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
Most stable form of carbon with
weak inter-planar coupling.
Anisotropic properties: good
(poor) propagation of phonons
and electrons in plane (along
c axis).
Quasi 2D structure: hexagonal
planes weakly coupled to one
another
Graphene is a single layer of
Graphite.
Experimental separation of graphene
has been a long-standing challenge.
Isolation of graphene: the “Scotch Tape” technique
AFM image of multilayer graphene on SiO2
Multilayer
graphene
under
ordinary
microscope
Schematic
experimental
setup
AFM image of
single layer
graphene
Scanning electron microscope
image of an experimental setup
Novoselov et al. Science 2004
Relevant Basics about graphene
Each unit cell has two electrons
from 2pz orbital leading to
delocalized p bond.
Honeycomb lattice
A
B
Two 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
Brillouin zone
K
K’
t2 t1
t3
K
K’
K
K’
Model for kinetic energy: electrons hop from any site to its nearest neighbor.
Diagonalize the Hamiltonian in momentum space to get energy bands
Energy dispersion: Two energy
bands with dispersion
There are two energy bands
(valence and conduction)
corresponding to energies
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.
Two inequivalent Fermi points
rather than a Fermi-line.
Dirac cone about the K and K’ points
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 vF.
Absence of large k scattering leads to two species of massless Dirac Fermions.
Helicity associated with Dirac electrons at K and K’ points.
Solution of Ha about K point:
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.
EF=0
Zero doping
Fermi point
EF>0
Finite doping
Fermi surface
EF 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) = r0 |E+EF|
Within RG, interactions are (marginally) irrelevant.
Potential barriers in graphene
t
r
V0
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 V0 >>E, T a monotonically decreasing
function of the dimensionless barrier strength.
Simple QM 102: A 2D massless Dirac electron in a potential barrier
t
r
V0
d
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.
Basic point: T is an oscillatory function of the dimensionless barrier strength.
Qualitatively different physics from that of the Schrödinger 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’s
interaction with
the lattice potential
E
Anderson insulators:
energy gap arising out of
electron’s interaction with
impurities
Conduction
EF
Valence
Mott insulators:
energy gap arising
out of electron-electron
interaction
Realization over the last few years:
3D Band insulators can have
interesting topological features.
Integer quantum Hall effect: bulk edge correspondence
B
Quantum Hall system
A system of planar
electrons in the presence
of a magnetic field
perpendicular to the plane.
Bulk
Classical picture
Quantum picture
B
E
Localized motion
In the bulk
Skipping
orbits
EF
k
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
B
spin up
Opposite motion and skipping orbits
for electrons with different spins
-B
E
Bulk levels
EF
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.
Topological insulators
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 .
Tabletop experiments studying
properties of Dirac spinors
in (2+1)D.
Schematic representation of single
Dirac cone on the surface of a
topological insulator
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
Geometric phase
P-B field:
Independent of
phase convention
of the wave function
Distribution of P-B field
over Brillouin zone of band
insulators: classification
Vector potential whose line
integral gives the P-B phase
for a closed path, the phase
is an integer multiple of 2p
Introduction to topological insulators: 2D
E
E
k
Time reversal invariant bulk-systems
with spin-orbit splitting.
For these bands, E1(k, )= E2(-k, )
k
Model: Graphene with
spin-orbit coupling
(Kane and Mele 2005)
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)
E=1 implies
Z2 invariant in 2D
odd number of
localized pairs of
states crossing
the Fermi level
at the edge.
Introduction to topological insulators: 3D
Consider three time-reversal invariant
planes and compute Z2 index of each
Three Z2 invariants to
characterize an insulator
Planes kx=0, ky=0 and kz=0
The crossing point of these bands
in 3D are specified by three integers
M = (n1b1,n2b2,n3b3)/2 where bi s are
reciprocal lattice vectors.
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
Fourth invariant and strong and weak topological insulators
Two classes of such
insulators: odd/even
number of crossing
within the half-tori
-1< kx, ky <1, 0< kz <1
Kane and Mele, Roy,
Balents and Moore
Odd crossing: odd number
of Dirac point robust against
TRS preserving perturbation
n0=1 --- Strong Topological Insulator
Even crossing: even number of
Dirac points--- Weak Topological
Insulators with n0 =0.
Helicity of Dirac electrons
Hamiltonian for the Dirac electrons on the surface
Solution for the eigenvalues and the eigenfunction of H
ky
kx
Unlike graphene, applying a Zeeman
field perpendicular to the surface opens
up a gap: massive Dirac particles.
Spin orientation around the
Fermi surface of Dirac
electrons on the surface of
a pristine topological insulator.
Properties: Applying a constant magnetic field
y
B
TI surface
x
Conventional materials
No orbital motion due to
an in-plane magnetic field.
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.
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
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.
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 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
Superconducting Junctions in graphene
Superconductivity and tunnel junctions
eV
N-I-N interface
Normal metal (N)
Measurement of
tunneling conductance
Normal metal (N)
Insulator (I)
eV
N-I-S interface
Normal metal (N)
Superconductor (S)
Insulator (I)
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
N
I
S
Andreev reflection
2e charge transfer
In conventional junctions, subgap
tunneling conductance is a
monotonically decreasing function
of the effective barrier strength Z.
Zero bias tunneling condutance
decays as 1/(1+2z^2)^2 with
increasing barrier strength.
BTK, PRB, 25 4515 (1982)
Graphene N-B-S junctions
Superconductivity is induced via
proximity effect by the electrode.
Effective potential barrier created by a gate
voltage Vg over a length d. Dimensionless
barrier strength:
Applied bias voltage V.
Dirac-Bogoliubov-de Gennes (DBdG) Equation
EF
Fermi energy
U(r)
Applied Potential = Vg 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?
Application of BTK formalism
rA
N
t
B
S
t’
r
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 rA for arbitrary
applied bias voltage V.
Obtain tunneling
conductance
using BTK formula
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.
Tunneling conductance maxima
occur at V0d/hvF=(n+1/2)p
Periodic oscillations of subgap tunneling
conductance as a function of barrier
strength and thickness.
Similar unconventional oscillatory behavior of graphene Josephson junctions.
Transmission resonance condition
Maxima of conductance occur
when r=0.
For subgap voltages, in the thin barrier limit, and for
eV << EF, it turns out that
1.g=0: Manifestation of Klein Paradox. Not seen
in tunneling conductance due to averaging over
transverse momenta.
r=0 and
hence G is
maximum if:
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.
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
Oscillations persists: so one expects
the oscillatory behavior both as
functions of VG and d to be robust.
Josephson Effect
S1
S2
The ground state wavefunctions
have different phases for S1 and S2
Thus one might expect a current
between them: DC Josephson Effect
Experiments: Josephson junctions [Likharev, RMP 1979]
S1
N
S2
S-N-S junctions or weak links
S1
B
S2
S-B-S or tunnel junctions
Josephson effect in conventional tunnel junctions
S1
B
S2
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 Ic and IcRN monotonically decrease as we go from KO to AB limit.
Graphene S-B-S junctions
Schematic Setup
EF
Fermi energy
U(r)
Applied Potential = V0 for 0>x>-d
D(r)
Superconducting pair-potential in
regions I and II as shown
Procedure:
1. Solve the DBdG equation in regions
I, II and B.
Question: How would
the Josephson current
behave as a function
of the gate voltage V0
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.
Ic and IcRN are p periodic bounded oscillatory functions of the effective barrier strength
IcRN is bounded with values between pD0/e for c=np and 2.27D0/e for
c=(n+1/2)p.
For c=np, IcRN reaches pD0/e:
Kulik-Omelyanchuk limit.
Due to transmission resonance of DBdG
quasiparticles, it is not possible to make
T arbitrarily small by increasing gate voltage
V0. Thus, these junctions never reach
Ambegaokar-Baratoff limit.
Junctions of Topological Insulators
Junctions involving two ferromagnets
Yokoyama et.
al (2009)
Induced magnetization on
the TI surface below F1
and F2.
Perfect junction
Unconventional dependence of
the tunneling conductance on the
azimuthal angle.
Junction with
strong barrier
One can obtain very large
magnetoresistance by tuning
mz of F1.
Junctions of Ferromagnet and superconductor
Induced s-wave
superconductivity in region
S and ferromagnetism in
region F.
Linder et al. (2010)
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.
Interferometry with Majorana Fermions
Beenakker et al (2009)
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)
Tuning conductance of topological insulator junctions
What the electron sees
Idea: Deposit a proximate ferromagnetic
thin film with a magnetization m0 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)
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.
Solve the transmission problem and compute the conductance
Beyond a critical magnetization
or equivalently below a critical
applied voltage, conductance
switches from oscillatory to
exponentially decaying function
of the junction width d.
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
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 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
STM with Topological Insulators
The spin quantization axis for the
Dirac electrons is already fixed along
z and can not be chosen along M
The two-component wavefunction of the
tip and the TI leads to a more complicated
matrix element.
May lead to azimuthal angle
dependence of G(V).
Tunneling conductance G
Spin independent part
Depends on local
value of Sz. Vanishes
for 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
Azimuthal angle dependence
Break the azimuthal symmetry
by putting an electric field.
Exact solution for the wavefunction
in the presence of an electric and
magnetic field for E<vF B
Prediction: The tunneling current
would depend on the direction
of the electric field.
The slope of G provides information
About the local relative phase of the
TI wavefunction
eV/E0 = 1.4 E/vFB =0.1
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 direction
Of Jz by varying a gate
voltage.
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
Wavefunction in region II
Wavefunction in region III
Boundary condition ( continuity of current across the junction)
Solve for reflection and transmission amplitudes and find G [ BTK formalism]
BTK formula for the
tunneling conductance
Reproduces the well-known
Zero-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
For d=x,
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.
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 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.
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.