Josephson Effect and JJ

A Josephson junction is the active nonlinear element in many superconducting circuits. It is usually made from two superconductors separated by a very thin barrier. Cooper pairs tunnel through the barrier, so the junction couples the phases of the two superconductors.

If the two superconducting phases are ${\theta }_1$ and ${\theta }_2$, the key variable is the gauge-invariant phase difference:

$$\phi ={\theta }_2-{\theta }_1-\frac{2\pi }{{\mathrm{\Phi }}_0}{\int }_1^2\mathbf{A}\cdot d\mathbf{l}$$

Most first-pass circuit explanations simply call this $\phi$.

Figure TODO

Recommended figure: cross-section of a Josephson junction, showing superconductor / thin insulator / superconductor and the phase difference $\phi$.

Image path used by this page: /figures/fundamentals/josephson-junction-structure.svg

Why Junctions Matter

A superconducting loop stores quantized flux, but by itself it does not give us an easy digital switch. A Josephson junction provides controlled phase motion:

• below its critical current, it supports supercurrent with zero DC voltage,
• near or above its critical current, the phase can move rapidly,
• phase motion produces voltage,
• a $2\pi$ phase slip corresponds to one flux quantum event.

This is the bridge from superconducting physics to SFQ pulses.

DC Josephson Effect

With no DC voltage across the junction, a supercurrent can flow:

$$I_s=I_c\sin \phi$$

where:

• $I_s$ is the supercurrent through the junction,
• $I_c$ is the critical current,
• $\phi$ is the phase difference.

This equation says the junction is not an ordinary resistor. It is a phase-controlled nonlinear element.

Josephson Energy

The Josephson current can be derived from a phase-dependent energy:

$$U_J(\phi )=-E_J\cos \phi$$

where:

$$E_J=\frac{{\mathrm{\Phi }}_0{I}_c}{2\pi }$$

The current follows from the slope of this energy:

$$I_s=\frac{2\pi }{{\mathrm{\Phi }}_0}\frac{\partial U_J}{\partial \phi }=I_c\sin \phi$$

This energy picture is useful because SQUIDs and SFQ gates are often explained by potential-energy wells. Bias current or flux tilts the potential so the phase can move from one stable point to another.

Figure TODO

Recommended figure: Josephson energy $U_J=-E_J\cos \phi$ and how bias current tilts the potential.

Image path used by this page: /figures/fundamentals/josephson-energy-potential.svg

AC Josephson Effect

When a voltage appears across the junction, the phase changes with time:

$$\frac{d\phi }{dt}=\frac{2\pi V}{{\mathrm{\Phi }}_0}$$

Equivalently:

$$V=\frac{{\mathrm{\Phi }}_0}{2\pi }\frac{d\phi }{dt}$$

So a rapidly changing phase produces a voltage pulse.

SFQ Pulse Area

For a single $2\pi$ phase slip:

$$\int Vdt=\frac{{\mathrm{\Phi }}_0}{2\pi }\int \frac{d\phi }{dt}dt=\frac{{\mathrm{\Phi }}_0}{2\pi }\mathrm{\Delta }\phi ={\mathrm{\Phi }}_0$$

This is why an SFQ pulse is precise. The pulse height and width can change, but the time integral is fixed by the phase change.

Figure TODO

Recommended figure: voltage pulse waveform with shaded area labeled ${\mathrm{\Phi }}_0$, plus a small note that the pulse shape may vary while the area is quantized.

Image path used by this page: /figures/fundamentals/sfq-pulse-area.svg

RCSJ Model

For circuit design, a Josephson junction is usually modeled by the RCSJ model: an ideal Josephson element in parallel with a capacitance and a resistance.

The current balance is:

$$I=I_c\sin \phi +\frac{V}{R}+C\frac{dV}{dt}$$

Using $V=({\mathrm{\Phi }}_0/2\pi )(d\phi /dt)$:

$$I=I_c\sin \phi +\frac{{\mathrm{\Phi }}_0}{2\pi R}\frac{d\phi }{dt}+\frac{{\mathrm{\Phi }}_{0}C}{2\pi }\frac{d^2\phi }{d{t}^2}$$

This is mathematically similar to a driven damped pendulum:

• phase $\phi$ behaves like the pendulum angle,
• capacitance gives inertia,
• resistance gives damping,
• bias current gives the drive,
• the Josephson sine term gives the nonlinear restoring force.

Damping and McCumber Parameter

The McCumber parameter is commonly written:

$${\beta }_c=\frac{2\pi I_c{R}^{2}C}{{\mathrm{\Phi }}_0}$$

Interpretation:

• ${\beta }_c\ll 1$: overdamped, slow but non-hysteretic,
• ${\beta }_c\approx 1$: near critical damping, useful for clean SFQ pulses,
• ${\beta }_c\gg 1$: underdamped, hysteresis and ringing can appear.

SFQ circuits often use shunt resistors to control damping so the junction switches quickly and then returns cleanly to the zero-voltage state.

Josephson Inductance

For small phase variations, the junction behaves like a nonlinear inductor. Differentiate the DC Josephson relation:

$$I_s=I_c\sin \phi$$

The small-signal Josephson inductance is:

$$L_J(\phi )=\frac{{\mathrm{\Phi }}_0}{2\pi I_c\cos \phi }$$

Near $\phi =0$:

$$L_J(0)=\frac{{\mathrm{\Phi }}_0}{2\pi I_c}$$

This is why junctions can be used not only as switches but also as nonlinear inductive elements.

Switching Picture

Use this mental model:

1. The junction is initially in a zero-voltage state.
2. Bias current or flux tilts the phase potential.
3. The phase escapes from one well and advances by about $2\pi$.
4. A voltage pulse appears through the AC Josephson relation.
5. A flux quantum is inserted into or removed from a superconducting loop.

This is not CMOS voltage-level switching. The information is in pulse events and stored flux states.

Beginner Pitfalls

• "Current above $I_c$ destroys the whole circuit." In circuit operation, the junction switches locally; the surrounding superconducting circuit can remain functional.
• "The voltage pulse height is the digital value." In SFQ, the important quantity is the pulse area and its arrival, not a static voltage level.
• "$2\pi$ is just math." It is the physical phase winding that connects Josephson switching to flux quantization.
• "A JJ is just an ideal switch." Real junction capacitance and damping strongly shape the pulse.

Training Exercise

1. Starting from $V=({\mathrm{\Phi }}_0/2\pi )(d\phi /dt)$, integrate both sides for a $2\pi$ phase change.
2. Derive $\int Vdt={\mathrm{\Phi }}_0$.
3. Starting from $U_J=-E_J\cos \phi$, show that $I_s=I_c\sin \phi$.
4. Write the RCSJ equation and label which term is Josephson current, resistive current, and capacitive current.
5. Explain why an underdamped junction can cause pulse interaction.

Next

Continue to SQUID Basics, where junctions and superconducting loops combine into the energy landscape used for SFQ logic.