subgap states in majorana wires piet brouwer dahlem center for complex quantum systems physics...
Post on 23-Dec-2015
215 Views
Preview:
TRANSCRIPT
Subgap States in Majorana Wires
Piet Brouwer
Dahlem Center for Complex Quantum SystemsPhysics DepartmentFreie Universität Berlin
Inanc AdagideliMathias DuckheimDganit MeidanGraham KellsFelix von OppenMaria-Theresa RiederAlessandro Romito
Aachen, 2013
excitations in superconductors
one fermionic excitation → two solutions of BdG equation
Eigenstate of HBdG at e = 0: Majorana fermion,
ue, ve: solution of Bogoliubov-de Gennes equation:
Eigenvalues of HBdG come in pairs ±e, with(particle-hole symmetry)
D: antisymmetric operator
-e
e
Overview• Spinless superconductors as a habitat for Majorana fermions
• Disordered spinless supeconducting wires
• Multichannel spinless superconducting wires
-e
e
• Semiconductor nanowires as a spinless superconductor
• Disordered multichannel superconducting wires
superconductor proximity effect
S N
eh
ideal interface:
Deifrhe(e) = a e-if
reh(e) = a eif
a = e-i arccos(e/D)
n/n
N
e (mV)
x1
x3
x2
Guéron et al. (1996)
x1 x2 x3
S NS
Mur et al. (1996)
I(nA
)S N S
spinless superconductors are topological
scattering matrix for Andreev reflection:
S is unitary 2x2 matrix
S
h
particle-hole symmetry:
combine with unitarity:
Andreev reflection is either perfect or absent
if e = 0
Béri, Kupferschmidt, Beenakker, Brouwer (2009)
e
e
scattering matrix for point contact to S
spinless p-wave superconductors
one-dimensional spinless p-wave superconductor
Majorana fermion end statesbulk excitation gap
Kitaev (2001)
spinless p-wave superconductor
superconducting order parameter has the form
SN D(p)eif(p)rhe
p
Andreev reflection at NS interface
Andreev (1964)
reh-p
*p-wave:
spinless p-wave superconductors
one-dimensional spinless p-wave superconductor
Majorana fermion end statesbulk excitation gap
Kitaev (2001)
spinless p-wave superconductor
superconducting order parameter has the form
SN D(p)eif(p)rhe
reh
p
-p
eih
e-ih
Bohr-Sommerfeld: Majorana bound state if
*
Always satisfied if |rhe|=1.
Proposed physical realizations• fractional quantum Hall effect at ν=5/2
• unconventional superconductor Sr2RuO4 • Fermionic atoms near Feshbach resonance
• Proximity structures with s-wave superconductors, and topological insulators semiconductor quantum well
ferromagnet
metal surface states
Moore, Read (1991)
Das Sarma, Nayak, Tewari (2006)
Gurarie, Radzihovsky, Andreev (2005)Cheng and Yip (2005)
Fu and Kane (2008)
Sau, Lutchyn, Tewari, Das Sarma (2009)Alicea (2010)
Lutchyn, Sau, Das Sarma (2010)Oreg, von Oppen, Refael (2010)
Duckheim, Brouwer (2011)Chung, Zhang, Qi, Zhang (2011)
Choy, Edge, Akhmerov, Beenakker (2011)Martin, Morpurgo (2011)
Kjaergaard, Woelms, Flensberg (2011)
Weng, Xu, Zhang, Zhang, Dai, Fang (2011)Potter, Lee (2010)
(and more)
Semiconductor proposal
Semiconducting wire with spin-orbit coupling, magnetic field
S
N B and SOI
Sau, Lutchyn, Tewari, Das Sarma (2009)Alicea (2010)
Lutchyn, Sau, Das Sarma (2010)Oreg, von Oppen, Refael (2010)
spin-orbit coupling
Zeeman fieldproximity coupling to superconductor
Semiconductor proposal
S
N B and SOIspin-orbit coupling
Zeeman fieldproximity coupling to superconductor
p
e
Semiconductor proposal
S
N B and SOI
p
e
e
p-pF
BpF
e
Semiconductor proposal
S
N B and SOI
p
e
p-pF
BpF
p-pF
BpF
ee
Semiconductor proposal
S
N B and SOI
p
e
p-pF
BpF
p-pF
BpF
ee e
p
Semiconductor proposal
S
N B and SOI
p
e
p-pF
BpF
p-pF
BpF
ee e
pD
spinless p-wave superconductor
Semiconductor proposal
S
N B and SOI
p
e
p-pF
BpF
p-pF
BpF
ee e
pD
spinless p-wave superconductor
B
p
-pF
Mourik et al. (2012)
Spinless superconductor
S
N B and SOI
p
e
p-pF
BpF
p-pF
BpF
ee e
pD
spinless p-wave superconductor
Effective description as a spinless superconductor
Spinless superconductor
spinless p-wave superconductor
e
0
Disorder-induced subgap states
spinless p-wave superconductor
topological phase persists for
Motrunich, Damle, Huse (2001)
e
with disorder:
at critical disorder strength:
density of subgap states:
Disorder-induced subgap states
Disorder-induced subgap states are localized in the bulk of the wire.
Localization length in topological regime: -1
e
n
2lel » xweak disorder
n
e2lel > x~
e
n
2lel < x~almost critical beyond critical
Power-law tail for density of states: if 2lel > x~
Disorder-induced subgap states
e0,max: log-normal distribution e1,min: distribution has algebraic tail near zero energy
Majorana state
fermionic subgap states
spinless p-wave superconductor
Finite L: discrete energy eigenvalues
L
algebraically small energy for large L
exponentially small energy for large L
beyond one dimension
If B D, ap: semiconductor model can be mapped to p+ip model
projection onto “spinless” transverse channels
12321 3
Tewari, Stanescu, Sau, Das Sarma (2012)Rieder, Kells, Duckheim, Meidan, Brouwer (2012)
semiconductor model:
Multichannel spinless p-wave superconducting wire
? ?
L
W
bulk gap:
coherence length
induced superconductivity is weak: and
D
p+ip
Multichannel spinless p-wave superconducting wire
? ?
L
W
bulk gap:
coherence length
induced superconductivity is weak:
Majorana end-states→
and
…Without D’py : chiral symmetry
p+ip
inclusion of py: effective Hamiltonian Hmn for end-states
Hmn is antisymmetric: One zero eigenvalue if N is odd,no zero eigenvalue if N is even.
D
0
Multichannel wire with disorder
? ?
L
W
bulk gap:
coherence length
p+ip
Multichannel wire with disorder
? ?
L
Wp+ip
disorder strength0
Series of N topological phase transitions at
n=1,2,…,N
Multichannel wire with disorder
? ?
L
Wp+ip
Without Dy’: chiral symmetry (H anticommutes with ty)
Topological number Qchiral .
Qchiral is number of Majorana states at each end of the wire.
Without disorder Qchiral = N.
With Dy’:
Topological number Q = ±1
Scattering theory
? p+ip
Without Dy’: chiral symmetry (H anticommutes with ty)
Topological number Qchiral .
Qchiral is number of Majorana states at each end of the wire.
Without disorder Qchiral = N.
With Dy’:
Topological number Q = ±1
N S
L
Fulga, Hassler, Akhmerov, Beenakker (2011)
Chiral limit
? p+ipN S
L
Basis transformation:
Chiral limit
? p+ipN S
L
if and only if
Basis transformation:imaginary gauge field
Chiral limit
? p+ipN S
L
Basis transformation:
if and only if
imaginary gauge field
Chiral limit
? p+ipN S
L
if and only if
“gauge transformation”
Basis transformation:imaginary gauge field
Chiral limit
? p+ipN S
L
if and only if
“gauge transformation”
Basis transformation:imaginary gauge field
Chiral limit
? p+ipN S
L
“gauge transformation”
Basis transformation:
N, with disorder
L
Chiral limit
? p+ipN S
L
Basis transformation:
“gauge transformation”
N, with disorder
L
Chiral limit
? p+ipN S
L
N, with disorder
L
: eigenvalues of
Chiral limit
? p+ipN S
L
N, with disorder
L
: eigenvalues of
Distribution of transmission eigenvalues is known:
with , self-averaging in limit L →∞
Series of topological phase transitions
Without Dy’: chiral limit
? ?
L
Wp+ip
topological phase transitions at
n=1,2,…,N
Qchiral
x/(N+1)l
disorder strength
Series of topological phase transitions
Without Dy’: chiral limit
? ?
L
Wp+ip
With Dy’:
topological phase transitions at
n=1,2,…,N
Series of topological phase transitions
? ?
L
Wp+ip
With Dy’:
topological phase transitions at
n=1,2,…,N
Dy’
/Dx’
(N+1)l /xdisorder strength
Series of topological phase transitions
? ?
L
Wp+ip
With Dy’:
topological phase transitions at
n=1,2,…,N
Dy’
/Dx’
(N+1)l /xdisorder strength
Conclusions• Majorana fermions may persist in the presence of disorder
and with multiple channels
• Disorder leads to fermionic subgap states in the bulk; Density of states has power-law singularity near zero energy.
• Multiple channels may lead to fermionic subgap states at the wire ends.• For multichannel p-wave superconductors there is a
sequence of disorder-induced topological phase transitions. The last phase transition takes place at l=x/(N+1).
disorder strength0
top related