1. Junctions of Dirac Materials K. Sengupta Indian Association for the Cultivation of Sciences, Kolkata

2. 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

3. Graphene

4. Graphite Graphite: 3D allotrope of carbon most commonly found in pencil tips Quasi 2D structure: hexagonal planes 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 graphene has been a long-standing challenge.

5. Isolation of graphene: the “Scotch Tape” technique Multilayer graphene under ordinary microscope AFM image of multilayer graphene on SiO 2 Schematic experimental setup Scanning electron microscope image of an experimental setup AFM image of single layer graphene Novoselov et al. Science 2004

6. Relevant Basics about graphene A B Each unit cell has two electrons from 2p z orbital leading to delocalized p bond. Honeycomb latticeTwo inequivalent sites A and B whose surrounding sites 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

7. t1t1 t2t2 t3t3 Brillouin zone 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

8. 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 other at 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

9. 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 massless Dirac eqn. with Dirac matrices replaced by Pauli matrices and c replaced by v F.

10. Zero doping Fermi point Finite doping Fermi surface E F can be tuned by an external gate voltage. DOS varies linearly with E for undoped graphene but is almost a constant at large doping. r(E) = r 0 |E+E F | E F =0E F >0 Helicity associated with Dirac electrons at K and K’ points. Electrons with E>0 around K point have their pseudospin along k where pseudospin refers to A-B space. For K’, pseudospin points opposite to k. Solution of H a about K point: Within RG, interactions are (marginally) irrelevant.

11. Potential barriers in graphene V0V0 t r d Simple Problem: What is the probability of the incident electron to penetrate the barrier? Solve the Schrodinger equation and match the boundary conditions Answer: where Basic point: For V 0 >>E, T a monotonically decreasing function of the dimensionless barrier strength.

12. Simple QM 102: A 2D massless Dirac electron in a potential barrier V0V0 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 if c=np. Transmission resonance condition for Dirac electrons. Katnelson et al. Nature Physics ( 2006).

13. Topological Insulators

14. Insulators History: Class of solids which have high electrical resistance. Classes of insulators Band insulators: energy gap arising out of electron’s interaction 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 EFEF Valence Conduction Realization over the last few years: 3D Band insulators can have interesting topological features.

15. 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 motion In the bulk Skipping orbits E k EFEF Quantum picture Additional chiral states at the edge: broken time-reversal symmetry (TRS)

16. 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 EFEF Bulk levels A pair of edge states carrying opposite spin and chirality. TRS is preserved and the states do not interact in absence of TRS breaking perturbation. B -B

17. 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 studied originally in context of high energy physics for describing properties of relativistic massless particles. Topological insulators Schematic representation of single Dirac cone on the surface of a topological insulator Tabletop experiments studying properties of Dirac spinors in (2+1)D.

18. 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 t 1 to t 2 Initial condition Vector potential whose line integral gives the P-B phase for 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

19. 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 hosts an effective Dirac theory. (Roy 2006) For these bands, E 1 (k, )= E 2 (-k, ) Z 2 invariant in 2D Model: Graphene with spin-orbit coupling (Kane and Mele 2005) E=1 implies odd number of localized pairs of states crossing the Fermi level at the edge.

20. Introduction to topological insulators: 3D Odd crossing: odd number of Dirac point robust against TRS preserving perturbation n 0 =1 --- Strong Topological Insulator Even crossing: even number of Dirac points--- Weak Topological Insulators with n 0 =0. The crossing point of these bands in 3D are specified by three integers M = (n 1 b 1,n 2 b 2, n 3 b 3 )/2 where b i s are reciprocal lattice vectors. Two classes of such insulators: odd/even number of crossing within the half-tori -1< k x, k y <1, 0< k z <1 Consider three time-reversal invariant planes and compute Z2 index of each Planes k x =0, k y =0 and k z =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

21. 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. kyky kxkx Unlike graphene, applying a Zeeman field perpendicular to the surface opens up a gap: massive Dirac particles.

22. 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 and has no effect on their motion Conventional materials The magnetic field couples to the electron spin: Zeeman effect, which changes the electron’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 by the electrons. Such a field can affect the motion of the electrons.

24. Spin-resolved ARPES: demonstration of spin-momentum locking Experimental uncertainties: for angle measurements and for magnitude measurements. D. Hsieh et al. Nature 2009

25. STM data on Sb (1,1,1) surface Indication of absence of scattering from the Step in measurement of 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

26. Superconducting Junctions in graphene

27. 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

28. NIS Andreev reflection 2e 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 condutance decays as 1/(1+2z^2)^2 with increasing barrier strength.

29. Graphene N-B-S junctions Superconductivity is induced via proximity effect by the electrode. Effective potential barrier created by a gate voltage V g over a length d. Dimensionless barrier strength: Dirac-Bogoliubov-de Gennes (DBdG) Equation E F Fermi energy U(r) Applied Potential = V g for 0>x>-d D(r) Superconducting pair-potential between electrons and holes at K and K’ points Question: How would the tunneling conductance of such a junction behave as a function of the gate voltage? Applied bias voltage V.

30. Application of BTK formalism Wavefunction of a Dirac quasiparticle in the normal region Amplitude of normal reflection Amplitude of Andreev reflection Match boundary conditions and eliminate p, q, m and n to find r and r A for arbitrary applied bias voltage V. Obtain tunneling conductance using BTK formula r rArA t t’ NSB is the critical angle of incidence for the electron at bias voltage eV

31. 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 maxima occur at V 0 d/hv F =(n+1/2)p Similar unconventional oscillatory behavior of graphene Josephson junctions.

32. Transmission resonance condition Maxima of conductance occur when r=0. For subgap voltages, in the thin barrier limit, and for eV << E F, 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.

33. Oscillations persists: so one expects the oscillatory behavior both as functions of V G 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 function of bias and gate voltages at fixed barrier width

34. Josephson Effect S1S1 S2S2 The ground state wavefunctions have different phases for S 1 and S 2 Thus one might expect a current between them: DC Josephson Effect Experiments: Josephson junctions [Likharev, RMP 1979] S1S1 S2S2 N S-N-S junctions or weak links S1S1 S2S2 B S-B-S or tunnel junctions

35. Josephson effect in conventional tunnel junctions S1S1 S2S2 B Formation of localized subgap Andreev bound states at the barrier with energy dispersion which depends on the phase difference of the superconductors. The primary contribution to Josephson current comes from these bound states. Kulik-Omelyanchuk limit:Ambegaokar-Baratoff limit: Both I c and I c R N monotonically decrease as we go from KO to AB limit.

36. Graphene S-B-S junctions E F Fermi energy U(r) Applied Potential = V 0 for 0>x>-d D(r) Superconducting pair-potential in regions I and II as shown Question: How would the Josephson current behave as a function of the gate voltage V 0 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

37. I c and I c R N are p periodic bounded oscillatory functions of the effective barrier strength For c=np, I c R N reaches pD 0 /e: Kulik-Omelyanchuk limit. I c R N is bounded with values between pD 0 /e for c=np and 2.27D 0 /e for c=(n+1/2)p. Due to transmission resonance of DBdG quasiparticles, it is not possible to make T arbitrarily small by increasing gate voltage V 0. Thus, these junctions never reach Ambegaokar-Baratoff limit.

38. Junctions of Topological Insulators

39. Junctions involving two ferromagnets Induced magnetization on the TI surface below F1 and F2. Perfect junction Junction with strong barrier Unconventional dependence of the tunneling conductance on the azimuthal angle. One can obtain very large magnetoresistance by tuning m z of F1. Yokoyama et. al (2009)

40. 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 spin whose chirality (dispersion) is determined by the sign (magnitude) of the magnetization in the F region. Linder et al. (2010)

41. Interferometry with Majorana Fermions Two Majorana modes with opposite chirality at the interface of S with M 1 (red) and M 2 (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)

42. Tuning conductance of topological insulator junctions Idea: Deposit a proximate ferromagnetic thin film with a magnetization m 0 along the plane. The film, due to its proximity to the surface of the topological insulators, 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

43. Classical picture of electron motion Weak applied magnetization: The Dirac electrons sees a weak “effective field” which curves their trajectory a bit but let them pass through. Strong applied magnetization: The Dirac electrons sees a strong “effective field” which curves their trajectory enough to 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.

44. 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 can be controlled using magnetization of the proximate ferromagnetic film or the applied voltage leading to its use as a magnetic/voltage controlled switch

45. 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

46. STM spectroscopy with a magnetized tip

47. Magnetic STM and Conventional magnetic materials STM current depends on the tunneling matrix element of an electron from the STM tip to the sample. The tunneling matrix element is Usually determined the Bardeen Tunneling 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 the relative orientation of the tip and the sample magnetization. Note that there is no dependence on the azimuthal angle of the tip magnetization

48. 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.

49. Tunneling conductance G Spin independent part Depends on local value of S z. Vanishes for a pristine TI Depends on the azimuthal angle of the tip; also vanishes for a pristine TI The local S z orientation can be measured by computing tunneling conductance with the tip magnetization along z and -z

50. Break the azimuthal symmetry by putting an electric field. Azimuthal angle dependence Exact solution for the wavefunction in the presence of an electric and magnetic field for E<v F B Prediction: The tunneling current would depend on the direction of the electric field. eV/E 0 = 1.4 E/v F B =0.1 The slope of G provides information About the local relative phase of the TI wavefunction

51. Coupling Dirac and conventional materials

52. 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

53. 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.

54. 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 direction Of J z by varying a gate voltage.

55. Junctions with triplet superconductor Triplet order parameter with spin state of the Cooper pair specified 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

56. 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]

57. BTK formula for the tunneling conductance Reproduces the well-known Zero-bias peak for NM-TSC Junctions.

58. 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 ensures the R, T etc and hence G depend on the 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 c 1 gets transmitted as a hole in region II, which, for Dirac electrons, sees a barrier of –c 2. Analytical expression of zero-bias conductance

59. Experiments? Microscopic treatment of Junction details?

60. 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 are evanascent Andreev modes The spin conductance shows a finite bias peak. The peak approaches zero bias in the Limits of large chemical potential/barrier.

61. Conclusion Band physics can give rise to interesting properties of low energy quasiparticles of a many-body system. Possibilities of studying aspects of Dirac Physics on a tabletop. Distinct class of condensed matter systems with unconventional low-energy properties 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.