Surface states in p-wave superconductors and odd-frequency pairing

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1 Surface states in p-wave superconductors and odd-frequency pairing Alexander Golubov University of Twente, The Netherlands ABS and Moscow Institute of Physics and Technology, Russia F/S junction Odd-frequency pairing Vortex Proximity

2 Collaboration - Y. Tanaka and K. Yada, Nagoya University, Japan - Y. Asano, Hokkaido University, Japan - S. Kashiwaya, AIST, Tsukuba, Japan - S.V. Bakurskiy and M. Yu. Kupriyanov, Moscow State University, Russia

3 Contents - What is odd-frequency pairing - Relation to Andreev bound states - Impurity scattering effect - Anomalous surface impedance - Effect of surface roughness

4 Chiral p-wave superconductors Sr 2 RuO 4 Maeno (1994) Topological index for chirality Spin-triplet p-wave Time-reversal symmetry broken Volovik

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

6 Odd-frequency pairing Fermi-Dirac statistics Symmetry of pair wave functions: (ω), (ω) f(ω) even-frequency superconductivity Momentum x Spin x Frequency Berezinskii (1974): Spin-triplet s-wave Balatsky&Abrahams (1992): Spin-singlet p-wave (ω), f(ω) ω odd-frequency superconductivity ω

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

8 Symmetry of the pair amplitude + symmetric, anti-symmetric Frequency (time) Spin Orbital Total ESE +(even) (singlet) +(even) BCS Cuprate ETO OTE +(even) (odd) + (triplet) + (triplet) (odd) +(even) 3 He Sr 2 RuO 4 OSO (odd) (singlet) (odd) 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

9 Odd-frequency pairing state is possible in inhomogeneous superconductors even for conventional even-frequency paring in the bulk Broken spin rotation symmetry or spatial invariance symmetry can induce odd-frequency pairing state: - ferromagnet/superconducor junctions: Bergeret,Volkov&Efetov, non-uniform systems: Junctions: Tanaka&Golubov, 2007; Eschrig&Lofwander, 2007 Vortices: Yokoyama et al., 2008; Tanuma et al., 2009)

10 How to produce long range odd-frequency triplet from singlet s-wave pairing in S/F hybrid system? OTE OTE (1) (2) S ESE singlet s-wave Bergeret, Volkov, Efetov(2001) F(Ferromagnet) :Diffusive We consider only s-wave pairing here. OTE S F 2 ESE singlet s-wave OTE F 1 F 2 OTE OTE Eschrig Lofwander (2008) (long range) F 1 OTE OTE (long range) ESE (Even-frequency spin-singlet even-parity) OTE (Odd-frequency spin-triplet even-parity) z One needs two ferromagnets to generate long range odd-frequency pairing y x

11 Contents - What is odd-frequency pairing - Relation of odd-ω to Andreev bound states - Impurity scattering effect - Anomalous surface impedance - Effect of surface roughness

12 Andreev reflection 1964 Andreev Normal metal electron Superconductor Cooper pair Fermi energy Energy gap hole

13 Andreev reflection and Andreev bound state Cooper pair electron hole Cooper pair S N(I) S Inner gap state (Andreev bound state) is realized by Andreev reflection. (McMillan Rowell, 1966) Josephson current can be expressed by the probability of Andreev reflection. (Furusaki, Tsukada, 1990)

14 Are Andreev bound states related to odd-frequency state? Yes

15 Andreev bound state (non-topological and topological) Andreev bound state with non zero energy (de Gennes, Saint James) Not edge state Mid gap (zero energy) Andreev bound state Surface Andreev bound state Edge state Topological Non topological L. Buchholtz & G. Zwicknagl (81):, J. Hara & K. Nagai : Prog. Theor. Phys. 74 (86) C.R. Hu : (94)

16 Odd-frequency pairing in the presence of nonzero energy ABS Fully transparent N S OSO (Odd-frequency spin-singlet odd-parity) ESE (Even-frequency spin-singlet even-parity) OSO component is generated from superconductor with ESE symmetry. Y. Tanaka, Y. Tanuma A.A. Golubov, PRB (2007)

17 Andreev bound state (non-zero) generated by quasiparticle interference in N Andreev Bound states condition (non zero energy) (de Gennes, Saint-James, McMillan Thomas Rowell) Fully transparent interface Odd-frequency pairing N S Even-frequency pairing Bound state in the junctions can be expressed by the generation of the odd-frequency Cooper pair (pair amplitude). Y. Tanaka, Y. Tanuma and A.A. Golubov, PRB (2007)

18 Superconducting Materials where MARS is observed YBa 2 CuO 7-δ (Geerk, Kashiwaya, Iguchi, Greene, Yeh,Wei..) Bi 2 Sr 2 CaCu 2 O y (Ng, Suzuki, Greene.) La 2-x Sr x CuO 4 (Iguchi) La 2-x Ce x CuO 4 (Cheska) Pr 2-x Ce x CuO 4 (R.L.Greene) Sr 2 RuO 4 (Mao, Meno, Kawamura,Laube) κ (BEDT-TTF) 2 X, X=Cu[N(CN) 2 ]Br (Ichimura) UBe 13 (Ott) CeCoIn 5 (Wei Greene) PrOs 4 Sb 12 (Wei) Superfluid 3 He (Okuda, Nomura, Higashitani, Nagai)

19 Andreev bound state in unconventional pairing Mid gap Andreev bound state (MARS) Normalized DOS 4 2 ー ε/ 0 Local density of state has a zero energy peak. (Sign change of the pair potential at the interface) Interface (surface) ー + + ー Tanaka Kashiwaya PRL (1995); Kashiwaya Tanaka Rep. Prog. Phys (2000) Buchholz, Zwicknagl(1981) Hara Nagai(1986); Hu(1994), Matsumoto Shiba(1995) Ohashi Takada(1995); Hatsugai and Ryu (2002)

20 Mid gap Andreev resonant (bound) state Electron-like quasiparticle Hole-like quasiparticle Cooper pair Odd-frequency Cooper pair (Odd-frequency pair amplitude) Tanaka, Golubov, Asano: Phys. Rev. Lett (2007), Phys. Rev. Lett (2007), Phys. Rev. Lett (2007)

21 Sr 2 RuO 4 (1.5K phase)/au junction fabrication Sr 2 RuO 4 vacuum cleavage Au deposition in-situ SIM image of junction Kashiwaya Furuta Maeno PRL c187-f7 (09/11/16) di/dv-v (T-dep) H = c187-f7 (09/11/16) di/dv-v (H-dep) T = 0.55 K di/dv (Ω -1 ) T = 0.55 K 0.80 K 1.00 K 1.30 K 1.60 K di/dv (Ω -1 ) H = G 1000 G 2000 G 3000 G 4000 G 5000 G V (mv) V (mv) Zero-bias conductance peak observation between Sr 2 RuO 4 (1.5Kphase) and Au Consistent with theoretical calculation based on chiral p-wave pairing Yamashiro, Tanaka and Kashiwaya, Phys. Rev. B. 56, 7847 (1997). Honerkamp and Sigrist, J. Low Temp. Phys (1998).

22 Feature of the Andreev bound states d xy wave Chiral p-wave Hu(94) Tanaka Kashiwaya (95) Tanaka Kashiwaya (97) Sigrist Honerkamp (98)

23 spin-triplet superconductor junctions Ballistic Normal metal Superconductor Impurity scattering (isotropic) Diffusive Normal metal (DN) Superconductor Proximity effect in aerogel, Higashitani, Nagato, and Nagai (2009)

24 Eilenberger Equation Quasiclassical approximation (ballistic) (1)Cooper pair is formed by two electrons on the Fermi surface (2)The effective pair potential is determined by the direction of the motion of electrons. Kopnin Theory of Nonequilibrium Superconductivity", Oxford University Press (2001).

25 Eilenbergerequation Quasiclassical Green s function (Matsubara representation) Pair potential Quasiparticle Cooper pair which does not exist in bulk Cooper pair in bulk Normal metal Odd-frequency Even-frequency superconductor Direction of motion + Direction of motion

26 Condition of the generation of odd-frequency pairing Spatial change of pair potential + nonzero quasiparticle state Odd-frequency pairing Normal metal superconductor General theory, Higashitani 2011

27 Underlying physics Near the interface, even and odd-paritypairing 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.

28 Contents - What is odd-frequency pairing - Relation to Andreev bound states - Impurity scattering effect - Anomalous surface impedance - Effect of surface roughness

29 Impurity scattering effect Diffusive Normal metal (DN) Impurity scattering (isotropic) Superconductor Only s-wave pair amplitude exists in DN (1)ESE (2)OTE ESE (Even-frequency spin-singlet even-parity

30 Usadel equation Available for diffusive limit Angular average Diffusive limit Diffusive normal metal region attached to superconductor Boundary condition available for unconventional superconductors Tanaka et al, PRL (2003), PRB (2004)

31 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

32 Odd frequency spin triplet s-wave (OTE) pair is induced in DN P x -wave case ー + Re f ( ε ) DN Im f ( ε ) New type of proximity effect Y.Tanaka, A.A.Golubov, Phys.Rev.Lett. 98, (2007)

33 Unconventional proximity effect Odd-frequency pairing at the interface includes s-wave component No proximity effect Odd-frequency pairing at the interface: Odd-parity

34 STS experiments Proximity effect via odd-frequency pairing Conventional proximity effect STS STS ZEP MARS No ZEP MARS Diffusive normal Metal (DN) Spin-Triplet superconductor Diffusive normal Metal (DN) Spin-Singlet superconductor LDOS in DN has a zero energy peak LDOS in DN has a gap DoS peak is robust again imputiry satterin

35 Anomalous proximity effect in nano-wire normal superconductor PRB (2013) Topological (Effective triplet p x -wave superconductor) LDOS in Nano wire (Disordered segment) Non Topological robust zero energy peak of LDOS in topological phase!! Similar anomalous proximity effect has been predicted in spin-triplet p-wave superconductor. Tanaka and Kashiwaya, PRB 2004

36 Odd-frequency pairing and LDOS LDOS and pair amplitude at ε=0 Even-frequency pairing Odd-frequency pairing LDOS Odd-frequency pair amplitude Odd-frequency pairing always exists in topological phase where Majorana mode (fermion) exists Y. Asano and Y. Tanaka PRB (2013)

37 Contents - What is odd-frequency pairing - Relation to Andreev bound states - Impurity scattering effect - Anomalous surface impedance - Effect of surface roughness

38 Surface impedance of a superconductor Response to electromagnetic field 1.0 ωτ = 0.1 Superconductor R & X 0.5 X 0.0 R T / T c R: surface resistance loss of microwave power due to normal carrier X: reactance response of superconducting carriers

39

40 Unusual electromagnetic response Tanaka, Asano, Golubov, Kashiwaya, PRB 72, R (05); PRB 73, (06) : even-freq. proximity F is real number odd-freq. proximity F is pure imaginary even freq. pairs odd freq. pairs : unusual relation

41 The result: local impedance of the n-layer N S singlet R < X triplet anomalous! Y. Asano, A.A. Golubov, Ya.V. Fominov and A.A. Golubov, PRL 107, (2011).

42 Contents - What is odd-frequency pairing - Relation to Andreev bound states - Impurity scattering effect - Anomalous surface impedance - Effect of surface roughness in p-wave superconductors

43 Model for surface roughness Eilenberger equations Self-Consistent equation (for p x -pairing) 0 L/ξ S 3 Clean p-wave superconductor d/l e Dirty spacer Ovchinnikov model: roughness parameter d/l e <

44 + + Pair potential _ + p x state: = 0 cos(θ) Chiral p x +i p y state: = x cos(θ)+i y sin(θ) 2 d/l=5 d/l=1 d/l=0.1 reflection 2 x)/t C 1 x)/t C 1 d/l=0.01 d/l=0.1 d/l=1 d/l=10 ( x, y ) x Y. Nagato, S. Higashitani, K. Nagai, J. Phys. Soc. Jpn. 80, (2011) S. Higashitani, S. Matsuo, Y. Nagato, et al., Phys. Rev. B 85, (2012) S. V. Bakurskiy, A. A. Golubov, M. Yu. Kupriyanov, K. Yada, Y. Tanaka, Phys. Rev. B 90, (2014) x

45 Pair amplitude f (p x -pairing) Normal metal superconductor f=f (θ) f=f 1 +f 2 Show average value <f>= f(θ) dθ Interface splits f into two components <f 1 > Odd-frequency component f 1 Even-frequency component f <f 2 > ω In a bulk only even-frequency component f 2 exists ω

46 Chiral p-wave state: pair amplitudes Angle-resolved pairing amplitudes at a rough surface (k y =k F sinθ) Odd-ω Even-ω Clean p-wave SC x=0 d/l=1 Dirty spacer

47 Angle-resolved surface density of states (k y = k F sinθ) Specular surface Rough surface - subgap spectrum broadened by surface roughness -additional subgap bound states appear: enhancement of low-energy DoS by disorder

48 Spatial distribution of angle-resolved DOS at k y = 0 Angle-averaged surface DOS: topologically protected midgap bound states x=-3ξ S L/ξ S x=0 Clean p-wave SC Dirty spacer

49 Spatial distribution of angle-resolved DOS at k y = 0 Angle-averaged surface DOS: enhancement of subgap DOS by disorder x=-3ξ S L/ξ S x=0 Clean p-wave SC Dirty spacer

50 boundary Rough (d/le=1) SDoS roughness Intermediate the boundary (d/le=0.1) of boundary Clean (d/le=0) px wave Angle-averaged surface DOS px+ i p y wave a) c) b) d) Enhancement of subgap DOS by disordered in chiral p=wave See alsos. Murakawa, et al., PRL (2009), JPSJ (2011) 3He-B with a wall coated by 4He 1 e) f) E/T C E/T C

51 Emergence of sharp edge in DoS at * < Y. Nagato, et al., JLTP (1998) W. Zhang, Phys. Lett. (1988) A. B. Vorontsov and J. A. Sauls, PRB (2003) This calculation: 2 Origin: suppression of y by surface disorder x)/t C 1 0 d/l=0.01 d/l=0.1 d/l=1 d/l=10 ( x, y ) x

52 Summary -Odd-frequency pairing state is realized in superconductors with broken translational or spin rotation symmetry -Manifestations of odd-frequency pairing: Andreev bound states, anomalous surface impedance, long-range proximity effects in S/F hybrids -Odd-frequency pairing is generated at rough surfaces in p-wave superconductors. Subgap surface bound states are robust with respect to disorder, as a consequence of odd-frequency pairing