Topological Matter
• 2D topological insulators (quantum spin-Hall effect)
• 2D topological superconductors (Majorana fermions)
• 1D topological superconductors
Topological Matter
• 2D topological insulators (quantum spin-Hall effect)
• 2D topological superconductors (Majorana fermions)
• 1D topological superconductors
E
EFnormal semiconductor,
or vacuumtopological
insulator
topological insulators (parity inversion)
E
EFodd parity(p-band)
even parity(s-band)
normal semiconductorgraphene :(
E
EFeven parity
(s-band)
odd parity(p-band)
HgTe/CdTe or InAs/GaSb quantum wells (2D),
Bi2Se3, Bi2Te3, SmB6 (3D)+ many, many more candidates
needed for parity inversion: strong spin-orbit coupling (heavy atoms)
E
EF
¿topology?
E
EF
© Justin Wells (Aarhus)
¿topology?
© Ju
stin
Wel
ls (
Aar
hus)
Hong Kong
China
E
EF
vacuum inverted band gapsemiconductor
topological insulator:insulating bulk + gapless surface
topological consequenceof band inversion
only a “bulk” inversion can remove the crossing (“topological protection”)
Stockholm, 3 September 1967
2D topological insulator(HgTe quantum well)
2D topological insulator
edg
e st
ate
s
ordinary insulator
x
y
© M
olen
kam
p
quantum spin Hall effect – zero-field analogue of thequantum Hall effect
¿quantized conductance?
valence band
conduction band
valence band
momentum
ener
gy
conduction band
topological insulator ordinary insulator
surface states
band gap
0
momentum
0
Fu, Kane & Mele | Moore & Balents | S.C. Zhang et al.
Kramers degeneracy protects the crossing at zero momentum
• 2D topological insulators (quantum spin-Hall effect)
• 2D topological superconductors (Majorana fermions)
• 1D topological superconductors
Topological Matter
Majorana fermions might emerge as midgap states in a superconductor
Majorana fermion at E=0(p-wave order needed to remove zero-point motion)
Volovik (1999)
Bogoliubov meets Majorana
Bogoliubov quasiparticle
superposition of electron and hole excitations in a superconductor
spinless bound state at zero energy is a Majorana fermion
particle = antiparticleobstacle: zero-point motion
bound states in a vortex core
however, zero-point motion prevents a bound state at E=0
Fu & Kane (2008): use Berry phase of massless electrons to eliminate the ½ phase shift
so Majorana fermion at E=0
electron-hole symmetry:
Bpseudospin rotates 2π ➟ Berry phase shift of π ➟ Landau levels shifted
2005: electrons in a carbon monolayer are massless (like photons) and have a “pseudospin”
pointing in the direction of motion
the problem with graphene
• each Majorana fermion is four-fold degenerate (2x spin + 2x valley)
• nonlocality broken if spins and/or valleys can mix
for full protection from decoherence, we need massless electrons without spin or valley degeneracy
3D topological insulators to the rescue
3D topological insulator(Bi2Se3, Bi2Te3)
massless Dirac Hamiltonian
all degeneracies removed
proximity to a superconductor transforms the surface of a 3D topological insulator into a 2D topological superconductor, with Majorana fermions in vortex cores
insulating bulk, metallic surface
Hsi
eh, H
asan
(200
9)
• 2D topological insulators (quantum spin-Hall effect)
• 2D topological superconductors (Majorana fermions)
• 1D topological superconductors
Topological Matter
1D topological superconductor(InSb nanowire)
in a superconductor, electron-hole symmetry would pin these states at E=0
Shockley states
Majorana fermions!
gap closes and reopens upon formation of a line defect, leaving behind a pair of bound states at the end points
Majorana-Shockley statesin a p-wave superconductor
p-wave from s-wave
© Jay Sau
Lutchyn, Sau & Das Sarma
Oreg, Refael & Von Oppen
Nb or Al superconductor
InAs nanowireMagnetic field
s-wave proximity effectRashba spin-orbit couplingZeeman spin polarization
+p-wave superconductor
�0.04 �0.02 0 0.02 0.04klso
�20
�10
0
10
20
E/E
so
µ = �5
−300 −200 −100 0 100 200 300Vbias [ µV ]
0
0.5
1
1.5
G[2e2/h]
model calculation: Wimmer & Akhmerov
superconductor (Nb)
semiconductor (InSb)
barrierbound state
experiment:Mourik, Zuo, Frolov, Plissard, Bakkers & Kouwenhoven(Science, April 2012)