Oddfrequencypairingin superconductingheterostructures Alexander Golubov Twente University The Netherlands

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

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-wave Even Parity d-wave Cuprate BCS Spin-triplet Cooper pair p-wave Odd Parity 3 He Sr 2 Ru. O 4

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 D(ω), D(w) f(ω) even-frequency superconductivity Symmetry of　pair　wave functions: w spin-singlet: s = odd s-, d-wave D(ω), f(ω) odd-frequency superconductivity p-, f-wave spin-triplet: s = even w p-, f-wave s-, d-wave

Symmetry of the pair amplitude + symmetric, - 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 3 He Sr 2 Ru. O 4 - 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 （１）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. Eschrig, T. Lofwander, Th. Champel, J. C. Cuevas and G. Schon J. Low Temp. Phys 147 457(2007)

Eilenberger equation (explicitly denote direction of motion) Pair potential Quasiparticle function Pair amplitudes Bulk state ballistic normal metal Form factor Only S N S

General properties（frequency) Superconductor is conventional even-frequency one. Even-frequency (real) bulk-component Spatial change of the pair potential Odd-frequency (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.

Mid gap Andreev resonant (bound) state (MARS) Interface (surface) ー ＋ ＋ ＋ ー Local density of state has a zero energy peak. (Sign change of the pair potential at the interface) ー 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 amplitude）

Odd-frequency pairing state in N/S junctions (N finite length) Bounds state are formed in the normal metal Y. Tanaka, Y. Tanuma and A. A. Golubov, Phys. Rev. B 76, 054522 (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 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) （Mc. Millan Thomas Rowell) Odd-frequency pairing Even-frequency pairing Bound states are due to the generation of the odd-frequency Cooper pair amplitude Y. Tanaka, Y. Tanuma and A. A. Golubov, PRB 76 054522 (2007)

Symmetry of the Cooper pair (No spin flip) Sign change Bulk state (MARS) (1) ESE (s, dx 2 -y 2 -wave) (1) • • Interface-induced symmetry (subdominant component ) No (2) (3) ESE (dxy-wave) ETO (px-wave) Yes ESE + (OSO) OSO +(ESE) OTE + (ETO) (4) ETO (py-wave) No ETO + (OTE) (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) Phys. Rev. Lett. 99 037005 (2007)

Contents （１）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 scattering (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 for diffusive limit Diffusive limit Angular average Diffusive normal metal region attached to superconductor

Even frequency spin singlet even parity (ESE) pair potential DN S Even frequency spin singlet s-wave (ESE) pair is induced in DN. ESE pair /ESE pair potential

Even frequency spin singlet　even parity 　(ESE) pair potential s-wave case DN S

Odd frequency spin triplet s-wave (OTE) pair is induced in DN Px-wave case ー ＋ DN New type of proximity effect Y. Tanaka, A. A. Golubov, Phys. Rev. Lett. 98, 037003 (2007)

Even frequency spin triplet odd parity (ETO) pair potential Px-wave case ー ＋ DN Py-wave case ＋ ー DN (no proximity& no MARS)

Summary of Proximity effect (diffusive normal metal) Symmetry of the pair potential Induced pair amplitude in DN (1) Even frequency spin singlet even parity (ESE) ESE (2) Even frequency spin triplet odd parity (ETO) OTE (3) Odd frequency spin triplet even parity (OTE) OTE (4) 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 to detect odd-frequency paring amplitude: measuring electrical conductivity 　　　　　　　　 Asano, Tanaka, Golubov, Kashiwaya, PRL 99, 067005 (2007) Sr 2 Ru. O 4　 　 OTE (Odd-frequency spin-triplet even-parity) ESE (Even-frequency spin-singlet even-parity) Ａｕ: I+ Ａｕ: V+ Ａｕ: IＡｕ: V- Kashiwaya, Maeno 2007 OTE proximity Zero energy peak ESE proximity (conventional) No Zero energy peak

Contents （１）What is odd-frequency pairing （２）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 s-wave pair _ + Ferromagnet Superconductor Bergeret, Efetov, Volkov, (2001) Eschrig, Buzdin, Kadigrobov, Fominov, Radovic Generation of the odd-frequency pair amplitude in ferromagnet

Spin triplet Cooper pairs in SF systems triplet Josephson effect in SF multilayered system with noncollinear ferromagnets and thin superconductors <<

Josephson current in S/HM/S Half metal : Cr. O 2 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 Parameters : exchange : spin-flip Y. Asano, Y. Tanaka, A. A. Golubov, Phys. Rev. Lett. 98, 107002 (2007)

Spin active interface SFS, S/HM/S self-averaging No sign change odd-frequency pairs SFS S/HM/S only

Odd-frequency pairs

Quasiparticle DOS in Half Metal SFS S/HM/S only even-odd mix pure odd Zero Energy Peak can be detected by 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 oddfrequency 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))