Fate of the Kondo impurity in a superconducting medium
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1 Karpacz, 2 8 March 214 Fate of the Kondo impurity in a superconducting medium T. Domański M. Curie Skłodowska University Lublin, Poland
2 Motivation
3 Physical dilemma to screen or not to screen?
4 Physical dilemma to screen or not to screen? Quantum impurity (dot) coupled to a metallic bath can form the Kondo state with itinerant electrons (at T < T K )
5 Physical dilemma to screen or not to screen? Quantum impurity (dot) coupled to a superconducting reservoir
6 Physical dilemma to screen or not to screen? Quantum impurity (dot) coupled to a superconducting reservoir encounters the following obstacles : = electron states near the Fermi level are depleted, = induced pairing competes with the Kondo physics.
7 Physical dilemma to screen or not to screen? Quantum impurity (dot) coupled to a superconducting reservoir
8 Quantum impurity embedded in a superconducting host
9 Electronic spectrum
10 Electronic spectrum S ε d +U ε d µ Γ S
11 Electronic spectrum S ε d +U ε d µ Γ S Γ S (ω) = 2π k V k 2 δ(ω ε k ) hybridization coupling
12 Microscopic model Anderson-type Hamiltonian Quantum impurity (dot)
13 Microscopic model Anderson-type Hamiltonian Quantum impurity (dot) Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d
14 Microscopic model Anderson-type Hamiltonian Quantum impurity (dot) Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d coupled with a superconductor
15 Microscopic model Anderson-type Hamiltonian Quantum impurity (dot) Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d coupled with a superconductor Ĥ = σ + k,σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d + Ĥ S ( V k ˆd σĉkσ + V k ĉ ˆd ) kσ σ
16 Microscopic model Anderson-type Hamiltonian Quantum impurity (dot) Ĥ QD = σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d coupled with a superconductor Ĥ = σ + k,σ ǫ d ˆd σ ˆd σ + U ˆn d ˆn d + Ĥ S ( V k ˆd σĉkσ + V k ĉ ˆd ) kσ σ where ( ĉ k ĉ k +h.c. ) Ĥ S = k,σ (ε k µ) ĉ kσĉkσ k
17 Uncorrelated QD exactly solvable U d = case
18 Uncorrelated QD exactly solvable U d = case ρ d (ω) / Γ S ω / Γ S 2
19 Uncorrelated QD exactly solvable U d = case ρ d (ω) / Γ S ω / Γ S 2
20 Uncorrelated QD exactly solvable U d = case ρ d (ω) / Γ S ω / Γ S 2
21 Uncorrelated QD exactly solvable U d = case ρ d (ω) / Γ S ω / Γ S 2
22 Uncorrelated QD exactly solvable U d = case ρ d (ω) / Γ S ω / Γ S 2
23 Uncorrelated QD exactly solvable U d = case ρ d (ω) / Γ S ω / Γ S 2
24 Uncorrelated QD exactly solvable U d = case ρ d (ω) / Γ S ω / Γ S 2
25 Uncorrelated QD exactly solvable U d = case ε d = ρ d (ω) / Γ S ω / Γ S 2 Appearance of the in-gap resonances (Andreev bound states) J. Barański and T. Domański, J. Phys.: Condens. Matter 25, (213).
26 Uncorrelated QD exactly solvable U d = case ρ d (ω) ε d = - Γ S / Γ S ω / Γ S 2 Appearance of the in-gap resonances (Andreev bound states) J. Barański and T. Domański, J. Phys.: Condens. Matter 25, (213).
27 Uncorrelated QD exactly solvable U d = case ω / Γ S.5 + Andreev bound states / Γ S Energies of the in-gap resonances (Andreev bound states) J. Barański and T. Domański, J. Phys.: Condens. Matter 25, (213).
28 Correlated QD singlet/doublet configurations Deep in a subgap regime ω QD can described by
29 Correlated QD singlet/doublet configurations Deep in a subgap regime ω QD can described by Ĥ QD = σ ǫ d ˆd σ ˆd σ + U d ˆn d ˆn d ( d ˆd ˆd + h.c. ) where the induced on-dot pairing gap d = Γ S /2.
30 Correlated QD singlet/doublet configurations Deep in a subgap regime ω QD can described by Ĥ QD = σ ǫ d ˆd σ ˆd σ + U d ˆn d ˆn d ( d ˆd ˆd + h.c. ) where the induced on-dot pairing gap d = Γ S /2. Eigen-states of this problem:
31 Correlated QD singlet/doublet configurations Deep in a subgap regime ω QD can described by Ĥ QD = σ ǫ d ˆd σ ˆd σ + U d ˆn d ˆn d ( d ˆd ˆd + h.c. ) where the induced on-dot pairing gap d = Γ S /2. Eigen-states of this problem: and doublet (spin 1 2 ) u v singlets (spin ) v + u
32 Correlated QD singlet/doublet configurations Deep in a subgap regime ω QD can described by Ĥ QD = σ ǫ d ˆd σ ˆd σ + U d ˆn d ˆn d ( d ˆd ˆd + h.c. ) where the induced on-dot pairing gap d = Γ S /2. Eigen-states of this problem: and doublet (spin 1 2 ) u v singlets (spin ) v + u The doublet-singlet quantum phase transition occur by varrying ε d, U d or Γ S.
33 Tunneling heterojunction
34 Physical situation N-QD-S scheme To probe the subgap states one can study the electron transport through a quantum dot (QD) coupled between the normal (N) and superconducting (S) electrodes
35 Physical situation N-QD-S scheme To probe the subgap states one can study the electron transport through a quantum dot (QD) coupled between the normal (N) and superconducting (S) electrodes V = µ N - µ S N Γ N QD Γ S V G S metallic lead QD superconductor
36 Physical situation N-QD-S scheme To probe the subgap states one can study the electron transport through a quantum dot (QD) coupled between the normal (N) and superconducting (S) electrodes 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.
37 Electronic spectrum
38 Electronic spectrum Components of N-QD-S heterostructure have the following spectra
39 Electronic spectrum Components of N-QD-S heterostructure have the following spectra N QD ε d +U S µ N ε d µ S Γ N Γ S
40 Electronic spectrum Components of 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.
41 Theoretical background: list of the applied techniques
42 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998)
43 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999)
44 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2)
45 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21)
46 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24)
47 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24) NRG Y. Tanaka, N. Kawakami, and A. Oguri, (27)
48 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24) NRG Y. Tanaka, N. Kawakami, and A. Oguri, (27) EOM revised T. Domański et al, (27)
49 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24) NRG Y. Tanaka, N. Kawakami, and A. Oguri, (27) EOM revised T. Domański et al, (27) NRG J. Bauer, A. Oguri, and A.C. Hewson, (27)
50 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24) NRG Y. Tanaka, N. Kawakami, and A. Oguri, (27) EOM revised T. Domański et al, (27) NRG J. Bauer, A. Oguri, and A.C. Hewson, (27) f-rg C. Karrasch, A. Oguri, and V. Meden, (28)
51 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24) NRG Y. Tanaka, N. Kawakami, and A. Oguri, (27) EOM revised T. Domański et al, (27) NRG J. Bauer, A. Oguri, and A.C. Hewson, (27) f-rg C. Karrasch, A. Oguri, and V. Meden, (28) QMC A. Koga (213)
52 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24) NRG Y. Tanaka, N. Kawakami, and A. Oguri, (27) EOM revised T. Domański et al, (27) NRG J. Bauer, A. Oguri, and A.C. Hewson, (27) f-rg C. Karrasch, A. Oguri, and V. Meden, (28) QMC A. Koga (213) CUT M. Zapalska, T. Domański (214)
53 Theoretical background: list of the applied techniques EOM R. Fazio and R. Raimondi (1998) slave bosons P. Schwab and R. Raimondi (1999) NCA A.A. Clerk, V. Ambegaokar, and S. Hershfield (2) IPT J.C. Cuevas, A. Levy Yeyati, and A. Martin-Rodero (21) constrained sb M. Krawiec and K.I. Wysokiński (24) NRG Y. Tanaka, N. Kawakami, and A. Oguri, (27) EOM revised T. Domański et al, (27) NRG J. Bauer, A. Oguri, and A.C. Hewson, (27) f-rg C. Karrasch, A. Oguri, and V. Meden, (28) QMC A. Koga (213) CUT M. Zapalska, T. Domański (214) other...
54 Relevant problems : issue # 1
55 Relevant problems : issue # 1 Coupling of QD with the metallic electrode:
56 Relevant problems : issue # 1 Coupling of QD with the metallic electrode: broadens the QD levels (by Γ N ) can induce the Kondo resonance (below T K ).
57 Relevant problems : issue # 2
58 Relevant problems : issue # 2 Both hybridizations Γ N and Γ S effectively lead to 75 Kondo peak ρ d (ω) [ 1/Γ S ] 5 25 Γ S ΓN ε d ε d + U ω / Γ S
59 Relevant problems : issue # 2 Both hybridizations Γ N and Γ S effectively lead to 75 Kondo peak ρ d (ω) [ 1/Γ S ] 5 25 Γ S ΓN ε d ε d + U ω / Γ S / interplay between the Kondo effect and superconductivity /
60 Correlated QD influence of ΓS /ΓN
61 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N
62 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N =
63 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 1
64 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 2
65 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 3
66 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 4
67 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 5
68 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 6
69 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Γ S /Γ N = 8
70 Correlated QD influence of Γ S /Γ N Spectral function obtained below T K for U =1Γ N ρ d (ω) [ 1 / Γ N ] T = 1-3 Γ N ε d = -1.5 Γ N ω / Γ N Superconductivity suppresses the Kondo resonance
71 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N
72 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
73 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
74 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 1 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
75 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 2 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
76 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 3 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
77 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 4 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
78 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 5 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
79 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 6 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
80 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 7 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
81 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 8 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28).
82 Correlated QD influence of Γ S /Γ N Andreev conductance G A (V ) for: U = 1Γ N Γ S / Γ N = 8 G A (V) [ 4e 2 /h ] Γ S /Γ N ev / Γ 5 N 1 8 T. Domański and A. Donabidowicz, PRB 78, 7315 (28). Kondo resonance slightly enhances the zero-bias Andreev conductance, especially for Γ S Γ N!
83 Interplay with the Kondo effect
84 Interplay with the Kondo effect R.S. Deacon et al, Phys. Rev. B 81, 12138(R) (21).
85 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, 12138(R) (21).
86 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, 12138(R) (21).
87 Outline of the procedure
88 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation
89 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation Ĥ Ĥ(l 1 ) Ĥ(l 2 )... Ĥ( )
90 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation Ĥ Ĥ(l 1 ) Ĥ(l 2 )... Ĥ( ) gradually eliminating the hybridization term.
91 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation Ĥ Ĥ(l 1 ) Ĥ(l 2 )... Ĥ( ) gradually eliminating the hybridization term. F. Wegner (1994); K.G. Wilson (1994) - inventors of this RG-like scheme
92 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation Ĥ Ĥ(l 1 ) Ĥ(l 2 )... Ĥ( ) gradually eliminating the hybridization term. F. Wegner (1994); K.G. Wilson (1994) - inventors of this RG-like scheme Hamiltonian at l = Ĥ QD + Ĥ N + ˆV QD N
93 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation Ĥ Ĥ(l 1 ) Ĥ(l 2 )... Ĥ( ) gradually eliminating the hybridization term. F. Wegner (1994); K.G. Wilson (1994) - inventors of this RG-like scheme Hamiltonian at < l < Ĥ QD (l)+ĥ N (l)+ ˆV QD N (l)
94 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation Ĥ Ĥ(l 1 ) Ĥ(l 2 )... Ĥ( ) gradually eliminating the hybridization term. F. Wegner (1994); K.G. Wilson (1994) - inventors of this RG-like scheme Hamiltonian at l = Ĥ QD ( ) + Ĥ N ( ) +
95 Outline of the procedure For studying an interplay between the induced pairing and electron correlations we use the continuous unitary transformation Ĥ Ĥ(l 1 ) Ĥ(l 2 )... Ĥ( ) gradually eliminating the hybridization term. F. Wegner (1994); K.G. Wilson (1994) - inventors of this RG-like scheme Hamiltonian at l = Ĥ QD ( ) + Ĥ N ( ) + M. Zapalska and T. Domański, (214) submitted.
96 Generalized Schrieffer-Wolff transformation
97 Generalized Schrieffer-Wolff transformation Transformation of the Hamiltonian is characterized by the set of flow equations
98 Generalized Schrieffer-Wolff transformation Transformation of the Hamiltonian is characterized by the set of flow equations ε d (l) l U d (l) l d (l) l V k (l) l J kp (l) l = 2 = 4 = 2 k k k η k (l)v kn (l), (1) η (2) k (l)v kn(l), (2) η (1) k (l)v kn(l), (3) [ ] = η k (l) ε d (l) ξ kn + U d (l) ˆn d,σ + η (1) k (l) d(l) + η (2) k (l) [ε d(l) ξ kn + U d (l)] ˆn d,σ p η kp (l)v pn (l) (4) = η (2) k (l)v pn(l) + η (2) p (l)v kn (l) (ξ kn ξ pn ) 2 J kp (l). (5)
99 Generalized Schrieffer-Wolff transformation Transformation of the Hamiltonian is characterized by the set of flow equations ε d (l) l U d (l) l d (l) l V k (l) l J kp (l) l = 2 = 4 = 2 k k k η k (l)v kn (l), (1) η (2) k (l)v kn(l), (2) η (1) k (l)v kn(l), (3) [ ] = η k (l) ε d (l) ξ kn + U d (l) ˆn d,σ + η (1) k (l) d(l) + η (2) k (l) [ε d(l) ξ kn + U d (l)] ˆn d,σ p η kp (l)v pn (l) (4) = η (2) k (l)v pn(l) + η (2) p (l)v kn (l) (ξ kn ξ pn ) 2 J kp (l). (5) with the initial condition J kp () = for the induced spin-exchange interaction H exch (l) = k,p J kp(l)ŝ d Ŝ kp.
100 Kondo temperature vs induced pairing
101 Kondo temperature vs induced pairing spin exchange coupling J kf p F / D d / D The effective exchange coupling J kf p F (l ) = 4U d V kf 2 U 2 d +(2 d) 2.
102 Kondo temperature vs induced pairing 1 T K / T K ( d =) Kondo temperature d / D Kondo temp. from the Bethe ansatz T K = 2D πk B exp { φ[2ρ(ε F )J kf p F ( )]}.
103 Kondo temperature vs induced pairing E - E+ ρ d (ω) d / D ω / D.2.3 The low-energy part of the spectral function obtained from the EOM technique.
104 Summary
105 Summary QD coupled between N and S electrodes:
106 Summary QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect /
107 Summary QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is sensitive to correlations / Kondo & charging effects /
108 Summary QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is sensitive to correlations / Kondo & charging effects / Interplay between the proximity and correlation effects shows up in a subgap Andreev transport by:
109 Summary QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is sensitive to correlations / Kondo & charging effects / Interplay between the proximity and correlation effects shows up in a subgap Andreev transport by: particle-hole splitting / when ε d µ S /
110 Summary QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is sensitive to correlations / Kondo & charging effects / Interplay between the proximity and correlation effects shows up in a subgap Andreev transport by: particle-hole splitting / when ε d µ S / zero-bias enhancement / below T K /
111 Summary QD coupled between N and S electrodes: absorbs the superconducting order / proximity effect / is sensitive to correlations / Kondo & charging effects / Interplay between the proximity and correlation effects shows up in a subgap Andreev transport by: particle-hole splitting / when ε d µ S / zero-bias enhancement / below T K /
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