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

2. Contents （１） What is odd-frequency pairing （２） Normal metal / Superconductor junctions （３） Ferromagnet/Superconductor junctions

3. Conventional Classification of Symmetry of Cooper pair Spin-singlet Cooper pair s-wave d-wave Spin-triplet Cooper pair Even Parity Odd Parity p-wave BCS Cuprate 3 HeSr 2 RuO 4

4. Pair amplitude (pair correlation) Exchange of two electrons Fermi-Dirac statistics

5. Pair amplitude Exchange of time Even-frequency pairing (conventional pairing) Odd-frequency pairing

6. Pauli principle Symmetry of pair wave functions: spin-singlet:  = odd spin-triplet:  = even even-frequency superconductivity odd-frequency superconductivity ()()   ( ω ), f( ω )  s-, d-wave p-, f-wave

7. 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 3 He Sr 2 RuO 4

8. 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)

9. 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

10. Contents （１） What is odd-frequency pairing (2) Ballistic normal metal junctions （ 3 ） Diffusive normal metal junctions (4 ） Ferromagnet/Superconductor junctions

11. 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. Schon J. Low Temp. Phys 147 457(2007) Ballistic Normal metal (semi-infinite) Superconductor (semi-infinite)

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

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

14. Normal metal spin-triplet p-wave superconductor

15. p x -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

16. 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

17. Mid gap Andreev resonant (bound) state (MARS) Interface (surface) ＋ー 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 ＋

18. Mid gap Andreev resonant (bound) state Electron-like quasiparticle Hole-like quasiparticle Cooper pair Odd-frequency Cooper pair （ Odd-frequency pair amplitude ）

19. 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

20. 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

21. 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)

22. (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,d x2-y2 -wave) ESE (d xy -wave) ETO (p x -wave) ETO (p y -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)

23. Contents （１） What is odd-frequency pairing (2) Ballistic normal metal junctions （ 3 ） Diffusive normal metal junctions (4 ） Ferromagnet/Superconductor junctions

24. Impurity scattering effect Tanaka and Golubov, PRL. 98, 037003 (2007) Ballistic Normal metal Superconductor Diffusive Normal 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)

25. Usadel equation Available for diffusive limit Angular average Diffusive  limit Diffusive normal metal region attached to superconductor

26. 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

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

28. DN ＋ー P x -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)

29. DN ＋ー Even frequency spin triplet odd parity (ETO) pair potential DN ＋ ー (no proximity& no MARS) P x -wave case P y -wave case

30. Symmetry of the pair potential Induced pair amplitude in DN  Even frequency spin singlet even parity (ESE) Even frequency spin triplet odd parity (ETO) Odd frequency spin triplet even parity (OTE) Odd frequency spin singlet odd parity (OSO)    ESE OTE ESE Summary of Proximity effect (diffusive normal metal)

31. 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

32. How to detect odd-frequency paring amplitude: measuring electrical conductivity Asano, Tanaka, Golubov, Kashiwaya, PRL 99, 067005 (2007) Sr 2 RuO 4 Ａｕ :I+ Ａｕ :V+ Ａｕ :I- Ａｕ :V- Kashiwaya, Maeno 2007 Zero energy peak No Zero energy peak OTE proximity ESE proximity (conventional) OTE (Odd-frequency spin-triplet even-parity) ESE (Even-frequency spin-singlet even-parity)

33. Contents （１） What is odd-frequency pairing （２） Ballistic normal metal junctions （ 3 ） Diffusive normal metal junctions （ 4 ） Ferromagnet/Superconductor junctions

34. _ Odd frequency spin-triplet s-wave pair Superconductor Ferromagnet 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 +

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

36. Josephson current in S/HM/S Half metal : CrO 2 Keizer et.al., Nature (2006) Eschrig et. al., PRL(‘03) Theory in the clean limit Spin active interface Bergeret et. al., PRL(‘01), Kadigrobov et. al., Europhys Lett.(‘01) Lofwander and Eschrig, Nature Physics (2008)

37. Furusaki, Physica B(‘92), Asano, PRB(‘01) Advantages SNS, SFS, S/HM/S Parameters : exchange : spin-flip Recursive GF Y.Asano, Y. Tanaka, A.A.Golubov, Phys.Rev.Lett. 98, 107002 (2007)

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

39. Odd-frequency pairs

40. SFS S/HM/S only even-odd mixpure odd Quasiparticle DOS in Half Metal Zero Energy Peak can be detected by tunneling spectroscopy

41. 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))