Odd frequency superconducting pairing heterostructures Alexander Golubov Twente University, The Netherlands Y. Tanaka Nagoya University, Japan Y. Asano Hokkaido University, Japan Y. Tanuma Akita University, Japan in

Contents What is odd-frequency pairing Normal metal / Superconductor junctions Ferromagnet/Superconductor junctions Conventional Classification of Symmetry of Cooper pair Spin-singlet Cooper pair s-wave Even Parity d-wave Cuprate BCS Spin-triplet Cooper pair p-wave

Odd Parity He Sr2RuO4 3 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 functions: pair (), f(() f() even-frequency superconductivity wave spin-singlet: = odd s-, f(d-wave (), f(f()

odd-frequency superconductivity p-, f-wave spin-triplet: = even p-, f(f-wave s-, d-wave Symmetry of the pair amplitude + f(symmetric, f( f(anti-symmetric Frequency time) Spin Orbital

singlet) +(even) Total ESE +(even) ETO +(even) + (triplet) (odd) OTE odd)

+ triplet) +(even) OSO odd) singlet) (odd) BCS Cuprate He Sr2RuO4 3

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 Previous studies about oddfrequency 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 oddfrequency pairing state: - ferromagnet/superconducor junctions - non-uniform systems Contents What is odd-frequency pairing (2) Ballistic normal metal junctions 3 Diffusive normal metal junctions (4 Ferromagnet/Superconductor junctions Ballistic junction Ballistic Normal metal (semi-infinite)

Superconductor (semi-infinite) Y. Tanaka, A. Golubov, S. Kashiwaya, and M. Ueda Phys. Rev. Lett. 99 037005 (2007) M. f(Eschrig, f(T. f(Lofwander, f(Th. f(Champel, f(J.C. f(Cuevas f(and f(G. f(Schon J. f(Low f(Temp. f(Phys f(147 f(457(2007) f( Eilenberger equation (explicitly denote direction of motion) Pair f(potential f( Quasiparticle f(function Pair f(amplitudes f( Bulk state ballistic normal

metal Form f(factor Only S S N 0 x General properties frequency) Superconductor is conventional even-frequency one. Even-frequency (real) bulk-component

f( Spatial change of the pair potential Odd-frequency f( (imaginary) Interface-induced component Normal metal spin-triplet p-wave superconductor Symmetry of the bulk pair potential is ETO (low-transparent) Pair potential (high-transparent)

px-wave component of ETO pair amplitude s-wave component of OTE pair amplitude ETO (Even-frequency spin-triplet odd-parity) OTE (Odd-frequency spin-triplet even-parity) Y. Tanaka, et al PRL 99 037005 (2007) Underlying physics Near the interface, even and odd-parity pairing states (pair amplitude) can mix due to the breakdown of the translational symmetry. Fermi-Dirac statistics 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. 4 Normalized f(D OS Mid gap Andreev resonant (bound) state

(MARS) 2 0 Interface f((surface) 0 1 Local density of state has a zero energy peak. (Sign change of the pair potential at the interface)

1 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) Mid gap Andreev resonant (bound) state Electron-like quasiparticle Cooper pair

Hole-like quasiparticle Odd-frequency Cooper pair Odd-frequency pair amplitud e Odd-frequency pairing state in N/S junctions (N finite length) Bounds f(state f(are f(formed f(in f(the f(normal f(metal Y. f(Tanaka, f(Y. f(Tanuma f(and f(A.A.Golubov, f(Phys. f(Rev. f(B f(76, f(054522 f((2007) Ratio of the pair amplitude in the N region (odd/even) At some energy, odd-frequency component can exceed over even frequency one. Odd-frequency f(pairing Even-frequency f(pairing

Hidden odd-frequency component in the s-wave superconductor junctions Ratio f(of f(the f(pair f(amplitude f(at f(the f(N/S f( interface f(and f(the f(bound f(state f(level f( Bound states condition (Z=0) McMillan Thomas Rowell) Odd-frequency f(pairing Even-frequency f(pairing Bound states are due to the generation of the odd-frequency Cooper pair amplitude Y. f(Tanaka, f(Y. f(Tanuma f(and f(A.A. f(Golubov, f(PRB f(76 f(054522 f((2007) Symmetry of the Cooper pair (No spin flip) Sign f(change Bulk f(state

(1) (MARS) Interface-induced f(symmetry (subdominant f(component f() (1) ESE f((s,dx2-y2 f(-wave) (2) (3) ESE f((dxy-wave) ETO f((px-wave)

Yes Yes ESE f(+ f((OSO) OSO f(+(ESE) OTE f(+ f((ETO) (4) ETO f((py-wave) No ETO f(+ f((OTE) (2) (3) No

(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) Phys. Rev. Lett. 99 037005 (2007) Contents What is odd-frequency pairing (2) Ballistic normal metal junctions 3 Diffusive normal metal junctions (4 Ferromagnet/Superconductor junctions Impurity scattering effect Tanaka and Golubov, PRL. 98, 037003 (2007) Ballistic Normal metal

Superconductor Impurity f(scattering f((isotropic) Diffusive Normal metal (DN) Superconductor 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 equation Available f(for f(diffusive f(limit

Diffusivelimit Angular average Diffusive f(normal f(metal f(region f(attached f(to f(superconductor Even frequency spin singlet even parity (ESE) pair potential DN S Even f(frequency f(spin f(singlet f( s-wave f((ESE) f(pair f(is f(induced f(in f(DN. ESE pair /ESE pair potential Even frequency spin singlet even parity (ESE) pair potential Even frequency spin singelt s wave

s-wave f(case 1 DN S swave 0 0.3 0 0.3 Odd frequency spin triplet s-wave (OTE) pair is

induced in DN Px-wave case Re f ( ) DN Im f ( ) New type of proximity effect Y.Tanaka, f(A.A.Golubov, f(Phys.Rev.Lett. f(98, f(037003 f((2007) Even frequency spin triplet odd parity (ETO) pair potential Px-wave case Even frequency spin triplet pwave

2 px DN Py-wave case 1 py DN

(no proximity& no MARS) 0 0.3 0 0.3 Summary of Proximity effect (diffusive normal metal) Symmetry of the pair potential Induced pair amplitude in DN Even frequency spin singlet even parity (ESE)

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

ESE How to detect triplet superconductor MARS (Mid gap Andreev resonance state) can penetrate into DN by proximity effect only for triplet superconductor junctions STS ZEP Diffusive normal Metal (DN) STS MARS Triplet superconductor LDOS in DN has a zero energy peak

No ZEP MARS Diffusive normal Singlet superconductor Metal (DN) LDOS in DN does not have a zero energy peak How f(to f(detect f(odd-frequency f(paring f(amplitude: f( measuring electrical conductivity Asano, Tanaka, Golubov, Kashiwaya, PRL 99, 067005 (2007) Sr2RuO4 OTE (Odd-frequency spin-triplet even-parity) ESE (Even-frequency spin-singlet even-parity)

:I+ :V+ :I :V- Kashiwaya, Maeno 2007 OTE proximity Zero energy peak f( ESE proximity (conventional) No Zero energy peak Contents What is odd-frequency pairing Ballistic normal metal junctions 3 Diffusive normal metal junctions 4 Ferromagnet/Superconductor junctions

Odd-frequency pair amplitude (not pair potential) is generated in ferromagnet junctions Odd frequency spin-triplet s-wave pair spin-singlet f(s-wave f(pair _ + Ferromagnet Superconductor Bergeret, Efetov, Volkov, (2001) Eschrig, f(Buzdin, f(Kadigrobov,Fominov, f(Radovic Generation of the odd-frequency pair amplitude in ferromagnet Spin triplet Cooper pairs in SF systems S

F df z d 2ds (x) f(/ f( F f S z

x x f(triplet f(Josephson f(effect f(in f(SF f(multilayered f(system f(with f(noncollinear f(ferromagnets f(and f(thin f(superconductors s in g le t: F trip le t: x D h << D

Josephson current in S/HM/S Half metal : CrO2 Spin active Keizer et.al., Nature (2006) Bergeret et. al., PRL(01), interface Kadigrobov et. al., Europhys Lett.(01) Theory in the clean limit Eschrig et. al., PRL(03) Lofwander and Eschrig, Nature Physics (2008) Recursive GF Furusaki, Physica B(92), Asano, PRB(01) Advantages

SNS, SFS, S/HM/S NS J J i i 1 J J 2 J 2 Parameters Vex : exchange VS : spin-flip Y.Asano, Y. Tanaka, A.A.Golubov, Phys.Rev.Lett. 98, 107002 (2007) Spin active interface V VS 2 SFS, S/HM/S

J J self-averaging f f , f 0 , f 3 f f No sign change odd-frequency pairs SFS f f0 , f , f 3 S/HM/S f only Odd-frequency pairs

f( 1.0 0.5 0.0 f( Pairing f(function f(/ f(f B < f(f0 f(> -0.5 < f(f f(> -1.0 -0.5 0.0 n f(/ f(0

0.5 1.0 Quasiparticle DOS in Half Metal S/HM/S f0 SFS f f , f , f 3 even-odd mix pure odd f( 3

SNS Zero Energy Peak 0 S/HM/S can be detected by tunneling spectroscopy f( LDOS f(at f(j f(= f(37 f(/ f(n 2 1 0 0.0

only 0.5 E f(/ f( 1.0 Conclusions (1) f(Ubiquitous f(presence f(of f(the f(odd-frequency f(pair f( f( PRL 99, 037005 (2007) (2) f(Odd-frequency f(pair f(amplitude f(is f(enhanced f(in f(the f(presence f( of f(the f(midgap f(Andreev f(resonant f(state f((PRL 99, 037005 (2007)). f( (3) f(Low f(energy f(Andreev f(bound f(states f(can f(be f(expressed f(in f( terms f(of f(odd-frequency f(pairing f((PRB 76 054522 (2007)) (4) f(The f(origin f(of f(the f(anomalous proximity effect in DN/spin-triplet p-wave junction f(is f(the f(generation f(of f(the f( odd-frequency f(pairing f(state. f( (PRB 70, 012507 (2004), PRL 98 037003 (2007) , PRL 99 067005 (2007))

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