Junctions of Dirac Materials

K. Sengupta

Indian Association for the Cultivation of Sciences, Kolkata

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

Graphene

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.

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

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

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

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

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.

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.

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.

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).

Topological Insulators

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.

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)

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

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.

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

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.

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

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.

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.

Spin-resolved ARPES: demonstration of spin-momentum locking

Experimental uncertainties: for angle measurements and for magnitude measurements.

D. Hsieh et al. Nature 2009

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

Superconducting Junctions in graphene

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

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.

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.

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

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.

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.

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

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

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.

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

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.

Junctions of Topological Insulators

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)

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)

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)

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

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.

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

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

STM spectroscopy with a magnetized tip

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

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.

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

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

Coupling Dirac and conventional materials

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

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.

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.

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

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]

BTK formula for the tunneling conductance

Reproduces the well-knownZero-bias peak for NM-TSC Junctions.

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

Experiments? Microscopic treatment of Junction details?

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.

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.