Procesy Andreeva w silnie skorelowanych układach fermionowych

Transcription

1 Kraków, 4 marca 2013 r. Procesy Andreeva w silnie skorelowanych układach fermionowych T. Domański Uniwersytet Marii Curie Skłodowskiej w Lublinie

2 Outline

3 Outline 1. Introduction / underlying idea /

4 Outline 1. Introduction / underlying idea / 2. Andreev transport via quantum dots / correlations versus superconductivity /

5 Outline 1. Introduction / underlying idea / 2. Andreev transport via quantum dots / correlations versus superconductivity / 3. Further extensions / quantum interference, dephasing, Cooper splitting, etc /

6 Outline 1. Introduction / underlying idea / 2. Andreev transport via quantum dots / correlations versus superconductivity / 3. Further extensions / quantum interference, dephasing, Cooper splitting, etc / 4. Andreev spectroscopy in bulk superconductors / probing the pair coherence /

7 Outline 1. Introduction / underlying idea / 2. Andreev transport via quantum dots / correlations versus superconductivity / 3. Further extensions / quantum interference, dephasing, Cooper splitting, etc / 4. Andreev spectroscopy in bulk superconductors / probing the pair coherence / 5. Andreev scattering in ultracold gasses / fermion vs molecular channels /

8 1. Introduction

9 Andreev reflections the main concept

10 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S

11 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S N S V Let us restrict to the subgap regime ev of an applied bias V.

12 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S N S electron

13 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S N S electron

14 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S N S electron

15 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S N S hole Cooper pair

16 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S N S hole Cooper pair

17 Andreev reflections the main concept Let us consider the process of electron tunneling from the normal conductor N (e.g. metallic lead) to the superconducting electrode S N S hole Cooper pair Such double-charge exchange is named the Andreev reflection (scattering).

18 Andreev reflections historical remark

19 Andreev reflections historical remark This anomalous transport channel allows for a finite subgap current across N-S interface even though the single-particle transmissions are forbidden. Its original idea has been suggested by

20 Andreev reflections historical remark This anomalous transport channel allows for a finite subgap current across N-S interface even though the single-particle transmissions are forbidden. Its original idea has been suggested by A.F. Andreev / P. Kapitza Institute, Moscow (Russia) / A.F. Andreev, Sov. Phys. JETP 19, 1228 (1964).

21 2. Andreev transport via quantum dot

22 Physical situation N-QD-S scheme Let us consider the quantum dot (QD) on an interface between the external metallic (N) and superconducting (S) leads

23 Physical situation N-QD-S scheme Let us consider the quantum dot (QD) on an interface between the external metallic (N) and superconducting (S) leads V = µ N - µ S N Γ N QD Γ S V G S metallic lead QD superconductor

24 Physical situation N-QD-S scheme Let us consider the quantum dot (QD) on an interface between the external metallic (N) and superconducting (S) leads V = µ N - µ S N Γ N QD Γ S V G S metallic lead QD superconductor This setup can be thought of as a particular version of the SET.

25 Physical situation energy spectrum

26 Physical situation energy spectrum Components of the N-QD-S heterostructure have the following spectra

27 Physical situation energy spectrum Components of the N-QD-S heterostructure have the following spectra N QD ε d +U S µ N ε d µ S Γ N Γ S

28 Physical situation energy spectrum Components of the N-QD-S heterostructure have the following spectra N QD ε d +U S µ N ε d µ S Γ N Γ S External bias ev = µ N µ S induces the current(s) through QD.

29 Microscopic model The correlation effects

30 Microscopic model The correlation effects Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d

31 Microscopic model The correlation effects Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d are expected to affect the transport properties of the system

32 Microscopic model The correlation effects Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d are expected to affect the transport properties of the system Ĥ = σ + k,σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d + Ĥ N + Ĥ S β=n,s ( V kβ ˆd σĉkσβ + V kβ ĉ kσ,β ˆd σ )

33 Microscopic model The correlation effects Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d are expected to affect the transport properties of the system Ĥ = σ + k,σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d + Ĥ N + Ĥ S β=n,s ( V kβ ˆd σĉkσβ + V kβ ĉ kσ,β ˆd σ ) where Ĥ N = k,σ (ε k,n µ N ) ĉ kσnĉkσn

34 Microscopic model The correlation effects Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d are expected to affect the transport properties of the system Ĥ = σ + k,σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d + Ĥ N + Ĥ S β=n,s ( V kβ ˆd σĉkσβ + V kβ ĉ kσ,β ˆd σ ) where ( ĉ k Sĉ k S +h.c. ) Ĥ S = k,σ (ε k,s µ S ) ĉ kσsĉkσs k

35 Relevant problems : issue # 1

36 Relevant problems : issue # 1 Hybridization of QD to the metallic lead is responsible for:

37 Relevant problems : issue # 1 Hybridization of QD to the metallic lead is responsible for:

38 Relevant problems : issue # 1 Hybridization of QD to the metallic lead is responsible for: ρ d (ω) [ 1 / Γ N ] Γ S = 0 T / Γ N = ω / Γ N a broadening of QD levels

39 Relevant problems : issue # 1 Hybridization of QD to the metallic lead is responsible for: ρ d (ω) [ 1 / Γ N ] Γ S = 0 T / Γ N = ω / Γ N a broadening of QD levels and...

40 Relevant problems : issue # 1 Hybridization of QD to the metallic lead is responsible for: ρ d (ω) [ 1 / Γ N ] Γ S = 0 T / Γ N = ω / Γ N a broadening of QD levels and...

41 Relevant problems : issue # 1 Hybridization of QD to the metallic lead is responsible for: ρ d (ω) [ 1 / Γ N ] Γ S = 0 T / Γ N = ω / Γ N a broadening of QD levels and appearance of the Kondo resonance below T K.

42 Relevant problems : issue # 2

43 Relevant problems : issue # 2 Hybridization of QD to the superconducting lead

44 Relevant problems : issue # 2 Hybridization of QD to S transmits the on-dot pairing (proximity effect).

45 Relevant problems : issue # 2 Hybridization of QD to S transmits the on-dot pairing (proximity effect). U =0 ε d = 0 ΓΓ S /Γ S N = ρ d (ω) ω/d

46 Relevant problems : issue # 2 Hybridization of QD to S transmits the on-dot pairing (proximity effect). U =0 ε d = 0 ΓΓ S /Γ S N = 1.0 ρ d (ω) ω/d

47 Relevant problems : issue # 2 Hybridization of QD to S transmits the on-dot pairing (proximity effect). U =0 ε d = 0 ΓΓ S /Γ S N = 2.0 ρ d (ω) ω/d

48 Relevant problems : issue # 2 Hybridization of QD to S transmits the on-dot pairing (proximity effect). U =0 ε d = 0 ΓΓ S /Γ S N = 5.0 ρ d (ω) ω/d

49 Relevant problems : issue # 2 Hybridization of QD to S transmits the on-dot pairing (proximity effect). U =0 ε d = 0 ΓΓ S /Γ S N = 10.0 ρ d (ω) ω/d

50 Relevant problems : issue # 2 Hybridization of QD to S transmits the on-dot pairing (proximity effect). 7.5 Γ S ρ d (ω) [ 1/Γ S ] Γ N ω / Γ S

51 Relevant problems : # 1 + 2

52 Relevant problems : # Hybridizations Γ N and Γ S are thus effectively leading to 75 Kondo peak ρ d (ω) [ 1/Γ S ] Γ S ΓN 0 ε d 0 ε d + U ω / Γ S

53 Relevant problems : # Hybridizations Γ N and Γ S are thus effectively leading to 75 Kondo peak ρ d (ω) [ 1/Γ S ] Γ S ΓN 0 ε d 0 ε d + U ω / Γ S / interplay between the Kondo effect and superconductivity /

54 Questions:

55 Questions: What relation does occur between superconductivity (transmitted onto the QD) and the Kondo effect?

56 Questions: What relation does occur between superconductivity (transmitted onto the QD) and the Kondo effect? Do they coexist or compete?

57 Questions: What relation does occur between superconductivity (transmitted onto the QD) and the Kondo effect? Do they coexist or compete? How do these effects show up in the charge current through N-QD-S junction?

58 Questions: What relation does occur between superconductivity (transmitted onto the QD) and the Kondo effect? Do they coexist or compete? How do these effects show up in the charge current through N-QD-S junction? Are there any particular features?

59 Formal aspects

60 Formal aspects To account for both, the proximity effect and the correlations, we have to deal with the Nambu (2 2 matrix) Green s function

61 Formal aspects To account for both, the proximity effect and the correlations, we have to deal with the Nambu (2 2 matrix) Green s function G d (τ, τ )= ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ )

62 Formal aspects To account for both, the proximity effect and the correlations, we have to deal with the Nambu (2 2 matrix) Green s function G d (τ, τ )= ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) In equilibrium its Fourier transform obeys the Dyson equation

63 Formal aspects To account for both, the proximity effect and the correlations, we have to deal with the Nambu (2 2 matrix) Green s function G d (τ, τ )= ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) In equilibrium its Fourier transform obeys the Dyson equation G d (ω) 1 = ω ε d 0 0 ω + ε d Σ 0 d (ω) ΣU d (ω)

64 Formal aspects To account for both, the proximity effect and the correlations, we have to deal with the Nambu (2 2 matrix) Green s function G d (τ, τ )= ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) In equilibrium its Fourier transform obeys the Dyson equation G d (ω) 1 = ω ε d 0 0 ω + ε d Σ 0 d (ω) ΣU d (ω) with Σ 0 d (ω) the selfenergy for U = 0

65 Formal aspects To account for both, the proximity effect and the correlations, we have to deal with the Nambu (2 2 matrix) Green s function G d (τ, τ )= ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) ˆT τ ˆd (τ) ˆd (τ ) In equilibrium its Fourier transform obeys the Dyson equation G d (ω) 1 = ω ε d 0 0 ω + ε d Σ 0 d (ω) ΣU d (ω) with Σ U d (ω) correction due to U 0.

66 Non-equilibrium phenomena

67 Non-equilibrium phenomena The steady current J L = J R is found to consist of two contributions J(V ) = J 1 (V ) + J A (V )

68 Non-equilibrium phenomena The steady current J L = J R is found to consist of two contributions J(V ) = J 1 (V ) + J A (V ) which can be expressed by the Landauer-type formula J 1 (V ) = 2e h dω T 1 (ω) [f(ω+ev, T) f(ω, T)] J A (V ) = 2e h dω T A (ω) [f(ω+ev, T) f(ω ev, T)] with the transmittance ( G r T 1 (ω) = Γ N Γ S 11 (ω) 2 + G r 12 (ω) 2 2 ) ω ReGr 11 (ω)gr 12 (ω)

69 Non-equilibrium phenomena The steady current J L = J R is found to consist of two contributions J(V ) = J 1 (V ) + J A (V ) which can be expressed by the Landauer-type formula J 1 (V ) = 2e h dω T 1 (ω) [f(ω+ev, T) f(ω, T)] J A (V ) = 2e h dω T A (ω) [f(ω+ev, T) f(ω ev, T)] with the transmittance T A (ω) = Γ 2 N G 12(ω) 2

70 Transport channels

71 Transport channels Qualitative features in the differential conductance G(V ) = J(V ) V

72 Transport channels Qualitative features in the differential conductance G(V ) = J(V ) V G G 12 G 22 G(V,T)/G 0 G A particle hole peaks zero bias anomaly ev/d

73 Transport channels Qualitative features in the differential conductance G(V ) = J(V ) V G G 12 G 22 G(V,T)/G 0 G A particle hole peaks zero bias anomaly ev/d T. Domański, A. Donabidowicz, K.I. Wysokiński, PRB 76, (2007).

74 Transport channels Qualitative features in the differential conductance G(V ) = J(V ) V G G 12 G 22 G(V,T)/G 0 G A particle hole peaks zero bias anomaly ev/d T. Domański, A. Donabidowicz, K.I. Wysokiński, PRB 76, (2007). We shall now focus on the subgap Andreev conductance.

75 Correlated QD effect of the asymmetry ΓS /ΓN

76 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N

77 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 0

78 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 1

79 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 2

80 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 3

81 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 4

82 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 5

83 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 6

84 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 8

85 Correlated QD effect of the asymmetry Γ S /Γ N Spectral function obtained below T K for U =10Γ N ρ d (ω) [ 1 / Γ N ] T = 10-3 Γ N ε d = -1.5 Γ N ω / Γ N Superconductivity suppresses the Kondo resonance

86 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N

87 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N T. Domański and A. Donabidowicz, PRB 78, (2008).

88 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

89 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 1 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

90 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 2 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

91 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 3 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

92 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 4 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

93 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 5 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

94 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 6 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

95 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 7 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

96 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 8 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008).

97 Correlated QD effect of the asymmetry Γ S /Γ N Andreev conductance G A (V ) for: U = 10Γ N Γ S / Γ N = 8 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 10 8 T. Domański and A. Donabidowicz, PRB 78, (2008). Kondo resonance slightly enhances the zero-bias Andreev conductance, especially for Γ S Γ N!

98 Experimental setup / University of Tokyo /

99 Experimental setup / University of Tokyo /

100 Experimental setup / University of Tokyo / QD : self-assembled InAs diameter 100 nm backgate : Si-doped GaAs

101 Experimental setup / University of Tokyo / T c 1K 152µeV QD : self-assembled InAs diameter 100 nm backgate : Si-doped GaAs

102 Experimental setup / University of Tokyo / T c 1K 152µeV QD : self-assembled InAs diameter 100 nm backgate : Si-doped GaAs R.S. Deacon et al, Phys. Rev. Lett. 104, (2010).

103 Interplay with the Kondo effect

104 Interplay with the Kondo effect R.S. Deacon et al, Phys. Rev. B 81, (R) (2010).

105 Interplay with the Kondo effect The zero-bias conductance peak is consistent with Andreev transport enhanced by the Kondo singlet state R.S. Deacon et al, Phys. Rev. B 81, (R) (2010).

106 Interplay with the Kondo effect The zero-bias conductance peak is consistent with Andreev transport enhanced by the Kondo singlet state We note that the feature exhibits excellent qualitative agreement with a recent theoretical treatment by Domanski et al R.S. Deacon et al, Phys. Rev. B 81, (R) (2010).

107 Summary / for part 2 /

108 Summary / for part 2 / QD coupled between N and S electrodes:

109 Summary / for part 2 / QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect /

110 Summary / for part 2 / QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is affected by the correlations / Kondo & charging effects /

111 Summary / for part 2 / QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is affected by the correlations / Kondo & charging effects / Interplay between the proximity and correlation effects is manifested in a subgap Andreev transport by:

112 Summary / for part 2 / QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is affected by the correlations / Kondo & charging effects / Interplay between the proximity and correlation effects is manifested in a subgap Andreev transport by: particle-hole splitting / when ε d µ S /

113 Summary / for part 2 / QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is affected by the correlations / Kondo & charging effects / Interplay between the proximity and correlation effects is manifested in a subgap Andreev transport by: particle-hole splitting / when ε d µ S / zero-bias enhancement / below T K /

114 3. Further extensions

115 Double QD between a metal and superconductor N QD 2 QD 1 t ε 2 +U 2 ε 2 ε 1 +U 1 S µ N ε 1 µ S Γ N Γ S Relevant issues: ( T-shape configuration) induced on-dot pairing (due to Γ S ) Coulomb blockade & Kondo effect (via U 1 and Γ N ) quantum interference (because of t)

116 Quantum interference in the particle and hole channels

117 Quantum interference in the particle and hole channels Fano-type lineshapes appear simultaneously at ±ε 2 ρ d1 (ω) Γ S / Γ N ω / Γ N / The case: U 1 = 0 and U 2 = 0 / J. Barański and T. Domański, Phys. Rev. B (2012).

118 Quantum interference in the particle and hole channels 0.2 Differential Andreev conductance G A = J A(V ) V G A (V) [4e 2 /h] t / Γ N = ev / Γ N -0.2 J. Barański and T. Domański, Phys. Rev. B 84, (2011).

119 Double QD decoherence effects V = µ N - µ S N Γ N QD 1 t QD 2 Γ S S Γ D D In this setup the floating lead (D) is responsible for a dephasing.

120 Quantum interference influence of the decoherence

121 Quantum interference influence of the decoherence ρ d1 (ω) Density of states ρ d1 (ω) -ε ρ d1 (ω) 0.0 Γ D / Γ N = ω / Γ N ε ω / Γ N J. Barański and T. Domański, Phys. Rev. B (2012).

122 Quantum interference influence of the decoherence Andreev transmittance T A (ω) T A (ω) T A (ω) ω / Γ N Γ S / Γ N = 8 Γ D = ω / Γ N J. Barański and T. Domański, Phys. Rev. B (2012).

123 Interplay with ferromagnetism Univ. of Basel group

124 Interplay with ferromagnetism Univ. of Basel group

125 Interplay with ferromagnetism Univ. of Basel group L. Hofstetter et al, Phys. Rev. Lett. 104, (2010).

126 Interplay with ferromagnetism Univ. of Basel group L. Hofstetter et al, Phys. Rev. Lett. 104, (2010). Effects of ferromagnetism and superconductivity

127 Interplay with ferromagnetism Univ. of Basel group L. Hofstetter et al, Phys. Rev. Lett. 104, (2010). Effects of ferromagnetism and superconductivity left Ni/Co/Pd trilayer ferromagnet QD InAs nanowire right Ti/Al bilayer superconductor

128 Three terminal junctions

129 Three terminal junctions

130 Three terminal junctions Crossed Andreev reflections tunable via gate voltages

131 Three terminal junctions Crossed Andreev reflections tunable via gate voltages J. Eldridge, M.G. Pala, M. Governale, J. König, Phys. Rev. B 82, (2010)

132 Cooper pair splitter

133 Cooper pair splitter

134 Cooper pair splitter Realization of the Cooper pair splitting in a microwave cavity

135 Cooper pair splitter Realization of the Cooper pair splitting in a microwave cavity A. Cottet, T. Kontos, and A. Levy Yeyati, Phys. Rev. Lett. 108, (2012)

136 QD spin-valve in three terminal junctions

137 QD spin-valve in three terminal junctions

138 QD spin-valve in three terminal junctions Idea of the spin valves using the Andreev reflections

139 QD spin-valve in three terminal junctions Idea of the spin valves using the Andreev reflections B. Sothmann, D. Futterer, M. Governale, J. König, Phys. Rev. B 82, (2010).

140 4. Bulk superconductors

141 Andreev spectroscopy for bulk superconductors

142 Andreev spectroscopy for bulk superconductors The subgap Andreev spectroscopy is also a valuable tool for studying various superconducting compounds.

143 Andreev spectroscopy for bulk superconductors The subgap Andreev spectroscopy is also a valuable tool for studying various superconducting compounds. K.Y. Yang et al, Phys. Rev. Lett. 105, (2010). For practical experimental realizations one can e.g. use an insulating barrier sandwiched between the conducting (N) and the probed superconductor (S).

144 Andreev spectroscopy for bulk superconductors The subgap Andreev spectroscopy is also a valuable tool for studying various superconducting compounds. STM tip E STM tip (constant DOS) O orbital CuO 2 plane (d wave DOS) BiO ev SrO E 0 Ω 0 CuO 2 Cu O Bi Sr Γ n Γ s S. Pilgram et al, Phys. Rev. Lett. 97, (2006). Other experimental realizations are also possible in the STM configuration, where the apex oxygen atoms play a role similar to QD in the N-QD-S setup.

145 Andreev spectroscopy for bulk superconductors The subgap Andreev spectroscopy is also a valuable tool for studying various superconducting compounds. 80 La 2-x Sr x CuO T (K) T C (onset) cover electrode T C (bulk) 2 STM ARPES >T c ARPES <T c Energy gaps i (mev) Doping x G. Koren, T. Kirzner, Phys. Rev. Lett. 106, (2011). Such Andreev spectroscopy has revealed the intriguing two-gap feature.

146 5. Ultracold gasses

147 Andreev spectroscopy for ultracold atoms

148 Andreev spectroscopy for ultracold atoms Proposal for the Andreev-type spectroscopy has been discussed also in a context of the superfluid ultracold fermion atom systems.

149 Andreev spectroscopy for ultracold atoms Proposal for the Andreev-type spectroscopy has been discussed also in a context of the superfluid ultracold fermion atom systems. A.J. Daley, P. Zoller, and B. Trauzettel, Phys. Rev. Lett. 100, (2008). The wave packet propagating along the 1-dimensional optical lattice can be scattered at an interaction boundary in the Andreev-type fashion.

150 Feshbach resonance [ local problem ] Ĥ loc (r) = σ +g ε(r) ĉ σ (r)ĉ σ(r) + E(r) ˆb (r)ˆb(r) (ˆb ) (r)ĉ (r)ĉ (r) + ĉ (r)ĉ (r)ˆb(r)

151 Feshbach resonance [ local problem ] Ĥ loc (r) = σ +g ε(r) ĉ σ (r)ĉ σ(r) + E(r) ˆb (r)ˆb(r) (ˆb ) (r)ĉ (r)ĉ (r) + ĉ (r)ĉ (r)ˆb(r) describes:

152 Feshbach resonance [ local problem ] Ĥ loc (r) = σ +g ε(r) ĉ σ (r)ĉ σ(r) + E(r) ˆb (r)ˆb(r) (ˆb ) (r)ĉ (r)ĉ (r) + ĉ (r)ĉ (r)ˆb(r) describes: ĉ ( ) σ (r) fermion atoms (open channel)

153 Feshbach resonance [ local problem ] Ĥ loc (r) = σ +g ε(r) ĉ σ (r)ĉ σ(r) + E(r) ˆb (r)ˆb(r) (ˆb ) (r)ĉ (r)ĉ (r) + ĉ (r)ĉ (r)ˆb(r) describes: ĉ ( ) σ (r) fermion atoms (open channel) ˆb( ) (r) molecules (closed channel)

154 Feshbach resonance [ local problem ] Ĥ loc (r) = σ +g ε(r) ĉ σ (r)ĉ σ(r) + E(r) ˆb (r)ˆb(r) (ˆb ) (r)ĉ (r)ĉ (r) + ĉ (r)ĉ (r)ˆb(r) describes: ĉ ( ) σ (r) fermion atoms (open channel) ˆb( ) (r) molecules (closed channel) resonantly interacting via:

155 Feshbach resonance [ local problem ] Ĥ loc (r) = σ +g ε(r) ĉ σ (r)ĉ σ(r) + E(r) ˆb (r)ˆb(r) (ˆb ) (r)ĉ (r)ĉ (r) + ĉ (r)ĉ (r)ˆb(r) describes: ĉ ( ) σ (r) fermion atoms (open channel) ˆb( ) (r) molecules (closed channel) resonantly interacting via: ˆb ĉ ĉ + h.c (feshbach resonance)

156 Feshbach resonance [ local problem ] Ĥ loc (r) = σ +g ε(r) ĉ σ (r)ĉ σ(r) + E(r) ˆb (r)ˆb(r) (ˆb ) (r)ĉ (r)ĉ (r) + ĉ (r)ĉ (r)ˆb(r) describes: ĉ ( ) σ (r) fermion atoms (open channel) ˆb( ) (r) molecules (closed channel) resonantly interacting via: ˆb ĉ ĉ + h.c (feshbach resonance) M.L. Chiofalo, S.J.J.M.F. Kokkelmans, J.N. Milstein, and M.J. Holland, Phys. Rev. Lett. 88, (2002).

157 Local solution [ exact ] ( u 2 G loc (iω n ) = [1 Z(T)] + iω n ε + v 2 iω n ε ) + Z(T) iω n ε

158 Local solution [ exact ] ( u 2 G loc (iω n ) = [1 Z(T)] + iω n ε + v 2 iω n ε ) + Z(T) iω n ε where

159 Local solution [ exact ] ( u 2 G loc (iω n ) = [1 Z(T)] + iω n ε + v 2 iω n ε ) + Z(T) iω n ε where ε energy of non-bonding state

160 Local solution [ exact ] ( u 2 G loc (iω n ) = [1 Z(T)] + iω n ε + v 2 iω n ε ) + Z(T) iω n ε where ε energy of non-bonding state Z(T) the spectral weight

161 Local solution [ exact ] ( u 2 G loc (iω n ) = [1 Z(T)] + iω n ε + v 2 iω n ε ) + Z(T) iω n ε where ε energy of non-bonding state Z(T) the spectral weight ε ± = E/2 ± (ε E/2) 2 + g BCS-like excitation energies

162 Local solution [ exact ] ( u 2 G loc (iω n ) = [1 Z(T)] + iω n ε + v 2 iω n ε ) + Z(T) iω n ε where ε energy of non-bonding state Z(T) the spectral weight ε ± = E/2 ± (ε E/2) 2 + g BCS-like excitation energies u 2, v 2 = 1 2 [1±(ε E/2)/ (ε E/2) 2 + g 2 ] BCS-like coefficients

163 Local solution [ exact ] ( u 2 G loc (iω n ) = [1 Z(T)] + iω n ε + v 2 iω n ε ) + Z(T) iω n ε where ε energy of non-bonding state Z(T) the spectral weight ε ± = E/2 ± (ε E/2) 2 + g BCS-like excitation energies u 2, v 2 = 1 2 [1±(ε E/2)/ (ε E/2) 2 + g 2 ] BCS-like coefficients T. Domański, Eur. Phys. J. B 33, 41 (2003); T. Domański et al, Sol. State Commun. 105, 473 (1998).

164 Approximate general solution [ near the unitary limit ] Ĥ = ( ) dr ˆT kin (r) + Ĥ loc (r)

165 Approximate general solution [ near the unitary limit ] Ĥ = ( ) dr ˆT kin (r) + Ĥ loc (r) A(k F,ω) T / T c ω / T. Domański, Phys. Rev. A 84, (2011).

166 Approximate general solution [ near the unitary limit ] Ĥ = ( ) dr ˆT kin (r) + Ĥ loc (r) A(k,ω) T = 0.76 T c k / k F ω / T. Domański, Phys. Rev. A 84, (2011).

167 Approximate general solution [ near the unitary limit ] Ĥ = ( ) dr ˆT kin (r) + Ĥ loc (r) A(k,ω) T = 1.24 T c k / k F ω / T. Domański, Phys. Rev. A 84, (2011).

168 Approximate general solution [ near the unitary limit ] Ĥ = ( ) dr ˆT kin (r) + Ĥ loc (r) A(k,ω) T = 1.47 T c k / k F ω / T. Domański, Phys. Rev. A 84, (2011).

169 Approximate general solution [ near the unitary limit ] Ĥ = ( ) dr ˆT kin (r) + Ĥ loc (r) A(k,ω) T = 2.06 T c k / k F ω / T. Domański, Phys. Rev. A 84, (2011).

170 Evidence for Bogoliubov QPs above Tc

171 Evidence for Bogoliubov QPs above T c D. Jin group (Boulder, USA) T/Tc = 0.74 T/Tc = 1.24 Spectral Weight (arb.) k = T/Tc = 1.47 T/Tc = Single-Particle Energy (E S/E F) Results for the ultracold 40 K atoms J.P. Gaebler et al, Nature Phys. 6, 569 (2010).

172 Conclusions

173 Conclusions Andreev spectroscopy :

174 Conclusions Andreev spectroscopy : is a suitable tool for probing the pair-coherence

175 Conclusions Andreev spectroscopy : is a suitable tool for probing the pair-coherence simultaneously exploring the particle and hole states

176 Conclusions Andreev spectroscopy : is a suitable tool for probing the pair-coherence simultaneously exploring the particle and hole states It can be applied to:

177 Conclusions Andreev spectroscopy : is a suitable tool for probing the pair-coherence simultaneously exploring the particle and hole states It can be applied to: nanoscopic objects / coupled to superconducting electrodes /

178 Conclusions Andreev spectroscopy : is a suitable tool for probing the pair-coherence simultaneously exploring the particle and hole states It can be applied to: nanoscopic objects / coupled to superconducting electrodes / bulk materials / superconductors, atomic superfluids, black holes (?) /