Edge states in graphene

Graphene offers more [http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.107.236806]:

In condensed matter systems, topology often gives rise to gapless excitations at the edge (in 2D) or the surface (in 3D). Such excitations in the 2D fractional quantum Hall state should manifest in the edge behaving as a Luttinger liquid, in which tunneling is determined by a universal power law related to an attribute—the filling factor—of the magnetic flux through, and the number of electrons in, the 2D state.

However, no such behavior has yet been observed at the edges of 2D semiconductor heterostructures, the most-studied quantum Hall systems. Theorists say that in these systems the conflicting interplay between the confinement potential, attracting each electron towards the center, and the Coulomb force, pushing them apart from each other, modifies the edge itself. This process—edge reconstruction—disturbs the universal Luttinger liquid picture in the experimentally accessible distance scales.

In a paper in Physical Review Letters, Zi-Xiang Hu, at Princeton University, and his colleagues tell us that we may, after all, be able to see chiral Luttinger behavior in another system in which fractional quantum Hall effect has been observed—graphene. In graphene, electrons are confined by metallic gates that are placed a specific distance away. By contrast, in semiconductors, electrons are confined by dopants. This one difference should make graphene less susceptible to edge reconstruction and reveal the fractional quantum Hall state. The authors say that experimentalists should therefore finally see the elusive universal edge behavior in the experimentally accessible state with filling factor 1/3. – Sami Mitra

The 5/2 engima in FQHE

Some progress in QHE. Despite the toil invested in this startling subject and estoric thoughts and concepts invoked, QHE, in my opinion, is far from being well understood. The simplicity of the phenomena makes strong contrast with the intricacy of theoretical context. This contrast makes me feel uncomfortable with the present understanding. Now here is an experiment that shows the inadequacy. Is the FQHE indeed so different from IQHE ?

larger mystery surrounding the fractional quantum Hall effect in the second Landau level. The other observed fractions, such as 2 + 1/3, 2 + 2/3, 2 + 2/5, 2 + 3/5, match those observed in the lowest Landau level, but detailed calculations show, surprisingly, that the model of weakly interacting composite fermions, which is successful for the explanation of the lowest Landau level fractions, is not adequate for these second Landau level fractions. A resolution of the second Landau level fractional quantum Hall effect is likely to lead to much exciting physics.

PRL 105, 096801 (2010)

Boundary matters: topological insulators

It has become a habit for those who profess in solid state physics to consider a crystal as a periodic array of atoms. In reality, this is, however only an approximation. Any solid is limited by its surfaces, which means the periodicity terminates at these bounds. Despite this, people still in most cases take them as infinite and unbounded, so that exact periodicity can be used to obtain exact solutions of some models. Usually, such solutions indeed provide very good descriptions of the sample, provided the bulk is dominant over the boundaries. One such artificial boundary condition goes under 'Von Karmen periodic condition', which yields Bloch waves.

Nevertheless, surface (not film, [1]) states can display many exotic properties that are not supported by bulky solutions. These properties may be related to, let's say, impurities, dangling bonds, surface tensions, structural reconstructions and et al. Due to these properties, modern computers could be made. As we know, transistors and diodes just make use of the properties of interfaces between two semiconductors.

Surface states have not ceased to surprise people. Some ten years ago, people found a novel type of conducting channels in two dimensional electron gas systems. Such systems under strong magnetic field exhibit the famous quantum Hall effects. It was later suggested that, such Hall states possess edge states that can conduct electricity along the edges of the 2D sample. These states are squeezed out of and split from the insulating bulk states, by magnetic field.

In 2005, Kane and his collaborators suggested that [2], such edge states could exist even without magnetic field. The considered graphene, namely a single graphite layer. In such materials, they guessed, there might be strong spin-orbit coupling. Such couplings can actually play a similar role as a magnetic field, and thus may create edge states. In this case, new complexity arises due to the spin degrees of freedom (see the figure).

All the as-described edge states are 1D objects. Most recently, a kind of 2D edge states was discovered existing in Bi2Te3 compounds. Such compounds have complex crystal structures and strong spin orbit coupling. These states are able to conduct spin and charges along the surface. So, one has this very gorgeous phenomenon: insulating bulk+metallic surface. They are termed 'topological insulators'. Why topology ? Topological properties are the properties that are invariant under continuous transformations of parameter space. In Bi2Te3, that is the number of edge states, which is conserved, however the shape of the sample is changed.

Although something has been learned of these new properties, a realistic and analytically tractable solution still awaits to show up. More experiments need be done to confirm and explore our understanding.

[1]A surface is linked with a bulk, but a film has its own bulk and surface (edges). Interface can be deemed as a special surface.
[2]PRL 95, 226801 (2005)

New surge in topological insulators

[1]PRL 105, 076801 (2010)
[2]PHYSICAL REVIEW B 82, 081305 R 2010
[3]Physics 3, 66 (2010)

The prediction [1] and experimental discovery [2, 3] of a class of materials known as topological insulators is a major recent event in the condensed matter physics community. Why do two- and three-dimensional topological insulators (such as HgTe/CdTe [2] and Bi2Se3 [3], respectively) attract so much interest? Thinking practically, these materials open a rich vista of possible applications and devices based on the unique interplay between spin and charge. More fundamentally,
there is much to enjoy from a physics point of view, including the aesthetic spin-resolved Fermi surface topology [3], the possibility of hosting Majorana fermions (a fermion that is its own antiparticle) in a solid-state system [4], and the intrinsic quantum spin Hall effect, which can be thought of as two copies of the quantum Hall effect for spin-up and spin-down electrons [5]. Now, an exciting new addition to the above list comes from two teams that are reporting the first experimental observation of quantized topological surface states forming Landau levels in the presence of a magnetic field. The two papers - one appearing in Physical Review Letters by Peng Cheng and colleagues at Tsinghua University in China, and collaborators in the US, the other, appearing as a Rapid Communication in Physical Review B, by Tetsuo Hanaguri at Japan’s RIKEN Advanced Science Institute in Wako and scientists at the Tokyo Institute of Technology - pave the way for seeing a quantum Hall effect in topological insulators.

Thermal Hall effect of magnons

Thermal Hall Effects refer to a class of phenomena in which thermal current carried by elementary excitations of a material is deflected by applying magnetic field. Magnons, as magnetic excitations have been predicted to show such behaviors when both magnetic field and temperature gradient are present. This has been just observed in Lu2V2O7, whose V element is magnetic and magnetic ordering occurs below about 100K[1].

[1]Science 329, 297 (2010)

The Hall effect usually occurs in conductors when the Lorentz force acts on a charge current in the presence of a perpendicular magnetic field. Neutral quasi-particles such as phonons and spins can, however, carry heat current and potentially exhibit the thermal Hall effect without resorting to the Lorentz force. We report experimental evidence for the anomalous thermal Hall effect caused by spin excitations (magnons) in an insulating ferromagnet with a pyrochlore lattice structure. Our theoretical analysis indicates that the propagation of the spin waves is
influenced by the Dzyaloshinskii-Moriya spin-orbit interaction, which plays the role of the vector potential, much as in the intrinsic anomalous Hall effect in metallic ferromagnets.

Curved space acts as gauge field in graphene

a, Distortion of a graphene disc which is required to generate uniform B_S. The original shape is shown in blue. b, Orientation of the graphene crystal lattice with respect to the strain. Graphene is stretched or compressed along equivalent crystallographic directions 100. Two graphene sublattices are shown in red and green. c, Distribution of the forces applied at the disc’s perimeter (arrows) that would create the strain required in a. The uniform colour inside the disc indicates strictly uniform pseudomagnetic field. d, The shown shape allows uniform B_S to be generated only by normal forces applied at the sample’s perimeter. The length of the arrows indicates the required local stress.

Among many remarkable qualities of graphene, its electronic properties attract particular interest owing to the chiral character of the charge carriers, which leads to such unusual phenomena as metallic conductivity in the limit of no carriers and the half-integer quantum Hall effect observable even at room temperature^{1, 2, 3}. Because graphene is only one atom thick, it is also amenable to external influences, including mechanical deformation. The latter offers a tempting prospect of controlling graphene’s properties by strain and, recently, several reports have examined graphene under uniaxial deformation^{4, 5, 6, 7, 8}. Although the strain can induce additional Raman features^{7, 8}, no significant changes in graphene’s band structure have been either observed or expected for realistic strains of up to ~15% (refs 9, 10, 11). Here we show that a designed strain aligned along three main crystallographic directions induces strong gauge fields^{12, 13, 14} that effectively act as a uniform magnetic field exceeding 10 T. For a finite doping, the quantizing field results in an insulating bulk and a pair of countercirculating edge states, similar to the case of a topological insulator^{15, 16, 17, 18, 19, 20}. We suggest realistic ways of creating this quantum state and observing the pseudomagnetic quantum Hall effect. We also show that strained superlattices can be used to open significant energy gaps in graphene’s electronic spectrum.