Graphene

1. Graphene Rachel Wooten Department of Physics Solid State Physics II March 6, 2008 Taught by Professor Dagotto rwooten1@utk.edu

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 6 183-191 (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. http://www.nature.com/nmat/journal/v6/n11/fig_tab/nmat2011_F1.html#figure-title

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 6 183-191 (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 http://en.wikipedia.org/wiki/Aromaticity -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 6 183-191 (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 http://www.eeel.nist.gov/812/effe.htm

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. 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 6 183-191 (March 2007) • 2. N. D. Mermin. Crystalline Order in Two Dimensions. Phys. Rev. 176, 1 250-253 • 3. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl & R. E. Smalley. C60: Buckminsterfullerene. Nature 318, 162-163 (1985). • 4. Sumio Iijima. Helical microtubules of graphitic carbon. Nature 354, 56-58 (1991). • 5. P. R. Wallace. The band theory of graphite. Phys. Rev. 71, 622-634 (1947). • 6. J. C. Slonczewski & P. R. Weiss. Band structure of graphite. Phys. Rev. 109, 272-279 (1958). • 7. A. Fasolino, J. H. Los & M. I. Katsnelson. Intrinsic ripples in graphene. Nature Materials 6, 858-861 (2007) • 8. K. S. Novoselov, et al. Electric field effect in atomically thin carbon films. Science306, 666-669 (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, 438 197-200 (2005) • 10. Adriaan M. J. Schakel. Relativistic quantum Hall effect. Phys. Rev. D 43, 4 1428-1431 (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, 2007. http://www.technologyreview.com/Nanotech/19097/

17. End • http://en.wikipedia.org/wiki/Graphite