1. Taught by Professor Dagotto
Graphene
Rachel Wooten
Department of Physics
Solid State Physics II
March 6, 2008
Taught by Professor Dagotto

2. Outline What is graphene? How it is made Properties
Electronic & physical properties
Relativistic charge carriers
Anomalous quantum Hall effect
Future Applications

3. What is graphene? 2-dimensional hexagonal lattice of carbon
sp2 hybridized carbon atoms
Basis for C-60 (bucky balls), nanotubes, and graphite
Among strongest bonds in nature
A. K. Geim & K. S. Novoselov. The rise of graphene. Nature Materials Vol (March 2007)

4. A Two dimensional crystal
In the 1930s, Landau and Peierls (and Mermin, later)showed thermodynamics prevented 2-d crystals in free state.
Melting temperature of thin films decreases rapidly with temperature -> monolayers generally unstable.
In 2004, experimental discovery of graphene- high quality 2-d crystals
Possibly, 3-d rippling stabilizes crystal
Representation of rippling in graphene. Red arrows are ~800nm long.

5. How to make graphene Strangely cheap and easy.
Either draw with a piece of graphite, or repeatedly peel with Scotch tape
Place samples on specific thickness of Silicon wafer. The wrong thickness of silicon leaves graphene invisible.
Graphene visible through feeble interference effect. Different thicknesses are different colors.

6. Samples of graphene
Graphite films visualized through atomic force microscopy.
Transmission electron microscopy image
c) Scanning electron microscope image of graphene.
A. K. Geim & K. S. Novoselov. The rise of graphene. Nature Materials Vol (March 2007)

7. Electrons in graphene Electrons in p-orbitals above and below plane
p-orbitals become conjugated across the plane
Electrons free to move across plane in delocalized orbitals
Extremely high tensile strength
-Graphene and graphite are great conductors along the planes.

8. Properties: charge carriers
Samples are excellent- graphene is ambipolar: charge carrier concentration continuously tunable from electrons to holes in high concentrations
A. K. Geim & K. S. Novoselov. The rise of graphene. Nature Materials Vol (March 2007)

9. Relativistic charge carriers
Linear dispersion relation- charge carriers behave like massless Dirac fermions with an effective speed of light c*~106. (But cyclotron mass is nonzero.)
Relativistic behavior comes from interaction with lattice potential of graphene, not from carriers moving near speed of light.
Behavior ONLY present in monolayer graphene; disappears with 2 or more layers.

10. Anomalous quantum Hall effect
Classical quantum Hall effect.
Apply B field and current. Charges build up on opposite sides of sample parallel to current.
Measure voltage: + and - carriers create opposite Hall voltages.
Quantum Hall effect
Classical Hall effect with voltage differences = integer times e2/h

11. Anomalous quantum Hall effect
Fractional Quantum Hall effect
Quantum Hall effect times rational fractions. Not completely understood.
Graphene shows integer QHE shifted by 1/2 integer
Non-zero conductivity as charge carrier dentsity -> zero.

12. Inset: xy in 2-layer graphite. Half-integer QHE unique to monolayer.
Hall conductivity xy (red) and resistivity xy vs. carrier concentration.
Inset: xy in 2-layer graphite.
Half-integer QHE unique to monolayer.
*Note non-zero conductivity as carrier concentrations approach zero.

13. Possible Applications
High carrier mobility even at highest electric-field-induced concentrations, largely unaffected by doping= ballistic electron transport over < m distances at 300K
May lead to ballistic room-temperature transistors.
GaTech group made proof of concept transistor- leaks electrons, but it’s a start.
Energy gap controlled by width of graphene strip.
Must be only 10s of nm wide for reasonable gap.
Etching still difficult consistently and random edge configuration causes scattering.

14. Even more applications?
Very high tensile strength
Replacement of nanotubes for cheapness in some applications: composite materials and batteries for improved conductivity
Hydrogen storage
Graphene based quantum computation? Low spin-orbit coupling-> graphene may be ideal as a q-bit.

15. In Conclusion
Graphene is a novel material with very unusual properties
Easy to make in lab; may prove easy and economical to manufacture (unknown).
Broad range of applications for future research.
Variety of possible practical applications.

16. Resources
1. A. K. Geim & K. S. Novoselov. The rise of graphene. Nature Materials Vol (March 2007)
2. N. D. Mermin. Crystalline Order in Two Dimensions. Phys. Rev. 176,
3. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl & R. E. Smalley. C60: Buckminsterfullerene. Nature 318, (1985).
4. Sumio Iijima. Helical microtubules of graphitic carbon. Nature 354, (1991).
5. P. R. Wallace. The band theory of graphite. Phys. Rev. 71, (1947).
6. J. C. Slonczewski & P. R. Weiss. Band structure of graphite. Phys. Rev. 109, (1958).
7. A. Fasolino, J. H. Los & M. I. Katsnelson. Intrinsic ripples in graphene. Nature Materials 6, (2007)
8. K. S. Novoselov, et al. Electric field effect in atomically thin carbon films. Science 306, (2004).
9. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, & A. A. Firsov. Two-dimensional gas of massless Dirac fermions in graphene. Nature, (2005)
10. Adriaan M. J. Schakel. Relativistic quantum Hall effect. Phys. Rev. D 43, (1991)
11. J. Hass, R. Feng, T. Li, X. Li, Z. Zong, W. A. de Heer, P. N. First & E. H. Conrad. Highly ordered graphene for two dimensional electroncs. Applied Physics Letters 89, (2006)
12. Prachi Patel-Predd. “Ultrastrong paper from graphene”. July 25,

17. End