Three-Dimensional Crystalline Modification of ... - ACS Publications

Subscriber access provided by ALBRIGHT COLLEGE

Surfaces, Interfaces, and Catalysis; Physical Properties of Nanomaterials and Materials 2

Three-Dimensional Crystalline Modification of Graphene in all-sp Hexagonal Lattices with or without Topological Nodal Lines Jian-Tao Wang, Yuting Qian, Hongming Weng, EnGe Wang, and Changfeng Chen

J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00844 • Publication Date (Web): 30 Apr 2019 Downloaded from http://pubs.acs.org on May 2, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry Letters

Three-dimensional Crystalline Modification of Graphene in all-sp2 Hexagonal Lattices with or without Topological Nodal Lines Jian-Tao Wang,∗,†,‡,¶ Yuting Qian,†,‡ Hongming Weng,†,¶,§ Enge Wang,∥,¶,§ and Changfeng Chen⊥ Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China, School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China , Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China , CAS Center for Excellence in Topological Quantum Computation, Beijing 100190, China, International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China, and Department of Physics and Astronomy, University of Nevada, Las Vegas, Nevada 89154, USA E-mail: [email protected]

∗ To

whom correspondence should be addressed National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China ‡ School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China ¶ Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China § CAS Center for Excellence in Topological Quantum Computation, Beijing 100190, China ∥ International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China ⊥ Department of Physics and Astronomy, University of Nevada, Las Vegas, Nevada 89154, USA † Beijing

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract The discovery of fullerenes, nanotubes and graphene has ignited tremendous interest in exploring additional all-sp2 carbon networks with novel properties. Here we identify by ab initio calculations a new series of three-dimensional crystalline modification of carbon in allsp2 bonding networks that comprises trigonal polycyclic benzenoid nanoflakes in a 2n2 (n ≥ 4) atom hexagonal cell. The resulting 32-, 50-, 72-, and 98-atom structures (termed as tr32, tr50, ¯ tr72, and tr98) in trigonal (P3m1) symmetry are characterized as the crystalline modification of (n × n × 1)-graphene in AA stacking, which are energetically more stable than or comparable to solid fcc-C60 and (5,5) carbon nanotube. Electronic band-structure calculations show that tr72 without 2d (1/3, 2/3, z) symmetric carbon atoms is a semiconductor, while tr32, tr50, and tr98 with 2d carbon atoms are topological nodal-line semimetals comprising nodal lines on the H-K-H ′ edge in the hexagonal Brillouin zone, as a three-dimensional extension of the Dirac point at the K-point in two-dimensional graphene. The present findings establish an additional crystalline modification of graphene in the all-sp2 carbon allotrope family and offer insights into its outstanding structural and electronic properties.

TOC

Network structure of tr32 carbon and its band decomposed charge density distributions between H-K points on nodal lines. Carbon is among the most versatile elements, and its half-filled valence electrons of 2s2 2p2 support four basic types of single, double, triple, and aromatic carbon-carbon bonds in sp3 -, sp2 - and sp-hybridized states, 1–9 as well as those in alkane, alkene, alkyne, and benzene-type arene structures. At ambient conditions, graphite is the thermodynamically most stable carbon allotrope. The polycyclic benzenoid carbon atoms form a two-dimensional (2D) benzenoid sp2 bonding network. 2


Page 2 of 20

Page 3 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Diamond is the second most stable carbon allotrope, which can be converted from graphite via slipping, buckling, and cross-linking of the carbon sheets under high-pressure and high-temperature conditions. 10–16 The polycyclic saturated carbon atoms form a very rigid three-dimensional (3D) carbon network in a methane-like tetrahedral sp3 bonding. In addition to graphite and diamond, during the last three decades, tremendous attention has been focused on synthesizing a variety of new carbon allotropes under laboratory conditions, 17 including the well-known carbon nanotube (CNT), 18 fullerenes, 19 graphene, 20 and graphdiyne. 21 The simplest sp-carbyne chain has been recently synthesized 22 despite its rather high energy of ∼ 1 eV/atom above that of graphite. Moreover, based on the ethene-type planar π -conjugation, 7 a three-dimensional three-connected (3D3C) crystalline modification of graphite (termed rh6 carbon with a small band gap of 0.47 eV) 23 in allsp2 bonding networks has been derived by ab initio calculations and confirmed experimentally in the milled fullerene soot. 24 Topological nodal line semimetals are a fascinating class of quantum materials that possess extraordinary electronic and transport properties. 25–39 They host nodal lines formed by band crossings near the Fermi level, and when these nodal lines are projected to a surface, there is topologically protected drumhead state nested inside of the nodal lines. Such surface drumhead flat bands are promising for generating high-temperature superconductors. 38 Recently 3D conductive interconnected graphene networks have been synthesized by chemical vapor deposition. 40 Theoretical studies suggested that these 3D graphene networks support topological nodal line semimetals, 41–50 with the nodal line as a closed ring or a line traversing the whole Brillouin zone (BZ). The topological semimetals with nodal lines that go through the whole BZ have been found in sp2 -sp3 hybrid network structures such as interpenetrated graphene network C6 (ign-C6 ), 41 simple orthorhombic C12 (so-C12 ), 46 and body-centered tetragonal C40 (bct-C40 ); 49 meanwhile, topological semimetals with closed nodal rings localized in a mirror plane of the BZ have been found in all-sp2 carbon network structures such as Mackay-Terrones carbon (MTC) crystal, 37 body-centered orthorhombic C16 (bco-C16 ), 43 and body-centered tetragonal C16 (bct-C16 ). 45 These works provide a new way to build viable 3D graphene network structures into novel topological semimetals.

3



In this Letter, we report ab initio calculations that identify a new type of 3D3C crystalline modification of carbon in all-sp2 bonding networks that comprises trigonal graphitic nanoflakes in ¯ trigonal (P3m1) symmetry in a 2n2 (n ≥ 4) atom unit cell. The resulting 32-, 50-, 72-, and 98-atom hexagonal structures (termed as tr32, tr50, tr72, and tr98 carbon) are topologically corresponding to a two-dimensional (n × n) graphene lattice and energetically more stable than solid fcc-C60 and comparable to (5,5)-CNT. Their dynamical stability have been verified by phonon mode analysis. Electronic band-structure calculations and symmetry analysis reveal that the 2d (1/3, 2/3, z) symmetric carbon atoms in tr32, tr50, and tr98 structures play a key role to establish nodal line semimetals, however, the absence of 2d atoms leads to a band gap of 0.71 eV in tr72 structure similar to that in rh6 carbon. 23 The topological nodal-line feature in tr32, tr50, and tr98 carbon was further confirmed by Berry phase calculations. Moreover, when the nodal lines are projected to the (100) surface, there is topologically protected drumhead state nested inside of the nodal lines. The calculations were carried out using the density functional theory as implemented in the Vienna ab initio simulation package (VASP). 51 The generalized gradient approximation (GGA) developed by Armiento-Mattsson (AM05) 52 were adopted for the exchange-correlation potential. The all-electron projector augmented wave (PAW) method 53 was adopted with 2s2 /2p2 treated as valence electrons. A plane-wave basis set with a large energy cutoff of 800 eV was used. Convergence criteria employed for both the electronic self-consistent relaxation and the ionic relaxation were set to 10−6 eV and 0.01 eV/Å for energy and force, respectively. A hybrid density functional method based on the Heyd-Scuseria-Ernzerhof scheme (HSE06) 54 was used to calculate electronic properties. Phonon calculations are performed using the phonopy code. 55 The spin-orbit coupling (SOC) were calculated using the standard GGA-PBE method. 56 Berry phase was calculated using the Wannier Tools package 57 based on a Wannier tight-binding model constructed by Wannier90. 58 We first present structural characterization of the newly constructed carbon phases. Figure 1a ¯ shows the tr32 carbon. It has a 32-atom hexagonal unit cell in trigonal (P3m1, D33d , No. 164) symmetry with equilibrium lattice parameters a = 9.4026 Å and c = 3.3907 Å, with carbon atoms occupying the 12 j (0.6683, 0.5927, 0.0936), 6i (0.6826, 0.8413, 0.7863), 2d (0.3333, 0.6667,

4


Page 4 of 20

Page 5 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


¯ No.164) Figure 1: Three-dimensional all-sp2 network structures of graphene in trigonal (P3m1, 2 symmetry in a 2n (n ≥ 4) atom hexagonal cell. (a) tr32 carbon structure with lattice parameters a = 9.4026 Å and c = 3.3907 Å; (b) tr50 carbon structure with lattice parameters a = 11.8896 Å and c = 3.4445 Å; (c) tr72 carbon structure with lattice parameters a = 14.3255 Å and c = 3.5442 Å; (d) tr98 carbon structure with lattice parameters a = 16.7985 Å and c = 3.5986 Å. The 2d (1/3, 2/3, z) symmetric atoms in tr32, tr50, and tr98 structure are marked by red dots (details on Wyckoff positions are given in Table S1 in Supporting Information). (e) The hexagonal Brillouin eΓ e Me (left) e Le Azone (BZ) with nodal lines on the H-K-H ′ edge. The projected (100) surface BZ Γ¯ ¯ ¯ ¯ ¯ and projected (110) surface BZ Γ-M-L-A-Γ (right) are shown relative to the bulk BZ Γ-M-L-A-Γ along the [100] and [110] crystalline directions, respectively.

Table 1: Calculated equilibrium structural parameters (space group, volume V0 per atom in Å3 , lattice parameters a and c, bond lengths dC−C in Å), total energy Etot , bulk modulus B0 , and electronic band gap Eg for tr32, tr50, tr72, and tr98 carbon along with rh6, fcc-C60 , graphite, and diamond at zero pressure, compared to available experimental data. 5,60 Structure ¯ Diamond (Fd 3m)

Method AM05 Exp 5 ¯ rh6 (R3m) AM05 ¯ tr32 (P3m1) AM05 ¯ tr50 (P3m1) AM05 ¯ tr72 (P3m1) AM05 ¯ tr98 (P3m1) AM05 ¯ fcc-C60 (Fm3) AM05 Exp 60 Graphite (P63 /mmc) AM05 Exp 5

V0 (Å3 ) 5.605 5.673 7.968 8.119 8.439 8.767 8.981 12.141 12.082 8.814 8.784

a (Å) 3.552 3.567 6.902 9.403 11.889 14.326 16.799 14.269 14.260 2.462 2.460

5

c (Å)

3.470 3.391 3.445 3.544 3.599

6.710 6.704

dC−C (Å) Etot (eV) 1.538 -9.018 1.544 1.359, 1.483 -8.550 1.374∼1.500 -8.711 1.382∼1.510 -8.813 1.379∼1.514 -8.884 1.382∼1.518 -8.923 1.395∼1.448 -8.684 1.456∼1.487 1.422 -9.045 1.420


B0 (GPa) Eg (eV) 451 5.36 446 5.47 299 0.47 298 Semimetal 288 Semimetal 278 0.71 272 Semimetal 198 280 286


0.7797), 6i (0.9114, 0.8227, 0.4397), and 6i (0.1597, 0.5799, 0.8421) Wyckoff positions denoted by C1 , C2 , C3 , C4 , and C5 , respectively. These carbon atoms form two zigzag-type (4 × 4) trigonal graphene nanoflakes; meanwhile, such carbon nanoflakes are cross-linked via ethene-type

π -conjugation (C1 =C1 ), making a corrugated graphene layer in a 3D3C all-sp2 network (see Figure S1 in the Supporting Information). It topologically corresponds to a (4 × 4) superlattice of graphene. Thus, tr32 carbon can be characterized as a 3D crystalline modification of graphene. There are six sets of distinct carbon-carbon bonds: two longer bonds of 1.500 Å (C4 -C4 ) and 1.467 Å (C1 -C2 ), corresponding to the C(sp2 )-C(sp2 ) single bond in 1,3-butadiene, and two shorter bonds of 1.374 Å (C2 =C4 ) and 1.386 Å (C1 =C1 ), corresponding to the C(sp2 )=C(sp2 ) double bond length in 1,3-butadiene, associated with ethene-type planar π -conjugation; 7 meanwhile, two aromatic bond length of 1.422 Å (C1 -C5 ) and 1.429 Å (C3 -C5 ) at the inner part are close the value of 1.422 Å in graphene. Similar bonding behaviors are also found in tr50, tr72, and tr98 carbon structures as shown in Figure 1b, 1c, and 1d (the detailed Wyckoff positions are given in Table S1 in the Supporting Information). From tr32 to tr98, the interlayer spacings are changed from 3.391 Å to 3.599 Å, slightly larger than graphite and close to the value reported in the milled fullerene soot. 24 These new carbon structures present a new series of 3D3C crystalline modification of graphene in a ¯ 2n2 (n ≥ 4) atom hexagonal unit cell in trigonal (P3m1) symmetry, and can be easily derived from a (n × n × 1) hexagonal graphene lattice, following the Wells’s approach. 59 Note that among these carbon structures, the 2d (1/3, 2/3, z) Wyckoff positions have maximal symmetry and play a key role to establish the topological semimetal state with nodal lines on the H-K-H ′ edge (see Figure 1e) in tr32, tr50, and tr98 carbon, while the absence of 2d positions induce a band gap in tr72 carbon (see Figure 4c) as in rh6 carbon. 23 These structures are consisting of interconnected graphitic nanoflakes and maybe fabricated by the polycyclic aromatic hydrocarbons (PAHs). Figure 2 presents calculated total energy versus volume per atom for tr32, tr50, tr72, and tr98 in comparison with data for rh6, 23 bco-C16 , 43 fcc-C60 , 60 (5,5)-CNT, and graphite in all-sp2 bonding networks and diamond in all-sp3 bonding networks. The energetic stability sequence is estimated

6


Page 6 of 20

Page 7 of 20

-8.4 -8.5

Energy (eV/atom)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


rh6 bco-C16 fcc-C60

-8.6 -8.7 -8.8

(5,5) -8.9

tr32 tr72

-9.0 Graphite

Diamond -9.1

5

6

7

tr50 tr98

8

9

10

11

12

13

14

3

Volume (Å /atom)

Figure 2: Energy versus volume per atom for tr32, tr50, tr72, and tr98 carbon in comparison with those for rh6, bco-C16 , fcc-C60 , (5,5)-CNT, diamond, and graphite. to be: rh6 < bco-C16 ≤ fcc-C60 < tr32 < tr50 < tr72 < (5,5)-CNT < tr98 < graphite and diamond. While tr32, tr50, tr72, and tr98 are slightly (0.10∼0.30 eV/atom) higher in energy than diamond and graphite, they are more stable than the reported rh6, bco-C16 , and fcc-C60 , and comparable to (5,5)-CNT for tr72 and tr98. The calculated equilibrium structural parameters are listed in Table 1. With increasing unit cell size from tr32 (n = 4) toward tr98 (n = 7), the equilibrium volume gradually rises toward that of graphite. By fitting the calculated total energy as a function of volume to Murnaghan’s equation of state, 61 we obtain the bulk modulus (B0 ) of 298, 288, 278, 272 GPa for tr32, tr50, tr72, and tr98, respectively, which are between 299 GPa for rh6 and 280 GPa for graphite in accordance to the atomic density. To assess dynamical stability, we have calculated phonon dispersion for tr32, tr50, tr72, and tr98 carbon using the phonopy code. 55 The obtained phonon band structures are shown in Figure 3. The highest phonon frequencies are 1608 cm−1 , 1614 cm−1 , 1628 cm−1 , and 1623 cm−1 for tr32, tr50, tr72, and tr98 carbon, respectively, which are close to the highest phonon frequency of 1610 cm−1 for graphite. 62 Throughout the entire Brillioun zone, no imaginary frequencies are observed, confirming the dynamical stability of these new all-sp2 crystalline allotropes of carbon. We next discuss the electronic properties. The electronic band structures and density of states (DOS) are calculated based on the hybrid density functional method (HSE06) 54 and plotted in Figure 4. The results show that tr72 carbon is a semiconductor with an indirect band gap of 7



1400

1400

1200

1200

-1

-1

Frequency (cm )

(c) 1600

Frequency (cm )

(a) 1600

1000 800 600

1000 800 600

400

400

200

200

0

Γ

A H

K Γ M

0

L

Γ

1600

(d) 1600

1400

1400

1200

1200

A H

K ΓM

L

K ΓM

L

-1

-1

Frequency (cm )

(b)

Frequency (cm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 20

1000 800 600

1000 800 600

400

400

200

200

0

Γ

A H

K ΓM

0

L

Γ

AH

Figure 3: Phonon band structures at zero pressure for tr32 (a), tr50 (b), tr72 (c), and tr98 (d). The highest phonon frequencies are close to the highest phonon frequency of 1610 cm−1 for graphite. 62 0.71 eV. The conduction band minimum is located along the Γ-A direction and the valence band maximum is located at the Γ point (see Figure 4c). On the other hand, for tr32, tr50, and tr98 carbon, the conduction band and valence band around the Fermi level are connected at the H and K points and form a nodal line along the H-K direction (see Figure 4a, 4b, 4d). Distinct from the reported nodal lines in ign-C6 , 41 so-C12 , 46 and bct-C40 49 that go through the whole BZ or nodal rings in MTC, 37 bco-C16 , 43 and bct-C16 45 that reside inside a mirror plane of the BZ, the nodal lines in tr32, tr50, and tr98 carbon are on the H-K-H ′ edge in the hexagonal BZ (Figure 1e). Moreover, band decomposed charge density distributions at and between the H, K points (see Figure S2 in the Supporting Information) reveal that the nodal lines are mainly coming from the pz orbitals of carbon atoms at 2d sites, like the Dirac point at the K point in 2D graphene composed by the pz orbitals of atoms at 2c (1/3, 2/3, 0) sites. Therefore, the nodal lines in tr32, tr50, and tr98 carbon are 3D extensions of the Dirac point at the K point in 2D graphene. These nodal-line behaviors can be understood based on the recently reported topological elec-

8


Page 9 of 20

Energy (eV)

(a) 2

2

1

1

0

0

-1

-1

-2

-2

Γ

A

H

K

Γ M

L

0

5

10

DOS (states/eV)

Energy (eV)

(b) 2

2

1

1

0

0

-1

-1

-2

-2

Γ

A H

K Γ M

0

L

5

10

15

20

DOS (states/eV)

Energy (eV)

(c) 2

2

1

1

0

0

-1

-1

-2

-2

Γ

A H

K Γ M

L

0

2

2

1

1

0

0

-1

-1

-2

-2

Γ

A H

K Γ M

L

5

10

15

20

DOS (states/eV)

(d)

Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

10

20

30

DOS (states/eV)

Figure 4: Calculated electronic band structures and density of states (DOS). (a) tr32, (b) tr50, (c) tr72, and (d) tr98. The Fermi level is set to zero eV. The tr72 carbon has an indirect band gap of 0.71 eV. The conduction band and valence band around the Fermi level in tr32, tr50, and tr98 carbon are connected to form nodal lines along the H-K direction.

9



tronic band theory. 63 The little group at the H and K point is D3 , and it is C3 for the point on the path H-K. The band-representations derived from the 2d Wyckoff positions are H3 (2) and K3 (2) at the H and K point, respectively (see Figure S3a in the Supporting Information). Both are 2D irreducible representations 63 and this indicates doubly degenerate nodal points at H and K; meanwhile, for a k-point on the H-K path, considering the additional time reversal symmetry and inversion symmetry of the system, two conjugate 1D irreducible representations P2 (1) and P3 (1) of the C3 group can be combined to form a 2D irreducible representation. This ensures the double degenerate nodes and protects the nodal line along H-K-H ′ . The other Wyckoff positions, such as 12 j and 6i, are in lower symmetry and they cannot guarantee the presence of doubly degenerate nodes along H-K (such as in tr72). When spin-orbit coupling (SOC) is included, it opens a gap of 10−4 eV along the nodal lines (see Figure S3b in the Supporting Information). Such a small SOC splitting is neglected in the following discussion. Furthermore, Berry phase along a closed path encircling the nodal lines 64 is calculated to be π (see Figure S4a in the Supporting Information). We also choose two lines (perpendicular to the x-z plane) in ky direction passing through the whole BZ inside or outside of the area between two separated nodal lines (see Figure S4b in the Supporting Information). Berry phase along these lines is calculated to be either π or 0, respectively. The quantized Berry phase further confirms the topological nodal-line feature in tr32, tr50, and tr98 carbon. We finally discuss the surface states of these nodal-line semimetals. Figure 5a and 5b show the surface band structures calculated using a nine-layer thick slab geometry along the [100] crystalline direction. The surface dangling bonds in Figure 5b are saturated with hydrogen atoms as shown eΓ e Me is set relative to the bulk BZ Γ-M-L-A-Γ, e Le Ain Figure 5d. The projected surface BZ Γincluding two projected nodal lines (left picture in Figure 1e). When the two nodal lines are projected onto the surface BZ, they can produce one topologically protected surface flat band around the Fermi level (red lines), either inside between the two HH’ nodal lines (region containing e point in Figure 5a) or outside (region containing the BZ boundaries MeL e in Figure 5b), the Γ depending on the termination of the surface without or with saturation by hydrogen atoms. 29

10


Page 10 of 20

Page 11 of 20

(b)

(a) 2

2

1

1

Energy (eV)

Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

0

-1

-1

-2

-2

~Γ M ~

~L A ~

~ Γ

~ M ~ Γ

~L A ~

~ Γ

Figure 5: Calculated surface band structures of tr32 using the standard GGA-PBE functional. (a,b) the (100) surface band structures obtained using a slab geometry along the [100] direction. The surface flat band can be inside (a) or outside (b) the surface projected nodal lines, depending on the termination of the surface without (a) or with (b) saturation by hydrogen atoms. (c,d) the partial e point in (a) and charge density isosurfaces (0.01 e/Å3 ) related to the (red) surface bands at the Γ e point in (b). the M The appearance of surface states in a nodal-line semimetal is arising from a quantized Berry phase along the periodical path perpendicular to the surface. The Berry phase is calculated and equal to π or 0 in different region (see Figure S4a, S4b in Supporting Information). Surface states are expected to appear in the surface BZ with π phase. However, whether the dangling bond is saturated with hydrogen atoms or not will modify the surface potential and change the position of the surface state, but will not eliminate it. Here, the hydrogenation saturates the dangling bond so that the surface state inside of the two nodal lines in Figure 5a is pulled into the body state and the surface state outside is dragged to Fermi level and appeared in Figure 5b. This indicates the robustness of topological surface states and both these two cases are topologically equivalent. In Figure 5c and 5d, the partial charge density isosurfaces related to the (red) energy bands near the e point in Figure 5a and M e point in Figure 5b are plotted. The electronic charges Fermi level at Γ are located on the surface carbon layers, confirming that the surface flat bands are indeed derived from the surface carbon atoms.

11



For comparison, we have also calculated the (110) surface band structures using a ten-layer thick slab geometry along the [110] crystalline direction for the termination of surface without or with saturation by hydrogen atoms (see Figure S5 in the Supporting Information). The projected ¯ Γ¯ is set relative to the bulk BZ Γ-M-L-A-Γ, but it contains only one projected ¯ M¯ L¯ Asurface BZ Γ¯ H) ¯ K)¯ A( ¯ direction (right picture in Figure 1e). As shown in Figure S4c, the nodal line along the Γ( Berry phase along kx direction passing through the BZ are all zero, which means there will be no topological surface states. This is consistent with the results in Figure S5. In summary, we have identified by ab initio calculations a new type of three-dimensional crystalline modification of carbon in all-sp2 bonding networks that comprises trigonal polycyclic ben¯ zenoid nanoflakes in trigonal (P3m1) symmetry. The resulting tr32, tr50, tr72, and tr98 carbon structures are topologically corresponding to a two-dimensional (n × n) superlattice of graphene and can be regarded as a three-dimensional crystalline modification of graphene in AA stacking. These new carbon structures are energetically more table than or comparable to solid fcc-C60 and (5,5) carbon nanotube. Electronic band structure calculations and symmetry analysis reveal that the maximal symmetric 2d (1/3, 2/3, z) carbon atoms in tr32, tr50, and tr98 structure play a key role to establish a zero-gap topological semimetal that hosts nodal lines on the H-K-H ′ edge in the hexagonal Brillouin zone, as a three-dimensional extension of the Dirac point at the K-point in two-dimensional graphene. However, the lower symmetric Wyckoff positions, such as 12 j and 6i, cannot guarantee the doubly degenerate nodes along the H-K direction and induce a band gap in tr72 structure as in cR6 carbon 7 and rh6 carbon. 23 Moreover, when the nodal lines in the bulk are projected onto the (100) surface Brillouin zone, they produce one topologically protected surface flat band around the Fermi level, either outside or inside of two projected nodal lines, depending on the termination of the surface with or without saturation by hydrogen atoms. The present results expand the realm of nodal manifolds in topological semimetals and improve the understanding of structural and electronic properties of these fascinating carbon structures.

12


Page 12 of 20

Page 13 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ASSOCIATED CONTENT Supporting Information Additional structural information for tr32, tr50, tr72, and tr98 carbon structures (Figure S1 and Table S1); band decomposed charge density distributions for tr32 (Figure S2); electronic band structures of tr32 with and without spin-orbit coupling (Figure S3); calculated Berry phase (Figure S4), and the calculated (110) surface band structures (Figure S5).

AUTHOR INFORMATION Corresponding Author [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This study was supported by the National Natural Science Foundation of China (Grants No. 11674364 and No. 11674369), the Ministry of Science and Technology of China (Grants No. 2016YFA0300600 and 2018YFA0305700), the Science Challenge Project (No. TZ2016004) and the Strategic Priority Research Program of Chinese Academy of Sciences (Grants No. XDB07000000 and XDB28000000). Y.Q. and H.W are also supported by K.C. Wong Education Foundation (Grant No. GJTD-2018-01).

References (1) Balaban, A. T. Carbon and its nets. Computers Math. Applic. 1989, 17, 397-346.

13



(2) Hoffmann, R.; Hughbanks, T.; Kertesz, M.; Bird, P. H. A hypothetical metallic allotrope of carbon. J. Am. Chem. Soc. 1983, 105, 4831-4832. (3) Karfunkel, H. R.; Dresslert, T. New hypothetical carbon allotropes of remarkable stability estimated by modified neglect of diatomic overlap solid-state self-consistent field computations. J. Am. Chem. Soc. 1992, 114, 2285-2288. (4) O’Keeffe, M.; Adams, G. B.; Sankey, O. F. Predicted new low energy forms of carbon. Phys. Rev. Lett. 1992, 68, 2325-2328. (5) Occelli, F.; Loubeyre, P.; Letoullec, R. Properties of diamond under hydrostatic pressures up to 140 GPa. Nat. Mater. 2003, 2, 151-154. (6) Ribeiro, F. J.; Tangney, P.; Louie, S. G.; Cohen, M. L. Structural and electronic properties of carbon in hybrid diamond-graphite structures. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 72, 214109. (7) Wang, J. T.; Chen, C. F.; Kawazoe, Y. New carbon allotropes with helical chains of complementary chirality connected by ethene-type π -conjugation. Sci. Rep. 2013, 3, 03077. (8) Belenkov, E. A.; Greshnyakov, V. A. Classification of structural modifications of carbon. Phys. Solid State 2013, 55, 1754-1764. (9) Kuc, A.; Seifert, G. Hexagon-preserving carbon foams: Properties of hypothetical carbon allotropes. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 214104. (10) Irifune, T.; Kurio, A.; Sakamoto, S.; Inoue, T.; Sumiya, H. Ultrahard polycrystalline diamond from graphite. Nature 2003, 421, 599-600. (11) Irifune, T.; Kurio, A.; Sakamoto, S.; Inoue, T.; Sumiya, H.; Funakoshi, K. Formation of pure polycrystalline diamond by direct conversion of graphite at high pressure and high temperature. Phys. Earth Planet. Inter. 2004, 143-144, 593-600.

14


Page 14 of 20

Page 15 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(12) Mao, W. L.; Mao, H. K.; Eng, P. J.; Trainor, T. P.; Newville, M.; Kao, C. C.; Heinz, D. L.; Shu, J. F.; Meng, Y.; Hemley, R. J. Bonding changes in compressed superhard graphite. Science 2003, 302, 425-427. (13) Li, Q.; Ma, Y. M.; Oganov, A. R.; Wang, H. B.; Wang, H.; Xu, Y.; Cui, T.; Mao, H. K.; Zou, G. T. Superhard monoclinic polymorph of carbon. Phys. Rev. Lett. 2009, 102, 175506. (14) Umemoto, K.; Wentzcovitch, R. M.; Saito, S.; Miyake, T. Body-centered tetragonal C4 : A viable sp3 carbon allotrope. Phys. Rev. Lett. 2010, 104, 125504. (15) Wang, J. T.; Chen C. F.; Kawazoe Y. Low-temperature phase transformation from graphite to sp3 orthorhombic carbon. Phys. Rev. Lett. 2011, 106, 075501. (16) Amsler, M.; Flores-Livas, J. A.; Lehtovaara, L.; Balima, F.; Ghasemi, S. A.; Machon, D.; Pailhes, S.; Willand, A.; Caliste, D.; Botti, S. San Miguel, A.; Goedecker, S.; Marques, M. A. L. Crystal structure of cold compressed graphite. Phys. Rev. Lett. 2012, 108, 065501. (17) Hirsch, A. The era of carbon allotropes. Nat. Mater. 2010, 9, 868-871. (18) Iijima, S. Helical microtubules of graphitic carbon. Nature 1991, 354, 56-58. (19) Kroto, H. W.; Heath, J. R.; Obrien, S. C.; Curl, R. F.; Smaliey, R. E. C60 : Buckminsterfullerene. Nature 1985, 318, 162-163. (20) Novoselov, K.; Geim, A.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric field effect in atomically thin carbon films. Science 2004, 306, 666-669. (21) Li, G.; Li, Y.; Liu, H.; Guo, Y.; Zhu, D. Architecture of graphdiyne nanoscale films, Chem. Commun. 2010, 46, 3256-3258. (22) Chalifoux, W. A.; Tykwinski, R. R. Synthesis of polyynes to model the sp-carbon allotrope carbyne. Nat. Chem. 2010, 2, 967-971. 15



(23) Wang, J. T.; Chen, C. F.; Wang, E. G.; Kawazoe, Y. A new carbon allotrope with six-fold helical chains in all-sp2 bonding networks. Sci. Rep. 2014, 4, 04339. (24) Calderon, H. A.; Estrada-Guel, I.; Alvarez-Ramirez, F.; Hadjiev, V. G.; Robles-Hernandez, F. C. Morphed graphene nanostructures: Experimental evidence for existence. Carbon, 2016, 102, 288-296. (25) Burkov, A. A.; Hook, M. D.; Balents, L. Topological nodal semimetals. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 235126. (26) Phillips, M.; Aji, V. Tunable line node semimetals. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 115111. (27) Fang, C.; Chen, Y.; Kee, H. Y.; Fu, L. Topological nodal line semimetals with and without spin-orbital coupling. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 081201. (28) Kim, Y.; Wieder, B. J.; Kane, C. L.; Rappe, A. M. Dirac line nodes in inversion-symmetric crystals. Phys. Rev. Lett. 2015, 115, 036806. (29) Yu, R.; Weng, H.; Fang, Z.; Dai, X.; Hu, X. Topological node-line semimetal and dirac semimetal state in antiperovskite Cu3 PdN. Phys. Rev. Lett. 2015, 115, 036807. (30) Heikkila, T. T.; Volovik, G. E. Nexus and Dirac lines in topological materials. New J. Phys. 2015, 17, 093019. (31) Xie, L. S.; Schoop, L. M.; Seibel, E. M.; Gibson, Q. D.; Xie, W.; Cava, R. J. A new form of Ca3 P2 with a ring of Dirac nodes. APL Mater. 2015, 3, 083602. (32) Mullen, K.; Uchoa, B.; Glatzhofer, D. T. Line of Dirac nodes in hyperhoneycomb lattices. Phys. Rev. Lett. 2015, 115, 026403. (33) Chan, Y.-H.; Chiu, C.-K.; Chou, M. Y.; Schnyder, A. P. Ca3 P2 and other topological semimetals with line nodes and drumhead surface states. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 205132. 16


Page 16 of 20

Page 17 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(34) Bzdusek, T.; Wu, Q.; Ruegg, A.; Sigrist, M.; Soluyanov, A. A. Nodal-chain metals. Nature 2016, 538, 75-78. (35) Yan, Z.; Wang, Z. Tunable Weyl points in periodically driven nodal line semimetals. Phys. Rev. Lett. 2016, 117, 087402. (36) Bian, G.; Chang, T. R.; Sankar R, R.; Xu, S.-Y.; Zheng, H.; Neupert, T.; Chiu, C.-K.; Huang, S.-M.; Chang, G.; Belopolski, I.; Sanchez, D. S.; Neupane, M.; Alidoust, N.; Liu, C.; Wang, B.; Jeng, H.-T.; Bansil, A.; Chou, F.; Lin, H.; Hasan, M. Z. Topological nodal-line fermions in spin-orbit metal PbTaSe2 . Nat. Commun. 2016, 7, 10556. (37) Weng, H.; Liang, Y.; Xu, Q.; Yu, R.; Fang, Z.; Dai X.; Kawazoe, Y. Topological node-line semimetal in three-dimensional graphene networks. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 045108. (38) Volovik, G. E. From standard model of particle physics to room-temperature superconductivity. Phys. Scr. 2015, T164, 014014. (39) Yang, B.; Zhou, H.; Zhang, X.; Liu, X.; Zhao, M. Dirac cones and highly anisotropic electronic structure of super-graphyne. Carbon 2017, 113, 40-45. (40) Chen, Z.; Ren, W.; Gao, L.; Liu, B.; Pei, S.; Cheng, H. Three-dimensional flexible and conductive interconnected graphene networks grown by chemical vapour deposition. Nat. Mater. 2011, 10, 424-428. (41) Chen, Y.; Xie, Y.; Yang, S. A.; Pan, H.; Zhang F.; Cohen, M. L.; Zhang, S. B. Nanostructured carbon allotropes with Weyl-like loops and points. Nano Lett. 2015, 15, 6974-6978. (42) Dong, X.; Hu, M.; He, J. L.; Tian, Y. J.; Wang, H. T. A new phase from compression of carbon nanotube with anisotropic Dirac fermions. Sci. Rep. 2015, 5, 10713. (43) Wang, J. T.; Weng, H.; Nie, S.; Fang, Z.; Kawazoe Y.; Chen, C. F. Body-centered orthorhombic C16 : A novel topological node-line semimetal. Phys. Rev. Lett. 2016, 116, 195501. 17



(44) Cheng, Y.; Du, J.; Melnik, R.; Kawazoe Y.; Wen, B. Novel three dimensional topological nodal line semimetallic carbon. Carbon 2016, 98, 468-473. (45) Cheng, Y.; Feng, X.; Cao, X. T.; Wen, B.; Wang, Q.; Kawazoe Y.; Jena, P. Body-centered tetragonal C16 : A novel topological node-line semimetallic carbon composed of tetrarings. Small 2017, 13, 1602894. (46) Wang, J. T.; Chen, C. F.; Kawazoe, Y. Topological nodal line semimetal in an orthorhombic graphene network structure. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 97, 245147. (47) Zhong, C. Y.; Chen, Y. P.; Yu, Z. M.; Xie, Y.; Wang, H.; Yang, S. A.; Zhang, S. B. Threedimensional pentagon carbon with a genesis of emergent fermions. Nat. Commun. 2017, 8, 15641. (48) Li, Z. Z.; Chen, J.; Nie, S.; Xu, L. F.; Mizuseki, H.; Weng, H.; Wang, J. T. Orthorhombic carbon oC24: A novel topological nodal line semimetal. Carbon 2018, 133, 39-43. (49) Wang, J. T.; Nie, S.; Weng, H.; Kawazoe, Y.; Chen, C. F. Topological nodal-net semimetal in a graphene network structure. Phys. Rev. Lett. 2018, 120, 026402. (50) Feng, X.; Wu, Q. S.; Cheng, Y.; Wen, B.; Wang, Q.; Kawazoe, Y.; Jena, P. Monoclinic C16 : sp2 -sp3 hybridized nodal-line semimetal protected by PT -symmetry. Carbon 2018, 127, 527532. (51) Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 1116911186. (52) Armiento, R.; Mattsson, A. E. Functional designed to include surface effects in self-consistent density functional theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 72, 085108. (53) Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 50, 17953-17979. 18


Page 18 of 20

Page 19 of 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(54) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid functionals based on a screened coulomb potential. J. Chem. Phys. 2003, 118, 8207-8215. (55) Togo, A.; Oba, F.; Tanaka, I. First-principles calculations of the ferroelastic transition between rutile-type and CaCl2 -type SiO2 at high pressures. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 134106. (56) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865-3868. (57) Wu, Q. S.; Zhang, S. N.; Song, H. F.; Troyer, M.; Soluyanov, A. A. WannierTools: an opensource software package for novel topological materials. Comput. Phys. Commun. 2018, 224, 405-416. (58) Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N. Wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 2008, 178, 685-699. (59) Wells, A. F. The geometrical basis of crystal chemistry. Acta Crystallogr. 1954, 7, 535-554. (60) David, W. I. F.; Ibberson, R. M.; Matthewman, J. C.; Prassides, K.; Dennis, T. J. S.; Hare, J. P.; Kroto, H. W.; Taylor, R.; Walton, D. R. M. Crystal structure and bonding of ordered C60 . Nature 1991, 353, 147-149. (61) Murnaghan, F. D. The compressibility of media under extreme pressures. Proc. Nat. Acad. Sci. U.S.A. 1944, 30, 244-247. (62) Maultzsch, J.; Reich, S.; Thomsen, C.; Requardt, H.; Ordejon, P. Phonon dispersion in graphite. Phys. Rev. Lett. 2004, 92, 075501. (63) Bradlyn, B.; Elcoro, L.; Cano, J.; Vergniory, M. G.; Wang, Z.; Felser, C.; Aroyo, M. I.; Andrei Bernevig, B. Topological quantum chemistry. Nature, 2017, 547, 298-305.

19



(64) Hyart, T.; Ojajarvi, R.; Heikkila, T. T. Two topologically distinct Dirac-line semimetal phases and topological phase transitions in rhombohedrally stacked honeycomb lattices. J. Low Temp. Phys. 2018, 191, 35-48.

20


Page 20 of 20

Three-Dimensional Crystalline Modification of ... - ACS Publications

Recommend Documents