Y. TanakaNagoya University, Japan

Y. AsanoHokkaido University, Japan

Y. TanumaAkita University, Japan

Alexander GolubovTwente University, The Netherlands

Odd 　 frequency 　 pairing 　 in

superconducting 　 heterostructures

Contents

（１） What is odd-frequency pairing

（２） Normal metal / Superconductor junctions

（３） Ferromagnet/Superconductor junctions

Conventional Classification of Symmetry of Cooper pair

Spin-singlet Cooper pair

s-waved-wave

Spin-triplet Cooper pair

Even Parity

Odd Parity

p-wave

BCSCuprate

3He Sr2RuO4

Pair amplitude(pair correlation)

Exchange of two electrons

Fermi-Dirac statistics

Pair amplitude

Exchange of time

Even-frequency pairing (conventional pairing)

Odd-frequency pairing 　

Odd-frequency pairing

Pauli principle

Symmetry of 　 pair 　 wave functions:

spin-singlet: = odd

spin-triplet: = even

even-frequency superconductivity

odd-frequency superconductivity

()

(ω), f(ω)

(ω), f(ω)

s-, d-wave

s-, d-wave

p-, f-wave

p-, f-wave

Symmetry of the pair amplitude

Frequency（ time) Spin

+(even)

odd)

+(even)

odd)

+(even)

Orbital

+ symmetric, anti-symmetric

ESE

ETO

OTE

OSO

singlet)

Total

(odd)+ (triplet)

+ triplet)

ESE (Even-frequency spin-singlet even-parity) ETO (Even-frequency spin-triplet odd-parity)OTE (Odd-frequency spin-triplet even-parity) Berezinskii

OSO (Odd-frequency spin-singlet odd-parity) Balatsky, Abrahams

singlet)

+(even)

(odd)

BCS

Cuprate

3HeSr2RuO4

Previous studies about odd-frequency pairing

Bulk state (Pair potential, Gap function) Berezinskii (1974)Balatsky Abrahams Schrieffer Scalapino(1992-1993)Zachar Kievelson Emery (1996)Coleman Mirranda Tsvelik (1997)Vojta Dagotto (1999)Fuseya Kohno Miyake (2003)

Junction (No pair potential)Induced odd-frequency pair amplitude in ferromagnet attached to spin-singlet s-wave superconductor Bergeret, Efetov, Volkov, (2001)

Odd-frequency pairing state

• Odd-frequency pairing state is possible even if we start from the conventional even-frequency paring state:

Broken spin rotation symmetry or spatial invariance symmetry can induce odd-frequency pairing state:

- ferromagnet/superconducor junctions

- non-uniform systems

Contents

（１） What is odd-frequency pairing

(2) Ballistic normal metal junctions

（ 3 ） Diffusive normal metal junctions

(4 ） Ferromagnet/Superconductor junctions

Ballistic junction

Y. Tanaka, A. Golubov, S. Kashiwaya, and M. Ueda Phys. Rev. Lett. 99 037005 (2007)

M. Eschrig, T. Lofwander, Th. Champel, J.C. Cuevas and G. SchonJ. Low Temp. Phys 147 457(2007)

BallisticNormal metal(semi-infinite)

Superconductor(semi-infinite)

Eilenberger equation

ballistic normal metal

S

Form factor

SN

x0

(explicitly denote direction of motion)

Bulk state

Quasiparticle function

Pair amplitudes

Only

Pair potential

General properties （ frequency)

Odd-frequency(imaginary) Interface-inducedcomponent

Even-frequency(real)bulk-component

Spatial change of the pair potential

Superconductor is conventional even-frequency one.

Normal metal spin-triplet p-wave superconductor

px-wave component of ETO pair amplitude

s-wave component of OTE pair amplitude

(high-transparent)

(low-transparent)

Y. Tanaka, et al PRL 99 037005 (2007)

ETO (Even-frequency spin-triplet odd-parity)OTE (Odd-frequency spin-triplet even-parity)

Pair potential

Symmetry of the bulk pair potential is ETO

Underlying physics

Near the interface, even and odd-parity pairing states (pair amplitude) can mix due to the breakdown of the translational symmetry.

The interface-induced state (pair amplitude) should be odd in frequency where the bulk pair potential has an even -frequency component since there is no spin flip at the interface.

Fermi-Dirac statistics

Mid gap Andreev resonant (bound) state

(MARS)

Interface (surface)

＋ー

–1 0 10

2

4

Nor

mal

ized

DO

S

Local density of state has a zero energy peak. (Sign change of the pair potential at the interface)

Tanaka Kashiwaya PRL 74 3451 (1995), Rep. Prog. Phys. 63 1641 (2000) Buchholz(1981) Hara Nagai(1986)Hu(1994) Matsumoto Shiba(1995)Ohashi Takada(1995)Hatsugai and Ryu (2002)

＋ー

ー＋

Mid　gap　Andreev　resonant　(bound)　state

Electron-like　quasiparticle

Hole-like　quasiparticle

Cooper　pair

Odd-frequency Cooper pair（ Odd-frequency pair amplitude ）

Y. Tanaka, Y. Tanuma and A.A.Golubov, Phys. Rev. B 76, 054522 (2007)

Odd-frequency pairing state in N/S junctions (N finite length)

Bounds state are formed in the normal metal

Ratio of the pair amplitude in the Nregion (odd/even)

At some energy, odd-frequency component can exceed over even frequency one.

Odd-frequency pairing

Even-frequency pairing

Hidden odd-frequency component in the s-wave superconductor junctions

Ratio of the pair amplitude at the N/S interface and the bound state level

Bound states condition (Z=0)（ McMillan Thomas Rowell)

Bound states are due to the generation of the odd-frequency Cooper pair amplitude

Odd-frequency pairing

Even-frequency pairing

Y. Tanaka, Y. Tanuma and A.A. Golubov, PRB 76 054522 (2007)

(1)

(2)

(3)

(4)

• ESE (Even-frequency spin-singlet even-parity)• ETO (Even-frequency spin-triplet odd-parity)• OTE (Odd-frequency spin-triplet even-parity)• OSO (Odd-frequency spin-singlet odd-parity)

Bulk state

ESE (s,dx2-y2 -wave)

ESE (dxy-wave)

ETO (px-wave)

ETO (py-wave)

Sign change(MARS)

No

Yes

Interface-induced symmetry(subdominant component )

Yes

No

ESE + (OSO)

OSO +(ESE)

OTE + (ETO)

ETO + (OTE)

Symmetry of the Cooper pair (No spin flip)

Phys. Rev. Lett. 99 037005 (2007)

(1) (2) (3) (4)

Contents

（１） What is odd-frequency pairing

(2) Ballistic normal metal junctions

（ 3 ） Diffusive normal metal junctions

(4 ） Ferromagnet/Superconductor junctions

Impurity scattering effectTanaka and Golubov, PRL. 98, 037003 (2007)

BallisticNormal metal

Superconductor

DiffusiveNormal metal

(DN)Superconductor

Impurity scattering (isotropic)Only s-wave pair amplitude exists in DN

(1)ESE (2)OTE

ESE (Even-frequency spin-singlet even-parity)OSO (Odd-frequency spin-singlet odd-parity)

Usadel equationAvailable for diffusive limit

Angular average

Diffusivelimit

Diffusive normal metal region attached to superconductor

DN S

Even frequency spin singlet even parity (ESE) pair potential

Even frequency spin singlet s-wave (ESE) pair is induced in DN.

ESE pair /ESE pair potential

DN S

Even frequency spin singlet 　 even parity 　 (ESE) pair potential

s-wave case

–0.3 0 0.30

1

Even frequency spin singelt s –wave

s–wave

DN

＋ー

Px-wave case

New type of proximity effect

Odd frequency spin triplet s-wave (OTE) pair is induced in DN

Y.Tanaka, A.A.Golubov, Phys.Rev.Lett. 98, 037003 (2007)

)(Re f)(Im f

DN

＋ー

Even frequency spin triplet odd parity (ETO) pair potential

–0.3 0 0.30

1

2

Even frequency spin triplet p–wave

px

py

DN

＋

ー

(no proximity& no MARS)

Px-wave case

Py-wave case

Symmetry of the pair potential Induced pair amplitude in DN

Even frequency spin singleteven parity (ESE)

Even frequency spin tripletodd parity (ETO)

Odd frequency spin tripleteven parity (OTE)

Odd frequency spin singletodd parity (OSO)

ESE

OTE

OTE

ESE

Summary of Proximity effect (diffusive normal metal)

How to detect triplet superconductor

MARS (Mid gap Andreev resonance state) can penetrate into DN by proximity effect only for triplet superconductor junctions

LDOS in DN has a zero energy peak

Diffusive normal Metal (DN)

STS

Triplet superconductor

MARS

Diffusive normal Metal (DN)

STS

Singlet superconductor

MARS

LDOS in DN does not have a zero energy peak

ZEP No ZEP

How to detect odd-frequency paring amplitude: measuring electrical conductivity 　　　　　　　　　　　　　　　　 Asano, Tanaka, Golubov, Kashiwaya, PRL 99, 067005 (2007)

Sr2RuO4 　

　

Ａｕ :I+

Ａｕ :V+

Ａｕ :I-

Ａｕ :V-Kashiwaya, Maeno 2007

Zero energy peak No Zero energy peak

OTE proximityESE proximity (conventional)

OTE (Odd-frequency spin-triplet even-parity)

ESE (Even-frequency spin-singlet even-parity)

Contents

（１） What is odd-frequency pairing

（２） Ballistic normal metal junctions

（ 3 ） Diffusive normal metal junctions

（ 4 ） Ferromagnet/Superconductor junctions

_

Odd frequency spin-triplet s-wave pair

SuperconductorFerromagnet

Bergeret, Efetov, Volkov, (2001)Eschrig, Buzdin, Kadigrobov,Fominov, Radovic

Generation of the odd-frequency pair amplitude in ferromagnet

Odd-frequency pair amplitude (not pair potential) is generated in ferromagnet

junctions

spin-singlet s-wave pair+

F FSd f d f2d s

x x

z z

triplet Josephson effect in SF multilayered system with noncollinear ferromagnets and thin superconductors

(x) /

x

FS

D

h

D<<

s in g le t: tr ip le t:

Spin triplet Cooper pairs in SF systems

Josephson　current　in　S/HM/S

Half metal : CrO2Keizer et.al., Nature (2006)

Eschrig et. al., PRL(‘03)Theory in the clean limit

Spin active interfaceBergeret et. al., PRL(‘01),

Kadigrobov et. al., Europhys Lett.(‘01)

Lofwander and Eschrig, Nature Physics (2008)

Furusaki, Physica B(‘92), Asano, PRB(‘01)

Advantages

SNS, SFS, S/HM/S

22

1

JJJ

JJSN

ii

Parameters

S

ex

V

V : exchange: spin-flip

Recursive　GF　

Y.Asano, Y. Tanaka, A.A.Golubov, Phys.Rev.Lett. 98, 107002 (2007)

Spin　active　interface2SVσV

JJ

self-averaging

SFS, S/HM/S

30 ,, ffff

ff

No sign change

SFS

3,, fff

0f

fS/HM/S

only

odd-frequency pairs

-1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

< f >

Pair

ing

func

tion

/ fB

< f0 >

n /

0

Odd-frequency　pairs

SFS

3,, fff

0f

f

S/HM/S

only

even-odd mix pure odd

Quasiparticle　DOS　in　Half　Metal

0.0 0.5 1.00

1

2

3

LD

OS

at j

= 37

/ n

0

S/HM/S

SNS

E /

Zero Energy Peak

can be detectedby tunneling spectroscopy

Conclusions(1) Ubiquitous presence of the odd-frequency pair PRL 99, 037005 (2007)

(2) Odd-frequency pair amplitude is enhanced in the presence of the midgap Andreev resonant state (PRL 99, 037005 (2007)).

(3) Low energy Andreev bound states can be expressed in terms of odd-frequency pairing (PRB 76 054522 (2007))

(4) The origin of the anomalous proximity effect in DN/spin-triplet p-wave junction is the generation of the odd-frequency pairing state.

(PRB 70, 012507 (2004), PRL 98 037003 (2007) , PRL 99 067005 (2007))

(4) Odd-frequency pairs carry supercurrent in S/HM/S junctions (PRL 98, 107002 (2007))