A suspended sheet of pure graphene – a plane layer of C atoms bonded together in a honeycomb lattice – is the “most two-dimensional” system imaginable.
A.J. Leggett
Such sheets have long been known to exist in disguised forms – in graphite (many graphene sheets stacked on top of one another), C nanotubes (a graphene sheet rolled into a cylinder) and fullerenes (buckyballs), which are small areas of a graphene sheet sewn together to form an approximately spherical surface.
Until 2004, it was generally believed (a) that an extended graphene sheet would not be stable against the effects of thermal and other fluctuations, and (b) that even if they were stable, it would be impossible to isolate them so that their properties could be systematically studied.
In 2004, André Geim et al. (University of Manchester, UK) demonstrated that both these beliefs were false: they created single graphene sheets by peeling them off a graphite substrate using scotch tape, and characterized them as indeed single-sheet by simple optical microscopy on top of a SiO2 substrate. Now it is done mostly by Raman spectroscopy. Subsequently it was found that small graphene sheets do not need to rest on substrates but can be freely suspended from a scaffolding; furthermore, bilayer and multilayer sheets can be prepared and characterized. As a result of these developments, the number of papers on graphene published in the last few years exceeds 3000.
Graphene is a very promising material both for applications and fundamental research.
The Nobel Prize in Physics 2010
Andre Geim Konstantin Novoselov
The trick: Finding the Graphene
• Using correct substrate
• And correct light frequency
• Interference effect makes monolayers show up in ordinary optical microscope
Blake et al (2007) arXiv:0705.0259 Si SiO
Graphene
What is graphene? • “Imagine a piece of paper but a million times thinner.
This is how thick graphene is. • Imagine a material stronger than diamond. This is how
strong graphene is [in the plane]. • Imagine a material more conducting than copper. This is
how conductive graphene is. • Imagine a machine that can test the same physics that
scientists test in, say, CERN, but small enough to stand on top of your table. Graphene allows this to happen.
• Having such a material in hand, one can easily think of many useful things that can eventually come out. As concerns new physics, no one doubts about it already...''
From a recently interview with Andre Geim.
Graphene is a honeycomb lattice of carbon atoms.
Graphite can be viewed as a stack
of graphene layers.
Carbon nanotubes are rolled-up cylinders of graphene
Fullerenes C60 are molecules
consisting of wrapped graphene by the introduction of pentagons on the hexagonal
lattice.
Carbon is the materia prima for life and the basis of all organic chemistry. Because of the flexibility of its bonding, carbon-based systems show an unlimited number of different structures with an equally large variety of physical properties.
From Castro Neto et al, 2009
The electronic structure of an isolated C atom is (1s)2(2s)2(2p)4; in a solid-state environment the 1s electrons remain more or less inert, but the 2s and 2p electrons hybridize.
Crystal structure
An alternative possibility is to form three sp2 orbitals, leaving over a more or less pure p-orbital. In that case the natural tendency is for the sp2 orbitals to arrange themselves in a plane at 120o angles, and the lattice thus formed is the honeycomb lattice.
One possible result is four sp3 orbitals, which naturally tend to establish a tetrahedral bonding pattern that soaks up all the valence electrons: this is precisely what happens in the best known solid form of C, namely diamond, which is a very good insulator (band gap about 5 eV).
Lattice structure of graphene, made out of two interpenetrating triangular lattices
a1 and a2 are the lattice unit vectors, and δi are the nearest-neighbor vectors.
Band structure
Corresponding Brillouin zone.
Energy vanishes at the points K, K’ , and those equivalent to them
The picture and derivations adapted from Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009)
Sketch of derivation (1)
Wallace, Phys. Rev. 71, 622 (1947)
Sketch of derivation (2)
Conic (Dirac) points
Let us put and expand the Hamiltonian in small
Coefficient can be dropped
Similarly
The Hamiltonian is, from a formal point of view, exactly that of an ultra-relativistic (or mass-less) particle of spin 1/2 (such as the neutrino), with the velocity of light c replaced by the Fermi velocity vF, which is a factor 300 smaller.
It is suggestive to express the Hamiltonian describing conic points in the form
- Pauli matrices
Thus there arises the prospect, which excites a lot of people, of finding analogs to many phenomena predicted to occur in a solid-state context.
Comment
However, it should be remembered that the Dirac excitations near K are not the antiparticles of those near K’; rather it is the two possible combinations of the excitations near one Dirac point on the A and B sub-lattices, with energies respectively, which are one another’s “antiparticles.”
Eigenfunctions for the vicinity of the K-point
Note that when q rotates once around the Dirac point, the phase of changes by , not by , as is characteristic for “spin-1/2” particles.
Important parameters for conic spectrum
Density of states at K-point:
Since there are 2 Dirac points
How one can introduce effective mass?
Doping shifts the Fermi level leading to creating a Fermi line.
This definition provides the cyclotron effective mass.
Klein tunneling
Due to the chiral nature of their quasiparticles, quantum tunneling in these materials becomes highly anisotropic, qualitatively different from the case of normal, non-relativistic electrons. Massless Dirac fermions in graphene allow a close realization of Klein’s gedanken experiment - unimpeded penetration of relativistic particles through high and wide potential barriers.
Schematic diagrams of the spectrum of quasiparticles in single-layer graphene. The spectrum is linear at low Fermi energies (<1eV). The red and green curves emphasize the origin of the linear spectrum, which is the crossing between the energy bands associated with crystal sublattices A and B.
The Fermi level (dotted lines) lies in the conduction band outside the barrier and the valence band inside it. The blue filling indicates occupied states.
The paradox: When the barrier increases, transmission becomes perfect!
Explanation: Absence of backscattering for
Because the magnitude v of the carrier velocity is independent of the energy, an electron moving along the field lines cannot be backscattered— since that would require v = 0 at the turning point. Adapted from Beenakker, 2007
An electron-like excitation continue under the barrier as a hole-like one Momentum changes its sign, but the (group) velocity is conserved. As a result – anisotropic scattering.
Result for high energies :
Under resonance conditions the barrier becomes transparent (interference effects) . More significantly, however, the barrier always remains perfectly transparent for angles close to the normal incidence φ = 0. The latter is the feature unique to massless Dirac fermions and is directly related to the Klein paradox in QED.
Klein tunneling is the mechanism for electrical conduction through the interface between p-doped and n-doped graphene.
n-p-n junction in graphene a) cross-sectional view of the device. b) electrostatic potential profile U(x) along the cross-section of a). The combination of a positive voltage on the back gate and a negative voltage on the top gate produces a central p-doped region flanked by two n-doped regions. c) Optical image of the device. The barely visible graphene flake is outlined with a dashed line and the dielectric layer of PMMA appears as a blue shadow. (Huard et al., 2007)
Minimal conductance of graphene
What happens with the conductance of graphene when the Fermi level reaches the conic point? Does it vanish or not?
Conductivity versus gate voltage
The conductance reaches a minimum corresponding to double value comparing to the value at the lowest quantum Hall plateau.
The band structure of graphene has two valleys, which are decoupled in the case of a smooth edge. In a given valley the excitations have a two-component envelope wave function
The two components of refer to the two sublattices in the two-dimensional honeycomb lattice of carbon atoms.
with the velocity of the massless excitations of charge and energy , the momentum operator, , the position, and a Pauli matrix. We choose the zero of energy such that the Fermi level is at .
Dirac equation:
Boundary conditions:
The transversal momenta are quantized as:
Minimal conductance of graphene
Tworzydlo et al., 2006
Level quantization and quantum Hall Effect in graphene
At a typical doping, electrons are degenerate at room temperature.
In the absence of magnetic field the tight binding Hamiltonian close to Dirac points is
In magnetic field one has to replace to get:
We have to find eigenvalues of this Hamiltonian versus magnetic field B
Let us assume that and choose the Landau gauge
It is also convenient to choose dimensionless variables
It is immediately seen that the energies are parameterized by the quantity
Measuring lengths in units of , one gets
The Hamiltonian can be rewritten as
Trick: let us apply the Hamilton operator twice. We have
Note that the operator is the Hamiltonian of a “dimensionless Harmonic oscillator” with eigenvalues Consequently,
Quantum numbers n correspond to Landau levels, but they are not equidistant. The number of states per the Landau level corresponds to the number of flux quanta through the cell,
Experimental check: gated structures
Manchester group
Room-temperature quantum Hall Effect in graphene
At B=45 T the Fermi level is located between n=0 and 1
At T= 300 K, the plateaus are seen at B < 20 T
What I have not discussed….
• Bilayer graphene • Epitaxial graphene and graphene stacks • Surface states in graphene and graphene stacks • Graphene nanoribbons • Flexural Phonons, Elasticity, and Crumpling • Disorder in Graphene: Ripples, topological lattice defects, impurity states,
localized states near edges, cracks, and voids; vector potential and gauge field disorder, coupling to magnetic impurities
• Mechanical properties • Potential device applications: Single molecule gas detection, Graphene nano-
ribbons as ballistic and FET transistor devices and components for integrated circuits; Transparent conducting electrodes, Graphene biodevices, NEMSs, components of lasers, etc…
100 GHz transistor from Wafer-Scale Epitaxial Graphene
Ph. Avouris group, IBM
Cutoff frequency of 100 gigahertz for a gate length of 240 nanometers. The high-frequency performance of these epitaxial graphene transistors exceeds that of state-of-the-art silicon transistors of the same gate length.
(A) Image of devices fabricated on a 2-inch graphene wafer and schematic cross-sectional view of a top-gated graphene FET.
(B) The drain current, ID, of a graphene FET (gate
length LG = 240 nm) as a function of gate voltage at drain bias of 1 V with the source electrode grounded. The device trans-conductance, gm, is shown on the right axis.
(C) The drain current as a function of VD of a
graphene FET (LG = 240 nm) for various gate voltages.
(D) Measured small-signal current gain |h21| as
a function of frequency f for a 240-nm-gate (◇) and a 550-nm-gate (△) graphene FET at VD = 2.5 V. Cutoff frequencies, fT, were 53 and 100 GHz for the 550-nm and 240-nm devices, respectively.
a, AFM images of the sample: large flat terraces on the surface of the Si-face of a 4H-SiC(0001) substrate with graphene after high-temperature annealing in an argon atmosphere. b, Graphene patterned in the nominally 2-μm-wide Hall bar configuration on top of the terraced substrate. c, Layout of a 7 × 7 mm2 wafer with 20 patterned devices. Encircled are two devices with dimensions L = 11.6 µm and W = 2 µm (wire bonded) and L = 160 µm and W = 35 µm. The contact configuration for the smaller device is shown in the enlarged image. To visualize the Hall bar this optical micrograph was taken after oxygen plasma treatment, which formed the graphene pattern, but before the removal of resist.
Nature Nanotechnology 5, 186 - 189 (2010)
Model device acting as a resistor standard
T=300 mK Accuracy ca. 3x10-9
First graphene touch screen
The process includes adhesion of polymer supports, copper etching (rinsing) and dry transfer-printing on a target substrate. A wet-chemical doping can be carried out using a set-up similar to that used for etching.
Nature Nanotechnology (2010)
Sustains strain up to 6%
Extremely promising for transparent electrodes
Example: Graphene Q-switched, tunable fiber laser Group by Ferrari, UK: preprint 2010
Graphene is used as a non-linear (saturated absorption) medium to create short optical pulses.
The authors have succeeded to make 2 μs pulses, tunable between 1522 and 1555 nm with up to 40nJ energy. This is a simple and low-cost light source for metrology, environmental sensing and biomedical diagnostics.
Studies of graphene is a very interesting and promising area, but a lot of things remains to be done to create graphene electronics.
Giant Faraday rotation
Graphene turns the polarization by several degrees in modest magnetic
fields.
This opens pathways to use graphene in fast tunable ultrathin infrared magneto-optical devices.
The polarization plane of the linearly polarized incoming beam is rotated by the Faraday angle θ
after passing through graphene on a SiC substrate in a perpendicular magnetic field.
Simultaneously, the polarization acquires a certain
ellipticity.
Crassee et al., Nature Physics 2010
Studies of graphene is a very interesting and promising area, but a lot of things remains to be done to create graphene
