Quantum Processes in Josephson Junctions & Weak Links J. A. - - PowerPoint PPT Presentation

CMS Colloquium, Los Alamos National Laboratory, December 9, 2015

Quantum Processes in Josephson Junctions & Weak Links

• J. A. Sauls

Northwestern University

• +i

e ∆

φ2 +i

e ∆

φ1

z

∆

2ξ

2a

Research supported by NSF grant DMR-1106315.

◮ Erhai Zhao, George Mason University

Tomas L¨

• fwander, Chalmers University

1 / 26

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 superfmuid, heavy

fermion, organic, high T c superconductors

• Band structure efgect – graphene, T
• pological

states

• T. Wehling, A Black-Schafger and A. V. Balatsky,

Dirac Materials, Adv Phys 2014

2 / 26

Dirac Fermions & Zero Energy Bound States

◮ Dirac Fermion coupled to a Scalar Bose Field

i¯ h∂t|ψ = (−i¯ hc α ·∇+βgΦ)|ψ

• α =
• σ
• σ
• β =

1 −1

• |ψ = col(ψ1,ψ2,ψ3,ψ4)

3 / 26

Dirac Fermions & Zero Energy Bound States

◮ Dirac Fermion coupled to a Scalar Bose Field

i¯ h∂t|ψ = (−i¯ hc α ·∇+βgΦ)|ψ

• α =
• σ
• σ
• β =

1 −1

• |ψ = col(ψ1,ψ2,ψ3,ψ4)

◮ Broken Symmetry State: Φ = Φ0 Mass: Mc2 = gΦ0 E± = ±

• c2|p|2 +(Mc2)2

3 / 26

Dirac Fermions & Zero Energy Bound States

◮ Dirac Fermion coupled to a Scalar Bose Field

i¯ h∂t|ψ = (−i¯ hc α ·∇+βgΦ)|ψ

• α =
• σ
• σ
• β =

1 −1

• |ψ = col(ψ1,ψ2,ψ3,ψ4)

◮ Broken Symmetry State: Φ = Φ0 Mass: Mc2 = gΦ0 E± = ±

• c2|p|2 +(Mc2)2

◮ Degenerate Vacuum States: Φ(x → ±∞) = ∓Φ0: ◮

“Zero Mode” Fermion with E = 0 confined on the the Domain Wall : “Topologically Protected” Zero Mode

• R. Jackiw and C. Rebbi, Phys. Rev. D 1976

3 / 26

Nambu-Dirac Fermions in Superconductors

◮ Bogoliubov-Nambu Equations - particle-hole coherence :

• − ¯

h2 2m∇2 − µ ¯ h2 2m∇2 + µ

u v

• +
• ∆

Ơ u v

• = ε

u v

• 4 / 26

Nambu-Dirac Fermions in Superconductors

◮ Bogoliubov-Nambu Equations - particle-hole coherence :

• − ¯

h2 2m∇2 − µ ¯ h2 2m∇2 + µ

u v

• +
• ∆

Ơ u v

• = ε

u v

• ◮ Separation of scales: ¯

h/pf ≪ ¯ hvf /∆ ≤ λ: u = Up eip·r/¯

h

/ h pf

x

λ

y p x p

vp

◮ Nambu-Dirac Spinors coupled to the (Bosonic) Cooper-Pair Field

¯ hvp ·∇r U −V

• +
• ∆(p,r)

Ơ(p,r) U V

• = ε

U V

• 4 / 26

Nambu-Dirac Fermions in Superconductors

◮ Bogoliubov-Nambu Equations - particle-hole coherence :

• − ¯

h2 2m∇2 − µ ¯ h2 2m∇2 + µ

u v

• +
• ∆

Ơ u v

• = ε

u v

• ◮ Separation of scales: ¯

h/pf ≪ ¯ hvf /∆ ≤ λ: u = Up eip·r/¯

h

/ h pf

x

λ

y p x p

vp

◮ Nambu-Dirac Spinors coupled to the (Bosonic) Cooper-Pair Field

¯ hvp ·∇r U −V

• +
• ∆(p,r)

Ơ(p,r) U V

• = ε

U V

• ◮ Zero Modes if ∆(x = −∞) = −∆(x = +∞)

along x = ˆ vp ·r

4 / 26

Electron-Hole Coherence & Zero-Energy Interface Bound States

◮ Andreev’s Equation for Coherent Electron-Hole States

¯ hvp ·∇r U −V

• +
• ∆(p,r)

Ơ(p,r) U V

• = ε

U V

• ◮ ∆(p) = ∆(ˆ

p2

x − ˆ

p2

y)

p

x

p

y

p − + + − − p

[110] reflection:

5 / 26

Electron-Hole Coherence & Zero-Energy Interface Bound States

◮ Andreev’s Equation for Coherent Electron-Hole States

¯ hvp ·∇r U −V

• +
• ∆(p,r)

Ơ(p,r) U V

• = ε

U V

• ◮ ∆(p) = ∆(ˆ

p2

x − ˆ

p2

y)

p

x

p

y

p − + + − − p

[110] reflection:

◮ ◮ Electron & Hole Bound State:

|ψ ∼

• |∆(p)|

1 i

• e−2|∆(p)||x|/¯

hvf

• 1.0
• 0.5

0.0 0.5 1.0

ε/2πΤc

0.0 1.0 2.0 3.0 4.0 5.0

N(p,x=0 ; ε)

◮ Tunneling into Surface States of HTC Superconductors, PRL 79:281–284 (1997), M. Fogelstr¨

• m, D. Rainer, & J. A. Sauls

5 / 26

Josephson Tunneling in Superconductors

◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963).

H = H1 +H2 +HtH

6 / 26

Josephson Tunneling in Superconductors

◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963).

H = H1 +H2 +HtH

H1 = ∑

kσ

ξkσc†

kσckσ + 1 2 ∑ kσ

• ∆kc†

kσc† −k−σ +∆∗ kc† −k−σckσ

• H2 = ∑

pσ

ξpσa†

pσapσ + 1 2 ∑ pσ

• ∆pa†

pσa† −p−σ +∆∗ pa† −p−σapσ

• HtH = ∑

p,k,σ

• tp,k a†

pσckσ +t∗ p,k c† kσapσ

• 6 / 26

Josephson Tunneling in Superconductors

◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963).

H = H1 +H2 +HtH

H1 = ∑

kσ

ξkσc†

kσckσ + 1 2 ∑ kσ

• ∆kc†

kσc† −k−σ +∆∗ kc† −k−σckσ

• H2 = ∑

pσ

ξpσa†

pσapσ + 1 2 ∑ pσ

• ∆pa†

pσa† −p−σ +∆∗ pa† −p−σapσ

• HtH = ∑

p,k,σ

• tp,k a†

pσckσ +t∗ p,k c† kσapσ

• ◮ I = e ˙

N2(t) = 2eIm ∑

p,k,σ

tp,k a†

pσ(t)ckσ(t)

6 / 26

Josephson Tunneling in Superconductors

◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963).

H = H1 +H2 +HtH

H1 = ∑

kσ

ξkσc†

kσckσ + 1 2 ∑ kσ

• ∆kc†

kσc† −k−σ +∆∗ kc† −k−σckσ

• H2 = ∑

pσ

ξpσa†

pσapσ + 1 2 ∑ pσ

• ∆pa†

pσa† −p−σ +∆∗ pa† −p−σapσ

• HtH = ∑

p,k,σ

• tp,k a†

pσckσ +t∗ p,k c† kσapσ

• ◮ I = e ˙

N2(t) = 2eIm ∑

p,k,σ

tp,k a†

pσ(t)ckσ(t)

◮

I = Ic(T) sin(∆φ)

6 / 26

Josephson Tunneling in Superconductors

◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963).

H = H1 +H2 +HtH

H1 = ∑

kσ

ξkσc†

kσckσ + 1 2 ∑ kσ

• ∆kc†

kσc† −k−σ +∆∗ kc† −k−σckσ

• H2 = ∑

pσ

ξpσa†

pσapσ + 1 2 ∑ pσ

• ∆pa†

pσa† −p−σ +∆∗ pa† −p−σapσ

• HtH = ∑

p,k,σ

• tp,k a†

pσckσ +t∗ p,k c† kσapσ

• ◮ I = e ˙

N2(t) = 2eIm ∑

p,k,σ

tp,k a†

pσ(t)ckσ(t)

◮

I = Ic(T) sin(∆φ) Ic(T) = 2e ×

• π2 N(0)2||t|2FS
• ×
• ∝DtH≪1Transmission Amplitude

∆tanh ∆ 2T )

• 6 / 26

a.c. Josephson Effects

◮ Supercurrent: Is = Ic(T) sin(φt)

7 / 26

a.c. Josephson Effects

◮ Supercurrent: Is = Ic(T) sin(φt) ◮ a.c. Josephson Equation: φt = 2e

h Vt

7 / 26

a.c. Josephson Effects

◮ Supercurrent: Is = Ic(T) sin(φt) ◮ a.c. Josephson Equation: φt = 2e

h Vt

◮ Dissipative Current: IOhmic =

• σ0 + σ1 cos(φt)
• V

◮ Phase-sensitive dissipation B. Josephson, Adv. Phys. (1965).

7 / 26

a.c. Josephson Effects

◮ Supercurrent: Is = Ic(T) sin(φt) ◮ a.c. Josephson Equation: φt = 2e

h Vt

◮ Dissipative Current: IOhmic =

• σ0 + σ1 cos(φt)
• V

◮ Phase-sensitive dissipation B. Josephson, Adv. Phys. (1965). ◮ What is the origin of phase-dependent dissipation?

7 / 26

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

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

◮ IQ = −i ∑

p,k,σ

• tp,k
• ξpσ a†

pσckσ −∆p a† pσc−k−σ

• −h.c.
• 8 / 26

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

◮ IQ = −i ∑

p,k,σ

• tp,k
• ξpσ a†

pσckσ −∆p a† pσc−k−σ

• −h.c.
• ◮ IQ = δT ×4πN(0)|t|2FS
• ∝DtH

∞

∆ dε

• − ∂ f

∂T

• thermal excitations

8 / 26

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

◮ IQ = −i ∑

p,k,σ

• tp,k
• ξpσ a†

pσckσ −∆p a† pσc−k−σ

• −h.c.
• ◮ IQ = δT ×4πN(0)|t|2FS
• ∝DtH

∞

∆ dε

• − ∂ f

∂T

• thermal excitations

 ε2 −∆2 cos(φ) ε2 −∆2  

• 8 / 26

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

◮ IQ = −i ∑

p,k,σ

• tp,k
• ξpσ a†

pσckσ −∆p a† pσc−k−σ

• −h.c.
• ◮ IQ = δT ×4πN(0)|t|2FS
• ∝DtH

∞

∆ dε

• − ∂ f

∂T

• thermal excitations

 ε2 −∆2 cos(φ) ε2 −∆2  

• Trouble!!

→ ∞ Failure of Linear Response Theory?

8 / 26

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

◮ IQ = −i ∑

p,k,σ

• tp,k
• ξpσ a†

pσckσ −∆p a† pσc−k−σ

• −h.c.
• ◮ IQ = δT ×4πN(0)|t|2FS
• ∝DtH

∞

∆ dε

• − ∂ f

∂T

• thermal excitations

 ε2 −∆2 cos(φ) ε2 −∆2  

• Trouble!!

→ ∞ Failure of Linear Response Theory?

8 / 26

Non-Perturbative Theory of Transport in Phase-Biased Josephson Junctions

◮ Phase Bias: φ = φ2 −φ1 ◮ Thermal Bias: δT = T2 −T1 ◮ Barrier Transmission: 0 < D ≤ 1 ◮ Mesoscopic Junction: ¯

h/pf ≪ a < ξ∆ +i

e ∆

φ2 +i

e ∆

φ1

z

2a

• 9 / 26

Non-Perturbative Theory of Transport in Phase-Biased Josephson Junctions

◮ Phase Bias: φ = φ2 −φ1 ◮ Thermal Bias: δT = T2 −T1 ◮ Barrier Transmission: 0 < D ≤ 1 ◮ Mesoscopic Junction: ¯

h/pf ≪ a < ξ∆ +i

e ∆

φ2 +i

e ∆

φ1

z

2a

• +i

e ∆

φ2 +i

e ∆

φ1

z

∆

2ξ

2a Josephson Phase New Electronic States Confined to the Interface !

9 / 26

Non-Perturbative Theory of Transport in Phase-Biased Josephson Junctions

◮ Phase Bias: φ = φ2 −φ1 ◮ Thermal Bias: δT = T2 −T1 ◮ Barrier Transmission: 0 < D ≤ 1 ◮ Mesoscopic Junction: ¯

h/pf ≪ a < ξ∆ +i

e ∆

φ2 +i

e ∆

φ1

z

2a

• +i

e ∆

φ2 +i

e ∆

φ1

z

∆

2ξ

2a Josephson Phase New Electronic States Confined to the Interface ! ◮ Energy & Phase-dependent Transmission: D D(ε,φ)

9 / 26

Fermion Bound States of a Josephson Weak Link or 2π Vortex

∆e−i

φ/2

∆ +i e

φ/2

Sharvin Contact ε±(φ) = ±|∆||cos(φ/2)|

p

∆ +i ϕ

π φ

f

e

• 1.00
• 0.50

0.00 0.50 1.00 0.5 1 1.5 2

φ/π

ε+/∆ ε−/∆

Fermi Level Gap Edge

10 / 26

Andreev → Riccati Equations: Electron-Hole Branch Conversion

◮ Andreev’s Equation for Coherent Electron-Hole States

¯ hvp ·∇r U −V

• +
• ∆(p,r)

Ơ(p,r) U V

• = ε

U V

• 11 / 26

Andreev → Riccati Equations: Electron-Hole Branch Conversion

◮ Andreev’s Equation for Coherent Electron-Hole States

¯ hvp ·∇r U −V

• +
• ∆(p,r)

Ơ(p,r) U V

• = ε

U V

• ◮ Electron-Hole Coherence Amplitudes:

γ(p,r;ε) = U/V ¯ γ(p,r;ε) = V/U

◮ Riccati Equation:

¯ hvp ·∇γ +2εγ +∆+∆∗γ2 = 0

11 / 26

Andreev → Riccati Equations: Electron-Hole Branch Conversion

◮ Andreev’s Equation for Coherent Electron-Hole States

¯ hvp ·∇r U −V

• +
• ∆(p,r)

Ơ(p,r) U V

• = ε

U V

• ◮ Electron-Hole Coherence Amplitudes:

γ(p,r;ε) = U/V ¯ γ(p,r;ε) = V/U

◮ Riccati Equation:

¯ hvp ·∇γ +2εγ +∆+∆∗γ2 = 0

◮ Nonequilibrium: γ(p,r;ε,t)

∂∆(R) −e , +vf +e , −vf +e , +vf −e , −vf ∂∆(R)

Amplitude: γ(p,r;ε,t) Amplitude: ¯ γ(p,r;ε,t) ”h-e and e-h branch conversion scattering”

11 / 26

Multiple-scattering of Coherent e-h excitations at a Boundary

Multiple Scattering from a Potential + Branch conversion scattering

12 / 26

Multiple-scattering of Coherent e-h excitations at a Boundary

Multiple Scattering from a Potential + Branch conversion scattering

◮ Interface Potential Scattering:

S-matrix S11 S12 S21 S22

• 12 / 26

Multiple-scattering of Coherent e-h excitations at a Boundary

Multiple Scattering from a Potential + Branch conversion scattering

◮ Interface Potential Scattering:

S-matrix S11 S12 S21 S22

• ◮ Andreev Scattering: Riccati

∂∆(R) −e , +vf +e , −vf +e , +vf −e , −vf ∂∆(R)

γ(p,r;ε,t) ¯ γ(p,r;ε,t)

12 / 26

Multiple-scattering of Coherent e-h excitations at a Boundary

Multiple Scattering from a Potential + Branch conversion scattering

◮ Interface Potential Scattering:

S-matrix S11 S12 S21 S22

• ◮ Andreev Scattering: Riccati

∂∆(R) −e , +vf +e , −vf +e , +vf −e , −vf ∂∆(R)

γ(p,r;ε,t) ¯ γ(p,r;ε,t) ... ...

ra

1

˜ γR

2

S11 S21 S12 S22 γR

2

˜ γR

2

S12 S22 S22 S12 ta

1

S†

11

S†

12

e γR

1

γR

2

h ta

1

ra

1

γR

2

ΓR

1

12 / 26

Multiple-scattering of Coherent e-h excitations at a Boundary

Multiple Scattering from a Potential + Branch conversion scattering

◮ Interface Potential Scattering:

S-matrix S11 S12 S21 S22

• ◮ Andreev Scattering: Riccati

∂∆(R) −e , +vf +e , −vf +e , +vf −e , −vf ∂∆(R)

γ(p,r;ε,t) ¯ γ(p,r;ε,t) ... ...

ra

1

˜ γR

2

S11 S21 S12 S22 γR

2

˜ γR

2

S12 S22 S22 S12 ta

1

S†

11

S†

12

e γR

1

γR

2

h ta

1

ra

1

γR

2

ΓR

1

◮ Bound-States Poles of the re-normalized S-matrix amplitudes

◮ Nonequilibrium Superconductivity Near Spin-Active Interfaces, PRB 70:134510 (2004), E. Zhao, T. L¨

• fwander & J. A. Sauls

12 / 26

Fermion Bound States of a Josephson Junction

D e ∆

φ/2 −i

e ∆

φ/2

R

+i

S = √ R i √ D i √ D √ R

• 13 / 26

Fermion Bound States of a Josephson Junction

D e ∆

φ/2 −i

e ∆

φ/2

R

+i

S = √ R i √ D i √ D √ R

• ε±( R ,φ) = ±∆
• cos2(φ/2)+ R sin2(φ/2)

0 < D ≤ 1 Potential + Andreev Scattering Gap in the Bound-State Dispersion

• 1.00
• 0.50

0.00 0.50 1.00 0.5 1 1.5 2

φ/π

ε+/∆ ε−/∆

Fermi Level

Gap Edge

D=1.0 0.8 0.4 0.2

13 / 26

Heat Transport through a Phase-Biased Josephson Junction

◮ Heat Current jQ = −κ δT ◮ Carriers = bulk quasiparticles

14 / 26

Heat Transport through a Phase-Biased Josephson Junction

◮ Heat Current jQ = −κ δT ◮ Carriers = bulk quasiparticles ◮ Nbulk(ε) = N(0)

ε √ ε2 −∆2

14 / 26

Heat Transport through a Phase-Biased Josephson Junction

◮ Heat Current jQ = −κ δT ◮ Carriers = bulk quasiparticles ◮ Nbulk(ε) = N(0)

ε √ ε2 −∆2

◮ vg(ε) = vf

√ ε2 −∆2 ε

14 / 26

Heat Transport through a Phase-Biased Josephson Junction

◮ Heat Current jQ = −κ δT ◮ Carriers = bulk quasiparticles ◮ Nbulk(ε) = N(0)

ε √ ε2 −∆2

◮ vg(ε) = vf

√ ε2 −∆2 ε Thermal Conductance κ(φ,T) = A

∞

∆ dε Nbulk(ε) [ε vg(ε)] D(ε,φ)

• − ∂ f

∂T

• ◮ D(ε,φ) = Quasiparticle Transmission Probability

14 / 26

Transmission Probability for a Phase-Biased Josephson Junction

D(ε,φ) = De→e(ε,φ) = De→h(ε,φ)

15 / 26

Transmission Probability for a Phase-Biased Josephson Junction

D(ε,φ) = De→e(ε,φ) = De→h(ε,φ) Excitations: ε ≥ ∆

◮ Direct Transmission:

De→e(ε,φ) = D (ε2 −∆2) (ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

15 / 26

Transmission Probability for a Phase-Biased Josephson Junction

D(ε,φ) = De→e(ε,φ) = De→h(ε,φ) Excitations: ε ≥ ∆

◮ Direct Transmission:

De→e(ε,φ) = D (ε2 −∆2) (ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

◮ Branch Conversion Transmission:

De→h(ε,φ) = DR (ε2 −∆2) ∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

15 / 26

Transmission Probability for a Phase-Biased Josephson Junction

D(ε,φ) = De→e(ε,φ) = De→h(ε,φ) Excitations: ε ≥ ∆

◮ Direct Transmission:

De→e(ε,φ) = D (ε2 −∆2) (ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

◮ Branch Conversion Transmission:

De→h(ε,φ) = DR (ε2 −∆2) ∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Thermal Conductance Limits:

◮ φ = 0

D(ε,φ = 0) = D BCS Thermal Conductivity

15 / 26

Transmission Probability for a Phase-Biased Josephson Junction

D(ε,φ) = De→e(ε,φ) = De→h(ε,φ) Excitations: ε ≥ ∆

◮ Direct Transmission:

De→e(ε,φ) = D (ε2 −∆2) (ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

◮ Branch Conversion Transmission:

De→h(ε,φ) = DR (ε2 −∆2) ∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Thermal Conductance Limits:

◮ φ = 0

D(ε,φ = 0) = D BCS Thermal Conductivity

◮ D = 1

D(ε,φ) = ε2 −∆2 ε2 −∆2 cos2(φ/2) Sharvin Limit

15 / 26

Transmission Probability for a Phase-Biased Josephson Junction

D(ε,φ) = De→e(ε,φ) = De→h(ε,φ) Excitations: ε ≥ ∆

◮ Direct Transmission:

De→e(ε,φ) = D (ε2 −∆2) (ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

◮ Branch Conversion Transmission:

De→h(ε,φ) = DR (ε2 −∆2) ∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Thermal Conductance Limits:

◮ φ = 0

D(ε,φ = 0) = D BCS Thermal Conductivity

◮ D = 1

D(ε,φ) = ε2 −∆2 ε2 −∆2 cos2(φ/2) Sharvin Limit

◮ D ≪ 1

D(ε,φ) = D(ε2 −∆2 cos2(φ/2)) ε2 −∆2

• Tunneling Limit

15 / 26

Transmission Probability for a Phase-Biased Josephson Junction

D(ε,φ) = De→e(ε,φ) = De→h(ε,φ) Excitations: ε ≥ ∆

◮ Direct Transmission:

De→e(ε,φ) = D (ε2 −∆2) (ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

◮ Branch Conversion Transmission:

De→h(ε,φ) = DR (ε2 −∆2) ∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Thermal Conductance Limits:

◮ φ = 0

D(ε,φ = 0) = D BCS Thermal Conductivity

◮ D = 1

D(ε,φ) = ε2 −∆2 ε2 −∆2 cos2(φ/2) Sharvin Limit

◮ D ≪ 1

D(ε,φ) = D(ε2 −∆2 cos2(φ/2)) ε2 −∆2

• Tunneling Limit

Tunneling Limit Essential Singularity

15 / 26

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

◮ IQ = −i ∑

p,k,σ

• tp,k
• ξpσ a†

pσckσ −∆p a† pσc−k−σ

• −h.c.
• ◮ IQ = δT ×4πN(0)|t|2FS
• ∝DtH

∞

∆ dε

• − ∂ f

∂T

• thermal excitations

 ε2 −∆2 cos(φ) ε2 −∆2  

• Trouble!!

→ ∞ Failure of Linear Response Theory?

16 / 26

Andreev’s Demon & Resonant Transmission

Direct Transmission De→e(ε,φ) = D (ε2 −∆2)(ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Branch Conversion De→h(ε,φ) = DR (ε2 −∆2)∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

17 / 26

Andreev’s Demon & Resonant Transmission

Direct Transmission De→e(ε,φ) = D (ε2 −∆2)(ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Branch Conversion De→h(ε,φ) = DR (ε2 −∆2)∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0

17 / 26

Andreev’s Demon & Resonant Transmission

Direct Transmission De→e(ε,φ) = D (ε2 −∆2)(ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Branch Conversion De→h(ε,φ) = DR (ε2 −∆2)∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0

17 / 26

Andreev’s Demon & Resonant Transmission

Direct Transmission De→e(ε,φ) = D (ε2 −∆2)(ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Branch Conversion De→h(ε,φ) = DR (ε2 −∆2)∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0

17 / 26

Andreev’s Demon & Resonant Transmission

Direct Transmission De→e(ε,φ) = D (ε2 −∆2)(ε2 −∆2 cos2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2 Branch Conversion De→h(ε,φ) = DR (ε2 −∆2)∆2 sin2(φ/2) [ε2 −∆2 (1−Dsin2(φ/2)]2

0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Bound State Resonance

17 / 26

Transmission Resonance for Heat Transport

D e ∆

φ/2 −i

e ∆

φ/2

R

+i

Iε(φ,T) = − κ(φ,T) δT, with δT = T2 −T1 κ(φ,T) = 4A

∞

∆ dε N (ε)

• εvg(ε)
• D(ε,φ) ∂ f

∂T

0.0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0

resonant transmission shallow bound state

• b
• b

18 / 26

Non-analyticity of the Thermal Conductance

◮ Tunneling Hamiltonian: κ tH = κ tH

BCS + κ tH

2 sin2(φ/2) ... But κ tH 2 → ∞

◮ Self-Consistent S-matrix for D ≪ 1: κ = κBCS −κ1 sin2(φ/2)ln

• sin2(φ/2)
• +κ2 sin2(φ/2)

◮ κ1,2 − − − →

D→0 DlnD ⇒ Finite, but Non-Analytic and Non-perturbative 0.2 0.4 0.6 0.8 1.0 T/Tc 0.0 0.5 1.0 D=0.01 D=0.005

0.0 0.5 1.0 φ /π 0.01 0.02 κ(φ) −κ0 [ N0vf TcS]

κ(φ) − κBCS κ1 term κ2 term

T=0.5 Tc D=0.01

κ1 / κ2

◮ Andreev Bound-State Formation is non-perturbativce

19 / 26

Phase-Tuneable Resonant Enhancement of the Heat Current

0.5 1

φ / π

0.5 1.0 1.5 2.0 2.5

D=0.10 D=0.25 D=0.50 D=0.75 D=0.90

0.5 1 1.5 2

ε/∆

1 2 3 4 5 6

D(ε,π) −−−−− D δεres

κ(φ)/κ(0)

20 / 26

Phase-Tuneable Resonant Enhancement of the Heat Current

0.5 1

φ / π

0.5 1.0 1.5 2.0 2.5

D=0.10 D=0.25 D=0.50 D=0.75 D=0.90

0.5 1 1.5 2

ε/∆

1 2 3 4 5 6

D(ε,π) −−−−− D δεres

κ(φ)/κ(0)

0.4 0.5 0.6 0.7 0.8 0.9 1 T/T c 0.5 1 1.5 κ(φ,Τ) / κN

φ = π φ = 0

D = 0.05

Andreev’s Demon Fermion Bound States “control” thermal transport

◮ φ = 0: κ ↓ for T < Tc. ◮ φ = π: D < 0.5 κ(T) ↑ below Tc. ◮ D 0.5: κ(φ) < κ(0)

20 / 26

Phase-Tuneable Resonant Enhancement of the Heat Current

0.5 1

φ / π

0.5 1.0 1.5 2.0 2.5

D=0.10 D=0.25 D=0.50 D=0.75 D=0.90

0.5 1 1.5 2

ε/∆

1 2 3 4 5 6

D(ε,π) −−−−− D δεres

κ(φ)/κ(0)

0.4 0.5 0.6 0.7 0.8 0.9 1 T/T c 0.5 1 1.5 κ(φ,Τ) / κN

φ = π φ = 0

D = 0.05

Andreev’s Demon Fermion Bound States “control” thermal transport

◮ φ = 0: κ ↓ for T < Tc. ◮ φ = π: D < 0.5 κ(T) ↑ below Tc. ◮ D 0.5: κ(φ) < κ(0)

Northwestern LT Group: Z. Jiang et al., PRB 72, 020502 (2005) 20 / 26

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

21 / 26

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

21 / 26

Superconducting-Ferromagnetic-Superconducting Junctions

e−iϑσz/2| α ∆ e−iφ/2 ∆ e+iφ/2 | α

S S N µa µb F

a b

F

L< ∼¯ hvf/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

Ferromagnetic Point Contacts Spin Filtering Spin-dependent Transmission: D↑ = D↓ Spin Mixing Spin Faraday Rotation Angle: ϑ

23 / 26

Ferromagnetic Point Contacts Spin Filtering Spin-dependent Transmission: D↑ = D↓ Spin Mixing Spin Faraday Rotation Angle: ϑ

◮ S11 = S22 =

• R↑eiϑ/2
• R↓e−iϑ/2
• ◮ S12 = S21 = i
• D↑eiϑ/2
• D↓e−iϑ/2
• ◮ R↑/↓ = 1−D↑/↓

◮ | ↑ → cos(ϑ/2)| ↑+sin(ϑ/2)| ↓ ◮ First principles theory of magnetically active interfaces S(D,ϑ,...)

◮ Nonequilibrium Superconductivity near Spin-Active Interfaces, PRB 70, 134510 (2004), Zhao, L¨

• fwander, JAS

23 / 26

Multiple Andreev reflection (MAR) Quantum Transport Theory of Spin and Charge in SFS Junctions: spin filtering + spin mixing + MAR

• E

N F b 2e ∆ F a eV 2e S2 S1 h e

¯ h 2, e

◮ e/h’s scatter inelastically: ε → ε +mωJ (mth order MAR). ◮ e/h’s can escape into leads for ε > ∆ ◮

mth order MAR: transports charge = m×2e, spin ¯ h/2

24 / 26

Long-range spin-transfer torque in SFNFS contacts (L ∼ 0.1−1.0µm)

S S N µa µb F

a b

F

L< ∼¯ hvf/π∆

◮ Spin-Transfer Torques:

• τ b(t) = τ b

+

∞

∑

k=1

• τ b

k,c cos(kωJt)+

τ b

k,s sin(kωJt)

• ◮ τ b

0z ∝ Nf (v f ¯

h)V, eV ≫ ∆

◮ MAR + spin-mixing +

µa × µb = sin(ψ)ˆ x τ b

0x

• 0.1

0.1 0.3 0.5 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 τb

0z [Nf vf A∆−

h/2] eV/∆

In-Plane d.c. Torque

θ = 0.146 π θ = π/3 θ = π/2 θ = 2π/3 θ = π

• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 τb

0x [Nf vf A∆−

h/2] eV/∆

Out-of-Plane d.c. Torque

D↑ = 0.95, D↓ = 0.6 T = 0.5 Tc, ψ = π/2 θ = 0.146 π θ = π/3 θ = π/2 θ = 2π/3 θ = π

25 / 26

Directions and Challenges

◮ Nano-scale SFS JJs with CNT and Single Molecular Magnets ◮ Circuit QED with Spin-Triplet Superconductors (Sr2RuO4, UPt3, ?) ◮ Interacting Classical or Quantum Magnets mediated via Long-Range Josephson Spin-Transfer Torques ◮ Arrays of SFNFS JJs for Voltage-Controlled Spin Transport (L ∼ 0.1−1.0µm)

26 / 26