EPL 335: LOW DIMENSIONAL PHYSICS

Term Paperon

GRAPHENE

Submitted by:

Aishwarya Kumar 2008PH10605

1.IntroductionGraphene is the name given to a flat monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice, and is a basic building block for graphitic materials of all other dimensionalities. It can be wrapped up into 0D fullerenes, rolled into 1D nanotubes or stacked into 3D graphite. The carbon-carbon bond length in graphene is about 0.142 nm. Graphene sheets stack to form graphite with an interplanar spacing of 0.335 nm, which means that a stack of 3 million sheets would be only one millimeter thick. Graphene is the basic structural element of some carbon allotropes including graphite, charcoal, carbon nanotubes, and fullerenes. It can also be considered as an indefinitely large aromatic molecule, the limiting case of the family of flat polycyclic aromatic hydrocarbons. Theoretically, graphene (or ‘2D graphite’) has been studied for sixty years, and is widely used for describing properties of various carbon-based materials.

Fig 1. Graphene can be wrapped up into 0D fullerenes, rolled into 1D nanotubes or stacked into 3D graphite.The charge carriers in graphene have a unique nature. Its charge carriers mimic

relativistic particles and are more easily and naturally described starting with the Dirac equation rather than the Schrödinger equation. Although there is nothing particularly relativistic about electrons moving around carbon atoms, their interaction with the periodic potential of graphene’s honeycomb lattice gives rise to new quasiparticles that at low energies E are accurately described by the (2+1)-dimensional Dirac equation with an eff ective speed of light vF ≈ 106 m/s. Th ese quasiparticles, called massless Dirac fermions, can be seen as electrons that have lost their rest mass m0 or as neutrinos that acquired the electron charge e. The relativistic like description of electron waves on honeycomb lattices has been known theoretically for many years, never failing to attract attention, and the experimental discovery of graphene now provides a way to probe quantum electrodynamics (QED) phenomena by measuring graphene’s electronic properties.

2.Properties

Electron diffraction patterns have shown the expected hexagonal lattice of graphene. Suspended graphene also showed "rippling" of the flat sheet, with amplitude of about one nanometer. These ripples may be intrinsic to graphene as a result of the instability of two-dimensional crystals, or may be extrinsic, originating from the ubiquitous dirt seen in all TEM images of graphene.

Graphene differs from most conventional three-dimensional materials. Intrinsic graphene is a semi-metal or zero-gap semiconductor. Understanding the electronic structure of graphene is the starting point for finding the band structure of graphite. It was realized early on that the E-k relation is linear for low energies near the six corners of the two-dimensional hexagonal Brillouin zone, leading to zero effective mass for electrons and holes.

Experimental results from transport measurements show that graphene has a remarkably high electron mobility at room temperature, with reported values in excess of 15,000 cm2V−1s−1. Additionally, the symmetry of the experimentally measured conductance indicates that the mobilities for holes and electrons should be nearly the same. The mobility is nearly independent of temperature between 10 K and 100 K, which implies that the dominant scattering mechanism is defect scattering. Scattering by the acoustic phonons of graphene places intrinsic limits on the room temperature mobility to 200,000 cm2V−1s−1 at a carrier density of 1012 cm−2. The corresponding resistivity of the graphene sheet would be 10−6 Ω·cm. This is less than the resistivity of silver, the lowest resistivity substance known at room temperature. However, for graphene on silicon dioxide substrates, scattering of electrons by optical phonons of the substrate is a larger effect at room temperature than scattering by graphene’s own phonons. This limits the mobility to 40,000 cm2V−1s−1.

Despite the zero carrier density near the Dirac points, graphene exhibits a minimum conductivity on the order of 4e2 / h. The origin of this minimum conductivity is still unclear. However, rippling of the graphene sheet or

ionized impurities in the SiO2 substrate may lead to local puddles of carriers that allow conduction. Several theories suggest that the minimum conductivity should be 4e2 / πh; however, most measurements are of order 4e2 / h or greater and depend on impurity concentration.

Recent experiments have probed the influence of chemical dopants on the carrier mobility in graphene. Doped graphene with various gaseous species (some acceptors, some donors), and found the initial undoped state of a graphene structure can be recovered by gently heating the graphene in vacuum. Even for chemical dopant concentrations in excess of 1012 cm−2 there is no observable change in the carrier mobility.

Due to its two-dimensional property, charge fractionalization (where the apparent charge of individual pseudoparticles in low-dimensional systems is less than a single quantum) is thought to occur in graphene. It may therefore be a suitable material for the construction of quantum computers using anyonic circuits.

Graphene's unique electronic properties produce an unexpectedly high opacity for an atomic monolayer, with a startlingly simple value: it absorbs πα ≈ 2.3% of white light, where α is the fine-structure constant. Recently it has been demonstrated that the bandgap of graphene can be tuned from 0 to 0.25 eV (about 5 micrometer wavelength) by applying voltage to a dual-gate bilayer graphene field-effect transistor (FET) at room temperature. It is further confirmed that such unique absorption could become saturated when the input optical intensity is above a threshold value. This nonlinear optical behavior is termed saturable absorption and the threshold value is called the saturation fluence. Graphene can be saturated readily under strong excitation over the visible to near-infrared region, due to the universal optical absorption and zero band gap.

Graphene is thought to be an ideal material for spintronics due to small spin-orbit interaction and near absence of nuclear magnetic moments in carbon. Electrical spin-current injection and detection in graphene was recently demonstrated up to room temperature.

By dispersing oxidized and chemically processed graphite in water, and using paper-making techniques, the monolayer flakes form a single sheet and bond very powerfully. These sheets, called graphene oxide paper have a measured tensile modulus of 32 GPa.The peculiar chemical property of graphite oxide is related to the functional groups attached to graphene sheets. They even can significantly change the pathway of polymerization and similar chemical processes. Graphene Oxide flakes in polymers also shown enhanced photo-conducting properties.

The near-room temperature thermal conductivity of graphene was recently measured to be between (4.84±0.44) ×103 to (5.30±0.48) ×103 Wm−1K−1. These measurements, made by a non-contact optical technique, are in excess of those measured for carbon nanotubes or diamond. It can be shown by using the Wiedemann-Franz law, that the thermal conduction is phonon-dominated. However, for a gated graphene strip, an applied gate bias causing a Fermi energy shift much larger than kBT can cause the electronic contribution to increase and dominate over the phonon contribution at low temperatures. The ballistic thermal conductance of graphene is isotropic. Despite its 2-D nature, graphene has 3 acoustic phonon modes. The two in-plane modes (LA, TA) have a linear dispersion relation, whereas the out of plane mode (ZA) has a quadratic dispersion relation. Due to this, the T2 dependent thermal conductivity contribution of the linear modes is dominated at low temperatures by the T1.5 contribution of the out of plane mode.

Graphene appears to be one of the strongest materials ever tested. Measurements have shown that graphene has a breaking strength 200 times greater than steel, a bulk strength of 130GPa.Using an atomic force microscope (AFM), the spring constant of suspended graphene sheets has been measured. Graphene sheets, held together by van der Waals forces, were suspended over silicon dioxide cavities where an AFM tip was probed to test its mechanical properties. Its spring constant was in the range 1-5 N/m and the Young's modulus was 0.5 TPa, which differs from that of the bulk graphite. These high values make graphene very strong and rigid. These intrinsic properties could lead to using graphene for NEMS applications such as pressure sensors and resonators.

3.Production of Graphene

3.1 Drawing

In 2004, the Russian researchers obtained graphene by mechanical exfoliation of graphite. They used cohesive tape to repeatedly split graphite crystals into increasingly thinner pieces. The tape with attached optically transparent flakes was dissolved in acetone and, after a few further steps, the flakes including monolayers were sedimented on a silicon wafer. Individual atomic planes were then hunted in an optical microscope. A year later, the researchers simplified the technique and started using dry deposition, avoiding the stage when graphene floated in a liquid. Relatively large crystallites (first, only a few micrometres in size but, eventually, larger than 1 mm and visible by a naked eye) were obtained by the technique. It is often referred to as a scotch tape or drawing method. The latter name appeared because the dry deposition resembles drawing with a piece of graphite. The key for the success probably was the use of high throughput visual recognition of graphene on a proper chosen substrate, which provides a small but noticeable optical contrast.

3.2 Epitaxial Growth

Yet another method of obtaining graphene is to heat silicon carbide to high temperatures (>1100 °C) to reduce it to graphene. This process produces a sample size that is dependent upon the size of the SiC substrate used. The face of the silicon carbide used for graphene creation, the silicon-terminated or carbon-terminated, highly influences the thickness, mobility and carrier density of the graphene.

Many important graphene properties have been identified in graphene produced by this method. For example, the electronic band-structure (so-called Dirac cone structure) has been first visualized in this material. Weak anti-localization is observed in this material and not in exfoliated graphene produced by the pencil trace method. Extremely large, temperature

independent mobilities have been observed in SiC epitaxial graphene. They approach those in exfoliated graphene placed on silicon oxide but still much lower than mobilities in suspended graphene produced by the drawing method. It was recently shown that even without being transferred graphene on SiC exhibits the properties of massless Dirac fermions such as the anomalous quantum Hall effect.

The weak van der Waals force that provides the cohesion of multilayer graphene stacks does not always affect the electronic properties of the individual graphene layers in the stack. That is, while the electronic properties of certain multilayered epitaxial graphenes are identical to that of a single graphene layer, in other cases the properties are affected as they are for graphene layers in bulk graphite. This effect is theoretically well understood and is related to the symmetry of the interlayer interactions.

Epitaxial graphene on silicon carbide can be patterned using standard microelectronics methods.

Graphene can also be epitaxially grown on metals. This method uses the atomic structure of a metal substrate to seed the growth of the graphene (epitaxial growth). Graphene grown on ruthenium doesn't typically yield a sample with a uniform thickness of graphene layers, and bonding between the bottom graphene layer and the substrate may affect the properties of the carbon layers. Graphene grown on iridium on the other hand is very weakly bonded, uniform in thickness, and can be made highly ordered. Like on many other substrates, graphene on iridium is slightly rippled. Due to the long-range order of these ripples generation of minigaps in the electronic band-structure (Dirac cone) becomes visible. High-quality sheets of few layer graphene exceeding 1 cm2 (0.2 sq in) in area have been synthesized via chemical vapor deposition on thin nickel films. These sheets have been successfully transferred to various substrates, demonstrating viability for numerous electronic applications. An improvement of this technique has been found in copper foil where the growth automatically stops after a single graphene layer, and arbitrarily large graphene films can be created.

4.Applications

4.1 Gas Detection

Graphene makes an excellent sensor due to its 2D structure. The fact that its entire volume is exposed to its surrounding makes it very efficient to detect adsorbed molecules. Molecule detection is indirect: as a gas molecule adsorbs to the surface of graphene, the location of absorption experiences a local change in electrical resistance. While this effect occurs in other materials, graphene is superior due to its high electrical conductivity (even when few carriers are present) and low noise which makes this change in resistance detectable.

The operational principle of graphene gas detectors is based on changes in their electrical conductivity, due to gas molecules adsorbed on graphene's surface and acting as donors or acceptors, similar to other solid-state sensors. However, the following characteristics of graphene make it possible to increase the sensitivity to its ultimate limit and detect individual dopants. First, graphene is a strictly two-dimensional material and, as such, has its whole volume exposed to surface adsorbates, which maximizes their effect. Second, graphene is highly conductive, exhibiting metallic conductivity and, hence, low Johnson noise even in the limit of no charge carriers, where a few extra electrons can cause notable relative changes in carrier concentration, n. Third, graphene has few crystal, which ensures a low level of excess (1/f) noise caused by their thermal switching. Fourth, graphene allows four-probe measurements on a single-crystal device with electrical contacts that are ohmic and have low resistance. All of these features contribute to make a unique combination that maximizes the signal-to-noise ratio to a level sufficient for detecting changes in a local concentration by less than one electron charge, e, at room temperature.

Graphene-based gas sensors allow the ultimate sensitivity such that the adsorption of individual gas molecules could be detected. Large arrays of such sensors would increase the catchment area , allowing higher sensitivity for short-time exposures and the detection of active (toxic) gases in as minute concentrations as practically desirable. The epitaxial growth of few-

layer graphene offers a realistic promise of mass production of such devices. Our experiments also show that graphene is sufficiently electronically quiet to be used in single-electron detectors operational at room temperature and in ultrasensitive sensors of magnetic field or mechanical strain, in which the resolution is often limited by 1/f noise.

4.2 Graphene Nanoribbons

Graphene nanoribbons (GNRs) are essentially single layers of graphene that are cut in a particular pattern to give it certain electrical properties. Depending on how the un-bonded edges are configured, they can either be in a zigzag or armchair configuration. Calculations based on tight binding predict that zigzag GNRs are always metallic while armchairs can be either metallic or semiconducting, depending on their width. However, recent density functional theory calculations show that armchair nanoribbons are semiconducting with an energy gap scaling with the inverse of the GNR width. Indeed, experimental results show that the energy gaps do increase with decreasing GNR width. Zigzag nanoribbons are also semiconducting and present spin polarized edges. Their 2D structure, high electrical and thermal conductivity, and low noise also make GNRs a possible alternative to copper for integrated circuit interconnects. Some research is also being done to create quantum dots by changing the width of GNRs at select points along the ribbon, creating quantum confinement.

4.3 Graphene Transistors

Due to its high electronic quality, graphene has also attracted the interest of technologists who see it as a way of constructing ballistic transistors. Graphene exhibits a pronounced response to perpendicular external electric fields, allowing one to build FETs (field-effect transistors In 2006, Georgia Tech researchers announced that they had successfully built an all-graphene planar FET with side gates. Their devices showed changes of 2% at cryogenic temperatures. The first top-gated FET (on-off ratio of <2) was demonstrated in 2007. Graphene nanoribbons may prove generally capable of replacing silicon as a semiconductor in modern technology.

Facing the fact that current graphene transistors show a very poor on-off ratio, researchers are trying to find ways for improvement. In 2008 a new switching effect in graphene field-effect devices was demonstrated. This switching effect is based on a reversible chemical modification of the graphene layer and gives an on-off ratio of greater than six orders of magnitude. These reversible switches could potentially be applied to nonvolatile memories.

In 2009 researchers at the Politecnico di Milano demonstrated four different types of logic gates, each composed of a single graphene transistor. In the same year, the Massachusetts Institute of Technology researchers built an experimental graphene chip known as a frequency multiplier. It is capable of taking an incoming electrical signal of a certain frequency and producing an output signal that is a multiple of that frequency. Although these graphene chips open up a range of new applications, their practical use is limited by a very small voltage gain (typically, the amplitude of the output signal is about 40 times less than that of the input signal). Moreover, none of these circuits was demonstrated to operate at frequencies higher than 25 kHz.

In February 2010, researchers at IBM reported that they have been able to create graphene transistors with an on and off rate of 100 gigahertz, far exceeding the rates of previous attempts, and exceeding the speed of silicon. The 240 nm graphene transistors made at IBM were made using extant silicon-manufacturing equipment, meaning that for the first time graphene transistors are a conceivable—though still fanciful—replacement for silicon.

5. Conclusion

Graphene is a rapidly rising star on the horizon of materials science and condensed-matter physics. This strictly two-dimensional material exhibits exceptionally high crystal and electronic quality, and, despite its short history, has already revealed a cornucopia of new physics and potential applications, which are briefly discussed here. Whereas one can be certain of the realness of applications only when commercial products appear, graphene no longer requires any further proof of its importancein terms of fundamental physics. Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of ‘relativistic’ condensed-matter physics, where quantum relativisticphenomena, some of which are unobservable in high-energy physics, can now be mimicked and tested in table-top experiments. More generally, graphene represents a conceptually new class of materials that are only one atom thick, and, on this basis, offers new inroads into low-dimensional physics that has never ceased to surprise and continues to provide a fertile ground for applications.

6.References

1. Progress Article Nature Materials 6, 183 - 191 (2007)doi:10.1038/nmat1849 : The rise of graphene, A. K. Geim & K. S. Novoselov

2. 100 GHz Transistors from Wafer Scale Epitaxial Graphene Y.-M. Lin*, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.-Y. Chiu, A. Grill, and Ph. Avouris

3. Detection of individual gas molecules adsorbed on graphene F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson & K. S. Novoselov

4. Z-shaped graphene nanoribbon quantum dot device , Z. F. Wang1, Q. W. Shi, Qunxiang Li, Xiaoping Wang, J. G. Hou, Huaixiu Zheng, Yao Yao, and Jie Chen

5. www.wikipedia.org

6. www.grapheneworld.org