Quantum Processes in Josephson Junctions & Weak Links. J. A. Sauls

Transcription

1 CMS Colloquium, Los Alamos National Laboratory, December 9, 2015 Quantum Processes in Josephson Junctions & Weak Links J. A. Sauls Northwestern University e +iφ 2 e +iφ a z 2ξ Research supported by NSF grant DMR Erhai Zhao, George Mason University Tomas Löfwander, Chalmers University 1 / 26

2 Preface on Dirac Materials Dirac materials Materials whose low energy electronic properties are a direct consequence of Dirac spectrum E = vk How do we design Dirac Materials? Can be a collective state: 3He superfluid, heavy fermion, organic, high Tc superconductors Band structure effect graphene, Topological states T. Wehling, A Black-Schaffer and A. V. Balatsky, Dirac Materials, Adv Phys / 26

3 Dirac Fermions & Zero Energy Bound States Dirac Fermion coupled to a Scalar Bose Field i h t ψ = ( i hc α + βgφ) ψ α = ( 0 ) σ σ 0 ( ) 1 0 β = 0 1 ψ = col(ψ 1,ψ 2,ψ 3,ψ 4 ) 3 / 26

4 Dirac Fermions & Zero Energy Bound States Dirac Fermion coupled to a Scalar Bose Field i h t ψ = ( i hc α + βgφ) ψ ( 0 ) σ σ 0 ( ) 1 0 β = 0 1 α = ψ = col(ψ 1,ψ 2,ψ 3,ψ 4 ) Broken Symmetry State: Φ = Φ 0 Mass: Mc 2 = gφ 0 E ± = ± c 2 p 2 + (Mc 2 ) 2 3 / 26

5 Dirac Fermions & Zero Energy Bound States Dirac Fermion coupled to a Scalar Bose Field i h t ψ = ( i hc α + βgφ) ψ ( 0 ) σ σ 0 ( ) 1 0 β = 0 1 α = ψ = col(ψ 1,ψ 2,ψ 3,ψ 4 ) Broken Symmetry State: Φ = Φ 0 Mass: Mc 2 = gφ 0 E ± = ± c 2 p 2 + (Mc 2 ) 2 Degenerate Vacuum States: Φ(x ± ) = Φ 0 : Zero Mode Fermion with E = 0 confined on the the Domain Wall : Topologically Protected Zero Mode R. Jackiw and C. Rebbi, Phys. Rev. D / 26

6 Nambu-Dirac Fermions in Superconductors Bogoliubov-Nambu Equations - particle-hole coherence : ( ) (u ) ( ) (u ) ( ) h2 2m 2 µ 0 0 u + h 0 2 2m 2 + µ v = ε 0 v v 4 / 26

7 Nambu-Dirac Fermions in Superconductors Bogoliubov-Nambu Equations - particle-hole coherence : ( ) (u ) ( ) (u ) ( ) h2 2m 2 µ 0 0 u + h 0 2 2m 2 + µ v = ε 0 v v Separation of scales: h/p f hv f / λ: u = U p e ip r/ h p y h/ p f v p p x λ x Nambu-Dirac Spinors coupled to the (Bosonic) Cooper-Pair Field ( ) ( )( ) ( ) U 0 (p,r) U U hv p r + V = ε (p,r) 0 V V 4 / 26

8 Nambu-Dirac Fermions in Superconductors Bogoliubov-Nambu Equations - particle-hole coherence : ( ) (u ) ( ) (u ) ( ) h2 2m 2 µ 0 0 u + h 0 2 2m 2 + µ v = ε 0 v v Separation of scales: h/p f hv f / λ: u = U p e ip r/ h p y h/ p f v p p x λ x Nambu-Dirac Spinors coupled to the (Bosonic) Cooper-Pair Field ( ) ( )( ) ( ) U 0 (p,r) U U hv p r + V = ε (p,r) 0 V V Zero Modes if (x = ) = (x = + ) along x = ˆv p r 4 / 26

9 Electron-Hole Coherence & Zero-Energy Interface Bound States Andreev s Equation for Coherent Electron-Hole ) States + (p) = (ˆp 2 x ˆp 2 y) hv p r ( U V ( 0 (p,r) (p,r) 0 )( U V ) = ε ( ) U V p y p + p + p x [110] reflection: 5 / 26

10 Electron-Hole Coherence & Zero-Energy Interface Bound States Andreev s Equation for Coherent Electron-Hole ) States + (p) = (ˆp 2 x ˆp 2 y) hv p r ( U V ( 0 (p,r) (p,r) 0 )( U V ) = ε ( ) U V p y p + p + p x [110] reflection: Electron & Hole Bound State: ψ ( ) 1 (p) i N(p,x=0 ; ε) e 2 (p) x / hv f ε/2πτ c Tunneling into Surface States of HTC Superconductors, PRL 79: (1997), M. Fogelström, D. Rainer, & J. A. Sauls 5 / 26

11 Josephson Tunneling in Superconductors B. Josephson, Phys. Lett. 1, 251 (1962). V. Ambegaokar & A. Baratoff, PRL (1963). H = H 1 + H 2 + H th 6 / 26

12 Josephson Tunneling in Superconductors B. Josephson, Phys. Lett. 1, 251 (1962). V. Ambegaokar & A. Baratoff, PRL (1963). H 1 = kσ H = H 1 + H 2 + H th ( ) ξ kσ c kσ c kσ k c kσ c k σ + k c k σ c kσ kσ H 2 = pσ ξ pσ a pσ a pσ pσ ( p a pσ a p σ + pa p σ a pσ ( ) H th = t p,k a pσ c kσ +tp,k c kσ a pσ p,k,σ ) 6 / 26

13 Josephson Tunneling in Superconductors B. Josephson, Phys. Lett. 1, 251 (1962). V. Ambegaokar & A. Baratoff, PRL (1963). H 1 = kσ H = H 1 + H 2 + H th ( ) ξ kσ c kσ c kσ k c kσ c k σ + k c k σ c kσ kσ H 2 = pσ ξ pσ a pσ a pσ pσ ( p a pσ a p σ + pa p σ a pσ ( ) H th = t p,k a pσ c kσ +tp,k c kσ a pσ p,k,σ I = e Ṅ 2 (t) = 2eIm t p,k a pσ (t)c kσ (t) p,k,σ ) 6 / 26

14 Josephson Tunneling in Superconductors B. Josephson, Phys. Lett. 1, 251 (1962). V. Ambegaokar & A. Baratoff, PRL (1963). H 1 = kσ H = H 1 + H 2 + H th ( ) ξ kσ c kσ c kσ k c kσ c k σ + k c k σ c kσ kσ H 2 = pσ ξ pσ a pσ a pσ pσ ( p a pσ a p σ + pa p σ a pσ ( ) H th = t p,k a pσ c kσ +tp,k c kσ a pσ p,k,σ I = e Ṅ 2 (t) = 2eIm t p,k a pσ (t)c kσ (t) p,k,σ ) I = I c (T ) sin( φ) 6 / 26

15 Josephson Tunneling in Superconductors B. Josephson, Phys. Lett. 1, 251 (1962). V. Ambegaokar & A. Baratoff, PRL (1963). H 1 = kσ H = H 1 + H 2 + H th ( ) ξ kσ c kσ c kσ k c kσ c k σ + k c k σ c kσ kσ H 2 = pσ ξ pσ a pσ a pσ pσ ( p a pσ a p σ + pa p σ a pσ ( ) H th = t p,k a pσ c kσ +tp,k c kσ a pσ p,k,σ I = e Ṅ 2 (t) = 2eIm t p,k a pσ (t)c kσ (t) p,k,σ ) (π 2 N(0) 2 t 2 FS I = I c (T ) sin( φ) I c (T ) = 2e ) } {{ } D th 1 Transmission Amplitude ( ) tanh 2T ) 6 / 26

16 a.c. Josephson Effects Supercurrent: I s = I c (T ) sin(φ t ) 7 / 26

17 a.c. Josephson Effects Supercurrent: I s = I c (T ) sin(φ t ) a.c. Josephson Equation: φ t = 2e h Vt 7 / 26

18 a.c. Josephson Effects Supercurrent: I s = I c (T ) sin(φ t ) a.c. Josephson Equation: φ t = 2e h Vt Dissipative Current: I Ohmic = ( σ 0 + σ 1 cos(φ t ) ) V Phase-sensitive dissipation B. Josephson, Adv. Phys. (1965). 7 / 26

19 a.c. Josephson Effects Supercurrent: I s = I c (T ) sin(φ t ) a.c. Josephson Equation: φ t = 2e h Vt Dissipative Current: I Ohmic = ( σ 0 + σ 1 cos(φ t ) ) V Phase-sensitive dissipation B. Josephson, Adv. Phys. (1965). What is the origin of phase-dependent dissipation? 7 / 26

20 Heat Transport through a Phase-Biased Josephson Junction Linear Response to a Thermal Bias Maki & Griffin, PRL (1965); Guttman et al. PRB 57, 2717 (1998) 8 / 26

21 Heat Transport through a Phase-Biased Josephson Junction Linear Response to a Thermal Bias Maki & Griffin, PRL (1965); Guttman et al. PRB 57, 2717 (1998) Heat Current: Tunneling Hamiltonian I Q = i p,k,σ {t p,k (ξ pσ a pσ c kσ p a pσ c k σ ) h.c. } 8 / 26

22 Heat Transport through a Phase-Biased Josephson Junction Linear Response to a Thermal Bias Maki & Griffin, PRL (1965); Guttman et al. PRB 57, 2717 (1998) Heat Current: Tunneling Hamiltonian I Q = i p,k,σ {t p,k (ξ pσ a pσ c kσ p a pσ c k σ ) h.c. ( I Q = δt 4πN(0) t 2 FS dε f ) }{{} T D th }{{} thermal excitations } 8 / 26

23 Heat Transport through a Phase-Biased Josephson Junction Linear Response to a Thermal Bias Maki & Griffin, PRL (1965); Guttman et al. PRB 57, 2717 (1998) Heat Current: Tunneling Hamiltonian I Q = i p,k,σ {t p,k (ξ pσ a pσ c kσ p a pσ c k σ ) h.c. ( I Q = δt 4πN(0) t 2 FS dε f ) ε2 2 cos(φ) }{{} T ε D th }{{} 2 2 }{{} thermal excitations } 8 / 26

24 Heat Transport through a Phase-Biased Josephson Junction Linear Response to a Thermal Bias Maki & Griffin, PRL (1965); Guttman et al. PRB 57, 2717 (1998) Heat Current: Tunneling Hamiltonian I Q = i p,k,σ I Q = δt 4πN(0) t 2 FS }{{} {t p,k (ξ pσ a pσ c kσ p a pσ c k σ ) h.c. D th ( dε f ) ε2 2 cos(φ) T ε }{{} 2 2 }{{} thermal excitations Trouble!! Failure of Linear Response Theory? } 8 / 26

25 Heat Transport through a Phase-Biased Josephson Junction Linear Response to a Thermal Bias Maki & Griffin, PRL (1965); Guttman et al. PRB 57, 2717 (1998) Heat Current: Tunneling Hamiltonian I Q = i p,k,σ I Q = δt 4πN(0) t 2 FS }{{} {t p,k (ξ pσ a pσ c kσ p a pσ c k σ ) h.c. D th ( dε f ) ε2 2 cos(φ) T ε }{{} 2 2 }{{} thermal excitations Trouble!! Failure of Linear Response Theory? } 8 / 26

26 Non-Perturbative Theory of Transport in Phase-Biased Josephson Junctions Phase Bias: φ = φ 2 φ 1 Barrier Transmission: 0 < D 1 Thermal Bias: δt = T 2 T 1 Mesoscopic Junction: h/p f a < ξ e +iφ 2 e +iφ 1 2a z 9 / 26

27 Non-Perturbative Theory of Transport in Phase-Biased Josephson Junctions Phase Bias: φ = φ 2 φ 1 Barrier Transmission: 0 < D 1 Thermal Bias: δt = T 2 T 1 Mesoscopic Junction: h/p f a < ξ e +i φ 2 e +i φ a z 2ξ Josephson Phase New Electronic States Confined to the Interface! 9 / 26

28 Non-Perturbative Theory of Transport in Phase-Biased Josephson Junctions Phase Bias: φ = φ 2 φ 1 Barrier Transmission: 0 < D 1 Thermal Bias: δt = T 2 T 1 Mesoscopic Junction: h/p f a < ξ e +i φ 2 e +i φ a z 2ξ Josephson Phase New Electronic States Confined to the Interface! Energy & Phase-dependent Transmission: D D(ε,φ) 9 / 26

29 Fermion Bound States of a Josephson Weak Link or 2π Vortex e i φ/2 e +iφ/2 Sharvin Contact ε ± (φ) = ± cos(φ/2) π 0 e +i ϕ p f φ ε + / Gap Edge 0.00 Fermi Level ε / φ/π 10 / 26

30 Andreev Riccati Equations: Electron-Hole Branch Conversion Andreev s Equation for Coherent Electron-Hole ) States + hv p r ( U V ( 0 (p,r) (p,r) 0 )( U V ) = ε ( ) U V 11 / 26

31 Andreev Riccati Equations: Electron-Hole Branch Conversion Andreev s Equation for Coherent ( Electron-Hole ) ( States U 0 (p,r) hv p r + V (p,r) 0 Electron-Hole Coherence Amplitudes: Riccati Equation: )( U V γ(p,r;ε) = U/V γ(p,r;ε) = V /U hv p γ + 2εγ + + γ 2 = 0 ) = ε ( ) U V 11 / 26

32 Andreev Riccati Equations: Electron-Hole Branch Conversion Andreev s Equation for Coherent ( Electron-Hole ) ( States U 0 (p,r) hv p r + V (p,r) 0 Electron-Hole Coherence Amplitudes: Riccati Equation: Nonequilibrium: γ(p, r; ε, t) )( U V γ(p,r;ε) = U/V γ(p,r;ε) = V /U hv p γ + 2εγ + + γ 2 = 0 +e, v f (R) e, v f e, +v f +e, +v f ) = ε Amplitude: γ(p, r; ε, t) Amplitude: γ(p, r; ε, t) ( ) U V (R) h-e and e-h branch conversion scattering 11 / 26

33 Multiple-scattering of Coherent e-h excitations at a Boundary Multiple Scattering from a Potential + Branch conversion scattering 12 / 26

34 Multiple-scattering of Coherent e-h excitations at a Boundary Multiple Scattering from a Potential + Branch conversion scattering Interface Potential Scattering: S-matrix ( ) S11 S 12 S 21 S / 26

35 Multiple-scattering of Coherent e-h excitations at a Boundary Multiple Scattering from a Potential + Branch conversion scattering Interface Potential Scattering: S-matrix ( ) S11 S 12 S 21 S 22 Andreev Scattering: Riccati +e, v f e, +v f e, v f +e, +v f γ(p,r;ε,t) (R) (R) γ(p,r;ε,t) 12 / 26

36 Multiple-scattering of Coherent e-h excitations at a Boundary Multiple Scattering from a Potential + Branch conversion scattering γ R 2 Interface Potential Scattering: S-matrix r a 1 S 11 S 21 S 22 S ( ) S11 S 12 γ R 2 S 21 S 22 Andreev Scattering: Riccati t a 1 S 12 S 22 S 22 S 12 γ R e, v f (R) γ R 2 e, +v f e, v f (R) Γ R 1 γ R 1 γ R 2 +e, +v f γ(p,r;ε,t) γ(p,r;ε,t) e h r a 1 S 11 t a 1 S / 26

37 Multiple-scattering of Coherent e-h excitations at a Boundary Multiple Scattering from a Potential + Branch conversion scattering γ R 2 Interface Potential Scattering: S-matrix r a 1 S 11 S 21 S 22 S ( ) S11 S 12 S 21 S 22 Andreev Scattering: Riccati +e, v f e, +v f e, v f +e, +v f γ(p,r;ε,t) (R) (R) γ(p,r;ε,t) Bound-States Poles of the re-normalized S-matrix amplitudes e h Γ R 1 t a 1 S 12 S 22 S 22 S 12 γ R 1 γ R 2 r a 1 S 11 t a 1 S 12 γ R 2 γ R 2 γ R 2... Nonequilibrium Superconductivity Near Spin-Active Interfaces, PRB 70: (2004), E. Zhao, T. Löfwander & J. A. Sauls 12 / 26

38 Fermion Bound States of a Josephson Junction R e iφ/2 D e+i φ/2 ( ) R i D S = i D R 13 / 26

39 Fermion Bound States of a Josephson Junction R e iφ/2 D e+i φ/2 ( ) R i D S = i D R ε ± ( R,φ) = ± cos 2 (φ/2) + R sin 2 (φ/2) 0 < D 1 Potential + Andreev Scattering Gap in the Bound-State Dispersion ε + / Gap Edge ε / Fermi Level D= φ/π 13 / 26

40 Heat Transport through a Phase-Biased Josephson Junction Heat Current j Q = κ δt Carriers = bulk quasiparticles 14 / 26

41 Heat Transport through a Phase-Biased Josephson Junction Heat Current j Q = κ δt Carriers = bulk quasiparticles ε N bulk (ε) = N(0) ε / 26

42 Heat Transport through a Phase-Biased Josephson Junction Heat Current j Q = κ δt Carriers = bulk quasiparticles ε N bulk (ε) = N(0) ε 2 2 ε v g (ε) = v 2 2 f ε 14 / 26

43 Heat Transport through a Phase-Biased Josephson Junction Heat Current j Q = κ δt Carriers = bulk quasiparticles ε N bulk (ε) = N(0) ε 2 2 v g (ε) = v f ε 2 2 Thermal Conductance ( κ(φ,t ) = A dε N bulk (ε) [ε v g (ε)] D(ε,φ) f ) T D(ε, φ) = Quasiparticle Transmission Probability ε 14 / 26

44 Transmission Probability for a Phase-Biased Josephson Junction D(ε,φ) = D e e (ε,φ) = D e h (ε,φ) 15 / 26

45 Transmission Probability for a Phase-Biased Josephson Junction D(ε,φ) = D e e (ε,φ) = D e h (ε,φ) Direct Transmission: Excitations: ε D e e (ε,φ) = D (ε2 2 ) (ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 15 / 26

46 Transmission Probability for a Phase-Biased Josephson Junction D(ε,φ) = D e e (ε,φ) = D e h (ε,φ) Direct Transmission: Excitations: ε D e e (ε,φ) = D (ε2 2 ) (ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Branch Conversion Transmission: D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 15 / 26

47 Transmission Probability for a Phase-Biased Josephson Junction Thermal Conductance Limits: D(ε,φ) = D e e (ε,φ) = D e h (ε,φ) Direct Transmission: Excitations: ε D e e (ε,φ) = D (ε2 2 ) (ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Branch Conversion Transmission: φ = 0 D(ε,φ = 0) = D BCS Thermal Conductivity D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 15 / 26

48 Transmission Probability for a Phase-Biased Josephson Junction Thermal Conductance Limits: D(ε,φ) = D e e (ε,φ) = D e h (ε,φ) Direct Transmission: Excitations: ε D e e (ε,φ) = D (ε2 2 ) (ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Branch Conversion Transmission: φ = 0 D(ε,φ = 0) = D BCS Thermal Conductivity D = 1 D(ε,φ) = ε 2 2 ε 2 2 cos 2 (φ/2) D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Sharvin Limit 15 / 26

49 Transmission Probability for a Phase-Biased Josephson Junction Thermal Conductance Limits: D(ε,φ) = D e e (ε,φ) = D e h (ε,φ) Direct Transmission: Excitations: ε D e e (ε,φ) = D (ε2 2 ) (ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Branch Conversion Transmission: φ = 0 D(ε,φ = 0) = D BCS Thermal Conductivity D = 1 D(ε,φ) = ε 2 2 ε 2 2 cos 2 (φ/2) D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Sharvin Limit D 1 D(ε,φ) = D (ε2 2 cos 2 (φ/2)) ε 2 2 Tunneling Limit 15 / 26

50 Transmission Probability for a Phase-Biased Josephson Junction Thermal Conductance Limits: D(ε,φ) = D e e (ε,φ) = D e h (ε,φ) Direct Transmission: Excitations: ε D e e (ε,φ) = D (ε2 2 ) (ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Branch Conversion Transmission: φ = 0 D(ε,φ = 0) = D BCS Thermal Conductivity D = 1 D(ε,φ) = ε 2 2 ε 2 2 cos 2 (φ/2) D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Sharvin Limit D 1 D(ε,φ) = D (ε2 2 cos 2 (φ/2)) Tunneling Limit ε 2 2 Essential Singularity 15 / 26

51 Heat Transport through a Phase-Biased Josephson Junction Linear Response to a Thermal Bias Maki & Griffin, PRL (1965); Guttman et al. PRB 57, 2717 (1998) Heat Current: Tunneling Hamiltonian I Q = i p,k,σ I Q = δt 4πN(0) t 2 FS }{{} {t p,k (ξ pσ a pσ c kσ p a pσ c k σ ) h.c. D th ( dε f ) ε2 2 cos(φ) T ε }{{} 2 2 }{{} thermal excitations Trouble!! Failure of Linear Response Theory? } 16 / 26

52 Andreev s Demon & Resonant Transmission Direct Transmission D e e (ε,φ) = D (ε2 2 )(ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 Branch Conversion D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] 2 17 / 26

53 Andreev s Demon & Resonant Transmission Direct Transmission D e e (ε,φ) = D (ε2 2 )(ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] Branch Conversion D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] / 26

54 Andreev s Demon & Resonant Transmission Direct Transmission D e e (ε,φ) = D (ε2 2 )(ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] Branch Conversion D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] / 26

55 Andreev s Demon & Resonant Transmission Direct Transmission D e e (ε,φ) = D (ε2 2 )(ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] Branch Conversion D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] / 26

56 Andreev s Demon & Resonant Transmission Direct Transmission D e e (ε,φ) = D (ε2 2 )(ε 2 2 cos 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] Branch Conversion D e h (ε,φ) = DR (ε2 2 ) 2 sin 2 (φ/2) [ε 2 2 (1 Dsin 2 (φ/2)] Resonance Bound State / 26

57 Transmission Resonance for Heat Transport R e iφ/2 D e+i φ/2 I ε (φ,t ) = κ(φ,t ) δt, with δt = T 2 T 1 κ(φ,t ) = 4A dε N (ε) [ εv g (ε) ] D(ε,φ) f T resonant transmission shallow bound state / 26

58 Non-analyticity of the Thermal Conductance Tunneling Hamiltonian: κ th = κ th BCS + κ th 2 sin 2 (φ/2)... But κ th 2 Self-Consistent S-matrix for D 1: κ 1,2 D 0 κ = κ BCS κ 1 sin 2 (φ/2)ln ( sin 2 (φ/2) ) + κ 2 sin 2 (φ/2) DlnD Finite, but Non-Analytic and Non-perturbative κ 1 / κ D=0.01 D=0.005 κ(φ) κ 0 [ N 0 v f T c S] T=0.5 T c D=0.01 κ(φ) κ BCS κ 1 term κ 2 term φ /π T/T c Andreev Bound-State Formation is non-perturbativce 19 / 26

59 Phase-Tuneable Resonant Enhancement of the Heat Current δε res 5 κ(φ)/κ(0) D(ε,π) D ε/ D=0.10 D=0.25 D=0.50 D=0.75 D= φ / π 20 / 26

60 Phase-Tuneable Resonant Enhancement of the Heat Current δε res 5 κ(φ)/κ(0) D(ε,π) D ε/ D=0.10 D=0.25 D=0.50 D=0.75 D=0.90 κ(φ,τ) / κ N φ = π φ = 0 D = φ / π T/T c Andreev s Demon Fermion Bound States control thermal transport φ = 0: κ for T < T c. φ = π: D < 0.5 κ(t ) below T c. D 0.5: κ(φ) < κ(0) 20 / 26

61 Phase-Tuneable Resonant Enhancement of the Heat Current δε res 5 κ(φ)/κ(0) D(ε,π) D ε/ D=0.10 D=0.25 D=0.50 D=0.75 D=0.90 κ(φ,τ) / κ N φ = π φ = 0 D = φ / π T/T c Andreev s Demon Fermion Bound States control thermal transport Z. JIANG AND V. CHANDRASEKHAR φ = 0: κ for T < T c. φ = π: D < 0.5 κ(t ) below T c. D 0.5: κ(φ) < κ(0) It can B, w quan loop. symm metri with the d ducti In Andr Northwestern LT Group: Z. Jiang FIG. 4. et Theal., solidprb symbols 72, are the thermal (2005) resistance of a house that interferometer at different magnetic fields, measured at T=40 mk. enha The solid line is a guide to the eye. The dashed line is the resistance 20 / 26 estim

62 The Josephson heat interferometer - Giazotto et al. Nature 492, 401 (2012) 21 / 26

63 The Josephson heat interferometer - Giazotto et al. Nature 492, 401 (2012) 21 / 26

64 Superconducting-Ferromagnetic-Superconducting Junctions e iϑσz/2 α F a F b e +iφ/2 e iφ/2 S µ a N µ b S α L < hv f /π Magnetic control of Charge Tuneable superconducting transition Spin-triplet pairing correlations π junctions Voltage control of Spin Spin valves - Spin supercurrents SC Spin-Transfer Torque Spin manipulation Nonequilibrium quantum transport theory in S/F heterostructures Spin-Polarized Supercurrents for Spintronics, Physics Today, Jan. 2011, M. Eschrig 22 / 26

65 Ferromagnetic Point Contacts Spin Filtering Spin Mixing Spin-dependent Transmission: D D Spin Faraday Rotation Angle: ϑ 23 / 26

66 Ferromagnetic Point Contacts Spin Filtering Spin Mixing Spin Faraday Rotation Angle: ϑ Spin-dependent Transmission: D D ( R ) e iϑ/2 0 S 11 = S 22 = 0 R e iϑ/2 R / = 1 D / ( D ) e iϑ/2 0 S 12 = S 21 = i 0 D e iϑ/2 cos(ϑ/2) + sin(ϑ/2) First principles theory of magnetically active interfaces S(D,ϑ,...) Nonequilibrium Superconductivity near Spin-Active Interfaces, PRB 70, (2004), Zhao, Löfwander, JAS 23 / 26

67 Multiple Andreev reflection (MAR) Quantum Transport Theory of Spin and Charge in SFS Junctions: spin filtering + spin mixing + MAR E o h 2, e 2e o ev h o 2e e o S 1 o N S 2 F a F b e/h s scatter inelastically: ε ε + mω J (mth order MAR). e/h s can escape into leads for ε > mth order MAR: transports charge = m 2e, spin h/2 24 / 26

68 Long-range spin-transfer torque in SFNFS contacts (L µm) S F a µ a N µ b F b S Spin-Transfer Torques: τ b (t) = τ0 b [ ] + τ k,c b cos(kω Jt) + τ k,s b sin(kω Jt) k=1 τ b 0z N f (v f h)v, ev L < hv f /π MAR + spin-mixing + µ a µ b = sin(ψ)ˆx τ b 0x τ b 0z [N f v f A h/2] θ = π θ = π/3 θ = π/2 θ = 2π/3 θ = π In-Plane d.c. Torque τ b 0x [N f v f A h/2] Out-of-Plane d.c. Torque θ = π θ = π/3 θ = π/2 θ = 2π/3 θ = π D = 0.95, D = 0.6 T = 0.5 T c, ψ = π/ ev/ ev/ 25 / 26

69 Directions and Challenges Nano-scale SFS JJs with CNT and Single Molecular Magnets Circuit QED with Spin-Triplet Superconductors (Sr 2 RuO 4, UPt 3,?) Interacting Classical or Quantum Magnets mediated via Long-Range Josephson Spin-Transfer Torques Arrays of SFNFS JJs for Voltage-Controlled Spin Transport (L µm) 26 / 26