Graphene and Applications to Electronic Transistors

Paul Lim

(Submitted as coursework for AP272, Stanford University, Winter 2007)

Fig. 1: Graphene atomic structure.

In recent years, there has been a lot of excitement generated by a material called graphene. Graphene is a single planar sheet of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. It forms the basic structure for all other graphitic materials including graphite, carbon nanotubes and fullerenes. What is driving the excitement in graphene is the discovery of its excellent electronic transport property, which we will discuss here. The high mobility of carriers in graphene make it an excellent material to use as a channel in ultrafast electronic transistors. In fact, estimates claim that such grpahene transistors can run about a thousand times faster than conventional silicon transistors in use today. As the use of silicon in electronic devices is quickly approaching its physical limits, the search for alternative materials like graphene has gained some urgency.

Graphene Structure

Graphene is basically a single atomic layer of graphite, and so is a 2-D structure. In the 1930's, Landau and Peierls showed that strictly 2-D crystals are thermodynamically unstable and therfore do not exist. They explained that a divergent contribution of thermal fluctuations in low-dimensional crystal lattices will lead to displacements of atoms that they become comparable to interatomic distances at any finite temperature. Mermin later extended this argument, which is supported by experimental results. Thus, it is assumed that any free-standing 2-D crystal structure would be unstable and quickly decompose.

The existence of graphene then presents a challenge to this argument. The apparent stability of the 2-D structure of graphene can be explained by arguing that the 2-D crystallites are quenched in a metastable state because they are extracted from a 3-D material. Another argument for its existence is that graphene becomes intrinsically stable by having minute crumpling in the third dimension to compensate for the thermal fluctuations. This crumpling is indeed observed in graphene sheets.

Electronic Properties of Graphene

In most materials, electron transport can be accurately described using the non-relativistic Schrodinger equation. Graphene however, behaves as a zero-gap semiconducor. At low energies near the vertices of the hexagonal Brillouin zone of graphene, the E-k relation is approximately linear

where vf is the Fermi velocity, about 1,000,000 m/s. This is still 300 times slower than the speed of light but is much faster than carrier velocities in conventional semiconductors. Becasue of the linearity of the E-k relation in these areas, electrons behave as massless, relativistic particles. These electrons are therefore best described by the Dirac equation for spin 1/2 particles. This is a direct consequence of graphene's crystal symmetry. Its honeycomb lattice is made up of two equivalent carbon sublattices, and cosine-like energy bands associated with the sublattices intersect at zero E near the edges of the Brillouin zone, giving rise to conical sections of the energy spectrum for |E| < 1 eV.

Experimental measurements have shown that electron mobility in graphene at room temperature is extremely high, at about 15000 square centimeters per volt-second, compared to only a few thousands for most semiconductors.

Graphene Transistors and Nano-Ribbons

With such excellent electronic transport properties, graphene is definitely being heavily investigated for use in electronic transistors. The problem is that because of its near ballistic electron transport properties, graphene is still quite conductive even with the presence of just a few electrons. So even though graphene can still be swtiched between different states of electrical conductivity (the so-called on/off states in electronic transistors), the ratio of the electrical current between these two states is not high. Graphene transistors have been found to have an on/off ratio of only 30. Compared to silicon based transistors whose off state is close to being an ideal insulator, graphene transitors will continue to conduct electrons even when "turned off". This leads to leakage of electric current and the waste of enormous amounts of energy. This would thus make graphene impractical for such use.

Current research in this application area is focusing on reducing this on/off ratio. The current trend is to carve the graphene sheets into very narrow ribbons that are only a few nanometers wide. Lithographic techniques are commonly used to make these nano-ribbons. Professor Hongjie Dai from the Department of Chemistry at Stanford University has a different approach using a solutions based technique that breaks up the graphene into long thin slices. Transistors made out of these ribbons had on/off ratios on the order of 100,000.

© 2007 Paul Lim. This work is distributed under the terms of a Creative Commons license. The author grants permission to copy, distribute and display the work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] K. S. Novoselov et al.,"Electric Field Effect in Atomically Thin Carbon Films," Science 306, 666 (2004).

[2] K. S. Novoselov et al., "Two-Dimensional Gas of Massless Dirac Fermions in Graphene," Nature 438, 197 (2005).

[3] N. D. Mermin, "Crystalline Order in Two Dimensions," Phys. Rev. 176, 250 (1968).

[4] H. Dai, et al., "Chemically Derived, Ultrasmooth Graphene Nanoribbon Semiconductors," Science 319, 1229 (2008).