2 )R (h/e R xy ) Ω (k xx

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Electronic Properties

of Nano-Carbon Systems

Carbon Nanotubes Graphene

Intercalated Graphite Networks

Individual Wires Siegmar Roth

s.roth@fkf.mpg.de

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Lecture 2:

10th March 2009

“Electronic Properties of

Graphitic Materials”

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This lecture has 77 viewgraphs, I plan questions and a break

at viewgraphs #22 and #43 Please interrupt at any time

to ask questions

In science questions are more important than the answers

Practice to ask questions!!

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This lecture is given from the

Kirchberg Winterschool IWEPNM

While we are following this lecture, the following talks are scheduled in the main auditory:

J. Hone: “Graphene Mechanics, …, NEMS”

K. Kern: “Doping Effects in Graphene”

J.C. Meyer: “Microscopic Studies of Graphene”

T.F. Heinze: “Optical Spectroscopy of Graphene”

E. Andrei: “Scanning Tunneling in Graphene”

A.C. Ferrari: “Raman Spectroscopy on Graphene”

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“nano”

APS: objects of which at least one of the three dimensions < 100 nm

Better: at least one of the dimensions must be smaller than a characteristic lenth governing the physics of the sample

2DEGs, Quantum Wires and Quantum Dots certainly are “nano”

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Surfaces are “nano”

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Graphitic momolayers are “nano”

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What is graphite?

What are graphitic materials?

What are their electronic properties?

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γραφειν

Photography program

Kalligraphy 1g = 1 gram Pornography grammer

Geography

Graphite

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Etymology tells us:

Graphite is black - otherwise we would not see what we write absorbs light

probally is metallic

Graphite is easy to cleave into chiplets

probably is a layered structure mono-atomic layers?

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The Crysatl Structure of Graphite

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Digression: Crystals

(cryos = cold)

Crystals: faces (regular outside) Æregular inside

periodic arrangement of atoms or molecules

Solid state physics is physics inside a crystal Ideal: crystals should be infinite

If low-dimensional, they should be

alone in the universe

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Questions?

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Ringberg Digression:

Special effects in surfaces and layers:

Hall Effect

Quantum Hall Effect (infinite 2-dim systems,

finite 2-dim systems, edge-effects …)

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Sample Current

Voltage ~ Resistance Hall Voltage Magnetic Field perpendicular to plaine

Hall Effect … Lorentz Force Coriolis Force

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Sweeping Magnetic Field

In practice often: sweep the gate

The gate voltage controls the electron density in the surface layer

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Sample Current

Voltage ~ Resistance Hall Voltage Magnetic Field perpendicular to plaine

Hall Effect … Lorentz Force Coriolis Force

gate changes

electron density in surface layer

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2126 2126

V_g(V) Rxy(h/e2 )

V_g(V) Rxx(kΩ)

T = 1.5 K B = 10 T

sample B:

HOPG

4b. Transport and Raman spectroscopy

on mono- and bilayers of graphite

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Electronic properties of graphitic carbons Graphite is a well-known electric conductor in industry

Contacts in electromotors Galvanic contacts:

Contacts in batteries Contacts for fuel cells Electro steel

Graphitic contact sprays

Graphitized beard hair of Edison’s

servant for filaments in light bulbs

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Why is graphite a metal?

two-dimenssional lattice

1 spare electron per lattice site (like alkali metals)

half-filled electron energy band

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What is special about the hexagonal graphite lattice?

Hexagonal Symmetry

Bi-atomic elementary cell (not a primitive lattice) Note: gaps in the electronic density of states

come from the interaction of the electrons with the crystalline lattice. A non-primitive lattice can create gaps, like in silicon or diamond

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Zero-Gap Semiconductor Linear Dispersion Relation Massless Dirac Equation

Effective Speed of Light ~c/300

“Pocket-QED”

(Quantum Electrodynamics):

Zitterbewegung

Klein Oscillations

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Non-primitive lattice

Two atoms in elementary cell

Superposition of two primitive triagonal lattices Triangular lattice in STM

Excitations on Lattice A only or on Lattice B only Symmetry breaking be edge effects

or by stacking (double and multi-layers, graphite crystals)

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Paso doble

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Questions?

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Graphene

Remember: “ ~ene” is for

conjugated double bonds

Graphene: graphitic monolayers

A theoretical model system since

many decades

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Graphene

Graphitic Monolayers Double-layer Graphene

Oligotichotic Graphene (Oligographene)

Thin graphitic flakes

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Graphene

Graphane

(Alkenes, Alkanes Graphene Ribbons Dangling Bonds

Dangling Bonds saturated by Hydrogen

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How do we make graphene?

Scotch Tape (“Nanomechanical Cleaving”) Defoliation of Graphite

“Epitaxy” on SiC

Epitaxy on Nickel Films

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Hye Jin Park’s Device:

Transparent Conductive Film

of Oligographene Transparency: ~90%, 1kΩ/sq

ITO

Graphene, Oligographene CNT Networks

Chicken Wire

Nano-holes in Copper Sheets (Ebbesen)

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“Fossils”, found by Viera

B. Noyes: Phys. Rev. 24, 190 (1924)

“The Variation of the Resistance of Carbon and Graphite with Temperature”

P.R. Wallace: Phys. Rev. 71, 622 (1947)

“Band Theory of Graphite”

D. Bowen : Phys. Rev. 76, 1878 (1949)

“Theory of Electrical Resistivity of Polycrystalline Graphite”

A.K. Dutta: Phys. Rev. 90, 187 (1953)

“Electrical Conductivity of Single Crystals of Graphite”

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“There is nothing new on earth”

Waves of fashion:

1940’s: Conjugated bonds (benzene, poly-enes, graphene)

1980’s: Layers structures: graphite

chalcogenides intercalation compounds 2000’s: Graphene

(Geim, Kim, …)

Science is a social endeavour, there are waves of fashion …

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Known from the period

of intercalation compounds

Anisotropic conductivity in graphite:

undoped: 10⁴ p-doped: 10⁶ n-doped: 10²

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Chen-Wei’s question at the last group meeting:

Why bother to peel-off single layers?

Can’t we simply take the top layer of a single crystal or even

of a HOPG sample?

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What is HOPG?

Highly Oriented Pyrolytic Graphite

Poycrystalline, but the c-axes are parallel Grain size up to several micrometers

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Linear Dispersion Relation

What is the dispersion relation?

Originally the relation between

velocity and wavelenght for a wave package If the velocity depends on the wavelength

(which is true for most waves in most media) the wave package disperses

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Dispersion relation for a free particle:

parabolic!

E = mv

²

E = p

²

/2m

in wave language

E = hν p = 2πhk k =1/ λ

ν = const. k

²

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Parabolic Dispersion Relation

DOS Dispersion Rel.

holes

electrons

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Linear Dispersion Relation:

Light in vacuum

Electrons in graphene

( influence of honeycomb lattice on electrons)

λ ν = c ν = c/λ 1/λ = k

ν = c k

c(electrons in graphene) = c(light in vacuum)/300

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Zero-Gap Semiconductor Dirac Point

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k lin.disp. rel. ν

Dirac Point

Charge Neutrality Pt

“Midgap”

Gating

Bipolar Transport

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Questions?

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Carbon Nanotubes:

Like graphene, but:

curvature

no edge (cylinders)

(detrimental edge effects in graphene

~ 10 nm on each side!)

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Nanotubes:

Rolling (Diameter) Srew-like rolling

helicity

structural indices (n, m)

Chirality (= handedness)

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Semiconducting Nanotubes Metallic Nanotubes

Single-Walled Nanotubes

Multi-Walled Nanotubes

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TEM of Multi-Walled Carbon Nanotube

Electrical conductance mainly in outer wall

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…. Graphene ___ Nanotube Nanotube:

lateral (axial) confinement

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Fullerene

Curved in two directions, Semiconductors

Bandgap ~ 1 eV

C

₆₀

C

₇₀

C

₈₀

higher Fullerenes

carbon nanotubes

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Fullerene Crystals

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Fulleren Molecules:

C

₆₀