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 φ 2 ∆ e ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� z ��������� ��������� ��������� ��������� +i φ 1 ��������� ��������� ��������� ��������� ∆ 2a e 2ξ ∆ Research supported by NSF grant DMR-1106315. ◮ Erhai Zhao, George Mason University Tomas L¨ ofwander, 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 opological 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 h ∂ t | ψ � = ( − i ¯ hc � α · ∇ + β g Φ ) | ψ � i ¯ � 0 � � 1 � � σ 0 � α = β = | ψ � = col ( ψ 1 , ψ 2 , ψ 3 , ψ 4 ) � σ 0 0 − 1 3 / 26

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

Dirac Fermions & Zero Energy Bound States ◮ Dirac Fermion coupled to a Scalar Bose Field h ∂ t | ψ � = ( − i ¯ hc � α · ∇ + β g Φ ) | ψ � i ¯ � 0 � � 1 � � σ 0 � α = β = | ψ � = col ( ψ 1 , ψ 2 , ψ 3 , ψ 4 ) � σ 0 0 − 1 � ◮ Broken Symmetry State: Φ = Φ 0 � Mass : Mc 2 = g Φ 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 1976 3 / 26

Nambu-Dirac Fermions in Superconductors ◮ Bogoliubov-Nambu Equations - particle-hole coherence : � �� u � �� u 2 m ∇ 2 − µ h 2 � � � u � − ¯ 0 ∆ 0 = ε + 2 m ∇ 2 + µ h 2 ¯ v ∆ † v v 0 0 4 / 26

Nambu-Dirac Fermions in Superconductors ◮ Bogoliubov-Nambu Equations - particle-hole coherence : � �� u � �� u 2 m ∇ 2 − µ h 2 � � � u � − ¯ 0 ∆ 0 = ε + 2 m ∇ 2 + µ h 2 ¯ v ∆ † v v 0 0 hv f / ∆ ≤ λ : � u = U p e i p · r / ¯ h ◮ Separation of scales: ¯ h / p f ≪ ¯ h p f / p y v p x p x λ ◮ Nambu-Dirac Spinors coupled to the (Bosonic) Cooper-Pair Field � U � � �� U � � U � ∆ ( p , r ) 0 h v p · ∇ r + = ε ¯ ∆ † ( p , r ) − V V V 0 4 / 26

Nambu-Dirac Fermions in Superconductors ◮ Bogoliubov-Nambu Equations - particle-hole coherence : � �� u � �� u 2 m ∇ 2 − µ h 2 � � � u � − ¯ 0 ∆ 0 = ε + 2 m ∇ 2 + µ h 2 ¯ v ∆ † v v 0 0 hv f / ∆ ≤ λ : � u = U p e i p · r / ¯ h ◮ Separation of scales: ¯ h / p f ≪ ¯ h p f / p y v p x p x λ ◮ Nambu-Dirac Spinors coupled to the (Bosonic) Cooper-Pair Field � U � � �� U � � U � ∆ ( p , r ) 0 h v p · ∇ r + = ε ¯ ∆ † ( p , r ) − V V V 0 ◮ Zero Modes if ∆ ( x = − ∞ ) = − ∆ ( x = + ∞ ) along x = ˆ v p · r 4 / 26

Electron-Hole Coherence & Zero-Energy Interface Bound States ◮ Andreev’s Equation for Coherent Electron-Hole States � U � � �� U � � U � ∆ ( p , r ) 0 h v p · ∇ r + = ε ¯ ∆ † ( p , r ) − V 0 V V p 2 p 2 ◮ ∆ ( p ) = ∆ ( ˆ x − ˆ y ) p y − p − p p + + x − [ 110 ] reflection: 5 / 26

Electron-Hole Coherence & Zero-Energy Interface Bound States ◮ Andreev’s Equation for Coherent Electron-Hole States � U � � �� U � � U � ∆ ( p , r ) 0 h v p · ∇ r + = ε ¯ ∆ † ( p , r ) − V 0 V V p 2 p 2 ◮ ∆ ( p ) = ∆ ( ˆ x − ˆ y ) p y − ◮ p ◮ Electron & Hole Bound State: − p p + � 1 � + x � e − 2 | ∆ ( p ) || x | / ¯ hv f | ψ � ∼ | ∆ ( p ) | i 5.0 − N( p,x=0 ; ε ) 4.0 [ 110 ] reflection: 3.0 2.0 1.0 0.0 -1.0 -0.5 0.0 0.5 1.0 ε/2πΤ c ◮ Tunneling into Surface States of HTC Superconductors , PRL 79:281–284 (1997), M. Fogelstr¨ om, 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 = H 1 + H 2 + H tH 6 / 26

Josephson Tunneling in Superconductors ◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963). H = H 1 + H 2 + H tH � � H 1 = ∑ ξ k σ c † ∆ k c † k σ c † − k − σ + ∆ ∗ k c † k σ c k σ + 1 2 ∑ − k − σ c k σ k σ k σ � � H 2 = ∑ ξ p σ a † ∆ p a † p σ a † − p − σ + ∆ ∗ p a † p σ a p σ + 1 2 ∑ − p − σ a p σ p σ p σ � � H tH = ∑ t p , k a † p σ c k σ + t ∗ p , k c † k σ a p σ p , k , σ 6 / 26

Josephson Tunneling in Superconductors ◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963). H = H 1 + H 2 + H tH � � H 1 = ∑ ξ k σ c † ∆ k c † k σ c † − k − σ + ∆ ∗ k c † k σ c k σ + 1 2 ∑ − k − σ c k σ k σ k σ � � H 2 = ∑ ξ p σ a † ∆ p a † p σ a † − p − σ + ∆ ∗ p a † p σ a p σ + 1 2 ∑ − p − σ a p σ p σ p σ � � H tH = ∑ t p , k a † p σ c k σ + t ∗ p , k c † k σ a p σ p , k , σ N 2 ( t ) � = 2 e Im ∑ t p , k � a † ◮ � I � = e � ˙ p σ ( t ) c k σ ( t ) � p , k , σ 6 / 26

Josephson Tunneling in Superconductors ◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963). H = H 1 + H 2 + H tH � � H 1 = ∑ ξ k σ c † ∆ k c † k σ c † − k − σ + ∆ ∗ k c † k σ c k σ + 1 2 ∑ − k − σ c k σ k σ k σ � � H 2 = ∑ ξ p σ a † ∆ p a † p σ a † − p − σ + ∆ ∗ p a † p σ a p σ + 1 2 ∑ − p − σ a p σ p σ p σ � � H tH = ∑ t p , k a † p σ c k σ + t ∗ p , k c † k σ a p σ p , k , σ N 2 ( t ) � = 2 e Im ∑ t p , k � a † ◮ � I � = e � ˙ p σ ( t ) c k σ ( t ) � p , k , σ ◮ � I � = I c ( T ) sin ( ∆ φ ) 6 / 26

Josephson Tunneling in Superconductors ◮ B. Josephson, Phys. Lett. 1, 251 (1962). ◮ V. Ambegaokar & A. Baratoff, PRL (1963). H = H 1 + H 2 + H tH � � H 1 = ∑ ξ k σ c † ∆ k c † k σ c † − k − σ + ∆ ∗ k c † k σ c k σ + 1 2 ∑ − k − σ c k σ k σ k σ � � H 2 = ∑ ξ p σ a † ∆ p a † p σ a † − p − σ + ∆ ∗ p a † p σ a p σ + 1 2 ∑ − p − σ a p σ p σ p σ � � H tH = ∑ t p , k a † p σ c k σ + t ∗ p , k c † k σ a p σ p , k , σ N 2 ( t ) � = 2 e Im ∑ t p , k � a † ◮ � I � = e � ˙ p σ ( t ) c k σ ( t ) � p , k , σ � ∆ � � � π 2 N ( 0 ) 2 |�| t | 2 � FS ◮ � I � = I c ( T ) sin ( ∆ φ ) I c ( T ) = 2 e × × ∆ tanh 2 T ) � �� � ∝ D tH ≪ 1 Transmission Amplitude 6 / 26

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

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

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

a.c. Josephson Effects ◮ Supercurrent: I s = I c ( T ) sin ( φ t ) ◮ a.c. Josephson Equation: φ t = 2 e h V t � � σ 0 + σ 1 cos ( φ t ) ◮ Dissipative Current: I Ohmic = 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 � � � � ◮ I Q = − i ∑ ξ p σ a † p σ c k σ − ∆ p a † − h . c . t p , k p σ c − k − σ p , k , σ 8 / 26