junctions of dirac materials k. sengupta indian association for the cultivation of sciences, kolkata
TRANSCRIPT
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Junctions of Dirac Materials
K. Sengupta
Indian Association for the Cultivation of Sciences, Kolkata
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Overview
1. Dirac materials: Introduction and basic properties
a) Graphene b) Topological Insulators
2. Superconducting Junctions of graphene
3. Junctions of Topological Insulators
4. Spin-polarized STM spectra
4. Coupling Dirac and conventional metals
5. Conclusion
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Graphene
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Graphite
Graphite: 3D allotrope of carbon most commonly found in pencil tips
Quasi 2D structure: hexagonalplanes weakly coupled to one another
Most stable form of carbon with weak inter-planar coupling.
Anisotropic properties: good(poor) propagation of phonons and electrons in plane (along c axis).
Graphene is a single layer of Graphite.
Experimental separation of graphenehas been a long-standing challenge.
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Isolation of graphene: the “Scotch Tape” technique
Multilayer grapheneunder ordinary microscope
AFM image of multilayer graphene on SiO2
Schematic experimentalsetup
Scanning electron microscope image of an experimental setup
AFM image of single layer graphene
Novoselov et al. Science 2004
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Relevant Basics about graphene
A
B
Each unit cell has two electronsfrom 2pz orbital leading to delocalized p bond.
Honeycomb lattice Two inequivalent sites A and B whose surroundingsites form a Y (inverted Y).
One can talk about probability of an electron on an A site or a B site: can be thought as up and down states of some fictitious spin: pseudospin.
Two component wavefunction
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t1t2t3
Brillouin zoneK’
K’
K’
K
K
K
Energy dispersion: Two energy bands with dispersion
Model for kinetic energy: electrons hop from any site to its nearest neighbor.
Diagonalize the Hamiltonian in momentum space to get energy bands
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There are two energy bands (valence and conduction)corresponding to energies
Two inequivalent Fermi points rather than a Fermi-line.
These two bands touch each otherat six points at the edges of the Brillouin zone
Two of these points K and K’are inequivalent; rest are related by translation of a lattice vector.
Dirac cone about the K and K’ points
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Absence of large k scattering leads to two species of massless Dirac Fermions.
Terminology Pauli matrix Relevant space
Pseudospin 2 by 2 matrix associated with two sublattice structure
Valley 2 by 2 matrix associated with two BZ points K and K’
Spin 2 by 2 matrix associated with the physical spin.
At each valley, we have a masslessDirac eqn. with Dirac matrices replaced by Pauli matrices and c replaced by vF.
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Zero dopingFermi point
Finite dopingFermi surface
EF can be tuned by an external gate voltage.
DOS varies linearly with E forundoped graphene but is almost a constant at large doping. r(E) = r0 |E+EF|
EF=0 EF>0
Helicity associated with Dirac electrons at K and K’ points.
Electrons with E>0 around K point have theirpseudospin along k where pseudospin refers to A-B space. For K’, pseudospin points opposite to k.
Solution of Ha about K point:
Within RG, interactions are (marginally) irrelevant.
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Potential barriers in graphene
V0
t
r
d
Simple Problem: What is the probability of the incident electron to penetrate the barrier?
Solve the Schrodinger equationand match the boundary conditions
Answer:
where
Basic point: For V0 >>E, T a monotonically decreasing function of the dimensionless barrier strength.
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Simple QM 102: A 2D massless Dirac electron in a potential barrier
V0
t
r
d
Basic point: T is an oscillatory function of the dimensionless barrier strength. Qualitatively different physics from that of the Schrödinger electrons.
For normal incidence, T=1.Klein paradox for Dirac electrons.Consequence of inability of a scalar potential to flip pseudospin
For any angle of incidence T=1 ifc=np. Transmission resonance condition for Dirac electrons.
Katnelson et al. Nature Physics ( 2006).
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Topological Insulators
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Insulators
History: Class of solids which have high electrical resistance.
Classes of insulators
Band insulators: energy gap arising out of electron’sinteraction with the lattice potential
Anderson insulators: energy gap arising out of electron’s interaction with impurities
Mott insulators: energy gap arising out of electron-electron interaction
E
EF
Valence
ConductionRealization over the last few years: 3D Band insulators can have interesting topological features.
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Integer quantum Hall effect: bulk edge correspondence
Bulk
B
Quantum Hall system
Classical picture
B
A system of planar electrons in the presence of a magnetic field perpendicular to the plane.
Localized motionIn the bulk
Skipping orbits
E
k
EF
Quantum picture
Additional chiral states at the edge: broken time-reversal symmetry (TRS)
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Spin Quantum Hall effect
Spin-orbit interaction provide opposite effective magnetic fields for up and down electrons spin up
Opposite motion and skipping orbits for electrons with different spins
E
EF
Bulk levels
Bulk levels
A pair of edge states carrying opposite spin and chirality.
TRS is preserved and the states do not interact in absenceof TRS breaking perturbation.
B
-B
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Topological Insulators: A special class of 3D band insulators with strong spin-orbit coupling.
Topological properties of bands in the bulk lead to presence of gapless electrons at the surface. [Balents and Moore (2007), Fu and Kane (2007), Roy (2009)]
The properties of these electrons are quite different from conventional electrons in solids.
Some properties of these electrons mimic those of massless Dirac electrons studiedoriginally in context of high energy physicsfor describing properties of relativisticmassless particles .
Topological insulators
Schematic representation of single Dirac cone on the surface of a topological insulator
Tabletop experiments studyingproperties of Dirac spinors in (2+1)D.
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Pancharatnam-Berry phase
Consider a Hamiltonian H which depends on slowly varying parameters ( slow compared to energy/time scale of the eigenstates of the Hamiltonian).
Schrodinger Equation
Let us consider the time evolution of a quantum system under H from t1 to t2
Initial condition
Vector potential whose line integral gives the P-B phasefor a closed path, the phase is an integer multiple of 2p
P-B field:
Geometric phase
Independent of phase convention of the wave function
Distribution of P-B field over Brillouin zone of band insulators: classification
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Introduction to topological insulators: 2D
Time reversal invariant bulk-systems with spin-orbit splitting.
k
E E
k
Corresponding to each band, there is a Chern integer; these change by 0,2,4.. when bands cross. The crossing point hostsan effective Dirac theory. (Roy 2006)
For these bands, E1(k, )= E2(-k, )
Z2 invariant in 2D
Model: Graphene with spin-orbit coupling(Kane and Mele 2005)
E=1 impliesodd number of localized pairs of states crossing the Fermi levelat the edge.
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Introduction to topological insulators: 3D
Odd crossing: odd numberof Dirac point robust againstTRS preserving perturbationn0=1 --- Strong Topological Insulator
Even crossing: even number of Dirac points--- Weak TopologicalInsulators with n0 =0.
The crossing point of these bandsin 3D are specified by three integersM = (n1b1,n2b2,n3b3)/2 where bi s arereciprocal lattice vectors.
Two classes of such insulators: odd/even number of crossing within the half-tori-1< kx, ky <1, 0< kz <1
Consider three time-reversal invariant planes and compute Z2 index of each Planes kx=0, ky=0 and kz=0
Three Z2 invariants to characterize an insulator
Fourth invariant and strong and weak topological insulators
The 2D surfaces of these 3D insulators hosts Dirac points: analog of edge states in 2D insulators. The positions of these Dirac points are determined by projection of M on the surface Brillouin zone
Kane and Mele, Roy, Balents and Moore
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Helicity of Dirac electrons
Hamiltonian for the Dirac electrons on the surface
Solution for the eigenvalues and the eigenfunction of H
Spin orientation around the Fermi surface of Dirac electrons on the surface of a pristine topological insulator.
ky
kx
Unlike graphene, applying a Zeemanfield perpendicular to the surface opens up a gap: massive Dirac particles.
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TI surface
y
x
B
Properties: Applying a constant magnetic field
No orbital motion due to an in-plane magnetic field.
Zeeman effect just provides a constant shift to electron momentum. It does not change the energy of these electrons andhas no effect on their motion
Conventional materials
The magnetic field couples to the electron spin: Zeemaneffect, which changes theelectron’s energy.
Surface of topological insulator
No orbital motion due to an in-plane magnetic field.
However, if the applied field changes in space, it’s presence is perceived bythe electrons. Such a field can affect the motion of the electrons.
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Spin-resolved ARPES: demonstration of spin-momentum locking
Experimental uncertainties: for angle measurements and for magnitude measurements.
D. Hsieh et al. Nature 2009
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STM data on Sb (1,1,1) surface
Indication of absence of scattering from theStep in measurementof G(r,E) above 230 meV.
Absence of scattering between spin up electrons with momentum k and spin down electrons with momentum -k
Manoharan et al. 2009
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Superconducting Junctions in graphene
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Superconductivity and tunnel junctions
Measurement of tunneling conductance
eV
Normal metal (N) Superconductor (S)
Insulator (I)
eV
Normal metal (N)
Insulator (I)
Normal metal (N)
N-I-N interface
N-I-S interface
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N I S
Andreev reflection2e charge transfer
Basic mechanism of current flow in a N-I-S junction
Andreev reflection is strongly suppressed in conventional junctions if the insulating layer provides a large potential barrier: so called tunneling limit
In conventional junctions, subgap tunneling conductance is a monotonically decreasing function of the effective barrier strength Z.
BTK, PRB, 25 4515 (1982)
Zero bias tunneling condutancedecays as 1/(1+2z^2)^2 with increasing barrier strength.
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Graphene N-B-S junctions
Superconductivity is induced viaproximity effect by the electrode.
Effective potential barrier created by a gate voltage Vg over a length d. Dimensionless barrier strength:
Dirac-Bogoliubov-de Gennes (DBdG) Equation
EF Fermi energyU(r) Applied Potential = Vg for 0>x>-dD(r) Superconducting pair-potential between electrons and holes at K and K’ points
Question: How wouldthe tunneling conductanceof such a junction behave as a function of the gate voltage?
Applied bias voltage V.
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Application of BTK formalism
Wavefunction of a Dirac quasiparticlein the normal region
Amplitude of normal reflection
Amplitude of Andreev reflection
Match boundary conditions and eliminate p, q, m and nto find r and rA for arbitrary applied bias voltage V.
Obtain tunneling conductance using BTK formula
r
rA t
t’N SB
is the critical angle of incidence for the electron at bias voltage eV
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Tunneling conductance of graphene NBS junctions
Central Result: In complete contrast to conventional NBS junction,Graphene NBS junctions, due to the presence of Dirac-like dispersion of its electrons, exhibit novel p periodic oscillatory behavior of subgap tunneling conductance as the barrier strength is varied.
Periodic oscillations of subgap tunneling conductance as a function of barrier strength and thickness.
Tunneling conductance maximaoccur at V0d/hvF=(n+1/2)p
Similar unconventional oscillatory behavior of graphene Josephson junctions.
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Transmission resonance condition
Maxima of conductance occurwhen r=0.
For subgap voltages, in the thin barrier limit, and foreV << EF, it turns out that
r=0 and hence G is maximum if:
1. g=0: Manifestation of Klein Paradox. Not seen in tunneling conductance due to averaging over transverse momenta.
2. b=0: Maxima of tunneling conductance at the gap edge: also seen in conventional NBS junctions.
3. c=(n+1/2)p: Novel transmission resonance condition for graphene NBS junction.
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Oscillations persists: so one expects the oscillatory behavior both as functions of VG and d to be robust.
Not so thin barrier
Zero bias tunneling conductance as a function of barrier width and gate voltage
Tunneling conductance as a functionof bias and gate voltages at fixed barrier width
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Josephson Effect
S1 S2
The ground state wavefunctionshave different phases for S1 and S2
Thus one might expect a current between them: DC Josephson Effect
Experiments: Josephson junctions [Likharev, RMP 1979]
S1 S2N
S-N-S junctions or weak links
S1 S2B
S-B-S or tunnel junctions
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Josephson effect in conventional tunnel junctions
S1 S2B
Formation of localized subgap Andreev bound states at the barrier with energy dispersionwhich depends on the phasedifference of the superconductors.
The primary contributionto Josephson current comesfrom these bound states.
Kulik-Omelyanchuk limit: Ambegaokar-Baratoff limit:
Both Ic and IcRN monotonically decrease as we go from KO to AB limit.
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Graphene S-B-S junctions
EF Fermi energyU(r) Applied Potential = V0 for 0>x>-dD(r) Superconducting pair-potential in regions I and II as shown
Question: How would the Josephson currentbehave as a functionof the gate voltage V0
Procedure:1. Solve the DBdG equation in regions I, II and B.
2. Match the boundary conditions at the boundaries between regions I and B and B and II.
3. Obtain dispersion for bound Andreev subgap states and hence find the Josephson current.
Schematic Setup
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Ic and IcRN are p periodic bounded oscillatory functions of the effective barrier strength
For c=np, IcRN reaches pD0/e: Kulik-Omelyanchuk limit.
IcRN is bounded with values between pD0/e for c=np and 2.27D0/e for c=(n+1/2)p.
Due to transmission resonance of DBdGquasiparticles, it is not possible to makeT arbitrarily small by increasing gate voltageV0. Thus, these junctions never reach Ambegaokar-Baratoff limit.
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Junctions of Topological Insulators
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Junctions involving two ferromagnets
Induced magnetization on the TI surface below F1 and F2.
Perfect junctionJunction with strong barrier
Unconventional dependence of the tunneling conductance on the azimuthal angle.
One can obtain very large magnetoresistance by tuning mz of F1.
Yokoyama et. al (2009)
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Junctions of Ferromagnet and superconductor
Induced s-wave superconductivity in region S and ferromagnetism in region F.
These junctions support localized subgap states which are equal superposition of electrons and holes: one such state per spin
One Majorana mode per spinwhose chirality (dispersion) is determined by the sign (magnitude) of the magnetization in the F region.
Linder et al. (2010)
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Interferometry with Majorana Fermions
Two Majorana modes with opposite chirality at the interface of S with M1 (red) and M2 (blue)
Putting an electron/hole at “a” converts it to two Majorana modes “b” and “c”
These modes propagate and recombine at “d” to form an electron or a hole.
The recombination depends on the relative phase picked by the Majorana modes during the propagation which depends on the parity of the vortex number in the superconductor.
For odd number of vortices, there is a 2e charge transfer to the superconductor leading to a finite G(V)
Beenakker et al (2009)
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Tuning conductance of topological insulator junctions
Idea: Deposit a proximate ferromagnetic thin film with a magnetization m0 alongthe plane.
The film, due to its proximity to the surface of the topologicalinsulators, induces a magnetization in region II
Same as applying a parallel magnetic field in region II.
The magnetization abruptly drops at the edges of region II. The electrons perceive this change.
The change in magnetization appears to the electrons as an “effective Magnetic field” in opposite directions at the edges ( along z or –z)
What the electron sees
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Classical picture of electron motion
Weak applied magnetization:The Dirac electrons sees a weak “effective field” which curves their trajectory a bitbut let them pass through.
Strong applied magnetization:The Dirac electrons sees a strong “effective field” which curves their trajectory enoughto cut off passage across the junction.
There is a critical magnetization beyond which no quasiparticles pass through the junction: such junctions can be switched on or off by tuning magnetization.
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Beyond a critical magnetization or equivalently below a critical applied voltage, conductance switches from oscillatory to exponentially decaying function of the junction width d.
Solve the transmission problem and compute the conductance
For reasonably thick barriers, the conductance of these junctions canbe controlled using magnetization of the proximate ferromagnetic film or the applied voltage leading to its use as a magnetic/voltage controlled switch
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Multiple junctions
Junctions with N magnetic regions(N=2 in the fig.)
N=2 N=3
Complex behavior of G as a function of z for small M
G approaches zero much faster than the single magnetic region for large M
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STM spectroscopy with a magnetized tip
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Magnetic STM and Conventional magnetic materials
STM current depends on the tunneling matrix element of anelectron from the STM tip to the sample.
The tunneling matrix element isUsually determined the BardeenTunneling formula
For a magnetized tip and a magnetic sample, one usually choose the spin quantization axis to be along the tip magnetization.
M and hence I depend on therelative orientation of the tipand the sample magnetization.Note that there is no dependenceon the azimuthal angle of the tipmagnetization
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STM with Topological Insulators
The spin quantization axis for the Dirac electrons is already fixed along z and can not be chosen along M
May lead to azimuthal angle dependence of G(V).
The two-component wavefunction of the tip and the TI leads to a more complicated matrix element.
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Tunneling conductance G
Spin independent part
Depends on local value of Sz. Vanishesfor a pristine TI
Depends on the azimuthal angle of the tip; also vanishes for a pristine TI
The local Sz orientation can be measured by computing tunneling conductance with the tip magnetization along z and -z
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Break the azimuthal symmetryby putting an electric field.
Azimuthal angle dependence
Exact solution for the wavefunctionin the presence of an electric andmagnetic field for E<vF B
Prediction: The tunneling current would depend on the direction of the electric field.
eV/E0 = 1.4 E/vFB =0.1
The slope of G provides informationAbout the local relative phase of theTI wavefunction
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Coupling Dirac and conventional materials
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Coupling a TI with Dirac electron on the surfaces ( regions I and II)with a metallic or ferromagnetic film
Boundary condition between Dirac and Schrodinger electrons
Current continuity
Linear conditions
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Generation of spin current in metals
Reflection from the barrier in region I and transmission into region II changes the direction of motion of the electron.
Spin-momentum locking ensures that such a scattering leads to redistribution of electron wavefunction amplitudes among different spin components
Possibility of generation of finite spin current along x through a metallic film which can be controlled by an external applied voltage.
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Generation of spin current in FM films
Control of magnitude of the spin current along the film magnetization
Non-monotonic behavior as a function of barrier Potential
Control over the directionOf Jz by varying a gate voltage.
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Junctions with triplet superconductor
Triplet order parameter with spin state of the Cooper pairspecified by the d vector
The junction is modeled by three barrier potentials: these are potential barriers seen electrons as they approach the junction from TI1, TI2, and the SC .
Wave functions for particles and holes in TI1
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Boundary condition ( continuity of current across the junction)
Wavefunction in region II
Wavefunction in region III
Solve for reflection and transmission amplitudes and find G [ BTK formalism]
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BTK formula for the tunneling conductance
Reproduces the well-knownZero-bias peak for NM-TSC Junctions.
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Zero-bias conductance
One can obtain an analytical solution for R, T etc from the boundary conditions when V=0
Rotational symmetry breaking in the spin space due to spin-momentum locking of the Dirac electrons ensuresthe R, T etc and hence G depend onthe orientation of the d vector.
For d=x, Constant G(0) independent of barrier strength
For d=z(y)
An electron approaching the barrier which sees a barrier c1 gets transmitted as a hole in region II, which, for Dirac electrons, sees a barrier of –c2.
Analytical expression of zero-bias conductance
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Experiments? Microscopic treatment of Junction details?
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Spin conductance
The spin current along x(in spin space)is the only finite along x
The main contribution to the spin conductance comes from finite angle of incidence where there areevanascent Andreev modes
The spin conductance shows a finite biaspeak. The peak approaches zero bias in the Limits of large chemical potential/barrier.
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Conclusion
Band physics can give rise to interesting properties of low energy quasiparticlesof a many-body system.
Possibilities of studying aspects of Dirac Physics on a tabletop.
Distinct class of condensed matter systems with unconventional low-energyproperties arising from a combination of Dirac physics and many-body phenomena such as superconductivity.
Large number of engineering/technological aspects: graphene may have important role for future LED films and transistors.
Some reviews: 1) Graphene: arXiv:0709.1163. 2) Topological insulators: arXiv:0912.2157.