DOCSLIB.ORG

HOT Graphene and HOT Graphene Nanotubes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
HOT Graphene and HOT Graphene Nanotubes Xu et al. Nanoscale Research Letters (2020) 15:56 https://doi.org/10.1186/s11671-020-3279-1 NANO REVIEW Open Access HOT Graphene and HOT Graphene Nanotubes: New Low Dimensional Semimetals and Semiconductors Lin-Han Xu1* , Shun-Qing Wu1, Zhi-Quan Huang2, Feng Zhang3, Feng-Chuan Chuang2, Zi-Zhong Zhu1* and Kai-Ming Ho3,4 Abstract We report a new graphene allotrope named HOT graphene containing carbon hexagons, octagons, and tetragons. A corresponding series of nanotubes are also constructed by rolling up the HOT graphene sheet. Ab initio calculations are performed on geometric and electronic structures of the HOT graphene and the HOT graphene nanotubes. Dirac cone and high Fermi velocity are achieved in a non-hexagonal structure of HOT graphene, implying that the honeycomb structure is not an indispensable condition for Dirac fermions to exist. HOT graphene nanotubes show distinctive electronic structures depending on their topology. The (0,1) n (n ≥ 3) HOT graphene nanotubes reveal the characteristics of semimetals, while the other set of nanotubes (1,0) n shows continuously adjustable band gaps (0~ 0.51 eV) with tube size. A competition between the curvature effect and the zone-folding approximation determines the band gaps of the (1,0) n nanotubes. Novel conversion between semimetallicity and semiconductivity arises in ultra-small tubes (radius < 4 Å, i.e., n < 3). Keywords: HOT graphene, Nanotubes, Electronic structures, First-principles calculations Introduction in sp2-bonded carbon systems are dependent on the Because of its bonding flexibility, carbon-based systems honeycomb structure. In the lower dimension, the carbon show an unlimited number of different structures with nanotube is a honeycomb structure rolled into a hollow an equally large variety of physical properties. These cylinder with nanometric diameter and μmlength[7–10]. physical properties are, in great part, the result of the di- As there is an infinite number of ways of rolling a sheet mensionality of these structures [1]. Graphene is a single into a cylinder, the large variety of possible helical geom- two-dimensional layer of carbon atoms bound in a hex- etries, defining the tube chirality, provides a family of agonal lattice structure [2] revealing a number of unique nanotubes with different diameters and microscopic struc- properties, such as massless carriers, high Fermi velocity tures [11–13]. The electronic and transport properties are [3], and Dirac cones [4, 5], which are characteristic of certainly among the most significant physical properties of two-dimensional Dirac fermions. The honeycomb lattice carbon nanotubes, and crucially depend on the diameter consisting of two equivalent carbon sublattices plays a and chirality [14–18]. Graphene nanotubes can be either crucial role in forming such intriguing properties [2]. semimetallic [14]orsemiconducting[19–21], with a band Enyashin and Ivanovskii [6] constructed 12 artificial 2D gap varying from zero to a few tenths of an eV, depending carbon networks but found no structures other than the on their diameter and chirality [10, 14, 16]. Furthermore, graphyne allotrope exhibit the graphene-like electronic the band gap of semiconducting tubes can be shown to be behavior. It seems to imply that the Dirac-like fermions simply related to the tube diameter. The semimetallic nanotubes also maintain the unique properties from gra- phene, such as massless carriers, high Fermi velocity [22], * Correspondence: [email protected]; [email protected] 1Department of Physics, Collaborative Innovation Center for Optoelectronic and Dirac cones [23]. Such remarkable results can be ob- Semiconductors and Efficient Devices, Jiujiang Research Institute, Xiamen tained from a variety of considerations, starting from the University, Xiamen, China so-called band-folding approach, based on knowledge of Full list of author information is available at the end of the article © The Author(s). 2020 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Xu et al. Nanoscale Research Letters (2020) 15:56 Page 2 of 11 the electronic properties of the graphene sheet, to the dir- Method of Calculation ect study of nanotubes using semiempirical tight-binding The present calculations on HOT graphene and HOT approaches [14, 16, 18, 23]. Comparing with more sophis- graphene nanotubes were performed by using a first- ticated ab initio calculations and available experimental principles method based on the density-functional results, finer considerations, such as curvature effects, kF theory (DFT) with the generalized gradient approxi- shifting [24, 25], σ-π hybridization [26] are introduced. mation (GGA) in the form of Perdew-Burke-Ernzerh Graphene and graphene-like materials [6] are considered (PBE) exchange-correlation functional [43], as imple- a revolutionary material for future generation of high- mented in the Vienna Ab initio Simulation Package speed electronic, radio frequency logic devices [27, 28], (VASP) [44, 45]. The wave functions were expanded thermally and electrically conductive reinforced compos- in plane waves up to a cutoff of plane wave kinetic ites [29, 30], catalyst [31], sensors [32–35], transparent energy of 520 eV. The Brillouin zone (BZ) integrals electrodes [27, 36], etc. basing on the unusual properties were performed by using a Monkhorst-Pack [46]sam- all above. Over the past few decades, carbon nanotubes pling scheme with a k-point mesh resolution of 2π × − have also shown great potential in logic circuits, gas stor- 0.03 Å 1. The unit cell basis vectors (representing age, catalysis, and energy storage because of their extraor- unit cell shape and size) and atomic coordinates were dinary electronic, mechanical, and structural properties fully relaxed in each system until the forces on all the [37–39]. Hence, the creation of new carbon allotropes (in- atoms were smaller than 0.01 eV/Å. cluding 2D and 1D) has been the focus of numerous theor- etical and experimental explorations because of their Results and Discussion fundamental scientific and technological importance [40]. Geometric and Electronic Structures of HOT Graphene However, completely clarifying the structures of these excit- The geometric structure of HOT graphene (Fig. 1a) ing carbon phases through current experimental technolo- shows a more complicated bonding situation than gra- gies is usually unrealistic due to their limited quantity, as phene. The variety of carbon polygons in HOT graphene well as the mixture of other phases. Theoretical prediction results in various carbon bonding characters. These is necessary and has yielded great success [31–35, 40–42]. polygons in HOT graphene share common edges with In this study, we designed a new allotrope of graphene each other, and the bonds can be distinguished by the that has two-dimensional Dirac fermions without an ex- two polygons they belong to. Therefore, in our research, clusively hexagonal structure. The new allotrope was con- they are named as 6–8 bonds, 4–8 bonds, 4–6 bonds, 6– structed with interlaced carbon hexagons, octagons, and 6 bonds, and 8–8 bonds. The 4–8 bonds and 6–8 bonds tetragons, and was named HOT graphene. HOT graphene have two different bond lengths: 1.44 Å and 1.47 Å for nanotubes were also constructed by rolling up HOT gra- 4–8 bonds; 1.41 Å and 1.48 Å for 6–8 bonds. The 4–6 phene sheet along with different directions. The electronic bonds, 6–6 bonds, and 8–8 bonds have unique bond property, curvature effect, kF shifting effect, etc. of HOT lengths of 1.44 Å, 1.46 Å, and 1.34 Å, relatively. Figure 1b graphene and nanotubes were calculated using ab initio shows the band structure and density of states (DOS) of calculations based on density function theory (DFT). HOT graphene with the corresponding BZ depicted in Fig. 1 (a) Geometry of HOT graphene; (b) Band structures and DOS of the HOT graphene; (c) the corresponding BZ of HOT graphene Xu et al. Nanoscale Research Letters (2020) 15:56 Page 3 of 11 Fig. 1c. The crossing point of energy bands at the Fermi calculation. Different rolling directions between (1,0) n and (0, level indicates semimetallicity of HOT graphene, which 1) n tubes result in the differences in geometry and bonding is confirmed by the vanishing DOS at the Fermi level. situation. Two tubes, (1,0)6 and (0,1)4, are depicted in Fig. 3d The Dirac point is located at (0, 0.0585, 0) adjacent to Γ. and e to describe the geometric differences between the two The 3D band structure (Fig. 2) presents the band sur- rolling directions. The arrangement of polygons along (1,0) faces near the Fermi level, where one can see the Dirac direction can be divided into two patterns: C4–C6–C8 (orange) cones formed by upper and lower conical surfaces meet- and C8–C6–C4 (blue) which is exactly opposite to each other. ing at two Dirac points exactly at the Fermi surface. The These two opposite patterns alternate along the circumferen- corresponding Fermi velocity (vF) of the Dirac fermions, tial direction of the tube. In direction (0,1), polygons arranging evaluated from the gradient of the linear dispersions of along the tube axis also have two patterns: C4–C8 (blue) and 5 the band structures, is 6.27 × 10 m/s, which is a little C6–C6 (orange). Two C4–C8 patterns alternate with one C6– 5 lower than 8.1 × 10 m/s [22] for graphene nanotube and C6 pattern along the circumferential direction. 5 8.6 × 10 m/s [47, 48] for graphene. The high vF implies To reveal the energy cost in rolling up a sheet into high mobility of carriers in the HOT graphene.
Load more

Recommended publications
  • The Electronic Properties of the Graphene and Carbon Nanotubes: Ab Initio Density Functional Theory Investigation
    International Scholarly Research Network ISRN Nanotechnology Volume 2012, Article ID 416417, 7 pages doi:10.5402/2012/416417 Research Article The Electronic Properties of the Graphene and Carbon Nanotubes: Ab Initio Density Functional Theory Investigation Erkan Tetik,1 Faruk Karadag,˘ 1 Muharrem Karaaslan,2 and Ibrahim˙ C¸ omez¨ 1 1 Physics Department, Faculty of Sciences and Letters, C¸ukurova University, 01330 Adana, Turkey 2 Electrical and Electronic Engineering Department, Faculty of Engineering, Mustafa Kemal University, 31040 Hatay, Turkey Correspondence should be addressed to Erkan Tetik, [email protected] Received 20 December 2011; Accepted 13 February 2012 AcademicEditors:A.Hu,C.Y.Park,andD.K.Sarker Copyright © 2012 Erkan Tetik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We examined the graphene and carbon nanotubes in 5 groups according to their structural and electronic properties by using ab initio density functional theory: zigzag (metallic and semiconducting), chiral (metallic and semiconducting), and armchair (metallic). We studied the structural and electronic properties of the 3D supercell graphene and isolated SWCNTs. So, we reported comprehensively the graphene and SWCNTs that consist of zigzag (6, 0) and (7, 0), chiral (6, 2) and (6, 3), and armchair (7, 7). We obtained the energy band graphics, band gaps, charge density, and density of state for these structures. We compared the band structure and density of state of graphene and SWCNTs and examined the effect of rolling for nanotubes. Finally, we investigated the charge density that consists of the 2D contour lines and 3D surface in the XY plane.
    [Show full text]
  • Nanotubes Electronics
    They are stronger than steel, but the most important uses for these threadlike macromolecules may be in faster, more efficient and more durable electronic devices by Philip G. Collins and Phaedon Avouris Nanotubes ElectronicsFOR early 10 years ago Sumio Iijima, sitting at an electron microscope at the NEC Fundamental Research Laboratory in Tsukuba, Japan, first noticed odd nano- scopic threads lying in a smear of soot. NMade of pure carbon, as regular and symmetric as crystals, these exquisitely thin, impressively long macromolecules soon became known as nanotubes, and they have been the object of intense scientific study ever since. Just recently, they have become a subject for engineering as well. Many of the extraordinary properties attributed to nanotubes—among them, superlative resilience, tensile strength and thermal stability—have fed fantastic predictions of microscopic robots, dent-resistant car bodies and earth- quake-resistant buildings. The first products to use nanotubes, however, exploit none of these. Instead the earliest applica- tions are electrical. Some General Motors cars already include plastic parts to which nanotubes were added; such plastic can be electrified during painting so that the paint will stick more readily. And two nanotube-based lighting and display prod- ucts are well on their way to market. In the long term, perhaps the most valuable applications will take further advantage of nanotubes’ unique electronic properties. Carbon nanotubes can in principle play the same role as silicon does in electronic circuits, but at a molecular scale where silicon and other standard semiconductors cease to work. Although the electronics industry is already pushing the critical dimensions of transistors in commercial chips be- low 200 nanometers (billionths of a meter)—about 400 atoms wide—engineers face large obstacles in continuing this miniaturization.
    [Show full text]
  • Transport, Criticality, and Chaos in Fermionic Quantum Matter at Nonzero Density
    Transport, criticality, and chaos in fermionic quantum matter at nonzero density a dissertation presented by Aavishkar Apoorva Patel to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2019 ©2019 – Aavishkar Apoorva Patel all rights reserved. Dissertation advisor: Professor Subir Sachdev Aavishkar Apoorva Patel Transport, criticality, and chaos in fermionic quantum matter at nonzero density Abstract This dissertation is a study of various aspects of metals with strong interactions between electrons, with a particular emphasis on the problem of charge transport through them. We consider the physics of clean or weakly-disordered metals near some quantum critical points, and highlight novel transport regimes that could be relevant to experiments. We then develop a variety of exactly-solvable lattice models of strongly interacting non-Fermi liquid metals using novel non-perturbative techniques based on the Sachdev-Ye-Kitaev models, and relate their physics to that of the ubiquitous “strange metal” normal state of most correlated- electron superconductors, providing controlled theoretical insight into the possible mechanisms behind it. Finally, we use ideas from the field of quantum chaos to study mathematical quantities that can provide evidence for the existence of quasiparticles (or the lack thereof) in quantum many-body systems, in the context of metals with correlated electrons. iii Contents 0 Introduction 1 0.1 The quantum mechanics of metals ............................... 1 0.2 “Strange” metals ........................................ 4 0.3 Metals beyond Fermi liquids .................................. 6 0.4 Many-body quantum chaos .................................. 14 1 Hyperscaling at the spin density wave quantum critical point in two dimensional metals 18 1.1 Introduction .........................................
    [Show full text]
  • Physics 215C: Particles and Fields Spring 2019
    Physics 215C: Particles and Fields Spring 2019 Lecturer: McGreevy These lecture notes live here. Please email corrections to mcgreevy at physics dot ucsd dot edu. Last updated: 2021/04/20, 14:28:40 1 Contents 0.1 Introductory remarks for the third quarter................4 0.2 Sources and acknowledgement.......................7 0.3 Conventions.................................8 1 Anomalies9 2 Effective field theory 20 2.1 A parable on integrating out degrees of freedom............. 20 2.2 Introduction to effective field theory.................... 25 2.3 The color of the sky............................. 30 2.4 Fermi theory of Weak Interactions..................... 32 2.5 Loops in EFT................................ 33 2.6 The Standard Model as an EFT...................... 39 2.7 Superconductors.............................. 42 2.8 Effective field theory of Fermi surfaces.................. 47 3 Geometric and topological terms in field theory actions 58 3.1 Coherent state path integrals for bosons................. 58 3.2 Coherent state path integral for fermions................. 66 3.3 Path integrals for spin systems....................... 73 3.4 Topological terms from integrating out fermions............. 86 3.5 Pions..................................... 89 4 Field theory of spin systems 99 4.1 Transverse-Field Ising Model........................ 99 4.2 Ferromagnets and antiferromagnets..................... 131 4.3 The beta function for 2d non-linear sigma models............ 136 4.4 CP1 representation and large-N ...................... 138 5 Duality 148 5.1 XY transition from superfluid to Mott insulator, and T-duality..... 148 6 Conformal field theory 158 6.1 The stress tensor and conformal invariance (abstract CFT)....... 160 6.2 Radial quantization............................. 166 6.3 Back to general dimensions......................... 172 7 Duality, part 2 179 7.1 (2+1)-d XY is dual to (2+1)d electrodynamics.............
    [Show full text]
  • Arxiv:Cond-Mat/0601372V5 [Cond-Mat.Str-El] 18 Apr 2006 .E Volovik E
    Quantum phase transitions from topology in momentum space G. E. Volovik1,2 1 Low Temperature Laboratory, Helsinki University of Technology, P.O.Box 2200, FIN-02015 HUT, Espoo, Finland 2 Landau Institute for Theoretical Physics, Kosygina 2, 119334 Moscow, Russia Many quantum condensed matter systems are strongly correlated and strongly interacting fermionic systems, which cannot be treated perturba- tively. However, physics which emerges in the low-energy corner does not depend on the complicated details of the system and is relatively simple. It is determined by the nodes in the fermionic spectrum, which are protected by topology in momentum space (in some cases, in combination with the vacuum symmetry). Close to the nodes the behavior of the system becomes univer- sal; and the universality classes are determined by the toplogical invariants in momentum space. When one changes the parameters of the system, the transitions are expected to occur between the vacua with the same symmetry but which belong to different universality classes. Different types of quantum phase transitions governed by topology in momentum space are discussed in this Chapter. They involve Fermi surfaces, Fermi points, Fermi lines, and also the topological transitions between the fully gapped states. The consid- eration based on the momentum space topology of the Green’s function is general and is applicable to the vacua of relativistic quantum fields. This is illustrated by the possible quantum phase transition governed by topology of nodes in the spectrum of elementary particles of Standard Model. 1 Introduction. There are two schemes for the classification of states in condensed matter physics and relativistic quantum fields: classification by symmetry (GUT scheme) and by momentum space topology (anti-GUT scheme).
    [Show full text]
  • Emergent Weyl Spinors in Multi-Fermion Systems
    Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 881 (2014) 514–538 www.elsevier.com/locate/nuclphysb Emergent Weyl spinors in multi-fermion systems G.E. Volovik a,b,M.A.Zubkovc,d,∗ a Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland b Landau Institute for Theoretical Physics RAS, Kosygina 2, 119334 Moscow, Russia c ITEP, B. Cheremushkinskaya 25, 117259 Moscow, Russia d University of Western Ontario, London, ON, N6A 5B7, Canada Received 17 December 2013; received in revised form 15 February 2014; accepted 17 February 2014 Available online 22 February 2014 Abstract In Ref. [1] Horaˇ va suggested, that the multi-fermion many-body system with topologically stable Fermi surfaces may effectively be described (in a vicinity of the Fermi surface) by the theory with coarse-grained fermions. The number of the components of these coarse-grained fermions is reduced compared to the original system. Here we consider the 3+ 1 D system and concentrate on the particular case when the Fermi surface has co-dimension p = 3, i.e. it represents the Fermi point in momentum space. First we demonstrate explicitly that in agreement with Horavaˇ conjecture, in the vicinity of the Fermi point the original system is reduced to the model with two-component Weyl spinors. Next, we generalize the construction of Horavaˇ to the situation, when the original 3 + 1 D theory contains multi-component Majorana spinors. In this case the system is also reduced to the model of the two-component Weyl fermions in the vicinity of the topologically stable Fermi point.
    [Show full text]
  • Chiral Anomaly and Transport in Weyl Metals
    TOPICAL REVIEW Chiral anomaly and transport in Weyl metals A.A. Burkov Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Abstract. We present an overview of our recent work on transport phenomena in Weyl metals, which may be connected to their nontrivial topological properties, particularly to chiral anomaly. We argue that there are two basic phenomena, which are related to chiral anomaly in Weyl metals: Anomalous Hall Effect (AHE) and Chiral Magnetic Effect (CME). While AHE is in principle present in any ferromagnetic metal, we demonstrate that a magnetic Weyl metal is distinguished from an ordinary ferromagnetic metal by the absence of the extrinsic and the Fermi surface part of the intrinsic contributions to the AHE, as long as the Fermi energy is sufficiently close to the Weyl nodes. The AHE in a Weyl metal is thus shown to be a purely intrinsic, universal property, fully determined by the location of the Weyl nodes in the first Brillouin zone. In other words, a ferromagnetic Weyl metal may be thought of as the only example of a ferromagnetic metal with a purely intrinsic AHE. We further develop a fully microscopic theory of diffusive magnetotransport in Weyl metals. We derive coupled diffusion equations for the total and axial (i.e. node-antisymmetric) charge densities and show that chiral anomaly manifests as a magnetic-field-induced coupling between them. We demonstrate that an experimentally-observable consequence of CME in magnetotransport in Weyl metals is a quadratic negative magnetoresistance, which will dominate all other contributions to magnetoresistance under certain conditions and may be regarded as a smoking-gun transport characteristic, unique to Weyl metals.
    [Show full text]
  • Momentum Space Topology in Standard Model and in Condensed Matter
    2469-1 Workshop and Conference on Geometrical Aspects of Quantum States in Condensed Matter 1 - 5 July 2013 Momentum space topology in Standard Model and in condensed matter Grigory Volovik Aalto University and Landau Institute ^Momentum+^ space topology in Standard Model & condensed matter a a Aalto University G. Volovik Landau Institute Geometrical Aspects of Quantum States in Condensed Matter, 1-5 July 2013 1. Gapless & gapped topological media 2. Fermi surface as topological object 3. Fermi points (Weyl, Majorana & Dirac points) & nodal lines * superfluid 3He-A, topological semimetals , cuprate superconductors , graphene vacuum of Standard Model of particle physics in massless phase * gauge fields, gravity, chiral anomaly as emergent phenomena; quantum vacuum as 4D graphene * exotic fermions: quadratic, cubic & quartic dispersion; dispersionless fermions; Horava gravity 4. Flat bands & Fermi arcs in topological matter * surface flat bands: 3He-A, semimetals , cuprate superconductors , graphene , graphite * towards room-temperature superconductivity * 1D flat band in the vortex core *Fermi-arc on the surface of topological matter with Weyl points 5. Fully gapped topological media * superfluid 3He-B, topological insulators , chiral superconductors, quantum spin Hall insulators, vacuum of Standard Model of particle physics in present massive phase, vacua of lattice QCD * Majorana edge states & zero modes on vortices ( planar phase , topological insulator & 3He-B ) 3+1 sources of effective Quantum Field Theories in many-body system & quantum vacuum Lev Landau I think it is safe to say that no one understands Quantum Mechanics Richard Feynman Thermodynamics is the only physical theory of universal content Albert Einstein effective theories Symmetry: conservation laws, translational invariance, of quantum liquids: spontaneously broken symmetry, Grand Unification, ..
    [Show full text]
  • Electronic Band Structure of Isolated and Bundled Carbon Nanotubes
    PHYSICAL REVIEW B, VOLUME 65, 155411 Electronic band structure of isolated and bundled carbon nanotubes S. Reich and C. Thomsen Institut fu¨r Festko¨rperphysik, Technische Universita¨t Berlin, Hardenbergstr. 36, 10623 Berlin, Germany P. Ordejo´n Institut de Cie`ncia de Materials de Barcelona (CSIC), Campus de la U.A.B. E-08193 Bellaterra, Barcelona, Spain ͑Received 25 September 2001; published 29 March 2002͒ We study the electronic dispersion in chiral and achiral isolated nanotubes as well as in carbon nanotube bundles. The curvature of the nanotube wall is found not only to reduce the band gap of the tubes by hybridization, but also to alter the energies of the electronic states responsible for transitions in the visible energy range. Even for nanotubes with larger diameters ͑1–1.5 nm͒ a shift of the energy levels of Ϸ100 meV is obtained in our ab initio calculations. Bundling of the tubes to ropes results in a further decrease of the energy gap in semiconducting nanotubes; the bundle of ͑10,0͒ nanotubes is even found to be metallic. The intratube dispersion, which is on the order of 100 meV, is expected to significantly broaden the density of states and the optical absorption bands in bundled tubes. We compare our results to scanning tunneling microscopy and Raman experiments, and discuss the limits of the tight-binding model including only ␲ orbitals of graphene. DOI: 10.1103/PhysRevB.65.155411 PACS number͑s͒: 71.20.Tx, 73.21.Ϫb, 71.15.Mb I. INTRODUCTION wall strongly alters the band structure by mixing the ␲* and ␴* graphene states.
    [Show full text]
  • Nodal Line Entanglement Entropy: Generalized Widom Formula from Entanglement Hamiltonians
    Nodal line entanglement entropy: Generalized Widom formula from entanglement Hamiltonians Michael Pretko Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: May 22, 2017) A system of fermions forming a Fermi surface exhibits a large degree of quantum entanglement, even in the absence of interactions. In particular, the usual case of a codimension one Fermi surface leads to a logarithmic violation of the area law for entanglement entropy, as dictated by the Widom formula. We here generalize this formula to the case of arbitrary codimension, which is of particular interest for nodal lines in three dimensions. We first rederive the standard Widom formula by calculating an entanglement Hamiltonian for Fermi surface systems, obtained by repurposing a trick commonly applied to relativistic theories. The entanglement Hamiltonian will take a local form in terms of a low-energy patch theory for the Fermi surface, though it is nonlocal with respect to the microscopic fermions. This entanglement Hamiltonian can then be used to derive the entanglement entropy, yielding a result in agreement with the Widom formula. The method is then generalized to arbitrary codimension. For nodal lines, the area law is obeyed, and the magnitude of the coefficient for a particular partition is non-universal. However, the coefficient has a universal dependence on the shape and orientation of the nodal line relative to the partitioning surface. By comparing the relative magnitude of the area law for different partitioning cuts, entanglement entropy can be used as a tool for diagnosing the presence and shape of a nodal line in a ground state wavefunction.
    [Show full text]
  • Nanotubes Electronics
    They are stronger than steel, but the most important uses for these threadlike macromolecules may be in faster, more efficient and more durable electronic devices by Philip G. Collins and Phaedon Avouris Nanotubes ElectronicsFOR early 10 years ago Sumio Iijima, sitting at an electron microscope at the NEC Fundamental Research Laboratory in Tsukuba, Japan, first noticed odd nano- scopic threads lying in a smear of soot. NMade of pure carbon, as regular and symmetric as crystals, these exquisitely thin, impressively long macromolecules soon became known as nanotubes, and they have been the object of intense scientific study ever since. Just recently, they have become a subject for engineering as well. Many of the extraordinary properties attributed to nanotubes—among them, superlative resilience, tensile strength and thermal stability—have fed fantastic predictions of microscopic robots, dent-resistant car bodies and earth- quake-resistant buildings. The first products to use nanotubes, however, exploit none of these. Instead the earliest applica- tions are electrical. Some General Motors cars already include plastic parts to which nanotubes were added; such plastic can be electrified during painting so that the paint will stick more readily. And two nanotube-based lighting and display prod- ucts are well on their way to market. In the long term, perhaps the most valuable applications will take further advantage of nanotubes’ unique electronic properties. Carbon nanotubes can in principle play the same role as silicon does in electronic circuits, but at a molecular scale where silicon and other standard semiconductors cease to work. Although the electronics industry is already pushing the critical dimensions of transistors in commercial chips be- low 200 nanometers (billionths of a meter)—about 400 atoms wide—engineers face large obstacles in continuing this miniaturization.
    [Show full text]
  • Arxiv:Cond-Mat/9503150V2 31 Mar 1995
    Fermi liquids and non–Fermi liquids H.J. Schulz Laboratoire de Physique des Solides Universit´eParis–Sud 91405 Orsay France Contents I Introduction 2 II Fermi Liquids 2 A TheFermiGas................................. 3 B Landau’s theory: phenomenological . .. 4 1 Basichypothesis .............................. 4 2 Equilibriumproperties........................... 5 C Conclusion ................................... 8 III Renormalization group for interacting fermions 9 A Onedimension................................. 10 B Twoandthreedimensions........................... 15 IV Luttinger liquids 19 A Bosonization for spinless electrons . .... 19 B Spin–1/2fermions ............................... 24 1 Spin–chargeseparation . .. .. 26 2 Physicalproperties............................. 27 3 Long–rangeinteractions . 29 4 Persistentcurrent ............................. 30 arXiv:cond-mat/9503150v2 31 Mar 1995 V Transport in a Luttinger liquid 31 A Conductance and conductivity . 31 B EdgestatesinthequantumHalleffect. 32 C Singlebarrier.................................. 35 D Randompotentialsandlocalization . .. 37 VI The Bethe ansatz: a pedestrian introduction 40 A TheHeisenbergspinchain .......................... 41 1 Bethe’ssolution .............................. 41 2 TheHeisenbergantiferromagnet . 44 3 The Jordan–Wigner transformation and spinless fermions . ...... 46 B TheHubbardmodel.............................. 47 1 Theexactsolution ............................. 47 2 Luttingerliquidparameters . 49 1 3 Transport properties and the metal–insulator
    [Show full text]