All-Silicon Topological Semimetals with Closed Nodal Line - The

Subscriber access provided by University of Rhode Island | University Libraries

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

All-Silicon Topological Semimetals with Closed Nodal Line Zhifeng Liu, Hongli Xin, Li Fu, Yingqiao Liu, Tielei Song, Xin Cui, Guojun Zhao, and Jijun Zhao J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b03345 • Publication Date (Web): 12 Dec 2018 Downloaded from http://pubs.acs.org on December 15, 2018

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

All-Silicon Topological Semimetals with Closed Nodal Line Zhifeng Liu,†, ‡ Hongli Xin,† Li Fu,† Yingqiao Liu,# Tielei Song,† Xin Cui,† Guojun Zhao, † Jijun Zhao‡, # †

School of Physical Science and Technology, Inner Mongolia University, Hohhot 010021, China ‡

#

Beijing Computational Science Research Center, Beijing 100094, China

Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China

Abstract: Owing to the natural compatibility with current semiconductor industry, silicon allotropes with diverse structural and electronic properties provide promising platforms for the next-generation Si-based devices. After screening 230 all-silicon crystals in the zeolite frameworks by first-principles calculations, we disclose two structurally stable Si allotropes (AHT-Si24 and VFI-Si36) containing open channels as topological node-line semimetals with Dirac nodal points forming a nodal loop in the kz=0 plane of Brillouin zone. Interestingly, their nodal loops protected by inversion and time-reversal symmetries are robust against SU(2) symmetry breaking due to very weak spin-orbit coupling of Si. When the nodal lines are projected onto the (001) surface, flat surface bands can be observed because of the nontrivial topology of the bulk band structures. Our discoveries extend the topological physics to the three-dimensional Si materials, highlighting the possibility to realize low-cost, nontoxic and semiconductor-compatible Si-based electronics with topological quantum states.

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

TOC Graphic:

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


Silicon is the backbone of modern microelectronics due to its natural abundance, nontoxicity, propensity for doping, high-temperature stability, and native oxide passivation layer.1,2 At ambient conditions, the ground state of Si crystal adopts the cubic diamond structure (d-Si) with sp3 hybridization, which is an indirect-bandgap (1.17 eV) semiconductor. Moreover, the tetrahedral bonding character of Si atoms leads to an exceptionally complex energy landscape, giving rise to numerous metastable allotropes with small energy differences from the d-Si phase. To date, many compact Si allotropes3-8 have been proposed and discovered in laboratory by compressing/decompressing the known phases or ultrafast laser-induced confined microexplosion.8 In addition, more and more low-density phases of Si have also been proposed and even synthesized.1,9-17 These allotropes with different arrangements of Si atoms possess diverse electronic band structures, offering an effective way toward the desired physical properties for Si-based electronics. For instance, a number of direct- or quasidirect-bandgap semiconducting allotropes of Si13-16 have been predicted by means of ab initio calculations, which could be utilized in the thin-film photovoltaic devices due to their modest bandgap and advanced optoelectronic properties beyond those of d-Si. Motivated by those theoretical studies, a recent experiment using a high-pressure precursor method reported the first synthesis of Si allotrope with quasidirect bandgap (1.3 eV), namely Cmcm-Si24.17 On the other hand, Sung et al. proposed the first metallic phase (P/6m-Si6) of Si in open framework, featuring as a superconductor with the critical temperature of ~12 K.18 Notably, among the known metastable allotropes, BC8-Si was once predicted to be a 3



semimetal,19,20 whereas recent experiment confirmed it is a narrow direct gap semiconductor21. Then a question arises naturally: is it possible to obtain a semimetal in the crystal structures of pure Si, especially a topological semimetal (TSM) as predicted in a previous toy model22 ? Ignited by the successful theoretical design23 and experimental observation24,25 of Weyl semimetals (WSMs), the study of TSMs has attracted significant attention recently.26-31 As new topological states of matter, TSMs are characterized by the non-accidental crossing between the conduction and valence bands near the Fermi level, and the linear energy dispersion as a function of momentum. The forming crossing-points (termed as nodes or nodal points) are either discrete or continuous. In the discrete case, according to the degeneracy of nodes, TSMs can be classified into three catalogs: (i) two-fold degenerate WSMs,23-26 (ii) triply degenerate nodal points semimetals (TDNPSMs),32-35 and (iii) four-fold degenerate Dirac semimetals (DSMs).36-39 For the continuous situations, the nodal points form a periodically continuous line running across the Brillouin zone (BZ) or a closed loop inside the BZ; thus the corresponding systems are called topological node-line semimetals (TNLSMs). Under specific symmetry operations, quantum phase transition may occur between these TSM states. Importantly, their nontrivial topology encoded in the bulk band structure gives rise to remarkable physical properties, such as high carrier mobility, novel surface sates (e.g., Fermi arcs and drumhead-like flat bands), and Chiral anomaly.26-29 In this regard, it is crucial to search for superb materials to realize the aforesaid TSM states for practical applications. After years of efforts, many 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


fascinating TSM materials have been proposed and even confirmed experimentally. 26-30

However, most of them not only contain heavy elements that are often expensive

or toxic, but also are hardly compatible with the current Si-based semiconductor industry. In this letter, we show by first-principles calculations that it is indeed possible to obtain the desired TSM in the all-silicon solids. Upon screening 230 candidate structures in zeolite framework, we identify two open-framework Si allotropes termed as AHT-Si24 and VFI-Si36, belonging to TSMs. With all sp3 bonding, these two structures show satisfactory stability, prevailing many other metastable Si allotropes. Both of them exhibit a novel TNLSM state with Dirac nodal points forming a closed loop within the kz = 0 plane of BZ, and the flat bands around the Fermi level projected on the (001) surfaces. Distinct from most of the known TNLSMs formed by heavy elements, the symmetry-protected nodal lines in these two phases are robust against the breaking of SU(2) spin rotation symmetry at temperature higher than 14 K because of the weak spin-orbit coupling (SOC) of Si element. The pioneering synthesis of Cmcm-Si2417 has highlighted the possibility of unveiling new Si allotrope with zeolite structure since that Cmcm-Si24 shares the same topology with the zeolite type code CAS in the database of International Zelolite Association (IZA).40 Interestingly, full examination for the lattice structures of the known Si allotropes reveals that the prominent Si clathrates9,41 i.e., Si46 and Si136, also correspond to the zeolite frameworks of MTN and MEP, respectively. Likewise, the recently predicted phase hP12-Si from particle swarm optimization algorithm 5



search,14 has identical framework of the zeolite CAN. Considering these facts and the four-fold-coordinated character of all zeolite frameworks, it is reasonable to search the possible TSM allotropes by screening the Si crystals constructed from the IZA database (see Figure S1 of the Supplemental Materials for a flow of screening). The atomic and electronic structures of these 230 all-Si candidate structures from the IZA database were computed using density functional theory as implemented in the Vienna ab initio simulation package.42 The Perdew-Burke-Ernzerhof functional43 within the generalized gradient approximation (GGA) was employed to treat the exchange-correlation interactions. The hybrid functional HSE0644 was further employed to validate the PBE band structures for the semimetallic Si phases. The core-valence interactions were described by the PAW method45 with a kinetic energy cutoff of 500 eV. All atoms were fully relaxed until total energy and atomic force were less than 10−6 eV and 10−3 eV/Å, respectively. Monkhorst-Pack k-point meshes with a uniform density of 2π × 0.01 Å−1 were generated to sample the BZ. To explore the nontrivial topological properties of the TSM phases, the maximally localized Wannier functions46 were constructed to obtain the tight-binding Hamiltonian. Upon relaxation, all these hypothetic zeolite-like Si framework structures are well preserved. With such abundant Si crystal lattices, a rich variety of electronic properties is observed from the calculated PBE band structures, ranging from direct and indirect semiconductors, metals, to semimetals. The key data are listed in Table S1 (included Figure S2) of the Supplemental Materials. For the plausible semimetallic systems, more accurate HSE06 calculations were conducted to validate the PBE band 6


Page 6 of 20

Page 7 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


structures. Finally, two zeolite-like Si structures, i.e., AHT-Si24 and VFI-Si36 with linear bands crossing near the Fermi level, are confirmed as semimetals. Hereafter, we will focus on AHT-Si24 as a representative.

17 Figure 1. (a) The primitive cell of AHT-Si24 with Cmcm ( D2h ) space group consisting

of two kinds of crystallographically inequivalent atoms: Si1 (red) and Si2 (blue) at 8f (0, 0.3504, 0.9361) and 16h (0.6829, 0.1504, 0.5639) Wyckoff positions, respectively. The lattice constants of AHT-Si24: a = 12.18 Å, b = 7.52 Å, c = 6.37 Å. (b) A perspective view from crystallographic c-axis for the structure of AHT-Si24. (c) The cohesive energy as a function of volume per Si atom for AHT-S24 and VFI-Si36 in comparison with some known Si allotropes. (d) Phonon dispersion of AHT-Si24 confirming its dynamic stability. 7



Figure 1(a) shows the crystal structure of AHT-Si24 which has two kinds of inequivalent Si atoms, marked as Si1 and Si2. All the atoms are four-fold coordinated, forming sp3-like hybridization (each Si1 atom bonds with two Si1 and two Si2 atoms, while every Si2 is connected to one Si1 and three Si2 atoms). The average bond angle is 107.58°, rather close to the perfect tetrahedral angle of 109.47° in d-Si phase. Similar to Cmcm-Si2417, AHT-Si24 is featured by periodic open channels constructed by ten-membered rings along the crystallographic c-axis [see Figure 1(b)]. This suggests that the AHT-Si24 phase could also be synthesized by the high-pressure precursor method17 like Cmcm-Si24. If a precursor Na-Si system, e.g., AHT-Na4Si24 (see Figure S3 and Table S2), is produced and retains its stability at ambient condition, the guest Na atoms could be removed from the open channels by a degassing process.17,18,47 The calculated cohesive energy of AHT-Si24 is 5.17 eV, which is less than that of d-Si by 0.25 eV/atom [Figure 1(c)], due to the ring strain in the open channels. In spite of this, it is more stable than the open-framework allotropes P/6m-Si6 (5.07 eV)18 and Si20-T (5.14 eV)13, and other synthesized compact allotropes of Si, including -Sn, Imma-Si, SH-Si, Cmca-Si and Si-VII, and Si-X (see Figure S4) at zero pressure. The dynamic stability of AHT-Si24 has been confirmed by phonon dispersion calculation, as shown in Figure 2(d). Moreover, its independent elastic constants C11, C12, C13, C22, C23, C33, C44, C55, and C66 are 134, 36, 29, 96, 24, 162, 27, 31, and 36 GPa, respectively, satisfying the mechanical stability criteria [i.e., Cii > 0 (i=1, 2, ..., 6), C11 + C22 + C33 + 2 (C12 + C13 + C23) > 0, C11 + C22  2C12 > 0, C11 + 8


Page 8 of 20

Page 9 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


C33  2C13 > 0, C22 + C33  2C23 > 0] for an orthorhombic crystal.48

Figure 2. (a) Bulk electronic band structure of AHT-Si24. (b) The corresponding bulk and (001)-surface BZ together with the high-symmetry k-points. (c) Orbital decomposed linear crossing bands around the nodal points. (d) Colorful band structure near the nodal points weighted by the contribution ratio, i.e.,  = WSi2 / (WSi1 + WSi2 ) , where W denotes the weight of the corresponding atom.

The electronic band structure of AHT-Si24 is presented in Figure 2(a). Along two perpendicular paths, namely ΓX and ΓY, the highest valence band (HVB) and lowest conduction band (LCB) exhibit Dirac linear dispersions in the vicinity of the Fermi energy, and cross at asymmetric Dirac nodal points D1 ( λ1sec 2θ /4 , λ1sec 2θ /4 , 9



0) and D2 ( λ2 /2 , λ2 /2 , 0) on the Fermi level. Here, 1 = 0.241 and 2 = 0.228 are the length ratios between Γ-D1 and Γ-X, and between Γ-D2 and Γ-Y, respectively;  = 31.69°is half of the angle between kx and ky vectors in the BZ. Note that the linear crossing also exists in the HSE06 band structure (Figure S5). To further explore the origin of band crossings in AHT-Si24, we calculated the orbital and atom decomposed band structures near D1 and D2. As shown in Figure 2(c) and 2(d), the crossing bands are mainly originated from the pxy states of Si2 atoms termed as Si2, pxy , and the s and pxy hybridized states of Si1 atoms denoted as Si1, s +pxy . Clearly, there exists exchange of energy ordering between Si2, pxy

and Si1, s +pxy , resulting in a

band crossing. Importantly, such band inversion is one of the key features for topological insulators49,50 as well as TNLSMs.27,51 To further reveal the pattern of the crossing nodal points in the entire BZ, we calculated the Fermi surface and 3D band structure of AHT-Si24. The results show that HVB and LCB cross with each other along a closed loop [Figure 3(a)], which locates on the kz = 0 plane of BZ due to the mirror reflection symmetry. Hence, the AHT-Si24 is a new member of topological node-line semimetals. A closer examination shows that the nodal loop is a rounded rectangle with the long edge parallel to ΓX line and the short one parallel to ΓY line [Figure 3(b)], which is a consequence of the orthorhombic crystalline symmetry. Naturally, one may wonder whether the low-energy Dirac Fermions along the two perpendicular directions exhibit anisotropic transport behavior? To explore this, the Fermi velocities, denoted as vF, have been evaluated by a linear fittings [ћvF ≈ dE(k)/dk] for the linear crossing bands. Along the 10


Page 10 of 20

Page 11 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


D1→X direction, i.e., the long side of the rounded rectangle, the Fermi velocities of the hole and electron carriers are 4.86×105 m/s and 5.44×105 m/s, respectively. Along the D2→Y direction, i.e., the short side of the rounded rectangle, the Fermi velocities of the hole and electron carriers are 7.76×105 m/s and 6.87×105 m/s, respectively. In other words, the carrier transport in AHT-Si24 is anisotropic. Remarkably, these velocities are comparable with that of graphene (9.42×105 m/s52) and even larger than that of silicene (5.31×105 m/s53).

Figure 3. (a) 3D band structure of the HVB and LCB for AHT-Si24. (b) The colorized contour plot of the energy band gap between HVB and LCB in the kz=0 plane. (c) The structure of 20-layer slab along the [001] crystalline direction containing 120 Si atoms. (d) The color mapped band structure for the slab using the surface as weight. 11



Page 12 of 20

According to the classification of TNLSMs,28 we infer that the AHT-Si24 belongs to type-B because its closed nodal line is protected by space inversion (P), time-reversal (T), and SU(2) spin-rotation symmetries. Specifically, if we enclose the nodal line with a ring, the independent Z2 index, termed as ζ1, should be topologically invariant. Here, the explicit expression of ζ1 is the Berry phase on the ring that interlocks A( k )  i

with



nocc

the

closed

un ( k )  k un ( k )

nodal

line:28

( 1) ζ1 =

 dkA( k )  dk

,

where

is the Abelian Berry connection. For AHT-Si24, ζ1 is

equal to 1, implying that the rounded-rectangle nodal line is topologically protected by the mentioned P, T and SU(2) symmetries. Usually, the nodal lines in TNLSMs may vanish with the breaking of symmetries. To investigate the possible topological phase transition, we carried out first-principles calculations of AHT-Si24 with the breaking of SU(2) symmetry (or by considering SOC). As known, in the presence of SOC, the band repulsion between opposite spins is usually non-negligible, making the closed nodal line unstable.27-29 For AHT-Si24, however, like the reported TNLSM materials formed by light elements,51,54-57 the computed SOC splitting is remarkably small (around 0.7 meV and 0.4 meV along Γ-X and Γ-Y lines, respectively), which can be ignored completely at temperature higher than 9 K. In this sense, the topological nodal line in AHT-Si24 is rather robust against SOC under ambient temperature. Thus the needed quantum phase transitions should resort to other kinds of symmetry breakings, such as P, T or both of them.27,28 For a TNLSM, another key feature is the existence of flat surface bands,27 which provides the possibility for experimental detection by angle-resolved photoemission 12


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


measurement.58 To visualize the topological surface states of AHT-Si24, we construct a 20-layer-thick slab along the [001] crystalline direction of primitive unit cell [see Figure 3(c)], whose surface dangling bonds are saturated by hydrogen atoms. Figure 3(d) illustrates the calculated band structure of the slab model, which is color mapped by the weight of the (001) surface. Around the Fermi level and projected  point, one can find the formation of the nearly flat surface state (i.e., drumhead-like surface state), which connects the projected Dirac nodal points and nestled inside the nodal line — a rounded rectangle. For practical application, such flat surface band could induce interesting physical effects, e.g., two-dimensional surface superconductivity with high critical temperature.59,60 We have also performed systematical calculations on VFI-Si36 phase, and the detailed results are shown in Figures. S6-S8 in the Supplemental Materials. Essentially, we can draw the same conclusions as those for AHT-Si24. As for VFI-Si36, the HVB and LCB linearly cross at the asymmetric points D1 along Γ-M and D2 along Γ-K, which are caused by the band inversion between Si1+Si2, pxy Si2, s +p xy

and

3 states. Due to the P63/mcm ( D6h ) symmetry, its nodal line winds into a

closed rounded-hexagon in the kz=0 plane. The computed SOC splitting is about 1.2 meV and 0.6 meV along Γ-K and Γ-M paths, respectively, which can be ignored at temperature higher than 14 K. Moreover, the projected flat surface bands can be found around the Fermi level as well. In summary, using first-principles calculations we have verified two Si allotropes with zeolite framework as topological semimetals, namely, AHT-Si24 and VFI-Si36. 13



Their structural stabilities were confirmed from cohesive energy, phonon dispersion and elastic constants. These two allotropes hold open channels along certain crystallographic axis, similar to the structural characteristics of the recently synthesized Cmcm-Si24; thus they can be possibly fabricated by the high-pressure precursor method. The calculated electronic band structures uncover that both AHT-Si24 and VFI-Si36 exhibit a novel TNLSM state, in which the Dirac nodal points with linear dispersion form a closed ring within the kz=0 plane of BZ. More importantly, the P and T symmetry-protected nodal lines, derived from the linear band crossings caused by Si-s and Si-pxy bands, are rather robust in the presence of SOC under room temperature. Furthermore the flat surface bands have been observed on their (001) surface, giving rise to exotic transport behavior (e.g., high-temperature surface superconductivity) and other strong correlation physical properties.61-63 Our prediction of TNLSM Si phases may pave a way to realize the all-silicon topological devices with low-cost, nontoxic and semiconductor- compatible features.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] ORCID Jijun Zhao: 0000-0002-3263-7159 Zhifeng Liu: 0000-0002-8151-6839 Notes

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


The authors declare no competing financial interest.

ACKNOWLEDGMENTS We acknowledge useful discussions with Prof. S.-W. Gao. This work is supported by the National Natural Science Foundation of China (11604165, 11574040) and Natural Science Foundation of Inner Mongolia (2016BS0104). The Supercomputing Center of Dalian University of Technology is acknowledged for providing computing resources.

Supporting Information Available: the flow of screening; the key data for all of the considered zeolite Si crystals; structure and stability of AHT-Na4Si24; energy vs. volume for some compact Si allotropes; electronic band structures from HSE06 calculation; atomic structure and stability of VFI-Si36; electronic band structure of VFI-Si36; nodal line and surface flat bands of VFI-Si36; simulated XRD patterns.

REFERENCES (1) Beekman, M.; Wei, K.; Nolas, G. S. Clathrates and Beyond: Low-Density Allotropy in Crystalline Silicon. Appl. Phys. Rev. 2016, 3, 040804. (2) Wippermann, S.; He, Y.; Vörös, M.; Galli, G. Novel Silicon Phases and Nanostructures for Solar Energy Conversion. Appl. Phys. Rev. 2016, 3, 040807. (3) Mujica, A.; Rubio, A.; Munoz, A.; Needs, R. J. High-Pressure Phases of Group-IV, III−V, and II−VI Compounds. Rev. Mod. Phys. 2003, 75, 863-912. (4) Besson, J. M.; Mokhtari, E. H.; Gonzalez, J.; Weill, G. Electrical Properties of Semimetallic Silicon III and Semiconductive Silicon IV at Ambient Pressure. Phys. Rev. Lett. 1987, 59, 473-476. 15



(5) Duclos, S. J.; Vohra, Y. K.; Ruoff, A. L. Experimental Study of the Crystal Stability and Equation of State of Si to 248 GPa. Phys. Rev. B 1990, 41, 12021-12028. (6) McMahon, M. I.; Nelmes, R. J. New High-Pressure Phase of Si. Phys. Rev. B 1993, 47, 8337-8340. (7) Wang, J. T.; Chen, C.; Mizuseki, H.; Kawazoe, Y. Kinetic Origin of Divergent Decompression Pathways in Silicon and Germanium. Phys. Rev. Lett. 2013, 110, 165503. (8) Rapp, L.; Haberl, B.; Pickard, C. J.; Bradby, J. E.; Gamaly, E. G.; Williams, J. S.; Rode, A. V. Experimental Evidence of New Tetragonal Polymorphs of Silicon Formed Through Ultrafast Laser-Induced Confined Microexplosion. Nat. Commun. 2015, 6, 7555. (9) Adams, G. B.; O’Keeffe, M.; Demkov, A. A.; Sankey, O. F.; Huang, Y.-M. Wide-Band-Gap Si in Open Fourfold-Coordinated Clathrate Structures. Phys. Rev. B 1994, 49, 8048-8053. (10) Pickard, C. J.; Needs, R. J. Hypothetical Low-Energy Chiral Framework Structure of Group 14 Elements. Phys. Rev. B 2010, 81, 014106. (11) Nguyen, M. C.; Zhao, X.; Wang, C.-Z.; Ho, K.-M. sp3-Hybridized Framework Structure of Group-14 Elements Discovered by Genetic Algorithm. Phys. Rev. B 2014, 89, 184112. (12) Amsler, M.; Botti, S.; Marques, M. A. L.; Lenosky, T. J.; Goedecker, S. Low-Density Silicon Allotropes for Photovoltaic Applications. Phys. Rev. B 2015, 92, 014101. (13) Xiang, H. J.; Huang, B.; Kan, E.; Wei, S. H.; Gong, X. G. Towards Direct-Gap Silicon Phases by the Inverse Band Structure Design Approach. Phys. Rev. Lett. 2013, 110, 118702. (14) Wang, Q.; Xu, B.; Sun, J.; Liu, H.; Zhao, Z.; Yu, D.; Fan, C.; He, J. Direct Band Gap Silicon Allotropes. J. Am. Chem. Soc. 2014, 136, 9826-9829. (15) Lee, I.-H.; Lee, J.; Oh, Y. J.; Kim, S.; Chang, K. J. Computational Search for Direct Band Gap Silicon Crystals. Phys. Rev. B 2014, 90, 115209. (16) Botti, S.; Flores-Livas, J. A.; Amsler, M.; Goedecker, S.; Marques, M. A. L. 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


Low-Energy Silicon Allotropes with Strong Absorption in the Visible for Photovoltaic Applications. Phys. Rev. B 2012, 86, 121204(R). (17) Kim, D. Y.; Stefanoski, S.; Kurakevych, O. O.; Strobel, T. A. Synthesis of an Open-Framework Allotrope of Silicon. Nat. Mater. 2015, 14, 169-173. (18) Sung, H.-J.; Han, W. H.; Lee, I.-H.; Chang, K. J. Superconducting Open-Framework Allotrope of Silicon at Ambient Pressure. Phys. Rev. Lett. 2018, 120, 157001. (19) Biswas, R.; Martin, R. M.; Needs, R. J.; Nielsen, O. H. Stability and Electronic Properties of Complex Structures of Silicon and Carbon Under Pressure: Density-Functional Calculations. Phys. Rev. B 1987, 35, 9559-9568. (20) Malone, B. D.; Sau, J. D.; Cohen, M. L. Ab initio Survey of The Electronic Structure of Tetrahedrally Bonded Phases of Silicon. Phys. Rev. B 2008, 78, 035210. (21) Zhang, H.; Liu, H.; Wei, K.; Kurakevych, O. O.; Le Godec, Y.; Liu, Z.; Martin, J.; Guerrette, M.; Nolas, G. S.; Strobel, T. A. BC8 Silicon (Si-III) is a Narrow-Gap Semiconductor. Phys. Rev. Lett. 2017, 118, 146601. (22) Shtyk, A.; Chamon, C. Topological Electronic Properties of Silicon. arXiv: 1705.10835v2 2018. (23) Weng, H.; Fang, C.; Fang, Z.; Bernevig, B. A.; Dai, X. Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides. Phys. Rev. X 2015, 5, 011029. (24) Lv, B. Q.; Weng, H. M.; Fu, B. B.; Wang, X. P.; Miao, H.; Ma, J.; Richard, P.; Huang, X. C.; Zhao, L. X.; Chen, G. F.; et al. Experimental Discovery of Weyl Semimetal TaAs. Phys. Rev. X 2015, 5, 031013. (25) Xu, S.-Y.; Belopolski, I.; Alidoust, N.; Neupane, M.; Bian, G.; Zhang, C.; Sankar, R.; Chang, G.; Yuan, Z.; Lee, C.-C.; et al. Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs. Science 2015, 349, 613-617. (26) Armitage, N. P.; Mele, E. J.; Vishwanath, A. Weyl and Dirac Semimetals in Three-Dimensional Solids. Rev. Mod. Phys. 2018, 90, 015001. (27) Weng, H.; Dai, X.; Fang, Z. Topological Semimetals Predicted from First-Principles Calculations. J. Phys.: Condens. Matter 2016, 28, 303001. 17



(28) Fang, C.; Weng, H.; Dai, X.; Fang, Z. Topological Nodal Line Semimetals. Chin. Phys. B 2016, 25, 117106. (29) Bernevig, A.; Weng, H.; Fang, Z.; Dai, X. Recent Progress in the Study of Topological Semimetals. J. Phys. Soc. Jpn. 2018, 87, 041001. (30) Weng, H.; Fang, C.; Fang, Z.; Dai, X. A New Member of the Topological Semimetals Family. Natl. Sci. Rev. 2017, 4, 798-799. (31) Zhang, X.; Jin, L.; Dai, X.; Liu, G. Topological Type-II Nodal Line Semimetal and Dirac Semimetal State in Stable Kagome Compound Mg3Bi2. J. Phys. Chem. Lett. 2017, 8, 4814-4819. (32) Weng, H.; Fang, C.; Fang, Z.; Dai, X. Coexistence of Weyl Fermion and Massless Triply Degenerate Nodal Points. Phys. Rev. B 2016, 94, 165201. (33) Weng, H.; Fang, C.; Fang, Z.; Dai, X. Topological Semimetals with Triply Degenerate Nodal Points in -Phase Tantalum Nitride. Phys. Rev. B 2016, 93, 241202. (34) Zhu, Z.; Winkler, G. W.; Wu, Q.; Li, J.; Soluyanov, A. A. Triple Point Topological Metals. Phys. Rev. X 2016, 6, 031003. (35) Sun, J.-P.; Zhang, D.; Chang, K. Coexistence of Topological Nodal Lines, Weyl Points, and Triply Degenerate Points in TaS. Phys. Rev. B 2017, 96, 045121. (36) Wang, Z.; Weng, H.; Wu, Q.; Dai, X.; Fang, Z. Three-Dimensional Dirac Semimetal and Quantum Transport in Cd3As2. Phys. Rev. B 2013, 88, 125427. (37) Wang, Z.; Sun, Y.; Chen, X.-Q.; Franchini, C.; Xu, G.; Weng, H.; Dai, X.; Fang, Z. Dirac Semimetal and Topological Phase Transitions in A3 Bi (A=Na, K, Rb). Phys. Rev. B 2012, 85, 195320. (38) Liu, Z. K.; Zhou, B.; Zhang, Y.; Wang, Z. J.; Weng, H. M.; Prabhakaran, D.; Mo, S.-K.; Shen, Z. X.; Fang, Z.; Dai, X.; et al. Discovery of a Three-Dimensional Topological Dirac Semimetal, Na3Bi. Science 2014, 343, 864-867. (39) Liu, Z. K.; Jiang, J.; Zhou, B.; Wang, Z. J.; Zhang, Y.; Weng, H. M.; Prabhakaran, D.; Mo, S. K.; Peng, H.; Dudin, P.; et al. A Stable Three-Dimensional Topological Dirac Semimetal Cd3As2. Nat. Mater. 2014, 13, 677-681. (40) Baerlocher, C.; McCusker, L. B.; Olson, D. H. Atlas of Zeolite Framework Types; Elsevier: Amsterdam; 2007. 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


(41) Gryko, J.; McMillan, P. F.; Marzke, R. F.; Ramachandran, G. K.; Patton, D.; Deb, S. K.; Sankey, O. F. Low-Density Framework Form of Crystalline Silicon with a Wide Optical Band Gap. Phys. Rev. B 2000, 62, R7707-R7710. (42) Kresse, G.; Furthmüller, J. Efficient Iterative Schemes for Ab initio TotalEnergy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169-11186. (43) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865-3868. (44) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid Functionals Based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207-8215. (45) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758-1775. (46) Marzari, N.; Vanderbilt, D. Maximally Localized Generalized Wannier Functions for Composite Energy Bands. Phys. Rev. B 1997, 56, 12847-12865. (47) Stefanoski, S.; Malliakas, C. D.; Kanatzidis, M. G.; Nolas, G. S. Synthesis and Structural Characterization of NaxSi136 (0 < x ≤

24) Single Crystals and

Low-Temperature Transport of Polycrystalline Specimens. Inorg. Chem. 2012, 51, 8686-8692. (48) Wu, Z.-j.; Zhao, E.-j.; Xiang, H.-p.; Hao, X.-f.; Liu, X.-j.; Meng, J. Crystal Structures and Elastic Properties of Superhard IrN2 and IrN3 From First Principles. Phys. Rev. B 2007, 76, 054115. (49) Qi, X.-L.; Zhang, S.-C. Topological Insulators and Superconductors. Rev. Mod. Phys. 2011, 83, 1057-1110. (50) Hasan, M. Z.; Kane, C. L. Colloquium: Topological Insulators. Rev. Mod. Phys. 2010, 82, 3045-3067. (51) 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 2015, 92, 045108. (52) Zhang, X.; Wang, A.; Zhao, M. Spin-Gapless Semiconducting Graphitic Carbon Nitrides: A Theoretical Design from First Principles. Carbon 2015, 84, 1-8. 19



(53) Drummond, N. D.; Zólyomi, V.; Fal'ko, V. I. Electrically Tunable Band Gap in Silicene. Phys. Rev. B 2012, 85, 075423. (54) Li, R.; Ma, H.; Cheng, X.; Wang, S.; Li, D.; Zhang, Z.; Li, Y.; Chen, X. Q. Dirac Node Lines in Pure Alkali Earth Metals. Phys. Rev. Lett. 2016, 117, 096401. (55) Wang, J. T.; Nie, S.; Weng, H.; Kawazoe, Y.; Chen, C. Topological Nodal-Net Semimetal in a Graphene Network Structure. Phys. Rev. Lett. 2018, 120, 026402. (56) Wang, J. T.; Weng, H.; Nie, S.; Fang, Z.; Kawazoe, Y.; Chen, C. Body-Centered Orthorhombic C16: A Novel Topological Node-Line Semimetal. Phys. Rev. Lett. 2016, 116, 195501. (57) Cheng, Y.; Feng, X.; Cao, X.; 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. (58) Bian, G.; Chang, T. R.; Sankar, R.; Xu, S. Y.; Zheng, H.; Neupert, T.; Chiu, C. K.; Huang, S. M.; Chang, G.; Belopolski, I.; et al. Topological Nodal-Line Fermions in Spin-Orbit Metal PbTaSe2. Nat. Commun. 2016, 7, 10556. (59) Kopnin, N. B.; Heikkilä, T. T.; Volovik, G. E. High-Temperature Surface Superconductivity in Topological Flat-Band Systems. Phys. Rev. B 2011, 83, 220503(R). (60) Volovik, G. E. From Standard Model of Particle Physics to Room-Temperature Superconductivity. Phys. Scr. 2015, 2015, 014014. (61) Kim, Y.; Wieder, B. J.; Kane, C. L.; Rappe, A. M. Dirac Line Nodes in Inversion-Symmetric Crystals. Phys. Rev. Lett. 2015, 115, 036806. (62) Yu, R.; Weng, H.; Fang, Z.; Dai, X.; Hu, X. Topological Node-Line Semimetal and Dirac Semimetal State in Antiperovskite Cu 3PdN. Phys. Rev. Lett. 2015, 115, 036807. (63) Zhang, C.-L.; Yuan, Z.; Bian, G.; Xu, S.-Y.; Zhang, X.; Hasan, M. Z.; Jia, S. Superconducting Properties in Single Crystals of the Topological Nodal Semimetal PbTaSe2. Phys. Rev. B 2016, 93, 054520.

20


Page 20 of 20

All-Silicon Topological Semimetals with Closed Nodal Line - The

Recommend Documents