$) conductivities over the
relaxation time ($\tau$). Copyright (2020) American Physical Society.
Reprinted (adapted) with permission from ref [186].

The temperature dependent specific heat, electrical conductivity and thermal
conductivity are presented in Fig. 27(a). It was noted that, at low
temperature below 15 K, specific heat $C(T)\sim T^{3}$, at $15<T<50$,
$C(T)\sim T^{2}$, and at high temperature above 50 K, $C(T)\sim T$. In
addition, the temperature dependent Lorenz number defined by $L=\kappa/(\sigma
T)$, as given in Fig. 27(b), was not a constant at temperature below 50 K,
which dramatically violates the Wiedemann-Franz law, and it shows a constant
at high temperature, indicating a Fermi liquid behavior. Therefore, the normal
state of Na-doped T-carbon superconductor exhibits a non-Fermi liquid behavior
at temperature below 50 K, and such a non-Fermi liquid normal state may be an
anomalous metal.

The effect of pressure on superconducting transition temperature Tc for Na-
doped T-carbon superconductor was also studied and the results are listed in
Table 4. As can be seen, with the increase of pressure, Tc increases owing to
the increase of both electron-phonon coupling strength $\lambda(\omega)$ and
spectral function $\alpha^{2}F(\omega)$. The increase of Tc with pressure for
Na-doped T-carbon superconductor results from an enhancement of the electron-
phonon coupling $\lambda$ due to the shift of the phonon spectrum to lower
frequencies, whereas for doped fullerenes A3C60 (A = K, Rb, Cs), the Tc is
rapidly decreased under pressure because of the sharp decrease of electronic
density of states under pressure caused by the rapid increase of the width of
the conduction band when C60 molecules are compressed together [309, 310,
311].

###  IV.7 Hydrogenated T-carbon molecules in interstellar medium

The 2175 Å UV extinction bump in interstellar medium (ISM) of the Milky Way
was firstly discovered in 1965 by Stecher [312]. Since then, several satellite
observatories have confirmed the existence of this bump feature in tens stars
of the Milky Way [313, 314, 315, 316]. Many subsequent studies on numerous
galaxies show that such a 2175 Å extinction bump feature in interstellar
medium (ISM) is almost ubiquitous and pronounced in the Milky Way [317, 316].
Carbonaceous materials are commonly regarded as the origin of the 2175 Å
extinction bump in ISM [317]. However, the exact physical origin of the bump
has been debated for a long time, which is still hitherto unclear after
intensive explorations of more than a half century. Although a few potential
carbonaceous candidates including graphite, nongraphic carbon, diamond and
polycyclic aromatic hydrocarbons were proposed to explain the physical origin,
a consensus is yet far to achieve. Therefore, at present a great puzzle still
remains in the interpretation of this mysterious bump in ISM. It is noteworthy
that, for T-carbon with anisotropic $sp^{3}$ hybrid bonds and low density, a
prominent peak appears at the wavelength of 2250 Å (not far from 2175 Å) in
its UV-visible optical absorption spectra [199]. Because T-carbon is more
readily formed under negative pressure [203], it is easier to form T-carbon or
its fragments in interstellar space with negative pressure circumstance. In
addition, neutral hydrogen is full of the interstellar space, thus we believe
that T-carbon would most likely exist in interstellar space as the
hydrogenated T-carbon (HTC) molecules or clusters [318].

Figure 28: (a) The structure of hydrogenated T-carbon (HTC) molecule (C40H16).
(b) Calculated molecular orbital energy levels of HTC molecule together with
the molecular orbital maps of the lowest unoccupied molecular frontier
orbitals (LUMOs) and the highest occupied molecular frontier orbitals (HOMOs).
The green and yellow regions represent the positive and negative phases of the
orbital wave function. The calculations show that the 2175 Å absorption peak
of HTC molecule is intrinsically generated from the excitation from the HOMO
(with the orbital symmetry $E$) to the 10th level (degenerate in energy with
the 11th and 12th levels with the orbital symmetry T2) above the LUMO (with
the orbital symmetry A1). Reprinted (adapted) with permission from ref [318].
Figure 29: Calculated UV absorption spectra of (a) HTC molecule and HTC
mixtures, (b) Fe2SiO4, (c) MgSiO3 and (d) graphite. The red curve represents
the overall superposed broadened absorption spectrum of HTC mixtures, where
the peak is positioned at 2175 Å. The straight solid lines are for HTC
molecule and dashed curves are for HTC mixtures at different absorbed
wavelengths labelled in the legend of (a). Red solid lines in (b), (c) and (d)
represent the smoothed spectral curves by Savitzky–Golay polynomial
regression. Reprinted (adapted) with permission from ref [318].
Figure 30: Fitting results for the normalized extinction curves of six stars
including (a) BD582292, (b) HD239689, (c) HD239742, (d) HD239722, (e)
HD283800, and (f) HD147889. Diamonds denote the observed data adapted from the
references as indicated. The red solid lines are fitting curves of the
observed data according to the equation
$I_{Fit}=\alpha_{1}I_{HTC}+\alpha_{2}I_{Graphite}+\alpha_{3}I_{MgSiO_{3}}+\alpha_{4}I_{Fe_{2}SiO_{4}}$,
where $\alpha_{i}$ ($i$=1,2,3,4) are linear fitting parameters, $I_{HTC}$,
$I_{Graphite}$, $I_{MgSiO_{3}}$, and $I_{Fe_{2}SiO_{4}}$ stand for the
absorption intensities of HTC mixtures, graphite, MgSiO3, and Fe2SiO4,
respectively. The other lines are marked in the legend of (a). Reprinted
(adapted) with permission from ref [318].

The most possible HTC molecule consists of 40 carbon and 16 hydrogen atoms
with $sp^{3}$ hybrid bonds as depicted in Fig. 28(a). A sharp absorption peak
at the wavelength of 2175 Å is found in the UV-visible absorption spectra as
shown in Fig. 29(a), which is stable and does not change with the calculation
method, indicating the possible physical origin of the 2175 Å UV extinction
bump. The absorption peak for HTC molecule is in accord with the excitation
from the highest occupied molecular orbitals (HOMO) to the 10th level above
the lowest unoccupied molecular orbitals (LUMO). In ISM there may exist HTC
mixtures, which might lead to the broadened absorption spectra with the
central peak kept at 2175 Å through a mixture model based on clusters of HTC
molecules. Since silicate [319], Mg/Fe-containing olivine [320, 321] and
graphite grains may exist in the interstellar dust, fayalite (Fe2SiO4),
enstatite (MgSiO3), and graphite grains should be taken into account when
calculating the UV absorption spectra. By fitting the UV extinction curves of
optional six stars [322, 323, 324] with linearly combining the calculated UV
absorption spectra of HTC mixtures, graphite, MgSiO3 and Fe2SiO4 [319, 325,
326], the UV extinction curves in these six stars are perfectly fitted as
shown in Fig. 30. To distinguish the contributing proportion of the four
ingredients in fitting the UV extinction feature bump in ISM, the column
density ratio is calculated. The column density ratio of carbonaceous dust
(HTC mixtures and graphite) and silicate dust (MgSiO3 and Fe2SiO4) in fittings
is about 0.24. And the fitting results indicate that the central peak at 2175
Å is mainly attributed to HTC clusters as shown in Fig. 30. Thus, HTC
molecules and clusters are demonstrated to play an important role in
explaining the UV extinction curves of interstellar dust in the Milky.

In summary, T-carbon has a wide range of potential applications due to its
unique structure and properties. The fluffy structure and low density make
T-carbon have good potential in various energy storage devices. As the battery
anode material, T-carbon shows high lithium-ion diffusion constant and lithium
capacity. As a hydrogen storage material, it displays high storage capacity.
High electron mobility and appropriate energy levels matched well with
perovskite also make T-carbon expected to be a great electron transport
material in solar cells. Furthermore, the band structure of T-carbon can be
effectively tuned by different doping schemes. Pb-doped T-carbon and B-N co-
doped T-carbon hold appropriate band gap, localized electronic states and good
optical absorption, which have potential applications in photocatalyst
materials. Ni-doped T-carbon with magnetic and half-metal properties can be
used in spintronics. Na-doped T-carbon shows a high superconducting transition
temperature of 11 K at ambient pressure, which can be enhanced to 19 K at 14
GPa. In addition, it is worth mentioning that the hydrogenated T-carbon
molecules may be related to the physical origin of the UV extinction feature
in interstellar media.

Besides those mentioned potential applications in this section, other
appealing properties of T-carbon are also accompanied with possible
applications. For instance, the high failure strain of T-carbon achieved by
global graphitization can be used in industrial and mechanical equipment with
high ductility requirements. As a potential thermoelectric material,
thermoelectric devices based on T-carbon are also very promising. New devices
based on the unique topological phonons of T-carbon can be used to explore
more novel physical effects of phonon topological states in energy information
science.

##  V Derivative structures from T-carbon

Inspired by T-carbon, many new carbon materials were predicted, which inherit
some excellent properties from T-carbon and even perform some better
properties than T-carbon. In the following, as examples, we will introduce
several derivatives of T-carbon, including TY-carbon [327], T-II carbon [328],
bct-C16 [329], $sp^{2}$-diamond [330], L-carbon [331], carboneyane [203],
cyclicgraphyne and cycligraphdiyne [241].

Figure 31: Several derivative structures of T-carbon: (a) TY-carbon obtained
by introducing $sp$ hybrid bonds into T-carbon and (b) its [111] view; (c)
T-II carbon derived by stacking two T-carbon together and (d) its [110] view;
(e) top and (f) side views of bct-C16, which is obtained by a structural phase
transition from T-carbon. Copyright (2012) by the American Physical Society.
(a) and (b) Reprinted (adapted) with permission from ref [327]. Copyright
(2013) Royal Society of Chemistry. (c) and (d) Reprinted (adapted) with
permission from ref [328]. (e) and (f) Reprinted (adapted) with permission
from ref [329].

###  V.1 TY-carbon

After the proposal of T-carbon, a new stable carbon allotrope named as
tetrayne-carbon (TY-carbon) was predicted by introducing $sp$ hybrid bonds
into T-carbon [327]. The main characteristics of TY-carbon come from the
introduced C-C triple bonds. By introducing $sp$ hybrid bonds to diamond,
another novel carbon allotrope coined as yne-carbon (Y-carbon) was obtained.
The structure of TY-carbon is shown in Figs. 31(a) and (b), and its main
properties are summarized in the following: (1) TY-carbon was obtained by
inserting two carbon atoms connected by an alkyne ($sp$) bond between two
neighboring C4 units in T-carbon, thus it has the same space group as
T-carbon; (2) The phonon spectra of TY-carbon and T-carbon are very similar
except for the influence of $sp$ bonds; (3) The XRD spectra of TY-carbon are
very similar to that of T-carbon; (4) TY-carbon has a direct band gap of 2.234
eV, and a phase transition from insulator to metal can occur with increasing
the bond length because of the dimerization of carbon atoms as a result of
Peierls instability [332]; (5) TY-carbon has a much lower density of
0.523$g/cm^{3}$ than that of T-carbon, implying its potential application in
hydrogen storage; (6) By injecting inert gas such as Xe into the cavity of TY-
carbon, it is expected to dramatically increase its stiffness [333].

###  V.2 T-II carbon

Li $et$ $al.$ designed a new carbon allotrope named as T-II carbon by moving
carbon atoms in T-carbon half of the lattice vector along the crystallographic
$a$-axis and then stacking it with T-carbon as shown in Figs. 31(c) and (d)
[328]. T-II carbon has the space group of $Pn\bar{3}m$ with each primitive
cell containing two C4 units. The specific characteristics of T-II carbon
include: (1) T-II carbon exhibits a larger stiffness constant $C_{11}$ (560
GPa) than that of T-carbon (198 GPa) due to the denser stacking and more
compact structure of the former, indicating the high incompressibility along
the principal axis of cubic T-II carbon. (2) T-II carbon is metastable, and it
is more stable than T-carbon under the pressure above 6.8 GPa. Thus it is
expected to fabricate T-II carbon under high-pressure condition. (3) T-II
carbon is a semiconductor with an indirect energy gap of about 0.88 eV; (4)
The calculated hardness of T-II carbon (27 GPa) is higher than that of
T-carbon (5.6 GPa); (5) The calculations on the ideal strength and electron
localization function show the bonds of T-II carbon can withstand larger shear
strain than those of T-carbon.

###  V.3 Bct-C16

By the molecular dynamics simulation, it was found that T-carbon underwent a
structural phase transition at about 500 K, where the carbon tetrahedron in
T-carbon will become a non-planar square, forming a new carbon material with
body-centered tetragonal lattice coined as bct-C16 as shown in Figs. 31(e) and
(f) [329]. In addition to high-temperature heating, the phase transition may
be induced by doping elements (Cu, Fe) to break the carbon tetrahedrons in
T-carbon. There are two kinds of non-planar rings in bct-C16 including the
ring composed of four carbon atoms with the bond length of 1.483 Å and the
ring consisting of eight carbon atoms with the bond length of 1.361 and 1.483
Å. Bct-C16 with quasi $sp^{2}$ binding feature was revealed to be a
topological nodal ring semimetal protected by the time reversal and inversion
symmetries. Because of weak and negligible spin-orbit coupling for carbon
atoms, the nodal ring in bct-C16 is stable.

Bct-C16 can be regarded as the three-dimensional extended structure of
octagraphene [334], which is a versatile structurally favorable periodic
$sp^{2}$-bonded carbon atomic planar sheet with $C_{4v}$ symmetry. The crystal
structure of octagraphene is the same as the top view of bct-C16 in Fig.
31(e). Octagraphene has attracted widespread attention due to its excellent
thermoelectric, optical and mechanical properties, and the potential
applications in high-temperature superconductivity and hydrogen storage, etc
[335, 336, 337, 338, 339, 340].

Figure 32: Two crystal structures inspired by T-carbon: (a) $sp^{2}$ diamond
obtained by substituting each C–C bond in diamond with a distorted ring
composed of six carbon atoms and (b) its [111] view; (c) L-carbon obtained by
replacing each carbon atom in h-diamond with one carbon tetrahedron and (d)
its [001] view. Copyright (2013) Royal Society of Chemistry. (a) and (b)
Reprinted (adapted) with permission from ref [330]. (c) and (d) Reprinted
(adapted) with permission from ref [331]

###  V.4 sp2-diamond

Inspired by the proposal of T-carbon, He $et$ $al.$ proposed another carbon
allotrope coined as $sp^{2}$-diamond by substituting the C-C bond in diamond
with a ring composed of six carbon atoms in a specific direction and position
as shown in Figs. 32(a) and (b) [330]. $sp^{2}$-diamond also has the
Fd$\bar{3}$m space group with carbon atoms occupying the Wyckoff position
(0.451, 0.451, 0.726). The interesting properties of sp2-diamond are listed
here: (1) $sp^{2}$-diamond has a low density (2.116 $g/cm^{3}$), about 1.5
times that of T-carbon, but lower than that of most three-dimensional carbon
materials. The very low density and porous structure indicate its potential
for hydrogen storage, catalysts, and molecular sieves. (2) Comparing the
phonon spectrum, PDOS and temperature dependent Helmholtz free energies of
$sp^{2}$-diamond to that of T-carbon, $sp^{2}$-diamond is dynamically stable;
(3) Unlike the most previously reported 3D carbon materials with $sp^{2}$
network, which are metals, $sp^{2}$-diamond is a semiconductor with a direct
band gap of 1.66 eV because in $sp^{2}$-diamond the 2$s$, 2$p_{x}$, 2$p_{y}$,
and 2$p_{z}$ orbitals are all involved in the hybridization.

###  V.5 L-carbon, C–c–C4, H–c–C4, H–2c–C4

Diamond has two different structures: c-diamond and h-diamond, which have two
and four carbon atoms in the primitive cell, respectively [341]. T-carbon was
obtained by replacing each carbon atom in c-diamond with one carbon
tetrahedron. Applying the same substitution rule to h-diamond, a new structure
similar to T-carbon coined as L-carbon was obtained [331]. Based on the C4
units substitution method, if half of the carbon atoms in the two kinds of
diamond are replaced by C4 units, several novel carbon materials can be
obtained, including C-c-C4 with half of carbon atoms in c-diamond substituted
by C4 units, H-c-C4 with the non-adjacent carbon atoms in h-diamond replaced
by C4 unit and H–2c–C4 with two adjacent carbon atoms in the primitive cell of
h-diamond replaced by C4 units. The crystal structure of L-carbon was shown in
Figs. 32(c) and (d). The density, cohesive energy, band gap, density of
vibrational modes and bulk modulus of L-carbon are similar to those of
T-carbon. Most of the characteristics of C-c-C4 are similar to H-c-C4 because
of their close structure of C4 units. In order to further understand the
micromechanism, the root-mean-square (RMS) of bond angles and bond lengths
were calculated for these materials, and the results show that the larger the
proportion of C4 units is, the greater the change of RMS of bond angles is,
i.e. the greater the strain of the material is. Because the same proportion of
C4 units in L-carbon and T-carbon, as well as C-c-C4 and H-c-C4, the strains
and several properties of the two are similar.

The electronic band structures of these four materials show that they are all
semiconductors, but different from T-carbon with the direct band gap, and they
exhibit indirect band gaps in the range of 3.2-4.7 eV. The Raman shifts of
four carbon allotropes in the high-frequency region are related to the local
vibration modes of C4 units, while in the medium frequencies are associated
with the collective vibration modes. Because of the different structures and
components of C4 units, these carbon allotropes possess different Raman
shifts.

Similar to T-carbon, these carbon materials also have low density due to the
substitution of C4 units, especially L-carbon (1.50 $g/cm^{3}$). It is worth
noting that L-carbon has a one-dimensional channel with a radius of 2.65 Å,
which can be used to screen very small molecules in gas mixtures.

Inspired by the structure of T-carbon, three stable counterparts cF-B8 [342],
cF-Al8 [343] and T-Si [344] were proposed by replacing the carbon atoms in
T-carbon with B, Al and Si atoms, respectively. The metal structure cF-B8 has
a low density of 0.9 g/cm3, which is close to that of water. cF-Al8 is a
semimetal with high plasticity and a very low density of 0.61 g/cm3. T-Si is a
metal silicon allotrope with a density of 0.869 g/cm3. The calculated results
of adsorption and migration of Li and Li+ in T-Si indicate its potential
application in the field of lithium ion batteries.

###  V.6 Carboneyane

Figure 33: (a) The geometrical structure of carboneyane and (b) its primitive
cell, where the carbon atoms are classified into three types C1, C2, C3 by
bonding nature, and the red, green and blue colored bonds represent triple,
double and single bonds, respectively. (c) Electronic band structures and
partial density of states (PDOS) of carboneyane. The blue and red colored
circles represent type-I and type-II band dispersions, respectively. (d)
Shapes of the two nodal lines obtained from DFT calculations. (e) A constant
energy slice at 0.16 eV, where the bright lines showing the drumhead surface
states. Copyright (2019) Elsevier. Reprinted (adapted) with permission from
ref [203].

In 2019, the first carbon allotrope with simultaneously all three kind of
hybrid bonds ($sp$, $sp^{2}$ and $sp^{3}$) was proposed, which is coined as
carboneyane, as shown in Fig. 33 [203]. It has the space group of $C2/m$ with
a low equilibrium density of 1.43 $g/cm^{3}$. There are eight carbon atoms in
one primitive cell and the ratio of $sp$, $sp^{2}$, and $sp^{3}$ hybrid atoms
is 1:2:1. Carboneyane is found to be feasible in experiments due to its
energetically, dynamically and kinetically stable structure and much lower
enthalpy than that of T-carbon. The calculations on the mechanical properties
of carboneyane show that it has good ductility. Due to the exotic bonding mode
and unique distribution of different hybrid bonds, carboneyane shows the
anisotropy in electrical transport.

Carboneyane is shown to be a topological metal with a pair of nodal lines
appealing in the (010) plane, which are protected by the mirror symmetry. The
nodal lines are hybrid with type-I and type-II Weyl points and are robust
against the negligibly weak SOC in carboneyane. The drumhead surface states
can be obtained in carbonyane as the feature of a nodal line semimetal. Owing
to the high specific capacity of Li, K, Mg atoms (558 mAh/g) and low energy
barrier of Li+ migration (0.094 eV) in carboneyane, it may exhibit great
potential applications in the storage and absorption of atoms as well as the
battery anode.

###  V.7 Cycligraphyne and Cyclicgraphdiyne

Figure 34: The crystal structures of cyclicgraphyne and (b) cyclicgraphdiyne,
and (c) and (d) their corresponding band structures and partial DOS. Reprinted
(adapted) with permission from ref [241].

The view from [110] direction of T-carbon shows a kagome-like lattice with the
triangles separated compared to kagome lattice [Fig. 3(c)], which leads to the
flat band above Fermi level in the path $L$-$\Gamma$ [Fig. 4]. If one and two
acetylene bonds are inserted between adjacent carbon triangles Fig. 3(c), two
new stable structures, coined as cyclicgraphyne and cyclicgraphdiyne [241],
can be obtained, respectively, as shown in Fig. 34, which have the space group
of $P6/mmm$. The calculated results show a flat band with a band width of
about 2 and 145 meV near Fermi level in the above two structures,
respectively, which mainly comes from the valley part of the total electron
charge density. By doping holes into cyclicgraphdiyne to make its flat band
partially occupied, a phase transition from non-magnetic to ferromagnetic
(half-metal) states can be obtained.

Both cyclicgraphyne and cyclicgraphdiyne have a crossing point between the
flat band and another parabolic band, in which the flat band is called
singular flat band (SFB) [345]. The system with SFB can develop anomalous
Landau level spectrum as the response of the magnetic field [346]. Since the
two structures only contain light carbon atoms, the spin-orbit coupling has
little effect on the band structure just like graphene. In addition, both the
ferromagnetism caused by the instability of the flat band and electron
correlation effect as a result of effective Coulomb interaction have
negligible effects on the bandwidth of the flat band in cyclicgraphdiyne
[346]. The perfect flat band structure of these two carbon allotropes makes
them promising candidates for the experimental realization of SFBs.

##  VI Prospects and outlook

Figure 35: Members of carbon family.

In the past decade since the proposal of T-carbon, various properties of
T-carbon have been widely investigated. However, besides these studies, there
are still many interesting properties of T-carbon that need to explore. The
future works on T-carbon may focus on the following aspects: how to promote
its application in the field of energy; how to find new T-carbon based
superconductors; and how to use the existing results on T-carbon to inspire
the research of other classical carbon structures or vice versa. Furthermore,
the experimental realization of T-carbon adds a new member to the carbon
family (see Fig. 35). As the massive production of T-carbon is feasible, one
may expect that T-carbon may have wide practical applications in near future,
which calls for more experimental explorations.

The possibility of improving the thermoelectric properties of T-carbon remains
to be further studied. The characteristics of a good thermoelectric material
can be loosely said as ”phonon-glass $\&$ electron-crystal”, which indicates
the materials with a low thermal conductivity as in a glass, and a high
electrical conductivity as in crystals [347]. To achieve this purpose, doping
elements in semiconductors is a wise and feasible method [348]. Precise
control and optimization of doping in the hollow structure of T-carbon should
be studied in depth in future. In addition, one can also improve the
thermoelectric performance of T-carbon by introducing nanotwin structures
[349], applying external field [220] or bond nanodesigning [350, 351].

Although T-carbon has a high hydrogen storage capacity, it still needs to be
improved. The effects of different surface areas, pore shapes, and pore
volumes on hydrogen storage in T-carbon should be investigated. It was
reported that carbon materials could induce chemisorption to enhance their
hydrogen storage capacity by dispersing metal catalysts onto the surface
[352]. Moreover, the ability of carbon materials to bind hydrogen molecules
can also be enhanced by doping with appropriate elements [353]. These methods
could be applied to promote the hydrogen storage capacity in T-carbon.

In addition to the applications in thermoelectrics and hydrogen storage, the
porous structure of T-carbon makes it an alternative cathode material in
lithium-ion battery, which requires further experimental attempts including
its cycle stability, power density, and reversible capacity. On the other
hand, the voids in T-carbon can be considered as molecular sieves for
filtering small molecules. The potential applications of T-carbon in other
energy fields, such as electrochemistry, adsorption, energy storage,
supercapacitors, aerospace, electronics and so on, are worthy of further
study.

Beyond the applications of T-carbon in energy fields, new T-carbon based