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


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


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


Antisymmetry of pairing amplitude

0 0 2,, ,, , ,R

iR R

ifF x xx xf x x i

triplet singlet


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


frequency, spin, orbital

even, singlet, even -> ESE odd, singlet, odd -> OSO even, triplet, odd -> ETO odd, triplet, even -> OTE


0 0 2,, ,, , ,R

iR R

ifF x xx xf x x i


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



even, singlet, even -> ESE odd, singlet, odd -> OSO even, triplet, odd -> ETO odd, triplet, even -> OTE


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


Classification of pairing amplitude: results

, ,

, , , , , ,R R Rbulk edge

f x x f x x f x x


-> ESE, OSO, ETO, OTE

, ,, , , ,

x x x xR Rbulk edge

f x x e f x x e


(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


Outline




Detection of OTE: idea

use crossed Andreev reflection


Detection of OTE: results

LAR EC CAR

ER

-> dilemma

2 20

tanh 2 ta 1nhCAR

ECT

dmT


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




Inspiring experiments

Hart et al. Nature Phys 2014

Pribiag et al. arXiv 2014

Hg(Cd)Te QWs InAs/GaSb QWs

Vorführender

Präsentationsnotizen

The SC is in both cases aluminum and the superconducting coherence length is much smaller than the width of the sample.

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


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


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

Vorführender


S_N: describes elastic scattering by the normal metal; S_A: describes AR

Josephson current


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

Vorführender


Omega_n=(2n+1) pi kB T = fermionic Matsubara frequencies

Josephson current


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

Vorführender


Omega_n=(2n+1) pi kB T = fermionic Matsubara frequencies

When is it relevant?


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?


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


L

020

8Re

iB

mA eB

ek TJ B

even-odd effect

0

AR oscB B

wL

Pribiag et al. arXiv 2014

Vorführender


Fig d: Each current initially has a sawtooth profile. The two currents tend to enhance or cancel each other.

Results: short junction L

strong suppression

0

AR oscB B

wL


Calado et al. arXiv 2015

Vorführender


Fig d: Each current initially has a sawtooth profile. The two currents tend to enhance or cancel each other.

Summary Lecture 2

Gauge-invariant phase


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:



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



0

22 2Ndl

winding number

, 0 0x

x , , x xj x y j x y

, 0 piecewise const.x



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

superconducting hybrids based on qsh systems lecture 2

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