Carbon Nanotubes (CNTs)

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1 Carbon Nanotubes (s) Seminar: Quantendynamik in mesoskopischen Systemen Florian Figge Fakultät für Physik Albert-Ludwigs-Universität Freiburg July 7th, 2010 F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

2 Outline 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

3 Outline Introduction Brief Description 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

4 Introduction Brief Description What is a? tube(s) made of graphene metal or semiconductor mechanically very stable length to diameter ratio: :1 high carrier mobility ideal interconnects in nanosized devices Figure: wikipedia F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

5 Outline Introduction History 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

Introduction History 1991: Discovery of Multiwall Nanotubes (MWNT) Iijima, NEC laboratories, Japan 1 arc discharge to produce fullerenes several concentric cylindrical nanotubes regular spaced by 3.

6 Introduction History 1991: Discovery of Multiwall Nanotubes (MWNT) Iijima, NEC laboratories, Japan 1 arc discharge to produce fullerenes several concentric cylindrical nanotubes regular spaced by 3.4Å (as in graphite) range of diameters: 1nm - 100nm length: µm Figure: Multiwall Nanotube, wordpress.com 1 Iijima, Nature 354, 56 (1991) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

Introduction History 1993: Discovery of Singlewall Nanotubes (SWNT) Bethune 2 ; Iijima and Ichihashi 3 made of a single graphene sheet diameter 1nm arc discharge method small diameter and crystalline

7 Introduction History 1993: Discovery of Singlewall Nanotubes (SWNT) Bethune 2 ; Iijima and Ichihashi 3 made of a single graphene sheet diameter 1nm arc discharge method small diameter and crystalline perfection of atomic network ultimate carbon-based 1D system Figure: Singlewall Nanotube, pinktentacle.com 2 Bethune, Nature 363, 605 (1993) 3 Iijima, Ichihashi, Nature 363, 603 (1993) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

8 Introduction History 1995: Discovery of Crystalline Ropes or Bundles of SWNT Guo 4 laser vaporization each rope contains tens to hundreds of tubes with similar diameter packed in a hexagonal configuration applications: tennis racket, bicycle frame,... Figure: Bundle of SWNT 4 Guo, Chem. Phys. Lett. 243, 49 (1995) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

9 Outline Introduction Experimental Production 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

Introduction Experimental Production Arc Discharge extracted from carbon soot of graphite electrodes I 100A produces SWNT and MWNT length

10 Introduction Experimental Production Arc Discharge extracted from carbon soot of graphite electrodes I 100A produces SWNT and MWNT length 50µm 30% efficiency (weight) Figure: Arc discharge, fkf.mpg.de F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

Introduction Experimental Production Laser Ablation pulsed laser vaporizes a graphite target high-temperature reactor with inert gas vaporized carbon condenses on cooler surfaces of the reactor

11 Introduction Experimental Production Laser Ablation pulsed laser vaporizes a graphite target high-temperature reactor with inert gas vaporized carbon condenses on cooler surfaces of the reactor primarily SWNT develop 70% efficiency diameter controllable by reaction temperature most expensive Figure: Laser ablation, chem.tue.nl F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

12 Introduction Experimental Production Chemical Vapor Deposition (CVD) substrate prepared with a layer of metal catalyst particles substrate at 700 C process gas and carbon-containing gas nanotubes grow at the sites of metal catalysts high efficiency diameter of tubes controlled by the size of metal particles commercial and industrial-scale production of MWNT Figure: CVD, epfl.ch F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

13 Outline Structure of s Labeling of SWNTs 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

14 Structure of s From Graphene to s Labeling of SWNTs geometrically: rolling up a graphene strip SWNT SWNT are labeled in terms of graphene lattice vectors structure specified and indexed by the chiral vector C h Chiral Vector C h C h = na 1 + ma 2 (1) where n,m integers and a 1,2 unit vectors of the hexagonal honeycomb lattice F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

15 Structure of s Labeling of SWNTs The Chiral Vector Chiral Vector C h = na 1 + ma 2 a 1 = a 2 = a = 3a C C Figure: Graphene honeycomb network and a (5,3) nanotube diameter d t = C h π = a π n 2 + nm + m 2 F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

16 Structure of s Achiral and Chiral Tubes Labeling of SWNTs zigzag tubes (n,0) (θ = 0 ) armchair tubes (n,n) (θ = 30 ) chiral tube n m 0 (0 < θ < 30 ) Figure: Atomic structures of zigzag, armchair and chiral nanotubes F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

com (Semi-)Metallic Nanotube: n m = 3l F.

17 Structure of s Labeling of SWNTs Periodic Table of Carbon Nanotubes 2009 QuantumWise A/S - (Semi-)Metallic Nanotube: n m = 3l F. Figge (University of Freiburg) Semiconducting Nanotube: n m = 3l ± 1, l integer Carbon Nanotubes (s) July 7th, / 37

18 Outline Electronic Properties Graphene 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

19 Electronic Properties Graphene Electronic Configuration of Graphene 2s 2 2p 2 2s 1 2p 1 x 2p 1 y 2p 1 z sp 2 Hybridization + 2p 1 z Figure: sp 2 Hybridization Figure: σ- and π-bonds and their energy bands F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

20 Electronic Properties Energy Band of Graphene Graphene Figure: Brillouin zone of graphene Figure: Electronic band structure of graphene. E F 0, Dotted line = Vaccum level. F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

Electronic Properties Graphene Tight-Binding Model of Graphene π-band approximation: σ- and σ -band neglected Interactions restricted to first nearest neighbors ( ) ( ) ( ) E ± (k) = ±γ 0

21 Electronic Properties Graphene Tight-Binding Model of Graphene π-band approximation: σ- and σ -band neglected Interactions restricted to first nearest neighbors ( ) ( ) ( ) E ± (k) = ±γ cos 2 k xa cos 2 k ya + 4cos 2 2 k ya = ±γ 0 α (k) (2) (3) Figure: dispersion relation of graphene F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

22 Outline Electronic Properties 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

23 Electronic Properties Periodic Boundary Conditions Allowed wave vectors around tube are quantized wave vector along tube axis remains continuous for infinite tubes Restrictions on wave function: Ψ k (r + C h ) = e ik C h Ψ k (r) (4) = Ψ k (r) (5) e ik C h = 1 (6) for k,r taken on tube surface F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

24 Electronic Properties Periodic Boundary Conditions Analysis of the neighborhood of the Fermi surface k = K + δk where K = (b 1 b 2 )/3 K C h = 1 3 (b 1 b 2 ) (na 1 + ma 2 ) where a i b j = 2π δ i j (7) = 2π 3 (n m) (8) Restrictions e ik C h = e ik C h e iδk C h (9) = e i 2π 3 (n m) e iδk C h (10) = 1 (11) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

25 Electronic Properties Metallic Nanotubes n m = 3l e ik C h = e i 2π 3 3l e iδk C h (12) = e iδk C h (13) = 1 (14) (15) Constraint on the wave function δk C h = 2πq (q integer) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

26 Electronic Properties Metallic Nanotubes: Dispersion Rel. Rewriting the dispersion relation in the general case: E 2 (k x,k y ) = γ0 2 α (k x,k y ) 2 (16) 2nd order perturbative expansion around K (17) ( 3 = γ0 2 a 2 ( δk 2 4 x + δky) 2 ( + O δk 3 )) (18) Close to E F : Linear energy-momentum relation E ± (δk) ± 3 2 aγ 0 δk where δk C h = 2πq (19) States arbitrarily close in energy to the Fermi level can be found for q = 0 system is metallic F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

27 Electronic Properties Metallic Nanotubes: Dispersion Rel. Figure: Dispersion relations for the graphene plane, together with that of a (5,5) armchair nanotube (bold lines) Figure: Band structure for a (5,5) armchair nanotube F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

28 Electronic Properties Metallic Nanotubes: Dispersion Rel. and DOS Figure: Band structure and DOS for a (5,5) armchair nanotube F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

29 Electronic Properties Semiconducting Nanotubes n m = 3l ± 1 e ik C h = e ±i 2π 3 e iδk C h = 1 E ± (δk) ± 3 2 aγ 0 δk = ± Energy Gap at Fermi Level: (20) ( ) 3 2π 2 ( 2 aγ 0 q ± k C h 3) 2 (21) E q=0 + ( k = 0 ) Eq=0 ( k = 0 ) = 2πaγ 0 3 Ch = E1 g (22) 1 d t = π C h (23) d t : zero-gap semiconductor (recovers graphene sheet) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

30 Electronic Properties Semiconducting Nanotubes: Dispersion Rel. Figure: Dispersion relations for the graphene plane, together with that of a zigzag nanotube (bold lines) Figure: Band structure for a (10,0) zigzag nanotube F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

31 Electronic Properties Semiconducting nanotubes: Dispersion Rel. and DOS Figure: Band structure and DOS for a (10,0) zigzag nanotube F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

32 Electronic Properties Band Structure and DOS Figure: Band structure and DOS for a (9,0) zigzag nanotube F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

33 Outline Electronic Properties Experimental Proof of Linear Energy Spectrum 1 Introduction Brief Description History Experimental Production 2 Structure of s Labeling of SWNTs 3 Electronic Properties Graphene Experimental Proof of Linear Energy Spectrum 4 Summary F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

Electronic Properties Scanning Tunneling Spectroscopy Experimental Proof of Linear Energy Spectrum measures tunnelling differential conductance (di/dv ) di/dv approx.

34 Electronic Properties Scanning Tunneling Spectroscopy Experimental Proof of Linear Energy Spectrum measures tunnelling differential conductance (di/dv ) di/dv approx. proportional to the local density of electronic states di dv (V,r) ev ε j <δ Ψ j (r) 2 (24) Figure: Experimental setup 5 5 Buchs, Bercioux, Ruffieux, P. Gröning, Grabert, O. Gröning, Phys. Rev. Lett. 102, (2009) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

Results Electronic Properties Experimental Proof of Linear Energy Spectrum Figure: Experimental Results 6 6 Lemay, Janssen, van den Hout, Mooij,

35 Results Electronic Properties Experimental Proof of Linear Energy Spectrum Figure: Experimental Results 6 6 Lemay, Janssen, van den Hout, Mooij, Bronikowski, Willis, Smalley Kouwenhoven, Dekker, Nature 412, 617 (2001) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

36 Summary Summary Production arc discharge laser ablation CVD Electronic Properties Metallic Nanotubes (n,m) : n m = 3l Linear dispersion relation Semiconducting Nanotubes (n,m) : n m = 3l ± 1 Energy Gap: Eg 1 = 2πaγ 0 3 Ch F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

37 Summary Literature Charlier, Blase, Roche, Electronic and transport properties of nanotubes, Review Of Modern Physics, 79, 677 (2007) Lemay, Janssen, van den Hout, Mooij, Bronikowski, Willis, Smalley Kouwenhoven, Dekker, Two-dimensional imaging of electronic wavefuntions in carbon nanotubes, Nature 412, 617 (2001) Buchs, Bercioux, Ruffieux, P. Gröning, Grabert, O. Gröning, Electron Scattering in Intrananotube Quantum Dots, Physical Review Letters 102, (2009) F. Figge (University of Freiburg) Carbon Nanotubes (s) July 7th, / 37

2 Symmetry. 2.1 Structure of carbon nanotubes

2 Symmetry. 2.1 Structure of carbon nanotubes 2 Symmetry Carbon nanotubes are hollow cylinders of graphite sheets. They can be viewed as single molecules, regarding their small size ( nm in diameter and µm length), or as quasi-one dimensional crystals