Graphene: a playground for physics
06/01/2008
Graphene, a two–dimensional carbon sheet, has attracted enormous interest because it appears to combine the desirable properties of nanotubes with a manufacturing–friendly planar structure. On closer inspection, the material offers unique challenges and opportunities of its own. Graphene’s electronic structure, while fascinating, is likely to force device manufacturers to reconsider some of their most basic assumptions.
Carbon is perhaps the most versatile element on earth. The endless library of carbon compounds defines the building blocks of life itself, from the simplest amino acids to the double helix of DNA. With four valence electrons, carbon can form long chains, decorated with a wide array of functional groups.
Typically, carbon atoms in such chains share two bonds with adjacent carbon atoms, leaving two bonds free to attach other components. If instead three of the four bonds reach other carbon atoms, the structural unit changes, from chains to hexagons. Isolated, the hexagons form benzene rings. Bonded together, they form a stable honeycomb network. For each atom, three valence electrons form σ and σ* orbitals, in the plane of the network. The fourth forms π and π* orbitals, perpendicular to the plane. The σ–electrons are tightly bound and do not carry current, while in large sheets the π and π* orbitals become valence and conduction bands.
Flat nanotubes should be simple, right?
Graphene is such a 2D carbon sheet. Stack many graphene layers on top of each other, and the mineral graphite results: the humble pencil lead, also used in lubricants, steelmaking, and high–voltage electrodes. It is a good conductor, able to withstand high temperatures and current densities.
Until recently, it was believed that single layers of graphite could not exist, that ripples due to thermal vibrations would cause them to roll up. Rolled–up sections of graphene sheet form either fullerenes or carbon nanotubes. Since Sumio Iijima of NEC Corp. called attention to these materials in 1991 [1], nanotubes have intrigued physicists and electronics researchers alike. The dense carbon network forming the surface of the nanotube confines electrons inside, as in a waveguide. Isolated from the effects of thermal vibrations and lattice defects, electrons in carbon nanotubes travel ballistically for the length of the tube. A nanotube transistor could potentially offer unprecedented switching speeds, while nanotube interconnects could surpass the scalability and conductivity of copper. In practice, however, nanotubes have yet to realize their enormous potential. While laboratory–scale proof–of–concept devices have shown impressive results, this performance has not yet been seen in practical circuits.
The tight carbon network that gives nanotubes such desirable properties also makes them unusually challenging to work with. For example, carbon nanotubes can be either metallic or semiconducting, depending on the orientation of the carbon network relative to the axis of the nanotube (Fig. 1). To control device placement and alignment, manufacturers would like to grow nanotubes with the desired conductivity in situ on the eventual substrate. Yet although researchers can measure the conductivity of individual nanotubes, and can preferentially remove those with the “wrong” characteristics, they cannot reliably grow a particular type.
Applications of metallic nanotubes have used bundles of several tubes to ensure a conductive path exists, but this approach has not yet achieved the nanotube density needed for low resistance wiring [2]. Nor can mixed bundles of metallic and semiconducting nanotubes be used in semiconductor applications, where a single metallic conductor might short circuit the device.
Even once researchers learn how to build arrays of semiconducting nanotubes with consistent properties, integrating them into functioning devices remains challenging. Low resistance contacts to nanotubes and other single molecules have so far eluded researchers: The contact properties tend to dominate those of the molecule being studied.
In light of these limitations, the isolation of graphene sheets in 2004 was hailed as a significant breakthrough [3]. As a planar material, graphene is at least potentially amenable to the standard sequence of deposition and patterning steps used to define conventional silicon circuits. The electrical properties of graphene depend on the orientation of current flow relative to the lattice, as in nanotubes, and also on the shape of the edge of the graphene ribbon. In a planar sheet, though, controlling the orientation is much easier???cutting a section with the desired structure. Other aspects of manufacturing are likely to be easier as well. Metallic graphene contacts to semi–conducting graphene channels could be patterned as a single unit.
Manufacturers could then connect to relatively large pads, rather than individual nanometer–scale features. The energy gap in graphene depends on ribbon width, suggesting that manufacturers could control threshold voltage without doping by adjusting the channel width. At the same time, experience with nanotubes should be transferable to graphene, saving years of experimentation with a completely new material.
That’s the promise of graphene and one reason why it has sparked so much interest. Yet graphene is not “just” a 2D carbon nanotube. Just as nanotubes have different properties from fullerenes, graphene sheets have unique and intriguing characteristics of their own.
Graphene’s uniqueness begins with the material’s electronic structure. As noted above, three of each carbon atom’s four possible bonds are tied to other carbon atoms in the planar hexagonal lattice. The fourth atom defines the π and π* orbitals, which form the conduction and valence bands. In high symmetry directions within the carbon sheet, however, the valence and conduction bands touch, reducing the band gap to zero.
As Fig. 2 shows, the hexagonal carbon lattice can be broken down into more primitive trigonal sublattices. A carbon atom on the A sublattice is at the center of a triangle formed by three atoms from the B sublattice, and vice versa. Each sublattice has its own potential field, and each contributes to the potential seen by electrons as they move along the sheet. As electrons hop between sublattices, they produce an effective magnetic field proportional to their momentum.
However, remember that the two sublattices are identical. Because of their symmetry, the momentum–dependent field vanishes at certain high symmetry points, known as Dirac points.** Since the split between the valence and conduction bands is a consequence of the change in field with momentum, the two bands touch at those points, making graphene a zero gap material with respect to conduction in those directions.
Similarly, the effective mass depends on the second derivative of energy with respect to the electron’s momentum vector. The second derivative of a linear function is zero, and so electrons in graphene have an effective mass of zero near the Dirac points. In high symmetry lattice directions, graphene is a zero–gap semi–conductor in which massless electrons travel at constant speed.
All of the above has been confirmed experimentally and has been anticipated by theory for years: Graphene has long been a theoretical model for 2D crystals. The unique properties of graphene excite theoretical physicists for another reason, too. It turns out that the quantum mechanics of electrons in graphene are identical to the quantum mechanics of massless relativistic particles, with the Fermi velocity (vF, about 106m/sec) taking the place of the speed of light (c, about 3 ?? 108m/sec). Graphene brings to benchtop equipment experiments that would otherwise require high energy particle accelerators.
Are relativistic transistors possible?
While theoreticians are fascinated, the consequences of this behavior for potential graphene electronics are not yet clear. For example, tunneling by relativistic particles is quite different from tunneling by normal electrons. For normal electrons, the probability of tunneling diminishes as the height of a potential barrier increases. Transistors are possible because source–drain tunneling can be reduced to approximately zero by applying a sufficient gate voltage.
Relativistic particles, in contrast, encounter the Klein Paradox [6]. According to the Klein Paradox, a barrier with height greater than the rest energy of the electron becomes completely transparent to relativistic electrons. In fact, the paradox illustrates one of the key ways in which relativistic electrons differ from their normal counterparts. While the Schrödinger equation describing the motion of non–relativistic particles treats electrons and holes separately, the relativistic Dirac equation treats them as two components of the same wave. The massless positive charge, the positron, is the antiparticle to the massless electron. A potential barrier that exceeds the rest energy of the electron, is repulsive to electrons, and therefore attractive to positrons. Positron states inside the barrier match the energy of the electron continuum outside, allowing electrons to tunnel through the barrier. Bizarre though it sounds, this prediction has been tested: Regardless of the applied gate voltage, graphene’s minimum conductivity remains approximately 4e2/h, where e is the electron charge and h is Planck’s constant [7].
Yet, as Geim and MacDonald note, the presence of a minimum conductivity is itself unexpected behavior [4]. The carrier density at the Dirac points approaches zero, so the conductivity should vanish as well, but it does not. The reason is not yet known, but one suggested theory illustrates the uniqueness of graphene’s electrons. Though graphene’s electrons behave in a relativistic way, they are do not actually move at the speed of light. At the relatively sedate speed of 106m/sec, interactions among electrons are much stronger.
The idea that an electron state might be present or absent in a particular region in space implies that the electron has been localized: It is behaving as a particle, rather than as a wave. It is possible that unusually strong interactions among electrons near the Dirac point suppress localization. With the electron behaving as a wave, conduction continues to occur even as the number of identifiable carrier states vanishes.
By the standards of conventional silicon devices, graphene’s behavior is downright weird, and it challenges the fundamental assumptions of conventional circuit design. It is not clear how a circuit might accommodate a material with no zero–conductivity “off” state, much less one with no gap between the valence and conduction bands.
However, these are properties of freestanding single–layer graphene. While it is the most perfect available example of a 2D crystal, and therefore, the most desirable material for fundamental experiments, other structures may be easy to manufacture and integrate.
Graphene was originally isolated by mechanical exfoliation of graphite flakes onto an oxidized silicon wafer. Since then, researchers at Georgia Institute of Technology have produced it by thermal decomposition of silicon carbide [8]. At high temperatures, the silicon evolves away, leaving a hexagonal carbon layer behind. This processes places a carbon support directly under the graphene layer. As a result, the A and B sublattices are no longer identical???one sublattice rests on top of an array of carbon atoms, while the other does not. Breaking the sublattice symmetry re–opens the band gap at the Dirac points.
Further modification of the band structure can be achieved by adding a second graphene layer. In a bilayer, interactions between the two layers substantially complicate the band structure. Bilayer graphene is also the only known material in which the application of an electric field affects the band structure, changing both the width of the band gap and the transparency of potential barriers. Though the Klein Paradox still affects some current directions in bilayer graphene, it may be more manageable than in the single layer material. As the number of layers increases, the structure becomes more complex; graphene structures with five or more layers have properties comparable to bulk graphite.
It may also be possible to make metal–insulator–metal junction devices in graphene. The Georgia Tech Group also demonstrated that oxidizing graphene sheets converts them from conducting to insulating, and used graphene oxide flakes to build all–graphene devices [9]. More generally, adsorption of gas molecules changes the conductivity of the sheet as well. This behavior suggests that graphene could be used in extremely sensitive gas detectors and may also offer an approach to device integration.
Though much has been made of the relationship between carbon nanotubes and graphene, it should be clear from this discussion that graphene presents substantial challenges for device engineers. Practical applications are probably a distance away. On the other hand, the unique behavior of graphene offers important opportunities as well. While conventional scaling may be less than straightforward, advanced concepts based on quantum dots and single electron devices may be easier to achieve in graphene than in other materials.
References
- Sumio Iijima, “Helical microtubules of graphitic carbon,” Nature 354, 1991,
pp. 56–58. - Z. Liu et al., “Densification of Carbon Nanotube Bundles for Interconnect Application,” IITC 2007, pp. 201–203.
- K.S. Novoselov et al., “Electric field effect in atomically thin carbon films,” Science, Vol. 306, 2004, pp. 666–669.
- A.K. Geim and A.H. MacDonald, “Graphene: exploring carbon flatland,” Physics Today, August 2007, pp. 35–41.
- S.Y. Zhou et al., “Substrate–induced band gap opening in epitaxial graphene,” Nature Materials, Vol. 6, 2007, pp. 770–775. Preprint at http://arxiv.org/abs/0709.1706.
- M.I. Katsnelson, K.S. Novoselov, and A.K. Geim, “Chiral tunneling and the Klein paradox in graphene,” Nature Physics, Vol. 2, 2006, pp. 620–625.
- J.B. Oostinga et al., “Gate–induced insulating state in bilayer graphene devices,” Nature Materials, Vol. 7, 2007, pp. 151–157. Preprint at http://arxiv.org/abs/0707.2487.
- W.A. de Heer et al., “Epitaxial graphene,” Solid State Communications, Vol. 143, 2007, pp. 92–100.
- X. Wu et al., “The epitaxial–graphene/graphene–oxide junction, An essential step towards epitaxial graphene electronics,” submitted, preprint at http://arxiv.org/abs/0712.0820.
In 2001, Katherine Derbyshire founded Thin Film Manufacturing, a consulting firm helping the industry manage the interaction between business forces and technology advances. She has engineering degrees from MIT and the U. of California, Santa Barbara. She can be reached at [email protected].
**Phrased more precisely, the momentum is defined by a 2??2 Hamiltonian matrix, the components of which vanish at the K and K’ corners of the first Brillouin zone. Readers interested in more detail should consult [4] for an accessible introduction to graphene physics, or [5] for a more rigorous analysis.