Generalized Josephson Junctions. Junctions with Resistive Channel

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Bertram Grant
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1 Generalized Josephson Junctions Outline 1. Junctions with Resistive Channel 2. RCSJ Model 3. DC Current Drive Overdamped Underdamped Junctions Return Current Dynamical Analysis 4. Pendulum Model October 25, 2005 Junctions with Resistive Channel Please see: Figure 9.1, page 450, from Orlo, T., K. Delin. 1

2 Tunneling between two superconductors Giaever Tunneling S-I-S G(v) Please see: p. 148 of Giaever's 1973 Nobel lecture: Josephson Tunneling Normal Superconducting Analogy Superconductor Superconducting Josephson Junction L J -1 Normal metal For a normal junction, the phase is constantly being driven back to zero so linearize near zero add a damping time for dc drive for dc drive 2

3 I C R n Product The condition is equivalent to Experimentally, For Nb at 2K, Please see: Figure 9.2, page 454, from Orlo, T., K. Delin. Please see: Figure 9.3, page 456, from Orlo, T., K. Delin. Capacitance of a Josephson Junction Please see: Figure 8.4, page 399, from Orlo, T., K. Delin. 3

4 Generalized Josephson Junction Please see: Figure 9.4, page 457, from Orlo, T., K. Delin. Therefore, RCSJ Model Please see: Figure 9.6, page 459, from Orlo, T., K. Delin. Therefore, 4

5 DC Current drive in the RSCJ Model Please see: Figure 9.6, page 459, from Orlo, T., K. Delin. Therefore, The equation of motion can be rewritten as where Josephson Time Constant Stewart-McCumber Parameter β c = Q 2 Overdamped Junction β c << 1 Please see: Figure 9.6, page 459, from Orlo, T., K. Delin. A. Static Solution: B. Dynamical Solution for i > I c This is periodic with period 5

6 Overdamped Junction β c << 1 Please see: Figure 9.7, page 462, from Orlo, T., K. Delin. The time averaged voltage is Use the voltage-phase relation, Please see: Figure 9.8, page 463, from Orlo, T., K. Delin. Therefore, Non-hysteretic Underdamped Junction β c >> 1 Please see: Figure 9.6, page 459, from Orlo, T., K. Delin. A. Static Solution: B. Dynamical Solution The phase changes quickly compared to RC, so the voltage is just from R C. Therefore, <v(t)> = i R Hysteretic Please see: Figure 9.9, page 464, from Orlo, T., K. Delin. 6

7 Junction with arbitrary β c Please see: Figure 9.6, page 459, from Orlo, T., K. Delin. A. Static Solution: B. Dynamical Solution Please see: Figure 9.10, page 464, from Orlo, T., K. Delin. Please see: Figure 9.11, page 465, from Orlo, T., K. Delin. Return Current Energy Loss per cycle = Energy supplied by source where V= IR τ = Φ 0 / (2 π I R), therefore So that Please see: Figure 9.11, page 465, from Orlo, T., K. Delin. 7

8 Dynamical Analysis where Please see: Figure , page 272, from Strogatz, S. Nonlinear Dynamics Chaos. 1st ed. Cambridge, MA: Perseus Books Group, January 15, ISBN: βc = 4 A A i/ic φ(t) Β C V(φ) <V>/ICR Β C f(t) φ(t) V(φ) V(φ) 8

9 β c =0.5 B A A i/i C C φ (t) B <V>/I C R C φ (t) φ (t) V(φ ) V(φ ) Pendulum Model for a Josephson Junction I app R + V Φ = o & ϕ 2 π C τ app l I c sinϕ - ϕ I 1 app = C V & + V + I c sinϕ R mg 2 τ app = ml & ϕ + D & ϕ + mgl sinϕ F = & ϕ + Γ & ϕ + sinϕ Single junction (RCSJ model) pendulum (damped) Coupled junctions can support non-linear excitations (breathers moving vortices) 9

10 Pendulum Model for a vortex. Please see: Figure 5.42, page 237, from Kardin, A. Introduction to Supercomputing Circuits. 1st ed. New York, NY: Wiley-Interscience, March 11, ISBN:

Chapter 3. The Voltage State

Chapter 3. The Voltage State AS-Chap. 3-1 Chapter 3 Physics of Josephson Junctions: The Voltage State 3. Physics of Josephson junctions: The voltage state AS-Chap. 3-2 For bias currents I > I s m Finite junction voltage Phase difference