Superconducting hybrids based on QSH systems
Lecture 2
Björn Trauzettel
11th Capri Spring School
Transport in Nanostructures
April 13-17, 2015 Capri, Italy
Pablo Burset (Uni Würzburg) François Crépin (Uni Würzburg)
Fabrizio Dolcini (PolyTech Torino) Florian Geissler (Uni Würzburg)
Ewelina Hankiewicz (Uni Würzburg) Naoto Nagaosa (Tokyo University) Yukio Tanaka (Nagoya University) Grigory Tkachov (Uni Würzburg)
Summary Lecture 1
0 0
0 0*B
e
dFM
hFM
G
H
HH
x
x
Outline
• Classification: Symmetry of pairing amplitude
• Transport signature: Crossed Andreev reflection in NSN junctions
• Doppler shift: Topological SQUID based on helical edge states
Model and setup
Crépin, Burset & BT arXiv 2015
BdG Hamiltonian
1 2zBd F xG yz xiv m x x xH
†
† † †
1
2, , ,
R
B
L
G
L
d
R
H Hdx
spin space
particle-hole space
Green’s functions
†
†
†
, ,
, ,
, , , , ,
R
A
M
G r r i t t r r
G r r i t t r r
G x x T x x
XeeX
Xhh
Xeh
Xhe
G
G
GG
G
X Xeh eh
X Xeh e
Xeh
h
G G
GG
G
( , )r x t
4x4 matrix
Scattering states: N side
1 1 1
2 2 2
3 3 3
4 4 4
e h e
e h e
e h
e h
ik x ik x ik x
e h e
ik x ik x ik x
h h e
ik x ik x
e h
ik x ik x
e h
x e a e b e
x e b e a e
x c e d e
x d e c e
e h
k k
McMillan Phys. Rev. 1968
Scattering states: S side
1 1 1
2 2 2
3 3 3
4 4 4
S Se h
S Se h
S S Se e e
S S Sh h e
ik x ik x
e h
ik x ik x
e h
ik x ik x ik x
e h h
ik x ik x ik x
h h e
x c e d e
x d e c e
x e a e b e
x e b e a e
2 2 2 2 S Se h
k k
McMillan Phys. Rev. 1968
Green’s functions from scattering states
, , , ,i i t tR RG x x dte G x t x t
,RBdG
w H x G x x x x
0
1lim , ,R R
z zF
G x x G x xiv
1
3 4
1 2
3 4
2
if ,
if
T T
RT T
A x A x x xG x x
A
x
xx x x
x
xx A
McMillan Phys. Rev. 1968
Pairing amplitude
ReeR
Rhh
Reh
Rhe
G
G
GG
G
Green‘s function
2R
R R
R RhR
eG
F
Fi
FF
F
Pairing amplitude
0 0 2,, ,, , ,R
iR R
ifF x xx xf x x i
triplet singlet
Crépin, Burset & BT arXiv 2015
Antisymmetry of pairing amplitude
0 0 2,, ,, , ,R
iR R
ifF x xx xf x x i
triplet singlet
Crépin, Burset & BT arXiv 2015
0 0
, , ,
, , ,
,
,R A
i i
R A
f x x f x
f x x
x
x x f
Classification of pairing amplitude
Tanaka, Sato & Nagaosa JPSJ 2012
0 0 2,, ,, , ,R
iR R
ifF x xx xf x x i
Berezinskii JETP Lett 1974
0 0
, , ,
,
,
, , ,R Ri i
R R
f
f x x f x
x x
x
x f x
0 0, , ,
, , , ,
,R A
i i
R A
f
f x x f x x
x x f x x
orbital
frequency
Classification of pairing amplitude
frequency, spin, orbital
even, singlet, even -> ESE odd, singlet, odd -> OSO even, triplet, odd -> ETO odd, triplet, even -> OTE
Tanaka, Sato & Nagaosa JPSJ 2012
0 0 2,, ,, , ,R
iR R
ifF x xx xf x x i
Berezinskii JETP Lett 1974
0 0
, , ,
,
,
, , ,R Ri i
R R
f
f x x f x
x x
x
x f x
0 0, , ,
, , , ,
,R A
i i
R A
f
f x x f x x
x x f x x
orbital
frequency
Classification of pairing amplitude
frequency, spin, orbital
even, singlet, even -> ESE odd, singlet, odd -> OSO even, triplet, odd -> ETO odd, triplet, even -> OTE
Tanaka, Sato & Nagaosa JPSJ 2012
0 0 2,, ,, , ,R
iR R
ifF x xx xf x x i
S and T mix if spin rotational symmetry is broken
(orbital) E and O mix if inversion is broken
-> NS junctions
Berezinskii JETP Lett 1974
Classification of pairing amplitude: results
, ,
, , , , , ,R R Rbulk edge
f x x f x x f x x
frequency, spin, orbital
-> ESE, OSO, ETO, OTE
, ,, , , ,
x x x xR Rbulk edge
f x x e f x x e
Crépin, Burset & BT arXiv 2015
(in region IV)
Classification of pairing amplitude: results
, ,
, , , , , ,R R Rbulk edge
f x x f x x f x x
3x LOTE
OTE OTE
ESE
0
0.5m
Crépin, Burset & BT arXiv 2015
Outline
• Classification: Symmetry of pairing amplitude
• Transport signature: Crossed Andreev reflection in NSN junctions
• Doppler shift: Topological SQUID based on helical edge states
Detection of OTE: idea
use crossed Andreev reflection
Crépin, Burset & BT arXiv 2015
Detection of OTE: results
LAR EC CAR
ER
-> dilemma
2 20
tanh 2 ta 1nhCAR
ECT
dmT
Crépin, Burset & BT arXiv 2015
Adroguer et al. PRB 2010
Way out …
221
1CAR EC
TI
GV
T
… add complexity
Way out …
221
1CAR EC
TI
GV
T
… add complexity
OTE
Outline
• Classification: Symmetry of pairing amplitude
• Transport signature: Crossed Andreev reflection in NSN junctions
• Doppler shift: Topological SQUID based on helical edge states
Inspiring experiments
Hart et al. Nature Phys 2014
Pribiag et al. arXiv 2014
Hg(Cd)Te QWs InAs/GaSb QWs
Setup & Hamiltonian
Tkachov, Burset, BT & Hankiewicz arXiv 2014
0
0 *
i x
yi xBdG
y
i x e
xi x
h xH
e h
0, 2
, 2
Lxx
Lx
0
00
, x2 2
, x2 2
Lx
L
Setup & Hamiltonian
Tkachov, Burset, BT & Hankiewicz arXiv 2014
0
0 *
i x
yi xBdG
y
i x e
xi x
h xH
e h
2x
SF x
v ih U xp
x
Setup & Hamiltonian
Tkachov, Burset, BT & Hankiewicz arXiv 2014
0
0 *
i x
yi xBdG
y
i x e
xi x
h xH
e h
2x
SF x
v ih U xp
x
shielding response of SC -> Doppler shift
0
2
2S x
eB
w
c
Bwp A
Tkachov & Falko PRB 2004 Tkachov & Richter PRB 2005
Rohlfing et al. PRB 2009
Josephson current: basics
Beenakker & Brouwer Chaos, Solitons & Fractals 1997
2e dJ
F
d
phase difference
free energy
0
2 ln 2 cosh -independent2B
B
k T dT
Fk
DOS
0
22 ln det 1
B A n N nn
e dJ k T S i S i
d 2 1
n Bn k T
Josephson current
Tkachov, Burset, BT & Hankiewicz arXiv 2014
0 220
2
2 2
22
,
1
i i
u
n n
n n n
Bn i
n
i
A B A B
A B A B
e eeJ
A B A
k T
e eB
B
0 0
0S
k L
Josephson current
Tkachov, Burset, BT & Hankiewicz arXiv 2014
0 220
2
2 2
22
,
1
i i
u
n n
n n n
Bn i
n
i
A B A B
A B A B
e eeJ
A B A
k T
e eB
B
0 0, ,
l uJ B J B
0 0
0S
k L
/
2
2
2 2
n FL v
nSF F Sn
n
p B
A
piv B
e
iv
B
B-dependent Andreev reflection
When is it relevant?
Tkachov, Burset, BT & Hankiewicz arXiv 2014
max ,2
F S
B
v pk T
0
**
2; min ,F
RB
AF
v vB
w k TB
quite small: a few mT
When is it relevant?
Tkachov, Burset, BT & Hankiewicz arXiv 2014
max ,2
F S
B
v pk T
0
**
2; min ,F
RB
AF
v vB
w k TB
How does the maximum current at zero field change as a function of B?
B=0 current-phase relation
0,max
,m
J B J B
0 0 0, , ,
u lJ B J B J B
Results: long junction
Tkachov, Burset, BT & Hankiewicz arXiv 2014
L
020
8Re
iB
mA eB
ek TJ B
even-odd effect
0
AR oscB B
wL
Pribiag et al. arXiv 2014
Results: short junction L
strong suppression
0
AR oscB B
wL
Tkachov, Burset, BT & Hankiewicz arXiv 2014
Calado et al. arXiv 2015
Summary Lecture 2
Gauge-invariant phase
Tkachov, Burset, BT & Hankiewicz arXiv 2014
4 4, , , , ,
y x x yB x y j x y B x y j x y
c c
0, 0, ,B x yB
,2
wB x B
bc imply: , , , , x x
B x y B x y j x y j x y
Ampère‘s law:
Gauge-invariant phase
Tkachov, Burset, BT & Hankiewicz arXiv 2014
4 4, , , , ,
y x x yB x y j x y B x y j x y
c c
0, 0, ,B x yB
,2
wB x B
Ampère‘s law:
type-II SC ->
j
00
2Agauge-invariant
phase gradient
Gauge-invariant phase
Tkachov, Burset, BT & Hankiewicz arXiv 2014
0
22 2Ndl
winding number
, 0 0x
x , , x xj x y j x y
, 0 piecewise const.x
Gauge-invariant phase
Tkachov, Burset, BT & Hankiewicz arXiv 2014
0
22 2Ndl
0
, 0 , 02 2
L L
integral over path 3
integrals over path 2&4 vanish
integral over path 1
* *, ,2 2 2 2 2
L Lw w w
0
0 22
w N