Electronic Properties
of Nano-Carbon Systems
Carbon Nanotubes Graphene
Intercalated Graphite Networks
Individual Wires Siegmar Roth
s.roth@fkf.mpg.de
Lecture 2:
10th March 2009
“Electronic Properties of
Graphitic Materials”
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!!
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”
“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”
Surfaces are “nano”
Graphitic momolayers are “nano”
What is graphite?
What are graphitic materials?
What are their electronic properties?
γραφειν
Photography program
Kalligraphy 1g = 1 gram Pornography grammer
Geography
Graphite
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?
The Crysatl Structure of Graphite
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
Questions?
Ringberg Digression:
Special effects in surfaces and layers:
Hall Effect
Quantum Hall Effect (infinite 2-dim systems,
finite 2-dim systems, edge-effects …)
Sample Current
Voltage ~ Resistance Hall Voltage Magnetic Field perpendicular to plaine
Hall Effect … Lorentz Force Coriolis Force
Sweeping Magnetic Field
In practice often: sweep the gate
The gate voltage controls the electron density in the surface layer
Sample Current
Voltage ~ Resistance Hall Voltage Magnetic Field perpendicular to plaine
Hall Effect … Lorentz Force Coriolis Force
gate changes
electron density in surface layer
2126 2126
Vg(V) Rxy(h/e2 )
Vg(V) Rxx(kΩ)
T = 1.5 K B = 10 T
sample B:
HOPG
4b. Transport and Raman spectroscopy
on mono- and bilayers of graphite
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
Why is graphite a metal?
two-dimenssional lattice
1 spare electron per lattice site (like alkali metals)
half-filled electron energy band
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
Zero-Gap Semiconductor Linear Dispersion Relation Massless Dirac Equation
Effective Speed of Light ~c/300
“Pocket-QED”
(Quantum Electrodynamics):
Zitterbewegung
Klein Oscillations
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)
Paso doble
Questions?
Graphene
Remember: “ ~ene” is for
conjugated double bonds
Graphene: graphitic monolayers
A theoretical model system since
many decades
Graphene
Graphitic Monolayers Double-layer Graphene
Oligotichotic Graphene (Oligographene)
Thin graphitic flakes
Graphene
Graphane
(Alkenes, Alkanes Graphene Ribbons Dangling Bonds
Dangling Bonds saturated by Hydrogen
How do we make graphene?
Scotch Tape (“Nanomechanical Cleaving”) Defoliation of Graphite
“Epitaxy” on SiC
Epitaxy on Nickel Films
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)
“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”
“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 …
Known from the period
of intercalation compounds
Anisotropic conductivity in graphite:
undoped: 104 p-doped: 106 n-doped: 102
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?
What is HOPG?
Highly Oriented Pyrolytic Graphite
Poycrystalline, but the c-axes are parallel Grain size up to several micrometers
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
Dispersion relation for a free particle:
parabolic!
E = mv
2E = p
2/2m
in wave language
E = hν p = 2πhk k =1/ λ
ν = const. k
2Parabolic Dispersion Relation
DOS Dispersion Rel.
holes
electrons
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
Zero-Gap Semiconductor Dirac Point
k lin.disp. rel. ν
Dirac Point
Charge Neutrality Pt
“Midgap”
Gating
Bipolar Transport
Questions?
Carbon Nanotubes:
Like graphene, but:
curvature
no edge (cylinders)
(detrimental edge effects in graphene
~ 10 nm on each side!)
Nanotubes:
Rolling (Diameter) Srew-like rolling
helicity
structural indices (n, m)
Chirality (= handedness)
Semiconducting Nanotubes Metallic Nanotubes
Single-Walled Nanotubes
Multi-Walled Nanotubes
TEM of Multi-Walled Carbon Nanotube
Electrical conductance mainly in outer wall
…. Graphene ___ Nanotube Nanotube:
lateral (axial) confinement
Fullerene
Curved in two directions, Semiconductors
Bandgap ~ 1 eV
C
60C
70C
80higher Fullerenes
carbon nanotubes
Fullerene Crystals
Fulleren Molecules:
C
60has 60 molecular orbitals They spread over about 10 eV (average spacing ~ 160 meV,
but there is a larger gap (~1eV) in the middle
In crystal the molecular orbitals
develop into bands of ~ 0.5 eV width The gap in the middle does not close
Æ Semiconductors
Fulleren crystals can be doped
(“intercalated”) to become degenerate semiconductors (metal-like),
they even become superconductors
with transition temperatures up to 40 K?
Endohedral Metallofullerene
Charge transfer from metal to carbon cage
“Doping from inside”
Dy@C82
(Dy@C82)n@nanotube Dy atom
Metallofullerene Peapod