superconducting hybrids based on qsh systems lecture 2

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