CNTbands First-Time User Guide

1. CNTbands First-Time User Guide Xufeng Wang Electrical and Computer Engineering Purdue University West Lafayette, IN 47906 Youngki Yoon Electrical Engineering and Computer Science University of California Berkeley, CA 94720

2. Table of Contents • Introduction 3 • The Origin: Graphene • Carbon Nanotubes (CNT) • Graphene Nanoribbons (GNR) • What Can Be Simulated by CNTbands? • Input (for GNR and CNT) 7 • Output 10 • Results Generated by CNTbands • What if You Just Hit “Simulate”? 13 • Examples of Simulation Runs 14 • Limitations of CNTbands 18 • References 19

3. Introduction to CNTbands • CNTbands v2.2 can simulate electronic band structure and density-of-states for carbon nanotubes (CNTs) and graphene nanoribbons (GNRs). • It also computes some basic parameters, such as nanotube diameter, number of hexagons in the unit cell, band gap, etc. CNTbands Avaliable on nanoHUB at http://nanohub.org/resources/cntbands-ext

4. The Origin: Graphene Graphene is a one-atom-thick planar sheet of carbon atoms that is densely packed in a honeycomb crystal lattice. Direct image of a single-layer graphene membrane (Red dots denote carbon atoms)1 Computer generated graphene visual showing the honeycomb lattice structure 2 Ref. 1: J.C.Meyer and all, “Direct imaging of lattice atoms and topological defects in graphene membranes”, Nano Lett., 2008, 8 (11), 3582-3586 Ref. 2: Youngki Yoon; James K Fodor; Jing Guo; Akira Matsudaira; Diego Kienle; Gengchiau Liang; Gerhard Klimeck; Mark Lundstrom (2006), "CNTbands,”

5. Carbon Nanotubes (CNT) • Carbon nanotubes (CNT) are basically graphene sheets rolled up in a certain direction. • “Chiral vector” or “chirality” describes how a graphene sheet is rolled up to form CNT. • CNT might be metallic or semiconducting, depending on its chirality. Rolling of a Carbon nanotube (CNT) from a graphene sheet

6. Graphene Nanoribbons (GNR) • Graphene Nanoribbons are thin strips of graphene. • The electronic property of GNR largely depends on its edge structure. • Since GNRs are “strips,” its length is defined via chirality as well. • Its width, or the “thinness” of strip, is usually defined using how wide it is in nanometers or angstroms. • GNRs’ chirality is defined slightly different from that of CNT. We will explain this later in detail. Cutting of a graphene nanoribbon (GNR) from a sheet of graphene

7. What Can Be Simulated?: CNTbands Parameters Two possible input interfaces For GNR For CNT Structure specific Common for both CNT and GNR

8. CNTbands Parameters Carbon Nanotube (CNT) Carbon Nanoribbon (GNR) Structure Carbon-Carbon spacing Tight-binding energy • Distance between two nearest neighboring carbon atoms (Red segment) • Usual value is 1.42 angstrom • Tight-binding energy, or hopping energy to be used • Put simply, it describes the coupling between a carbon atom (red) and its nearest neighbors (blue)

9. Chirality for CNT • CNT is formed by rolling a graphene sheet in a certain direction. This direction is specified by chirality. • Two vectors a1 & a2 can describe this rolling direction completely. • From the starting unit cell to the other unit cell, the vectors will connect to each other after rolling. A path can be drawn (shown in red) using vectors a1 and a2 only. • The amount of a1 and a2 vectors used in the path are m and n accordingly. The chirality is then denoted as (m,n). a1 A B a2 A Atom A&B. Together they form an unit cell a1 Two chiral vectors a1&a2. An unit cell consisting of two atoms

10. Output: Molecular Structure and Unit Cell • CNTbands display a full 3-D molecular structure of simulated material based on the parameters you specify. • A full 3-D unit cell is also displayed. • Each green sphere denotes a carbon atom, and the white stick denotes bonding between two carbon atoms. A CNT Molecular Structure A CNT Unit Cell A GNR Molecular Structure A GNR Unit Cell

11. Output: E-K diagram and lowest subbands • The E-k diagram describes the energy-wave momentum relationship for carriers within the first Brouillon zone. • Each continued line is an allowed level of energy for carriers, or a subband. The E-k diagram thus describes the “bandstructure” of the studied material. • Subbands closest to the equilibrium Fermi level (denoted E = 0 here) are of particular interest, since they are usually the levels giving rise to current. In CNTbands, these subbands are extracted and outputted as “Lowest subbands”. E-k diagram Zoomed in E-k diagram Lowest subbands

12. Output: Density of States vs. Energy • Density of states vs. energy tells us how many allowed states there are at a certain energy. • Each state can accommodate up to 2 electrons having different spins. This is the Pauli Exclusion Principle. • Notice that this output tells us the availability of states but nothing about the occupancy of these states. Density of States vs. Energy

13. What if You Just Hit “Simulate”? By default, a carbon nanotube with chirality (7,7) will be simulated via Pz orbital method. (7,7) CNT Molecular Structure is presented in 3D formats The bandstructure reveals the metalic nature of such CNT samples

14. Example #1: Semiconducting and Metallic CNTs • CNTs can be metallic or semiconducting depending on its chirality (m,n). • If the CNT’s chirality difference (m-n) is a multiple of 3 (includes zero), the CNT is metallic; otherwise, it is semiconducting. CNTband simulation results — ”Lowest Subbands” metallic semiconducting (7,7) CNT (4,2) CNT

15. Example #1 (larger view) CNTband simulation results — ”Lowest Subbands” metallic (7,7) CNT

16. Example #1 (larger view) CNTband simulation results — ”Lowest Subbands” semiconducting (4,2) CNT

17. Example #2: Evaluating transport property Although CNTbands is unable to simulate transport phenomena, this simulation tool can still give us great insight regarding the transport property of a certain sample. Simulation example for a (4,0) GNR sample: CNTband simulation results — Local Density of States At near equilibrium, the carriers that give rise to current are located around Fermi level (red line). In this result, the lack of states near Fermi level indicates it will not conduct well at zero bias, but if Fermi level is biased to around 1eV, it will conduct.

18. Limitations of CNTbands Every great tool has its limitations. Users are advised to be aware of the capability of CNTbands or, in general, what CNTbands cannot do. Please consider the following: • CNTbands is a bandstructure calculation tool for CNT and GNR. It does not treat transport phenomena. For users interested in the latter, OMEN has such capabilities. • The structure is ideal and intrinsic; there is no defect, doping, passivation, nor bending, etc. • The CNT structure is single walled; the GNR is single-layered.

19. References • Related papers and resources are listed on CNTbands tool page: Youngki Yoon; James K Fodor; Jing Guo; Akira Matsudaira; Diego Kienle; Gengchiau Liang; Gerhard Klimeck; Mark Lundstrom (2006), "CNTbands," DOI: 10254/nanohub-r1838.3. • For more exercises involving CNTbands, please see: James K Fodor; Jing Guo (2007), "Introduction to CNTbands," http://nanohub.org/resources/2843. • For detailed learning on CNT and GNR, these are highly recommended: Supriyo Datta (2006), "Quantum Transport: Atom to Transistor," http://nanohub.org/resources/1490. James K Fodor; Seokmin Hong; Jing Guo (2007), "Bandstructure of Carbon Nanotubes and Nanoribbons," http://nanohub.org/resources/2762.